American Journal of Oper ations Research, 2011, 1, 203213 doi:10.4236/ajor.2011.14023 Published Online December 2011 (http://www.SciRP.org/journal/ajor) Copyright © 2011 SciRes. AJOR 203 Multiobjective Stochastic Linear Programming: An Overview A. Segun Adeyefa, Monga K. Luhandjula* Department of Deci sio n Sci e n ces , Uni versi t y of So ut h Africa, Pretoria, South Africa Email: *luhanmk@unisa.ac.za Received June 15, 2011; revised July 10, 2011; accept ed July 29, 2011 Abstract Many Optimization problems in engineering and economics involve the challenging task of pondering both conflicting goals and random data. In this paper, we give an uptodate overview of how important ideas from optimization, probability theory and multicriteria decision analysis are interwoven to address situations where the presence of several objective functions and the stochastic nature of data are under one roof in a linear optimization context. In this way users of these models are not bound to caricature their problems by arbitrarily squeezing different objective functions into one and by blindly accepting fixed values in lieu of imprecise ones. Keywords: Linear Programming, Multiobjective Programming, Stochastic Programming, Expected Value Optimality/Efficiency, Variance Optimality/Efficiency, Expected Value/Standard Deviation Efficiency, Tammer Optimality, Minimum Risk Optimality/Efficiency, Optimality/Efficiency in Probabilities 1. Introduction Many concrete real life problems may be put into a Lin ear Programming framework (see e.g., [18]). For some of these problems, the Decision maker has to ponder conflicting objective functions. Such competing goals cannot be arbitrarily squeezed within the narrow frame work of a unique objective function, without running the risk of invalidating all implications that are supposed to be drawn from the analysis. Simple examples (see e.g., [913]) are in line with the endorsed paradox [14] and the Arrow’s impossibility Theorem [15], where there are no good ways of aggregating conflicting criteria into a sin gle one. This has given rise to the field of Multiobjective Programming (MOP). For discussions on Multiobjective Programming problems, the reader may consult [1621]. Over and above the presence of several conflicting goals, the above mentioned problems may involve some level of uncertainty about the values to be assigned to various parameters. In this connection the noted phi losopher Nietzche was quoted as saying, “No one is gifted with immaculate perception”. False certainty is bad science and it could be danger ous if it stunts articulation of critical choices. Interested readers are referred to [2232] for problems where uncer tainty should be accommodated in an optimization setting. Uncertainty presents unique difficulties in constrained optimization problems, because the Decision makers are faced with doubtful situations, requiring an analysis of multiple outcomes in different states of nature. When the uncertainty in question is stochastic in nature, then we enter the field of Multiobjective Stochastic Linear Pro gramming (MOSLP); the subject matter of this paper. In such a turbulent environment, the notion of “opti mum optimorum” no longer applies. One has, then, to resort to the notion of satisficing solution, based upon Simon’s bounded rationality principle [33]. Methods for singling out a compromise solution in a MOSLP problem have been developed in the literature, leading to three main trends, namely: the hard, the soft and the metaheuristics. For the first trend, we refer the reader to [3437]. For the second one, the reader may consult [38,39]. Examples of the third trend may be found in [4042]. Within each group, the original problem may be either reduced to a single objective stochastic program (stochas tic approach) or converted to a deterministic multiobjec tive program (multiobjective approach). A third alternative is to combine in an appropriate manner a technique of single objective Stochastic Programming with a technique
A. S. ADEYEFA ET AL. 204 of Multiobjective Programming (hybrid approach). For the sake of space, this review focuses on the hard trend. An interested reader is referred to [4347] where he may find details about the other two trends. The methodological line followed in this overview consists of discussing upstream, existing solution con cepts and placing extant results in a coherent and com putational framework. Some existing applications are then listed downstream. We also take a step towards comparing the approaches mentioned. Such a comparison may help in designing a Decision Support System for MOSLP. The above men tioned extension is outside the scope of the present paper, and has therefore, been left for further research. Despite the purely mathematical nature of many works in the field of MOSLP as illustrated in [4850], research in this field has been suggested by a specific class of con crete, reallife problems. Such a class of problems includes reservoir operation [51], coal mining [11], water resource management [52] and transportation planning [14]. The paper is organized as follows. In the next section, we give a mathematical formulation of the problem at hand and discuss related solution concepts. Section 3 deals with some mathematical results in connection with MOSLP. Section 4 is devoted to a discussion of meth odological aspects of MOSLP along with a comparison of the above mentioned approaches. In Section 5, we point out some existing applications. We end with a number of concluding remarks along with suggestions for further developments in this field. 2. Problem Formulation and Solution Concepts 2.1. Problem Formulation A Multiobjective Stochastic Linear Programming pro blem is a problem of the type: 1 () ,, min K xD cxc x (1) where ():; 0 n DxAxbx 1,, K cc are ndimensional random vectors defined on a probability space , ,, A and b are respectively m × n and m × 1 random matrices defined on the same probability space. As an example of a concrete problem that may be put into the form of (1), we mention the automated manufac turing system in a production planning situation, with several objective functions, where the costs and time of production are known only stochastically [53]. For other problems that may be modelled in the same way as (1), we may mention reconfigurable manufac turing systems [40], distributed energy resources planning [54], water use planning [55], manufacturing planning [56], power systems planning [5759] energy and reserves markets [60] and multiproduct batch plant design [61]. Owing to the presence of conflicting goals and the randomness surrounding data, the mathematical program described in (1) is an illstated problem. Therefore, neither the notion of feasibility nor that of optimality is clearly defined for this problem. One, then, has to resort to the Simon’s bounded rationality principle [33] and seek for a satisficing solution instead of an optimal one. Before discussing some existing solution concepts for this problems along with some related mathematical results and methodological approaches, let us attempt to provide some meaning to problem (1). 2.2. Transformation of the Feasible Set One generally transform D to a deterministic set, say according to the rules used in Stochastic Pro gramming (see e.g. [6264]). D Some commonly used deterministic counterparts of D are listed below: 1) :; n DxEA xEbx 0 where stands for the expected value. 2) :; n DxPAxb x 0 where is a probability level predefined by the Deci sion maker. 3) 11 ,, m mi i D i D where for each fixed =1, ,im :, n iiii i DxPAxb x 0 here i are probability levels apriori fixed by the Decision maker and i A , i b are respectively the row of th i A and the component of th i b . 4) probability 1Dx: n,Qx , with iv where inf ; , if ; if qy Qx y where q is a penalty cost, W is a recourse matrix and :; m yWybAxy 0 In the next section, we discuss some existing solution Copyright © 2011 SciRes. AJOR
205 A. S. ADEYEFA ET AL. concepts for MOSLP problems. 3. Solution Concepts for Multiobjective Stochastic Linear Programming Problems To avoid complications unrelated to our subject, we as sume that involved random data have known distributions with finite expected values and variances. 3.1. Expected Value and Variance Optimalities Consider the following deterministic mathematical pro grams: min xDEc x (2) min xD Vc x (3) with E and V denoting the expected value and the variance respectively. Definition 3.1 If * is an optimal solution for Pro gram (2), ((3)) then * is called an expected value (a variance) optimal solution for problem (1), when D is a transformation of D obtained through technique of stochastic optimization. Where c is an aggregation of 1,, K cc based on techniques of multiattribute utility theory [65]. From now on and V stand respectively for the set of expected value and variance optimal solutions for problem (2). A shortcoming of the above defined solution concepts is that, the expected value and the variance do not exhaust the information contained in the distributions of involved random variables [34]. To overcome this draw back, other solution concepts have been proposed. We discuss some of them in the next three subsections. 3.2. Tammer and Minimum Risk Optimalities and Optimality in Probability Definition 3.2 * is a Tammer optimal solution for Problem (1), if there is no D such that * :1Pcxcx and * :<Pcxcx >0 when D is a transformation of D obtained through technique of stochastic optimization. Here is a pro bability level predefined by the Decision maker. For details on this solution concept, we invite the reader to consult [66]. Definition 3.3 * is an minimum risk optimal solution for Problem (1), if * is an optimal solution for the following program: max xDPc x (4) when D is a transformation of D obtained through technique of stochastic optimization. Where is an aspiration level apriori fixed by the Decision maker. An interested reader is referred to [67] for key facts about the minimum risk solution concept. Definition 3.4 * is a optimal solution in proba bility for Problem (1), if there is such that * ** ,x is optimal for the program: ,min subject to = xD Pc x R D (5) when D is a transformation of obtained through technique of stochastic optimization. Where is a probability level predefined by the Decision maker. A reader interested to know more about this solution concept is referred to [68]. 3.3. Expected Value and Variance Efficiencies Consider the following deterministic multiobjective pro grams: 1 minK xD Ec xcx ,,E (6) 1 minK xD Vc xcx ,,V (7) 11 ,,, ,, min xD K Ec xEcxc x cx K (8) where stands for the standard deviation. Definition 3.5 * is called an expected value, a variance or an expected value/standard deviation effi cient solution for problem (1), If * is efficient for Programs (6), (7) or (8) respectively, when is a transformation of D D obtained through technique of stochastic optimization. The sets of expected value, variance and expected value/standard deviation efficient solutions for Program (1) are denoted by , V and E respectively. The concept of expected value weak efficiency, variance weak efficiency and expected value/standard deviation weak efficiency and those of expected value proper efficiency, variance proper efficiency and ex pected value/standard deviation proper efficiency are obtained by replacing “efficiency” by “weak efficiency” and by “proper efficiency” respectively. Copyright © 2011 SciRes. AJOR
A. S. ADEYEFA ET AL. 206 In the sequel w (), p E ( V ) and w E ( E ) denote the sets of expected value weakly (properly) efficient solutions, variance weakly (properly) efficient solutions and expected value/standard deviation weakly (properly) efficient solutions for program (1) respectively. 3.4. Minimum Risk Efficiency and Efficiency in Probabilities Minimum risk efficiency is defined as follows. Definition 3.6 * is an 1,, *minimum risk efficient solution for problem (1), if is efficient for the multiobjective program: 1 1,, max KK xD Pc xPcx (9) when D is a transformation of D obtained through technique of stochastic optimization. Here 1,, are aspiration levels apriori fixed by the Decision maker. Characterizations of minimum risk efficiency with aspiration levels, may be found elsewhere [69]. As in the case of expected value efficiency, the concepts of 1,, minimum risk weak efficiency and 1 ,, minimum risk proper efficiency may be obtained by respectively replacing “efficiency” by “weak efficiency” or “proper efficiency” in the above definition. In what follows 1,, RK , 1,, w RK and 1 ,, p RK denote the sets of 1,, ,,  minimum risk efficient solutions, 1 1,, mini mum risk weakly efficient solutions and  minimum risk properly efficient solutions for Program (1) respectively. For efficiency with given probabilities we give the following definition. Definition 3.7 * is a 1,, efficient solution in probability for problem (1), if there is 1 ** =,, * such that ** ,x is efficient for the mathematical program: 1 , ,, min subject to ,=1, , K K xD kkk Pc xkK R (10) when D is a transformation of D obtained through technique of stochastic optimization. Where 1,, are probability levels that are apriori fixed by the Decision maker. An interested reader may consult [14] for a thorough discussion on this efficiency concept. Concepts of 1,, ,, weak efficiency in pro bability and 1 proper efficiency in pro bability may also be obtained in a way similar to the one in which minimum risk weak and proper efficiencies were obtained. From now on 1,, TK , 1,, w TK and 1,, P TK denote the sets of 1,, 1,, effi cient solutions in probability, 1,, weakly efficient solutions in probability and  properly efficient solutions in probability for Program (1) respectively. In the next section we present some theoretical results related to problem (1). 4. Related Mathematical Results Most stochastic constraint transformations yield noncon vexity on resulting deterministic feasible sets. This pre cludes the application of existing powerful convex opti mization algorithms (see e.g. [70,71]). It is therefore, relevant to know when a deterministic counterpart of D is convex. The following four propositions; the proofs of which may be found in [72], provide some insights to this issue. Proposition 4.1 0D , 1D , , 0; =1,, i Dim 1;1, , i Dimiv D and are convex sets. Proposition 4.2 Consider Problem (1) and suppose that A is a fixed matrix with maximal rank. Then ():(); 1,, n iiii i DxFAxi m are convex for every probability distribution i of i b . Proposition 4.3 Assume that the probability space under consideration is discrete, that is, 1,, and 0 ll Pp , 1, ,lL . Let l L * max 1:1 l pl then the set ll D is convex for any l * > l D and is convex for any * >, * where and * l are real numbers. Proposition 4.4 Suppose that the probability space under consideration is 1,, and suppose that => ll pP 0 if and only if 1, ,lN r N. Assume also that only one element exists such that o l min ll olN pp , then the sets D and l D are convex for every 1 >1 l p where 1\{ } ll lN l o minpp . The next two results established in [73,74], bridge the gap between solution concepts based on the first two moments (Proposition 4.5) and establish a connection between a minimum risk efficient solution with aspiration levels and an efficient solution with given probabilities (Proposition 4.6). Proposition 4.5 1) EV E Copyright © 2011 SciRes. AJOR
207 A. S. ADEYEFA ET AL. 2) w EV E 3) ww w EV E AsProposition 4.6sume that the probability distribu tions of the random vectors 1,, K cc are con tinuous and strictly increasing 1,, . Then for any K , * 1,, RK x if a only if nd * 1,, TK x , where k kk Pc x ; 1, ,kK Moreover, we have: 1 Proposition 4.7 1 , 1 1,, ,, , ,, KK K RK KT B R K with 1,,:0,1; 1,, Kk B kK Wellknown characterizations of proper efficiency have been explored to relate optimality and efficiency of program (1). This is the subject matter of the next two propositions. Proposition 4.8 If * is an expected value optimal solution for problem (2 then * ), is an expected value properly efficient solution for program (6). That is, E Proposition 4.9 If is a convex set and ions, then D , k c x ; =1,kK are convex functE * is an the expected v palueroperly efficient solution for multiobjective program (6), if and only if, * is an expected value optimal solution for the problem ). That is, (2 = P E An interested reader is referred to [14] for more details on ogical Approaches for Solving hed in the previous sections have served tochastic Approach ethod described in [38] for olving problem (1), using the stochastic approach. For this matter. . Methodol5Multiobjective Stochastic Linear Programs e ideas discussT as guidelines in implementing efficient techniques for solving Multiobjective Stochastic Linear Programming problems. In what follows we outline a method within each of the three existing approaches namely, the stochastic approach, the multiobjective approach and the hybrid one. .1. S5 In this section we present a m s this method the following assumptions should be met: i A , =1,,im; b and k c , =1, ,kK are normally distributed random vectors. k , , interval =1,kK are strictly positive real numbers in the 0,1 such that =1 K . =1 k k the follo notations are used: Moreovewing 1) r, =hxAx ,b iii 1,, =,im. 2) denotes the cumulative distribution function of the standard normal random variable. ion of 3) 1 and 2 q are weights associated with the ex pected value and the standard deviat q c res pectiv. 4) ely 1 =,, m where i , =1, ,im pro bability le on m are bed by the Decisiaker for constraints satisfaction vels prescri . A stepwise description of the method is as follows: Step 1. Read k , =k1, ,; K k c , =1, ,kK; ,x , 1, ,im i h ; i , =1,,im Step 2. Find K =1 =k k k cc D by Step 3. Replace 1 (),0,1,,;0 vn i ii h hx imx :,DxRE x Step 4. Solve the mathematical program: 12 min v xD qEcqc (11) Let * Step 5. Stop. be a solution of (11). an transforms the original prngle objective problem, that has been pu As cbe seen, this algorithm oblem into a si t in the deterministic form (11), using the expected value model approach [63]. The solution * obtained is an expected value/stan dard deviation efficient solution for problem (1) as de fin olving problem (1) include, decomposition m ach n the multiobjective proach. For this method, we need ed in §3.1. Other techniques closely related to the stochastic approach for s ethod [7577], chanceconstrained method [4,78], si mulation based techniques [7981], two stage method [61] and multistage method [82]. 5.2. Multiobjective Appro Here we outline a method withi ap k , =1, ,kK; such that >0 k , 11 K k k as in §5.1. Copyright © 2011 SciRes. AJOR
A. S. ADEYEFA ET AL. 208 : Read The steps of the method are as follow Step 1. k , =1, ,kK; k c , A=1, ,kK; ; b . Step 2. Replace D by :0DxEA xEbx n R ;0 (12) Step 3. Find: 1,, K Ec Ec Step 4. Solve the mathematical program: (13) Let min Kk Ecx =1 k xD k * Step 5. Stop. be a solution of (13) is mtackle randomness, while St objective functions. The so In thethod, Steps 2 and 3 ep 4 als with multiplicity of de lution * obtained is an expected value efficient solu tion for MOSLP problem (1) as defined in §3.3. For a more thorough discussion of other methods for solving MOSLP problem (1) based on the multiobjective approach, the reader is referred to [8389]. 5.3. Hybrid Approach In this section, we describe for solving MOSLP proble a hybrid method due to [90], m (1). This method is based on the assumptions given in §5.1. The following nota tions are used in the sequel. 1) , =1,, S; t , =1, ,tT; u , u , =1, ,uU denote positive,and two sided deviations from targets negative , =1,, S; t , =1, u ,tT; , =1, ,uU respectively. S, T and U are respectively the total number of positive, negative and twosided deviationsm targets fro , t anu d . 2) , =1,, S; t , =1, ,tT; u , =1, ,uU are probability levels apriori fixed by the Decision maker. Herhe steps of the method. Step 1. Read S, T, U, e are t , , =1,, S; t , t , =1, ,tT; u , u , =1, ,uU; k c , =1, ,K; k ,hx , =1,,im i Step 2. Put D the following form in: 1 1 1 : )0 s cx g ( , 1,,; 1()0, 1,,; 1 0 2 vi s sss t t ttt u k uu Dx Ex sSEcx cxg tTEcx cxg nc 1 1,,;, ,0 1,,;0,0, 0 i ii st uUEhx hx im x Step 3. Solve the mathematical program: 1 T 11 min vi US uus t xD ust (14) Let * Step 4. Stop. be a solution of (14). It is clear that this method combines the goal program ming technique for solving a multiobjective program with the chanceconstrained method for soing a sto chastic optimization problem. Other methods pertaining to the hybrid approach may be 5.4. Comparison of Different Approach Thinn while comparing the abed above are as follow: nt than requires lv found in [9193]. es e ma lessons that can be draw pproaches outlinove described a 1) The stochastic approach takes into account depen dencies between objective functions, whereas the multi objective approach does not (see for example [94]). This makes the stochastic approach closer to reality. Therefore, the stochastic approach is more effective for finding solutions to a MOSLP problem than the multiobjective pproach. a 2) The multiobjective approach is more efficie he stochastic approach, in the sense that it t fewer computations. These computations are easier to handle than those required by the stochastic approach. (see e.g., [49,58,95]). 3) The hybrid approach combines the strengths of the stochastic and the multiobjective approaches. Conse quently, the hybrid approach could perform better than either of the other two approaches for a given problem. Interested readers may consult [96] for a substantiation of this claim. 4) Methods pertaining to the hybrid approach create more flexibility in allowing the Decision maker to specify his preferences (see e.g., [91]). Nevertheless, it is the nature and the structure of the problem that determines which approach to use. In what follows, we briefly discuss some applications of Multiobjective Stochastic Linear Programming to concrete reallife problems. 6. Applications 6.1. Applications of the Stochastic Approach Production planning problems, lend themselves better to Copyright © 2011 SciRes. AJOR
A. S. ADEYEFA ET AL. 209 matter of fact, th ns and la wer system security problem preventive maintenance scheduling ing [99], hydrothermal electricity res ant esign [61]. problems [86], ar rtation network design problem [85] and . 6.3.s within the Hybrid Approach s, it is be ch, to e thods described in this field are valuable re caricature the underlying problem by itional (deterministic) Paraphrasing Howard [110], the scientific approach to es, along with the meaning of in of MOSLP. To cater best for a br logical aspects and applications) ha ration Research techniques ignore managerial ne been us nagement needs have evolved and are more co uided an cy ob velopments in this field we m to ltiobjective Fuzzy Linear Pro gr der uncertainty. the use of the stochastic approach. As a e structure of these problems dictates that one starts dealing with the multiplicity of objective functio ter tackles the randomness in data [97]. Some other applications of the stochastic approach to MOSLP problems include po [98], power plant [75], capacity plann generation [100], deployment of roadway incident ponse vehicles [101] and multiproduct batch pl d 6.2. Applications along the Multiobjective Approach Water resource planning and management e most appropriately dealt with using the multiobjective approach. Random parameters are first transformed into appropriate fixed data, before the conflicting goals are sorted out. The literature is rich in models using the multiobjective approach. We list a few of them: Water use planning [55], workforce scheduling model [102], transpo nuclear generation of electricity problem [57,103] Application To significantly bridge the dangerous gap between the problems of designing reliable portfolio assets and the mathematical programming models used to solve them, the Decision maker should be able to consider different objective functions and incorporate imprecision into the model. Owing to the complexity of such problem st to couple different techniques in an appropriate way to solve them. There are several good papers using this approa which the reader may refer. The papers [96,104109] ar some of them. 7. Concluding Remarks Multiobjective Stochastic Linear Programming is a worthwhile topic. It provides a glimpse into what it means to jostle with the complicated issue (which is nevertheless useful for applications) of combining ran domness and multiplicity of objectives into an optimiza tion setting. Me sources for those facing optimization problems in volving conflicting goals and random parameters and wishing not to blindly replacing it with a trad optimization problem. decision making and problem solving has demonstrated that, it can provide efficient tools to those few who have the resources and the will to use it. The new challenge is to provide this help at an affordable price to all who could benefit from it. There is a rich array of methods that can be used to deal with both Multiobjective Programming and Stochas tic Programming problems. This paper has somewhat demonstrated that, the Howards view applies to Multi objective Stochastic Programing. Nevertheless, theoretical and computational issu troduced solution concepts, play a crucial role in such a turbulent environment. In this paper we have presented the main principle of MOSLP. We have also indicated that there are concrete realizations in this field. We have also discussed oppor tunities and limitations oad readership, the paper has the following distinctive features: 1) It is organized towards the techniqueoriented for mat in contrast to the theoretically speculative one. 2) Practical aims take precedence over mathematical niceties. 3) The basic ideas (solution concepts, related mathe matical results, methodo ve been presented in an understandable manner. 4) The paper is filled with references for those whose appetite have been sufficiently wetted. Kirby [111] has argued that the main objections against Operation Research techniques are as follows: 1) Ope eds (perversion criticism). 2) Operation Research methods have already ed wherever they were needed (obsolescence criticism). 3) Ma mplex than those which Operation Research caters for (inadequacy criticism). 4) Operation Research’s practice has been misg d has undermined the confidence managers had in it (counterperformance criticism). This paper makes some contributions towards reme dying the above mentioned perversion and inadequa jections. Among lines for further de ay mention: 1) Extension of the theory and methods outlined here the nonlinear cases. 2) Comparison of Multiobjective Stochastic Linear Programming with Mu amming [28,50,112]. 3) Design of a userfriendly Decision Support System for Multiobjective Programming un 4) Incorporation of both randomness and fuzziness Copyright © 2011 SciRes. AJOR
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