Expert Systems with Applications 38 (2011) 10384–10388

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Expert Systems with Applications

journal homepage: www.elsevier .com/locate /eswa

A fuzzy interactive method for a class of bilevel multiobjectiveprogramming problem q

Yue Zheng a,b, Zhongping Wan a,⇑, Guangmin Wang c

a School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Chinab College of Mathematics and Information Sciences, Huanggang Normal University, Huanggang 438000, Chinac School of Economics and Management, China University of Geosciences, Wuhan 430074, China

a r t i c l e i n f o a b s t r a c t

Keywords:Multiobjective optimizationBilevel programmingInteractive methodDecision making

0957-4174/$ - see front matter � 2011 Published bydoi:10.1016/j.eswa.2011.02.069

q Supported by the National Science Foundation of CSocial Science Foundation of Ministry of Education (N⇑ Corresponding author.

E-mail address: zpwan-whu@126.com (Z. Wan).

In this paper, we address a class of bilevel multiobjective programming problem where the lower level isa linear multiobjective optimization problem. We use the concepts of satisfactoriness as well as multiob-jective optimization at the upper level and a measurement function at the lower level, to develop aninteractive method for solving such a problem. The final solution of the proposed method is always effi-cient to the upper level when the lower level achieves satisfaction. It may be more significant in practicesince bilevel programming problem is hierarchical, and the upper level decision making is in the domi-nant position. Finally, an illustrative numerical example is given to demonstrate the feasibility of the pro-posed method.

� 2011 Published by Elsevier Ltd.

1. Introduction

Bilevel programming problem (BLPP), which is a special case ofmultiple level programming problem (MLPP), involves two optimi-zation problems where the constraint region of the first-level(upper level) problem is implicitly determined by another sec-ond-level (lower level) optimization problem. The upper level deci-sion maker (also called the leader) makes a decision first andthereafter the lower level decision maker (also called the follower)chooses his/her strategy according to the leader’s action. Therefore,the leader’s decision making can be able to influence the behaviorof the follower without completely controlling the follower’s strat-egy. At the same time, the leader may be affected by the follower’saction. In brief, each decision maker independently seeks its owninterest, but is affected by the action of the other decision maker.

BLPP has been developed and researched by many authors. Themonographs and surveys can refer to Bard (1998), Dempe (2002,2003), Luo, Pang, and Ralph (1996), Shimizu, Ishizuka, and Bard(1997), Colson, Marcotte, and Savard (2005, 2007), Wang, Wan,and Wang (2007), Vicente and Calamai (1994), Wen and Hsu(1991), and so on. Many papers have been addressed bilevel singleobjective programming problems, but not many exist, which tacklebilevel multiobjective problems.

Elsevier Ltd.

hina (No. 70771080) and theo. 10YJC630233).

Shi and Xia (1997) presented an interactive algorithm based onthe concepts of satisfactoriness and direction vector for nonlinearbilevel multiobjective problem. Thereafter, Shi and Xia (2001) dis-cussed the situation with multiple interconnected decision makers.Nishizaki and Sakawa (1999) considered a bilevel multiobjectivelinear programming problem from three kinds of situations (i.e.,optimistic anticipation, pessimistic anticipation and anticipationarising from the past behavior of the follower) based on anticipa-tion of the decision maker at the upper level. Following this ap-proach, Nishizaki, Sakawa, and Katagiri (2003) addressed thedeterministic case and investigated some extensions using con-cepts from stochastic programming. Teng, Li, and Li (2000) pro-posed a genetic algorithm. Abo-Sinna (2001) and Osman et al.(2004) presented an approach via using fuzzy set theory for solvingbilevel and multiple level multiobjective problem, respectively. Inaddition, Bonnel (2006) gave necessary optimality conditions to aso-called semivectorial bilevel programming problem (i.e., bilevelprogramming problem with a scalar-valued optimization problemon the upper level and a vector-valued problem on the lower level).Such a problem had been considered also by Bonnel and Morgan(2006). They proposed a solution method based on a penalty meth-od but no numerical results were reported. Recently, Abo-Sinna andBaky (2007) presented an interactive balance space approach formulti-level multiobjective programming problem who extendedthe work of Shi and Xia (1997). Ankhili and Mansouri (2009) pre-sented an exact penalty method for solving a class of semivectorialbilevel programming problem where the upper level objectivefunction was concave and the lower level was a multiobjectiveoptimization problem. Eichfelder (2010) developed a numerical

Y. Zheng et al. / Expert Systems with Applications 38 (2011) 10384–10388 10385

method for solving nonlinear non-convex multiobjective bileveloptimization problems. Besides, Deb and Sinha (2009a, 2009b,2009c), and Sinha and Deb (2008) discussed bilevel multi-objectiveoptimization problems based on evolutionary multi-objectiveoptimization (EMO) principles. Based on these, Deb and Sinha(2009d) proposed a viable and hybrid evolutionary-local-searchbased algorithm, and presented challenging test problems.

Now, suppose that the final preferred solution using the algo-rithm of Shi and Xia (1997) is xðkÞ; �yðkÞ

� �which is an efficient solu-

tion to the lower level problem and (x(k),y(k)) is an efficientsolution to the upper level problem and xðkÞ; �yðkÞ

� �– xðkÞ; yðkÞ� �

.The above assumption is reasonable because an efficient solutionfor the lower level problem (i.e., the final preferred solution) maynot be efficient for the upper level problem according to the algo-rithm steps (see Shi & Xia, 1997). In other words, there is a possi-bility that the final solution of their method is preferred to thelower level but not the efficient one for the upper level, althoughalgorithm stops until the upper level decision maker satisfies thissolution. A similar situation also appears (Abo-Sinna & Baky,2007), i.e., the final solution may be efficient for the third levelbut not for the first or/and second level in the case of three-level multiobjective problem. However, we note that the upperlevel decision making is in the dominant position. So, in thispaper, we develop an interactive method which use the conceptsof satisfactoriness as well as multiobjective optimization at theupper level and a measurement function (see (8) below) at thelower level for solving bilevel multiobjective problem, and whosefinal solution is always efficient to the upper level problem. First,we transform the upper level problem into a multiobjective opti-mization problem. Then we use the e-constraint method to get aset of efficient solutions that are satisfactory in rank order tothe upper level decision maker. Finally, the lower level examinethese efficient solutions one by one with the help of the measure-ment function until a best satisfactory solution is reached. Theproposed method may be more significant in practice since bilevelprogramming problem is hierarchical and the upper level decisionmaking is in the dominant position.

The paper is organized as follows. In Section 2, we review thebasic notions of multiobjective optimization problem. In Section3, we give the problem formulation. A new interactive methodfor solving bilevel multiobjective problem is presented in Section4. Finally, an illustrative numerical example is given to demon-strate the proposed algorithm in Section 5, while the conclusionis given in Section 6.

2. Basic notions in multiobjective optimization

Consider a multiobjective optimization problem:

max f ðxÞ ¼ ðf1ðxÞ; . . . ; fmðxÞÞT ;s:t: x 2 X # Rn;

ð1Þ

where x 2 Rn and X denote the decision making variable and feasibleset, respectively. f : Rn ? Rm, fi(x) (i = 1,2, . . . ,m) are continuous real-valued functions over X.

Definition 2.1. A point x� 2 X is called a weakly efficient solutionof (1) if there does not exist another �x 2 X such that fið�xÞ > fiðx�Þ forall i = 1,2, . . . ,m.

Definition 2.2. A point x� 2 X is called an efficient solution of (1) ifthere does not exist another �x 2 X such that fið�xÞP fiðx�Þ for alli = 1,2, . . . ,m, and fjð�xÞ > fjðx�Þ for at least one index j 2 {1,2, . . . ,m}.

In what follows, we review a well-known scalarization tech-nique, the so-called e-constraint method (Chankong & Haimes,

1983) which is effective for solving multiobjective optimizationproblem. It is based on a scalarization where one of the objectivefunctions is maximized while all the other objective functionsare bounded from above by means of additional constraints,

ðPe�kÞ max f kðxÞ;

s:t: f iðxÞP ei; i – k;

x 2 X;

where e�k = (e1, . . . ,ek�1,ek+1, . . . ,em)T 2 Rm�1 and k 2 {1,2, . . . ,m}.Throughout the paper, we assume that e�k is always chosen

such that ðPe�kÞ is feasible. Then, we give a main theorem about

the e-constraint method (Chankong & Haimes, 1983):

Theorem 2.1. For any e�k 2 Rm�1, the two following statements hold.

(1) If x� 2 X is an optimal solution of ðPe�kÞ, then x⁄ is weakly

efficient.(2) A solution x� 2 X is efficient if and only if it is an optimal solu-

tion of ðPe�kÞ for every k = 1, 2, . . . ,m, where ei = fi(x

⁄) fori = 1,2, . . . ,m, i – k.

3. Problem formulation

In this paper, we consider the following bilevel multiobjectiveprogramming problem of the form:

ðBMOPÞ maxx;y

Fðx; yÞ;

s:t: A1xþ B1y 6 b1;

x P 0;

where y solves;

maxy

Cy;

s:t: A2xþ B2y 6 b2;

y P 0;

ð2Þ

where x 2 Rn1 and y 2 Rn2 . The decision variables of problem (BMOP)are divided into two classes, namely the upper level decision vari-able x and the lower level decision variable y. F : Rn1 � Rn2 ! Rm1 ;

F ¼ ðF1; F2; . . . ; Fm1 Þ; C 2 Rm2�n2 ; A1; A2; B1; B2; b1 and b2 are ofappropriate dimensions.

Let S = {(x,y)jA1x + B1y 6 b1,A2x + B2y 6 b2,x P 0,y P 0}, X ={xjA1x + B1y 6 b1,x P 0}, S(x) = {yjB2y 6 b2 � A2x,y P 0}. Denote byR(x) the set of the efficient solution of the lower level problemfor any fixed x.

Definition 3.1. For a fixed (x,y) 2 S, if y is an efficient solution tothe lower level problem, then (x,y) is a feasible solution to theproblem (BMOP).

Definition 3.2. If (x,y) is a feasible solution to the problem (BMOP)and x is an efficient solution to the upper level problem for fixed y,then (x,y) is an efficient solution to the problem (BMOP).

Remark 1. If the leader’s and follower’s objectives are overly con-flicting, their solutions will very possibly only be in favor of the fol-lower and the follower might have more power on makingdecisions than the leader who actually has the authority to enforcehis/her decision power to the follower. This paradox does not sat-isfy intuitively for the leader’s rational choice (Lai, 1996). So, weassume that the decision makers between the two levels arecooperative throughout the paper. If there exists a cooperative

10386 Y. Zheng et al. / Expert Systems with Applications 38 (2011) 10384–10388

relationship between them, it is not rational for the decisionmakers to choose a non-efficient solution.

4. Interactive method for (BMOP)

4.1. Upper level problem

First, we consider the upper level problem:

maxx;yðF1ðx; yÞ; F2ðx; yÞ; . . . ; Fm1 ðx; yÞÞ;

s:t: ðx; yÞ 2 S:ð3Þ

Obviously, problem (3) can be rewritten as the following multi-objective optimization problem:

maxzðF1ðzÞ; F2ðzÞ; . . . ; Fm1 ðzÞÞ;

s:t: z 2 S;

z ¼ ðx; yÞ:ð4Þ

To build membership functions, goals and tolerances should bedetermined first. We should find the individual best solutions FL

j

and individual worst solutions FUj for each objective of (4), where

FLj ¼ max

z2SFjðzÞ; FU

j ¼ minz2S

FjðzÞ; j ¼ 1;2; . . . ;m1: ð5Þ

For the sake of simplicity, we denote Fj(z) by Fj. To facilitatecomputation for obtaining solutions, we use the following linearmembership function l(Fj) (j = 1,2, . . . ,m1) to describe the fuzzygoals of the upper level decision makers.

lðFjÞ ¼

0; if Fj < FUj ;

FUj �Fj

FUj �FL

j; if FU

j 6 Fj 6 FLj ;

1; if Fj > FLj ;

8>>><>>>:

ð6Þ

where FLj and FU

j denote the values of the objective function Fj(z)such that the degrees of the membership function are 1 and 0,respectively.

Let lj(z) denote l(Fj). Then, we give the following definitions:

Definition 4.1. If z⁄ is an efficient solution of (4), then lj(z⁄) isdefined as the satisfactoriness of z⁄ for objective function Fj(z).

Definition 4.2. For l ¼ ðl1;l2; . . . ;lm1Þ given in advance by the

leader, if an efficient solution z⁄ satisfies lj(z⁄) P lj (j = 1,2, . . . ,m1), then z⁄ is the efficient solution corresponding to the sat-isfactoriness l.

Set the satisfactoriness lj and if l(Fj) P lj, then we can obtainwith (6),

FjðzÞ ¼ FUj þ FL

j � FUj

� �lðFjÞP FU

j þ FLj � FU

j

� �lj: ð7Þ

According to the definitions and theorems above, the steps ofthe algorithm for solving (4), using the e-constraint method, areas follows:

Algorithm 1

Step 1. Set a set of satisfactoriness (si0,si1, . . . ,sij) (where sij

means the j satisfactoriness of objective i of the upperlevel problem), i = 1,2, . . . ,m1, j = 1,2,. . .. Let si = si0 andj = 0 at the beginning.

Step 2. Set ei ¼ FUi þ FL

i � FUi

� �si ði ¼ 1;2; . . . ;m1Þ based on

(7) and solve problem ðPe�kÞ. Suppose that z⁄ is an

optimal solution to (4).

Step 3. If z⁄ is an efficient solution, then go to Step 4. Other-wise go to Step 5.

Step 4. If the leader is satisfied with z⁄, then stop. Otherwisego to Step 5.

Step 5. Adjust the satisfactoriness. Let si = sij+1 (i = 1,2, . . . ,m1)and go to Step 2.

Remark 2. The upper level decision maker may achieve a set ofefficient solutions according to different satisfactoriness and thenput these solutions in rank order referring to the satisfactoriness.

4.2. Lower level problem

Define g : S ? R based on the concept of Benson (1984):

gðx; yÞ ¼maxw

eT Cw� eT Cy;

s:t: Cw P Cy;w 2 SðxÞ;

ð8Þ

where e ¼ ð1;1; . . . ;1ÞT 2 Rm2 .Then, we have the following results which are similar to Benson

(1984) and Dauer and Fosnaugh (1995).

Theorem 4.1. For (x,y) 2 S, g(x,y) P 0, and "x 2 X, R(x) = {yjg(x,y) = 0, y 2 S(x)}.

Theorem 4.2. "x 2 X, y1, y2 2 S(x), we have

(1) If Cy1 P Cy2, then g(x,y1) 6 g(x,y2).(2) If Cy1 P Cy2 and Cy1 – Cy2, then g(x,y1) < g(x,y2).

Remark 3. For a fixed x 2 X, if y1 is efficient to the lower levelproblem, then g(x,y1) = 0; and if y1 is not efficient, then g(x,y1) >0. So, the function g(x,y) can examine whether a solution isefficient or not. In addition, it can be seen from Theorem 4.2 thatthe closer a point (x,y) possibly get to the efficient set R(x), thesmaller the value of g(x,y) is. As a consequence, this function canbe regarded as a measurement between a point and the efficientset.

Now, we present an algorithm for solving problem (BMOP) asfollows:

Algorithm 2

Step 1. Let k = 0 and solve problem (3) using Algorithm 1 toobtain a set of efficient solutions that are acceptableto the leader. Rank these solutions based on the satis-factoriness as following:

ðx0; y0Þ � ðx1; y1Þ � � � � � ðxp; ypÞ:

Step 2. Let (x⁄,y⁄) = (xk,yk). If g(x⁄,y⁄) = 0, then (x⁄,y⁄) is anefficient solution and stop. Otherwise go to Step 3.

Step 3. From Theorem 4.1, it is easy to check that (x⁄,y⁄) isnot efficient to the lower level problem. If the lowerlevel decision maker satisfies this solution, however,then it is the final satisfactory solution and stop.Otherwise go to Step 4.

Step 4. Let k = k + 1 and go to Step 2.

Remark 4. In Step 2 of Algorithm 2, it may be possible thatg(x⁄,y⁄) – 0 for all the solutions. In this case, we may give a e > 0in advance. Algorithm 2 stops if g(x⁄,y⁄) < e.

Table 2Computational results of (9) using algorithm of Shi and Xia (1997) with different r.

r (x,y) (F1,F2) (l(F1),l(F2))

1 (2.8,0.2) (�2,�5.4) (0.17,0.2)0.8 (2.6,0.4) (�1,�4.8) (0.33,0.4)0.6 (2.4,0.6) (0,�4.2) (0.5,0.6)0.4 (2.2,0.8) (1,�3.6) (0.67,0.8)

Y. Zheng et al. / Expert Systems with Applications 38 (2011) 10384–10388 10387

5. Numerical example

To illustrate the feasibility of the proposed method, we considerthe following example:

maxx;y

Fðx; yÞ ¼ ð�xþ 4y;�2xþ yÞ;

s:t: 0 6 x 6 3;where y solves;max

yCy ¼ ð�y;�2yÞ;

s:t: � x� y 6 �3;� xþ 2y 6 0;2xþ y 6 12;� 3xþ 2y 6 �4;y P 0:

ð9Þ

Step 1 Finding individual optimal solutions by solving (5), we get

FL1; F

U1

� �¼ ð3;�3Þ and FL

2; FU2

� �¼ ð�3;�6Þ.

Table 1Comput

l

0.50.60.70.80.9

The upper level problem can be rewritten as the followingmultiobjective optimization problem:

max ð�xþ 4y;�2xþ yÞ;s:t: � x� y 6 �3;

� xþ 2y 6 0;2xþ y 6 12;� 3xþ 2y 6 �4;0 6 x 6 3;y P 0:

ð10Þ

Construct problem ðPe�kÞ:

max �xþ 4y;

s:t: � 2xþ y P 3l� 6 ¼ e2;

� x� y 6 �3;� xþ 2y 6 0;2xþ y 6 12;� 3xþ 2y 6 �4;0 6 x 6 3;y P 0:

ð11Þ

ati

e2

�����

We obtain a set of efficient solutions for solving the aboveproblem through changing the satisfactoriness of objectiveF2 as shown in Table 1.

Step 2: Examine which solution is efficient. e = 0.9 is givenin advance in Remark 4. Finally, we can obtain thatg(2.2,1.1) 6 0.9 and the upper level decision maker satis-fies this solution. In other words, (2.2,1.1) is the final effi-cient solution. Now, the upper level decision maker’ssatisfactoriness is (0.87,0.9).

onal results of problems (11) and (10).

Optimal solutions of(11)

Efficient solutions of(10)

(F1,F2)

4.5 (3,1.5) (3,1.5) (3,�4.5)4.2 (2.8,1.4) (2.8,1.4) (2.8,�4.2)3.9 (2.6,1.3) (2.6,1.3) (2.6,�3.9)3.6 (2.4,1.2) (2.4,1.2) (2.4,�3.6)3.3 (2.2,1.1) (2.2,1.1) (2.2,�3.3)

We use the algorithm of Shi and Xia (1997) for solvingproblem (9) and give some numerical results in Table 2.It can be seen that there are different efficient solutionswith different r. Numerical results show that the finalsolution of Shi and Xia (1997) is always efficient to thelower level problem for the fixed upper level decision var-iable. However, the final solution of the proposed methodis efficient to the upper level problem. So, it may be moresignificant in practice since bilevel programming problemis hierarchical, and the upper level decision making is inthe dominant position.

6. Conclusion

In this paper, we present an interactive method for solving aclass of bilevel multiobjective programming problem. The pro-posed method uses the concepts of satisfactoriness as well as mul-tiobjective optimization at the upper level and a measurementfunction at the lower level. This method mainly has the followingcharacteristics:

� The problem considered in this paper is solved through twostages. In the first stage, the upper level problem is transformedinto a multiobjective optimization problem. Then, we achieve aset of efficient solutions. In the second stage, we use a measure-ment function to check which solution is efficient.� The final solution of the proposed method is always efficient to

the upper level problem. However, the solution of Shi and Xia(1997) is always efficient to the lower level problem.� The measurement function has some special property.

References

Abo-Sinna, M. (2001). A bi-level non-linear multi-objective decision making underfuzziness. Journal of Operational Research Society of India, 38(5), 484–495.

Abo-Sinna, M., & Baky, I. (2007). Interactive balance space approach for solvingmulti-level multi-objective programming problems. Information Sciences, 177,3397–3410.

Ankhili, Z., & Mansouri, A. (2009). An exact penalty on bilevel programs with linearvector optimization lower level. European Journal of Operational Research, 197,36–41.

Bard, J. F. (1998). Practical bilevel optimization: Algorithms and applications.Dordrecht: Kluwer Academic.

Benson, H. P. (1984). Optimization over the efficient set. Journal of MathematicalAnalysis and Applications, 98, 562–580.

Bonnel, H. (2006). Optimality condition for the semivectorial bilevel optimizationproblem. Pacific Journal of Optimization, 2(3), 447–468.

Bonnel, H., & Morgan, J. (2006). Semivectorial bilevel optimization problem: Penaltyapproach. Journal of Optimization Theory and Applications, 131(3), 365–382.

Chankong, V., & Haimes, Y. (1983). Multiobjective decision making theory andmethodology. New York: Elsevier.

Colson, B., Marcotte, P., & Savard, G. (2005). Bilevel programming: A survey. 4OR: AQuarterly Journal of Operations Research, 3, 87–107.

Colson, B., Marcotte, P., & Savard, G. (2007). An overview of bilevel optimization.Annals of Operations Research, 153, 235–256.

Dauer, J. P., & Fosnaugh, T. A. (1995). Optimization over the efficient set. Journal ofGlobal Optimization, 7, 261–277.

Deb, K., & Sinha, A. (2009a). Constructing test problems for bilevel evolutionarymulti-objective optimization. In IEEE congress on evolutionary computation, 2009(pp. 1153–1160).

Deb, K., & Sinha, A. (2009b). Solving bilevel multi-objective optimization problemsusing evolutionary algorithms. In Lecture notes in computer science. Evolutionarymulti-criterion optimization (Vol. 5467, pp. 110–124). Springer.

10388 Y. Zheng et al. / Expert Systems with Applications 38 (2011) 10384–10388

Deb, K., & Sinha, A. (2009c). An evolutionary approach for bilevel multi-objectiveproblems. In Communications in computer and information science. Cutting-edgeresearch topics on multiple criteria decision making (Vol. 35, pp. 17–24). BerlinHeidelberg: Springer.

Deb, K., & Sinha, A. (2009d). An efficient and accurate solution methodology forbilevel multi-objective programming problems using a hybrid evolutionary-local-search algorithm. Technical Report Kangal Report No. 2009001. Availablefrom: <http://www.iitk.ac.in/kangal/reports.shtml>.

Dempe, S. (2002). Foundations of bilevel programming. Nonconvex optimization and itsapplications series (Vol. 61). Dordrecht: Kluwer Academic.

Dempe, S. (2003). Annottated bibliography on bilevel programming andmathematical problems with equilibrium constraints. Optimization, 52, 333–359.

Eichfelder, G. (2010). Multiobjective bilevel optimization. MathematicalProgramming, 123(2), 419–449.

Lai, Y. J. (1996). Hierarchical optimization: A satisfactory solution. Fuzzy Sets andSystems, 77, 321–335.

Luo, Z. Q., Pang, J. S., & Ralph, D. (1996). Mathematical programs with equilibriumconstraints. Cambridge: Cambridge University Press.

Nishizaki, I., & Sakawa, M. (1999). Stackelberg solutions to multiobjective two-levellinear programming problems. Journal of Optimization Theory and Applications,103(1), 161–182.

Nishizaki, I., Sakawa, M., & Katagiri, H. (2003). Stackelberg solutions tomultiobjective two-level linear programming problems with random variablecoefficients. Central European Journal of Operations Research, 11, 281–296.

Osman, M., Abo-Sinna, M., Amer, A., & Emam, O. (2004). A multi-level nonlinearmulti-objective decisionmaking under fuzziness. Journal of Applied Mathematicsand Computation, 153(1), 239–252.

Shimizu, K., Ishizuka, Y., & Bard, J. F. (1997). Nondifferentiable and two-levelmathematical programming. Dordrecht: Kluwer Academic.

Shi, X., & Xia, H. (1997). Interactive bilevel multi-objective decision making. Journalof the Operational Research Society, 48, 943–949.

Shi, X., & Xia, H. (2001). Model and interactive algorithm of bi-level multi-objectivedecision-making with multiple interconnected decision makers. Journal ofMulticriteria Decision Analysis, 10, 27–34.

Sinha A., & Deb, K. (2008). Towards understanding evolutionary bilevel multi-objective optimization algorithm. Technical Report Kangal Report No. 2008006,Kanpur, India: Department of Mechanical Engineering, Indian Institute ofTechnology Kanpur. Available from: <http://www.iitk.ac.in/kangal/reports.shtml>.

Teng, C., Li, L., & Li, H. (2000). A class of genetic algorithms on bilevel multi-objective decision making problem. Journal of Systems Science and SystemsEngineering, 9(3), 290–296.

Vicente, L. N., & Calamai, P. H. (1994). Bilevel and multilevel programming: Abibliography review. Journal of Global Optimization, 5, 1–23.

Wang, G., Wan, Z., & Wang, X. (2007). Bibliography on bilevel programming.Advances in Mathematics, 36(5), 513–529 (in Chinese).

Wen, U. P., & Hsu, S. T. (1991). Linear bilevel programming problems – A review.Journal of Operational Research Society, 42, 125–133.