application of the fuzziness concept to the systems to be studied may play a significant ..... and the choice of the approach is a prerogative of the decision maker.
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Fuzzy Preference Modeling and Its Application to Multiobjective Decision Making P. YA. EKEL*t AND M. R. SILVA$ G r a d u a t e P r o g r a m in Electrical E n g i n e e r i n g Pontifical Catholic University of M i n a s Gerais Av. D o m Jose G a s p a r , 500 30535-610, Belo Horizonte, M G , Brazil ekel@pucminas, br marinapuc©hotmail, com F. SCHUFFNER NETO D e p a r t m e n t of Electronics E n g i n e e r i n g a n d T e l e c o m m u n i c a t i o n s Pontifical Catholic University of M i n a s Gerais Av. D o m Jose G a s p a r , 500 30535-610, Belo Horizonte, M G , Brazil levisto©ig, com. br R. M. PALHARES D e p a r t m e n t of Electronics E n g i n e e r i n g Federal University of M i n a s Gerais Av. A n t o n i o Carlos, 6627 31270-010, Belo Horizonte, M G , Brazil palhares¢cpdee, ufmg. br Abstract--This paper reflects results of research into the construction and analysis of (X, R) models within the framework of a general approach to solving optimization problems with fuzzy coefficients. This approach involves a modification of traditional mathematical programming methods and is associated with formulating and solving one and the same problem within the framework of mutually interrelated models. The use of the approach allows one to maximally cut off dominated alternatives. The subsequent contraction of the decision uncertainty region is based on reducing the problem to models of multiobjective choosing alternatives in a fuzzy environment with the use of fdzzy preference relation techniques for analyzing these models. Three techniques for fuzzy preference modeling are discussed in the paper. The first technique is based on the construction of membership functions of subsets of nondominated alternatives with simultaneous considering of all criteria (fuzzy preference relations). The second technique is o f a lexicographic character and consists of step-by-step introducing of fuzzy preference relations. The third technique is based on aggregating membership functions of subsets of nondominated alternatives corresponding to each preference relation. These techniques have served for developing a corresponding system for multiobjective decision making (MDMS). C-t-+ windows of the MDMS axe presented for input, output, and some intermediate procedures. The results of the paper are of a universal character and axe already being used to solve power engineering, naval engineering, and management problems. (~) 2006 Elsevier Ltd. All rights reserved. tAuthor to whom all correspondence should be addressed. :~Reseaxch of these authors was supported by the CNPq Grant No. 550408-2002/9. 0898-1221/06/$ - see front matter (~) 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2006.08.012
Typeset by A A~-TEX
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U e y w o r d s - - U n c e r t a i n t y factor, Fuzzy coefficients, Interrelated models, Multiobjective decision making, Fuzzy preference relations.
1. I N T R O D U C T I O N In the process of posing and solving a wide range of problems related to the design and control of complex systems, one inevitably encounters different kinds of uncertainty [1]. Taking into account the uncertainty in shaping the mathematical models should be inherent to the practice of systems analysis. This serves as a means for increasing the adequacy of these models and, as a result, their credibility and the factual effectiveness of solutions based on their analysis. At present, investigators have doubts about the validity or, at least, the expediency of including the uncertainty within the framework of models that are shaped by traditional approaches. In general, these approaches do not ensure an adequate or sufficiently rational consideration of the uncertainty throughout the entire spectrum of its manifestations. Considering this, the application of the fuzziness concept to the systems to be studied may play a significant role in overcoming the indicated difficulties. Use of this concept opens a natural path to giving up "excessive" precision, which is inherent in the traditional approaches to constructing models while preserving reasonable rigor. Besides, operation with a fuzzy parameter space allows one not only to be oriented toward the contextual or intuitive aspect of qualitative analysis as a fully substantiated process, but, by means of fuzzy set theory [2,3], also to use this approach as a reliable source for obtaining quantitative information. Finally, fuzzy set theory allows one to reflect more adequately the essence of the decision-making process. In particular, since the "human" factor has a noticeable effect in making decisions, it is expedient to use the important linguistic aspect of fuzzy set theory [2,3]. Nevertheless, in solving problems under condition of uncertainty it is necessary to exert maximum efforts in seeking the possibilities for overcoming the uncertainty. This is done, for example, by using information of informal character (based on experience, knowledge, and intuition of specialists) or, in the general case, by aggregating information arriving from various sources of both formal and informal nature [41. Here we are essentially speaking of the fact that the characteristic of uncertain information (usually specified by intervals) may and should be supplemented by specifically adopted, well-founded assumptions as to differentiated reliability of different values of uncertain parameters. Such supplementation represents a generalization of the interval specification of information and serves as a technique for removing the uncertainty, but it requires the use of the corresponding apparatus. The apparatus of fuzzy set theory can serve as the latter. Its utilization in problems of complex system optimization offers advantages of both fundamental nature (based on the possibility of validly obtaining the more effective, less "cautious solutions") and computational character [4,5]. When using fuzzy set theory, certain fundamental problems arise in the comparison of alternatives on the basis of fuzzy values of objective functions, consideration of constraints containing fuzzy coefficients, development of fundamental principles, and concrete methods of solving associated optimization problems. Below we discuss some approaches to solving these problems and propose ways of implementing these approaches as applied to discrete programming models with fuzzy coefficients both in the objective function and constraints with the use of modification of the algorithms of discrete optimization and contraction of arising decision uncertainty regions on the basis of procedures of multiobjective choosing alternatives in a fuzzy environment. 2. P R O B L E M
FORMULATION
AND
TRANSFORMATION
Numerous problems related to the design and control of complex systems may be formulated as follows: maximize f ( x l , • • •, x~),
(1)
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181
subject to the constraints
~ j ( X l , . . . , x n ) C bd,
j = 1,...,m,
(2)
where the objective function (1) and constraints (2) include fuzzy coefficients, as indicated by the tilde (-) symbol. Given a maximization problem (1),(2), we can state the following problem: (3)
minimize/(Xl,..., x,~),
subject to the same constraints (2). An approach [1] to handling the constraints such as (2) involves replacing each of them by a finite set of non fuzzy constraints. Depending on the sense of the problem, it is possible to convert constraints (2) to equivalent nonfuzzy analogs
gj(xl,. ..,x~) ~ bj,
j = 1 , . . . , d ' >_ m,
(4)
g j ( x l , . . . , x , ) >_ bj,
j = 1,. . . , d " > m.
(5)
or
Problems with fuzzy coefficients only in the objective functions can be solved by modifying traditional optimization methods. As an example, we consider the analysis of fuzzy discrete optimization models. 3. A N A L Y S I S COEFFICIENTS
OF IN
MODELS WITH FUZZY OBJECTIVE FUNCTIONS
The desirability of allowing for constraints on the discreteness of variables in the form of increasing or decreasing discrete sequences
(6)
xs~,ps,,T~,... ,si = 1,... ,ri
has been validated in [6,7]; here Psi, T ~ , . . . are characteristics required to form objective functions, constraints, and their increments that correspond to the 8 t h standard value of the variable x~. It permits one to formulate the maximization problem as follows. Assume we are given discrete sequences of type (6). From these sequences it is necessary to choose elements such that the objective maximize / (xst, p,~, rs~, • • •, xs., p,., ~-~.,...)
(7)
is met while satisfying the constraints
gj (x81,Psl,T81,... ,xs,,,ps.,%,~,...) ~ bj,
j ----I,... ,d'.
(8)
Given (6)-(8), we can formulate a problem of minimization as followh: minimize / (x81, Psi, ~'s~, • • •, xs., p~., 7-8.,...),
(9)
while satisfying the constraints
gj(xs,,ps,,Ts,,...,xs=,ps~,~-8=,...)>bj,
j = 1,...,d".
(I0)
The solution of problems (6)-(8) and (6), (9), (10) is possible on the basis of modifying the generalized algorithms of discrete optimization [6,7]. These algorithms are based on a combination of formal and heuristic procedures and allow one to obtain quasioptimal solutions after a small
P. YA. EKEL et al.
182
number of steps, thus overcoming the NP-completeness. Considering t h a t the minimization problem is more difficult than the maximization problem [6,7], we shall describe the algorithm of solving problem (6), (9), (10) on the basis of the results of [7]. Assume t h a t at step t, variable xi is at its discrete level x(t~) with its parameters at the respective levels p~0, T(! ), . .Introducing ~(t) . . s, = ix(t) t s , , p s(t) , , r&(t) , . . . } and assuming that constraints (10) are already normalized as ,
,
•
g~0) (081,. .. ,0s~)
= gj
(0~,, . . . ,0~o) -~jb
>>_b,
j = l , . .. , d " ,
with the use of a normalized factor b > 0, we can write the following algorithm. 1. The components of the constraint increment vector {AG~ t)} are evaluated as
~G~')= Z~g~,),
~ ~ I('),
j ~ J(~).
01)
J In
(11),
~(~) g3~ = [gj(os,,... , 0 ~ + 1 , . . . ,o,o)- g J (o.~" " .,o~,," ..,o,o)]b~tl) b '
j E fit),
i E I (t),
where j ( t ) is the set of the constraints at the t th step (for t = 1 we have j E J d , , Jd,, is the initial set of constraints); I (t) is the set of variables at the t th step (for t = 1 we have i ~ I~, I~ is the initial set of variables); b~.t - l ) = b~°) = bi. 2. The components of the increment vector of the objective function { A ] (t) } are calculated as
models within the framework of a general approach to solving optimization problems with fuzzy coefficients has been considered. This approach is associated with modifying traditional mathematical programming methods and consists of formulating and solving one and the same problem within the framework of mutually interrelated models. The use of the approach allows one to maximally cut off dominated alternatives. The subsequent contraction of the decision uncertainty region is based on reducing the problem to models of multiobjective choosing alternatives in a fuzzy environment with the use of fuzzy preference relation techniques for analyzing these models. Three techniques for fuzzy preference modeling are discussed in the paper. The first technique is based on the construction
Fuzzy Preference Modeling
195
of membership functions of subsets of nondominated alternatives with simultaneous considering all criteria (fuzzy preference relations). The second technique is of a lexicographic character and consists of step-by-step introducing of fuzzy preference relations. The third technique is based on aggregating membership functions of subsets of nondominated alternatives corresponding to each preference relation. The choice of the technique is a prerogative of a decision maker. The techniques have served for developing a corresponding system for multiobjective decision making (MDMS). C + ÷ windows of the MDMS have been presented for input, output, and some intermediate procedures. All techniques have been illustrated by considering an example. The results of the paper are of a universal character and are already being used to solve power engineering, naval engineering, and management problems.
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