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A Procedure to Find Discrete Representations of theEfficient Set with Specified Coverage ErrorsSerpil Sayin,

To cite this article:Serpil Sayin, (2003) A Procedure to Find Discrete Representations of the Efficient Set with Specified Coverage Errors.Operations Research 51(3):427-436. http://dx.doi.org/10.1287/opre.51.3.427.14951

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A PROCEDURE TO FIND DISCRETE REPRESENTATIONS OF THEEFFICIENT SET WITH SPECIFIED COVERAGE ERRORS

SERPIL SAYINKoç University, College of Administrative Sciences and Economics, Rumeli Feneri Yolu, Sarıyer, 80910 Istanbul, Turkey, ssayin@ku.edu.tr

An important issue in multiple objective mathematical programming is finding discrete representations of the efficient set. Because discretepoints can be directly studied by a decision maker, a discrete representation can serve as the solution to the multiple objective problem athand. However, the discrete representation must be of acceptable quality to ensure that a most–preferred solution identified by a decisionmaker is of acceptable quality. Recently, attributes for measuring the quality of discrete representations have been proposed. Althoughdiscrete representations can be obtained in many different ways, and their quality evaluated afterwards, the ultimate goal should be to findsuch representations so as to conform to specified quality standards. We present a method that can find discrete representations of theefficient set according to a specified level of quality. The procedure is based on mathematical programming tools and can be implementedrelatively easily when the domain of interest is a polyhedron. The nonconvexity of the efficient set is dealt with through a coordinateddecomposition approach. We conduct computational experiments and report results.

Received January 2000; revisions received May 2001, March 2002; accepted March 2002.Subject classifications: Programming, multiple criteria: efficient set. Discrete representations.Area of review: Optimization.

1. INTRODUCTION

The Multiple Objective Linear Programming (MOLP) prob-lem can be written as:

(MOLP) Maximize Cx� subject to x ∈ X�where X ⊆ �n is a polyhedral set, C = �c1� � � � � cpT is thep×n matrix of objective function coefficients, p � 2� Theobjective functions of problem MOLP are usually conflict-ing in nature and efficient solutions are the ones that conveythe trade-off information to a decision maker (DM) whoprefers more to less in each objective.

Definition 1. xo ∈ �n is an efficient solution for theMOLP problem if xo ∈ X and there exists no x ∈ X suchthat Cx � Cxo with cjx > cjx

o for some j ∈ �1� � � � � p�.Let XE denote the set of efficient solutions of problem

MOLP. A most-preferred solution to the MOLP problemshould belong to XE� Vector maximization methods proposeto identify all of the efficient solutions of the MOLP prob-lem (see, for example, Yu and Zeleny 1975, Ecker et al.1980, Armand and Malivert 1991, Sayin 1996), which hasproven to be a computationally prohibitive task. All of thesealgorithms utilize the fact that XE is a union of efficientfaces of X and except for the algorithm in Sayin (1996),they enumerate all of the efficient extreme points of X inthe process.Finding only the extreme point efficient solutions of the

MOLP problem helps the drawbacks of identifying all ofXE to a certain extent as, for example, in Evans and Steuer(1973). Although the task is still complex and computation-ally demanding, the extreme point efficient solutions consti-tute a discrete representation of the efficient set. However,this set does not seem to constitute a good representation ofthe efficient set, as it usually contains too many points, not

necessarily uniformly spread across XE . This has motivatedthe search for discrete representations that consist of effi-cient points different than the extreme points. In an earlyattempt, Steuer and Harris (1980) have proposed generat-ing random convex combinations of extreme points of anefficient face to create a more representative sample. Byadjusting the distribution underlying the random generationof the points and by filtering away the ones that are tooclose to each other, they aim at generating a representativeand uniform discrete set. Although this method is designedto control the quality of the representation it delivers, itsmajor drawback is that it requires the efficient faces to beknown, and all of the extreme points that define the facesneed to be enumerated prior to obtaining the discrete rep-resentation.Along the way there have been propositions to generate

more manageable subsets of the efficient extreme points ofthe MOLP problem, such as those by Steuer (1976) andArmann (1989). As methods were proposed, a consensuson the quality attributes that define good representative setsseemed to evolve. The error made in representing XE , thedistances among the points in the representation, and thenumber of points in the representation were mentioned asfactors that determine the quality of a discrete representa-tion in Steuer and Harris (1980). However, the approachesproposed to generate a subset of the efficient set usually donot seem to guarantee that the obtained samples of pointsare globally representative and uniform subsets of the effi-cient set.While seeking discrete representations of XE based on

efficient faces and efficient extreme points, another impor-tant problem is the potential distortions in the quality of therepresentation when mapped into the outcome space. Afterall, the quantities of utmost relevance for the DM are theobjective function values, i.e., outcome space values. It has

0030-364X/03/5103-0427 $05.001526-5463 electronic ISSN 427

Operations Research © 2003 INFORMSVol. 51, No. 3, May–June 2003, pp. 427–436

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been argued in Dauer (1987) that the efficient set collapseswhen mapped into the outcome space. That is, when XEis mapped into the outcome space, it is possible that manyefficient points map into the same outcome. If a discreterepresentation is obtained based on the information of thedecision space and then mapped into the outcome space,some points may map into the same or nearby points. Thismay distort the quality of the representation at hand. Thus,it becomes important to take this phenomenon into accountwhen finding a discrete representation of XE .An attempt at finding a discrete representation of the

efficient set without explicitly enumerating the efficientextreme points or the efficient faces is given by Bensonand Sayin (1997). The global shooting procedure applies tomore general problems than MOLP, works in the outcomespace, and returns a discrete sample of points from theefficient set. The procedure is simple and computationallytractable for the MOLP case; however, it can control thequality of the representations it generates only indirectly.Since the efficient set is a union of efficient faces of X

in the MOLP case, and efficient faces are polyhedra, onecan see that finding discrete representations of XE is equiv-alent to finding discrete representations of a union of poly-hedra. Finding representations of polyhedra is a generalproblem that has been studied with various motivations.In particular, there are studies that probabilistically gener-ate uniformly distributed points in a polyhedron defined byconstraints (see, for example Smith 1984 and referencestherein). There are also proposed methods for generatinguniformly distributed points on the boundary of a poly-hedron, such as the one given by Boender et al. (1991).Although these approaches pose very interesting ideas, thealgorithms are not readily applicable to the problem of gen-erating a discrete sample of points from the efficient set dueto their assumptions, for example, on the full dimensional-ity of the set under study. Moreover, the results concentrateon the limiting distribution of the points generated usingsuch techniques: thus, it is hard to anticipate how uniformand representative a sample they would create with a smallnumber of points. Finally, it may not be straightforwardto devise mechanisms to account for the collapsing of XEwhen mapped into the outcome space. Therefore, since ouractual task is to generate uniform samples from a polyhe-dron that is mapped by a vector-valued objective function,and the constraints that define the image polyhedron are notreadily available and are difficult to obtain, these methodswill have to be modified to generate points in the imagepolyhedron.In this study we propose a mathematical-programming-

based approach for obtaining a representation of the effi-cient set. This approach is based on the fact that theefficient set is a union of efficient faces. The procedurereceives the efficient set represented as a collection of max-imally efficient faces as an input. The entire collection ofmaximally efficient faces can be obtained via enumerationtechniques such as the one in Sayin (1996). It is also possi-ble to sample from among the collection of efficient faces

heuristically with less computational effort as proposed byBenson and Sayin (1993). The procedure is motivated bythe tools given in Sayin (2000) to evaluate the quality of agiven representation. The major advantage of our approachis that it is designed to generate representations that areof specified quality. Moreover, the procedure performs thecomputations in the outcome space and hence avoids com-plications that might arise due to collapsing of the efficientset. To our knowledge, this is the first method for generat-ing discrete representations of the efficient set that worksin the outcome space, does not require generation of theentire set of efficient extreme points, and has a direct con-trol over the quality of the generated representations at thesame time.In the next section, we establish the preliminaries, state

our method for finding discrete representations, discussits properties, and present an example. In §3, we presentresults of experiments that highlight certain computationalaspects of the approach by first focusing on delivering arepresentation of a single face, and then incorporating acomparison to an implementation that delivers a represen-tation of the efficient set. Section 4 contains our concludingremarks.

2. REPRESENTING THE EFFICIENT SET

2.1. Preliminaries

Let YE = �y ∈ �p � y = Cx for some x ∈ XE� denote theimage of the efficient set XE under the transformation of theobjective function coefficient matrix C. When we are seek-ing a representative set of points within XE , we are equiv-alently looking for a representation of XE in the outcomespace, a representation of YE , and it is actually the qualityof the image of the representation with respect to the imageof the efficient set that is more relevant. In other words,as we are looking for a discrete representation D of XE ,we are more interested in the properties of CD as a dis-crete representation of YE , where CD denotes the image ofthe representative set D under the mapping C. Henceforth,when we refer to the quality of a discrete representation Dof XE , it should be understood that indeed the quality ofthe representation in the outcome space is implied and thetwo expressions ‘D representing XE’ and ‘CD represent-ing YE’ are used interchangeably. We assume that YE is nota singleton and thus is a continuous set, since otherwiseseeking its discrete representations would not be meaning-ful. Moreover, we also assume that YE is either originally abounded set or an appropriately truncated set, since other-wise seeking a discrete representation of an unbounded YEwould lead to a practically problematic case.Quantifying the quality of a discrete representation is

necessary, but is not straightforward. It has been men-tioned, for example in Armann (1989) and Benson andSayin (1997), that for D to be accepted as a good repre-sentation of XE it needs to contain a reasonable number ofpoints, should not miss large portions of YE , and should notcontain points that are very close to each other. In Sayin

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(2000), measures are proposed to quantify these criteria.The number of points contained in a representation is thesimplest to deal with as it is already measurable. To mea-sure the coverage error, the following definition is made. Apoint x in XE is considered to be represented by an elementy of D whose image is closest to its image. The distancebetween the images of the two points measures the error inrepresenting the point x. Then the coverage error for XEis defined as the largest such error; i.e., it is the error thatis associated with a worst-represented element of XE . Theerror made in representing XE by D, the coverage error,can be mathematically written as:

� =maxx∈XE

miny∈D

d�Cx�Cy�

where d��� �� denotes the distance between the two vectors.In multiple criteria decision making, the Tchebycheff dis-tance (also referred to as the l� distance) has received wideacceptance. The Tchebycheff distance marks the maximumcoordinatewise distance between two vectors as the overalldistance between the two. This is especially attractive inour case because we are measuring the coverage error, andthus we are referring to the Tchebycheff distance wheneverwe refer to the distance function denoted by d��� ��, i.e. forx� y ∈ �n, d�x� y�=maxi=1��n �xi−yi�.Similarly, it is proposed in Sayin (2000) that the unifor-

mity of a representation can be measured by the distancebetween a pair of closest elements of D. Thus, the unifor-mity level can be expressed as

�= miny�z∈D�y �=z

d�Cy�Cz��

The number of points in the representative set, the cov-erage error, and the uniformity level are the attributes thatdetermine the quality of a representation. Smaller numberof points, lower coverage error, and higher uniformity levelare desirable. Therefore, if the number of points in a dis-crete representation is predetermined, one would aspire tominimize the coverage error and maximize the uniformitylevel.Whenever a discrete representation of a set is available it

is possible to compute these quantities to be able to judgewhether the representation is of an acceptable quality forthe DM. Computing the coverage error is not trivial; how-ever, for the MOLP problem it becomes possible to com-pute the coverage error by solving a number of mixed 0-1programming problems, as XE is a union of efficient facesof X as shown in Sayin (2000). Computing the uniformitylevel is always simple. Moreover, when a given represen-tation does not meet the quality expectations due to a lowuniformity level, that is, whenever there are points that aretoo close to each other, an approach such as the filteringapproach of Steuer and Harris (1980) can be used to elim-inate some of these points. However, when the coverageerror of a representation turns out to be high, it is not easyto propose a similar simple way of improving the error.

Therefore, it is important to incorporate the quality mea-sures while finding a discrete representation so as to guar-antee a specified quality level. The proposed procedure isdesigned to control both the coverage error and the unifor-mity level of the representation it delivers. In the followingsection, the main idea of the procedure is outlined and theresults that validate the approach are given. For clarity ofpresentation, the issues regarding the implementation of theprocedure are discussed in a subsequent section.

2.2. The Procedure

Suppose that we would like to obtain N distinct points inXE so that their collection D constitutes a good represen-tation of XE , or we would like to obtain as many pointsas necessary until a specified target coverage error � isreached. Consider the following iterative procedure. Start-ing with one arbitrary point in the representation, supposethat we find a point in XE that is worst-represented. Then,adding this worst-represented point to the representationand seeking a (next) worst-represented point, we repeat thisprocedure until a representation that consists of a speci-fied number of points is constructed, or a target coverageerror is reached. The procedure, labeled REPR, that findsdiscrete representations of the efficient set can be mathe-matically expressed as follows:

Step 0. Let N be the number of points to be obtainedand set � = 0, or let � be the target coverage error to beattained and set N =�. Let x1 ∈ XE be an arbitrary point.Set CD = �y1 = Cx1�� Set i = 1, and go to Step i.Step i. Solve problem �Pi� given by:

�Pi� �∗i =max �i

s.t. �i � d�y� yj�� j = 1� � � � � i�

y ∈ YE�Let y∗ denote an optimal solution to problem �Pi�� Set� = �∗i . If � > � and i < N , set i = i+ 1, yi = y∗, setCD = CD ∪ �yi�, and go to Step i. Else if � � � andi < N , set K = i and STOP. CD = �y1� � � � � yK� is a rep-resentation of desired quality. Else (i = N ) set K = i andSTOP. CD = �y1� � � � � yK� is a discrete representation ofXE that has a coverage error of �.

The procedure is simple, and to see its validity one needsto make the following observation. By definition, the cov-erage error of CD as a representation of YE is given by�=maxy∈YE minj=1��K d�y� y

j�� which is exactly what prob-lem �PK� delivers, and therefore �

∗K is the coverage error

of the representation.Another feature of procedure REPR is that the uni-

formity level of the representation generated is auto-matically available and is equal to �∗K−1. To see this,just observe that the uniformity level of CD is givenby �=mini�j=1��K� i �=j d�yi� yj�=mini=2��K� j=1��i−1 d�yi� yj�=mini=1��K−1 �∗i = �∗K−1� Intuitively, since the solutionobtained at a particular Step i is added to the representation

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at the end of the step, the minimum of the point-to-pointdistances within the representation is always available. Bypicking the worst-represented or most distant points andadding them to the representation, the procedure controlsthe uniformity level of the representation indirectly whileconcentrating on the coverage error.We would like to mention here that it is possible to

express the constraints of problem �Pi� that involve theTchebycheff distance by using binary variables. In order tosimplify the presentation of the approach, we will delay thediscussion of the issues surrounding this formulation until§2.3.The following properties are easy to see, yet important

for the functionality of procedure REPR.

Theorem 1. The sequence of optimal values to problem�Pi� satisfy �

∗1 � �

∗2 � · · ·� �∗K�

Proof. Observe that as a new point yi+1 is added to therepresentative set from iteration i to i+ 1� the problem�Pi+1� remains the same as �Pi� except for the addition of anew constraint �i � d�y� y

i+1�, thus making the feasible setof problem �Pi� more restrictive and the optimal objectivefunction value the same or smaller. �

Lemma 1. When N is set to � and � set to 0 in the proce-dure, it will generate an infinite sequence of points �yi��i=1�

Proof. Suppose that N = � and � = 0 in the procedure.Notice that the only way for the procedure to stop is toobtain �∗i = 0 at some Step i. Clearly the procedure willnot stop at the end of Step 1 as YE is not a singleton byassumption, and there exists y ∈ YE such that d�y� y1� > 0,which implies �∗1 > 0.Let yi be obtained at the end of Step i− 1 of the pro-

cedure, yielding D = �y1� � � � � yi� for Step i. Assume that�i−1 > 0. Since YE is a continuous set, there exists y ∈ YEsuch that y �= yj for j = 1� � � � � i. Then d�y� yj� > 0 forj = 1� � � � � i. Therefore, minj=1��i d�y� yj� > 0. Since y ∈ YEwas arbitrarily chosen,

�∗i =maxy∈YE

minj=1��i

d�y� yj�� minj=1��i

d�y� yj� > 0�

Since N = � and �i > �� the procedure will not stop atStep i. Since this is true for any i, we can conclude that theprocedure will not terminate at a finite number of iterationswith the settings N = � and � = 0, and will create aninfinite sequence of points �yj��j=1� �

Lemma 2. At any Step i�K� for any �� �∗i � d�yi+1� yj�� �

for any j ∈ �1� � � � � i�.Proof. Let i �K denote any step of the procedure and let� � �∗i . Suppose, to the contrary, that d�y

i+1� yj � < � forsome j ∈ �1� � � � � i�. Then clearly d�yi+1� yj � < �∗i � Sinceyi+1 is the optimal solution to problem �Pi�, it satisfies�∗i =minj=1��i d�yi+1� yj�� which leads to the conclusion thatd�yi+1� yj � <minj=1��i d�yi+1� yj�� which is a contradiction.The result follows. �

Corollary 1. At any Step i�K� for any j� k ∈ �1� � � � � i�and any � � �∗i � d�y

j� yk�� ��

Proof. Let i � K denote a step of the procedure and let� � �∗i be arbitrary. Without loss of generality, assume thatj < k� j� k ∈ �1� � � � � i�� Since �∗i � �∗k−1, � � �∗k−1 is true.By the above lemma, then, d�yk� yj�� � follows. �

Theorem 2. Let �yi� be a sequence of optimal solutionsgenerated by the procedure with the setting N = � and� = 0. Let ��∗i � be the associated sequence of optimal val-ues. Then limi→� �∗i = 0�

Proof. Since the distance between two points, d�x� y� isbounded below by 0, and �∗i = maxy∈YE minj=1��i d�y� y

j�,the nonincreasing sequence of �∗i ’s is bounded from belowand thus should converge (Theorem 3.14 in Rudin 1976).To see that the sequence actually converges to 0, sup-pose, to the contrary, that lim��∗i �i→� = � > 0� Let B��x�=�y ∈ �p �d�x� y� < ��. Since YE is compact by assumption,there exists a finite number of points �x1� � � � � xM� such thatYE ⊆

⋃Mj=1B�/2�x

j�� Because �yi� is an infinite sequence ofpoints, there exists j ∈ �1� � � � �M� such that at least twoyi’s, say yi1� yi2 with i1 < i2 without loss of generality, liein B�/2�x

j �� Because yi1 ∈ B�/2�xj � and yi2 ∈ B�/2�xj �, wehave d�yi1� xj � < 1

2 � and d�yi2� xj � < 1

2 �� Then d�yi1� yi2��

d�yi1� xj �+d�yi2� xj � < �. Because �� �∗i2 is clear from thefact that the �i’s is a nonincreasing sequence, this contra-dicts that d�yi1� yi2� � � proposed by the above corollary.Therefore we conclude that limi→� �∗i = 0� �

Theorem 2 points out an important property of procedureREPR. We see that the coverage error of the representationobtained by our method will keep improving as the repre-sentative points build up. This property is in line with theexpectation that by including more points in a representa-tion it should be possible to improve its coverage error to adesired level. Thus, the procedure can guarantee a coverageerror less than or equal to a particular level when allowedto include a large enough number of points in the discreterepresentation.The following example demonstrates how the procedure

works. To keep the illustration simple, we assume that theefficient set XE consists of the single face F defined by theconstraints F = ��x1� x2� ∈ R2 �0 � x1 � 10� 0 � x2 � 4�and we have y1 = f1�x1� x2� = x1, y2 = f2�x1� x2� = x2, orin other words, the objective function coefficient matrix isthe identity matrix, and YE = XE = F .Figure 1 demonstrates the representative set created by

the procedure with the setting N = 10� It is also possi-ble to trace the sequence of points as they were obtainedby the procedure. Note that for a particular problem �Pi�,i = 1� � � � �10, there are usually multiple optimal solutions.Depending on which of the alternative optimal solutionsis chosen, different representations may evolve. The cover-age error progresses as �∗1 = 10� �∗2 = 5� �∗3 = �∗4 = �∗5 =4� �∗6 = �∗7 = 2�5� �∗8 = �∗9 = �∗10 = 2� The uniformity levelof the representation is �= �9 = 2� The progression of thepoints added to the discrete representation also hints that

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Figure 1. Example of a representative set with N = 10points.

✲

✻

y2

y1

4

10•

y1 =(10,0)•

y2 =(0,0)

•y3 = (5,4)

•y4 =(9,4)

•y5 =(1,4)

•y6 =(5,0)

•y7 =(7.5,1.5)

•y8 =(2.5,1.5)

• y9 =(0.5,2)

• y10 =(3,3.5)

the procedure tends to obtain points from the boundary ofthe face before it starts finding points from the interior. Asthe procedure tries to locate a most distant point from theexisting ones, it is pushed towards the boundary of the faceat the beginning stages.

2.3. Implementation Issues

There are two main issues that need to be addressed to beable to apply procedure REPR practically. These stem fromthe difficulty of solving problem �Pi� due to the nonlin-earity of the Tchebycheff function that appears in the con-straint set, and the nonconvexity of the efficient set itself.The Tchebycheff function can be dealt with by introducingbinary variables as proposed in Savin (2000).Let dj denote the Tchebycheff distance between point y

and yj� j = 1� � � � � i� Let ep ∈ �p denote a vector of 1’s, Mdenote a sufficiently large number. Then problem �Pi� canbe written as problem �P ′

i �:

�P ′i � max �i

s�t� �i−dj � 0� j = 1� � � � � i� (1)

−epdj +y+uj = yj� j = 1� � � � � i� (2)

epdj +y−oj = yj� j = 1� � � � � i� (3)

uj −Mtj � 0� j = 1� � � � � i� (4)

oj −Msj � 0� j = 1� � � � � i� (5)

eTk tj + eTk sj � 2p−1� j = 1� � � � � i� (6)

y−Cx = 0� (7)

x ∈ XE� (8)

uj� oj � 0� j = 1� � � � � i� (9)

tj � sj ∈ �0�1�� j = 1� � � � � i� (10)

Since the procedure works in the outcome space, andXE is given in the decision space as expressed by con-straint (8), constraint (7) controls the transformation. Theadditional constraints given by (2)–(7) help compute thecoverage error which can be written as

�i =maxy∈YE

mini=1��j

maxk=1��p

�yk−yjk��

To see why the variable dj measures the distance betweeny and yj , first note that by constraints (2) and (3),

dj � yk−yjk� k = 1� � � � � p� (11)

and

dj � yjk−yk� k = 1� � � � � p� (12)

Moreover, the slack and surplus variables uj and oj in con-straints (2) and (3) are introduced to ensure that at least oneinequality among the 2p in (11) and (12) holds as an equal-ity. This is controlled through the related binary variablesdefined by constraints (4) and (5). Note that the binary vari-ables tj � sj are defined as indicator variables whose compo-nents identify whether the associated component of a uj� oj

is positive or not. Constraint (6), allowing at most 2p− 1positive components in a pair of sj and tj , also forcesone among the 2p inequalities (11) and (12) to hold asan equality. Thus dj = maxk=1��p �yk− yjk� is ensured. Notealso that constraint (1) helps establish �i as the minimumof the dj� j = 1� � � � � i. The objective function maximizes�i, ensuring the definition.Note that uj� oj� sj � tj ∈�p, and thus �P ′

i � has 2pi binaryvariables, n+p+ �2p+ 1�i+ 1 continuous variables, and�4p+ 2�i+ p constraints excluding the definition of thedomain of interest XE . Thus, the size of the formulationincreases as i grows. It should also be noted that K of theseproblems are solved by procedure REPR and the total com-putational effort is more than that is required to solve oneof these problems. Our computational results show that thenumber of binary variables, and therefore K and p, are thefactors that influence computational effort the most. Formost realistic cases, as both K and p have to be reason-ably small, the procedure is expected to result in affordablecomputation times.Although the formulation works in the outcome space,

it does so indirectly. This is important because given XE ,finding YE may not be a simple task, as demonstrated inDauer and Saleh (1992). Moreover, because of the way for-mulation �P ′

i � is written, whenever an optimal solution y∗

is obtained and added to the set of representative points,a corresponding decision variable solution, say x∗, is alsoavailable without additional effort. Thus, as we build a dis-crete representation CD of YE in the outcome space, a dis-crete representation D in the decision space whose imageis CD is obtained.Another issue that needs to be clarified is finding an

initial point y1 = Cx1 to include in the representation sothat problem �P ′

1� can be solved. An idea that is easy toincorporate into the procedure is to take an arbitrary pointy0 ∈ �p, not necessarily in YE , and to apply problem �P ′

0�to generate y1� where �P ′

0� is given by

�P ′0� max �0

s�t� �0−dj � 0� j = 1� � � � � i�

�2�− �10��

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Figure 2. Alternative representation starting with y1 =�5�2�.

✲

✻y2

y1

4

10

• y1 =(5,2)

•y2 =(0,4)

• y3 =(10,4)

•y4 = (9,0)

• y5 =(1,0)

• y6 =(2.5,2.5) • y7 =(7.5,2.5)

• y8 =(5,0)

• y9 =(9.5,2)

• y10 =(5,4)

Although applying �P ′0� does not have a particular intu-

itive appeal, y1 obtained as the optimal solution of thisproblem will belong to YE and can be used to initialize theprocedure.It is true that the choice of the initial point will affect the

subsequent points that the procedure will find and includein the representation. Figure 2 shows the previous exam-ple when the procedure is initialized with y1 = �5�2�. Therepresentative points are different than the ones obtainedbefore. The progression of the coverage error is as follows.�∗1 = �∗2 = 5, �∗3 = �∗4 = 4, �∗5 = �∗6 = 2�5, �∗7 = �∗8 = �∗9 =�∗10 = 2� Although the representative points are different,with N = 10, the procedure achieves the same coverageerror and the same uniformity level as before.Dealing with a nonconvex efficient set as the domain of

interest in problem �Pi� remains to be a difficult issue. Solv-ing problem �Pi� as it is would require the development ofa special solution procedure. Instead, we propose a heuris-tic modification of REPR that relies on a decompositionof the efficient set. The decomposition is based on the factthat the efficient set for problem MOLP is a union of effi-cient faces, each of which is in turn a convex polyhedron.A face F of X, being a polyhedron itself, can be char-

acterized in a number of ways. (For an informative dis-cussion on faces of convex sets, see §18 in Rockafellar1970, for example.) First, F can be expressed as a convexcombination of its extreme points if it is bounded and itsextreme points are known. Second, given the set of con-straints and nonnegativity restrictions that define X, a facecan be characterized by a subset of these constraints andnonnegativity restrictions that are binding for all elementsof F as in Yu and Zeleny (1975) and Sayin (1996). Third,F can be defined as the optimal solution set to a linear pro-gramming problem as in Ecker et al. (1980), which againlets it be expressed through constraints. Because our pro-cedure is based on a mathematical programming formula-tion, we require that F be expressed via constraints. Thatis, we rule out the first characterization above and assumethat F is expressed as described in the second or third char-acterization, and thus F can be easily expressed via linearconstraints.

To apply procedure REPR, we propose to solve prob-lem �Pi� for the individual faces that make up XE so as toobtain their discrete representations. These individual rep-resentations then can be put together to form a representa-tion of the efficient set. When we restrict our attention to aface F , problem �P ′

i � becomes a mixed 0-1 programmingproblem with linear constraints. However, when each effi-cient face is treated independently, bringing the pieces ofthe representation together to constitute an overall repre-sentation for the efficient set raises further issues for con-sideration. The coverage error for the entire representationwill equal the largest coverage error across individual faces.However, one could predict almost with certainty that theuniformity level of the overall presentation will be inferiorto those of the individual faces, since there might be pointson separate faces that are close to each other. Filtering suchpoints might be an idea to improve the quality of the repre-sentation, but this might cause an increase in the coverageerror because some points will be eliminated. To addressthis problem in a systematic way we propose the followingapproach, which is based on a coordinated decompositionidea. We propose that efficient faces should still be consid-ered one by one and sequentially, yet not independently. Ina sequential manner, representations should be constructedwithin a face, taking into consideration all of the pointsincluded in the face representations that have been formedbefore. Here is a formal definition of the procedure that isbased on the decomposition idea.

Procedure REPR-D:Step 0. Let Fl� l= 1� � � � �L denote the collection of dis-

tinct efficient faces that make up the efficient set. Let � bethe target coverage error. Initialize the uniformity level �as a large number. Set l = 1. Let x1 ∈ Fl be an arbitrarypoint. Set CD = �y1 = Cx1�� Set i = 1, and go to Step 1.Step 1. Solve problem �PDi � given by:

�PDi � �∗i =max �i

s�t� �i � d�y� yj�� j = 1� � � � � i�

y ∈ CFl�where CFl denotes the image of the face Fl� Use formula-tion �PD′i �, which is a counterpart of formulation �P

′i � with

Fl as the domain of interest to obtain an optimal solution y∗.

If �∗i > �, set i = i + 1� yi = y∗�CD = CD ∪ �yi�. If�∗i−1 < �� set �= �∗i−1 and go to Step 1�If �∗i � �, set �l = �∗i � This is the coverage error for

face Fl�If l < L� set l= l+1 and go to Step 1. Otherwise, STOP.

CD = �y1� � � � � yK� is a discrete representation of YE thathas a coverage error equal to max��1� � � � � �L�, which isless than or equal to �� and a uniformity level of ��

Procedure REPR-D is one way of implementing proce-dure REPR; however, there are a number of differencesin the implementation as opposed to the original state-ment. Unlike procedure REPR, procedure REPR-D is not

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designed to accommodate the choice of working with a pre-specified number of points to obtain as a representation, asallocating these to individual faces is not straightforward.Instead, the procedure iterates to reach the desired level ofcoverage in a face and then progresses to the next face inthe list to do the same until all of the faces have a cover-age error equal to or below the desired level. This way, theoverall coverage error is guaranteed to be within the spec-ifications given at the beginning of the procedure. As pro-cedure REPR-D iterates from one face to the other in noparticular order of the efficient faces, the coverage error forthe entire efficient set is not necessarily the last coverageerror computed. Rather, it is the maximum of the individualcoverage errors across faces. Likewise, the uniformity levelof the representation delivered by REPR-D is not availableas a biproduct, and it is traced explicitly whenever a pointis added to the representation and is deliverable at the endof the procedure. Finally, we would like to point out thatthere might be other ways of devising an implementableversion of procedure REPR and we wanted to limit ourattention to one straightforward one in order not to lose ourfocus.The example that appears in Figure 3 has three variables,

six constraints, three objective functions and four efficientfaces. This problem was originally given in Dessouky et al.(1986). The efficient faces of the problem are faces F1 � � � F4as labeled in the figure. The faces were taken into consider-ation in this order to generate the representation. When runwith a desired coverage error of 2.0, procedure REPR-Dgenerates 18 representative points in 18.5 CPU secondsusing the code and machine specifications described in the

Figure 3. Example of a representative set obtainedusing procedure REPR-D.

✲

✻

��

��

��

��

��

��

��

��

��✠ x2

x1

x3

F1

F2

F3

F4

F6

F5

•

•

•

•

•

•

•• •

• •

•

•

•

•

•

••

next section with a coverage error of 1.97. The uniformitylevel of the representation turns out to be equal to 2.0.

3. COMPUTATIONAL EXPERIMENTS

In this section, we report the results of two separate sets ofcomputational experiments. To apply procedure REPR-D,the collection of efficient faces of the problem shouldbe available. The set of maximally efficient faces can beobtained using the implementation of the algorithm givenin Sayin (1996). However, finding the efficient set is a com-putationally demanding task, and therefore the efficient setin its entirety cannot be obtained for large problems. Inpractice, what can be done for large problems is to use aheuristic method to obtain a sample of efficient faces ratherthan the entire set. Therefore, for the purposes of this study,we first apply procedure REPR by randomly generatinga polyhedron which we assume to be a face of X� Thisway, we can generate and solve relatively large problems.We use this set of experiments to investigate the computa-tional aspects of procedure REPR and the impact of variousparameters on its performance when the domain of interestis restricted to a face. For the second set of experiments,we restrict our attention to smaller-size problems for whichthe entire efficient set is obtained using the code of Sayin(1996). In this set we focus on the additional issues thatthe decomposition idea of procedure REPR-D brings. Thecode was developed in C, and the CPLEX Callable Library(CPLEX 1999) was used to solve the mixed 0-1 linear pro-gramming problems. The computational experiments wereconducted on a Sun Ultra Enterprise 2 Server that runsunder the operating system SunOS 5.5.1.The first set of experiments is designed to assess the

computational performance of procedure REPR to obtain arepresentative sample of N points from a domain of inter-est that is given by a polyhedral F expressed in the formF = �x ∈ �n � Ax � b� x � 0�, where A / �m×n is the con-straint coefficients matrix and b ∈ R �m. Thus, � is set to0 in the procedure. In the test problems, the elements ofthe constraint matrix A, the right-hand-side vector b, andthe objective function coefficient matrix C were randomlygenerated integers from the discrete uniform distribution inthe intervals �1�20�, �1�000�2�500� and �−10�10�, respec-tively. A 25% zero density was provided in the matrix A.In this set of experiments, the parameters that are inves-

tigated are primarily the ones that affect the size of theformulation �P ′

i �. These are the number of constraints thatdefine F � �m�, the number of decision variables (n), and thenumber of objective functions (p), in addition to the num-ber of points sought (N ), which determines the number ofproblems �P ′

i � solved and affects their sizes.We created problems in three different �m×n combina-

tions. For each combination, we also attempted to gener-ate three different sets of problems with p = 2, p = 4, andp = 6. For each problem class thus obtained, we createda set of 30 test problems. In all of the test problems, the

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initial representative point that is needed to start the proce-dure, y1, was obtained by solving a problem �P0� as sug-gested in §2.3, where y0 ∈ �p is arbitrarily chosen as theorigin. We use this approach as a means of obtaining afeasible point in CF , and the computational effort requiredto solve problem �P0� implemented through a mixed 0-1programming formulation similar to �P ′

i � is included in thereported computation times. We report results for three dif-ferent N values of 5�10 and 15.In Table 1 we report the average CPU times for the test

problems in each category. As indicated in the table, theCPU times exceeded 2,000 seconds in three problem cate-gories. It is possible to observe the computational trends inTable 1. The average CPU times increase as the problemsize increases, that is, as �m and n increase. However, theeffect of the number of objective functions p on CPU timesis more pronounced, possibly because p is a factor thatdetermines the number of binary variables in the formula-tion �P ′

i � which has 2pi binary variables. For example, forp= 2 and N = 15, the last formulation �P ′

N � has 60 binaryvariables. The number of additional constraints that corre-spond to constraint sets (2)–(6) also depends on p and N ,amounting to 150 additional constraints for problem �P ′

N �for the above category.The CPU times increase rapidly as the number of points

to be generated, N , increases. Again, this is expectedbecause N determines how many mixed 0-1 programmingproblems �P ′

i � will be solved. Moreover, the number ofbinary variables in �P ′

i � grows, as i grows which is boundedabove by N . For p = 6 and N = 15, the last formula-tion �P ′

N � has 180 binary variables and 450 additionalconstraints.In addition to the computational performance of proce-

dure REPR, we also collected information on how the cov-erage error progresses as the number of points includedin the representation increases. As an indication of this, inTable 2 we report the averages of the percent reduction incoverage error, which can be expressed as

�1− �N�1

×100�

This quantity is the reduction in the coverage error whenN points are present in the representation as opposed to

Table 1. Performance of the procedure: CPU time (sec).

p× �m×n N = 5 N = 10 N = 15

2 × 40× 50 2�6 9�2 21�44 × 40× 50 10�2 81�4 516�16 × 40× 50 23�1 188�6 1�497�42 × 80× 100 18�1 52�0 106�24 × 80× 100 56�2 290�7 1�219�5∗6 × 80× 100 98�5 465�7 >2,0002 × 100 × 150 47�9 116�9 213�74 × 100 × 150 142�5 660�3 >2,0006 × 100 × 150 223�5 944�7 >2,000

∗One problem in this group exceeded 2,000 seconds and isexcluded from the average.

the initial coverage error, expressed as a percentage of theinitial error. Thus, higher quantities indicate more improve-ment in coverage error and are desirable.As expected, the reduction in coverage error increases as

N increases for all problem categories. Furthermore, onecan observe that the incremental reduction in coverage errorobtained by moving from N = 10 points to N = 15 pointsis much less than that obtained by moving from N = 5points to N = 10 points. In other words, there is a dimin-ishing effect of additional points included in the represen-tation as far as the reduction in the coverage error is con-cerned. Combined with the observation that the CPU timesincrease as N increases, this information may suggest thatrepresentative points beyond N = 10, for example, may notbe worth the additional effort.Table 2 suggests that approximately the same level of

reduction in coverage error is attained with the same num-ber of representative points for different n and �m categorieswith the same p value. This is probably due to the fact thatthe coverage error is measured in the outcome space. More-over, there is a clear decrease in the reduction of coverageerror for a given N value as p increases. For example, ifwe take the category with n= 50 and �m= 40, and chooseN = 5, the average reduction in coverage error for the cat-egory with p = 2 is higher than that of p = 4, which is inturn higher than that of p = 6. Again, this may be relatedto the fact that the coverage error is measured in the out-come space, and a higher p means a higher dimensionaloutcome space.In the second set of computational experiments, we wish

to observe the computational performance of procedureREPR-D when a list of efficient faces is provided and seehow it compares to the version that focuses on a singleface. In other words, we would like to see how differentit is to be representing an efficient set that is a union ofpolyhedra rather than representing a single polyhedron thatcorresponds to a face. Let the feasible region of problemMOLP be given by X = �x ∈ �n �Ax � b�x � 0�� whereA / m× n is the constraint matrix. The test problems aregenerated using a scheme similar to that of the first setof experiments. The number of variables n is set equal to10 and the number of constraints m is set equal to 15 tokeep the computation times of generating the efficient set

Table 2. Progression of the reduction in coverage error.

p× �m×n N = 5 N = 10 N = 15

2 × 40 × 50 63�3% 75�3% 80�8%4 × 40 × 50 47�9% 59�4% 63�7%6 × 40 × 50 46�0% 55�2% 60�0%2 × 80 × 100 60�4% 74�9% 79�5%4 × 80 × 100 48�3% 58�9% 62�9%6 × 80 × 100 45�6% 53�9% *2 × 100 × 150 59�5% 74�9% 79�1%4 × 100 × 150 48�6% 58�6% 62�1%6 × 100 × 150 46�0% 53�8% *

∗Not available due to excessive CPU time requirements.

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under control. We used three different p values of 2, 4,and 6, respectively and created 30 problems in each cat-egory. Specifying a reasonable stopping criterion in ran-dom experiments of this type is not trivial, as the problemindeed does not correspond to a real one where expecta-tions can be set more realistically or at least iteratively. Inour case, we used a target value of � = 0�3× �∗1 to pro-vide some scaling, and limited the number of points to begenerated by 20. Because �∗1 represents the first coverageerror computed within the face that is treated first, the stop-ping rule based on �∗1 becomes inappropriately low whenthere are other faces that actually are “larger” in size thanthe first face. In such cases, the second stopping criteria ofN = 20 takes effect and the algorithm may terminate with-out visiting all of the efficient faces. Once a test problemis solved and the number of points in the representation isdetermined, the same problem is solved so as to obtain thesame number of points, this time from the entire feasibleregion X. The information generated as a result of this sec-ond solution does not have a physical meaning in terms ofgenerating the efficient set, but it helps us to position theresults of the first set of experiments better.For each category, Table 3 displays information on the

number of points obtained (K), the number of efficientfaces (L), percent of faces visited by procedure REPR-D,the CPU time, and the reduction in coverage error compar-ing the highest coverage error encountered across faces dur-ing the iterations of the procedure and the resulting overallcoverage error reported by the procedure. The additionalnumber of constraints that correspond to constraint sets(2)–(6) in problem �PD′K � average 121, 331.2, and 454.13for categories p = 2, 4, and 6, respectively. When we lookat the representations obtained for the efficient set, wenotice that the reduction in coverage error is quite satisfac-tory in relation to the number of points used. As expected,CPU times tend to increase significantly as p increases.When we compare these results to those whose domain ofinterest is the entire polyhedron X, we notice that CPUtimes for the latter are higher and reduction in coverageerrors is less across all categories. Both of these observa-tions are possibly related to the additional constraints thatdefine the efficient faces which make the problems solvedby the procedure more restrictive and thus easier whendealing with lower dimensional faces rather than the entirepolyhedron. Based on this comparison, we can build sometrust towards the results of the first set of experiments inthe sense that what we have reported as CPU times andreduction in coverage error could possibly be close if it

Table 3. Comparison of REPR-D running with efficient faces as opposed to a single polyhedron.

Category REPR-D run with the efficient set REPR-D run with X

p× �m×n K L Faces Covered (%) CPU (sec) Reduction in � CPU(sec) Reduction in �

2 × 15 × 10 12.1 5.3 87% 2.4 87% 17.7 74%4 × 15 × 10 18.4 20.1 55% 86.8 78% 642.0 73%6 × 15 × 10 17.5 14.6 58% 125.2 81% 656.7 72%

were possible to obtain and work with the efficient set forthose large problem sizes.

4. CONCLUSION

The procedure we propose to find a discrete representationof an efficient set is a deterministic method based on math-ematical programming tools. The way it is originally stated,it can be implemented in two different ways. The DM mayspecify a number of points to inspect, and the procedurewill deliver a discrete representation that contains the exactnumber requested, and the coverage error of the represen-tation will be reported. Alternatively, the DM is allowed tospecify a coverage error and the procedure will deliver adiscrete representation of as many points as needed so thatthe DM’s specification is met. In either case, the unifor-mity level of the representation is available as a result ofthe computations made by the algorithm. All computationsof the procedure are carried out in the outcome space in aneffort to offset the negative effects of possible collapsingof the efficient set. Each point in the discrete representa-tion is available in the outcome space, along with a set ofdecision variables that map into it under the transformationof the objective functions.Because the procedure works iteratively, it always guar-

antees a good spread of points regardless of the number ofrepresentative points requested. That is, even with three orfive points in the representation, the procedure ensures thatthese points are not close to each other at all. It is alsoimportant to know that any coverage error is attainable ifone does not limit the number of points to be included inthe representation.The computational results reported in the previous sec-

tion show that the CPU time requirement of obtaining adiscrete representation depends on the size of the originalproblem, and is quite sensitive to the number of objec-tives. However, it seems possible to obtain, for example,five points even for problems with 150 variables withinreasonable time when the number of objectives is small.Therefore, it seems possible to obtain a small number ofrepresentative points even for big problems.The coordinated decomposition approach proposed to

implement the procedure requires the collection of efficientfaces as an input. Because finding the efficient set is a com-putationally demanding task for problems of realistic size,forming a collection of efficient faces heuristically mightbe proposed. More computational experiments need to beconducted to establish whether this is a viable approach.

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The coordinated decomposition idea accumulates all ofthe representative points obtained and includes them in thesubproblems to be solved. The number of representativepoints is a major contributor to CPU time requirements.Although the procedure is not intended to be run in realtime, the simpler option of representing each face individ-ually with a smaller number of points and bringing themtogether by filtering away the close ones may be explored.Again, further computational experiments would be neces-sary to observe the characteristics of such an approach.Our procedure may also be used in an interactive mode

with the DM. Rather than obtaining many representativepoints all at once, it is possible to obtain a small set ofrepresentative points from the efficient set and ask for theDM’s input. Because our approach is based on mathemat-ical programming techniques, it is quite flexible in incor-porating additional requirements as constraints, and thus itmay be possible to concentrate on certain areas, or to avoidcertain areas of the efficient set based on the informationprovided by the DM. There are a number of questions thatneed to be addressed while building an interactive algo-rithm based on our procedure, such as the content of theinformation to be presented to and collected from the DM.The prospects of such an approach need to be establishedthrough further research.Finally, the validation of the approach remains incom-

plete both computationally and behaviorally without appli-cation to a real problem. Application of the procedure toa real-life problem can be considered for future work. Itcould be possible to observe the computational behaviorof the approach and better judge the validity of the com-putational experiments only after such a study. Moreover,it would be possible to observe a DM’s reaction to theapproach, and maybe to its possible variations, and fine-tune the procedure along these guidelines.

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Armann, R. 1989. Solving multiobjective programming problemsby discrete representation. Optimization 20 483–492.

Benson, H., S. Sayin. 1993. A face search heuristic algorithmfor optimizing over the efficient set. Naval Res. Logist. 40103–116., . 1997. Towards finding global representations of theefficient set in multiple objective mathematical programming.Naval Res. Logist. 44 47–67.

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Ecker, J., N. Hegner, I. Kouada. 1980. Generating all maxi-mal efficient faces for multiple objective linear programs.J. Optim. Theory Appl. 30 353–381.

Evans, J., R. Steuer, R. 1973. A revised simplex method for linearmultiple objective problems. Math. Programming 5 54–72.

Rockafellar, R. 1970. Convex Analysis. Princeton UniversityPress, Princeton, NJ.

Rudin, W. 1976. Principles of Mathematical Analysis, 3rd edition.McGraw–Hill, New York.

Sayin, S. 1996. An algorithm based on facial decomposition forfinding the efficient set in multiple objective linear program-ming. Oper. Res. Lett. 19 87–94.. 2000. Measuring the quality of discrete representations ofefficient sets in multiple objective mathematical program-ming. Math. Programming 87 543–560.

Smith, R. L. 1984. Efficient Monte Carlo procedures for generat-ing points uniformly distributed over bounded regions. Oper.Res. 32(6) 1296–1308.

Steuer, R. 1976. Multiple objective linear programming with inter-val criterion weights. Management Sci. 23(3) 305–316., F. Harris. 1980. Intra-set point generation and filtering indecision and criterion space. Comput. Oper. Res. 7 41–53.

Yu, P., M. Zeleny. 1975. The set of all nondominated solutionsin linear cases and a multicriteria simplex method. J. Math.Anal. Appl. 49 430–468.

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