Using ranking functions in multiobjective fuzzy linear programming 1
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1 Fuzzy Sets and Systems 111 (2000) Using ranking functions in multiobective fuzzy linear programming 1 J.M. Cadenas a;, J.L. Verdegay b a Departamento de Informatica, Inteligencia Articial y Electronica, Universidad de Murcia, Espinardo, Murcia, Spain b Departamento de Ciencias de la Computacion e Inteligencia Articial, Universidad de Granada, Granada, Spain Received October 1998 Abstract Multiobective mathematical programming problems, in particular the vector optimization problems, dene a well known and studied area because of its relevance to numerous practical applications. In this paper vector optimization problems with a fuzzy nature are considered. In these problems usually it is assumed that all the obective functions involved come from the same decision maker. The problem considered here assumes, however, that the obective functions can be dened by dierent decision makers, and that the coecients in each of these obective functions are fuzzy numbers. Hence, solution methodologies for these multiobective fuzzy mathematical programming problems, using dierent ordering methods ranking fuzzy numbers, are proposed. As an illustration a bi-obective model for land use is presented. c 2000 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy linear programming; Vector optimization problem; Fuzzy numbers; Ordering methods 1. Introduction Making decisions involving multiple obectives is a daily task for a lot of people in the more diverse elds, and hence multiple obective decision making problems have dened a very well-studied topic in the general area of decision making theory. In particular, multiobective decision making problems which can be modeled as mathematical programming (MP) problems also are very well known. The involvement of dierent kinds of fuzziness in these problems is a matter which also has received Corresponding author. 1 A rst version of this paper was presented in the VII IFSA World Congress (Prague, 1997). Work supported by proect 1FD C03-01 and -02. a great deal of work since the early 1980s, as it is very frequent that decision makers have some lack of precision in stating some of the parameters involved in the model [3,4,6,7,11,12,15]. In this paper we consider a linear multiobective mathematical programming problem in which the coecients dening the obective functions are given as fuzzy numbers, and where moreover, each obective can be dened by a dierent decision maker, with which the respective ways of comparing the fuzzy numbers involved are to be taken into account in order to give operational methodologies for solving the problem. Consequently in Section 2, the classical vector optimization problem is briey surveyed as the conceptual basis. Then fuzzy multiobective optimization /00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved. PII: S (98)
2 48 J.M. Cadenas, J.L. Verdegay / Fuzzy Sets and Systems 111 (2000) problems are introduced and described. In Section 4, the focus is on the methodologies to solve the case of fuzzy coecients in the obective functions. Finally in Section 5, an illustrative example for land use, from [8], is presented and then addressed in accordance with the previously shown methodology approach. 2. Multiobective mathematical programming problems Typically, a MP problem is concerned with solving a model like, f(x) x X; (1) where x is an N-vector of decision variables, f is a real-valued function, usually called obective function, and X is a constraint set, X = {x=x R N ;h i (x)=0;i=1;2;:::;p; g (x)60; =1;2;:::;q}; where h i and g are real-valued functions dened on X. In order to solve the problem one needs to nd x X such that, f(x )6f(x); x X and then x is called a global optimum of (1). There are, however, many practical situations which involve multiple obectives. An important class of multiobective decision problems is the well-known vector optimization problem (VOP) or multiobective optimization problem. Key works on this subect are due to Pareto [10], introducing the concept of noninferior solution, Kuhn and Tucker [9], giving necessary and sucient conditions for non-inferiority, Zadeh [14], in referring to a solution of a VOP as noninferior, Chankong and Haimes [5], providing an account of multiobective theories and methodologies, and many others. A VOP is addressed as [f 1 (x);:::;f n (x)] x X: In particular, when the linear case is considered, that is, when f 1 ;:::;f n ;h i ;g are linear, as it will be in this paper, the model becomes a linear VOP which is typically stated as [c 1 x; c 2 x;:::;c n x] Ax6b; x 0 (2) where c ; =1;:::;n is an N vector of cost coecients, A an m N-coecients matrix of constraints (the technological matrix), and b an m-vector of demand (resource) availability. In the following, for the sake of simplicity, we will refer this linear VOP simply as VOP. Solving a VOP implies nding its set of noninferior solutions, that is, the set of x such that there exists no other x X such that f (x)6f (x ) for all =1;:::;n with strict inequality for at least one. From a practical point of view, however, one needs to relate this concept to an operational one, and the bestknown way is to characterize the noninferior solutions as optimal solutions. There are two main approaches to do it: the weighting approach and the constraint approach. By means of the rst approach [5], the weighting problem is dened for some vector of weights as, n w (c x) =1 Ax6b; x 0; (3) where w 0 and n =1 w = 1. In this linear case, as it is well known, all noninferior solutions can be found by solving (3). The auxiliary problem corresponding to the second approach is usually called the kth-obective -constraint problem, [5], and it is addressed as follows: c k x c x6 ;=1;:::;n; k; Ax6b; x 0; (4) where =( 1 ;:::; k 1 ; k+1 ;:::; n ). In this case all noninferior solutions can be found by solving the constraint problem (4).
3 J.M. Cadenas, J.L. Verdegay / Fuzzy Sets and Systems 111 (2000) Fuzzy multiobective optimization problem Since the pioneer papers by Bellman and Zadeh in 1970 [1] and later by Zimmermann in 1978 [15] a great deal of work has been devoted to solve the fuzzy multiobective optimization (FMO) problem. In almost all of the cases, and in a parallel way to the classical VOP, the research has been oriented towards the characterization of noninferior solutions in this fuzzy case. In a fuzzy environment, however, we have dierent possibilities to address an FMO problem, extending in all the cases and generalizing the conventional VOP. These extensions and generalizations are as follows: 3.1. Fuzziness in the constraints Two dierent models can be considered. In the rst the fuzzication of (2) leads to the following model: [c 1 x; c 2 x;:::;c n x] (5) Ax6 g b; x 0; where the symbol 6 g indicates, as usual, that there exist membership functions i : R [0; 1]; i=1;:::;m expressing for each x R N the accomplishment degree of the ith constraint. In the second the fuzzication of (2) is translated into both the coecients of the technological matrix and the right-hand side. Then the model is dened as, [c 1 x; c 2 x;:::;c n x] (6) A f x6 f b f ;x 0 with A f an m N-matrix of fuzzy numbers, b f an m- vector of fuzzy numbers, and the symbol 6 f standing for a fuzzy relation ranking fuzzy numbers, [2]. In the following, the set of real fuzzy numbers will be denoted by F(R). Thus a real fuzzy number will be denoted by A f. When, in particular, we consider triangular fuzzy numbers these will be represented by A f =(A; A; A) with a clear enough meaning Fuzziness in the obective functions As in the above case, here also two dierent possibilities arise. On the one hand, we can suppose that the coecients in the obective functions are given by fuzzy numbers. Then the corresponding model can be dened as [c f 1 x; cf 2 x;:::;cf nx] (7) Ax6b; x 0; where each c f ; =1;:::;n is an N-vector of fuzzy numbers. On the other hand, and as originally stated by Zimmermann [15] the existence of fuzzy goals can be assumed. Then the problem is dened as Find x R N such that c x 6 g z ;=1;:::;n; Ax 6 g b; x 0; (8) where 6 g has the same meaning as in (5) and z s are aspirations levels xed, together with its respective membership functions, by the decision maker. As far as solution methodologies are concerned, on the one hand FMO problems (5) and (6) can be solved, in a similar way, by transforming its respective fuzzy constraint sets into conventional ones, to obtain auxiliary classical models like (2). Thus for instance, if (5) is considered and linear membership functions on the constraints are assumed, it is easy to obtain the following formal problem: [c 1 x; c 2 x;:::;c n x] Ax6 1 (); (9) x 0; [0; 1]; where 1 is an m-vector constituted by the inverse of the membership functions i, i =1;:::;m. Hence for each [0; 1], (9) is a VOP like (2). On the other hand (8) can be elegantly solved by means of the well known Zimmermann s approach [15]. The case of fuzzy coecients in the obective functions, as stated in (7), is however not so straightforward to solve, and therefore it will be focused in the next section. 4. The case of fuzzy coecients in the obective functions Let us consider (7) and, for the sake of simplicity, assume that all the fuzzy numbers giving the fuzzy
4 50 J.M. Cadenas, J.L. Verdegay / Fuzzy Sets and Systems 111 (2000) costs of the obective functions have a triangular shape. From a theoretical point of view to solve (7) a constraint-based approach can be used. Then the parallel problem to (4), in a rst step, is dened as c f 1 x c f x6 ;=2;:::;n; Ax6b; x 0: (10) But some of the constraints in (10) do not make much sense as they involve fuzzy values, real values and a conventional inequality relation comparing these values. It seems therefore more appropiate to propose the following model: c f 1 x c f x6 f f ;=2;:::;n; Ax6b; x 0 (11) with a clear enough meaning from (6). About this problem (11) several considerations are to be made. (a) As it is clear (11) generalized in a trivial way problem (4). In fact, when the fuzzy numbers involved in (11) become crisp numbers the relation 6 f becomes the conventional inequality relation 6 between real numbers. (b) The relation 6 f ranking fuzzy numbers can be any one in the wide list of this kind of relations, [13], but what is much more important, is that besides we can also consider dierent ranking functions. More concretely, we can assume that the n obective functions come from dierent decision makers. Then, as it is clear that not all the decision makers involved have to agree on the same comparison criterion for fuzzy numbers, the problem (11) permits to represent these dierent ways of comparison by introducing the different ranking functions in the respective constraint obective. (c) Although here we are considering the linear case, it is not theoretically inconvenient to extend and to dene its nonlinear counterpart. From this point of view it becomes clear that when we have an FMO problem like (7), model (11) provides us with a theoretical framework to nd a solution (or a set of solutions), but not with a sole auxiliary problem solving (7) as we will show in the following. In fact, to give an operational form to (11) we will resort to the dierent existent indices of comparison between fuzzy numbers [2,13]. In this way, as it is usual, if I : F(R) [0; 1] is an index to compare fuzzy numbers, then A f ;B f F(R); A f 6B f I(A f )6I(B f ) and therefore according to the index I to be used, a dierent auxiliary model to (11) is to be obtained. As a trivial illustration, let us consider two triangular fuzzy numbers A f =(A; A; A) and B f =(B; B; B) and the ranking relation dened as A f 6 T B f A6B, then (11) becomes c 1 x c x6 ;=2;:::;n; Ax6b; x 0 which is obviously identical to (4). In general, the auxiliary model solving (11), and consequently (7), is dened as I(c f 1 x) I(c f x)6i(f );=2;:::;n; Ax6b; x 0 or much more generally as, I 1 (c f 1 x) I (c f x)6i ( f );=2;:::;n; Ax6b; x 0; (12) (13) where I, =1;:::;n is the index with which the th decision maker involved in the problem wants to compare the fuzzy numbers in the obective function provided by him (her). Therefore if we assume, in the easiest hypothesis, that all the decision makers agree on the use of a common comparison index, the following auxiliary models could be obtained from (12) as some examples. (a) From the rst index of Yager [13], dened as 1 F 1 (A f 0 )= g(z) A f(z)dz 1 0 A f(z) ;
5 J.M. Cadenas, J.L. Verdegay / Fuzzy Sets and Systems 111 (2000) where the weight g(z) is a measure of the importance of the value z (if we assume linear weights, that is g(z)=z, then F 1 (A f ) represents the center of gravity of the fuzzy set A f ), the model obtained is (c 1 + c 1 + c 1 )x (c + c + c )x6( + + ); Ax6b; x 0; =2;:::;n: But taking into account that + is, in any case, a constant real number, R, to be xed by the decision maker, this model becomes (c 1 + c 1 + c 1 )x (c + c + c )x6( + ); Ax6b; x 0; =2;:::;n: (b) From the index of Adamo, [13], dened as F (A f ) = max{z= A f(z) } for a given threshold [0; 1], we obtain (c 1 (c 1 c 1 ))x (c (c c ))x6 ( ); Ax6b; x 0; [0; 1]; =2;:::; n: As above, taking into account that is a constant real number, R, to be xed by the decision maker, the problem becomes (c 1 (c 1 c 1 ))x (c (c c ))x6 +(1 ); Ax6b; x 0; [0; 1]; =2;:::; n and similarly for other comparison indices. It is important to remark that, although the former VOP is linear, we will not be able to obtain a linear VOP in all the cases and this is so because of the nonlinear nature of some of the possible comparison indices to be used. This is the case for instance when we select the second index of Yager [13], that it is dened as F 2 (A f ) = max z S min(z; A f(z)): In this case the auxiliary model is obtained as c 1 x (c 1 c 1 )x +1 c x (c c )x+1 6 ( )+1 ;=2;:::;n; Ax6b; x 0: As it is evident, when each decision maker uses a dierent comparison index, numerous possibilities to concrete (13) appear. Consider, for instance, the fuzzy biobective optimization problem dened as [c f 1 x; cf 2 x] 2x 1 x 2 4; 5x 1 +2x 2 15; x 1 +2x 2 6; x i 0 with c f 1 =(3f ;1 f ), c f 2 =(2f ;3 f ), and 3 f =(3;1;5), 1 f =(1;0;2), 2 f =(2;0;3). Suppose there are involved two decision makers, each one dening a membership function and each one xing, as comparison index, the rst index of Yager and the index of Adamo, respectively. Then the corresponding version of (13), if, for example, an amplitude equal to 1 is xed for the fuzzy value 2,is 1 3 (9x 1 +3x 2 ) (3 )x 1 +(5 2)x 2 2 +(1 ); 2x 1 x 2 4; 5x 1 +2x 2 15; x 1 +2x 2 6; x i 0 which can be easily solved for each [0; 1]. 5. Illustrative example Let us consider the generic LP model for land use proposed by Greenberg [8]. Accordingly, we will
6 52 J.M. Cadenas, J.L. Verdegay / Fuzzy Sets and Systems 111 (2000) assume i producers and markets. There will be k different methods of production, s soils, h chemicals and p products. The variables, representing activities on the production and the distribution are, respectively: (a) X ipk, which allocates land in region i to make product p by method k, and (b) T pi to represent the transports of product p from region i to market. Some of the constraints which can be taken into account are as follows: For land use, as the total area to be used will be limited, we will have part in the corresponding obective functions are dicult to x with precision because of dierent reasons: climatological aspects, level of the demands, etc., and therefore it is clear these coecients can be modeled as fuzzy numbers. Let us denote (cx) f ipk the production costs in each case, and (ct) f pi the transportation benets for the dierent carriers. Then, the two obective functions can be modeled as (cx) f ipk X ipk i; L i = X ipk : and As far as the balance between production and sales, i.e. the amounts to be transported to the dierent markets, is concerned, if R ipk denotes the rate of product p per unit in the zone i, using the method k, R ipk X ipk T pi =0: k It is also assumed that the demand for product p in each market (denoted by d p ) satises T pi d p : Finally, as far as damage is concerned, two dierent constraints are to be considered. First, D i 6 a psk is X ipk ; p; s; k where a psk is the rate of soil damage in producing p with soil class s, and is will be equal to 1 if region i has soil class s, and 0 otherwise. Second, C ih b phk X ipk ; where b phk is the chemical h used by, or produced from, method k to make p, and it is to be seen as a constraint on the damage because of the nal quality of the products. As it is evident, we could consider some kind of fuzziness in any of these restrictions, but that is not the main aim in this example but to focus on the case of fuzzy costs in the obective. To this purpose, at least two dierent obective functions can be approached. First concerning one the production costs, and second regarding the distribution benets. It is apparent that any of the coecients taking Max (ct) f pi T pi: p; i; Because of the linearity of these two functions, we can clearly address the obective as (cx) f ipk X ipk ; (ct) f pi T pi p; i; i; and therefore the model is formulated as (cx) f ipk X ipk ; (ct) f pi T pi p; i; i; Li= X ipk ; R ipk X ipk k T pi d p ; Di 6 p; s; k a psk is X ipk ; T pi =0; C ih b phk X ipk ; X ipk ;T pi 0; is = {0; 1}: As is evident, producers and carriers have dierent obectives which may be even contradictory. Consequently, if each of the i producers wants to use the comparison index I i, for the fuzzy coecients corresponding to his (her) products and lands, and the carriers want to use the comparison indices I, respectively, then according to (13), the resulting
7 J.M. Cadenas, J.L. Verdegay / Fuzzy Sets and Systems 111 (2000) problem solving this model for land use is nally addressed as I i (cx) f pk X ipk ; i; I ((ct) f pi )T pi I ( f pi ); p; i; p; i; Li = X ipk ; R ipk X ipk k T pi d p ; Di 6 p; s; k a psk is X ipk ; T pi =0; C ih b phk X ipk ; X ipk ;T pi 0; is = {0; 1} which is general enough, and even could be much more general if fuzzy constraints were assumed. 6. Conclusion Multiple obective decision making problems when they can be formulated as mathematical programming problems have been considered. When a fuzzy environment is assumed, dierent vector optimization problems appear. In this case fuzzy costs dening the obective functions have been considered. Also, it has been assumed that each obective is given by a different decision maker. Then several models solving the former fuzzy multiobective optimization problem have been proposed and, as particular cases of conventional vector optimization problems, also solved. As an illustration, a biobective linear programming problem for land use, involving fuzzy coecients in the obective functions, has been described. This last model, based on a former generic problem in [8], is to be seen as a rst approximation to the use of fuzzy mathematical programming models for controlling environmental quality in dierent contexts (air, water or land). References [1] R.E. Bellman, L.A. Zadeh, Decision making in a fuzzy environment, Management Sci. 17 (B) 4 (1970) [2] L. Campos, J.L. Verdegay, Linear programming problems and ranking of fuzzy numbers, Fuzzy Sets and Systems 32 (1989) [3] Ch. Carlsson, R. Fuller, Fuzzy multiple criteria decision making. Recent developments, Fuzzy Sets and Systems 78 (1996) [4] S. Chanas, Fuzzy programming in multiobective linear programming: a parametric approach, Fuzzy Sets and Systems 29 (1989) [5] V. Chankong, Y.Y. Haimes, Multiobective Decision Making, Theory and Methodology, North-Holland, Amsterdam, [6] M. Delgado, J. Kacprzyk, J.L.Verdegay, M.A. Vila (Eds.), Fuzzy Optimization: Recent Advances, Physica-Verlag, Wurzburg, [7] M. Delgado, J.L.Verdegay, M.A. Vila, A Possibilistic Approach for Multiobective Programming Problems Eciency of Solutions, in: R. Slowinski, J. Teghem (Eds.), Stochastic versus Fuzzy Approaches to Multiobective Mathematical Programming under Uncertainty, D. Reidel Publ. Co., Dordrecht, 1990, pp [8] H.J. Greenberg, Mathematical programming models for environmental quality control, Oper. Res. 43(4) (1995) [9] H.W. Kuhn, A.W. Tucker, Nonlinear programming, Proc. 2nd Berkeley Symp. on Mathematics, Statistics and Probability. University of California Press, Berkeley, 1951, pp [10] V. Pareto, Cours deconomie Politique, Rouge, Laussanne, Switzerland, [11] M. Sakawa, Fuzzy sets and interactive multiobective optimization, Applied Information Technologies Series, Plenum Press, New York, [12] R. Slowinski, J. Teghem (Eds.), Stochastic versus Fuzzy Approaches to Multiobective Mathematical Programmming under Uncertainty, D. Reidel Publ. Co., Dordrecht, [13] X. Wang, E. Kerre, On the classication and the dependencies of the ordering methods, in: D. Ruan (Ed.), Fuzzy Logic Foundation and Industrial Applications, International Series in Intelligent Technologies, Kluwer, Dordrecht, 1996, pp [14] L.A. Zadeh, Optimality and nonscalar-valued performance criteria, IEEE Trans. Automat. Control AC-8 (1963) [15] H.J. Zimmermann, Fuzzy programming and linear programming with several obective functions, Fuzzy Sets and Systems 1(1) (1978)
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