Expert Systems with Applications 41 (2014) 2957–2963

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Expert Systems with Applications

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

Interactive fuzzy stochastic multi-level 0–1 programming using tabusearch and probability maximization

0957-4174/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.eswa.2013.10.027

⇑ Corresponding author.E-mail address: sakawa@hiroshima-u.ac.jp (M. Sakawa).

Masatoshi Sakawa ⇑, Takeshi MatsuiFaculty of Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan

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

Keywords:Multi-level programmingRandom variablesInteractive fuzzy programmingProbability maximizationTabu search0–1 Programming

In this paper, we consider interactive fuzzy programming for multi-level 0–1 programming problemsinvolving random variable coefficients both in objective functions and constraints. Following the proba-bility maximization model together with the concept of chance constraints, the formulated stochasticmulti-level 0–1 programming problems are transformed into deterministic ones. Taking into accountvagueness of judgments of the decision makers, we present interactive fuzzy programming. In the pro-posed interactive method, after determining the fuzzy goals of the decision makers at all levels, a satis-factory solution is derived efficiently by updating satisfactory levels of the decision makers withconsiderations of overall satisfactory balance among all levels. For solving the transformed deterministicproblems efficiently, we also introduce novel tabu search for general 0–1 programming problems. Anumerical example for a three-level 0–1 programming problem is provided to illustrate the proposedmethod.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction For two-level linear programming problems or multi-level ones

Decision making problems in hierarchical managerial or publicorganizations are often formulated as multi-level programmingproblems where there are more than two decision makers(DMs): the upper level DM and the lower level DMs (Shimizu,Ishizuka, & Bard, 1997; Sakawa & Nishizaki, 2009). To describethe concept of the Stackelberg solution, consider a two-level pro-gramming problem. There are two DMs; each DM completelyknows objective functions and constraints of the two DMs, andthe DM at the upper level (leader) first make a decision and thenthe DM at the lower level (follower) specifies a decision so as tooptimize an objective function with full knowledge of the decisionof the leader. According to the rule, the leader also make a decisionso as to optimize the leader’s objective function. Then a solutiondefined as the above-mentioned procedure is called the Stackel-berg solution (Shimizu et al., 1997; Sakawa & Nishizaki, 2009).

When the Stackelberg solution is employed, it is assumed thatthere is no communication between the two DMs, or they do notmake any binding agreement even if there exists such communica-tion. However, the above assumption is not always reasonablewhen we model decision making problems in a decentralized firmas a two-level programming problem in which top management isa leader and an operation division of the firm is a follower becauseit is supposed that there exists cooperative relationship betweenthem.

such that decisions of decision makers in all levels are sequentialand all of the decision makers essentially cooperate with eachother, Lai (1996) and Shih, Lai, and Lee (1996) proposed fuzzyinteractive approaches. In their methods, the decision makersidentify membership functions of the fuzzy goals for their objectivefunctions, and in particular, the decision maker at the upper levelalso specifies those of the fuzzy goals for the decision variables.The decision maker at the lower level solves a fuzzy programmingproblem with a constraint with respect to a satisfactory degree ofthe decision maker at the upper level.

Unfortunately, however, there is a possibility that the methodsof Lai (1996) and Shih et al. (1996) lead a final solution to an unde-sirable one because of inconsistency between the fuzzy goals of theobjective function and those of the decision variables. In order toovercome the problem in their methods, by eliminating the fuzzygoals for the decision variables, Sakawa et al. have proposed inter-active fuzzy programming for two-level or multi-level linear pro-gramming problems to obtain a satisfactory solution for decisionmakers (Sakawa, Nishizaki, & Uemura, 1998; Sakawa, Nishizaki,& Uemura, 2000). Extensions to two-level linear fractional pro-gramming problems (Sakawa et al., 2000) and decentralized two-level linear programming problems (Sakawa & Nishizaki, 2002;Sakawa, Nishizaki, & Uemura, 2002; Sakawa, 2002) have also beenconsidered. A recent survey paper of Sakawa and Nishizaki (2012)is devoted to reviewing and classifying the numerous major papersin the area of so-called multi-level programming.

In actual decision making situations, however, we must oftenmake a decision on the basis of vague information or uncertain data.

2958 M. Sakawa, T. Matsui / Expert Systems with Applications 41 (2014) 2957–2963

For such decision making problems, it is natural to consider that theuncertainty in real world decision making problems is often ex-pressed by a fusion of fuzziness and randomness rather than eitherfuzziness or randomness (Sakawa, Nishizaki, & Katagiri, 2011;Sakawa, Yano, & Nishizaki, 2013). For handling not only the decisionmaker’s vague judgments in multiobjective problems but also therandomness of the parameters involved in the objectives and/orconstraints, Sakawa and his colleagues incorporated their interac-tive fuzzy satisficing methods for deterministic problems (Sakawa,1993; Sakawa & Yano, 1985; Sakawa, Yano, & Yumine, 1987;Sakawa, 2001) into multiobjective stochastic programming prob-lems, through the introduction of several stochastic programmingmodels such as expectation optimization (Sakawa, Kato, &Nishizaki, 2003; Sakawa & Kato, 2008; Sakawa, 2013), variance min-imization (Sakawa & Kato, 2008), probability maximization(Sakawa & Kato, 2002; Sakawa, Kato, & Katagiri, 2004; Sakawa &Kato, 2008; Sakawa, 2013) and fractile criterion optimization(Sakawa & Kato, 2008; Sakawa, 2013), to derive a satisficing solutionfor a decision maker (DM) from among Pareto optimal solution sets.

Recently published two books of Sakawa et al. (2011, 2013) aredevoted to introducing the latest advances in the field of multiob-jective optimization under both fuzziness and randomness on thebasis of authors’ continuing research works. Special stress is placedon interactive decision making aspects of fuzzy stochastic multiob-jective programming for human-centered systems under uncer-tainty in most realistic situations when dealing with bothfuzziness and randomness.

Furthermore, in real world decision making situations, it is of-ten found that decision variables in multiobjective stochastic pro-gramming problems are not continuous but rather discreteBecause multiobjective stochastic programming problems withdiscrete decision variables are difficult to solve strictly, it becomesimportant to develop highly efficient approximate computationalmethods. From this observation, to deal with practical sizes of sto-chastic multi-level 0–1 programming problems, as an efficientmeta-heuristics, it is required to introduce an efficient methodfor solving general 0–1 programming problems.

Under these circumstances, in this paper, we consider multi-le-vel 0–1 programming problems with random variable coefficientsin both objective functions and constraints. The main contributionof this paper is to provide a novel decision making methodologyincluding a new model, solution concept and solution algorithmto deal with more realistic problems in the real world, by simulta-neously considering various concepts such as hierarchy structure,fuzziness, randomness, 0–1 decision variables and interactive fuz-zy programming, while most of previous papers dealt with eitherof the concepts or a part of them.

Following the concept of chance constraints, stochastic con-straints are transformed into deterministic ones. Adopting theprobability maximization model, the minimization of each sto-chastic objective function is replaced with the maximization ofthe probability that each objective function is less than or equalto a certain value. Under some appropriate assumptions for distri-bution functions, the formulated stochastic multi-level 0–1 pro-gramming problems are transformed into deterministic ones. Inour interactive method, after determining the fuzzy goals of theDMs at all levels, a satisfactory solution is derived efficiently byupdating the satisfactory degrees of the DMs at the upper levelswith considerations of overall satisfactory balance among all lev-els. For solving the transformed deterministic problems efficiently,we also propose a novel tabu search method by extending tabusearch based on strategic oscillation for multidimensional 0–1knapsack problems (Hanafi & Freville, 1998) into general 0–1programming problems. An illustrative numerical example for athree-level 0–1 programming problem is provided to demonstratethe feasibility of the proposed method.

2. Stochastic multi-level 0–1 programming problems anddeterministic equivalents

In this paper, we consider stochastic multi-level 0–1 program-ming problems where each of the DMs at all levels takes overallsatisfactory balance among all levels into consideration and triesto optimize each objective function. Such a stochastic multi-level0–1 programming problem is formulated as

minimizeDM1ðLevel 1Þ

z1ðxÞ ¼ c11ðxÞx1 þ � � � þ c1KðxÞxK

..

. ...

minimizeDMKðLevel KÞ

zKðxÞ ¼ cK1ðxÞx1 þ � � � þ cKKðxÞxK

subject to A1x1 þ � � � þ AK xK 6 bðxÞx1 2 f0;1gn1 ; . . . ; xK 2 f0;1gnK

9>>>>>>>>>=>>>>>>>>>;

ð1Þ

where xl, l = 1, . . . ,K, is an nl-dimensional 0–1 decision variable col-umn vector, Al, l = 1, . . . ,K are m � nl coefficient matrices, and clj(x),l = 1, . . . ,K, j = 1, . . . ,K are nj-dimensional Gaussian random variablerow vectors with mean vectors �clj and covariance matrices

Vlpq ¼ v lhphq

� �¼ Cov½clhp ðxÞ; clhq ðxÞ�� �

; p ¼ 1; . . . ;K; q ¼ 1; . . . ;K ,

and they are independent of each other, and b(x) is a random var-iable vector whose joint distribution function is F(�).

Observing that the stochastic multi-level 0–1 programmingproblem (1) contains random variable coefficients, it should beemphasized here that solution methods for ordinary mathematicalprogramming problems cannot be applied directly. Consequently,we first deal with the constraints in (1) as chance constraints(Charnes & Cooper, 1959) which mean that the constraints needto be satisfied with a certain probability (satisficing level) and over.Namely, replacing constraints in (1) by chance constraints with asatisficing level b, the problem can be transformed as:

minimizeDM1ðLevel 1Þ

z1ðxÞ ¼ c11ðxÞx1 þ � � � þ c1KðxÞxK

..

. ...

minimizeDMKðLevel KÞ

zKðxÞ ¼ cK1ðxÞx1 þ � � � þ cKKðxÞxK

subject to Prfai1x1 þ � � � þ aiK xK 6 biðxÞgP bi; i ¼ 1; . . . ;mx1 2 f0;1gn1 ; . . . ; xK 2 f0;1gnK

9>>>>>>>>>=>>>>>>>>>;ð2Þ

where aij is the ith row vector of Al, l = 1, . . . ,K, and bi(x) is the ithelement of b(x).

Following the concept of chance constraints, the first constraintin (2) can be rewritten as:

Prfai1x1 þ � � � þ aiK xK 6 biðxÞgP bi

() 1� Prfai1x1 þ � � � þ aiK xK P biðxÞgP bi

() 1� Fiðai1x1 þ � � � þ aiK xKÞP bi

() Fiðai1x1 þ � � � þ aiK xKÞ 6 1� bi

() ai1x1 þ � � � þ aiK xK 6 F�i ð1� biÞ

where F�i is a pseudo-inverse function of Fi.Letting b̂i ¼ F�i ð1� biÞ, problem (2) can be represented as:

minimizeDM1ðLevel 1Þ

z1ðxÞ ¼ c11ðxÞx1 þ � � � þ c1KðxÞxK

..

. ...

s minimizeDMKðLevel KÞ

zKðxÞ ¼ cK1ðxÞx1 þ � � � þ cKKðxÞxK

subject to A1x1 þ � � � þ AK xK 6 b̂

x1 2 f0;1gn1 ; . . . ; xK 2 f0;1gnK

9>>>>>>>>>=>>>>>>>>>;

ð3Þ

where b̂ ¼ ðb̂1; b̂2; . . . ; b̂mÞT.

M. Sakawa, T. Matsui / Expert Systems with Applications 41 (2014) 2957–2963 2959

It should be noted here that the constraint of (3) is no longerstochastic but becomes deterministic through the idea of chanceconstraint.

Charnes and Cooper (1963) also considered three types of deci-sion rules for optimizing objective functions with random vari-ables: (i) the minimum or maximum expected value model, (ii)the minimum variance model, and (iii) the maximum probabilitymodel, which are referred to as the expectation model, the vari-ance model, and the probability model, respectively. Moreover,Kataoka (1963) and Geoffrion (1967) individually proposed thefractile model.

For instance, let the objective function represent a profit. If theDM wishes to simply maximize the expected profit without caringabout the fluctuation of the profit, the expectation model to opti-mize the expectation of the objective function is appropriate. Onthe other hand, if the DM hopes to decrease the fluctuation ofthe profit as little as possible from the viewpoint of the stabilityof the profit, the variance model to minimize the variance of theobjective function is useful. In contrast to these two types of opti-mizing approaches, as satisficing approaches, the probability mod-el and the fractile model have been proposed.

When the DM wants to maximize the probability that the profitis greater than or equal to a certain permissible level, probabilitymodel is recommended. In contrast, when the DM wishes to opti-mize such a permissible level under a given threshold probabilitywith respect to the achieved profit, the fractile model will beappropriate. In this paper, assuming that the DM wants to maxi-mize the probability that the profit is greater than or equal to a cer-tain permissible level for safe management, we adopt theprobability maximization as a decision making model.

In the probability maximization model, the minimization ofeach of objective function zl(x) in (3) is substituted with the max-imization of the probability that zl(x) is less than or equal to a cer-tain permissible level hl under the chance constraints. Throughprobability maximization, problem (3) can be rewritten as:

maximizeDM1iLevel 1j

Prfz1ðx1; . . . ; xK ;xÞ 6 h1g

..

. ...

maximizeDMKiLevel Kj

PrfzKðx1; . . . ; xK ;xÞ 6 hKg

subject to A1x1 þ � � � þ AK xK 6 b̂

x1 2 f0;1gn1 ; . . . ; xK 2 f0;1gnK :

9>>>>>>>>>=>>>>>>>>>;

ð4Þ

In this formulation, it is significant to note that all elements of clj(x),l = 1, . . . ,K, j = 1, . . . ,K are assumed to be Gaussian random variablewith mean �clj and covariance matrices Vlpq, p = 1, . . . ,K, q = 1, . . . ,K.

With these assumptions, it can be easily understood that

ðhl � ð�cl1x1 þ � � � þ �clK xKÞÞ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxT

1; . . . ; xTKÞVlðxT

1; . . . ; xTKÞ

Tq

is a random

variable following the standard Gaussian distribution with mean0 and variance 1. Hence, we can rewrite the objective functionsin (4) as follows.

Pr zlðx1; . . . ;xK ;xÞ6hlf g¼Pr cl1ðxÞx1þ�� �þclKðxÞxK 6hlf g

¼Prcl1ðxÞx1þ���þclKðxÞxK �ð�cl1x1þ���þ�clK xKÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

xT1; . . . ;x

TK

� �Vl xT

1; . . . ;xTK

� �Tq

8><>:

6hl�ð�cl1x1þ�� �þ�clK xKÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

xT1; . . . ;x

TK

� �Vl xT

1; . . . ;xTK

� �Tq

9>=>;

¼/lhl�ð�cl1x1þ�� �þ�clK xKÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

xT1; . . . ;x

TK

� �Vl xT

1; . . . ;xTK

� �Tq

0B@

1CA

In this way, (4) can be equivalently transformed into the determin-istic multi-level programming problem

maximizeDM1iLevel 1j

p1ðx1; . . . ; xKÞ ¼ /1h1�ð�c11x1þ���þ�c1K xK Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

xT1;...;xT

Kð ÞV1 xT1;...;xT

Kð ÞTq

0@

1A

..

. ...

maximizeDMKiLevel Kj

pKðx1; . . . ; xKÞ ¼ /KhK�ð�cK1x1þ���þ�cKK xK Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

xT1;...;xT

Kð ÞVK xT1;...;xT

Kð ÞTq

0@

1A

subject to x 2 X

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

ð5Þ

where X denotes the feasible region satisfying the constraints of (4).

3. Interactive fuzzy programming

Thus far, having formulated the stochastic multi-level 0–1 pro-gramming problem (1), through the ideas of chance constraintsand probability maximization, the formulated problem (1) hasbeen transformed into the deterministic multi-level 0–1 program-ming problem (5). It is now appropriate to consider the vague nat-ure of human judgments through the introduction of fuzzy goalsfor the objective functions in the multi-level 0–1 programmingproblem (5).

Considering the imprecise nature of the DMs’ judgments for theprobabilities pl(x), l = 1,2, . . . ,K in the deterministic multi-level 0–1programming problem (5), it seems natural to assume that theDMs have fuzzy goals such as ‘‘pl(x) should be substantially greaterthan or equal to some specific value.’’ Then, (5) can be rewritten as:

maximizeDM1iLevel 1j

l1ðp1ðx1; . . . ; xKÞÞ

..

. ...

maximizeDMKiLevel Kj

lKðpKðx1; . . . ; xKÞÞ

subject to x 2 X

9>>>>>>=>>>>>>;

ð6Þ

where ll(�) is a membership function to quantify a fuzzy goal for thel th objective function in (5) and it is assumed to be nondecreasing.

Although the membership function does not always need to belinear, for the sake of simplicity, we adopt a linear membershipfunction. To be more specific, if the DM feels that pl(x) should begreater than or equal to at least pl,0 and pl(x) P pl,1(>pl,0) is satisfac-tory, the linear membership function ll(pl(x)) is defined as:

llðplðxÞÞ ¼

0 ;llðplðxÞÞ < pl;0

llðplðxÞÞ�pl;0pl;1�pl;0

; pl;0 6 llðplðxÞÞ 6 pl;1

1 ;llðplðxÞÞ > pl;1

8>><>>:

ð7Þ

and it is depicted in Fig. 1.Zimmermann (1978) suggested a method for assessing the

parameter values of the linear membership function. In his meth-od, the parameter values pl,1,l = 1, . . . ,K are determined as

p1;1 ¼ p1;max ¼ p1 x11;max; . . . ; x1

K;max

� �¼ max

xT1 ;...;x

TKð ÞT2X

p1ðx1; . . . ; xKÞ

..

.

pK;1 ¼ pK;max ¼ pK xK1;max; . . . ; xK

K;max

� �¼ max

xT1 ;...;x

TKð ÞT2X

pKðx1; . . . ; xKÞ

and the parameter values pl,0,l = 1, . . . ,K are specified as

p1;0 ¼ p1 x11;min; . . . ; x1

K;min

� �

..

.

pK;0 ¼ pK xK1;min; . . . ; xK

K;min

� �

1.0

0

1.0

µl (pl (x))

pl (x)pl,0 pl,1

Fig. 1. Linear membership function.

2960 M. Sakawa, T. Matsui / Expert Systems with Applications 41 (2014) 2957–2963

where xl;min ¼ x1l;min; . . . ; xK

l;min

� �; l ¼ 1; . . . ;K is an optimal solution to

the following problem

maximize plðx1; . . . ; xKÞ ¼ /lhl�ð�cl1x1þ���þ�clK xK Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

xT1 ;...;x

TKð ÞVl xT

1 ;...;xTKð ÞT

q0@

1A

subject to x 2 X:

9>>=>>;

ð8Þ

From the monotonicity of the distribution function /l(�), problem(8) is equivalent to:

maximize ZPl ðxÞ ¼

hl�ð�cl1x1þ���þ�clK xK ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxT

1 ;...;xTK ÞVlðxT

1 ;...;xTK Þ

Tp

subject to x 2 X

9=;; l ¼ 1; . . . ;K: ð9Þ

Then, by setting the parameters as described above, the linearmembership functions (7) is identified.

To derive an overall satisfactory solution to the membershipfunction maximization problem (6), we first find the maximizingdecision of the fuzzy decision proposed by Bellman and Zadeh(1970). Namely, the following problem is solved for obtaining asolution which maximizes the smaller degree of satisfaction be-tween those of all DMs:

maximizex2X

minl¼1;...;K

ll ZPl ðxÞ

� �n oð10Þ

Solving problem (10), we can obtain a solution which maximizesthe smaller satisfactory degree between those of all DMs.

The following two conditions are considered for deriving anoverall satisfactory solution.

Termination conditions of the interactive process.

(1) For all k = 1, . . . ,K � 1, DMk’s satisfactory degree is largerthan or equal to the minimal satisfactory level d̂Dk

specifiedby DMk.

(2) For all k = 1, . . . ,K � 1, the ratio dk of satisfactory degrees isin the closed interval the lower and the upper bounds ofwhich are specified by DMk.

Condition (1) means DMk’s required condition for solutionsproposed by DM (k + 1). Condition (2) is provided in order to keepoveral satisfactory balance among all the level.

Unless the conditions are satisfied simultaneously, DMk,k = 1, . . . ,K, needs to update the minimal satisfactory level d̂Dk

. Sup-pose that the DMs from at the (q + 1) th level to at the (K � 1) thlevel, i.e., DM (q + 1), DM (q + 2), . . ., and DM (K � 1), satisfy theproposed solution but DMq does not satisfy it. Then DMq, DM(q + 1), . . ., and DM (K � 1) need to update their minimal satisfac-tory levels d̂Dk

; k ¼ q; qþ 1; . . . ;K � 1. For any two levels adjacentto each other, giving a DM at an upper level serious consideration,a DM at a lower level should update the minimal satisfactory level.

Now we are ready to propose interactive fuzzy programmingfor deriving a satisfactory solution by updating the satisfactory de-gree of the DM at the upper level with considerations of overallsatisfactory balance among all the levels.

Computational procedure of interactive fuzzy programming.Step 1: Ask the decision maker at the upper level, DM1, to subjec-

tively determine a satisficing level b 2 (0,1) for con-straints. Go to Step 2.

Step 2: In order to determine permissible levels hl, l = 1,2, . . . ,K,the following problems are solved to find the minimumvalues zE

l;min and zEl;M of objective functions zE

l ðxÞ under thechance constraints with satisficing levels bi, i = 1,2, . . . ,m.

minimize �cl1x1 þ � � � þ �clK xK

subject to x 2 X

�; l ¼ 1; . . . ;K ð11Þ

If the set of feasible solutions to these problems is empty, the satis-ficing levels bi, i = 1,2, . . . ,m must be reassessed and return to step 1.Otherwise, let zE

l;min be optimal objective function values to (11). AskDM1 to determine permissible levels hl, l = 1,2, . . . ,K for objectivefunctions in consideration of zE

l;min and zEl;M . Go to Step 3.

Step 3: Solve the following problems to find the maximum valuespl,max and pl,M of objective functions pl(x) under the chanceconstraints with satisficing levels bi, i = 1,2, . . . ,m.

maximize /lhl�ð�cl1x1þ���þ�clK xK Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

xT1 ;...;x

TKð ÞVl xT

1 ;...;xTKð ÞT

q0@

1A

subject to x 2 X

9>>=>>;

ð12Þ

From the monotonicity of the distribution function /l(�), problem(12) is equivalent to:

maximize ZPl ðxÞ ¼

hl�ð�cl1x1þ���þ�clK xK ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixT

1 ;...;xTKð ÞVl xT

1 ;...;xTKð ÞT

qsubject to x 2 X

9>=>;; l ¼ 1; . . . ;K

ð13Þ

Then, identify the linear membership function llðZPl ðxÞÞ;

l ¼ 1;2; . . . ;K of the fuzzy goal for the corresponding objective func-tion. Go to step 4.Step 4: Solve the following corresponding maxmin problem.

maximizex2X

minl¼1;...;K

llðZPl ðxÞÞ

n oð14Þ

Go to step 5.Step 5: Ask DM1 to subjectively set the minimal satisfactory level

d̂1. Then, solve the following maxmin problem.

maximizex2X

minl¼2;...;K

ll ZPl ðxÞ

� �n o

subject to l1ðp1ðxÞÞP d̂1

9=; ð15Þ

Set k:¼2 Ck0:¼1. Go to step 6.Step 6: Ask DMk to set the membership function lDk

ðDkðxÞÞ for theratio Dk ¼ ðlkþ1ðZ

Pkþ1ðxÞÞÞ=ðlkðZ

PkðxÞÞÞ of satisfactory

degrees and the minimal satisfactory level d̂Dk. Solve the

following maxmin problem.

maximizex2X

minl¼kþ1;...;K

ll ZPl ðxÞ

� �n o

subject to l1 ZP1ðxÞ

� �P d̂1

lD2ðD2ðxÞÞP d̂D2

..

.

lDkðDkðxÞÞP d̂Dk

9>>>>>>>>>>=>>>>>>>>>>;

ð16Þ

Repeat this step until k = K � 1.Step 7: If the current solution satisfies the termination conditions,

DMK � k0 accepts it, and K � k0 = 1, then the procedurestops and the current solution is determined to be a

M. Sakawa, T. Matsui / Expert Systems with Applications 41 (2014) 2957–2963 2961

satisfactory solution. Otherwise, ask DMK � k0 to updatethe minimal satisfactory level d̂DK�k0

. If K � k0 = 1, ask DM1to update the minimal satisfactory level d̂1. Go to step 8.

Step 8: Solve the following problem, and return to step 5.

maximize vsubject to x 2 X; 0 6 v 6 1

l1 ZP1ðxÞ

� �P d̂1

lD2ðD2ðxÞÞP d̂D2

..

.

lDK�1ðDK�1ðxÞÞP d̂DK�1

PK�1l¼K�k0þ1D̂llK�k0þ1ðZ

PK�k0þ1ðxÞÞP v

..

.

D̂K�1D̂K�2lK�2 ZPK�2ðxÞ

� �P v

D̂K�1lK�1 ZPK�1ðxÞ

� �P v

lK ZPKðxÞ

� �P v

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

ð17Þ

It should be noted here that, in the proposed interactive fuzzy pro-gramming method, it is required to solve the 0–1 programmingproblems (11), (12), (14), (15), (16) and (17) for practical ones. Inview of this, as an efficient meta-heuristics, the tabu search (TS) ap-proaches (Glover, 1989, 1990; Glover & Laguna, 1997) seem to bepromising.

4. Tabu search for general 0–1 programming problems

For solving the 0–1 programming problems in the proposedinteractive fuzzy programming method, in this paper, we extendtabu search based on strategic oscillation for multidimensional0–1 knapsack problems (Hanafi & Freville, 1998) into general 0–1programming problems.

Consider a general 0–1 programming problem formulated as:

minimize f ðxÞsubject to giðxÞ 6 0; i ¼ 1; . . . ;m

x 2 f0;1gn

9>=>; ð18Þ

where f(�) and gi(�), i = 1, . . . ,m are convex or nonconvex real-valuedfunctions and x = (x1, . . . ,xn)T is an n-dimensional column vector of0–1 decision variables.

The tabu search proposed in Hanafi and Freville (1998) madeuse of the property of multidimensional 0–1 knapsack problemsthat the improvement or disimprovement of the objective func-tion value corresponds with the decrease or increase of the de-gree of feasibility. From the property, it is clear that the optimalsolution to multidimensional 0–1 knapsack problems exists inthe area near the boundary of the feasible region which is calledthe promising zone. Thus, the search direction in multidimen-sional 0–1 knapsack problems can be controlled by checkingthe change of the objective function value. In the case of general0–1 programming problems, observing that the monotone rela-tion between the objective function value and the degree of fea-sibility no longer holds, the promising zone does not always existnear the boundary of the feasible region. Considering that thepromising zone originally means the area which include an opti-mal solution, we define the promising zone for general 0–1 pro-gramming problems as neighborhoods of local optimalsolutions. Thus, in order to use not only the change of the objec-tive function value but the degree of feasibility, we introduce theindex of surplus of constraints d(x) and that of slackness of con-straints �(x) defined as:

dðxÞ ¼Xi2Iþ

diðxÞ ¼Xi2Iþ

giðxÞ

�ðxÞ ¼Xi2I��iðxÞ ¼

Xi2I�� giðxÞ

where I+ = {ijgi(x) > 0, i 2 {1, . . . ,m}} and I� = {ijgi(x) < 0,i 2 {1, . . . ,m}}, Furthermore, let Djf(x) denote the change of f(x) bysetting xj :¼ 1 � xj. Similarly, Djd(x),Djdi(x),Dj�(x) and Dj�i(x) are de-fined for xj:¼1 � xj. In addition, we assign the feasible solution to x�,and update x� when the feasible solution is updated.

Computational procedure of tabu search for general 0–1 pro-gramming problems.

Step 1: INITIALIZATIONGenerate an initial solution x at random. Initialize the tabulist (TL). Set the tabu term (TT), the depth (D), the maxi-mum number of oscillation (Omax) and the oscillationcounter O:¼1. If the initial solution x is feasible, let theincumbent solution be �x :¼ x and the incumbent value be�z :¼ f ð�xÞ, and go to Step 2. Otherwise, let �z :¼ þ1, and goto Step 5. TL is updated when a value of a decision variableis changed.

Step 2: TS_ADD

(2–1)Let x0:¼x, J:¼{1,2, . . . ,n}.(2–2)If there exists j⁄ such that j⁄:¼argmin{Djf(x0)jDj-⁄:¼argmin{Djf(x0)jDjf(x0) 6 0andDjd(x0) = 0andj 2 J}, go toStep 2–3. Otherwise, obtain the solution x0 in the promisingzone, and go to Step 3.(2–3)If f ðx0Þ þ Dj� f ðxÞ 6 �z, let x0j� :¼ 1� x0j� ; �x :¼ x0;�z :¼ f ð�xÞand J:¼J � {j⁄}, and return to Step 2–2. If f ðx0Þ þ Dj� f ðxÞ > �zand j⁄ R TL, let x0j� :¼ 1� x0j� and J:¼J � {j⁄}, and return to Step2–2. If f ðx0Þ þ Dj� f ðxÞ > �z and j⁄ 2 TL, let J:¼J � {j⁄}, and returnto Step 2–2.

Step 3: TS_COMPLEMENT

(3–1) For the current solution x0, let x00:¼x0, j:¼1.(3–2) Let x⁄:¼x0.(3–3) If j R TL, let x�j :¼ 1� x�j . Then, if d(x⁄) = 0, performTS_ADD for x⁄, and go to Step 3–4. If d(x⁄) > 0, performTS_PROJECT for x⁄, and go to Step 3–4. If j 2 TL, go to Step3–4.(3–4) If x⁄ obtained in Step 3–3 satisfies f(x⁄) 6 f(x00), letx00:¼x⁄. In addition, if x⁄ satisfies f ðx�Þ 6 �z, let �x :¼ x� and�z :¼ f ð�xÞ. Let j:¼j + 1, and go to Step 3–5.(3–5) If j > n, obtain the new solution x⁄ in the promisingzone, and go to Step 4. Otherwise, return to Step 3–2.

Step 4: TS_INFEASIBLE_ADD

(4–1) Introduce logical variable called near-feasible. Near-feasible represents the approximate feasibility of the solu-tion x. For example, in the tth search, it is assumed that sis remainder when t is divided by m and gs+1(x) is the(s + 1) th constraint. If gs(x) 6 0, near-feasible is true. Letx000:¼x00 and J:¼{1,2, . . . ,n}.(4–2) If there exists j⁄ such that j⁄:¼argmin{Djf(x000)jDj-

f(x000) 6 0andj R TL andj 2 J}, go to Step 4–3. Otherwise, letnear-feasible be false, and go to Step 5.(4–3) Let y⁄:¼x000 and x�j� :¼ 1� x�j� . If d(x⁄) = 0 and f ðx�Þ 6 �z forthe solution x⁄, let x000j� :¼ 1� x000j� ; �x :¼ x000 and J:¼J � {j⁄}, andreturn to Step 4–2. If d(x⁄) > 0 and near-feasible is true forthe solution x⁄, let x000j� :¼ 1� x000j� and J:¼J � {j⁄}, and returnto Step 4–2. If d(x⁄) > 0 and near-feasible is false for the solu-tion x⁄, let J:¼J � {j⁄}, and return to Step 4–2.

Step 5: TS_PROJECT

(5–1) Let x0:¼x and J:¼{1,2, . . . ,n}.(5–2) If there exists j⁄ such that j⁄:¼argmin{Djf(x0)jDj-

d(x0) 6 0andj 2 J}, go to Step 5–3. Otherwise, go to Step 5–4.

2962 M. Sakawa, T. Matsui / Expert Systems with Applications 41 (2014) 2957–2963

(5–3) (I) In the case of j⁄ 2 TL, let x⁄:¼x0 and x�j� :¼ 1� x�j� .Then, if d(x⁄) = 0 and f ðx�Þ 6 �z, let �x :¼ x�; �z :¼ f ð�xÞ andx0:¼x⁄, and go to Step 6. Otherwise, let J:¼J � {j⁄}, and returnto Step 5–2.(II) In the case of j⁄ R TL, let x⁄:¼x0 and x�j� :¼ 1� x�j� . Then, ifd(x⁄) = 0 and f ðx�Þ 6 �z, let �x :¼ x�; �z :¼ f ð�xÞ and x0:¼x⁄, andgo to Step 6. If d(x⁄) = 0 and f ðx�Þ > �z, let x0:¼x⁄, and go toStep 6. If d(x⁄) > 0, let x0:¼x⁄ and J:¼J � {j⁄}, and return toStep 5–2.(5–4) Let J:¼{1,2, . . . ,n}. If there exists j⁄ such that j⁄:¼arg-⁄:¼argmin{Djd(x0)jDjd(x0) 6 0andj 2 J}, go to Step 5–5. Other-wise, go to Step 5–6.(5–5) (I) In the case of j⁄ 2 TL, let x⁄:¼x0 and x�j� :¼ 1� x�j� .Then, if d(x⁄) = 0 and f ðx�Þ 6 �z, let �x :¼ x�; �z :¼ f ð�xÞ andx0:¼x⁄, and go to Step 6. Otherwise, let J:¼J � {j⁄}, and returnto Step 5–4.(II) In the case of j⁄ R TL, let x⁄:¼x0 and x�j� :¼ 1� x�j� . Then, ifd(x⁄) = 0 and f ðx�Þ 6 �z, let �x :¼ x�; �z :¼ f ð�xÞ and x0:¼x⁄, andgo to Step 6. If d(x⁄) = 0 and f ðx�Þ > �z, let x0:¼x⁄, and go toStep 6. If d(x⁄) > 0, let x0:¼x⁄ and J:¼J � {j⁄}, and return toStep 5–4.(5–6) Let J:¼{1,2, . . . ,n}. If there exists j⁄ such thatj� :¼ argminfDjf ðx0Þjxj – xyj andj 2 Jg, go to Step 5–7. Other-wise, obtain the solution x0 in the promising zone, and goto Step 6.(5–7) (I) In the case of j⁄ 2 TL, let x⁄:¼x0 and x�j� :¼ 1� x�j� .Then, if d(x⁄) = 0 and f ðx�Þ 6 �z, let �x :¼ x�;�z :¼ f ð�xÞ and x0:¼x⁄,and go to Step 6. Otherwise, let J:¼J � {j⁄}, and return to Step5–6.(II) In the case of j⁄ R TL, let x⁄:¼x0 and x�j� :¼ 1� x�j� . Then, ifd(x⁄) = 0 and f ðx�Þ 6 �z, let �x :¼ x�;�z :¼ f ð�xÞ and x0:¼x⁄, andgo to Step 6. If d(x⁄) = 0 and f ðx�Þ > �z, let x0:¼x⁄, and go toStep 6. If d(x⁄) > 0, let x0:¼x⁄ and J:¼J � {j⁄}, and return toStep 5–6.

Step 6: TS_COMPLEMENTPerform the same operations as Step 3. Substitute Step 4with Step 7 in Step 3–5.

Step 7: TS_DROP

(7–1) Let x000:¼x00,J:¼{1,2, . . . ,n} and d:¼1(7–2) If there exists j⁄ such that j⁄:¼argmin{Djf(x000)jDj-⁄:¼argmin{Djf(x000)jDjd(x000) = 0andDj-

⁄:¼argmin{Djf(x000)jDjd(x000) = 0andDj-

:¼argmin{Djf(x000)jDjd(x000) = 0andDj�(x000) P 0and j 2 Jand jno-j 2 JandjnotinTL}, go to Step 7–4. Otherwise, go to Step 7–3.(7–3) If there exists j⁄ that j⁄:¼argmin{Djf(x000)jDj-⁄:¼argmin{Djf(x000)jDjd(x000) = 0andDj-

Fig. 2. Tabu search for general 0–1 programming.

⁄:¼argmin{Djf(x000)jDjd(x000) = 0andDj-

:¼argmin{Djf(x000)jDjd(x000) = 0andDj-

:¼argmin{Djf(x000)jDjd(x000) = 0andDj-

{Djf(x000)jDjd(x000) = 0andDj�(x000) P 0andj 2 Jandj 2 TL}, go toStep 7–4. Otherwise, go to Step 8.(7–4) Let x000j� :¼ 1� x000j� ; J :¼ J � fj�g and d:¼d + 1, go to Step 7–5.(7–5) If d > D, go to Step 8. Otherwise, return to Step 7–2.

Step 8: TERMINATED CONDITIONLet O:¼O + 1. If O > Omax, then stop. Otherwise, return to

Step 2.

The search procedure of the proposed tabu search for general0–1 programming problems is illustrated in Fig. 2.

5. Numerical example

As an example for a stochastic multi-level 0–1 programmingproblem, consider the following three-level problem:

minimizeDM1ðLevel 1Þ

c11ðxÞx1 þ c12ðxÞx2 þ c13ðxÞx3

minimizeDM2ðLevel 2Þ

c21ðxÞx1 þ c22ðxÞx2 þ c23ðxÞx3

minimizeDM3ðLevel 3Þ

c31ðxÞx1 þ c32ðxÞx2 þ c33ðxÞx3

subject to A1x1 þ A2x2 þ A3x3 6 bðxÞx1 2 f0;1gn1 ; . . . ; x3 2 f0;1gn3

9>>>>>>>>>=>>>>>>>>>;

ð19Þ

where x1 = (x1, . . . ,x15)T,x2 = (x16, . . . ,x30)T,x3 = (x31, . . . ,x45)T; eachentry of 15-dimensional row constant vectors cij,i,j = 1,2,3, and eachentry of 3 � 15 coefficient matrices A1,A2, and A3 are random

In step 1 of the interactive fuzzy programming, DM1 specifiessatisficing levels bi,i = 1,2, . . . ,9 as:

ðb1;b2;b3;b4;b5;b6;b7;b8;b9ÞT

¼ ð0:95;0:80;0:85;0:90;0:90;0:85;0:85;0:95;0:80ÞT :

For the specified satisficing levels bi,i = 1,2, . . . ,9, in step 2, min-imal values zE

l;min and maximal values zEl;max of objective functions

Ef�zlðx1; x2; x3Þg under the chance constraints are calculated. By con-sidering these values, the DMs subjectively specify permissiblelevels.

In step 3, maximal values pl,max of pl(x1,x2,x3) are calculated. As-sume that the DMs identify the linear membership function whoseparameter values are determined by the Zimmermann method(Zimmermann, 1978).

In step 4, the maxmin problem is solved. The obtain result isshown at the column labeled ‘‘1st’’ in Table 1.

In step 5, for the obtained optimal solution, then, the ratio ofsatisfactory degrees D1 is equal to 0.9837. Since DM1 is not

Table 1Interaction process.

Interaction 1st 2nd 3rd 4th

d̂1 0.7500 0.7500 0.8000

d̂D10.8000

d̂D20.8000

l1 ZP1ðxÞ

� �0.7160 0.7772 0.7696 0.8095

l2 ZP2ðxÞ

� �0.7043 0.6122 0.6923 0.6585

l3 ZP3ðxÞ

� �0.6856 0.6618 0.6118 0.6001

D1(x) 0.9837 0.7877 0.8996 0.8135D2(x) 0.9734 1.0801 0.8837 0.9123

lD1ðD1ðxÞÞ 0.9325

lD2ðD2ðxÞÞ 0.9767 0.8775

M. Sakawa, T. Matsui / Expert Systems with Applications 41 (2014) 2957–2963 2963

satisfied with this solution, DM1 sets the minimal satisfactory leveld̂1 to 0.75. (15) for d̂1 ¼ 0:75 is solved. For the obtained optimalsolution to (15), l1ðZ

P1ðxÞÞ ¼ 0:7772;l2ðZ

P2ðxÞÞ ¼ 0:6122, and

l3ðZP3ðxÞÞ ¼ 0:6618, shown at the column labeled ‘‘2nd’’ in Table 1.

In step 6, DM2 sets the membership function lD2ðD2ðxÞÞ for the

ratio D2 of satisfactory degrees and the minimal satisfactory levelas d̂D2 ¼ 0:80. (16) for d̂D2 ¼ 0:80 is solved. The obtained result isshown at the column labeled ‘‘3rd’’ in Table 1. For the obtainedoptimal solution to (16), l1ðZ

P1ðxÞÞ ¼ 0:7696;l2ðZ

P2ðxÞÞ ¼

0:6923;l3ðZP3ðxÞÞ ¼ 0:6118 and lD2

ðD2ðxÞÞ ¼ 0:9767.In step 7, since the ratio of satisfactory degrees D2 is greater

than d̂D2 ¼ 0:80, the condition of termination of the interactive pro-cess is fulfilled. Then, DM1 is asked whether he is satisfied with theobtained solution. Since DM1 is not satisfied, and he updates theminimal satisfactory level d̂1 from 0.75 to 0.80 in order to improvel1ðZ

P1ðxÞÞ and sets d̂D1 ¼ 0:80.

In step 8, (17) for d̂1 ¼ 0:80 and d̂D2 ¼ 0:8837 is solved. The ob-tained result is shown at the column labeled ‘‘4th’’ in Table 1. Forthe obtained optimal solution to (17), l1ðZ

P1ðxÞÞ ¼ 0:8095;

l2ðZP2ðxÞÞ ¼ 0:6585;l3ðZ

P3ðxÞÞ ¼ 0:6001 and lD1

ðD1ðxÞÞ ¼ 0:9325.In step 6, since the current solution satisfies all termination

conditions of the interactive process and DM1 is satisfied withthe current solution, the satisfactory solution is obtained and theinteraction procedure is terminated.

6. Conclusion

With simultaneously considering hierarchy structure, random-ness and fuzziness as well as discreteness of decision variables of-ten appeared in actual decision making, in this paper, we firstformulated multi-level 0–1 programming problems with randomvariable coefficients in both objective functions and constraints.For tackling the formulated problems, it has been assumed thatDMs concern about the probabilities that each of the objectivefunction values is smaller than or equal to a certain permissible le-vel. Using the probability maximization model in chance con-strained programming, the stochastic multi-level 0–1programming problems were transformed into deterministic 0–1programming ones under some appropriate assumptions for distri-bution functions. Taking into account vagueness of judgments ofthe DMs, interactive fuzzy programming has been proposed. Inthe proposed interactive method, after determining the fuzzy goalsof the DMs at all levels, a satisfactory solution is derived efficientlyby updating the satisfactory levels of the DMs at the upper levelswith considerations of overall satisfactory balance among all lev-els. It should be noted here that, in the proposed interactive fuzzyprogramming method, it is required to solve 0–1 programmingproblems which becomes difficult for practical ones. In view ofthis, we extended tabu search based on strategic oscillation formultidimensional 0–1 knapsack problems into general 0–1 pro-gramming problems. Through the proposed novel tabu search forgeneral 0–1 programming problems, the transformed determinis-tic problems to derive an overall satisfactory solution can beeffectively solved. An illustrative numerical example for a three-le-vel 0–1 programming problem was provided to demonstrate thefeasibility of the proposed method. However, further computa-tional experiences should be carried out for several types ofnumerical examples. From such experiences the proposed compu-tational method must be revised. As a subject of future work, appli-cations of the proposed method to the real world decision makingsituations should be considered in the near future. Extensions toother stochastic programming models will be considered else-where. Especially, it would be interesting to construct models opti-mizing not only the probability levels but also other parameterssuch as aspiration levels. Also extensions to multi-level 0–1

programming problems involving fuzzy random variable coeffi-cients and/or random fuzzy coefficients will be required in the nearfuture.

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