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Interactive stability of multiobjective integer nonlinear programming problems
Applied Mathematics and Computation 176 (2006) 230–236 www.elsevier.com/locate/amc
Interactive stability of multiobjective integer nonlinear programming problems Mervat Mohamed Kamel ElShafei Department of Mathematics, Faculty of Science, Helwan University, Ain Helwan, Cairo, Egypt
Abstract In this paper we present a solution method for fuzzy multiobjective integer nonlinear programming (FMOINLP) problems and the stability of this solution. An interactive stability compromise programming method for solving (FMOINLP) problems by using the compromise weights from the pay-off table of membership function for each objective function is presented. 2005 Elsevier Inc. All rights reserved. Keywords: Integer programming; Multiobjective nonlinear programming; Fuzzy parameters; Fuzzy number; Membership functions; Branch-and bound method; Stability set of the first kind; Interactive decision making; Parametric programming; Interactive decision making; Compromise weights; Interactive compromise programming
1. Introduction In practice most decision problems have multiple objectives conflicting among themselves. The solution for such problems can only be obtained by trying to get compromises based on the information provided by the decision maker (DM). Several methods have been developed to solve multiobjective decision making (MODM) problems [1]. In [2,3] some of these methods are based on prior information required from the DM. This information may be in the form of desired achievement levels of the objective functions and the raking of the levels indicating their importance, such as in goal programming. It may also be in the form of weights showing the importance of the objectives. The disadvantage with these methods is that the DM cannot easily provide this prior information since he has no idea about the solution process of the problem. Other methods, called interactive methods, have been developed in order to overcome this disadvantage. There are two categories of interactive methods. Interactive methods of the first type require the DM to provide some trade-offs among the attained values of the objective functions in order to determine the new solution [4]. The interactive methods of the second type require the DM to provide some preference information by comparing the various efficient solutions in the space of the objective functions or the decision variables [5]. The quantity and complexity of the information required from the DM in such methods are important factors
E-mail address: [email protected] 0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.09.046
M.M.K. ElShafei / Applied Mathematics and Computation 176 (2006) 230–236
231
1
0
P1
P2
P3
P4
Fig. 1. Membership function of fuzzy number ~p.
affecting the chances of reaching the best compromise solution. In [6] an interactive linear multiple objective method, called interactive compromise programming (ICP), is introduced. The notions of the solvability set, stability set of the first kind and the stability set of the second, and analysed these concepts for parametric convex nonlinear programming problems were introduced in [7]. In [8,10] an interactive stability for solving multiobjective nonlinear programming (MONLP) problems was introduced. This paper, is presented an interactive stability compromise programming (ISCP) method for solving fuzzy multiobjective integer nonlinear programming (FMOINLP) problems by using the compromise weights from the pay-off table of the membership function for each objective function (Fig. 1). 2. Problem formulation Let us consider the following fuzzy multiobjective integer nonlinear programming problem (FMOINLP): ðFMOINLPÞ
max subject to
ðf1 ðx; ~ a1 Þ; f2 ðx; ~ ak Þ; . . . ; fk ðx; ~ak ÞÞ ~ x 2 X ðbÞ ¼ fx 2 Rn jgj ðx; ~bj Þ 6 0; j ¼ 1; . . . ; m; and x integerg;
~i Þ, i = 1, . . . , k and gj ðx; ~ where fi ðx; a bj Þ, j = 1, . . . , m are convex real valued functions which ~ai ¼ ~ ~ ~ ~ ai2 ; . . . ; ~ aipi Þ and bj ¼ ðbj1 ; bj2 ; . . . ; bjqj Þ represent respectively a vector of fuzzy parameters involved in ð~ ai1 ; ~ the objective function fi ðx; ~ ai Þ and in the constraint function gj ðx; ~bj Þ. These fuzzy parameters are assumed to be characterized as the fuzzy numbers. A real fuzzy number ~p is a convex continuous fuzzy subset of the real line whose membership function l~p ðpÞ is defined by [9]: (1) (2) (3) (4) (5) (6)
A continuous mapping from R1 to the closed interval [0, 1], l~p ðpÞ ¼ 0 for all p 2 (1, p1], strictly increasing on [p1, p2], l~p ðpÞ ¼ 1 for all p 2 [p2, p3], strictly decreasing on [p3, p4], l~p ðpÞ ¼ 0 for all p 2 [p4, 1).
Assume that ~ air ði ¼ 1; . . . ; k; r ¼ 1; . . . ; pi Þ and ~ bjs ðj ¼ 1; . . . ; m; s ¼ 1; . . . ; qj Þ in that (FMOINLP) are fuzzy numbers whose membership function are l~air ðair Þ and l~bjs ðbjs Þ respectively. Definition 1 (a-level set). The a-level set of fuzzy numbers ~air and ~bjs is defined as the ordinary set La ð~a; ~bÞ for which the degree of their membership exceeds the level a: La ð~ a; ~ bÞ ¼ fða; bÞ j l~air ðair Þ P a; i ¼ 1; . . . ; k; r ¼ 1; . . . ; pi ; l~bjs ðbjs Þ P a; j ¼ 1; . . . ; m; s ¼ 1; . . . ; qj g.
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for a certain degree a, the (FMOINLP) can be understood as the following nonfuzzy a-multiobjective mixedinteger nonlinear programming problem (a-MOMINLP) 0 ða-MOMINLPÞ
0
max
ðf1 ðx; a1 Þ; f2 ðx; a2 Þ; . . . ; fk ðx; ak ÞÞ
subject to x 2 X ¼ fx 2 Rn jgj ðx; bj Þ 6 0; j ¼ 1; . . . ; m; ða; bÞ 2 La ð~a; ~bÞ and x integerg. In the (a-MOMINLP) 0 , the parameters (a, b) are treated as decision variables rather than constants. Problem (a-MOMINLP) 0 can be rewritten as the following form: ða-MOMINLPÞ
max subject to
ðf1 ðx; a1 Þ; f2 ðx; a2 Þ; . . . ; fk ðx; ak ÞÞ x 2 X ¼ fx 2 Rn jgj ðx; bj Þ 6 0; j ¼ 1; . . . ; m; Ai 6 ai 6 Bi ; i ¼ 1; . . . ; k; A0j 6 bj 6 B0j ; j ¼ 1; . . . ; m and x integerg,
where (Ai, Bi) and ðA0j ; B0j Þ are lower and upper bounds on ai and bj (i = 1, . . . , k, j = 1, . . . , m) respectively. Definition 2 (a-pareto optimal solution). (x*, a*) 2 X is said to be an a-pareto optimal solution to the (a-MOMINLP), if and only if there does not exist another x 2 X, ða; bÞ 2 La ð~a; ~bÞ such that: fi(x, ai) 6 fi(x*, a*), i = 1, . . . , k with strictly inequality holding for at least one i, where the corresponding values of parameters (a*, b*) are called a-level optimal parameters. Problem (a-MOMINLP) will be treated using the method called interactive stability compromise programming (ISCP) [10]. Zeleny [1] has suggested that the set of efficient solutions can be reduced by introducing the ‘‘compromise set’’ concept. Let fi(x, ai) be the i the objective function and fiu ðx; ai Þ be the maximum possible value of fi(x, ai) and fiL ðx; ai Þ be the minimum possible value of fi(x, ai) found under the given constraints, respectively. To obtain the compromise solution of the (a-MOMINLP) problem, find the solution which has a minimum distance with respect to the ideal solution fiu ðx; ai Þ. This idea requires normalization of the objective functions and appropriate choice for the distance measure. The solutions found in this way are a reduced set of all efficient solutions. The set of compromise solutions may be large, and also the choice of weights by the DM may be difficult. These difficulties could be reduced by combining the basic ideas for the methods of compromise programming and compromise weights. 3. Compromise weights The interactive compromise programming (ICP) is based on two main ideas. First, the DM could state his preference among some alternative solutions more easily if values of objective functions were measured on the same scale varying between zero and one. This could be done by employing ‘‘the membership functions for the objective functions’’ concept in the compromise programming. In order to elicit a membership function lfi ðx; ai Þ, from the DM for each of the objective functions fi(x, ai) in (a-MOMINLP) problems, first calculate the individual minimum fiL and maximum fiu of each objective function fi(x, ai) under the given constraints. By taking account of the calculated individual minimum and maximum of each objective function together with the rate of increase of membership of satisfaction, DM must determine his subjective membership function lfi ðx; ai Þ which is a strictly monotone increasing function with respect to fi(x, ai). Here, it is assumed that lfi ðx; ai Þ ¼ 0 lfi ðx; ai Þ ¼ 1
or ! 0 if f i ðx; ai Þ 6 fi0 or ! 1 if f i ðx; ai Þ P
and
fi1 ;
where fic represents the value of fi(x, ai) such that the value of membership function lfi ðx; ai Þ is c, fi0 is an unacceptable level for fi(x, ai) and fi1 is a completely desirable level for fi(x, ai) within fiL and fiu . The DM can select his membership function in a subjective manner by considering the rate of increase of membership satisfaction among the following four types of functions [7]:
M.M.K. ElShafei / Applied Mathematics and Computation 176 (2006) 230–236
233
1. Linear membership function lfi ðxÞ ¼
fi ðxÞ fi0 ; fi1 fi0
i ¼ 1; 2; . . . ; k.
2. Exponential membership function lfi ðxÞ ¼ d i ½1 expfai ðfi ðxÞ fi0 Þ=ðfi1 fi0 Þg; where di > 1, ai > 0 or di < 0, ai < 0. 3. Hyperbolic membership function 1 1 lfi ðxÞ ¼ tanhðai ðfi ðxÞ ei ÞÞ þ ; 2 2 where ai > 0 or ai < 0. 4. Hyperbolic inverse membership function 1 lfi ðxÞ ¼ d i tanh1 ðai ðfi ðxÞ ei ÞÞ þ ; 2
i ¼ 1; . . . ; k;
i ¼ 1; . . . ; k;
i ¼ 1; . . . ; k;
where di > 0, ai > 0, or ai < 0. In this paper, the following definition of the membership functions is used for scaling: lfi ðx; ai Þ ¼
fi ðx; ai Þ fiL ; fiu fiL
ð1Þ
where fi(x, ai) are the objective functions, fiu are the maximum possible values of fi(x, ai), i = 1, . . . , k, and fiL are the minimum possible values of fi(x, ai) satisfying the constraints x 2 X. The lfi ðx; ai Þ are defined as the membership functions of fi(x, ai) to the maximum possible value fi(x, ai). The scalarization problem is maxðlfi ðx; ai ÞÞ, x 2 X, i = 1, . . . , k, is proposed as the following problem: max
lf kþ1 ðx; aÞ ¼
k X
ki lfi ðx; ai Þ
i¼1
subject to x 2 X .
ð2Þ
The second main idea, one of the main drawbacks of the interactive methods is the difficulty of getting the weights of the objective functions from the DM even if the values of objective functions are presented to him on the same scale. In this method, the compromise weights of objective functions can be obtained by means of the pay-off fuzzy matrix D of order k · k of which k successive columns show the effects of i instrument vector xi , ai on the membership objective functions f ðxk ; ak ÞÞ. D ¼ ð lf ðx1 ; a1 Þ; . . . ; l The compromise weights ki, i = 1, 2, . . . , k, can be obtained from the normalized version of the pay-off fuzzy matrix D as in the form: ki ¼
D01 L0 ; L; D01 L0
i ¼ 1; 2; . . . ; k;
ð3Þ
where L is the unit vector and L0 is the transpose unit vector. Also D 0 is the transpose pay-off fuzzy matrix D. 4. Stability set of the first kind Definition 3 (The solvability set). The solvability set of scalarization problem (a-MOMINLP)k which are formulated as:
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M.M.K. ElShafei / Applied Mathematics and Computation 176 (2006) 230–236
ða-MOMINLPÞk
k X
max
ki fi ðx; ai Þ
i¼1
subject to
x 2 X;
where k ¼ ðk1 ; k2 ; . . . ; kk Þ 6¼ 0;
ki P 0; i ¼ 1; . . . ; k
and
k X
ki ¼ 1
i¼1
is defined by ( k
H¼
k 2 R j max
k X
) ki fi ðx; ai Þ exists; k P 0 .
i¼1
Definition 4 (The stability set of the first kind). Suppose that H 5 / with a corresponding optimal point ðx; a; bÞ, then the stability set of the first kind of problem (a-MOMINLP)k corresponding to ðx; a; bÞ is defined by ( ) k k X X k f ðx; ai Þ ¼ max ki fi ðx; ai Þ . Sðx; a; bÞ ¼ k 2 H i¼1 i i i¼1 It is clear that the stability set of the first kind is the set of all parameters corresponding to an optimal solution 2 Rk , l 0 2 Rk, g 2 Rm and g0 2 Rm such of the scalarization problem. Let k 2 Sðx; a; bÞ then there exist u 2 Rm , l 0 0 ; l ; that (x; a; b; u; l g; g ) solve the Kuhn–Tucker problem: k X i¼1
m X
ofi ðx; ai Þ ki þ oxr
j¼1
uj
ogj ðx; bj Þ ¼ 0; oxr
r ¼ 1; . . . ; n;
ofi ðx; ai Þ i l 0i ¼ 0; i ¼ 1; . . . ; k; ki þl oai ogj ðx; bj Þ þ gj g0j ¼ 0; j ¼ 1; . . . ; m; uj obj bj Þ 6 0; j ¼ 1; . . . ; m; gj ðx; ð ai Bi Þ 6 0; j B0 Þ 6 0; ðb j
ðAi ai Þ 6 0; i ¼ 1; . . . ; n; 0 j Þ 6 0; j ¼ 1; . . . ; m; ðAj b
uj gj ðx; bj Þ ¼ 0;
j ¼ 1; . . . ; m;
i ð ai Bi Þ ¼ 0; l
i ¼ 1; . . . ; k;
0i ðAi l
ai Þ ¼ 0; gj ðbj B0j Þ ¼ 0;
i ¼ 1; . . . ; k;
g0j ðA0j bj Þ ¼ 0;
j ¼ 1; . . . ; m;
uj ¼ 0;
gj ¼ 0;
uj P 0; gj P 0; i ¼ 0; l
0j ¼ 0; l
0i l
0i l
P 0;
P 0;
j ¼ 1; . . . ; m;
g0j ¼ 0; g0j P 0;
for j 2 J f1; 2; . . . ; kg; for j 2 f1; . . . ; kg J ;
for i 2 M f1; 2; . . . ; kg; for i 2 f1; . . . ; kg; M.
M.M.K. ElShafei / Applied Mathematics and Computation 176 (2006) 230–236
235
Consider the system of equations X ogj ðx; bj Þ ofi ðx; ai Þ þ ¼ 0; r ¼ 1; . . . ; n; uj ox ox r r i¼1 j2f1;...;kgJ ofj ðx; bj Þ þ gj g0j ¼ 0; j 2 f1; . . . ; kg J ; uj obj ofi ðx; ai Þ þ li l0i ¼ 0; i 2 f1; . . . ; kg M. ki oai k X
ki
ðIÞ
It represents n + c + s linear equations in k + 3c + 2s unknowns ki, li, l0i , i = 1, . . . , k, and uj, gj, g0j , j = 1, . . . , c, which can be solved explicitly. Suppose that ki P 0, i = 1, . . . , k, uj P 0, j = 1, . . . , c, solve the above system of equations, then it is clear ; l0 ; that ðx; a; b; u; l g; g0 Þ solve the Kuhn–Tucker problem, where uj ¼ uj , j = 1, . . . , c, uj ¼ 0, j = c + 1, . . . , m, and hence k 2 Sðx; a; bÞ. Let us define the set N ðk; u; l; l0 ; g; g0 Þ ¼ fðk; u; l; l0 ; g; g0 Þ 2 R3kþ3c jðk; u; l; l0 ; g; g0 Þ solves the system (I)}. Then Sðx; a; bÞ ¼ fk 2 Rm j ðk; u; l; l0 ; g; g0 Þ 2 N ðk; u; l; l0 ; g; g0 Þg.
ðIIÞ
If gj ðx; bj Þ < 0; ð ai ; Bi Þ < 0; ðAi ai Þ < 0; bj B0j < 0 and A0j bj < 0, then it is easy to see that, Sðx; a; bÞ can be written in the following form: ( ) X k ofi ðx; ai Þ k Sðx; a; bÞ ¼ k 2 R k ¼ 0; r ¼ 1; 2; . . . ; n; k P 0 . i¼1 i oxr 5. A solution algorithm The steps of the algorithm can be summarized as follows: Step 1: Set a certain degree a = a* 2 [0, 1]. Step 2: Elicit a membership function for each fuzzy number in the formulated problem (FMOINLP) satisfying assumption (1–6). Then, determine the a-level set of the fuzzy numbers. Step 3: Convert the (FMOINLP) in the form of problem (a-MOMINLP). Step 4: Determine fiu for all i = 1, . . . , k as follows: max subject to Step Step Step Step Step
5: 6: 7: 8: 9:
f i ðx; ai Þ x2X
Find the solution of this problem by ignoring the integer condition, and use lingo program. If the solution is an integer then go to step (8), otherwise using branch and bound method [11]. If the integer solution has not been reached go to step (7), otherwise go to step (8). No integer feasible solution exists and go to step (18). The solutions of this problem are xiu, aui and fiu which are known as ‘‘ideal solution’’. Determine fiL for all i = 1, . . . , k as min
f i ðx; ai Þ
subject to
x 2 X.
Find the solution of this problem by ignoring the integer condition, and use lingo program. Step 10: Go to step (5), step (6) and step (7) after that go to step (11). Step 11: The solutions are xiL, aLi and fiL which are knows as the ‘‘anti-ideal solution’’. Step 12: Determine the membership functions corresponding the solution xiu, aiu, i = 1, . . . , k as lfi ðx; ai Þ ¼
fi ðx; ai Þ fiL fiu fiL
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The table of fuzzy matrix can be arranged as follows: lf
x1
x2
...
xk
fu
f1 f2 .. . fk
lf 1 1 lf 1 2
lf 2 1 lf 2
... ...
lf k 1 lf k 2
f1u f2u
lf 1
lf 2
...
lf k
fku
2
k
k
k
where lf j is the value of lfi in xj ; aji . i Step 13: The compromise weights ki, i = 1, . . . , k can be found from ki ¼
ðD01 L0 Þ ; LD01 L0
i ¼ 1; . . . ; k.
Step 14: By using these weights, we establish the new composite function to obtain the new alternative compromise solution, xk+1, from the problem max
lf kþ1 ðx; aÞ ¼
k X
ki lfi ðx; ai Þ
i¼1
subject to
x 2 X.
Step 15: Determine the stability set of the first kind corresponding to this solution as in relations I and II. Step 16: Determine the membership objective function of the new solution of the problem in step 14, lf kþ1 . Add this column of the table in step 12. Step 17: Ask the DM whether he prefers one solution strictly over all the other the k solutions. If he does, go to step 18, otherwise ask him his least preferred solution among all the others. Then replace this preferred solution by the new found in step 16 and go to step 13. Step 18: stop.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
M. Zeleny, Multiple Criteria Decision Making, McGraw-Hill, New York, 1982. J.M. Ignizio, Goal Programming and Extensions, Heath, Lexington, MA, 1976. S.M. Lee, Goal Programming for Decision Analysis, Auerbach, Philadelphia, PA, 1972. A.M. Geoffrion, J.S. Dyer, A. Feinberg, An interactive approach for multi-criteria optimization with an application to the operation of an academic department, Manage. Sci. 19 (1972) 357–368. S. Zionts, J. Wallenius, An interactive programming method for solving the multiple criteria problem, Manage. Sci. 22 (1976) 652– 663. R. Evren, Interactive compromise programming, J. Oper. Res. Soc. 38 (2) (1987) 163–172. M. Osman, A. EL-Benna, Stability of multiobjective nonlinear programming problems with fuzzy parameters, Math. Comput. Simul. 35 (1993) 321–326. M. Kassem, Interactive stability of multiobjective nonlinear programming problems with fuzzy parameters in the constraints, Fuzzy Sets Syst. 73 (1995) 235–243. M. Sakawa, Fuzzy Sets and Interactive Multiobjective Optimization, Plenum Press, New York, 1993. M. Kassem, Interactive stability of vector optimization problems, Eur. J. Oper. Res. 134 (2001) 616–622. S. Rao, Optimization Theory and Application, Indian Institute of Technology, Kanpur, 1977.
Interactive stability of multiobjective integer nonlinear programming problems Mervat Mohamed Kamel ElShafei Department of Mathematics, Faculty of Science, Helwan University, Ain Helwan, Cairo, Egypt
Abstract In this paper we present a solution method for fuzzy multiobjective integer nonlinear programming (FMOINLP) problems and the stability of this solution. An interactive stability compromise programming method for solving (FMOINLP) problems by using the compromise weights from the pay-off table of membership function for each objective function is presented. 2005 Elsevier Inc. All rights reserved. Keywords: Integer programming; Multiobjective nonlinear programming; Fuzzy parameters; Fuzzy number; Membership functions; Branch-and bound method; Stability set of the first kind; Interactive decision making; Parametric programming; Interactive decision making; Compromise weights; Interactive compromise programming
1. Introduction In practice most decision problems have multiple objectives conflicting among themselves. The solution for such problems can only be obtained by trying to get compromises based on the information provided by the decision maker (DM). Several methods have been developed to solve multiobjective decision making (MODM) problems [1]. In [2,3] some of these methods are based on prior information required from the DM. This information may be in the form of desired achievement levels of the objective functions and the raking of the levels indicating their importance, such as in goal programming. It may also be in the form of weights showing the importance of the objectives. The disadvantage with these methods is that the DM cannot easily provide this prior information since he has no idea about the solution process of the problem. Other methods, called interactive methods, have been developed in order to overcome this disadvantage. There are two categories of interactive methods. Interactive methods of the first type require the DM to provide some trade-offs among the attained values of the objective functions in order to determine the new solution [4]. The interactive methods of the second type require the DM to provide some preference information by comparing the various efficient solutions in the space of the objective functions or the decision variables [5]. The quantity and complexity of the information required from the DM in such methods are important factors
E-mail address: [email protected] 0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.09.046
M.M.K. ElShafei / Applied Mathematics and Computation 176 (2006) 230–236
231
1
0
P1
P2
P3
P4
Fig. 1. Membership function of fuzzy number ~p.
affecting the chances of reaching the best compromise solution. In [6] an interactive linear multiple objective method, called interactive compromise programming (ICP), is introduced. The notions of the solvability set, stability set of the first kind and the stability set of the second, and analysed these concepts for parametric convex nonlinear programming problems were introduced in [7]. In [8,10] an interactive stability for solving multiobjective nonlinear programming (MONLP) problems was introduced. This paper, is presented an interactive stability compromise programming (ISCP) method for solving fuzzy multiobjective integer nonlinear programming (FMOINLP) problems by using the compromise weights from the pay-off table of the membership function for each objective function (Fig. 1). 2. Problem formulation Let us consider the following fuzzy multiobjective integer nonlinear programming problem (FMOINLP): ðFMOINLPÞ
max subject to
ðf1 ðx; ~ a1 Þ; f2 ðx; ~ ak Þ; . . . ; fk ðx; ~ak ÞÞ ~ x 2 X ðbÞ ¼ fx 2 Rn jgj ðx; ~bj Þ 6 0; j ¼ 1; . . . ; m; and x integerg;
~i Þ, i = 1, . . . , k and gj ðx; ~ where fi ðx; a bj Þ, j = 1, . . . , m are convex real valued functions which ~ai ¼ ~ ~ ~ ~ ai2 ; . . . ; ~ aipi Þ and bj ¼ ðbj1 ; bj2 ; . . . ; bjqj Þ represent respectively a vector of fuzzy parameters involved in ð~ ai1 ; ~ the objective function fi ðx; ~ ai Þ and in the constraint function gj ðx; ~bj Þ. These fuzzy parameters are assumed to be characterized as the fuzzy numbers. A real fuzzy number ~p is a convex continuous fuzzy subset of the real line whose membership function l~p ðpÞ is defined by [9]: (1) (2) (3) (4) (5) (6)
A continuous mapping from R1 to the closed interval [0, 1], l~p ðpÞ ¼ 0 for all p 2 (1, p1], strictly increasing on [p1, p2], l~p ðpÞ ¼ 1 for all p 2 [p2, p3], strictly decreasing on [p3, p4], l~p ðpÞ ¼ 0 for all p 2 [p4, 1).
Assume that ~ air ði ¼ 1; . . . ; k; r ¼ 1; . . . ; pi Þ and ~ bjs ðj ¼ 1; . . . ; m; s ¼ 1; . . . ; qj Þ in that (FMOINLP) are fuzzy numbers whose membership function are l~air ðair Þ and l~bjs ðbjs Þ respectively. Definition 1 (a-level set). The a-level set of fuzzy numbers ~air and ~bjs is defined as the ordinary set La ð~a; ~bÞ for which the degree of their membership exceeds the level a: La ð~ a; ~ bÞ ¼ fða; bÞ j l~air ðair Þ P a; i ¼ 1; . . . ; k; r ¼ 1; . . . ; pi ; l~bjs ðbjs Þ P a; j ¼ 1; . . . ; m; s ¼ 1; . . . ; qj g.
232
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for a certain degree a, the (FMOINLP) can be understood as the following nonfuzzy a-multiobjective mixedinteger nonlinear programming problem (a-MOMINLP) 0 ða-MOMINLPÞ
0
max
ðf1 ðx; a1 Þ; f2 ðx; a2 Þ; . . . ; fk ðx; ak ÞÞ
subject to x 2 X ¼ fx 2 Rn jgj ðx; bj Þ 6 0; j ¼ 1; . . . ; m; ða; bÞ 2 La ð~a; ~bÞ and x integerg. In the (a-MOMINLP) 0 , the parameters (a, b) are treated as decision variables rather than constants. Problem (a-MOMINLP) 0 can be rewritten as the following form: ða-MOMINLPÞ
max subject to
ðf1 ðx; a1 Þ; f2 ðx; a2 Þ; . . . ; fk ðx; ak ÞÞ x 2 X ¼ fx 2 Rn jgj ðx; bj Þ 6 0; j ¼ 1; . . . ; m; Ai 6 ai 6 Bi ; i ¼ 1; . . . ; k; A0j 6 bj 6 B0j ; j ¼ 1; . . . ; m and x integerg,
where (Ai, Bi) and ðA0j ; B0j Þ are lower and upper bounds on ai and bj (i = 1, . . . , k, j = 1, . . . , m) respectively. Definition 2 (a-pareto optimal solution). (x*, a*) 2 X is said to be an a-pareto optimal solution to the (a-MOMINLP), if and only if there does not exist another x 2 X, ða; bÞ 2 La ð~a; ~bÞ such that: fi(x, ai) 6 fi(x*, a*), i = 1, . . . , k with strictly inequality holding for at least one i, where the corresponding values of parameters (a*, b*) are called a-level optimal parameters. Problem (a-MOMINLP) will be treated using the method called interactive stability compromise programming (ISCP) [10]. Zeleny [1] has suggested that the set of efficient solutions can be reduced by introducing the ‘‘compromise set’’ concept. Let fi(x, ai) be the i the objective function and fiu ðx; ai Þ be the maximum possible value of fi(x, ai) and fiL ðx; ai Þ be the minimum possible value of fi(x, ai) found under the given constraints, respectively. To obtain the compromise solution of the (a-MOMINLP) problem, find the solution which has a minimum distance with respect to the ideal solution fiu ðx; ai Þ. This idea requires normalization of the objective functions and appropriate choice for the distance measure. The solutions found in this way are a reduced set of all efficient solutions. The set of compromise solutions may be large, and also the choice of weights by the DM may be difficult. These difficulties could be reduced by combining the basic ideas for the methods of compromise programming and compromise weights. 3. Compromise weights The interactive compromise programming (ICP) is based on two main ideas. First, the DM could state his preference among some alternative solutions more easily if values of objective functions were measured on the same scale varying between zero and one. This could be done by employing ‘‘the membership functions for the objective functions’’ concept in the compromise programming. In order to elicit a membership function lfi ðx; ai Þ, from the DM for each of the objective functions fi(x, ai) in (a-MOMINLP) problems, first calculate the individual minimum fiL and maximum fiu of each objective function fi(x, ai) under the given constraints. By taking account of the calculated individual minimum and maximum of each objective function together with the rate of increase of membership of satisfaction, DM must determine his subjective membership function lfi ðx; ai Þ which is a strictly monotone increasing function with respect to fi(x, ai). Here, it is assumed that lfi ðx; ai Þ ¼ 0 lfi ðx; ai Þ ¼ 1
or ! 0 if f i ðx; ai Þ 6 fi0 or ! 1 if f i ðx; ai Þ P
and
fi1 ;
where fic represents the value of fi(x, ai) such that the value of membership function lfi ðx; ai Þ is c, fi0 is an unacceptable level for fi(x, ai) and fi1 is a completely desirable level for fi(x, ai) within fiL and fiu . The DM can select his membership function in a subjective manner by considering the rate of increase of membership satisfaction among the following four types of functions [7]:
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233
1. Linear membership function lfi ðxÞ ¼
fi ðxÞ fi0 ; fi1 fi0
i ¼ 1; 2; . . . ; k.
2. Exponential membership function lfi ðxÞ ¼ d i ½1 expfai ðfi ðxÞ fi0 Þ=ðfi1 fi0 Þg; where di > 1, ai > 0 or di < 0, ai < 0. 3. Hyperbolic membership function 1 1 lfi ðxÞ ¼ tanhðai ðfi ðxÞ ei ÞÞ þ ; 2 2 where ai > 0 or ai < 0. 4. Hyperbolic inverse membership function 1 lfi ðxÞ ¼ d i tanh1 ðai ðfi ðxÞ ei ÞÞ þ ; 2
i ¼ 1; . . . ; k;
i ¼ 1; . . . ; k;
i ¼ 1; . . . ; k;
where di > 0, ai > 0, or ai < 0. In this paper, the following definition of the membership functions is used for scaling: lfi ðx; ai Þ ¼
fi ðx; ai Þ fiL ; fiu fiL
ð1Þ
where fi(x, ai) are the objective functions, fiu are the maximum possible values of fi(x, ai), i = 1, . . . , k, and fiL are the minimum possible values of fi(x, ai) satisfying the constraints x 2 X. The lfi ðx; ai Þ are defined as the membership functions of fi(x, ai) to the maximum possible value fi(x, ai). The scalarization problem is maxðlfi ðx; ai ÞÞ, x 2 X, i = 1, . . . , k, is proposed as the following problem: max
lf kþ1 ðx; aÞ ¼
k X
ki lfi ðx; ai Þ
i¼1
subject to x 2 X .
ð2Þ
The second main idea, one of the main drawbacks of the interactive methods is the difficulty of getting the weights of the objective functions from the DM even if the values of objective functions are presented to him on the same scale. In this method, the compromise weights of objective functions can be obtained by means of the pay-off fuzzy matrix D of order k · k of which k successive columns show the effects of i instrument vector xi , ai on the membership objective functions f ðxk ; ak ÞÞ. D ¼ ð lf ðx1 ; a1 Þ; . . . ; l The compromise weights ki, i = 1, 2, . . . , k, can be obtained from the normalized version of the pay-off fuzzy matrix D as in the form: ki ¼
D01 L0 ; L; D01 L0
i ¼ 1; 2; . . . ; k;
ð3Þ
where L is the unit vector and L0 is the transpose unit vector. Also D 0 is the transpose pay-off fuzzy matrix D. 4. Stability set of the first kind Definition 3 (The solvability set). The solvability set of scalarization problem (a-MOMINLP)k which are formulated as:
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ða-MOMINLPÞk
k X
max
ki fi ðx; ai Þ
i¼1
subject to
x 2 X;
where k ¼ ðk1 ; k2 ; . . . ; kk Þ 6¼ 0;
ki P 0; i ¼ 1; . . . ; k
and
k X
ki ¼ 1
i¼1
is defined by ( k
H¼
k 2 R j max
k X
) ki fi ðx; ai Þ exists; k P 0 .
i¼1
Definition 4 (The stability set of the first kind). Suppose that H 5 / with a corresponding optimal point ðx; a; bÞ, then the stability set of the first kind of problem (a-MOMINLP)k corresponding to ðx; a; bÞ is defined by ( ) k k X X k f ðx; ai Þ ¼ max ki fi ðx; ai Þ . Sðx; a; bÞ ¼ k 2 H i¼1 i i i¼1 It is clear that the stability set of the first kind is the set of all parameters corresponding to an optimal solution 2 Rk , l 0 2 Rk, g 2 Rm and g0 2 Rm such of the scalarization problem. Let k 2 Sðx; a; bÞ then there exist u 2 Rm , l 0 0 ; l ; that (x; a; b; u; l g; g ) solve the Kuhn–Tucker problem: k X i¼1
m X
ofi ðx; ai Þ ki þ oxr
j¼1
uj
ogj ðx; bj Þ ¼ 0; oxr
r ¼ 1; . . . ; n;
ofi ðx; ai Þ i l 0i ¼ 0; i ¼ 1; . . . ; k; ki þl oai ogj ðx; bj Þ þ gj g0j ¼ 0; j ¼ 1; . . . ; m; uj obj bj Þ 6 0; j ¼ 1; . . . ; m; gj ðx; ð ai Bi Þ 6 0; j B0 Þ 6 0; ðb j
ðAi ai Þ 6 0; i ¼ 1; . . . ; n; 0 j Þ 6 0; j ¼ 1; . . . ; m; ðAj b
uj gj ðx; bj Þ ¼ 0;
j ¼ 1; . . . ; m;
i ð ai Bi Þ ¼ 0; l
i ¼ 1; . . . ; k;
0i ðAi l
ai Þ ¼ 0; gj ðbj B0j Þ ¼ 0;
i ¼ 1; . . . ; k;
g0j ðA0j bj Þ ¼ 0;
j ¼ 1; . . . ; m;
uj ¼ 0;
gj ¼ 0;
uj P 0; gj P 0; i ¼ 0; l
0j ¼ 0; l
0i l
0i l
P 0;
P 0;
j ¼ 1; . . . ; m;
g0j ¼ 0; g0j P 0;
for j 2 J f1; 2; . . . ; kg; for j 2 f1; . . . ; kg J ;
for i 2 M f1; 2; . . . ; kg; for i 2 f1; . . . ; kg; M.
M.M.K. ElShafei / Applied Mathematics and Computation 176 (2006) 230–236
235
Consider the system of equations X ogj ðx; bj Þ ofi ðx; ai Þ þ ¼ 0; r ¼ 1; . . . ; n; uj ox ox r r i¼1 j2f1;...;kgJ ofj ðx; bj Þ þ gj g0j ¼ 0; j 2 f1; . . . ; kg J ; uj obj ofi ðx; ai Þ þ li l0i ¼ 0; i 2 f1; . . . ; kg M. ki oai k X
ki
ðIÞ
It represents n + c + s linear equations in k + 3c + 2s unknowns ki, li, l0i , i = 1, . . . , k, and uj, gj, g0j , j = 1, . . . , c, which can be solved explicitly. Suppose that ki P 0, i = 1, . . . , k, uj P 0, j = 1, . . . , c, solve the above system of equations, then it is clear ; l0 ; that ðx; a; b; u; l g; g0 Þ solve the Kuhn–Tucker problem, where uj ¼ uj , j = 1, . . . , c, uj ¼ 0, j = c + 1, . . . , m, and hence k 2 Sðx; a; bÞ. Let us define the set N ðk; u; l; l0 ; g; g0 Þ ¼ fðk; u; l; l0 ; g; g0 Þ 2 R3kþ3c jðk; u; l; l0 ; g; g0 Þ solves the system (I)}. Then Sðx; a; bÞ ¼ fk 2 Rm j ðk; u; l; l0 ; g; g0 Þ 2 N ðk; u; l; l0 ; g; g0 Þg.
ðIIÞ
If gj ðx; bj Þ < 0; ð ai ; Bi Þ < 0; ðAi ai Þ < 0; bj B0j < 0 and A0j bj < 0, then it is easy to see that, Sðx; a; bÞ can be written in the following form: ( ) X k ofi ðx; ai Þ k Sðx; a; bÞ ¼ k 2 R k ¼ 0; r ¼ 1; 2; . . . ; n; k P 0 . i¼1 i oxr 5. A solution algorithm The steps of the algorithm can be summarized as follows: Step 1: Set a certain degree a = a* 2 [0, 1]. Step 2: Elicit a membership function for each fuzzy number in the formulated problem (FMOINLP) satisfying assumption (1–6). Then, determine the a-level set of the fuzzy numbers. Step 3: Convert the (FMOINLP) in the form of problem (a-MOMINLP). Step 4: Determine fiu for all i = 1, . . . , k as follows: max subject to Step Step Step Step Step
5: 6: 7: 8: 9:
f i ðx; ai Þ x2X
Find the solution of this problem by ignoring the integer condition, and use lingo program. If the solution is an integer then go to step (8), otherwise using branch and bound method [11]. If the integer solution has not been reached go to step (7), otherwise go to step (8). No integer feasible solution exists and go to step (18). The solutions of this problem are xiu, aui and fiu which are known as ‘‘ideal solution’’. Determine fiL for all i = 1, . . . , k as min
f i ðx; ai Þ
subject to
x 2 X.
Find the solution of this problem by ignoring the integer condition, and use lingo program. Step 10: Go to step (5), step (6) and step (7) after that go to step (11). Step 11: The solutions are xiL, aLi and fiL which are knows as the ‘‘anti-ideal solution’’. Step 12: Determine the membership functions corresponding the solution xiu, aiu, i = 1, . . . , k as lfi ðx; ai Þ ¼
fi ðx; ai Þ fiL fiu fiL
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The table of fuzzy matrix can be arranged as follows: lf
x1
x2
...
xk
fu
f1 f2 .. . fk
lf 1 1 lf 1 2
lf 2 1 lf 2
... ...
lf k 1 lf k 2
f1u f2u
lf 1
lf 2
...
lf k
fku
2
k
k
k
where lf j is the value of lfi in xj ; aji . i Step 13: The compromise weights ki, i = 1, . . . , k can be found from ki ¼
ðD01 L0 Þ ; LD01 L0
i ¼ 1; . . . ; k.
Step 14: By using these weights, we establish the new composite function to obtain the new alternative compromise solution, xk+1, from the problem max
lf kþ1 ðx; aÞ ¼
k X
ki lfi ðx; ai Þ
i¼1
subject to
x 2 X.
Step 15: Determine the stability set of the first kind corresponding to this solution as in relations I and II. Step 16: Determine the membership objective function of the new solution of the problem in step 14, lf kþ1 . Add this column of the table in step 12. Step 17: Ask the DM whether he prefers one solution strictly over all the other the k solutions. If he does, go to step 18, otherwise ask him his least preferred solution among all the others. Then replace this preferred solution by the new found in step 16 and go to step 13. Step 18: stop.
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M. Zeleny, Multiple Criteria Decision Making, McGraw-Hill, New York, 1982. J.M. Ignizio, Goal Programming and Extensions, Heath, Lexington, MA, 1976. S.M. Lee, Goal Programming for Decision Analysis, Auerbach, Philadelphia, PA, 1972. A.M. Geoffrion, J.S. Dyer, A. Feinberg, An interactive approach for multi-criteria optimization with an application to the operation of an academic department, Manage. Sci. 19 (1972) 357–368. S. Zionts, J. Wallenius, An interactive programming method for solving the multiple criteria problem, Manage. Sci. 22 (1976) 652– 663. R. Evren, Interactive compromise programming, J. Oper. Res. Soc. 38 (2) (1987) 163–172. M. Osman, A. EL-Benna, Stability of multiobjective nonlinear programming problems with fuzzy parameters, Math. Comput. Simul. 35 (1993) 321–326. M. Kassem, Interactive stability of multiobjective nonlinear programming problems with fuzzy parameters in the constraints, Fuzzy Sets Syst. 73 (1995) 235–243. M. Sakawa, Fuzzy Sets and Interactive Multiobjective Optimization, Plenum Press, New York, 1993. M. Kassem, Interactive stability of vector optimization problems, Eur. J. Oper. Res. 134 (2001) 616–622. S. Rao, Optimization Theory and Application, Indian Institute of Technology, Kanpur, 1977.