F U Z Z Y L I N E A R P R O G R A M M I N G A N D R E S E R V O I R M A N A G E M E N T B y Evans Kaseke B . A . , T H E U N I V E R S I T Y O F W A L E S , 1981 M . A . , T H E U N I V E R S I T Y O F W A T E R L O O , 1983 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L E M N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S ( C I V I L E N G I N E E R I N G D E P A R T M E N T ) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A August 1987 © E v a n s Kaseke, 1987 In presenting this 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 freely 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 or her representatives. It is understood that copying or publication of this thesis for f i n a n c i a l gain shall not be allowed without my written permission. Department of C i v i l Engineering The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: 26 August 1987 Abstract The presence of imprecision in parameter specification of water re-sources management problems leads to the formulation of fuzzy program-ming models. This thesis presents the formulation of a two-reservoir system problem as a fuzzy L . P . model. The aim is to determine if larger mone-tary benefits, over and above the usual benefits, can be obtained from the system. The other a im is to determine if desired industrial and domestic water allocations, as well as outflows for selected periods can be achieved. The problem is formulated as a conventional L . P . model. Then selected water allocations and outflows are fuzzified resulting in a fuzzy L . P . model. The alternative fuzzy L . P . model is also presented. Monetary benefits larger than those from the conventional L . P . were obtained through the fuzzy L . P . model. The desired water allocations and outflows were also realised for selected periods. Sensitivity information was obtained for fuzzy and non-fuzzy constraints. The alternative fuzzy L . P . model did not give additional valuable information than that obtained from the ini t ial fuzzy L . P . model. ii C ontents page A b s t r a c t i i L i s t o f Tab le s v L i s t o f F i g u r e s v i A c k n o w l e d g e m e n t v i i 1 I n t r o d u c t i o n 1 1.1 Thesis Objectives 2 1.2 Thesis Outline 2 2 C o n v e n t i o n a l a n d F u z z y L i n e a r P r o g r a m m i n g 4 2.1 Conventional Linear Programming 4 2.1.1 What Is Linear Programming 4 2.1.2 General Problem Formulation 4 2.1.3 Duality In Linear Programming 6 2.2 Fuzzy Linear Programming 6 2.2.1 Decision-Making In A Fuzzy Environment 7 2.2.2 General Framework Of Fuzzy Mathematical Programming . . 8 2.2.3 Fuzzy Linear Programming Model 9 2.3 Altarnative Fuzzy Formulation 12 2.4 Crisp Constraints 13 2.5 Summary 14 3 H e n r y H a l l a m a n d P r i n c e E d w a r d R e s e r v o i r s 15 3.1 Background Information 15 3.1.1 Introduction 15 3.1.2 Physical Environment 17 3.1.3 Interaction And Water Uses 19 3.1.4 Data Sources 20 iii 4 L.P. and F.L.P. Formulation of the Two-Reservoir Problem 21 4.1 Linear Programming Formulation 21 4.1.1 Variable Definition 21 4.1.2 Objective A n d Constraints Formulation 22 4.2 Fuzzification Of The Two-Reservoir Problem 25 4.2.1 Fuzzy Constraints Formulation 26 4.2.2 Fuzzy Constraints 27 4.2.3 Membership Function Evaluation Constraints 35 4.2.4 Remaining Non-Fuzzy Constraints 36 4.2.5 Complete Formulation A n d Solution 36 4.3 Alternative Fuzzy Formulation 36 4.4 Summary 38 5 Discussion of Results 39 5.1 Conventional L . P . Results 39 5.2 Comparison Of Conventional L . P . A n d Fuzzy L . P . Results 41 5.2.1 Alternative Fuzzy Formulation 43 5.3 Sensitivity Analysis 44 5.3.1 Conventional L . P 44 5.3.2 Fuzzy L . P 45 6 Conclusions 47 Bibliography 52 Appendix A 54 Appendix B 58 Appendix C 63 Appendix D 68 Appendix E 75 Appendix F 82 iv List o f Tables 5.1 Prince Edward Reservoir (figures in 10 6 m 3 ) 40 5.2 Henry Ha l l am Reservoir (figures in 10 6 m 3 ) 40 5.3 L . P . and F . L . P . results (volumes in 10 6 m 3 and GOAL in Z$10 6 ) 42 v List of Figures 3.1 Locations of Harare and Chitungwiza [thornton, J . A . , (ed.), 1982]. . 16 3.2 Locations of Henry Hal lam and Prince Edward reservoirs [same source as F i g . 3.1] 16 3.3 Aspects of Harare's climate (Kay, G . , et al., 1977) 18 4.1 Goal 's membership function 26 4.2 Membership function for P A D 4 28 4.3 Membership function for PAI4 29 4.4 Membership function for P A D 5 31 4.5 Membership function for PAI5 32 4.6 Membership function for H 0 2 33 4.7 Membership function for H 0 4 34 4.8 ti diagram for increasing values 37 4.9 ti diagram for decreasing values 38 v i Acknowledgement 1 would like to express my gratitude to my advisor, Dr. W. F. Caselton, for providing me with his invaluable assistance and guidance throughout the duration of the research period and the writing of this thesis. Thanks are extended to Dr. A. Russell for reading and commenting on this thesis. I am indebted to the Canadian Commonwealth Scholarship and Fellowship Ad-ministration for financially supporting my studies at U.B.C.. This thesis would not have been realised without the assistance with data for the models from the Hydrological Branch in Zimbabwe - thanks. Finally, I would like to thank Jim Mattison, Paul Jacobs and Dr. S. 0 . Russell for their valuable advice and encouragement in completing this thesis. vii C hapter 1 Intro duction The development and adoption of optimisation techniques for the planning, designing and management of complex water resources systems has advanced sig-nificantly over the last two decades. When formulated, complex water resources systems problems may involve hundreds, if not thousands, of decision variables and constraints, but they can often be solved by using appropriate Operations Research techniques. High-speed computers further aid decision makers in finding the sensi-tivity of the optimal solutions to many aspects of the problems under scrutiny. Although it has been established that there is no general algorithm for solv-ing reservoir operation problems, optimisation techniques have been successfully applied to these types of problems. Aspects such as data availability, objectives, constraints, etc., determine the choice of technique to be applied. Mathematical programming methods have been used to determine optimal allocations of scarce water resources with the aim of meeting specified objectives. In the mathematical modelling of decision- making processes, there exists the problem of parameter im-precision. Probabilistic decision-theories handle risk and uncertainty in modelling decision-making, but, it is still felt that such theories are inadequate for dealing with uncertainty which can be characterised as inexactness, ill-definedness, vagueness, or in short: fuzziness [Kickert, W.J.M., 1978]. With the realisation that probability alone is inadequate to describe reality in sit-1 y uations clouded by doubts over the inexactness of concepts, the search, promotion, assessment and practical application of appropriate systems analysis techniques has been an on going concern. A recent development in this connection is the solution of multiple-objective problems using fuzzy programming [Kickert, W . J .M. , 1978]. The application of fuzzy mathematical programming to the management of a connected system of reservoirs in Zimbabwe is at the core of this thesis. The notion of fuzziness may be introduced into decision variables, constraints and goals, but fuzzy mathematical programming limits its fuzziness to the elements of goals and constraints, only. Fuzzy goals and constraints are characterised by their aspiration levels or membership functions, which are also known as degradation allowances or leeways. 1.1 Thesis Objectives The first and principal general objective is to assess the usefulness of fuzzy linear programming as a tool in managing water resourses problems, especially, as an aid in decision making. The second objective of this thesis is to determine if larger monetary benefits can be obtained from water sales to domestic and industrial consumers in Harare, a major industrial centre in Zimbabwe. The third objective is to determine if higher and lower domestic and industrial water allocations, as well as lower water releases or outflows can be achieved for specific periods. The fourth objective is to determine if the live storage of Prince Edward reservoir could be maintained at or above 70% of its full storage capacity (i.e., at all times) as a requirement for sustaining a healthy fishery and weed control by flooding. 1.2 Thesis Outline To accomplish the above stated tasks, Chapter 2 discusses conventional and fuzzy linear programming. Chapter 3 describes the physical and economic aspects of 2 Henry Hallam and Prince Edward reservoirs. Chapter 4 is a formulation of the two-reservoir problem both as conventional and as fuzzy linear programming models. Chapter 5 has a comparison of results from the two models and an interpretation of the sensitivity information. Chapter 6 has general comments and concluding remarks on fuzzy linear programming. Henry Hallam and Prince Edward reservoirs have been picked for this study because they are the most critical of Harare's water supply system of four reser-voirs, which also include Lakes Mcllwaine and Robertson. If poorly managed, these reservoirs could run dry in a very short time. The consequences will be a loss of a valuable water supply supplement to the City of Harare, the loss of an economically viable fishery, and the resurgence of the weed problem in Prince Edward reservoir which has to be controlled by higher water levels. Thus, their is need to balance all of the demands made on the reservoirs. 3 Chapter 2 Conventional and Fuzzy Linear Programming 2.1 Conventional Linear Programming 2.1.1 What Is Linear Programming Linear programming (L.P.) is a mathematical technique which permits a decision-maker to mathematically define all possible (i.e. feasible) solutions to a problem and then find the one that best optimises a particular objective, or a weighted mix of objectives [Dantzig, G.B., 1963]. Best, is emphasized since the problem being solved is, in most cases, an approximation of the real world problem. Thus, the resulting solution is not really the best solution but rather an approximation of it (Sheer, D.P., 1979]. L.P. is specifically concerned with solving a special type of problem, i.e., one in which all relations among the variables are linear in both the constraints and the function to be optimised. 2.1.2 General Problem Formulation In order to use L.P., the problem to be solved must be described in a specific and comprehensive manner. This is called the format of an L.P. problem. The format has two distinct parts: an objective function and a set of constraints, which are both linear combinations of variables. The objective function is a mathematical description of the benefit that is to be maximised or a description of the cost or 4 impact that is to be minimised. The constraints are mathematical descriptions of things that must be true if a solution is to be realistic or feasible. The general problem may be stated as follows: [Hadley, G., 1962: Haimes, Y.Y., 1977]; Given a set of m linear inequalities or equations with n variables, non-negative values of these variables must be determined, which satisfy the constraints and maximise some linear function of the variables. The concepts in the preceeding paragraphs may be expressed mathematically in this form: Find a vector xl = (xi,x2, - • •,xn) which maximises or minimises the the following linear function. f(x): f(x) = C1X1 + c2x2 + ••• cnxn or f(x) = Y , c j X i (2.1) subject to the restrictions Xj > 0 where./ = 1,2, • • •, n (2.2) and the linear constraints aii^i + a12x2 + V alnxn {>,=,<} h a21Xx + a22x2 H h a2nxn {>,=,<} b2 (2.3) a-m\xx + am2x2 + • • • + amnxn {>,=,<} bm where a t J, bj, and Cj are given constants, for: j = 1,2, • • •, n i — 1,2, •••,m. 5 Thus, in this format of n variables and m constraints, c j 5 bj and a,.,- are known coefficients, while Xj are the unknown decision variables. It is the values of the Xj variables that are sought in a solution. Linear programming packages are now available on almost all digital computer systems. L.P. has also been extended in other directions, such as parametric pro-gramming, integer programming, convex programming, stochastic linear program-ming and multi-stage linear programming [Ackoff, R.L., (ed.), 1961; Hillier, F.S., et. al., 1974; Yeh W.W.G., 1985]. 2.1.3 Duality In Linear Programming The duality concept has important applications in linear programming as well as other fields such as economics. With each linear programming problem, called the primal, is associated another problem called the dual. The dual is an equivalent problem whose minimum involves the same policy as the maximisation or primal problem. It provides valuable information regarding the effect of the active con-straints on the optimal solution [Hall, W.A., et.al., 1970; Loucks, D.P., et.al., 1981; Goodman, A.S., 1984]. This information is useful when carrying out sensitivity analysis. It is often provided as an option in standard L.P. computer packages. 2.2 Fuzzy Linear Programming In many decision problems, the objective functions and contraints are clouded with uncertainty. The decision maker may no longer be able to specify exactly his constraints and his objectives, either because it is just unrealistic or because he wants to give himself some leeway, and these quantities can only be stated in such terms as " much bigger ", " near to ", " very small " , etc.. In such situations, neither the methods of determinism nor those of probability are satisfactory for taking these uncertainties into account [Szidarovszky, F., et al., 1986]. 6 More recently, fuzzy theory has been advanced as a technique for handling un-certainties in decision making situations [Bellman, R . E . and Zadeh, L.A., 1970; Kickert, W.J.M., 1978]. This theory has been applied to linear programming, and the result is fuzzy linear programming. The particular kind of fuzzy decision making presented in this chapter describes the decision situation defined as fuzzy program-ming. This also involves a set of decision variables; a set of constraints on these variables; an objective function which orders the alternatives according to their de-sirability; and seeks an optimal fuzzy solution [Chang, L.L., 1975: Ostasiewicz, W., 1982: Tanaka, H, et al., 1975: Rodder, W, et al., 1977: Rodder, W, 1975]. 2 . 2 . 1 Decision-Making In A Fuzzy Environment It is necessary to look at the particular kind of a fuzzy decision situation as depicted in the theory. Consideration will be given to a special model of the problem " maximise an objective function subject to constraints ", that is, the " linear programming model ": Maximise Z = clx such that Ax < b x > 0 (2.4) where c and x are n-vectors, b is an m-vector, and A is an m x n matrix. In conven-tional linear programming , A, b and c describe the relevant state , x represents the decision variables, Z is the event resulting from the combination of the state and the decision variables, while the utility function is expressed by the requirement to maximise Z. Thus, the objective function "maximise Z = c'x" of (2.4) is the utility function. A fuzzy situation is similarly comprised of decision variables, constraints and goals. 7 The notion of fuzziness can be introduced into all of these basic elements. The elements of A, b or c can be fuzzy numbers rather than crisp numbers. Here, crisp numbers are defined as precise numbers. The constraints can be given as fuzzy sets rather than crisp inequalities. The objective function can either be a fuzzy set or a fuzzy function. The theory of fuzzy mathematical programming restricts fuzziness to constraints and goals, only. Only constraints and goals constitute categories of of alternatives whose boundaries are not sharply defined [Kickert, W.J.M., 1978; Zimmermann, H.J., 1985]. The decision variables are considered to remain deter-ministic, although a mechanism will be explored to fuzzify certain decision variables which are themselves associated with aspirations. The fuzzy objective function, as well as the fuzzy constraints are characterised by their membership functions [Bellman, R.E. and Zadeh, L.A., 1970]. Since the aim is to satisfy the objective function as well as the constraints, a fuzzy decision is considered to be the intersection of fuzzy constraints and goals. An important feature of fuzzy theory is that the relationship between constraints and objective functions in a fuzzy environment is fully symmetric, i.e., there is no longer a differ-ence between the former and the latter [Zimmermann, H.J., et.al., 1984]. Finally, the solution or the decision can either be a crisp solution (i.e., a maximising solution) or a fuzzy set. Only the crisp solution will be considered in this thesis. Obtaining the fuzzy set solution involves a more complex formulation and substantially larger computational loads. 2.2.2 G e n e r a l F r a m e w o r k O f F u z z y M a t h e m a t i c a l P r o g r a m -m i n g The symmetric model described in this section is based on Bellman and Zadeh's (1970) approach which assumes that objectives and constraints in an ill-structured situation can be represented by fuzzy sets. Let X be a set of possible alternative decision actions. A fuzzy goal, gj(x), is a fuzzy subset on X which is characterised by 8 its membership function Vgj(x), j — 1,2, • • • , n, x £ X, of the objective (utility) function or goal. Multiple fuzzy goals are possible but only a single fuzzy goal is considered here. A fuzzy constraint C is a subset on X and is characterised by its membership function Vci(x), i — l,2,---,m, x 6 X, which defines the decision space. The fuzzy decision D resulting from the fuzzy goal G and the fuzzy constraint C is the intersection of both, such that: D = G n C and is characterised by its membership function Vd(x): Vd{x) = Vgj{x) * Va(x) where i' = 1, 2, • • •, ra, and j = 1, 2, • • •, rc, and '*' denotes an appropriate, possibly context dependant, aggregator, or connective [Zimmermann, H.J., 1985]. Let M be the set of points x £ X for which Vd(x) attains its maximum if it exists. Thus, M is the maximising decision. If Vd(x) possesses a unique maximum at xm, then the maximising decision is also a uniquely defined crisp decision which may be interpreted as the action belonging to all the fuzzy sets satisfying all constraints and goals with the highest possible minimum degree of membership. 2.2.3 Fuzzy Linear Programming Model Zimmermann, H.J. (1976) undertook the first extension of L.P. into fuzzy lin-ear programming (F.L.P.). He fuzzified the usual L.P. model, as in model (2.4). Assuming that the decision maker's goal can be expressed as a fuzzy set and the solution space is defined by constraints which can be' modelled by fuzzy sets, the appropriate model would be: Find x 9 such that Clx = Z Ax = b (2.5) x > 0 A, C, b and x are as denned in model (2.4), but = denotes the fuzzified version of <, and has the linguistic interpretation of "essentially smaller (or larger) than or equal to". Z represents an aspriration level of the decision maker. (2.5) is fully symmetric with respect to the objective function and the con-straints. Reflecting thus, redefine: * = ( « ) » = ( ! ) and (2.5) then becomes Find x such that Bx = b (2.6) x > 0 Each of the rows of (2.6) will have its own individual membership function Vi[x) and this is interpreted as the degree to which £ fulfills (satisfies) the fuzzy inequality (Bx)i = bi. Here, denotes the ith row of (2.6). The resultant membership function of the fuzzy set "decision" of the complete model (2.6) is defined by: Vd(x) = mini=li...im+1Vi(x) (2.7) In order to operationalise the fuzzy inequality concept of =, we must specify the function Vi{x). This function should at least satisfy the following external condi-tions: 0 if the objective function and constraints are strongly violated, or 1 if they 10 are well satisfied, i.e., satisfied in a crisp sense. The membership function increases monotonically from 0 to 1 over the tolerance interval [6,, 6j + d:], i.e.: ' = 1 if {Bx)i < bi ViX I £ [0,1] i f b{ < (Bx)i < ^ + di > *'=l,---,m+l (2.8) = 0 if {Bx)i > bi + di In this work we adopted the simplest type of membership function. This is one which is assumed to be linearly decreasing over the tolerance interval [6,, bi + d,]: 1 if {Bx)i < bi if bi < { B x ) i < bi + di > i = 1, , m (2.9) 0 if ( B x ) i > bi + di The d, are subjectively chosen constants of admissible violations of the constraints and the goal constraint. By dropping 1 from the (Bx)i function and making these substitutions b[ = bi/di and B[ = S./d, the problem of maximising Vd(x) in equation (2.7) can be written as: max z >o m i n ™ ' + (6| — (B'x)i) (2.10) or, equivalently, maximise Vd(x), where Vd(x) is the membership function and x > 0 of the decision (solution) set. In order to solve this problem, we introduce a new variable 'A', which essentially corresponds to Vd(x) in (2.7). This problem is now equivalent to the conventional L.P. formulation of a maxmin problem [Wagner, H.M., 1969]: Maximise A subject to the constraints A < b ' i - ( B ' x ) i X , x > 0 (2.11) 11 or, conventionally stated with decision variables on the right-hand-side,: Maximise A such that Xdi + (Bx)i < bi + di where i = 1,2, • • •, m + 1 0 < A < 1 A,x > 0 (2.12) If the optimal solution to (2.12) is the vector A0, x ° , then x ° is the maximising solution to model (2.5), with the membership function as specified in (2.9). This maximising solution can be found by solving one standard (crisp) L.P. with only one more variable and one more constraint than model (2.4). This makes this approach computationally very efficient. Also added to model (2.11) to facilitate fuzzy L.P. in practice is an equality constraint for each of the fuzzified constraints is defined as follows: di where V, is an introduced variable, and is a measure of our position between 0 and 1. The purpose of this equality constraint is simply to evaluate the individual status of all of the fuzzy constraints in the optimal solution as well as the fuzzy goal. 2.3 Altarnative Fuzzy Formulation Slightly modified versions of models (2.10) and (2.11) result if the membership functions are defined as follows: A variable i — 1, 2, • • •, m + 1, 0 < ti < di is adopted and defined as a measure of the degree of violation of the i t h constraint. The membership function of the i t h row is now: Vi{x) = U/di where tt = (Bx)% - 6, (2.14) 12 Paired constraints, with d, and 6, as right-hand-side coefficients, are introduced for each of the fuzzified constraints, therefore, the final formulation would be: Maximise A such that Xdi + ti < di where i = 1,2, • • •, m + 1 ( B i ) , - U < k U < di X,x,t > 0 (2.15) Because term Ad, is always nonnegative, the constraint 2, < di is redundant. This model is still larger than model (2.12). However, model (2.15) is claimed to be ad-vantageous, especially, when performing sensitivity analysis on 6, and d, separately [Hamacher, H., et al., 1978]. 2.4 Crisp Constraints So far, the constraints and the objective function were considered fuzzy. If some of the constraints are crisp, that is, (Dx)i < bi, then these constraints can also be added to models (2.11) and (2.15). For example, model (2.11) would become: Maximise A such that r Xdi + (Bx)i < bi + dt Dx < bt A < 1 x,A, > 0 (2.16) 13 2.5 Summary Conventional L.P. and general problem formulation have been reviewed. The re-view of fuzzy linear programming included decision making in a fuzzy environment, and the general framework of fuzzy mathematical programming. The discussion of fuzzy linear programming included some variants of the basic concept. The main advantage of the fuzzy model over the non-fuzzy model is that, the decision maker is not forced into precise formulation for mathematical reasons, even though he may only be able or is willing to describe his problem in fuzzy terms. Membership functions which increase or decrease linearly between 68 and 6, + di, or bi and 6, — d,-, respectively, can be handled quite easily and have been fouhd to be adequately realistic (Zimmermann, H.J., 1985]. 14 C hapter 3 Henry Hallam and Prince Edward Reservoirs 3.1 Background Information 3.1.1 Introduction The growth of population in Greater Harare (650 000 in 1984) and its dormitory town of Chitungwiza (275 000 in 1984), and also that of its manufacturing industry, have resulted in greater demands for water [Nelson, H.D., (ed.), 1985]. With an environment where rain is very unreliable, as evidenced by the droughts of 1982 to 1985 and 1986 to 1987, this calls for management of water stored in Henry Hallam and Prince Edward reservoirs, Lakes Mcllwaine and Robertson (see Figures 3.1 and 3.2). Mathematical programming models are useful when attempting to balance out the various demands on these insufficient and fluctuating water resources. This chapter focuses on the description of physical and economic environment in which these reservoirs are situated. The application of fuzzy mathematical pro-gramming will be limited to Henry Hallam and Prince Edward reservoirs, hence, the description is also restricted to them. These two reservoirs have been singled out for F.L.P. modelling and subsequent analysis because they are managed as a separate system in real life: they form a smaller system when compared to the two downstream lakes, thus, they are easier to handle. The other reason for singling 15 Figure 3.1: Locations of Harare and Chitungwiza [thornton, J. A., (ed.), 1982]. Figure 3.2: Locations of Henry Hallam and Prince Edward reservoirs [same source as Fig. 3.1]. 16 them out is that they are more vulnerable to the impact of very late rains or pro-longed droughts or severe drawdowns so that they need careful management.The following sections deal with the reservoirs' physical environment, water uses, and data sources for the reservoirs' models. 3.1.2 Physical Environment Henry Hallam and Prince Edward reservoirs are situated in the upper reaches of the Hunyani catchment, at an altitude of 1550m above sea level. The mean annual temperature is about 18°c, with highs of 28°c occurring at the peak of the hot season in October. Four seasons, essentially based on the considerations of temperatures and rainfall, are recognised. These are, the hot-dry season from September to mid-November; the hot-rainy season from mid-November to the end of March; the post-rainy season from April to mid- May; and the cool-dry season from mid-May to the end of August. Humidity is high during the rainy season but can be very low during the hot-dry season (see Figure 3.3 on Harare's climatic aspects). Most of the rain comes in the form of heavy thunderstorms which occur from mid-November until March. The annual rainfall averages 800mm although this figure varies from year to year. Mid-May to about the end of October is the annual drought season when insignificant to no rain at all, is experienced. Evaporation from free water surfaces averages 1990mm per year [Kay, G, et. al., 1977]. For example, Lake Mcllwaine with a useful capacity of 247 * 10 6 m3, evaporates at a rate of 40 * 10 6 m 3 per year when it is three quarters full. Generally, 2 5 % of each reservoir's full storage capacity is lost to evaporation each year. River inflows to the reservoirs are moderate to large during the rain season, and are negligible during the annual drought season. The coefficient of variation of the annual series run-off of the Hunyani and the Ruwa rivers flowing into these two reservoirs is around 80% [Thornton, J.A., (ed.), 1982]. Thus, the variation of 17 18 inflow from a wet year to a dry year is very large. An example is, 123 * 10 6 m 3 (in February, 1981) and 0.92* 10 6 m 3 (in February, 1984), as per records of station C-81 on the Hunyani river and just upstream of Henry Hallam reservoir [Hydrological Summaries, 1985]. 3.1.3 Interaction And Water Uses The four reservoirs on the Hunyani river store water which meets Harare's demands.Their sizes are as follows: Henry Hallam (9.026* 106 m 3), Prince Edward (3.380* 10 6 m 3), Lake Mcllwaine (250*10 6 m 3) and Lake Robertson (490*10 6 m 3). Although most of the water for Harare and downstream consumers is supplied by the last two, large reservoirs, Henry Hallam and Prince Edward reservoirs are still an inseparable component of the overall strategy for meeting the city's water requirements. However, climatic as well as geological factors still pose real problems in securing adequate supplies for the city. The rainfall is highly unreliable so that river flows and run-off are also unreliable. Geologically, there are no large aquifers in the environs of Harare which can sustain large yields, and there are no nearby sites suitable for damming. Thus, the management of water in storage is crucial to sustaining of the the city's economic and domestic activities, as well as its overall well-being. Henry Hallam and Prince Edward store supplementary water supplies for the city. They are operated jointly in that Henry Hallam stores water which is released into Prince Edward from which all abstractions are made. The releases from the former reservoir replenish what is withdrawn from from the latter reservoir. From Prince Edward, water is pumped to Harare and Chitungwiza for domestic industrial supplies. Prince Edward reservoir's level is maintained at approximately 0.5m to 1.0m below the spillway, or at about 80% of the full storage capacity, i.e., a volume of 19 around 2.704 * 10 6 m3. Two reasons for maintaining a more or less constant water level from mid-May to mid-November (i.e. the annual drought season) are: • There is a significant fishery whose yields' economic value approximates Z$150 000 (Zimbabwean dollars) per year. • This volume is considered effective in controlling weed growth by flooding. Generally, rains of 650mm in the season are sufficient to replenish what is presently lost to withdrawals and evaporation from the reservoirs. But the precarious nature of the climatic factors call for the careful management of these water resources if the the various demands are to be met. 3.1.4 D a t a S o u r c e s The sources of data for the model are: 1. Water consumption data from Harare City Council's Department of Finance. This data shows monthly domestic as well as industrial consumption. The respective revenues are -ZS0.173 per mz and .£$0,248 per m3. A value of Z$0.100 per m 3 of water in storage is assumed for fish benefits. 2. Hydrological records of inflows were taken from the Hydrological Summaries of 1985, which are published by the Ministry of Water Development's Hydro-logical Branch. For inflows to Henry Hallam, gauged records from stations C-81 (Hunyani - Henry Hallam U/S G/W) and C-82 (Ruwa - Henry Hallam U/S G/W) were used. Records from C-2 (Hunyani - Prince Edward U/S G/W) show outflows from Henry Hallam, which are also the total inflows to Prince Edward. Station C-3 (Hunyani - Prince Edward D/S G/W) shows outflows from Prince Edward. 3. Evaporation records from station CE-9 (Kutsaga Tobacco Research Station) were used since this station is the nearest to the two reservoirs. 20 Chapter 4 L.P. and F.L.P. Formulation of The Two-Reservoir Problem 4.1 Linear Programming Formulation 4.1.1 Variable Definition The time span being considered in the formulation is from October 1981 to the end of March 1983; and is divided into six periods of three months each. The decision variables used in the model are as follows: • Water sales to domestic consumers are represented by PADl, • • • , PAD6, and the revenue is Z$0.173/ m 3 . • Water sales to industrial consumers are represented by PAH, • • • , PAI6, and the revenue is Z$0.248/ m 3 . Inputs, outputs and losses are specified by the following: • Inflows to Henry Hallam are given as INHl, • • • , INH6. • Evaporation from Henry Hallam is given as EHl, • • • , EH6. • Outflows from Henry Hallam are given as HOI, • • • , H06. • Inflows to Prince Edward are given as INP1, • • • , INP6. • Evaporation from Prince Edward is given as EPl, • • • , EP6. 21 • Outflows from Prince Edward are given as POl, • • • , P06. The following derived variables are also included: • The live storage contents of Henry Hallam are given as STHl, • • • , STH6. • The live storage contents of Prince Edward are given as STPl, • • • , STP6. • Fish dis-benefits are related to FXlMIN, • • • , FX6MIN which reflect storage volume deficiencies (MIN infers minus volume) which reduce fish harvests. The assumed value of Z$0.100/ m 3 of water in storage is adopted. N.B. - all figures for water volumes are in 10 6 m 3 or in millions of cubic metres. 4.1.2 Objective And Constraints Formulation A . Object ive Funct ion Formulat ion The two-reservoir operation problem involves finding the optimal sequence of ab-stractions that maximises the total monetary benefits for the six periods. Thus, the objective involves: - the total domestic water sales benefits = CxPADl + ••• + CyPADS = 0.173P^£>1 +••• + ().173PAD6 - the total industrial water sales benefits = C2PAI1 + : • • + C2PAI6 = 0.248P.4/1 + • • • + 0.248P.4J6 total fish disbenefits by failing to maintain Prince Edward at least greater than or equal to 2.366 * 10 6 m 3 = -C3FX1MIN C3FX6MIN = -0.1FX1MIN 0.1FX6MIN 22 The objective function can be crisply stated as: Maximise Z = 0.173PAD1 + ••• + 0.173P AD6 + Q.248PAI1 + ••• + 0.248PAJ6 0.1FX1MIN 0.1FX6MIN For details, see computer program listing in Appendix (A). B. System Operat ing Constra int Formulat ion A l l constraints for input data, such as, inflows, evaporation, outflows and abstrac-tions were entered as inequalities (see Appendix (A)) to facilitate RHS coefficient ranging although this form of sensitivity output was not actually exploited. For each of the two reservoirs, and for each period i, there is a set of three constraints, namely, the storage capacity constraint, the continuity constraint and the storage calculation constraint. For Prince Edward, there is an additional fourth constraint for the fish benefits. 1. Storage Capacity Constraint For each reservoir, and at the beginning of each period, there is a storage volume ( 5 T H { or S T P i ) , where i = period 1, 6. The constraints on outflows, abstractions and evaporation losses limit these aspects to available water in that period, and force a spill if the available water exceeds the reser-voir capacity. Therefore, for each period i, the storage capacity constraint is Thus, the respective storage capacity constraints for Henry Hallam and Prince Edward are as follows: as follows: Storage at the beginning of period, + + Inflow; - Outflow; - Abstractions; - Evaporation, < [Storage Capacity] S T H i + INH, - H O i - E H i < 9.026 23 and STP, + IN^ + HOi - POi - PAD, - PAIt - EP{ < 3.380 See Appendix (A) for these constraints, labelled CSHi and C 5 P , , respec-tively. 2. Continuity Constraints These constraints include a mass balance of inflows, outflows, abstractions and evaporation losses and equate the final storage volume in period i (which is also the initial storage volume in periodi+i) to the intial storage volume i plus inflow, minus abstractions and evaporation losses. Thus, for each period i, the continuity constraint is as follows: Storage at the beginning of period, + + Inflow; - Outflow, Abstractions, Evaporation; Storage at the end of the \ t h period, which is also the beginning of the i th+1 period Therefore, the respective continuity constraints for Henry Hallam and Prince Edward are: STHi + IN Hi - HOi - EHi - STHi+l = 0 and STPi + IN Pi + HOi - POi - PADi - PAIi - EPt - STPi+1 = 0 See Appendix (A) for these constraints, labelled CSTHi and CSTPi , re-spectively. 3. Storage Deficiency Constraints Prince Edward reservoir is very rich in fish whose value per year is about Z$150 000. Hence, the level of this reservoir is always maintained at 0.5m to 1.0m below the spillway, or at least 80% of full storage capacity. For this 24 model, it is safe to assume that adverse effects on the fishery occur if the storage volume in any period falls below 70% of the full storage capacity, i.e., below 2.366 * 106 m 3 . This constraint will show by how much the reservoir is above or below this 70% specification. The constraint is given as follows: FXiMIN - FXiPLUS + O.bSTP, + 0.5STPl+1 = 2.366 FXiMIN > 0 shows that the average storage volume is below the 70% mark by FXiMIN * 106 m 3 , while FXiPLUS > 0 shows that the average storage volume is above the 70% mark by FXiPLUS * 106 m 3 . FXiPLUS > 0 implies fixed positive benefits from the fishery with the existence of the optimum storage conditions, while FXiMIN > 0 implies reduced benefits from the fishery since below optimum storage conditions are prevailing. These disbenefits are assumed proportional to FXiMIN. If the average storage volume in period i is less than 70%, then FXi PLUS will be 0 and FX{ MIN will be positive. Also, if the average storage volume in period »is over the 70% mark, then FXi MIN will be 0 and FXi PLUS will be positive. See Appendix (A) for these constraints, labelled CFPi. 4.2 Fuzzification Of The Two-Reservoir Problem The adopted fuzzy objective is to achieve overall monetary benefits or goal (Zg) from water sales as closely as possible while minimising the maximum violation of any of the fuzzy constraints. The fuzzy constraints are used to determine if certain water allocation tradeoffs can be achieved during periods four and five. There are also fuzzy constraints to determine if lowered outflows can be achieved for periods two and four. Constraints pertaining to storage capacity limits and continuity requirements will be retained as crisp constraints. 25 4.2.1 Fuzzy Constraints Formulation The two-reservoir problem is formulated for the above stated objectives using the procedure outlined below. In many instances the conventional L . P . solution wi l l be used as a guide in formulating the fuzzy membership function. The reader wi l l find reference to Table 5.3 which summarises the conventional L . P . solution useful in this regard. 1. Fuzzy Goal Constraint The original, conventional objective function is turned into a fuzzy goal con-straint with the goal (Zg) as the r i g h t - h a n d - s i d e coefficient. Then the mem-bership function reflecting how far we are to this goal is defined as follows: 1 Figure 4.1: Goal's membership function where: = 1 w h e n ZcijXj > Zg Vg(x) I = w h e n Zg<Ect]x,<Zg-dg \ (4.1) = 0 w h e n Y,cijxj < Zg ~ dg Specification O f Parameters: ZS5.400 mill ion are the lowest benefits derived from the reservoir during the critical drought period of 1983/85 over a similar time span of eighteen months 26 as that considered in our model. This is a less desirable situation such that benefits above the L.P. result of Z$5.749 million are more preferred. The choice of an upper bound of Z$5.900 million is arbitrary but is consistent with our desire to derive even bigger benefits from the system. The figures below are in Z$106 . Zg = 5.900 dg = 0.500 Zg - dg = 5.400 YlcijXj = Original non-fuzzy objective function In order to evaluate the actual monetary benefits, the variable "GOAL" is incorporated into the fuzzy formulation as follows: "^CijXj = G O A L which becomes ^ djXj - G O A L = 0 The fuzzy goal constraint together with all other fuzzy constraints are refor-mulated back to conventional L.P. for solution by introducing a new variable 'A' which is represented by xl in the F.L.P. problem (see Appendix (B)) where A < all Vi(x) . The proper L.P. form of the fuzzy goal constraint is: G O A L - 5.400 , . , , _ r > xl which becomes G O A L - 0.500x1 > 5.400 0.500 4.2.2 Fuzzy Constraints The fuzzy constraints are formulated for selected critical periods only. During periods 4 and 5, domestic water demands are high. Having established the normal water demands from the results of the conventional L.P. run, see Appendix (D), a water manager might be interested in ascertaining if larger domestic and industrial allocations are possible without affecting the fishery requirements (in period 4). If 27 the rains are late, such that there is less water in storage by the end of November to meet known demands at this time of the year, the water manager may be inter-ested in determining if lower allocations are possible without going down to critical allocations which are known to have adverse effects on domestic and commercial activities. It should be noted that the subsequent fuzzy constraints impact directly upon certain abstractions and discharges. In effect this demonstrates a direct fuzzification of decision variables. 1. Domestic Supplies In Period 4 - (PAD4) During the rainy season, (mid-November to mid-May), water demands are lower than during the hot, dry season. The advent of the annual drought season results in increased demands for water. Therefore, the aim is to deter-mine how high the supplies could be increased. The membership function is defined as follows: bj - ^ PADA where: Figure 4.2: Membership function for PAD4 Vi[x) = 1 when PADA > bi = P^A-ib.-di) w h e n b i < p A D 4 < 6 i _ d i = 0 when PADA <bx- di (4.2) 28 Specification O f Parameters: 1.830 * 106 m 3 is the smallest consumption record for a period similar to that of PAD4 during the critical drought of 1983/85. Higher consumptions would be more desirable, hence the aspiration towards 2.200 * 106 m 3 , which is higher than the L.P. result of 1.900 * 10G m 3 , is more preferred as it will increase the overall benefits. The figures below are in 106 m 3 . 6i = 2.200 dx = 0.370 bi-d! = 1.830 Introducing xl and reformulating back to normal L.P. for solution: PADA - (bj. - di) > xl which becomes PADA - 0.370x1 > 1.830 2. Industrial Supplies In Period 4 - (PAI4) The aim is to determine how high these supplies can be increased to in the the dry season, i.e., in period 4. The membership function is defined as follows: Vt(x) 6; — di bi Figure 4.3: Membership function for PAI4 • PA 14 where: 29 = 1 when PAI4 > b2 V2(x) I = PAi4-{b,-d2) w h e n ^ < p A l 4 <b2_dl \ (4.3) = 0 when PAH < b2 - d2 Specification O f Parameters: The figure of 2.580 * 106 m 3 is the lowest record for a period similar to that of PAI4 during the critical drought period of 1983/85. The aspiration towards 2.900 * 106 m 3 , which is higher than the L.P. result of 2.650 * 106 m 3 , is more preferred. It is an arbitrary choice consistent with our desire to get bigger benefits from the system. The figures below are in 106 m 3 . b2 = 2.900 d2 = 0.320 b2 - d2 = 2.580 Introducing xl and reformulating back to conventional L.P. for solution: PAI4 ~(b2-d2) , . , , - > xl which becomes PAI4 - 0.320x1 > 2.580 d2 3. Domestic Supplies In Period 5 - (PAD5) Towards the end of the annual drought period, the normal water demands may be difficult to meet due to low water volumes in storage, or due to the lateness of rains which replenish all that has been consumed. It is possible to restrict water use by prohibiting the watering of vegetable gardens and lawns, hence, allocating less than is usually supplied. Thus, the aim is to determine how low the domestic supplies could be in period 5. The membership function is defined as follows: 30 PADS Figure 4.4: Membership function for P A D 5 where: V3(x) = & _ (bz+dz)-PAD5 when PADb < b3 when 63 < PADb < b3 + d3 (4.4) = 0 when PADS > b3 + d3 Specification Of Parameters: 1.780 * 106 m 3 is a high consumption figure from a similar period to that of PAD5, during the 1979/81 years of abundant water. Consumption lower than the L.P. result of 1.775 * 106 m 3 , without reaching a critical low of 1.700 * 106 mz from the 1983/85 drought period would be desirable. Figures below are in 106 m 3 . 63 = 1-700 d3 = 0.080 b3 + d3 = 1.780 Introducing xl and reformulating back to conventional L.P. for solution: (63 + d3) - P^ 7J>5 > i l which becomes PADb + 0.080x1 < 1.780 4. Industrial Supplies In Period 5 - (PAI5) As is the case with domestic supplies above it may be difficult to meet the nor-mal industrial consumers' water demands. It is possible to allocate less than 31 is usually supplied. This could be achieved by asking those industries that use raw water in their manufacturing operations to supplement with ground water. The aim is to determine how far the industrial supplies could be de-creased. The membership function is defined as follows: 1 V Figure 4.5: Membership function for PAI5 64 + dA •PAI'o where: = 1 when PAIS < b4 V4{x) { = (*«+<M-PA/5 w h e n ^ < p A j 5 < b i + d 4 (4.5) 0 when PAIS > b4 + d4 Specification Of Parameters: The criteria used to choose the lower and upper limits is similar to that applied to PAD5 above. The figures below are in 106 m 3 . 64 = 2.455 d4 = 0.196 64 + d4 = 2.651 Introducing xl and reformulating back to normal L.P. for solution: (64 + d4) - PAIS d4 > xl which becomes PAIS + 0.196x1 < 2.651 5. Henry Hallam Outflows In Period 2 - (H02) Assuming that outflows are controlled by a gate, the aim is to determine if outflows can be decreased in order to reduce wastage. For period 2, a large amount of water is flowing through but the reservoir is not full at the begin-ning of period 3. It is desirable to find if lower outflows are possible so that we remain with more water in storage, while not affecting abstractions and the storage volume of Prince Edward. The membership function is defined as follows: 1 1 k H ° 2 Figure 4.6: Membership function for H02 where: = 1 when H02 < 65 V5(x) { = ^+d'-)-H02 when 65 < H02 < 65 + d5 \ (4.6) = 0 when H02 > b5 + d5 Specification Of Parameters: The outflow of 41.760 * 106 m 3 is the highest figure recorded for a similar period as that for H02, over Henry Hallam's spillway in 1980. An outflow slightly less than the L.P. result of 41.752 * 106 m 3 without the arbitrarily chosen lower bound would result in less wastage of water. More water would 33 be retained in storage. The figures below are in 10 6 m 3 65 = 41.585 <k = 0.175 65 + d B = 41.760 Introducing xl and reformulating back to the orginal L.P. for solution: (65 + <k) - H02 ^ ^ becomes H02 + 0.175x1 < 41.760 6. Henry Hallam Outflows In Period 4 - (HQ4) The aim is to determine if outflows can be reduced so that more water is retained in storage during this dry season, while satisfying abstractions and fisheries requirements for Prince Edward. More water in storage means that if the rains are late in period 5, water demands will not be seriously affected. The membership function is defined as follows: 1 Ve(x) be 66 + ^ 6 -H04 where: Figure 4.7: Membership function for H04 Ve(x) = 1 when H04 < b6 = (b<+d<)-HOi when 66 < H04 < 66 + 4 = 0 when H04 > b& + d6 (4.7) 34 Specification O f Parameters: The upper bound of 6.510 * 106 m 3 is a high outflow for a period similar to that for H04, in 1980. The lower bound is an arbitrary choice consistent with our desire to retain more water in storage by releasing less than the L.P. result of 6.500* 106 m 3 . Outflows below the lower bound are known to be insufficient for sustaining demands made on Prince Edward Reservoir downstream. 66 = 6.420 d6 = 0.090 66 + d6 = 6.510 Introducing xl and reformulating back to conventional L.P. for solution: [be + de) - HOA — > xl becomes HOA + 0.090x1 < 6.510 a 6 4.2.3 Membership Function Evaluation Constraints For each of the fuzzy constraints formulated in the preceeding section, a Vj-(x) equality constraint is added. The value of each VJ-(x) for the fuzzy optimal solution is then computed within the F.L.P. model and provided in the solution. This gives the position of V,(x) between 0 and 1. For all cases whose values are increasing, the mathematical expression is: = Vi[x) where n = — d,-. di Thus, for domestic and industrial supplies in period 4 (PAD4 and PAI4), as shown in the preceeding section, a new variable V{ is introduced for each fuzzy constraint. The respective linear equalities are: PADA - 0.370F, = 1.830 35 and PAI4 - 0.320V2 = 2.580 For the rest of the last four cases, i.e., domestic and industrial supplies in period 5 and Henry Hallarn's outflows in periods 2 and 4, whose values are decreasing, the mathematical expression is: v'-ZaijXj = r}l = h i + d i di The respective V,(x) linear equalities are as follows: PADS + O.O8OV3 = 1.780 PAIS + 0.196K, = 2.651 H02 + 0.175V5 = 41.760 H04 + 0.090V6 = 6.510 4.2.4 Remaining Non-Fuzzy Constraints The rest of the non-fuzzy constraints are retained without alteration in the final F.L.P. formulation. 4.2.5 Complete Formulation And Solution The objective of the fuzzy linear programming problem is simply to maximise xl. Solution of the reformulated L.P. version of the F.L.P. problem was obtained using UBC:LIP linear programming package. See Appendix (B) for full input listing to UBC:LIP. 4.3 Alternative Fuzzy Formulation As formulated in Section 2.3, each fuzzy constraint discussed in Section 4.2 may be replaced by a pair of fuzzy constraints so that more sensitivity information can be 36 extracted for the b{ and dt, individually [Hamacher, H. , et al., 1978]. The variable U is introduced, which measures the degree of violation of the ith fuzzy constraint. For domestic and industrial supplies of period 4, the membership function is depicted as follows: bi-di bi Figure 4.8: f, diagram for increasing values For domestic supplies- (PAD4) the pair of constraints.is: 0.370x1 + <i < 0.370 and P ADA + tY = 2.200 For industrial supplies (PAH), the pair of constraints is: 0.320x1 + t2 < 0.320 and PAIA + t2 = 2.900 For domestic and industrial supplies in period 5 and Henry Hallam's outflows in periods 2 and 4, the membership function is as follows: Thus, the pair of fuzzy constraints for domestic supplies in period 5 (PAD5) is: 0.080x1 + h < 0.080 and PADS + ts = 1.700 for industrial supplies in period 5 (PAI5), the pair of constraints is: 0.196x1 + t4 < 0.196 and PAIS + t4 = 2.455 37 Figure 4.9: i, diagram for decreasing values for Henry Hallam's outflows in period 2 (H02),the pair of constraints is: 0.175x1 + U < 0-175 and H02 + h = 41.585 and for the same resrvoir's outflows in period 4 (H04), the pair of constraints is: 0.090x1 + U < 0.090 and HOA + t6 = 6.420 To run this model, the objective is still to maximise xl, which is the minimum of the Vi(x). See Appendix (C) for the paired fuzzy constraints model. 4.4 Summary The variables and constraints used to formulate the two-reservoir problem have been presented. The formulation of the problem as a conventional L.P. model has been described with the appropriate operational constraints being given particular attention. The fuzzification of the problem has been described with special atten-tion being focused on the practical implications of the membership functions. One variant of the F.L.P. formulation which is claimed to enrich the sensitivity output has also been described. The two fuzzy models would be expected to give the same answers. 38 C hapter 5 Discussion of Results The results from the conventional L.P. model are presented first. These results are then compared with those from the fuzzy L.P. model. Lastly, sensitivity analysis is discussed. 5.1 Conventional L.P. Results The conventional L.P. model gives a maximum value of the objective function of: Z = Z$5.749025 million. The optimal values of the decision variables are shown in Table 5.1. These results are satisfactory since they reflect the trend of the real world situation whereby higher water consumptions occur during the annual drought periods 1, 3, 4, and 5. Domestic water consumption is the main cause for the higher water consumption figures. Lower consumptions occur during the rain season as shown by the figures of periods 2 and 6, since the rains remove the necessity for watering gardens, etc.. For a healthy fishery on Prince Edward reservoir, the live storage must be main-tained at 70% of the full storage capacity. The model met this requirement. The FXiPLUS variables' values depict the desirable situation whereby the amount of water in storage is maintained above the 70% limit at all times (see Table 5.1 for values). This is a desirable situation, especially, during the annual drought season 39 - Period Period Period Period Period Period Period - 1 2 3 4 5 6 7 P A D i 1.850 1.700 1.800 1.900 1.775 1.700 -P A I i 2.650 2.550 2.650 2.650 2.650 2.550 -IN Pi + H Oi 5.647 44.286 7.552 6.640 5.177 4.697 -E P % 0.245 0.198 0.161 0.213 0.247 0.197 -P O , 0.724 39.356 3.325 1.800 0.200 0.250 -S T P i 2.720 2.898 3.380 2.996 3.073 3.380, 3.380 F X i P L U S 0.443 0.773 0.822 0.669 0.861 1.014 -Table 5.1: Prince Edward Reservoir (figures in 106 m 3) - Period Period Period Period Period Period Period - 1 2 3 4 5 6 7 IN H i 7.522 45.295 7.006 3.021 6.849 8.102 -E H i 0.669 0.544 0.454 0.589 0.679 0.545 -H O i 5.500 41.752 7.384 6.500 4.225 4.440 -S T H i 4.512 5.865 8.864 8.032 3.964 5.909 9.026 Table 5.2: Henry Hallam Reservoir (figures in 106 ra3) when it is necessary to keep the weeds flooded as a viable weed control measure. The releases from Henry Hallam reservoir (see Table 5.2 for values of variables H O i ) into Prince Edward reservoir help the latter meet the abstraction, fisheries and weed control demands placed on it. The releases from Prince Edward (see PO, values in Table 5.1) are for downstream conservation purposes, in order to sustain the river's aquatic life. The live storage volume of Henry Hallam is allowed to have large fluctuations since it only acts as a storage facility for Prince Edward reservoir. There are no fishery benefits or weed control requirements directly associated with this reservoir. The S T H i variable values over the six periods show these swings quite clearly (see Table 5.2). For the rest of the results see Appendix (D). 40 5 . 2 Comparison O f Conventional L .P . And Fuzzy L .P . Results The fuzzy L.P. model yielded an optimal solution that conforms with the in-tended aspirations. The maximum degree of overall satisfaction, whereby A* = 0.82503, is achieved. The optimal values of the fuzzified domestic (PAD4 and PAD5) and industrial (PAI4 and PAI5) variables, as well as outflows (H02 and H04) are as shown in Table 5.3. This is the maximum solution which yields overall benefits of: Zg = ZS5.812 516 million. When the fuzzy solution is compared with the non-fuzzy solution, larger monetary benefts and water allocations were realised through the fuzzy model. For example, industrial supplies (PAI4) of up to 2.9 * 106 m 3 are possible when compared to the non-fuzzy figure of 2.650 * 106 m 3 . During period 5, when there is a likelihood of water shortages due to low volumes in storage, coupled by the lateness of rains, it is possible to lower allocations to the figures shown in Table 5.3. For example, domestic supplies (PAD5) can be lowered to 1.714 * 106 m 3 (fuzzy value), from a figure of 1.775 * 106 m 3 (non-fuzzy value). Lower outflows have also been achieved for Henry Hallam reservoir for the assessed periods, (see Table 5.3 for the fuzzy and non-fuzzy values of H02 and H04). The implications of achieving these lower outflows is that more water remains in storage. By lowering outflows in period 4, more water that is retained in storage is a desirable contingency for meeting higher water demands in drier period 5. Optimum volumes are maintained in storage for meeting Prince Edward's fishery and weed control requirements. For the rest of the results pertaining to the fuzzy optimal solution, see Appendix (E). Given the optimal solution to the fuzzy model, we were able to maximise the minimum value of V,(x). That is, with an overall value of A* = 0.82503, our desire 41 Fuzzified Constraint L.P. Non-Fuzzy Solution F.L.P. Fuzzy Solution Infeasible Feasible Desired Direction vt = 0 V{ = 1 di G O A L Overall Goal 5.749 5.813 0.825 5.400 5.900 0.500 PAD4 Domestic Supplies 1.900 2.200 1.0 1.830 2.200 0.370 PAI4 v2 Industrial Supplies 2.650 2.900 1.0 2.580 2.900 0.320 PAD5 v3 Domestic Supplies 1.775 1.714 0.825 1.780 1.700 0.080 PAI5 vA Industrial Supplies 2.650 2.489 0.825 2.651 2.455 0.196 H02 Outflow 41.752 41.614 0.825 41.760 41.585 0.175 H04 v6 Outflow 6.500 6.435 0.825 6.510 6.420 0.090 A* = x l 0.825 Table 5.3: L.P. and F.L.P. results (volumes in 106 m 3 and GOAL in Z$106) 42 to minimise the worst underachievement of any of our goals was attained. The V,-values (see Table 5.3 or Appendix (E)) reflect the degree of achievement for each fuzzy constraint. For the fuzzy constraints on variables PAD4 and P A H where we are attempting to maximise allowable water allocations, the V^(i), i.e., V\ and V2, values are 1.0, indicating that the highest level of our aspirations for PAD4 and P A H has been achieved. Recalling the definition of our membership function in Section 4.2.2, for PAD4, the situation is as follows: YaHx-i ~ bi 0 7 PADA = 2.2 and PAIA = 2.9 For the rest of the fuzzified variables (PAD5, PAI5, H02 and H04), the Vi(x), i.e., V3 to V6, an intermediate value of 0.82503 is achieved. For these variables, and recalling the definition of our membership functions, the situation is as follows: b{ < Y aijxj < ^ + di Similarly, our overall goal of increasing the monetary benefits achieved an interme-diate value of Vg = 0.825 where: Z 9 < Y.C<JX) ^ Z 9 - d9-5 . 2 . 1 A l t e r n a t i v e F u z z y F o r m u l a t i o n The results, i.e., of the goal function, the variables, A*, etc., correspond numer-ically to those from the single fuzzy constraint model. However, it also gives the i, values for each of the fuzzy constraints. ^ measures the degree of violation of each of the respective constraints. For PAD4 and P A H whose V, = 0, thus, perfectly achieving the aspiration levels, their ti and t2 = 0. For the decreasing variables, whose Vi = 0.825, their values are as follows: t3 = 0.014, t4 — 0.034, £5 = 0.005 and t§ — 0.047. See Appendix (F) for full solution output. 43 The dual solution variables corresponding to constraints of the type: (Bx)i - U < bi provide unconditional interpretation of the sensitivity of A* to bt. The numerical results agreed with the results obtained in the initial fuzzy formulation when the assumption outlined in Section 5.4.2 was applied. Dual solution variables for constraints of the form: Xdi + ti < di could not be interpreted as representing the sensitivity of A* to di due to the presence of d, on both the LHS and RHS of these constraints. One condition where this interpretation would be possible would be when A* = 0. 5.3 Sensitivity Analysis 5.3.1 Conventional L.P. Sensitivity analysis is a well documented area in conventional L.P., and is linked to the concept of duality. A sensitivity analysis examines the effect small changes in the model inputs and model parameters on the values of the model outputs and the objective function. To each resource t, there corresponds a dual variable y, which by its dimensions, is a price or cost to be associated with one unit of resource i. Thus, " the values of the variables in the dual optimal solution can be interpreted as shadow prices indicating the (marginal) increase of the optimal value of the primal objective function as a result of (marginal) increases of the respective components of the right-hand-side of the primal problem " [Hamancher, H, et.al, 1978]. The dual solution variable y,- for conventional constraints gives: Vi = SZ*/6b, 44 where 6Z~ is a change in overall optimal benefits due to an increase in the value of the right-hand-side coefficient denoted by <56,. This conventional interpretation of the dual solution variables is applicable to all non-fuzzy inequality constraints appearing in both regular non-fuzzy L.P. as well as the fuzzy L.P. formulations. This is well documented in the linear programming literature and will not be discussed here. It should be noted, however, that, in the case of a non-fuzzy inequality constraint in a fuzzy L.P. formulation, the sensitivity is with respect to A* and not the economic or benefits goal. Thus, the shadow price interpretation of the dual solution vector does not apply directly to the F.L.P. problem of the two reservoirs. 5.3.2 Fuzzy L.P. For the fuzzy constraints, the dual solution variable gives: 6X* 6X* V i ~ 6{b{ + di) °T 6{bi - di) where d, is the maximum allowable violation of the aspiration level 6;. In a typical fuzzy constraint, we have: Xdi + { B x ) i = bi + di or Xdi + { B x ) i = bi - di and di appears on both LHS and RHS of the constraints. As variation must be confined to the RHS of a single constraint for the above interpretation of the dual solution vector to be correct, it is necessary to assume that all change is confined to bi alone. Under this restriction, then: 6X* m = ~6bt 45 for all fuzzy constraints. The bi values for the fuzzy constraints on PAD4 and PAI4, whose values tending to be decreased, have no effect on the optimal value of A, i.e. A*, since they have corresponding 0 values in the dual solution vector. In fact, having both achieved a Vi = 1.0, they have both significantly exceeded A* = 0.82503. The bi values for PAD5 and PAI5, namely 6 3 and 6 4 , whose values are also tending to be decreased, have a pronounced effect on A*. The dual solution gives: — = +0.308 and — = +0.441 003 064 Thus, for example, a 0.1*106 m 3 increase in the value of b4 (which falls within the range specified by the RHS coefficient ranging output) would result in an increase in A* from 0.8250 to 0.8691. Additionally, this increase in 6 4 , and consequent increase in A* would also yield a higher value of the goal. The corresponding change in total benefits would be from Z$5.8125 * 106 to Z$5.8346 * 106. The dual solution variable dg corresponding to the fuzzy goal constraint has a value of -1.7794 (after the sign is reversed for a > constraint). Thus, for example, a Z$0.l * 106 increase in bg will result in a reduced A" of 0.6472. With this value of A' and the revised membership function for the goal constraint, the corresponding value of the goal is now Z$5.8236* 106. This represents an INCREASE of Z$0.0111*: 106 benefits over the original fuzzy solution of Z$5.8125 * 106. •-46 Chapter 6 C onclusions It is recognised that in most mathematical programming models that handle real world problems, such as the ones experienced in the field of water resources management, some degree of imprecision is encountered when specifying parame-ters. The realisation of the presence of such imprecision has lead to the formulation of fuzzy programming models. The two-reservoir problem of Henry Hallam and Prince Edward, presented in this thesis closely resembles a real-world situation where the water resources manager could be faced with the problem of deciding "how much more shall I give the consumers ?" or "by how much shall I cut their allocations without affecting their living standards and their commercial activities ?". In both cases, he will also be required to maintain adequate live storages at all times in order to maintain a healthy fishery and keep the weeds flooded. The conventional L.P. model was able to produce results that approximate the real world Henry Hallam and Prince Edward reservoir situation, i.e., the various consumption figures by the periods, the outflows, inflows, the varying live storage volumes, evaporation, etc. However, this model is characterised by a crisply or precisely defined mathematical model. The main objective of the thesis was to re-formulate the problem in a non-crisp, or fuzzy form and then consider, for example, if larger monetary benefits could be obtained from the system, i.e., over and above the conventional L.P. model's optimal benefit of Z$5.749025 * 106. The other objec-47 tive was to determine how high and how low water allocations and outflows could be increased to or be decreased to, while still maintaining a healthy fishery and weed control. This was readily accomplished by the application of fuzzy mathematical pro-gramming. Essentially, the approach involved fuzzification of the original conven-tional L.P.'s objective function and selected water consumption and outflow con-straints. The result was an equivalent fuzzy goal and fuzzy constraints from which a fuzzy solution was to be derived. The F.L.P. problem was then translated to an equivalent MAXMINform and finally transformed into a conventional L.P. problem by adopting a new variable A which was to be maximised. The formulation developed here for the two reservoir case in Zimbabwe demon-strates the use of fuzzy constraints in connection with discharge and abstraction quantities which are of prime importance in resource management. Thus it demon-strates that the concept of fuzziness can be successfully applied to quantities which are normally designated as decision variables as well as more conventional con-straining factors such as reservoir levels, etc.. It should be noted that quite often in practice, constraints on such factors as reservoir storage may be more appropriately represented as crisp constraints even in a F.L.P. formulation. An outstanding feature of F.L.P. is that, the decision maker is no longer com-pelled to state the constraints in exact terms, as is required when using conventional L.P.. The admission of imprecision into problem formulation is valuable in situa-tions where boundaries are not sharply defined, but only exist as boundary regions. In the two-reservoir problem looked at in the preceeding chapters, the benefit goal as well as the constraints on domestic and industrial water demands were described in terms of desirable upper and lower limits. For example, for the domestic sup-plies in period 4 (PAD4), the upper limit is 2.200 * 106 m 3 and the lower limit is 1.200 * 106 m 3 . In this particular case, the upper bound is favoured since the aim 48 is to determine if larger water allocations are possible. A knowledge of linear programming is essential to developing, solving and in-terpreting the results of an F.L.P. approach to a practical resource management problem. An appreciation of the fuzzy aspects and their manifestation as individ-ual membership functions does not require an understanding of programming. The simple graphic form of the membership functions, as presented in Chapter 4, can easily be understood by interested parties with very little explanation. The nego-tiability of the quantities represented in fuzzy constraints is a more natural idea in resource management than the far less realistic implications of a conventional crisp L.P. constraint. Thus, although F.L.P. might be regarded as a more advanced form of L.P. in some respects, it might present less of a challenge when soliciting input from a lay public and when explaining results to the same groups. Given the linear forms of the fuzzy membership functions as shown in the two-reservoir problem, F.L.P. can be readily converted into a conventional L.P. problem without significantly altering the size of the problem. Therefore, F.L.P. can be easily solved like any other conventional L.P. model. The treatment of an objective which has been transformed into a goal constraint with its own membership function is identical to the treatment of constraints. Although not explored in this thesis, the representation of goals by constraints also opens the opportunity to formulating and solving multiple goal problems. The choice of which parts of the model remain non-fuzzy, which ones are to be fuzzified, and the assumed character of the goal and the fuzzy constraints all rest with the decision maker. The linear form of the membership function V;(x), is questionable. However, one can look at the membership function as a form of arbitrary weighting and hence, side step this criticism. It should be pointed out that the under achievement of a single goal or constraint is capable of having a major impact on the solution, since the attempt is to minimise the maximum 49 under achievement. Fuzzy constraints will often reflect subjective feelings so that this could be a significant problem if any fuzzy constraint membership functions are erroneously specified. Obtaining good information on the aspiration levels and the allowable degradation is vital to the credibility of any results obtained (as is the case with most alternative techniques dealing with uncertainty and inexactness). The monetary benefits realised from the F.L.P. formulation were 1.2% higher than those from the conventional L.P. formulation. Also achieved were higher and lower water allocations, and lower outflows, as intended. In some instances signifi-cant differences in allocations were noted between the conventional and fuzzy L.P. solutions. An acceptable value of A* was also obtained, indicating that the lowest achievement of our goals was adequate. Also computed, through the introduction of equality constraints, were the Vi(x) values which are evaluations of the achievement levels for each of the fuzzy con-straints. The inclusion of these equality constraints is not strictly necessary for F.L.P. solution. The presence of these values in the F.L.P. output was found to be invaluable at all stages of developing an F.L.P. solution. Their non dimensional nature and confinement to the same 0 to 1.0 interval permits their immediate and direct comparison in the F.L.P. output. These constraints did not affect the fuzzy optimal solution. With the F.L.P. model slightly larger in size than the L.P. form, the computa-tional load is only slightly heavier but, the computer package running on A M D A H L mainframe returns solutions virtually instantly. The L.P. computer software pack-age (UBC:LIP) used also gives sensitivity information on the fuzzy as well as the non-fuzzy constraints. Experience with the example developed here suggests that sensitivity information concerning the fuzzy constraints, specifically the member-ship function parameter 6,, is more difficult to interprete than in conventional L.P.. The implications of increasing or decreasing a bi value upon the V,*(x) and A* 50 values are not self evident and yet, an understanding of their implications might be essential to judging whether an increase or a decrease in 6, is appropriate under the circumstances. Furthermore, as was demonstrated for sensitivity of A* to bg in the example, a negative sensitivity to bg can still result in an increase in the net benefits reflected by the value of the fuzzy goal. The alternative F.L.P. formulation proposed by Hamacher, which was also ap-plied to the two-reservoir example here, provided little additional information of practical value. The values of the variables in the output were not found to be as useful as the V{(x) values in the original F.L.P. formulation. It was confirmed that sensitivity of the bi could be obtained from the original F.L.P. formulation. Sensitivity of the di was only provided under the special condition of A" = 0. This would not normally be the optimal result in a case of practical interest so that this special form of di sensitivity might only be useful for diagnostic purposes during the early formulation stages. 51 B ibliography Ackoff, R.L. (editor) - Progress in Operations Research, Vol.1. Pub. by Wiley, New York, 1961. Bellman, R .E . and Zadeh, L .A. - Decision making in a fuzzy environment. In Management Science, Vol.17, No.4, 1970, pp.141 - 164. Chang,L.L. - Interpretation and execution of fuzzy programmes. In Zadeh, L.A. et al., (editors), New York, U.S.A., 1975, PP.191 - 218. Dantzig, G.B. - Linear Programming and Extensions. Pub. by Princeton Univ. Press, Princeton, N.J. , 1963. Goodman, A.S. - Principles of Water Resources Planning. Pub. by Prentice-Hall Inc., Englewood Cliffs, N.J., 1984, pp.355 - 363. Hadley, G. - Linear Programming. Pub. by Addison-Wesley, Reading, Mass., U.S.A., 1962. Haimes, Y. Yacov - Hierarchical Analysis of Water Resources Systems. Pub. by McGraw-Hill Inter. Book Co., New York, U.S.A., 1977, pp.19 - 23. Hall, W.A. and Dracup, J.A. - Water Resources Systems Engineering. Pub. by McGraw-Hill Book Co., New York, 1970, pp.52 - 74. Hamacher, H. , et al., - Sensitivity Analysis in fuzzy linear programming. In Fuzzy Sets and Systems, Vol. 1, 1978, pp.269 - 281. Hillier, F.S. and Lieberman, G.J. - Introduction to Operations Research (2nd Ed.). Pub. by Holden-Day, San Francisco, 1974. Kay, G. , et al., (ed.) - Salisbury: A geographic survey of the capital of Rhode-sia. Pub. by Hodder and Stoughton, London, 1977, pp.1 - 3. Kickert, W.J .M. - Fuzzy theories on decision making, Vol. 3. Pub. by Martinus Nijhoff Soc. Sci. Div., London, 1978, Chapter 2. Loucks, D.P., et al. - Water Resources Systems Planning and analysis. Pub. by Prentice-Hall Inc., Englewood Cliffs, N.J., 1981, pp.44 - 56. Ministry of Water Resources and Development - Hydrological Summaries. Pub. by Zimbabwe Govt. Printers, Harare, 1985, pp.115 - 116. Nelson, H.D., (Ed.) - Zimbabwe: a country study. Pub. by U.S. Govt. Pri. Office, Washington D.C. , 1983, pp.85 - 86. 52 [16] Ostasiewicz, W. - A new approach to fuzzy programming. In Fuzzy Sets and Systems, Vol. 7, 1982, pp.139 - 152. [17] Rodder, W. - On 'and' and 'or' connectives in fuzzy set theory. Working paper 75/07, R W T H Aachen. Presented at E U R O I, Brussels, 1975. [18] Rodder, W., et al. - Analyse, Beschreibung und Optimierung von unscharf formulierten problemen. In Z. Operations Res., Vol. 22, 1977, pp.1 - 18. [19] Sheer, D.P. - A non-technical introduction to linear programming and its use in reservoir operations. In Proceed, of the Nat. Work, on Res. Sys. Oper., 1979, pp.100 - 106. [20] Szidarovszky, F. , et al. - Techniques of multi-objective decision making in systems management. Pub. by Elsevier Sci. Pub. Co. Inc., New York, 1986, Section 8.1. [21] Tanaka, H. , et al. - Fuzzy programmes and their execution [Art. No. 28]. In Zadeh, L .A . , et al., (Ed.) - Fuzzy set and their application to cognitive decision processes, New York, 1975. [22] Thornton, J.A.(ed.) - Lake Mcllwaine. Pub. by Dr. W. Junk Pub., The Hague, 1982, pp.23 - 28. [23] Wagner, H . M . - Principles of Operations Research. Pub. by Prentice-Hall, New York, 1966, p.558. [24] Yeh William W - G . - Reservoir Management and Operations Models: A state-of-the-art review. In Water Resourses Research, Vol. 21, No. 12, Dec. 1985, pp.1797 - 1818. [25] Zimmermann, H.J. - Description and optimisation of fuzzy systems. In Int. J. General Systems, Vol. 2, 1976, pp.209 - 215. [26] Zimmermann, H.J. - Fuzzy programming and linear programming with several objective functions. In Fuzzy Sets and Systems, Vol. 1, 1978, pp.45.- 55. [27] Zimmermann, H.J. , et al., (Ed.) - Fuzzy sets and decision analysis, Vol. 20. Pub. by North-Holland Pub. Co., Amsterdam, 1984, pp.112 - 116. [28] Zimmermann, H.J. - Application of fuzzy set theory to mathematical program-ming. In Information Sciences, Vol. 36, No. 1, 1985, pp.29 - 58. 53 Appendix A Conventional L . P . input. 1 t - Henry Hallam and Prince Edward r e s e r v o i r s problem 2 / These two reservoirs store supplImentary water supplies to Harare. 3 / Henry Hal lam's water 1s released Into Prince Edward for abstraction. 4 c • 142. v = 74, rhs.obj, dual 5 / Monetary benefits from water sales are being maximised. 6 / Sales are to domest1c(pad1 - pad6) and I n d u s t r i a l ( p a l 1 - pal6) 7 / consumers. 8 / Fish benefits are shown by fxlmln to fx6m1n If Prince Edward's 9 / storage volume 1s maintained above 2366 thousand cubic metres. 10 max 11 obj 0.173pad1 0.173pad2 0.173pad3 0.173pad4 0.173pad5 0.173pad6 12 obj 0.248pa11 0.248pa12 0.248pa13 0.248pat4 0.248pa*5 0.248pai6 13 obj -0,Ifxlmln -0.1fx2m1n -0.1fx3m1n -0.1fx4m1n -0.1fx5m1n -0.1fx6m1n 14 / The time span being considered In the model 1s from October 1981 to 15 / end of march 1983. 16 / This time span Is divided Into s i x periods of three months each. 17 / VARIABLES USED. 18 / 1nh1 to 1nh6 are Henry Hal lam's Inflows for the s i x periods, resp.. 19 / ehl to eh6 are Henry Hal lam's respective evaporation f i g u r e s . 20 / Inpl to 1np6 are Prince Edward's Inflows for the s i x periods, resp.. 21 / epl to ep6 are Prince Edward's respective evaporation f i g u r e s . 22 c l h u l 1nh1 < 7.528 23 c l h l 1 1nh1 > 7.522 24 c l p u l Inpl < 0.152 25 d p i 1 1np1 > 0. 147 26 cehul eh1 < 0.669 27 cehl1 eh1 > 0.665 28 cepul epl < 0.245 29 cepl1 epl > 0.242 30 c1hu2 1nh2 < 45.302 31 d h l 2 1nh2 > 45.295 32 C ( p u 2 1np2 < 2.538 33 c1pl2 1np2 > 2.534 34 cehu2 eh2 < 0.544 35 cehl2 eh2 > 0.541 36 cepu2 ep2 < 0.198 37 cepl2 ep2 > 0.194 38 c1hu3 1nh3 < 7.O06 39 d h l 3 1nh3 > 7.002 40 C l p u 3 1np3 < 0.168 41 c i p l 3 1np3 > 0.164 42 cehu3 eh3 < 0.454 43 cehl3 eh3 > 0.451 44 cepu3 ep3 < 0.163 45 cepl3 ep3 > 0.160 46 d h u 4 1nh4 < 3.025 47 d h 1 4 1nh4 > 3.021 48 c1pu4 1np4 < 0.143 49 C l p l 4 1np4 > 0.140 50 cehu4 eh4 < 0.589 51 cehl4 eh4 > 0.586 52 cepu4 ep4 < 0.213 53 c e p l 4 ep4 > 0 . 2 1 0 54 c1hu5 1nh5 < 6.853 55 d h l 5 1nh5 > 6.849 56 dpu5 1np5 < 0.952 57 c t p l 5 1np5 > 0.949 58 cehu5 eh5< 0.679 54 59 ceh15 eh5 > 0.675 60 cepu5 ep5 < 0.245 61 cepl5 ep5 > 0.242 62 C l h u 6 inh6 < 8.105 63 d h l 6 1nh6 > 8. 102 64 dpu6 1np6 < 0.260 65 C l p 1 6 1np6 > 0.257 66 cehu6 eh6 < 0.545 67 cehl6 eh6 > 0.542 68 cepu6 ep6 < 0.197 69 cepl6 ep6 > 0.194 70 / sth1 to sth6 are Henry Hallam's actual stored volumes f o r the 71 / s i x periods, r e s p e c t i v e l y . 72 / s t p l to stp6 are Prince Edward's actual stored volumes for the 73 / s i x periods, r e s p e c t i v e l y . 74 chus s t h l < 4.515 75 chls s t h l > 4.512 76 cpus S tp1 < 2.723 77 cpl s s t p l > 2.720 78 / pad1 to pad6 are domestic water supplies abstracted from Prince 79 / Edward for the six respective periods. 80 cupl padl < 1.850 81 d p i padl > 1.650 82 cup2 pad2 < 1.70O 83 clp2 pad2 > 1.300 84 cup3 pads < 1.800 85 c1p3 pad3 > 1. 2O0 86 cup4 pad4 < 1.900 87 clp4 pad4 > 1.650 88 cup5 pad5 < 1.775 89 clp5 pad5 > 1.20O 90 cup6 pad6 < 1.700 91 clp6 pad6 > 1.lOO 92 / pa11 to pa16 are Industrial water supplies abstracted from Prince 93 / Edward for the six respective periods. 94 cup7 pa 11 < 2.650 95 clp7 p a l l > 2.450 96 cup8 pa 12 < 2.550 97 C l p 8 pa 12 > 2.350 98 cup9 pa 13 < 2.650 99 clp9 pa 13 > 2.450 100 cup 10 pa 14 < 2.650 101 clp10 pa14 > 2.370 102 cup11 pais < 2.650 103 clp11 pa15 > 2.455 104 cup12 pa16 < 2.550 105 C l p 1 2 pa16 > 2.350 106 / po1 to po6 are Prince Edward's respective outflows. 107 cup13 po1 < 0.724 108 C l p 1 3 po1 > 0.721 109 cup14 po2 < 39.500 110 Clp14 po2 > 0 111 cup 15 p63 < 3.330 112 C l p 1 5 po3 > 3.325 113 c u p 16 po4 < 1.800 114 clp16 po4 > 1.795 115 cup17 po5 < 0.200 116 Clp17 po5 > 0.197 55 117 cup 18 poS < 0.250 118 Clp18 po6 > 0.248 119 / hoi to ho6 are Henry Hal 1 am's respective outflows. 120 cuhl ho1 < 5.500 121 c l h l ho1 > 0 122 cuh2 ho2 < 41.752 123 clh2 ho2 > 0 124 cuh3 ho3 < 7.384 125 Clh3 ho3 > 7.20O 126 cuh4 ho4 < 6.50O 127 clh4 ho4 > 6.350 128 cuh5 ho5 < 4.500 129 clh5 ho5 > 0 130 Cuh6 ho6 < 4.440 131 Clh6 ho6 > 2.400 132 / PERIOD ONE 133 / csh2 and csp2 are storage capacity constraints f o r Henry Hallam 134 / and Prince Edward, re s p e c t i v e l y . 135 csh2 s t h l 1nh1 - ho1 - eh1 < 9.026 136 csp2 s t p l 1np1 ho1 - po1 - padl - pa11 - epl < 3.380 137 / cch1 and ccpl are co n t i n u i t y constraints for Henry Hallam and Prince 138 / Edward , res p e c t i v e l y . 139 cch1 s t h l 1nh1 - ho1 - ehl > 0 140 ccpl s t p l 1np1 ho1 - po1 - padl - p a l l - ep1 > 0 141 / csth2 and cstp2 are storage capacity c a l c u l a t i o n s at the end of 142 / period one which 1s also the beginning of period two. 143 csth2 s t h l tnhl - ho1 - eh1 - sth2 = 0 144 cstp2 s t p l 1np1 ho1 - po1 - padl - p a l l - ep1 - stp2 = 0 145 / c f p l Is a f i s h b e n e f i t constraint for Prince Edward. It w i l l show us 146 / to have p o s i t i v e f i s h b enefits 1f stored volume 1n the respective 147 / period 1s above 2366 , or negative benefits i f volume 1s below 2366. 148 c f p l fxlmln - f x l p l u s 0.5stp1 0.5stp2 = 2.366 149 / PERIOD TWO 150 / csh3 and csp3 are storage capacity constraints for H.H and P.E., 151 / res p e c t i v e l y . 152 csh3 sth2 1nh2 - ho2 - eh2 < 9.026 153 csp3 stp2 1np2 ho2 - po2 - pad2 - pa12 - ep2 < 3.380 154 / cch2 and ccp2 are co n t i n u i t y constraints for H.H. and P.E. , resp.. 155 cch2 Sth2 1nh2 - ho2 - eh2 > 0 156 ccp2 stp2 1np2 ho2-- po2 - pad2 - pa12 - ep2 > 0 157 / csth3 and cstpS are storage capacity c a l c u l a t i o n s for H.H. and P.E. 158 / , respectively. 159 csth3 sth2 1nh2 - ho2 - eh2 - Sth3 = 0 160 cstp3 stp2 1np2 ho2 - po2 - pad2 - pa12 - ep2 - stp3 = 0 161 /cfp2 Is a f t s h benefit constraint for P.E. 162 cfp2 fx2m1n - fx2plus 0.5stp2 0.5stp3 = 2.366 163 / PERIOD THREE 164 / csh4 and csp4 are storage capacity constraints for H.H. and P.E. , 165 / res p e c t i v e l y . 166 csh4 Sth3 1nh3 -ho3 - eh3 < 9.026 167 csp4 stp3 1np3 ho3 - po3 - pad3 - pa 13 - ep3 < 3.380 168 / cch3 and ccp3 are c o n t i n u i t y constraints for H.H. and P.E. , resp.. 169 cch3 sth3 inh3 - ho3 -eh3 > 0 170 ccp3 stp3 1np3 ho3 - po3 - pad3 - pa13 - ep3 > 0 171 / csth4 and cstp4 are storage capacity c a l c u l a t i o n s for H.H. and P.E., 172 / respectively. 173 csth4 sth3 inh3 - ho3 - eh3 - sth4 = 0 174 cstp4 stp3 1np3 ho3 - po3 - pad3 - pa13 - ep3 - stp4 = 0 56 175 / cfp3 is f i s h benefit constraint for P.E. 176 Cfp3 fx3min - fx3plus 0.5stp3 0.5stp4 • 2.366 177 / PERIOD FOUR 178 / csh5 and csp5 are storage capacity c o n s t r a i n t s for H.H. and P.E. 179 / respectively. 180 csh5 sth4 1nh4 - ho4 - eh4 < 9.026 181 csp5 stp4 1np4 ho4 - po4 - pad4 - pa 14 - ep4 < 3.380 182 / cch4 and ccp4 are co n t i n u i t y c o n s t r a i n t s for H.H. and P . E. , resp 183 cch4 sth4 1nh4 - ho4 - eh4 > 0 184 ccp4 stp4 1np4 ho4 - po4 - pad4 - pa 14 - ep4 > 0 185 / csthS and cstp5 are storage capacity c a l c u l a t i o n s f o r H .H. & P.E. 186 / respectively. 187 csth5 sth4 1nh4 - ho4 - eh4 - sth5 = 0 188 cstp5 stp4 1np4 ho4 - po4 - pad4 - pa14 - ep4 - stp5 = 0 189 / cfp4 Is a f i s h benefit constraint f o r P.E. 190 cfp4 fx4m1n - fx4plus 0.5stp4 0.5stp5 « 2.366 191 / PERIOD FIVE 192 / csh6 and csp6 are storage capacity c o n s t r a i n t s f o r H.H. & P.E. . 193 / re s p e c t i v e l y . 194 csh6 sth5 1nh5 - ho5 - eh5 < 9.026 195 csp6 Stp5 1np5 ho5 - po5 - pad5 - pa 15 - ep5 < 3.380 196 / cch5 and ccp5 are co n t i n u i t y c o n s t r a i n t s for H.H. & P.E resp.. 197 cch5 sth5 1nh5 - ho5 - eh5 > 0 198 ccp5 stp5 1np5 ho5 - po5 - pads - pa 15 - ep5 > 0 199 / csth6 and cstp6 are storage capacity c a l c u l a t i o n s for H .H. & P.E. 200 / respectively. 201 csth6 Sth5 1nh5 - ho5 - eh5 - sth6 « 0 202 cstp6 stp5 1np5 ho5 - po5 - pad5 - pa 15 - ep5 - stp6 • 0 203 / cfp5 Is a f i s h benefit constraint f o r P.E. 204 cfp5 fx5m1n - fx5plus 0.5stp5 0.5stp6 = 2.366 205 / PERIOD SIX 206 / csh7 and csp7 are are storage capacity constraints f o r H.H . & P.E 207 / re s p e c t i v e l y . 208 csh7 sth6 1nh6 - ho6 - eh6 < 9.026 209 csp7 stp6 1np6 ho6 - po6 - pad6 - pa 16 - ep6 < 3.380 210 / cch6 and ccp6 are co n t i n u i t y c o n s t r a i n t s for H.H. & P.E . , resp.. 211 cch6 sth6 1nh6 - ho6 - eh6 > 0 212 ccp6 stp6 1np6 ho6 - po6 - pad6 - pa 16 - ep6 > 0 213 / csth7 and cstp7 are storage capacity c a l c u l a t i o n s for H .H. & P.E. 214 / respectively. 215 csth7 sth6 inh6 - ho6 - eh6 - sth7 = 0 216 cstp7 stp6 1np6 ho6 - po6 - pad6 - pa 16 - ep6 - stp7 = 0 217 / cfp6 Is a f i s h benefit constraint f o r P.E. 218 Cfp6 fx6min - fx6plus 0.5stp6 0.5stp7 = 2.366 57 Appendix B Fuzzy L.P. input. 1 t = Henry Hallam and Prince Edward r e s e r v o i r s problem - FLP 2 / This Is the fuzzy version of the o r i g i n a l LP formulation - i . e . -3 / DAMS 1.DAT . This 1s also the s i n g l e fuzzy formulation . 4 c « 150, v = 82, rhs, obj, d u a l 5 / The goal function i s being maximised while selected varlables'values 6 / are e i t h e r being Increased or decreased without adverse e f f e c t s 7 / o n r e l a t e d v a r i a b l e s , or o v e r a l l perfomance of the whole system. 8 / The objective 1s to maximise lamda - x1 -. This Is the minimum 9 / of the VI that i s to be maximised. In other words we are maximising 10 / the minimum p a r t i a l v i o l a t i o n of a constraint or f a i l u r e to meet a 11 / goal. 12 / The equality constraints - VI - are a measure of the values' 13 / p o s i t i o n between 0 and 1.... 14 max 15 obj x1 16 / The time span being considered 1n the model Is from October 1981 to 17 / end of march 1983. 18 / This time span Is divided Into six periods of three months each. 19 / VARIABLES USED. 20 / 1nh1 to 1nh6 are Henry Hal lam's Inflows f o r the s i x periods, resp.. 21 / ehl to eh6 are Henry Hal lam's respective evaporation f i g u r e s . 22 / 1np1 to 1np6 are Prince Edward's Inflows for the s i x periods, resp.. 23 / epl to ep6 are Prince Edward's respective evaporation f i g u r e s . 24 / ALL FIGURES ARE MILLIONS e.g. 7.525 m i l l i o n cubic metres of water. 25 c1hu1 1nh1 < 7.528 26 d h l 1 1nh1 > 7.522 27 c1pu1 1np1 < 0.152 28 d p i 1 1np1 > 0. 147 29 c e h u l eh1 < 0.669 30 c e h l 1 eh1 > 0.665 31 cepul epl < 0.245 32 cepl 1 epl > 0.242 33 C l h u 2 1nh2 < 45.302 34 C l h l 2 1nh2 > 45.295 35 C l p u 2 1np2 < 2.538 36 dp12 1np2 > 2.534 37 cehu2 eh2 < 0.544 38 c e h l 2 eh2 > 0.541 39 c epu2 ep2 < 0. 198 40 c e p l2 ep2 > 0.194 41 C l h u 3 1nh3 < 7.006 42 c i h l 3 1nh3 > 7.002 43 C l p u 3 1np3 < 0.168 44 d p l 3 1np3 > 0. 164 45 cehu3 eh3 < 0.454 46 cehl3 eh3 > 0.451 47 cepu3 ep3 < 0.163 48 cepl3 ep3 > 0.160 49 C l h u 4 1nh4 < 3.025 50 c i h l 4 1nh4 > 3.021 51 Cipu4 1np4 < 0.143 52 d p l 4 1np4 > 0. 140 53 cehu4 eh4 < 0.589 54 cehl4 eh4 > 0.586 55 cepu4 ep4 < 0.213 56 cepl4 ep4 > 0.210 57 dhu5 1nh5 < 6.853 58 c l h l S inh5 > 6.849 58 59 C l p u 5 1np5 < 0 . 9 5 2 60 C l p 1 5 1np5 > 0 . 9 4 9 61 cehu5 eh5 < 0 . 6 7 9 62 c e h l 5 eh5 > 0 . 6 7 5 63 cepuS ep5 < 0 . 2 4 5 64 c e p l 5 ep5 > 0 . 2 4 2 65 c1hu6 1nh6 < 8 . 105 66 c 1 h ! 6 1nh6 > 8 .102 67 c1pu6 1np6 < 0 . 2 6 0 68 C l p 1 6 1np6 > 0 . 2 5 7 69 cehu6 eh6 < 0 . 5 4 5 70 c e h l 6 eh6 > 0 . 5 4 2 71 cepu6 ep6 < 0 . 1 9 7 72 c e p l 6 ep6 > 0 . 1 9 4 73 / s t h l t o s t h 6 a r e H e n r y H a l l a m ' s a c t u a l s t o r e d v o l u m e s f o r t h e 74 / s i x p e r i o d s , r e s p e c t i v e l y . 75 / s t p l t o s t p 6 a r e P r i n c e E d w a r d ' s a c t u a l s t o r e d v o l u m e s f o r t h e 76 / s i x p e r i o d s , r e s p e c t i v e l y . 77 chus s t h l < 4 . 5 1 5 78 c h l s s t h l > 4 . 5 1 2 79 c p u s s t p l < 2 . 7 2 3 8 0 c p l s s t p l > 2 . 7 2 0 81 / pad1 t o pad6 a r e d o m e s t i c w a t e r s u p p l i e s a b s t r a c t e d f rom P r i n c e 82 / Edward f o r t h e s i x r e s p e c t i v e p e r i o d s . 83 c u p l pad1 < 1 . 8 5 0 84 d p i pad1 > 1 . 650 85 cup2 pad2 < 1 . 700 86 C l p 2 pad2 > 1 . 300 87 cup3 pad3 < 1.800 88 c1p3 pad3 > 1 . 2 0 0 89 cup4 pad4 < 2 . 2 0 0 90 Cfpu4 pad4 - 0 . 3 7 0 x 1 > 1.830 / * * * * » f u z z y c o n s t r a i n t - pad4 *** 91 C f p v l pad4 - 0 . 3 7 0 V 1 = 1 .830 / « » * * « e q u a l i t y c o n s t r a i n t - v1 ** 92 c f p u 5 pad5 0 . 0 8 0 x 1 < 1.780 / * • « • • f u z z y c o n s t r a i n t - pad5 *** 93 c f p v 3 pad5 0 . 0 8 0 v 3 = 1.780 / * * « « « e q u a l i t y c o n s t r a i n t - v3 ** 94 C l p 5 pad5 > 1 . 7 0 0 95 cup6 pad6 < 1 . 700 96 C l p 6 pad6 > 1 . 100 97 / p a l l t o pa16 a r e I n d u s t r i a l w a t e r s u p p l i e s a b s t r a c t e d f rom P r i n c e 98 / Edward f o r t h e s i x r e s p e c t i v e p e r i o d s . 99 cup7 pa 11 < 2 . 6 5 0 lOO c l p 7 p a l l > 2 . 4 5 0 101 cup8 pa 12 < 2 . 5 5 0 102 C l p 8 pa12 > 2 . 3 5 0 103 cup9 pa 13 < 2 . 6 5 0 104 c l p 9 pa13 > 2 . 4 5 0 105 cup10 pa14 < 2.900 106 c f l p t O pa<4 - 0 . 3 2 0 x 1 > 2 . 5 8 0 / * * « « * f u z z y c o n s t r a i n t - p a i 4 • * * 107 c f p v 2 pa 14 - 0 . 3 2 0 V 2 = 2 . 5 8 0 / * « * * * e q u a l i t y c o n s t r a i n t - v2 ** 108 c fup11 pa 15 0 . 196x1 < 2 . 6 5 1 / * * » » » f u z z y c o n s t r a i n t - pa15 *** 109 c f p v 4 pa 15 0 . 1 9 6 v 4 = 2 . 6 5 1 / « « » * « e q u a l i t y c o n s t r a i n t - v4 ** 110 c1p11 pa 15 > 2 . 4 5 5 111 cup 12 pa 16 < 2 . 5 5 0 1 12 c l p 1 2 pa 16 > 2 . 3 5 0 113 / po1 t o po6 a r e P r i n c e Edward ' s 3 r e s p e c t i v e o u t f l o w s . 114 cup 13 po1 < 0 . 7 2 4 1 15 c l p 1 3 pot > 0 . 7 2 1 116 cup 14 po2 < 39.500 59 117 C l p 1 4 po2 > 3 9 . 2 5 0 118 cup 15 po3 < 3 . 3 3 0 1 19 C l p 1 5 po3 > 3 . 3 2 5 120 cup 16 po4 < 1.800 121 C l p 1 6 po4 > 1 . 795 122 cup 17 po5 < 0 . 2 0 0 123 C l p 1 7 po5 > 0 . 1 9 7 124 cup 18 po6 < 0 . 2 5 0 125 C l p 1 8 po6 > 0 . 2 4 8 126 / h o i t o ho6 a r e H e n r y H a l l a m ' s r e s p e c t i v e o u t f l o w s . 127 c u h l ho1 < 5 . 5 0 0 128 c l h l ho1 > 0 129 c f u h 2 ho2 0 . 1 7 5 x 1 < 4 1 . 7 6 0 / « • » * » f u z z y c o n s t r a i n t - ho2 * * * * * 130 c f p v 5 ho2 0 . 1 7 5 v 5 = 4 1 . 7 6 0 / * » * * * e q u a l i t y c o n s t r a i n t - v5 * * * 131 C l h 2 ho2 > 4 1 . 5 8 5 132 cuh3 ho3 < 7 . 3 8 4 133 C l h 3 ho3 > 7 . 2 0 0 134 c f hu4 ho4 0 . 0 9 0 x 1 < 6 . 5 1 0 / * * * * • f u z z y c o n s t r a i n t - ho4 *** 135 c f pv6 ho4 0 . 0 9 0 V 6 » 6 . 5 1 0 / * * * « * e q u a l i t y c o n s t r a i n t - v6 ** 136 c l h 4 ho4 > 6 . 4 2 0 137 cuh5 ho5 < 4 . 5 0 0 138 c l h 5 ho5 > 0 139 cuh6 ho6 < 4 . 4 3 7 140 c l h 6 ho6 > 2 . 4 0 0 141 /**•**************************************************************** 142 / * * * * * * c f g l and c f g 2 a r e f u z z y g o a l c o n s t r a i n t s - I . e . - t h e « * « « « 143 / * * * * * * o r i g i n a l o b j e c t i v e f u n c t i o n has been t u r n e d I n t o a • » » * * 144 / « * * * * • c o n s t r a i n t so t h a t t h e o v e r a l l b e n e f i t s c a n be » « * * * 145 / * * * * * * m a x i m i s e d . . . . » » « * * 146 c f g l 0 .173pad1 0 . 1 7 3 p a d 2 0 . 1 7 3 p a d 3 0 .173pad4 0 . 1 7 3 p a d 5 147 c f g l 0 . 173pad6 0 .248pa11 0 . 2 4 8 p a 1 2 0 . 2 4 8 p a i 3 0 . 2 4 8 p a 1 4 148 c f g l 0 . 2 4 8 p a 1 5 0 . 2 4 8 p a i 6 - 0 . 1 f x 1 m 1 n - 0 . 1 f x 2 m 1 n - O . 1 f x 3 m 1 n 149 c f g l - 0 . 1 f x 4 m 1 n - 0 . 1 f x 5 m 1 n - 0 . 1 f x 6 m 1 n - g o a l = O 150 c f g 2 g o a l - 0 . 5 0 0 x 1 > 5 . 4 0 0 151 / • . f t * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 152 / f t * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 153 / PERIOD ONE 154 / c s h 2 and c s p 2 a r e s t o r a g e c a p a c i t y c o n s t r a i n t s f o r H e n r y H a l l a m 155 / and P r i n c e E d w a r d , r e s p e c t i v e l y . 156 c sh2 s t h l 1nh1 - ho1 - eh1 < 9 . 0 2 6 157 c s p 2 s t p l 1np1 ho1 - po1 - p a d l - pa11 - e p l < 3 . 3 8 0 158 / c e h l and c c p l a r e c o n t i n u i t y c o n s t r a i n t s f o r H e n r y H a l l a m and P r i n c e 159 / Edward , r e s p e c t i v e l y . 160 c e h l s t h l i n h l - ho1 - eh1 > 0 161 c c p l s t p l 1np1 ho1 - p o l - p a d l - p a i l - e p l > 0 162 / c s t h 2 and c s t p 2 a r e s t o r a g e c a p a c i t y c a l c u l a t i o n s a t t h e e n d o f 163 / p e r i o d one w h i c h i s a l s o t h e b e g i n n i n g o f p e r i o d t w o . 164 c s t h 2 s t h l 1nh1 - ho1 - eh1 - s t h 2 = 0 165 c s t p 2 s t p l 1np1 ho1 - po1 - p a d l - p a i l - e p l - s t p 2 = 0 166 / c f p l 1s a f i s h b e n e f i t c o n s t r a i n t f o r P r i n c e E d w a r d . I t w i l l show us 167 / t o have p o s i t i v e f i s h b e n e f i t s i f s t o r e d vo lume i n t h e r e s p e c t i v e 168 / p e r i o d 1s above 2366 . o r n e g a t i v e b e n e f i t s i f v o l u m e I s b e l o w 2 3 6 6 . 169 c f p l f x l m i n - f x l p l u s 0 . 5 s t p 1 0 . 5 s t p 2 = 2 . 3 6 6 170 / PERIOD TWO 171 / c s h 3 and c s p 3 a r e s t o r a g e c a p a c i t y c o n s t r a i n t s f o r H . H a n d P . E . , 172 / r e s p e c t i v e l y . 173 c s h 3 s t h 2 1nh2 - ho2 - eh2 < 9 . 0 2 6 174 c sp3 s t p 2 1np2 ho2 - po2 - pad2 - pa12 - ep2 < 3 . 3 8 0 60 175 / cch2 and ccp2 are c o n t i n u i t y constraints for H.H. and P, . E . , resp. 176 cch2 sth2 inh2 - ho2 - eh2 > 0 177 ccp2 stp2 1np2 ho2 - po2 - pad2 - pa 12 - ep2 > 0 178 / csth3 and cstpS are storage capacity c a l c u l a t i o n s for H. .H. and P.E 179 / , re s p e c t i v e l y . 180 csth3 sth2 1nh2 - ho2 - eh2 - sth3 = 0 181 cstp3 stp2 1np2 ho2 - po2 - pad2 - pa12 - ep2 • - stp3 = 0 182 /cfp2 Is a f i s h benefit constraint for P.E. 183 cfp2 fx2m1n - fx2plus 0.5stp2 0.5stp3 = 2.366 184 / PERIOD THREE 185 / csh4 and csp4 are storage capacity constraints for H.H. and P.E. . 186 / r e s p e c t i v e l y . 187 csh4 sth3 1nh3 -ho3 - eh3 < 9.026 188 csp4 stp3 1np3 ho3 - po3 - pad3 - pa 13 - ep3 < 3.380 189 / cch3 and ccp3 are con t i n u i t y constraints f o r H.H. and P. . E. . resp. 190 cch3 sth3 1nh3 - ho3 -eh3 > 0 191 ccp3 stp3 1np3 ho3 - po3 - pad3 - pa 13 - ep3 > 0 192 / csth4 and cstp4 are storage capacity c a l c u l a t i o n s f o r H. H. and P.E 193 / r e s p e c t i v e l y . 194 csth4 Sth3 1nh3 - ho3 - eh3 - sth4 = 0 195 cstp4 stp3 1np3 ho3 - po3 - pad3 - pa 13 - ep3 • - Stp4 = 0 196 / cfp3 1s f i s h benefit constraint for P.E. 197 Cfp3 fx3m1n - fxSplus 0.5stp3 0.5stp4 = 2.366 198 / PERIOD FOUR 199 / csh5 and csp5 are storage capacity constraints f o r H.H. and P.E. , 200 / r e s p e c t i v e l y . 201 csh5 sth4 1nh4 - ho4 - eh4 < 9.026 202 csp5 Stp4 1np4 ho4 - po4 - pad4 - pa 14 - ep4 < 3.380 203 / cch4 and ccp4 are con t i n u i t y constraints for H.H. and P. E. , resp. 204 cch4 sth4 tnh4 - ho4 - eh4 > 0 205 ccp4 stp4 1np4 ho4 - po4 - pad4 - pa 14 - ep4 > 0 206 / csth5 and cstp5 are storage capacity c a l c u l a t i o n s f o r H. H. & P.E. 207 / res p e c t i v e l y . 208 csth5 Sth4 1nh4 - ho4 - eh4 - sth5 = 0 209 cstp5 Stp4 inp4 ho4 - po4 - pad4 - pa14 - ep4 -- Stp5 = 0 210 / cfp4 1s a f i s h benefit constraint for P.E. 211 Cfp4 fx4m1n - fx4plus 0.5stp4 0.5stp5 = 2.366 212 / PERIOD FIVE 213 / csh6 and csp6 are storage capacity constraints f o r H.H. & P.E. , 214 / res p e c t i v e l y . 215 csh6 sth5 1nh5 - ho5 - eh5 < 9.026 216 csp6 Stp5 1np5 ho5 - po5 - pad5 - pa15 - ep5 < 3.380 217 / cch5 and ccpS are con t i n u i t y constraints for H.H. & P.E. resp.. 218 cch5 sth5 1nh5 - ho5 - eh5 > 0 219 ccp5 stp5 1np5 ho5 - po5 - pad5 - pa 15 - ep5 > 0 220 / csth6 and cstp6 are storage capacity c a l c u l a t i o n s for H. H. a P.E. 221 / res p e c t i v e l y . 222 csth6 sth5 1nh5 - ho5 - eh5 - sth6 = 0 223 cstp6 stpS 1np5 ho5 - po5 - pad5 - pa15 - ep5 -- Stp6 = 0 224 / Cfp5 1s a f i s h benefit constraint f o r P.E. 225 C f p 5 fx5min - fx5plus 0.5stp5 0.5stp6 = 2.366 226 / PERIOD SIX 227 / csh7 and csp7 are are storage capacity constraints for H.H . a P.E. 228 / res p e c t i v e l y . 229 csh7 sth6 1nh6 - ho6 - eh6 < 9.026 230 csp7 stp6 1np6 ho6 - po6 - pad6 - pa16 - ep6 < 3.380 231 / cch6 and ccp6 are c o n t i n u i t y constraints for H.H. a P.E. resp.. 232 cch6 sth6 1nh6 - ho6 - eh6 > 0 61 233 ccp6 stpG 1np6 ho6 - poG - padS - pa 16 - ep6 > 0 234 / csth7 and cstp7 are storage capacity c a l c u l a t i o n s for H.H. & P.E. , 235 / res p e c t i v e l y . 236 csth7 sth6 1nh6 - ho6 - eh6 - sth7 = 0 237 cstp7 stp6 1np6 ho6 - po6 - pad6 - pa 16 - ep6 - stp7 • 0 238 / cfp6 (s a f i s h benefit c o n s t r a i n t for P.E. 239 Cfp6 fx6m1n - fx6plus 0.5stp6 0.5stp7 - 2.366 62 Append i x C Al te rnat ive Fuzzy L.P. input. 1 t = Henry Hallam and Prince Edward re s e r v o i r s problem - FLP(P) 2 / This Is the fuzzy version of the o r i g i n a l LP formulation - I.e. -3 / DAMS 1.DAT. This 1s the paired fuzzy formulation using BI and D1 . 4 c = 150, v • 88. rhs, obj. dual, t o l = 1.OE-8 5 / The goal function Is being maximised while selected v a r i a b l e s f o r 6 / f u z z l f I c a t I o n are e i t h e r being Increased or decreased without 7 / adverse e f f e c t s on the operation of the whole system. 8 / N.B.-- This formulation should give same r e s u l t s as the s i n g l e 9 / fuzzy formulation - I.e. - DAMS2.DAT . 10 / Also Included are equality constraints for each of the v a r i a b l e s 11 / being fuzz i f led. These constraints c a l c u l a t e the po s i t i o n s of 12 / each of the VI - 1.e. - between 0 and 1 ... 13 / The objective Is to maximise lambda - x l . This Is the minimum 14 / of the VI that i s to be maximised. 15 / The t1 va r i a b l e gives a measure of the v i o l a t i o n of the respective 16 / constraints that have been f u z z l f i e d . 17 max 18 obj x1 19 / The time span being considered in the model i s from October 1981 to 20 / end of march 1983. 21 / This time span 1s divided Into s ix periods of three months each. 22 / VARIABLES USED. 23 / mhl to 1nh6 are Henry Hal lam's Inflows for the s i x periods, resp.. 24 / eh1 to eh6 are Henry Hal lam's respective evaporation f i g u r e s . 25 / mp1 to 1np6 are Prince Edward's inflows for the s i x periods, resp.. 26 / epl to ep6 are Prince Edward's respective evaporation f i g u r e s . 27 / ALL FIGURES ARE IN MILLIONS e.g. 7.525 m i l l i o n cubic metres of 28 / water. 29 d h u l 1nh1 < 7. .528 30 c l h l 1 1nh1 > 7. 522 31 c i p u l inpl < 0. 152 32 d p i 1 Inpl > 0. . 147 33 cehul eh1 < O. 669 34 c e h l 1 ehl > 0. 665 35 cepul epl < 0. ,245 36 c e p l 1 epl > 0. 242 37 Clhu2 1nh2x < 45.302 38 d h l 2 1nh2 > 45.295 39 dpu2 1np2 < 2. .538 40 c i p 1 2 1np2 > 2. .534 41 cehu2 eh2 < 0. .544 42 cehl 2 eh2 > 0. .541 43 cepu2 ep2 < 0. 198 44 cepl 2 ep2 > 0. 194 45 cihu3 1nh3 < 7. 006 46 clh13 1nh3 > 7. 002 47 cipu3 1np3 < 0 168 48 d p l 3 1np3 > 0. 164 49 cehu3 eh3 < 0. ,454 50 cehl 3 eh3 > O. 451 51 cepu3 ep3 < 0. 163 52 c e p l 3 ep3 > 0. , 160 53 dhu4 1nh4 < 3 025 54 c i h l 4 1nh4 > 3. ,021 55 C1pu4 1np4 < 0. , 143 56 C i p l 4 1np4 > 0. . 140 57 cehu4 eh4 < 0. .589 58 c e h l 4 eh4 > 0. .586 63 59 cepu4 ep4 < 0.213 60 cepl4 ep4 > 0.210 61 clhuS 1nh5 < 6.853 62 c1h15 1nh5 > 6.849 63 c1pu5 1np5 < 0.952 64 c1p15 1np5 > 0.949 65 cehu5 eh5 < 0.679 66 cehl5 eh5 > 0.675 67 cepu5 ep5 < 0.245 68 cepl5 ep5 > 0.242 69 C l h u 6 1nh6 < 8.105 70 dh16 1nh6 > 8. 102 71 c1pu6 1np6 < 0.260 72 c1pl6 1np6 > 0.257 73 cehu6 eh6 < 0.545 74 cehl6 eh6 > 0.542 75 cepu6 ep6 < 0.197 76 cepl6 ep6 > 0.194 77 / s t h l to sth6 are Henry Hal lam's actual stored volumes f o r the 78 / s i x periods, r e s p e c t i v e l y . 79 / s t p l to stp6 are Prince Edward's actual stored volumes f o r the 80 / six periods, r e s p e c t i v e l y . 81 chus s t h l < 4.515 82 c h l s s t h l > 4.512 83 cpus s t p l < 2.723 84 cp l s S t p l > 2.720 85 / padl to pad6 are domestic water supplies abstracted from Prince 86 / Edward f o r the s i x respective periods. 87 cupl padl < 1.850 88 c l p l padl > 1.650 89 cup2 pad2 < 1.700 90 C l p 2 pad2 > 1.300 91 cupS pad3 < 1.800 92 clp3 pad3 > 1.200 93 / »«*»**»**»*«*»*«* pad4 and pad5 ***«********««***»*»*«»»»«** 94 c f p v l pad4 - 0.370V1 = 1.830 /«***» fuzzy e q u a l i t y 95 c f s a l 0.370x1 t i < 0.370 /«**« p a d 4 - d1 - s e n s i t i v i t y 96 c f s b l pad4 t1 < 2.200 /**** p a d 4 - b1 - s e n s i t i v i t y 97 cfpv3 pad5 0.080v3 = 1.780 /**»** fuzzy e q u a l i t y 98 cfsaS 0.080x1 t3 < 0.080 /****« pad5 - d3 - s e n s i t i v i t y 99 cfsb3 pad5 - t3 < 1.700 /«**** p a d 5 - b3 - s e n s i t i v i t y 100 / **************************************************************** 101 cup6 pad6 < 1.700 102 clp6 pad6 > 1.100 103 / p a l l to pa16 are Industrial water supplies abstracted from Prince 104 / Edward for the six respective periods. 105 cup7 p a l l < 2.650 106 d p 7 p a l l > 2.450 107 cup8 pa 12 < 2.550 108 clp8 pa12 > 2.350 109 cup9 pa 13 < 2.650 110 c1p9 pa13 > 2.450 111 / ****************** pa14 and pa15 ************************ 112 cfpv2 pa14 - 0.320v2 » 2.580 /***** fuzzy e q u a l i t y 113 cfsa2 0.320x1 t2 < 0.320 /***»* p a 1 4 - d 2 - s e n s i t i v i t y 114 cfsb2 pa14 t2 < 2.900 /*•*** p a i 4 - b2 - s e n s i t i v i t y 115 cfpv4 pa15 0.196v4 = 2.651 /***** fuzzy e q u a l i t y 116 cfsa4 0.196x1 t4 < 0.196 /***** p ais - d4 - s e n s i t i v i t y 64 117 c f s b 4 pa1S - t 4 < 2 . 4 5 5 / « * . « * p a i 5 - b4 - s e n s i t i v i t y 1-13 / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 119 cup 12 pa16 < 2 . 5 5 0 120 C l p 1 2 pa16 > 2 . 3 5 0 121 / po1 t o po6 a r e P r i n c e E d w a r d ' s r e s p e c t i v e o u t f l o w s . 122 c u p I S p o l < 0 . 7 2 4 123 c l p 1 3 p o l > 0 . 7 2 1 124 cup 14 po2 < 3 9 . 5 0 O 125 C l p 1 4 po2 > 3 9 . 2 5 0 126 cup 15 po3 < 3 . 3 3 0 127 c l p 1 5 po3 > 3 . 3 2 5 128 cup16 po4 < 1.80O 129 C l p 1 6 po4 > 1.795 130 cup17 po5 < 0 . 2 0 0 131 c l p 1 7 po5 > 0 . 1 9 7 132 cup18 po6 < 0 . 2 5 0 133 C l p 1 8 po6 > 0 . 2 4 8 134 / ho1 t o ho6 a r e Henry H a l l a m ' s r e s p e c t i v e o u t f l o w s . 135 c u h l ho1 < 5 . 5 0 0 136 c l h l h o i > 0 137 / * * * * * * * * * * * * * * * * * * * * * * ^o2 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 138 C f p v 5 ho2 0 . 1 7 5 V 5 = 4 1 . 7 6 0 / * * * * * * * * f u z z y e q u a l i t y 139 C f s a 5 0 . 1 7 5 x 1 t 5 < 0 . 1 7 5 / * * * . * * » ho2 - d5 - s e n s i t i v i t y 140 C f s b 5 ho2 - t 5 < 4 1 . 5 8 5 / « « « * * * * n o 2 - b5 - s e n s i t i v i t y 141 / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 142 cuh3 ho3 < 7 . 3 8 4 143 c l h 3 ho3 > 7 .2O0 144 / * * * * * * * * * * * * * * * * * * * * * ho4 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 145 C f p v 6 ho4 0 . 0 9 v 6 = 6 . 5 1 0 / . * * * * * * . f u z z y e q u a l i t y 146 c f s a 6 0 . 0 9 0 x 1 t 6 < 0 . 0 9 0 / * * * « * ho4 - d6 - s e n s i t i v i t y 147 C f s b 6 ho4 - t 6 < 6 . 4 2 0 / * * « * * ho4 - b6 - s e n s i t i v i t y 148 / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 149 CUh5 ho5 < 4 . 5 O 0 150 c l h 5 ho5 > 0 151 CUh6 ho6 < 4 . 4 3 7 152 C l h 6 ho6 > 2 . 4 0 0 153 / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 154 / « « * * * * C f g i and c f g 2 a r e f u z z y g o a l c o n s t r a i n t s - I . e . - t h e * * * * * 155 / • * * * * » o r i g i n a l o b j e c t i v e f u n c t i o n has been t u r n e d I n t o a * * * * * 156 / * * « * * « c o n s t r a i n t so t h a t t h e o v e r a l l b e n e f i t s c a n be * * * * * 157 / « * » * * * m a x i m i s e d . . . . * * * * * 158 c f g l 0 .173pad1 0 . 1 7 3 p a d 2 0 . 1 7 3 p a d 3 0 . 1 7 3 p a d 4 0 . 1 7 3 p a d 5 159 C f g l 0 . 1 7 3 p a d 6 0 . 2 4 8 p a i 1 0 . 2 4 8 p a 1 2 0 . 2 4 8 p a 1 3 0 . 2 4 8 p a 1 4 160 c f g l 0 . 2 4 8 p a 1 5 0 .248pa<6 - 0 . 1 f x 1 m 1 n - 0 . 1 f x 2 m 1 n - 0 . 1 f x 3 m 1 n 161 c f g l - 0 . 1 f x 4 m 1 n - 0 . 1 f x 5 m 1 n - 0 . 1 f x 6 m 1 n - g o a l * 0 162 C f g 2 g o a l - 0 . 5 0 0 x 1 > 5 . 4 0 0 163 / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 1g4 / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 165 / PERIOD ONE 166 / c s h 2 and c s p 2 a r e s t o r a g e c a p a c i t y c o n s t r a i n t s f o r H e n r y H a l l a m 167 / and P r i n c e E d w a r d , r e s p e c t i v e l y . 168 c s h 2 s t h l 1nh1 - h o i - eh1 < 9 . 0 2 6 169 c s p 2 s t p l 1np1 ho1 - p o l - p a d l - p a l l - e p l < 3 . 3 8 0 170 / c e h l and c c p l a r e c o n t i n u i t y c o n s t r a i n t s f o r H e n r y H a l l a m a n d P r i n c e 171 / Edward , r e s p e c t i v e l y . 172 c e h l s t h l 1nh1 - ho1 - eh1 > 0 173 c c p l s t p l 1np1 ho1 - p o l - p a d l - p a l l - e p l > 0 174 / c s t h 2 and c s t p 2 a r e s t o r a g e c a p a c i t y c a l c u l a t i o n s a t t h e end o f 65 175 / period one which 1s also the beginning of period two. 176 csth2 s t h l 1nh1 - hoi - eh1 - sth2 » 0 177 cstp2 s t p l 1np1 ho1 - po1 - pad1 - pa11 - epl - stp2 = 0 178 / c f p l 1s a f i s h benefit constraint for Prince Edward. It w i l l show us 179 / to have p o s i t i v e f i s h benefits If stored volume In the respective 180 / period 1s above 2366 , or negative benefits If volume Is below 2366. 181 c f p l fxlmln - f x l p l u s 0.5stp1 0.5stp2 = 2.366 182 / PERIOD TWO 183 / csh3 and csp3 are storage capacity c o n s t r a i n t s for H.H and P.E., 184 / re s p e c t i v e l y . 185 csh3 sth2 1nh2 - ho2 - eh2 < 9.026 186 csp3 stp2 1np2 ho2 - po2 - pad2 - pa12 - ep2 < 3.380 187 / cch2 and ccp2 are co n t i n u i t y constraints f o r H.H. and P.E. , resp.. 188 cch2 sth2 1nh2 - ho2 - eh2 > 0 189 ccp2 Stp2 1np2 ho2 - po2 - pad2 - pa12 - ep2 > 0 190 / csth3 and cstp3 are storage capacity c a l c u l a t i o n s for H.H. and P.E. 191 / . re s p e c t i v e l y . 192 csth3 sth2 1nh2 - ho2 - eh2 - sth3 = 0 193 cstp3 stp2 1np2 ho2 - po2 - pad2 - pa 12 - ep2 - stp3 = 0 194 /cfp2 Is a f i s h benefit constraint for P.E. 195 Cfp2 fx2m1n - fx2plus 0.5stp2 0.5stp3 = 2.366 196 / PERIOD THREE 197 / csh4 and csp4 are storage capacity constraints f o r H.H. and P.E. , 198 / re s p e c t i v e l y . 199 csh4 sth3 1nh3 -ho3 - eh3 < 9.026 200 csp4 stp3 1np3 ho3 - po3 - pad3 - pa13 - ep3 < 3.380 201 / cch3 and ccp3 are c o n t i n u i t y constraints f o r H.H. and P.E. . resp.. 202 cch3 sth3 1nh3 - ho3 -eh3 > 0 203 ccp3 stp3 1np3 ho3 - po3 - pad3 - pa 13 - ep3 > 0 204 / csth4 and cstp4 are storage capacity c a l c u l a t i o n s for H.H. and P.E.. 205 / r e s p e c t i v e l y . 206 csth4 sth3 1nh3 - ho3 - eh3 - sth4 = 0 207 cstp4 stp3 1np3 ho3 - po3 - pad3 - pa13 - ep3 - stp4 = 0 208 / cfp3 Is f i s h benefit constraint for P.E. 209 C f p 3 fx3m1n - fx3plus 0.5stp3 0.5stp4 • 2.366 210 / PERIOD FOUR 211 / csh5 and csp5 are storage capacity constraints f o r H.H. and P.E. . 212 / respectively. 213 csh5 sth4 1nh4 - ho4 - eh4 < 9.026 214 csp5 stp4 1np4 ho4 - po4 - pad4 - pa 14 - ep4 < 3.380 215 / cch4 and ccp4 are c o n t i n u i t y constraints for H.H. and P.E. , resp.. 216 cch4 sth4 1nh4 - ho4 - eh4 > 0 217 ccp4 stp4 1np4 ho4 - po4 - pad4 - pa 14 - ep4 > 0 218 / csth5 and cstp5 are storage capacity c a l c u l a t i o n s for H.H. & P.E. , 219 / re s p e c t i v e l y . 220 csth5 sth4 inh4 - ho4 - eh4 - sth5 = 0 221 cstp5 stp4 1np4 ho4 - po4 - pad4 - pa14 - ep4 - stp5 = 0 222 / cfp4 1s a f i s h benefit constraint for P.E. 223 Cfp4 fx4m1n - fx4plus 0.5stp4 0.5stp5 = 2.366 224 / PERIOD FIVE 225 / csh6 and csp6 are storage capacity constraints f o r H.H. & P.E. , 226 / re s p e c t i v e l y . 227 csh6 sth5 1nh5 - ho5 - eh5 < 9.026 228 csp6 stp5 1np5 ho5 - po5 - pad5 - pa 15 - ep5 < 3.380 229 / cch5 and ccp5 are c o n t i n u i t y constraints for H.H. a P.E. . resp.. 230 cch5 sth5 1nh5 - ho5 - eh5 > 0 231 ccp5 stp5 1np5 ho5 - po5 - pad5 - pa15 - ep5 > 0 232 / csth6 and cstp6 are storage capacity c a l c u l a t i o n s for H.H. a P.E. . 66 233 / re s p e c t i v e l y . 234 csth6 sth5 1nh5 - ho5 - eh5 - sth6 = 0 235 cstp6 stp5 1np5 ho5 - po5 - pad5 - pa 15 - ep5 - stp6 » 0 236 / cfp5 Is a f i s h benefit constraint f o r P.E. 237 Cfp5 fx5m1n - fx5plUS 0.5stp5 0.5stp6 = 2.366 238 / PERIOD SIX 239 / csh7 and csp7 are are storage capacity c o n s t r a i n t s f o r H.H. & P.E. . 240 / re s p e c t i v e l y . 241 csh7 sth6 1nh6 - ho6 - eh6 < 9.026 242 csp7 stp6 1np6 ho6 - po6 - pad6 - pa16 - ep6 < 3.380 243 / cch6 and ccp6 are c o n t i n u i t y c o n s t r a i n t s f o r H.H. & P.E. , resp.. 244 cch6 sth6 1nh6 - ho6 - eh6 > 0 245 ccp6 stp6 1np6 ho6 - po6 - pad6 - pa 16 - ep6 > 0 246 / csth7 and cstp7 are storage capacity c a l c u l a t i o n s for H.H. & P.E. , 247 / re s p e c t i v e l y . 248 csth7 sth6 1nh6 - ho6 - eh6 - sth7 = 0 249 cstp7 stp6 1np6 ho6 - po6 - pad6 - pa 16 - ep6 - stp7 = 0 250 / cfp6 Is a f i s h benefit constraint f o r P.E. 251 Cfp6 fx6m1n - fx6plus 0.5stp6 0.5stp7 = 2.366 67 Append ix D Convent ional L.P. output. 1 1HENRY HALLAM AND PRINCE EDWARD RESERVOIRS PROBLEM 2 ONon-default parameters s p e c i f i e d : 3 4 The number of vari a b l e s = 74 5 The number of constraints • 142 6 Prin t the dual s o l u t i o n 7 Perform rhs ranging 8 Perform objective function ranging 9 Maximize the objective function 10 0 23 pages of additional memory are acquired to solve the problem. 11 12 The tableau as entered: 13 Number of Constraints = 142 14 In e q u a l i t i e s • 124 15 Equal 1 t i e s = 18 16 Number of Variables => 74 17 The sol u t i o n Is now f e a s i b l e . 18 -Maximum value of the objec t i v e function = 5.749025O0OO0 19 20 1The Solution Basis: HENRY HALLAM AND PRINCE EDWARD RESERVOIRS PROBLEM 21 Variable Value 22 23 PA01 1.85000000000 24 PA02 1.70000000000 25 PAD3 1 .80000000000 26 PAD4 1 . 9COOOOOOOOO 27 PAD5 1 .77500000000 28 PAD6 1 . 7C>C)O0OO0O0O 29 PAI1 2.65C)OOO0O0OO 30 PAI2 2.55000000000 31 PAI3 2.65000000000 32 PAI4 2.65000000000 33 PAI5 2.650OCOO0O0O 34 PAI6 2.55000000000 35 INH1 7.522C>C<XX>O0O 36 INP1 O. 147000000000 37 EH1 0.669000000000 38 EP1 0.245000000000 39 INH2 45.2950000000 40 INP2 2.53400000000 41 EH2 0.544000000000 42 EP2 O. 198CXX)OO0OO0 43 INH3 7.00600000000 44 INP3 0. 168000000000 45 EH3 0.454000000000 46 EP3 O. 161000000000 47 INH4 3.02100000000 48 INP4 0. 140000000000 49 EH4 0.589000000000 50 EP4 O.213000000000 51 INH5 6.84900000000 52 INP5 0.952000000000 53 EH5 0.679000000000 54 EP5 0.245000000000 55 INH6 8. 10200000000 56 INP6 0.257000000000 57 EH6 0.545000000000 58 EP6 0.197000000000 68 59 STH1 4.512COOOOOOO 60 STP1 2.72000000000 61 P01 0.7240O0O0OO0O 62 P02 39. 3560000000 63 P03 3.32500000000 64 P04 i . 80CXXIO0OOO0 65 P05 0.2COCO0000000 66 P06 0.25O0O0O0OOOO 67 H01 5.50000000000 68 H02 41.7520000000 69 HO 3 7 .38400000000 70 H04 6.50000000000 71 H05 4.225OO0OO0O0 72 H06 4.44O00O0OOO0 73 STH2 5.86500000000 74 STP2 2.89800000000 75 FX1PLUS 0. 443000000000 76 STH3 8 .86400000000 77 STP3 3.38000000000 78 FX2PLUS 0. 773000000000 79 STH4 8.032OOOOO0O0 80 STP4 2.99600000000 81 FX3PLUS 0.822000000000 82 STH5 3.964OOO0OOOO 83 STP5 3.073O0O0OO0O 84 FX4PLUS 0.668500000000 85 STH6 5.90900000000 86 STP6 3 . 38COOOOOOOO 87 FX5PLUS 0.860500000000 88 STH7 9.02600000000 89 STP7 3.38CICOOOOOOO 90 FX6PLUS 1 .014COOOOOOO 91 Slack CIHU1 6.OOOOOOOOOOOOE -03 92 Slack CIPU1 4.999999999999E -03 93 Slack CEHL 1 4 . C>C)0O00OO0OO0E -03 94 Slack CEPL 1 3. C<XXXX)0OO0O0E -03 95 Slack CIHU2 6.999999999998E -03 96 Slack CIPU2 4.000000000005E -03 97 Slack CEHL2 3. OOCKXXXJOOOOOE -03 98 Slack CEPL2 3.999999999998E -03 99 Slack CIHL3 4 . CIOOOOOOOOOOOE -03 100 Slack CIPL3 3.999999999998E -03 101 Slack CEHL3 3. C<XXXXXXXX>OOE -03 102 Slack CEPU3 2.000000000004 E -03 103 Slack CEPL3 9.999999999957E -04 104 Slack CIHU4 4 . OOOOOOOOOOOOE -03 105 Slack CIPU4 3. OCOCOOOOOOOOE -03 106 Slack CEHL4 3 . C>OC>OCKXKX>OOOE -03 107 Slack CEPL4 3. C)0O000O00OO0E -03 108 Slack CIHU5 4 . OOOOOOOOOOOOE -03 109 Slack CIPL5 3.OOOOOOOOOOOOE -03 110 Slack CEHL5 4 . OOCOOOOOOOOOE -03 111 Slack CEPL5 3 . OOOOOOOOOOOOE -03 112 Slack CIHU6 3. OOC)COOOOOOOOE -03 1 13 Slack CIPU6 3. C>C<X>0O0O0OO0E -03 1 14 Slack CEHL6 3 . OOOOOOOOOOOOE -03 1 15 Slack CEPL6 3 . CXXXXXXJOOOOOE -03 1 16 Slack CHUS 3. CIOCOOOOOOOOOE -03 69 117 Slack CPUS 3.000000000000E 118 Slack CLP 1 0.200000000000 119 Slack CLP2 0.400000000000 120 Slack CLP3 0.60COO0000000 121 Slack CLP4 0.250000000000 122 Slack CLP5 0. 575000000000 123 Slack CLP6 0.6CI00CO000000 124 Slack CLP7 0.200000000000 125 Slack CLP8 0.200000000000 126 Slack CLP9 0.200000000000 127 Slack CLP 10 O. 280000000000 128 Slack CLP 1 1 0.195000000000 129 Slack CLP 12 0.20CKXXXXXX300 130 Slack CLP13 3. O C i O O O O O O O O O O E 131 Slack CUP 14 0. 144000000000 132 Slack CLP 14 39.3560000000 133 Slack CUP 15 5. OCIOOOCXXXXXXDE 134 Slack CLP16 5. OOOOOOOOOOOOE 135 Slack CLP 17 3.OOOOOOOOOOOOE 136 Slack CUP 18 4.440892098501E 137 Slack CLP18 2.0C)0C)C)OC)OO0O0E 138 Slack CLH1 5.50000000000 139 Slack CLH2 41 .7520000000 140 Slack CLH3 0. 1840CIOO0O0O0 141 Slack CLH4 0. 150000000000 142 Slack CUH5 O. 275000000000 143 Slack CLH5 4.22500000000 144 Slack CLH6 2 .04000000000 145 Slack CSH2 3. 16100000000 146 Slack CSP2 0.482COOOOOOOO 147 Slack CCH1 5. 86500000000 148 Slack CCP1 2.89800000000 149 Slack CSH3 0. 162000000000 150 Slack CCH2 8 .86400000000 151 Slack CCP2 3 . 38000000000 152 Slack CSH4 0. 994000000000 153 Slack CSP4 0. 384000000000 154 Slack CCH3 8.03200000000 155 Slack CCP3 2.99600000000 156 Slack CSH5 5.06200000000 157 Slack CSP5 0.307000000000 158 Slack CCH4 3.96400000000 159 Slack CCP4 3.07300000000 160 Slack CSH6 3.11700000000 161 Slack CCH5 5.90900000000 162 Slack CCP5 3.38000000000 163 Slack CCH6 9.02600000000 164 Slack CCP6 3. 38000000000 165 -The Reduced Costs: 166 Variable Value 167 168 FX1MIN O. 10OOOOOOOOOO 169 FX2MIN 0. 1OO0OOOO0OOO 170 FX3MIN 0. 10OOOOOOOOOO 171 FX4MIN 0. 1OOOOOOOOOOO 172 FX5MIN 0. 1OOOOOOOOOOO 173 FX6MIN 0. 1OOOOOOOOOOO 174 -03 70 175 IDual Solution Vector: HENRY HALLAM AND PRINCE EDWARD RESERVOIRS PROBLEM 176 Constraint Value 177 178 CIHL1 0.0 179 CIPL1 0.0 180 CEHU1 0.0 181 CEPL) 1 0.0 182 CIHL2 0.0 183 CIPL2 0.0 184 CEHU2 0.0 185 CEPU2 0.0 186 CIHU3 0.0 187 CIPU3 0.0 188 CEHU3 0.0 189 CIHL4 0.0 190 CIPL4 0.0 191 CEHU4 0.0 192 CEPU4 0.0 193 CIHL5 0.0 194 CIPU5 0.0 195 CEHU5 0.0 196 CEPU5 0.0 197 CIHL6 0.0 198 CIPL6 0.0 199 CEHU6 0.0 200 CEPU6 0.0 201 CHLS 0.0 202 CPLS 0.0 203 CUP 1 O. 173000000000 204 CUP2 0.173000000000 205 CUP3 0.173000000000 206 CUP4 0.173000000000 207 CUP5 0.17300OO0OO0O 208 CUP6 0. 173000000000 209 CUP7 0.248000000000 210 CUP8 0.248000000000 211 CUP9 0.248000000000 212 CUP 10 0.248000000000 213 CUP 11 0.248000000000 214 CUP 12 0.248000000000 215 CUP 13 0.0 216 CLP15 0.0 217 CUP 16 0.0 218 CUP 17 O.O 219 CUH1 0.0 220 CUH2 0.0 221 CUH3 0.0 222 CUH4 0.0 223 CUH6 0.0 224 CSP3 0.0 225 CSP6 0.0 226 CSH7 0.0 227 CSP7 0.0 228 ONote that the dual s o l u t i o n vector excludes e q u a l i t i e s . 229 o*** Objective ranging i s not possible when the tableau Includes e q u a l i t i e s . 230 1R1ght Hand Side Ranging: HENRY HALLAM AND PRINCE EDWARD RESERVOIRS PROBLEM 231 Constraint Lower Bound Rhs Upper Bound 232 71 233 CIHU1 7.522000 7.528000 • I n f i n i t y 234 CIHL 1 7.521000 7.522000 7.524000 235 CIPU1 0. 1470000 0.1520000 • I n f i n i t y 236 CIPL1 -2.0122792E-15 0.1470000 0.1520000 237 CEHU1 0.6670000 0.6690000 0.67OO0OO 238 CEHL1 -Inf m i ty 0.665O0O0 0.6690000 239 CEPU1 0.2420000 0.2450000 1.131000 240 CEPL 1 - I n f i n i t y 0.2420000 0.2450000 241 CIHU2 45.29500 45.30200 • I n f i n i t y 242 CIHL2 45.29400 45.29500 45.29700 243 CIPU2 2.534000 2.538000 • I n f i n i t y 244 CIPL2 4.6629367E-15 2.5340O0 2.538000 245 CEHU2 0.5420000 0.5440000 0.5450000 246 CEHL2 - I n f i n i t y 0.5410000 0.5440000 247 CEPU2 0.1940OOO 0.198O0O0 39.55400 248 CEPL2 - I n f i n i t y 0.1940000 0.1980000 249 CIHU3 7.OO50OO 7.006000 7.008000 250 CIHL3 - I n f i n i t y 7.002000 7.0O60OO 251 CIPU3 0.1670OOO 0.1680000 0.1700000 252 CIPL3 - I n f i n i t y 0.164O0O0 0.325OOO0 253 CEHU3 0.452OO0O 0.4540000 O.4550OOO 254 CEHL3 - I n f i n i t y 0.4510000 0.4540000 255 CEPU3 -2.557000 0.1630000 • I n f i n i t y 256 CEPL3 - I n f i n i t y O.1600O00 0.16100O0 257 CIHU4 2.011000 3.025000 • I n f i n i t y 258 CIHL4 3.020000 3.021000 3.02300O 259 CIPU4 3.0000000E-03 0.1430000 • I n f i n i t y 260 CIPL4 0.1390000 0.14OO0O0 0.1420000 261 CEHU4 0.587OOO0 O.589O0OO 0.590O0OO 262 CEHL4 - I n f i n i t y 0.5860000 0.5890O0O 263 CEPU4 0.2110000 O.213OO0O 0.2140OOO 264 CEPL4 -Inf1ni ty O.21OO0O0 0.213OO0O 265 CIHU5 4.O00OOO0E-03 6.853000 • I n f i n i t y 266 CIHL5 6.8480OO 6.849OO0 6.8510O0 267 CIPU5 0.951OOO0 0.9520O00 0.954O0O0 268 CIPL5 -Inf1n1ty 0.9490000 0.9520000 269 CEHU5 0.6770O0O O.6790O0O 0.680OOOO 270 CEHL5 - I n f i n i t y 0.6750O0O 0.679OOO0 271 CEPU5 0.2430OO0 0.2450OOO 0.246O0O0 272 CEPL5 - I n f i n i t y 0.2420000 0.2450OOO 273 CIHU6 8.102OO0 8.105000 •Inf1n1ty 274 CIHL6 8.101000 8.102000 8.104000 275 CIPU6 3.0OOOO0OE-03 0.26OOOO0 • Inf tnlty 276 CIPL6 0.2550000 0.2570000 0.2570000 277 CEHU6 0.5430000 0.5450000 0.5460000 278 CEHL6 - I n f i n i t y 0.5420000 0.545OO0O 279 CEPU6 0.1970OOO 0.197O0O0 0.1990OOO 280 CEPL6 - I n f i n i t y 0.1940000 0.197OO0O 281 CHUS 4.512000 4.5150OO • I n f i n i t y 282 CHLS 4.5110OO 4.512000 4.5140O0 283 CPUS 2.720000 2.723000 •Inf1n1ty 284 CPLS 2.277000 2.720OO0 2.723O0O 285 CUP 1 1.706OOO 1 .850OOO 2.736000 286 CLP 1 - I n f i n i t y 1.65O0O0 1.850000 287 CUP2 1.5560OO 1.7O0OO0 41.05600 288 CLP2 -Inf1ni ty 1 .30O0O0 1.7000O0 289 CUP3 1.7980O0 1 .80OO0O 1.801OOO 290 CLP3 - I n f i n i t y 1.20O00O 1.8O0OO0 72 291 CUP4 1.898000 1.900000 1.901000 292 CLP4 - I n f i n i t y 1.650000 1.900000 293 CUP5 1.773000 1.7750O0 1.776000 294 CLP5 - I n f i n i t y 1 . 200000 1.775000 295 CUP6 1.700000 1.700000 1.702000 296 CLP6 - I n f i n i t y 1.10OOO0 1.700000 297 CUP7 2.506000 2.650000 3.536000 298 CLP7 - I n f i n i t y 2.450000 2.650000 299 CUP8 2.406000 2.550OO0 41.9O60O 300 CLP8 - I n f i n i t y 2.350000 2.55OOO0 301 CUP9 2.648000 2.650OOO 2.651000 302 CLP9 - I n f i n i t y 2.45O0OO 2.65OO0O 303 CUP 10 2.648O0O 2.650000 2.651O0O 304 CLP 10 - I n f i n i t y 2.370OOO 2.650OO0 305 CUP 11 2.648000 2.650OOO 2.651000 306 CLP 11 - I n f i n i t y 2.455000 2.650000 307 CUP 12 2.550000 2.550000 2.5520OO 308 CLP12 - I n f i n i t y 2.35O0OO 2.55OO0O 309 CUP 13 0.7210000 0.7240000 1.610000 310 CLP 13 -Inf1n1ty 0.7210000 0.7240000 311 CUP 14 39.35600 39.50000 • I n f i n i t y 312 CLP 14 - I n f i n i t y -0.0 39.35600 313 CUP 15 3.3250O0 3.330OOO • I n f i n i t y 314 CLP15 3.323O00 3.325000 3.326000 315 CUP 16 1.798000 1 .800OO0 1.801000 316 CLP 16 - I n f i n i t y 1.795000 1.800000 317 CUP 17 0.1980OOO 0.20OO0O0 0.2010000 318 CLP17 - I n f i n i t y 0.1970000 O.2OOO0OO 319 CUP 18 0.25OOO0O 0.2500O0O • I n f i n i t y 320 CLP18 - I n f i n i t y 0.2480O0O 0.2500000 321 CUH1 5.4980O0 5.500000 5.501000 322 CLH1 - I n f i n i t y -0.0 5.5OO0O0 323 CUH2 41.75000 41 .75200 41.75300 324 CLH2 - I n f i n i t y -0.0 41.75200 325 CUH3 7.200000 7.384000 7.691000 326 CLH3 - I n f i n i t y 7.200000 7.384000 327 CUH4 6.350000 6.500000 6.807000 328 CLH4 - I n f i n i t y 6.35O0OO 6.500000 329 CUH5 4.225000 4.5OO0OO • I n f i n i t y 330 CLH5 - I n f i n i t y -0.0 4.225000 331 CUH6 4.4380O0 4.440OO0 4.440O0O 332 CLH6 - I n f i n i t y 2.400000 4.440000 333 CSH2 5.865000 9.026000 • I n f i n i t y 334 CSP2 2.898000 3.380000 • I n f i n i t y 335 CCH1 - I n f i n i t y -0.0 5.8650OO 336 CCP1 - I n f i n i t y -0.0 2.8980OO 337 CSH3 8.8640O0 9.026O0O • I n f i n i t y 338 CSP3 3.379000 3.380000 3.382000 339 CCH2 - I n f i n i t y -0.0 8.864000 340 CCP2 - I n f i n i t y -0.0 3.38OOO0 341 CSH4 0.1620000 9.026000 • I n f i n i t y 342 CSP4 -4.O040O0 3.380OO0 • InfIni ty 343 CCH3 - I n f i n i t y -0.0 8.032000 344 CCP3 - I n f i n i t y -0.0 2.996000 345 CSH5 3.964000 9.026000 • Inf i n i t y 346 CSP5 3.073000 3.380000 • I n f i n i t y 347 CCH4 - I n f i n i t y -0.0 3.964000 348 CCP4 - I n f i n i t y -0.0 3.07SOOO 73 349 CSH6 5.909000 9.026000 • I n f i n i t y 350 CSP6 3.378000 3.380OOO 3.380000 351 CCH5 - I n f i n i t y -0.0 5.9090OO 352 CCP5 - I n f i n i t y -0.0 3.380000 353 CSH7 9.024000 9.026000 9.027000 354 CSP7 3.380000 3.380000 3.382000 355 CCH6 - I n f i n i t y -0.0 9.026000 356 CCP6 - I n f i n i t y -0.0 0.8605000 357 ONotO that r i g h t hand side ranging Is not done f o r equal 111es. 358 0 74 Append ix E Fuzzy L.P. output . 1 1HENRY HALLAM AND PRINCE EDWARD RESERVOIRS PROBLEM - FLP 2 ONon-default parameters s p e c i f i e d : 3 4 The number of variables = 82 5 The number of constraints = 150 6 Print the dual s o l u t i o n 7 Perform rhs rangtng 8 Perform objective function ranging 9 Maximize the objective function 10 O 26 pages of additional memory are acquired to solve the problem. 11 12 The tableau as entered: 13 Number of Constraints « 150 14 Ine q u a l i t i e s « 125 15 Equal1t1es = 25 16 Number of Variables = 82 17 The s o l u t i o n i s now f e a s i b l e . 18 -Maximum value of the objective function • 0.825032714135 19 20 1The Solution Basis: HENRY HALLAM AND PRINCE EDWARD RESERVOIRS PROBLEM - FLP 21 Variable Value 22 23 X1 0.825032714135 24 INH1 7.522COOOOOOO 25 INP1 0. 147000000000 26 EH1 0.665000000000 27 EP 1 0.245000000000 28 INH2 45.2950000000 29 INP2 2.53400000000 30 EH2 0.5440CO000O0O 31 EP2 0. 198000000000 32 INH3 7.00200000000 33 INP3 0. 164000000000 34 EH3 0.4540OOOO0O00 35 EP3 0. 163000000000 36 INH4 3.02100000000 37 INP4 0.140000000000 38 EH4 0.589000000000 39 EP4 0.2130COOOOOOO 40 I NHS 6.84900000000 41 INP5 0.949000000000 42 EHS 0.679000000000 43 EPS 0.245000000000 44 INH6 8. 10200000000 45 INP6 0.257000000000 46 EH6 0.545000000000 47 EP6 O. 197000000000 48 STH1 4.51200O00O0O 49 STP1 2.72000000000 50 PAD 1 1 .85000000000 51 PAD2 1 .70000000000 52 PAD3 1 . 8CIO0O0OO0O0 53 PAD4 2.20000000000 54 V1 1 . OOOOOOOOOOO 55 PAD5 1.71399738287 56 V3 0.825032714135 57 PAD6 1.70000000000 58 PA II 2.650O000O0O0 75 59 PAI2 2.55000000000 60 PAI3 2. 65000000000 61 PAI4 2 .90000000000 62 V2 1 .OOOOCKXXXJOO 63 PAI5 2.48929358803 64 V4 0 .825032714135 65 PAI6 2.55000000000 66 POI 0 .724000000000 67 P02 39.5000000000 68 P03 3.330OOOOOOO0 69 P04 1.80000000000 70 P05 0 .197000000000 71 P06 0 .248COOOOOOOO 72 H01 5.50000000000 73 H02 41.6156192750 74 V5 0 .825032714135 75 HO 3 7. 38400000000 76 H04 6.43574705573 77 V6 0 .825032714135 78 HO 5 4 . 50000000000 79 H06 4.36563366925 80 GOAL 5.81251635707 81 STH2 5.86900000000 82 STP2 2.89800000000 83 FX1PLUS 0 .443000000000 84 STH3 9.00438072497 85 STP3 3.09961927503 86 FX2PLUS 0 .632809637513 87 STH4 8.16838072497 88 STP4 2.70461927503 89 FX3PLUS 0 .536119275026 90 STH5 4.16463366925 91 STP5 2. 16736633075 92 FX4PLUS 6 .999280289020E -02 93 STH6 5.83463366925 94 STP6 2.97107535986 95 FX5PLUS O .203220845305 96 STH7 9.02600000000 97 STP7 2.89870902910 98 FX6PLUS 0 .568892194478 99 Slack CIHU1 6 .OOOOOOOOOOOOE -03 100 Slack CIPU1 5 .0O0C>O0OOO0O3E -03 101 Slack CEHU1 4 .OOOOOOOOOOOOE -03 102 Slack CEPL 1 3 .O000O0OO0O14E -03 103 Slack CIHU2 6 .999999999998E -03 104 Slack CIPU2 4 .000000000002E -03 105 Slack CEHL2 3 .OOOOOOOOOOOOE -03 106 S1 ack CEPL2 3 .999999999998E -03 107 Slack CIHU3 4 . COOOOOOOOOOOE -03 108 Slack CIPU3 4 .OOOOOOOOOOOOE -03 109 Slack CEHL3 3 .OOOOOOOOOOOOE -03 110 Slack CEPL3 3 . C>COOOOOOOOOOE -03 1 1 1 Slack CIHU4 4 . OCICKXXIOOOOOOE -03 112 Slack CIPU4 3 . OCOCKXXXXXXfOE -03 1 13 Slack CEHL4 3 .OOOOOOOOOOOOE -03 114 Slack CEPL4 3 .OOOOOOOOOOOOE -03 115 SI ack CIHU5 4 .OOOOOOOOOOOOE -03 116 Slack CIPU5 3 .OOOOOOOOOOOOE -03 76 117 Slack CEHL5 4. OOOOOOOOOOOOE -03 118 Slack CEPL5 3. OOOOOOOOOOOOE -03 119 Slack CIHU6 3. COOOOOOOOOOOE -03 120 Slack CIPU6 3. OOOOOOOOOOOOE -03 121 SI ack CEHL6 3. OOOOOOOOOOOOE -03 122 Slack CEPL6 3. OOOOOOOOOOOOE -03 123 Slack CHUS 3. OOOOOOOOOOOOE -03 124 Slack CPUS 3. CKXWOOOOOOOOE -03 125 Slack CLP 1 0. 200000000000 126 Slack CLP2 0. 400000000000 127 Slack CLP3 0. 600000000000 128 Slack CFPU4 6. 473789576992E -02 129 Slack CLP5 1 . 399738286917E -02 130 Slack CLP6 0. 600000000000 131 Slack CLP7 0. 200000000000 132 Slack CLP8 0. 200000000000 133 Slack CLP9 0. 200000000000 134 Slack CFLP10 5. 598953147669E -02 135 Slack CLP 11 3. 429358802947E -02 136 Slack CLP12 0. 200000000000 137 Slack CLP13 2. 999999999999E -03 138 Slack CLP 14 0. 250000000000 139 Slack CLP15 5. OOOOOOOOOOOOE -03 140 Slack CLP 16 5. OOOOOOOOOOOOE -03 141 Slack CUP 17 3. OOOOOOOOOOOOE -03 142 Slack CUP 18 2. OOOOOOOOOOOOE -03 143 Slack CLH1 5 i.50000000000 144 Slack CLH2 3. 061927502632E -02 145 Slack CLH3 0. 184000000000 146 Slack CLH4 1 . 574705572782E -02 147 Slack CLH5 4 .50000000000 148 Slack CUH6 7. 136633075413E -02 149 Slack CLH6 1 .96563366925 150 Slack CSH2 3 I. 15700000000 151 Slack CSP2 0. 482000000000 152 Slack CCH1 5 i. 86900000000 153 Slack CCP1 2 :.89800000000 154 Slack CSH3 2. 161927502631E -02 155 Slack CSP3 0. 280380724974 156 Slack CCH2 9 i . 00438072497 157 Slack CCP2 3 1.09961927503 158 Slack CSH4 0. 857619275026 159 Slack CSP4 0. 675380724974 160 Slack CCH3 8 .16838072497 161 Slack CCP3 2 .70461927503 162 Slack CSH5 4 .86136633075 163 Slack CSP5 1 .21263366925 164 Slack CCH4 4 .16463366925 165 Slack CCP4 2 .16736633075 166 Slack CSH6 3 .19136633075 167 Slack CSP6 0. 408924640145 168 Slack CCH5 5 .83463366925 169 Slack CCP5 2 .97107535986 170 Slack CSP7 0. 481290970899 171 Slack CCH6 9 . 02600000000 172 Slack CCP6 2 .89870902910 173 -The Reduced Costs: 174 Variable Value 77 175 176 FXIMIN 0 . 177794213865 177 FX2MIN 0 . 177794213865 178 FX3MIN 0 . 177794213865 179 FX4MIN 0 . 177794213865 180 FX5MIN 0 . 177794213865 181 FX6MIN 0 . 177794213865 182 183 1Dual Solution Vector: I 184 Constraint Value 185 186 CIHL1 2 .220651336129E- 17 187 CIPL1 1 . 233695186739E-17 188 CEHL 1 3 .081772803794E- 34 189 CEPU1 -2 .467390373477E- 18 190 CIHL2 2 .220651336129E- 17 191 CIPL2 -7 .402171120431E- 17 192 CEHU2 2 .220651336129E- 17 193 CEPU2 -5 . 181519784302E-17 194 CIHL3 -5 . 136288006324E-34 195 CIPL3 -7. .402171120431E- 17 196 CEHU3 7 .402171120431E- 17 197 CEPU3 -7. .402171120431E- 17 198 CIHL4 -5. . 1362880O6324E-34 199 CIPL4 2 344020854803E- 17 200 CEHU4 4 934780746954E- 17 201 CEPU4 2 344020854803E- 17 202 CIHL5 -5, . 136288006324E-34 203 CIPL5 2. 467390373477E- 17 204 CEHU5 -5. .1362880O6324E- 34 205 CEPU5 4. 934780746954E- 17 206 CIHL6 -5. ,136288006324E- 34 207 CIPL6 2 467390373477E- 17 208 CEHU6 -5. .136288O06324E- 34 209 CEPU6 -2. 467390373477E- 17 210 CHLS 2. 220651336129E- 17 211 CPLS -9. 129344381B65E- 17 212 CUP 1 0. 307583989987 213 CUP2 0. 307583989987 214 CUP3 0. .307583989987 215 CUP4 0. 307583989987 216 CFPU5 0. 307583989987 217 CUP6 0. 307583989987 218 CUP7 0. 440929650385 219 CUP8 0. 440929650385 220 CUP9 0. 440929650385 221 CUP 10 0. 440929650385 222 CFUP11 0. 440929650385 223 CUP 12 0. 440929650385 224 CUP 13 -2. 467390373477E- 18 225 CUP 14 -1. 184347379269E- 16 226 CUP 15 -4. 934780746954E- 17 227 CUP 16 2. 344020854803E- 17 228 CLP17 -2. 467390373477E- 17 229 CLP18 -2. 467390373477E- 17 230 CUH1 7. 648910157779E- 17 231 CFUH2 1 . 043994319176E- 16 232 CUH3 2. 467390373477E- 17 HENRY HALLAM AND PRINCE EDWARD RESERVOIRS PROBLEM - FLP 78 233 CFHU4 1 . 647889100041E-16 234 CUH5 3. 084237966846E-19 235 CFG2 1 .77794213865 236 CSH7 -5. 1362880O6324E-34 237 ONote that the dual solution vector excludes e q u a l i t i e s . 23S 0**« Objective ranging i s not possible when the tableau Includes equa 239 1R1ght Hand Side Ranging: HENRY HALLAM AND PRINCE EDWARD RESER' 240 OA i Constraint Lower Bound Rhs Upper Bound 1 242 CIHU1 7.522000 7.528000 • I n f i n i t y 243 CIHL1 6.384216 7.522000 7.528000 244 CIPU1 -0.2910000 0.1520000 • I n f i n i t y 245 CIPL1 7.70O7197E-O2 0.1470000 0.1520000 246 CEHU1 -3.495634 0.6690OOO • I n f i n i t y 247 CEHL 1 0.6433807 0.6650000 0.6690000 248 CEPU1 0.2420000 O.245O0OO 0.3149928 249 CEPL 1 - I n f i n i t y 0.2420000 0.2450000 250 CIHU2 39.43300 45.30200 • I n f i n i t y 251 CIHL2 44.15722 45.2950O 45.30200 252 CIPU2 2.534000 2.5380OO • I n f i n i t y 253 CIPL2 2.464007 2.534000 2.538000 254 CEHU2 0.5410OOO 0.5440OOO 1.681784 255 CEHL2 - I n f i n i t y 0.5410000 0.5440000 256 CEPU2 0.1940000 0.1980OOO 0.2679928 257 CEPL2 -Inf ini ty O.19400O0 0.1980OO0 258 CIHU3 7.O02OOO 7.006000 • I n f i n i t y 259 CIHL3 5.864216 7.O02OOO 7.006000 260 CIPU3 2.8OO0OO0E-02 0.1680O0O • I n f i n i t y 261 CIPL3 9.4O07197E-02 0.1640000 0.1680000 262 CEHU3 0.4510000 0.4540000 1.591784 263 CEHL3 - I n f i n i t y 0.4510OOO O.454O0OO 264 CEPU3 0. 16000OO 0.1630000 0.2329928 265 CEPL3 - I n f i n i t y O. 1600000 0. 1630OOO 266 CIHU4 3.021000 3 .025000 • I n f i n i t y 267 CIHL4 1.883216 3.021000 3.025000 268 CIPU4 0.139O0OO 0. 1430COO • Inf ini ty 269 CIPL4 1.4394220E-05 0.14OOOOO 0.1430000 270 CEHU4 0.586O0O0 0.58900O0 1.726784 271 CEHL4 - I n f i n i t y 0.5860000 0.5890OO0 272 CEPU4 0.210O000 O.2130OOO 0.3529856 273 CEPL4 - I n f i n i t y 0.21OOOO0 0.2747379 274 CIHU5 6.8490OO 6.853000 • I n f i n i t y 275 CIHL5 5.711216 6.849000 6.853000 276 CIPU5 0.9490000 0.9520OOO • I n f i n i t y 277 CIPL5 0.5425583 0.949OOOO O.9520O0O 278 CEHU5 0.6750000 0.6790000 1.816784 279 CEHL5 - I n f i n i t y O.675O0O0 O.6790000 280 CEPU5 0.2420COO 0.245O0OO 0.6514417 281 CEPL5 - I n f i n i t y 0.2420000 0.245O0O0 282 CIHU6 8.1020OO 8.105000 • I n f i n i t y 283 CIHL6 6.964216 8. 1020O0 8.1O50OO 284 CIPU6 0.2570OO0 0.260O0O0 • I n f i n ! t y 285 CIPL6 -0.0 0.2570OOO O.2600000 286 CEHU6 0.5420OO0 0.5450000 1.682784 287 CEHL6 - I n f i n i t y 0.5420000 0.5450000 288 CEPU6 0.1940O00 0.1970000 1.334784 289 CEPL6 - I n f i n i t y 0.1940OOO 0.19700OO 290 CHUS 4.512000 4.5150OO • I n f i n i t y 79 291 CHLS 3.374216 4.512O0O 4. 515000 292 CPUS 2.72O0OO 2.723O0O • I n f i n i t y 293 CPLS 2.650O07 2.72O0OO 2. 723O0O 294 CUP1 1.650000 1.850000 1. 915557 295 CLP 1 - I n f i n i t y 1.650000 1 . ,850000 296 CUP2 1.433941 1.700000 1 . 765557 297 CLP2 -Inf1n1ty 1 .3O0O00 1 . 7O0O00 298 CUP3 1.389687 1 .800000 1 . 865557 299 CLP3 -Inf i m ty 1.200O0O 1. 8OOO0O 300 CUP4 2. 126948 2.200000 2. 323299 301 CFPU4 - I n f i n i t y 1.83OOO0 2. .655033 302 CFPU5 1.765649 1.78OOO0 1 . 847668 303 CLP5 - I n f i n i t y 1.700000 1. ,713997 304 CUP6 1.174060 1.7O00O0 2. ,101642 305 CLP6 -Inf1n1ty 1.1OOOOO 1 . 700000 306 CUP7 2.450000 2.650000 2. ,713804 307 CLP7 - I n f i n i t y 2.450OOO 2. ,650000 308 CUP8 2.35OOO0 2.55O0O0 2. 613804 309 CLP8 - I n f i n i t y 2.350000 2. ,550000 310 CUP9 2.450000 2.650000 2. ,713804 311 CLP9 - I n f i n i t y 2.450000 2. ,650000 312 CUP 10 2.834813 2.900000 3. 017240 313 CFLP10 - I n f i n i t y 2.58O0O0 3. .405033 314 CFUP11 2.613462 2.651000 2. 828003 315 CLP 11 - I n f i n i t y 2.455COO 2. .489294 316 CUP 12 2.350OO0 2.550000 2. 830178 317 CLP12 - I n f i n i t y 2.350000 2. ,355000 318 CUP 13 0.7210000 0.7240000 0.7939928 319 CLP13 - I n f i n i t y O.721O0OO 0.724OO0O 320 CUP 14 39.25000 39.50000 39.56999 321 CLP14 - I n f i n i t y 39.250OO 78.75000 322 CUP 15 3.3250O0 3.33OO0O 3. 399993 323 CLP15 - I n f i n i t y 3.3250OO 3. 330O0O 324 CUP 16 1.7950OO 1 .8000O0 1 . 939986 325 CLP 16 - I n f i n i t y 1.795000 3. 595000 326 CUP 17 0. 19700O0 0.2000000 • I n f i n i t y 327 CLP 17 -0.0 0. 1970OO0 O.2OO0OO0 328 CUP 18 0.2480O0O O.25O0OOO • I n f i n i t y 329 CLP 18 -0.0 0.2480000 0.2500000 330 CUH1 5.478381 5.5OO0O0 5. ,780381 331 CLH1 - I n f i n i t y -0.0 5. 5O00O0 332 CFUH2 41.73838 41.76000 41.90438 333 CLH2 - I n f i n i t y 41.585CO 41.61562 334 CUH3 7.314007 7.384O0O 7. 792925 335 CLH3 - I n f i n i t y 7.20OOO0 14.58400 336 CFHU4 6.494253 6.510000 6. 584253 337 CLH4 - I n f i n i t y 6.420O0O 6. ,435747 338 CUH5 4.428634 4.50OO0O 4. ,908925 339 CLH5 -Inf1nlty -0.0 4. 5OOO0O 340 CUH6 4.365634 4.437O0O •Inf1n1ty 341 CLH6 -Inf1n1ty 2.40OOOO 4. ,365634 342 CFG2 5.330516 5.4OO0OO 5. .864038 343 CSH2 5.869O0O 9.026O0O • I n f i n i t y 344 CSP2 2.898O0O 3.380000 • I n f i n i t y 345 CCH1 - I n f i n i t y -0.0 5. 869O0O 346 CCP1 - I n f i n i t y -0.0 2. 898000 347 CSH3 9.O23O0O 9.O26OO0 • Inf i m ty 348 CSP3 0.73OOO0O 3.380O00 • I n f i n i t y 80 349 CCH2 - I n f i n i t y -0.0 9.004381 350 CCP2 -Inf1n1ty -0.0 3.33O0O0 351 CSH4 8.168381 9.026000 • I n f i n i t y 352 CSP4 2.704619 3.380000 • I n f i n i t y 353 CCH3 - I n f i n i t y -0.0 8. 168381 354 CCP3 - I n f i n i t y -0.0 2.704619 355 CSH5 4.164634 9.026000 • I n f i n i t y 356 CSP5 3.182O0O 3.38OOO0 + Inf i n i ty 357 CCH4 - I n f i n i t y -0.0 9.026000 358 CCP4 - I n f i n i t y -0.0 2.167366 359 CSH6 5.834634 9.026000 • I n f i n i t y 360 CSP6 3.373000 3.38OOO0 • I n f i n i t y 361 CCH5 - I n f i n i t y -O.O 5.834634 362 CCP5 - I n f i n i t y -0.0 2.971075 363 CSH7 8.954634 9.026000 10.16378 364 CSP7 2.971075 3.380000 • Inf i n l t y 365 CCH6 - I n f i n i t y -0.0 9.026O0O 366 CCP6 - I n f i n i t y -0.0 2.898709 367 ONote that rig h t hand side ranging Is not done f o r equal 11les. 368 0 81 Appendix F Alternative Fuzzy L.P. output. 1 1HENRY HALLAM AND PRINCE EDWARD RESERVOIRS PROBLEM - FLP(P) 2 ONon-default parameters s p e c i f i e d : 3 4 The number of vari a b l e s » 88 5 The number of constraints = 150 6 Prin t the dual s o l u t i o n 7 Perform rhs ranging 8 Perform objective function ranging 9 The tolerance = 1.00E-08 10 Maximize the obje c t i v e function 11 0 28 pages of additional memory are acquired to solve the problem. 12 13 The tableau as entered: 14 Number of Constraints = 150 15 Ine q u a l i t i e s = 125 16 Equal1t1es - 25 17 Number of Variables = 88 18 The solution i s now f e a s i b l e . 19 -Maximum value of the objective function •• 0.825032714135 20 21 1The Solution Basis: HENRY HALLAM AND PRINCE EDWARD RESERVOIRS PROBLEM - FLP(P) 22 Variable Value 23 24 X1 0.825032714135 25 INH1 7.52200000000 26 INP1 0.147000000000 27 EH1 0.669000000000 28 EP1 0.245000000000 29 INH2 45.2950000000 30 INP2 2.53400000000 31 EH2 0.544000000000 32 EP2 0. 198000000000 33 INH3 7.00200000000 34 INP3 O. 164000000000 35 EH3 O.454000000000 36 EP3 O. 163000000000 37 INH4 3.02100000000 38 INP4 0. 140000000000 39 EH4 0.589000000000 40 EP4 0.213000000000 41 INH5 6.84900000000 42 INP5 0.949000000000 43 EH5 0.679C>0O0OO0O0 44 EP5 0.245000000000 45 INH6 8.10200000000 46 INP6 0.257000000000 47 EH6 0.545000000000 48 EP6 0.197000000000 49 STH1 4.5120O00O0O0 50 STP 1 2.72000000000 51 PAD1 1 .85000000000 52 PAD2 1 .70000000000 53 PADS 1 .80000000000 54 PAD4 2.20000000000 55 V1 1.OOOOOOOOOOO 56 PAD5 1.71399738287 57 V3 0.825032714135 58 T3 1.399738286917E-02 82 59 PAD6 1 . 70000000000 60 PA11 2 .65000000000 61 PAI2 2 .55000000000 62 PAIS 2.65000000000 63 PAI4 2.90000000000 64 V2 1.OOOOOOOOOOO 65 PAIS 2.48929358803 66 V4 0.825032714135 67 T4 3.429358802947E -02 68 PAI6 2.55000000000 69 P01 0.724000000000 70 P02 39.5000000000 71 P03 3.33000000000 72 P04 1.79500000000 73 P05 0.197000000000 74 P06 0.248000000000 75 H01 5.50000000000 76 H02 41.5900000000 77 V5 0.971428571429 78 T5 5.OOOOOOOOOOO1E -03 79 H03 7.38400000000 80 H04 6.43574705573 81 V6 0.825032714135 82 T6 1.574705572782E -02 83 H05 4 .50000000000 84 H06 4.38725294427 85 GOAL 5.81251635707 86 STH2 5 .86500000000 87 STP2 2.89800000000 88 FX1PLUS 0. 443000000000 89 STH3 9 .02600000000 90 STP3 3 .07400000000 91 FX2PLUS 0.620000000000 92 STH4 8.19000000000 93 STP4 2.67900000000 94 FX3PLUS 0.510500000000 95 STH5 4.18625294427 96 STP5 2.14674705573 97 FX4PLUS 4.687352786390E -02 98 STH6 5.85625294427 99 STP6 2.95045608483 100 FX5PLUS 0.182601570278 101 STH7 9.02600000000 102 STP7 2.89970902910 103 FX6PLUS 0.559082556965 104 Slack CIHU1 6. OOOOOOOOOOOOE -03 105 Slack CIPU1 4.999999999999E -03 106 Slack CEHL 1 4 . uOCKXKiOOOOOOE -03 107 Slack CEPL 1 3.OOOOOOOOOOOOE -03 108 Slack CIHU2 6.999999999998E -03 109 Slack CIPU2 4 .000000000005E -03 110 Slack CEHL2 3 . OOOOOOOOOOOOE -03 111 Slack CEPL2 3.999999999998E -03 112 Slack CIHU3 4 . OOOOOOOOOOOOE -03 1 13 Slack CIPU3 4.OOOOOOOOOOOOE -03 114 Slack CEHL3 3.OOOOOOOOOOOOE -03 115 Slack CEPL3 3 . OOOOOOOOOOOOE -03 1 16 Slack CIHU4 4.00C>O00O00OO0E -03 83 117 Slack CIPU4 3. OOOOOOOOOOOOE -03 118 Slack CEHL4 3. OOOOOOOOOOOOE -03 119 Slack CEPL4 3. OOOOOOOOOOOOE -03 120 Slack CIHU5 4. OOOOOOOOOOOOE -03 121 Slack CIPU5 3. OOOOOOOOOOOOE -03 122 Slack CEHL5 4. OOOOOOOOOOOOE -03 123 Slack CEPL5 3. OOOOOOOOOOOOE -03 124 Slack CIHU6 3. OOOOOOOOOOOOE -03 125 Slack CIPU6 3. OOOOOOOOOOOOE -03 126 Slack CEHL6 3. OOOOOOOOOOOOE -03 127 Slack CEPL6 3. OOOOOOOOOOOOE -03 128 Slack CHUS 3. 00CKXXX)O0OO0E -03 129 Slack CPUS 3. OOOOOOOOOOOOE -03 130 Slack CLP 1 0. 200000000000 131 Slack CLP2 0. 4CO0CO000000 132 Slack CLP3 0. 600000000000 133 Slack CFSA1 6. 473789576992E -02 134 Slack CLP6 0. 600000000000 135 Slack CLP7 0. 20OCKXK3OOOO0 136 Slack CLP8 0. 2COC<)OOO0OO0 137 Slack CLP9 0. 200000000000 138 Slack CFSA2 5. 598953147669E -02 139 Slack CLP12 0. 200000000000 140 Slack CLP 13 2. 999999999999E -03 141 Slack CLP 14 0. 250000000000 142 Slack CLP 15 5. COCOCOOOOOOOE -03 143 Slack CUP 16 5. OOOOOOOOOOOOE -03 144 Slack CUP 17 3. OOOOOOOOOOOOE -03 145 Slack CUP18 2. OOOOOOOOOOOOE -03 146 Slack CLH 1 5 >. 50000000000 147 Slack CFSA5 2. 561927502631E -02 148 Slack CLH3 0. 184000000000 149 Slack CLH5 4 . 50000000000 150 Slack CUH6 4. 974705572782E -02 151 Slack CLH6 1 .98725294427 152 Slack CSH2 3 I. 16100000000 153 Slack CSP2 0. 482000000000 154 Slack CCH1 5 i. 86500000000 155 Slack CCP1 2 !.89800000000 156 Slack CSP3 0. 306000000000 157 Slack CCH2 9 1.02600000000 158 Slack CCP2 3 1.07400000000 159 Slack CSH4 0. 836000000000 160 Slack CSP4 O. 701000000000 161 Slack CCH3 8 I. 19000000000 162 Slack CCP3 2 !. 67900000000 163 Slack CSH5 4 .83974705573 164 Slack CSP5 1 .23325294427 165 Slack CCH4 4 l. 18625294427 166 Slack CCP4 2 !. 14674705573 167 Slack CSH6 3 I. 16974705573 168 Slack CSP6 0. 429543915171 169 Slack CCH5 s i. 85625294427 170 Slack CCP5 2 :. 95045608483 171 Slack CSP7 0. 480290970899 172 Slack CCH6 9 1.02600000000 173 Slack CCP6 2 1.89970902910 174 -The Reduced Costs: 84 175 Variable Value 176 177 TI 0 .307583989987 178 T2 0 .440929650385 179 FX1MIN 0 .177794213865 180 FX2MIN 0 .177794213865 181 FX3MIN 0 .177794213865 182 FX4MIN 0 .177794213865 183 FX5MIN 0 .177794213865 184 FX6MIN 0 .177794213865 185 186 1Dual Solution Vector: I 187 Constraint Value 188 189 CIHL1 8 . 181513994111E-17 190 C1PL1 5. , 122302415338E-17 191 CEHU1 9 035231063334E- 17 192 CEPU1 5. ,122302415338E- 17 193 CIHL2 8. 608372528723E- 17 194 CIPL2 -4. .268585346115E- 18 195 CEHU2 8. 608372528723E- 17 196 CEPU2 4. .268585346115E- 18 197 CIHL3 8. 537170692231E- 18 198 CIPL3 2 . 988009742281E-17 199 CEHU3 1 . 024460483068E- 16 200 CEPU3 4. 695443880727E- 17 201 CIHL4 8. 537170692231E- 18 202 CIPL4 4 . 268585346115E- 18 203 CEHU4 3. 414868276892E- 17 204 CEPU4 4. ,268585346115E- 18 205 CIHL5 8. 537170692231E- 18 206 CIPL5 2. .134292673058E- 17 207 CEHU5 8. 537170692231E- 18 208 CEPU5 4. 695443880727E- 17 209 CIHL6 8. 537170692231E- 18 210 CIPL6 1 . 280575603835E- 17 211 CEHU6 8. .537170692231E- 18 212 CEPU6 1 . 280575603835E- 17 213 CHLS 8. 608372528723E- 17 214 CPLS 4. 268585346115E- 18 215 CUP 1 0. 307583989987 216 CUP2 0. .307583989987 217 CUP3 0. 307583989987 218 CFSB1 0. 307583989987 219 CFSA3 0. 307583989987 220 CFSB3 O. 307583989987 221 CUP6 0. 307583989987 222 CUP7 0. 440929650385 223 CUP8 0. 440929650385 224 CUP9 0. 440929650385 225 CFSB2 0. 440929650385 226 CFSA4 0. 440929650385 227 CFSB4 0. 440929650385 228 CUP 12 0. 440929650385 229 CUP 13 7. 683453623008E- 17 230 CUP 14 4. 268585346115E- 18 231 CUP 15 4. 695443880727E- 17 232 CLP16 -4. 268585346115E- 18 HENRY HALLAM AND PRINCE EDWARD RESERVOIRS PROBLEM - FLP(P) 85 233 CLP 17 -2. 134292673058E-17 234 CLP18 -1 . 280575603835E-17 235 CUH1 3. 806830861936E-17 236 CFSB5 -1 . 848892746612E-32 237 CUH3 1 . 542118983423E-18 238 CFSA6 5. 140396611411E-18 239 CFSB6 5. 140396611411E-18 240 CUH5 -2. 313178475135E-17 241 CFG2 1 .77794213865 242 CSH3 7. 754655459499E-17 243 CSH7 8. 537170692231E-18 244 ONote that the dual s o l u t i o n vector excludes e q u a l i t i e s . 245 0*** Objective ranging is not possible when the tableau Includes e q u a l i t i e s . 246 1R1ght Hand Side Ranging: HENRY HALLAM AND PRINCE EDWARD RESERVOIRS PROBLEM 247 Constraint Lower Bound Rhs Upper Bound 248 249 CIHU1 6.928000 7.528000 • I n f i n i t y 250 CIHL1 7.517000 7.5220O0 7.528O0O 251 CIPU1 O.1470OOO 0.1520000 + I n f i n i ty 252 CIPL1 0.1001265 0.1470000 0.1520000 253 CEHU1 0.6650000 0.6690OO0 O.674OOO0 254 CEHL 1 - I n f i n i t y O.6650OOO 0.67O0OOO 255 CEPU1 0.2420000 0.2450O0O 0.2918735 256 CEPL 1 - I n f i n i ty 0.242O0OO 0.2450000 257 CIHU2 39.43700 45.30200 • I n f i n i t y 258 CIHL2 45.290O0 45.295O0 45.302O0 259 CIPU2 2.534000 2.538000 + I n f i n i ty 260 CIPL2 2.487126 2 . 5340O0 2.53SOOO 261 CEHU2 0.541OO0O 0.5440000 0.5490000 262 CEHL2 - I n f i n i t y 0.5410OOO 0.544O0O0 263 CEPU2 0.1940OO0 0.198O0OO 0.2448735 264 CEPL2 - I n f i n i t y 0.194OO0O 0.1980000 265 CIHU3 7.OO20OO 7.O060O0 + Inf1ni ty 266 CIHL3 5.883835 7.O020OO 7.0O6OOO 267 CIPU3 2.80OO0O0E-02 0.1680O00 + Inf i n i ty 268 CIPL3 0.1171265 0.164OO0O 0.1680000 269 CEHU3 0.4510000 0.4540000 1.572165 270 CEHL3 - I n f i n i t y 0.4510000 0.4540000 271 CEPU3 0.1600O0O O.1630000 O.2098735 272 CEPL3 - I n f i n i t y 0. 160O0OO 0. 1630OOO 273 CIHU4 3.021OOO 3.025000 • I n f i n i t y 274 CIHL4 1.902835 3.021000 3.025000 275 CIPU4 0.1400000 0.1430000 +I n f i n l t y 276 CIPL4 4.6252944E-02 0.14OO0OO 0.1430OOO 277 CEHU4 0.5860000 0.5890000 1.707165 278 CEHL4 -Infln1ty 0.586OOO0 0.589OOOO 279 CEPU4 0.2100000 0.2130O0O O.3067471 280 CEPL4 - I n f i n i t y 0.210OOO0 0.2130000 281 CIHU5 6.849000 6.853000 • I n f i n i t y 282 CIHL5 5.730835 6.8490O0 6.853OO0 283 CIPU5 0.949OOO0 0.952OO0O + Inf i n i ty 284 CIPL5 0.5837969 0.9490000 0.9520OOO 285 CEHU5 0.6750OO0 0.67900OO 1.797165 286 CEHL5 -Inf ini ty 0.67500O0 0.679O0O0 287 CEPU5 0.2420000 O.245O0OO 0.6102031 288 CEPL5 -Inf i n i ty 0.242O0OO 0.2450000 289 CIHU6 8.102OO0 8. 1050O0 • I n f i n i t y 290 CIHL6 6.983835 8.102000 8.1050OO 86 291 CIPU6 0.2570OO0 0.2600000 • I n f i n i t y 292 CIPL6 -0.0 0.257OO0O 0.26OOOOO 293 CEHU6 0.5420000 0.5450OO0 1.663165 294 CEHL6 - I n f i n i t y 0.5420000 0.5450000 295 CEPU6 0.194OO00 0.1970O0O 1.315165 296 CEPL6 - I n f i n i t y 0.1940OO0 0.19700O0 297( CHUS 4.512000 4.515000 • I n f i n i t y 298 CHLS 4.507000 4.512OO0 4.515000 299 CPUS 2.72O0O0 2.7230OO • I n f i n i t y 300 CPLS 2.673126 2.720000 2.723000 301 CUP 1 1.650000 1.850000 1.896234 302 CLP 1 - I n f i n i t y 1.650000 1.850000 303 CUP2 1.394000 1.700000 1.746234 304 CLP2 - I n f i n i t y 1.300000 1.700000 305 CUP3 1.344390 1.800000 1.846234 306 CLP3 - I n f i n i t y 1.200000 1.800000 307 CFSA1 -0.4550327 0.37O0OOO • I n f i n i t y 308 CFS81 1.830000 2.200OO0 2.291222 309 CFSA3 6.5649498E-02 8.OOO0OOOE-02 0.1476677 310 CFSB3 1.244390 1.700000 1.767668 311 CUP6 1.175153 1.700000 2.175954 312 CLP6 - I n f i n i t y 1.100000 1.104000 313 CUP7 2.45OOO0 2.65OO0O 2.695962 314 CLP7 - I n f i n i t y 2.450000 2.650OOO 315 CUP8 2.35OOO0 2.550O0O 2.595962 316 CLP8 - I n f i n i t y 2.35O0O0 2.55O0OO 317 CUP9 2.450000 2.65O0O0 2.695962 318 CLP9 -Inf1n1ty 2.45OOO0 2.65O0OO 319 CFSA2 0.2640105 0.32O0OOO • I n f i n l t y 320 CFSB2 2.580000 2.9O0OO0 2.990169 321 CFSA4 0.1584623 0.196O0OO 0.3730034 322 CFSB4 1.987081 2.455000 2.632003 323 CUP 12 2.350OO0 2.55OO0O 2.882016 324 CLP12 - I n f i n i t y 2.35O0OO 4.900000 325 CUP 13 0.7210000 0.724OOO0 0.7708735 326 CLP13 - I n f i n i t y 0.7210OO0 O.724OOO0 327 CUP 14 39.250OO 39.500O0 39.54687 328 CLP 14 - I n f i n i t y 39.250OO 78.75000 329 CUP 15 3.325000 3.33OOO0 3.376874 330 CLP 15 -InfIn1ty 3.325O0O 3.330000 331 CUP 16 1.795OO0 1.800000 • I n f i n i t y 332 CLP 16 1.365456 1.795000 1.8OO0O0 333 CUP 17 O. 1970OOO 0.20OOOOO •Inf1n1ty 334 CLP17 -0.0 0. 1970000 0.20O00OO 335 CUP 18 0.2480000 0.2500000 • I n f i n i t y 336 CLP 18 -O.O 0.2480OO0 0.25OOOO0 337 CUH1 5.474381 5.500000 5.5O5OO0 338 CLH1 -Inf1nlty -0.0 5.5OOOO0 339 CFSA5 0.1493807 0.1750OOO •Inf1n1ty 340 CFSB5 41.55938 41.58500 41.59000 341 CUH3 7.337126 7.384OO0 7.813544 342 CLH3 - I n f i n i t y 7.200O0O 14.584O0 343 CFSA6 7.4252944E-02 9.O0OO0OOE-02 0.1642529 344 CFSB6 6.370253 6.420O0O 6.494253 345 CUH5 4.450253 4.5OO0O0 4.929544 346 CLH5 - I n f i n i t y -0.0 4.500000 347 CUH6 4.387253 4.437000 •Inf1n1ty 348 CLH6 -Inf m i ty 2.400OO0 4.387253 87 349 CFG2 5.317660 5.40O0OO 5.864038 350 CSH2 5.865000 9.026OO0 • I n f i n i t y 351 CSP2 2.898000 3.380OOO + Inf i n i t y 352 CCH1 - I n f i n i t y -0.0 5.865O0O 353 CCP1 - I n f i n i t y -0.0 2.898000 354 CSH3 9.000381 9.0260O0 9.031000 355 CSP3 0.7300000 3.38OOO0 • Inf t n l t y 356 CCH2 - I n f i n i t y -0.0 9.026000 357 CCP2 - I n f i n i t y -0.0 3.074000 358 CSH4 8.190000 9.026000 • I n f i n i t y 359 CSP4 2.679000 3.380000 • I n f i n i t y 360 CCH3 - I n f i n i t y -0.0 8.190O00 361 CCP3 - I n f i n i t y -0.0 2.679000 362 CSH5 4.186253 9.026000 • I n f i n i t y 363 CSP5 3.182000 3.380000 • I n f i n i t y 364 CCH4 - I n f i n i t y -0.0 9.026000 365 CCP4 - I n f i n i t y -0.0 2.146747 366 CSH6 5.856253 9.026000 • I n f i n i t y 367 CSP6 3.373000 3.380OO0 • Inf i n l t y 368 CCH5 - I n f i n i t y -0.0 5.856253 369 CCP5 - I n f i n i t y -0.0 2.950456 370 CSH7 8.976253 9.026000 10.14417 371 CSP7 2.950456 3.380000 • I n f i n i t y 372 CCH6 - I n f i n i t y -0.0 9.026000 373 CCP6 -Inf m i ty -0.0 2.899709 374 ONote that r i g h t hand side ranging Is not done f o r equal 111es. 375 O 88
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Fuzzy linear programming and reservoir management Kaseke, Evans 1987
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Title | Fuzzy linear programming and reservoir management |
Creator |
Kaseke, Evans |
Publisher | University of British Columbia |
Date Issued | 1987 |
Description | The presence of imprecision in parameter specification of water resources management problems leads to the formulation of fuzzy programming models. This thesis presents the formulation of a two-reservoir system problem as a fuzzy L.P. model. The aim is to determine if larger monetary benefits, over and above the usual benefits, can be obtained from the system. The other aim is to determine if desired industrial and domestic water allocations, as well as outflows for selected periods can be achieved. The problem is formulated as a conventional L.P. model. Then selected water allocations and outflows are fuzzified resulting in a fuzzy L.P. model. The alternative fuzzy L.P. model is also presented. Monetary benefits larger than those from the conventional L.P. were obtained through the fuzzy L.P. model. The desired water allocations and outflows were also realised for selected periods. Sensitivity information was obtained for fuzzy and non-fuzzy constraints. The alternative fuzzy L.P. model did not give additional valuable information than that obtained from the initial fuzzy L.P. model. |
Subject |
Reservoirs -- Zimbabwe Reservoirs -- Management Linear programming |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-07-20 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0062617 |
URI | http://hdl.handle.net/2429/26708 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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