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

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<ul><li><p>1</p><p>intcoel togingf theafted e</p><p>numerical example for a three-level 01 programming problem is provided to illustrate the proposed</p><p>1. Introduction</p><p>rarchicas muthane loweNishizion, coDMs;</p><p>so as to optimize the leaders objective function. Then a solution</p><p>there 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 rmas a two-level programming problem in which top management isa leader and an operation division of the rm is a follower becauseit is supposed that there exists cooperative relationship betweenthem.</p><p>objective function and those of the decision variables. In order toinating the fuzzye proposedi-level line</p><p>gramming problems to obtain a satisfactory solution for dmakers (Sakawa, Nishizaki, &amp; Uemura, 1998; Sakawa, Nis&amp; Uemura, 2000). Extensions to two-level linear fractiongramming problems (Sakawa et al., 2000) and decentralized two-level linear programming problems (Sakawa &amp; Nishizaki, 2002;Sakawa, Nishizaki, &amp; 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.</p><p>In actual decision making situations, however, we must oftenmake a decision on the basis of vague information or uncertain data.</p><p> Corresponding author.</p><p>Expert Systems with Applications 41 (2014) 29572963</p><p>Contents lists availab</p><p>w</p><p>.eE-mail address: sakawa@hiroshima-u.ac.jp (M. Sakawa).dened as the above-mentioned procedure is called the Stackel-berg solution (Shimizu et al., 1997; Sakawa &amp; Nishizaki, 2009).</p><p>When the Stackelberg solution is employed, it is assumed that</p><p>overcome the problem in their methods, by elimgoals for the decision variables, Sakawa et al. havactive fuzzy programming for two-level or mult0957-4174/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.eswa.2013.10.027inter-ar pro-ecisionhizaki,al pro-knows objective functions and constraints of the two DMs, andthe DM at the upper level (leader) rst make a decision and thenthe DM at the lower level (follower) species 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 decision</p><p>problem with a constraint with respect to a satisfactory degree ofthe decision maker at the upper level.</p><p>Unfortunately, however, there is a possibility that the methodsof Lai (1996) and Shih et al. (1996) lead a nal solution to an unde-sirable one because of inconsistency between the fuzzy goals of theDecision making problems in hieorganizations are often formulatedproblems where there are more(DMs): the upper level DM and thIshizuka, &amp; Bard, 1997; Sakawa &amp;the concept of the Stackelberg solutgramming problem. There are twomethod. 2013 Elsevier Ltd. All rights reserved.</p><p>al managerial or publiclti-level programmingtwo decision makersr level DMs (Shimizu,aki, 2009). To describensider a two-level pro-each DM completely</p><p>For two-level linear programming problems or multi-level onessuch 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 species those of the fuzzy goals for the decision variables.The decision maker at the lower level solves a fuzzy programmingconsiderations of overall satisfactory balance among all levels. For solving the transformed deterministicproblems efciently, we also introduce novel tabu search for general 01 programming problems. AInteractive fuzzy stochastic multi-level 0search and probability maximization</p><p>Masatoshi Sakawa , Takeshi MatsuiFaculty of Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan</p><p>a r t i c l e i n f o</p><p>Keywords:Multi-level programmingRandom variablesInteractive fuzzy programmingProbability maximizationTabu search01 Programming</p><p>a b s t r a c t</p><p>In this paper, we considerinvolving random variablebility maximization modemulti-level 01 programmvagueness of judgments oposed interactive method,factory solution is derive</p><p>Expert Systems</p><p>journal homepage: wwwprogramming using tabu</p><p>eractive fuzzy programming for multi-level 01 programming problemsfcients both in objective functions and constraints. Following the proba-ether with the concept of chance constraints, the formulated stochasticproblems are transformed into deterministic ones. Taking into accountdecision makers, we present interactive fuzzy programming. In the pro-r determining the fuzzy goals of the decision makers at all levels, a satis-fciently by updating satisfactory levels of the decision makers with</p><p>le at ScienceDirect</p><p>ith Applications</p><p>lsevier .com/locate /eswa</p></li><li><p>witFor such decisionmaking 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, &amp; Katagiri, 2011;Sakawa, Yano, &amp; Nishizaki, 2013). For handling not only the decisionmakers 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 satiscing methods for deterministic problems (Sakawa,1993; Sakawa &amp; Yano, 1985; Sakawa, Yano, &amp; Yumine, 1987;Sakawa, 2001) into multiobjective stochastic programming prob-lems, through the introduction of several stochastic programmingmodels such as expectation optimization (Sakawa, Kato, &amp;Nishizaki, 2003; Sakawa&amp;Kato, 2008; Sakawa, 2013), variancemin-imization (Sakawa &amp; Kato, 2008), probability maximization(Sakawa &amp; Kato, 2002; Sakawa, Kato, &amp; Katagiri, 2004; Sakawa &amp;Kato, 2008; Sakawa, 2013) and fractile criterion optimization(Sakawa&amp;Kato, 2008; Sakawa, 2013), to derive a satiscing solutionfor a decisionmaker (DM) from among Pareto optimal solution sets.</p><p>Recently published two books of Sakawa et al. (2011, 2013) aredevoted to introducing the latest advances in the eld 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.</p><p>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 difcult to solve strictly, it becomesimportant to develop highly efcient approximate computationalmethods. From this observation, to deal with practical sizes of sto-chastic multi-level 01 programming problems, as an efcientmeta-heuristics, it is required to introduce an efcient methodfor solving general 01 programming problems.</p><p>Under these circumstances, in this paper, we consider multi-le-vel 01 programming problems with random variable coefcientsin 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, 01 decision variables and interactive fuz-zy programming, while most of previous papers dealt with eitherof the concepts or a part of them.</p><p>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 01 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 efciently 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 efciently,we also propose a novel tabu search method by extending tabusearch based on strategic oscillation for multidimensional 01knapsack problems (Hana &amp; Freville, 1998) into general 01</p><p>2958 M. Sakawa, T. Matsui / Expert Systemsprogramming problems. An illustrative numerical example for athree-level 01 programming problem is provided to demonstratethe feasibility of the proposed method.2. Stochastic multi-level 01 programming problems anddeterministic equivalents</p><p>In this paper, we consider stochastic multi-level 01 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-level01 programming problem is formulated as</p><p>minimizeDM1Level 1</p><p>z1x c11xx1 c1KxxK</p><p>..</p><p>. ...</p><p>minimizeDMKLevel K</p><p>zKx cK1xx1 cKKxxKsubject to A1x1 AKxK 6 bx</p><p>x1 2 f0;1gn1 ; . . . ; xK 2 f0;1gnK</p><p>9&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;=&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;;</p><p>1</p><p>where xl, l = 1, . . . ,K, is an nl-dimensional 01 decision variable col-umn vector, Al, l = 1, . . . ,K are m nl coefcient matrices, and clj(x),l = 1, . . . ,K, j = 1, . . . ,K are nj-dimensional Gaussian random variablerow vectors with mean vectors clj and covariance matrices</p><p>Vlpq v lhphq </p><p> Covclhp x; clhq x </p><p>; p 1; . . . ;K; q 1; . . . ;K ,and they are independent of each other, and b(x) is a random var-iable vector whose joint distribution function is F().</p><p>Observing that the stochastic multi-level 01 programmingproblem (1) contains random variable coefcients, it should beemphasized here that solution methods for ordinary mathematicalprogramming problems cannot be applied directly. Consequently,we rst deal with the constraints in (1) as chance constraints(Charnes &amp; Cooper, 1959) which mean that the constraints needto be satised with a certain probability (satiscing level) and over.Namely, replacing constraints in (1) by chance constraints with asatiscing level b, the problem can be transformed as:</p><p>minimizeDM1Level 1</p><p>z1x c11xx1 c1KxxK</p><p>..</p><p>. ...</p><p>minimizeDMKLevel K</p><p>zKx cK1xx1 cKKxxKsubject to Prfai1x1 aiKxK 6 bixgP bi; i 1; . . . ;m</p><p>x1 2 f0;1gn1 ; . . . ; xK 2 f0;1gnK</p><p>9&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;=&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;;2</p><p>where aij is the ith row vector of Al, l = 1, . . . ,K, and bi(x) is the ithelement of b(x).</p><p>Following the concept of chance constraints, the rst constraintin (2) can be rewritten as:</p><p>Prfai1x1 aiKxK 6 bixgP bi() 1 Prfai1x1 aiKxK P bixgP bi() 1 Fiai1x1 aiKxKP bi() Fiai1x1 aiKxK 6 1 bi() ai1x1 aiKxK 6 Fi 1 biwhere Fi is a pseudo-inverse function of Fi.</p><p>Letting b^i Fi 1 bi, problem (2) can be represented as:minimizeDM1Level 1</p><p>z1x c11xx1 c1KxxK</p><p>..</p><p>. ...</p><p>sminimizeDMKLevel K</p><p>zKx cK1xx1 cKKxxK</p><p>subject to A1x1 AKxK 6 b^</p><p>9&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;=&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;;</p><p>3</p><p>h Applications 41 (2014) 29572963x1 2 f0;1gn1 ; . . . ; xK 2 f0;1gnK</p><p>where b^ b^1; b^2; . . . ; b^mT.</p></li><li><p>witIt should be noted here that the constraint of (3) is no longerstochastic but becomes deterministic through the idea of chanceconstraint.</p><p>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.</p><p>For instance, let the objective function represent a prot. If theDM wishes to simply maximize the expected prot without caringabout the uctuation of the prot, the expectation model to opti-mize the expectation of the objective function is appropriate. Onthe other hand, if the DM hopes to decrease the uctuation ofthe prot as little as possible from the viewpoint of the stabilityof the prot, the variance model to minimize the variance of theobjective function is useful. In contrast to these two types of opti-mizing approaches, as satiscing approaches, the probability mod-el and the fractile model have been proposed.</p><p>When the DM wants to maximize the probability that the protis 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 prot, the fractile model will beappropriate. In this paper, assuming that the DM wants to maxi-mize the probability that the prot is greater than or equal to a cer-tain permissible level for safe management, we adopt theprobability maximization as a decision making model.</p><p>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:</p><p>maximizeDM1iLevel 1j</p><p>Prfz1x1; . . . ; xK ;x 6 h1g</p><p>..</p><p>. ...</p><p>maximizeDMKiLevel Kj</p><p>PrfzKx1; . . . ; xK ;x 6 hKg</p><p>subject to A1x1 AKxK 6 b^x1 2 f0;1gn1 ; . . . ; xK 2 f0;1gnK :</p><p>9&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;=&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;;</p><p>4</p><p>In this formulation, it is signicant 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.</p><p>With these assumptions, it can be easily understood that</p><p>hl cl1x1 clKxK=xT1; . . . ; xTKVlxT1; . . . ; xTKT</p><p>qis a random</p><p>variable following the standard Gaussian distribution with mean0 and variance 1. Hence, we can rewrite the objective functionsin (4) as follows.</p><p>Pr zlx1; . . . ;xK ;x6hlf gPr cl1xx1 clKxxK 6hlf g</p><p>Pr cl1xx1clKxxK cl1x1clKxKxT1; . . . ;x</p><p>TK</p><p> Vl xT1; . . . ;x</p><p>TK</p><p> Tq8&gt;:</p><p>6 hlcl1x1 clKxKxT1; . . . ;x</p><p>TK</p><p> Vl xT1; . . . ;x</p><p>TK</p><p> Tq9&gt;=&gt;;</p><p>0 1</p><p>M. Sakawa, T. Matsui / Expert Systems/lhlcl1x1 clKxKxT1; . . . ;x</p><p>TK</p><p> Vl xT1; . . . ;x</p><p>TK</p><p> TqB@ CAIn this way, (4) can be equivalently transformed into the determin-istic multi-level programming problem</p><p>maximizeDM1iLevel 1j</p><p>p1x1; . . . ; xK /1 h1c11x1c1K xK xT1;...;xT</p><p>K V1 xT1 ;...;xTK Tq</p><p>0@</p><p>1A</p><p>..</p><p>. ...</p><p>maximizeDMKiLevel Kj</p><p>pKx1; . . . ; xK /K hKcK1x1cKK xK xT1;...;xT</p><p>K VK xT1 ;...;xTK Tq...</p></li></ul>

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