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ISSN 0701-3086

May 2001

DISCUSSION PAPER Generating a Representative Subset of the Efficient Frontier in Multiple Criteria Decision Making Esra Koktener Karasakal and Murat Koksalan

WORKING PAPER 01-20

This working paper should not be quoted or reproduced without the written consent of the authors.

GENERATING A REPRESENTATIVE SUBSET OF THE EFFICIENT FRONTIER IN MULTIPLE CRITERIA DECISION MAKING

ESRA KÖKTENER KARASAKAL1,2 AND MURAT KÖKSALAN1,3 1

Department of Industrial Engineering, Middle East Technical University 06531, Ankara, Turkey 2

Faculty of Administration, University of Ottawa Ottawa, ON K1N 6N5, Canada

3

Krannert School of Management, Purdue University West Lafayette, IN 47907, USA

Keywords: Multiple criteria, discrete representation of efficient frontier.

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ABSTRACT In this paper, we address the problem of generating a discrete representation of the efficient frontier in multiple criteria decision making where the solution space is defined by a set of constraints. We find a surface that approximates the shape of the efficient frontier. Utilizing the surface we generate a set of discrete points that is representative of the efficient frontier. Our experience on randomly generated problems demonstrates that the approach performs well in terms of both the quality of the representation and the computation time regardless of the problem size.

Many researchers have addressed Multiple Criteria Decision Making (MCDM) problems for several decades due to the recognition that most real-life problems involve multiple conflicting criteria. An MCDM problem having a continuous solution space may be defined as “Maximize” z =f(x) s.t. x∈X where f(x)={f1(x),…, fm(x)} is an m-vector of objective functions, fi(x) represents the ith objective function, z is the criterion vector, x is the decision vector and X ⊆ Rn represents the feasible decision space. A decision vector, xk∈X is said to be efficient if and only if there does not exist xj ∈ X such that fi(xj) ≥ fi(xk) for all i with a strict inequality for at least one i. Otherwise, xk is said to be inefficient. The set of efficient solutions is called the efficient frontier (efficient set). Following the classification scheme proposed by Hwang and Masud (1979), the approaches developed to solve MCDM problems can be classified into three groups according to the timing of obtaining preference information from the decision maker (DM). Each approach requires either a priori, or progressive, or a posteriori articulation of preferences of the DM. The procedure presented in this paper belongs to the class of techniques that requires a posteriori articulation of preferences.

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Some approaches in this class generate all efficient extreme points or all efficient faces in the decision space (e.g., Evans and Steuer, 1973; Ecker and Kouada, 1978; Yu and Zeleny, 1975; Ecker, Hegner and Kouada, 1980; Armand and Malivert, 1991), and others generate all efficient faces in the criterion space (e.g. Dauer and Saleh, 1992; Dauer and Gallagher, 1996; Benson, 1998). Since the decision space has a large number of extreme points and edges compared to those of the criterion space in general (Dauer, 1987; Benson 1995), operating in the criterion space has a computational advantage over operating in the decision space. Generating all efficient faces or all efficient extreme points is computationally demanding. There would be just too many efficient extreme points in a moderate size problem to make such an approach practical. Even when all efficient faces or all extreme points can be generated, selecting the most preferred solution remains a difficult problem to be solved and may cause information overload for the DM (Steuer, 1986, p. 245). Furthermore, the set of efficient extreme points may not represent the set of efficient solutions well. Finding a discrete set of points that represent all parts of the efficient frontier is desirable. It is important to be able to find this set with a reasonable amount of computational effort. Research in this area has been limited. Steuer and Harris (1980) proposed using random convex combinations of the extreme points of an efficient face to generate a representative sample. The procedure requires generating all efficient extreme points and this is not practical for larger problems. Benson and Sayin (1997) proposed a global shooting procedure that seeks to find global representations of the efficient sets by constructing a special simplex that contains the feasible criterion space. The quality of the representation obtained by this approach is problem dependent as it is related with the structure of the efficient frontier. Sayin (1999) proposed an approach for generating a representative subset from the efficient frontier while guaranteeing a specified level of quality. The approach operates on a given face of the solution space and requires the generation of all efficient faces before creating the representative points. Furthermore, it requires solving a mixed integer program and the computational effort needed for generating a representative sample may be excessive. None of the existing approaches is satisfactory in terms of both the quality of the representation and the practicality of the computational effort. In this paper, we develop an approach that is designed to quickly find a good discrete representation of the efficient frontier regardless of the size of the problem. Once such a representation is found, it can be considered

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as an approximation of the original problem. The best of the discrete set of solutions can be searched for using a discrete MCDM method (such as Koksalan and Sagala, 1995).

The

neighborhood of the best discrete solution is likely to contain the overall best solution provided that a representative discrete set had been generated. If the DM wishes to search further, he/she can conduct a localized searched in this neighborhood using a free-search approach (such as Korhonen and Wallenius, 1988). In the proposed approach, we fit a surface that approximates the shape of the efficient frontier. We select a set of approximately evenly spaced reference points to represent the fitted surface. We project these reference points onto the efficient frontier in the gradient direction of the fitted surface to obtain the representative points on the efficient frontier. Since the fitted surface approximates the efficient frontier, we expect the projections of these reference points to be approximately uniform over the efficient frontier and to be a good representation. In Section 1, we present some background information. We develop the approach in Section 2 and demonstrate its performance on several example problems in Section 3. We report our computational experiments in Section 4 and conclude in Section 5.

1. THEORETICAL BACKGROUND Sayin (2000) argued that a good representation of the efficient set should contain points from every portion of the efficient set (i.e., every part of the efficient set should be covered well) and the representative points should be at the same distance to each other (i.e., representative points should be uniformly distributed over the efficient set). She provided two measures for assessing the quality of a given discrete subset of the efficient set: coverage error and uniformity level. Sayin (2000) defined the coverage error as the distance (in terms of L∞ metric) of the efficient point that is farthest from the representative point closest to it. Specifically, let ZE denote the efficient set of the feasible criterion space Z and FZ represent an efficient face of Z. Let D be a discrete subset of the efficient face FZ. For any point z ∈ FZ, there exists a point y ∈ D such that the distance between z and y is less than or equal to the coverage error, ε. Thus, the coverage error, ε, is given by

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ε = Max Min z − y z∈FZ

where

z− y

y∈D

∞

∞

is the Tchebycheff distance between z and y.

Sayin (2000) proposed an

approach based on a mixed integer programming formulation for computing the coverage error of the representative points. Sayin (1999) also proposed an iterative procedure using this formulation to generate a representative discrete subset from an efficient face. The procedure requires solving a mixed integer program with 2mN binary variables at each iteration (where m is the number of criteria and N is the number of representative points generated so far). Sayin (2000) defined the uniformity level, δ, as the minimum distance between the representative points, yj and yk ∈D, as

δ=

Min k

y ,y ∈D, j ≠ k j

y j − yk

∞

.

It is desired to have any point in the efficient set not any farther than a certain threshold to its closest point in the representative subset (coverage).

It is also desired to have every

representative point at least at a certain distance to its closest point in the representative subset (uniformity).

2. AN APPROACH In this section, we first give an overview of the proposed approach. We then develop procedures for approximating the efficient frontier by a surface, selecting approximately equidistant points from the fitted surface, and generating a representative sample from the efficient frontier using the selected points.

2.1 An Overview of the Approach Koksalan (1999) argued that Lp surfaces may well represent the efficient frontier on discrete problems. He fitted Lp distance curves and demonstrated that they work well on two bicriteria scheduling problems. We use the idea of fitting an Lp surface. We consider a compact feasible criterion space and try to fit a surface that approximates the shape of the efficient frontier well. The fitted surface is

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such that selecting a discrete representative set on it is easier than selecting directly on the original efficient frontier. In order to illustrate how the efficient frontier is approximated by a weighted-Lp surface, consider a two criteria problem shown in Figure 1a. The feasible criterion region is the shaded area and the efficient frontier is the union of the two line segments γ{q,r} and γ{r,s} where

γ{z1,z2} denotes the set of convex combinations of z1 and z2.

We select approximately

equidistant points on the fitted surface as representative points of the fitted surface, and call these the reference points (see Figure 1b). When the fitted surface approximates the efficient frontier well, we expect the projection of these reference points onto the efficient frontier in the gradient direction to give us a representative set of points that are approximately uniformly distributed over the efficient frontier (see Figure 1c). The resulting projections on the efficient frontier are selected as a representative subset of the efficient frontier (see Figure 1d). z2

z2 weighted-Lp surface

q

r

s

z1

z1

(a)

(b)

z2

z2

z1

z1

(c)

(d) Figure 1. An Illustration

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2.2 Fitting a Surface to Approximate the Efficient Frontier Since selecting equidistant representative points from the efficient frontier is a difficult task, we approximate the efficient frontier by a surface on which we can select representative points more easily. We try to construct a weighted-Lp surface that approximates the efficient frontier of the feasible criterion region. We use the following weighted-Lp function as an approximating surface.

m  Lp(z1,…, zm) = ∑ λi ( z i − qi* ) p   i =1 

1 /p

=d

where λ=(λ1,…, λm) t is a nonnegative vector of weights and q* =(q1*,…, qm*) t denotes the nadir criterion vector corresponding to the efficient solutions, i.e., qi* = Min{fi(x), x∈XE} where XE denotes the set of efficient solutions of the feasible decision space X. d corresponds to the weighted-Lp distance between any point on the fitted surface and q*. Since d can be scaled with the help of λ, in our calculations we take d =1 in order to reduce the number of parameters to be estimated. We first generate a number of efficient points to obtain information about the shape of the efficient frontier. These points need not be uniformly distributed over the efficient frontier and need not have a prespecified coverage error. We then construct a weighted-Lp surface whose sum of squares of distances (errors) from the generated efficient points is the minimum. To generate the efficient solutions we solve a series of equal-weighted Tchebycheff parametric programs (P1) by systematically restricting the objectives. Specifically, in each program we assign one of the objectives, f i ( x ) , to a certain value and find the efficient solution closest to the ideal point (in terms of Tchebycheff distance). We assign each objective i to k different values between its ideal and nadir levels using: q i* + ∆i,v = qi* +

v ( s i* − q i* ) ( v = 0 , 1, 2,..., k − 1) k −1

where s*=(s1*, …,sm*)t denotes the ideal criterion vector, i.e., si* = Max{fi(x), x∈X}. To find the efficient solutions, we solve the following Tchebycheff program for each of the k values (v=0,…,k-1) of the m objectives (j=1,…, m).

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(P1) Min α-∑ ε f i ( x )

(1)

i≠ j

s.t. α ≥ ( s i* − f i ( x ))

for all i ≠ j

(2)

f j ( x ) = q j* + ∆ j,v

(3)

x∈ X

(4)

where X represents the feasible decision space, fi(x) represents the ith objective function, and ε is a small positive constant used to avoid inefficient solutions. Let f

j,v

be the vector of objective

function values obtained by solving (P1), j=1,…,m, v=0,…,k-1. Constraints (2) ensure that the weighted Tchebycheff distance between f(x) and s* is at most

α. Minimization of objective function (1) ensures that an efficient feasible point closest to the ideal point s* according to the Tchebycheff metric is found as the optimal solution. The values for the bounds of objectives in constraint (3) are chosen from the efficient ranges of the objectives. We construct a payoff table by individually solving for each of the objective functions and recording the values of the other objective functions. The jth column of the table, aj, corresponds to the values of the objectives obtained by solving the problem using only the jth objective function. The table yields the ideal criterion values and a rough estimate of the nadir criterion values of objectives. We scale the criterion values by the approximate range of each criterion, si* − qi* , over the efficient frontier in order not to be biased in favor of the criteria with larger efficient ranges. Initially, we use equal weights λ=(λ1,…, λm) t = (1,…,1) t in the weighted-Lp surface. We try to find an estimator of p that determines the degree of “ovalness” of the weighted-Lp surface for the initial value of λ. We first find an efficient point b that is closest to the ideal criterion vector according to the weighted Tchebycheff metric whose contours move in the diagonal direction (1,…,1)t. We then find the value of p such that the Lp surface passes through point b (see Figure 2).

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z2

Weighted-Lp surface s* Weighted Tchebycheff contours

b

z1

Figure 2. Fitting a Weighted-Lp Surface In the weighted-Lp function, we set λi =1 and zi= bi (for i=1,…,m), and obtain the following equality

m p ∑ (bi − qi* )   i =1 

1 /p

= 1.

We solve the above equation using a bisection method to find an initial estimator of p. Next, we update the λ vector using the obtained p value. We try to find the λ vector for which the weighted-Lp surface best represents mk+m+1 efficient points f

j,v

, (j=1,…,m,

v=0,…,k-1), aj (j=1,…,m), and b. More specifically, we compute the new λ value for the given p value in such a way that the sum of squares deviation of the weighted-Lp surface from the

mk+m+1 efficient solutions is minimum. That is, we find the value of λ that minimizes m

E = ∑ (1 − ∑ λi ( z ij − q i* ) p ) 2 j

i =1

where zj = ( z1j ,…, z mj ) denotes one of the mk+m+1 efficient points used to fit a surface. Differentiating the above equation with respect to λi and equating to zero, we obtain R λ- S =0 where mk + m +1  ( z1j − q1* ) 2 p ∑  j =1  R= M mk + m +1 j  ( z1 − q1* ) p ( z mj − q m* ) p  ∑  j =1

 − q1* ) p ( z mj − q m* ) p   M  mk + m +1 2p j  ( z m − q m* ) ∑  j =1 

mk + m +1

L

∑ (z j =1

M L

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j 1

mk + m +1 S =  ∑ ( z1j − q1* ) p L  j =1

 ( z − q m* )  ∑ j =1 

mk + m +1

j m

t

p

The value of the λ vector is L λm ] = R −1 S .

[ λ1

It can be easily shown that the columns of R form a linearly independent set, i.e., matrix R is invertible.

2.3 Generating Reference Points

After fitting the weighted-Lp surface, we select reference points that are approximately evenly spaced on the fitted surface. We first illustrate our procedure for selecting these points with two and three criteria problems and then generalize it for more criteria. Consider the example of Figure 3. If we partition [q, s] and connect the corresponding points on the curve with n line segments, we obtain a polygonal path that approximates the curve. The length of the jth line segment in L2-metric (shown in Figure 3) is T

j −1

T

= (∆z1j ) 2 + (∆z 2j ) 2 .

j 2

The length of the curve can be approximated by n

∑T j =1

j −1

n

T

j 2

= ∑ (∆z1j ) 2 + (∆z 2j ) 2 . j =1

The approximation improves as the subintervals of [q, s] become smaller.

z2

.

q =T0

∆z11

.

∆ z 12

T1

.

T2

.

s = T3

z1

Figure 3. A Polygonal Path Approximating the Curve

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It is possible to divide the weighted-Lp curve into equal-length arcs with more precision using a computationally demanding approach. However, the above approximation is sufficient for our purposes as we just need approximately evenly spaced points. We first find the maximum and minimum values of each objective over the efficient frontier (the minimum for each objective is demonstrated at the origin in Figure 3 for the sake of simplicity). We use an adjacent separator parameter, ∆z , to locate adjacent points that are approximately ∆z units apart. We define the following line passing from T 0 ∇z1 ( z1 − T10 ) + ∇z 2 ( z 2 − T20 ) = 0 where ∇z = (∇z1 ,∇z 2 ) is the normal vector of the line.

Weighted-Lp function is continuous at every point of the closed interval [q, s] and differentiable at every point of its interior (q, s). In order to stay close to the curve, we define ∇z i as the average of the gradients at T 0 and T 0 where T 0 is a point on the curve close to T 0 (specifically T 0 is such that Lp( T 0 )=1 and T1 0 = T10 + ∆z ). ∂L p (T 0 ) ∇z i =

∂z i

+ 2

∂L p (T 0 ) ∂z i

.

We move ∆z units along the line to find T 1 . If T 1 is not on the curve, then we project it onto the curve in the orthogonal direction to the z1-axis and denote the new point as T 1 . We find the other approximately equidistant points similarly until the whole curve is traversed. Consider next the three criteria problem shown in Figure 4 (where the minimum for each objective is again selected at the origin for the sake of simplicity). In this case we first divide the weighted-Lp curve that lies on the z1z3- plane (where criterion 2 is set to its minimum value) into equal-length arcs as in the two criteria case and project the endpoints of the arcs onto the z1-axis. Similarly, we partition the curve lying on the z2z3-plane into equal-length arcs and project the end points of the arcs onto the z2-axes. Let the number of projections on the z1-axis be n1 and on the z2-axis be n2. Using n1 values of objective 1 and n2 values of objective 2, we form n1n2 points on the z1z2-plane. We project each point onto the weighted-Lp surface by fixing objectives 1 and 2 at their corresponding values and maximizing objective 3 over the weighted-Lp surface. The projection of each of the n1n2 combinations of points that lie under the fitted weighted-Lp surface 11

(that is the points that satisfy ( λ1 ( z1 − q1* ) p + λ2 ( z 2 − q 2* ) p )1 / p ≤1) will yield an efficient solution on the weighted-Lp surface. z3 Weighted-Lp function

z1

z2

Figure 4. Weighted-Lp Surface in a Three Criteria Problem The procedure developed for selecting points on the weighted-Lp curve for two and three criteria problems can be generalized for m criteria problems as follows. Step 1. Divide the curve lying in the zizm-plane into approximately equal-length arcs using a step size of ∆z for i=1,…,m-1. Step 2. Project the endpoints of the arcs on the zizm-plane onto the zi-axis for all i=1,…,m-1. (Let ni be the number of projections on the zi -axis.) Step 3. Find n1n2…nm-1 points on the z1z2…zm-1 –hyperplane as the intersection of points projected to each of the axes. Step 4. Project each of these n1n2…nm-1 points onto the weighted-Lp surface to find the value of zm corresponding to the combination under consideration. The adjacent separator parameter, ∆z , used in Step 1 plays an important role in generating the targeted number of points over the fitted surface. The number points generated depends on the value of ∆z as well as the fitted weighted-Lp surface. In order to approximately generate a predetermined number of points, it is necessary to vary the value of ∆z iteratively. Initially we set ∆z to a specific value and then update it by a bisection–like method until the desired number of points is generated (see Karasakal, 2000 for details).

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Another issue in Step 1 that needs further discussion is selecting the starting reference point on the curve lying on the zizm-plane. Consider the two criteria example in Figure 5. Suppose we want to select three reference points on the curve and let L be the length of the curve. Let q be the efficient point that has the minimum value, and s be the efficient point that has the maximum value in the first criterion (see Figure 5a). In case 1, we select points q, s and the midpoint of the curve as the reference points. The distance between any two adjacent reference points is L/2 and any point on the curve is at most at a distance of L/4 from the nearest reference point. In case 2, we select two points that are at a distance of L/6 from points q and s, respectively, as well as the midpoint of the curve as the reference points. Then the distance between any two adjacent reference points is L/3 (see Figure 5b) and any point on the curve is at most at a distance L/6 from the nearest reference point. Projecting the points of case 2 onto the efficient frontier is likely to produce representative points with a smaller coverage error compared to that of case 1. We use the idea of Case 2 to partition the weighted-Lp curve that lies in the zizm-plane into approximately equal-length arcs. z2

z2

.

q=T0

L/6 L/2

.

.

T0

L/3

.

T1

T1

L/3

.

L/2

.

s=T2

T2

L/6

z1

z1

(a) Case 1

(b) Case 2

Figure 5. Selecting Reference Points on the Weighted-Lp Surface

2.4 Generating Representative Points

After approximating the efficient frontier by a surface on which we can select representative points (i.e. reference points) more easily, we need to project these points onto the efficient frontier in order to find approximately evenly spaced representative points. In this section, we explain how to project these reference points onto the efficient frontier.

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Let Lp(z1,…,zm) = c where c is any constant in the range of the weighted- Lp function be a level surface of the weighted Lp function. Proposition 1: Consider two level surfaces of a weighted-Lp function, Lp(z1,…,zm) = c1 and Lp(z1,…,zm) = c2. Projection of equidistant points from one level surface in the gradient direction produces equidistant points on the other level surface. Proof: Let F1(z1,…,zm) = Lp(z1,…,zm) -c1 and F2(z1,…,zm) = Lp(z1,…,zm) -c2. Proof follows from the fact that F1(z1,…,zm) and F2(z1,…,zm) have the same partial derivatives. g Proposition 1 implies that projecting in the gradient direction onto the efficient frontier generates approximately evenly spaced points if weighted-Lp function approximates the shape of the efficient frontier well. Let (g1,…gm) be the gradient of the weighted-Lp surface at point T, i.e., ( g1 ,..., g m ) = (

∂L p (T ) ∂z1

,...,

∂L p (T ) ∂z m

)

Reference points on the weighted-Lp surface can be projected onto the efficient frontier by an achievement scalarizing function (Wierzbicki, 1980), which can be formulated as (P2). (P2) m

Min α − ∑ ε f i ( x )

(5)

i =1

s.t. α ≥ (Ti − f i ( x ))

1 gi

i = 1,..., m

( 6)

x∈ X

(7 )

where X is the feasible criterion space, ε is a small positive constant used to avoid inefficient solutions, T=(T1,…Tm) is a reference point and ( g1 ,..., g m ) is the gradient vector at T. Let the optimal solution be x* and let z*=f(x*). Achievement scalarizing function always finds a feasible efficient point. If moving from the reference point in the gradient direction results in an inefficient solution, the achievement

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scalarizing function avoids it by changing the direction towards the efficient frontier. However, this may have an adverse effect on the uniformity level of the points. In order not to reduce the uniformity level, we eliminate such points from our representative set. Let φi be the step length for criterion i (i=1,…,m) defined as φi =

Ti − f i ( x* ) . gi

The gradient direction leads to an efficient solution if φ1 ≅ φ2 ≅ ... ≅ φm ≅ α . We therefore, place the point obtained by (P2) into the representative set only if φ1 ≅ φ2 ≅ ... ≅ φm = α .

3. EXAMPLES

In this section, we generate representative subsets of the efficient frontier on several example problems using both the proposed approach and Sayin’s approach (1999). In the example problems of this section and the computational experiments of the next section we scale the criteria values so that each criterion has the same range of [0,1]. The coverage error and the uniformity level can, therefore, be interpreted in the absolute sense and their values are comparable over different problems. A coverage error of 0.10, for example, implies that for any efficient solution, in any criterion, there exists a representative point that is not farther than 10% of the range of values of that criterion. Similarly, a uniformity level of 0.10 implies that there is no representative point within 10% of the range of values of any criterion for any given representative point. Based on this scaling, both the coverage error and uniformity level take values between 0 and 1. Smaller coverage error values and larger uniformity level values are desirable.

Their attainable values are directly related with the total number of

representative points to be selected.

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Example 1. Consider the following problem from Benson and Sayin (1997).

Max z1 = x1 Max z 2 = x 2 Max z 3 = x3 s.t. 6 x1 + 15 x 2 + 10 x3 ≤ 210 5 x1 + 8 x 2 + 12 x3 ≤ 152 22 x1 + 29 x 2 + 28 x3 ≤ 458 24 x1 + 16 x 2 + 11 x3 ≤ 312 x1 + 4 x3 ≤ 40 8 x1 + x3 ≤ 72 x1 , x 2 , x3 ≥ 0 The efficient frontier is the union of four efficient faces γ{(0,4,10), (8,2,8), (0,10,6)} ∪

γ{(8,2,8), (5,12,0), (0,10,6)} ∪ γ{(8,2,8), (9,6,0), (5,12,0)} ∪ γ{(0,10,6), (5,12,0), (0,14,0)} where γ(z1,…,zn) is the set of all convex combinations of z1,…, zn. 20 representative points were generated using both the proposed procedure and Sayin’s procedure (1999) (see Figures 6a and 6b). Notice that, we underestimated the nadir point but it does not seem to effect the quality of the representation generated by the proposed procedure. Coverage error and uniformity level of both representations were calculated using the methods proposed by Sayin (2000) (see Table 1).

(0,4,10)

(8,2,8) (0,10,6)

(0,14,0) (5,12,0)

(9,6,0)

(a) Proposed Procedure

(b) Sayin’s Procedure

Figure 6. Graph of the Representation for Example 1

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Table 1. Results of Example 1 Proposed Sayin’s

Coverage Error 0.198 0.221

Uniformity Level 0.148 0.031

Example 2. The following problem is from Sayin (2000).

Max z1 = x1 Max z 2 = x 2 Max z 3 = x3 s.t. 4 x1 + 8 x 2 + x3 ≤ 24 8 x1 + 4 x 2 + x3 ≤ 24 x3 ≤ 8 x1 , x 2 , x3 ≥ 0 The efficient frontier is the union of two efficient faces γ{(0,2,8), (4/3,4/3,8), (2,2,0), (0,3,0)} ∪ γ{(4/3,4/3,8), (2,2,0), (3,0,0), (2,0,8)}. 21 representative points were generated using both the proposed procedure and Sayin’s procedure (see Figures 7a and 7b). Coverage error and uniformity level of the representations are summarized in Table 2.

(0,2,8)

(4/3,4/3,8) (2,0,8)

(0,3,0)

(3,0,0) (2,2,0)

(a) Proposed Procedure

(b) Sayin’s Procedure

Figure 7. Graph of the Representation for Example 2

17

Table 2 Results of Example 2 Proposed Sayin’s

Coverage Error 0.229 0.223

Uniformity Level 0.133 0.060

Example 3. The following problem is from Ecker et al. (1980).

Max z1 = 4 x1 + x 2 + 2 x3 Max z 2 = x1 + 3 x 2 − x3 Max z 3 = − x1 + x 2 + 4 x3 s.t. x1 + x 2 + x3 ≤ 3 2 x1 + 2 x 2 + x3 ≤ 4 x1 − x 2 ≤ 0 x1 , x 2 , x3 ≥ 0 The efficient frontier is the union of two efficient faces γ{(2,6,2), (5,4,0), (6.5,0,8), (5,1,9)} ∪ γ{(5,1,9), (6.5,0,8), (6,-3,12)}. 20 representative points were generated using both Sayin’s procedure and the proposed procedure. Since the payoff table resulted in the overestimation of the nadir point, the coverage error of our representation was found to be 0.368 that is significantly higher than that of Sayin’s procedure (0.159). When the correct value of the nadir point was used, we obtained a good representation (with 21 points) as shown in Figure 8a. Representation (with 21 points) obtained by Sayin’s procedure is depicted in Figure 8b. Coverage error and uniformity of the representations are summarized in Table 3. Our observations based on the three example problems are summarized below. •

Coverage errors of the methods are close to each other.

•

Uniformity level of the proposed method is better (higher) than that of Sayin’s method. The proposed method takes a global approach and produces points approximately uniformly distributed over the efficient frontier. Sayin’s method takes a local approach and considers faces separately.

•

If the nadir point is overestimated, then the proposed procedure may miss some parts of the efficient frontier as in Example 3. This may cause an increase in the coverage error of the representative set. Heuristics (such as Korhonen, Salo and Steuer, 1997) can be used to

18

better estimate the nadir points. However, underestimation of the nadir point does not seem to cause any problems and does not seem to reduce the quality of our representation as can be seen in Example 1.

(6,-3,12)

(5,1,9) (6.5,0,8)

(2,6,2)

(5,4,0)

(a) Proposed Procedure

(b) Sayin’s Procedure

Figure 8. Graph of the Representation for Example 3 Table 3. Results of Example 3 Proposed Sayin’s

Coverage Error 0.144 0.130

Uniformity Level 0.081 0.049

4. COMPUTATIONAL RESULTS

We compare the performance of the proposed procedure to that of Sayin (1999) in terms of three measures, namely coverage error, uniformity level and computation-time requirements. Both the proposed method and Sayin’s method were coded in Borland C++ Version 5.0 and mathematical programming models were built with the general algebraic modeling language GAMS Version 2.25 (Brooke et al., 1992). Linear programming models were solved with

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MINOS Version 5.3 and mixed integer linear programming models were solved with OSL Version 2. The computational experiments were conducted on a Pentium II 233 MHz personal computer with 32 MB random access memory (RAM). In our experiments, we used randomly generated polytopes that are expressed in the form {z∈ℜm, z=Cx, Ax≤ b, x≥0} where A is an n×m matrix and b∈ℜn. We used one of the models proposed by Steuer (1994) for random problem generation. Our experimental study consisted of two problem size parameters. These are the number of objectives (m) and the number of constraints (n). The levels of parameter m are (2,3,4,5) and those of parameter n are (5,10,30). 10 randomly generated problems are solved for each parameter combination. In the proposed method, we fit a weighted-Lp surface that approximates the efficient frontier. To fit the surface we use mk+m+1 initially generated efficient solutions, and we set k=5 in our experiments.

In order to give more insight into the Lp surface approximations, minimum,

maximum and average p-values obtained in each problem set are reported in Table 4. Results show an increase in p-values as the number of constraints increases over problems with a given number of criteria. Increase is relatively high over the problems with 2 criteria. We observed a decrease in the p-values as the number of criteria increases over problems with a given number of constraints. The gap between the p-values is especially high for problems with 2 criteria and problems with 3 criteria. Table 4. Minimum, Maximum and Average p-values of the Fitted Weighted-Lp Surfaces # of Criteria 2 3 4 5

Min. 1.250 1.250 1.250 1.250

5 Max. 5.000 1.875 1.875 1.562

Avg. 2.719 1.625 1.531 1.468

# of Constraints 10 Min. Max. Avg. 2.500 7.500 4.625 1.562 3.125 2.094 1.562 2.188 1.797 1.406 1.719 1.547

Min. 5.000 1.875 1.875 1.562

30 Max. 11.250 3.750 2.500 2.188

Avg. 6.813 2.500 2.031 1.797

We measured the quality of the solutions obtained using the coverage error and uniformity level measures proposed by Sayin (2000). The computation of the coverage error requires solving a mixed integer linear program with 2mN binary variables for each face (where m is the

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number of criteria and N is the number of points that represent the given face). To calculate the true coverage error of a given face, some of the representative points located on the other faces should be included in the program as well as the representative points lying on the face under consideration. To determine such points, we first calculate the coverage error of the face ( εˆ ) using the points located on the given face. The points that are located on the other faces and are at most at a Tchebycheff distance of εˆ to the nearest efficient point of the given face may affect the coverage error of a given face. In measuring the results of problems with 2 criteria, we considered all such points and calculated the true coverage errors. In the other problem sets, since increase in the number of representative points brings a substantial computational burden, we cut down the number of representative points to be considered by using the following heuristic approach. We limit the number of binary variables to be used to 120. We select the representative points to be considered in the following order until the binary variable limit is reached. We first consider the representative points that are on other faces but at most at a Tchebycheff distance of

εˆ from the worst represented point on the given face. We keep selecting such points in the order they are processed. We then consider the representative points that are at most at a Tchebycheff distance of εˆ from the nearest efficient point of the given face. Of these points, we select the ones that are not close to any representative points on the given face (that is the Tchebycheff distances to their respective closest representative points on the face are at least εˆ ). For large-size problems, since we could not use all the points that may affect the coverage error of a given face, the calculated average error is only an upper bound on the true coverage error. In order to have a better understanding, we also calculated a lower bound for the coverage error as follows. We computed the Tchebycheff distance between the worst represented point of a face and the representative points not considered in the calculation of the coverage error of the given face. The minimum of such distances is a lower bound on the coverage error. For a given problem, we expect that as the number of representative points increases, the number of representative points that affect the coverage error of a face but are disregarded in calculating upper bounds on the coverage error due to the computational restrictions increases, and this may worsen the quality of the upper bounds used in the comparison of the two methods. We aimed to generate a small number points (3 –5 points) per face in each problem. Table 5 presents the average number of representative points generated for each problem type (10

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randomly generated problems were solved for each cell). Notice that there is a small difference between the targeted number (N) and the observed number. Since in the proposed approach, we try to obtain the targeted number of representative points by experimentation, it is difficult to exactly attain the targeted number of points.

We defined a range for the number of

representative points to be generated as N ±0.1N and tried to generate a number of representative points in this range. Table 5. Average Number of Representative Points Generated* # of Criteria 2 3 4 5

5

21.1 24.9 24.8 25.5

(20) (25) (25) (25)

# of Constraints 10 20.5 (20) 34.5 (35) 40.6 (40) 44.5 (45)

30 26.1 (25) 44.8 (45) 60.2 (60) 64.1 (65)

( ): targeted number of representative points * average of 10 problems per cell

For each problem type, average coverage error of both methods are presented in Table 6. In measuring the results of problems with 2 criteria, we calculated the true coverage errors and for these problems the average coverage errors of the proposed method are slightly better. Table 6. Average Coverage Error Obtained by the Proposed Method and Sayin’s Method* # of Criteria 2 3 4 5

5 Proposed 0.046 0.213 0.379 0.479

Sayin 0.049 0.217 0.374 0.442

# of Constraints 10 Proposed Sayin 0.050 0.053 0.228 0.216 0.419 0.376 0.450 0.401

30 Proposed 0.044 0.223 0.417 0.480

Sayin 0.053 0.197 0.349 0.413

* average of 10 problems per cell

Sayin’s method produced slightly better results than the proposed method for larger number of criteria. Average coverage errors that are calculated for problems with 3, 4 and 5 criteria are upper bounds on the true coverage errors. For problem sets with 30 constraints and 4 and 5 criteria, the gap between the results obtained by two methods is larger than those for the other

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problem sets. We calculated lower bounds on the coverage errors for such problems in order to obtain some insight on the range of the coverage error. Lower bounds for the proposed method and Sayin’s method are 0.157 and 0.272, respectively for the problem set with 4 criteria and 30 constraints; and 0.230 and 0.308, respectively for the problem set with 5 criteria and 30 constraints. Lower bounds for the proposed algorithm seem to be lower than those of Sayin’s. Sayin’s method tends to generate boundary points in the initial representative points because the faces are considered separately. On the other hand, the proposed algorithm tries to avoid generating boundary points as representative points as much as possible. Thus, the points that are located on the other faces and that are close to the boundaries may have more effect on the coverage error of the given face in the proposed method. In problems with 4 and 5 criteria, results of both methods show a gradual increase in the coverage error.

The sizes of the faces increase with the number of criteria, and 3 to 5

representative points per face are not sufficient to attain a reasonable coverage for four and five criteria problems. We cannot increase the number of representative points much since the size of the mixed integer program that needs to be solved for Sayin’s method and for calculating the coverage error becomes prohibitive. To demonstrate the effect of increasing the number of representative points, we conducted a small experiment with 3-criteria, 10-constraint problems. Using 20, 35 and 50 representative points and solving 10 problems for each case, we obtained average coverage errors of 0.274, 0.228 and 0.184, respectively, with the proposed method.

As expected, there is a drastic

improvement in the coverage error with increasing number of representative points. In Table 7, the average uniformity levels of both methods for all problems sets are reported. Averages of the proposed method for 10 problems outperform Sayin’s results with significant margins. Our experimental study reveals that the computation time (including the initial effort for obtaining the weighted-Lp surface) required by the proposed algorithm is about 5% -15% of that required by Sayin’s algorithm.

Since Sayin’s algorithm requires solving mixed integer

programs, its computation time increases exponentially as the problem size (especially the number of criteria) increases (see Figure 9).

We observed a significant increase in the

computation time when 60 and 65 representative points were generated for the problem sets with 4 or 5 criteria and 30 constraints. We observed a linear increase in the computation time of the

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proposed algorithm as the number of points and the size of the problems (i.e. the number of criteria and the number of constraints) increase (see Figure 9). This is expected since the proposed method solves linear programs whose size does not increase as the number of the representative points increases. Table 7. Average Uniformity Level Obtained by the Proposed Method and Sayin’s Method* # of Criteria 2 3 4 5

5 Proposed 0.056 0.111 0.146 0.316

Sayin 0.036 0.015 0.020 0.014

# of Constraints 10 Proposed Sayin 0.055 0.038 0.102 0.013 0.132 0.006 0.167 0.018

30 Proposed 0.047 0.093 0.125 0.136

Sayin 0.013 0.007 0.006 0.006

*average of 10 problems per cell

250,0

Sayin's 350,0

CPU Time (Sec)

150,0

100,0

Sayin's

900,0

Proposed

Proposed

800,0

300,0

CPU Time (Sec)

Proposed

200,0

CPU Time (Sec)

1000,0

400,0

Sayin's

250,0 200,0 150,0 100,0

50,0

700,0 600,0 500,0 400,0 300,0 200,0

50,0

100,0 0,0

0,0 2

3

4

Criteria

(a) 5 Constraints

5

2

3

4

5

0,0 2

Criteria

(b) 10 Constraints

3

4

5

Criteria

(c) 30 constraints

(Solution times are given in sec. for a Pentium II 233 MHz PC with 32 MB RAM)

Figure 9. Average Computation Times of the Proposed Method and Sayin’s Method

5. CONCLUSIONS

We developed an approach that finds a discrete representation of the efficient frontier of MCDM problems that have a continuous solution space. These problems are harder to solve compared to the discrete multiple criteria decision making problems.

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We aimed to

approximately transform the continuous solution space problem into a discrete multiple criteria decision making problem by sampling representative solutions from the efficient frontier. There has been limited research in the literature to sample a representative set of solutions from the efficient set. These methods either produce low quality representations in terms of coverage and uniformity or produce high quality representations but are computationally cumbersome. We developed a fast approximation approach that may be especially useful in large problems where generating a representative subset with a specified level of quality requires excessive computational effort. We compared the performance of the proposed procedure to that of Sayin (1999) in terms of three measures, namely coverage error, uniformity level and computation-time requirements. The coverage errors of both methods are close to each other. In problems that have a single efficient face, Sayin’s method guarantees a discrete representation having the minimum coverage error for the given number of representative points. However, for problems with two or more efficient faces, it is possible to find discrete representations having smaller coverage errors than those found by Sayin’s method with a given number of representative points because it considers the faces separately and does not take into account the representative points on the other faces. The uniformity levels obtained by the proposed method are much higher than (which is desirable) those of Sayin’s method because the proposed method considers the entire efficient set. In order to improve its uniformity levels, Sayin’s method can be modified in such a way that all representative points that may affect the coverage error of a given face are considered while a new representative point on a given face is generated. This modification may bring additional improvement in terms of the coverage error. However, since including a new representative point brings 2m additional binary variables to the mixed integer program solved in Sayin’s approach, considering all the representative points that may affect the coverage error of a given face may be computationally prohibitive. In the proposed approach, computation time increases linearly with the number of representative points whereas it increases exponentially in Sayin’s method that requires solving mixed integer programs. Reasonable size real-life problems would require a large number of discrete points to represent the efficient frontier well. This does not create any computational difficulty for the proposed approach.

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An increased number of discrete points naturally reduce the coverage error, as it is possible to represent more parts of the efficient frontier. In the proposed procedure, there is also a secondary benefit of increasing the number of discrete points.

As the number of points

increases, the length of line segments that are used to approximate the weighted-Lp surface decreases. This results in a better approximation of the weighted-Lp surface and thus results in generation of approximately equidistant points on the weighted-Lp surface. The performance of the proposed approach is expected to improve as the number of points increases for a given problem. The proposed method can be used to generate a much larger number of representative points without difficulty. Since the method has been robust in terms of the coverage error and the uniformity level on the problems we tested, we expect it to perform well with larger number of representative points as well. The calculation of the coverage error requires the solution of a mixed integer program (Sayin, 2000) that becomes computationally prohibitive with increasing number of representative points. An area for future research is to develop a new measurement technique that is suitable for large representative samples. Experiments can then be conducted to test the performance of the proposed procedure with larger number of representative points on large size problems. Our method is applicable, in theory, to any multiple criteria problem where the solution space is defined by a set of constraints. In practice, however, there are difficulties in solving the resulting mathematical problems when the solution space is not polyhedral. The same difficulty is shared with other approaches that try to operate with non-polyhedral solution spaces.

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