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UBC Theses and Dissertations

Allocation of outdoor recreation across a mine waste section Rouse, Clayton Karl 1983

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A L L O C A T I O N A CROSS A OF OUTDOOR M I N E WASTE R E C R E A T I O N S E C T I O N by C L A Y T O N KARL ROUSE B . S c . ( A g r . ) , The U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1981 A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S FOR T H E D E G R E E OF MASTER O F S C I E N C E i n T H E F A C U L T Y OF GRADUATE S T U D I E S D e p a r t m e n t o f R e s o u r c e Management S c i e n c e We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A September 1983 (c) C l a y t o n K a r l Rouse, 1983 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s thesis for s c h o l a r l y purposes may be granted by the head of my department or by his or her representatives. It i s understood that copying or p u b l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Resource Management Science The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date September 9, 19 83 7 Q ^ ABSTRACT i i The objective of t h i s thesis was to t e s t multi-attribute u t i l i t y analysis as a method to help a planner allocate outdoor recreational a c t i v i t i e s across a mine waste section. The mine waste examined was a hypothetical 10 km x 10 km section. The physical features of the mine waste resemble the coal mine waste of the Elk River Valley i n southeastern B r i t i s h Columbia. Nine a c t i v i t i e s were chosen fo r examination. These were t r a i l b i k i n g , four-wheel d r i v i n g , snowmobiling, downhill s k i i n g , cross-country s k i i n g , snowshoeing, hiking, horseback r i d i n g and recreational vehicle camping. The a c t i v i t i e s were grouped into sixteen land uses. A resident of the Elk River Valley was chosen to represent the interests of each a c t i v i t y user group. These interests were described as preferences for attributes of the mine waste. Mul t i - a t t r i b u t e u t i l i t y analysis was used to develop the nine representatives' preference structures for the mine waste a t t r i -butes. The results of the analysis were used to develop an objective function which measured how well a recreation plan for the mine waste s a t i s f i e d the user groups' i n t e r e s t s . A computer program was developed to evaluate the objective function. Using t h i s program, a recreation land use plan was produced for the hypothetical mine waste section which maximized the value of the objective function. Two l i m i t a t i o n s of the multi-attribute u t i l i t y analysis were i d e n t i f i e d i n t h i s study. The f i r s t was the large time commitment required by user groups to structure t h e i r preferen-ces for the mine waste a t t r i b u t e s . This resulted i n user groups becoming t i r e d with the preference assessment procedure. The second l i m i t a t i o n was that the assessed preferences did not take into account the cost to the user groups of obtaining each a t t r i -bute l e v e l . These two factors may influence user groups* pre-ference structures for the mine waste a t t r i b u t e s . Accepting these l i m i t a t i o n s , multi-attribute u t i l i t y anal-y s i s i n t h i s study was successful i n breaking the large outdoor recreation planning problem into smaller problems where user groups' objectives and associated attributes were i d e n t i f i e d . The analysis enabled user groups to systematically a r t i c u l a t e and understand t h e i r preferences for each of the a t t r i b u t e s . Using t h i s information, a planner was able to i s o l a t e agreements and differences i n the preferences of the user groups, which provided a firm basis on which to begin a process of c o n f l i c t resolution. A planner i s then able to incorporate these results with other information on the mine waste development area to develop a feasible outdoor recreation land use plan. Thesis Supervisor i v TABLE OF CONTENTS P a ? e Abstract i i Table of Contents i v L i s t of Tables v i i i L i s t of Figures ix Acknowledgement x i CHAPTER 1. Statement of the Problem 1 1.1 Introduction 1 1.2 Objective of the Study 4 1.3 Organization of the Study 4 CHAPTER 2. Methodology - Part 1 Multi-Attribute U t i l i t y Analysis 8 2.1 Literature Review of Multi-Attribute U t i l i t y Analysis Development and Applications 8 2.2 Overview of Multi-Attribute U t i l i t y Analysis 9 2.3 Choice of Recreational A c t i v i t i e s 10 2.4 Choice of Recreational A c t i v i t y Attributes 11 2.5 Determination of Numerical Ranges for the Attributes 12 2.6 Assessment of Attribute U t i l i t y Functions 14 2.61 Assumptions of the Assessment Procedure 17 2.62 Strengths and Weaknesses of the Assessment Procedure 22 2.63 A Decision Maker's Attitude Toward Risk 27 2.7 V e r i f i c a t i o n of Independence Properties of Attributes 30 2.71 U t i l i t y Independence 30 V Page 2.72 P r e f e r e n t i a l Independence 32 2.73 Additive Independence 33 2.8 Assessment of Scaling Constants for the Attributes 35 2.81 Consistency Checks 37 2.82 Evaluating the Scaling Constant Against Which Other Attributes are Traded for the M u l t i p l i c a -t i v e U t i l i t y Function 39 2.83 Strengths and Weaknesses of the Attribute Scaling Constant Assessment Procedures 40 2.9 Choice of a Multi-Attribute U t i l i t y Function 42 2.91 Additive U t i l i t y Function 45 2.92 M u l t i p l i c a t i v e U t i l i t y Function 47 2.10 Development of an Equation to Scale the Recreational A c t i v i t i e s 50 CHAPTER 3. Results and Discussion - Part 1 Multi-Attribute U t i l i t y Analysis 52 CHAPTER 4. Methodology - Part 2 Maximization of the Objective Function 55 4.1 Introduction 55 4.2 Grouping the Recreational A c t i v i t i e s into Land Uses 55 4.3 Bounding of the Study Area to be Evaluated 59 4.4 Calculation of Attribute Levels from the 10 km x 10 km Mine Waste Section 60 4.41 Calculation of Travel Time from Town 60 4.42 Calculation of Length of T r a i l 61 4.43 Calculation of Average Slope and Snow Depth 64 4.44 Calculation of the Number of Co n f l i c t s per Hour Between A c t i v i t i e s 65 v i Page 4.45 Calculation of the Distance to the Nearest Drinking Water Source 67 4.5 Evaluating the Mine Waste Plan 70 CHAPTER 5. Results - Part 2 Maximization of the Objective Function CHAPTER 6. Discussion - Part 2 Maximization of the Objective Function CHAPTER 7. Summary and Conclusions REFERENCES CITED 89 APPENDIX 1. Attributes of the Recreational A c t i v i t i e s ... 94 APPENDIX 2. Dialogue to Fami l i a r i z e the Decision Maker with the Terminology and Motivation for the Assessment of His U t i l i t y Function 101 APPENDIX 3. Dialogue for Assessing Attribute U t i l i t y Functions and Ve r i f y i n g Attribute U t i l i t y Independence 103 APPENDIX 4. Dialogue for Obtaining Tradeoffs, Consistency Checks and Ve r i f y i n g P r e f e r e n t i a l Independence 107 APPENDIX 5. Attribute U t i l i t y Functions I l l A. 5.1 T r a i l b i k i n g 112 A. 5.2 Four-Wheel Driving 114 A. 5.3 Snowmobiling 116 A.5.4 Downhill Skiing 118 A. 5.5 Cross-Country Skiing 120 A.5.6 Snowshoeing 122 A.5.7 Hiking 124 A. 5.8 Horseback Riding 125 A.5.9 Summer Motorized Camping 12 7 v i i Page APPENDIX 6. A t t r i b u t e T r a d e o f f s f o r A s s e s s i n g S c a l i n g Cons tant s 129 A . 6.1 T r a i l b i k i n g 130 A . 6 . 2 Four -Whee l D r i v i n g 133 A . 6. 3 Snowmobil ing 136 A . 6.4 D o w n h i l l S k i i n g 139 A . 6 . 5 C r o s s - C o u n t r y S k i i n g 141 A . 6 . 6 Snowshoeing 144 A . 6 . 7 H i k i n g 147 A . 6.8 Horseback R i d i n g 148 A . 6 . 9 Summer M o t o r i z e d Camping 15 0 APPENDIX 7. A d d i t i v e U t i l i t y F u n c t i o n s 152 APPENDIX 8. R e l a t i v e Importance o f the A t t r i b u t e s t o Each D e c i s i o n Maker 154 APPENDIX 9. The N L P . S A l g o r i t h m 156 APPENDIX 10. E x p l a n a t i o n o f the NLP.S A l g o r i t h m 16 8 APPENDIX 11. The N L P . S A l g o r i t h m F l o w c h a r t 174 APPENDIX 12 . Procedure f o r U s i n g the NLP.S A l g o r i t h m 176 APPENDIX 13. F i l e C o n t a i n i n g the Data P o i n t s o f the 31 A t t r i b u t e U t i l i t y F u n c t i o n s 181 APPENDIX 14. The OUTPUT.S A l g o r i t h m 183 APPENDIX 15 . Data on Snow D e p t h , S lope and D r i n k i n g Water Sources f o r the Land Use P l a n 185 APPENDIX 16. The U T I L V A L . S A l g o r i t h m 187 APPENDIX 17. Procedure f o r U s i n g the U T I L V A L . S A l g o r i t h m 197 APPENDIX 18. U t i l i t y V a l u e s Generated by the U T I L V A L . S A l g o r i t h m f o r Each A c t i v i t y f o r Each G r i d Square 200 APPENDIX 19 . The OUTSPLIT.S A l g o r i t h m 203 APPENDIX 20. Procedure f o r U s i n g the OUTSPLIT.S A l g o r i t h m 206 v i i i LIST OF TABLES Table Page 1 Data for percentage households p a r t i c i p a t i n g i n recreational a c t i v i t i e s of the Elk River Valley and t h e i r s c a l i n g constants 51 2 Land uses chosen for analysis i n t h i s study 58 3 T r a i l length i n r e l a t i o n to the number of g r i d squares 62 4 F i l e containing the data points of the 31 attribute u t i l i t y functions 182 5 Data on snow depth, slope and drinking water sources for the land use plan 186 6 U t i l i t y values generated by the UTILVAL.S algorithm for each a c t i v i t y for each g r i d square .. 201 ix LIST OF FIGURES Figure Page 1 Mine waste section examined i n t h i s study 6 2 The Elk River Valley 7 3 Choice between two outcomes 14 4 Graph representing a u t i l i t y function 15 5 Attribute u t i l i t y function i l l u s t r a t i n g a r i s k neutral attitude 27 6 Concave increasing and decreasing u t i l i t y functions i n d i c a t i n g a r i s k averse attitude 2 8 7 Convex increasing and decreasing u t i l i t y functions i n d i c a t i n g a r i s k seeking attitude 29 8 Graph to t e s t for u t i l i t y independence between attributes X^ and 30 9 Lottery between attribute l e v e l X, and a 50-50 gamble between P and Q 31 10 Graphs to t e s t for p r e f e r e n t i a l independence between the attribute p a i r (X, ,X2) and the attribute X 3 .... 32 11 Tradeoff between attributes X 2 6 and X 2 7 35 12 Lottery to determine k 0 l, 39 Z b 13 Method s p e c i f i c a t i o n chart for multiple attribute decision problems 44 14 Objective function to determine the u t i l i t y of a land use plan with outdoor recreational a c t i v i t i e s 51 15 T r a i l c i r c u i t used by the a c t i v i t i e s 61 16 T r a i l length i n r e l a t i o n to the number of g r i d squares 63 17 An example of a f i l e with slope values (%) 64 18 An example of a f i l e with snow depth values (cm) .. 64 19 Land use plan with the square under examination K at I,J coordinates (6,3) and drinking water source at I,J coordinates (3,7) 67 Figure x Page 20 An example of a f i l e containing drinking water sources 6 8 21 Three dimensional representation of the plan evaluated i n t h i s study 72 22 Grid squares with high u t i l i t i e s for cer t a i n a c t i v i t i e s 73 23 Land uses r e s u l t i n g from the high u t i l i t y a c t i v i t i e s presented i n Figure 22 74 24 Starting plan for the evaluation 75 25 The land use plan with the highest function value 80 26 A c t i v i t y allocations for the s t a r t i n g land use plan 81 27 A c t i v i t y allocations for the best land use plan generated 82 28 The NLP.S algorithm flowchart 175 ACKNOWLEDGEMENT I g r a t e f u l l y acknowledge my great debt to my committee for t h e i r support i n making t h i s thesis possible. I am very grat e f u l to Dr. L.M. Lavkulich, my thesis supervisor, for his guidance, advice and resources made a v a i l -able to me throughout t h i s t h e s i s . I am also very grat e f u l to Dr. A.D. Chambers for his guidance and advice throughout the study. I express my gratitude to Dr. A.H.J. Dorcey for his guidance during the thesis and h e l p f u l suggestions towards making i t a success. I also express sincere appreciation to Dr. R. Hilborn for introducing me to the methods used i n t h i s thesis and his patience, a v a i l a b i l i t y and devotion to the study. Aside from my committee, I would l i k e to express my grat-itude to Mr. John F l i n n who f a i t h f u l l y assisted with the computer programming i n t h i s thesis and to Mrs. Joyce Hollands for her support and advice. I would l i k e to express my appreciation to the residents of the Elk River Valley, B.C., who par t i c i p a t e d i n the assessments i n t h i s study for t h e i r patience and i n t e r e s t . I am also very g r a t e f u l to Dr. J. Dick and Ms. L. Bailey of the Planning Branch, Ministry of Environment i n V i c t o r i a , B.C., for t h e i r time and resources. F i n a l l y , I would l i k e to thank the personnel of the companies, clubs, associations, agencies and int e r e s t groups, too numerous to mention here, who made i n f o r -mation available to me during t h i s t h e s i s . 1 CHAPTER 1 STATEMENT OF THE PROBLEM 1.1 Introduction While an extraordinary amount of technical data and i n f o r -mation has been compiled on the b i o l o g i c a l reclamation of sur-face mined lands, there has been l i t t l e e f f o r t i n Canada to analyse the e x i s t i n g information i n r e l a t i o n to other land uses. Surface mining temporarily a l t e r s the topography of an area, displacing a l l vegetation and leaving the area i n long, successive p a r a l l e l ridges or p i l e s of fractured rock and s o i l material. This land base can be u t i l i z e d for both motorized and non-motorized outdoor recreational a c t i v i t i e s . On natural areas, ecological damage from motorized recreation can be ex-tensive (Baldwin and Stoddard 19 73, Bury et. a l . 1976, Geological Society of America 1977, Webb et. a l . 1977, Sheridan 1979); however, mine waste has already been d r a s t i c -a l l y disturbed and ec o l o g i c a l damage by motorized recreation i s minimal except that caused by dust, noise and vehicle exhaust fumes. Non-motorized recreational a c t i v i t i e s such as cross-country s k i i n g , hiking, horseback r i d i n g and snowshoeing may also be conducted on the mine waste base, f o r such areas usually o f f e r challenging t e r r a i n and are very often located i n mountainous regions which afford spectacular views. Many sur-face mined areas i n the United States have been developed for off-road vehicle use, camping, picnicking and other recreational a c t i v i t i e s (Higgins 1973, O'Neill 1973, Timmons 1973, U.S.D.A. 1973) . 2 When planning a mine waste area f o r outdoor recreation, a planner must decide where to locate various recreational a c t i v -i t i e s . He must incorporate into his decision, information on factors such as cost of development, projected use of the mine waste, adjacent development projects, ownership, environmental impacts from development and recreational user groups' i n t e r e s t s . This thesis i s concerned with how a planner develops a rec-reation plan which s a t i s f i e s the most recreational user groups' i n t e r e s t s . I t i s assumed i n t h i s study that a user group's interests can be described by i t s preferences for attributes of the mine waste. In t h i s t h e s i s , a decision maker i s defined as a represent-ative of a recreational user group, whose preference structure for-mine waste attributes i s assumed to be representative of the user group. A planner w i l l almost always find that although a decision maker prefers one course of action when one mine waste attribute such as slope i s considered, he w i l l prefer another course of action when a d i f f e r e n t attribute i s considered. Seldom i s one course of action preferred for every a t t r i b u t e . A planner, therefore, must f i n d a method to analyse these at t r i b u t e trade-o f f s to develop the recreation plan. There are two main sets of methods for addressing the a t t r i -bute tradeoff issue. One set requires a decision maker to i n -formally weigh tradeoffs i n his mind. The other set of methods requires a decision maker to formalize e x p l i c i t l y his preference structure for attributes and uses t h i s to evaluate attribute tradeoffs. To date, formal methods have not been extensively applied 3 to mine waste recreation areas. In t h i s study, the formal method of multi-attribute u t i l i t y analysis developed by Keeney and R a i f f a (1976) i s used to address the tradeoff issue between mine waste a t t r i b u t e s . There are nine recreational a c t i v i t i e s examined i n t h i s study. The a c t i v i t i e s are grouped into 16 land uses. The mine waste examined i n t h i s study i s a hypothetical 10 km x 10 km section. The physical features of the mine waste resemble the coal mine waste of the Elk River Valley i n south-eastern B r i t i s h Columbia. The 10 km x 10 km section i s divided i n t o 100 g r i d squares. Each g r i d square i s assigned one rec-reation land use. The mine waste section i s i l l u s t r a t e d i n Figure 1. The Elk River Valley, i l l u s t r a t e d i n Figure 2, has char-a c t e r i s t i c a l l y hot summers and mild to severe winters with heavy snowfall at elevations near 2300 metres (B.C. Research 1976; 1980). There are currently three surface coal mining companies a c t i v e l y operating i n the study area. These are B.C. Coal Limited, Fording Coal Limited and Crowsnest Resources Limited. Three towns i n the Valley accomodate the mines' employees and t h e i r f a m i l i e s . These are Fernie, Sparwood and E l k f o r d . The waste rock from these coal mines has present and future outdoor recreation p o t e n t i a l , and for t h i s reason, the Elk River Valley was chosen for study. There are 9 decision makers i n t h i s study. Each decision maker i s a resident of the Elk River Valley who frequently par-t i c i p a t e s i n one of the outdoor a c t i v i t i e s examined i n t h i s study. Multi-attribute u t i l i t y analysis i s used to assess pre-ference structures of these decision makers for attributes of the mine waste i n t h i s study. The results of the analysis are used i n a computer program to help a planner best alloc a t e out-door recreation a c t i v i t i e s to the mine waste section. 1.2 Objective of the Study The objective of t h i s study i s to t e s t m u l t i - a t t r i b u t e u t i l i t y analysis as a method to help a planner best a l l o c a t e outdoor recr e a t i o n a l a c t i v i t i e s across a mine waste section. 1.3 Organization of the Study The study i s divided into two parts: (i) Part 1 - M u l t i - a t t r i b u t e u t i l i t y analysis; ( i i ) Part 2 - Maximization of the objective function developed from multi- a t t r i b u t e u t i l i t y analysis i n Part 1. Part 1 begins with a l i t e r a t u r e review of multi-attribute u t i l i t y analysis development and applications. An overview of Keeney and Raiffa's (1976) multi- a t t r i b u t e u t i l i t y analysis methodology used i n t h i s study i s then presented. The method-ology of the m u l t i - a t t r i b u t e u t i l i t y analysis i s then presented s t a r t i n g with a discussion of how the recreational a c t i v i t i e s were chosen for t h i s study, followed by discussions on choice of the recreational a c t i v i t y a t t r i b u t e s and determination of t h e i r numerical ranges. The methodology f o r assessing a t t r i b u t e u t i l i t y functions i s then presented, followed by discussions of the assessment procedure's assumptions, the strengths and weak-nesses of the procedure i t s e l f and on decision makers' attitudes 5 towards r i s k . The methodology fo r v e r i f y i n g independence prop-e r t i e s of attributes i s then presented, followed by the method-ology f o r assessing a t t r i b u t e s c a l i n g constants. A discussion of the strengths and weaknesses of the s c a l i n g constant assess-ment prodecure i s then presented followed by methodologies f o r choosing a m u l t i - a t t r i b u t e u t i l i t y function and development of an equation to scale the a c t i v i t i e s . Part 1 i s concluded by a presentation and discussion of the r e s u l t s of multi-attribute u t i l i t y analysis i n t h i s study. Part 2 of the study consists of using a computer program, which evaluates the objective function developed i n Part 1 , to maximize the objective function value of a l t e r n a t i v e recreation plans on the hypothetical mine waste section. Part 2 begins with an introduction to the methodology to evaluate the objective function followed by a methodology to group the recreational act-i v i t i e s examined i n t h i s study into land uses. A more detailed discussion on bounding the mine waste section i s then presented, followed by methodologies to calculate a t t r i b u t e l e v e l s from the mine waste section. A discussion of the methodology for eval-uating the mine waste section i s then presented. Part 2 i s con-cluded by a presentation of the results of the mine waste rec-reation plan evaluation, followed by a discussion of the r e s u l t s . F i g . 1. Mine waste section examined i n t h i s study. S 7 8 CHAPTER 2 METHODOLOGY - PART 1 Multi-Attribute U t i l i t y Analysis 2.1 Literature Review of Multi-Attribute U t i l i t y Analysis  Development and Applications The use of u t i l i t y theories i n decision problem analysis has become popular i n recent years (Fishburn 1966). Among these theories, the von Neumann-Morgenstern (1947) expected u t i l i t y theory i s one of the more popular theories (Fishburn 1964, Larsson 1977). Ramsey (1931, as c i t e d by MacCrimmon and Lars-son 1975) and l a t e r von Neumann and Morgenstern (1947) developed a set of assumptions or axioms of " r a t i o n a l behaviour" that, when s a t i s f i e d by a decision maker, would make i t possible to empir-i c a l l y assess a u t i l i t y function for him. Furthermore, von Neumann and Morgenstern showed that i f the assumptions were accepted, a decision maker would be compelled to choose the max-imization of expected u t i l i t y as the decision c r i t e r i o n i n risky situations (Fishburn 1964). In recent times, multi-attribute u t i l i t y models or functions based on the von Neumann-Morgenstern axioms have been developed and used to assess the u t i l i t y of alternative courses of action with multiple attributes (Lee 1971, Keeney 1972a, Huber 1974). Keeney and R a i f f a (1976) have developed a multi-attribute u t i l i t y analysis based on the axioms. This multi-attribute u t i l i t y analysis has been successfully applied i n research stud-ies on water resource project evaluation (Shih and Dean undated, as c i t e d by Morris 1971) , analysing patient management decisions as applied to c l e f t palate (Krischer 1974), forest pest manage-ment (Bell 1975, as c i t e d by Keeney and R a i f f a 1976), salmon management on the Skeena River (Hilborn and Walters 1977), s i t i n g energy f a c i l i t i e s (Keeney 1980) and determining salmon coho po l i c y i n Oregon (Walker 1982). P r a c t i c a l applications of the multi-attribute u t i l i t y anal-y s i s have been conducted i n developing the major a i r p o r t f a c i l -i t i e s of the Mexico City metropolitan area (Keeney 1973), structuring corporate preferences for multiple objectives (Keeney 1975) , evaluating environmental impacts at proposed nuclear power plant s i t e s (Keeney and Robillard 1977) and for evaluating proposed pump storage f a c i l i t i e s for power generation (Keeney 1979) . Keeney and Raiffa's multi-attribute u t i l i t y analysis i s used i n t h i s study. 2.2 Overview of Multi-Attribute U t i l i t y /Analysis Multi-attribute u t i l i t y analysis i n t h i s study i s composed of the following steps: (i) Choice of the recreational a c t i v i t i e s ; ( i i ) Choice of recreational a c t i v i t y a ttributes; ( i i i ) Determination of numerical ranges of the a t t r i b u t e s ; (iv) Assessment of attribute u t i l i t y functions; (v) V e r i f i c a t i o n of independence properties of a t t r i b u t e s ; (vi) Assessment of scaling constants for the a t t r i b u t e s ; (v i i ) Choice of a multi-attribute u t i l i t y function; ( v i i i ) Development of an equation to scale the recreational a c t i v i t i e s . 2.3 Choice of Recreational A c t i v i t i e s 10 In 1980, the Ministry of Environment Planning Branch con-ducted a study to determine current l e v e l s of p a r t i c i p a t i o n i n land and water-based recreational a c t i v i t i e s by Southeast Coal Block residents (Nessman and Bailey 1981). The following land-based recreational a c t i v i t i e s were i d e n t i f i e d i n t h e i r study: (i) T r a i l b i k i n g (TB) (vi) Snowshoeing (SHOE) ( i i ) Four-Wheel Driving (4x4) (vii) Hiking (HIKE) ( i i i ) Snowmobiling (SNOW) ( v i i i ) Horseback Riding (HORSE) (iv) Downhill Skiing (DOWN) (ix) Summer Motorized (v) Cross-country Skiing (X-C) Camping (CAMP) This c l a s s i f i c a t i o n i s used i n thi s study. 11 2.4 Choice of Recreational A c t i v i t y Attributes An a t t r i b u t e i s a c h a r a c t e r i s t i c or property that con-tributes to the success or f a i l u r e of a recreatonal a c t i v i t y (Holloway 1979). When discussing a t t r i b u t e s , Keeney and R a i f f a (1976) state the following: An a t t r i b u t e should be both comprehensive and measurable. An a t t r i b u t e is comprehensive i f , by knowing the level of an a t t r i b u t e in a p a r t i c u l a r s i t u a t i o n , the decision maker has a clear understanding of the extent that the associated objective is achieved. An a t t r i b u t e is measurable i f i t is reasonable both (a) to obtain a p r o b a b i l i t y d i s t r i b u t i o n for each a l t e r n a t i v e over the possible l e v e l s of the a t t r i b u t e - or in extreme cases to assign a point value - and (b) to assess the decision maker's preferences for d i f f e r e n t possible l e v e l s of the a t t r i b u t e , for example, in terms of a u t i l i t y function or, in some circumstances, a rank ordering. Attributes which are both comprehensive and measurable chosen for the nine recreational a c t i v i t i e s are presented and discussed i n Appendix 1. 12 2.5 D e t e r m i n a t i o n o f N u m e r i c a l Ranges f o r t h e A t t r i b u t e s A f t e r t h e a t t r i b u t e s have b e e n c h o s e n , t h e n u m e r i c a l r a n g e s f o r them a r e a s s e s s e d . T h i s i s a c c o m p l i s h e d by a s s e s s i n g t h e h i g h e s t and l o w e s t p o s s i b l e v a l u e s o f t h e a t t r i b u t e s t h a t a d e c i s i o n maker w i l l e n c o u n t e r on t h e mine w a s t e a r e a . The r a n g e o f t r a v e l t i m e f o r a l l t h e a t t r i b u t e s was c h o s e n t o be between 0 and 4 h o u r s . The 10 km x 10 km g r i d s e c t i o n a l l o w s a maximum t r a v e l t i m e o f 23.6 m i n u t e s t o and f r o m any a r e a w i t h i n t h e s e c t i o n ; however, t h e l o c a t i o n o f t h e town c a n be o u t s i d e t h e s e c t i o n and f r o m t h e a u t h o r ' s e x p e r i e n c e i n t h e E l k R i v e r V a l l e y , 4 h o u r s w o u l d be t h e maximum t r a v e l t i m e t h e r e s i d e n t s o f t h e E l k R i v e r V a l l e y w o u l d d r i v e t o a r e c r e a t i o n a l a r e a w i t h i n t h e V a l l e y . The r a n g e o f l e n g t h o f t r a i l was c h o s e n t o be between 0 and 20 km b e c a u s e 20 km i s v e r y c l o s e t o t h e t r a i l l e n g t h f o r 9 g r i d s q u a r e s (18.4 km) w h i c h i s t h e maximum number o f g r i d s q u a r e s t h e o b j e c t i v e f u n c t i o n a l g o r i t h m i n P a r t 2 w i l l examine a t any one t i m e t o c a l c u l a t e t r a i l d i s t a n c e . The r a n g e o f a v e r a g e s l o p e o f an a r e a was c h o s e n t o be between 0 and 50 p e r c e n t f o r a l l t h e a c t i v i t i e s e x c e p t f o r f o u r -w h e e l d r i v i n g and d o w n h i l l s k i i n g w h i c h have s l o p e r a n g e s between 0 and 60 p e r c e n t and 0 and 120 p e r c e n t r e s p e c t i v e l y . A l t h o u g h some a v e r a g e s l o p e s e n c o u n t e r e d i n t h e E l k R i v e r V a l l e y a r e i n e x c e s s o f 120 p e r c e n t , t h e c h o s e n maximum s l o p e v a l u e s a r e r e a l -i s t i c l i m i t i n g v a l u e s f o r t h e a c t i v i t i e s ; t h e r e f o r e , any v a l u e s o f s l o p e o v e r t h e s e maximum v a l u e s a r e assumed t o have a c o r -r e s p o n d i n g u t i l i t y o f z e r o . The a v e r a g e w i n t e r snow d e p t h o f t h e E l k R i v e r V a l l e y r a n g e s 13 from 0 to 244 cm or 8 feet. These ranges were chosen for the winter a c t i v i t i e s . The ranges of distance to a drinking water source for the a c t i v i t i e s hiking and camping were chosen to be between 0 and 10 km because 10 km i s the maximum length or width of the 10 km x 10 km g r i d section. 14 2.6 Assessment of Attribute U t i l i t y Functions In t h i s study a recre a t i o n a l a c t i v i t y i s composed of several a t t r i b u t e s . An alternative i s defined as an amount of an a t t r i b u t e . For example, the range of an attribute X may be from 0 to 10. An alternative may be any l e v e l from 0 to 10. for instance, 6 or 7. An outcome i s defined as a choice among several a l t e r n a t i v e s . For example, assume a decision maker i s faced with a gamble where he has a chance of receiving a l t e r -natives 0 or 10 with equal p r o b a b i l i t y . Whichever alternative he receives, 0 or 10, i s the outcome of the gamble. The von Neumann-Morgenstern expected u t i l i t y theory states that for a given set of al t e r n a t i v e s , A^, A2-..A n), with preference rankings A Q ^ A^ ^ A n, a decision maker can specify a p r o b a b i l i t y p such that the following outcomes are i n d i f f e r e n t : (i) Certain outcome (receive A^ for sure) ( i i ) Risky outcome (a) a p(A^) p r o b a b i l i t y of receiving Afi (best possible alternative) (b) a l-p(A^) p r o b a b i l i t y of receiving A Q (worst possible alternative) where U(A R) = 1 and U(A Q) =0 U = u t i l i t y . The choice betv/een the outcomes can be represented by Figure 3. F i g . 3 . Choice between two outcomes. 1 5 The e x p e c t e d u t i l i t y o f t h e r i s k y outcome i s p U ( A n ) + l - p U ( A o ) . I f t h e outcomes a r e i n d i f f e r e n t , t h e u t i l i t y o f t h e c e r t a i n o u t -come i s e q u a l t o t h e e x p e c t e d u t i l i t y o f t h e r i s k y outcome. The c h o i c e between t h e two outcomes i s c a l l e d a l o t t e r y . The r i s k y outcome i s c a l l e d a gamble. The v a l u e A^ where t h e gamble and t h e c e r t a i n outcome a r e i n d i f f e r e n t i s c a l l e d t h e c e r t a i n t y e q u i v a l e n t o f t h e l o t t e r y . The method u s e d i n t h i s s t u d y f o r a s s e s s i n g a t t r i b u t e u t i l i t y f u n c t i o n s i s t h e f i x e d p r o b a b i l i t y method (Keeney and R a i f f a 1 9 7 6 ) . I t i s b a s e d on t h e von Neumann-Morgenstern t h e o r y . The f i x e d p r o b a b i l i t y method f i x e s t h e v a l u e s o f p and 1-p a t 0.5. The e x p e c t e d u t i l i t y o f t h e gamble i s D i f f e r e n t gambles a r e s e t up i n l o t t e r i e s w i t h e x p e c t e d u t i l i t i e s o f 0.25, 0.5 and 0.75 and c e r t a i n t y e q u i v a l e n t v a l u e s a r e t h e n a s s e s s e d . The u t i l i t y f u n c t i o n c a n be drawn by p l o t t i n g t h e 0.5U(A Q) + 0 . 5 U ( A n ) . c e r t a i n t y e q u i v a l e n t v a l u e s on a g r a p h as i l l u s t r a t e d by t h e example i n F i g u r e 4. 1.00H U T I L I T Y 0.75H 0.50H 0 .25H 0.00 A. n F i g 4. G r a p h r e p r e s e n t i n g a u t i l i t y f u n c t i o n . 16 The fixed p r o b a b i l i t y method employs an interview tech-nique based on that used by Keeney (1977b) and Keeney (1980) for assessing decision maker's attribute u t i l i t y functions. The interview procedure begins with the assessor f a m i l i a r -i z i n g the decision maker with the terminology and motivation for the assessment. The concept of u t i l i t y theory should be explained i n simple terms so the decision maker r e a l i z e s the purpose of assessing his preference structure for attributes and i s motivated to think hard about his feelings concerning various outcomes (Keeney and R a i f f a 1976). An example of d i a -logue i n t h i s study to f a m i l i a r i z e the decision maker with the terminology and motivation for the assessment i s presented i n Appendix 2 for the a c t i v i t y horseback r i d i n g . The next step i n the interview procedure i s for the assessor to ask the decision maker a series of simple hypothet-i c a l questions using the fixed p r o b a b i l i t y method to obtain the decision maker's preferences over attribute levels r e s u l t i n g i n a u t i l i t y function (Keeney 1977b). An example of dialogue used i n t h i s study to assess the u t i l i t y function for the horse-back r i d i n g attribute t r a v e l time i s presented i n Appendix 3. 2.61 Assumptions of the Assessment Procedure 17 The fix e d p r o b a b i l i t y method of att r i b u t e u t i l i t y function assessment i s based on the axioms of the von Neumann-Morgenstern expected u t i l i t y theory. I f a decision maker accepts these assumptions, then i t i s possible to assign a single r e a l number c a l l e d a u t i l i t y i n the set of possible outcomes of l o t t e r i e s involving a t t r i b u t e l e v e l s . From these numbers, the expected u t i l i t y can be calculated for any choice considered by the assumptions. A decision maker i s then compelled to prefer the outcome with the highest expected u t i l i t y and to be i n d i f f e r e n t between outcomes with equal expected u t i l i t i e s . The following discussion presents the von Neumann-Morgenstern axioms with corresponding c r i t i c i s m s . Symbolism ^ represents "preferred to" ^ represents "less preferred than" r^J represents " i n d i f f e r e n t to" represents a 50-50 gamble between alternatives A 1 and A ? represents a p r o b a b i l i t y between 0 and 1 Assumption 1 For any two alternatives A-j^  and A 2, one and only one of the following r e l a t i o n s i s true: A1 ; A2 A l } A2' A2 V A l ° r A l ^  A2 18 This assumption implies that any two alter n a t i v e s are d i r e c t l y comparable. Either one i s preferred to the other or the two are equally preferred. This assumption has been c r i t i c i z e d by Lee (1971) on the grounds that r e a l i s t i c a l l y , when outcomes are valued about equally, a decision maker believes he prefers A^ at one moment, A 2 the next moment and shortly thereafter cannot make up his mind. Is he r e a l l y i n d i f f e r e n t between A^ and A 2? I f t h i s i s true, then one of these three r e l a t i o n s cannot be true unless r e f e r -ring to a moment i n time. Because a decision maker cannot demon-strate preference inconsistency i n a "moment", the assumption cannot be tested e m p i r i c a l l y . Many researchers believe t h i s uncertainty i s a factor which shapes the u t i l i t y function and not outside of i t . Any increased aversion towards r i s k w i l l make the u t i l i t y function more concave; any enjoyment of gamb-l i n g (risk seeking, discussed i n Section 2.63) w i l l make the u t i l i t y function more convex (Kauder 1965) . Assumption 2 Given the three a l t e r n a t i v e s , A^, A 2 and A^, i f A 1 ^ A 2 and A 2 ^ A 3, then A l ^  A 3 • This assumption states that the preference r e l a t i o n (^) i s t r a n s i t i v e ; i f a decision maker prefers A^ to Aj and ^  to A^, then he w i l l prefer to h^. 19 Luce and R a i f f a (1957) state that Assumption 2 can be c r i t i c i z e d because i t does not conform to behaviour that res-u l t s when decision makers are presented with a sequence of paired comparisons. Decision makers only have vague l i k e s and d i s l i k e s and can e r r i n reporting them; however, when decision makers are made aware of these i n t r a n s i t i v i t i e s , they w i l l often r e a l i g n t h e i r responses to a t r a n s i t i v e ordering. Assumption 3 I f A, > A 2 then A 1 > pAlf- (l-p ) A 2 f o r any p, O Z p Z l . This assumption states that i f A^ i s preferred to A 2, then a decision maker prefers the c e r t a i n outcome A^ to a gamble which could give him A^ at best or might give him a less valued outcome Assumption 4 If A x ^ A 2 for any p, O ^ p ^ l . then pA^- (l-p)A 2 This assumption states that i f A^ ^ i s less preferred than A 2, a decision maker prefers a gamble which would give him A^ at worst and possibly the more preferred A 2, to the certain outcome A . This i s the dual of Assumption 3. 20 A s s u m p t i o n 5 I f A i ^ A 2 y A 3 ' t n e n t h e r e e x i s t s a p s u c h t h a t p A 1 ; ( l - p ) A 3 > A, T h i s a s s u m p t i o n s t a t e s t h a t i f i s p r e f e r r e d t o A 2 , and A 2 i s p r e f e r r e d t o A^ * t h e n t h e r e i s some gamble i n v o l v i n g A^ and A 3 t h a t i s p r e f e r r e d t o A 2 . A s s u m p t i o n 6 I f A^ ^ A 2 ^ A 3 ' t n e n t h e r e e x i s t s a p s u c h t h a t p A 1 ? ( l - p ) A 3 T h i s a s s u m p t i o n s t a t e s t h a t i f A 3 i s p r e f e r r e d t o A 2 , and A 2 i s p r e f e r r e d t o A^ , t h e n A 2 i s p r e f e r r e d t o some gamble i n v o l v i n g A^ and A 3 . T h i s i s t h e d u a l o f A s s u m p t i o n 5. L u c e and R a i f f a (1957) s t a t e t h a t a l t h o u g h A s s u m p t i o n 5 and A s s u m p t i o n 6 seem r e a l i s t i c , t h e r e a r e e x a m p l e s where t h e y a r e n o t u n i v e r s a l l y a p p l i c a b l e . T h ey c i t e an example where most p e o p l e p r e f e r $1.00 t o 1* t o d e a t h , b u t w o u l d one be i n d i f f e r e n t b e t w e e n IC and a gamble i n v o l v i n g $1.00 and d e a t h ? The gamble w o u l d be p r e f e r r e d i f t h e c h a n c e o f d e a t h was v e r y low, f o r exa m p l e 1 0 1 0 0 0 , b e c a u s e o f i t s low p r o b a b i l i t y o f o c c u r r i n g . A l t h o u g h t h e s e a s s u m p t i o n s may n o t be u n i v e r s a l l y a p p l i c a b l e , few a p p l i c a t i o n s h a v e e x t r e m e a l t e r n a t i v e s s u c h as d e a t h . No e x t r e m e a l t e r n a t i v e s a r e c o n t a i n e d i n t h i s s t u d y . 21 Assumption 7 ^ f d - p J A j j ^ jV-pJA^-pA^ This assumption states that the arrangement of alter n a t i v e s i n a gamble does not a f f e c t t h e i r preference. Assumption 8 I f A x nd A 3, then i^pA1; (1-p) A 2j £pA 3 ; (1-p) A 2j f o r any p and A 2. This assumption states that i f A^ appears i n any gamble, and i f A 3 i s i n d i f f e r e n t to A^, then i f A 3 i s substituted f o r A 1 i n the gamble, the two gambles w i l l be i n d i f f e r e n t . 22 2.62 Strengths and Weaknesses of the Assessment Procedure  Strengths of the Assessment Procedure Festinger (1957) showed i n his theory of cognitive disson-ance, that the more d i f f i c u l t y a decision maker has i n making a decision, the greater i s the tendency fo r him to j u s t i f y the decision by increasing the attractiveness of his decision and by decreasing the attractiveness of the rejected a l t e r n a t i v e s . In general, decision makers w i l l also avoid information that does not support t h e i r decision. These behaviourisms r e s u l t i n less emphasis on o b j e c t i v i t y and more p a r t i a l i t y i n the decision (Festinger 1964). I t i s therefore desirable to simplify a large decision problem, reducing the number of things a decision maker must keep i n perspective at the same time. This w i l l r e s u l t i n reduced cognitive s t r a i n on the decision maker and less bias i n the decision. The main strength of multi-attribute u t i l i t y analysis i s that i t breaks a large decision problem into smaller problems. The a t t r i b u t e u t i l i t y function assessment procedure further reduces the problem to choices among att r i b u t e levels i n simple hypothetical l o t t e r i e s ; therefore, the cognitive s t r a i n on the decision maker i s greatly reduced. Another strength of the assessment procedure i s that i t i s based on a u t i l i t y theory, providing sound procedures for formal-i z i n g and integrating judgments and preferences of decision makers. Assumptions of the methodology can also be e x p l i c i t l y stated. The assessment procedure allows for decision makers' prefer-e n c e s t o be i n t e g r a t e d i n t o a l o g i c a l framework so r e g u l a t o r y a u t h o r i t i e s and o t h e r p o l i t i c a l b o d i e s c a n f u l l y s e e where t h e d a t a came f r o m and how and why t h e r e s u l t i n g m o d e l was c o n -s t r u c t e d . The a s s e s s m e n t p r o c e d u r e r e s u l t s i n a n u m e r i c a l a s s e s s m e n t w h i c h i n t u r n c a n be u s e d t o d e v e l o p an o b j e c t i v e f u n c t i o n f o r a mine w a s t e r e c r e a t i o n p l a n . B e c a u s e t h e o b j e c t i v e f u n c t i o n has a n u m e r i c a l v a l u e , i t c a n be e v a l u a t e d and m a x i m i z e d by c o m p u t e r r o u t i n e s . Weaknesses o f t h e A s s e s s m e n t P r o c e d u r e The m a i n r e s u l t o f t h e v o n Neumann-Morgenstern e x p e c t e d u t i l i t y t h e o r y i s t h a t d e c i s i o n makers s h o u l d f i r s t c h o o s e o u t -comes i n gambles i n v o l v i n g a t t r i b u t e l e v e l s w h i c h have t h e h i g h -e s t e x p e c t e d u t i l i t y . E x p e r i m e n t s by Edwards (1953; 1954) showed t h a t d e c i s i o n m akers a p p e a r t o have t h e i r own n o t i o n s o f p r o b a b i l i t y s u c h t h a t t h e y w i l l a c t n o t i n a c c o r d a n c e w i t h t h e t r u e p r o b a b i l i t i e s d e s -c r i b e d by t h e gamble, b u t i n a c c o r d a n c e w i t h t h e i r own e s t i m a t e s o f t h e p r o b a b i l i t i e s (Churchman 1 9 6 1 ) . I n t h e s e e x p e r i m e n t s , d e c i s i o n makers p r e f e r r e d gambles w i t h c e r t a i n c o m b i n a t i o n s o f p r o b a b i l i t i e s o v e r o t h e r gambles e v e n t h o u g h t h e i r p r e f e r r e d g a m b l e s h a d l o w e r e x p e c t e d v a l u e s . The a s s e s s m e n t p r o c e d u r e u s e d i n t h i s s t u d y f i x e s t h e p r o b a b i l i t i e s i n e a c h gamble a t 0.5 t h e r e f o r e , p r e f e r e n c e s f o r p r o b a b i l i t y c o m b i n a t i o n s a r e e l i m i n * -a t e d . The q u e s t i o n s t i l l r e m a i n s w h e t h e r t h e d e c i s i o n maker's e s t i m a t e s o f t h e p r o b a b i l i t i e s i n t h e gambles (0.5) a r e i n a c c o r d a n c e w i t h t h e t r u e p r o b a b i l i t i e s d e s c r i b e d by t h e gamble. 24 Another weakness of the assessment procedure i s that a decision maker i s assuming that there i s no cost to him from choosing cert a i n attribute levels i n the l o t t e r i e s . For example, a decision maker may place a ce r t a i n value on a l e v e l of an att r i b u t e ; however, t h i s value may change i f he must share the cost of developing the mine waste to obtain t h i s attribute l e v e l . The assessment procedure does not take into account cost and other factors which may influence the decision maker's prefer-ences for at t r i b u t e l e v e l s . Three studies have shown a high c o r r e l a t i o n between d i r e c t assessment of u t i l i t i e s by asking decision makers to d i r e c t l y evaluate items and the Keeney-Raiffa fixed p r o b a b i l i t y method of u t i l i t y assessment. Von Winterfedlt (1971, as c i t e d by Fischer 1973) conducted a study where decision makers were asked to d i r e c t l y evaluate the attractiveness of hypothetical apartments described by fourteen a t t r i b u t e s . Then the decision makers were asked to assess t h e i r u t i l i t y functions for the attributes by the fixed p r o b a b i l i t y method. A mean c o r r e l a t i o n of R = 0.84 was obtained between the d i r e c t assessment and fixed p r o b a b i l i t y methods when the additive u t i l i t y model was used with the u t i l i t y functions (Section 2.91). Fischer (1972, c i t e d by Fischer 1973) conducted a study on the attractiveness of cars with eight a t t r i b u t e s . A median cor r e l a t i o n of R = 0.93 between the d i r e c t assessment method and the fixed p r o b a b i l i t y method using the additive u t i l i t y model was obtained. Fischer (1973, c i t e d by Fischer 1973) i n another experiment involving decision makers' preferences for jobs involving three attributes found a median c o r r e l a t i o n of R = 0.93 between the d i r e c t assess-ment and fixed p r o b a b i l i t y methods using the additive u t i l i t y m o d e l . The r e s u l t s i l l u s t r a t e a good p r e d i c t i v e n e s s o f t h e f i x e d p r o b a b i l i t y method o f a c t u a l d e c i s i o n makers* p r e f e r e n c e s ; h owever, t h e s e s t u d i e s i n v o l v e d s i m p l e p r o b l e m s w i t h s i t u a t i o n s f a m i l i a r t o t h e d e c i s i o n m a k e r s . The w e i g h t o f e m p i r i c a l e v i d -e n c e u s i n g von Neumann-Morgenstern l o t t e r i e s i n v o l v i n g c h o i c e s among l o t t e r y t i c k e t s , a t v a r i o u s o d d s , f o r s m a l l amounts o f money, c o n c l u d e s t h a t most d e c i s i o n makers c h o o s e i n a way t h a t i s r e a s o n a b l y c o n s i s t e n t w i t h t h e axioms o f t h e t h e o r y . T h ey behave as t h o u g h t h e y were m a x i m i z i n g t h e e x p e c t e d v a l u e o f u t i l i t y as t h o u g h t h e u t i l i t i e s o f s e v e r a l a l t e r n a t i v e s c a n be m e a s u r e d (Edwards 1954, as c i t e d b y Simon 1 9 5 9 ) . When t h e e x p e r i m e n t s a r e e x t e n d e d t o more r e a l - l i f e s i t u a t i o n s , i t i s n o t c l e a r t h a t d e c i s i o n makers behave i n a c c o r d a n c e w i t h t h e u t i l i t y a x i o m s . T h e r e i s some i n d i c a t i o n t h a t when t h e c h o i c e s a r e s i m p l e , where t h e d e c i s i o n maker c a n s e e and remember when he i s b e i n g c o n s i s t e n t , t h e d e c i s i o n maker w i l l behave as t o m a x i m i z e h i s e x p e c t e d u t i l i t y . As t h e c h o i c e s become more c o m p l i c a t e d , he becomes much l e s s c o n s i s t e n t (May 1954, as c i t e d by Simon 1959, D a v i d s o n an d Suppes 1957, as c i t e d by Simon 1 9 5 9 ) . The a s s e s s m e n t p r o c e d u r e does n o t t a k e i n t o a c c o u n t t h e r e f e r e n c e e f f e c t d e s c r i b e d by T v e r s k y (1977) , whereby a t t r i b u t e s a r e o f t e n p e r c e i v e d and e v a l u a t e d w i t h r e s p e c t t o some r e f e r e n c e p o i n t o r a d a p t a t i o n l e v e l , p r o v i d e d by p a s t and p r e s e n t e x p e r -i e n c e . Outcomes t h a t l i e above t h e r e f e r e n c e p o i n t a r e p e r c e i v e d as p o s i t i v e ; t h o s e b e l o w n e g a t i v e . Changes i n p r e f e r e n c e s f r o m s h i f t s i n t h e r e f e r e n c e p o i n t a r e t e r m e d r e f e r e n c e e f f e c t s . The manner i n w h i c h t h e p r o b l e m i s p r e s e n t e d d e t e r m i n e s a d e c i s i o n maker's r e f e r e n c e p o i n t w h i c h , i n t u r n , d e t e r m i n e s a d e c i s i o n maker's u t i l i t y f u n c t i o n . P e o p l e a r e more s e n s i t i v e t o n e g a t i v e changes that p o s i t i v e changes and the wording of the question can therefore have an e f f e c t on the reference point location. Tversky showed that for attributes involving sensory or percept-ual q u a l i t i e s , s e n s i t i v i t y to change i n att r i b u t e l e v e l s decreases as a decision maker moves from his reference l e v e l . Another shortcoming of the assessment procedure i s the large time commitment required to complete the assessment. A study by Dennis (1979) showed that i f decision makers were not used to the formal procedure required by the assessment, they were reluctant to p a r t i c i p a t e f u l l y i n the assessment. 27 2.63 A Decision Maker's Attitude Toward Risk The functional form of a decision maker's at t r i b u t e u t i l i t y function i s determined by his basic attitudes towards r i s k . The following discussion on r i s k i s condensed from Keeney and Raiffa's (1976) chapter on unidimensional u t i l i t y theory. The u t i l i t y function i l l u s t r a t e d i n Figure 5 i s a straight l i n e u t i l i t y function for an attribute X with le v e l s ranging from 0 to 10. I t i s referred to i n the following discussion. The ex-pected outcome of a 50-50 gamble i s defined as the average of the two reference l e v e l s i n the gamble. Thus, f o r a gamble between 0 and 10, the expected outcome i s 5. A decision maker's r i s k premium i s defined as the expected outcome of a gamble minus the certainty equivalent of a l o t t e r y involving the gamble. I t i s the amount of an attribute a.decision maker i s w i l l i n g to f o r f e i t from the expected outcome to avoid the r i s k s associated with a p a r t i c u l a r gamble. The u t i l i t y function i n Figure 5 has a r i s k premium of zero over a l l l e v e l s of attribute X. A decision maker with such a u t i l i t y function i s termed r i s k neutral or, i n other words, he does not prefer to either avoid r i s k s or seek r i s k s associated with the gamble. Fi g . 5 . Attribute u t i l i t y function i l l u s t r a t i n g a r i s k neutral attitude. If a decision maker's certainty equivalent of a l o t t e r y i s less than the expected outcome of the gamble, f o r example a cer-t a i n t y equivalent of 4 i n Figure 5, then the r i s k premium i s p o s i t i v e and equal to 1. The decision maker's attitude r e f l e c t s that he prefers to avoid r i s k s associated with the gamble and i s w i l l i n g to f o r f e i t some amount, i n t h i s case 1 unit, of attribute X to avoid accepting the gamble. This attitude i s termed r i s k averse. I f a r i s k premium i s p o s i t i v e for a l l l o t t e r i e s over the range of X, f o r increasing and decreasing u t i l i t y functions, a decision maker i s said to be r i s k averse and h i s u t i l i t y function w i l l always be concave as i l l u s t r a t e d i n Figure S-X A F i g . 6- Concave increasing and decreasing u t i l i t y functions i n d i c a t i n g a r i s k averse attitude. A decision maker i s increasingly r i s k averse i f (1) he i s r i s k averse, and (2) his r i s k premium for any l o t t e r y increases as the reference l e v e l s of the attribute i n the l o t t e r y increase. The r i s k premium a decision maker i s w i l l i n g to pay to avoid the gamble increases with increasing reference values i n the gamble. His u t i l i t y function becomes more concave as X increases. A decision maker i s decreasingly r i s k averse i f Cl) he i s r i s k averse and (2) his r i s k premium f o r any l o t t e r y decreases as the reference le v e l s of the att r i b u t e i n the l o t t e r y increase. The r i s k premium a decision maker i s w i l l i n g to pay to avoid the lo t t e r y decreases with increasing reference values i n the gamble. His u t i l i t y function becomes less concave as X increases. If a decision maker's certainty equivalent of a l o t t e r y i s greater than the expected outcome of the gamble, for example a certainty equivalent of 6 i n Figure 5 , then the r i s k premium i s negative, equal to -1. The decision maker's attitude r e f l e c t s that he prefers to take r i s k s associated with the gamble and i s not w i l l i n g to f o r f e i t units of attribute X to avoid these r i s k s . This attitude i s termed r i s k seeking. If a decision maker's r i s k premium i s negative f o r a l l l o t t e r i e s over the range of X, for increasing and decreasing u t i l i t y functions, he i s said to be r i s k seeking and his u t i l i t y function w i l l be con-vex as i l l u s t r a t e d i n Figure 7. Y X Y X Fi g . 7. Convex increasing and decreasing u t i l i t y functions i n d i c a t i n g a r i s k seeking attitude. A decision maker i s increasingly r i s k seeking i f (1) he i s r i s k seeking and (2) his r i s k premium for any l o t t e r y decreases as the reference levels of the attribute i n the lo t t e r y increase The r i s k premium a decision maker i s w i l l i n g to pay to avoid the gamble decreases with increasing reference values i n the gamble. His u t i l i t y function becomes more convex as X increases. A decision maker i s decreasingly r i s k seeking i f (1) he i s r i s k seeking and (2) his r i s k premium for any l o t t e r y increases as the reference le v e l s of the attribute i n the l o t t e r y increase The r i s k premium a decision maker i s w i l l i n g to pay to avoid the gamble increases with increasing reference values i n the gamble. His u t i l i t y function becomes less convex as X increases. 30 2.7 V e r i f i c a t i o n of Independence Properties of Attributes Multi- a t t r i b u t e u t i l i t y functions are v a l i d only when cer-t a i n independence properties concerning attributes are true (Keeney 1974; 1977a; 1977b, Keeney and R a i f f a 1976). The independence properties of concern to t h i s methodology are u t i l i t y independence, p r e f e r e n t i a l independence and additive independence. 2.71 U t i l i t y Independence Keeney (1970; 1972a; 1974; 1977a) and Keeney and R a i f f a (1976) showed that given the set of attributes (X^, X 2 , . . • x n ) / then X^ i s u t i l i t y independent of the other attributes i f the preference order for l o t t e r i e s over X^, given the other a t t r i -bute leve l s are fixed, does not depend on the l e v e l where those attributes are fixed. To test for u t i l i t y independence between attributes X., and X 0, Figure 8 can be used. P • -• Q 1 P -• Q 0 0 X 1 F i g . 8. Graph to test for u t i l i t y independence between attributes X, and X_. 31 The attribute X 2 i s fixed (eg. at P). Lotteries are then set up between the 50-50 gamble of P and Q and one of the other attribute Xn levels as i l l u s t r a t e d by Figure 9. Fi g . 9 . Lottery between attribute l e v e l X^ and'a 50-50 gamble between P and Q. The levels of X^ are changed u n t i l the l e v e l of X^ i s found where the decision maker i s i n d i f f e r e n t between the two a l t e r -natives. The l e v e l of X^ i s then fixed at P 1 and the procedure i s repeated. I f the decision maker chooses the same l e v e l of X^ to be i n d i f f e r e n t between the two al t e r n a t i v e s , then X^ i s said to be u t i l i t y independent of X^. An example of dialogue i n t h i s study used to v e r i f y u t i l i t y independence between two horseback r i d i n g attributes i s presented i n Appendix 3. 32 2.72 P r e f e r e n t i a l Independence Keeney (1972a; 1974; 1977a) and Keeney and R a i f f a (1976) showed that given a set of attributes (X^, X 2 » . . . X n ) , the p a i r of attributes (X^rX^) i s said to be p r e f e r e n t i a l l y independent of other attributes i f value tradeoffs between X 1 and X 2 do not depend on the levels of the other a t t r i b u t e s . To test for p r e f e r e n t i a l independence, Figure 10 can be used. X, X^ at fixed value = A 4 X, X 3 at fixed value = B .Indifference Points .Indifference Points X, X, F i g . 10. Graphs to test for p r e f e r e n t i a l independence between the attribute p a i r (X l fX 2) and the attribute X^. Whi le f i x i n g the l e v e l of X 3 at some value A, a point (X l fX 2) i i chosen and the decision maker i s asked to choose a l e v e l X 2 ' for a d i f f e r e n t l e v e l of X-L = X 1 1 so that he would be i n d i f f e r e n t between the two points. X 3 i s then fixed at a d i f f e r e n t l e v e l B and the procedure i s repeated. I f the proportional change i n indifference points does not very s i g n i f i c a n t l y , then X^ and X 2 are said to be p r e f e r e n t i a l l y independent of X 3. Given the set of attributes ( X ^ X 2,...X n), then X 3 through X R would be fixed to see i f ( X 1 , X 2 ) was p r e f e r e n t i a l l y independent of X 3 through X n < An example of dialogue i n t h i s study used to v e r i f y prefer-e n t i a l independence i s presented i n Appendix 4. 33 2.73 Additive Independence Attributes X^ and X 2 are additive independent i f the follow-ing l o t t e r i e s are found to be i n d i f f e r e n t to a decision maker: C(x 1 ,x 2) L 2 < - o (X 1',X 2') °-5^(X 1,X 2') These l o t t e r i e s must be i n d i f f e r e n t over a l l ranges of X 1 and X2_ (Keeney and Raif f a 1976) . To i l l u s t r a t e , take the attributes t r a v e l time and length of t r a i l X 2 6 and X 2 7, for the a c t i v i t y horseback r i d i n g . The range of t r a v e l time i s between 0 and 240 minutes. The range of length of t r a i l i s between 0 and 20 km. The following l o t t e r i e s are <> ^(0,0) 5^(0,20) °.,$-\(240,20) In l o t t e r y , the expected value of the gamble i s 0.5U(240) + 0.5U(0) + 0.5U(0) + 0.5U(20) or 0.5(0) + 0.5(0) + 0.5(1) + 0.5(1) which i s equal to 1.0. Notice that i n there i s a 50-50 chance of getting both attributes at t h e i r best values and both at t h e i r worst values. In l o t t e r y L 2 the expected value of the gamble i s 0.5U(0) + 0.5U(0) + 0.5U(240) + 0.5U(20) or 0.5(1) + 0.5(0) + 0.5(0) + 0.5(1) 34 which i s equal to 1. Therefore and have the same ex-pected value. Notice that i n the decision maker w i l l always receive one attribute at i t s best l e v e l and one at i t s worst. If a l l the attributes for an a c t i v i t y are additive i n -dependent, mutual u t i l i t y independent and p r e f e r e n t i a l l y i n -dependent then the additive u t i l i t y function i s the appropriate u t i l i t y function for the a c t i v i t y . 35 2 . 8 Assessment of Scaling Constants for the Attributes To assess the r e l a t i v e importance of each at t r i b u t e to a recreational a c t i v i t y , s c a l i n g constants k^ must be assessed for each attribute X.. 1 Scaling constants can be determined by choosing an a t t r i -bute, for example X 2 6, and examining tradeoffs between X 2g and the other a t t r i b u t e s . From these tradeoffs, equations are dev-eloped for s c a l i n g constants i n terms of other s c a l i n g con-stants. The following example w i l l be used to i l l u s t r a t e the technique (Keeney and Rai f f a 1 9 7 6 ) . Given the following tradeoff, represented by Figure 1 1 , between attributes X 2g and X 2 7 for the a c t i v i t y horseback r i d i n g , the decision maker chooses points A and B to be i n d i f f e r e n t . 2 4 0 - , 180 J B • ( 1 3 , 1 8 0 ) Travel Time (minutes) 1 2 0 J X 26 604 0 0 10 15 20 X 27 Length of T r a i l (km) . 1 1 . Tradeoff between attributes X_, and X 27* Because points A and B are i n d i f f e r e n t , the following equation i s true: k 2 7 U 2 7 ( 4 . 5 ) + k 2 6 U 2 6 ( 0 ) = k 2 7 U 2 7 ( 1 3 ) + k 2 6U 2 6(180) (7) From the u t i l i t y functions f o r attributes X 2g and X 2 7 i n Appendix 5, U 2 6(0) = 1.00 U 2 ? ( 4 . 5 ) = 0.130 U 2 6(180) = 0.250 u 2 7 ( 1 3 ) = 0.750 Substituting into equation (7) , k 2 7(0.130) + k 2 6(1.00) = k 2 ?(0.750) +k 2 g(0.250) k 2 6(0.750) = k 2 7(0.620) therefore, k 2 g = 0.827k 2 7 This procedure i s continued for a l l the attribute tradeoff combinations. 37 2.81 Consistency Checks Before assessing s c a l i n g constants for the at t r i b u t e s , the decision maker i s asked to rank the attributes i n order of importance to the success of the a c t i v i t y i n question. Assessed s c a l i n g constant values should be consistent with the decision maker's ranking of the at t r i b u t e s . Several tradeoffs between the attributes should be examined and calculations made to check the ra t i o s of the two s c a l i n g constants for each tradeoff. For example, i f for one tradeoff between attributes X_, and X„_, the r a t i o of scal i n g constants 26 27 k26^ k27 a n d f o r a n o t h e r tradeoff, k 2 g / k 2 7 i s 3.0, then where the r a t i o i s 3.0, the decision maker considers the a t t r i -bute X 2g to be 3 times as important as attribute X 2 7, and where the r a t i o i s 0.75, the decision maker considers X„^ to be 0.75 2 6 times as important as X 2 7. From the ra t i o s i t i s c l e a r that the decision maker i s not consistent i n his preferences. I f the ratio s are not reasonably close between the tradeoffs, then a decision maker should be t o l d how he i s inconsistent with his assessed attribute u t i l i t y functions and asked whether he wants to change one of his tradeoffs. I f he does not want to change one of his tradeoffs, then other tradeoffs between the same attributes should be conducted to obtain some consistency i n the scali n g constant r a t i o s . An example of dialogue i n t h i s study used to assess the sca l i n g constants with consistency checks between tradeoffs i s presented i n Appendix 4. Another method for checking the decision maker's consistency for attribute preference i s to examine the scal i n g constant equat-ions that do not involve the attribute against which other attributes are traded. From these equations, several values for each scaling constant are assessed. To check for consistency, the s c a l i n g constant f o r the at t r i b u t e against which other attributes are traded i s equated to 1.0 and the other scaling constants are assessed. The d i f f e r e n t values for an i n d i v i d u a l s c a l i n g constant should not vary s i g n i f i c a n t l y . For example, the s c a l i n g constant k f o r the a c t i v i t y horseback 26 r i d i n g i s equated to 1.0. The scaling constant k 2g i s expressed i n the following equations: k28 = k 2 6 / 0 , 6 4 4 = 1 , 5 5 k28 = k 2 7 / 0 ' 6 5 5 = 1 - 5 8 where k 2 g = 1.00 and k 2 7 = 1.03. The value 1.63 does not vary s i g n i f i c a n t l y from the 1.55 value of k 2g; therefore, the decision maker i s said to be consistent i n his r e l a t i v e preference f o r the attribute X ? Q. 39 2.82 Evaluating the Scaling Constant Against Which Other A t t r i - butes are Traded for the Multiplicative U t i l i t y Function In the course of determining the scaling constant equations, a l l the attributes are traded off against one attribute, for example X 2 g for the activity horseback riding. Therefore the value of the scaling constant k 0 / r i s not known. To determine the value of k 26' t* i e decision maker i s asked to choose probabilities p and q i n the lottery illustrated in Figure 12, so that he i s indifferent between the gamble and the certain outcome. r X_ at i t s best level A l l other attributes at their worst level •o.^'"— A ^ attributes at their best level ^ ^ S N N — A l l attributes at their worst level Fig. 12. Lottery to determine k . The u t i l i t y of the attributes at their best level i s 1.0. The u t i l i t y of the attributes at their worst level i s 0. Therefore, the expected u t i l i t y of the gamble i s p(l.O) + q(0) = p For example, let p = 0.3 and q = 0.7. The expected u t i l i t y of the gamble i s 0.3 and because the outcomes in the lottery are indifferent, k 2 g = 0.3. k 2 g i s then substituted into the other scaling constant equations to determine the other constants. 40 2.83 Strengths and Weaknesses of the Attribute Scaling Constant  As se s sment Procedure s Without the scaling constant assessment procedures described i n Sections 2.8 to 2.82, a decision maker must keep a l l the attributes of an a c t i v i t y i n perspective at the same time when he i s conducting a tradeoff between two of the a t t r i b u t e s . The strength of the assessment procedure i s that i t breaks t h i s large tradeoff problem into several simpler tradeoff problems between two attributes at a time with a l l other attributes being held constant. This decreases the cognitive s t r a i n on the decision maker. Weaknesses of the Procedures While conducting consistency checks, the analyst must examine tradeoffs to see i f the scaling constants produced are consistent. For every set of tradeoffs he must perform the c a l c u l a t i o n described by Equation 7 i n Section 2.8. This c a l -culation i s very time consuming and decision makers may be r e l u c t -ant to devote the time to complete the assessment. To reduce t h i s time factor i n t h i s study, Equation 7 was programmed into a pocket c a l c u l a t o r so the analyst only had to enter the four a t t r i -bute l e v e l u t i l i t y values for each tradeoff to assess the scaling constant. If scaling constants are not consistent, the analyst must either conduct many tradeoffs u n t i l he receives several consis-tent values or he must prompt the decision maker to be more con-s i s t e n t with his u t i l i t y functions i n the tradeoffs. The c r i t i c -41 ism i n the l a t t e r case i s that the analyst becomes involved i n the attribute tradeoff procedure, which may bias the r e s u l t i n g s c a l i n g constant values. The l o t t e r y to determine the unknown scaling constant (Figure 12), for the m u l t i p l i c a t i v e u t i l i t y function (Section 2.82), requires the decision maker to keep a l l the attributes i n perspective at the same time which increases the cognitive s t r a i n on the decision maker; hence, the accuracy of the res-ponses to t h i s l o t t e r y may be questionable. 42 2.9 Choice of a Multi-Attribute U t i l i t y Function MacCrimmon (1973) presented an overview of methods for deal-ing with multiple objective/multiple a t t r i b u t e decision problems and a method s p e c i f i c a t i o n chart (Figure 13). Tracing through the chart, the purpose of the decision problem i s normative (trying to improve the course of action) rather than descriptive (trying to describe the course of action). A d i r e c t assessment of preferences from the fixed p r o b a b i l i t y method described i n Section 4.5 i s assumed to be v a l i d and r e l i a b l e . In t h i s study only one decision maker's choices are being considered for each recreational a c t i v i t y . The success of each recreational a c t i v i t y w i l l not solely determined by only the best (Maximax A.3.b) or worst (Maximin A.3.a) a t t r i b u t e . The alternative courses of action (recreational a c t i v i t i e s ) w i l l be chosen from a l i s t , rather than designed. The tracing results i n a choice between f i v e method types. The tradeoff method has the disadvantage of obtaining a decision maker's preferences by d i r e c t l y asking him which may r e s u l t i n the decision maker unable to verbalize his true preferences. Multi-dimensional scaling with i d e a l points (D.2.a) does not take into account i n t r a - a t t r i b u t e preferences. This study i s con-cerned with a decision maker's preferences between attributes of a recreational a c t i v i t y as well as preferences between varying levels of each a t t r i b u t e . The i n t e r and i n t r a - a t t r i b u t e weighting methods (A.2.b,c, and d) demand numerical inputs representing a decision maker's i n t e r and i n t r a - a t t r i b u t e preferences. Such numerical weighting values derived from Keeney and Raiffa's (1976) fixed p r o b a b i l i t y method have been successfully used i n 43 the additive (A.2.b) and m u l t i p l i c a t i v e (A.2.d) weighting models (Keeney 1973; 1975; 1977a; 1979; 1980, Krischer 1974, Hilborn and Walters 1977, Keeney and Rob i l l a r d 1977, Keeney et. a l . 1978, Walker 1982) . These models were chosen for use i n t h i s thesis because they have been employed successfully i n other multi-attribute decision problems, are part of Keeney and Raiffa's (1976) multi-attribute u t i l i t y analysis and t h e i r r e s u l t s can be used to develop an objective function for a recreation plan which can be used i n an optimization algorithm (Fishburn 1968). Keeney (197 4) showed that given the attribute set (X^, X 2,...X n), n =5: 3, i f X^ i s u t i l i t y independent of (X 2,...X n), and at t r i b u t e pairs are p r e f e r e n t i a l l y independent of other a t t r i b u t e s , then a multi-attribute u t i l i t y function i s either additive i f the attributes are also additive independent or m u l t i p l i c a t i v e i f the attributes are not additive independent. 44 F i g . 13. Method s p e c i f i c a t i o n chart for multiple a t t r i b u t e decision problems (MacCrimmon 1973). MULTIPLE OBJECTIVE/MULTIPLE ATTRIBUTE DECISION METHODS WEIGHTING METHODS 1. Inferred Preference a. linear Regression b. Analysis of Variance c. Quasi-linear Regression 2. Directly Assessed Preferences: General Aggregation a. Trade-offs b. Simple Additive Weighting c. Hierarchical Additive Weighting d. Quasi-Additive Weighting 3. Directly Aesessed Preferences: Specialised Aggregation a. Max loin b. Maxima* SEQUENTIAL ELIMINATION METHODS 1. Alternative versus Standard: Comparison Across Attributes a. Disjunctive and Conjunctive Constraints 2. Alternative versus Alternativet Comparison Across Attributes a. Dominance 3. Alternative versus Alternative: Comparison Across Alternatives a. Lex i cog raphy b. Elimination by Aspects MATHEMATICAL PROGRAMMING METHODS 1. Global Objective Function a. Linear Programming 2. Goals in Constraints a. Goal Programming 3. Local Objectives: Interactive a. Interactive, Multi-criterion Programming SPATIAL PROXIMITY METHODS 1. Iso-preference Graphs a. indifference Map 2. Ideal Points a. Multi-dimensional, non-metric Scaling 3 . Graphical Preferences a. Graphical Overlays METHOD SPECIFICATION CHART IS THE PURPOSE NORMATIVE RATHER THAN DESCRIPTIVE? HILL A DIRECT ASSESSMENT OF PREFERENCES BE VALID AND RELIABLE? IS A PROCESS MODEL DESIRED? HAS THIS TYPE OF SITUATION OCCURRED FREQUENTLY BEFORE7 ARE THERE MULTIPLE DECISION MAKERS KITH Fi.TCTTwr. PprrrprwEs? WILL THE RESULT OF IMPLEMENTING THE ALTERNATIVES BE DETERMINED BY ONLY THE BEST (OR WORST) ATTRIBUTE VALUES? ^^TBS ARE ALTERNATIVES TO STANDARDS THAN TO EACH COMPARED RATHER OTHER? NO YES ARE THE ALTERNATIVES TO BE DESIGNED RATHER THAN CHOSEN FROM A LIST? WHAT IS THE MOST VALID KIND OF PREFERENCE INFORMATION? WHAT IS THE MOST VALID KIND OF PREFERENCE INFORMATION? r Global Goal a Local Inter-Cojective and Tradeoffs Attribute deviation! • Weight* 2.91 . Additive U t i l i t y Function 45 I f attributes X^...Xn are found to be additive independent, u t i l i t y independent and p r e f e r e n t i a l l y independent, then the following additive u t i l i t y function can be used to determine the u t i l i t y for the a c t i v i t y (Keeney and Ra i f f a 1976): n u(x x , x 2 , . . . x n ) = k i u i ( x i ) (1) i = l where U and U. are u t i l i t y functions scaled from 0 to 1 and k. are at t r i b u t e s c a l i n g constants O Z k . Z l , where the scaling constants sum to 1.0. For example, the following scaling constant equations were assessed for the attributes of horseback r i d i n g : (i) k 2 6 = 0.969k 2 7 ( i i ) k 2 6 = 0.644k 2 8 28 The must equal 1.0 for the additive function to be v a l i d . i=26 The scaling constants are determined by equating k 2g to 1.0 and subst i t u t i n g k 2g into the other scaling constant equations. The r e s u l t i n g s c a l i n g constants are the following: (i) k 2 6 = 1.0 ( i i ) k 2 7 = 1.03 ( i i i ) k 2g = 1.55 The sum of the sca l i n g constants i s 3.58. 46 Scaling to sum to 1.0, (i) k 2 6 = 1.0/3.58 = 0.279 ( i i ) k 2 7 = 0.288 ( i i i ) k 2 g = 0.433 The additive u t i l i t y function U f o r the attributes ( X 2 g , X 2 7, X 2 g) of the a c t i v i t y horseback r i d i n g i s the following: U = 0.279U 2 6(X 2 6) + 0.288U 2 ?(X 2 7) + 0.433U 2 g(X 2 8) where i s the u t i l i t y value of the attribute i at l e v e l X^. 47 2.92 M u l t i p l i c a t i v e U t i l i t y F u n c t i o n I f a t t r i b u t e s X^...X n o f an a c t i v i t y a r e n o t a d d i t i v e i n -d e p e n d e n t b u t a r e m u t u a l l y u t i l i t y i n d e p e n d e n t and p r e f e r e n t -i a l l y i n d e p e n d e n t , t h e n t h e m u l t i p l i c a t i v e u t i l i t y f u n c t i o n i s t h e a p p r o p r i a t e u t i l i t y f u n c t i o n t o u s e (Keeney and R a i f f a 1 9 7 6 ) . The m u l t i p l i c a t i v e u t i l i t y f u n c t i o n i s i l l u s t r a t e d by e q u a t i o n ( 2 ) : n 1 + K U ( X i r X 2,...X n) = | | (1 + K k i U i ( X i ) ) (2) i = l where U and U^ a r e u t i l i t y f u n c t i o n s s c a l e d f r o m 0 t o 1, k^ a r e a t t r i b u t e s c a l i n g c o n s t a n t s 0 ^  k^ ^ 1, K i s a n o n - z e r o s c a l i n g c o n s t a n t , and t h e ^  k^ ? 1.0. I n o r d e r f o r t h e m u l t i p l i c a t i v e f u n c t i o n t o be v a l i d , t h e s c a l i n g c o n s t a n t s must n o t sum t o 1.0; t h e r e f o r e , s c a l i n g c o n -s t a n t s a r e n o t s c a l e d t o 1.0 as i n t h e a d d i t i v e c a s e . Assume t h a t t h e f o l l o w i n g s c a l i n g c o n s t a n t e q u a t i o n s were a s s -e s s e d f o r t h e a t t r i b u t e s o f h o r s e b a c k r i d i n g : ( i ) k 2 6 = 0 . 3 5 0 k 2 ? ( i i ) k 2 6 = 0 . 4 0 0 k 2 8 The s c a l i n g c o n s t a n t k 2 g i s d e t e r m i n e d by t h e method i n S e c t i o n 4.9. The o t h e r s c a l i n g c o n s t a n t s a r e d e t e r m i n e d b y s u b s t i t u t i n g k 2 g i n t o t h e e q u a t i o n s . The r e s u l t i n g s c a l i n g c o n s t a n t s a r e t h e f o l l o w i n g ( t h e i r sum n o t e q u a l t o 1 . 0 ) : ( i ) k 2 6 = 0.300 ( i i i ) k 2 8 = 0.750 ( i i ) k 2 7 = 0.857 T h e r e f o r e t h e m u l t i p l i c a t i v e f u n c t i o n U f o r t h e a t t r i b u t e s (X~c, X 0 - , X O Q ) o f t h e a c t i v i t y h o r s e b a c k r i d i n g i s t h e f o l l o w i n g Zb 2.1 Zo 28 1 + K U ( X 2 6 , . . . X 2 8 ) = (1 + Kk ± u i ( x i ) ) i=26 o r 28 1 + K = (1 + K k ± ) (Keeney 1974) i=26 w h i c h i m p l i e s (3) 1 + K = (1 + K k 2 6 ) ( 1 + K k 2 ? ) ( 1 + K k 2 8 ) (4) o r 1 - K k 2 6 k 2 7 k 2 8 + K ( k 2 6 k 2 7 + k 2 g k 2 8 + k 2 ? k 2 8 ) + ( k 2 6 + k 2 7 + k 2 8 } I n t h e example ( i ) k 2 6 = 0.300 ( i i ) k , 7 = 0 . 8 5 7 ( i i i ) k 2 g = 0.750 ^27 S u b s t i t u t i n g i n t o e q u a t i o n (4) 0.193K^ + 1.13K + 0.910 = 0 K = -0.964 I f t h e r e a r e two r o o t s t o t h e e q u a t i o n , K X - 1 i s t h e s o l u t i o n (Keeney 1 9 7 4 ) . F o r t h e example, K i s s u b s t i t u t e d i n t o e q u a t i o n (3) t o y i e l d t h e f o l l o w i n g m u l t i p l i c a t i v e u t i l i t y f u n c t i o n f o r t h e a t t r i b u t e s 49 (X2g,...X2g) of the horseback r i d i n g a c t i v i t y : 28 u ( x 2 6 , . . . x 2 8 ) (1 - 0.964k j LU i(X j L)) - 1 1=26 (5) -0.964 Substituting the k^ values into equation (5), the u t i l i t y function can be reduced to equation (6) u(x 2 g,...x 2 8) -((1 - 0 .289U 2 6 ( X2 6 ) ) ( 1 - 0 . 8 2 6 U 2 ? ( X 2 7 ) ) ( 1 - 0 . 7 2 3 U 2 8 ( X 2 8 ) ) ) - 1 - 0.964 (6) whe re U\ i s the u t i l i t y value of the attribute l e v e l X^. 50 2.10 Development of an Equation to Scale the Recreational A c t i v i t i e s From the multi-attribute u t i l i t y analysis, decision makers' preferences for each attribute of an a c t i v i t y are assessed and scaled to y i e l d an additive or m u l t i p l i c a t i v e u t i l i t y function for each a c t i v i t y . To develop an objective function for a mine waste recrea-ti o n plan, a c t i v i t i e s must be scaled as to t h e i r r e l a t i v e im-portance to the residents of the Elk River Valley. I f a c t i v i t y u t i l i t i e s are simply added, many mine waste areas may be a l l o -cated less preferred a c t i v i t i e s which may not benefit the major-i t y of the Elk River Valley residents. A s c a l i n g system i s needed to r e f l e c t the residents' preferences for each a c t i v i t y . Nessman and Bailey (1981) conducted a mail survey of out-door recreation p a r t i c i p a t i o n i n the study area. Data for the percentage of households p a r t i c i p a t i n g i n each recreational a c t i v i t y considered i n t h i s study were c o l l e c t e d . These data are the only available recent data on outdoor recreation par-t i c i p a t i o n i n the study area. They were used i n t h i s study to indicate the r e l a t i v e importance of the a c t i v i t i e s to the r e s i -dents of the Elk River Valley. The percentages were scaled between 0 and 1 and both are presented i n Table 1. The scaling factors were then used to develop the objective function pre-sented i n Figure 14 for an outdoor recreation plan on the mine waste section i n t h i s study. This scaling system for a c t i v i t i e s does not take into account changes in preferences for a c t i v i t i e s over time. The system would have to be updated regularly as people with 51 varying preferences for a c t i v i t i e s migrate to and leave the Elk River Valley Recreational A c t i v i t y % Households P a r t i c i p a t i n g Scaling Constants T r a i l b i k i n g 14. 8 14.8 320.6 = 0. 046 Four-Wheel Driving 35. 7 0. 111 Snowmobiling 26. 8 0. 084 Downhill Skiing 42. 8 0. ,133 Cross-Country Skiing 26. .5 0. .087 Snowshoeing 20, .6 0, .064 Hiking 46 .8 0 .146 Horseback Riding 25 .1 0 .078 Summer Camping 81 .5 0 .254 320 .6 1 .00 Table 1. Data f o r percentage households p a r t i c i p a t i n g i n recreational a c t i v i t i e s of the Elk River Valley and t h e i r s c a l i n g constants. V a n - ° ' 0 4 6 UTB + . ° - 1 1 1 0 4 x 4 + ° - 0 8 4 USNOW + ° - 1 3 3 UDOWN + ° ' 0 8 7 UX-C + 0 ' 0 6 4 USHOE + ° ' 1 4 6 UHIKE + ° - 0 7 8 UH0RSE + ° ' 2 5 4 U CAMP F i a 14. Objective function to determine the u t i l i t y of a land use plan with outdoor recreational a c t i v i t i e s . 52 CHAPTER 3 RESULTS AND DISCUSSION - PART 1 Multi-Attribute U t i l i t y Analysis Results (i) In the course of assessing a t t r i b u t e u t i l i t y functions, mutual u t i l i t y independence and p r e f e r e n t i a l independence were v e r i f i e d between a l l the a t t r i b u t e s ; ( i i ) Assessed attribute u t i l i t y functions for the recreational a c t i v i t i e s are presented i n Appendix 5; ( i i i ) Attribute tradeoffs and the assessed scaling constant equations are presented i n Appendix 6; (iv) Improper interview questions resulted from the assessor's incomplete understanding of the additive independence lo t t e r y presented i n Section 2.73, at the time of the interviews; therefore, additive independence was not es-tablished between the attributes and a proper selection of u t i l i t y equations could not be obtained. To demon-strate the analysis and evaluation of the mine waste, the attributes were assumed to be additive independent and the additive u t i l i t y model used. The additive u t i l i t y functions for the recreational a c t i v i t i e s are presented i n Appendix 7; (v) The r e l a t i v e importance of each attribute of an a c t i v i t y to a decision maker i s presented i n Appendix 8. Discussion The multi-attribute u t i l i t y analysis for each decision maker took, on average, two hours to complete. More time would have been useful to recheck decision makers' answers to i n t e r -view questions; however, two hours was the maximum any decision maker i n t h i s study was w i l l i n g to devote to the analysis. The following observations were noted during the assessment of decision makers' attribute u t i l i t y functions: (i) At the beginning of the assessment, decision makers' answers to the l o t t e r i e s contradicted themselves. Dec-i s i o n makers would answer i n accordance with what they thought the analyst wanted as an answer from the tone of analyst's question. The analyst pointed out the contra-d i c t i o n which c l a r i f i e d the assessment procedure for the decision makers and t h i s r e s u l t was not encountered again during the assessment; ( i i ) Decision makers would begin to lose i n t e r e s t i n the ass-essment due to the r e p e t i t i v e nature of the interview questions and would begin to answer the questions without thinking hard about the hypothetical situations posed by the l o t t e r i e s , answering very quickly. At t h i s stage the analyst must try to restore the decision maker's in t e r e s t i n the l o t t e r i e s by putting enthusiasm and emphasis into the interview questions. I t i s noted that the decision makers i n t h i s study did not have a vested i n t e r e s t i n the r e s u l t i n g recreation plan. I t would seem l o g i c a l to predict that a decision maker who would use the r e s u l t i n g plan would be more attentive during the assessment because 54 of t h i s vested i n t e r e s t . The attribute scaling constant assessments were a l l con-ducted l a t e r i n the interview because the procedure uses the assessed u t i l i t y functions. About half way through the s c a l -ing constant assessment, decision makers became t i r e d and would respond very quickly to tradeoffs without thinking hard about what they were saying. To get the decision maker to think hard about each tradeoff, the analyst would repeat the decision maker's responses to the decision maker, to v e r i f y that the response was one i n which the decision maker r e a l l y believed. For a l l the questions i n the multi-attribute u t i l i t y anal-y s i s , i t i s e s s e n t i a l that the analyst remain impartial i n the phraseology of the questions. Decision makers are very quick to change t h e i r responses i f the analyst words the interview question i n such a way that the decision maker thinks a c e r t a i n response would be preferred by the analyst. The average slope of a mine waste area i n the Elk River Valley proved to be the most important attribute to the decision makers for a l l the a c t i v i t i e s which had slope as an a t t r i b u t e , except for snowshoeing. The attribute u t i l i t y functions were drawn as straight l i n e s between assessed certainty equivalent points for ease of c a l c u l a t i o n of u t i l i t y values for the objective function, maxi-mized i n Part 2. 55 CHAPTER 4 METHODOLOGY - PART 2 Maximization of the Objective Function 4.1 Introduction The intent i n t h i s study was to use a computer optimiza-t i o n routine to produce a recreation plan which would maximize the objective function developed i n Part 1. The objective function i s non-linear. A c t i v i t i e s are grouped into 16 land uses as discussed i n Section 1.1. Land uses 1 - 1 6 are INTEG-ERS; therefore, an INTEGER optimization routine must be used for for non-linear functions. At the present time, there i s l i t t l e software available at the University of B r i t i s h Columbia for such optimization. The software that i s available i s r e l a t i v e l y new, w i l l not guarantee successful solutions and i s extremely expensive to use. Since an optimization routine was not avail a b l e , an algo-rithm c a l l e d NLP.S (a filename) which evaluates the objective function value of a recreation plan within the 10 km x 10 km mine waste section was developed. The algorithm i s presented i n Appendix 9, explained i n Appendix 10 and summarized by a flowchart i n Appendix 11. The procedure for using i t i s pres-ented i n Appendix 12. 4.2 Grouping the Recreational A c t i v i t i e s into Land Uses Many of the mine waste areas are desirable for both summer and winter recreational a c t i v i t i e s . Many of the roads i n the 5 6 Elk River Valley, which are e a s i l y t r a v e l l e d during the dry summer months, become blocked with deep snow from late f a l l to early summer. As a r e s u l t , these areas are only accessible to snowmobilers, snowshoers and cross-country s k i e r s . The f i r s t d i v i s i o n of the a c t i v i t i e s i s therefore between summer and winter a c t i v i t i e s . Serious incompatability exists between the summer motor-ized a c t i v i t i e s t r a i l b i k i n g and four-wheel dri v i n g and the non-motorized a c t i v i t i e s hiking, horseback r i d i n g and camping. T r a i l b i k e s and four-wheel drive vehicles create noise and safety hazards from using the same roads and areas as hikers, horseback r i d e r s and campers (Brewer and Fulton 1974; as c i t e d by Brander 1974). Hikers and horseback riders also c o n f l i c t with t r a i l b i k e s and four-wheel drive vehicles by impeding the t r a f f i c of these off-road vehicles (Chilman 1979). Incompatability also exists between the winter motorized a c t i v i t y snowmobiling and the non-motorized a c t i v i t i e s cross-country s k i i n g and snowshoeing. Lund and Williams (1974) state that snowmobiles leave the t r a i l i n a condition that i s not sa t i s f a c t o r y for cross-country s k i e r s . Cross-country skiers and snowshoers cannot toler a t e the noise from snowmobiles and may be endangered by speeding snowmobiles (Hoene undated, Selles 1973, Al l a n 1975, Jubenville 1978). When cross-country skiers or snowshoers use the same t r a i l as snowmobilers, they impede snowmobile t r a f f i c (Allan 1975), and decrease enjoyment for the snowmobiler because the snowmobiler i s con-stantly worried about s t r i k i n g a cross-country skier or snow-shoer (Lund and Williams 1974). Zoning by a c t i v i t y i s a common technique used to control recreational user group c o n f l i c t . C o n f l i c t i n g a c t i v i -t i e s are separated s p a t i a l l y , assigning areas for motorized uses that are d i s t i n c t from areas reserved for non-motorized uses (O'Riordan 1974, McCall and McCall 1977, Jubenville 1978, Knudson 1980). In t h i s study, t r a i l b i k i n g and four-wheel d r i v i n g are separated from hiking, horseback r i d i n g and camping. Snowmobiling i s separated from cross-country s k i i n g and snowshoeing. Minor incompatability exists between t r a i l b i k e r s and four-wheel dr i v e r s . One can impede the other's t r a f f i c r e s u l t i n g i n safety hazards. In t h i s study, t r a i l b i k i n g and four-wheel dri v i n g are separated for certain land uses, and because the c o n f l i c t i s minor, and both a c t i v i t i e s frequently occur to-gether, are grouped for other land uses. A "no a c t i v i t y " land use was also selected to be a summer or winter land use. A planner wishes to maximize the objective function. Rather than assign a land use with a low u t i l i t y to a g r i d square, decreasing the function value for the plan, the planner should have the option of assigning a "no a c t i v i t y " land use to the gr i d square under examination, which doesn't a f f e c t the objective function value of the plan. Land uses composed of both summer and winter a c t i v i t i e s chosen for analysis i n t h i s study are presented i n Table 2. 58 LAND USES LAND USE * SUMMER ACTIVITIES WINTER ACTIVITIES 1 TRAILBIKING SNOWMOBILING 2 TRAILBIKING X-C/SNOWSHOEING 3 TRAILBIKING NO ACTIVITY 4 4X4 SNOWMOBILING 5 4X4 X-C/SNOWSHOEING 6 4X4 NO ACTIVITY 7 TRAILBIKING/4X4 SNOWMOBILING 8 TRAILBIKING/4X4 X-C/SNOWSHOEING 9 TRAILBIKING/4X4 NO ACTIVITY 0 HIKING/HORSEBACK RIDING/CAMPING SNOWMOBILING 1 HIKING/HORSEBACK RIDING/CAMPING X-C/SNOWSHOEING 2 HIKING/HORSEBACK RIDING/CAMPING NO ACTIVITY 3 NO ACTIVITY SNOWMOBILING 4 NO ACTIVITY X-C/SNOWSHOEING 5 NO ACTIVITY NO ACTIVITY 6 NO ACTIVITY DOWNHILL SKIING T a b l e 2 . L a n d u s e s c h o s e n f o r a n a l y s i s i n t h i s s t u d y . 59 4.3 Bounding of the Study Area to be Evaluated There i s a danger i n maximizing the objective function of a mine waste area within a larger area of not examining s p i l l o v e r e f f e c t s which may have important impacts; however, i f the mine waste area to be analysed i s not bounded i n some way, i t w i l l remain hopelessly i n t r a c t a b l e (Keeney and R a i f f a 1976). It i s desirable to examine as large an area as possible without becoming too general i n the analysis. The number of variables the non-linear monitor fo r non-linear optimization routines at the University of B.C. used to run the NLP.S program i n t h i s study can examine, i s r e s t r i c t e d to 100. The mine waste i n t h i s study i s bounded into a 10 km x 10 km square section. The section i s further divided into 100 g r i d squares, each 1 km x 1 km, and representing one variable or land use. A 10 km x 10 km square section i s reasonably large for the study area and detailed results can s t i l l be obtained. 60 4.4 Calculation of Attribute Levels from the 10 km x 10 km  Mine Waste Section 4.41 Calculation of Travel Time from Town The location of a town can be inside or outside the 10 km x 10 km mine waste section. A main highway runs from north to south through the study area. Secondary roads branch east and west o f f the highway. The t o t a l t r a v e l time to a g r i d square from a town i s therefore the sum of the highway and secondary road t r a v e l times. The speed l i m i t on the highway i s 80 km/hour. Average speed i s approximately 70 km/hour. Average speed on secondary roads i s approximately 40 km/hour. The following relationships were developed for the two t r a v e l times: Travel Time on Highway = (Distance t r a v e l l e d (km))(70 km/h) ( 1 ) (minutes) 60 minutes Travel Time on Second ary Roads (minutes) Total t r a v e l time from town to a g r i d square = (1) + (2). In t h i s study, roads running from north to south are assumed to be highways and roads running east to west are assumed to be secondary roads. (Distance t r a v e l l e d (km))(40 km/h) 60 minutes 4.4 2 Calculation of Length of T r a i l 61 The t r a i l c i r c u i t i l l u s t r a t e d i n Figure 15 was chosen to be c h a r a c t e r i s t i c of c i r c u i t s used by many t r a i l b i k e r s , four-wheel d r i v e r s , snowmobilers, cross-country s k i e r s , snow-shoers, hikers and horseback r i d e r s . F i g . 15. T r a i l c i r c u i t used by the a c t i v i t i e s . This t r a i l c i r c u i t was chosen to simplify the complexity of t r a i l patterns for use i n the computer model. The spider-web type t r a i l pattern, which t h i s t r a i l pattern cl o s e l y resembles, i s commonly used by t r a i l b i k e r s and snowmobilers (Hetherington 1979). For the other a c t i v i t i e s , t h i s t r a i l c i r c u i t i s not un-reasonable and offers the following combinations of routes: (a) 1 W L 4 (g) 3 L' W L 4 (b) 1 W L W 3 (h) 3 L« W L W1 3 (c) 1 w L W1 L' W L 4 (i) 4 W 3 (d) 2 L 4 (j) 4 W» L 1 W L 4 (e) 2 L W 3 (k) 4 W L 1 W L W (f) 2 L W L' W L 4 L = l e n g t h o f g r i d s q u a r e c o m b i n a t i o n - 200 m e t r e s . W = W i d t h o f g r i d s q u a r e c o m b i n a t i o n - 400 m e t r e s . The l e n g t h o f t r a i l f o r t h e c i r c u i t was c a l c u l a t e d t o be t h e sum o f L + W + L ' +W 1 + 1 + 2 + 3 + 4 and was u s e d i n t h i s s t u d y as t h e t r a i l l e n g t h f o r a s q u a r e o r r e c t a n g u l a r p a t t e r n o f g r i d s q u a r e s . The sum can be r e d u c e d t o e q u a t i o n 3: T r a i l l e n g t h f o r a s q u a r e o r r e c t a n g u l a r = 2 L + 2W + 2 / L 2 + W2 p a t t e r n o f g r i d s q u a r e s ^ T a b l e 3 d e s c r i b e s t r a i l l e n g t h i n r e l a t i o n t o t h e number o f g r i d s q u a r e s . G r i d S q u a r e C o m b i n a t i o n # S q u a r e s T r a i l L e n g t h (km) (Row x Column) l x l 1 4.8 1 x 2 2 8.6 1 x 3 3 12.5 2 x 2 4 11.6 2 x 3 6 15.2 3 x 3 9 18.4 T a b l e 3. T r a i l l e n g t h i n r e l a t i o n t o t h e number o f g r i d s q u a r e s . The a l g o r i t h m i n t h i s s t u d y (NLP.S) i s l i m i t e d t o o n l y b e i n g a b l e t o examine one g r i d s q u a r e a t a t i m e and t a k i n g i n t o a c c o u n t 8 g r i d s q u a r e s s u r r o u n d i n g t h e e x a m i n e d g r i d s q u a r e ; t h e r e f o r e , t h e maximum l e n g t h o f t r a i l t h a t c an be c a l c u l a t e d by t h e a l g o r i t h m i s t h e t r a i l l e n g t h f o r 9 g r i d s q u a r e s . From T a b l e 3, 18.4 km o f t r a i l i s a v a i l a b l e f o r 9 g r i d s q u a r e s and 4.8 km f o r 1 g r i d s q u a r e . U s i n g t h e s e d a t a , a s t r a i g h t l i n e r e l a t i o n s h i p i l l u s t r a t e d by F i g u r e 16 and e q u a t i o n 63 4 were developed between t r a i l length and the number of g r i d squares. 18.4J T r a i l Length (km) 4.8-, , , 0 1 9 10 # Grid Squares Fi g . 16. T r a i l length i n r e l a t i o n to the number of g r i d squares. Equation of the l i n e : T r a i l Length = m(# Grid Squares) + b Maximum value of t r a i l length = 18.4 = 9m + b Minimum value of t r a i l length = 4.8 = lm + b Subtracting equations, 13.6 = 8m 1.7 = m 3.1 = b Therefore, T r a i l Length (km) = 1.7(Nl) + 3.1 (4) where Nl = number of g r i d squares (up to 9). 64 4.43 Calculation of Average Slope and Snow Depth Average slope and snow depth values are measured for each area represented by a g r i d square and stored i n a f i l e f o r use l a t e r i n the program. The values are stored i n li n e s of 10 values, each l i n e representing one row on the g r i d square. For example the f i l e s f or slopes and snow depths w i l l have the same format as the f i l e s i n Figures 17 and 18 respectively. Line (Row) 1 2 3 4 5 6 7 8 9 10 050,050,050,050,060,040,010,000,000,060, 050,050,050,050,050,045,040,000,000,000, 060,065,070,070,060,050,080,090,095,060, 070,065,040,030,030,035,055,055,045,050, 010,010,000,005,005,000,010,020,030,035, 060,040,020,020,025,043,045,045,050,050, 090,080,070,080,090,060,070,060,055,050, 100,100,090,080,090,050,060,050,045,045, 110,120,100,090,080,040,050,040,035,035, 130,140,130,120,110,090,080,07 0,080,045, F i g . 17. An example of a f i l e with slope values (%). Line (Row) 1 2 3 4 5 6 7 8 9 10 183,213,213,213,183,183,152,122,000,152, 183,213,213,198,183,152,122,000,122,152, 213,213,213,19 8,152,122,000,122,122,152, 213,213,213,183,152,122,000,122,122,152, 213,213,213,183,152,122,137,152,168,183, 213,213,183,152,122,000,137 ,16 8 ,16 8,137 , 183,183,183,169,16 8,152,152,168,168,183, 183,183,152,168,152,137,122,122,122,122, 152,152,183,152,152,137,152,16 8,152,152, 183,183,152,168,168,152,137,137,122,122, F i g . 18. An example of a f i l e with snow depth values (cm). 65 4.44 Calculation of the Number of C o n f l i c t s per Hour  Between A c t i v i t i e s There are three land uses (7, 8 and 9) with c o n f l i c t i n g a c t i v i t i e s . The c o n f l i c t occurs between t r a i l b i k i n g and four-wheel d r i v i n g . I t i s d i f f i c u l t to estimate how many c o n f l i c t s per hour w i l l occur on a hypothetical mine waste plan; however, four c o n f l i c t s per hour or 1 c o n f l i c t every 15 minutes seemed reasonable from the author's experience observing the two a c t i v i t i e s , for one g r i d square allocated both a c t i v i t i e s . This relationship for one g r i d square i s used i n the following example to develop an equation (equation 5) which determines the number of c o n f l i c t s per hour i n g r i d squares. Example: Case 1: One square allocated two c o n f l i c t i n g a c t i v i t i e s . A c t i v i t y #1 = s o l i d l i n e ( ) A c t i v i t y #2 = dashed l i n e ( ) Q = Square under examination. The number of c o n f l i c t s i n the square under examination i s 4 c o n f l i c t s per hour. O 66 Case 2: A c t i v i t y #1 allocated 2 contiguous g r i d squares A c t i v i t y #2 allocated 3 contiguous g r i d squares O = Square under examination. A decision maker engaging i n A c t i v i t y #1 w i l l now be inside the square under examination 1/2 of the time. A decision maker engaging i n A c t i v i t y #2 w i l l now be inside the square under examination 1/3 of the time; hence, there w i l l be 1/2 x 1/3 as many c o n f l i c t s i n the square under examination. The number of c o n f l i c t s per hour i s now equal to 4 x 1/2 x 1/3 = 2/3 c o n f l i c t s per hour for the square under examination. The r e l -ationship represented by equation 5 f o r the number of c o n f l i c t s per hour i n the square being examined i s the following: 4 Number of C o n f l i c t s per Hour = (5) (NI)(N2) where NI = # squares allocated a c t i v i t y #1 N2 = # squares allocated a c t i v i t y #2 Equation 5 i s used i n the NLP.S algorithm. 4.45 Calculation of the Distance to the Nearest Drinking Water  Source Figure 19 i s used i n the following discussion to explain the c a l c u l a t i o n . F i g . 19. Land use plan with the square under examination K at I,J coordinates (6,3) and drinking water source at I,J coordinates (3,7). In Figure 19, (I,J) i s the location of square K being examined. (IL,JL) i s the loc a t i o n of the drinking water source. DX i s the distance from square K to the drinking water source. The distance DX i s calculated by a diagonal from square K to the water source. In Figure 19, I-IL i s the v e r t i c a l length of the r i g h t angle t r i a n g l e formed between square K and the water source. The following equation can therefore be used to calculate DX: DX = v / ( I " I L ) 2 + (J - J L ) 2 The computer reads the location of the water sources from a f i l e . The squares are numbered from 1 to 100. Square 001 corresponds to the I,J location (1,1); square 100 corresponds to the I , j lo c a t i o n (10,10). Whenever a drinking water source i s present i n a g r i d square, the number of the square i s l i s t e d i n the f i l e as i l l u s t r a t e d by the f i l e i n Figure 20. AN EXAMPLE OF A FILE CONTAINING GRID SQUARES WHICH HAVE A DRINKING WATER SOURCE FILE - LAKES 1 0 0 7 2 0 1 8 3 0 2 7 4 0 3 7 5 0 4 6 6 0 5 5 7 0 6 4 8 0 7 4 9 0 8 4 1 0 0 9 5 F i g . 20. An example of a f i l e containing drinking water sources. 69 The g r i d s q u a r e number f r o m 1 t o 100 must be c o n v e r t e d t o an ( I . J ) l o c a t i o n . To i l l u s t r a t e , i f s q u a r e 027 has a d r i n k i n g w a t e r s o u r c e , t h e n i t s c o r r e s p o n d i n g ( I , J ) l o c a t i o n i s Row 3, Column 7 o r ( 3 , 7 ) . To d e t e r m i n e t h e I l o c a t i o n f o r g r i d s q u a r e 027, t h e f o l l o w i n g e q u a t i o n i s u s e d : I L = ( L A K E S ( I I ) - 1) + 1 10 where I L = t h e I l o c a t i o n o f t h e w a t e r s o u r c e and where L A K E S ( I I ) i s t h e g r i d s q u a r e number r e a d by t h e c o m p u t e r . F o r t h e e xample, I L = 2 7 - 1 + 1 = 2 6 = 2 . 6 + 1 10 10 The a l g o r i t h m u s e s INTEGERS; t h e r e f o r e , a l l f r a c t i o n s a r e t r u n -c a t e d . Hence I L = 2 + 1 = 3. To d e t e r m i n e t h e J l o c a t i o n f o r g r i d s q u a r e 027, t h e f o l l o w i n g e q u a t i o n i s u s e d : J L = L A K E S ( I I ) - ( I L - 1 ) ( 1 0 ) where J L = t h e J l o c a t i o n o f t h e w a t e r s o u r c e . F o r t h e e xample, J L = 27 - ( 3 - 1 ) ( 1 0 ) = 27 - 20 = 7. To c a l c u l a t e t h e d i s t a n c e DX, l e t s q u a r e K be a t ( 6 , 3 ) . T h e r e f o r e , I = 6, J = 3, I L = 3, J L = 7. Hence d i s t a n c e DX i s DX = / ( I - I L ) 2 + ( J - J L ) 2 = / (6 - 3 ) 2 + (3 - 7 ) 2 4 2 = / 2 5 = 5 km. 70 4.5 Evaluating the Mine Waste Section The hypothetical mine waste section to be evaluated i n t h i s study i s graphically i l l u s t r a t e d i n Figure 21. Data for the mine waste on average snow depths, average slopes and drinking water source locations are presented i n Appendix 15. The town i s located at g r i d square (-20.0,7.0), outside the 10 km x 10 km section. There are two steps i n the evaluation. The f i r s t step i s to f i n d a s t a r t i n g land use plan to generate a value for the objective function. Since there are 16^"^ possible land use plans, and no optimization routine available at t h i s time, i t i s desirable to save time by s t a r t i n g with a plan which has a high objective function value. To generate a plan with a high func-t i o n value, an algorithm c a l l e d UTILVAL.S was developed which pr i n t s u t i l i t y values for each a c t i v i t y for every g r i d square. The algorithm i s presented i n Appendix 16, and the procedure f o r using i t i n Appendix 17. The values produced by UTILVAL.S for the mine waste section are presented i n Appendix 18. From these values, a c t i v i t i e s with high u t i l i t i e s are allocated to respect-ive g r i d squares. Figure 22 i l l u s t r a t e s g r i d squares which pro-duce high u t i l i t y values for certain a c t i v i t i e s . These a c t i v i -t i e s can then be grouped into land uses presented i n Figure 23. Combinations of these land uses were evaluated and the land use plan presented i n Figure 2 4 had the highest function value of 0.862886567996. This plan was chosen as a s t a r t i n g plan. The second step i n the evaluation i s to change land uses i n the g r i d squares with the goal of increasing the value of the objective function. When a land use a l l o c a t i o n plan has been 71 generated, where any change i n a land use for any g r i d square, or blocks of g r i d squares, decreases the function value for the plan, the best land use plan based on decision makers' prefer-ences between attributes and at t r i b u t e levels has been generated, without an optimization routine. F i g . 21. Three dimensional representation of the p l a evaluated i n t h i s study (Town i s at row -20 and column 7.0 or (-20.0,7.0)). 73 74 Fig.23. Land uses r e s u l t i n g from the high u t i l i t y a c t i v i t i e s presented i n Figure 22. LAND U S E » 2 3 4 5 6 7 B 9 10 15 16 SUMMER A C T I V I T I E S T R A 1 L B I K I N G T R A I L B I K I N G T R M L B I K I N G 4 X 4 T R A I L B I K I N G / 4 X 4 T B A I L B 1 K I N G / 4 X 4 T R A I L B I K I N G / 4 X 4 H I K I N G / H O R S E B A C K R I D I N G / C A M P I N G H I K I N G / H O R S E B A C K R I D I N G / C A M P I N G H I K I N G / H O R S E B A C K R I D I N G / C A M P I N G NO A C T I V I T Y NO A C T I V I T Y NO A C T I V I T Y NO A C T I V I T Y W I N T E R A C T I V I T I E S S N 0 W M 0 6 I L I N G X - C / S N O W S H O E I N G NO A C T I V I T Y S N O W M O B I L I N G X - C / S N O W S H O E I N G NO A C T I V I T Y S N O W M O B I L I N G X - C / S N O W S H O E I N G NO A C T I V I T Y S N O W M O B I L I N G X - C / S N O W S H O E I N G NO A C T I V I T Y S N O W M O B I L I N G X - C / S N O W S H O E I N G NO A C T I V I T Y D O W N H I L L S K I I N G 6 8 AO 1 2 3 ll 4 5 ei F i g . 2 4 . S t a r t i n g p l a n f o r t h e e v a l u a t i o n . 75 LAND USE * 15 16 SUMMER A C T I V I T I E S T R A I L B I K I N G T R A I L B I K I N G T R A I L B I K I N G 4 X 4 4 X 4 4 X 4 T R A I L B I K I N G / 4 X 4 TRAI LB I K I N G / 4 X 4 T R A 1 L B I K 1 N G / 4 X 4 HIKING/HORSEBACK R I D I N G / C A M P I N G HIKING/HORSEBACK R I O I N G / C A M P I N G HIKING/HORSEBACK R I D I N G / C A M P I N G NO ACT IVI TV NO A C T I V I T Y NO A C T I V I T Y NO A C T I V I T Y 8 WINTER A C T I V I T I E S SNOWMOBILING X-C/SNOWSHOEING NO A C T I V I T Y SNOWMOBILING X-C/SNOWSHOEING NO A C T I V I T Y SNOWMOBILING X-C/SNOWSHOEING NO A C T I V I T Y SNOWMOBILING X-C/SNOWSHOEING NO A C T I V I T Y SNOWMOBILING X-C/SNOWSHOEING NO A C T I V I T Y DOWNHILL S K I I N G 9 "tO 11 v l 2 _ F = 0.862886567996 76 CH/APTER RESULTS - PART 2 Maximization of the Objective Function The following land use plans were generated by changing land uses i n the s t a r t i n g plan developed i n Section 4.5. These land use plans correspond to the 100 g r i d square section i n Figure 21. The top l e f t corner land use corresponds to g r i d square (1,1); the bottom right corner land use corresponds to g r i d square (10,10). The f i r s t land use plan presented i s the s t a r t i n g plan developed i n Section 4.5. The plans that follow the s t a r t i n g plan sequentially increase i n objective function value. The symbol "F" represents "objective function value". Each plan presented represents 5 land use changes which increase the function value. For example, the second land use plan presented has a function value of 0.863583195736. Five of i t s land uses are d i f f e r e n t than the s t a r t i n g plan. 11 11 15 11 11 11 12 12 6 6 14 15 1 11 11 11 12 12 6 6 15 15 1 11 11 11 12 12 6 6 6 6 6 11 11 11 12 12 6 6 6 6 9 11 11 3 3 6 e 6 15 15 1 3 3 11 3 6 6 6 15 15 1 3 1 1 6 6 6 3 15 15 1 3 1 4 6 6 16 3 15 15 1 12 1 4 6 6 16 3 15 15 2 12 12 4 6 6 3 3 11 11 15 11 11 11 12 12 6 6 14 15 1 11 11 11 12 12 6 6 15 15 1 11 11 11 12 12 6 6 6 6 6 11 11 11 12 12 6 6 6 6 9 11 11 3 3 6 6 6 15 15 1 3 3 11 3 6 6 6 15 15 1 3 1 1 6 6 6 15 15 15 1 3 1 4 6 - 6 16 3 15 15 1 3 10 4 6 6 16 15 15 15 2 12 12 4 6 6 15 3 STARTING PLAN F • 0.862886567996 0.863583195736 77 1 1 1 1 1 5 1 1 1 1 1 1 1 2 1 2 6 6 1 4 1 5 1 1 1 1 1 1 1 1 2 1 2 6 6 1 5 1 5 1 1 1 1 1 1 1 1 2 1 2 6 6 6 6 6 1 1 1 1 1 1 1 2 1 2 6 6 6 6 9 1 1 1 1 3 3 6 6 6 1 5 1 5 1 3 3 1 1 3 6 6 6 1 5 1 5 1 3 1 1 6 6 6 1 5 1 5 1 5 1 3 1 4 6 6 1 6 3 1 5 1 5 1 1 2 1 0 4 6 6 1 6 1 5 1 5 1 5 2 1 2 1 2 4 6 6 1 5 3 i 1 1 1 1 1 5 1 1 1 1 1 1 1 2 1 2 6 6 1 4 1 5 1 1 1 1 1 1 1 1 2 1 2 6 6 1 5 1 5 1 1 1 1 1 1 1 1 2 1 2 6 6 6 6 6 1 1 1 1 1 1 1 2 1 2 6 6 6 6 g 1 1 1 1 3 3 6 6 6 1 5 1 5 1 3 3 1 1 3 6 6 6 1 5 1 5 1 3 1 1 6 6 6 1 5 1 5 1 5 1 3 1 4 6 6 1 6 3 1 5 1 5 1 3 1 2 1 0 4 6 6 1 6 1 5 1 5 6 1 2 1 2 1 2 4 6 6 1 5 3 4 1 1 1 1 1 5 1 1 1 1 1 1 1 2 1 2 6 6 1 4 1 5 1 1 1 1 1 11 1 2 1 2 6 6 1 5 3 3 1 1 1 1 1 1 1 2 1 2 6 6 6 6 6 1 1 1 1 1 1 1 2 1 2 6 6 6 6 1 5 1 1 1 1 2 3 6 6 6 1 5 1 5 1 3 3 1 1 3 6 6 6 1 5 1 5 1 3 1 1 6 6 6 1 5 1 5 1 5 1 3 1 4 6 6 1 6 3 1 5 1 5 1 3 1 2 1 0 4 6 6 1 6 1 5 1 5 6 1 2 1 2 1 2 4 6 6 1 5 3 11 1 1 1 5 1 1 3 1 4 1 2 1 2 6 6 1 2 1 5 1 3 2 1 1 1 1 1 2 1 2 6 6 1 5 3 3 1 1 1 1 1 1 1 2 1 2 6 6 6 6 6 1 1 1 1 1 1 1 2 1 2 6 6 6 6 1 5 1 1 1 1 2 3 6 6 6 1 5 1 5 1 3 3 1 1 3 6 6 6 1 5 1 5 1 3 1 1 6 6 6 1 5 1 5 1 5 1 3 1 4 6 6 1 6 3 1 5 1 5 1 3 1 2 1 0 4 6 6 1 6 1 5 1 5 6 1 2 1 2 1 2 J 4 • 6 6 1 5 3 1 2 1 2 1 5 3 3 V 1 4 1 5 1 2 6 6 1 2 1 5 1 3 2 1 1 1 1 1 2 1 2 6 6 1 5 1 5 3 1 1 1 1 1 1 1 2 1 2 6 6 6 6 6 1 1 1 1 1 1 1 2 1 2 6 6 6 6 1 5 1 1 1 1 2 3 6 6 6 1 5 1 5 1 3 3 1 1 3 6 6 6 1 5 1 5 1 3 1 1 6 6 6 1 5 1 5 1 5 1 3 1 4 6 6 1 6 3 1 5 1 5 1 3 1 2 1 0 4 6 6 1 6 1 5 1 5 6 1 2 1 2 1 2 4 6 6 1 5 3 F - 0.863822679413 F = 0.864386478268 F = 0.866497449359 F = 0.870043828838 F = 0.873877838249 78 12 12 15 3 3 14 15 14 6 6 12 15 13 2 11 11 1 1 15 6 6 15 15 3 11 11 11 12 1 6 6 6 6 6 11 11 11 12 13 6 6 6 6 15 1 11 12 3 6 6 6 15 15 1 3 3 11 3 6 6 6 15 15 1 3 1 1 6 6 6 15 15 15 1 3 1 4 6 6 16 3 15 15 13 12 10 4 6 6 16 15 15 6 12 12 12 4 6 6 15 3 12 12 15 3 3 14 15 14 6 6 12 15 13 2 11 1 1 11 15 6 6 15 15 3 11 11 1 1 12 1 6 6 6 6 6 11 11 1 1 12 13 6 6 6 6 15 1 11 12 1 6 6 6 15 15 1 3 3 1 1 1 6 6 6 15 15 1 3 1 1 6 6 6 15 15 •15 1 3 1 4 6 6 16 3 15 15 13 12 13 4 6 6 16 15 15 6 12 12 15 6 6 6 15 3 12 12 15 3 3 14 15 14 6 6 12 15 13 2 1 1 11 1 1 15 6 6 15 15 3 11 11 1 1 12 1 6 6 6 6 6 1 1 11 11 12 13 6 6 6 6 15 1 11 12 1 6 6 6 15 15 1 3 3 1 1 1 6 6 6 15 15 1 1 1 1 6 6 6 15 15 15 1 1 1 4 6 6 16 3 15 15 13 15 13 4 6 6 16 15 15 6 15 15 15 6 6 6 15 3 15 15 15 3 3 14 15 14 6 6 12 15 13 2 11 11 11 15 6 6 15 15 3 11 11 11 12 1 6 6 6 6 6 11 11 11 12 13 6 6 6 6 15 1 10 12 1 6 6 6 15 15 1 3 3 10 1 6 6 6 15 15 1 1 1 1 6 6 6 15 15 15 1 1 1 6 6 6 16 3 15 15 13 13 13 4 6 6 16 15 15 6 15 15 15 6 6 6 15 3 A 15 15 15 3 3 14 15 14 6 6 12 15 13 2 11 11 11 15 6 15 15 15 3 11 11 11 12 1 6 6 6 6 6 11 11 11 12 13 6 6 6 6 15 1 10 12 1 6 6 6 15 15 1 3 3 10 1 6 6 6 15 15 1 1 1 1 6 6 6 15 15 15 1 1 1 6 6 6 15 15 15 15 13 13 13 6 6 6 16 15 15 6 15 15 15 6 6 6 15 3 F = 0.878617576695 F = 0.879539610232 F = 0.880765256587 F = 0.881204715644 BEST SOLUTION FOUND F = 0.884954632831 From the evaluation, the f i n a l plan generated had an objective function value of 0.884954632831. The plan i s graphically presented i n Figure 25. An algorithm c a l l e d OUTSPLIT.S, presented i n Appendix 19, was developed to take a land use plan as i n Figure 25, and s p l i t the land uses into a c t i v i t y a l l o c a t i o n s for each g r i d square. The procedure for using the algorithm i s presented i n Appendix 20. Figures 26 and 27 i l l u s t r a t e where the a c t i v i -t i e s are allocated i n the s t a r t i n g land use plan and f i n a l land use plan respectively. 80 Fig.25. The land use plan with the highest function value. L A N D U S E * SUMMER A C T I V I T I E S W I N T E R A C T I V I T I E S 8 9 15 16 T R A I L B I K I N G T R A I L B I K I N G T R A I L B I K I N G 4 X 4 4 X 4 4 X 4 T R A I L B 1 K I N G / 4 X 4 T R A I L B I K I N G / 4 X 4 T R A 1 L B I K I N G / 4 X 4 H I K I N G / H O R S E B A C K R I O I N G / C A M P I N G H I K I N G / H O R S E B A C K R I O I N G / C A M P I N G H I K I N G / H O R S E B A C K R I D I N G / C A M P I N G NO A C T I V I T Y NO A C T I V I T Y NO A C T I V I T Y NO A C T I V I T Y •\ | 15_\ 2 | l 5 _ _ 3 1 l c 15 X 6 A 1 l 1 £. \ 6 5 1 © \ 6 S N O W M O B I L I N G X - C / S N O W S H O E I N G NO A C T I V I T Y S N O W M O B I L I N G X - C / S N O W S H O E I N G NO A C T I V I T Y S N O W M O B I L I N G X - C / S N O W S H O E I N G NO A C T I V I T Y S N O W M O B I L I N G X - C / S N O W S H O E I N G NO A C T I V I T Y S N O W M O B I L I N G X - C / S N O W S H O E I N G NO A C T I V I T Y O O W N H I L L S K I I N G F = 0.884954632831 7 " l 5 8 9 J°_ n i i i i 15_ H 11 12. 12 13 10 12 15 15 15 15 81 Fig.26 . A c t i v i t y a l l o c a t i o n s for the s t a r t i n g land use plan. 0 = A c t i v i t y not allocated to g r i d square 1 = A c t i v i t y allocated to g r i d square SUMMER A C T I V I T I E S T r a i l b l k l n g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 b 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 4 x 4 s 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 ' 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 < 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 WINTER A C T I V I T I E S Snowmobi l1ng 0 0 0 0 0 0 0 0 0 0 ' 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 Snowshoe1ng/X-C S k i i n g . 1 1 0 1 1 1 0 0 0 0 i 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 ' 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 H i k i n g / H o r s e b a c k R i d i n g / C a m p i n g 1 1 0 1 1 1 1 1 0 0 0 0 0 1- 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 D o w n h i l l S k i i n g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 p 82 F i g . 2 7 . A c t i v i t y a l l o c a t i o n s f o r t h e b e s t l a n d u s e p l a n g e n e r a t e d . 0 = A c t i v i t y n o t a l l o c a t e d t o g r i d s q u a r e 1 = A c t i v i t y a l l o c a t e d t o g r i d s q u a r e SUMMER A C T I V I T I E S T r a l l b i k l n g 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 o 0 o 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 x 4 s 0 0 o 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 o 0 0 1 1 1 0 0 0 1 0 0 0 1 1 1 0 0 WINTER A C T I V I T I E S S n o w n o b f 1 I n g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S n o w s h o e l n g / X - C S k i i n g 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 H t k i n g / H o r s e b a c k R l d t n g / C e m p I n g 0 0 0 0 0 0 0 o 0 0 1 0 0 0 1 1 1 0 0 0 o 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Downhl11 S k i ( n g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 83 CHAPTER 6 DISCUSSION - PART 2 Maximization of the Objective Function From the s t a r t i n g plan to the f i n a l plan, the function value increased by 0.022068064835 or approximately 2.5 percent. The change of any land use or blocks of land uses i n the f i n a l plan decreased the function value; therefore, the f i n a l land use plan i s thought to be the best plan generated i n the absence of a computer optimization routine. The "no a c t i v i t y " land use (#15) almost doubled from 14 i n the s t a r t i n g plan to 27 i n the f i n a l plan. This occured because every other land use lowers the average u t i l i t y value for one or more of the respective a c t i v i t i e s which i n turn lowers the objective function value of the plan. The "no a c t i v i t y " land use does not lower an a c t i v i t y u t i l i t y . The number of g r i d squares allocated t r a i l b i k i n g decreased from 24 i n the s t a r t i n g plan to 19 i n the f i n a l plan. T r a i l -biking, as expected, was allocated i n the f i n a l plan to g r i d squares with slope values of 0 to 30 percent. This slope range corresponds to high u t i l i t y values for t r a i l b i k i n g and slope i s the most important a t t r i b u t e . The locations of t r a i l b i k i n g did not change greatly from the s t a r t i n g plan to the f i n a l plan. The number of g r i d squares allocated four-wheel dri v i n g decreased by 1, from 32 i n the s t a r t i n g plan to 31 i n the f i n a l plan. The locations of four-wheel driving did not change greatly from the s t a r t i n g plan. The four-wheel dri v i n g a c t i v i t y , as expected, was r e s t r i c t e d to areas with slope values of 35 to 55 percent. These slope values correspond to high u t i l i t y values 84 for four-wheel dr i v i n g and slope i s the most important a t t r i b u t e . The f i n a l plan did not contain land uses 5, 7, 8 or 9. During the evaluation, land uses 4 and 6 always yielded higher function values than land use 5. This i s explained by the con-f l i c t i n g u t i l i t y values f o r slope between four-wheel d r i v i n g and cross-country skiing/snowshoeing. The common attributes of the three a c t i v i t i e s are t r a v e l time, length of t r a i l and slope. The u t i l i t y functions for t r a v e l time and length of t r a i l are si m i l a r for the three a c t i v i t i e s . High slope values have high u t i l i t y values for four-wheel d r i v i n g while the reverse i s true for cross-country s k i i n g and snowshoeing. Slope i s the most important attribute for four-wheel dr i v i n g and cross-country skiing while not as important to snowshoeing. The c o n f l i c t created from these opposing u t i l i t i e s for slope results i n a decrease i n either the four-wheel dr i v i n g or the cross-country s k i i n g and snowshoeing u t i l i t i e s for a given slope value. Land use 4 represents four-wheel dr i v i n g and snowmobiling which prefer s i m i l a r slope values. Land use 6 represents only four-wheel d r i v i n g . Land uses 4 and 6 w i l l always y i e l d higher objective function values for the plan than land use 5 because they have si m i l a r u t i l i t y functions for slope. Land uses 7, 8 and 9 were never chosen for the f i n a l plan. This i s due to the c o n f l i c t between t r a i l b i k i n g and four-wheel d r i v i n g . Having the a c t i v i t i e s together decreases the u t i l i t y value for both a c t i v i t i e s . Land uses 1 - 6 separate the two a c t i v i t i e s and always y i e l d higher objective function values for the plan because the c o n f l i c t i s not present. This r e s u l t could have been predicted before the evaluation; land uses 7 - 9 were included only for i n t e r e s t . 85 The number o f g r i d s q u a r e s a l l o c a t e d s n o w m o b i l i n g i n c r e a s e d f r o m 13 i n t h e s t a r t i n g p l a n t o 19 i n t h e f i n a l p l a n . Snowmobil-i n g , as e x p e c t e d , was a s s i g n e d t o g r i d s q u a r e s w i t h s l o p e v a l u e s r a n g i n g f r o m 0 t o 25 p e r c e n t and snow d e p t h v a l u e s r a n g i n g f r o m 91 t o 122 cm. T h e s e r a n g e s c o r r e s p o n d t o h i g h u t i l i t y v a l u e s f o r t h e s e a t t r i b u t e s . Some s n o w m o b i l i n g was a s s i g n e d t o g r i d s q u a r e s w i t h snow d e p t h s o f 0 cm w h i c h i s n o t l o g i c a l . G r i d s q u a r e s w i t h snow d e p t h s o f 0 cm c o r r e s p o n d t o r i v e r l o c a t i o n s . A r i v e r p o s e s s a f e t y h a z a r d s t o s n o w m o b i l i n g ; t h e r e f o r e , m o d i f i c a t i o n o f t h e e v a l u a t i o n a l g o r i t h m i s n e c e s s a r y t o p r e v e n t s n o w m o b i l i n g f r o m b e i n g a s s i g n e d t o g r i d s q u a r e s w i t h l a k e s o r r i v e r s . D o w n h i l l s k i i n g a l l o c a t i o n d e c r e a s e d by 1 g r i d s q u a r e f r o m 2 g r i d s q u a r e s i n t h e s t a r t i n g p l a n . I t was a s s i g n e d t o t h e g r i d s q u a r e w h i c h has a s l o p e v a l u e o f 60 p e r c e n t , t h e most i m p o r t a n t a t t r i b u t e t o d o w n h i l l s k i i n g a c o r r e s p o n d i n g t o a u t i l i t y o f 1.0. C r o s s - c o u n t r y s k i i n g and s n o w s h o e i n g a l l o c a t i o n s d e c r e a s e d f r o m 19 g r i d s q u a r e s i n t h e s t a r t i n g p l a n t o 12 i n t h e f i n a l p l a n . T h e s e a c t i v i t i e s p r e f e r , on a v e r a g e , 67 - 152 cm o f snow. T h e s e a c t i v i t i e s were a l l o c a t e d t o g r i d s q u a r e s w i t h t h i s r a n g e o f snow d e p t h s . B o t h a c t i v i t i e s p r e f e r s l o p e v a l u e s o f 0 t o 15 p e r c e n t and as e x p e c t e d , were a l l o c a t e d t o s q u a r e s w i t h t h e s e s l o p e v a l u e s . H i k i n g , h o r s e b a c k r i d i n g and camping d e c r e a s e d f r o m 2 8 i n t h e s t a r t i n g p l a n t o 15 i n t h e f i n a l p l a n . They a r e m a i n l y c o n c e n t r a t e d a r o u n d t h e r i v e r w h i c h s e r v e s as a d r i n k i n g w a t e r s o u r c e f o r h i k i n g and c a m p i n g . H o r s e b a c k r i d i n g p r e f e r s s l o p e v a l u e s o f 5 t o 20 p e r c e n t . Camping p r e f e r s s l o p e v a l u e s o f 0 t o 10 p e r c e n t . The a l l o c a t i o n o f t h e t h r e e a c t i v i t i e s was r e s t r i c t e d t o g r i d s q u a r e s w i t h s l o p e v a l u e s o f 0 p e r c e n t , 86 c o r r e s p o n d i n g t o a u t i l i t y o f 1.0 f o r camping and a u t i l i t y v a l u e o f 0.0 f o r h o r s e b a c k r i d i n g . The r e a s o n why t h e s e a c t i v i t i e s were n o t a l l o c a t e d t o g r i d s q u a r e s w i t h h i g h e r s l o p e v a l u e s i s b e c a u s e t h e a c t i v i t y camping has t h e h i g h e s t s c a l i n g f a c t o r i n t h e o b j e c t i v e f u n c t i o n (0.254) w h i l e h o r s e b a c k r i d i n g has a s c a l i n g f a c t o r o f 0.078. H i k i n g has a s c a l i n g f a c t o r o f 0.146; t h e r e f o r e , a r e a s w i t h s l o p e v a l u e s f a v o u r a b l e t o c a m p i n g w i l l be c h o s e n f o r t h e l a n d u s e s c o n t a i n i n g t h e s e t h r e e a c t i v i t i e s . The a c t i v i t i e s were g r o u p e d t o g e t h e r b e c a u s e o f t h e i r compat-a b i l i t y ; i n f u t u r e t h e y s h o u l d be s e p a r a t e d . CHAPTER 7 SUMMARY AND CONCLUSIONS 87 In summary, the i n t e r e s t s of recreational user groups which w i l l use a mine waste area must be incorporated into a mine waste recreation plan. In t h i s study, a user group's i n t e r e s t s are defined as the user group's preferences for mine waste a t t r i b u t e s . The mine waste examined i n t h i s study was a hypothetical 10 km x 10 km section. The physical features of the mine waste resemble the coal mine waste of the Elk River Valley i n southeastern B r i t i s h Columbia. Nine recreational a c t i v i t i e s were chosen for analysis. These were grouped into 16 land uses. A resident of the Elk River Valley was chosen to represent the interests of each of the a c t i v i t y user groups. Multi-a t t r i b u t e u t i l i t y anal-y s i s was used to develop the nine representatives' preference structures for the mine waste a t t r i b u t e s . The results of the analysis were used to develop an objective function which meas-ured how well a recreation plan for the mine waste s a t i s f i e d the user groups' i n t e r e s t s . A computer program was developed to evaluate the objective function. Using t h i s program, a recrea-t i o n land use plan was produced for the hypothetical mine waste section which maximized the objective function. In conclusion, a planner i n t h i s study i s faced with the large problem of s a t i s f y i n g the most recreational user groups' in t e r e s t s i n an outdoor recreation plan for mine waste. Multi-a t t r i b u t e u t i l i t y analysis was employed to address t h i s objective. Two major l i m i t a t i o n s of the analysis were i d e n t i f i e d . The f i r s t was the large time commitment required by user groups to struc-ture t h e i r preferences for mine waste a t t r i b u t e s . Given time and budget constraints for a mine waste recreation development project, a planner must seriously consider whether the analysis i s j u s t i f i e d . Also user groups tend to become t i r e d with the preference assessment procedures. This may a f f e c t the shapes of the r e s u l t i n g u t i l i t y functions. The second l i m i t a t i o n was that the assessed preferences did not take into account the cost to the user groups of obtaining each attribute l e v e l . This cost may indeed influence the user groups' preferences over attribute l e v e l s . Accepting these l i m i t a t i o n s , multi-attribute u t i l i t y anal-y s i s i s successful i n breaking the large outdoor recreation planning problem into smaller problems, where user groups' objectives and associated a t t r i b u t e s , i n d i c a t i n g the extent that the objectives are achieved, can be i d e n t i f i e d . The analysis enables user groups to systematically a r t i c u l a t e and understand t h e i r preferences f o r each of the a t t r i b u t e s . Using t h i s i n f o r -mation i n a computer program, a planner i s able to i s o l a t e agree-ments and differences i n the judgments and preferences of the user groups, providing a firm basis on which to f a c i l i t a t e communication between the user groups and the planner, and to begin a constructive process of c o n f l i c t resolution. 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An o u t l i n e o f t h e b a s i c c r i t e r i a n e e d e d t o d e v e l o p a t r a i l b i k e s y s t e m . In U.S.D.I. H e r i t a g e C o n s e r -v a t i o n and R e c r e a t i o n S e r v i c e . P l a n n i n g f o r t r a i l b i k e r e c r e a t i o n P a r t I I . W a s h i n g t o n , D.C. APPENDIX Attributes of the Recreational A c t i v i t i e s 95 ATTRIBUTES OF THE RECREATIONAL ACTIVITIES T r a i l b i k i n g A t t r i b u t e s : = Travel time by road to t r a i l b i k i n g area (minutes) X 2 = Length of t r a i l available for t r a i l b i k i n g (km) X^ = Average slope of t r a i l b i k i n g area (%) X^ = # Encounters per hour with 4x4s A study by Wells (1979) showed that when developing a t r a i l -bike system, t r a i l b i k e access and length of t r a i l must be con-sidered. In t h i s study, access was assumed to be part of any t r a i l b i k e development and t r a v e l time to the area was chosen as a more appropriate a t t r i b u t e . Studies of t r a i l b i k e r i d e r preferences have shown that variety of topography i s one of the most desired features of a t r a i l b i k i n g area (Bury and Fillmore 1975, Chilman 1979). Average slope was chosen as an attribute to measure the variety of topography on an area of mine waste. The mine waste i s com-posed of uniform material consisting of various sizes of f r a c t -ured rock and therefore was not used as an at t r i b u t e to measure the variety of topography. Incompatability may ex i s t between t r a i l b i k e s and other recreational uses of a land area (Chilman 1979). Four-wheel drivers are a p o t e n t i a l source of motorized c o n f l i c t with t r a i l b i k e r s i n the study area and therefore was chosen as an a t t r i b u t e . The units of c o n f l i c t were chosen to be the number of encounters per hour with the c o n f l i c t i n g a c t i v i t y . An encounter i s defined as a single incident where a 4x4 impedes 96 the t r a f f i c of a t r a i l b i k e r . Four-wheel Driving A t t r i b u t e s : Xj. = Travel time by road to 4x4 area (minutes) Xg = Length of road available for 4x4 (km) X 7 = Average slope of 4x4 area (%) Xg = # Encounters per hour with t r a i l b i k e s As with t r a i l b i k i n g and other off-road motorized a c t i v i t i e s , access, length of t r a i l , varied topography and c o n f l i c t s with other recreational uses of the same land base are attributes that must be considered when planning an area for four-wheel dri v i n g (Arychuk et. a l . 1979). Therefore, the attributes chosen for the a c t i v i t y four-wheel d r i v i n g were the same as for t r a i l -b iking. An encounter i s defined as a single incident where a t r a i l b i k e impedes the t r a f f i c of a four-wheel drive vehicle. Snowmobiling At t r i b u t e s : Xg = Travel time by road to snowmobiling area (minutes) X ^ = Length of t r a i l available for snowmobiling (km) X l l = A v e r a 9 e winter snow depth of snowmobiling area (cm) X^2 = Average slope of snowmobiling area (%) Kuehn (1971) , Helmker (1975) and Arychuk et. a l . (1979) have shown that access to a snowmobiling area and length of t r a i l are important parameters that contribute to the success of snowmobiling. As with t r a i l b i k i n g , access was assumed to be part of any snowmobiling development and t r a v e l time was chosen as a more appropriate a t t r i b u t e . Snow depth and varied topography have been documented as being other important factors for snowmobiling (Hetherington 1971, Arychuk et. a l . 1979). Average slope was chosen as an attribute to measure the varying topography of a mine waste area. Downhill Skiing At t r i b u t e s : = Travel time by road to b a s e l i f t (minutes) X ^ = Average slope of s k i i n g area (%) X^j. = Average winter snow depth of skiing area (cm) The time a s k i e r must t r a v e l to the b a s e l i f t of a s k i h i l l has been reported as the most important factor i n determining a skier's preference for a s k i area for single day t r i p s i n the midwestern United States (Leuschner 1970). Newby and L i l l e y (1980) report s i m i l a r r e s u l t s . Leuschner (1970) reports that the physical qu a l i t y of a s k i slope i s the second most important factor i n determining s k i area preference i n the midwestern United States. The physical qu a l i t y of a s k i slope i n t h i s study i s described by the attributes average slope and the average winter snow depth of the s k i area. As i s character-i s t i c of msot downhill skiing areas i n B r i t i s h Columbia, i t was assumed i n t h i s study that any downhill skiing development would be r e s t r i c t e d to downhill s k i i n g only; therefore, no attributes describing c o n f l i c t s were required for t h i s a c t i v i t y . 98 Cross-Country Skiing A t t r i b u t e s : X = Travel time by road to s k i i n g area (minutes) 16 X = Length of t r a i l available for s k i i n g (km) X ^ g = Average slope of s k i i n g area (%) X19 = A v e r a 9 e winter snow depth of s k i i n g area (cm) Newby and L i l l e y (1980) report t r a v e l distance as being a s i g n i f i -cant factor i n choosing to use a cross-country s k i i n g area. In t h i s study t r a v e l time was used as an attribute rather than t r a v e l distance because i n proctice u t i l i t y assessment i n t e r -views, decision maker's found i t easier to estimate how long they would prefer to t r a v e l rather than how f a r . Cross-country skiers require varied topography (Lund and Williams 1974, A l l a n 1975) which i s represented by the a t t r i -bute average slope. Length of t r a i l i s also a major factor i n determining whether to use a s k i area (Newby and L i l l e y 1980). Snow depth w i l l contribute s i g n i f i c a n t l y to the q u a l i t y of a s k i area and was therefore chosen as an a t t r i b u t e . Snowshoeing At t r i b u t e s : X 2 Q = Travel time by road to snowshoeing area (minutes) X21 = L e n 9 t h o f t r a i l available for snowshoeing (km) X22 = A v e r a 9 e winter snow depth of s k i i n g area (cm) X = Average slope of s k i i n g area (%) The attributes for snowshoeing were chosen to be those of cross-country skiing because the a c t i v i t i e s require s i m i l a r areas 99 and physical features of the land base. Hiking A t t r i b u t e s : = Travel time by road to hiking area (minutes) X 2 5 = Distance of hiking area from a drinking water source (km) Travel time to a hiking area was chosen as an attribute because a hiker generally considers how long he must t r a v e l to an area i n deciding whether he w i l l use the area. Distance from a drinking water source was chosen as an attribute because many hikers cannot hike for very long or very f a r without drinking water. Horseback Riding A t t r i b u t e s : X = Travel time by road to horseback r i d i n g area (minutes) 2 0 X 2 7 = Length of t r a i l available for horseback r i d i n g (km) X_ = Average slope of horseback r i d i n g area (%) 2. o Travel time to a horseback r i d i n g area was chosen as an attribute because a ri d e r considers how long he must t r a v e l to and area when deciding whether to use the area. Length of t r a i l and slope were chosen as attributes because both help dictate a decision maker's decision whether to use the horseback r i d i n g area and both a f f e c t the r i d i n g experience. Summer Motorized Camping At t r i b u t e s : X = Travel time by road to camping area (minutes) 2 9 X = Distance of camping area from a drinking water source (km) X ^ = Average slope of camping area (%) Travel time to a camping area was chosen as an attr i b u t e because campers generally have varying preferences for how long i t takes them to get to the camping area (Brockman and Merriam 1973) . Drinking water supply i s necessary for a successful camping experience (Can. Govt. Office of Tourism 1972, Brockman and Merriam 197 3). The slope of the camping area cannot be too great or many recreational vehicles cannot park properly and campers w i l l f i n d i t d i f f i c u l t to sleep; therefore, average slope was chosen to be an attribute (Can. Govt. Of f i c e of Tourism 1972) . 1 0 1 APPENDIX Dialogue to Familiarize the Decision Maker with the Terminology and Motivation for the Assessment of His U t i l i t y Function 102 DIALOGUE TO FAMILIARIZE THE DECISION MAKER WITH THE TERMINOLOGY AND MOTIVATION FOR THE ASSESSMENT OF HIS UTILITY FUNCTION D = Decision Maker I: I ' l l t e l l you b r i e f l y what I am t r y i n g to accomplish i n t h i s interview. As you know, much of the land on e i t h e r side of the Elk Valley i s being mined for coal. Some day most of t h i s v a l l e y w i l l eventually be mined and there w i l l be quite a large area of mine waste on both sides of the v a l l e y . What i s happening now i s we are revegetating the waste p i l e s with grasses and other plant species and f e r t i l i z i n g them. However, i n the future, these areas may have some demand on them to be used for outdoor recreation such as t r a i l b i k i n g , horseback r i d i n g , snowmobiling, camping and other a c t i v i t i e s . Now assume the valley was make up of mostly mine waste, and the government decided to b u i l d a f i r s t class horseback r i d i n g area on i t . Now, as you know, horseback r i d i n g has many factors that make i t enjoyable or rotten such as a sunny horseback r i d i n g area versus a rainy area. What I want to do i s , for a factor l i k e say how long you would t r a v e l to get to t h i s horseback area, f i n d your prefer-ences for various lengths of t r a v e l time and the same for other factors. Do you understand? D: Yes, I think so. I: O.K., I w i l l represent your preferences for various levels of a factor by what we c a l l a u t i l i t y . ( I l l u s t r a t i n g ) I f we have a graph, with the horizontal axis as the factor, say length of t r a i l , the v e r t i c a l scale w i l l be a u t i l i t y scale. U t i l i t y i s just another word for preference. So, the least preferred t r a i l distance w i l l have a value of 0 and the most preferred t r a i l distance w i l l have a value of 1. So, a f t e r I ask you questions we w i l l develop your preference rela t i o n s h i p for the factors. The r e s u l t i n g curve may have any shape. I t may look l i k e t h i s ( i l l u s t r a t i n g ) or t h i s or t h i s - that i s what I am t r y i n g to f i n d out. Do you get the idea behind what I'm t r y i n g to do? Symbolism A c t i v i t y I = Interviewer Horseback Riding D: Yes, I think so. APPENDIX Dialogue for Assessing Attribute U t i l i t y Functions and V e r i f y i n g Attribute U t i l i t y Independence 104 DIALOGUE FOR ASSESSING ATTRIBUTE UTILITY FUNCTIONS AND VERIFYING ATTRIBUTE UTILITY INDEPENDENCE Symbolism Attribute A c t i v i t y I = Interviewer Travel Time (X o c) Horseback Riding D = Decision Maker I: I would l i k e to ask you questions about hypothetical s i t u a t -ions and t h i s i s the type of question I w i l l ask you. I ' l l give you a choice between two alt e r n a t i v e s . One w i l l be where the government says they are going to give you t h i s alternative for sure, but you may say you don't l i k e i t . If you don't l i k e i t , you can then go to a r b i t r a t i o n where you may end up with an alternative that i s better than t h e i r o f f e r or with an alternative that i s worse than t h e i r o f f e r . In other words, i f you don't want t h e i r o f f e r , you can go for t h i s 50-50 gamble. What I want to do i s pose these choices to you, changing the government's o f f e r u n t i l I f i n d the value of what the government w i l l give you where you are i n d i f f e r e n t between i t and the 50-50 gamble. We'll do i t for the factors l i k e t r a v e l time, length of t r a i l etc. Do you understand the basic idea? D: I think so. I: It w i l l become clearer as we go on. To show you how t h i s works, l e t ' s take t r a v e l time to a r i d i n g area by road. I f the government were going to b u i l d you a r i d i n g area 2 hours from town, you have the option of taking t h e i r o f f e r or i f you don't l i k e i t you have the option of going to a r b i t r a -t i o n where you may end up with the best t r a v e l time which would be right outside your house as you stated e a r l i e r or you may end up with having to drive 4 hours. Would you take the government's o f f e r for sure or would you go for a 50-50 gamble between the 4 hours and 0 hours? D: I'd say i f i t ' s a 2 hour drive then that's o.k. I: Would you go for the gamble or take the government's o f f e r at 2 hours? D: I would go for the 2 i n that case. I: What i f they were going to give you 3 hours for sure? D: That's getting to be too long. That's s t a r t i n g to get up there. I: Would you take the chance then? 105 D: Yes, I would then. I: What i f t h e i r o f f e r were 2.5 hours? Would you take i t or go to a r b i t r a t i o n and try to get 0 or end up with 4 hours t r a v e l time? D: It's hard to say. I: Would you be i n d i f f e r e n t between the two choices when the government's o f f e r i s at 2.5? D: Yes, that's about r i g h t . I: Assume that 2.5 i s your indiffernece point where the other factors length of t r a i l and slope are at t h e i r best l e v e l s . Would you be i n d i f f e r n e t at 2.5 i f they were at t h e i r worst levels? D: I guess so. I: Would i t be true to say that your indifference point does not depend on the levels of the other factors? D: Yes, I guess so. ( V e r i f i c a t i o n of u t i l i t y independence) I: O.k. I f we had another choice here but t h i s time the government offers you an area that takes 3 hours to get to. You can take t h e i r o f f e r or go to a r b i t r a t i o n where you may end up with 2.5 hours or 4 hours t r a v e l l i n g time. D: I'd t r y and get i t shortened up. I: Would you go for the gamble then? D: Yes. I: What i f t h e i r choice were say 2.7 hours? D: Well, 2.7 versus 2.5, - I'd probably be better o f f with 2.7 for sure. I: What i f they gave you 3 hours for sure? D: Well, i t ' s one mroe hour to 4 but I may get down to 2.5. I think I'd probably take a chance and try and p u l l i t back to 2.5. I: What i f t h e i r o f f e r were 3.5 hours t r a v e l l i n g time? D: For sure I'd take the gamble and try and shorten i t . I: What i f t h e i r o f f e r were say 3.2 hours? D: I'd try and get the 2.5. I: So you would go for the 2.7 for sure but at 3 you would go 106 for the gamble. Would you be i n d i f f e r e n t at about 2.8 or 2.9 hours? D: I'd probably be i n d i f f e r e n t at 3. I: And you would go f o r the gamble at 3.2? D: That's r i g h t , sure. I: Does your choice of 3 depend on the leve l s of the other factors? D: No, not r e a l l y . ( V e r i f i c a t i o n of u t i l i t y independence) I: Now, i f the government were going to give you an area 1 hour from town for sure, you could accept t h e i r o f f e r or go to a r b i t r a t i o n where you could end up with 0 hours t r a v e l l i n g time or 2% hours. Which one would you take? D: I'd take the 1 hour. I: What i f 1% hours was t h e i r o f f e r ? D: In that case I'd take a chance on a r b i t r a t i o n and try to get the 0 hours. I: So between 1 and 1% you would be i n d i f f e r e n t between the choices? D: Yes. I: Would t h i s choice change with varying levels of the other factors? D: No. ( V e r i f i c a t i o n of u t i l i t y independence) 107 APPENDIX Dialogue for Obtaining Attribute Tradeoffs, Consistency Checks and V e r i f y i n g Attribute P r e f e r e n t i a l Independence DIALOGUE FOR OBTAINING ATTRIBUTE TRADEOFFS, CONSISTENCY CHECKS AND VERIFYING ATTRIBUTE PREFERENTIAL INDEPENDENCE Symbolism A c t i v i t y I = Interviewer Horseback Riding D = Decision Maker The Decision Maker's Ranking of the Attributes I: Now i f you were asked to rank the factors t r a v e l time, length of t r a i l and slope, i n order of preference to you for horse-back r i d i n g , how would you rank them? D: That's t r a v e l time, t r a i l length and slope? D: I guess slope would be the most l i m i t i n g factor aft e r a while so I'd say i t would be the most important. I: Then what? D: I'd say that t r a i l distance would be next. Travel time i s n ' t that important. Tradeoff #1: Travel Time vs. Length of T r a i l vs. X~_) I: What I would l i k e to do now i s f i n d out how to weight each of the factors numerically as to t h e i r r e l a t i v e importance to you. We do t h i s by trading o f f between the factors. Let's say that the government was going to locate t h i s r i d i n g area where you had to drive 0 hours to get i t , or right outside your house and with i t you would get 4.5 km of t r a i l . How much further would you drove to get 13 km of t r a i l with 20 km as your most preferred distance? D: I think I'd go about 3 more hours to get 13 because 4.5 i s n ' t very much for a day t r i p . I: O.k., i f you had to t r a v e l 1.5 hours to get an area with 11 km of t r a i l , to get 20 km of t r a i l , how much longer would you I: Yes. 109 drive? D: Probably another hour. I: Just a second while I do a quick c a l c u l a t i o n (consistency check). (The c a l c u l a t i o n i s to check the r a t i o s of the k scaling constant values from equation (page ). For the two pairs of tradeoffs, the ratios should be reasonably close. In t h i s case, k 2 7 w a s 0.827 f ° r the 4.5 km to 13 km tradeoff while k26^ k27 w a s ^.50 for the 11 to 20 km trade-o f f . The r a t i o s are c l e a r l y not very close. To t r y and bring the ra t i o s closer together, the decision maker i s asked to confirm his tradeoffs.) I: Now you said that you would t r a v e l from 0 to 3 hours to go from 4.5 km to 13 km, i s that right? D: Yes. I: /And you would t r a v e l one hour more to go from 11 km to 2 0 km t r a i l length? D: Yes, between an hour and an hour and a h a l f . I: So about an hour an a quarter? D: No, I'd say I'd drive another hour and a h a l f . I: O.k. , l e t me do a quick c a l c u l a t i o n (consistency check) D: (The k 0 / r/k 0_ r a t i o f o r the 11 km to 20 km tradeoff was Zb 2 / lowered to 1.11 which i l l u s t r a t e s more consistency between the tradeoffs.) Testing for P r e f e r e n t i a l Independence I: For the tradeoffs we just did between t r a v e l time and length of t r a i l - would your answers be influenced by varying slope values? D: What do you mean? I: O.k., l e t ' s examine a tradeoff we just d id. You said that you would t r a v e l 0 to 3 hours to go from a 4.5 km to 13 km t r a i l distance. Let's say that the r i d i n g area has a 0% slope. Would you s t i 1 drive the 3 hours to go from 4.5 km to 13 km i f the slope were 40%? D: Yes, I think so. I: So varying slope values don't r e a l l y influence the tradeoff? D: No. (This confirms that t r a v e l time and length of t r a i l are 110 p r e f e r e n t i a l l y independent of slope) Tradeoff #2: Travel Time vs. Average Slope of Area (X vs X~a) ~ ~ ~ ~ ~ ~ ~ — — — — — — — — - — — — — — — — — — — — — — — — — — — — — — — _ . 26 2 8 I: Now, i f you had to drive 1 hour to get a 20% slope, i f 10% slope i s your best value, how much longer would you drive to get the 10%? D: O.k., you said 1 f o r 20? I: Yes. D: Between 10 and 20 i s n ' t that great - I wouldn't consider more than 2 hours. I: So i f you had to t r a v e l 2.5 hours to get a 30% slope, how much longer would you drive to get i t down to 20%? D: You said 2.5 for 30? I: Yes, 2.5 for 30. D: I'd go up to the 4 hours. I: O.k., l e t me do another c a l c u l a t i o n . (The ra t i o s of k„,/k_ were close to each other (0.718 and 26 28 0.644 i l l u s t r a t i n g that the decision maker i s consistent i n his tradeoffs.) Testing for P r e f e r e n t i a l Independence I: Would length of t r a i l a f f e c t how much t r a v e l time you would give up for varying slopes? D: No, I don't think so. ( V e r i f i c a t i o n of p r e f e r e n t i a l independ-ence of t r a v e l time and slope on length of t r a i l . ) I l l APPENDIX Attribute U t i l i t y Functions 112 113 Four-Wheel Driving 116 Downhill Skiing 120 1 2 1 Snowshoeing Hiking i i i i i i i i i — n - (15,1) 1 "T 1 T to tn d - ]\ -CD (J> (14) V(45,0.75) -[LITY 0.64 -I—1 1— -O _ A (12,0.5) Jo (75,0.5) -CM CO _A(10,0.25) \ u 2 0 , 0 . 2 5 ) -£ d -(0,0) ) 1 1 1 1 V x \ ( 2 4 0 , 0 ) 1 1 0.0 40.0 80.0 120.0 IBO.O 200.0 240.0 2B0.0 TRAVEL T IME BY ROAD TO HIKING AREA IMJNUTESJ H o r s e b a c k R i d i n g 126 • "i 1 1 1 1 1 1 1 1 1 1 T 1 1 (10, X) -ca CD 5(7.5) CDU6.0.75) _ >" * •— _ _ 1-1 •=> _ l i—t 1 — -5 (6.5,0.5) OU23,0.5) O rs* CO _ o - f (3. 5,0.25) t ^ 33,0.25) -to C3 (0,0) V (50,0) a r i i i i i i i i i i I 1 > i 1 1 6.0 16.0 24.0 32.0 4O.0 48.0 56.0 AVERAGE SLOPE OF HORSEBACK RIDING AREA (%) A. .9 Summer M o t o r i z e d Camping 128 T - 1 ! 1 1 1 I 1 1 1 1 1 I I 5 C ><0,1> -OD © ( 4 , 0 . 7 5 ) -LITY 0.64 -UTI 0.48 ( D ( 8 , 0 . 5 ) -o \ U 5 , 0 . 2 5 ) -IS o - ^ " " ^ • ^ ( 5 0 , 0 ) >= 1 1 1 1 1 1 1 1 1 0.0 B.0 16.0 2 4 . 0 2 2 . 0 40 .0 4 0 . 0 5 6 . 0 RVERAGE SLOPE OF CAflPING AREA (%) APPENDIX A t t r i b u t e T r a d e o f f s f o r A s s e s s i n g S c a l i n g C o n s t a n t s 130 ATTRIBUTE TRADEOFFS FOR ASSESSING SCALING CONSTANTS A. .1 T r a i l b i k i n g to UJ => S L cr =. LU ? CC C M ex m <x cc • -o ->-CO „ Ul ! JZ LLl > CC cc 1 X r i i i T 1 1 1 1 1 " "I 1 T 1 0 (9 ) = 0 .580 U(15) - 0 .925 - U(15) - 0 .770 TJ(75) «= 0 .580 - (20 ,180 ) • -o k x = 0 . 5 5 1 k 2 - - U(9) - 0 .580 _ U(30) « 0 .820 U(20) - 1.00 - TJU80) - 0 .140 - • k1 m 0 . 6 1 8 k 2 - 0 (15 ,75 ) - ' Average • (9 ,30 ) k j = 0 . 5 8 5 k 2 ° ( 9 , 1 5 ) i i i t i i i i I l 1 1 . . J 1 8.8 ia4 12.0 13.6 15.2 16.8 1 1 4 20.0 X-2 LENGTH OF T R A I L A V A I L A B L E FOR T R A I L B I K I N G KID to L U •-1 ffi cc UJ „ £ = * S u m R CC O S o CX o >-m UJ CC 5 CC I X — i 1 1 1 r 1 T —I'" I I i i l l ! UOO) - 0.650 U(120) - 0.335 - U(40) - 0.340 UC15) - 0.925 - ° k l - 0.525k 3 - • (15,180) U(30) - 0.650 _ U(30) - 0.810 U(15) - 1 .00 - 11(180) - 0 .140 - O (30,120) • * i • 0.522k 3 - A v e r a g e k l " 0.524k 3 • ( 30 ,30 ) - 0 ( 4 0 , 1 5 ) 1 1 1 , _ l L 1 1 J . I i I i i i l 14.0 1B.0 22.0 26.0 30.0 34.0 38.0 42.0 4E.D X-3 AVERAGE SLOPE OF TRAILBIKING AREA ») 1 3 1 to cr LU tt g <• (2.60) o z l-l U(2) • 0 .200 U(60) - 0 .650 U(3) - 0 .100 U(30) - 0 .810 e> k . 0 . 6 2 5 k . O I s >-m • (3 ,45 ) O ( 3 , 3 0 ) (4 ,30 ) D(3 ) - 0 .100 U(45) - 0 .725 U(4) - 0 .000 0 (30 ) - 0 .810 1 .18k , Average LU > 1 cr cr 0 . 9 0 3 k . x p J L _1_ 2.0 2.4 2.8 3.2 3.6 4.D 4.4 4.8 X - 4 « ENCOUNTERS PER HOUR UITH 4X4S 5.2 V ID ""* rf LD Z H fc! °-m B CX CC o LU -J a> _ J M cr > CX rg T 1 1 1 1 1 1 1 1 1 1 I 1 ~ cr • (25 ,11 ) ( 4 0 , 1 5 . 5 ) o O (35 ,11 ) U(35) = 0 .500 U ( l l ) •= 0 .670 U(40) " 0 .340 0 ( 1 5 . 5 ) •» 0 .840 o k „ 0 . 9 4 1 k . U(15) - 1.00 U ( 6 . 5 ) - 0 .410 0(25) - 0.795 U ( l l ) - 0 .670 • k „ = 0 . 7 8 8 k , O Average 13 Z UJ CM I ( 1 5 , 6 . 5 ) J_ 14.0 J _ _1_ _ l _ _i_ _1_ _ l _ i f iO 72.0 2« . l 31.0 34.0 38.0 42.0 X - 3 AVERAGE S L O P E OF T R A I L B I K I N G AREA [%) 0 . 8 6 5 k , 132 cr cc »-» CC E O U. U I m us cr e _ i n cr > cr <o cr cc o :r i— CO » Z H" LD CM 1 X oo (1 ,9 ) -i 1 i r • (3 ,20 ) ( 4 , 1 5 . 5 ) A O (2 ,11 ) LO 1.4 IB 2.2 2.6 3.D 3.4 3.8 X - 4 fl ENCOUNTERS PER HOUR U ITH 4X4S 0 (2 ) - 0 .200 0 (11 ) = 0 .670 U(4 ) " 0 .000 U ( 1 5 . 5 ) - 0 .840 o k „ 1 . 1 8 k . 0 (1 ) - 0 .500 0 (9 ) - 0 .580 0 (3 ) - 0 .100 O(20) - 1.00 • k . 0 . 9 5 2 k . Average k 2 = 1 . 0 7 k 4 cc cr CD Z n CD cr cc LU CL. o _ J to U i o cr cc u i > cr ro i x 4 ( 1 ,30 ) O (2 ,30 ) ( 3 . 1 5 ) • J I l _ ( 4 , 25 ) 6 0 (2 ) - 0.200 0 (30 ) - 0.650 0 (4 ) - 0 .00 0 (25 ) - 0 .795 _| o k 3 - 1 . 3 8 k 4 0 (1 ) - 0 .500 0 (30 ) - 0 .650 0 (3 ) • 0 .100 0 (15 ) - 1.00 1.0 1.4 -.9 2.2 26 3.0 3.4 3.8 X -4 tt ENCOUNTERS PER HOUR UITH 4X4 'S • k 3 - 1 . 1 4 k 4 J Average k 3 - 1 . 2 6 k 4 133 Four-Wheel Driving CO R UJ cc UJ or cr X D CC o CC a CO — UJ UJ > cc cc L O 1 X -i 1 1 r -i 1 r 6- ( 6 . 5 , 1 2 0 ) (7 ,10 ) ( 1 2 . 5 , 1 8 0 ) O ( 1 2 . 5 , 9 0 ) ( I i • i ' I L \ s 1.5 B.5 a5 10-5 115 12-5 13-5 X -6 LENGTH OF ROAD AVAILflBLE FOR 4X4 (KID U ( 6 . 5 ) » 0 .470 U (120) - 0 .500 U U 2 . 5 ) - 0 .695 U(180) - 0 .190 o k. 0 . 7 2 6 k , U (7 ) - 0 .500 O(10) • 0 .880 U ( 1 2 . 5 ) - 0 .695 U(90) » 0 .575 • k , = 0 . 6 3 9 k , Average 0 . 6 8 3 k , LU § t- r« n z M n o cc ™ LU CC cc x I o o cc O (35,150) % (45 ,180 ) J 0 (20 ) • 0 .180 0 (0 ) - 1.00 0 (35 ) - 0 .500 0 (150 ) - 0 .315 o k 5 » 0 . 4 6 7 k ? 0 (35 ) - 0-500 0 (15 ) - 0 .815 0 (45 ) - 0 .895 0 (180 ) - 0 .190 LU • k c 0 . 6 3 2 k , UJ CD > CC CC in 2 X (20 ,0 ) I • ( 35 ,15 ) J I L. 20.3 J I I 1 L 24.0 28.0 32.0 3B.D 40.0 44.0 X-7 AVERAGE SLOPE OF 4X4 AREA IT.) 4B.0 52.0 Average 0 . 5 5 0 k ? 134 to 3 LU f2 t— — =3 Z M E R o tx » LU CC tx x 5 T 1 1 1 I 1 1 " ' 1 1 ' 1 1 P ,180) (2,150) D o C C -O c c >-C D R L U ^ > c c x 1.0 0 (3 ,90 ) (4 ,30 ) J 1 1 * -U ( l ) - 0 .830 U(180) • 0 .190 0 (3 ) - 0 .250 0 (90 ) » 0 .575 ,4 18 2.2 2.6 3-D 3-4 3.9 X-8 H ENCOUNTERS PER HOUR UITH TRAILBIKES o k< 4.2 1 .51k 8 0 (2 ) - 0 .500 0 (150 ) - 0 .315 0 (4 ) - 0 .00 0 (30 ) - 0 .720 • k 5 - 1 .23kg A v e r a g e k 5 - 1 . 3 7 k 8 £ r x £ c c o LO -_J m cr _ J M -3 C t i n > -" c r o c r u> o to z LL) _ J CO X O (10 ,18 ) T 1 r I ( 35 ,20 ) 8.0 O (30,11) (50,9) I ' l l I * J 1 1 L. 16.0 24.0 32.0 40J 48J) 56.0 X-7 AVERAGE SLOPE OF 4X4 AREA l« 64.0 0 (10 ) 0 (18 ) 0 (30 ) 0 (11 ) o k , • • 0 .090 = 0 .940 > 0 .320 = 0 .640 0 . 7 6 7 k n 0(35) - 0 .500 0 (20 ) - 1 .00 0 (50 ) - 1 .00 0 (9 ) - 0 .565 • k f i - 1 . 1 5 k , A v e r a g e k , - 0 . 9 5 9 k , 135 R r x s r g h U_ co ID UJ -_J CD CC _ J M ™ cr > cc D CC (o U-o :x => 1— C! z UJ X -i r 0. ( 2 ,13 ) (2 ,9 ) 1 T 0 ( 4 , 2 0 ) • ( 4 , 1 5 . 5 ) _ 1 _ _ 1 _ J_ JL 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 X-8 fl ENCOUNTERS PER HOUR UITH TRfllLBIKES U(2) U (13) U (4 ) U (20) 0 .500 0 .695 0 .000 1.00 o k , " 1 .64k U(2) 0 (9 ) 0 (4 ) 0 ( 1 5 . 5 ) - 0 .805 8 0 .500 0 .565 0 .000 k 6 « 2 . 0 8 k 8 Average 1.86k 8 CC LU CC CC 5 O a o LU * 0. o _ J * » o LU ft" CO CC cc LU CC V 0 (0 ,25 ) • (1 ,35 ) (3 ,45 ) (3,40)-6 0 (0 ) - 1.00 0 (25 ) - 0 .220 0 (3 ) - 0 .250 0 (40 ) - 0 .790 o k, - 1 . 3 2 k B 0 (1 ) - 0 .830 0 ( 3 5 ) - 0 .500 0 (3 ) - 0 .250 0 (45 ) - 0 .895 • k, - 1 . 4 7 k g i x A v e r a g e k, - 1 . 4 0 k 8 J _ J _ _ l _ " (LD 0.4 0.8 1 2 1 6 2.D 2 « 2-8 X-8 » ENCOUNTERS PER HOUR UITH TRfllLBIKES 1 3 6 Snowmobiling CO LU CX _ UJ ° . cc cx -to z t—i a _J rf H " 2 m o o R z s C O -o 5 cc >-CO o d LU r -cr cc co I X - i — f r (9,160) (4.5,90) (11,60)0 _ l _ (6.5,30) _ l _ _1_ _ J _ _ l _ _ l _ _ L 45 S5 6.5 -7-5 8.5 3.5 1D.S 1LS X - 1 0 LENGTH OF T R A I L AVA I LABLE FOR S N 0 U H 0 B I L I N G ( K I D U(6.5) = 0.410 U(30) - 1.00 0(11) - 0.775 0(60) - 0.695 o kg - 1.20k 1 Q 0(4.5) - 0.280 0(90) - 0.530 0(9) - 0.625 0(180) » 0.160 0.932k 10 Average 1.07k 10 E -cr LU CC o CX N v> O z CO £ o -JC o Z c 1/1 R o ~ o a =». o **• CC * >-CO LU °. LU > , cr cc co i X id —I 1— (91,180) • (152,150) ( 2 1 3 , 6 0 ) 0 ( 244 ,60 ) ( • i I J 1 1 L-0(91) • 1.00 0(180) - 0.160 0(213) - 0.200 0(60) - 0.695 o k. 1.50k 11 0(152) - 0.760 0(150) - 0.250 0(244) » 0.00 0(60) - 0.695 • k 9 - 1.71k u A v e r a g e 1.61k 11 30 0 110 0 O0J5 1S0J 170.0 1S0.D 2I0J 230.0 2510 X-11 AVERAGE UINTER SN0U DEPTH OF SNOUHOBILING AREA (CM) 137 „ g to <•> LU I— =5 Z o ex LU CC o z m g o « n 3 O Z e CO d C M O CC =>. g l >-m i 1 1 1 1 1 1 1 1 1 1 1 1 i r 0 ( 3 0 , 2 4 0 ) (10 ,180) 0 ( 3 0 , 1 5 0 ) TZ » LU o > E a — c c co j _ J _ _ l _ J _ J _ J _ 0 (30 ) - 1.00 0 (150 ) - 0 .250 U(40) - 0 .680 0 (60 ) - 0.695 O k . ! 0 J 14 D 18.0 22.0 26.0 30.0 34.0 38.0 X-12 AVERAGE SLOPE OF SNOUHOBILING AREA IX) 0.719k 12 0 (10 ) - 0 .840 0 (180 ) " 0 .160 0 (30 ) • 1.00 0 (240 ) - 0 .000 • k B - 1 .00k 12 Average k 9 - 0 . 8 6 0 k 1 2 42.0 CD R O z LO o c c H o LU CO erf CX _ _ J CX * 3 M <X CC U. O z LU L ( 91 ,11 ) (91 ,2 ) m ( 152 ,20 ) J _ _1_ _ l _ J L J _ J _ ( 2 1 3 , 9 ) D J _ J _ J _ " "ano 110.0 m o BO.O m o isao 210-O 230.0 X-11 AVERAGE UIN7ER SN0U DEPTH OF SN0UH0BILING AREA 0(91) - 1.00 0 (2 ) » 0 .120 0(213) - 0 .200 0 (9 ) - 0 .625 o k 10 1.58k 11 0 (91 ) - 1.00 U ( 1 D - 0 .775 0 (152 ) - 0 .760 0 (20 ) - 1.00 k 1 Q - 1 . 0 7 k n Average "10 1.33k 11 (ClU 138 z _J CD =». g s O z CO p cc S o Lu LU _ J =. CO 0 cc M _J M CX cc-. se _ i "~ M CC CC I— o IN LU -O CD c Z CD LU >< 5 Q ( 1 0 , 6 . 5 ) _ l L_ (20 ,4 )^ J _ • ( 3 0 , 6 . 5 ) (40 ,20 ) J _ ao 14 D 18.0 22.0 26-0 30.0 34 J 38.0 X-12 AVERAGE SLOPE OF SNOVJflOBILING AREA i%) U(10) - 0 .840 U ( 6 . 5 ) = 0 .410 U(20) » 0 .933 U(4) •= 0 .250 o k 10 0 .581k 12 U(30) - 1 .00 U ( 6 . 5 ) - 0 .410 U(40) - 0 .680 U(20) • 1 .00 "10 0 .542k 12 average "10 0 .562k 12 UJ in CC S CC o J § t - l n CD O § 5 i s CO u. O o M Cu LU o g i CO Eg UJ rr LU 5=8 L 1 x . ( 3 0 , 2 4 4 ) O ( 3 0 , 1 5 2 ) ( 1 0 , 9 1 ) ( 4 0 , 9 1 ) J I L_ 0 (1 0 ) - 0 . 8 4 0 0 (9 1 ) • 1 .00 0 (3 0 ) - 1 .00 0 (152 ) - 0 . 7 6 0 O k j j » 0 . 6 6 7 k 1 2 0 ( 3 0 ) - 1 .00 0 (244 ) - 0 .000 0 (4 0 ) » 0 .6B0 0 (9 1 ) - 1 .00 • k u • 0 . 3 2 0 k 1 2 A v e r a g e ao 14 0 « . 0 22.B 26.0 30.0 34.0 3BJ) X-12 AVERAGE SLOPE OF SNOUnOBILlNG AREA (» 0 . 4 9 4 k 1 2 139 Downhill Skiing 05 P LU — L U M —I = LU CO E CX CD • ~ CX O cr >- °. CD S3 LU H LU > cc cc ro ( 35 ,30 ) J L _1_ J L • (50 ,60 ) J _ ~ i — * r ( 60 ,180 ) (60 ,150 ) 0 J _ _1_ 34.0 38.0 42.3 46.0 50.0 54.0 SB.0 62.0 X-14 AVERAGE SLOPE OF D0VJNH3LL SKIING AREA ll) U(50) - 0 .885 U(60) - 0 .415 U(60) - 1.00 U(150) = 0 .250 o k 13 0.697k 14 U(35) - 0 .620 U(30) - 0 .720 TJ(60) - 1 .00 U<180) - 0 .165 *13 0.685k 14 Average "13 0 .691k 14 LU m ID F CX m cr o cc m l LU LU > cr cc ro i x P ( 1 2 2 , 4 5 ) .(122,30) ' ' O ( 1 5 2 , 9 0 ) ( 1 8 3 , 9 0 ) J _ JL J L J _ J L J L J L J L 120 0 140.0 160.0 180.0 200.0 220.0 240.0 260.0 X-15 AVERAGE UINTER SN0U DEPTH OF SKIING AREA ICfl) 0(122) - 0.570 U(45) - 0.615 0(152) - 0.785 U(90) « 0.415 o k j j - 1.08k1 5 U(122) « 0.570 D(30) • 0.720 0(183) - 0.940 0(80) • 0.415 • * i 3 " l-21 kl5 A v e r a g e k,, - 1 . 1 5 k " 1 3 1 5 140 cr LU CC <x =». o M z M M *3 O o o 0-o _ J CO LU oo a cc LU S 3 — ( « » — i r (122 ,60 ) • (152 ,45 ) r- 0 (107 ,40 ) J_ J_ ( 244 ,30 ) -1 1 L n 'lOILD 12O.0 140.0 160.0 1BQ.0 2O0.0 220.3 2410 X-15 AVERAGE UINTER SN0U DEPTH OF SKIING AREA (CH3 0 (107 ) •= 0.435 0 (40 ) " 0 .770 0 (244 ) - 1.00 0 (30 ) » 0 .430 O k 1 4 - 1 . 6 6 k 1 5 0 (122 ) - 0 .570 0 (60 ) - 1 .00 0 (152 ) - 0 .795 0 (45 ) - 0 .830 • k 14 1.32k 15 Average l 1 4 1.49k 15 141 A. .5 cross-Country Skiing LO = = 3 LD or ( X L O R co a cx o cc R io >-CO LU s° LU > C X R cc » 1 X P (9 ,15 ) _ J L (12 ,10 ) _1 L _ — i 1 r -(20 ,120) O (20 ,45 ) J _ J L J _ J _ J _ 5e e 10.4 12.0 13.6 1S.2 16.6 18.4 20.0 X-17 LENGTH OF TRAIL AVAILABLE FOR SKIING IKfU U(9) » 0 .250 U(15) - 0 .810 U(20) » 1.00 U(120) = 0 .200 o k 16 1.23k 17 U{12) - 0 .440 U(10) - 0 .870 U(20) - 1.00 U(45) - 0 .500 "16 1 .51k 17 Average "16 1.37k 17 C O U J 5- N LU CC o Z o M R C O cc o cc >- c=> CO D! > c r c c i -i r (3 ,180) L O (3 .120 ) (35 ,10 ) (50 ,0 ) • ' i I m i 1 L. U(3) » 0 .800 C (120 ) - 0 .200 TJ(35) - 0 .160 U(10) « 0 .870 o k 16 0 .955k 18 U(3) » 0 .800 U(180) - 0 .100 U(50) - 0 .00 U(0) » 1-00 « k - 0 .889k 16 IB k i 6 - 0 . 9 2 2 k 1 8 0.0 8 0 16.0 24.0 32.0 40.0 48.0 56.0 X-18 AVERAGE SLOPE OF SKIING AREA W) 64.0 142 Z C C ( X t-l R cr o cc m U J > cr cc i X - i—i 1 r O (122 ,15 ) (152 ,10 ) J * 1 L. i r ~i r i r (244 ,35 ) • ( 2 4 4 , 2 0 ) « J . 0 (122 ) - 0 .790 0 (15 ) - 0.B10 U(244) » 1.00 0 (35 ) - 0 .600 o k 16 1.00k 19 0 (152 ) - 0 .840 0 (10 ) » 0 .870 0 (244 ) - 1.00 0 (20 ) = 0 .745 16 1.28k 19 Average k l 6 •= 1 . 1 4 k 1 9 12D 0 136 0 152.0 168.D 1B4.0 200.0 216.0 232.0 X-19 AVERAGE UINTER SN0U DEPTH OF SKIING AREA (CH) 248.0 -i 1 r (50,20) O O z t-i •« M CO ^ -CO cc UJ _J CO cr _ J t M cr -cc cr £2 cc U-o , »-CO z UJ O ( 3 0 , 1 5 . 5 ) • (30,11) 1 X (25,9) _1_ JL 24.0 28J3 32J3 36.0 40JJ 44JJ 4BJ 52.0 X-18 AVERAGE SLOPE OF SKIING AREA (X) U(30) U ( 1 5 . 5 ) D(50) 0 (20 ) 0 .215 0 .625 •= 0 .00 = 1.00 o k 17 0.573k 18 0 (25 ) - 0 .290 0 (9 ) - 0 .250 D(30) - 0 .215 0 (11 ) - 0 .370 "17 0.625k 18 A v e r a g e "17 0.599k 18 143 LO 2 M o M CD ^ -to CC o _ ii UJ _J C O C X _ J o t—i » cx -C X cc U-o = X * \-L D z LU I = cc i X i r 20) (61 ,16 ) O (152 ,16 ) JL (244 , 6 . 5 ) i ' l_ J _ _J_ J _ '6D0 JDDO 1410 1810 220.0 2610 30a0 3410 X-19 flVERflGE UINTER SN0U DEPTH OF SKIING AREA (CM3 U(61) - 0 .450 U(20) - 1.00 U(152) - 0 .840 U(16) - 0 .650 o k 1 ? - l . l l k 1 9 0 (61 ) - 0 .450 U(16) » 0 .650 0 (244 ) - 1.00 0 ( 6 . 5 ) = 0 .180 • k 17 1 .17k 19 Average k 1 7 = 1 . 1 4 k l 9 cr o LL) •» r r ^ cr O z _ to 0 -O O C T C C L U > C E C D I X (0 ,10 ) (61 ,10 ) I I L O ( 1 5 2 , 3 5 ) , ( 244 ,25 ) 0 (0 ) - 0 .00 U(10) «= 0 .640 0 (152 ) •» 0 .840 U(35) 0 .160 O k ] ^ • 1 . 7 5 k j 9 U(61) 0 (10 ) U (244) 0 (25 ) 0 .450 0 .640 1.00 0 .290 • k l 8 " 1-5'}kl9 Average k l 8 1.66k 19 0.0 40.0 80.0 120.0 I6O.0 2OD.0 240.0 280.0 320.D X-19 FlVERflGE UINTER SN0U DEPTH OF SKIING flRER (CM) 144 Snowshoeing t o CC o LU o-CC f LX N U> 2 H °. S i to O z =. to 8 o o = LX ci O £! CC >-CO o LU cS n » LU =». c r 5 cc o CN i 1 i X ( 2 0 , 1 8 0 ) ( 2 0 , 1 8 0 ) 8 J. ( 1 3 , 6 0 ) J _ J _ O ( 9 . 1 5 ) t i l l I 1 8 . 8 10.4 B J ) 1 3 - 6 15-2 M L B 1B.4 2 1 0 X-21 LENGTH OF TRAIL AVAILABLE FOR SNOUSH0EING KID U ( 9 ) » 0 . 6 7 0 U ( 1 5 ) » 1 . 0 0 0 ( 2 0 ) = 1 . 0 0 0 ( 1 8 0 ) = 0 . 1 7 0 o k 2 0 0 . 3 9 8 k 2 1 0 ( 1 3 ) - 0 . 8 0 0 0 ( 6 0 ) - 0 . 6 7 0 U ( 2 0 ) - 1 . 0 0 0 ( 1 8 0 ) - 0 . 1 7 0 o k 2 0 0 . 4 0 0 k " •20 2 1 Average 0 . 3 9 9 k 2 1 CO LU 3 Z =>. cc U Ul cc <=. cr ? C M O z CO o D CX a 1 cc >-CD Ul cr o cc » t— o CM I X O 1 1 1 1 • 1 1 1 1 1 — ~ l i r i i -a ( 1 5 2 , 1 8 0 ) -- O ( 9 1 , 1 2 0 ) -( 2 1 3 , 6 0 ) m-i I i i I O ( 1 5 2 , 1 5 ) i I l _ 1 1 i i I l 0 ( 9 1 ) - 1 . 0 0 0 ( 1 2 0 ) 0 . 3 7 0 0 ( 1 5 2 ) - 0 . 6 8 0 0 ( 1 5 ) 1 . 0 0 ° k 2 0 ' 0 . 5 0 8 k 0 ( 1 5 2 ) <I 0 . 6 8 0 U ( 1 B 0 ) m 0 . 1 7 0 0 ( 2 1 3 ) m 0 . 3 4 0 0 ( 6 0 ) - 0 . 6 7 0 2 2 2 0 0 . 6 8 0 k 2 2 Average K20 0 . 5 9 4 k 2 2 '8B0 104 D 120.0 136.0 152.0 168.0 184.0 200.0 216.0 X-22 AVERAGE UINTER SN0U DEPTH OF SN0USH0EING AREA ICfl) 145 . . rs» CO " CL) I— Z> Z o . C I N cc <=>. cc ? , o z l-l UJ _ o S co-R O z CO p o 1 • cc O 1 cc >-CD U J > cr cc i— o t-i ( 10 ,240 ) 0 ( 2 0 , 1 5 0 ) J L ( 30 ,60 ) 0 (30 ,15 ) _1 I l_ U(20) = 0 .660 U(150) = 0 .250 U(30) - 0 .250 U(15) - 1.00 o k 10.0 14.0 18.0 22.0 26.0 3O.0 34.0 38.0 X-23 RVERRGE SLOPE OF SN0USH0EING RRER (X) 20 0 .547k 2 3 U(10) - 0 .850 U(240) •» 0 .00 U(30) •= 0 .250 U(60) - 0 .670 20 0 .896k 2 3 Average "20 42.0 0.722k 23 2Z ^ o L0 -Z t—i LU O zr ™. CO = 3 Q Z CO CC o o -u_ LU _ J CD I D CC a? _I CC > tt „ _J M CC CC U -o L3 LU ^ rvi - i 1 1 1 1 1 1 1 1 1 1 i ~ O (152 ,11 ) (183 ,11 ) 0 ( 1 2 2 , 9 ) ( 1 2 2 , 6 . 5 ) _ ! I L _J_ _J_ U(122) U(9) U(152) 0 (11 ) o k 2 1 " 11(122) 0 ( 6 . 5 ) Yitta 130.0 140.0 150.0 16D.0 DUD 180J 130.0 X-22 RVERRGE UINTER SN0U DEPTH OF SN0USH0EING RRER IDU 0 .820 0 .670 0 .680 0 .750 1.75k 22 0 .820 0 .575 0 (183 ) - 0 .560 U ( l l ) 0 .750 k 2 1 - 1 . 4 9 k 2 2 Average *21 = 1.62k 22 146 L D ™ z n LU ° s co 2 rj o z CO cr u> o _ L L . LU 5*3 c r 2 _ J M cr > cr c M cr cc co = LU 0 0 CM 1 T 1 1 1 1 1 1 1 r (40 ,20) -6 ( 35 ,13 ) 6- (10 ,9 ) ( 1 5 , 6 . 5 ) _1_ J _ JL _1_ J _ J L 10J3 14.11 18.0 22.0 26.0 30.0 34.0 38.0 X-23 AVERAGE SLOPE OF SNOUSHOEING AREA (50 U(10) U(9) U(40) U(20) o k 21 0 .850 0 .670 0 .125 1.00 2 .20k 23 U(15) » 0 .780 U ( 6 . 5 ) - 0 .575 U(35) - 0 .190 U(13) - 0 .800 • k 21 2 .62k 23 Average k 2 1 - 2 . 4 1 k 2 3 1 1 1 —1 1 • 1 1 1 • 1 1 • i • - U(25) » 0 . 500 U(152) • 0 .680 (10 ,183 ) U(50) » 0 .00 - U(91) « 1.00 - o k j 2 « 1 . 5 6 k 2 3 - U(10) - 0 .850 - 0 (25 ,152 ) D(183) « 0 .560 U(25) « 0 .500 U(122) - 0 .820 • k 2 2 - 1 . 3 5 k 2 3 - • (25 ,122 ) Average k 2 2 = 1 . 4 6 k 2 3 i i i i i i i i ( 50 ,91 ) , , o , , , 8.0 16.0 24.0 32.0 40.0 48.0 S6.0 64.0 X - 2 3 AVERAGE SLOPE OF SNOUSHOEING AREA (X) 72.0 147 .7 Hiking R 1 1 1 1 1 1 1 r— —1 1 T" 1 1 1 1 1 1 a •—• o to 2; -t— Z I—1 "= E § n -CX LU cc CX «=. -LD £ z TO HIKI 220.0 (- ( 0 , 2 4 0 ) -ROAD Q ( 1 , 1 8 0 ) ->-m -LU R -t— TRAVEL 100.0 - -CM C3 1 0 ' X 10 _ O ( 5 , 6 0 ) ( 7 , 3 0 ) V p I l l ' ' 1 1 1 1 1 1 .—1 0 0 0 8 16 2.4 3.2 4.0 4.8 5.6 6.4 7.2 X-25 DISTANCE OF HIKING AREA FROM DRINKING WATER SOURCE (KM) U ( l ) - 0 .75 U(180) » 0 .125 U(5) - 0 .195 U(60) - 0 .635 o k 2 4 « 1 . 0 9 k 2 5 U(0) - 1.00 0 (240 ) = 0 .00 0 (7 ) - 0 .120 0 (30 ) - 0 .875 • k 2 4 = 1 . 0 1 k 2 5 Average "24 1 .05k 25 148 Horseback Riding LO cs. H E o o r r 6 0 S " cn cx 2 V M M Q 1-i CC ^ 5 CQ LU LO CC =. 9 8 CC rsj O " CC >-CD o LU § O (13 ,180 ) (20 ,180 ) • (11 ,90 ) LU 2 > c i L LO ( 4 . 5 , 0 ) J _ J _ J _ J _ J _ 16.0 ie.a 20.0 - _ L_o_l I " 1 1  7° 40 6 0 B.0 U.0 «-0 U-» ~ X : 2 7 LENGTH OF TRAIL AVAILABLE FOR HORSEBACK RIDING IKMJ 11(4.5) U(0) 0 (13 ) U(180) ° k 2 6 = U ( l l ) 0 (90 ) 0 (20 ) 0 (180 ) • 0 .130 • 1.00 • 0 .750 * 0 .250 O.B27k 27 0 .500 0 .700 1.00 0 .250 k 2 6 - i . u k 2 7 Average k 2 6 - 0 . 9 6 9 k 2 ? CO LU cr LU cc cr o 2 f—I • <_J cr co UJ CO cc o X 8 ' ( 20 ,240 ) O 2 cr o cc >- ^ m ; r LU r: ( 10 ,180 ) (30 .150) > a cc C D I X ( 20 ,60 ) 30.0 34.0 3B.0 !0 0 14.0 18.0 22.0 26.D ' X-28 AVERAGE SLOPE OF HORSEBACK RIDING AREA (X) 0 (10 ) = 1.00 0 (180 ) - 0 .250 0 (20 ) = 0 .605 0 (60 ) - 0 .800 o k 26 0 .718k 28 0 (20 ) - 0 .605 0 (240 ) • 0 .00 0 (30 ) - 0 .320 0 (150 ) - 0 .500 26 0 .570k 28 Average 26 0 .644k 2B 149 CO z 8* ^ (_> tx °. m I D LU *"" to CC O X => c c = o U-LU CO rJ c c -_ J M a £ 5 C X C C r-CNI -1—W r ( 20 ,20 ) (30 ,20 ) (10,9) rt ( 1 0 , 4 . 5 ) 1 L . 1_ ULO 14.0 1B.0 22.0 26.0 30.0 34.0 38.0 X-28 AVERAGE SLOPE OF HORSEBACK RIDING AREA (30 42.0 TJ(10) U ( 4 . 5 ) U(30) U (20) o k 27 1 .00 0 .130 - 0 .320 - 1 .00 «= 0 .782k 28 U(10) -U(9) -U(20) -U(20) -1.00 0 .250 0 .605 1.00 k 2 ? = 0 .527k 29 Average "27 0 .655k 29 150 A. .9 Summer Motorized Camping - t (0 ,180) cr o LU CM Cc m cr O 1-cr C_) o cr o cc >- 5 m ° LU L U > C X cr CM I X 0 ( 1 , 1 2 0 ) _ l _ ( 4 ,60 ) O I 1_ ( 5 ,60 ) 0.0 X-30 U(4) - 0 .220 U(60) = 0 .690 0 (1 ) - 0 .750 0 (120 ) - 0 .375 o k 29 1.68k 30 U(5) = 0 .180 U(60) «= 0 .690 U(0) - 1 .00 0 (180 ) - 0 .160 "29 1.55k 30 Average *29 " 1 - « * 3 0 0 9 1 6 2 . 4 3.2 4.D 4 .8 5 .6 6 .4 DISTANCE OF CAMPING AREA FROM DRINKING WATER SOURCE (KM) LO LU tt 1 LU »*> c c ( X D_ TZ CE <-> o _ c f ( 1 r\> I— CM CO. C E O C C >-CO ( 5 ,240 ) S P- ( 0 ,180 ) LLi £ «=» c c s CM I x (10,60) _1_ ( 30 ,60 ) _J_ 0.0 4.0 8.0 12.0 1&D 23.0 2 4.0 28.0 X-31 AVERAGE SLOPE OF CAMPING AREA IXJ O(0) - 1 .00 0 (180 ) •= 0 .160 0 (10 ) = 0 .425 U(60) - 0 .690 O k 2 9 " 1 . 0 8 k 3 1 0 (30 ) • 0 .145 0 (60 ) » 0 .690 0 (5 ) » 0 .680 0 (240 ) - 0 .00 • k 29 0 .775k 31 Average k „ - 0 . 9 2 8 k 3 1 151 LU U CC fM Z 5 r o O CO CC cu „ co cc cc u_ cx (X co D_ I N n -cr u LL U (X t— CO n • o CO 1 X 4.0 -I 1 1 r T r j _ • (10,2) 0 (10,1.5) _1 L (20,0) _l I I 6 1 L-• (30,0.5) _l I I L. 0(10) - 0.425 U(1.5) « 0.620 U(20) - 0.220 U(0) - 1.00 8 9 12.0 16.0 20.0 24.0 28.0 32.0 36.0 X-31 AVERAGE SLOPE OF CAHPING AREA ('/•) 40.0 o k 30 0.539k 31 0(10) - 0.425 U(2) - 0.500 U(30) - 0.145 0(0.5) - 0.880 l30 0.737k 31 Average 30 0.638k 31 APPENDIX Additive U t i l i t y Functions 153 ADDITIVE UTILITY FUNCTIONS Symbolism O i(X i) - Ut i l i t y of attribute i at attribute level X. Trailbiking (TB) U T B - 0.175 V1(X1) + 0.298 l»2(X2) + 0.333 U3(X3> + 0.194 U4(X4> Four-wheel Driving (4x4) U 4 x 4 - 0.200 U 5(X g) + 0.291 0 6(X 6) + 0.363 U 7(X ?) + 0.146 Ugttg) Snowmobiling (SNOW) USNOW " 0 , 2 6 9 W + 0 , 2 5 2 U10(X10> + 0 - 1 6 7 " l l ^ l l 1 + 0 , 3 1 2 U12 < X12' Downhill Skiing (DOWN) °DOWN " 0 , 3 0 1 U13(X13> + 0 , 4 3 7 U14(X14> + 0 , 2 6 2 " l S ^ l S * Cross-Country Skiing (X-C) U x_ c - 0.271 D 1 6(X 1 6) + 0.198 U17<X17> + 0.293 UjglXjg) + 0.238 U 1 9(X l g) Snowshoeing (SHOE) USHOE " 0 , 1 5 2 U20(X20> + 0 , 3 8 2 U21(X21> + ° - 2 5 5 U22 ( X22 ) + 0 , 2 1 1 °23(X23> Hiking (HIKE) UHIKE " 0 , 5 1 2 °24(X24> + 0 , 4 8 8 U25(X25> Horseback Riding (HORSE) "HORSE • 0 , 2 7 9 U26 ( X26» + 0 , 2 8 8 °27 < X27> * 0 , 4 3 3 U28(X28> Summer Motorized Camping (CAMP) "CAMP " 0 , 3 7 1 u29 ( x29» + 0 , 2 2 9 u3o(x30> + ° - 4 0 0 " a i ^ a i * 154 APPENDIX R e l a t i v e Importance o f t h e A t t r i b u t e s t o each D e c i s i o n Maker RELATIVE IMPORTANCE OF THE ATTRIBUTES TO EACH DECISION MAKER Symbolism ^ represents "more important t o the dec i s ion maker than" T r a i l b i k i n g Slope Length of T r a i l ^» C o n f l i c t s with 4x4s T r ave l Time Four-wheel Dr i v ing Slope ia . Length of T r a i l ^ Trave l Time i^. C o n f l i c t s with T r a i l b i k e s Snowmobiling Slope ^ a . T rave l Time i » Length o f T r a i l l ^ . Snow Depth Downhill Sk i ing Slope ^ Trave l Time ^ Snow Depth Cross-country Sk i ing Slope Trave l Time i^. Snow Depth ~-~ Length of T r a i l Snowshoeing Length of T r a i l i i . Snow Depth ^ Slope ^ Trave l Time Hik ing Trave l Time ^ Distance from Drinking Hater Source Horseback R id ing Slope Length of T r a i l i i . T rave l Time Summer Motorized Camping lope J^. T rav l Ti e i ^ Distance from Dr inking Hater Source APPENDIX The NLP.S Algorithm 157 THE NLP.S ALGORITHM (Computer Language = FORTRAN) 1 FUNCTION XDFUNC(XX.NUM) 2 C 3 C F u n c t i o n 'XDFUNC' e v a l u a t e s the o b j e c t i v e f u n c t i o n 4 C f o r a l a n d use p l a n . A l t h o u g h we a c t u a l l y want 5 C t o maximize the u t i l i t y f u n c t i o n , the a v a i l a b l e 6 C s o f t w a r e o n l y a l l o w s f u n c t i o n s t o be mi n i m i z e d . 7 C To get around t h i s , the l a s t command i n t h i s 8 C a l g o r i t h m s e t s XDFUNC = -XDFUNC. 9 C 10 C V a r i a b l e s u s e d : 11 C XX - The mine waste p l a n t o be e v a l u a t e d . 12 C Each a r r a y element r e p r e s e n t s one g r i d square 13 C and c o n t a i n s a number between 1 & 16, depending 14 C on the l a n d use c l a s s a s s i g n e d t o t h a t square. 15 C X - Rounded-off v e r s i o n of a r r a y XX. We r e q u i r e 16 C i n t e g e r s t o e v a l u a t e the u t i l i t y f u n c t i o n . 17 C NUM - Dummy v a r i a b l e ; d i m e n s i o n of XX. SCALE - U t i l i t y s c a l e f a c t o r s f o r each of the 9 a c t i v i t i e s . C u m u l a t i v e u t i l i t y f o r each a c t i v i t y . 21 C N - Number of square s a s s i g n e d t o each a c t i v i t y . TRAILS - Keeps t r a c k o f the maximum t r a i l a v a i l a b l e f o r e a c h a c t i v i t y f o r the c u r r e n t p l a n . 18 C 19 C 20 C U 22 C " r INFLAG - 0/1 " a g to'deierm'i ne whether the f u n c t i o n It. c has been p r e v i o u s l y c a l l e d . I f not c a l l e d 26 C 2j Q p r o c e e d i n g . 28 C 29 IMPLICIT REAL*8 (A-H.O-Z) in p u t d a t a must be re a d i n f i r s t b e f o r e 30 DIMENSION XX(100) 31 DIMENSION SCALE(9), U ( 9 ) . N(9), T R A I L S O ) 32 INTEGER X(100) 33 DATA SCALE/0.046,0.111,0084,0.133,0.087.0.064,0.146, 34 + 0.078.0.254/ 35 DATA INFLAG/1/ 36 COMMON /MAXTRL/ TRAILS 37 C 38 C Get i n p u t d a t a from f i l e s f i r s t time through. 39 C 40 IF (INFLAG.EQ.1) CALL GETDAT 41 INFLAG » 0 42 C 43 C Round o f f a l l 100 v a r i a b l e s . 44 C 45 DO 10 1=1.100 46 X ( I ) = XX(I)+0.5 47 10 CONTINUE 48 C 49 C E v a l u a t e t h e u t i l i t i e s t w i c e . The f i r s t t ime J u s t d e t e r m i n e s the maximum t r a i l d i s t a n c e f o r each a c t i v i t y f o r the g i v e n p l a n , u The second time e v a l u a t e s the a c t u a l u t i l i t i e s 53 C u s i n g t h e s e maximum t r a i l d i s t a n c e s . 54 C 55 DO 15 1=1.9 56 TRAILS(I) • 0.0 57 15 CONTINUE 58 DO 501 LOOP - 1,2 59 C 60 C I n i t i a l i z e a l l a r r a y s . 50 C 51 C 52 C 61 C 62 63 64 DO 20 1=1.9 U ( l ) - 0.0 N(I ) - 0 6 5 20 CONTINUE 158 66 DO 500 K=1.100 67 C Land use c lasses should be between 1 & 16. 68 IF ((X(K).GT.16.5).OR.(X(K).LT.0.5)) GOTO 499 69 C 70 C Convert K to row (I) and column (J ) . 71 C K •= 1 --> Top l e f t square. 72 C K = 100 --> Bottom r ight square. 73 C 74 I «* (K-1 )/10 + 1 75 J = K - ( I - 1 ) * 1 0 76 C 77 C M0D1 = 0 --> Land use plans 1,2.3 in square K. 78 C 1 --> 4,5.6 79 C 2 --> 7,8.9 80 C 3 --> 10,11.12 81 C 4 --> 13,14.15 82 C 83 M0D1 = (X(K)-1)/3 84 C 85 C M0D2 = 1 --> Land use plans 1,4,7,10,13 1n square K. 86 C 2 --> 2,5,8,11,14 87 C 3 --> 3,6,9,12,15 88 C 89 M0D2 = X(K) - 3*M0D1 90 C 91 IF (M0D1.EQ.O) GOTO 30 92 IF (M0D1.E0.1) GOTO 40 93 IF (M0D1.EQ.2) GOTO 50 94 IF (M0D1.E0.3) GOTO 60 95 IF (M0D1.GE.4) GOTO 70 96 30 CONTINUE 97 C 98 C TB - Increment a c t i v i t y 1. 99 C 100 U(1) = U(1) + U1(X,I,J.NUM) 101 N(1) = N(1) + 1 102 GOTO 100 103 40 CONTINUE 104 C 105 C 4x4 - Increment a c t i v i t y 2. 106 C 107 U(2) = U(2) + U2(X,I,J.NUM) 108 N(2) = N(2) + 1 109 GOTO 100 110 50 CONTINUE 111 C 112 C TB/4x4 - Increment a c t i v i t i e s 1 & 2. 113 C 114 U(1) = U(1) + UI(X.I.J.NUM) 115 U(2) = U(2) + U2(X,I,J.NUM) 1 16 N( 1) = N(1) + 1 117 N(2) - N(2) + 1 118 GOTO 100 119 60 CONTINUE 120 C 121 C Hiking/Camping/Horse - Increment a c t i v i t i e s 7,8 & 9. 122 C 123 U(7) - U(7) •»• U7(X,I. J.NUM) 124 U(8) = U(8) + U8(X.I,J.NUM) 125 U(9) = U(9) + U9(X,I,J.NUM) 126 N(7) •= N(7) + 1 127 N(8) « N(8) + 1 128 N(9) = N(9) + 1 129 GOTO 100 130 70 CONTINUE 131 C 132 C No A c t i v i t y . 133 C 134 GOTO 100 135 100 CONTINUE 136 C 137 IF (X(K).E0.16) GOTO 110 138 IF (M0D2.EQ.1) GOTO 120 139 IF (M0D2.E0.2) GOTO 130 140 IF (M0D2.EQ.3) GOTO 140 141 110 CONTINUE 159 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 120 130 140 C C C 499 500 501 C C C C C C C 200 C C C C C C C C C C C C C C C c c c Downhill Sk i ing - Increment a c t i v i t y 4. U(4) = U(4) + U4(X,I.J,NUM) N(4) = N(4) + 1 GOTO 499 CONTINUE Snowmobiling - Increment a c t i v i t y 3. U(3) = U(3) + U3(X.I,J.NUM) N(3) = N(3) + 1 GOTO 499 CONTINUE X/C Sk i ing, Snowshoeing - Increment a c t i v i t i e s 5 & 6. U(5) = U(5) + U5(X,I.J,NUM) U(6) = U(6) + U6(X,I,J.NUM) N(5) = N(5) + 1 N(6) = N(6) + 1 GOTO 499 CONTINUE No A c t i v i t y . GOTO 499 CONTINUE CONTINUE CONTINUE Now, sum up the to ta l u t i l i t y . XDFUNC = O.O DO 200 1=1,9 U(I)/N(I) i s the average u t i l i t y of a c t i v i t y I over a l l squares a l l oca ted to the a c t i v i t y . IF (N(I).NE.O) XDFUNC = XDFUNC + SCALE(I)*U(I)/N(I) CONTINUE Convert maximizing to minimizing funct ion. XDFUNC RETURN END -XDFUNC FUNCTION XDG(X,N.I) This funct ion suppl ies the opt imizing rout ine with lower constra ints fo r each of the 100 var iab les . In th i s program, the constra int is 1 < XI for a l l 100 va r i ab le s . IMPLICIT REAL*8 (A-H.O-Z) DIMENSION X(N) XDG =0 .5 RETURN END FUNCTION XDH(X.N.I) This funct ion suppl ies the opt imiz ing rout ine with upper constra ints for each of the 100 var iab les In t h i s program, the constra int i s Xi < 16 for a l l 100 var i ab le s . IMPLICIT REAL*8 (A-H.O-Z) DIMENSION X(N) XDH =16.5 RETURN END 160 217 C 218 C 219 C 220 FUNCTION XDX(X,N,I) 221 C 222 C This funct ion suppl ies the opt imizing routine with 223 C any Impl ic i t var iab les required. For t h i s program, 224 C there are no i m p l i c i t var iab les ; th i s is a dummy 225 C funct ion to s a t i s f y the rou t ine ' s requirements and 226 C i s never c a l l e d . 227 C 228 IMPLICIT REAL*8 (A-H.O-Z) 229 DIMENSION X(N) 230 XDX = 0.0 231 RETURN 232 END 233 C 234 C 235 C 236 FUNCTION UI(X.I.d.N) 237 C 238 C Functions U1.U2 U9 evaluate the u t i l i t y of 239 C the corresponding a c t i v i t y in the square ( I .d). 240 C The array X i s used to determine surrounding 241 C a c t i v i t i e s for eva luat ing c o n f l i c t s . 242 C The input arrays SNOW, SLOPE, & TRAILS are used 243 C to evaluate the phys ica l q u a l i t i e s of square ( I .d). 244 C 245 C 246 C ACTIVITY #1 TRAILBIKING 247 C 248 C 249 IMPLICIT REAL*8 (A-H.O-Z) 250 DIMENSION SN0W( 10, 10) , SLOPE ( 10,10),LAKES( 100) , TRAILSO) 251 INTEGER X(100) 252 C 253 C Array K stores the m u l t i p l i c a t i v e factors for U-TB. 254 C 255 REAL K(5) 256 COMMON /DATA/ SNOW,SLOPE,TOWNI.TOWNd,LAKES,NLAKES 257 • COMMON /MAXTRL/ TRAILS 258 DATA K/0.175,0.298,0.333,0.194/ 259 C 260 C N1 counts the number of adjo in ing TB squares. 261 C N2 counts the number of adjo in ing 4x4 squares. 262 C 263 N1 = 1 264 IdSO = 10*(I-1)+d 265 N2 = O 266 IF ((X(IdS0).GE.7).AND.(X(IdS0).LE.9)) N2 = 1 267 IM1 = I - 1 268 IP1 = 1 + 1 269 dM1 = d - 1 270 dPI = d • 1 271 DO 10 IWM1.IP1 272 DO 10 dd=dM1,dP1 273 C 274 C Ensure that we are looking at a square ins ide 275 C the 10x10 planning area. 276 C 277 IF (( I I .LT.1).0R.(I I.GT.10).OR.(dd.LT.1).OR.(dd.GT.10)) GOTO 10 278 C 279 C Don't process square (I,d) 280 C 281 IF ((I I.EQ.I).AND.(dd.EQ.d)) GO TO 10 282 ISO = 10*(11-1) + dd 283 M0D1 = (X(ISQ)-1)/3 284 IF ((MOD1.E0.O).0R.(M0D1.E0.2)) Nl = N1 + 1 285 IF ((MOD1.EO.1).OR.(M0D1.E0.2)) N2 = N2 + 1 286 10 CONTINUE 287 C 288 C Ca lcu la te t rave l time at 70 KM/H on highway and 289 C 40 KM/H on t r a i l s . e d to run along columns. 290 C Highways are assumed to run along columns d. 291 C T r a i l s are assumed to run along rows I. 161 292 C 293 DHWY = DABS(TOWNI-I) 294 DTRL = DABS(TOWNJ-J) 295 X1 = DHWY*70./60. + DTRL»40./60. 296 C 297 C Ca lcu la te t r a i l d i s tance. 298 C 299 X2 = 1.7*N1+3.1 SCO C 301 C T r a i l d i s tance ut i11ty is assigned the maximum 302 C t r a i l d i s tance for the p lan. 303 C 304 IF (X2.LT.TRAILS(1)) X2=TRAILS(1) 305 IF (X2.GT.TRAILS(1)) TRAILS(1)=X2 306 C 307 C Average slope value for square K. 308 C 309 X3 = SLOPE(I.J) 310 C 311 C If land use plans 7.8,9. H c o n f l i c t s are 4/(N1*N2) 312 C Otherwise, there are no c o n f l i c t s . 313 C 314 IF ((X(IJS0).GE.7).AND.(X(IJS0).LE.9)) X4 = 4.0/(N1*N2) 315 IF ((X(IJS0).LT.7).0R.(X(IJS0).GT.9)) X4 = 0.0 316 U1 = K(1)*UF(1,X1) + K(2)*UF(2,X2) + K(3)*UF(3 , X3) + 317 + K(4)*UF(4,X4) 318 RETURN 319 END 320 C 321 C 322 C ACTIVITY #2 FOUR-WHEEL DRIVING 323 C 324 C 325 C 326 FUNCTION U2(X,I,J.N) 327 IMPLICIT REAL*8 (A-H.O-Z) 328 DIMENSION SN0W( 10, 10) , SLOPE( 10, 10) , LAKES( 100) , TRAILSO) 329 INTEGER X(100) 330 REAL K(5) 331 COMMON /DATA/ SNOW,SLOPE,TOWNI,TOWNJ,LAKES,NLAKES 332 COMMON /MAXTRL/ TRAILS 333 DATA K/0.200,0.291.0.363,O.146/ 334 N1 = 1 335 N2 = O 336 IJSO = 10*(I-1) + J 337 IF ((X(IJS0).GE.7).AND.(X(IJS0).LE.9)) N2 = 1 338 IM1 = I - 1 339 IP1 = 1 + 1 340 JM1 = J - 1 341 JP1 = J + 1 342 DO 10 II=IM1,IP1 343 DO 10 JJ=JM1,JP1 344 IF (( I I .LT.1).0R.( I I .GT.1O).0R.(JJ.LT.1).0R.(JJ.GT.1O)) GOTO 10 345 IF (( I I .EO.I).AND.(JJ.EO.J)) GO TO 10 346 ISO = 10*(II-1) + JJ 347 M0D1 = (X(IS0)-1)/3 348 IF ((M0D1.EQ.0).0R.(M0D1.EQ.2)) N2 = N2 + 1 349 IF ((MOD 1.EO.1).OR.(MOD 1.EO.2)) N1 = N1 + 1 350 10 CONTINUE 351 C 352 C Travel at 70 KM/H on highway and 40 KM/H on t r a i l . 353 C 354 DHWY = DABS(TOWNI-I) 355 DTRL = DABS(TOWNJ-J) 356 X5 = DHWY*70./60. + DTRL*40./60. 357 X6 = 1.7*N1+3.1 358 IF (X6.LT.TRAILS(2)) X6=TRAILS(2) 359 IF (X6.GT.TRAILS(2)) TRAILS(2)=X6 360 X7 = SLOPE(I.J) 361 IF ((X(IJS0).GE.7).AND.(X(IJS0).LE.9)) X8 • 4.0/(N1»N2) 362 IF ((X(IJS0).LT.7).0R.(X(IJS0).GT.9)) X8 = 0.0 363 U2 = K(1)*UF(5,X5) + K(2 )*UF(6.X6) + K(3)*UF(7,X7) + 364 + K (4 )»UF(8 .X8 ) 365 RETURN 366 END 162 367 C 368 C 369 C 370 C 371 C 372 C 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 1 396 C 397 c 398 c 399 400 401 402 403 404 405 c 406 c 407 c 408 409 c 410 411 412 413 414 415 c 416 c 417 c 418 c 419 c 420 c 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 ACTIVITY #3 SNOWMOBILING FUNCTION U3(X.I,J.N) IMPLICIT REAL*8 (A-H.O-Z) DIMENSION SN0W(1O.1O),SLOPE(1O.1O).LAKES(1OO), TRAILSO) INTEGER X(100) REAL K(5) COMMON /DATA/ SNOW,SLOPE,TOWNI.TOWNJ,LAKES,NLAKES COMMON /MAXTRL/ TRAILS DATA K/O.269,0.252,0.167,0.312/ N1 = 1 IM1 = I - 1 IP1 = 1 + 1 «JM1 = J - 1 JP1 = J + 1 DO 10 II=IM1,IP1 DO 10 JJ=JM1,JP1 IF ( ( I I .LT. 1) .OR.(II.GT.10).OR.(JJ.LT.1).OR. (JJ.GT. 10)) GOTO 10 IF (( I I .EO.I).AND.(JJ.EO.J)) GO TO 10 ISO = 10*(II-1) + J J M0D1 = (X( ISQ)-l)/3 M0D2 = X(ISO) - M0D1*3 IF (X(ISO).EQ.16) GOTO 10 IF (M0D2.EO.1) Nl = Nl + 1 10 CONTINUE Travel at 70 KM/H on highway and 40 KM/H on t r a i l . DHWY = DABS(TOWNI-I) DTRL = DABS(TOWNd-J) X9 = DHWY*70./60. + DTRL*40./60. X10 = 1.7*N1 +3.1 IF (X10.LT.TRAILSO)) X 10=TRAILS( 3 ) IF (X 10. GT. TRAILSO)) TRAILS( 3 )=X 10 Average snow depth value fo r square K. X11 = SNOW(I.J) X12 = SLOPE(I.J) U3 = K(1)*UF(9,X9) + K(2)*UF(10.X10) + K(3)*UF(11.X11) + + K(4)*UF(12.X12) RETURN END ACTIVITY #4 DOWNHILL SKIING FUNCTION U4(X,I.J.N) IMPLICIT REAL*8 (A-H.O-Z) DIMENSION SNOW(10.10).SL0PE(10,10),LAKES(IOO). TRAILSO) INTEGER X(100) REAL K(5) COMMON /DATA/ SNOW.SLOPE.TOWNI,TOWNJ,LAKES.NLAKES COMMON /MAXTRL/ TRAILS DATA K/O.301.0.437.0.262.0.0/ N1 = O IM1 « I - 1 IP1 = 1 + 1 JM1 = J - 1 JP1 * d + 1 DO 10 II=IM1.IP1 DO 10 dd=JM1.JP1 IF (( I I .LT.1).0R.( I I .GT.10).OR.(JJ.LT.1).DR.(JJ.GT.10)) GOTO 10 IF ((II.EQ.I).AND.(Od.EQ.J)) GO TO 10 ISO « 10*(II-1) + JJ IF (X(ISQ).EQ.16) N1 = Nl + 1 10 CONTINUE 163 441 C 442 C 443 C 444 445 446 447 448 449 450 451 452 C 453 C 454 C 455 C 456 C 457 C 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 1 480 C 481 c 482 c 483 484 485 486 487 488 489 490 491 492 493 494 495 c 496 c 497 c 498 c 499 c 500 c 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 Travel at 70 KM/H on highway and 40 KM/H on t r a i l . DHWY = DABS (TOWNI-I ) DTRL = DABS(TOWNJ-J) X13 = DHWY*70./60. + DTRL*40./60. X14 = SLOPE(I.J) X15 = SNOW(I.J) U4 = K(1)*UF(13.X13) + K(2)*UF(14.X14) + K(3)•UF(15,X15) RETURN END ACTIVITY #5 CROSS-COUNTRY SKIING FUNCTION U5(X,I.J.N) IMPLICIT REAL*8 (A-H.O-Z) DIMENSION SNOW(10,10).SLOPE(10,10),LAKES(100). TRAILSO) INTEGER X(100) REAL K(5) COMMON /DATA/ SNOW.SLOPE,TOWNI.TOWNJ,LAKES,NLAKES COMMON /MAXTRL/ TRAILS DATA K/O.271.0.198.0.293,0.238/ N1 = 1 IM1 = I - 1 IP1 = 1 + 1 JM1 = J - 1 JP1 = J + 1 DO 10 II=IM1.IP1 DO 10 JJ=JM1,JP1 IF (( I I .LT.1).OR.( I I .GT.10).OR.(JJ.LT.1).OR.(JJ.GT.10)) GOTO 10 IF (( I I .EQ.I).AND.(JJ.EO.J)) GO TO 10 ISO = 10*(II-1) + JJ M0D1 = (X(ISQ)-1)/3 M0D2 = X(ISQ)-3*M0D1 IF (M0D2.EQ.2) N1 = N1 + 1 0 CONTINUE Travel at 70 KM/H on highway and 40 KM/H on t r a i l . DHWY = DABS(TOWNI-I) DTRL = DABS(TOWNJ-J) X16 = DHWY*70./60. + DTRL*40./60. X17 = 1.7*N1 +3.1 IF (X17.LT. TRAILSO)) X 17=TRAI LS( 5 ) IF (X17.GT.TRAILSO)) TRAILS(5)=X17 X18 = SLOPE(I.J) X19 = SNOW(I.J) U5 « K(1)*UF(16.X16) + K(2)*UF(17.X17) + K(3)*UF(18,X18) + + K(4)*UF(19.X19) RETURN END ACTIVITY *6 SNOWSHOEING FUNCTION U6(X,I,J.N) IMPLICIT REAL*8 (A-H.O-Z) DIMENSION SNOW(10.10).SL0PE(10.10),LAKES(100), TRAILSO) INTEGER X(100) REAL K(5) COMMON /DATA/ SNOW,SLOPE,TOWNI.TOWNJ,LAKES.NLAKES COMMON /MAXTRL/ TRAILS DATA K/O.152.0.382.0.255,0.211/ N1 = 1 IM1 = I - 1 IP1 = 1 * 1 JM1 = J - 1 JP1 = J + 1 DO 10 II=IM1,IP1 DO 10 JJ=JM1,JP1 164 516 IF ( ( I I .LT.1).0R.( I I .GT.10).0R.(JJ.LT.1).0R.(JJ.GT.10)) GOTO 517 IF (( I I .EQ.I).AND.(JJ.EQ.J)) GO TO 10 518 ISO • 10*(II-1) + OJ 519 M0D1 = (X(IS0)-D/3 520 M0D2 = X(IS0)-3*M0D1 521 IF (M0D2.EQ.2) NI = N1 + 1 522 10 CONTINUE 523 C Travel at 70 KM/H on highway and 40 KM/H on t r a i l . 524 C 525 C 526 DHWY = DABS(TOWNI-I) 527 DTRL = DABS(TOWNJ-J) 528 X20 • DHWY*70./60. + DTRL*40./60. 529 X21 = 1 .7*N1 + 3.1 530 IF (X21 .LT.TRAILS(6)) X21 =TRAILS(6) 531 IF (X21.GT.TRAILS(6)) TRAILS(6)=X21 532 X22 = SNOW(I.J) 533 X23 = SLOPE(I.J) 534 U6 = K(1)*UF(20.X20) + K(2 ) *UF(21,X21) + K(3 ) *UF(22,X22) + 535 + K(4)*UF(23,X23) 536 RETURN 537 END 538 C 539 C 540 C ACTIVITY HI HIKING 541 C 542 C 543 C 544 FUNCTION U7(X,I.J.N) 545 IMPLICIT REAL*8 (A-H.O-Z) 546 DIMENSION SNOW(10.10),SLOPE(10,10).LAKES(100) 547 INTEGER X(100) 548 REAL K(5) 549 COMMON /DATA/ SNOW,SLOPE,TOWNI.TOWNJ.LAKES,NLAKES 550 DATA K/0.512.0.488,0.0.0.0/ 551 C Travel at 70 KM/H on highway and 40 KM/H on t r a i l . 552 C 553 C 554 DHWY = DABS(TOWNI-I) 555 DTRL = DABS(TOWNJ-J) 556 X24 = DHWY*70./60. + DTRL*40./60. 557 X25 = 1OOOOOO.0 558 C 559 C Ca lcu la te the distance to nearest dr ink ing water source. 560 C 561 IF (NLAKES.EQ.O) GOTO 11 562 DO 10 11=1,NLAKES 563 IL = (LAKES(II) - D/10 + 1 564 JL = LAKES(II) - ( IL-1)*10 565 DX = DSQRT(1.D0*(I-IL)*(I-IL) + ( J - J L ) * ( J - J L ) ) 566 IF (XD.LT.X25) X25 = XD 567 10 CONTINUE 568 11 CONTINUE 569 U7 = K(1)*UF(24.X24) + K(2)*UF(25,X25) 570 RETURN 571 END 572 C 573 C 574 C ACTIVITY #8 HORSEBACK RIDING 575 C 576 C 577 C 578 FUNCTION U8(X.I,J.N) 579 IMPLICIT REAL*8 (A-H.O-Z) 580 DIMENSION SN0W(10.10).SL0PE(10.10).LAKES(100), TRAILSO) 581 INTEGER X(100) 582 REAL K(5) 583 COMMON /DATA/ SNOW,SLOPE.TOWNI,TOWNJ,LAKES.NLAKES 584 COMMON /MAXTRL/ TRAILS 585 DATA K/0.279,0.288.0.433,0.0/ 586 N1 = 1 587 IM1 = I - 1 588 IP1 = 1 + 1 589 JM1 = J - 1 590 JP1 = J + 1 591 DO 10 II=IM1,IP1 592 DO 10 JJ=JM1,JP1 165 593 594 595 596 597 598 10 599 C 600 C 601 C 602 603 604 605 606 607 608 609 610 611 612 C 613 C 614 C 615 C 616 C 617 C 618 619 620 621 622 623 624 625 C 626 C 627 C 628 629 630 631 632 C 633 C 634 C 635 636 637 638 639 640 641 1< 642 1 643 644 645 646 647 C 648 C 649 C 650 651 C 652 C 653 C 654 C 655 C 656 657 658 659 C 660 c 661 c 662 c 663 664 665 666 c IF ( ( I I .LT.1).0R.( I I .GT.10).0R.(JJ.LT.1).0R.(JJ.GT.10)) GOTO 10 IF (( I I .EQ.I).AND.(JJ.EQ.J)) GO TO 10 ISO = 10*(II-1) + J J M0D1 = (X(IS0)-D/3 IF (M0D1.EQ.3) N1 » NI + 1 CONTINUE Travel at 70 KM/H on highway and 40 KM/H on t r a i l . DHWY = DABS(TOWNI-I) DTRL = DABS(TOWNJ-d) X26 = 0HWY*7O./6O. + DTRL*40./60. X27 = 1.7*N1 +3.1 IF (X27.LT.TRAILS(8)) X27=TRAILS(8) IF (X27.GT.TRAILS(8)) TRAILS(8)=X27 X28 = SLOPE(I.J) U8 = K(1 )*UF(26.X26) + K(2)*UF(27.X27) + K(3)*UF(28,X28) RETURN END ACTIVITY #9 SUMMER MOTORIZED CAMPING FUNCTION U9(X,I,J,N) IMPLICIT REAL*8 (A-H.O-Z) DIMENSION SN0W(10.10).SLOPE(10,10).LAKES( 100) INTEGER X(100) REAL K(5) COMMON /DATA/ SNOW.SLOPE,TOWNI.TOWNJ,LAKES.NLAKES DATA K/0.371.0.229.0.400.0.0/ Travel at 70 KM/H on highway and 40 KM/H on t r a i l . DHWY = DABS(TOWNI-I) DTRL = DABS(TOWNJ-J) X29 = 0HWY*70./60. + DTRL*40./60. X30 = 1000000.0 Ca lcu la te the distance to nearest dr ink ing water source. IF (NLAKES.EO.O) GOTO 11 DO 10 11=1.NLAKES IL = (LAKES(II) - 1)/10 + 1 JL = LAKES(II) - ( IL-1)*10 DX = D S Q R T ( 1 . D O * ( I - I L ) « ( I - I L ) + ( J - J L ) * ( J - J L ) ) IF (XD.LT.X30) X30 = XD CONTINUE CONTINUE X31 = SLOPE ( I .J) U9 = K(1)*UF(29.X29) + K(2 )*UF(30.X30) + K(3 )*UF(31.X31) RETURN END FUNCTION UF(I,X) This funct ion evaluates the u t i l i t y fo r one of the the 31 a t t r i b u t e s . I i s the a t t r i b u t e number. IMPLICIT REAL*8 (A-H.O-Z) DIMENSION UTIL(31,9).XR(31.9) COMMON /UTILTY/ UTIL.XR Find a range of input X-values which surround the a t t r i b u t e value. DO 10 J=1.9 IF (X.LE.XR(I.J)) GOTO 20 10 CONTINUE 166 6S7 C If no range 1s found, assign the last u t i l i t y . 668 C 669 UF = UTIL(I,9) 670 RETURN 671 20 CONTINUE 672 IF (d.EQ.1) GOTO 30 673 JM1 = J-1 674 C 675 C U t i l i t y 1s assigned by a s t ra ight l i ne approximation 676 C between the surrounding data po ints . 677 C 678 UF = UTIL(I.dMI) + (X-XR(I.JM1))/(XR(I,d)-XR(I,dM1))* 679 + (UTIL(I,d)-UTIL(I.JM1)) 680 RETURN 681 30 CONTINUE 682 C 683 C If the a t t r i b u t e value 1s less than the f i r s t input 684 C X-value, ass ign the f i r s t u t i l i t y . 685 C 686 UF = UTIL(I, 1 ) 687 RETURN 688 END 689 C 690 C 691 C 692 SUBROUTINE GETDAT 693 C 694 C This subroutine reads the Input data from f i l e s . 695 C 696 IMPLICIT REAL*8 (A-H.O-Z) 697 DIMENSION SNOW(10,10). SLOPE(10,10), LAKES(100) 698 DIMENSION UTIL(31,9) , XR(31,9) 699 COMMON /DATA/ SNOW,SLOPE,TOWNI,TOWNJ,LAKES,NLAKES 700 COMMON /UTILTY/ UTIL.XR 701 C 702 C I n i t i a l i z e a l l arrays f i r s t . 703 C 704 DO 5 1 = 1,10 705 DO 5 J=1.10 706 SNOW(I.J) =0 .0 707 5 SLOPE(I.J) = 0.0 708 DO 6 1=1.100 709 6 LAKES(I) = O 710 NLAKES = O 711 C 712 C Read snow depths from unit 1 and slopes from unit 2. 713 C 714 DO 10 1 = 1. 10 715 READ (1,100,END=8) (SNOW(I.d).J=1.10) 716 8 CONTINUE 717 READ (2,100,END=9) (SLOPE(I,J),J=1,10) 718 9 CONTINUE 719 100 FORMAT (10F8.2) 720 10 CONTINUE 721 C 722 C Read squares conta in ing water from unit 3. 723 C Note : Top row 1s #001 - #010. 724 C Bottom row 1s #091 - /MOO. 725 C 726 DO 20 1=1.100 727 READ (3.200.END=25) LAKES(I) 728 20 CONTINUE 729 200 FORMAT (13) 730 I = 101 731 25 CONTINUE 732 NLAKES = 1-1 733 C 734 C Read u t i l i t y graph points from unit 4. 735 C 736 DO 30 1=1.31 737 READ (4,300.END = 30) (XR(I.J) ,UTIL(I,d).J=1.9) 738 30 CONTINUE 167 739 300 FORMAT (18F8.2) 740 C 741 C Read po s i t i on of town from terminal ( i n t e r a c t i v e l y ) . 742 C Note : Input should be in real numbers (e.g. 1.0,3.0). 743 C 744 WRITE (6,400) 745 READ (5,401) TOWNI,TOWNJ 746 400 FORMAT (' ENTER COORDINATES OF TOWN (1,1 IS TOP LEFT CORNER', 747 + ' OF GRID BOX)' ) 748 401 FORMAT (2F10.3) 749 RETURN 750 END End of f i l e 168 APPENDIX Explanation of the NLP.S Algorithm 169 EXPLANATION OF THE NLP.S ALGORITHM This appendix explains the NLP.S algorithm which evaluates the objective function developed i n Part 1 for a recreation land use plan within the 10 km x 10 km section of mine waste. The following discussion elaborates on certain sections, of the algorithm, referred to by t h e i r l i n e number i n Appendix Lines 33-34 These l i n e s contain the a c t i v i t y scaling factors deter-mined for the objective function i n Section 2.10. Lines 40-41 For a recreation land use plan, the subroutine GETDAT (line 692) gets the data on snow depths, slope values, drinking water source locations and attribute u t i l i t y functions from t h e i r respective f i l e s the f i r s t time through the algorithm. Lines 45-47 For a recreation land use plan, the variables (land uses) are rounded o f f . These l i n e s make the program an INTEGER program; therefore a non-linear optimization routine using REAL numbers cannot be used. Lines 55-58 The algorithm calculates the maximum t r a i l distance for each a c t i v i t y the f i r s t time through the algorithm and uses these t r a i l distances to calculate the a c t i v i t y u t i l i t i e s the second time through the algorithm. The maximum t r a i l distance found for each a c t i v i t y i s used to calculate the u t i l i t y for 17 0 the t r a i l distance a t t r i b u t e . The following example i l l u s -trates why the maximum t r a i l distance i s used: Example For example, l e t one a l l o c a t i o n of t r a i l b i k i n g have a t r a i l distance of 3 km. Let another have a t r a i l distance of 10 km. The u t i l i t y for t r a i l b i k i n g for the plan w i l l include the average u t i l i t y of the attribute t r a i l distance. For t h i s example, i t w i l l be the average of the u t i l i t y of 3 km and the u t i l i t y of 10 km; hence, having an a l l o c a t i o n with a small t r a i l distance w i l l decrease the u t i l i t y of an a l l o c a t i o n with a large t r a i l distance. To get around t h i s problem, the algo-rithm uses the maximum t r a i l distance found for an a c t i v i t y a l l o c a t i o n and uses i t for the t r a i l distance u t i l i t y c a l c u l -ations . Lines 74-75 K i s the g r i d square under examination by NLP.S. Lines 83 and 89 th X(K) i s the land use for the K square. These l i n e s count the land uses assigned to each g r i d square. Lines 100-171 These l i n e s calculate how many squares have each a c t i v i t y u n t i l a l l 100 g r i d squares have been examined. Lines 175-176 These l i n e s add the t o t a l u t i l i t y for each of the 9 a c t i v -i t i e s . 171 Lines 181-182 These li n e s evaluate the objective function developed i n Section 2.10. Lines 186-187 These li n e s change the sign of the objective function from p o s i t i v e to negative. Most available software for optimizing routines minimize the objective function. The algorithm i s set up to give the minimum function value for future use when s o f t -ware becomes available for non-linear INTEGER optimization. Lines 190-216 If an optimization routine i s used, these l i n e s constrain each of the 100 variables or g r i d squares to be between land uses 1 and 16. Line 25 8 This l i n e supplies the scaling constants for the additive u t i l i t y function for t r a i l b i k i n g . Lines 263-272 These li n e s count the number of adjoining t r a i l b i k i n g and four-wheel d r i v i n g squares for the c o n f l i c t c a l c u l a t i o n (land uses 7, 8 and 9) i n li n e s 314-315. Lines 293-295 These li n e s calculate the t r a v e l times to each g r i d square using the equations developed i n Section 4.41. Line 299 These lines calculate.the t r a i l distance of t r a i l b i k i n g a llocations using the equation developed i n Section 4.42. 172 Line 309 This l i n e assigns a slope value to square K from the f i l e with slope values. Lines 314-315 These l i n e s calculate the number of c o n f l i c t s between t r a i l b i k i n g and four-wheel d r i v i n g using the equation developed i n Section 4.44. Line 316 This l i n e calculates the u t i l i t y of t r a i l b i k i n g for the square under examination, K, using the attribute levels deter-mined by li n e s 293-315. Lines 322-649 These l i n e s calculate the u t i l i t y values for the remaining a c t i v i t i e s i n the same way as for t r a i l b i k i n g . Lines 562-567 These li n e s calculate the distance of the g r i d square under examination, K, to the nearest drinking water source f o r the a c t i v i t y hiking. These l i n e s use the equations developed i n Section 4.45. Lines 696-738 Subroutine GETDAT reads the data from f i l e s containing snow depth values and slope values (lines 714-720), drinking water source locations (lines 726-732), and u t i l i t y function data points (lines 736-738). Lines 739-750 The location of the town i s input to the algorithm. The user i s prompted using the non-linear monitor to enter the I,J coordinates of the town i n RE/AL numbers. The town may be any-where inside or outside the g r i d square. 174 APPENDIX The NLP.S Algorithm Flowchart FIG.28. THE NLP.S ALGORITHM FLOWCHART CALLING SEQUENCE NA:HLHON XDFUNC (dllt mine) ( O b j t c t i v F u n o t i o n ) XDFUNC C A L L SUBROUTINE GETDAT I GET DATA FROM F I L E S ROUND OFF XX-ARRAY ( b i e a u i t mutt work with i n t a g t r t ) - r START OF COOP I I N I T I A L I Z E ARRAYS T 1) FOR A GIVEH LARD USE PLAR, TBE ALGORITHM EVALUATES THE UTILITY FOR EACH ACTIVIT1 AMD TBS OBJECTIVE FUHCTIOR FOR TBE PLAB; Z) E IS TBE GRID SQUARE VBDER tXAMIBATIOR; I) M(K) IS TBE LAHD USE FOR TBE Ktk SQUARE; 4) TRE FIRST TIME THROUGH TBE LOOP EVALUATES THE OBJECTIVE FURCTIOR VBILE SEARCHIMG FOR MAXIMUM TRAIL LEHCTHS ASSOCIATED WITH EACH ACTIVITY, THE SECOHD TIME THROUGH THE LOOP CALCULATES THE ACTUAL UTILITY USIKG THESE TRAIL DISTAMCES. YES YES _ ADD T R A I L B I K I N G U T I L I T Y FOR SQUARE K TO U y g . INCREMENT SQUARE COUNTER FOR T R A I L B I K I N G . ADD 1x4 U T I L I T Y FOR SQUARE K TO U|, x l |. INCREMENT SQUARE COUNTER FOR 1x4. > P YES ^ ADD T R A I L B I K I N G AND H X l U T I L I T I E S FOR SQUARE K TO U T , AND U,„,. INCREMENT SQUARE COUNTERS FOR THE TWO A C T I V I T I E S . > — • YES ADD H I K I N G , HORSEBACK RIDING AND CAMPING U T I L I T I E S FOR SQUARE K TO U H U E . U | < 0 R S E AND U £ 4 H p . INCREMENT SQUARE COUNTER FOR THE THREE A C T I V I T I E S ADD SNOWMOBILING U T I L I T Y FOR SQUARE K TO U s | | 0 t | , INCREMENT SQUARE COUNTER FOR SNOWMOBILING. ADD DOWNHILL S K I I N G U T I L I T Y FOR SQUARE K TO Ugp,,,. INCREMENT SQUARE COUNTER FOR DOWNHILL S K I I N G . ADD X-C S K I I N G AND SNOWSHOEING U T I L I T I E S FOR SOUARE K TO V c U S H O E ' INCREMENT SQUARE COUNTER FOR THE TWO A C T I V I T I E S . O B J E C T I V E FUNCTION " • OBJECTIVE FUNCTION - - JKJUJ KJUJ K j U ^ J THE AVERAGE UTILITY FOR ACTIVITY i AMOHC ALL SQUARES ALLOTTED THE ACTIVITY; THE BVMBER OF SQUARES ALLOTTED ACTIVITY i. T ] APPENDIX Procedure for Using the NLP.S Algorithm 177 PROCEDURE FOR USING THE NLP.S ALGORITHM The monitor for non-linear function optimization at the University of B r i t i s h Columbia can be used to obtain objective function values f o r land use plans. The following steps i l l u s t r a t e how to use the monitor with the NLP.S algorithm: Step 1 The algorithm NLP.S must be compiled into an object f i l e . The f i r s t step i s to create an object f i l e , f or example NLP.O. NLP.S i s compiled into the object f i l e by the following command: $RUN *FTN SCARDS=NLP.S SPUNCH=NLP.O Step 2 Step 2 i s to create a f i l e to store the land use plan, for example, the f i l e S0LN1. Step 3 Step 3 i s to c a l l the monitor using NLP.O by the following command: $RUN NLP.0+NA:NLMON 1=SN0W 2=SL0PES 3=LAKES 4=UTILITY where: NA:NLMON c a l l s the monitor; SNOW = the f i l e with snow depth values for the plan under examination. The f i l e with snow depths corresponds to unit 1; SLOPES = the f i l e with slope values for the plan under examination. The f i l e with slope values corresponds to unit 2; LAKES = the f i l e with drinking water source locations. The f i l e with drinking water source locations corresponds to unit 3; UTILITY = the f i l e with attribute u t i l i t y function data points. The f i l e with attribute u t i l i t y func-t i o n data points corresponds to unit 4 and are 178 presented i n Appendix 13. The following steps i l l u s t r a t e what i s displayed on the terminal and what i s required as input: DISPLAY: /READY Step 4 The next step i s to input the values of the 100 variables or g r i d squares to evaluate the objective function for the plan. Land uses (INTEGERS 1 to 16) are entered from square 0 01 to square 100 u n t i l 100 land uses have been entered. A land use may be repeated by the m u l t i p l i c a t i o n sign *. For example, to i n i t i a l i z e 100 g r i d squares, the input may be 100*3 for 100 g r i d squares allocated land use 3, or 9,11,3,4,96*3 f o r squares 001 to 004 allocated land uses 9,11,3 and 4 respectively and squares 005 to 100 allocated land use 3. The following commands i n i t -i a l i z e the land uses: DISPLAY: /READY INPUT: INIT P DISPLAY: ? INPUT: 9,11,3 DISPLAY: ? 4,96*3 DISPLAY: /READY Step 5 The next step i s to p r i n t the function value for the plan by the following command: DISPLAY: /READY 179 INPUT: PRINT F DISPLAY: ENTER COORDINATES OF TOWN (1,1 i s i n top l e f t corner of g r i d box): INPUT: -20.0,7.0 (an example town location) DISPLAY: Function value F = -0.7869958477 89 (an example value) Step 6 To store the plan i n the created f i l e SOLNl, the following commands are entered: DISPLAY: /READY INPUT: ASSIGN 20=SOLNl DISPLAY: /READY INPUT: SAVE PAR ON 20 DISPLAY: /READY Step 7 To change land uses of g r i d squares, the following commands are entered: DISPLAY: /READY INPUT: MODIFY DISPLAY: /ENTER PARAMETER INDEX AND NEW VALUE, OR NULL LINE: INPUT: 47 8 (change square 47 to land use 8) DISPLAY: /ENTER PARAMETER INDEX AND NEW VALUE, OR NULL LINE Press "BREAK" on the keyboard to stop entering new values. DISPLAY: /READY INPUT: PRINT F DISPLAY: Function value F = -0.8887879546 87 (new function value) DISPLAY: /READY Step 8 To stop the monitor, enter MTS or STOP. This w i l l transfer the user to the MTS mode. Step 9 To view the land use plan on the terminal screen, the following commands are entered: DISPLAY: # INPUT: $RUN OUTPUT.0 1=S0LN1 Step 10 To p r i n t the land use plan on the XEROX 9700 p r i n t e r , the following commands are entered: DISPLAY: # INPUT: $RUN OUTPUT.0 1=S0LN1 6=*PRINT* For steps 9 and 10, the f i l e OUTPUT.0 i s an object f i l e containing the compiled f i l e OUTPUT.S presented i n Appendix 14. OUTPUT.S i s an algorithm which takes the land use plan i n machine language from SOLNl and presents i t for viewing on the terminal screen. To compile OUTPUT.S int o f i l e OUTPUT.O, the following command i s entered: DISPLAY: # INPUT: $RUN *FTN SCARDS=OUTPUT.S SPUNCH=OUTPUT.0 181 APPENDIX F i l e Containing the Data Points of the 31 Attribute U t i l i t y Functions 182 TABLE 4. FILE CONTAINING THE DATA POINTS OF THE 31 ATTRIBUTE UTILITY FUNCTIONS ATTRIBUTE UTILITY FUNCTION DATA POINTS FILE = UTILITYFCNS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 0, 1 , 0.0. 0.0. 0. 1 , O. 1 . 0,0, 0.0. O. 1 , 0,0. 0.0, 0,0, 0.0. 0, 1 , 0.0, 0,0, 40,0.75,90,0.5,135,0.25,240,0.100000.0.0 5.5.0.25.7,0.5,13,0.75,20,1,100000.1.0 8,15.1.27,0.75,35.0.5.43.0.25.50.0.100000,0.0 1,0.5,1.5,0.25,4,0.100000,0.0 20,0.75,120.0.5,160,0.25,240,0,100000,0.0 3,0.25.7,0.5,14,0.75,20,1.100000,1.0 5,38,0.75,50,1,53,0.75,55.0.5,57,0.25,60,0,100000.0 5,3,0.25,4,0,100000,0.0 5,20,0.75,30,1,50,0.75,95,0.5. 10,0.75.20.1, 44.0.25,52.0.5,64,0.75.91, 28,0.25,35, 1.5,0.75.2, 13,0.25,17. 4,0.25,8,0 91 ,0. 20.0. 0,0 0.0 0,0 0. 1 0,0 O. 1 0, 1 0,0 0.0 0,0 0, 1 0. 1 150.0.25.240.0,100000,0 100000,1.0 1.155,0.75,190,.5,206,.25,244,0,100000,0 8,17,0.92,30,1,39.0.75,43,0.5,47,0.25,50,0,100000,0.00 25,0.75.60,0.5,150,0.25,240.0,100000,0.0 25,0.25,37,0.5,43.0.75,60,1,70.0.75,75.0.5,85,0.25,120,0,100000.0 25,113,0.5,143,0.75,244,1,1O0OOO,1.0 75,45,0.5,90,0.25,240,0,1OOOOO,0.0 .9,0.25,13,0.5.18,0.75,20,1.100000,1.0 .75,3,1,6,0.75.15,0.5.27,0.25,50,0,100000,0.0 .30,0.25.67.0.5,91.0.75,244,1,100000,1.0 .3.5,0.25,7,.5.11..75,15,1,45.0.75,90,0.5,150.0.25,240, ,3,0.25,4.5,0.5,11,0.75,20.1,100000,1.0 .30,0.25,67.0.5,79.0.75,91,1.137,0.75,198.0.5.221,0.25,244,O,100000.0.0 ,17.0.75.25.0.5.30.0.25.50.0,100000.0.0 , 10,0.25,12.0.5,14.0.75,15,1,45,0.75,75,0.5,120.0.25.240.0. , 1,0.75,2.0,0.5,3.5.0.25,10,0,100000,0.0 ,75,0.75,150,0.5,180,0.25,240,0.100000.0.0 ,9,0.25,11,0.5,13,0.75.20,1,100000,1 .3.5.0.25,6.5.0.5,7.5.0.75,10,1.16.0.75,23.0.5,33,0.25,50,0.1OO000. .3,0.25.10.0.5,24.0.75,30,1,50.0.75,90,0.5, " " " ,1,0.75,2,0.5,3.0.25.10,0,100000,0 ,4,0.75,8,0.5,15.0.25,50.0.100000,0.0 ,0,100000,0.0 100000.0 ,0.0 150,0.25,240,0,1O0000.0 APPENDIX The OUTPUT.S Algorithm THE OUTPUT.S ALGORITHM (Computer Language = FORTRAN) 184 1 REAL*8 X(100) 2 INTEGER M(100) 3 C 4 C Read land uses from f i l e . 5 C 6 READ (1) N,(X(I).I=1,N) 7 C 8 C Round off rea l s to integers. 9 C 10 DO 10 1=1.100 11 10 M(I) = X(I) + 0.5 12 C 13 C Write the integer array to unit 14 C 15 WRITE (6,10O0) M 16 1000 FORMAT (/,/.10(/.1014)) 17 STOP 18 END End of f i l e APPENDIX Data on Snow Depth, Slope and Drinking Water Sources for the Land Use Plan TABLE 5. DATA ON SNOW DEPTH, SLOPE AND DRINKING WATER SOURCES FOR THE LAND USE PLAN AVERAGE SNOW DEPTH VALUES FOR THE LAND USE PLAN (cm) FILE = SNOW 244,244,091.152,137,122,000.091,091,152, 244,213,122,152,137,122,122,000,091.152, 213,198,168,137,122,122,OOO,107.122.122. 168,183,152,137,122.122,OOO,107,122,122. 198,183,152,122,122,000,122.107,122,152, 107,091.091,000,000,107,107,152,168,152 , 107,091,091.000.107,122,152,168,183,213, 107,091.091,OOO,107,122,152,168,183,213, 122,107,091,000,091,122,152,168,198,213 , 107,091,076,000,000,076,137,168,198,213, AVERAGE SLOPE VALUES FOR THE LAND USE PLAN (%) FILE = SLOPES 000,OOO,100,015,010,000,000.000,050,050, 000,095,025.O15,000.000.OOO,000,050.025, 090,020,O1O,OOO.000.OOO,OOO.010,050,050, 050,050,050, OOO, 000, OOO. 000.025,050,050, 050,050.040,010,000,000.015,050,050,050, 090,090,015,000,000,000,010.050,050.050, 090,070,020,000,000,015,050.050,050,035, 090,070,020,000,000,040.050,050,055,020, 080,060,025,000,000.040,050,050,060,040, 070,050,000,000,000,040,050,055.030,010, GRID SQUARES CONTAINING A DRINKING WATER SOURCE - PLAN (Gr id squares for th i s f i l e are numbered from 1 to 100. Square (1,1) is number 001; square (10,10) Is number 10O) FILE • LAKES 1 007 2 018 3 027 4 037 5 046 6 055 7 064 8 074 9 084 10 095 APPENDIX The UTILVAL.S Algorithm 188 THE UTILVAL.S ALGORITHM (Computer Language = FORTRAN) 1 C Function 'XDFUNC' evaluates the object ive funct ion 2 C that 1s to be minimized. Although we ac tua l l y want 3 C to maximize the u t i l i t y funct ion, the ava i l ab le 4 C opt imiz ing softaware only allows funct ions to be 5 C minimized. To get around th i s , the last command 6 C 1n th i s funct ion sets XDFUNC = -XDFUNC. 7 C 8 C Var iab les used : 9 C XX - The mine waste plan to be evaluated. 10 C Each array element represents one g r i d square 11 C and contains a number between 1 & 16, depending 12 C on the land use c las s assigned to that square. 13 C X - Rounded-off vers ion of array XX. We require 14 C integers to evaluate the u t i l i t y funct ion. 15 C NUM - Dummy va r i ab le ; dimension of XX. 16 C SCALE - U t i l i t y sca le factors for each of the 17 C 9 a c t i v i t i e s . 18 C U - Cumulative u t i l i t y for each a c t i v i t y . 19 C N - Number of squares assigned to each a c t i v i t y . 20 C TRAILS - Keeps track of the maximum t r a i l a va i l ab le 21 C for each a c t i v i t y for the current plan. 22 C INFLAG - 0/1 f l a g to determine whether the funct ion 23 C has been previous ly c a l l e d . If not c a l l e d . 24 C input data must be read in f i r s t before 25 C proceeding. 26 C 27 IMPLICIT REAL'S (A-H.O-Z) 28 DIMENSION XX(100) 29 DIMENSION TRAILSO) 30 DIMENSION UT(10,10) 31 INTEGER X(100) 32 DATA INFLAG/1/ 33 COMMON /MAXTRL/ TRAILS 34 c 35 c Get input data from f i l e s f i r s t time through. 36 c 37 IF (INFLAG.E0.1) CALL GETDAT 38 INFLAG = 0 39 c 40 c Round of f a l l 100 var i ab le s . 41 c 42 DO 10 1=1,100 43 X(I) = XX(I)+0.5 44 10 CONTINUE 45 15 CONTINUE 46 WRITE (5.3000) 47 READ (6,3010) IACT 48 3000 FORMAT ('Which a c t i v i t y do you wish to analyze? ' ) 49 3010 FORMAT (11) 50 IF (IACT.E0.O) STOP 51 IF ((IACT.LT.0).0R.(IACT.GT.9)) GOTO 15 52 DO 500 K=1.100 53 C Convert K to row (I) and column (J ) . 54 C 55 C K = 1 --> Top l e f t square. 56 C K = 100 --> Bottom r ight square. 57 C 58 I = (K -O /10 + 1 59 d = K - ( I-1)*10 60 GOTO (21,22.23.24.25,26.27,28.29),IACT 61 21 CONTINUE 62 UT(I.J) = U K X . I . J . N ) 63 GOTO 500 64 22 CONTINUE 65 UT(I.J) = U 2 ( X . I . J .N ) 66 GOTO 500 67 23 CONTINUE 68 UT(I ,J) - U 3 ( X , I , J .N ) 69 GOTO 500 70 24 CONTINUE 189 71 UT(I.J) = U4(X,I.J.N) 72 GOTO 500 73 25 CONTINUE 74 UT(I ,J) • U5(X,I.J,N) 75 GOTO 500 76 26 CONTINUE 77 UT(I.J) = U6(X.I.J.N) 78 GOTO 500 79 27 CONTINUE 80 UT(I.J) = U7(X.I.J.N) 81 GOTO 500 82 28 CONTINUE 83 UT(I .J) = U8(X.I.J.N) 84 GOTO 500 85 29 CONTINUE 86 UT(I .J) = U9(X.I.J.N) 87 GOTO 500 88 500 CONTINUE 89 WRITE (8.4000) IACT 90 4000 FORMAT ( ' 1 ' . ' Summary of U t i l i t y Functions f o r ' . 91 + ' A c t i v i t y ' .11,/) 92 4010 FORMAT (10( / . ' ' .10F8.5)) 93 GOTO 15 94 END 95 C 96 FUNCTION XDG(X.N.I) 97 C 98 C This funct ion suppl ies the opt imizing routine 99 C with lower const ra int s for each of the 100 var iab les 100 C In th i s program, the constra int is 1 < Xi for a l l 101 C 100 var i ab le s . 102 C 103 IMPLICIT REAL*8 (A-H.O-Z) 104 DIMENSION X(N) 105 XDG = 1.0 106 RETURN 107 END 108 C 109 C 1 10 c 1 1 1 FUNCTION XDH(X,N,I) 1 12 c 113 c This funct ion suppl ies the optimizing routine 1 14 c with upper constra ints for each of the 100 var iab les 1 15 c In th i s program, the constra int is XI < 16 for a l l 1 16 c 100 var i ab le s . 1 17 c 1 18 IMPLICIT REAL*8 (A-H.O-Z) 119 DIMENSION X(N) 120 XDH • 16.0 121 RETURN 122 END 123 c 124 c 125 c FUNCTION XDX(X.N.I) 126 127 c This funct ion suppl ies the opt imizing rout ine with 128 c 129 c any i m p l i c i t var iab les required. For th i s program. 130 c there are no Impl ic i t var iab les ; th i s is a dummy 131 c funct ion to s a t i s f y the rou t ine ' s requirements and 132 c i s never c a l l e d . 133 c IMPLICIT REAL*8 (A-H.O-Z) 134 135 DIMENSION X(NJ 136 XDX « 0.0 137 RETURN 138 END 139 c 140 c 141 c FUNCTION UKX.I.J.N) 142 143 144 c c Functions U1.U2 U9 evaluate the u t i l i t y of 145 c the corresponding a c t i v i t y in the square ( I . J ) . 1 9 0 146 C The array X Is used to determine surrounding 147 C a c t i v i t i e s for eva luat ing c o n f l i c t s . 148 C The input arrays SNOW, SLOPE, & TRAILS are used 149 C to evaluate the phys ica l q u a l i t i e s of square ( I , J ) . 150 C 151 C 152 C ACTIVITY #1 TRAILBIKING 153 C 154 C 155 IMPLICIT REAL*8 (A-H.O-Z) 156 DIMENSION SN0W( 10.10). SL0PE( 10.10). LAKES( 100). TRAILSO) 157 INTEGER X(100) 158 C 159 C Array K stores the m u l t i p l i c a t i v e factors for U-TB. 160 C 161 REAL K(5) 162 COMMON /DATA/ SNOW,SLOPE.TOWNI,TOWNJ,LAKES,NLAKES 163 COMMON /MAXTRL/ TRAILS 164 DATA K/O.175,0.298,0.333,0.194/ 165 C 166 C N1 counts the number of adjo in ing TB squares. 167 C N2 counts the number of adjo in ing Hike/Horse/Camp squares. 168 C 169 N1 = 1 170 IJSO = 10*(I-1)+J 171 N2 = 0 172 IF ((X(IJS0).GE.7).AND.(X(IJS0).LE.9)) N2 = 1 173 IM1 = I - 1 174 IP1 = 1 + 1 175 JM1 = J - 1 176 JP1 = J + 1 177 DO 10 II=IM1,IP1 178 DO 10 JJ=JM1,JP1 179 C 180 C Ensure that we are looking at a square ins ide 181 C the 10x10 planning area. 182 C 183 IF ((I I.LT.1 ) .OR.(II.GT.10) .OR.(JJ.LT.1).OR.(JJ.GT.10)) GOTO 10 184 C 185 C Don't process square ( I .J) 186 C 187 IF (( I I .EQ.I).AND.(JJ.EQ.J)) GO TO 10 188 ISO • 10*(II-1) + JJ 189 M0D1 = (X( IS0) -D/3 190 IF ((M0D1 .EO.O) .OR. (M0D1 .E0..2)) N1 = N1 + 1 191 IF ((MOD 1.EO.1).OR.(MOD 1.EO•2)) N2 = N2 + 1 192 10 CONTINUE 193 C 194 C Ca lcu la te t rave l time at 70 KM/H on highway and 195 C 40 KM/H on t r a i l . 196 C Highways are assumed to run along columns J . 197 C T r a i l s are assumed to run along rows I. 198 C 199 DHWY = DABS(TOWNI-I) 200 DTRL = DABS(TOWNJ-J) 201 X1 = DHWY*70./60. + DTRL*40./60. 202 C 203 C Ca lcu la te t r a i l d is tance. 204 C 205 X2 « 1.7*N1+3.1 206 C 207 C T r a i l d i s tance u t i l i t y Is assigned the maximum 208 C t r a i l d istance for the p lan. 209 C 210 IF (X2.LT.TRAILS( 1)) X2=TRAILS(1) 211 IF (X2.GT.TRAILSO)) TRAILS(1)=X2 212 C 213 C Average slope value for square K. 214 C 215 X3 - SLOPE(I.J) 216 C 217 C If land use plans 7.8,9. # c o n f l i c t s are 4/(N1*N2) 218 C Otherwise, there are no c o n f l i c t s . 219 C 220 IF ((X(IJS0).GE.7).AND.(X(IJS0).LE.9)) X4 - 4.0/(N1*N2) 221 IF ((X( IJS0).LT.7).0R.(X(IJS0).GT.9)) X4 - 0.0 222 U1 - K(1)*UF(1.X1) + K(3)*UF(3,X3) + K(4)*UF(4,X4) 191 223 RETURN 224 END 225 C 226 C 227 C ACTIVITY #2 FOUR-WHEEL DRIVING 228 C 229 C 230 C 231 FUNCTION U2(X.I,J,N) 232 IMPLICIT REAL*8 (A-H.O-Z) 233 DIMENSION SN0W(1O.1O),SL0PE(1O.10).LAKES(1OO). TRAILS(9) 234 INTEGER X(100) 235 REAL K(5) 236 COMMON /DATA/ SNOW.SLOPE.TOWNI.TOWNJ.LAKES.NLAKES 237 COMMON /MAXTRL/ TRAILS 238 DATA K/0.200.0.291,0.363.0.146/ 239 N1 = 1 240 N2 * 0 241 IJSO = 10*(I-1) + J 242 IF ((X(IJS0).GE.7).AND.(X(IJS0).LE.9)) N2 = 1 243 IM1 = I - 1 244 IP1 = 1 + 1 245 JM1 = J - 1 246 JP1 = J + 1 247 DO 10 II=IM1.IP1 248 DO 10 JJ=JM1,JP1 249 IF (( I I .LT.1).OR.( I I .GT.10).OR.(JJ.LT.1).OR.(JJ.GT.10)) GOTO 250 IF (( I I .EO.I).AND.(JJ.EO.J)) GO TO 10 251 ISO = 1 0 » ( I I - 1 ) + JJ 252 M0D1 = (X(IS0)-1)/3 253 IF ((M0D1.EQ.O).0R.(M0D1.EQ.2)) N2 = N2 + 1 254 IF ((M0D1.EQ.1).0R.(M0D1.EQ.2)) N1 = N1 + 1 255 10 CONTINUE 256 c 257 c Travel time at 70 KM/H on highway and 40 KM/H on t r a i l . 258 c 259 DHWY = DABS(TOWNI-I) 260 DTRL = DABS(TOWNJ-J) 261 X5 = DHWY*70./60. + DTRL*40./60. 262 X6 = 1.7*N1+3.1 263 IF (X6.LT.TRAILS(2)) X6=TRAILS(2) 264 IF (X6.GT.TRAILS(2)) TRAILS(2)=X6 265 X7 = SLOPE(I.J) 266 IF ((X(IJS0).GE.7).AND.(X(IJS0).LE.9)) X8 = 4.0/(N1*N2) 267 IF ((X(IJSQ).LT.7).0R.(X(IJSQ).GT.9)) X8 = 0.0 268 U2 = K(1 )*UF(5,X5) + K(3)*UF(7,X7) + K(4 )»UF(8.X8) 269 RETURN 270 END 271 c 272 c 273 c ACTIVITY #3 SNOWMOBILING 274 c 275 c 276 c 277 FUNCTION U3(X,I,J.N) 278 IMPLICIT REAL*8 (A-H.O-Z) 279 DIMENSION SN0W(1O,1O),SL0PE(10.1O),LAKES(1OO). TRAILS(9) 280 INTEGER X(100) 281 REAL K(5) 282 COMMON /DATA/ SNOW.SLOPE.TOWNI.TOWNJ,LAKES.NLAKES 283 COMMON /MAXTRL/ TRAILS 284 DATA K/0.269.0.252,0.167,0.312/ 285 N1 « 1 286 IM1 = I - 1 287 IP1 - I + 1 288 JM1 = J - 1 289 JP1 « d + 1 290 DO 10 II-IM1.IP1 291 DO 10 JJ=JM1,JP1 292 IF ( ( I I .LT.1).0R.( I I .GT.1O ) .0R . (JJ.LT .1) .0R . (dJ.GT.1O)) GOTO 293 IF (( I I .EO.I).AND.(JJ.EO.J)) GO TO 10 294 ISO = 10*(II-1) + J J 295 M0D1 = (X(IS0)-1)/3 296 M0D2 = X(ISO) - M0D1*3 297 IF (X(IS0).E0.16) GOTO 10 298 IF (M0D2.E0.1) NI • NI • 1 299 10 CONTINUE 192 300 C 301 C Travel time at 70 KM/H on highway and 40 KM/H on t r a i l . 302 C 303 DHWY = DABS(TOWNI-I) 304 DTRL = DABS(TOWNJ-J) 305 X9 « DHWY*70./60. + 0TRL*40./60. 306 X10 - 1.7*N1 +3.1 307 IF (X10.LT.TRAILSO)) X10=TRAILS(3) 308 IF (X10.GT.TRAILS(3)) TRAILS(3)=X10 309 C 310 C Average snow depth value for square K. 311 C 312 XII • SNOW(I.J) 313 C 314 X12 = SLOPE(I.J) 315 U3 = K(1 )*UF(9.X9) + K(3)*UF(11,X11) + K(4)*UF(12.X12) 316 RETURN 317 END 318 C 319 C 320 C ACTIVITY #4 DOWNHILL SKIING 321 C 322 C 323 C 324 FUNCTION U4(X,I.J.N) 325 IMPLICIT REAL*8 (A-H.O-Z) 326 DIMENSION SNOW(10.10).SL0PE(1O.10).LAKES(IOO), TRAILSO) 327 INTEGER X(100) 328 REAL K(5) 329 COMMON /DATA/ SNOW,SLOPE,TOWNI,TOWNJ,LAKES.NLAKES 330 COMMON /MAXTRL/ TRAILS 331 DATA K/O.301,0.437,0.262.0.0/ 332 N1 = O 333 IM1 = I - 1 334 IP 1 = I + 1 335 JM1 = J - 1 336 JP1 = J + 1 337 DO 10 II=IM1,IP1 338 DO 10 JJ=JM1,JP1 339 IF ( ( I I .LT.1).0R.( I I .GT.10).0R.(JJ.LT.1).0R.(JJ.GT.10)) GOTO 10 340 IF (( I I .EO.I).AND.(JJ.EO.J)) GO TO 10 341 ISO = 10*(II-1) • J J 342 IF (X(ISO).EO.16) N1 = N1 + 1 343 10 CONTINUE 344 C 345 C Travel time at 70 KM/H on highway and 40 KM/H on t r a i l . 346 C 347 DHWY = DABS(TOWNI-I) 348 DTRL = DABS(TOWNJ-J) 349 X13 = DHWY*70./60. + DTRL*40./60. 350 X14 = SLOPE(I.J) 351 X15 = SNOW(I.J) 352 U4 = K( 1)*UF(.13,X13) + K(2 )*UF( 14. X 14) + K( 3)»UF( 15 . X 15) 353 RETURN 354 END 355 C 356 C 357 C ACTIVITY #5 CROSS-COUNTRY SKIING 358 C 359 C 360 C 361 FUNCTION U5(X.I.J.N) 362 IMPLICIT REAL*8 (A-H.O-Z) 363 DIMENSION SNOW( 10. 10).SLOPE( 10, 10),LAKES( 100). TRAILSO) 364 INTEGER X(100) 365 REAL K(5) 366 COMMON /DATA/ SNOW,SLOPE.TOWNI.TOWNJ,LAKES.NLAKES 367 COMMON /MAXTRL/ TRAILS 368 DATA K/O.271,0.198.0.293,0.238/ 369 N1 = 1 370 IM1 = I - 1 371 IP1 = 1 + 1 372 JM1 = J - 1 373 JP1 = J + 1 374 DO 10 II-IM1.IP1 375 DO 10 JJ-JM1.JP1 193 388 389 390 391 376 IF (( I I .LT.1).OR.( I I .GT.10).OR.(JJ.LT.1).OR.(JJ.GT.10)) GOTO 10 377 IF (( I I .EQ.I).AND.(JJ.EQ.J)) GO TO 10 378 ISO = 10*(II-1) + JJ 379 M0D1 = (X(IS0)-1)/3 380 M0D2 = X(IS0)-3*M0D1 381 IF (M0D2.E0.2) N1 = N1 + 1 382 10 CONTINUE 383 C 384 C Travel time at 70 KM/H on highway and 40 KM/H on t r a i l . 385 C 386 DHWY = DABS(TOWNI-I) 387 DTRL = DABS(TOWNJ-J) X16 = DHWY*70./60. + DTRL*40./60. X17 = 1.7*N1 +3.1 IF (X17.LT.TRAILS(5)) X 17=TRAILS(5) IF (X17.GT.TRAILS(5)) TRAILS(5)=X17 392 X18 = SLOPE(I.J) 393 X19 = SNOW(I.J) 394 U5 = K( 1 )*UF(16.X16) + K(3)*UF(18.X18) + K(4 )*UF(19.X19) 395 RETURN 396 END 397 C 398 C 399 C ACTIVITY #6 SNOWSHOEING 400 C 401 C 402 C 403 FUNCTION U6(X,I,J.N) 404 IMPLICIT REAL*8 (A-H.O-Z) 405 DIMENSION SNOW(10.10).SLOPE(10.10).LAKES(100). TRAILSO) 406 INTEGER X(100) 407 REAL K(5) 408 COMMON /DATA/ SNOW.SLOPE,TOWNI.TOWNJ.LAKES.NLAKES 409 COMMON /MAXTRL/ TRAILS 410 DATA K/O.152,0.382.0.255.0.211/ 411 N1 = 1 412 IM1 = I - 1 413 IP1 = 1 + 1 414 JM1 = J - 1 415 JP1 = J + 1 416 DO 10 II=IM1,IP1 417 DO 10 JJ=JM1,JP1 418 IF (( I I .LT.1).OR.( I I .GT.10).OR.(JJ.LT.1).OR.(JJ.GT.10)) GOTO 10 419 IF (( I I .EO.I).AND.(JJ.EO.J)) GO TO 10 420 ISO = 10*(II-D + JJ 421 M0D1 = (X(IS0)-1)/3 422 M0D2 = X ( I S0 ) -3»M0D1 423 IF (M0D2.EQ.2) N1 = N1 + 1 424 10 CONTINUE 425 C 426 C 427 C 428 DHWY = DABS(TOWNI-I) 429 DTRL • DABS(TOWNJ-J) X20 = DHWY*70./60. + DTRL*40./60. X21 = 1 .7»N1 + 3.1 IF (X21 .LT.TRAILSO)) X21 =TRAILS(6) IF (X21 .GT.TRAILSO)) TRAILSO) =X2 1 434 X22 = SNOW(I.J) 435 X23 = SLOPE(I.J) 436 U6 = K (1 )»UF(20.X20) * K(3)*UF(22.X22) + K(4)*UF(23,X23) 437 RETURN 438 END 439 C 440 C 441 C ACTIVITY #7 HIKING 442 C 443 C 444 C 445 FUNCTION U7(X.I.J,N) 430 431 432 433 Travel time at 70 KM/H on highway and 40 KM/h on t r a i l . 446 447 448 INTEGER X(100) IMPLICIT REAL*8 (A-H.O-Z) DIMENSION SNOW(10.10),SLOPE(10.10),LAKES(100) 450 COMMON^/DATA/ SNOW.SLOPE.TOWNI.TOWNJ.LAKES.NLAKES 451 DATA K/O.512.0.488.0.0.0.0/ 194 453 C Travel time at 70 KM/H on highway and 40 KM/H on t r a i l . 454 C 455 DHWY = DABS(TOWNI-I) 45S DTRL = DABS(TOWNd-J) 457 X24 » DHWY*70./G0. + DTRL*40./60. 458 X25 = 1000000.0 459 IF (NLAKES.EO.0) GOTO 11 460 DO 10 11=1,NLAKES 461 IL = (LAKES(II) - D/10 + 1 462 JL - LAKES(II) - ( IL-1)*10 463 DX = DSQRT(1.DO*(I-IL)*(I-IL) + ( J -dL ) * ( J - JL ) ) 464 IF (XD.LT.X25) X25 = XD 465 10 CONTINUE 466 11 CONTINUE 467 U7 = K(1 )*UF(24,X24) + K(2)*UF(25,X25) 468 RETURN 469 END 470 C 471 C 472 C ACTIVITY #8 HORSEBACK RIDING 473 C 474 C 475 C 476 FUNCTION U8(X.I.J.N) 477 IMPLICIT REAL*8 (A-H.O-Z) 478 DIMENSION SNOW(10.10),SLOPE(10,10),LAKES(100). TRAILSO) 479 INTEGER X(100) 480 REAL K(5) 481 COMMON /DATA/ SNOW,SLOPE,TOWNI,TOWNJ,LAKES.NLAKES 482 COMMON /MAXTRL/ TRAILS 483 DATA K/0.279,0.288,0.433,0.0/ 484 N1 = 1 485 IM1 = I - 1 486 IP1 = 1 + 1 487 JM1 = J - 1 488 JP1 = J + 1 489 DO 10 II=IM1,IP1 490 DO 10 JJ=JM1,JP1 491 IF ( ( I I .LT.1).0R.( I I .GT.1O).0R.(JJ.LT.1).0R.(JJ.GT.1O)) GOTO 10 492 IF (( I I .EO.I).AND.(JJ.EO.J)) GO TO 10 493 ISO = 10*(II-1) + JJ 494 M0D1 = (X(IS0)-1)/3 495 IF (M0D1.E0.3) N1 = N1 + 1 496 10 CONTINUE 497 C 498 C Travel time at 70 KM/H on highway and 40 KM/H on t r a i l . 499 C 500 DHWY = DABS(TOWNI-I) 501 DTRL = DABS(TOWNJ-J) 502 X26 = DHWY*70./60. + DTRL*40./60. 503 X27 = 1 .7»N1. + 3.1 504 IF (X27.LT.TRAILS(8)) X27=TRAILS(8) 505 IF (X27.GT.TRAILS(8)) TRAILS(8)=X27 506 X28 = SLOPE(I.J) 507 U8 = K(1)*UF(26,X26) 508 RETURN 509 END 510 C 511 C 512 C ACTIVITY #9 SUMMER MOTORIZED CAMPING 513 C 514 C 515 C 516 FUNCTION U9(X.I,J,N) 517 IMPLICIT REAL'S (A-H.O-Z) 518 DIMENSION SNOW(10,10),SLOPE(10.10),LAKES(100) 519 INTEGER X(100) 520 REAL K(5) 521 COMMON /DATA/ SNOW,SLOPE.TOWNI.TOWNJ,LAKES.NLAKES 522 DATA K/0.371,0.229,0.400,0.0/ 523 C 524 C Travel time at 70 KM/H on highway and 40 KM/h on t r a i l . 525 C 526 DHWY « DABS(TOWNI-I) 527 DTRL = DABS(TOWNJ-d) 195 528 529 530 531 532 533 534 535 536 1C 537 1 538 539 540 541 542 C 543 C 544 C 545 546 C 547 C 548 C 549 c 550 c 551 552 553 554 c 555 c 556 c 557 c 558 559 560 1 561 c 562 c 563 c 564 565 566 2 567 568 569 c 570 c 571 c 572 c 573 574 575 576 3 577 c 578 c 579 c 580 c 581 582 583 584 c 585 c 586 c 587 588 c 589 c 590 c 591 592 593 594 595 596 c 597 c 598 c 599 600 X29 = DHWY*70./60. + DTRL*40./60. X30 = 1000000.0 IF (NLAKES.EO.0) GOTO 11 DO 10 11=1.NLAKES IL = (LAKES(II) - 1)/10 • 1 JL = LAKES(II) - ( IL-1)*10 DX = DSQRT(1.DO*(I-IL)*(I-IL) + ( J - J L ) * ( J - J L ) ) IF (XD.LT.X30) X30 = XD CONTINUE CONTINUE X31 = SLOPE ( I .J) U9 = K(1)*UF(29,X29) + K(3 ) *UF(31.X31) RETURN END FUNCTION UF(I.X) This funct ion evaluates the u t i l i t y for one of the the 31 a t t r i b u t e s . I Is the a t t r i b u t e number. IMPLICIT REAL*8 (A-H.O-Z) DIMENSION UTIL(31.9).XR(31,9) COMMON /UTILTY/ UTIL.XR Find a range of input X-values which surround the a t t r i b u t e value. DO 10 d=1,9 IF (X.LE.XR(I.J)) GOTO 20 CONTINUE If no range is found, assign the last u t i l i t y . UF = UTIL(I,9) RETURN CONTINUE IF (J.EO.1) GOTO 30 JM1 = J-1 U t i l i t y is assigned by a s t ra ight l i ne approximation between the surrounding data po ints . UF = UTIL(I,JM1) + (X-XR(I,JM1))/(XR(I,J)-XR(I,JM1))* + (UTIL(I.J) -UTIL(I,JM1)) RETURN 30 CONTINUE If the a t t r i b u t e value i s less than the f i r s t Input X-value, assign the f i r s t u t i l i t y . UF = UT ILU . 1) RETURN END SUBROUTINE GETDAT This subroutine reads the intut data from f i l e s . IMPLICIT REAL*8 (A-H.O-Z) DIMENSION SNOW(10,10), SLOPE(10.10). LAKES(100) DIMENSION UTIL(31.9) , XR(31,9) COMMON /DATA/ SNOW,SLOPE,TOWNI.TOWNJ,LAKES,NLAKES COMMON /UTILTY/ UTIL.XR I n i t i a l i z e a l l arrays f i r s t . DD 5 1 = 1.10 DO 5 J -1 .10 1 9 6 601 SNOW(I.J) = 0.0 602 5 SLOPE(I.J) = 0.0 603 DO 6 1=1.100 604 6 LAKES(I) = O 605 NLAKES = 0 606 C 607 C Read snow depths from uni t 1 and slopes from uni t 2. 608 C 609 DO 10 1 = 1. 10 610 READ (1,100.END=8) (SN0W(I,J).J=1. 10) 611 8 CONTINUE 612 READ (2.100.END=9) (SLOPE(I,J).J =1.10) 613 9 CONTINUE 614 100 FORMAT (10F8.2) 615 10 CONTINUE 616 C 617 C Read squares conta in ing water from unit 3. 618 C Note : Top row is #001 - #010. 619 C Bottom row 1s #091 - #100. 620 C 621 DO 20 1=1,100 622 READ (3.200,END=25) LAKES(I) 623 20 CONTINUE 624 200 FORMAT (13) 625 I = 101 626 25 CONTINUE 627 NLAKES = 1-1 628 C 629 C Read u t i l i t y graph points from unit 4. 630 C 631 DO 30 1=1.31 632 READ (4.300,END=30) (XR(I,J),UTIL(I,J).J=1.9) 633 30 CONTINUE 634 300 FORMAT (18F8.2) 635 C 636 C Read p o s i t i o n of town from terminal ( i n t e r a c t i v e l y ) . 637 C Note : Input should be 1n real numbers (e.g. 1.0,3.0). 638 C 639 WRITE (6,400) 640 READ (5,401) TOWNI,TOWNJ 641 400 FORMAT (' ENTER COORDINATES OF TOWN (1,1 IS IN TOP LEFT CORNER' 642 + . ' OF GRID SQUARE)') 643 401 FORMAT (2F10.3) 644 RETURN 645 END End of f i l e 197 APPENDIX Procedure for Using the UTILVAL.S Algorithm 198 PROCEDURE FOR USING THE UTILVAL.S ALGORITHM Using UTILVAL.S, the user i s able to examine the u t i l i t y values for each a c t i v i t y for every g r i d square of a recreation plan. The algorithm does not include t r a i l lengths i n the u t i l i t y calculations and c o n f l i c t attributes have a u t i l i t y value of 1.0 (no c o n f l i c t s because a l l g r i d squares are allocated the same a c t i v i t y ) . The following steps i l l u s t r a t e how to use UTILVAL.S: Step 1 An object f i l e i s created, f o r example UTILVAL.O. The algorithm UTILVAL.S i s compiled into UTILVAL.O by the following command: $RUN *FTN SCARDS=UTILVAL.S SPUNCH=UTILVAL.O Step 2 A f i l e i s created, for example f i l e OUT, for storing the re s u l t i n g u t i l i t y values. Step 3 The following command begins the UTILVAL.S program: $RUN UTILVAL.O 1=SN0W1 2=SL0PES1 3=LAKES 4=UTILITY 8=OUT where units 1, 2, 3, and 4 correspond to f i l e s containing snow depths, slope values, drinking water source locations and a t t r i -bute u t i l i t y function data points respectively. Unit 8 contains the f i l e which stores the re s u l t i n g u t i l i t y values. The following steps i l l u s t r a t e what i s displayed on the terminal and what i s required as input using UTILVAL.S: Step 4 DISPLAY: ENTER COORDINATES OF TOWN (1,1 i s i n top l e f t corner of g r i d box): INPUT: -20.0,7.0 (an example town location) DISPLAY: WHICH ACTIVITY DO YOU WISH TO ANALYZE? Step 5 The following numbers correspond to the respective a c t i v i -t i e s : 1 = T r a i l b i k i n g 6 = Snowshoeing 2 = Four-Wheel Driving 7 = Hiking 3 = Snowmobiling 8 = Horseback Riding 4 = Downhill Skiing 9 = Summer Motorized Camping 5 = Cross-Country Skiing 0 = To get out of program (MTS) DISPLAY: WHICH ACTIVITY DO YOU WISH TO ANALYZE? INPUT: 1 (an example a c t i v i t y ) DISPLAY: WHICH ACTIVITY DO YOU WISH TO ANALYSE? (The u t i l i t y values are being stored i n f i l e OUT) INPUT: 0 DISPLAY: # (MTS mode) Step 6 The following command l i s t s the u t i l i t y values on the screen $LIST OUT The following command p r i n t s the values with the XEROX 9700: $LIST OUT *PRINT* 200 APPENDIX U t i l i t y Values Generated by the UTILVAL.S Algorithm for Each A c t i v i t y for Each Grid Square TABLE U T I L I T Y VALUES GENERATED BY FOR EACH ACTIVITY FOR 201 6. THE UTILVAL.S ALGORITHM EACH GRID SQUARE Summary of U t i l i t y Functions for A c t i v i t y 1 (TRAILBIKING) 0.60423 0.60496 0.60295 0.33728 0.33528 0.63432 0.33400 0.33473 0.33272 0.33345 0.33145 0.33218 0.33017 0.33090 0.32890 0.32962 0.32762 0.32835 0.32634 0.32707 0.33929 0.67302 0.60164 0.67174 0.64753 0.60386 0.33546 0.60259 0.44865 0.64571 0.66591 0.60004 0.62994 0.59876 0.62867 0.59748 0.59270 0.59621 0.59420 0.59493 0.65154 0.60787 0.60587 0.60660 0.60459 0.60532 0.60332 0.60405 0.60204 0.60277 0.60076 0.60149 0.59949 0.66682 0.59821 0.44701 0.59694 0.44573 0.59566 0.44446 0.60860 0.60787 0.60733 0.60660 0.60605 0.64972 0.60477 0.60127 0.67010 0.33637 0.64662 0.33509 0.33455 0.33382 0.33327 0.33254 0.33199 0.33127 0.33072 0.32999 0.34074 0.34002 0.33947 0.60236 0.33819 0.33746 0.33692 0.33619 0.33564 0.33491 0.33436 0.33364 0.33309 0.49886 0.33181 0.62940 0.33054 0.44428 0.54779 0.63933 Summary of U t i l i t y Functions for A c t i v i t y 2 (FOUR-WHEEL DRIVING) 0. 29175 0. 29208 0. 29242 0. 34137 0. 32549 0 0. 29117 0. 29150 0. 37286 0. 34078 0. 29250 0 0. 29058 0. 35574 0. 32366 0. 29158 0. 29192 0 0. 65300 0. ,65333 0. 65367 0. ,29100 0. 29133 0 o. .65242 0. ,65275 0. ,57746 O. 32283 0. ,29075 0 0. .28883 0. .28917 0. 33812 0. 28983 0. ,29017 0 o. .28825 O 28858 0. .35374 0. .28925 0. 28958 0 0. .28767 0 .28800 0. .35315 0 .28867 0. ,28900 0 0 .28708 0, .28742 0 .36878 0 .28808 0. . 28842 0 0 .28650 0 .64983 0 .28717 0 .28750 0 .28783 0 29342 0.29375 0.29342 0.65608 0.65575 29283 0.29317 0.29283 0.65550 0.37319 29225 0.29258 0.32466 0.65492 0.65458 29167 0.29200 0.37269 0.65433 0.65400 .29108 0.34003 0.65408 0.65375 0.65342 29050 0.32324 0.65350 0.65317 0.65283 .33853 0.65325 0.65292 0.65258 0.47075 57671 0.65267 0.65233 0.47050 0.35349 .57612 0.65208 0.65175 0.28842 O.57546 .57554 0.65150 0.46967 0.40451 0.31991 Summary of U t i l i t y Functions for A c t i v i t y 3 (SNOWMOBILING) 0.50851 0.50403 O.41695 0.65530 0.64959 0.51636 0.29633 0.70697 0.66315 0.63542 0.30026 0.62386 0.64700 0.64775 0.65305 0.37202 0.35637 0.39397 0.65559 0.66089 0.32098 0.35244 0.60454 0.68348 O.66370 0.41099 0.42367 0.70855 0.51075 0.51300 0.40707 0.41975 0.71479 0.50683 0.66564 0.40315 0.41582 0.71087 0.50291 0.66171 0.38944 0.40146 0.71654 0.49899 0.66823 0.39530 0.40798 0.63663 0.49506 0.49730 0.632B7 0.48161 0.65310 0.40798 0.37267 0.64072 0.63624 0.49394 0.41582 0.68292 0.64856 0.49730 0.68037 0.40345 0.40793 0.65641 0.50515 0.71900 0.41129 0.41578 0.51748 0.69281 0.42444 0.41410 0.39228 0.67180 0.69607 0.39284 0.37314 0.38836 0.69113 0.39116 0.37146 0.35132 O.55996 0.61907 0.38724 0.36753 0.34740 0.58017 0.61515 0.38332 0.36361 0.31425 0.49794 0.60825 0.38918 0.35969 0.62233 0.55115 Summary of U t i l i t y Functions for A c t i v i t y 4 (DOWNHILL SKIING) 0.48022 0.48166 0.34902 0.49041 0.45106 0.47772 0.53708 0.47848 0.48790 0.40485 0.54875 0.53421 0.47249 0.40O91 0.36959 0.79615 0.80731 0.78864 0.39840 0.36708 0.81310 0.80480 0.68652 0.40684 0.36457 0.41246 0.36626 0.33960 0.20998 0.21142 0.40995 0.59786 0.35894 0.20747 0.32204 0.40745 0.59535 0.35643 0.20497 0.31954 0.51268 0.74973 0.37577 0.20246 0.26939 0.63654 0.63532 0.25322 0.19995 0.20138 0.37604 0.22725 0.29089 0.66219 0.79760 0.37353 0.37497 0.22288 0.65969 0.53160 0.37102 0.22181 O.37721 0.74233 0.74089 0.36852 0.21930 0.44025 0.73982 0.73839 0.21536 0.43299 0.70123 0.73731 0.78756 0.32599 0.37112 0.78792 0.79686 0.78506 0.42654 0.78685 0.79579 0.80408 0.64966 0.63161 0.78434 0.79328 0.83371 0.53426 0.62910 0.78183 0.79077 0.87306 0.71748 0.53064 0.76039 0.82040 0.58832 0.48555 202 Summary of U t i l i t y Functions for A c t i v i t y 5 (CROSS-COUNTRY SKIING) 0.63796 0.63977 0.36233 0.53436 0.57102 0.63480 0.40480 0.45668 0.53120 0.60042 0.39984 0.51179 0.57314 0.59545 0.59142 0.37917 0.38681 0.37657 0.59229 0.58826 0.38768 0.38365 0.40525 0.55074 0.58510 0.34913 0.34471 0.49302 0.38958 0.39138 0.34597 0.34155 0.45934 0.38641 0.57294 0.34281 0.33839 0.45618 0.38325 0.56978 O.34548 0.34145 0.42249 0.38009 0.56040 0.33648 0.33207 0.51644 0.37693 0.37874 0.59955 0.41080 0.58750 0.36594 0.38786 0.59639 0.59820 0.40584 0.36278 0.47015 0.59323 0.40448 0.55484 0.37167 0.36987 0.59007 0.40132 0.44994 0.36851 0.36671 0.39635 0.51546 0.36132 0.36535 0.37521 0.57791 0.54716 0.37566 0.3B008 0.37205 0.50733 0.37431 0.37872 0.38275 0.44038 0.38952 0.37115 0.37556 0.37959 0.50543 0.38636 0.36798 0.37240 0.38226 0.41813 0.33395 0.35899 0.36924 0.44279 0.57032 Summary of U t i l i t y Functions for A c t i v i t y 6 (SNOWSHOEING) 0.34590 0.34674 0.39159 0.47746 0.50950 0.34442 0.22019 0.45265 0.47599 0.53905 O.21787 0.39876 0.47246 0.53673 0.55836 0.28932 0.27449 0.30773 0.53525 0.55688 0.25649 0.27301 0.33263 0.52353 0.55540 0.36034 0.38336 0.54866 0.34104 0.34189 0.35886 0.38188 0.52119 0.33957 0.57324 0.35738 0.38040 0.51971 0.33809 0.57176 0.33512 0.35675 0.48527 0.33661 0.59246 0.35443 0.37744 0.50960 0.33513 0.33598 0.56216 0.35097 0.60512 0.39328 0.31301 0.56068 0.56153 0.34864 0.39180 0.41703 0.55920 0.34801 0.54896 0.34736 0.34652 0.55773 O.34653 0.47301 0.34588 0.34504 0.34421 0.51055 0.36604 0.34440 O.30710 0.57556 0.54537 0.30731 0.28974 0.30562 0.50675 0.30667 0.28911 0.27259 0.25405 0.36719 0.30520 0.28763 0.27111 0.35148 0.36571 0.30372 0.28615 0.25396 0.23791 0.32751 0.31792 0.28467 0.30523 0.39003 Summary of U t i l i t y Functions for A c t i v i t y 7 (HIKING) 0.94240 0.93742 0.93244 0.92747 0.92249 0.91751 0.91253 0.90756 0.90258 0.89760 0.94524 0.94027 0.93529 0.93031 0.92533 0.92036 0.91538 0.91040 0.90542 0.90044 0.94809 0.94311 0.93813 0.93316 0.92818 0.92320 0.91822 0.91324 0.90827 0.90329 0.95093 0.94596 0.94098 0.93600 0.93102 0.92604 0.92107 0.91609 0.91111 0.90613 0.95378 0.94880 0.94382 0.93884 0.93387 0.92889 0.92391 0.91893 0.91396 0.90898 0.95662 0.95164 0.94667 0.94169 0.93671 0.93173 0.92676 0.92178 0.91680 0.91182 95947 95449 94951 94453 93956 93458 92960 92462 91964 91467 95662 95164 94667 94169 93671 93173 92676 92178 91680 91 182 95378 94880 94382 93884 93387 92889 92391 91893 91396 90898 0.95093 0.94596 0.94098 0.93600 0.93102 0.92604 0.92107 0.91609 0.91111 0.90613 Summary of U t i l i t y Functions for A c t i v i t y 8 (HORSEBACK RIDING) 0.25249 0.25311 0.25373 0.59715 0.68797 0.25141 0.25203 0.44750 0.59606 0.25389 0.25032 0.51384 0.68456 0.25218 0.25280 0.24924 0.24986 0.25048 0.25110 0.25172 O.24815 0.24877 0.31307 0.68301 0.25063 0.24707 0.24769 0.59110 0.24893 0.24955 0.24598 0.24660 0.51012 0.24784 0.24846 O.24490 O.24552 0.50903 0.24676 O.24738 0.24381 0.24443 0.43990 0.24567 0.24629 O.24273 0.24335 0.24397 0.24459 0.24521 0.25559 O.25621 0.25559 O.25497 O.25435 0.25451 0.25513 0.25451 0.25389 0.44812 0.25342 0.25404 0.68642 0.25280 0.25218 0.25234 0.25296 0.44719 0.25172 0.25110 0.25125 0.59467 0.25125 0.25063 0.25001 0.25017 0.68379 0.25017 0.24955 0.24893 0.59188 0.24970 0.24908 0.24846 0.34336 0.31168 0.24862 0.24800 0.24738 0.50965 0.31059 0.24753 0.24691 0.24629 0.30935 0.30951 0.24645 0.24583 0.38593 0.67759 Summary of U t i l i t y Functions for A c t i v i t y 9 (SUMMER MOTORIZED CAMPING) 0.97681 0.96651 0.55620 0.64590 0.70702 0.99485 0.58454 0.64566 0.66393 0.95362 0.59614 0.68494 0.76370 0.98197 0.97166 0.59072 0.59382 0.59691 1.00000 0.98969 0.58531 0.5B841 0.62007 0.76602 0.99768 0.57990 0.58300 0.68609 0.98918 0.99227 0.57449 0.57759 0.66639 0.98377 0.98686 0.56908 0.57217 0.66098 0.97836 0.98145 0.56367 0.56676 0.64128 0.97295 0.97604 0.55826 0.56135 O.96445 0.96754 0.97063 0.92528 0.91498 0.92528 0.53559 0.54590 0.94332 0.93301 0.94332 0.55362 0.63536 0.96135 0.95105 0.73278 0.57166 0.58197 0.97939 0.96908 0.65082 O.58969 0.60000 0.99742 0.68712 0.59742 0.59768 0.59459 0.99536 0.76988 0.59536 0.59227 0.58918 0.68995 0.59304 0.58995 0.58686 0.62663 0.61311 0.58763 0.58454 0.58145 0.66407 0.60770 0.58222 0.57913 0.57604 0.60152 0.60229 0.57681 0.57372'0.62777 0.73897 APPENDIX The OUTSPLIT.S Algorithm THE OUTSPLIT.S ALGORITHM (Computer Language = FORTRAN) This program s p l i t s the output matrix from a land use plan into the ind iv idual a c t i v i t i e s . The output shows the 10x10 planning area for each a c t i v i t y with each square marked by a zero (0) i f the a c t i v i t y i s not a l l o c a t e d to that square and one (1) i f the a c t i v i t y is a l l o ca ted to that square. Var iab les used: X - Double p rec i s i on real matrix to contain the land use values. M - Integer array to contain the rounded values of X. 0UT1, 0UT2. 0UT3, 0UT4, 0UT5. 0UT6 -Six 100 element arrays which w i l l be set to 0/1 for each of the s ix c lasses of a c t i v i t i e s . REAL*8 X(100) INTEGER M(100) INTEGER 0UT1(100),0UT2(100),0UT3(100),0UT4(100),0UT5(100) INTEGER 0UT6(100) Read values from land use plan f i l e . READd) N, (X(I),I = 1,N) Round var iab les to integer. DO 10 1=1.100 M(I) = X(I)+0.5 CONTINUE Analyze each of the 100 var iab les in the land use plan. DO 20 K=1,100 0UT1(K) = 0 0UT2(K) = 0 0UT3(K) » 0 0UT4(K) = 0 0UT5(K) = 0 0UT6(K) = 0 M0D1 = (M(K)-1)/3 M0D2 • M(K) - 3»M0D1 Is TB a l located? IF ((MOD 1.EO.0).OR.(MOD 1.EO.2)) 0UT1(K) = 1 Is 4x4 a l located? IF ((M0D1.E0.1).0R.(M0D1.E0.2)) 0UT2(K) = 1 Is Hiking/Horseback Riding/Camping a l located? IF (M0D1.EQ.3) 0UT3(K) = 1 Is Snowmobiling a l located? IF ((M0D2.EQ.1).AND.(M(K).NE.16)) 0UT4(K) « 1 205 e i c 62 C Is Snowshoelng/X-C Sk i ing a l located? 63 C 64 IF (MOD2.EQ.2) 0UT5(K) = 1 65 C 66 C Is Downhill Sk i ing a l located? 67 C 6B IF (M(K).EQ.16) OUT6(K) « 1 69 20 CONTINUE 70 C 71 C Begin output. 72 C 73 WRITE (6.1000) 74 1000 FORMAT ( '1 ' .T44. 'LAND USE PLAN ALLOCATION BY ACTIVITY'. 75 + / . / . / , 76 + /.T24,'SUMMER ACTIVITIES',T84,'WINTER ACTIVITIES', 77 + / . T 2 4 , ' ,T84, ' , 78 + / , / , T 2 4 , ' T r a i l b i k i n g ',T86,'Snowmobl11ng') 79 DO 30 1 = 1. 10 80 11 = 10*(I-1) + 1 81 12 = 11+9 B2 WRITE (6.1010) (OUT 1(d),d=11.12),(0UT4(d).d=11,12) 83 1010 FORMAT (/,10X,1014,20X.1014) 84 30 CONTINUE 85 WRITE (6.1020) 86 1020 FORMAT(///.T24, ' 4x4s ',T82,'Snowshoeing/X-C Sk i ing ' 87 DO 40 1=1,10 88 11 = 10*(I-1) + 1 89 12 = 11+9 90 WRITE (6.1010) (0UT2(d),d=I1.12),(0UT5(d).d=I1,12) 91 40 CONTINUE 92 WRITE (6,1030) 93 1030 FORMAT ( ' 1 ' , / . 94 + '1',T24,'SUMMER ACTIVITIES',T84,'WINTER ACTIVITIES', 95 + / , T 2 4 , ' ' . T 8 4 , ' . 96 + / , / ,T17, 'Hik ing/Horseback R1ding/Camp1ng',T84,'Downhl11 97 +Sk11ng') 98 DO 50 1 = 1, 10 99 11 = 10*(I-1) + 1 100 12 = 11+9 101 WRITE (6,1010) (0UT3(J),d=I1.12).(0UT6(d),d=I1,12) 102 50 CONTINUE 103 STOP 104 END End of f i l e APPENDIX P r o c e d u r e f o r U s i n g the OUTSPLIT.S A l g o r i t h m 207 PROCEDURE FOR USING THE OUTSPLIT.S ALGORITHM The a l g o r i t h m OUTSPLIT.S a l l o w s the u s e r t o see which a c t i v i t i e s are a l l o c a t e d t o each g r i d square f o r a g i v e n r e c r e a t i o n l a n d use p l a n . The f o l l o w i n g s t eps i l l u s t r a t e to pr oc e dure f o r u s i n g O U T S P L I T . S : Step 1 An o b j e c t f i l e i s c r e a t e , f o r example , O U T S P L I T . O . The a l g o r i t h m OUTSPLIT.S i s c o m p i l e d i n t o OUTSPLIT.O by the f o l l o w -i n g command: $RUN *FTN SCARDS=OUTSPLIT.S SPUNCH=OUTSPLIT.0 Step 2 To run OUTSPLIT.S and t o p r i n t the r e s u l t s on paper w i t h the XEROX 9700, the f o l l o w i n g command i s e n t e r e d : $RUN OUTSPLIT.O 1=S0LN1 6=*PRINT* where SOLNl i s a f i l e w i t h the l a n d use p l a n which i s t o be s p l i t i n t o a c t i v i t y a l l o c a t i o n s . 

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