An interactive method using genetic algorithm for multi-objective optimization problems modeled in fuzzy environment

Expert Systems with Applications 38 (2011) 1659–1667

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

An interactive method using genetic algorithm for multi-objective optimization problems modeled in fuzzy environment Kusum Deep a, Krishna Pratap Singh a,⇑, M.L. Kansal b, C. Mohan c a

Department of Mathematics, Indian Institute of Technology, Roorkee 247 667, Uttrakhand, India Water Resources Development and Management, Indian Institute of Technology, Roorkee 247 667, Uttrakhand, India c Ambala College of Engineering and Applied Research, Ambala, Haryana, India b

a r t i c l e

i n f o

Keywords: Fuzzy multi-objective optimization Interactive methods Nonlinear programming

a b s t r a c t In this paper, an interactive approach based method is proposed for solving multi-objective optimization problems. The proposed method can be used to obtain those Pareto-optimal solutions of the mathematical models of linear as well as nonlinear multi-objective optimization problems modeled in fuzzy or crisp environment which reasonably meet users aspirations. In the proposed method the objectives are treated as fuzzy goals and the satisfaction of constraints is considered at different a-level sets of the fuzzy parameter used. Product operator is used to aggregate the membership functions of the objectives. To initiate the algorithm, the decision maker has to specify his(er) preferences for the desired values of the objectives in the form of reference levels in the membership space. In each iterative phase, a single objective nonlinear (usually nonconvex) optimization problem has to be solved. It is solved using real coded genetic algorithm, MI-LXPM. Based on its outcomes, the decision maker has the option to modify, if felt necessary, some or all of the reference levels in the membership function space before initiating the next iterative phase. The algorithm is stopped where user’s aspirations are reasonably met. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction In fuzzy optimization, we search for the best possible solution which can be achieved in the presence of incomplete or imprecise or vagueness in information. Mathematical models of majority of real life optimization problems are in fact fuzzy in nature even though these are often assumed to be crisp for the convenience of their solution. Many of these are also multi-objective optimization problems in which more than one conflicting objectives are to be optimized satisfying the same set of constraints. In a multi-objective optimization problem, an optimal solution which simultaneously optimizes all the objectives, and that too when the problem is modeled in a fuzzy environment, is rarely possible. In such situations one usually tries to search for a solution which is as close to the customer’s (decision maker’s) expectations as possible. Search of such a ‘satisfying solution’ requires solving the multi-objective fuzzy optimization problem iteratively in an interactive manner wherein the decision maker (DM) is initially asked to specify his(er) preferences and expectations. Based on ⇑ Corresponding author. Tel.: +91 1332 285339; fax: +91 1332 275360. E-mail addresses: [email protected] (K. Deep), [email protected] (K.P. Singh), [email protected] (M.L. Kansal), [email protected] (C. Mohan). 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.07.089

these, the problem is solved and the DM is provided with a possible solution. If the DM is satisfied with this solution the problem ends there, otherwise s(he) is asked to modify his(er) earlier preferences in the light of the results obtained. This iterative procedure is continued till a solution is achieved which is as close to DM’s expectations as possible. In this paper, we propose an interactive method for solving multi-objective optimization problems modeled in fuzzy environment. Fuzzy multi-objective optimization problems have been studied in literature by several authors. In Bellman and Zadeh (1970) famous model, the objectives and the constraints are treated equally as fuzzy goals. Later on Zimmermenn (1978), Ravi, Reddy, and Zimmermann (2000), Huanga, Gub, and Duc (2006) and some other authors (Kahraman, 2008) have used this approach for the solution of optimization problems. Mohan and Nguyen (1999) have used this approach to solve optimization problems in which the parameters involved in the problem are imprecise/uncertain (fuzzy numbers are used to represent such parameters). Sakawa and Yano (1989), Sakawa and Yauchi (2001), Mohan and Nguyen (1998) and Mohan and Verma (2003) have considered only the objectives as fuzzy whereas the constraints are used as satisfying criteria. Most of the solution techniques currently available in literature either aggregate the objectives into a single objective using weighted sum approach (Mohan & Verma, 2003) or max–min

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(or min–max) approach (Mohan & Nguyen, 1998, 1999; Nguyen, 1996; Sakawa & Yano, 1989; Sakawa & Yauchi, 2001). Both of these approaches have their limitations. For instance the min (or max) aggregation operator is noncompensatory in nature. Other aggregation operators, have also been suggested in literature. One of these are the product operator (Zimmermenn, 1978). However, due to nonlinearity of the product operator (even when one uses liner membership functions to define fuzzy goals for each objectives), and the fact that for its effective use it is desirable that the objectives being aggregated be first normalized, it has not found much favor with the users. Cheng and Li (1996) have proved that the solution with product operator (using normalized objective functions) is always Pareto optimal. Ravi et al. (2000) have also used product operator for the aggregation of multi objectives, but their method is not interactive, therefore, it produce only one Pareto-optimal solution and all the possible solutions are not explored using their approach. In the present study, we have preferred product operator in the proposed interactive decision making process to aggregate the membership values of the objective functions. In order to solve the resulting crisp equivalent of the original fuzzy-multi objective optimization problem which is nonlinear (usually nonconvex also) optimization problem, a robust global optimization technique is needed. Keeping this in view, we have used a recently developed real code genetic algorithm technique, MI-LXPM (Deep, Singh, Kansal, & Mohan, 2009) to solve the resulting single objective crisp nonlinear optimization problem in each iterative phase. In MILXPM algorithm parameters free penalty approach is used for constraints handling and it is applicable to the problems in which have restrictions on decision variables to be integer or integer or mixedinteger. Sakawa and Yauchi (2001) have also used genetic algorithm (namely GENOCOP-III) for this purpose. However, in their technique constraint handling depends on extra parameters which have to be defined for each problem. In interactive methods, reference point methodology provides foundation for solving multi-objective optimization problems. In reference point methods DM have to specify either aspiration points (desirable levels of achievement) or reservation points (least levels of achievement which must be ensured), Buchanan and Gardiner (2003). Wierzbicki (1980), Sakawa and Yano (1989), Steuer et al. (1993), Steuer, Silverman, and Whisman (1998, 1999), Sakawa and Yauchi (2001), and Mohan and Verma (2003) have used aspiration points as a reference points. Reeves and MacLeod (1999) have used both reservation points as well as aspiration points, simultaneously, as reference points, however, only those problems are solved in which parameters or crisp. In the proposed interactive method, we have used reservation point as reference points in the form of an extra constraint for each objective in the optimization problem. In each interactive phase, DM has the option to modify the specified reservation values for some or all objective functions on the basis of their knowledge or outcomes of previous iteration. The rest of the paper is organized as follows. Methodology adopted for solving multi-objective optimization problems in fuzzy environment is discussed in Section 2. The proposed method for their solution is next described in Section 3. Use of the proposed method in solving some test examples is illustrated in Section 4. Conclusions based on the present study are finally drawn in Section 5. A brief account of the MI-LXPM algorithm as used by us is given in an Appendix. 2. Multi-objective fuzzy optimization problems A multi-objective fuzzy optimization problem (MOFPP) may be expressed as:

Fuzzy min f i ðx; a~i Þ i 2 I1 ; Fuzzy max f i ðx; a~i Þ i 2 I2 ; Fuzzy equal f i ðx; a~i Þ i 2 I3 ; I1 [ I2 [ I3 ; i ¼ 1; 2; . . . ; k; ~ Þ 6 0; j ¼ 1; 2; . . . mg; ~ ¼ fx 2 Rn jg ðx; b subject to x 2 XðbÞ j j

ð1Þ

~ ¼ ðb ~ ;b ~ ;...;b ~ Þ represent ~ i ¼ ða ~i1 ; a ~i2 ; . . . ; a ~ipi Þ and b where a j j1 j2 jqi respectively, the vectors of fuzzy parameters involved in the objec~ Þ, (a ~i Þ and the constraint functions g j ðx; b ~ir and tive functions fi ðx; a j ~js are assumed to be fuzzy numbers whose membership functions b are la~ir ðair Þ and lb~js ðbjs Þ, respectively). It is not necessary that all the parameters appearing in the problem be fuzzy. Some of the parameters can also be crisp. For the sake of convenience in notation, constraints of the type ci 6 xi 6 di, which specify bounds on the values of the decision variables, as well as integer restrictions, if any, imposed on the decision variables xi for some or all indices i, are not mentioned separately. In the mathematical model (1) the fuzzy goals have the following meanings. Fuzzy min means that the value of the objective function should be substantially reduced. Fuzzy max means that the value of the objective function should be substantially increased. Fuzzy equal means that the objective function should be kept in vicinity of the specified value as closely as possible. ~ is ~; bÞ For a chosen value of a (0 6 a 6 1), a-level set of ða defined as:

~ ¼ fða; bÞ jl ða Þ P a; ~; bÞ ða ir ~ir a a jlb~js ðbjs Þ P a;

i ¼ 1; 2; . . . ; k;

j ¼ 1; 2; . . . ; m;

r ¼ 1; 2; . . . ; pi ;

s ¼ 1; 2; . . . qj g;

~ is the cartesian product of the corresponding a-level sets ~; bÞ i.e. ða a ~ For simplicity we shall use the following ~ and b. of fuzzy numbers a notation

ai ¼ ðai1 ; ai2 ; . . . ; aipi Þ; a ¼ ða1 ; a2 ; . . . ; ak Þ; ~2 ; . . . ; a ~k Þ; ~ ¼ ða ~1 ; a a

bj ¼ ðbj1 ; bj2 ; . . . ; bjpj Þ; b ¼ ðb1 ; b2 ; . . . ; bm Þ; ~ ¼ ðb ~1 ; b ~2 ; . . . ; b ~m Þ: b

In case the DM wishes to take into account only those possible values of the fuzzy parameters whose membership value (or occurrence) is not less than some specified value of parameter a, MOFPP may be written as (1) along with:

~ : ~; bÞ ða; bÞ 2 ða a

ð2Þ 0

(we shall refer to (1) along with (2) as (1) . In our present study, we ~ also as decision variables ~; bÞ shall treat the fuzzy parameters ða taking their values from the corresponding a-level sets. Fuzzy objectives will be treated as fuzzy goals based on the traditional approach used in fuzzy programming with vagueness. For solving fuzzy optimization problem (1)0 , as a first step the following four single-objective optimization problems (two for minimization and two for maximization) have to be solved separately for each objective function:

Min=Max

f i ðx; aÞ i 2 f1; 2; . . . ; kg

~ once for a ¼ a0 ~; bÞ subject to x 2 XðbÞ and ða; bÞ 2 ða a and once for a ¼ a1 : These provide four individual values for each fi, namely, fmin,a at

a = a0 and a = a1, and fmax,a at a = a0 and a = a1 (the values of a0 and a1 are to be chosen by the decision maker between 0 and 1). Looking at these maximum and minimum values of fi at a0 and a1 levels and the values which s(he) would like these objectives to achieve, the DM may indicate his(er) choice of the membership functions to be assigned to each of the fuzzy objectives. The obtained membership functions are denoted by lfi ; i 2 1; 2; . . . ; k. The crucial part in MOFPP is the identification of appropriate aggregation function

K. Deep et al. / Expert Systems with Applications 38 (2011) 1659–1667

which well represents decision maker’s fuzzy preferences. As mentioned earlier, we have preferred product aggregation operator, lD() written as:

lD ðlf ðx; a~Þ; aÞ ¼ ðlf1 ðx; a~1 Þ  lf2 ðx; a~2 Þ      lfk ðx; a~k Þ; aÞ

ð3Þ

which does not give preferential treatment to any of the objectives. Once lD() is formulated, the MOFPP reduces to the following single objective nonlinear optimization problem:

Max

lD ðlf ðx; aÞ; aÞ

subject to x 2 XðbÞ ¼ fx 2 Rn j g j ðx; bj Þ 6 0; j ¼ 1; 2; . . . ; mg; ~ : ~; bÞ ða; bÞ 2 ða a

ð4Þ The optimization technique based on reservation level (threshold level which must be satisfied) as suggested by Reeves & Macleod (1999) has been used for interactive decision making. In this  fi ,whose satmethod, DM specifies reservation (threshold) value, l isfaction s(he) would like to be ensured for the membership functions lfi ; i ¼ 1; 2; . . . ; k. This is incorporated in the model as an additional constraint. For the specified value of a and specified threshold membership values lfi , the Pareto-optimal solution of the a-MOFPP, is now obtained by solving the following single objective crisp nonlinear optimization problem using suitable nonlinear optimization technique.

lD ðlf ðx; aÞ; aÞ  fi ðx; ai Þ P 0; i ¼ 1; 2; . . . k; subject to lfi ðx; ai Þ  l

Max

x 2 XðbÞ ¼ fx 2 Rn jg j ðx; bj Þ 6 0; j ¼ 1; 2; . . . ; mg; ~ ¼ fða; bÞjl ða Þ P a; i ¼ 1; 2; . . . ; k; ~; bÞ ða; bÞ 2 ða a

~ir a

ir

r ¼ 1; 2; . . . ; pi ; jlb~js ðbjs Þ P a;

1661

asked to specify the values which (s)he would like these objectives to achieve. Based on these minimum, maximum and aspired values of the objectives, DM may indicate the nature of the membership functions which (s)he would prefer to be used for each objective (linear, quadratic, exponential, hyperbolic, etc.) In the second phase, the membership functions of the objectives are transformed into a single objective using product operator. DM is next asked to specify reservation value (for each membership function and satisfaction level for fuzzy parameter a). It is used to create an additional constraint for each objective. In the third phase, the single objective nonlinear optimization problem thus framed is numerically solved using real coded genetic algorithm, MI-LXPM. Solution thus obtained in terms of membership value is used to find the values achieved by different objectives and these are provided to DM. Based on these results and on the basis of his(er) preferences or technical knowledge, (s)he may like to modify reservation level for some or all membership functions and go in for another iteration of the algorithm. This iterative process is to be continued till DM feels satisfied with the obtained results. The proposed method can summarized in algorithmic form as under: Computational algorithm Initialization Calculate the individual minimum and maximum values for ~i ) for two suitably chosen value of a each objective function fi(x, a (say a = a0 and a = a1, 0 6 a0, a1 6 1) and provide the solutions to DM. Keeping in view these results and the values that (s)he would like each objective to achieve, DM may indicate his(er) preference for the membership functions to be used for each objective and the desired threshold values for membership functions lfi and a which (s)he would like to be archived. Set iteration counter l = 1 and proceed to perform the following iterative steps Iterative steps

j ¼ 1; 2; . . . ; m; s ¼ 1; 2; . . . ; qj ð5Þ

This is deterministic equivalent of problem (1)0 . In this context we list here for ready reference certain results which ensure that the optimal solution of (5) is a Pareto-optimal solution of (1). Definition 1. x* 2 Rn is a global(local) a-Pareto optimal solution to  ~ such that (x*, a*, b*) is a ~; bÞ MOFPP (1) if there exist ða ; b Þ 2 ða a solution to problem (5); and there does not exist another feasible solution (x, a, b) to problem (5) (only in a neighborhood N(x*, a*, b*) in case of local) such that lfi ðx; ai Þ P lfi ðx ; ai Þ for all i with at least one inequality holding strictly.

Step 1: Using reservation value of a as specified by DM to solve the corresponding crisp single objective nonlinear optimization problem (5)using a suitable nonlinear optimization technique (such as MI-LXPM algorithm) and use the obtained a-Pareto optimal solution to provide the DM with achieved values of the objective functions.

Theorem 1. If (x*, a*, b*) is a global (local) optimal solution to (5) for some lfi ; i ¼ 1; 2; . . . ; k then x* is global(local) a-Pareto solution to the original MOFPP (1) (Cheng & Li, 1996). The optimal solution of crisp optimization problem (5) provides the values of the objectives of fuzzy optimization problem (1). If DM is satisfied with results, the process ends otherwise s(he) is asked to modify some or all of his(er) preferences in the light of these results and resulting problem (5) is solved again. In the next section we outline our proposed method for solving multi-objective optimization problems modeled in fuzzy environment. 3. The proposed interactive method The proposed interactive method works in three phases. In the first phase, each objective is separately solved for maximization and minimization (at two different a-level sets chosen by user between 0 and 1), using MI-LXPM algorithm. Keeping in view these maximum and minimum values of the objectives, the DM is next

Fig. 1. Flow chart of the proposed interactive method.

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K. Deep et al. / Expert Systems with Applications 38 (2011) 1659–1667

Step 2: Stop if DM is satisfied with solution. Else (if DM is not satisfied with any of the achieved objective function values) then (s)he be asked to suitably modify the reservation values of some or all the membership functions, or a and step (1) be repeated. The flow chart of the proposed algorithm is shown in Fig. 1. A brief account of the MI-LXPM algorithm as used by us in solving single objective crisp nonlinear optimization problem (5) is given for ready reference in Appendix.

iteration the DM specifies threshold value and value of a). Results of Mohan & Nguyen (1998) (f1 = 4768.47, f2 = 4507.159, f3 = 10,242.686 with a = .7) for this problem have been taken as the target solution. Initial step Solution of individual min and max problems at a = 0 and a = 1.0 levels are:

f1 ðxÞ : f min;1 ¼ 2985:5 at a ¼ 0;

f min;1 ¼ 3225:0 at a ¼ 1;

f max;1 ¼ 5925:18 at a ¼ 0;

f max;1 ¼ 5433:27 at a ¼ 1;

f2 ðxÞ : f min;2 ¼ 3484:97 at a ¼ 0; 4. Numerical examples

f max;2 ¼ 7982:0 at a ¼ 0;

In this section, we demonstrate the use of the proposed algorithm in solving five test examples taken from literature. Of these five problems whereas the first three are multi-objective problems molded in fuzzy environment, the last two are multi-objective problems modeled in crisp environment. These have been included to demonstrate the effectiveness of the proposed algorithm in solving multi-objective problems in crisp environment as well. In such problems DM has not to specify the aspired value of a. In order to give a better perspective to DM for deciding the necessary modifications which should be made in the values of input parameters supplied by him, in each iteration (s)he may be provided with more than one possible alternative solutions which or achievable on the basis of the input data. In these examples we have generally provided the DM with three sets of possible alternative Pareto optimal solutions in each iteration on the basis of which (s)he is asked to modify, if necessary, values of input parameters (this is shown in detail in Example 1). In each example we have chosen the solution given in the source as the target solution, which DM desires to be achieved. Example 1. This example has been taken from Sakawa & Yano (1989). It has also been analyzed by Mohan & Nguyen (1998). The problem is:

~1 Þ ¼ ðx1 þ 5Þ2 þ a ~11 x22 þ 2ðx3  a ~12 Þ2 ; Fuzzy min f 1 ðx; a ~2 Þ ¼ a ~21 ðx1  45Þ2 þ ðx2 þ 15Þ2 þ 3ðx3 þ a ~22 Þ2 ; Fuzzy min f 2 ðx; a

f3 ðxÞ : f min;3 ¼ 7142:5 at a ¼ 0;

0 6 xi 6 10:0;

1

2

f2 : hyperbolic ðf20 ; f20:5 ; f21 ; c2 Þ ¼ ð6900; 4600; 3900; 4400Þ; f3 : ðleftÞ ðexponential ðf30 ; f30:5 ; f31 ÞL ¼ ð7800; 8200; 10000Þ; ðrightÞ hyperbolic inverse ðf30 ; f30:5 ; f31 ; c3 ÞR ¼ ð13300; 11000; 10000; 12000Þ: These are same as used by Mohan & Nguyen (1998). The analytical expression for the membership functions become:

8 1; f1 < 3300; > > > < 1:031797ð1  expð3:479672 l f1 ¼ > ðf1  5400Þ=2100ÞÞ; 3300 6 f1 < 5400; > > : 0; f1 P 5400: 8 1; f2 < 3900; > > > < 0:5  1:3111397tanh lf2 ¼ > ð0:001182ðf2  4400ÞÞ þ 0:652146; 3900 6 f2 < 6900; > > : 0; f2 P 6900:

lf3

ÞLR ðpL ; pR ; p; p

Left

Right

~11 a ~12 a ~21 a ~22 a ~31 a ~32 a ~ b

(4.0, 4.0, 0.2, 0.3) (50.0, 50.0, 1.5, 2.0) (2.0, 2 .0, 0.15, 0.2) (20.0, 20.0, 1.8, 2.5) (3.0, 3.0, 0.1, 0.15) (5.0, 5.0, 0.3, 0.35) (1.0, 1.0, 0.1, 0.1)

L E E L E L E

E E L E L L E

~ b 12 ~ b

(1.0, 1.0, 0.2, 0.2)

E

E

(1.0, 1.0, 0.15, 0.15)

E

L

11

13

The problem has three nonlinear fuzzy objectives and one nonlinear fuzzy constraint besides the bounds on the values of the unknown variables. We have solved this example using the same decision making approach as used in the source (at each

f max;3 ¼ 13; 077:6 at a ¼ 1:

f1 : exponential ðf10 ; f10:5 ; f11 Þ ¼ ð5400; 5000; 3300Þ;

i ¼ 1; 2; 3:

~ p

f min;3 ¼ 7550:0 at a ¼ 1;

Looking at these results, and his expectation for the desired objective function values, suppose the DM shows the following preferences for the shapes and parameter values of the membership functions.

3

~11 ; . . . ; a ~32 and The membership functions for the fuzzy numbers a ~11 ; . . . ; b ~13 used by Sakawa & Yano (1989) are listed below, where b L and E denote linear and exponential membership functions, respectively.

f max;3 ¼ 7002:59 at a ¼ 1;

f max;3 ¼ 13; 996:5 at a ¼ 0;

~3 Þ ¼ a ~31 ðx1 þ 20Þ2 þ a ~32 ðx2  45Þ2 þ ðx3 þ 15Þ2 ; Fuzzy equal f 3 ðx; a ~1 Þ ¼ b ~11 x2 þ b ~12 x2 þ b ~13 x2 6 100; subject to g ðx; b 1

f min;2 ¼ 3875:0 at a ¼ 1;

8 0; > > > > > 1:026020ð1  exp > > > > < ð3:674562ðf3  7800Þ=2200ÞÞ; ¼ 1 > 0:4103 tanh > > > > > ð0:000469ðf3  12000ÞÞ þ 0:291036; > > > : 0;

f3 < 3300; 7800 6 f3 < 10000;

:

10000 6 f3 < 13000; f3 P 13000:

Further suppose that in order to initialize the interactive solution process, DM specifies the minimal reservation values for the  f1 ¼ l  f2 ¼ l  f3 ¼ 0:3 membership function of each objective as l and would like to consider only those values of the fuzzy parameters which yield a membership grade a P 0.9. The iterative algorithm is now initiated. Results of successive iterations are presented in Table 1.Three sets of results are provided to the DM after each iteration. On the basis of the results after first iteration, suppose, DM decides to change reservation value for the second objective as lf2 ¼ 0:55, keeping all others values same as in the first iteration. Iterative Process is repeated again and results presented to DM. In the third iteration solution for l f1 ¼ 0:6; l f2 ¼ 0:55; l f3 ¼ 0:7 and a = 0.9, in all the three outcomes it is observed that the level of satisfaction is infeasible. Therefore, realizing that attaining a = 0.9 is not possible, DM modifies this value of a and say sets it as a = 0.7, keeping others values same as in third iteration. The outcome of this fourth iteration is x1 = 8.128, x2 = 5.623, x3 = 1.739, f1 = 4868.84, f2 = 4436.18 and

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K. Deep et al. / Expert Systems with Applications 38 (2011) 1659–1667 Table 1 Solution of Example 1. Iteration-1  f1 ¼ l  f2 ¼ l  f3 ¼ 0:3 and a = 0.9 DM’s specifications l Solutions

Iteration-2  f1 ¼ 0:3, l  f2 ¼ 0:55, l  f3 ¼ 0:3 and a = 0.9 DM’s specifications l Solutions

I

II

III

I

II

III

lf1 lf2 lf3

0.52745

0.611879

0.618251

0.419443

0.57232

0.559301

0.4861

0.413113

0.419111

0.569767

0.556374

0.561415

0.9225

0.955287

0.952932

0.864014

0.655153

0.677189

f1 f2 f3 x1 x2 x3

4967.96 4619.01 10055.6 7.49826 6.18916 1.60907

4857.45 4723.29 10028.9 7.50152 6.24371 2.2799

4848.22 4714.39 10030.7 7.43053 6.2736 2.21369

5085.12 4506.86 10113 7.81779 6.17244 1.05461

4911.78 4524.46 10485.9 8.148440.3 5.43976 1.76428

4928.65 4517.82 10431.3 8.10836 5.4832 1.70634

Iteration-3  f1 ¼ 0:6, l  f2 ¼ 0:55, l  f3 ¼ 0:7 and a = 0.9 DM’s specifications l Solutions

Iteration-4  f1 ¼ 0:6, l  f2 ¼ 0:55, l  f3 ¼ 0:7 and a = 0.7 DM’s specifications l Solutions

I

II

III

I

II

lf1 lf2 lf3

Infeasible

Infeasible

Infeasible

0.614258

0.627144

0.603878

–

–

–

0.581872

0.559867

0.624122

–

–

–

0.760495

0.801567

0.70626

f1 f2 f3 x1 x2 x3

– – – – – –

– – – – – –

– – – – – –

4854.02 4491.02 10260.5 7.71232 5.63014 1.70078

4835.1 4519.86 10194.7 7.83219 5.85576 2.05171

4868.84 4436.18 10365.5 8.12774 5.62268 1.73861

f3 = 10365.5 at a level 0.7. This solution is close to DM’s expectation and so the algorithm is stopped.

f1 ðxÞ : f min;1 ¼ 0:001 at a ¼ 0;

III

f min;1 ¼ 0:000734 at a ¼ 1;

f max;1 ¼ 19:3581 at a ¼ 0; f2 ðxÞ : f min;2 ¼ 0:0011 at a ¼ 0;

Example 2

f max;2 ¼ 16:6863 at a ¼ 0;

Max ~f 1 ¼ ~c11 x1 þ ~c12 x2 þ 2x4 Max ~f 2 ¼ x1 þ x3 þ 3x4 subject to 3x1 þ 3x2 þ 4x3 þ 3x4 6 18;

f1: linear with ðf10 ; f11 Þ ¼ ð1:0; 18:0Þ. f2: linear with ðf20 ; f21 Þ ¼ ð1:0; 16:0Þ.

~32 x2 þ a ~33 x3 þ a ~34 x4 6 ð60; 50=9ÞRR ; ~31 x1 þ a a

0 6 x1 ; x2 ; x3 6 4; 0 6 x4 6 6;

Then the explicit expressions of the membership functions become

where fuzzy parameters are defined as:

~c11 ¼ ð4; 4:5; 5=9; 5=9ÞLR ;

~c12 ¼ ð3; 3; 2=9; 3=9ÞLR ;

~31 ¼ ð6; 6:5; 3=9; 3=9ÞLR ; a

~32 ¼ ð7:5; 8; 5=9; 6=9ÞLR ; a

~33 ¼ ð15; 16; 10=9; 10=9ÞLR ; a

~34 ¼ ð10; 11; 5=9; 5=9ÞLR ; a

~42 ¼ ð15; 16; 10=9; 10=9ÞLR ; a

~43 ¼ ð9:3; 10; 5=9; 3=9ÞLR : a

This problem is taken from Rommelfanger (1990). It was also studied by Nguyen (1996). Problem has two linear objectives of which one is fuzzy. Besides bounds on the variables the problem has four linear constraints of which three are fuzzy. We have used the proposed interactive method to solve this problem. Results reported in Nguyen (1996) (X = (1.815822, 1.422582, 0, 2.7411194) with f1 = 17.0134, f2 = 10.039406) are taken as the target solution. The interactive solution process is summarized below. Initialization The minimum and maximum objective function values for the individual objectives obtained for a = 0 and a = 1 are:

f max;2 ¼ 15:8989 at a ¼ 1:

Looking at these results, and his expectations for the objective functions values, suppose DM indicates the following preferences for the membership functions:

9x1 þ 4x2 þ 4x3 þ 6x4 6 ð35; 40=9ÞRR ;

~42 x2 þ a ~43 x3 þ 5x4 6 ð50; 60=9ÞRR ; 8x1 þ a

f max;1 ¼ 17:8334 at a ¼ 1; f min;2 ¼ 0:0015 at a ¼ 1;

l f1

l f2

8 0; > < ¼ ðf1  1:0Þ=ð18:0  1:0Þ; > : 1; 8 0; > < ¼ ðf2  1:0Þ=ð16:0  1:0Þ; > : 1;

f1 < 1:0; 1:0 6 f1 < 18:0; f1 P 18:0: f2 < 1:0; 1:0 6 f2 < 16:0; f2 P 16:0:

Next suppose that DM prefers to start the interactive solution process with minimum threshold membership values as 0.3 for lf1 and lf2 and would like to consider only those values of the fuzzy parameters which yield a membership grade P0.9. With these objectives in mind the interactive solution process is initiated by setting initial reservation (threshold) membership values for the  f1 ¼ l  f2 ¼ 0:3 and a = 0.9. objectives as l At each iteration the DM is provided with three possible alternative solutions based on which (s)he decides his(er) next course of action. Results are given in Table 2(solutions preferred by DM at each iteration are only shown). After first iteration the solution preferred by DM as shown in Table 2 is lf1 ¼ 0:735, lf2 ¼ 0:931,

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K. Deep et al. / Expert Systems with Applications 38 (2011) 1659–1667

~11 , . . . , a ~35 and The membership functions for the fuzzy numbers a ~11 ; . . . ; b ~81 are all assumed triangular with values listed below b where L, M, R, respectively denote left spread, model value and right spread.

Table 2 Solution of Example 2. Iteration

I

II

III

DM’s specifications

a

0.9 0.3 0.3

0.9 0.8 0.3

0.9 0.85 0.3

lf1 lf2

0.734544

0.813781

0.862787

0.930825

0.71191

0.652581

f1 f2 x1 x2 x3 x4

13.4873 14.9624 0.3655 0.7178 0.311573 4.8448

14.8343 11.6787 0.8639 1.3767 0.11302 3.5295

l f1 l f2 Solutions

15.6674 10.7887 1.0161 1.5829 0.06287 3.2157

with objective functions values as f1 = 13.4873, f2 = 14.9624 achieved at x1 = 0.3655, x2 = 0.7178, x3 = 0.0312 and x4 = 4.8448. Looking at this result suppose DM wants to further improve the value of the first objective function. For this, suppose (s)he decides to increase the threshold value of the first membership function and sets the new threshold value for the first membership function as lf1 ¼ 0:8, keeping the other values of the parameters unchanged. The solution obtained with this changed data is shown in column 2 of Table 2. On the basis of outcome of iteration 2 suppose DM wants to further increase membership function value of objective 1. For this suppose (s)he sets new threshold value for the first objective as lf1 ¼ 0:85, keeping the membership value for the other objectives unchanged. The problem is solved again yielding the result: lf1 ¼ 0:8628, lf2 ¼ 0:6526, with objective functions values as f1 = 15.6674, f2 = 10.7887 achieved at x1 = 1.0161, x2 = 1.5829, x3 = 0.06287 and x4 = 3.2156. DM is reasonably satisfied with the result as it is not far from the target solution.

L

M

R

~11 a ~12 a ~13 a

14.5

14

13.5

16.5

16

15.5

3.7

4

4.4

~14 a ~15 a

4.5

5

5.5

6.4

7

7.5

~21 a ~22 a

4.5

5

5.5

2.7

3

3.3

~23 a ~24 a

6.5

7

7.5

4.5

4

3.6

~25 a ~31 a

10.7

10

9.3

2.8

3

3.3

~32 a ~ a33

12.8

12

11.2

3.5

4

4.5

~34 a ~35 a

5.6

6

2.8

3

l f1

~1 Þ ¼ 7x21  x22 þ x1 x2 þ a ~11 x1 þ a ~12 x2 þ 8ðx3  10Þ2 min f 1 ðx; a

l f2

2

2

~13 ðx4  5Þ þ ðx5  3Þ þ 2ðx6  1Þ þ þa

~14 x27 a

~15 ðx8  11Þ2 þ 2ðx9  10Þ2 þ x210 þ 45 þa ~2 Þ ¼ ðx1  5Þ2 þ a ~21 ðx2  12Þ2 þ 0:5x43 þ a ~22 ðx4  11Þ2 min f 2 ðx; a ~23 x26 þ 0:1x47 þ a ~24 x6 x7 þ a ~25 x6 þ 0:2x55 þ a  8x7 þ x28 þ 3ðx9  5Þ2 þ ðx10  5Þ2 ~3 Þ ¼ x31 þ ðx2  5Þ2 þ a ~31 ðx3  9Þ2 þ a ~32 x3 þ 2x34 þ a ~33 x25 min f 3 ðx; a ~34 x27 þ a ~35 ðx7  2Þx28  x9 x10 þ 4x39 þ ðx6  5Þ2 þ a þ 5x1  8x1 x7 ~11  4ðx2  3Þ2  2x2 þ 7x4 þ b ~12 x5 x6 x8 þ 120 P 0; subject to b 3 ~21 x2  ðx3  6Þ2 þ b ~22 x4 þ 40 P 0;  5x21 þ b ~31 x1 x2 þ b ~32 x5  6x5 x6 P 0;  x21  2ðx2  2Þ2 þ b ~41 ðx1  8Þ2  2ðx2  4Þ2  3x2 þ b ~42 x5 x8 þ 30 P 0; b 5 ~ ðx  8Þ2 þ b ~ x P 0; 3x1  6x2 þ b 51 9 52 10 ~62 x7 þ 9x8 6 105; ~61 x1 þ 5x2 þ b b ~ x þb ~ x þ 2x 6 0; 10x1 þ b 71 2 72 7 8 ~81 x10 6 12;  8x1 þ 2x2 þ 5x9 þ b  5:0 6 xi 6 10:0;

i ¼ 1; 2; . . . ; 10:

M

R

~ b 11 ~ b

3.3

3

2.6

2.2

2

1.8

~ b 21 ~ b

8.6

8

7.4

1.8

2

2.2

~ b 31 ~ b

1.7

2

2.4

14.8

14

13.2

12

22

32

~ b 41 ~ b

0.9

0.5

0.8

1.0

~ b 51 ~ b

12.8

12

11.2

6.5

7

7.5

~ b 61 ~ b

3.6

4

4.4

62

3.4

3

2.6

8.4

8

6.4

~ b 71 ~ b 72

18.3

17

3.2

~ b 81

2.2

2

42

52

0.2 1.2

7.6 16 1.8

The solution to the above model as reported in the source is f1 = 535.736, f2 = 610.764, f3 = 582.128, which was obtained for a = 0.8. We use these objective function values as the target values which DM would like to be achieved. As earlier the maximum and the minimum values of each objective are first computed and based on these the linear membership functions are assigned to each of the objectives. These are:

Example 3. This example is taken from Sakawa & Yauchi (2001). Problem has three fuzzy nonlinear objectives and eight nonlinear fuzzy constraints besides bounds on variables. The problem is:

2

L

l f3

8 > < 1; ¼ 1  ðf1  1500Þ=ð1500  80Þ; > : 0; 8 > < 1; ¼ 1  ðf2  3500Þ=ð3500  200Þ; > : 0; 8 1; > < ¼ 1  ðf3  3000Þ=ð3000  50Þ; > : 0;

f1 < 80; 80 6 f1 < 1500; f1 P 1500: f2 < 200; 200 6 f2 < 3500; f2 P 3500: f3 < 50; 50 6 f3 < 3000; f3 P 3000:

For initiating the iterative process, suppose DM specifies reservation values for membership function of each objective as l f1 ¼ l f2 ¼ l f3 ¼ 0:3 and value of a as 1.0. The results of successive iterations are shown in Table 3. After the first iteration the preferred solution (out of the three solutions provides to DM) is f1 = 329.668, f2 = 859.042, f3 = 540.683. On the basis of this, suppose DM changes the minimum threshold value for the second and the third objec f2 ¼ 0:88; l  f3 ¼ 0:83 and goes for the next iteration. It tives to l was observed that after third iteration solution obtained becomes infeasible for the specified threshold values. Therefore, DM modifies value of a as a = 0.9, keeping minimum threshold values for the objectives same as in the third iteration. After fourth iteration solution achieved is f1 = 461.455, f2 = 595.994, f3 = 512.327, which completely dominates the target solution (Sakawa & Yauchi, 2001). Example 4 (Industrial water pollution in an artificial river basin). The problem is of industrial water pollution in an artificial river basin. The mathematical formulation of the water quality and economic objectives and constraints as given in Monarchi, Kisiel, & Duckstein (1975) for this problem is:

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K. Deep et al. / Expert Systems with Applications 38 (2011) 1659–1667 Table 3 Solution of Example 3. Iteration

I

II

DM’s specifications a 1.0 l f1 0.3 l f2 0.3 l f3 0.3

III

IV

1.0 0.3 0.88 0.83

1.0 0.70 0.88 0.83

0.9 0.70 0.88 0.83

0.668795

Infeasible

0.73137

Solutions

lf1 lf2 lf3

0.824178

f1 f2 f3 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10

0.800290

0.880000

–

0.88002

0.833667

0.830000

–

0.843279

329.668 859.042 540.683 1.44615 4.13212 4.83581 3.40002 0.03654 1.17028 0.06467 8.70675 6.22116 8.34648

550.310 596.000 551.500 0.92588 4.03443 2.76364 4.49156 0.088354 0.806027 0.305449 8.912000 6.010340 9.847730

– – – – – – – – – – – – –

461.455 595.994 512.327 0.96222 4.69128 3.61375 4.44664 0.327995 0.842474 0.663863 8.41848 6.06557 9.91179

f1 ðxÞ : fmin;1 ¼ 4:7500;

f max;1 ¼ 6:3404;

f2 ðxÞ : fmin;2 ¼ 2:0000;

f max;2 ¼ 6:7956;

f3 ðxÞ : fmin;3 ¼ 5:1000;

f max;3 ¼ 6:5970;

f4 ðxÞ : fmin;4 ¼ 0:3413;

f max;4 ¼ 7:5000;

f5 ðxÞ : fmin;5 ¼ 0:0000;

f max;5 ¼ 9:6824;

f6 ðxÞ : fmin;6 ¼ 1:6272;

f max;6 ¼ 11:3636:

Based on these results suppose DM prefers to use linear membership functions for each of these objectives as:

l f1

l f2

l f3

l f4 Max f 1 ðXÞ ¼ 4:75 þ ð5:68  105 Þð4:00  104 Þðx1  0:3Þ Max f 2 ðXÞ ¼ 2:0 þ ð1:31  105 Þð4:0  104 Þðx1  0:3Þ þ ð2:85  105 Þð1:28  105 Þðx2  0:3Þ

l f5

þ ð3:15  105 Þð2:80  104 Þðw1  0:3Þ þ ð5:53  105 Þð4:80  104 Þðw2  0:3Þ 4

5

Max f 3 ðXÞ ¼ 5:10 þ ð0:442  10 Þð4:0  10 Þðx1  0:3Þ þ ð0:764  105 Þð1:28  105 Þðx2  0:3Þ 4

5

þ ð0:771  10 Þð2:80  10 Þðw1  0:3Þ þ ð1:600  105 Þð4:80  104 Þðw2  0:3Þ Max f 4 ðXÞ ¼ 0:2  104 ½3:75  105  0:6ð59:0=ð1:09  x21 Þ  59:0Þ  103 

l f6

8 f1 < 4:75; > < 0; ¼ ðf1  4:75Þ=ð6:34  4:75Þ; 4:75 6 f1 < 6:34; > : 1; f1 P 6:34: 8 f2 < 2:0; > < 0; ¼ ðf2  2:0Þ=ð6:75  2:0Þ; 2:0 6 f2 < 6:75; > : 1; f2 P 6:75: 8 0; f3 < 5:1; > < ¼ ðf3  5:1Þ=ð6:59  5:1Þ; 5:1 6 f3 < 6:59; > : 1; f3 P 6:59: 8 f4 < 0:40; > < 0; ¼ ðf4  0:40Þ=ð7:0  0:40Þ; 0:40 6 f4 < 7:0; > : 1; f4 P 7:0: 8 1; f5 < 0:5; > < ¼ 1:0  ðf5  0:5Þ=ð9:5  0:5Þ; 0:5 6 f5 < 9:5; > : 0; f5 P 9:0: 8 1; f6 < 1:63; > < ¼ 1:0  ðf6  1:63Þ=ð11:0  1:63Þ; 1:63 6 f6 < 11:0; > : 0; f6 P 11:0:

Further suppose DM sets reservation values for these member f1 ¼ l  f2 ¼ l  f3 ¼ l  f4 ¼ l  f5 ¼ l  f6 ¼ 0:3. Results of ship functions as l successive iterations are shown in Table 4. Solution obtained after third iteration is f1 = 6.03746; f2 = 5.2331; f3 = 6.13984; f4 = 6.11886; f5 = 2.46134; f6 = 1.67587 at x1 = 0.866663, x2 = 0.899952 and x3 = 0.829807, which is reasonably close to the specified target solution.

Min f 5 ðXÞ ¼ ð2:4  103 Þð0:75Þð532:0=ð1:09  x22 Þ  532:0Þ Min f 6 ðXÞ ¼ ð3:33  103 Þð0:75Þð450:0=ð1:09  x23 Þ  450:0Þ subject to 1:0 þ ð8:30  107 Þð4:0  104 Þðx1  0:3Þ þ ð1:45  107 Þð1:28  105 Þðx2  0:3Þ 5

4

þ ð3:49  10 Þð9:57  10 Þðx3  0:3Þ þ ð7:30  107 Þð2:80  104 Þðw1  0:3Þ þ ð1:62  106 Þð4:80  104 Þðw2  0:3Þ þ ð7:33  105 Þð3:57  104 Þðw3  0:3Þ P 3:5; 0:3 6 xi 6 1:0; i ¼ 1; 2; 3; where wi ¼ 0:39ð1:39  x2i Þ; i ¼ 1; 2; 3: This is a multi-objective optimization problem modeled in crisp environment. It has six objective of which first two are linear and the last four nonlinear. It is subject to one nonlinear constraint besides bounds on the variables. The solution to this problem as reported in Nguyen (1996) is: X = (0.85996, 0.901043, 0.829162) with f1 = 6.022230, f2 = 5.229127, f3 = 6.139136, f4 = 6.187845, f5 = 2.485508, f6 = 1.668431. We have considered these objective function values as the targets which DM would like to be achieved. Initialization Since the problem is in crisp environment value of parameter a need not be specified by the DM. We first calculate the individual minimum and maximum values of each objective. These are:

Table 4 Solution of Example 4. Iteration

I

II

III

DM’s specifications l f1 l f2 l f3 l f4 l f5 l f6

0.3 0.3 0.3 0.3 0.3 0.3

0.3 0.3 0.3 0.3 0.75 0.3

0.3 0.3 0.3 0.85 0.75 0.3

lf1 lf2 lf3 lf4 lf5 lf6

0.851820

0.85186

0.809722

0.736934

0.697347

0.680653

0.751512

0.712572

0.697881

0.809208

0.809133

0.866494

0.698034

0.764197

0.769254

0.997135

0.994826

0.995105

f1 f2 f3 f4 f5 f6 x1 x2 x3

6.10439 5.50044 6.21975 5.74077 3.06671 1.65684 0.896294 0.923352 0.828166

6.10447 5.3124 6.16173 5.74028 2.50432 1.67848 0.896156 0.901882 0.830032

6.03746 5.2331 6.13984 6.11886 2.46134 1.67587 0.866663 0.899952 0.829807

Solutions

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K. Deep et al. / Expert Systems with Applications 38 (2011) 1659–1667

Table 5 Solution of Example 5. Iteration

I

DM’s specifications l f1 0.30 l f2 0.30 l f3 0.30 l f4 0.30 l f5 0.30

II

III

IV

0.30 0.30 0.55 0.30 0.30

0.30 0.30 0.565 0.30 0.25

0.30 0.30 0.568 0.20 0.20 0.77147

Solutions

lf1 lf2 lf3 lf4 lf5

1.00000

0.78112

0.74587

1.00000

0.56748

0.53119

0.54022

0.43601

0.55817

0.56621

0.56926

0.77122

0.31274

0.30006

0.27616

1.00000

0.30227

0.25366

0.24028

f1 f2 f3 f4 f5 x1 x2 x3

1.84695 102.055 8.66814 0.67188 129.046 139.997 600.129 89.9930

1.72339 94.9447 10.2562 0.41513 70.5769 129.671 1279.38 89.9165

1.71105 94.4366 10.3607 0.40835 66.7853 127.820 1303.78 89.9831

1.72002 94.5630 10.4003 0.39465 65.7420 129.606 1347.72 89.8685

Example 5. This problem is given in Tyagi (1988) and it has also been investigated Nguyen (1996). It is the mathematical model of optimization of the characteristics of vacuum moulding process (V-process). Quality of casting made by V-process is influenced by several characteristics, such as the bulk density f1, mould hardness of moulding sand f2, moulding impact strength f3, tensile strength f4 and Brinell hardness f5. The parameters affecting the values of these characteristics most are: sand grain size x1, vibration frequency x2 and vibrating time x3. The mathematical model of the problem as given in Tyagi (1988) is:

Max f 1 ðxÞ ¼ 1:303  0:385  103 x1 þ 0:132  103 x2  0:002x3 þ 0:305  104 x21 þ 0:255  107 x22 þ 0:909  105 x23  0:197  105 x1 x2 þ 0:139  104 x1 x3 þ 0:15  106 x2 x3 Max f 2 ðxÞ ¼ 107:243  0:306x1  0:005x2  0:077x3 þ 0:0024x21 þ 0:158  105 x22 þ 0:0014x23  0:345  104 x1 x2  0:672  103 x1 x2 Max f 3 ðxÞ ¼ 1:94 þ 0:155x1 þ 0:014x2 þ 0:05x3  0:37  103 x21  0:66  106 x22  0:39  104 x23  0:687  104 x1 x2 þ 0:27  104 x1 x3  0:14  104 x2 x3 Max f 4 ðxÞ ¼ 0:101 þ 0:863  103 x1  0:234  103 x2 þ 0:014x3 þ 0:176  104 x21 þ 0:93  107 x22  0:565  105 x23  0:922  106 x1 x2  0:484  104 x1 x3  0:209  105 x2 x3 Max f 5 ðxÞ ¼ 31:37  0:201x1 þ 0:0043x2 þ 0:019x3 þ 0:007x21  0:79  105 x22 þ 0:0025x23  0:35  103 x1 x2 þ 0:69  104 x1 x3  0:84  104 x2 x3 subject to 72 6 x1 6 140; 600 6 x2 6 2400; 30 6 x3 6 900: This is a crisp multi-objective problem with five nonlinear objectives subject to bounds on variables only. Solution reported in Tyagi (1988) is X = (87.916, 240, 48.295) with f1 = 1.481914, f2 = 93.747307, f3 = 2.975748, f4 = 0.473874, f5 = 67.063423. Solution obtained by Nguyen (1996) was X = (126.728760, 1303.496338, 90) with f1 = 1.704476, f2 = 94.211901, f3 = 10.389298, f4 = 0.408736, f5 = 65.576790. We have considered the solution obtained by Nguyen (1996) as the target solution which the DM would like to be achieved.

In order to initialize the interactive solution process suppose the DM specifies the minimal reservation values for the member f1 ¼ l  f2 ¼ l  f3 ¼ l  f4 ¼ l  f5 ¼ 0:3. ship function of each objective as l The results of successive iterations are reported in Table 5. Solution obtained after first iteration is f1 = 1.84695, f2 = 102.055, f3 = 8.66814, f4 = 0.67188, f5 = 129.046, which completely dominates the solution reported in Tyagi (1988). However, it does not completely dominate the solution reported in Nguyen (1996). Therefore, we decided to go in for more iterations to find a solution close to Nguyen’s solution. After 4th iteration solution obtained is f1 = 1.72002, f2 = 94.563, f3 = 10.4003, f4 = 0.39465, f5 = 65.742, with x1 = 129.606, x2 = 1347.72, x3 = 89.8685. This solution is quite close to Nguyen’s solution. 5. Conclusions In this paper, an interactive method has been proposed for solving multi-objective optimization problems modeled in fuzzy environment. Using this interactive method it is possible to obtain a solution to the multi-objective problem which is as reasonably close to the user’s expectations as possible. The method solves the multi-objective optimization problem by treating the objectives as fuzzy goals and the fuzzy constraints as crisp constraints which are to be satisfied at specified a-level of the fuzzy parameters involved. Whereas most of the authors usually consider the user specified aspiration levels as reference levels, we have preferred to use these as reservation levels (threshold levels whose satisfaction has to be ensured). Method uses product operator, a compensatory aggregation operator, to aggregate the membership functions of different objectives. Product operator assigns equal importance to each membership function. However, it produces nonlinearity in the formulation of the single objective and therefore, requires an efficient nonlinear optimization algorithm to solve nonconvex, nonlinear optimization problem formulated in each iterative phase. We have used a suitably modified version of real coded genetic algorithm, MI-LXPM, to solve these optimization problems. This algorithm tries to search for the global optimal solution of nonconvex,nonlinear optimization problems. Our experience of using this algorithm on some test examples has shown that its performances is satisfactory. User is, however, at liberty to use any other nonlinear optimization algorithm which searches global optimal solution in nonconvex cases. Our experience of using the proposed algorithm in solving five test examples has shown that the performance of the proposed method is quite satisfactory. It has been observed that in case user starts with a too higher aspiration value of a (examples 1 and 3) then he may ends up with an infeasible solution. In such cases he should reduce the aspired value of a. Besides problems modeled in fuzzy environment, the proposed method can also be used to solve multi-objective optimization problems modeled in crisp environment (examples 4 and 5). In such cases user has not to specify the value of parameter a. Acknowledgements One of the author Krishna Pratap Singh would like to thank Council for Scientific and Industrial Research (CSIR), New Delhi, India, for providing the financial support vide Grant No. 09/ 143(0504)/2004-EMR-I. Appendix Genetic algorithms are widely known global optimization evolutionary techniques. Recently, Deep et al. (2009) have proposed a new real coded genetic algorithm, MI-LXPM, for constrained optimization problems. It is applicable to the problems which have

K. Deep et al. / Expert Systems with Applications 38 (2011) 1659–1667

restrictions on decision variables to be integer or real or both. In this algorithm Laplace crossover, Power mutation and Tournament selection operators are used. Computational steps of MI-LXPM algorithm are as follows: 1. Generate a population of random individuals (it may be 5–10 times of number of variables) within the domain prescribed by the bounds on variables i.e. generated points should satisfy xLi 6 xi 6 xUi ; i ¼ 1; 2; . . . n. n is number of decision variables. 2. Apply truncation procedure for each variable which has integer restrictions. 3. Evaluate fitness value of each individual (chromosome) in the population. 4. Check the stopping criteria (here it is number of generations). If satisfied stop; else goto 5. 5. Apply tournament selection on existing population to select individual in mating pool. Here tournament size 3 has been taken. 6. Apply Laplace crossover and Power mutation to all individuals in the mating pool, with probability of crossover Pc = 0.8 and probability of mutation Pm = 0.002 respectively, to create new population. 7. Increase generation++; existing population new population; goto 2. Note: In proposed interactive method, at each interactive phase, to get three solutions, we run above algorithm three times. In all the three run input value (threshold value and alpha value) is same. In each run for given input value we started with new random solution, so for different run for same input value we may get different solution. Since genetic algorithm is a probabilistic technique, so it may possible, in some run, it converges to local optima (which rarely happen). Hence, we run three times at each iteration of proposed interactive method. Apart from that, three solutions in each iteration give more insight of the problem to DM to chose his(er) next preference value. References Bellman, R., & Zadeh, L. A. (1970). Decision making in a fuzzy environment. Management Science, 17B, 141–164. Buchanan, J., & Gardiner, L. (2003). A comparison of two reference point methods in multiple objective mathematical programming. European Journal of Operational Research, 149, 17–34.

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