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Optimum turnout spacing on forest haul roads Anderson, Dennis Ivar 1980

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OPTIMUM TURNOUT SPACING ON FOREST HAUL ROADS by DENNIS IVAR ANDERSON B.S.F., University of B r i t i s h Columbia, 1975 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF FORESTRY in the FACULTY OF GRADUATE STUDIES Department of Forestry We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1980 © Dennis Ivar Anderson, 1980 In presenting t h i s thesis in p a r t i a l fulfilment 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 available for reference and study. . I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publication 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 cf Forestry The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date j 2 k ^ ^ - ^ - ^ ^ _ — ABSTRACT Mathematical models are developed to determine the optimum spacing of turnouts and to predict the time l o s t due to the acceleration and deceleration of the vehicle and the time the vehicle spends in the. turnout. Previous a r t i c l e s have not completely defined a method of deriving or measuring the delay i n the turnout attributable to turnout spacing. The concept of the expected F-factor, a measurement of the expected delay i n the turnout, i s introduced. The expected F-factor i s the expected distance the loaded vehicle i s from the empty vehicle, once the empty vehicle has come to a complete halt i n the turnout, divided by the turnout spacing. . Two forms of the expected F-factor equation were developed. The results shew that the t o t a l expected delay time attributable to turnout spacing may be a s i g n i f i c a n t part of the t r a v e l empty time ( i . e . , 20 percent) but i t s sig n i f i c a n c e i s reduced when compared to the round t r i p time. The optimum turnout spacing model i s concerned with minimizing the sum of the turnout construction and maintenance costs and the cost of delays a t t r i b u t a b l e to turnout spacing.. I f the res u l t s of the optimum turnout spacing model are used in the i n i t i a l design of the.road network then the t o t a l potential savings can be important. Implementation of the optimum turnout spacing model can be achieved with the .uti l i z a t i o n of tables. . i i i These tables can be u t i l i z e d as a guide i n the design and construction of forest haul roads. TABLE OF CONTENTS PAGE LIST OF TABLES v i i LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . y i i i ACKNOWLEDGEMENT X 1.0 INTBODOCTION ........................................ . 1 2.0 DEVELOPMENT OF THE OPTIMUM SPACING MODEL ............ 5 2.1 FORMATION OF THE PROBLEM . . 5 2.2 THE ASSUMPTIONS OF THE MODEL ..................... 7 2.3 THE TURNOUT DELAY TIME 10 2.4 THEORETICAL DEVELOPMENT OF THE EXPECTED F-FACTOR . . 14 2.4.1 ONE LOADED VEHICLE MEETING ONE EMPTY VEHICLE 16 2.4.2 ONE EMPTY VEHICLE MEETING A FLEET OF LOADED VEHICLES 19 2.4.3 A FLEET OF LOADED VEHICLES MEETING A FLEET OF EMPTY VEHICLES 24 2.4.4 HEADWAY PROBABILITY DISTRIBUTIONS ........... . 30 2.4.4.1 UNIFORM ARRIVAL DISTRIBUTION ............. 32 2.4.4.2 EXPONENTIAL HEADWAY DISTRIBUTION ........ 33 2.4.4.3 ERLANG HEADWAY DISTRIBUTION ............ 34 2.4.4.4 PEARSON TYPE III HEADWAY DISTRIBUTION ... 35 2.5 DEVELOPMENT OF THE COST EQUATION 37 2.6 DISCOUNTING OF THE COST EQUATION ................. 40 3.0 THE ARRIVAL DISTRIBUTION OF LOGGING TRUCKS 42 3.1 ARRIVAL DATA FROM TWO OPERATIONS ................. 42 3.2 ANALYSIS OF THE ARRIVAL DATA 44 4 . 0 S I M U L A T I O N F O B T H E V E R I F I C A T I O N O F T H E E X P E C T E D F -F A C T O R E Q U A T I O N S . . 5 1 4 . 1 T H E S I M U L A T I O N M O D E L S 5 2 5 . 0 T H E C O S T V A R I A B L E S A N D T H E M O D I F I C A T I O N O F T H E E X P E C T E D F - F A C T O R 5 9 5 . 1 T H E C O S T O F T R U C K T R A N S P O R T A T I O N .. 5 9 5 . 2 T H E T U R N O U T C O N S T R U C T I O N A N D M A I N T E N A N C E C O S T S . . . 6 2 5 . 3 M O D I F I C A T I O N O F T H E E X P E C T E D F - F A C T O R E Q U A T I O N S . . . 6 6 6 . 0 T H E U S E A N D T E S T I N G O F T H E M O D E L 6 8 6 . 1 T H E U S E O F T H E M O D E L 6 8 6 . 2 S E N S I T I V I T Y A N A L Y S I S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5 6 . 2 . 1 V E L O C I T Y O F T H E L O A D E D V E H I C L E 8 7 6 . 2 . 2 V E L O C I T Y O F T H E E M P T Y V E H I C L E . . . . . . . . . . . . . . . 9 3 6 . 2 . 3 T R A F F I C F L O W R A T E 9 4 6 . 2 . 4 E X P E C T E D U S E F U L L I F E O F T H E R O A D 9 6 6 . 2 . 5 T U R N O U T C O N S T R U C T I O N C O S T . . . . . . . . . . . . . . . . . . . 9 7 6 . 2 . 6 A D J U S T E D T R U C K H A U L I N G C O S T . . . . . . . . . . . . . . . . . . 9 8 6 . 2 . 7 T H E A C C E L E R A T I O N A N D D E C E L E R A T I O N O F T H E E M P T Y V E H I C L E 9 9 6 . 2 . 8 T H E D I S C O U N T R A T E 1 0 1 6 . 2 . 9 T H E M A I N T E N A N C E C O S T , , 1 0 3 6 . 2 . 1 0 T H E D E R I V A T I V E O F T H E E X P E C T E D F - F A C T O R . . . . . 1 0 5 6 . 2 . 1 1 T H E L E N G T H O F T H E L O A D E D V E H I C L E . . . . . . . . . . . 1 0 6 6 . 2 . 1 2 T H E H E A D W A Y P R O B A B I L I T Y D I S T R I B U T I O N S . . . . . . 1 0 8 6 . 2 . 1 3 T H E T U R N O U T S P A C I N G A N D T H E O P T I M U M T U R N O U T S P A C I N G 1 10 7 . 0 D I S C U S S I O N A N D C O N C L U S I O N S . . . 1 1 5 7 . 1 D I S C U S S I O N 1 1 5 7 . 2 C O N C L U S I O N S . 1 1 9 v i L I T E R A T U R E C I T E D . 1 2 2 A P P E N D I C E S . 1 2 4 1 R O A D S T A N D A R D S S U R V E Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 5 2 A B B R E V I A T I O N S , S Y M B O L S , A N D U N I T S . . ; i i . . . . . . . . . . . . . . 1 2 8 3 A C C E L E R A T I O N A N D D E C E L E R A T I O N O F A V E H I C L E . 1 3 1 4 A N A L Y S I S O F H E A D W A Y D I S T R I B U T I O N S . . . . . . . . . . . . . . . . . . 1 3 8 5 S I M U L A T I O N O F T H E F - F A C T O R '. . . ' . 1 5 1 6 T H E L E N G T H O F T H E V E H I C L E 1 7 5 7 D E R I V A T I V E S O F T H E E X P E C T E D F - F A C T O R E Q U A T I O N S . 1 8 2 8 O P T I M U M T U R N O U T S P A C I N G C O M P U T E R P R O G R A M . . , , . . . . . . . 1 8 6 9 G R A P H I C A L R E S U L T S O F T H E S E N S I T I V I T Y A N A L Y S I S 1 9 8 1 0 C O N V E R S I O N F A C T O R S . . . . . 2 2 2 v i i LIST OF TABLES TABLE PAGE I Goodness Of F i t T e s t s — C o a s t a l Study ............... . 47 II Goodness Of F i t T e s t s — I n t e r i o r Study—Both Scales .. 50 III F-factor Simulation — The Interaction Between A Single Loaded Vehicle And A Single Empty Vehicle ... 53 IV F-factor Simulation — An Empty Vehicle Meeting Loaded Vehicles Eased On Equations 2.18, 2.19, 2.30, And 2.31 56 V F-factor Simulation — An Empty Vehicle Meeting Loaded Vehicles Based On Equations 2.18, 2.19, 2.33, And 2. 34 . 57 VI F-factor Simulation — Interaction Between Two Fleets Of Vehicles Based On Equations 2.18, 2.19, 2.30, And 2. 3 1 .............. 58 VII Truck Hauling Costs ................................. 61 VIII Excavation Costs Per Turnout 64 IX Turnout Maintenance Costs .......................... 65 v i i i LIST OF FIGURES FIGURE PAGE 1 Interaction Between An Empty Vehicle And A Loaded Vehicle .. 17 2 Diagram Showing The Physical Situation Leading To The C r i t i c a l Headway 20 3 Case 3--An Empty Vehicle.Stops At Turnout S 3 Since Next Turnout Is Occupied ................................... 25 4 Case 4—An Empty Vehicle Remains In Turnout S^ Since Next Turnout Is Occupied 29 5 Frequency Histogram Of Headways For Coastal Study (5-minute Intervals) ..................................... 46 6 Frequency Histogram Of Headways For Interior Study (1-minute Intervals) 48 7 Frequency Histogram Of Headways For I n t e r i o r Study (5-minute Intervals) ......... 49 8 Effect Of T r a f f i c Flow Rate On The Total Expected Delay Time For Various Turnout Spacings ....................... 69 9 Effect Of The Turnout Spacing On The Total Expected Delay Time For Various T r a f f i c Flow Rates ............. 70 10 Effect Of The Velocity Of The Empty Vehicle On The Total Expected Delay Time For Various V e l o c i t i e s Of The Loaded Vehicle 71 11 Effect Of The Velocity Of The Loaded Vehicle On The Total Expected Delay Time For Various V e l o c i t i e s Of The ix Empty Vehicle 72 12 Effect Of The Velocity Of The Loaded Vehicle On The Optimum Turnout Spacing For Various V e l o c i t i e s Of The Empty Vehicle ......... 78 13 Effect Of The Velocity Of The Empty Vehicle On The Optimum Turnout Spacing For Various V e l o c i t i e s Of The Loaded Vehicle 79 14 Effect Of The T r a f f i c Flow Bate On The Optimum Turnout Spacing 80 15 Effect Of The Turnout Construction Cost On The Optimum Turnout Spacing 81 16 Effect Of The Adjusted Truck Hauling Cost On The Optimum Turnout Spacing 82 17 Effect Of The Expected Useful L i f e Of The Road On The Optimum Turnout Spacing ............................... 83 18 Effect Of The Turnout Spacing On The Cost Function Based On Equation 2.38 84 19 Effect Of Perturbations To The Velocity Of The Loaded Vehicle On The Total Expected Delay Time 90 20 Effect Of Perturbations To The Velocity Of The Loaded Vehicle On The Optimum Turnout Spacing• w............... . 91 21 Effect Of Perturbations To The Velocity Of The Loaded Vehicle On The Cost Difference , 92 22 Effect Of Deviations From The Optimum Turnout Spacing On The Maximum Cost Difference ..113 23 Effect Of Deviations From The Optimum Turnout Spacing On The Average Cost Difference 114 X ACKNOWLEDGEMENT I would l i k e to acknowledge the assistance of my supervisor, Mr. G.G. Young, throughout my graduate studies program and in the development of t h i s thesis. I would l i k e to thank M r . . H . . J o l l i f f e . f o r the h e l p f u l suggestions and constructive c r i t i c i s m s i n the i n i t i a l development of the thesis. Furthermore, my gratitude i s also extended to Mr..D. Smith of FEEIC and Dr. P.L..Cottell f or being on my committee. F i n a l l y I would l i k e to thank Mr..L..Henkelman for writing the GPSSV computer program used to simulate the in t e r a c t i o n between a f l e e t of empty vehicles and a f l e e t of loaded vehicles. 1 1.0 INTRODUCTION Logging haul roads are an i n t e g r a l part of the harvesting system. Since hauling costs account for approximately 10 to 20 percent of the t o t a l logging cost, the design of the road networks should be analysed before and during the harvesting process. The hauling cost components are interdependent.. As a company invests more c a p i t a l into road maintenance and construction the log hauling costs should decrease. Consequently, there i s a trade off between the i n i t i a l investment in the road and the r e s u l t i n g hauling cost. This trade o f f i s reflected i n the road standards that are used for the various sections of the road network. Forest companies and government agencies include various features and service conditions of the road in th e i r assembling of the components of road standards. The road standards r e f l e c t the required physical and service c h a r a c t e r i s t i c s of the haul roads.. Some of the components that can be considered for i n c l u s i o n i n a s p e c i f i c road standard are: design speed, maximum adverse and favourable gradients, subgrade width, frequency of turnouts, and right-of-way width.. The decision as to which components should be included in the road standards can be complex and there are no d e f i n i t e c r i t e r i a for solving t h i s problem. The potential design elements must be evaluated to determine which should be included in the road standards. Environmental conditions and user factors should be considered. 2 User factors include use of the road for future timber development, u t i l i z a t i o n of the road by others (e.g., mining and a g r i c u l t u r e ) , and government regulations. Single-lane roads or double-lane roads can be u t i l i z e d i n the extraction of timber.. Single-lane roads are less expensive to construct but the cost of truck delays i n turnouts may be greater than the added investment of constructing and maintaining a double-lane. road. Furthermore, the potential increase in the speed of the vehicles on a double-lane road must be considered. Since road construction and maintenance costs increase as the.turnout spacing i s decreased, there i s a t h e o r e t i c a l optimum range of turnout spacing. It may be desirable to determine t h i s optimum range prior to deciding on whether to b u i l d a single-lane or double-lane road. There have been few publications concerned with the topic of optimum turnout spacing on forest haul roads. 11*^(1965) determined the c r i t i c a l t r a f f i c flow at which the number of lanes should be switched from one to two but he did not calculate or derive an expression to determine the expected turnout delay time. The United States Forest Service developed a manual tc predict the cost of truck and t r a i l e r transportation (Byrne et a l . , 1947). The system involved a very simple method to estimate the l o s t time in minutes per mile. The method assumed there i s a delay i n the turnout equivalent to the time required for a loaded vehicle to t r a v e l one-half turnout spacing but i t f a i l e d to j u s t i f y t h i s assumption. Porpaczy and Waelti(1976) produced an a r t i c l e with similar objectives as I l ' i n . This paper, as i s the case: for the previous a r t i c l e s , does not have a well defined method of 3 deriving or estimating turnout delay times. Since Il'in(1965) and Porpaczy and Waelti(1976) have written papers concerned with determining the number of lanes required for log transportation, t h i s study w i l l concentrate on the problem of deciding on the spacing of turnouts on a s i n g l e -lane road. The major objective of the study i s to develop a method to determine the optimum turnout spacing as a function of t r a f f i c flow rate, speeds of the empty and loaded vehicles, "expected useful l i f e of the road", turnout costs, and trucking costs. . A second objective is to analyse the s e n s i t i v i t y of the solution and test whether turnout spacing has any economic sign i f i c a n c e with respect to the road design problem.. During the f a l l of 1976, the author conducted a mail and telephone survey of road standards (Appendix 1).. Results of t h i s survey revealed that there was extensive variation in the length of turnouts (30 to 150 feet) and the spacing of turnouts (2 to 11 per mile). . This reinforces the need to investigate the spacing of turnouts. The t o t a l time a logging truck spends in turnouts can be measured, but i t i s d i f f i c u l t to determine the portion of t h i s time that i s d i r e c t l y attributable to turnout spacing.. For a number of reasons a logging truck may wait longer i n a given turnout than i s necessary. For example, i f the logging truck operator expects there w i l l be delays further along i n the road network he may decide, to wait longer in a turnout than i s required.. This paper develops a method to calculate the delay time s p e c i f i c a l l y attributable to turnout spacing.. Once the t o t a l delay time attributable to turnout spacing has been determined, the cost of the delays must be evaluated. 4 The relevant costs are those that are functions of the turnout spacing: the turnout construction cost, the turnout maintenance cost, and the delay cost a t t r i b u t a b l e to turnout spacing. Optimum turnout spacing can then be derived from the cost function.. Simulation models are u t i l i z e d to v e r i f y and te s t the s e n s i t i v i t y of the delay time, costing, and optimum turnout spacing models. Throughout the text Imperial units are u t i l i z e d . A table to convert Imperial units into the International System of Units i s located in Appendix 10. 5 2-.0 DEVELOPMENT OF THE OPTIMUM SPACING MODEL 2sJ. FOEMATION OF THE PROBLEM The d e t e r m i n a t i o n of the optimum number of t u r n o u t s per u n i t d i s t a n c e of road i s concerned with minimizing the sum of turnout c o n s t r u c t i o n and maintenance c o s t s and the c o s t of delays i n t u r n o u t s . The c o n s t r u c t i o n and maintenance c o s t s can be p r e d i c t e d from e x i s t i n g company records or manuals but the delay c o s t i s not so e a s i l y o b t a i n e d . The delay c o s t could be viewed as the t r u c k h a u l i n g c o s t ( d o l l a r s per u n i t time period) m u l t i p l i e d by the estimated time the v e h i c l e s must spend i n the t u r n o u t s . . T h i s estimated time c o u l d be determined by time s t u d i e s , s i m u l a t i o n , or mathematical formulas. I f the a c t u a l time spent i n the t u r n o u t s was u t i l i z e d i n the c a l c u l a t i o n of the delay c o s t an o v e r - e s t i m a t i o n of the delay c o s t would r e s u l t . T h i s occurs s i n c e some of the time spent i n the t u r n o u t i n c l u d e s delays not a t t r i b u t a b l e t o turnout s p a c i n g . I f a d r i v e r r e a l i z e s h i s v e h i c l e must l a t e r queue at the l a n d i n g or a t the dump, then he may d r i v e slower and wait longer i n t u r n o u t s than i f he d i d not expect a delay at these l o c a t i o n s . Furthermore, the t o t a l delay time would i n c l u d e delay s i t u a t i o n s t h a t would not i n f l u e n c e the round t r i p time, s i n c e the empty v e h i c l e would experience a delay a t the l a n d i n g r e g a r d l e s s of the delay i n the t u r n o u t . Consequently, t h i s approach t o determining delay c o s t s would have to e l i m i n a t e the d e l a y s not a t t r i b u t a b l e to turnout s p a c i n g . . Another approach i s to view delay c o s t s i n terms of the 6 number of round t r i p s a v e h i c l e can complete i n a day.. The i n i t i a l s t e p i s to determine the number of complete round t r i p s a given v e h i c l e can achieve f o r a p a r t i c u l a r t u r n o u t s p a c i n g . The i n i t i a l arrangement can be determined from p r e v i o u s experience or from an estimate o f a good turnout s p a c i n g arrangement. . A s i m u l a t i o n , based on d i s p a t c h i n g r u l e s , t r a f f i c behaviour, l o a d i n g times, and unloading times, can be u t i l i z e d to determine the number of complete round t r i p s per day. From t h i s s t a r t i n g arrangement the turno u t spacing can be i n c r e a s e d or decreased t o determine whether the v a r i a t i o n w i l l a l t e r the number o f complete round t r i p s . T h i s process can be repeated f o r a reasonable range of turnout s p a c i n g s . A comparison of the t o t a l t r a n s p o r t a t i o n c o s t per. c u n i t i s then r e q u i r e d to determine the best turnout spacing arrangement. This i s an adequate method provided one can a c c u r a t e l y f o r e c a s t the t r u c k d i s p a t c h i n g schedule. Since i n r e a l i t y t h i s i s not the case, the number of complete round t r i p s per v e h i c l e per day cannot be c o n s i s t e n t l y determined. The b e t t e r of the two approaches appears t o be the f i r s t method provided the delays i n the system not a t t r i b u t a b l e t o turnout s p a c i n g are e l i m i n a t e d . . Since a v e h i c l e may have to queue at the l a n d i n g r e g a r d l e s s of the tu r n o u t s p a c i n g an adjust e d h a u l i n g c o s t i s u t i l i z e d . The approach t o t h i s study w i l l be to l e t the delay c o s t be equal to the adjusted h a u l i n g c o s t m u l t i p l i e d by the turnout spacing delay times t h a t are a t t r i b u t a b l e t o turno u t s p a c i n g . 7 2 . 2 THE ASSUMPTIONS OF THE MODEL The nature of the log hauling process dictates that r i g h t -of-way p r i o r i t y belongs to the logging trucks. Therefore the only vehicles that w i l l be considered i n the delay process are the logging trucks themselves and i t w i l l therefore be assumed that the other vehicles w i l l not hinder the progression of the logging trucks. Furthermore, delays at the landing and dump w i l l not be considered as they are not solely related to turnout spacing but depend on the loading rate, the unloading rate, the dispatching procedure, the t o t a l number of vehicles, and the number of loaders., Depending on the st a r t i n g up and shutting down mode of the company, there may be no turnout delays during the f i r s t few hours and l a s t few hours of the "hauling" day. The various dispatching p o l i c i e s can be reflected i n the model by l i m i t i n g the number of hauling hours per day to the " c o n f l i c t " hours.. The " c o n f l i c t " hours refer to that portion of the day where there i s movement of both loaded and empty vehicles over a section of road.. I t s h a l l be assumed that the empty vehicle w i l l always stop at the turnout which w i l l y i e l d the least delay. A constant turnout construction cost i s assumed. Variation i n turnout construction costs could be p a r t i a l l y accounted for by dividing the road section to be analysed into subsections, each with a p a r t i c u l a r turnout construction cost. The discounting of costs i s included in the model but amortization of road costs w i l l not be considered., A fixed t r a f f i c flow rate w i l l be assumed.. 8 F u r t h e r l i m i t a t i o n s w i l l be added to the model by c o n s t r a i n i n g some of the speed c h a r a c t e r i s t i c s of the v e h i c l e s . The f o l l o w i n g assumptions s h a l l be i n c l u d e d : 1. a l l of the loaded v e h i c l e s t r a v e l at the same cons t a n t v e l o c i t y . 2 . A l l of the empty v e h i c l e s t r a v e l at the same cons t a n t v e l o c i t y except when d e c e l e r a t i n g i n t o a tu r n o u t . . 3 . The s p a c i n g of turno u t s w i l l be s u f f i c i e n t t o always allow the empty v e h i c l e t o a c c e l e r a t e to i t s designated speed before having t o d e c e l e r a t e i n t o a tu r n o u t . . 4. The empty v e h i c l e w i l l a c c e l e r a t e or d e c e l e r a t e at a constant r a t e . The model w i l l be developed i n fo u r p a r t s : 1. d e t e r m i n a t i o n of the t o t a l expected delay time per v e h i c l e per u n i t d i s t a n c e of road 2 . d e t e r m i n a t i o n of the expected delay time i n the turnout 3. d e t e r m i n a t i o n of the cost equations 4. d e t e r m i n a t i o n of the s o l u t i o n s to the c o s t equations. The t o t a l expected delay time per v e h i c l e per un i t d i s t a n c e equation w i l l c o n s i s t of the delay caused by the a c c e l e r a t i o n and d e c e l e r a t i o n of the empty v e h i c l e e n t e r i n g and l e a v i n g t u r n o u t s , plus an e x p r e s s i o n f o r the expected delay i n the t u r n o u t . The model to d e s c r i b e the delay time i n the t u r n o u t i s developed i n the f o l l o w i n g s e c t i o n . The general c o s t equation i s : Cost = turno u t c o n s t r u c t i o n c o s t + delay c o s t a t t r i b u t a b l e to turnout spacing + t u r n o u t maintenance c o s t . . The optimum turnout spacing i s determined by t a k i n g the f i r s t derivative of the cost function with respect to the turnout spacing and u t i l i z i n g a search technique to f i n d the values f o r which t h i s derivative i s equal to zero. Furthermore, these values must be checked to determine whether- they calculate a maximum or minimum value of the cost function. 10 2_. 3 THE TORNOOT DELAY TIME In the previous section the general formulation and basic assumptions of the model were outlined. The f i r s t phase i s to determine the delay time per vehicle per unit distance of road. The time required for a vehicle, t r a v e l l i n g at a constant speed, to traverse a unit distance of road i s the inverse of the velocity of the vehicle. The empty vehicle w i l l experience an expected "n" turnout delays while progressing along t h i s unit length of road. The t o t a l time required to travel t h i s section i s the uninterrupted t r a v e l time plus the time spent i n the turnouts plus the time l o s t i n the acceleration and deceleration of the empty vehicle. From the basic theory of the r e c t i l i n e a r motion of a p a r t i c l e (Meriam, 1971) i t can be easily shown that the stopping distance required for an empty vehicle, t r a v e l l i n g at velocity V2 , i s : Ds = V|/2a 0 2.1 and i t s corresponding stopping time i s : T5 = \ /a D 2^2 A l i s t i n g of the symbols u t i l i z e d i n the formulas i s located i n Appendix 2.. S i m i l a r l y , the acceleration distance (D^) and acceleration time (Tfl) are: D « = V | / 2 a A ^ and T A = \ ' a A ^ The next step i s to derive an expression f o r the delay time i n the turnout. This delay time i s not a constant but 11 depends on the distance the loaded vehicle i s from the empty vehicle when the empty vehicle comes to a complete stop i n the turnout. The delay i n the turnout can be represented by the r a t i o of the distance between the two vehicles and the turnout spacing. This r a t i o w i l l be referred to as the F-factor. Thus: F = D/S where: F = F-factor D = distance between the two vehicles S = distance between turnouts. The turnout spacing i s assumed to be uniform.. The F-factor i s u t i l i z e d rather than distance or time since i t gives a better concept of the e f f e c t that turnout spacing has on delays. Furthermore, previous a r t i c l e s (Byrne et a l . , 1947) on the subject have u t i l i z e d the same r a t i o to represent t h i s delay. The t h e o r e t i c a l development of an equation for c a l c u l a t i n g the expected F-factor w i l l be discussed in the next section. U t i l i z i n g the above concepts an expression can be derived to calculate the expected delay time per turnout incident.. The expected delay time per turnout incident consists of the time l o s t due to vehicle acceleration and deceleration: + - Vl "1 1 a A 2 a f l . a0 2 a D . 2 and the expected time the empty vehicle spends in the turnout: (SF)/V, The expected delay time per turnout incident (t) i s therefore: t = V2 (a f l+a D)/(2a f la D) + (SF)/V, 2^5 where : 12 V, = velocity of loaded vehicle F = expected F-factor. Since there are n expected delays per unit distance then the t o t a l expected delay time per unit distance per vehicle (T~) i s : T = n[ (V,, {a f l+a D}/{2a Aa 0}) + (SF)/VJ ] 2^6 The n delays consist of: 1. The c o n f l i c t s with the i n i t i a l number of loaded vehicles in the unit distance, which i s equal to the:unit distance of road times the t r a f f i c flow rate divided by the velocity of the loaded vehicle, plus 2. The number of loaded vehicles entering the:unit distance while the empty vehicle traverses the unit distance of road, which i s equal to the delay time plus the uninterrupted t r a v e l time multiplied by the t r a f f i c flow rate. The expression for n thus becomes: H 1 SF n v , + H - V, + n — + 2 a A a D or n = H H J, + v. 1 -HSF HV£ (a^+aD) 2 a A a 0 - I 2.7 where : H = t r a f f i c flow rate of loaded vehicles ( i . e . , vehicles ( per hour (vph)). Substituting equation 2.7 into equation 2.6 and simplifying yields an expression for the.total expected delay time that i s 13 not d i r e c t l y a function of n. \ (a„ +a0) SF" T = H H — + — 2a„ a 0 V, J 1 -HSF HV2 (a„+a0) 2 a « a o - l 2.8 Once the t o t a l delay time per unit distance per vehicle has been calculated the t o t a l delay cost per unit distance can be calculated by multiplying equation 2.8 by the t r a f f i c flow rate and the hauling cost.. The use of equation 2.8 i n determining the optimum turnout spacing w i l l be discussed i n section 2.5.. 14 2..4 THEORETICAL DEVELOPMENT OF THE EXPECTED F-FACTOR The c a l c u l a t i o n of the expected F-factor involves an analysis of the interactions between two separate t r a f f i c streams on a single-lane road. The objective i s to translate the r e a l t r a f f i c s ituation into a workable mathematical model. Several approaches are available. Fluid-flow analogies involve the p r i n c i p l e that the movement of t r a f f i c w i l l behave l i k e the flow of f l u i d s . The result i s the description of the t o t a l t r a f f i c network rather than the c o n f l i c t s between i n d i v i d u a l vehicles.. Wohl and Martin(1967) suggested that these methods are only applicable for high t r a f f i c densities. . Since log hauling systems involve low t r a f f i c densities another approach must be found. Car-following theories involve i n t e r v e h i c l e r e l a t i o n s h i p s . In general, i t i s assumed that the speed c h a r a c t e r i s t i c s of each vehicle depend to some extent on the speed c h a r a c t e r i s t i c s of each of the preceeding vehicles; thus, a driver's response i s constrained by the. surrounding vehicles and the c h a r a c t e r i s t i c s of the road (Wohl and Martin, 1967). This approach i s too detailed for use i n t h i s study. , T r a f f i c engineers have u t i l i z e d probability theory to analyse delay situations for merging t r a f f i c , i ntersection control, and l e f t - t u r n storage areas (Wohl and Martin(1967), Drew (1968)). The turnout delay problem i s similar to the merging t r a f f i c delay situation where one t r a f f i c stream constrains the movement of another t r a f f i c stream. The model developed herein w i l l u t i l i z e : probability 15 theory. . This approach involves the in t e r a c t i o n between the "average" loaded vehicle and the "average" empty vehicle.. In t h i s approach there are certain inherent assumptions.. Namely, a uniform velocity and rate of acceleration of the vehicles i s assumed. It w i l l be further assumed that the length of the vehicles w i l l be neglected as well as the length of the turnouts.. The ef f e c t of these assumptions on the:model w i l l be discussed l a t e r . The three situations that w i l l be discussed are the meeting of one loaded and one empty vehicle, the interactions between a single empty and a f l e e t of loaded vehicles, and the c o n f l i c t s between a group of empty and a group of loaded vehicles. , 16 2. 4.1 ONE LOADED VEHICLE MEETING ONE EMPTY VEHICLE The s i t u a t i o n pertaining to the int e r a c t i o n between one empty vehicle and one loaded vehicle i s i l l u s t r a t e d i n Figure 1.. A backwards approach i s u t i l i z e d for solving t h i s problem. The f i r s t phase i s to determine where the two vehicles w i l l meet (point B) i f the empty vehicle does not u t i l i z e a turnout.. Once t h i s has been accomplished the location where the empty vehicle must begin to decelerate i n order to stop at the c r i t i c a l turnout (S c) i s determined.. At th i s moment the empty vehicle i s at point C and the loaded vehicle, having backtracked an equal time i n t e r v a l , i s at point A. As i l l u s t r a t e d in the diagram, the c r i t i c a l distance (X C F) i s defined as the distance from the c r i t i c a l turnout where the two vehicles would meet i f the empty vehicle did not p u l l into the c r i t i c a l turnout. Furthermore, the loaded vehicle w i l l be exactly the F-factor times the turnout spacing from the c r i t i c a l turnout when the empty vehicle stops at t h i s turnout. The loaded vehicle must t r a v e l the distance A-Sc minus the F-factor times the turnout spacing in the time the empty vehicle decelerates. Thus v i V 2 / a o = v, < x c r + D s ) / V 2 + Xcf- " F S or X C f = [2FSV 2a D + V V|]/[2a D(V ( +Va) ] 2.9 17 _Q(2 Q Q Q Projected meeting point Turnout locations S„ Distances Times -Deceleration -No Deceleration |4-v, (X C F * D S ) / \ > K x c ^ K — D S ^ | 1 I I I | < ( X C P + D 5 ) / \ » ^ - ( X C F + D 5 ) / V ^ where h = stopping distance TS = stopping time S J = turnout locations F = F-factor Xcr = c r i t i c a l distance V , = velocity of the loaded vehicle v z = velocity of the empty vehicle Figure 1 Interaction between an empty vehicle and a loaded vehicle 18 I t i s reasonable to assume that the location of point B i s described by an uniform d i s t r i b u t i o n on the i n t e r v a l 0-S. We w i l l now introduce F m„ y which i s defined as the largest F-factor before the empty vehicle can t r a v e l to the next turnout.. The c r i t i c a l distance corresponding to a F-factor equal to zero i s denoted by . Since 1re minus X„„ equals the turnout spacing J C O C Frr>Ay. C O ^ r 3 an expression for F m I W can be derived. F ^ = <V. + W V Z ^ 1 0 The density function of the F-factor can be determined since for each c r i t i c a l distance there corresponds an unique F-factor and the density of the c r i t i c a l distance formula i s uniform over the i n t e r v a l 0-S. Thus Pr(F,< F <F-+SF) = (X C ( F. + S f ) -XCF. )/S = SFVa /(V, *\ ) 2. 11 where : F = a given F-factor <$F = a small change i n the F-factor. Once the density function of the F-factor has been determined the expected F-factor can be determined by integrating the density function times the F-factor between zero and F m A ) ( . . F = ' rn\Kf> p ^ \ F/(V( * \ )£F \ F 2 ~ 2\ 2.12 This expected F-factor i s referred to as the simple F-factor to distinguish i t from the expected F-factor equations developed in the next section. Those equations are referred to by t h e i r corresponding headway probability functions.. A simulation model developed in Chapter 4 i s u t i l i z e d to confirm the simple F-factor equation. . 19 2.4 . 2 ONE EM£II VEHICLE MEETING A FLEET OF LOADED VEHICLES In the previous section i t was assumed each turnout incident was independent of any other.. In t h i s section the turnout events w i l l be p a r t i a l l y dependent on each other.. This situation involves the investigation of the meeting of an empty vehicle and a f l e e t of loaded vehicles. I t i s not necessary to analyse the i n t e r a c t i o n between the f i r s t loaded vehicle and the empty vehicle since t h i s has been accomplished i n the previous section. Once a loaded vehicle has passed the empty vehicle, stopped i n the turnout, two situations may arise: 1. the next loaded vehicle may be close enough to prohibit the empty vehicle from advancing to the next turnout or 2 . the empty vehicle may proceed. An analysis of the headway d i s t r i b u t i o n of the loaded vehicles i s e s s e n t i a l to solve t h i s problem.. Drew(1968) defines headway as "the i n t e r v a l of time between successive vehicles moving i n the same lane and measured from head to head as they pass a point on the road." There exists a c r i t i c a l headway(hc) where the empty vehicle can just t r a v e l to the next turnout and experience no delay i n that turnout. The c r i t i c a l headway must be determined p r i o r to the c a l c u l a t i o n of the probability of each of the above sit u a t i o n s occuring. I n i t i a l l y , the f i r s t loaded vehicle and the empty vehicle are positioned at turnout S c (Figure 2 ) . The second loaded vehicle i s at point B. 2 0 /O Ol / 1 K V Q Q - Q _ J L Turnout lo c a t i o n s S. B l - T - T"T" - 0 Distances Times -acceleration -No acceleration K D — | ^ D f l ^ | I ' M i K—T. ^ I where : O = location of the empty vehicle 0 = location of the loaded vehicles h c = c r i t i c a l headway Situation pertaining tc the empty vehicle experiencing no delay « i n turnout S^  since i t comes to a complete halt at the turnout as the loaded vehicle passes the turnout. Figure 2 Diagram showing the physical s i t u a t i o n leading to the c r i t i c a l headway 21 The time required for the second loaded vehicle to t r a v e l distance D( i s equivalent to the time required for the empty vehicle to t r a v e l one turnout spacing. Consequently, T, = TS+T,, + (S-DS-DA)/V,, = (2Sa f ta D* V|{a f l*a 0})/(2a Aa 0Vj,) 2^3 and D, = V, (2Sa Aa D*V|{a f t+a D})/(2a Aa DV z) 2.14 Since vehicle arrangements are generally measured in time units the c r i t i c a l headway can be e a s i l y calculated as: h c = T.+S/V, = [2Sa f la 0(V, +\)+\ V|(a f l +a D) ]/[ 2aft a0V, \ ] 2. 15 and the pro b a b i l i t y that a headway i s less than the c r i t i c a l headway i s : Pr(h<h c) = I g(h) cSh 2. 16 Jo where: g(h) = headway probability density function. The expected F-factor (F^) for headways less than the c r i t i c a l headway varies with respect to the expected headway for headways less than the c r i t i c a l headway. Thus: V, /S ,2. 17 The expected F-factor (F, ) for headways greater than the c r i t i c a l headway i s the weighted average of functions s i m i l a r to F z except that the l i m i t s of the integrals vary. This i s because an empty vehicle can proceed one, two, or more turnout spacings before being forced into a turnout. The re s u l t i s that the l i m i t s of the integr a l s vary with respect to the turnout spacing and the number of turnouts the empty vehicle \ hg(h)£h 22 passes before being forced into a turnout.. The simple F-factor w i l l be u t i l i z e d to estimate F, since the actual function i s complex and i t s l i m i t s are d i f f i c u l t to determine. Thus: F, = (V, +V,, )/(2V a) 2. 18 A simulation model, developed i n Chapter H, i s u t i l i z e d to verify the accuracy' of t h i s method. The expected F-factor equation for the meeting of one empty vehicle and a f l e e t of loaded vehicles consists of two parts: 1. the component that represents the average F-factor attributable to the delay sit u a t i o n where the headway i s less than the c r i t i c a l headway and, 2. the component that represents the delay sit u a t i o n where the headway i s greater than the c r i t i c a l headway. The probability of each of these situations occuring must be calculated, as well as the expected F-factor attributable to each of the situ a t i o n s . Consequently, the expected F-factor equation i s : F = Pr(h>h c)F, + Pr(h<h c)F^ ls.ll where : h = headway F, = the expected F-factor for headways greater than the c r i t i c a l headway F^ = the expected F-factor for headways less than the c r i t i c a l headway. The f i r s t turnout incident of an empty vehicle's t r i p i s independent of the headway probability d i s t r i b u t i o n and the expected F-factor r e s u l t i n g from t h i s incident i s the simple F-factor.. The expected F-factor representing the rest of the 23 turnout incidents the empty vehicle encounters on i t s t r i p , i s given by equation 2.19. . The t o t a l number of t r i p s per day and the t o t a l number of turnout incidents per day must be determined p r i o r to ca l c u l a t i n g the expected F-factor.. The expected F-factor i s composed of two parts: L.The portion that represents the f i r s t turnout incident on a t r i p which i s equal to the simple F-factor times the number of t r i p s per vehicle divided by the t o t a l number of turnout incidents. plus 2. The portion representing the remaining turnout incidents on the t r i p which i s equal to equation 2.19 times the remaining number of turnout incidents divided by the t o t a l number of turnout incidents. Thus: F = [Pr(h>h c)F, +Pr(h<h c)F 2 ][Q 3/(Q 3+I) ] +[ (V *Vt)/{2Vz) ][I/(Q 3+I) ] 2^20 where: Q 3 = Hd-I = number of headways d = number of " c o n f l i c t " hours per day I = number of t r i p s per day. Four headway probability d i s t r i b u t i o n s and t h e i r respective expected F-factors are discussed i n section 2.4.4. The expected F-factor equations derived i n t h i s section w i l l be referred to by t h e i r respective headway probability d i s t r i b u t i o n . Chapters 4 and 6 w i l l discuss the r e l a t i v e e f f e c t of these equations compared to the simple F-factor and the average F-factor resulting from the meeting of two f l e e t s of vehicles.. 24 2.4.3 A - FLEET OF LOADED VEHICLES MEETING A FLEET OF EMPTY VEHICLES In t h i s section the turnout incidents between a f l e e t of loaded and a f l e e t of empty vehicles w i l l be examined.. The four cases that w i l l cause an empty vehicle to u t i l i z e a pa r t i c u l a r turnout are: Case 1. An empty vehicle may stop at a turnout because a loaded vehicle i s approaching. Case 2. An empty vehicle may remain in a given turnout since a loaded vehicle i s so close as to prohibit the empty vehicle from advancing to the next turnout. Case 3. An empty vehicle i s required to halt at a par t i c u l a r turnout since the next turnout i s occupied (Figure 3). Case.4. An empty vehicle cannot proceed to the next turnout but remains at i t s present turnout since the next turnout i s occupied (Figure 4). In the f i r s t two cases, s i t u a t i o n A, the empty vehicle i s forced to use a particular turnout since a loaded vehicle i s approaching while i n the l a t t e r two cases, si t u a t i o n B, the empty vehicle i s required to use a par t i c u l a r turnout since the next turnout i s occupied and a loaded vehicle is approaching. Since s i t u a t i o n A has been discussed in section 2.4.2, the following discussion w i l l primarily involve s i t u a t i o n B. 2 5 _ 0 j Q 0_ Q O O J L O O Turnout locations S„ I-C B 1 s. ' (D 1 Distances where: O = empty vehicle 9 = loaded vehicle O = potential location of empty vehicle ^ = location of set 1 empty vehicles Sj = turnout locations Figure 3 Case 3—An. empty vehicle stops at turnout S, since o next turnout i s occupied 26 Figure 3 i l l u s t r a t e s the case in which an empty vehicle, t r a v e l l i n g at a constant v e l o c i t y , i s delayed at a turnout since one empty vehicle or a group of empty vehicles (set 1) i s occupying the next turnout. Herein t h i s case w i l l be referred to as a Case 3 s i t u a t i o n . If set 1 did not occupy turnout S c then the empty vehicle would be able to stop at turnout S c instead of being forced to u t i l i z e turnout .. Maintaining t h i s s i t u a t i o n then the empty vehicle would be located between points A and B when the loaded vehicle i s between points C and E. The expected F-factor for the Case 3 s i t u a t i o n i s : ¥3 = F, + S/S + SV, /(SVZ) = F, +(V, +V Z)/V 2 2.21 where: F 3 = expected F-factor for a Case 3 s i t u a t i o n . Equations 2.12 and 2.19 can be used to determine an expected F-factor representing the meeting of an empty vehicle and a f l e e t of loaded vehicles. U t i l i z i n g t h i s expected F-factor the expected distance the empty vehicle, stopped at , i s from the loaded vehicle can be determined. . The expected F-factor equation i s the sum of equation 2.18 and the distance between turnouts S and S divided by the turnout spacinq plus the distance the loaded vehicle can t r a v e l in the time the empty vehicle can t r a v e l one turnout spacing divided by the turnout spacing. . Since F, has been approximated by equation 2. 18 F 3 becomes: F 3 = 3(V, +V Z)/(2V Z) 2. 22 After the loaded vehicle has passed both groups of empty vehicles and each vehicle has accelerated to i t s o r i g i n a l speed the distance between the empty vehicle and the group of empty vehicles i s the i n t e r v a l distance (D T).. The i n t e r v a l distance 27 i s the turnout spacing plus the distance an empty vehicle can t r a v e l , at a constant velocity, in the time a loaded vehicle can t r a v e l one turnout spacing. Dj = S (1 + \ /V, ) 2. 23 As i l l u s t r a t e d in Figure 4, Case 4 i s the si t u a t i o n that stems from an empty vehicle waiting i n a given turnout since the adjacent turnout i s occupied and a loaded vehicle i s approaching. I f the headway between any two loaded vehicles i s less than the c r i t i c a l headway then the expected F-factor attributable to the second loaded vehicle i s equivalent to the expected F-factor of equation 2.17.. The expected F-factor equation representing the meeting of two f l e e t s of vehicles i s composed of the probability that each of the above cases w i l l occur, times t h e i r respective expected F-factor. The expected F-factor equation representing the meeting of two f l e e t s of vehicles i s therefore: F = Pr(h>hc|A)F, +Pr(h<hc|A) F a+Pr(h<h c|B)F a • Pr(h>hc|B) (F( +1+V, /Vz) 2.24 where: A = conditional event that an empty vehicle i s required to u t i l i z e a turnout because a loaded vehicle i s approaching B = conditional event that an empty vehicle must u t i l i z e a turnout because an empty vehicle or a group of empty vehicles already occupy the adjacent turnout and a loaded vehicle i s approaching. The expected F-factor for each of the above cases can be e a s i l y determined but the probability of each of the events occurring i s d i f f i c u l t to determine. I t would not be neccessary to 2 8 calculate these p r o b a b i l i t i e s provided i t can be shown that the prob a b i l i t y of a conditional event B i s r e l a t i v e l y small. Consequently, i t i s desirable to determine an upper l i m i t of the probability of event B. I f the two f l e e t s of vehicles are independent of each other, then the probability of the number of empty and loaded vehicles a r r i v i n g during any given time i n t e r v a l , t, can be determined.. Since for low t r a f f i c flow rates t h i s probability i s r e l a t i v e l y small then the expected F-factor equation representing the meeting of a single empty vehicle and a f l e e t of loaded vehicles w i l l adequately describe the expected F-factor resulting from the interactions between two f l e e t s of vehicles. . This statement i s supported by a simulation model developed in Chapter 4 . 29 _0_0_ O 0 0 Turnout locations S, o Turnout locations S. 0 I-Turnout locations S„ , s, - I -s . " 1 $ — -°4 Time 1 t=0 1 | t = S/Vf ——I I t=h where : O = empty vehicles • = loaded vehicles ^ = location of set 1 vehicles S; = turnout locations Figure 4 Case 4 — An empty vehicle remains in turnout S 3 since next turnout i s occupied 30 2.4.4 HEADWAY PROBABILITY DISTRIBUTIONS The t h e o r e t i c a l technique described i n section 2.4.2 requires some knowledge of the p r o b a b i l i s t i c aspects of t r a f f i c behaviour. There have been no studies undertaken to determine the headway d i s t r i b u t i o n of logging trucks; however, t r a f f i c engineers have conducted studies to determine the a r r i v a l d i s t r i b u t i o n of low t r a f f i c flow along highways and r u r a l roads.. Various authors have attempted to describe the headway or a r r i v a l d i s t r i b u t i o n using various probability d i s t r i b u t i o n s . These have included the negative binomial d i s t r i b u t i o n , the Pearson Type III d i s t r i b u t i o n , the Erlang d i s t r i b u t i o n , and the Pearson Type I d i s t r i b u t i o n . Descriptions of these d i s t r i b u t i o n s may be found i n Wohl and Martin (1967) , Haight (1 963) , and Drew(1968). Gerlough (1955) tabulated the a r r i v a l frequency of f i e l d data for various r u r a l and urban roads in the: United States. The attempt to f i t a Poisson d i s t r i b u t i o n to four sets of data by performing goodness of f i t tests proved to be unsuccessful at the f i v e percent acceptance l e v e l . . A cause f o r the f a i l u r e of the tests may have been the fact that the Poisson d i s t r i b u t i o n assumes there can be more than one a r r i v a l during a time i n t e r v a l equivalent to the minimum headway. The minimum headway i s the t h e o r e t i c a l minimum allowable time i n t e r v a l between the fronts of successive vehicles. At very low t r a f f i c flows t h i s assumption may prove to be i n s i g n i f i c a n t . Wohl and Martin(1967) u t i l i z e d a shifted exponential curve to examine the frequency d i s t r i b u t i o n of vehicle headways. In 31 his empirical example (1000 vph) the shi f t e d exponential curve f i t s the data better than the non-shifted curve for lower headways while the reverse held true for larger headways.. The shifted exponential curve states that: Pr(h>t) =£-u<t-Y> 2.25 where: t = time i n t e r v a l % = minimum headway. Schuhl(19 55) developed a p a r t i a l l y shifted exponential curve of the form: Pr(h>t) = e _ < <t-f>/<h,-f>> +£-ct/k£ 2.26 where : h, = mean headway for constrained flow h^ = mean headway for free flow. This equation allows the speed c h a r a c t e r i s t i c s of some of the vehicles to be influenced by the c h a r a c t e r i s t i c s of other vehicles.. This equation to some extent involves car-following theory. Wohl and Martin argued that since a l l vehicles are constrained at higher flows then equation 2.26 can be restated as: Pr(h>t) = Q-K-L-fi/ib-t)) 2.27 Wohl and Martin tested equation 2.27 on two sets of data and concluded that t h i s equation was acceptable at the ten percent l e v e l of s i g n i f i c a n c e , for a data set with a flow rate of 500 vph, but was rejected for a data set with a flow rate of 1000 vph. Consequently, a general exponential equation does not adequately describe these sets of data. Since these authors were not able to consistently f i t data to a probability d i s t r i b u t i o n several d i s t r i b u t i o n s w i l l be 32 analysed.. In the analysis i t w i l l be assumed that the empty vehicle w i l l wait i n a turnout u n t i l a headway i s larger than the c r i t i c a l headway. The probability that the time i n t e r v a l between two loaded vehicles i s greater than the c r i t i c a l headway must be determined.. This i s equivalent to the probability of there being no a r r i v a l s during t h i s time period. The expected F-factor formulas for headways less than the c r i t i c a l headway w i l l be derived. The probability d i s t r i b u t i o n s that w i l l be analysed are the uniform a r r i v a l d i s t r i b u t i o n , the exponential headway d i s t r i b u t i o n , the Erlang headway d i s t r i b u t i o n , and the Pearson Type III headway d i s t r i b u t i o n . 2. 4 . 4 . 1 UNIFORM A R I I I M DISTRIBUTION The uniform probability d i s t r i b u t i o n has a constant value of l / ( i - j ) over i t s i n t e r v a l ( i , j ) . F e l l e r ( 1 966) showed that for an uniform a r r i v a l d i s t r i b u t i o n the probability that the the headway i s greater than the c r i t i c a l headway i s : Pr(h>h c) = (1-h c) m for 0<hc<1 where : m = number of headways. It was determined in Section 2.4.2 that the number of headways i s given by Q-j. Therefore the above equation may be rewritten as: Pr(h>h c) = ( 1 - h c ) Q 3 2.28 The density function of t h i s d i s t r i b u t i o n i s the derivative of the p r o b a b i l i t y that the headway i s less than the c r i t i c a l 3 3 headway.. Thus: 3[Pr(h<h c) ] g(h) = = - Q 3 ( 1 - h ) Q 3 - i The expected F-factor formula for headways les s than the c r i t i c a l headway i s derived as follows: SF^ Q hg(h)<Sh V. g{h)6h S Q3 ( [ l-hc ] Q 3 + * - 1 ) (Q., + 1) ( [ 1 - h c 1 -2 . 2 9 2 . 4 . 4 . 2 EXPONENTIAL HEADWAY DISTRIBUTION The gap density function of the exponential headway d i s t r i b u t i o n i s : >>[Pr(h<hc) ] g(h) = = Q 3e- Q j h c>h 3 This gap density function can be integrated between zero and the c r i t i c a l headway to determine the probability of a headway being less than the c r i t i c a l headway, yi e l d i n g : Pr(h<h c) = 1-e- Q3 hc. 2 . . 3 0 Furthermore, the expected F-factor equation for headways l e s s than the c r i t i c a l headway can be derived from the gap density function. S F 2 hg(h)6h _ er Q 3 K/Q 3 h-i>]^c v, !l c g(h)6h -e-Q^h V F 2 = V, [ e - Q 3 V , C ( - Q 3 h C - 1 ) + 1 ] / [ S £ - Q 3 ( e - ^ - 1 ) 3 ] 2 . 3 1 34 2 . 4 . 4 . 3 ERLANG HEADWAY DISTRIBUTION The d i s t i n c t i o n between the gamma and Erlang probability functions i s that the Erlang function i s r e s t r i c t e d to posit i v e integer values for alpha. , The gap density function of the Erlang d i s t r i b u t i o n i s : g(h) = {oLQ3f h * - i e - * Q ^ / ( o C - 1 ) ! Wohl and Martin(1967) showed that the probability of a headway being less than the c r i t i c a l headway i s : Pr(h<h c) = j0ht[7v(7.h)oL-»e-Xh Sh] / [ (oc-1)! ] = 1-e-*Kt- fettle)1 / i ! 2 . 3 2 The logging vehicle headway frequency d i s t r i b u t i o n s produced from data collected by Smith and Tse(1977) and Boyd and Young(1969) have been shown to be l e f t skewed (Chapter 4 ) . Consequently, the expected F-factor equations w i l l be developed for functions that are l e f t skewed. . I f alpha equals two then the p r o b a b i l i t y of a headway being less than the c r i t i c a l headway i s : Pr(h<h c) = 1-e - M c- (1+Xh c ) 2 . 3 3 where: A = Q-jOC- = 2 Q 3 g(h) = A 2 h e ~ A W and the expected F-factor i s derived as follows: hg(h)£h = Ofi rht g(h)6h F a = -e - ^ c ( - ? v 2 h f - 2 X h c -2) A • 2A" 2 . 3 4 1-e- XV>c(1+Ah c) S i m i l a r l y , the expected F-factors and probability functions can 3 5 be calculated for other values of alpha. 2 . 4 . 4 . 4 PEARSON TYPE III HEADWAY DISTRIBUTION The gap density function of the Pearson Type III d i s t r i b u t i o n i s : g(h) = ( b a [ h - c f - » e - b < K - c > ) / r<a) where : c = minimum value of h mean = a/b + c variance = a/b 2 ]7(a) = gamma function = (a-1) ! for a = 0 , 1 , 2 , . . . . For the same reasons as cit e d for the Erlang headway d i s t r i b u t i o n the value of 'a' w i l l be set to two.. The gap density function becomes: g(h) = b 2 (h-c)e _ b < K _ C > c<h< oo The p r o b a b i l i t y that the headway i s less than the c r i t i c a l headway i s : Pr(h<h c) =j^ Cg(h)£h = e-bch-o (-bh- 1+cb)]^c Pr(h<h c) = i-e-*>chc-o (bl^+1-cb) 2 . 3 5 and the expected F-factor for headways less than the c r i t i c a l headway i s derived as follows: F S j h c h g(h ) c S h [ e ~ b t K ~ C ) (-bhz-2h-2/b+chb+c) ] ] ^ V, J*c g(h)5h [e-bcVv-c> (-bh-Hcb) ] [ c + 2/b+e-b(ht -c) (-bh|-2h c-2/b+cbh c+c) ] [ i-e-bcK c-o (bh c+1-cb) ] 3 7 2. 5 DEVELOPMENT OF THE COST EQOATION The only costs that are needed i n the cost equation are those that are functions of the turnout spacing: the turnout construction cost, the turnout maintenance cost, and the delay cost attributable to turnout spacing. The general cost equation i s : C T = Cc/S + C^/S + Q( HMT 2. 37 where : C T = t o t a l cost attributable to turnout spacing per unit distance of road C c = turnout construction cost C M = turnout maintenance cost H = t r a f f i c flow rate M = adjusted truck hauling cost Q( = expected useful l i f e of the road S = distance between turnouts f = delay time per vehicle per unit distance of road. Since a vehicle may have to queue at the landinq regardless of the turnout spacing an adjusted hauling cost i s u t i l i z e d . A necessary condition for determining the optimum turnout spacing i s that the f i r s t derivative of the cost function with respect to the turnout spacing i s equal to zero. Once the derivative has been determined then a search technique can be used to locate t h i s zero. . Furthermore, when the value of the second derivative cf the cost function i s positive, the di r e c t i o n of concavity of the function i s upwards. Provided the value of the second derivative i s - positive t h i s zero represents a 38 minimum value. . Some of the numerical analysis techniques available are bisection, regula f a l s i , secant, and Newton's method.. For further reference consult Conte and deBoor(1972) or Arden and A s t i l l ( 1 S 7 0 ) . . When discounting of maintenance costs i s not used the cost equation becomes: C T = C/S + Q( HMT where: c = c M +c c 2.38 Three forms of the cost equation s h a l l be u t i l i z e d in determining the optimum turnout spacing. In the f i r s t case i t s h a l l be assumed that the turnout spacing i s non-uniform and the expected F-factor i s a function of the turnout spacing.. In the second case the expected F-factor w i l l be a function of the turnout spacing but the turnout spacing w i l l be assumed to be uniform.. The t h i r d case w i l l assume uniform turnout spacing but w i l l u t i l i z e the simple F-factor.. I f the turnout spacing i s non-uniform and the expected F-factor i s a function of turnout spacing then the cost equation becomes: y(s) + Q, HM H H — + — J.V, V2 y(S)F V" +— V, K 1-Hy(S)F E\ K J - i 2.39 where : y (S) = turnout spacing function K = 2 a A a D / (afl+a0) and the optimum turnout spacing occurs when: 39 0 = [ Y (S) ]2 (HQ, H2(V, +Vi) jy (S) aCy(s) ] '&F b[y (S) ]• .dS. T - H 2 F 2 R 2 V 2 C as d[y<s) j I- y(S) 2 a[y (S) T CV, V a FKH[ HVa -K ] C[ R2-2HV Z K+ H 2 V | ] 2.40 By m a i n t a i n i n g the expected F - f a c t o r equation to be a f u n c t i o n of the turnout spacing and a l l o w i n g the t u r n o u t spacing t o be uniform then the co s t equation becomes: C T = -+Q HM S "H H" "SF V,f — + — — +— 1 Lv, v2J Lv, K J HSF HV2 - l 2.41 and the optimum t u r n o u t spacing occurs when: 0 = S2[MQ i H 2 K 2 ( F + s||-) (V, *VZ )-R2H2F2CVa ] + S[ 2CV2 HFKV, (K-HV2 ) ] + C l f l z (2HV^ K-K 2-H 2 V 2 ) 2.42 The c o s t equation i n v o l v i n g the simple F - f a c t o r equation and uniform turnout s p a c i n g i s : C H H " "SF V^ " HSF n\~ - 1 = -+Q, HM — + — 1 — — s V Lv, K J L V, K J 2.43 and the optimum turno u t spacing occurs when: 0 = S 2 [ MQ( H 2K 2F (V, +V2 ) - R 2 H 2 F 2 C V Z ] + S[ 2CVe HFKV, (K-HV 2)] + C V 2 V Z (2HVj, K - R 2 - H 2 V|) 2.44 Since the simple F - f a c t o r i s u t i l i z e d i n t h i s equation, the qua d r a t i c formula can be used t o f i n d the r o o t s of the equation.. The use and the s e n s i t i v i t y of the equations w i l l be discussed i n Chapter 6. 40 2.6 DISCOUNTING OF THE COST EOJJATION Turnout construction costs are expenses of the present while delay costs are incurred over the l i f e of the road.. In order to relate present costs to future costs some form of discounting i s required.. The present worth formula for equal payments made in the future states that: PW = [ Qz (1 + i)™-1 1 / [ i (1+i) m ] 2.45 where: PW = present worth Q,£ = value of the constant cost i = i n t e r e s t rate over period m m = l i f e of project. This formula can be incorporated into the cost functions developed i n section 2.5 by allowing: Z 2 = [ (1 + i ) m -1 ] / [im (1+i) m ] 2.46 and incorporating t h i s equation into the delay component of the cost functions. Since equation 2.46 i s independent of the turnout spacing, the inclusion of discounting into the derivative of the cost function w i l l r e s u l t i n an additional constant.. This process executed on equations 2.43 and 2.44 resu l t s i n : ~~ - - l 2.47 c "H H " SF V —+ Q HMZ2 — + — — + — S Lv, v aJ Lv, K J 1--HSF E\ V, K J and the optimum turnout spacing occurs when: 0 = S2[MQ/ HZRZFZ^  (V, +V2 ) -K2 H 2 F 2 C V 2 ] + S[ 2CV,, HFKV, (K-HV2 ) ] + CVf\ (2E\ K - K 2 - H 2 V | ) 2^ 48 In the development of t h i s transformation process i t was 41 assumed that the maintenance cost i s not discounted and there i s a constant t r a f f i c flow volume per year. In r e a l i t y the t r a f f i c flow volume per year i s not a constant but i s variable from year to year. Chapter 6 w i l l discuss the s e n s i t i v i t y of t h i s assumption. 42 3.0 THE ARRIVAL DISTRIBUTION OF LOGGING TRUCKS la.! ARRIVAL DATA FROM TWO OPERATIONS There have been no studies to determine the headway d i s t r i b u t i o n of logging trucks, but there have been two studies that produced basic data that may be suitable i n the analysis of i n t e r a r r i v a l times. . The two study areas were located i n coastal B r i t i s h Columbia (B.C.) and in north c e n t r a l B.C.. The coastal study, u t i l i z i n g five-minute time i n t e r v a l s , measured the time the loaded logging trucks arrived at the log dump. The sample consisted of 201 a r r i v a l s over a eight-day period with an average flow rate of approximately two-and-one-half vehicles per hour. These data are quite. r e s t r i c t i v e for determining the headway d i s t r i b u t i o n of vehicles since there i s only one frequency class for vehicle headways of between zero and f i v e minutes. The . i n t e r i o r operation consists of vehicles t r a v e l l i n g on a company haul road and u t i l i z i n g one of two government weigh scales. The f i r s t scale was open between the hours of 6 A.M. and 5 P.M. while the.second scale was open u n t i l a l l of the hauling operations had ceased for the day. . The loaded vehicles would stop at t h e . f i r s t weigh scale provided i t was open.. Since the a r r i v a l time at the various scales was recorded to the nearest minute these data appear to be more useful f o r headway analysis than do the data from the coastal operation.. Observations were recorded for the month of February 1976 with a t o t a l sample size of 813 observations and 4 3 an average flow rate of approximately three vehicles per hour. For a more detailed description of the two studies consult Boyd and Young(1969) and Smith and Tse(1977).. The i n t e r a r r i v a l times of the two studies w i l l be analysed in section 3.2.. 4 4 1.2 ANALYSIS OF THE ARRIVAL DATA The primary purpose of the following analysis was to determine whether the a r r i v a l frequency of logging trucks f i t s a known probability function.. Since the expected F-factor equations involve headways instead of a r r i v a l times the data were grouped into headway frequency classes rather than a r r i v a l frequency classes. . A yj- goodness of f i t test was u t i l i z e d to examine the closeness of f i t of the data to various hypothesized probability d i s t r i b u t i o n s . This technique i s used to compare the expected frequency, for a given d i s t r i b u t i o n , to the observed frequency. The chi-square value i s determined as: X 2 = £ [ ( 0 ; - e , ) 2 / e . ] 3 .1 where: O; = observed frequency of the i c e l l e; = expected frequency of the i c e l l k = t o t a l number of c e l l s . . The i n i t i a l step i s to determine the c e l l widths and the upper l i m i t of the f i r s t c e l l . Once t h i s has been accomplished a frequency histogram of the observed data can be determined. The type:of d i s t r i b u t i o n s most l i k e l y to f i t the observed data can be estimated from these graphs.. A computer program written in the programming language BASIC for the Hewlett Packard(HP) 9830A cal c u l a t o r i s used i n t h i s analysis. The headway frequency histogram of the coastal study i s i l l u s t r a t e d i n Figure 5. This graph shows that the frequency of the headways i s l e f t skewed. Goodness of f i t tests were performed for the exponential, shifted exponential, and Erlang 45 p r o b a b i l i t y d i s t r i b u t i o n s . The goodness of f i t c a l c u l a t i o n s are l o c a t e d i n Appendix 4 while Table I summarizes the r e s u l t s . The r e s u l t s i n d i c a t e t hat none of the t e s t e d d i s t r i b u t i o n s f i t the observed data. F i g u r e 6 i l l u s t r a t e s the headway frequency d i s t r i b u t i o n f o r the north c e n t r a l i n t e r i o r study. T h i s f i g u r e groups the headways i n t o one-minute i n t e r v a l s while F i g u r e 7 uses c l a s s widths of f i v e minutes.. Both of these graphs i n d i c a t e t h a t a skewed d i s t r i b u t i o n may f i t the observed data. Table I I summarizes the r e s u l t s of the goodness of f i t t e s t s . . As was the case f o r the c o a s t a l study, none of the t e s t e d p r o b a b i l i t y f u n c t i o n s , with c l a s s widths o f one minute, f i t the observed data.. When the c l a s s widths were i n c r e a s e d to f i v e minutes the e x p o n e n t i a l and s h i f t e d e x p o n e n t i a l f u n c t i o n s were not r e j e c t e d but these f u n c t i o n s do not adequately d e s c r i b e the data f o r headways l e s s than the c r i t i c a l headway. The goodness of f i t t e s t s f a i l e d f o r the one-minute c l a s s i n t e r v a l s because there were too few o b s e r v a t i o n s f o r the lower headways(i.e., headways l e s s than three minutes). I f the values of the expected F - f a c t o r formulas do not change r a d i c a l l y with r e s p e c t to v a r i o u s p r o b a b i l i t y d i s t r i b u t i o n s then one or more of the t e s t e d , p r o b a b i l i t y d i s t r i b u t i o n s may be s u i t a b l e f o r use i n the expected F - f a c t o r formulas. The s e n s i t i v i t y of the expected F - f a c t o r models w i l l be d i s c u s s e d i n Chapters 4 and 6. . 46 E-4- in &9 I Ef . 1 / 1 ui m raf: + U , m + ut eg in x ut m x rvi x in F l >-z ca E3 U l DC X rvi rvi ui ca Figure 5 Frequency histogram of headways (5-minute intervals) for coastal study 4 7 Table I Goodness Of F i t T e s t s — C o a s t a l Study I T T 1 1 1 — 1 7 | T | P r o b a b i l i t y I D i s t . E e j e c t of Hypoth •|Number* I c f I C l a s s e s -+ C l a s s | Width| (min) | + LCL • (min) | Computed| ! xz j I T a b l e l I J* I lot'. 0 0 5 | - + 1 1 1 1 1 | 2 8 . 3 | 1 1 1 2 5 . 2 | 1 1 1 1 | Exponential! Yes 1 2 5 ( 1 4 ) 5 | 2 . 5 1 7 6 . 3 | 12 I E r l a n g (oc=2) | Yes | 2 5 (1 2) | 5 1 2 . 5 1 6 2 . 7 | 10 I S h i f t exponi 1 1 1 ! ! < : 0 . 1 min | Yes 1 2 5 ( 1 4 ) | 5 | 2 . 5 1 7 5 . 2 I 12 1 1 1 2 8 . 3 | 1 1 1 2 8 . 3 | 1 1 1 2 8 . 3 | | : 0 . 5 min j Yes 1 2 5 ( 1 4 ) | 5 1 2 . 5 | 7 1 . Q I 12 I : 1 . 0 min I Yes 1 2 5 ( 1 4 ) | 5 i 2 . 5 | 6 5 . 8 | 12 • R e j e c t i o n of hypothesis * The pare n t h e s i z e d number i s the f i n a l number of frequency c l a s s e s w hile the non-parenthesized number i s the o r i g i n a l number of frequency c l a s s e s • Lower c l a s s l i m i t + The number of degrees of freedom 4 8 Figure 6 Frequency histogram of headways for i n t e r i o r study (1-minute intervals) 4 9 rvi ra ra ra ra m EjLTl V-ZJ z ra ms-ec 3 az u cgx ui ra x ra m ra rvi ra CD rvi 10 x X in rvi m ra >-Z c g ca UJ ce u. ra P4 x U l in m m ra Figure 7 Frequency histogram of headways for (5-minute intervals) i n t e r i o r study 50 T a b l e I I Goodness Of F i t T e s t s - = I n t e r i o r S t u d ^ - B o t h S c a l e s P r o b a b i l i t y D i s t . . | Re j e c t I of I Hypoth • | Number*! | o f | | C l a s s e s I -+ + C l a s s j W i d t h | (min) | + LCL • (min) | Computed| ! ^ ! + + + V | T a b l e | 1 X2 1 |«r.005| - + - 1 E x p o n e n t i a l I Yes | 8 0 ( 5 3 ) | 1 | 0 . 5 1 9 2 . 0 | 51 1 5 3 . 7 1 E r l a n g (cC=2) I Yes | 8 0 ( 4 8 ) | 1 1 0 . 5 1 6 1 7 . 9 I 46 | 5 3 . 7 1 S h i f t ex pen i i 1 I I 1 1 : 0 . 1 min I Yes | 8 0 ( 5 3 ) | 1 I 0 . 5 I 8 7 . 5 | 51 1 5 3 . 7 1 : 0 . 5 min Yes I 8 0 ( 52 ) | 1 | 0 . 5 I 7 2 . 2 I 5 0 1 5 3 . 7 | E x p o n e n t i a l Yes 1 2 1 ( 1 7 ) | 5 1 0 . 5 I 3 3 . 7 I 15 1 3 2 . 8 | E r l a n g (oC=2) I Yes 1 2 1 ( 1 3 ) | 5 1 0 . 5 | 1 8 3 . 5 | 11 1 2 6 . 8 | S h i f t expen I I ! I I i ! : 0 . 1 min I No 1 2 1 ( 1 7 ) | 5 1 0 . 5 I 2 9 . 4 I 15 1 3 2 . 8 | : 0 . 5 min I No | 2 1 ( 1 6 ) | 5 1 0 . 5 I 1 7 . 6 | 14 1 3 1 . 3 | E x p o n e n t i a l No 1 2 0 ( 1 6 ) | 5 1 5 . 5 I 2 0 . 7 I 14 1 3 1 . 3 I • R e j e c t i o n c f h y p o t h e s i s * The p a r e n t h e s i z e d va lue i s the f i n a l number o f f requency c l a s s e s w h i l e the n o n - p a r e n t h e s i z e d number i s the o r i g i n a l number of frequency c l a s s e s • Lower c l a s s l i m i t + The number o f degrees o f freedom 51 iUO_ SIMULATION FOR THE VERIFICATION OF THE EXPECTED F-FACTOR EQUATIONS Simulation can be an valuable tool in the v e r i f i c a t i o n of the accuracy of mathematical equations. In t h i s instance i t w i l l be used to determine the accuracy and l i m i t a t i o n s of the expected F-factor equations. The technical description of the three simulation models i s discussed in Appendix 5 while the features and re s u l t s of the models w i l l be analysed in the following section. 5 2 4. 1 THE SIMULATION MODELS A discrete, c r i t i c a l event stochastic simulation model was written i n the general purpose programming language FORTRAN for the IBM 370 Model 168 computer to confirm or rej e c t the simple F-factor equation. The program simulates the meeting of one empty vehicle and one loaded vehicle. I n i t i a l l y , the empty and loaded vehicles are randomly separated by a distance of between f i v e and ten miles. The position where the two vehicles would meet i f the vehicles t r a v e l unhindered i s determined.. The f i n a l phase was to move the empty vehicle to the f i r s t turnout i t can safely u t i l i z e . Simultaneously, the loaded vehicle was backtracked for an equal time i n t e r v a l . The F-factor can be eas i l y determined by ca l c u l a t i n g the distance the loaded vehicle i s from the empty vehicle, stopped in the turnout, and dividing t h i s value by the turnout spacing.. This procedure i s repeated for the desired number of repetitions and the sample mean i s determined. The program i s executed either nine or twenty times and the mean and standard deviation of the samples are determined.. A t - t e s t i s performed with the n u l l hypothesis being that the mean of the average F-factors i s equal to the simple F-factor. . The speed and turnout spacing combinations were a r b i t r a r i l y chosen. . The results of the simulations are documented i n Table I I I . The results of the simulations, with a one percent l e v e l of si g n i f i c a n c e , were not rejected seventy-three percent of the time. . 5 3 Table III F-factor Simulation The Interaction Between A Single Loaded Vehicle And A Single Empty Vehicle Based On Equation 2 . 1 2 1 T |Vehicle Speed|Turn 2 r 1 1 Spac Empty j (mph) I I I r Sample* Loaded (mph) + + + + 1 0 0 0 0 (9) | 10 | 2 0 | 0 . 1 I 0 . 7 5 1 7 | 5 . 9 | 0 . 7 5 0 0 | No 5 1 0 0 0 0 ( 9 ) | 10 i 2 0 JO. 5 | 0 . 7 5 2 4 | 4 . 5 ( 0 . 7 5 0 0 1 No s 1 0 0 0 0 ( 9 ) | 3 0 | 3 0 1 0 . 0 5 I 0 . 9 9 9 5 1 3 . 0 | 1 . 0 0 0 0 1 NO 5 1 0 0 0 0 (9) | 3 0 | 3 0 | 0 . 15 | 1 . 0 0 2 5 1 4 . 4 | 1 . 0 0 0 0 1 NO s 1 0 0 0 0 (9) | 3 0 | 4 0 | 0 . 3 | 0 . 8 9 2 3 | 5 . 2 1 0 . 8 7 5 0 1 Yes 1 0 0 0 0 (9) | 10 | 3 5 I 0 . 15 | 0 . 6 3 9 1 | 3 . 8 | 0 . 6 4 2 9 | Yes 6 1 0 0 0 0 ( 2 0 ) | 10 | 3 5 I 0 . 2 | 0 . 6 4 5 3 | 3 . 4 | 0 . 6 4 2 9 | Yes 1 0 0 0 (20) | 3 0 | 4 0 | 0 . 2 | 0 . 8 6 8 4 | 1 4 . 5 | 0 . 8 7 5 0 | No 1 0 0 0 (20) | 3 0 | 4 0 I 0 . 3 | 0 . 9 0 2 6 | 1 4 . 1 1 0 . 8 7 5 0 | Yes 1 0 0 0 (20) | 10 | 3 5 | 0 . 15 | 0 . 6 3 8 6 | 7 . 8 | 0 . 6 4 2 9 | Yes* 1 0 0 0 (20) | 10 | 3 5 1 0 . 3 | 0 . 6 3 8 2 | 1 0 ; 3 | 0 . 6 4 2 9 | No i L— . — i —1 L ______ _ J _ L Ave. F S. D. 3 xlO - 3 Eq. • F |Rejection | of I Hypothesis |oC= 0 . 0 5 1 The sample s p e c i f i c a t i o n s : where the non-parenthesized number i s the sample size and the parenthesized number i s the: number of samples 2 The distance between turnouts i n miles 3 Standard deviation * The equation value of the expected F-factor 5 Hypothesis was not rejected for a 10% l e v e l of s i g n i f i c a n c e 6 Hypothesis was not rejected f o r a 1% l e v e l of sig n i f i c a n c e 5 4 A s i m i l a r simulation model was used to test the v a l i d i t y of the expected F-factor equation r e s u l t i n g from a single empty vehicle i n t e r a c t i n g with a f l e e t of loaded vehicles.. This model includes the p r o b a b i l i s t i c aspects of flow behaviour by having the headway d i s t r i b u t i o n of the loaded vehicles conform to a predetermined probability d i s t r i b u t i o n . . The model simulates the meeting of a single empty vehicle and a f l e e t of loaded vehicles. The empty vehicle progresses along an endless road u n t i l i t meets a loaded vehicle.. I t i s then backtracked to the f i r s t turnout i t can safely u t i l i z e . The model allows the empty vehicle to wait in the turnout u n t i l one or more loaded vehicles have passed.. The experience of the model i s collected by the same method and the expected F-factor equations tested i n the same manner as the previous model. Simulations were conducted for the exponential and Erlang (alpha=2) headway frequency d i s t r i b u t i o n s . The r e s u l t s are tabulated in Tables IV and V. With a one percent l e v e l of sig n i f i c a n c e the exponential F-factor equation was within acceptable l i m i t s ninety percent of the time while the estimate of the Erlang (alpha=2) F-factor equation was always within acceptable l i m i t s . . The range of t r a f f i c flow for which the pro b a b i l i t y form of the expected F-factor equations adequately describe the meeting of two f l e e t s of vehicles must yet be determined. Consequently, a simulation model was developed to predict the expected F-factor r e s u l t i n g from the meeting of two f l e e t s of vehicles. „ Since t h i s model i s more complex than the previous models a General Purpose.Simulation System V (GPPSV) program was used to model t h i s s i t u a t i o n . . The c h a r a c t e r i s t i c s of the 5 5 model are outlined i n Appendix 5 . The r e s u l t s of the simulation runs are tabulated i n Table VI.. The equations were tested by the same method as the previous models.. The r e s u l t s indicate that the exponential F-factor equation i s s u f f i c i e n t l y accurate for a t r a f f i c flow rate of 60 vph or les s and therefore equation 2.19 i s s u f f i c i e n t l y accurate for a l l pr a c t i c a l levels of t r a f f i c flow involved in log hauling. A reason for the f a i l u r e of the t - t e s t , with a flow rate of 100 vph, i s that the number of conditional event B delay situations has increased to the point where they s i g n i f i c a n t l y a f f e c t the expected F-factor. 56 Table IV F-factor Simulation -- An Empty. Vehicle Meeting Loaded Vehicles Based On Eguations~2. 18, 2* 11M. 2. 30 x And"2^3Jl Sample size = 10000 Number of samples = 9 Acceleration = 19759 mph2 " C o n f l i c t " hours = 11 Exponential headway d i s t r i b u t i o n I T 1 1 1 1 T 1 |Vehicle Speed|Distancel T r a f f i c I Ave. | S. D. H Expect | Rejection | R T — T Between | Flow I F | X10-3| F | of | |Loaded|Empty |Turnouts| Bate | j | | Hypothesis| | (mph) | (mph) | (miles) j (vph) + r + 4 oc = 0.05 | | 10 | 20 | 0. 1 | 1 |0.7476| 3.06|0.7507| Yes3 | I I I | 2 I0.7492J 3.1410.7514| No | I I I I 4 |0.7523| 4.46|0. 7526| No2 | I I I | 10 | 0. 7521 | 4.18|0.7543| NO2 | I I I I 60 ) 0.7023 | 5.02|0. 7034| No2 | I I I I 100 |0.6247| 3. 16|0.6269| No | I 30 | 30 | 0.05 | 1 | 0.9980 | 5.42| 1. 0020| No | I I I 1 2 | 1.00561 4.23| 1.00421 No2 | I I I 1 4 1 1.0131| 3.23|1.0085| Yes j I I I 1 10 | 1.02151 6. 07| 1. 0208| No2 | I I I 1 60 | 1.09221 6. 03| 1.09711 Yes* | 1 I I 1 100 1 1.1317| 5. 3811. 1300 I No2 | | 40 | 30 | 0. 2 | 1 I 1. 1697| 10.66|1.1684| No2 | I I I 1 2 1 1. 1694| 6.03|1.1703| No2 | I I I 1 4 |1.1690| 3. 24| 1. 17371 Yes | I I I 1 10 | 1. 1804| 5.26|1.1817| No2 | I I I | 60 |1.1503| 5.59|1.1558J Yes* | I I I 1 100 | 1.0690J 6.03|1.07181 No2 | | 30 | 40 | 0.3 | 1 |0.8737| 4.76|0.8767| No2 | I I 1 1 2 |0.8799| 5.85|0. 8785| No2 | | | | | 4 |0.8811| 4.07|0.8815| No2 | I I I 1 10 10.8841| 5.18|0.8873| No2 | I I I 1 60 |0.8138| 7. 80|0.8165| No2 | I I I | 100 |0.7062| 4.86|0.7054| No2 j 1 1 1 L _ _ _ _ _ _ _ X L 1 L i 1 Standard deviation 2 Hypothesis was not rejected f o r a 10% l e v e l of si g n i f i c a n c e 3 Hypothesis was not rejected for a 1% l e v e l of sig n i f i c a n c e 57 Table V - - - F - f a c t o r S i m u l a t i o n -- IS Empty Loaded V e h i c l e s Based On Equations 2. 18, 2. 19, Sample s i z e = 10000 Number of samples = 9 A c c e l e r a t i o n = 19759mph2 " C o n f l i c t " hours = 11 Erlang(alpha=2) headway d i s t r i b u t i o n Vehicle Meeting 2.33. And~2.34 |Vehicle Speed|Distance T r a f f i c 1 Ave. | r 1 1 Between Flew 1 F | |Loaded|Empty ITurnouts Kate I (mph) | (mph) | (miles) l + +  (vph) + r | 30 | 40 | 0.3 1 |0.8740| I I I 1 2 |0.6756| I I I 4 | 0. 8784 | I I I i 10 |0.8974| I I I 1 60 | 0.9762| I I I 1 100 |0. 8317| | 40 | 30 | 0.2 ! 1 | 1. 1679| I l l 2 I 1. 1647| I I I 1 4 | 1. 1692| I I I 1 10 |1.1829| I I I 1 60 | 1. 32431 I I I 1 100 | 1.2811 | 4.46|0. 8753| No 2 5.44|0. 8761| No 2 6.71|0.8793| No 2 4.79|0. 8973| No 2 5.67|0. 9733| No 2 6.96|0. 8345| No 2 I 6. 13| 1. 1668| No 2 6. 18| 1.. 1674| No 2 5.14|1. 1695| No 2 6.18|1. 182 51 No 2 9.39|1.3259| No 2 7.94|1.2856| No 2 S.D. 1|Expect|Rejection X10 - 3 | F I of Hypothesis oc = 0.05 H  3 j pI I * Standard d e v i a t i o n 2 Hypothesis was not rejected for a 10% l e v e l of si g n i f i c a n c e 58 Table VI F-factor Simulation -- Interaction Between Two Fleets Of - Vehicles Based On Equations 2.187 2^19 x 2.30 x And 2. 3,1 Empty vehicle speed = 40 mph Loaded vehicle speed = 30 mph Acceleration = 19759 mph2 Exponential headway d i s t r i b u t i o n Ave. . Sample Size r 9718 24150 37174 35390 i N umber|Distance| Of |Between Samples | Turnouts| |(miles) I 4 I 0.3 I 10 I 5 I 0. 3 | 20 I 5 I 0.3 | 60 I 5 I 0. 3 | 100 Ave. |S.P. 1|Expect|Rejection F |X10-3| F j of I I I I I I I I I I 0.88421 1. 03|0. 8873| I I I 0.87571 1.24|0.8873| I I I 0.84831 1. 70|0. 8165| I I I 0.90981 4. 57|0.7054| I Hypothesis | oL= 0.05 No2 No2 Yes* Yes * Standard deviation 2 Hypothesis was not rejected f o r a 10% l e v e l of sig n i f i c a n c e 3 Hypothesis was not rejected for a 1% l e v e l of significance 59 5.0 THE COST VARIABLES AHD THE MODIFICATION OF THE EXPECTED F-FACTOR 5.1 THE COST OF TRUCK TRANSPORTATION The cost equations developed i n Chapter 2 require an estimate of trucking costs. This cost involves such cost elements as the wages of the operator, vehicle licences, permits, insurance, depreciation, repair, maintenance, f u e l , and o i l . An estimate of these cost elements can be obtained from past records, production manuals, appraisal manuals, equipment ren t a l manuals, and technical reports. The Journal of Logging Management p e r i o d i c a l l y publishes equipment rental rates prepared by the Truck Loggers Association, Vancouver, B.C.. (Journal Of Logging Management, 1978). . Their calculations include depreciation (straight l i n e ) , i n t e r e s t (simple 15 percent), insurance ($1.30 per $100.00 of investment), repair and maintenance (55 percent of depreciation), t i r e s , l u b r i c a t i o n , wages of the operator (International Woodworkers of America wage plus a cost of l i v i n g provision plus 30 percent p a y r o l l loading), overhead (10 percent of t o t a l ) , and p r o f i t (10 percent of t o t a l ) . The renta l rate i s based on a ten-hour day but there are no provisions for fuel costs. Rental rates are calculated f o r two truck classes (twelve and f i f t e e n foot bunks) and three coastal zones. . The United States Bureau of Land Management produces a manual to predict the truck hauling costs for Oregon and 60 Washington (United States Bureau of Land Management, 1977). The cost predicted for a White truck (Model 4964) and Peerless t r a i l e r includes a fixed cost ($6.59 per hour), an operating cost ($9.32 per hour), a driver's wage ($11.62 per hour) and an overhead cost ($2.75 per hour). Smith and Tse(1977) subdivided trucking costs into in-use costs and t r a v e l l i n g costs. In-use costs include fixed costs (depreciation, i n t e r e s t , opportunity charges, and insurance) and the wage of the driver while the t r a v e l l i n g costs include such items as f u e l , o i l , t i r e s , r epairs, and maintenance. Smith tabulated the hauling costs for three truck classes (eight, ten, and twelve foot bunks) . Table VII summarizes the predicted hauling costs from these three sources. The re s u l t s of these a r t i c l e s cannot be d i r e c t l y compared since the costs apply to three d i s t i n c t operating areas (north central i n t e r i o r of B.C., coastal B.C., and northwestern United States). Consequently, a cost element from one area may be higher or lower than that cost element for the other two areas. These costs are given not as a hard and fast rule but as a guideline for the cost elements that should be included in the calculation of the adjusted hauling rate.. 6 1 Table VII Truck Hauling Costs Source | Zone | Bunk | Hourly Hauling Cost j Size | (feet) + 1 T ~ T " I Straight|Overtime| | Time | | 1 ($) 1 ($) 1 + r +-Delay Time ($) Smith and | I 8 I 24. 19» | | Tse(1977) | I 10 1 12 I 27.12* | | | 32. 241 | | Journal cf Logging| Basic 2 I 12 | 34. 553 | | Management (1978) | A 2 | 12 1 37. 053 | j ! B 2 1 12 \ 39. 553 | | ! Basic 1 15 | 45. 453 | j ! A 1 15 | 48.643 | | ; B I 15 | 52.053 | | United States | Bureau of Land | Management (1977) | | | 30.28 | 32.26 | 20.03 1 Based on average hourly cost between 'small' log and 'large' log loads. 'Large' sawlogs had a minimum butt diameter of fourteen inches. 2-The zones refer to i s o l a t i o n areas where: A = up coast areas with poor access B = i s o l a t e d parts of Dean and Rivers PSYU's and Queen Charlottes etc. 3-These are the rental rates without including the ten percent p r o f i t margin. . 62 5_i.2 THE TOBNOOT CONSTBUCTION AND MAINTENANCE COSTS Boad construction costs can be obtained from company records* manuals, and various publications but seldom do they categorize turnout construction costs. Since turnouts are usually located where the road can be r e l a t i v e l y e a s i l y widened the actual turnout construction cost i s somewhat less than that anticipated for a comparable stretch of road. Sauder and Nagy(1977) estimated the road construction costs for three road classes (B.C.. Forest Service - 3, 4a, 4b). The c a l c u l a t i o n s included the cost estimation of the subgrade construction (cable shovel, bulldozer and d r i l l ) , the b a l l a s t i n g (from rock quarries and gravel pits) , and the surfacing. The B.C.. Forest Service D i s t r i c t s have developed appraisal manuals (B.C. Forest Service(1975), B.C.. Forest Service (1977)) to estimate road construction costs under a variety of conditions (terrain, s o i l s , road c l a s s e s ) . . These manuals can be used to calculate the cost of a s t a t i o n of road and then apply a portion of t h i s figure to the turnout construction cost. The United States Bureau of Land Management(1977) determined the turnout excavation costs for two classes of turnouts (a f i f t y - f o o t turnout plus two twenty-five-foot approaches and a one hundred-foot turnout plus two f i f t y - f o o t approaches) under a variety of s o i l conditions and side slope classes. The t o t a l turnout construction cost included the excavation cost plus the grading and surfacing costs. The 63 r e s u l t s of t h e i r costing study are shown i n Table:VIII.. Turnout maintenance costs can be estimated from road maintenance costs. The cost per unit surface area i s not equivalent since the turnouts would experience less wear than the main road surface. The B.C.. Forest Service(1977) estimated the machine rates, production rates, and maintenance schedule for mainline and spur roads. Table IX i l l u s t r a t e s the projected turnout maintenance costs based on these figures. These figures must be altered to r e f l e c t the d i f f e r e n t deterioration rates between the turnout surface and the roadway surface. 64 Table VIII Excavation Costs Per Turnout* 14-foot Subgrade (10-foot Usable Width)* % Side Slope T Common Excavation T -I Sock Excavation Cost/ |Avg.,Cut Turnout|at Center |Line-Ft.. 1 I Avg. .Cu Yards/ Turnout 0 i $ 7.10| 1.3 | 28 |$ 50.70| 1.3 | 26 10 | 7. 10 | 1.3 | 28 | 50.70| 1.3 | 26 20 | 8. 15| 2.0 | 32 | 103. 251 2.0 I 53 30 | 12.45| 2.7 | 49 | 197.001 2. 8 | 101 40 | 13.45 | 3.5 I 53 | 138. 451 3.5 I 71 50 | 21.85| 4.7 | 86 | 206.701 4.7 | 106 60 | 79.00| 8.0 | 311 | 461. 701 8. 0 | 255 70 | 171.20| 12.0 | 674 | 970.701 12.0 | 509 80 | 208.80| 13.2 | 822 | 1146.601 13. 8 | 588 90 | 262. 15| 14.8 | 1032 | 1368. 90| 15.0 | 702 100 j 316.251 17.0 | 1245 | 1618.501 17. 0 | 830 x. i_ 1 1 Cost/ IAvg..Cut |Ave. Cu. Turnout |at Center!Yard/ |Line-Ft. |Turnout i Standard approaches lengths: 50-foot turnout plus two 25-foot Side Slope r "T" I r -20-foot Subgrade (12-foot Usable Width) 2 — 1  Common Excavation | Rock Excavation Cost/ Turnout Avg. Cut at Center Line-Ft. T 1 T 1 Avg. Cu| Cost/ IAvg. Cut |Ave..Cu. Yards/ I Turnout |at Center|Yard/ Turnoutl |Line-Ft..|Turnout I 1 0 | $ 1 9 . 5 5 1 1 . 7 | 7 7 | $ 2 4 3 . 7 5 | 1 . 0 | 1 2 5 1 10 | 1 9 . 5 5 | 1 . 7 I 77 | 2 4 3 . 75 J 1 . 0 | 1 2 5 I 20 | 2 6 . 4 0 | 3 . 0 | 104 | 3 5 6 . 8 5 | 2 . 5 | 183 | 30 | 3 0 . 2 2 | 3 . 1 | 119 I 3 7 2 . 4 5 ) 3 . 1 | 191 I 40 | 5 2 . 601 4 . 0 | 2 0 7 | 4 6 6 . 0 5 | 4 . 0 | 2 3 9 I 50 | 54 .10| 5 . 7 | 2 1 3 | 4 0 7 . 5 5 I 5 . 6 ! 2 0 9 | 60 | 2 6 9 . 0 0 | 1 0 . 1 | 1 0 5 9 | 1 7 0 0 . 4 0 ! 1 0 . 1 | 8 7 2 | 7 0 | 4 3 6 . 4 0 | 1 4 . 0 | 1 7 18 | 2 5 6 0 . 3 5 ! 1 4 . 0 | 1 3 1 3 I 80 | 5 4 7 . 10 | 1 6 . 0 | 2 1 5 4 | 3 1 2 0 . 0 0 1 1 6 . 0 | 1 6 0 0 I 90 | 6 7 4 . 3 5 1 1 8 . 0 | 2 6 5 5 | 3 5 0 2 . 2 0 | 1 8 . 0 ! 1 7 9 6 I 100 | 7 9 4 . 5 0 1 2 0 . 0 | 3 1 2 8 | 4 0 9 3 . 0 5 1 2 0 . 0 | 2 0 9 9 2 Standard lengths: 100-foot turnout plus two approache s * From United States Bureau of Land Management(1977) 50-foot 65 Table IX Turnout Maintenance C o s t s 1 2 Machine (Grader) T r Rate per S h i f t Cat 12 Cat 14 Cat 16 $227 277 334 Road Class Maintenance Schedule in days 3 r Mainline Spur Mainline Spur Mainline Spur 20 40 20 40 20 40 Annual Maintenance Cost per Turnout 2 miles|3 miles / s h i f t | / s h i f t $11.448|$ 7.632 | 3.816 13.9701 9. 313 I 4.657 16.844| 11.229 | 5.615 4 milesl / s h i f t $ 2.862 3. 492 4.211 1 assume a road width of sixteen feet and a turnout area of seven hundred f i f t y square feet. Calculations based on four turnouts per mile, two hundred operating days per year and eighty-five percent a v a i l a b i l i t y . 2 Based on B.C. Forest Service (1977) 3 These costs can be readily adjusted to determine the costs for various maintenance periods. 66 5.3 MODIFICATION OF THE EXPECTED F^FACTOE E<2DATIONS The expected F-factor equations, developed in Chapter 2 , assume there i s no minimum waiting time i n a turnout but i n r e a l i t y there i s a minimum delay period. In t r a f f i c analysis the merging of t r a f f i c from a ramp into the main t r a f f i c stream may be c l a s s i f i e d as an ideal or forced merge., An id e a l merge does not cause the main t r a f f i c stream to reduce speed or change lanes to allow the merging vehicle to enter the main t r a f f i c stream but a forced merge results i n one of these phenomena to occur. A p a r a l l e l can be drawn with regards to a logging truck u t i l i z i n g a turnout where an i d e a l p u l l - i n w i l l not cause the loaded vehicle to reduce speed to allow the empty vehicle t o u t i l i z e the turnout but a forced p u l l - i n w i l l cause the loaded vehicle to reduce speed., In the c a l c u l a t i o n of the F-factor i t has been assumed the vehicles t r a v e l at a constant speed except when the empty vehicle decelerates into a turnout and accelerates from a turnout. The operator of the empty vehicle may be able to increase his speed over a section of road so as to change the c l a s s i f i c a t i o n of a p u l l - i n from being a forced p u l l - i n to an i d e a l p u l l - i n . Provided the operator of the empty vehicle knows the loc a t i o n of the loaded vehicles a modification of the expected F-factor should only apply to a forced p u l l - i n . The model does not account for the length of the turnout. . If the length of the turnout i s accounted for then there i s a section of road where there are two lanes such that the empty vehicle and loaded vehicle can pass each other. This though 67 would not s i g n i f i c a n t l y a l t e r the c l a s s i f i c a t i o n o f a p u l l - i n . In c o n c l u d i n g , i f the operator does not know the l o c a t i o n of the other v e h i c l e s then a m o d i f i c a t i o n of the expected F-f a c t o r equation may be r e q u i r e d . However, i n t h i s study i t w i l l be assumed the operator knows the l o c a t i o n of the other v e h i c l e s . . 68 6.0 THE USE AND TESTING OF THE MODEL 6.1 THE USE OF THE MODEL The solution of the optimum turnout spacing model r e s u l t s from the evaluation of a complex set of equations and the u t i l i z a t i o n of a search technique. This solution can be used as a guideline f o r determining turnout spacing and can be used as an aid i n determining when the road should be switched from being single-lane to double-lane.. Furthermore, a section of the model, equation 2.8, can be u t i l i z e d to determine the t o t a l expected delay time per vehicle per unit distance:of road.. Figures 8, 9, 10, and 11 i l l u s t r a t e the ef f e c t the independent variables, t r a f f i c flow, turnout spacing, v e l o c i t y of the loaded vehicles, and ve l o c i t y of the empty vehicles, have on the t o t a l expected delay time per vehicle per unit distance of road. It i s assumed the rate of acceleration i s 11855 mph2, the rate of deceleration i s 19759 mph2, and the headway frequency d i s t r i b u t i o n i s exponential.. The formulas to determine the rates of acceleration and deceleration are developed i n Appendix 3. Unless i t i s stated otherwise, i t i s assumed the velo c i t y of the loaded vehicle i s 20 mph, the veloc i t y of the empty vehicle i s 25 mph, the turnout spacing i s 0.1 miles, and the t r a f f i c flow rate i s 4 vph.. Furthermore, the t o t a l expected delay time (T) i s expressed as a percentage of the t o t a l t r a v e l empty time. Figure 8 Effect of t r a f f i c flow rate on the t o t a l exp delay time for various turnout spacings ected 7 0 U . U J X „ ten i—a. S J - U - K u , -tr Lrl L/1 m m Lrl rvi P J ih Lrl m S I m S3 Lrl rvi Ln rvi si x: ID Z <*; r c a . Ln in 2 a : Ln sa sa Lrl is _ J > - u i > -r x r x u j f e * > » — U J i - j £ c n z : • U J — z u - o s x : — j - t a t — u i r -Figure 9 E f f e c t of the turnout spacing on the t o t a l expected delay time for various t r a f f i c flow rates 7 1 Figure 10 E f f e c t of the velocity of the empty vehicle on the t o t a l expected delay time for various v e l o c i t i e s of the loaded vehicle 7 2 Figure 11 E f f e c t of the velocity of the loaded vehicle on the t o t a l expected delay time f o r various v e l o c i t i e s of the empty vehicle 73 Figure 8 i l l u s t r a t e s the e f f e c t of the t r a f f i c flow rate (1 to 20 vph) on the t o t a l expected delay time for various turnout spacings (0.05 to 0.35 miles). The t o t a l expected delay time {%) increases as the t r a f f i c flow rate and the turnout spacing increase, though the marginal e f f e c t decreases as the t r a f f i c flow rate increases. For most log hauling s i t u a t i o n s , where the t r a f f i c flow rate i s less than 5 vph, the t o t a l expected delay time i s less than 20 percent of the t r a v e l empty time. Figure 9 i l l u s t r a t e s the e f f e c t that the turnout spacing has cn the t o t a l expected delay time for various t r a f f i c flow rates. The t o t a l expected delay time (%) increases and the marginal e f f e c t decreases as the turnout spacing increases. The t o t a l expected delay time i s less than 15 percent of the t r a v e l empty time provided the t r a f f i c flow rate i s l e s s than 5 vph and there i s a minimum of 5 turnouts per mile. . Figure 10 i l l u s t r a t e s the eff e c t that the velocity of the empty vehicle (15 to 45 mph) has on the t o t a l expected delay time for various v e l o c i t i e s of the loaded vehicle (10 to 40 mph). The t o t a l expected delay time (%) and the marginal effect increase as the velocity of the empty vehicle increases. The t o t a l expected delay time i s less than 20 percent of the travel empty time* Figure 11 depicts the e f f e c t that the velo c i t y of the loaded vehicle (10 to 35 mph) has on the t o t a l expected delay time f o r various v e l o c i t i e s of the empty vehicle (20 to 40 mph). The t o t a l expected delay time and the absolute value of the f i r s t derivative of the function decrease as the velocity of the loaded vehicle increases., Based on these four figures, the general trends that an increase or decrease i n one 74 of the independent variables have on the t o t a l expected delay time can be predicted. Since the cost function i s complex and a search technique i s required to locate the optimum turnout spacing, i t i s advantageous to develop a computer program to determine the optimum turnout spacing. Appendix 8 documents a computer program that can be u t i l i z e d to determine the optimum turnout spacing.. Besides determining the optimum solution the program calculates the costs, based on equation 2.38, fo r the optimum turnout spacing, 50 percent of the optimum turnout spacing, 100 percent of the optimum turnout spacing, and 200 percent of the optimum turnout spacing. A further discussion on the s e n s i t i v i t y of the optimum solution to fluctuations in the equation's variables i s located i n section 6.2. . Figures 12, 13, 14, 15, 16, and 17 i l l u s t r a t e the e f f e c t that the independent variables, the velocity of the loaded vehicle, the velocity of the empty vehicle, the t r a f f i c flow rate , the turnout construction cost, the adjusted truck hauling cost, and the expected useful l i f e of the road, have on the optimum turnout spacing. In the ca l c u l a t i o n of the optimum turnout spacing i t i s assumed the rate of acceleration i s 19759 mph2, the rate of deceleration i s 11855 mph2, the number of " c o n f l i c t " hours per day i s 5, the number of operating days per year i s 200, and the headway frequency d i s t r i b u t i o n i s exponential. Unless i t i s stated otherwise, i t i s assumed the velocity of the leaded vehicle i s 25 mph, the velocity of the empty vehicle i s 40 mph, the t r a f f i c flow rate i s 4 vph, the turnout construction cost i s $100, the adjusted truck hauling cost i s $15 per hour, and the expected useful l i f e of the road 75 i s 20 years. Figure 12 depicts the effect that the velocity of the loaded vehicle (10 to 45 mph) has on the optimum turnout spacing f o r various v e l o c i t i e s of the empty vehicle (25 to 40 mph). The optimum turnout spacing increases but the marginal effect decreases as the v e l o c i t y of the loaded vehicle increases. For any given velocity of the loaded vehicle, the optimum turnout spacing increases as the velocity of the empty vehicle increases. Figure 13 i l l u s t r a t e s the e f f e c t that the velocity of the empty vehicle (14 to 45 mph) has on the optimum turnout spacing for various values of the velocity of the loaded vehicle (10 to 35 mph). The same general relationships between the optimum turnout spacing and the independent variables, as shown i n Figure 12, are evident i n Figure 13. The e f f e c t that the t r a f f i c flow rate (1 to 20 vph) has on the optimum turnout spacing i s i l l u s t r a t e d in Figure 14. The optimum turnout spacing decreases as the t r a f f i c flow rate increases. The general shape of the curve appears to be hyperbolic and asymptotic to the x-axis and y-axis.. Figure 15 depicts the e f f e c t the turnout construction cost ($50 to $1000) has on the optimum turnout spacing. The optimum turnout spacing increases and the marginal e f f e c t decreases as the turnout construction cost increases. The.effect that the adjusted hauling cost ($1 to $45 per hour) has on the optimum turnout spacing i s i l l u s t r a t e d i n Figure 16. The general shape of the curve appears to be hyperbolic. The optimum turnout spacing decreases as the adjusted hauling cost increases. 76 Figure 17 i l l u s t r a t e s the ef f e c t that the expected useful l i f e of the road (1 to 25 years) has on the optimum turnout spacing. The optimum turnout spacing decreases as the expected useful l i f e of the road increases.. The general shape of the curve can be depicted by a hyperbolic function.. Based on the six figures, the general e f f e c t that changes to one of the independent variables has on the optimum turnout spacing can be predicted. Figure 18 i l l u s t r a t e s the cost function, based on equation 2.38. In t h i s diagram i t i s assumed: 1.,aA = 19759 mph2 2. a D = 19759 mph2 3. 5 " c o n f l i c t " hours per day 4. 200 operating days per year 5. L = 50 feet 6. the driver's reaction time i s 2 seconds 7. H = 4 vph 8. V, = 20 mph 9. V2 = 25 mph 10. C = $100 11. M = $15 per hour 12. ,Q] = 20 years 13. exponential headway frequency d i s t r i b u t i o n . . The cost function curve begins r e l a t i v e l y steeply but f l a t t e n s as the.optimum turnout spacing i s approached.. The portion to the r i g h t of the optimum turnout spacing i s not has steep as the portion to the l e f t of the optimum. A further discussion of the e f f e c t that perturbations tc the turnout spacing has on 7 7 the c o s t equation i s d i s c u s s e d i n s e c t i o n 6.2.13., 7 8 Figure 12 E f f e c t of the velocity of the loaded vehicle on the optimum turnout spacing for various v e l o c i t i e s of the empty vehicle 79 i u. a. • = >- ui ta ui tsa ui eg i - u i m m r v j IN — — IS U1 fS Lrl t g Lrl IS m rvj LD m rn ID eg ui w x m m IM rvi S O — r -— Z « ^ U J h - E K C C U J L U U . Q r - U l u Figure 1 3 E f f e c t of the velocity of the empty vehicle optimum turnout spacing for various v e l o c i t i e s of the vehicle on loa the aed 80 Figure 14 Ef f e c t of the t r a f f i c flow rate on the optimum turnout spacing 81 Figure 15 E f f e c t of the turnout construction cost on optimum turnout spacing the 8 2 U l X X ES X m v. in a vv in rvi i n z rc x —; ra v> rvi = CC Ul I ra i n eg ra rvi r-ca n ut ra x m ui x i - m x r _ — i -— z v u f— CC EC LU o.na.u. Qr-Ulw ra in m ra r-ra tn ui ra m m ts ra rvi Figure 16 E f f e c t of the adjusted truck hauling cost on the optimum turnout spacing 8 3 Figure 17 E f f e c t of the expected useful l i f e of the road the optimum turnout spacing on 84 EJ CD eg rg rg tg LO eg eg Lrt eg eg x eg es m eg eg rvj eg eg eg eg eg eg eg eg eg eg eg ra eg eg eg eg eg in ta Lrt eg Lrt eg Lrt eg eg X X n rvi rvi —• — Lrt eg Ul C£ DZ r- _1 Ul Ul I OS I nzaui — v»i —_»n.:e Figure 18 E f f e c t of the turnout spacing on the.cost function based on equation 2,38 85 6. 2 SENSITIVITY ANALYSIS The previous chapters have o u t l i n e d a method to c a l c u l a t e the optimum turnout spacing but the formulas u t i l i z e d assume t h a t there are no e r r o r s i n the independent v a r i a b l e s of the formulas. . G e n e r a l l y , the so c a l l e d " t r u e " values of the v a r i a b l e s are unknown and estimates of these values are required . . A measure of the s t a b i l i t y of the f u n c t i o n s with respect to v a r i a t i o n s i n the value of the v a r i a b l e s i s d e s i r a b l e . . This s e c t i o n w i l l i n c l u d e a s e n s i t i v i t y a n a l y s i s of the optimum turnout spacing f u n c t i o n , expected F - f a c t o r f u n c t i o n , t o t a l expected delay time f u n c t i o n , and cost f u n c t i o n with respect to the independent v a r i a b l e s , some of the assumptions, and headway d i s t r i b u t i o n s . The general format of t h i s s e c t i o n i s t o discuss the e f f e c t t h a t the independent v a r i a b l e s , the assumptions of the model, and the headway d i s t r i b u t i o n s have on the f u n c t i o n s . Simulation i s the b a s i c modelling t o o l used i n the s e n s i t i v i t y a n a l y s i s since the equations are g e n e r a l l y too complex to allow f o r a d i r e c t comparison method. D i s c r e t e , d e t e r m i n i s t i c s i m u l a t i o n models were w r i t t e n i n BASIC f o r the HP9830A desktop c a l c u l a t o r to solve the expected F - f a c t o r , the t o t a l expected delay time, the optimum turnout spacing, and the cost f u n c t i o n f o r a wide range of values of the independent v a r i a b l e s . The basic method i s to loop the independent v a r i a b l e s between a lower and an upper l i m i t i n constant or v a r i a b l e increments. The independent v a r i a b l e being t e s t e d i s looped i n a s i m i l a r manner but the f u n c t i o n i s a l s o evaluated f o r perturbed values 8 6 of the independent variable. The difference between the actual and the perturbed values of the function are calculated.. Once the experience of the simulation i s assembled into groups, the average, the standard deviation, and the maximum value of each group are determined.. Graphs of some of the r e s u l t s of the simulations are located i n Appendix 9 . . 87 6.2.1 VELOCITY OF THE LOADED VEHICLE There i s a potential error between the estimated and true value of the average velocity of the loaded vehicle. It w i l l be assumed that the estimated value w i l l be within 10 mph of the:actual value. Therefore the s e n s i t i v i t y analysis w i l l be based on a measurement error of 10 mph or less. I t can be shown that for the simple F-factor the rate of change in the expected F-factor (6F) for a given perturbation i n the velocity of the loaded vehicle (<5v( ) i s : Sf = 1/(2VZ) This formula can be u t i l i z e d as an estimation of the rate of change for the expected F-factor involving a single empty vehicle meeting a f l e e t of loaded vehicles. This formula, though, does not account for the t o t a l expected delay time error. . Since the t o t a l expected delay time function i s complex a simulation model i s u t i l i z e d to determine the percentage difference in the t o t a l expected delay time T) for various perturbations in the velocity of the loaded vehicle. The results of the simulation, as i l l u s t r a t e d i n Figure 19, show that the error may be large but i t coincides with a large error in the value of the velocity of the loaded vehicle. The assumptions, the independent variables, and the range of values of these variables are: 1. The headway frequency d i s t r i b u t i o n i s exponential 2. art = a 0 = 19759 mph2 3. S = 0.05, 0. 10, 0. 15, 0. 20, 0.25, 0. 30, or 0.35 miles 4. V, = 10, 20, 30, or 40 mph 8 8 5. \ = 10, 20, 30, or 40 mph 6. H = 2, 6, or 10 vph.. The v e l o c i t i e s of the vehicles are further constrained by: V( < • 10 mph \ < V, + 20 mph. As i l l u s t r a t e d i n Figure 20, there i s a smaller percent difference in the optimum turnout spacing (£s*) than i n the i n i t i a l percent perturbation i n V, . . The difference between the percentage values, <£V, -Ss*r decreases as the perturbation of V, i s decreased. As was the case in the previous simulation the same independent variables and the i r range of values are u t i l i z e d i n t h i s simulation. The extra variables required are: 1. Q( = 2 or 10 years 2. C = $100, $600, or $1000 3. M = $25 or $45 per hour 4. 200 operating days per year 5. 5 " c o n f l i c t " hours per day 6. The stopping distance i s less than one-half of the turnout spacing. By themselves large perturbations in V, appear to be s i g n i f i c a n t , but the e f f e c t that these perturbations have on the cost equations i s the r e a l question.. This relationship i s i l l u s t r a t e d i n Figure 21. . The values of the independent variables are the same as for the optimum turnout spacing simulation. The cost difference i s les s than $50 per vehicle per year per mile for values of V, greater than 20 mph. Velocity perturbations based on a true v e l o c i t y of 10 mph generally r e s u l t in a cost difference of less than $100 per vehicle per year per mile. This value transformed into d o l l a r s 89 per cunit, based on 25 cunits per t r i p , r e s u l t s i n a cost difference of less than $0.01 per cunit-mile., 90 i i 1 + 1 1 1 I 1 •+• z 1 1 a I + — » 1 I i— cc —• ta x cc a. UJ x Z2 x 1— LV. x OS a — U UJ X CL > EC CCS X CL U l _J X U l > a ui t _ x • lu. a • u i U J K> Z U l EK UJ U . U . U l _ l < ~ ' > -XUJXUI l - O . J E " - 2 Q X L J - K c _ — r - UJ <_ r - w Figure 19 E f f e c t of perturbations to the velocity of the loaded vehicle on the t o t a l expected delay time Figure 20 E f f e c t of perturbations to the velocity of loaded vehicle on the optimum turnout spacing the Figure 21 E f f e c t of perturbations to the velocity of the loaded vehicle on the cost difference 93 6.2.2 VELOCITY OF THE EMPTY VEHICLE I t w i l l be assumed that the potential maximum error i n the measurement of the velocity of the empty vehicle w i l l be the same as the maximum error for the velocity of the loaded vehicle, 10 mph. The rate of change i n the simple F-factor for a given perturbation i n the vel o c i t y of the empty vehicle (&VZ) i s : SF = -V, /(2V|) This rate of change formula can be u t i l i z e d to predict the potential rate of change for the expected F-factor involving the meeting of an empty vehicle and a f l e e t of loaded vehicles. Simulations to predict the error i n the ca l c u l a t i o n of the t o t a l expected delay time (6f) , the optimum turnout spacing (£s*) / and the cost eguation (<ScT) were conducted in a s i m i l a r manner as i n the case cf the velocity of the loaded vehicle. The r e s u l t s of these simulations were s i m i l a r to the case involving perturbations to the velocity of the loaded vehicle: 1. the percentage difference i n the t o t a l expected delay time and the percentage difference in the optimum turnout spacing were s l i g h t l y lower than i n the case:involving the velocity of the loaded vehicle 2. the cost difference was s l i g h t l y higher than i n the case involving the velocity of the loaded vehicle (Appendix 9). 94 6. 2.3 TRAFFIC FLOW RATE In the s e n s i t i v i t y analysis of the t r a f f i c flow rate (H) the perturbations that were used are 1, 2, and 4 vph.. The functions F, T, , and C T are tested for actual t r a f f i c flow rates of 1, 2, 4, 6, 8, and 10 vph. The rest of the independent variables and their range of values are the same as for the simulations involving the s e n s i t i v i t y analysis of the velocity of the loaded vehicle. The expected F-factor resulting from the meeting of one empty vehicle and a f l e e t of loaded vehicles experienced a change of less than 0.01, for any of the tested flow rates. This difference i s i n s i g n i f i c a n t . The percent error i n the calcu l a t i o n of the t o t a l expected delay time (&T) can be approximated by: oT = (100)£H/H As can be shown from this approximation formula the error created can be large. The results of the simulation to determine the percent difference in the optimum turnout spacing have small standard deviations. If the perturbation i s l e s s than 100 percent of the actual flow rate then the maximum value of w i l l be approximately 7 0 percent of the actual optimum turnout spacing. Maintaining t h i s as a maximum perturbation then the cost difference function (oCT) has a larger maximum value than that experienced in the case of the s e n s i t i v i t y analysis of either V( or (based on a maximum perturbation of 100 percent).. The maximum value of r J C f , based on a 100-percent perturbation of the t r a f f i c flow rate, i s approximately $400 per vehicle per mile per year but the average value i s 9 5 a p p r o x i m a t e l y $ 8 5 . 9 6 6,2.4 EXPECTED USEFUL LIFE OF THE ROAD The expected useful l i f e of the road (Q() is analysed f o r 5, 10, 15, 20, and 25 years with perturbations («JQ ) of 2 .5r 5, and 10 years. The remaining independent variables i n the simulation models u t i l i z e the same range of values as the simulations to test the s e n s i t i v i t y of the velocity of the loaded vehicle. The results of the simulation involving the percent difference in the optimum turnout spacing have small standard deviations for the various groupings of Q( and oRQ) . As c$Q, increases Js^ increases. Provided Q( can be predicted to within 100 percent of the actual value then the maximum value of 6 s * i s approximately 35 percent of the actual optimum turnout spacing. This maximum value decreases to 21 percent for a perturbation of 50 percent and 12 percent for a perturbation of 25 percent. The r e s u l t s of the cost difference simulation had a larger c o e f f i c i e n t of variation than the results of the <£s* simulation. By maintaining the assumption that the maximum error i n the estimation of Q( i s 100 percent then the maximum cost difference i s approximately $35 per year per vehicle per mile. I f the perturbation i s 50 percent then the maximum value of ScT i s $20. The accuracy of the estimation of Q( i s not as c r i t i c a l as in the sit u a t i o n s involving the velocity of the loaded vehicle, the velocity of the empty vehicle, nor the t r a f f i c flow rate, since for a given percent perturbation i n the expected useful l i f e of the road the maximum value of JcT i s l e s s . 97 6.2.5 TURNOUT CONSTRUCTION COST The s e n s i t i v i t y analysis of the turnout construction cost included costs of $100, $250, $500, and $1000 with perturbations of $50, $100, and $200. The oS* simulation produced s i m i l a r results as the model involving the s e n s i t i v i t y analysis of the expected useful l i f e of the road.. A 100-percent perturbation in C resulted in aS% being approximately 34 percent of the actual optimum turnout spacing while a 40-percent perturbation resulted i n <$S* being approximately 17 percent.. The simulation to determine Sc^ produced results that are not as s i g n i f i c a n t as the previous &C{ simulations; i . e . , velocity of the loaded vehicle, velocity of the empty vehicle, t r a f f i c flow rate, and expected useful l i f e of the road. A 100-percent perturbation of C res u l t s in a cost difference of less than $30 per year per vehicle per mile. 98 6.2.6 ADJUSTED TEOCK HAOLING COST The simulations to determine the difference i n the optimum turnout spacing and the cost difference were conducted with adjusted truck hauling costs of $10, $15, $25, $35, and $45 per hour and with perturbations of $2.5, $5, and $10 per hour. Both of these simulations u t i l i z e d the same basic assumptions and range of values of the variables as the simulations involving the analysis of the velocity of the loaded vehicle. There was a small standard deviation within the various groups, M and CSM, of the simulation to determine <5s^ . For a given percent perturbation of the independent variable, the r e s u l t i n g £s* (%) value i s less i n the case involving the adjusted truck hauling cost than i n the case involving any of the previously analysed variables.. A 100-percent perturbation i n the adjusted hauling cost r e s u l t s i n 6s* being approximately 34 percent of the actual optimum turnout spacing while a 50-percent perturbation r e s u l t s i n 6s% being approximately 20 percent. The r e s u l t s of the simulations to determine the cost difference had a larger c o e f f i c i e n t of variation than the simulation to determine the percent difference i n the optimum turnout spacing. In the case involving an adjusted hauling cost of $10 per hour and a 100-percent perturbation, the maximum cost difference was approximately $40 per vehicle per mile per year while the average difference was approximately $14.. The e f f e c t of a perturbation to the adjusted hauling cost on the cost i s not as s i g n i f i c a n t as in the case involving the perturbation of the turnout construction cost. 99 6.2.7 THE ACCELERATION AND DECELERATION OF THE EMPTY VEHICLE Perturbations to the rate of deceleration of the empty vehicle w i l l not s i g n i f i c a n t l y a f f e c t the expected F-factor, provided the perturbations can be predicted to within 50 percent of the actual rate of deceleration.. A 50-percent perturbation w i l l cause the maximum error i n the expected F-factor to be le s s than 0.01 or 20 percent of the actual F-factor. This summary i s based on a simulation model that u t i l i z e d the same range of values of the variables as was u t i l i z e d in the analysis of the velocity of the loaded vehicle. A simulation model was u t i l i z e d to determine the e f f e c t that perturbations to either the rate of acceleration or deceleration have on the t o t a l expected delay time.. The assumptions, constraints, and range of values of the independent variables u t i l i z e d i n the program were: 1. V, = 10, 15, 20, 25, 30, 35, or 40 mph 2. V a = 10, 15, 20, 25, 30, 35, or 40 mph 3. V, < V2 + 10 mph 4. V2 < V, + 20 mph 5. S = 0.05, 0. 10, 0. 15, 0.20, 0.25, 0. 30, or 0.35 miles 6. H = 2, 4, 6, 8, or 10 vph 7. afl or a D = 14555, 24259, or 33963 mph2 8. exponential headway d i s t r i b u t i o n . The rate of acceleration or deceleration being tested was assumed to have an actual rate of 14555, 19407, 24259, 29111, or 33963 mph2 while the perturbations were 2426, 4852, and 9704 mph2.. The development of formulas to determine the rate of 100 acceleration and deceleration are outlined in Appendix 3. The simulation r e s u l t s were grouped according to t h e i r respective rates of acceleration or deceleration and t h e i r perturbations. A 66-percent perturbation w i l l cause the maximum value of £T to be less than 20 percent of the t r a v e l empty time. The simulation indicated that the percent change i n the t o t a l expected delay time w i l l be: less than one-half of the perturbation, expressed i n percent. A simulation model was developed to examine the e f f e c t that perturbations to the rate of acceleration or deceleration have on the optimum turnout spacing and cost functions.. The range of the values of the variables and the assumptions that were u t i l i z e d i n the model were e s s e n t i a l l y the same as those u t i l i z e d i n the s e n s i t i v i t y analysis of the ve l o c i t y of the loaded vehicle (Section 6.2.1). It was assumed that the rate of acceleration eguals the rate of deceleration, 14555 or 24259 mph2. The perturbations were 2426, 4852, and 9704 mph2 and the maximum perturbation was 80 percent. Since the maximum value of rjs* was less than 7 percent of the actual optimum turnout spacing the e f f e c t that perturbations to the rate of acceleration or deceleration have on the optimum turnout spacing function i s i n s i g n i f i c a n t . 10 1 6.2,8 THE DISCOUNT RATE A simulation model was developed to determine the e f f e c t that the discount rate has on the optimum turnout spacing function and the cost function. The simulation results were grouped with respect to the i r corresponding discount rate (2.5, 5.0, 7.5, 10.0, or 12.5 percent per annum) and expected useful l i f e of the road (2, 5, 10, or 20 years). The range of values of the variables and the assumptions u t i l i z e d i n the simulation were: 1. V, = 10, 20, 30, or 40 mph 2. = 10, 20, 30, or 40 mph 3. V2 < V; + 20 mph 4. V, < + 10 mph 5. H = 3 or 10 vph 6. C = $100, $600, or $1100 7. M = $25 or $45 per hour 8. a„ = a 0 = 19759 mph2 9. exponential headway frequency d i s t r i b u t i o n 10. 200 operating days per year 11..5 " c o n f l i c t " hours per day. The marginal e f f e c t of cSs* (%) decreased as the expected useful l i f e of the road increased or as the discount rate increased (Appendix 9)., The maximum value of was approximately 50 percent of the optimum turnout spacing which corresponded to a situation involving a discount rate of 12.5 percent and an expected useful l i f e of the road of 20 years. When the discount rate was decreased to 7.5 percent the maximum value of 102 cSs^  became approximately 34 percent. The r e s u l t s of the cost d i f f e r e n c e (CNC-J-) s i m u l a t i o n had a l a r g e r c o e f f i c i e n t of v a r i a t i o n than the <$S* si m u l a t i o n . . The marginal e f f e c t of , f o r maximum and average v a l u e s , decreased as the discount r a t e or the expected u s e f u l l i f e of the road increased. The maximum value of &C-f was only $ 3 5 per v e h i c l e per mile per year which was based on a discount r a t e of 1 2 . 5 percent and an expected u s e f u l l i f e of the road of 2 0 years. For lower discount rates ( i . e . , 5 percent) the e f f e c t of the discount r a t e on the cost f u n c t i o n i s not very s i g n i f i c a n t . 103 6.2.9 THE MAINTENANCE COST A s i m u l a t i o n model was developed t o determine the e f f e c t t h a t maintenance c o s t s have on the c o s t f u n c t i o n and optimum turnout s p a c i n g f u n c t i o n . . The r e s u l t s of the s i m u l a t i o n were grouped with r e s p e c t to t h e i r c orresponding maintenance c o s t s ($0.05, $1.00, $2.50, $5.00, $7.50, $10.00, or $12.50 per annum) and expected u s e f u l l i f e of the road (2, 5, 10, 15, 20, or 25 years) before the maximum and average values were determined. . The values of the independent v a r i a b l e s and the assumptions of the model were the same as those u t i l i z e d i n the a n a l y s i s of the v e l o c i t y of the loaded v e h i c l e (Section 6.2.1). The marginal e f f e c t o f oS* (%) , f o r v a r i o u s maintenance c o s t s , decreased as the expected u s e f u l l i f e of the road i n c r e a s e d . Since the c o e f f i c i e n t of v a r i a t i o n f o r c$S* was approximately one, f o r each group, there.was a l a r g e spread i n the values of <$S*. To have a maximum value of <5s* egual to approximately 50 percent of the a c t u a l optimum t u r n o u t spacing, a maintenance c o s t of $7.50 per annum i s r e g u i r e d over a 25-year p e r i o d . T h i s same maximum value i s achieved f o r a maintenance c o s t of $10 over 20 years or $12.50 over 15 years (Appendix 9). The grouped r e s u l t s of the c o s t d i f f e r e n c e , had a l a r g e r c o e f f i c i e n t of v a r i a t i o n than the oS* s i m u l a t i o n . When aS* i s l e s s than 50 percent then the value c f 'bCj i s l e s s than $300 per year per mile or approximately $0.02 per c u n i t (based on 25 c u n i t s per l o a d over a 10-mile h a u l ) . Based on a maintenance c o s t of $10 per annum the value of 6cr i n c r e a s e s to approximately $0.03 per c u n i t when cS S* i s i n c r e a s e d to 104 approximately 60 percent.. In the case of a maintenance cost of $5 per annum over a 20 year period the maximum value of 6cT would be approximately $0.01 per cunit. 105 6.2.10 THE DERIVATIVE OF THE EXPECTED F-FACTOR In the determination of the optimum turnout spacing the f i r s t derivative of the cost function with respect to the turnout spacing must be equal to zero.. Since the expected F-factor i s a variable i n the cost function the f i r s t derivative of the expected F-factor i s included i n the optimum turnout spacing function. A simulation model i s used to determine the ef f e c t that the f i r s t derivative of the expected F-factor equation has on the optimum turnout spacing function and the resulting cost function. The same values of the independent variables were used i n t h i s simulation, as were u t i l i z e d i n section 6.2.1 except that: 1. H = 1, 2, 4, 6, 8, or 10 vph 2. The shifted exponential and Pearson Type III (a=2) headway d i s t r i b u t i o n s were used. For a t r a f f i c flow rate of less than 4 vph the value of J s * was 20 percent of the optimum turnout spacing.. The maximum value became approximately 51 percent when the t r a f f i c flow rate was increased to 10 vph. The corresponding maximum cost difference was only $27 per year per mile per vehicle.. Generally, the inclusion of the f i r s t derivative of the expected F-factor i n the c a l c u l a t i o n of the optimum turnout spacing i s i n s i g n i f i c a n t . 106 6.2.11 THE LENGTH OF THE LOADED VEHICLE Throughout the development of the model i t has been assumed that the loaded vehicle has no length. Some of the equations developed i n Chapter 2 can be modified to include the ef f e c t of the length of the loaded vehicles (Appendix 6). Furthermore, the appendix includes a summary of the r e s u l t s of a simulation program written to v e r i f y the accuracy of the expected F-factor equations which account for the length of the loaded vehicle. This simulation determined the average F-factor f o r delay situations involving the s h i f t e d exponential and the Pearson Type III(a=2) headway d i s t r i b u t i o n s . A simulation model was developed to compare the exponential tc the shifted exponential F-factor equation and the Erlang(alpha=2) tc the Pearson Type III(a=2) F-factor equation., The simulation assumed: 1. L =60 feet 2. a D = 19759 mph2 3. V, = 10, 20, 30, or 40 mph 4. V£ = 10, 20, 30, or 40 mph 5. V 2 < V( + 20 mph 6. V, < \ * 10 mph 7. S < V|/a D 8. S = 0. 10, 0. 15, 0.20, 0. 25, 0.30, or 0.35 miles 9. H =1, 2, 4, 6, 8, or 10 vph The r e s u l t s of the simulation indicated that the effect of the length of the vehicle on the expected F-factor was approximately: 107 i f = L/S where: L = e f f e c t i v e length of the vehicle. For a turnout spacing of 0.1 miles the difference i n the expected F-factor i s approximately 0.11. The e f f e c t of the length of the vehicle on the optimum turnout spacing proved to be inconsequential.. In the simulation u t i l i z e d to determine t h i s conclusion i t was assumed: 1. Q( = 2 or 10 years 2. M = $25 or $45 per hour 3. C = $100, $600, or $1100 4. 200 operating days per year 5. 5 " c o n f l i c t " hours per day. The maximum value of S s * , f o r the case involving the sh i f t e d exponential headway d i s t r i b u t i o n , was les s than 2 percent of the actual optimum turnout spacing while for the case involving the Pearson Type I I I (a=2) headway d i s t r i b u t i o n the maximum value of SS* was less than 5 percent. Consequently, the length of the vehicle w i l l not s i g n i f i c a n t l y affect the optimum turnout spacing nor the cost equation. 108 6.2.12 THE HEADWAY PROBABILITY DISTRIBUTIONS A simulation model was developed to compare the expected F-factors developed from various headway probability d i s t r i b u t i o n s . The range of values of the independent variables were e s s e n t i a l l y the same as for the simulation involving the e f f e c t of perturbations to the velocity of the loaded vehicles on the expected F-factors (Section 6.2.1).. The exceptions were that the length of the loaded vehicle was 60 feet and the t r a f f i c flow rate was 1, 2, 3, 4, 6, or 10 vph. The simulation compared the simple, exponential, Erlang (alpha=2), shifted exponential, and Pearson Type III (a=2) F-factor equations. The maximum difference between the simple F-factor and the exponential F-factor was 0.0527 at a t r a f f i c flow rate of 10 vph. This value dropped to 0.0054 when the rate was decrease to 4 vph._ A larger maximum difference occured between the simple F-factor and the Erlang (alpha-2) F-factor, the values being 0.091 for a rate of 10 vph and 0.027 for a rate of 4 vph. The maximum difference between the exponential and Erlang (alpha=2) F-factors was the smallest of those tested. This difference was of the order of magnitude of 10 - 3.. By incorporating the ef f e c t of the length of the vehicle into the expected F-factor equations there was e s s e n t i a l l y no difference i n the r e s u l t s . A s i m i l a r simulation model was developed to determine the effect that various headway d i s t r i b u t i o n s have on the optimum turnout spacing function and the cost function., The values of the variables u t i l i z e d i n the simulation were e s s e n t i a l l y the 109 same as those used in the analysis of the velocity of the loaded vehicles except that the t r a f f i c flow rate was 1, 2, 4, 6, or 10 vph. The res u l t s were grouped with respect to t h e i r F-factors, simple, exponential, and Erlang(alpha=2), before being compared. The maximum value of &S* was less than 3 percent of the actual optimum turnout spacing and the maximum value of <Sc^. was approximately $11 per vehicle per mile per year. Consequently, the type of headway d i s t r i b u t i o n u t i l i z e d w i l l not s i g n i f i c a n t l y a l t e r the optimum turnout spacing nor cost equation. 110 6.2,13 THE TURNOHT SPACING AND THE OPTIMOM TURNOUT SPACING There i s po t e n t i a l l y a small error i n the measurement of the average turnout spacing.. A simulation model was developed to determine the e f f e c t that perturbations to the turnout spacing (0.05, 0.10, and 0.20 miles) have on the t o t a l expected delay time.. The perturbations were added to the true values of the turnout spacing (0.05, 0. 10, 0. 15, 0.20, 0.25, 0. 30, and 0.35 miles). The range of values of the independent variables u t i l i z e d i n the simulation was e s s e n t i a l l y the same as those u t i l i z e d in section 6.2.1. Graphs were constructed to i l l u s t r a t e the eff e c t that perturbations to the turnout spacing have on the t o t a l expected delay time (Appendix 9). The t o t a l expected delay time and the absolute value of the f i r s t derivative of the function decrease as the turnout spacing increases. Furthermore, the t o t a l expected delay time increases as the size of the perturbation increases.. If i t i s assumed that the maximum perturbation of the turnout spacing i s 20 percent then the maximum difference in the t o t a l delay time i s approximately 30 percent of the tr a v e l empty time.. A simulation model was developed to i l l u s t r a t e the e f f e c t that deviations from the optimum turnout spacing (25% to 300%) have on the cost, equation. The range of values of the independent variables was similar to those u t i l i z e d in section 6.2.1 except: 1. headway frequency d i s t r i b u t i o n was the s h i f t e d exponential 2. _L = 60 feet 3. 25 cunits per t r i p 111 4..M = $5, $10, $15, or $20 per hour 5. Q = 5, 15, or 25 years 6. H = 1, 4, or 8 vph. The cost difference was measured i n do l l a r s per cunit-mile. The experience of the model, averages, standard deviations, and maximum values, was recorded with respect to the t r a f f i c flow rate, the adjusted truck hauling cost, and the expected useful l i f e of the road. The values of the cost difference function, in d o llars per cunit-mile, did not vary s i g n i f i c a n t l y with respect to the t r a f f i c flow rate.. Figures 22 and 23 depict the ef f e c t that deviations from the optimum turnout spacing have on the average and maximum values of the cost equation with respect to the expected useful l i f e of the road and the adjusted truck hauling cost.. The t r a f f i c flow rate was set equal to 1 vph. The cost difference and the marginal effect increased as the perturbations to the optimum turnout spacing increased. Furthermore, the cost difference increased as the adjusted hauling cost increased but decreased as the expected useful l i f e of the road increased. The c o e f f i c i e n t of variations of the grouped results are stable.. The average c o e f f i c i e n t of variation was 0.68 while the maximum value was 0.72.. For each of the groups, the maximum cost difference was approximately three times the siz e of the average cost difference while the maximum cost difference i s approximately four and one-half times the value of the standard deviation of the cost difference.. The maximum value of the cost difference, given an adjusted truck hauling cost of $20 per hour, an expected useful l i f e of the road of 5 years, and a 100-percent perturbation, i s approximately $0,007 112 per cunit-mile while the average value i s approximately $0,00 3 per cunit-mile. 113 Lu ui in uixas i n i - n Z3 U J Lu>-U J I — U ) r£» V » » U . I E U J — a C L _ I E K X Ul U l Lrl Lrl Ln Ln Ln rvi rvi 1 J 1 1 i n * -+- X 1 UJ 1 1 1 1 i — * + X 1 re J i l l LK * - * - X 1 1 1 1 1 LO • i l l z _ J K K K K E C x x x x a : E S C S E S E S ta E S E S ts <-» • • • -u i ca Ln ca rvi r- ** **** ** E 3 rvi 3" m m Ui z Ul E K Ul Ln t U l tr x Ui _ _J E S — i o: z rs «v> CJ — O Q. V^ r- IS Figure 22 E f f e c t of deviations from the optimum turnout spacing on the maximum cost difference 1.14 _j —« U.-IU1 ui x ce U l r - X n ui Lu >-ui r - U I _ « V L C U l — • • I K X U l U l U l U l U l U l U l U l rvi rvi rvi ui ui x as La z i i i i * - r X I I I I 1 * 4- X I 1 1 1 1 * + X I I I 1 I X x x x x X s s s s rararara v: carsicara KJ . . . . Z3 Lrt IS U l 53 ns rvi ra zr in rvi rvi m cn zr m ra r- rvi x m rvi bJ z UJ DC U l in tr x ui i — i— in ca — UJ ca inu. J - Z - S • —zt_u i_ i— • v> d — _, o. v>»- ra Figure 23 Ef f e c t of deviations from the optimum turnout spacing on the average cost difference 1 1 5 7.0 DISC0SSION AND CONCIOSIONS I i i DISCOSSION A model has been developed to determine the optimum turnout spacing which i s a function of v e l o c i t i e s of the vehicles, t r a f f i c flow rate, expected useful l i f e of the road, acceleration rate, deceleration rate, turnout construction cost, turnout maintenance cost, and adjusted hauling cost.. The results of the model should be u t i l i z e d as a guide and not as the absolute spacing of the turnouts. The road network, company policy, and terrain w i l l tend to di c t a t e the actual location of the turnouts. Figures 12, 13, 14, 15, 16, and 17 i l l u s t r a t e the relationships between the independent variables and the optimum turnout spacing. The optimum turnout spacing increased with increasing turnout construction cost, velocity of the empty vehicle, and velocity of the loaded vehicle. Optimum turnout spacing decreased as the expected useful l i f e of the road, t r a f f i c flow rate, and adjusted hauling cost increased.. Based on these. figures, an overestimation of the independent variables w i l l not cause as dramatic a change i n the optimum turnout spacing as an underestimation of the independent variables. The absolute value of the slope:of the curves decrease as the value of the independent variable increases. Consequently, there i s not as dramatic an affect to the optimum turnout spacing, provided the independent variable i s overestimated rather than underestimated. This though, does 116 not imply that the values of the independent variables should be overestimated. a s e n s i t i v i t y analysis showed that a given percent perturbation to each of the independent variables w i l l a f f e c t the optimum turnout spacing d i f f e r e n t l y . Generally, the t r a f f i c flow rate was the most sensitive variable followed by velocity of the loaded vehicle and velocity of the empty vehicle. . The turnout construction cost, expected useful l i f e of the road, and adjusted hauling cost caused about the same change to the optimum turnout spacing. These variables were not as sensitive as the t r a f f i c flow rate, velocity of the empty vehicle, and velocity of the loaded vehicle.. These s e n s i t i v i t y relationships cannot be compared unless the potential accuracy of the estimation of the variables i s considered. Generally the t r a f f i c flow rate, speed of the vehicles, and expected useful l i f e of the road are easier to predict than the turnout construction cost and adjusted hauling cost. Consequently, the confidence in the i n i t i a l estimation of the variables must be considered. Generally, the optimum turnout spacing was not s i g n i f i c a n t l y affected by the discount rate, headway frequency d i s t r i b u t i o n assumptions, rate of acceleration, rate of deceleration, and the length of the vehicle. The inclusion of the maintenance cost i n the determination of the optimum turnout spacing can be s i g n i f i c a n t ( i . e . , a 50 percent change in the optimum turnout spacing for a maintenance cost of $7.50 per annum over a 25-year span). The t o t a l expected delay time can be a s i g n i f i c a n t part of the t r a v e l empty time (Figures 8, 9, 10, and 11).. This significance i s dramatically reduced when the t o t a l cycle time 117 i s considered. I f the travel empty time i s one-quarter of the t o t a l time then for a given T of 20 percent of the travel empty time the corresponding e f f e c t of T on the t o t a l round t r i p time i s 5 percent. . By comparing the t o t a l expected delay time to the t o t a l cycle time rather than to the travel empty time the significance of the delay time i s reduced f o u r f o l d . Figures 22 and 23 depict the e f f e c t that deviations from the optimum turnout spacing have on the cost (dollars per cunit-mile), Road construction and log hauling costs are highly variable within and between d i f f e r e n t regions. . Sauder and Nagy(1977) have shown that the cost of road construction and hauling i s of the order of magnitude of $10 per cunit, based on a haul distance of approximately 13 miles. This cost estimation i s i n the order of $1 per cunit-mile. Based on a maximum deviation from the optimum turnout spacing of 100 percent (Figure 22), the maximum poten t i a l savings are about $0.01 per cunit-mile, $0.13 per cunit, or about 1 percent of the t o t a l road construction and hauling cost. These calculations, though, can be misleading. A small percentage of a large number may be s i g n i f i c a n t . The t o t a l potential savings must be considered rather than the potential savings per cunit -mile. . Based on a t r a f f i c flow rate of 4 vph, a 10-mile haul, 25 cunits per t r i p , 5-year time period, 200 working days per year, and 5 " c o n f l i c t " hours per day, the t o t a l potential savings are $50000.. Consequently, the potential savings become important. For a si t u a t i o n with a t r a f f i c flow rate of 4 vph, an 100-mile haul, 15-year time period, 200 working days per year, 5 " c o n f l i c t " hours per day, velocity of the loaded vehicle of 20 mph, vel o c i t y of the empty vehicle of 40 mph, an 118 adjusted hauling cost of $25 per hour, and a turnout construction and maintenance cost of $100 per turnout the optimum turnout spacing i s 510 feet.. The t o t a l potential savings, based on a 100 percent deviation from the optimum turnout spacing i s $68190. The implementing of the optimum turnout spacing model can be achieved with the u t i l i z a t i o n of tables.. ft table approach neglects the necessity of the repeated evaluation of a complex set of equations. Consequently, the tables can be incorporated into the road standards. Future areas of research that can be investigated are: 1. Develop a method to predict the adjusted hauling cost, turnout construction cost, and turnout maintenance cost. 2. Incorporate the effect that loaded logging trucks have of the t r a v e l time of vehicles other than logging trucks. 3. Develop a model that involves the interaction between f l e e t s of vehicles, non-uniform turnout spacing, da i l y variation in t r a f f i c flow rates, and variation in v e l o c i t i e s of vehicles. . 119 7. 2 CONCLUSIONS & model was developed to allow an engineer to determine optimum turnout spacing for s p e c i f i c conditions and determine e f f e c t of turnout spacing on hauling cost.. Simulation models were used to tes t the s e n s i t i v i t y of optimum turnout spacing to perturbations of the independent variables. Generally, optimum turnout spacing was most sensitive to the t r a f f i c flow rate followed by the speeds of the loaded and empty vehicles. Optimum turnout spacing was les s sensitive to turnout construction cost, expected useful l i f e of the road, and adjusted hauling cost. The r e s u l t s from a simulation revealed that a 10 0 percent increase of the t r a f f i c flow rate w i l l cause a maximum decrease of 70 percent of the optimum turnout spacing. Much of t h i s r e l a t i o n s h i p i s due to the e f f e c t of increasing t r a f f i c flow rate on t o t a l expected delay time.. The simulations determined that f o r r e a l i s t i c conditions (e.g., 5 vph and 0 . 3 miles between turnouts) the expected delay time was 20 percent of the t r a v e l empty time.. The percent deviation i n the t o t a l expected delay time was also found to vary s i g n i f i c a n t l y (approximately proportionally) with the percent deviation in the t r a f f i c flow rate.. Based on the concept of uniform spacing of turnouts, the potential savings of u t i l i z i n g the model as a guide in the design of the road network can be determined.. For example, i f an engineer uses the optimum turnout spacing rather than a 100 percent deviation from the optimum, then the re s u l t i n g potential savings can be as high as 1 percent of the 120 transportation cost.. In the si t u a t i o n c i t e d i n Section 7.1 t h i s 1 percent potential savings represents approximately $10000 per year over a 10-mile haul at 4 vph. Since off-road haul distances can ea s i l y approach 30 miles and wood can be hauled over several such routes simultaneously, potential savings through optimum turnout spacing could be i n the order of $90000 per year.. The concept of the expected F-factor was developed and u t i l i z e d to estimate the expected delay of a truck in a turnout. This i s a measure of the expected separation distance between the loaded vehicle and the empty vehicle as a proportion of the turnout spacing when the empty vehicle has come to a complete halt i n the'turnout. Previous a r t i c l e s have not completely defined a method of deriving or measuring t h i s delay. Two forms of the expected F-factor equation were developed: 1. F = (V, +V2)/(2V/j) : f o r one empty vehicle meeting one loaded vehicle 2. F = Pr(h>h c)F | + Pr(h<h c)F^ : for one empty vehicle meeting a f l e e t These equations were tested by simulation and found to be r e a l i s t i c . The simulations showed that the interaction between a f l e e t of empty vehicles and a f l e e t of loaded vehicles was adequately described by the expected F-factor equation representing the meeting of an empty vehicle and a f l e e t of loaded vehicles. The expected F-factor equation for one vehicle meeting a fl e e t requires the use of a headway d i s t r i b u t i o n function.. Two sets of i n t e r a r r i v a l time data were analysed to determine i f 121 the headway d i s t r i b u t i o n of logging trucks f i t s a known probability d i s t r i b u t i o n . . The frequency histogram of the headways was l e f t skewed but was found to follow neither an exponential nor Erlang d i s t r i b u t i o n . ft simulation model showed that the type of headway d i s t r i b u t i o n u t i l i z e d , exponential or Erlang (<<= 2) , did not s i g n i f i c a n t l y a l t e r the expected F-factor. Since the data appeared to follow a d i s t r i b u t i o n s imilar to the Erlang, and the F-factor i s r e l a t i v e l y i n s e n s i t i v e to the type of headway d i s t r i b u t i o n , i t i s concluded that the derived F-factor w i l l adequately model a r e a l i s t i c hauling s i t u a t i o n . Previous authors have written papers concerned with determining the number of lanes required for log transportation.. They did not calculate or derive an expression to determine the expected turnout delay time.. Since t h i s i s now available, the topic of determining the number of lanes required for log transportation should be reinvestigated.. ft method should be developed to predict the adjusted hauling cost, turnout construction cost, and turnout maintenance cost. The e f f e c t that loaded logging trucks have on the travel time of vehicles other than logging trucks should also be investigated. 122 LITEBATOBE CITED Arden, B.W. and A s t i l l , K.N. 1970. Numerical Algorithms: Prig ins and Applications.. Addison-Wesley Publishing Company , Beading, Mass. 308 pp. Boyd, C.W. and Young, G.G. .1969. A Study of Canfor's Logging Trucks at Harrison M i l l s . (Unpublished). Faculty of Forestry, University of B . C . , Vancouver, B.C..62 pp. B.C. Forest Service. 1975. Prince George D i s t r i c t - Forest Eoad Manual. B.C. Forest Service, V i c t o r i a , B.C. 121 pp.. B.C. Forest Service..1977. Appraisal Manual - Vancouver Forest D i s t r i c t . B.C. Forest Service, V i c t o r i a , B.C. 45 pp.. Byrne, J.J., Nelson, B.J. and Googins, P.H. 1947..Cost of Hauling Logs by Motor Truck and T r a i l e r . P a c i f i c Northwest Forest S Bange Experiment Station. Portland, Oregon. .112 pp. Conte, S.D. and deBoor, C. 1972..Elementary Numerical Anaysis: An Algorithmic Approach. 2nd ed. McGraw-Hill Book Company, New York..396 pp. Drew, D.R. .1968. T r a f f i c Flow Theory and Control. McGraw-Hill Book Company, New York. 467 pp. F e l l e r , W. 1966. An Introduction to Probability Theory and i t s Application - Volume I^.John Wiley S Sons, Inc., New York..461pp. Gerlough, D.L.. 1955. Use of Poisson Di s t r i b u t i o n in Highway T r a f f i c . In Poisson and T r a f f i c . pp.. 1-58. The Eno Foundation for Highway T r a f f i c Control, Saugatuck, Connecticut. Haight, F.A. 1963..Mathematical Theories of T r a f f i c Control.. Academic Press, New York. .242 pp.. Matson, T.M., Smith, S.S. and Hurd, F.W.T 1955.. T r a f f i c Engineering. McGraw-Hill Book Company, New York. 647 pp., Meriam, J.L. 1971. Dynamics. 2nd..ed. John Wiley & Sons, Inc. New York. 480 pp. Porpaczy, L.J. and Waelti, H. 1976. How to se l e c t forest road standards. Canadian Forest Industries 96 (12): 33, 36-37. Sauder, B.J. and Nagy, M.M.. 1977. Coast Logging: Highlead versus Long-reach Alternatives. Forest Engineering Researh In s t i t u t e of Canada. Technical Beport No. TB-19. Vancouver, B.C.. 51 pp. Schuhl, A. 1955. The Probability Theory Applied to 123 Dist r i b u t i o n of Vehicles on Two-lane Highways. In Poisson and T r a f f i c . pp. 59-75. The Eno Foundation For Highway T r a f f i c Control, Saugatuck, Connecticut. Smith, D.G. and Tse, P.P. 1977. Logging Trucks: Comparison of Productivity and Costs. Forest Engineering Research Ins t i t u t e of Canada. Technical Report No. TR-18. Vancouver, B.C..43 pp. Truck Loggers Assocation. . 1978. Equipment Rental Rates. Journal of Logging Management 9(1):1306-07. , United States Bureau of Land Management. . 1977. Timber Production Costs: Schedule 20._,Bureau of Land Management, Oregon State O f f i c e , Portland, Oregon. 524 pp. Wohl, H. and Martin, B.V. .1967. T r a f f i c Systems Analysis For Engineers and Planners. McGraw-Hill Book Company. New York. 558 pp. . 124 APPENDICES APPENDIX 1 BOAD STANDARDS SORVEY 126 THE UNIVERSITY OF BRITISH COLUMBIA 2075 WESBROOK MALL VANCOUVER,B.C.,CANADA V6T 1W5 MacMILLAN BUILDING FACULTY OF FORESTRY Dear S i r : I am a graduate student at the University of B r i t i s h Columbia and currently doing a study on road standards.. The objective of t h i s project i s to evaluate the design elements used i n the determination of road standards and road c l a s s i f i c a t i o n . I t would be appreciated i f you could forward the road sp e c i f i c a t i o n s your company employs in the construction of i t s forest roads.. Some of the design elements under consideration could be: 1. Road types ( i . e . Main, secondary, branch and spur) 2..Design speed 3. Minimum sight distance (horizontal, vertical) 4. Curve radius 5. Adverse and favourable grade 6. Subgrade width 8. Surface material 9. Ditch width and depth (rock or soil) 10. Maximum surfacing depth 11. Right-of-way width 12. Turnouts (length, width and number per mile) 13. Culverts (type) 14. Compaction (equipment u t i l i z e d and degree of compaction) 15. Use of roads (winter, summer or year-round) Thank you for your co-operation. Yours t r u l y , Dennis I. . Anderson DIA/ns Graduate Studies 127 Road Standards Survey Specifications | L Response r B r i t i s h Columbia (Total=16) Rest of Canada (Total=30) (United States | (Total=15) Speed I 12 (75) 23(77) I 8(53) Right-of-way | 10(63) 30(100) I 9(60) Subgrade width | 11(69) | 21 (70) | 11(73) Surface width | 14 (87) | 29(97) I 14(93) Surface depth | 2(13) 21 (70) I 11(73) Road Gradient | | | -Favourable | 13 (81) j 23(77) 1 13(87) -adverse | 13(81) | 23 (77) I 13(87) Curve radius | 13(81) | 19 (63) 1 11(73) Sight distance | 4(25) | 6(20) I M27) -Horizontal | 3(19) | 16(53) I 3(20) - V e r t i c a l | 1(6) j 1 (3) I 1 (7) Superelevation | 1 (6) j 7(23) I 1 (7) Ditch depth | 12 (75) | 18 (60) | 12(80) width | 6(37) | 10 (33) ! 10(67) Turnouts length | 8(50) | 3(10) 1 9(60) width | 6(37) | 3(10) 1 8(53) frequency| 9(56) 5(17) 1 9(60) Back slope | 3(19) | 12(40) | 6(40) F i l l slope | 3(19) | 9(30) 1 6(40) Cross slope | 2(13) | 4(13) 1 3(20) Load capacity | 5(31) J 10(33) 1 1(7) Survey response | | -sample size | 27 | 53 | 66 -response | 19(70) 30 (57) | 28 (42) * Parenthesized values represent percentages of: 1. the affirmative response, or 2. the sample size i n the cases involving the "survey response" 128 APPENDIX 2 ABBREVIATIONS. SYMBOLS^ AND ONITS I ROMAN SYMBOLS AND ABBREVIATIONS a,a f t,a D acceleration and deceleration where subscripts A and D refer to acceleration and deceleration C c turnout construction cost C M turnout maintenance cost C T t o t a l cost D,Dfl,Ds distance where subscript A and S refer to stopping distance and acceleration distance D^  i n t e r v a l distance, the distance between two sets of empty vehicles a f t e r the loaded vehicle has passed both groups of empty vehicles and each vehicle has accelerated to i t s o r i g i n a l speed d number of " c o n f l i c t " hours per day F,F L the F-factor, which i s the distance the loaded vehicle i s from the empty vehicle, once the empty vehicle has come to a complete halt in the turnout, divided by the turnout spacing. The subscript L refers to the F-factor when one accounts for the length of the vehicle F the expected F-factor F, the expected F-factor for headways greater than the c r i t i c a l headway f2 the expected F-factor for headways les s than the c r i t i c a l headway F-j the expected F-factor for a Case 3 turnout delay s i t u a t i o n 129 the maximum F-factor r e s u l t i n g from the i n t e r a c t i o n between one empty vehicle and one loaded vehicle c o e f f i c i e n t of f r i c t i o n road gradient, i n c l i n e , or grade acceleration of gravity General Purpose Simulation System V t r a f f i c flow rate, the number of vehicles passing a point per time unit headway, which i s the time i n t e r v a l between successive vehicles measured from front to front vehicle headway where the subscripts 1 and 2 r e f e r to constrained and free flow c r i t i c a l headway where the subscript CL refers to the i n c l u s i o n of vehicle length i n the c a l c u l a t i o n of the c r i t i c a l headway 2 a A a D / (afl+aD) the length of the loaded vehicle adjusted hauling cost ( i . e . , d o llars per hour) the number of expected delays while the empty vehicle travels a unit distance of road present worth "the expected useful l i f e of the road" the number of headways during the " c o n f l i c t " hours the minimum time gap between the back of the f i r s t vehicle and the front of the second vehicle distance between turnouts the c r i t i c a l turnout turnout locations 130 T,T A,T 5 time where the s u b s c r i p t s 4 and S r e f e r to the a c c e l e r a t i o n time and the stopping time f t o t a l expected delay time per u n i t d i s t a n c e per empty v e h i c l e t time or headway V, v e l o c i t y of loaded v e h i c l e v e l o c i t y c f empty v e h i c l e W g r o s s weight of the v e h i c l e X C F c r i t i c a l d i s t a n c e Z 2 a present worth f u n c t i o n I I Other Symbols oi, parameters of the E r l a n g headway d i s t r i b u t i o n 'Tj minimum headway ^ 2 c h i - s g u a r e value ^ base of the n a t u r a l l o grithums 131 APPENDIX 3 ACCELERATION AND DECELERATION OF A VEHICLE The braking force required to stop a vehicle involves the weight of the vehicle (W), the road gradient or i n c l i n e (0), and the c o e f f i c i e n t of f r i c t i o n (f) between the t i r e s and the road surface. In t h i s instance i t w i l l be assumed there i s a constant road gradient, a constant c o e f f i c i e n t of f r i c t i o n , and a braking force measured along the plane of the i n c l i n e . . By equating these conditions along the plane of the i n c l i n e the braking force equation can be obtained: Braking force = Wa( /g = Wfcose- Wsin© where: a( = deceleration (feet per second per second) g = acceleration of gravity.. I f angle 0 i s r e l a t i v e l y small then sin© i s approximately equal to tan© and cos8 approaches one., The equation can be rewritten as: a ; = g (f-tane) or a D = 79036 (f-G/100) where : a^ = deceleration (mph2). To account for adverse and favourable gradients the equation can be adjusted to: a D = 79036 (f±G/100) A.1 I f Q i s r e l a t i v e l y large then: a 0 = 79036 (fcos6± s i n e ) a D = 79036 (100f ±G)/(G 2 + 1002) o-s A. 2 132 where : D" = horizontal stopping distance D* = i n c l i n e stopping distance Fp = f r i c t i o n a l force N = normal force W = weight of the vehicle © = angle of the i n c l i n e The next step i s to determine the error between equations A.1 and A.2. A variable epsilon(G) can be introduced such that: (100f±G )/(G2 + 1002) o,s = (f ± G / 1 0 0 M 1 + £ ) or 0 = £2 (G 2 + 10()2) • 2£(G 2 + 10C)2) +G2 Solutions for epsilon are i l l u s t r a t e d i n the table below. In r e a l i t y the c o e f f i c i e n t of f r i c t i o n i s proportional to speed. . Therefore, a c o e f f i c i e n t of f r i c t i o n should be chosen such that the stopping distance or time i s correct (See table below). 133 If the braking distance i s defined as being along the horizontal plane then: Ds = Ds cos © where : Dg = horizontal braking distance D* = i n c l i n e braking distance the deceleration equation becomes: a D = 79036 (f±G/100) and epsilon equals zero. The acceleration of a vehicle i s limited by the horsepower of the vehicle and the forms of resistance to vehicle movement: r o l l i n g , a i r , gradient, engine, and i n e r t i a (Matson et a l . (19 55) and Byrne et a l . (1947)) . Matson et al.(1955) showed that the residual horsepower available for acceleration i s : HPn = HP_ - HP_ - HPA_ - HP„„ . A T R O L L AIR. GftAOt = HP f-(WV Z{R^±20G) + 0.0026K, AV|) /375 where: A = f r o n t a l area of vehicle i n sguare feet HPfl = re s i d u a l horsepower HP A l R = power to overcome a i r resistance HP, = power to overcome gradient resistance HP„ = power to overcome r o l l i n g resistance R O L L HPT = t o t a l horsepower available Kj = streamlining factor R R = r o l l i n g resistance i n pounds/ton W = vehicle weight in tons and the po t e n t i a l acceleration i s : 134 a f l = (14819 HPfl) / (WVg,) The above formula calculates the potential acceleration but seldom i s the potential acceleration equivalent to the actual acceleration. Consequently, the acceleration formula should be modified to account for t h i s deviation. If the acceleration time or distance i s known then the acceleration can be e a s i l y determined from basic dynamics, equations 2.3 or 2.4. S i m i l a r i l y , the stopping distance or time tables developed i n t h i s appendix can be u t i l i z e d , provided the c o e f f i c i e n t of f r i c t i o n becomes a c o e f f i c i e n t of t r a c t i o n . . In the development of the model the table approach w i l l be u t i l i z e d i n favour of the p o t e n t i a l acceleration formula approach. Consequently, the tables are used i n a similar manner as for the case involving deceleration. . Table Of Epsilon VS Road Gradient [%) Road Gradient (%) 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 Epsilon 0.000049604 0.000199725 0.000449010 0.000798332 0.001245908 0.001791597 0.002434749 0.003174604 0.004009128 0.004938118 0.005959723 0.007072758 0.008274801 0.009565637 0.010942270 0.012403550 0.013946550 0.015570130 0.017271420 0.019049160 0.020899980 0.022822160 0.024812970 0.026870420 0.028991170 1 3 6 T a b l e O f S t o p p i n g T i m e s — S e c o n d s V e l o c i t y C o e f f i c i e n t O f F r i c t i o n P l u s O r M i n u s R o a d G r a d i e n t iph) 0 . 0 5 0 . 10 0 . 15 0 . 2 0 0 . 2 5 0 . 3 0 0 . 3 5 5 4 . 6 2 . 3 1 . 5 1 . 1 0 . 9 0 . 8 0 . 7 10 9 . 1 4 . 6 3 . 0 2 . 3 1 . 8 1 . 5 1. 3 15 1 3 . 7 6 . 8 4 . 6 3 . 4 2 . 7 2 . 3 2 . 0 20 1 8 . 2 9 . 1 6 . 1 4 . 6 3 . 6 3 . 0 2 . 6 2 5 2 2 . 8 1 1 . 4 7 . 6 5 . 7 4 . 6 3 . 8 3 . 3 30 2 7 . 3 1 3 . 7 9 . 1 6 . 8 5 . 5 4 . 6 3 . 9 3 5 3 1 . 9 1 5 . 9 1 0 . 6 8 . 0 6 . 4 5 . 3 4 . 6 4 0 3 6 . 4 1 8 . 2 1 2 . 1 9 . 1 7 . 3 6 . 1 5 . 2 4 5 4 1 . 0 2 0 . 5 1 3 . 7 1 0 . 2 8 . 2 6 . 8 5 . 9 5 0 4 5 . 5 2 2 . 8 1 5 . 2 1 1 . 4 9 . 1 7 . 6 6 . 5 5 5 5 0 . 1 2 5 . 1 1 6 . 7 1 2 . 5 1 0 . 0 8 . 4 7 . 2 6 0 5 4 . 7 2 7 . 3 1 8 . 2 1 3 . 7 1 0 . 9 9 . 1 7 . 8 Table Of Stopping D i s t a n c e s — F e e t 137 V e l o c i t y C o e f f i c i e n t Of F r i c t i o n Plus Or Minus Road Gradient i p h ) 0 . 0 5 0 . 10 0 . 1 5 0 . 20 0 . 2 5 0 . 3 0 0 . 3 5 5 1 7 . 8 . 6 . 4 . 3 . . 3 . 2 . 10 6 7 . 3 3 . 2 2 . 1 7 . 1 3 . , 1 1 . 1 0 . 15 1 5 0 . 7 5 . 5 0 . . 3 8 . 3 0 . . 2 5 . 2 1 . 2 0 2 6 7 . 1 3 4 . 8 9 . 6 7 . 5 3 . 4 5 . . 3 8 . 2 5 4 1 8 . . 2 0 9 . . 1 3 9 . 1 0 4 . . 8 4 . . 7 0 . . 6 0 . 3 0 6 0 1 . 3 0 1 . 2 0 0 . . 1 5 0 . 1 2 0 . 1 0 0 . 3 6 . 3 5 8 1 8 . . 4 0 9 . 2 7 3 . . 2 0 5 . . 1 6 4 . 1 3 6 . 1 1 7 . 4 0 1 0 6 9 . 5 3 4 . 3 5 6 . 2 6 7 . 2 1 4 . 1 7 8 . 1 5 3 . 4 5 1 3 5 3 . 6 7 6 . . 4 5 1 . . 3 3 8 . 2 7 1 . . 2 2 5 . . 1 9 3 . 5 0 1 6 7 0 . 8 3 5 . 5 5 7 . 4 1 8 . . 3 3 4 . 2 7 8 . 2 3 9 . 5 5 2 0 2 1 . . 1 0 1 0 . . 6 7 4 . . 5 0 5 . 4 0 4 . . 3 3 7 . . 2 8 9 . 6 0 2 4 0 5 . 1 2 0 2 . 8 0 2 . 6 0 1 . . 4 8 1 . 4 0 1 . . 3 4 4 . 138 APPENDIX 4 ANALYSIS OF HEADWAY DISTRIBUTIONS The analysis of headway d i s t r i b u t i o n s involved data from two studies, a coastal study (Canadian Forest Products Limited) and a northern i n t e r i o r study (Northwood Pulp and Timber Limited). The analysis involved the determination of the observed frequency, the ca l c u l a t i o n of the t r a f f i c flow rate, and ~y* goodness of f i t tests. . The coastal study used frequency classes of f i v e minutes while the i n t e r i o r study used frequency classes of one and f i v e minutes.. The goodness of f i t te s t s employed the use of two computer programs, U.B.C. FREQ and a program written i n BASIC for the HP9830A c a l c u l a t o r . . A b r i e f documentation of the BASIC program follows. With the use of "REM" statements i n the program the.model i s e a s i l y explained. The basic sections of the program are: 1.. Input of the lower c e l l l i m i t , the c e l l i n t e r v a l , and the number of c e l l s 2.. Calculation of the c e l l boundaries 3.. Input of the parameters of the d i s t r i b u t i o n to be tested 4., Determining observed frequency 5. Determining expected frequency 6. Regrouping of the c e l l s such that there i s a minimum of fi v e expected values i n any c e l l 7. Calculation of the chi-square value.. 139 Variable A C D E F N N1 N2 N3 V1 V2 V3 Proqram Variables Function Chi-square value frequency of the observed and expected data frequency of observed data frequency of expected values c e l l boundaries i n i t i a l number of c e l l s c e l l width i n i t i a l value number of c e l l s a f t e r regrouping t o t a l number of vehicles vehicles per hour number of hauling hours per day n o Program L i s t i n g 5 REM GOODNESS OF FIT TEST 10 DIM F(80) ,C(2,80) ,Q$(40) ,X(2,80) ,D(80) ,B$(1) ,E(80) 15 REM FREQUENCY OF OBSERVED DATA 20 DATA 30 DATA 60 DISP "INPUT DATE"; 70 INPUT Q$ 80 PRINT LIN2,Q$,LIN2 90 DISP"INPUT PROBABILITY DISTRIBUTION"; 100 INPUT Q$ 110 A=0 120 MAT C=ZER 130 DISP"INPUT NUMBER OF CLASSES 6 INTERVAL"; 140 INPUT NfN1 150 DISP"INPUT INITIAL VALUE"; 160 INPUT N2 165 REM CALCULATION OF CELL BOUNDARIES 170 FOR 1=0 TO N-1 180 F (1 + 1)=I*N1+N2 190 NEXT I 200 F(N) = 200 210 DISP"INPUT # VEHICLES, VEH./HR.,DAY LENGTH"; 220 INPUT 71,72,73 230 DISP"IS DATA TO BE ENTERED"; 240 INPUT B$ 141 245 REM DETERMINING THE FREQUENCY OF THE OBSERVED DATA 250 FOR J = 1 TO N 260 IF B$="Y" THEN 300 270 C (1,J) = D (J) 280 GO TO 310 290 DISP"INPUT OBSERVED VALUE»J; 300 INPUT C(1,J) 310 NEXT J 320 C1=C3=0 325 REM DETERMINING THE FREQUENCY OF THE EXPECTED VALUES 330 FOR J=1 TO N 335 REM THE PROBABILITY FUNCTION 340 C (2,J) =V1* (1- (1 + 2*F(J) *V2/60) *EXP(-2*F (J) *V2/60) ) 350 C2=C(2,J) 360 C (2 rJ) =C {2,3) -C1 370 C1=C2 380 C3=C3+C (2, J) 390 NEXT J 400 PRINT LIN3,"THE DISTRIBUTION IS "Q$,LIN3 410 PRINT"NUMBER OF CLASSES = "N 420 PRINT"CLASS INTERVAL = "N1 430 PRINT"INITIAL VALUE = "N2 440 PRINT"TOTAL NUMBER OF VEHICLES = "V1 450 PRINT"VEHICLES PER HOUR = "V2 460 PRINT"LENGTH OF DAY = "V3 470 PRINT LIN2 480 FOR 1=1 TO N 490 PRINT J;C(1 rJ) ;C(2 fJ) 500 NEXT J 510 PRINT LIN2 515 REM GOODNESS OF FIT TEST 560 MAT D=ZER 565 MAT E=ZER 570 N3=1 575 REM REGROUPING OF THE CELLS SUCH THAT THERE IS 576 REM A MINIMUM FREQUENCY OF FIVE EXPECTED VALUES 577 REM IN EACH CELL 580 FOR 1=1 TO N 590 D (N3) =D(N3) +C(2 rI) 595 E (N3) =E (N3) +C (1,1) 600 IF D(N3)<5 THEN 620 610 N3=N3+1 620 NEXT I 630 IF D(N3)>=5 THEN 660 640 N3=N3-1 645 E (N3) =E (N3)+E (N3+1) 650 D(N3) =D(N3)+D(N3+1) 660 FOR 1=1 TO N3 670 C (2,1) =D (I) 675 C(1,1)=E (I) 680 NEXT I 690 A=0 695 REM CALCULATION OF THE CHI-SQUARE VALUE 700 FOR 1=1 TO N3 710 A=A*(C (1,1)-C (2,1) )f 2/C (2,1) 720 NEXT I 730 PRINT LIN1 740 PRINT"CELL NUMBER OBSERVED EXPECTED"LIN1 143 750 FOE J=1 TO N3 760 PRINT J;C(1,J) ;C(2,J)-770 NEXT J 780 PRINT LIN1 790 PRINT"THE CHI-SQUARE VALUE IS "A 800 PRINT LIN2 810 END 144 Coastal Study Observed Headway Frequency-Distribution Class Mark | Frequency| ! Class Mark |Frequency (minutes) 4 l -— + - (minutes) 1-25 I 2 | | 65 | 5 5 1 1 6 | I 70 | 1 10 I 60 | I 75 I o 15 I 31 | I 80 I o 20 I 17 | I 85 I 2 25 I 9 I I 90 I o 30 I 14 | I 95 I o 35 | 7 | I 100 I o 40 I 5 | ! 105 I o 45 I 6 | ! 110 | 1 50 I ^ | i 115 I o 55 I 2 I i 117.5 + I 6 60 I 5 | I Total | 193 145 T r a f f i c Flow Rate flow* : 201 vehicles/83.4 hours = 2.41 vehicles/hour or 193 vehicles/83.4 hours = 2.32 vehicles/hour * There was a t o t a l of 201 vehicles observed over the 8-day period but there were only 193 headway observations. The actual flow rate l i e s between the two calculated values. 146 Goodness Of F i t Te s t s — R e s u l t s Probability Dist. . Program Number* Of Classes Class Width (min) [ - J - -+ +- -+ +- -+ - + H IPoisson | EREQ 1 2 5 ( 1 1 ) | 5 1 0 | 3 1 6 I 9 1 2 3 . 6 | |Binomial | FREQ 1 2 5 ( 1 0 ) | 5 1 0 | 3 2 6 I 8 1 2 2 . 0 | | Neg bino m. . | FBEQ 1 2 5 ( 1 8 ) | 5 1 0 | 120 | 15 1 3 2 . 8 1 |Exponential! BASIC | 2 5 ( 1 4 ) | 5 1 2 . 5 J 7 6 . 3 | 12 1 2 8 . 3 1 | Erlang (oC=2) | Basic 1 2 5 ( 1 2 ) J 5 I 2 . 5 | 6 2 . 7 | 10 1 2 5 . 2 | |Shift expon| | : 0 . 1 min | B a s i c | 2 5 ( 1 4 ) | 5 1 2 . 5 1 7 5 . 2 I 12 1 2 8 . 3 ! | : 0 . 5 min | B a s i c 1 2 5 ( 1 4 ) | 5 1 2 . 5 J 7 1 . 0 I 12 1 2 8 . 3 | | : 1 . 0 min | B a s i c 1 2 5 ( 1 4 ) | 5 1 2 . 5 | 6 5 . 8 I 12 1 2 8 . 3 | L L _ J i — _ j i i _ x , LCL • (min) Computed r Table 00 5 * The parenthesized number i s the f i n a l number of frequency classes while the non-parenthesized number i s the number of frequency classes.. + The number of degrees of freedom • Lower cl a s s l i m i t 1 4 7 I n t e r i o r Study - F i r s t Weigh Scale Observed Headway Frequency Distribution Class Hark (min) Freq. Class Mark (min) Freq. L._ - + + + - -f+ — + + + +~ 0 . 2 5 | 3 | | 2 2 | 11 | | 44 1 2 | 1 66 | 0 1 | 54 I I 2 3 | 11 1 1 4 5 1 3 | 1 6 7 | 0 2 I 46 | | 24 | 7 1 1 46 1 5 | | 68 | 3 3 I 29 I I 2 5 | 9 | | 4 7 1 6 | | 6 9 | 2 4 I 27 I I 2 6 | 12 | | 4 8 1 2 | 1 70 | 0 5 I 3 6 I I 2 7 | 7 1 1 49 1 0 | 1 71 | 0 6 I 34 | | 2 8 | 2 1 1 5 0 1 1 I I 72 | 0 7 I 2 5 I I 2 9 | 6 1 1 51 I 3 | I 73 | 1 8 | 2 0 I I 30 | 4 I 1 5 2 I 0 | I 74 | 1 9 I 18 t I 31 | 10 1 1 5 3 I 3 | 1 75 | 0 10 | 26 I I 32 | 4 1 1 54 I 4 | | 76 | 2 11 I 18 I I 3 3 | 5 1 1 5 5 I 0 | 1 7 7 | 1 12 | 14 | | 34 | 6 1 1 56 I 0 | 1 78 | 0 13 | 16 I I 3 5 | 8 1 1 5 7 I 2 | 1 79 | 0 14 | 12 | | 3 6 | 3 1 1 5 8 I 0 | | 7 9 . 5 + 1 5 15 | 2 8 I I 3 7 | 6 | | 5 9 I 0 | | | 16 | 13 I I 3 8 | 8 | | 6 0 | 0 | | | 17 | 11 I I 3 9 | 3 | | 61 I 0 | | | 18 | 12 I I 4 0 | 7 I I 6 2 | 0 | | | 19 | 14 I I 4 1 | 3 I I 6 3 I 2 | | | 2 0 | 8 1 i 4 2 | 5 I I 6 4 I 0 | | | 21 1 9 | | 4 3 | 2 I 1 6 5 I 2 | | Total| 6 6 1 i _ _ J - J L _ -i-i 1 L 1 L_ Class Mark (min) Freq. Class Mark (min) Freq, T r a f f i c Flow Rate flow* = 681 vehicles/183.5 hours = 3.71 vehicles/hour or 661 vehicles/183.5 hours = 3.60 vehicles/hour * There was a t o t a l of 681 vehicles observed but there were only 661 headway observations. The actual flow rate l i e s between the calculated values. 1 4 8 Goodness Of F i t T e s t s — R e s u l t s P r o b a b i l i t y D i s t . . Program Number* Of C l a s s e s C l a s s Width (min) \- f — -+ -+ --+- -+ IPoisson | FREQ 1-80 ( 2 2 ) ! 1 I o. 5 1 + 5 0 0 | 20 1 4 0 . 0 IBinoiaial | 1 L | Neg bino m. . | FBEQ | 8 0 ( 2 0 ) ! 1 I o. 5 ! + 5 0 0 | 18 | 3 7 . 2 FREQ | 80 ( 5 9 ) ! 1 I 0 . 5 1 74 I 5 6 I 5 3 . 7 | E x p o n e n t i a l | BASIC | 80 ( 47 ) ! 1 I 0 . 5 1 7 6 . 8 I 4 5 1 5 3 . 7 | E r l a n g (cC=2) | BASIC | 8 0 ( 4 1 ) I 1 | 0 . 5 1 4 1 4 . 2 I 39 1 5 3 . 7 | S h i f t exponl | : 0 . 1 min | BASIC | 80 ( 47 ) j 1 I 0 . 5 1 7 4 . 3 | 4 5 1 5 3 . 7 1 : 0 . 5 min | BASIC | 8 0 ( 4 6 ) ! 1 1 0 . 5 | 5 5 . 1 | 4 4 1 5 3 . 7 | E x p o n e n t i a l ! BASIC I 1 7 ( 1 5 ) | 5 | 0 . 5 1 2 8 . 6 I 13 | 2 9 . 8 | E r l a n g (o(.=2) | EASIC I 1 7 ( 1 1 ) | 5 I 0 . 5 1 2 0 2 . 6 ! -9 | 2 3 . 6 | S h i f t expon| j : 0 . 1 min | BASIC 1 1 7 ( 1 5 ) I 5 I o. 5 ! 2 4 . 5 I 13 1 2 9 . 8 | : 0 . 5 min | BASIC I 17 ( 14 ) | 5 | 0 . 5 1 1 3 . 7 I 12 | 2 8 . 3 J. LCL • Value (min) Computed r v T a b l e l |oC=. 0 0 5| * The pa r e n t h e s i z e d value i s the f i n a l number of frequency c l a s s e s while the non-parenthesized number i s the i n i t i a l number of c l a s s e s + The number o f degrees of freedom • Lower c l a s s l i m i t 149 Interio r Study - Both Weigh Scales Observed Headway Frequency Distribution Class Mark (min) " T T " Freq. Class Mark (min) Freq. • T T " Class Mark (min) + + -+ — T + - — + + -1 +- -1 0. 25 | 3 1 I 22 I 11 I I 44 I 2 | 1 66 | 2 1 1 | 60 | | 23 I 11 II 45 I 3 | 1 67 | 0 1 2 1 53 | | 24 I 9 | | 46 | 8 | 1 68 | 3 1 3 1 30 | 1 25 I 1 0 | | 47 1 6 | | 69 | 2 1 4 i 32 | I 26 1 1 6 | | 48 1 4 | J 70 | 0 1 5 1 42 | 1 27 1 7 | | 49 1 1 I | 71 | 0 1 6 I 35 | | 28 1 4 | | 50 I 1 I I 72 | 1 | 7 1 28 | | 29 1 9 M 51 I 4 | I 73 | 1 | 8 1 22 I | 30 1 4 | | 52 | 1 | I 74 | 1 | 9 1 20 | 1 31 1 1 0 | | 53 I 4 | I 75 | 0 I 10 | 26 | I 32 i 5 | | 54 | 4 | | 76 | 3 I 1 1 | 21 1 | 33 1 7 | | 55 | 0 | | 77 | 1 | 12 | 17 I | 34 1 1 0 | | 56 I 2 | | 78 | 0 I 13 | 16 | 1 35 1 1 2 | | 57 | 3 | 1 79 | 0 I 14 | 18 | 1 36 1 5 | | 58 I 1 I | 79.5+1 10 | 15 | 28 | 1 37 1 7 | | 59 1 0 | | | 1 6 | 16 | | 38 1 8 | | 60 1 0 | | | 17 | 13 | I 39 1 4 | | 61 1 0 | | | 18 | 14 | | 40 1 8 | | 62 1 0 | j j 19 | 14 | 1 41 1 6 | | 63 1 2 | | | 20 | 9 1 | 42 1 8 || 64 I 0 | | j 21 | 11 1 | 43 1 4 | | 65 1 2 | | Total| 775 | i 1 . i ~ L _ _ _ — L J x _ — i Freq. • T T " Class Mark (min) Freq. T r a f f i c Flow Rate flow* : 814 vehicles/252.2 hours = 3.23 vehicles/hour or 775 vehicles/252.2 hours = 3.07 vehicles/hour * There was a t o t a l of 814 vehicles observed but there were only 775 headway observations'. The actual flow rate l i e s between the calculated values. 150 Goodness Of F i t Tes t s — R e s u l t s I T T 1 T 1 1 T 1 Probability! Dist. . | Program !Number* lOf |Classes f Class] Width! (min)| + LCL • Value (min) Computed! % ' j +• If* |Table| I ^ 2 I K=- 0 0 5 | - + 1 Poisson | FREQ ! 80 ( 2 3 ) ! ] 0 . 5 • 5 0 0 ! 21 1 4 1 . 4 | Binomial | FREQ | 8 0 ( 2 1 ) 1 ! 0 . 5 + 5 0 0 | 19 1 3 8 . 6 | Neg bino m. . | FREQ | 8 0 ( 6 4 ) 1 ! 0 . 5 8 5 . 3 | 6 1 | 5 3 . 7 1 Exponential! BASIC | 8 0 ( 5 3 ) 1 I 0 . 5 9 2 . 0 | 51 1 5 3 . 7 1 Erlang (cd=2) | BASIC | 8 0 ( 4 8 ) 1 ! 0 . 5 6 1 7 . 9 1 46 1 5 3 . 7 I S h i f t exponj : 0 . 1 min | BASIC | 8 0 ( 5 3 ) 1 ! 0 . 5 8 7 . 5 J 51 1 5 3 . 7 1 : 0 . 5 min | BASIC | 8 0 ( 52 ) i I 0 . 5 7 2 . 2 | 50 1 5 3 . 7 | Exponential! BASIC | 21 ( 1 7 ) 5 1 0 . 5 1 3 3 . 7 1 15 1 3 2 . 8 I Erlang (ct=2) | BASIC | 2 1 ( 1 3 ) 1 5 | 0 . 5 I 1 8 3 . 5 I 11 1 2 6 . 8 | Shift exponj : 0 . 1 min | BASIC | 21 ( 17 ) I 5 | 0 . 5 2 9 . 4 I 15 1 3 2 . 8 I : 0 . 5 min | BASIC 1 2 1 ( 1 6 ) 5 ! 0 . 5 I 1 7 . 6 | 14 1 3 1 . 3 ! Exponential! BASIC | 2 0 ( 1 6 ) 5 J 5 . 5 2 0 . 7 | 14 1 3 1 . 3 | * The parenthesized value i s the f i n a l number of frequency classes while the non-parenthesized number i s the i n i t i a l number of classes + The number of degrees of freedom • Lower class l i m i t 151 APPENDIX 5 SIMULATION OF THE f-FACTOR The purpose of the simulation models was to test the v a l i d i t y and determine the l i m i t a t i o n s of the expected F-factor equations, equations 2.12, 2.18, and 2.19., The programs that were developed consisted of the i n t e r a c t i o n between two single vehicles, the meeting of a single empty vehicle and a f l e e t of loaded vehicles, and the in t e r a c t i o n between two f l e e t s of vehicles. . The models do not r e f l e c t a l l of an operation's turnout delay times since many of the inherent delays i n the system are excluded.. The models assume that the empty vehicle w i l l always u t i l i z e the turnout which w i l l y i e l d the least delay.. Furthermore the models assume the empty vehicle has a constant rate of acceleration, the turnout spacing i s uniform, and the turnouts have no length.. The s p e c i f i c c h a r a c t e r i s t i c s of each of the models w i l l be discussed i n their corresponding sections. Interaction Between A Loaded Vehicle And An Empty Vehicle The simple F-factor equation to be tested i s : F = (V, +\ )/{2Vt) I n i t i a l l y , the empty vehicle i s positioned opposite turnout S 0 and the loaded vehicle i s randomly located (uniform d i s t r i b u t i o n over the i n t e r v a l (5,10)) at a distance D1 away. If both vehicles were to proceed at t h e i r corresponding v e l o c i t i e s they would meet at a point that i s at a distance 152 distance X from turnout S .. Thus: X = (V2 DIJ/CV/ *\) The f i r s t turnout (INTE) that the empty vehicle could po t e n t i a l l y use can be readily determined as: INTE = X/S where : INTE i s an integer number.. Referring to the figure below, the model tests whether or not the empty vehicle can safely u t i l i z e turnout INTE. I f the vehicle cannot safely use t h i s turnout, then the model w i l l check p r i o r turnouts u n t i l i t finds the f i r s t turnout that the empty vehicle can safely use. Now the F-factor can be e a s i l y determine d. 153 I 1 D1 10 where : X = p o t e n t i a l meeting point D1 = i n i t i a l d i s t a n c e v e h i c l e s are apart I i — I I D i s t a n c e ( m i l e s ) 0 X 5 Turnout l o c a t i o n s S INTE Turnout l o c a t i o n s Distances I (D m~ I 1 - T 4 + T 3 i Distances l<-Y6-^ | where: # = l o c a t i o n of loaded v e h i c l e 0= l o c a t i o n of empty v e h i c l e The v a r i a b l e names r e f e r t o the v a r i a b l e names used i n the computer program.. t i m e , | A S — " I 1 0 INTE Y 1 r - ^ r l 1 - T 4 ^ D3 X Y2-J-H k Y 1 >| ' 154 Program Variables Variable Function A1 rate of acceleration D1 i n i t i a l distance vehicles are apart D3 stopping distance F1 c o e f f i c i e n t of f r i c t i o n F9 F-factor for a single meeting INTE f i r s t turnout that empty vehicle t r i e s to use N the sample size S distance between turnouts S3 the average F-factor for the sample TOTAL the sum of the F-factors T3 the stopping time 14 time for empty vehicle to t r a v e l distance Y1 V1 speed of loaded vehicle V2 speed of empty vehicle X potential meeting point y1 distance from X to beginning of deceleration zone Y2 distance loaded vehicle i s from turnout when empty vehicle begins to decelerate Y6 distance loaded vehicle i s from turnout when empty vehicle has stopped in turnout 155 Program L i s t i n g C C SIMULATION OF ONE EMPTY VEHICLE MEETING ONE LOADED VEHICLE C F1=0.25 AT=79036.3636*F1 Q1=BAND(SCLCCK (0.0) ) N=10000 Z1=N C C THE VEHICLE'S CHABACTEEISTICS C V1=10. V2=35. S=0 . 2 C C THE STOPPING FUNCTIONS C D3=V2**2/(2.*A1) T3=V2/A1 WRITE (6, 101) 101 FORMAT('I') C C THE NUMBER OF SAMPLES C DO 1 11=1,20 156 TOT AL=0. C C THE SAMPLE SIZE C DO 2 1=1,N C C INITIAL DISTANCE THE VEHICLES AEE APART C D1=FRAND (0. 0) *5. +5. C C DETERMINE TURNOUT THAT EMPTY VEHICLE CAN SAFELY PULL INTO C X=V2*D1/(V1+V2) INTE=X/S Y1= X-S*INTE+D3 T4= Y1/V2 Y2=T4*V1+Y1-D3 T5= Y2/V1 C C CAN EMPTY VEHICLE SAFELY PULL INTO THIS TURNOUT C 4 IF(T5.GT.T3) GO TO 3 C C TRY TO STOP EMPTY VEHICLE AT PREVIOUS TURNOUT C Y1=Y1 + S Y2=Y2+V1*S/V2+S T5= Y2/V1 GO TO 4 157 C C CALCULATION OF THE F-FACTOR C 3 Y6= (T5-T3) *V1 F9 -Y6/S C C SUMATION OF THE F-FACTOR C 2 TOTAL=TOTAL+F9 C C THE AVERAGE F-FACTOR C S3=T0TAL/Z1 1 WRITE ( 6 , 100) S3 100 FORM AT (' «,5X,F8.4) RETURN END 158 Interaction Between An Empty Vehicle And A-Fleet Of Loaded Vehicles The F-factor equations to be tested are equations 2.18 and 2.19. The four headway d i s t r i b u t i o n s that are required are the exponential, s h i f t e d exponential, Erlang(alpha=2), and Pearson Type III(a=2).. The equations required for these:distributions are: 1. . Exponential: Pr(h<h c) = i - e - Q ^ < -\ = v, [ e - Q 3 ^ (-Q 3h c-1)+1] / [-so,., ( e - ^ A - i ) ] F, = (V, +V L)/(2V 2) 2. . Shifted exponential: Pr(h<hCJ.) = 1-e-Qs < K-*r > n= * i [ ^ + R ^ - e - ^ < K c - R T > ( h c + ^ + ^ ) ] / [S(1-e-^<K £-R T)) ] 3. Erlan g (alpha - 2) : Pr(h<h c) = 1-e-2<?^(1 + 2Q 3h c) F = V, [e- Z^»c(-2Q 3h2-2h c-1/Q 3) +1/Q3] / {S[ 1 1 + 20- h c ) ]} 4.. Pearson Type III(a=2): Pr(h<hCL) = 1 - e - b t W [ b (h c-R T) +1 ] Fa = e - bck-RT>[ b (-h*- ^  + R rh c +*^) -2hc -|-^*H T ]+[ R, • 1-e-kC  h<--RT >[ b (h c-R T) +1 ] This model i s sim i l a r to the one that has been developed i n the previous section.. In t h i s model the functions to determine vehicle headways are: 159 1. Exponential: Y = [-log(//) ]/ A * V where: JU = uniform (0,1) variable mean of the exponential d i s t r i b u t i o n = minimum headway and 2. Erlang (alpha=2) and Pearson Type III (a=2) : Y = [s=] (-log(//)/A + 1 The c r i t i c a l headway must be determined to check i f an empty vehicle can proceed from i t s current turnout. The empty vehicle progresses along a continuous road.. 160 Program Variables Variable Function Al rate of acceleration D1 i n i t i a l distance vehicles are apart D2 a distance function similar to D1 D3 stopping distance F1 c o e f f i c i e n t of f r i c t i o n F9 F-factor H t r a f f i c flow rate INTE f i r s t turnout empty vehicle t r i e s to u t i l i z e LENG the length of the loaded vehicle N sample size Q t o t a l number of vehicles during the day R alpha variable of Erlang function S distance between turnouts S3 average F-factor of the sample TOTAL sum of the F-factors of the sample T1 c r i t i c a l headway T2 loaded vehicle headway T3 stopping time T4 time of empty vehicle to trave l distance Y1 T5 time for loaded vehicle to reach turnout VI speed of loaded vehicle V2 speed of empty vehicle X potential meeting point XLAH a parameter in the gamma function Yl distance from X where deceleration zone begins Y2 distance loaded vehicle i s from turnout when empty vehicle begins to decelerate the F-factor times the turnout spacing 162 Proqram L i s t i n g C C C SIMULATE AN EMPTY VEHICLE AND A FLEET OF LOADED VEHICLES C REAL H (18) , LENG , MIN DATA H/1.,2.,4. , 10.,6 0., 100., 1.,2.,4.,10. ,6 0., 100. , 1. , 12. ,4. ,10. ,60. , 100./ C C THE VEHICLE'S CHARACTERISTICS C S=0. 3 V1 = 30. V2=40. D0=11. F1=0.25 A1=79036.3636*F1 LENG=60./5280. MIN=LENG/V1/D0+0.000050505051 C C THE CRITICAL HEADWAY C T1= (S*A1*(V1 + V2) +V1*V2**2)/(A1*V1*V2*D0) +LENG/ (V1 *D0) Q1=RAND(SCLOCK(0.0) ) N=10000 C 163 C THE STOPPING FUNCTIONS C D3=V2**2/(2.*A1) T3=V2/A1 WRITE (6, 101) 101 FORM AT (* 1') C C THE NUMBER OF SAMPLES C DO 1 11=1,18 TOT AL=0. Q3=D0*H (11) C C THE SAMPLE SIZE C DO 2 1=1,N IF(I.GT. 1.5) GO TO 3 C C THE INITIAL DISTANCE THE FIRST TWO VEHICLES ARE APART C D1= FRAND (0.0) *5. +5. D2=D1 GO TO 4 C C CALCULATION OF THE VEHICLE HEADWAY C 3 D4=FRAND (0.0) T2=-1.*ALOG (D4) /Q3 C 164 C IS VEHICLE SPACE HEADWAY LESS THAN THE VEHICLE'S LENGTH C IF(T2.LT.MIN) GO TO 3 D2=D1+T2*V1*D0 C C IS THE VEHICLE HEADWAY LESS THAN THE CRITICAL HEADWAY C IF(T2.GT.T1) GO TO 5 C C CALCULATE F-FACTOR FOR HEADWAYS LESS THAN CRITICAL HEADWAY C Y6=T2*V1*D0+LENG F9=Y6/S D1=D2 GO TO 6 5 D2=D2-(X-Y1+D3) 4 IF(I.GT. 1.5) GO TO 7 C C DETERMINING MEETING POINT OF THE FIRST ENCOUNTER C X=V2*D2/ (V1+V2) GO TO 8 C C DETERMINING MEETING POINT FOR THE OTHER ENCOUNTERS C 7 X=(V1*D2-V1*V2**2/(2. *A1) ) /(V1 + V2) C C FIND FIRST TURNOUT EMPTY VEHICLE WILL TRY TO PULL INTO C 165 8 INTE-X/S Y1=X-S*INTE+D3 T4=Y1/V2 Y2=T4*V1+Y1-D3 T5=Y2/V1 C C CAN EMPTY VEHICLE SAFELY POLL INTO THIS TURNOUT C 10 IF(T5.GT.T3) GO TO 9 C C TRY TO STOP EMPTY VEHICLE AT THE PREVIOUS TURNOUT C Y1=Y1 + S Y2=Y2+V1*S/V2+S T5= Y2/V1 GO TO 10 C C CALCULATE THE F-FACTOR C 9 Y6= (T5-T3) *V1+LENG F9=Y6/S IF(I.LT. 1.5) GO TO 6 D1=D2 C C SUMATION OF THE F-FACTORS C 6 TOTAL=TOTAL+F9 2 CONTINUE Z1=N 166 C C CALCULATION OF THE AVERAGE F-FACTOR C S3=TOTAL/Z1 1 WRITE(6,100) H(I1),S3 100 FORMAT (• • r5X,F6.1 f5X,F8.4) WRITE (6, 101) RETURN END 167 Interaction Between Two Fleets Of Vehicles The purpose of t h i s model i s to determine the range of t r a f f i c flow rates for which the expected F-factor equations are v a l i d . The basic equation to be tested i s : F = Pr(h>h c)F, + Pr(h<h c)F a Most of the testing of the model w i l l involve the exponential F-factor equation. The simulation model, written in GPSSV, has been formulated such that i t i s r e s t r i c t e d to certain combinations of vehicle speeds, turnout spacings, and acceleration rates. Generally, the empty vehicle must t r a v e l faster than the loaded vehicle. The figures on the following pages i l l u s t r a t e the formulation of the simulation model. . 168 Time Parameters Of The Em_p_ty_ Vehicles Decision Point , 4T T _ „ , , , _ _ I I I I 1 Turnout locations Dump I S; , S- S; , S; _ I j , J I , J t l J+u J+3 Times* K &^)<-B^l—C-^-B^I I i | ^ _ D ^ A = ( 2aS-V|) 60/ (2aV2 ) B = (V2 /a) 60 C = (Sa-V2) 60/(a\ ) D = (S/V2 ) 60 S. = turnout locations Decision Point •i i — \ — i i Turnout locations Sz I S, Destination I 1 ' Times* , K E ^ I i u — F a • i Distances K 4S 1^ E = [ (2aS + Vf) /(2aV2 ) ]60 + (3S/VZ ) 60 F = [ (2aS+V2) /(2aV2 ) ]60 + (3S/VZ ) 60 •the times are in minutes 169 Time Parameters Of The Loaded Vehicles I I - - - I I I i I Turnout locations Dump S- S; , S- . S;., I r I J J * i J*3 Times* N R >| R = (S/V( ) 60 S.- = turnout locations J Decision Point , , ,,.^ T_..,__. r , Turnout locations S2 ' S, Destination I I I ' I I I I ' I Times* K Q ^ Z - N ^ Y ^ r £ Q = [ (aS+V|)/(aVE ) - (S/V, ) ]60 X = [ (4S/V( ) - (aS+Vf)/(aV2 ) ]60 Y = [ (aS+V|)/(aVa) - (S/V, ) ]60 Z = [ (2S/V, ) - (aS+V|)/(aV2 ) ]60 •times are in minutes 170 Time parameters for the empty vehicles & = time to tr a v e l from dump to f i r s t decision point B = time to decelerate into turnout from decision point C = time to accelerate and t r a v e l at constant velocity to next decision point D = t r a v e l time between decision points at constant velocity E = t r a v e l time to accelerate and tr a v e l at constant velocity from f i r s t turnout to landing F = time to t r a v e l to landing (at constant velocity) from decision point Time parameters for loaded vehicles E = time to tr a v e l from l a s t turnout to Q = time from decision point to turnout X = time from landing to f i r s t decision at constant velocity y = same as Q Z = time from turnout to decision point dump point t r a v e l l i n g 171 Proqram L i g t i n q $EUN *GPSSV SPRINT--00- PAE=SIZE=B SIMULATE * * TUENOUT SIMULATION * MODEL BY LAEEY A. .HENKELMAN * GBADOATE STUDIES * FOBESTBY, UBC * NOVEMBEB, 1977 * # TURNOUTS 8 MAX # VEHICLES-2 * TIME UNITS=1/1000 MINUTES * * STORAGE DEFINITIONS STORAGE S1-S40,2 * * VARIABLE DEFINITIONS 1 VARIABLE PH1-1 V1 TAKES 1 FBOM PEESENT TURNOUT * * FUNCTION DEFINITIONS VEHICLE HEADWAY DISTRIBUTIONS EXPON FUNCTION BN1,C24 0,0/. 1 , .104/. 2,. 222/. 3,.335/. 4,. 509/. 5,. 69/. 6,. 915 . 7 , 1. 2 / . 7 5 , 1.38/. 8, 1.6/.84, 1.8 3/. 88, 2. 12/. 9,2. 3 172 .92,2.52/. 94,2.81/. 95,2. 99/. 96 ,3.2/. 97, 3.5 .98,3.9/.99,4.6/.995,5.3/.998,6.2/.999,7/1,8 * EXPO FUNCTION RN2,C24 0,0/. 1,. 104/. 2,. 2 22/. 3,. 335/. 4,. 509/. 5, . 69/. 6, . 91 5 . 7, 1. 2 / . 7 5, 1.38/. 8, 1. 6/.84, 1. 83/. 88 , 2. 12/. 9,2.3 .92,2.52/.94,2.81/.95,2.99/.96,3.2/.97,3.5 . 98,3.9/. 99,4.6/.995,5.3/.998,6.2,. 999,7/1,8 * THE LETTERS IN COL 62 OF THE ADVANCE BLOCKS REFER TO THE * THE TIME PARAMETERS ON THE ACCOMPANYING FIGURE * ********* ***************************************************** * * MODEL SEGMENT 1 - EMPTY TRUCKS * RMULT 31,743 RANDOM NUMBER GENERATOR SEED GENERATE 1000,FN$EXPON GENERATE AN EMPTY TRUCK ASSIGN 1,40,PH STARTING AT TURNOUT # 40 ADVANCE 389 GO TO 1ST. DECISION PT. DDD GATE LR PH1,AAA IF SHUT, USE TURNOUT, AAA TEST NE PH1,1,EEE IF LAST TURNOUT GO TO EEE GATE SNF VI,AAA NEXT FULL? PULL IN NOW.. BBB ADVANCE 450 DOESN'T USE TURNOUT ASSIGN 1-,1,PH CONSIDER NEXT TURNOUT TR ANSFER ,DDD LOOP TRUCKS CANNOT PROCEED, MUST TURNOUT AAA ADVANCE 121 DECELERATE INTO TURNOUT ENTER PH1 , 1 TRUCK ENTERS TURNOUT 173 CCC LINK LEAVE ADVANCE ASSIGN TEST NE TRANSFER PH1r FIFO PH1 , 1 329 1-,1,PH PH1 ,0,FFF , DDD TRUCK QUEUES IN TURNOUT EMPTY TRUCK LEAVES TURNOUT GOES TO NEXT DECISION PT.. CONSIDER NEXT TURNOUT IF LAST TURNOUT GO TO FFF LOOP * LAST TURNOUT, PROCEED TO LANDING. FFF ADVANCE 1861 THE LAST TURNOUT TRANSFER ,GGG EEE ADVANCE 1861 THE LAST TURNOUT GGG TERMINATE ******************************************** * * MODEL SEGMENT 2 - LOADED TRUCKS GENERATE ASSIGN LOGIC S ADVANCE LOGIC S ADVANCE LOGIC R UNLINK 1000,FN$EXPO 1,3,PH 1 1229 2 571 1 1 rCCC rALL GENERATE A LOADED VEHICLE REMEMBER GATE 3 SHUT GATE 1 TO EMPTY TRUCKS GO TO 1ST DECISION POINT X SHUT GATE 2 TO EMPTY TRUCKS TRAVEL TO 1ST TURNOUT Y PASS TURNOUT 1, SO OPEN GATE RELEASE ALL TRUCKS AT 1 * GENERALIZED TURNOUT HHH ADVANCE 29 ADVANCE TEST NE PH1,41,III LOGIC S PH1 TO THE NEXT DECISION PT. IS THIS THE LAST TURNOUT ? CLOSE GATE REMEMBERED IN PH1 174 ADVANCE LOGIC E UNLINK ASSIGN T E A N S F E E 571 V1 V 1 , C C C , A L L 1+r 1 rPH ,HHH LOOP GO TO THE NEXT TOENODT Q OPEN GATE AS PASSING TURNOUT BELEASE LINEUP AT TURNOUT THINK ABOUT NEXT TURNOUT * LAST TURNOUT BEFORE DUMP I I I ADVANCE 571 LOGIC R UNLINK ADVANCE TERMINATE 40 4 0 , C C C , A L L 600 GO TO L A S T TURNOUT (#40) OPEN GATE 40 AS PASS R E L E A S E LINEUP AT GATE 40 GO TO DUMP ************************************************************** * MODEL SEGMENT 3 - TIMING * GENERATE 2000000 TERMINATE 1 * ********* ************************** *************************** * CONTROL CARDS START 1 END 175 APPENDIX 6 THE LENGTH OF THE VEHICLE Some of the equations developed i n t h i s thesis can be modified to i l l u s t r a t e the effect of the length of the vehicle. This length w i l l not affect the time l o s t due to vehicle acceleration and deceleration but i t w i l l a l t e r the expected F-factor equations. The simple F-factor equation i s adjusted to allow for the loaded vehicle to t r a v e l one length of the loaded vehicle. Consequently, the simple F-factor equation w i l l become: where : L = the length of the loaded vehicle. In the case of a single empty vehicle meeting a f l e e t of loaded vehicles the c r i t i c a l headway i s increased, since the empty vehicle must wait the extra delay time created by the length of the loaded vehicle. Consequently, the c r i t i c a l headway w i l l become: The expected F-factor and the probability equations are accordingly modified.. The formulas can be further modified i f i t i s assumed there i s a minimum gap between vehicles. . With a shifted exponential headway d i s t r i b u t i o n the modified gap density function becomes: A. 3 h C L = [2Sa Aa 0 (V, * \ ) +V, Vf (a A +a0) +2LaA a D \ ] / [2a f la 0V, V£ ] A .4 176 ^[Pr(t<h)] _ 0 , g ( t ) - = Q e-Q 3<t -RT--t-> 3t 3 1 where : RT = the minimum time gap between the back of the f i r s t loaded vehicle and the front of the next vehicle RT+L/V, = minimum headway between vehicles.. The probability that a headway i s less than the c r i t i c a l headway i s : Pr(h<h C L) = 1 - e ~ Q 3 <V~*T- -^J = 1-e-^ (^c-RT> A . 5 and the expected F-factor for headways less than the c r i t i c a l headway i s : V, F 2 = L/V, +RT + 1 / C 3 - e - q 3 < K£-RT > (ht+L/V, +1/Q3) A. 6 A shifted Erlang (alpha=2) headway d i s t r i b u t i o n i s developed from a Pearson Type III density function.. I f the parameter "a" i s set to two then the gap density function becomes: g(t) = bz ( t - c ) e - b c t - c > c<t< 0 0 where: c = RT+L/V, b = 2 / (1/Q3-RT-L/V( ) The probability that the headway i s less than the c r i t i c a l headway i s : Pr(h<h C L) =j^ C Lb 2 (h-c ) e- b ( K- c>6h = 1-e- b t l r'c-^T>[b(h c-R T)+1] A^7 and the expected F-factor f o r headways less than the c r i t i c a l headway i s : 177 F 2 = e-b(hc-RT) [ b ( . n | . i ! ^ + E T h c + IrL) -21V VRT M % + ' 1-e' b ( h t" R T ) C b(h c-H f) +1 ] A . 8 178 Simulation Of The F-factor For The Shifted Exponential And Pearson Type I I I Headway Distributions The e f f e c t i v e length i s that portion of the length of the loaded vehicle that w i l l cause the empty vehicle to remain i n a turnout while the loaded vehicle i s passing the empty vehicle. A simulation model was used to confirm or r e j e c t the s h i f t e d exponential and Pearson Type II I (a= 2) F-factor equations.. The computer program written to simulate the meeting of an empty vehicle and a f l e e t of loaded vehicles was modified to include the effect of the length of the loaded vehicle on the expected F-factor (Chapter 4). This version of the program required a minimum headway of the length of the loaded vehicle divided by i t s velocity plus a reaction time. The length of the loaded vehicle was assumed to be 60 feet while the driver's reaction time was approximately 2 seconds. The tables below tabulate the average F-factor and expected F-factor of the simulations with a sample size of 10000 repeated 9 times. . A t - t e s t was u t i l i z e d to test the n u l l hypothesis that the expected F-factor i s equivalent to the average F-factor of the samples. The F-factor equation developed for the s h i f t e d exponential headway case was not rejected at the one percent l e v e l of significance and was only rejected, at a f i v e percent l e v e l of s ignificance, 16.7 percent of the time. The F-factor equation for the Pearson Type III(a=2) headway case was rejected s l i g h t l y more often than the F-factor equation f o r the shifted exponential headway case. It can be concluded that these two expected F-factor 179 equations adequately describe the interaction between a single empty vehicle and a group of loaded vehicles, for t h e i r respective headway d i s t r i b u t i o n s . . 180 Table: For The E^f actor Simulation Of The Interaction Between A Single Empty Vehicle And A Fleet Of Loaded Vehicles Based On Equations 2 . 18 . 2 . 1 9 , A. 5 . And"A. 6 Sample size = 1 0 0 0 0 Number of samples = 9 Acceleration = 1 9 7 5 9 mph2 "Conflict'* hours = 11 S h i f t = (L/V, + 2 seconds) L = 6 0 feet Shifted exponential headway d i s t r i b u t i o n Vehicle Speed|Distance | T r a f f i c 1 Ave. | S.D.1! Expect| Rejection 1 -| Between | Flow 1 F | x 1 0 " 3 | F I of Loaded|Empty JTurnouts | Rate j | | | Hypothesis (mph) |(mph) |(miles) 1 (vph) | | I I oL = 0 . 0 5 40 | 3 0 | 0 . 2 | 1 | 1. 2 2 2 6 | 4 . 6 8 | 1 . 2 2 5 8 1 No 1 1 I 2 | 1 . 2 2 7 2 1 5 . 6 9 | 1 . 2 2 8 3 1 No2 I | i 4 | 1 . 2 3 3 0 | 8 . 6 4 | 1 . 2 3 3 0 | No2 I | I 10 | 1 . 2 4 2 7 1 7 . 3 9 | 1 . 2 4 4 8 | No2 I | | 60 | 1. 2 4 5 3 | 4 . 451 1 . 2 4 9 8 | Yes 3 1 1 | 100 | 1 . 1 8 6 4 1 8 . 6 8 | 1 . 1 8 6 6 ! No2 30 | 4 0 | 0 . 3 | 1 | 0 . 9 1 2 5 | 3 . 6 0 | 0 . 9 1 5 0 | No | 2 | 0 . 9 1 8 6 | 4 . 0 7 1 0 . 9 1 7 3 ! No2 | 4 | 0 . 9 1 7 3 | 4 . 6 6 | 0 . 9 2 1 3 ! Yes 3 | | | 10 | 0 . 9 3 0 2 | 5 . 5 8 ! 0 . 9 2 9 9 | No2 1 1 | 60 | 0 . 8 7 9 4 | 5 . 6 5 | 0 . 8 8 1 0 J No2 1 1 | 1 00 I 0 . 7 7 7 9 1 4 . 1 4 | 0 . 7 8 2 0 | Yes 3 10 | 20 | 0 . 1 I 1 | 0 . 8 6 2 4 | 2 . 721 0 . 8 6 4 7 | Yes 3 j | i 2 | 0 . 8 6 3 5 | 2 . 84 f 0 . 8 6 5 8 | No 1 | I 4 J O . 8 6 7 3 | 3 . 0 9 | 0 . 8 6 7 8 | No2 I | I 10 | 0 . 8 7 1 9 | 3 . 9 7 | 0 . 8 7 2 0 1 No2 | j | 60 | 0 . 8 3 7 8 | 3 . 2 3 | 0 . 8 4 0 0 | No | | | 100 | 0 . 7 7 4 2 1 3 . 4 7 ! 0 . 7 7 4 7 J No2 1 Standard deviation 2 Hypothesis was not rejected f o r a 10% l e v e l of s i g n i f i c a n c e 3 Hypothesis was not rejected for a 1% l e v e l of significance 181 Table For The F-factor Simulation Of The Interaction Between A Single Empty Vehicle and A Fleet Of Loaded Vehicles Based On Equations 2 . 1 8 , 2 . 1 9 , A. 7, And A ..8 Sample size = 10000 ' Number of samples = 9 Acceleration = 1 9 7 5 9 mph2 " C o n f l i c t " hours = 11 S h i f t = (L/V, + 2 seconds) L = 60 feet Pearson Type III headway d i s t r i b u t i o n (Vehicle Speed|Distance T r a f f i c ( Ave..| r 1 T Between Flow 1 F | I Loaded|Empty (Turnouts Rate ) (mph) j (mph) | (miles) r — H — • + — (vph) + r I 40 ( 30 | 0 . 2 1 | 1. 2 2 0 7 | 1 1 1 2 J 1 . 2 2 6 8 | I I I 4 | 1. 2 2 5 9 | 1 1 1 10 | 1 . 2 3 3 0 ( I I I 60 | 1. 4 0 1 2 | 1 . 1 1 100 | 1 . 3 9 0 9 J | 30 | 4 0 ( 0 . 3 1 | 0 . 9 1 0 0 1 1 1 1 2 | 0 . 9 1 2 2 | i 1 1 4 I 0 . 9 151 | 1 1 1 10 | 0 . 9 3 3 1 | 1 1 1 60 | 1 . 0 3 1 0 1 1 1 1 100 | 0 . 9 0 9 7 | S.D X 1 0 6 . 6 . 8 . 5 . 6 . 7 . 4 . 5 . 3 . 6 . 5 . 4 . 11 Expect - 3 , F I I I I 3 3 | 1 . 2 2 3 6 6 2 f 1 . 2 2 4 2 2 9 | 1 . 2 2 6 3 3 8 1 1 . 2 3 9 5 8 0 | 1 . 4 0 5 7 6 2 ( 1- 3 7 3 1 I 4 4 | 0 . 9 1 3 1 7 4 | 0 . 9 1 4 0 8 4 | 0 . 9 1 7 2 0 9 | 0 . 9 3 5 6 0 5 | 1 . 0 2 6 0 7 9 | 0 . 8 7 2 7 Rejection of Hypothesis oL = 0 . 0 5 + H No2 No2 No2 Yes No Yes No No2 Noz Noz Yes Yes 1 Standard deviation 2 Hypothesis was not rejected for a 10% l e v e l of s i g n i f i c a n c e 182 APPENDIX-? DERIVATIVES OF THE EXPECTED F-FACTOR EQUATIONS There are several derivatives of the expected F-factor equations and components of the equation that are required i n the optimization routines.. Many of these derivatives are l i s t e d on the following pages.. 1. C r i t i c a l headway h c = [2Sa Aa 0(V, +V 2) +V ( Vf(a f l+a D) ] / (2a,aDVi V, ) Sht V, +Vj, >^S V( V2 A. 9 2. Exponential F-factcr Fz = D/S V^, +V2 F = 2 V, ( i - e _ ^ h c j D (Q-, + 1) V, +\ 2V2 (Q, + 1) dF f ~dS L V, *\ 2V2 dD as a h c 3s Q3 Q Q-^jhc D 3hc as D ( i - e - Q ^ c ) S2 (0^  + 1); A'_10 D = v, u e - Q A (-Q-hc- i j + i ] / [-Q3 ] as -v, Q | ( | ^ ) e - Q ^ c e - ^ ( - Q 3 h c - i ) + i ] v, Qf ( | ^ ) e - Q ^ c h ( as [-Q, ( e - ^ - i ) ] 2 -Q-7 (e-9j K c-i) A. 11 1 8 3 3 . E r l a n g (alpha=2) F - f a c t o r F = V + V, 2 V, \ D j £ - * h c (1+Xhc) +-(1-£- A^[ 1 + Ah c ]) ( Q 5 + 1 ) v, +va [ 2 V 2 ( Q 3 + 1 ) A= 2 Q 3 d F Bs 2V, / 2>hc' + — as / \ [ i - e - x h t (i+Ahc) ]\ D [ 1 - e - A l n - (1 + A h c ) ] /DX2h c e - X M/ coh^ S / \ 9S V S2 s \ ( Q , + 1 ) A. 12 D = V, [e - X t l - ( - A h f - 2 h c - 2 / A ) + 2 A 1 / t 1 - e - A h c (1 + Ah c) ] 2>D c)hc Q-^a ( A h | + 2 h c + 2 / A ) - 2 / A [SS J/ \ [ 1 - e - X ^ (1+Ah c) v (1-Ah c)/ A. 13 Simple F - f a c t o r i n c l u d i n g the v e h i c l e ' s l e n g t h F = (V/ ) / ( 2 V 2 ) + L/S (dF L/c}S ) = -L/S2 A. 14 184 185 6. . Pearson Type III(a=2) F-factor Let a = 2 b = 2/(1/Q_-Rf -L/V, ) c = Rf+L/V( Then 186 APPENDIX 8 OPTIMUM TURNOUT SPACING COMPUTER PROGRAM Program Variables Variable Function A1 deceleration A2 acceleration B b parameter of Pearson Type III d i s t r i b u t i o n C "cost functions" D1 number of " c o n f l i c t " hours per day D2 number of operating days per year F1 c o e f f i c i e n t of f r i c t i o n F2 simple F-factor including the vehicle's length F3 expected F-factor F4 prime of the expected F-factor F7 c o e f f i c i e n t of acceleration H t r a f f i c flow rate i n vph K acceleration i n mph2 L the length of the vehicle in feet L1 the length of the vehicle i n miles L2 minimum gap between vehicles in seconds L3 minimum gap between vehicles i n days M adjusted hauling cost in dollars per hour P1,P2, P3 components of the cost function OJ expected useful l i f e of the road i n years Q3 number of headways during the " c o n f l i c t " hours 187 R 1 - R 0 components of expected F-factor and prime of expected F-factor S turnout spacing V1 speed of loaded vehicle i n mph V2 speed of empty vehicle i n mph Y turnout spacing Z prime of "cost function" 188 Program L i s t i n g 50 EEM PROGRAM FOR THE CALCULATION OF OPTIMUM SPACING 100 REM PROGRAM DEVELOPED SEPT..1978 BY D.I..ANDERSON 150 DIM Y (4) ,Z (4) ,C(4,3) ,A$ (82) rH (6) ,S (4,3) 200 A$ (1,40)="DISTRIBUTION OPT. SPACING S* - 50% " 250 A$(41,80)=" S* + 50% S* + 200%" 300 MAT Z=ZER 350 DISP "INPUT THE VEHICLE'S LENGTH-FEET"; 400 INPUT L 450 L1=LZ5280 500 DISP "MINIMUM GAP BETWEEN TRUCKS—SEC. "; 550 INPUT L3 600 DISP "INPUT NUMBER OF CONFLICT HOURS PER DAY"; 650 INPUT D1 700 L2=L3/3600/Dl 750 DISP "NUMBER OF WORKING DAYS PER YEAR"; 800 INPUT D2 850 DISP "INPUT COEFFICIENT OF FRICTION"; 900 INPUT F1 925 DISP "INPUT COEFFICIENT OF ACCELERATION"; 930 INPUT F7 950 A1=79036.363636*F1 960 A2=79036.363636*F7 1000 DISP "INPUT TRAFFIC FLOW IN VPH"; 1050 INPUT H 1100 DISP "LOADED VEHICLE'S SPEED IN MPH"; 1150 INPUT V1 1200 DISP "EMPTY VEHICLE'S SPEED IN MPH"; 189 1250 IN POT V2 1300 DISP " ENTER COST PER TURNOUT (DOLLARS) n. i 1350 INPOT C 1400 DISP " TRUCK RENTAL RATE IN $/HR"; 1450 INPOT M 1500 DISP " EXPECTED LIFE OF THE ROAD IN YRS it. 1550 INPOT Q1 1600 PRINT LIN3"CALCULATION OF OPTIMUM SPACING AND COST FUNCTION 1650 PRINT LIN 1"NUMBER OF HOURS PER DAY = "D1 1700 PRINT "NUMBER OF WORKING DAYS PER YEAR - "D2 1750 PRINT "COEFFICIENT OF FRICTION "F1 1775 PRINT "COEFFICIENT OF ACCELERATION = «F7 1800 PRINT "THE LENGTH OF THE VEHICLE = "L" FEET" 1850 PRINT "VEHICLES PER HOUR = "H 1900 PRINT "MINIMUM GAP BETWEEN VEHICLES = "L3» SECONDS" 1950 PRINT "VELOCITY OF LOADED VEHICLE = "V1 " MPH" 2000 PRINT "VELOCITY OF EMPTY VEHICLE = it v2" MPH" 2050 PRINT "COST PER TURNOUT = «C" DOLLARS" 2100 PRINT "TRUCK RENTAL RATE = "M" DOLLARS/HOUR" 2150 PRINT "EXPECTED LIFE OF THE ROAD = "Q1" YEARS" 2200 Q3=H*D1-1 2250 FOR S1=1 TO 3 2300 S3=V1+V2 2350 K= (2*A1*A2)/(A1 + A2) 2400 Y (1) =0.00001 2450 Y (3) =10.5 2500 Y (2) = (Y (1) +Y (3) ) /2 2550 FOR 11=1 TO 3 2600 GOSOB S1 OF 7450,7450,6100 190 2650 NEXT 11 2700 IF Z(1)*Z(3)<0 THEN 2900 2750 Y (1) =F6*0.7 2800 Y (3) =F6*1.3 2850 Y (2) = <Y (1) +Y (3) ) /2 2900 Y4=Y (1) 2950 Y5=Y (3) 3000 Y6=2 3050 11=2 3100 IF ABS (Y (1)-Y (3) ) <0. 0001 THEN 3700 3150 IF Z(1)*Z(2)>0 THEN 3450 3200 Y (3) =Y (2) 3250 Y (2) = (Y (3) +Y (1) ) /2 3300 Z (3) =Z (2) 3350 GOSUB S1 OF 7450,7450,6100 3400 GOTO 3100 3450 Y (1) =Y (2) 3500 Y (2) = (Y (3) +Y (1) ) /2 3550 Z (1) =Z (2) 3600 GOSOB S1 OF 7450,7450,6100 3650 GOTO 3100 3700 IF S1=3 THEN 4000 3750 IF S1=2 THEN 3900 3800 GOSOB 4950 3850 GOTO 3950 3900 GOSOB 4950 3950 NEXT S1 4000 GOSOB 4950 4050 STANDARD 4100 PRINT LIN3 4150 PRINT A$ 4200 PRINT LIN1 4250 PRINT "SIMPLE" 4300 PRINT " SPACING (FEET) "S (1, 1) ; S (2, 1) ; S (3,1) ; S (4,1) 4350 PRINT " COST ($/M/VEH) "C (1, 1) ;C (2, 1) ;C (3, 1) ; C (4, 1) 4400 PRINT " " 4450 PRINT "SHIFTED EXPONENTIAL" 4500 PRINT " SPACING (FEET) "S (1, 2) ; S (2, 2) ; S (3, 2) ; S (4 ,2) 4550 PRINT " COST ($/M/VEH) "C (1 , 2) ; C (2, 2) ;C (3, 2) ; C (4, 2) 460 0 PRINT " " 4650 PRINT "PEARSON TYPE I I I " 4700 PRINT " SPACING (FEET) "S (1 , 3) ; S (2, 3) ; S (3 r 3) ; S (4, 3) 4750 PRINT " COST ($/M/VEH) "C (1 ,3) ; C (2,3) ;C (3, 3) ; C (4, 3) 4800 PRINT LIN2 4850 END 4900 REM S* 4950 J=1 5000 S (J, S1) =Y (2) *5280 5050 C(J, S1)=C/Y(2) +Q1*H*M*D2*D1*P3/P1* (H/V1 + H/V2) 5100 J=2 5150 Y (2) =Y (2) *0.5 5200 REM 50% OF S* 5250 S(J r S1)=Y (2) *5280 5300 GOSOB S1 OF 7450,7450,6100 5350 C (J, S1) =C/Y (2) +Q1 *H*M*D2*D 1 *P3/P 1* (H/V1+H/V2) 5400 J=3 5450 Y (2) =Y (2) *3 5500 REM 1 5055 OF S* 192 5550 S (J, S1) =Y (2) *5280 5600 GOSOB S1 OF 7450,7450,6100 5650 C(J, S1)=C/Y(2) +Q1 *H*M*D2*D1*P3/P1* (H/V1 + H/V2) 5700 J=4 5750 Y (2) =Y (2) *4/3 5800 EEM 200% OF S* 5850 S (J, S1) =Y (2) *5280 5900 GOSOB S1 OF 74 50,7450,6100 5950 C (J, S1)=C/Y(2) +Q1*H*M*D2*D1*P3/P1* (H/V1+H/V2) 6000 EETLTBN 6050 STOP 6100 EEM SOLUTION OF 1VSX FOE THE PEARSON TYPE III DISTEIBUTION 6150 EEM CALCOLATION OF THE EXPECTED F-FACTOR 6200 R1=(2*Y (11) *A1*A2*S3+V1*V2t2* (A1 + A2) ) / (2*A 1*A2*V1*V2*D1) 6250 F2=S3/(2*V2)+L1/Y(I1) 6300 B=2/(1/Q3-L1-L2) 6350 R2=EXF(-B*(R1-L2) ) 6375 Q6=2*Bl-2/B-L1/V1/D1+L2 640 0 R5=R2*(B*(-RIt2-R1*L1/V1/D1+L2*R1+L1*L2/V1/D1)-Q6) 6450 R6=R2*(E* (R1-L2)+1) 6500 R3=V1*D1/Y (11) * (E5+L2 + L1/V1/D1+2/B) 6550 F3=(E3+E6*F2) *Q3/(H*D1) +F2/(H*D1) 6600 GOSOB 7000 6650 EEM THE FIEST DEEIVATIVE OF THE COST EQUATION 6700 P 1 = 1 -H*Y (11) *F3/V1-H*V2/K 6750 P2=F3/V1 + Y (11) *F4/V1 6800 P3=V2/K+Y(11)*F3/V1 6825 Q7=H/V1+H/V2 6850 Z (11) =-C/Y (11) f 2+ (M*H*Q1*D2*D1) * (P2/P 1+H*P2*P3/P1t 2) *Q7 193 6900 RETURN 6950 STOP 7000 EEM CALCULAION OF THE PEIME OF THE F-FACTOE 7050 R4=S3/(V1*V2*D1) 7075 Q8=Q 3*V1/ (H* Y (11) t 2 7100 E7=-(E2*(B*(-Blt2+L2*E1)-2*E1-2/B+L2)+L2+L1/V1/D1+2/B)*Q8) 7150 E8=Q3*V1/ (H*Y (II) ) *R2*R4*Bt2* (Rlt2-L2*R1) 7 200 R9=Q3/(H»D1) *R4*R2*Bf2* (-E1+L2) *S3/(2*V2) 7250 R0=-L1/(H*D1*Y (11)^2) 7300 F4=R7+R8+R9+R0 7350 RETURN 740 0 STOP 7450 EEM EXPECTED F-FACTOE FOB SIMPLE AND SHIFTED EXPONENTIAL 7500 F2=S3/(2*V2)+L1/Y (II) 7550 IF S1=1 THEN 7950 760 0 B1 = (2*Y(I1) *A1*A2*S3+V1*V2f 2* (A1 + A2) ) /(2*A1*A2*V1*V2*D1) 7650 B2=EXP(-Q3*(E1-L2) ) 770 0 B3=( (L1/V1/D1+L2+ 1/Q3) -B2* (B1 + L 1/V 1/D1 + 1/Q3) ) *V 1*D 1/Y (11) 7750 F3=(B2*F2 + E3) *Q3/(H*D1)+F2/(H*D1) 7800 GOSOB 8350 7850 EEM CALCOLATION OF THE FIRST DERIVATIVE OF THE COST EQOATIO 7900 GOTO 8050 7950 F3=F2 8000 F4=-L1/Y (I1)t 2 8050 P1=1-H*Y(11)*F3/V1-H*V2/K 8100 P2=F3/V1 + Y (11) *F4/V1 8125 Q6=H/V1+H/V2 8150 P3=V2/K+Y(11)*F3/V1 820 0 Z (11) =-C/Y(I1) t2+ (M*H*Q1*D2*D1) * (P2/P1+H*P2*P3/P1t2) *Q6 8250 BETUEN 830 0 STOP 8350 EEM CALCULAION OF THE PEINE OF THE F-FACTOE 8400 B4 = S3/(V1*V2*D1) 8450 B5=B3/V1/D1*Y(I1) 8500 B6=-V1*Q3*E5/H/Y(11)f2 8550 B7=V1*Q3t2*B4*B2/H/Y (II)* (BUL 1/V 1/D 1) 8600 B8=Q3*B2/H/D1*(Q3*B4*F2+L1/Y(I1)|2) 8650 E9=L 1/(H*Dl*Y(I1)t2) 8700 F4=B6+B7-E8-B9 8750 EETOBN 8800 STOP 195 Operating Instructions Once, the computer program has been loaded into the memory of the calculator the program i s started by pressing "RUN" then "EXECUTE". Once th i s has been accomplished the screen w i l l display various questions.. See the following pages for examples. 196 RUN INPUT THE VEHICLE'S LENGTH-FEET?50 MINIMUM GAP EETWEEN TRUCKS-SEC.?2 INPUT NUMBER OF CONFLICT HOURS PER DAY?5 NUMBER OF WORKING DAYS PER YEAR7200 INPUT COEFFICIENT OF FRICTION?.25 INPUT COEFFICIENT OF ACCELERATION?.25 INPUT TRAFFIC FLOW IN VPH?1 LOADED VEHICLE'S SPEED IN MPH?15 EMPTY VEHICLE'S SPEED IN MPH?30 ENTER COST PER TURNOUT (DOLLARS)?250 TRUCK RENTAL RATE IN $/HR?15 EXPECTED LIFE OF THE ROAD IN YEARS? 10 CALCULATION OF THE OPTIMUM SPACING AND 'COST FUNCTION' NUMBER OF HOURS PER DAY = 5 NUMBER OF WORKING DAYS PER YEAR = 200 COEFFICIENT OF FRICTION = 0. 25 COEFFICIENT OF ACCELERATION = 0. 25 THE LENGTH OF THE VEHICLE = 50 FEET VEHCILE'S PER HOUR = 1 MINIMUM GAP BETWEEN VEHICLE •S = 2 SECONDS VELOCITY OF LOADED VEHICLE = 15 MPH VELOCITY OF EMPTY VEHICLE = 30 MPH COST PER TURNOUT = 250 DOLLARS TRUCK RENTAL RATE = 15 DOLLARS PER HOUR EXPECTED LIFE OF THE ROAD = 10 YEARS DISTRIBUTION OPT. SPACING S* - 50% S* + 50% S* + 200% SIMPLE SPACING(FEET) 2956.42 1478.2 1 4434.64 5912. 85 COST ($/M/VEH) 912.73 11 39 . 1 6 990.40 1 149.22 SHIFTED EXPONENTIAL SPACING(FEET) 2955. 58 1477.79 4433.37 5911.16 COST (S/H/VEH) 913.20 1 139.64 990.77 1 149.23 PEARSON TYPE III SPACING (FEET) 2951. 35 1475.67 4427.02 5902. 70 COST ($/M/VEH) 913,28 1 140.41 991.46 1151.90 EON INPOT THE VEHICLE'S LENGTH-FEET?50 MINIMUM GAP EETWEEN TBOCKS-SEC.?2 INPOT NOMBEE OF CONFLICT HOOES PEE DAY?5 NOMBEE OF WORKING DAYS PEE YEAB7200 INPOT COEFFICIENT OF FEICTION?.25 INPOT COEFFICIENT OF ACCELEBATION?.25 INPOT TBAFFIC FLOW IN VPH?4 LOADED VEHICLE'S SPEED IN MPH?20 EMPTY VEHICLE'S SPEED IN MPH?25 ENTEB COST PEE TOENOOT (DOLLABS)?100 TRUCK RENTAL BATE IN $/HB?15 EXPECTED LIFE OF THE EOAD IN YEABS? 20 CALCOLATION OF THE OPTIMOM SPACING AND 'COST FUNCTION' NUMBEB OF HOUES PEB DAY = 5 NUMBER OF WOBKING DAYS PEE YEAE = 200 COEFFICIENT OF FBICTION = 0. 25 COEFFICIENT OF ACCELEBATION = 0. 25 THE LENGTH OF THE VEHICLE = 50 FEET VEHCILE'S PER HOUE = 4 MINIMUM GAP BETWEEN VEHICLE •S = 2 SECONDS VELOCITY OF LOADED VEHICLE = 20 MPH VELOCITY OF EMPTY VEHICLE = 25 MPH COST PEE TUENOUT = 100 DOLLAES TEUCK EENTAL BATE = 15 DOLLAES PEB HOUB EXPECTED LIFE OF THE EOAD = 20 YEABS DISTBIBUTION OPT. SPACING S* - 50% S* + 50% S* + 200% SIMPLE SPACING(FEET) 371.21 185.61 556.82 742.42 COST ($/M/VEH) 3582. 59 4298. 54 3824.07 4311.96 SHIFTED EXPONENTIAL SPACING(FEET) 369.94 184.97 554.91 739.89 COST ($/M/VEH) 3592.66 4311.46 3834.32 4322.90 PEABSON TYPE I I I SPACING(FEET) 370.79 185.39 556.18 741.58 COST ($/M/VEH) 3584. 39 4301. 40 3827.34 4318.89 198 APPENDIX 9 GRAPHICAL RESULTS OF THE SENSITIVITY ANALYSIS Simulation models were developed to determine the e f f e c t perturbations to the independent variables, headway frequency d i s t r i b u t i o n s , and some on the assumptions have on the expected F-factor, t o t a l expected delay time, optimum turnout spacing, and cost. The difference between the actual and the perturbed values of the functions were calculated. . These differences were grouped according to the true value of the independent variable and the perturbed values or according to the assumptions. The average, standard deviation, and the maximum value of each group were determined. Graphs of some of the re s u l t s of the simulations are located on the following pages.. 270 + 2H3 + 2IE + IB9 + THE EFFECT THAT PERTURBRTIDN5 TD THE VELOCITY DF THE EMPTY VEHICLE HAVE ON THE C05T FUNCTION COST IE2 DIFFERENCE IN DOLLARS PER YEAR 135 PER MILE PER VEHI CLE 108 ai + SM + 27 + 0 PERTURBATION CMPH) AVERAGE MAXIMUM VELOCITY OF EMPTY VEHCILE CMPH) TRAFFIC FLDM RATE (VPH) TRAFFIC FLOW RATE (VPH) EXPECTED USEFUL LIFE DF THE RDRD CYEHR5) EXPECTED USEFUL LIFE UF THE RDP.D CYEP.F.5J g EFFECT THHT PERTURBP.T 10N5 TD THE TURNOUT CONSTRUCTION COST HAVE ON THE OPTIMUM TURNOUT SPACING TURNOUT CONSTRUCTION COST C$) TRUCK HAUL INE COST C J / H R ) TRUCK HAULING C05T U/HR) -I o EFFECT THAT PERTURBATIONS TO THE COEFFICIENT OF 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 . COEFFICIENT OF FRICTION HH THE EFFECT THAT THE DI5CDUNT RATE FOR VARIOUS TIME PERIODS HA5 ON THE COST FUNCTION EXPECTED USEFUL LIFE DF THE ROAD C YEARS > B0 - r 72 + EH + EFTECT THRT MR INTENRNCE COSTS FOR VARIOUS TIME PERIODS HAVE ON THE OPTIMUM TURNOUT SPACING EE DIFFERENCE IN OPTIMUM TURNOUT 5PRCING HB + H0 + 32 2H + IE + B 0 RVERRGE MRXIMUM -+-+-+-S 10 IS 20 EXPECTED USEFUL LIFE OF THE RORD (YERR5) TURNOUT MAINTENANCE COST C$/YEAR) 12.S 7.S 2.5 B.S B0 -r EFFECT THAT MAINTENANCE C05T5 FDR VARIOUS TIME TURNOUT PERIODS HAVE ON THE OPTIMUM TURNOUT SPACING MAINTENANCE COST (•/YEAR) 72 + EH + AVERAGE MAXIMUM -+-+-+-SE + EXPECTED USEFUL LIFE DF THE ROAD CYEARS) EXPECTED USEFUL LIFE DF THE ROAD CYEHR5) EXPECTED USEFUL LIFE OF THE RORD CYERR5) EFFECT THAT THE PRIME DF THE F-FRCTDR FDR VRRIDU5 TRAFFIC FLDW RRTE5 HA5 ON THE OPTIMUM TURNOUT SPACING TRAFFIC FLOW RRTE CVPH) 33 E F F E C T THBT T H E PRIME OF T H E F - F A C T D R FOR V R R I 0 U 5 T R A F F I C FLOW RRTE5 HR5 ON THE C05T FUNCTION HERDWRY D I S T R I B U T I O N 3 0 COST D I F F E R E N C E IN DDLLHR5 PER YEAR PER M I L E PER V E H I C L E 27 2H + R V E R H E E MAXIMUM - + - + - + -E = 5 H I F T E D E X P O N E N T I A L HEADWAY D I S T R I B U T I O N P = PEARSON T Y P E III CA=2) HEADWAY D I S T R I B U T I O N H E T R R F F 1 C FLOW RflTE CVPH) E F F E C T HEADWAY D I S T R I B U T I O N S FOR VARIOUS T R A F F I C FLOW R A T E S HRVE ON T H E OPTIMUM TURNOUT S P A C I N G HEADWAY D I S T R I B U T I O N COMPARISON T R A F F I C FLOW R A T E C V P H ) TURNOUT 5PRCINE C MILE5 ) APPENDIX - 10 CONVERSION FACTORS Imperial Units SI (Metric) Units 1 1 foot 0. 3048 metres 1 mile = 1. 609344 kilometres 1 cun i t = 2. 831685 metres 3 1 pound = 0. 45359237 kilograms 1 ton = 1. 016047 tonne 1 f o o t 2 = 0. 09 290304 metres 2 1 horsepower = 745. 7 watts 1 International System of Units 

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