# A Comparison of Benson’s Outer Approximation Algorithm with an Extended Version of Multiobjective Simplex Algorithm

Authors: Paschal B. Nyiam; Abdellah Salhi
Journal: Advances in Operations Research
 (2021)

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## Abstract

The multiple objective simplex algorithm and its variants work in the decision variable space to find the set of all efficient extreme points of multiple objective linear programming (MOLP). Other approaches to the problem find either the entire set of all efficient solutions or a subset of them and also return the corresponding objective values (nondominated points). This paper presents an extension of the multiobjective simplex algorithm (MSA) to generate the set of all nondominated points and no redundant ones. This extended version is compared to Benson’s outer approximation (BOA) algorithm that also computes the set of all nondominated points of the problem. Numerical results on nontrivial MOLP problems show that the total number of nondominated points returned by the extended MSA is the same as that returned by BOA for most of the problems considered.


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## Body

## 1. Introduction

Multiobjective linear programming seeks to optimize two or more linear objective functions subject to a set of linear constraints with a view of obtaining either all the efficient solutions or nondominated points or a subset of them, or a most preferred solution depending on the approach adopted. MOLP has been studied over the years because of its relevance in practice.Indeed, many decision-making problems that arise in the real world involve more than one objective function. Consequently, it has been widely applied in many fields and has become a useful tool in decision-making.Formally, it can be written as(1)minc1Tx=f1⋮cqTx=fqsubject tox∈X=x∈ℝn:Ax=b,b∈ℝm,x≥0.We noted in [1] that “in practice, MOLP is typically solved by the Decision-Maker (DM) in conjunction with the analyst who looks for a most preferred solution in the feasible region X. This is because optimizing all the objective functions at the same time is not possible due to their conflicting nature. Consequently, the concept of optimality is replaced with that of efficiency. Therefore, the purpose of MOLP is to obtain either all the efficient extreme points or nondominated extreme points or a subset of them, or a most preferred point depending on the purpose for which it is needed.”Many algorithms have been suggested for the problem. Most of them are based on the simplex method for linear programming. Prominent among them is the multiobjective simplex algorithm (MSA) and its variants. According to Eiselt and Sandblom [2], Evans and Steuer [3], Philip [4], and Zeleny [5] all derived generalized versions of the simplex method known as MSA. This algorithm works in the decision variable space to find the entire set of all efficient solutions. However, it was noted in [6] that finding the nondominated points instead of the efficient set is more important for the DM.The aim of this paper is to extend the MSA of Evans and Steuer [3] whose explicit form can be found in [7] to generate the whole set of nondominated points. We shall then compare this extended version with the original one and with the primal variant of Benson’s outer approximation (BOA) algorithm [8] which is an objective space based method that also computes the set of all nondominated points of the problem.This paper is organized as follows: Section2 is the motivation. Section 3 introduces MOLP and basic notation. Section 4 is a brief review of the relevant literature. We present MSA and its extended version in Sections 5 and 6, respectively. Section 7 discusses two scalarization techniques. BOA is presented in Section 8. Section 9 presents experimental results obtained with the different algorithms. Finally, a conclusion is presented in Section 10.


## 2. Motivation

From the outset, MSA and BOA are not comparable since one is decision space based while the other is objective space based; one computes efficient solutions and the other nondominated points. However, if one can generate nondominated points from MSA, then this can be performed.It is well known [6–16] that, in practice, decision-makers prefer to base their choice of a most preferred point on the objective values (nondominated points) rather than on the efficient solutions. This means that algorithms such as MSA which return only the set of efficient solutions are less favourable to the DM, say, compared to BOA which works in the objective space and returns the nondominated set. It is therefore desirable to generate the nondominated set from the efficient set returned by MSA. In other words, we extend it. This extended variant of MSA becomes as desirable as those computing the nondominated set for the DM. However, this can only be decided after a comparison with such an algorithm. Here, we suggest comparing with BOA on a number of nontrivial MOLP instances. Clearly, extended MSA becomes very attractive given that it gives all of the efficient solutions and nondominated points.


## 3. Notation and Definitions

An alternative and compact formulation of (1) is as follows:(2)minCxsubject toAx=bx≥0,where C is a q×n criterion matrix consisting of the rows ck,k=1,2,…,q, A is an m×n constraint matrix, and b∈ℝm is the right-hand side vector. The feasible set in the decision space is X=x∈ℝn:Ax=b,x≥0 and, in the objective space, it is Y=Cx:x∈X. The set Y is also referred to as the image of X [1].A nondominated point in the objective space is the image of an efficient solution in the decision space, and the set of all nondominated points forms the nondominated set [6].An efficient solution to the problem is a solution that cannot improve any of the objective functions without reducing at least one of the other objectives. A weakly efficient solution is the one that cannot improve all the objective functions simultaneously, [17]. Let x^∈X be a feasible solution of (2) and let y^=Cx^:(i)
x^ is called efficient if there is no x∈X such that Cx≤Cx^ and Cx≠Cx^;  correspondingly, y^=Cx^ is called nondominated(ii)
x^ is called weakly efficient if there is no x∈X such that Cx<Cx^; and y^=Cx^ is called weakly nondominated [7]The set of all efficient solutions and the set of all weakly efficient solutions of (2) are denoted by XE and XWE, respectively [10]. YN=Cx:x∈XE and YWN=Cx:x∈XWE are the nondominated and weakly nondominated sets in the objective space of (2), respectively.The nondominated faces in the objective space of the problem constitute the nondominated frontier and the efficient faces in the decision space of the problem constitute the efficient frontier [1].


## 4. Literature Review

As stated earlier, Eiselt and Sandblom [2] note that Evans and Steuer [3], Philip [4], and Zeleny [5] all derived generalized versions of the simplex method known as MSA for generating the entire efficient decision set XE of the problem. That of Philip [4] first determines if an extreme point is efficient and subsequently checks if it is the only one that exists. If not, the algorithm finds them all. This MSA approach, however, may fail at a degenerate vertex. In [18], Philip modified it to overcome this difficulty.The MSA of Evans and Steuer [3] also generates the set of all efficient solutions and unbounded efficient edges of the problem; see also Algorithm 7.1, page 178 of [7]. The algorithm first confirms that the problem is feasible and has efficient extreme points. Thereafter, it computes all of them by moving from one efficient extreme point to an adjacent efficient extreme point, until all of the efficient extreme points have been computed. An LP test problem is solved to determine the pivots that lead to efficient extreme points. The algorithm is implemented as software called ADBASE [19].The MSA variant of Zeleny [5] also uses an LP test problem to determine the efficiency of extreme points. But here, vertices are tested for efficiency after they have been obtained unlike in [3] where the test problem determines pivots leading to efficient vertices.Yu and Zeleny [20, 21] used the approach in [5] to generate the set of all efficient solutions and presented a formal procedure for testing the efficiency of extreme points.The efficient solutions are derived from the efficient faces, in a top-to-bottom search strategy. Numerical illustrations with three objectives were used to demonstrate the effectiveness of the method. In a similar paper, Yu and Zeleny [22] applied their approach expanded in [21] to parametric linear programming. Two basic forms of the problem and two computational approaches for generating the entire efficient set were presented: the direct decomposition approach that decomposes the parametric space into subspaces associated with extreme points and the indirect algebraic approach. From a numerical experience point of view, the indirect algebraic approach was superior to the direct decomposition method.In [23], Isermann proposed a variant of the MSA in [3] that solves fewer LPs when determining the entering variables. The algorithm first establishes whether an efficient solution for the problem exists and solves a test problem to determine pivots leading to efficient vertices. It was implemented as a software called EFFACET in Isermann and Naujoks [24].The MSA of Gal [25] generates the set of all higher-dimensional faces and all efficient vertices of the problem. This approach is meant to address the problem of determining efficient faces and higher-dimensional faces that were not resolved in [3, 4]. Here, efficient solutions are computed using a test problem. The algorithm also determines higher-dimensional efficient faces for degenerate problems which were only discussed in [5, 23] but were not solved. The efficient faces are computed in a bottom-to-top search strategy unlike what was suggested in [20, 21].Steuer [26] used the MSA of Evans and Steuer [3] to solve parametric and nonparametric problems. Different approaches for determining an initial efficient extreme point as well as different LP test problems were also considered. Efficient solutions were computed through the direct decomposition of the weight space into finite subsets that provided optimal weights corresponding to efficient solutions.In [7], Ehrgott also used the MSA of Evans and Steuer [3] to solve MOLP problem instances with two and three objective functions. Ecker and Kouada [27] also proposed a variation on the MSA of Evans and Steuer [3]. They noted that algorithms usually started from an initial efficient extreme point and moved to an adjacent one following the solution of an LP problem. The proposed method does not require the solution of any LP problem to test for the efficiency of extreme points and the feasible region need not be bounded. The algorithm enumerates all efficient extreme points and appears to have a computational advantage over other methods.In a different paper, Ecker et al. [28] presented yet another variant of MSA. The algorithm first determines the maximal efficient faces incident to a given efficient vertex (i.e., containing the efficient vertex) and ensures that previously generated efficient faces are not regenerated. This is done following a bottom-to-top search strategy as in [25], which dramatically improves computation time. The proposed approach was illustrated with a degenerate example given in [21], to demonstrate its applicability. It was computationally more efficient than the method in [21].The MSA of Armand and Malivert [29] determines the set of efficient extreme points even for degenerate MOLPs. The approach follows a bottom-to-top search strategy and utilizes a lexicographic selection rule to choose the leaving variables which proves effective when solving degenerate problems. It was tested successfully on a number of degenerate problems. A numerical example with five objectives and eight constraints which was solved in [21] was also used to demonstrate its effectiveness. The proposed MSA was superior to that in [21].Rudloff et al. [30] suggested a MSA which works in the decision variable space but does not generate all the efficient extreme points unlike the algorithm in [3]. Instead, it finds a subset of efficient extreme points based on the idea of Löhne [31]. That is, a subset of efficient extreme points and directions that allows computing the whole efficient frontier. The algorithm was compared with BOA [8] which also provides a solution based on the idea in [31] and with Evans and Steuer’s MSA [3]. Numerical experiments show that the proposed method is superior to Benson’s algorithm for nondegenerate problems. However, that of Benson’s outperforms it for highly degenerate ones.In [1], we presented the results of a more detailed computational investigation of the MSA in [30] and BOA [8] using existing small, medium, and realistic MOLP instances to evaluate the robustness and quality of the most preferred nondominated point (MPNP) returned by these two algorithms which was not discussed or considered in [30]. Also presented in [1] was a formal procedure for the computation of a MPNP of the problem. Numerical results on the robustness, efficiency, and quality of a MPNP show that BOA outperforms PSA in terms of the quality of a MPNP it returns and robustness, as well as confirming what was reported in [30] that BOA is computationally more efficient than PSA on highly degenerate problems, while PSA is superior to BOA computationally on nondegenerate MOLP problems.Of all these variants, it was noted in [32] that, that of Evans and Steuer [3] is the most popular and successful for computing all efficient extreme points of the problem.Apart from MSA and its variants that work in the decision variable space to find the entire set of all efficient solutions, there are algorithms that work in the objective space to find the set of nondominated points of the problem. Prominent among them is Benson’s outer approximation algorithm [8]. As was noted in [1], the author who presented an account of decision space based methods proposed an algorithm for computing the nondominated set in the objective space of the problem. According to him, his method is the first of its kind. It was motivated by the observation that many efficient extreme points map onto the same nondominated point in the objective space; Decision-Makers prefer to base their choice of a most preferred point on the nondominated set rather than the efficient set; and moreover, the dimension of the objective space is much smaller than that of the decision space, [12]. Therefore, finding the nondominated set instead of the efficient set is also more important for the DM [6]. The algorithm was compared with the MSA of Evans and Steuer [3]. Results show that the average number of nondominated solutions returned by BOA is less than the average number of efficient solutions returned by MSA in all the problems considered. In a similar paper, a further analysis of the objective space based methods was presented in Benson [9]. Here, it was shown that the algorithm in [8] also computes the weakly nondominated points, thereby enhancing the usefulness of the method as a decision aid [1].Before Benson’s proposal, Dauer and Liu [12] suggested a procedure for obtaining the nondominated points and edges in the objective space. It was noted that not all efficient extreme points necessarily map onto the nondominated points and the procedure analyzes a simpler structure.In [10], Benson suggested a hybrid method for solving an MOLP in the objective space. The method involves partitioning the objective space into simplices that lie in each face so as to compute the nondominated set. This idea was earlier presented in [33]. The method is quite similar to his algorithm in [8]. The difference between them is in the way in which the nondominated extreme points are generated. While a vertex enumeration approach is utilized in [8], a simplicial partitioning method is used in the latter [1].We also noted in [1] that a modification of the algorithm of Benson [8] was presented in [15]. While in [8], a bisection approach that requires the solution of more than one LP is required in one step; here, solving only one LP gives the desired effect and in the process improves computation time. Shao and Ehrgott [16] suggested an approximate dual variant of the algorithm of Benson [8] for generating approximate nondominated points of the problem. The proposed method was tested on the beam intensity optimization problem of radiotherapy treatment planning for which approximate nondominated points were generated. Numerical results show that the method is faster than solving the primal problem directly.Shao and Ehrgott [15] modified the algorithm of Benson [8] and presented it in explicit form in [31]. This modified version solves two LPs in each iteration during the process of computing the nondominated set. In [34], Löhne introduced the MATLAB implementation of this modified version called BENSOLVE-1.2, for generating all the nondominated extreme points and directions of the problem.Csirmaz [35] also presented an improved version of the algorithm of Benson [8] where only one LP and a vertex enumeration problem is solved in each iteration while in [8], two steps, as well as two LPs, are required to be solved in order to determine a unique boundary point and supporting hyperplane of the image; here, the two steps are merged into one and solving one LP does both tasks and dramatically improves computation time. The algorithm was used to compute all the nondominated points of the polytope defined by a set of Shannon inequalities on four random variables so as to map their entropy region [1]. Numerical results show the applicability of the algorithm to medium and large instances.Similarly, Hamel et al. [36] introduced new variants of the algorithm in [8] that solves only one LP problem in each iteration. Numerical experiments reveal a reduction in computation time.We noted in [1] that Löhne et al. [14] presented an extension of the primal and dual variants of the algorithm of Benson [8] to solve convex vector optimization problems approximately in the objective space.


## 5. The Multiobjective Simplex Algorithm

The MSA of Evans and Steuer [3] described in this section can be found in [7]. We consider this algorithm because of its popularity (see [32]), and because most of the MSA algorithms discussed earlier are either based on or are variants of it. It works in the decision variable space to find the entire set of all efficient solutions.In an MOLP problem, only one of the following situations can occur: the problem can be infeasible, meaning that the feasible setX is empty X=∅; the problem may be feasible, that is X≠∅ but may not have efficient solutions, that is, XE=∅; or it is feasible and has efficient solutions, that is XE≠∅. This algorithm handles these situations in three phases: in the first phase, it finds an initial basic feasible solution or stop with the conclusion that X=∅; in the second phase, it finds an initial efficient basis or stop with the conclusion that XE=∅; and, in the final phase, it pivots among efficient bases to determine all efficient extreme points of the problem [7].The algorithm starts by solving two auxiliary LPs to determine whether the problem is feasible and to verify that it has efficient solutions. If the feasible regionX and the efficient set XE are not empty, a weighted sum LP is solved to determine an initial efficient basis B. Its implementation stores a list of efficient bases L1 to be processed, a list L2 of efficient bases for output, and a list of efficient nonbasic variables NE. An LP test problem is solved to determine pivots that lead to efficient bases. The algorithm pivots from an initial efficient basis to an adjacent efficient basis until the list L1 of efficient bases to be processed is empty. The algorithm terminates and returns a list L2 of efficient bases from where all efficient extreme points are computed.Before we present the pseudocode of MSA, we first explain the notation used.A, b, C: the problem dataL1: list of efficient bases to be processedL2: list of efficient bases for outputeT=1,…,1∈ℝqI: the identity matrix of proper orderX: the feasible setXE: the set of efficient solutionsB: the efficient basisNE: list of efficient nonbasic variablesN: the set of nonbasic variablesB′: the new basisA and b: the updated constraint matrix and RHS vector, respectivelyR: the nonbasic part of the reduced cost matrixrj: a column of R corresponding to the nonbasic variable being tested for efficiency


### 5.1. Illustration of MSA

Consider the following MOLP adapted from [37]:(3)minf1=−x1minf2=−x2subject to6x1+10x2≤60,x1≤7,x2≤5,x1,x2≥0.The efficient solutions found using a MATLAB implementation of Algorithm1 are x1=7.0,1.8T, x2=1.6,5.0T, x3=1.6,5.0T, and x4=7.0,1.8T, where x1=x11,x21T,…,x4=x14,x24T∈XE. The algorithm is prone to generating more efficient solutions due to the way it operates and due to the fact that MSA may find the same efficient solutions in more than one iteration, as in this case; x1=x4 and x2=x3 are repetitive of what has already been found. Solutions x3 and x4 are redundant and would be of little or no use to the DM. The feasible region in the decision variable space is shown in Figure 1.Algorithm 1: Multiobjective Simplex Algorithm [7].
(0)
Input:A,b,C:Problem data(1)
Initialize:  SetL1⟵∅,L2⟵∅;Phase I:solve the LPmineTz:Ax+Iz=b,x,z≥0.If the optimal valueof this LP is nonzero,STOP,X=∅; Otherwisex0is a basic feasible solution of MOLPPhase II:solve the LPminuTb+wTCx0:uTA+wTC≥0,w≥e.If itis infeasible,STOP,XE=∅;Otherwiseu^,w^is an optimal solution; Find an optimal basisBof the LPminw^TCx:Ax=b,x≥0; SetL1⟵B,L2⟵∅.;(2)
whileL1≠∅(3)
ChooseB∈L1,L1⟵L1\B,L2⟵L2∪B;(4)
ComputeA˜,b˜,andRaccording toB;(5)
NE⟵N;(6)
forallj∈N(7)
Solve the LPmaxeTv:Ry−rjσ+Iv=0;y,σ,v≥0.(8)
If this LP is unboundedNE⟵NE\j;(9)
forallj∈NE(10)
foralli∈B(11)
ifB′⟵B\i∪jis feasible,B′∉L1∪L2 then;(12)
L1⟵L1∪B′;(13)
endif(14)
endfor(15)
endfor(16)
endfor(17)
endwhile(18)
Output:  L2:List of efficient bases.Figure 1
Edge connecting the two efficient points in the decision variable space.


## 5.1. Illustration of MSA

Consider the following MOLP adapted from [37]:(3)minf1=−x1minf2=−x2subject to6x1+10x2≤60,x1≤7,x2≤5,x1,x2≥0.The efficient solutions found using a MATLAB implementation of Algorithm1 are x1=7.0,1.8T, x2=1.6,5.0T, x3=1.6,5.0T, and x4=7.0,1.8T, where x1=x11,x21T,…,x4=x14,x24T∈XE. The algorithm is prone to generating more efficient solutions due to the way it operates and due to the fact that MSA may find the same efficient solutions in more than one iteration, as in this case; x1=x4 and x2=x3 are repetitive of what has already been found. Solutions x3 and x4 are redundant and would be of little or no use to the DM. The feasible region in the decision variable space is shown in Figure 1.Algorithm 1: Multiobjective Simplex Algorithm [7].
(0)
Input:A,b,C:Problem data(1)
Initialize:  SetL1⟵∅,L2⟵∅;Phase I:solve the LPmineTz:Ax+Iz=b,x,z≥0.If the optimal valueof this LP is nonzero,STOP,X=∅; Otherwisex0is a basic feasible solution of MOLPPhase II:solve the LPminuTb+wTCx0:uTA+wTC≥0,w≥e.If itis infeasible,STOP,XE=∅;Otherwiseu^,w^is an optimal solution; Find an optimal basisBof the LPminw^TCx:Ax=b,x≥0; SetL1⟵B,L2⟵∅.;(2)
whileL1≠∅(3)
ChooseB∈L1,L1⟵L1\B,L2⟵L2∪B;(4)
ComputeA˜,b˜,andRaccording toB;(5)
NE⟵N;(6)
forallj∈N(7)
Solve the LPmaxeTv:Ry−rjσ+Iv=0;y,σ,v≥0.(8)
If this LP is unboundedNE⟵NE\j;(9)
forallj∈NE(10)
foralli∈B(11)
ifB′⟵B\i∪jis feasible,B′∉L1∪L2 then;(12)
L1⟵L1∪B′;(13)
endif(14)
endfor(15)
endfor(16)
endfor(17)
endwhile(18)
Output:  L2:List of efficient bases.Figure 1
Edge connecting the two efficient points in the decision variable space.


## 6. The Extended Multiobjective Simplex Algorithm

As part of the initialization step (line 1 of Algorithm2), we have included the set of efficient extreme points XE and that of nondominated points YN. In the second phase, as the algorithm finds an initial efficient basis B by solving a weighted sum LP, the algorithm also finds a corresponding efficient basic feasible solution and appends it to the set of efficient solutions XE←x¯. The first nondominated point is also computed from CTx¯ and appended to the nondominated set YN←CTx¯.Algorithm 2: Extended Multiobjective Simplex Algorithm.
(0)
Input: A,b,Cdata of MOLP problem(1)
Initialize:  setL1⟵∅,L2⟵∅,XE⟵∅,YN⟵∅;Phase I: solve the LP mineTz:Ax+Iz=b,x,z≥0. If the optimal value of LP is not zero; STOP; X=∅. Otherwise, x0 is a basic feasible solution of MOLP.Phase II : solve the LP minuTb+wTCx0:uTA+wTC≥0,w≥e. If it is infeasible STOP, XE=∅. Otherwise, u^,w^ is an optimal solution. Find optimal basis B and basic feasible solution x¯ of LP min w^TCx:Ax=b,x≥0; SetL1⟵B,L2⟵∅,XE⟵x¯,YN⟵CTx¯.(2)
whileL1≠∅do(3)
ChooseB∈L1,L1⟵L1\B,L2⟵L2∪B;(4)
ComputeA˜,b˜,andRaccording toB;(5)
NE⟵N;(6)
forallj∈N(7)
Solve the LPmaxeTv:Rz−rjσ+Iv=0;z,σ,v≥0.(8)
If this LP is unboundedNE⟵NE\j;(9)
forallj∈NE(10)
foralli∈B(11)
ifB′⟵B\i∪jis feasible,B′∉L1∪L2,letx¯′be its basic solution; then(12)
L1⟵L1∪B′;(13)
XE⟵XE∪x¯′;(14)
YN⟵YN∪CTx¯′;(15)
endif(16)
endfor(17)
endfor(18)
endfor(19)
endwhile(20)
Output:XE:the efficient setYN:the non dominated set.As the algorithm iterates, a new efficient basisB′ is obtained after each pivot and the corresponding efficient basic feasible solution x¯′ (line 11 of Algorithm 2) is found and added to the set of efficient solutions XE (line 13). Likewise, the corresponding nondominated points are also found at each iteration and added to the nondominated set YN (line 14). This continues until the set of efficient bases L1 to be processed is empty. The algorithm returns the set of all efficient extreme points and the corresponding nondominated points (line 20).Before we present Algorithm2 as the extended MSA in pseudocode form, we first state here that the structure of the algorithm and the used notation remain the same as those in Algorithm 1. The additional components are x¯, x¯′, XE, and YN which stand for the efficient basic feasible solution, the new efficient basic feasible solution, the set of efficient solutions, and the corresponding set of nondominated points for output.


### 6.1. Illustration of the Extended MSA

We modified and extended the MATLAB implementation of Algorithm1 and used it to solve problem 3 of Section 5.1.The efficient solutions found arex1=7.0,1.8T, x2=1.6,5.0T, and the corresponding nondominated points are f1=−7.0,−1.8T and f2=−1.6,−5.0T, respectively, where x1=x11,x21T, x2=x12,x22T∈XE, and f1=f11,f21T, f2=f12,f22T∈YN. Note here that, the efficient extreme points x1 and x2 and the corresponding nondominated points f1 and f2 returned are devoid of redundant points. The algorithm is designed to avoid returning redundant nondominated points unlike the original version. The feasible region in the decision space is the same as in Figure 1.


### 6.2. Limitations of Extended MSA

As it is already known that the efficiency of the simplex-type algorithm is better when the problem is nondegenerate [30]. Like other simplex-type algorithms, the extended MSA or EMSA is not an exception. The extended MSA may exhibit one or more redundancies whenever the problem is degenerate. The first is that EMSA may find the same efficient solutions in more than one iteration or find different efficient solutions that leads to the same nondominated point. This can be seen clearly in Table 1, as the number of efficient solutions returned by EMSA is the same as that returned by the original MSA, though this is corrected in the computation of the corresponding nondominated points as the nondominated points are sorted after they are generated, leading to the same number of nondominated points returned by EMSA with that returned by BOA. In other words, EMSA may find redundant efficient solutions but the nondominated points found are devoid of redundant ones. Secondly, EMSA may also be sensitive to the number of nondominated points in a particular problem as can also be seen in the result table, as larger instances with more than four objective functions tend to take more than 3 days to return the required nondominated points. We note here that some of these problems most especially those from Bensolve and MOPLIB [38] are numerically ill-posed and highly challenging MOLP instances with difficult structures.Table 1
Comparative results for individual problem.
AlgorithmMSAEMSABOAProb.OriginnmqNESNESNNPNNP1Ehrgott, 200633355332Zeleny, 198222255333”2421616224”2433636335”2626464336”3331515557”53366448”52211119”64236361110”74336365511iMOLPe232884312”33412123313”3538888101014”33319197715”43318188816”423886617”4434444111118”33337375519”151029898111120”15103254254283721Steuer, 1986101535050151422”55228285523”44310103324”554154154141425”108420962096516326”54326269927”684560560131328”7644848123629”7641521529930”88610801080565631”8832082085532”88364641133”5547474121234”663304304171735”5542022029936”101043,0723,0726637”883608608131338”674440440252139”12164∗——60140”10145∗——13241Steuer, 198676340403342”77356567743”6641281285544”664168168101045”10145∗——47146”10145∗——12847”77360606648Bensolve-1.21001012∗——3249Bensolve-2.05315∗——2250”36362828231851”64642292292571452”100100211021102992053”3433433x——1,36854MOPLIB302112∗——155”53226356156155255256”532213∗——2552(∗) Aborted after 3 days of running time. (x) Out of memory.


## 6.1. Illustration of the Extended MSA

We modified and extended the MATLAB implementation of Algorithm1 and used it to solve problem 3 of Section 5.1.The efficient solutions found arex1=7.0,1.8T, x2=1.6,5.0T, and the corresponding nondominated points are f1=−7.0,−1.8T and f2=−1.6,−5.0T, respectively, where x1=x11,x21T, x2=x12,x22T∈XE, and f1=f11,f21T, f2=f12,f22T∈YN. Note here that, the efficient extreme points x1 and x2 and the corresponding nondominated points f1 and f2 returned are devoid of redundant points. The algorithm is designed to avoid returning redundant nondominated points unlike the original version. The feasible region in the decision space is the same as in Figure 1.


## 6.2. Limitations of Extended MSA

As it is already known that the efficiency of the simplex-type algorithm is better when the problem is nondegenerate [30]. Like other simplex-type algorithms, the extended MSA or EMSA is not an exception. The extended MSA may exhibit one or more redundancies whenever the problem is degenerate. The first is that EMSA may find the same efficient solutions in more than one iteration or find different efficient solutions that leads to the same nondominated point. This can be seen clearly in Table 1, as the number of efficient solutions returned by EMSA is the same as that returned by the original MSA, though this is corrected in the computation of the corresponding nondominated points as the nondominated points are sorted after they are generated, leading to the same number of nondominated points returned by EMSA with that returned by BOA. In other words, EMSA may find redundant efficient solutions but the nondominated points found are devoid of redundant ones. Secondly, EMSA may also be sensitive to the number of nondominated points in a particular problem as can also be seen in the result table, as larger instances with more than four objective functions tend to take more than 3 days to return the required nondominated points. We note here that some of these problems most especially those from Bensolve and MOPLIB [38] are numerically ill-posed and highly challenging MOLP instances with difficult structures.Table 1
Comparative results for individual problem.
AlgorithmMSAEMSABOAProb.OriginnmqNESNESNNPNNP1Ehrgott, 200633355332Zeleny, 198222255333”2421616224”2433636335”2626464336”3331515557”53366448”52211119”64236361110”74336365511iMOLPe232884312”33412123313”3538888101014”33319197715”43318188816”423886617”4434444111118”33337375519”151029898111120”15103254254283721Steuer, 1986101535050151422”55228285523”44310103324”554154154141425”108420962096516326”54326269927”684560560131328”7644848123629”7641521529930”88610801080565631”8832082085532”88364641133”5547474121234”663304304171735”5542022029936”101043,0723,0726637”883608608131338”674440440252139”12164∗——60140”10145∗——13241Steuer, 198676340403342”77356567743”6641281285544”664168168101045”10145∗——47146”10145∗——12847”77360606648Bensolve-1.21001012∗——3249Bensolve-2.05315∗——2250”36362828231851”64642292292571452”100100211021102992053”3433433x——1,36854MOPLIB302112∗——155”53226356156155255256”532213∗——2552(∗) Aborted after 3 days of running time. (x) Out of memory.


## 7. Scalarization Techniques

We now present two basic scalarization approaches that play an important part in the implementation of BOA as was discussed in [1]. These approaches are weighted sum scalarization and scalarization by a reference variable. As contained in [31], scalarization is one of the most important methods used in MOLP.In the weighted sum method, a new objective function based on theq-linear objectives is obtained by assigning nonnegative weights wi∈ℝq to each of the objectives. The weighted sum of the objectives is ∑i=1qwicix=wTCx. For each vector w∈ℝq, w≥0, we obtain a scalar linear program(4)minwTCxsubject toAx≥bThe weights are usually normalized so thateTw=1, with eT=1,…,1. The dual of (4) is(5)maxbTusubject toATu=CTw,u≥0.In the method of scalarization by a reference variable, theq objectives are associated with a common reference variable z and the i-th objective is restrained from being larger than the reference variable and a fixed real number yi, that is, c1x≤y1+z,c2x≤y2+z,…,cqx≤yq+z.The reference variablez is the objective function that has to be minimized. By setting e=1,…,1T, we obtain for each vector y∈ℝq the scalar linear program:(6)minzsubject toAx≥bCx−z.e≤yThe dual program is(7)maxbTu−yTwsubject toATu−CTw=0eTw=1u,w≥0as in [31]. The above two scalarization techniques are fundamental for the implementation of BOA which is discussed in Section 8 [1].


## 8. Benson’s Outer Approximation Algorithm

We hereby present a description of BOA as we did earlier in [1]. This version of BOA is due to [15]. It can be found in [31]. It works in the objective space to compute the nondominated set and directions of the problem. The algorithm is regarded as a primal-dual method as it also solves the dual problem. But here, our interest is only in the solution of the primal problem. The algorithm first constructs an initial polyhedron Y0 (outer approximation) containing the upper image Y in the objective space and an interior point p^ of the image is determined by solving equation (4). The inequality representation of Y0 is also determined by solving equation (5). The algorithm then constructs a sequence of decreasing polytopes Y0⊇Y1⊇⋯⊇Yk=Y. The vertices of each polytope Yk and their inequality representation are stored in each iteration. Then, for each vertex v of the polytope, the algorithm confirms if it is on the boundary of Y. If the vertices are on the boundary of Y, the problem is solved. The outer vertices of Y are among the vertices of Yk. Otherwise, for any vertex v of Yk that is not on the boundary of Y, the algorithm connects this vertex to the interior point p^ and finds the intersection y of this line with the boundary of Y by solving equation (6). Then, a supporting hyperplane adjacent to y is constructed by solving equation (7). This hyperplane is added to Yk to provide a smaller approximation. The algorithm is repeated in the same manner until the vertices of Yk coincide with the boundary of Y. The algorithm returns the set of vertices on the boundary of Y as the nondominated points Y¯ and directions Yh¯ of the problem.The used notation in the pseudocode of BOA is presented below.A, b, C: problem dataPh: the homogeneous problemD∗h: the homogeneous dual problem Th: the solution of the homogeneous dual problemp^: an interior pointT¯: a set of solutions of the dual problemYkd: the inequality representation of the current polytopek: the iteration counterYkp: the representation by vertices(y^, z): an optimal solution to P2(y)δ(0 < δ < 1): a unique value that determines the intersection or boundary point yRv: the LP that finds the unique value δThe command solve: solves an LPvert(): function that returns the vertices of a polytopeYkY¯: the set of nondominated vertices(Y¯h): the set of extreme directions [1]


### 8.1. Illustration of BOA

We consider again problem (3) of Section 5.1. The nondominated points found using a MATLAB implementation of Algorithm 3 are f1=−7.0,−1.8T and f2=−1.6,−5.0T where f1 and f2∈YN. These points are shown in Figure 2.Algorithm 3: Benson’s Outer Approximation Algorithm [31].
(0)
Input:A,b,C:Problem dataa solution0,Y¯htoPh;a solutionT¯htoD∗h;(1)
Initialize:    p^⟵PsolveP10+e;(2)
T¯⟵solveD1w,w|u,w∈T¯h;(3)
whilez=0do(4)
Ykd⟵D∗u,w|u,w∈T¯;(5)
Ykp⟵vertYd;(6)
Y¯⟵∅;(7)
fori=1toYpdo(8)
v⟵Ykpi;(9)
y^,z⟵solveP2y;(10)
Y¯⟵Y¯∪y^;(11)
ifz≠0then(12)
x,δ⟵solveRv,0<δ<1;(13)
y⟵δv+1−δp^;(14)
u,w⟵solveD2y;(15)
T¯⟵T¯∪u,w;(16)
endif;(17)
endfor;(18)
endwhile(19)
Output:Y¯,Yh¯: nondominated set and directions;T¯:a solution to dual.Figure 2
The edge joining the two nondominated points in the objective space [1].


## 8.1. Illustration of BOA

We consider again problem (3) of Section 5.1. The nondominated points found using a MATLAB implementation of Algorithm 3 are f1=−7.0,−1.8T and f2=−1.6,−5.0T where f1 and f2∈YN. These points are shown in Figure 2.Algorithm 3: Benson’s Outer Approximation Algorithm [31].
(0)
Input:A,b,C:Problem dataa solution0,Y¯htoPh;a solutionT¯htoD∗h;(1)
Initialize:    p^⟵PsolveP10+e;(2)
T¯⟵solveD1w,w|u,w∈T¯h;(3)
whilez=0do(4)
Ykd⟵D∗u,w|u,w∈T¯;(5)
Ykp⟵vertYd;(6)
Y¯⟵∅;(7)
fori=1toYpdo(8)
v⟵Ykpi;(9)
y^,z⟵solveP2y;(10)
Y¯⟵Y¯∪y^;(11)
ifz≠0then(12)
x,δ⟵solveRv,0<δ<1;(13)
y⟵δv+1−δp^;(14)
u,w⟵solveD2y;(15)
T¯⟵T¯∪u,w;(16)
endif;(17)
endfor;(18)
endwhile(19)
Output:Y¯,Yh¯: nondominated set and directions;T¯:a solution to dual.Figure 2
The edge joining the two nondominated points in the objective space [1].


## 9. Experimental Results

In this section, we provide numerical results to study the number of efficient solutions (NES) and to compare the number of nondominated points (NNP) returned by Algorithms2 and 3. Table 1 shows the numerical results for a collection of 56 problems from the literature; these problems range from small to moderate size MOLP instances and a few large instances. Problem 1 is taken from Ehrgott [7]. Problems 2 to 10 were taken from Zeleny [39]. Problems 11 to 20 are test problems from the interactive MOLP explorer (iMOLPe) of Alves et al. [40]. Problems 21 to 47 are taken from Steuer [26]. Problem 48 is a test problem in Bensolve-1.2 of Löhne [34], while problems 49 and 53 are test problems in Bensolve-2.0 of Löhne and Weißing [41]. Problems 50 to 52 are obtained using a script in Bensolve-2.0 of Löhne and Weißing [41] that is used to generate problem 53 with the same number of variables and constraints. Finally, problems 54 to 56 are test problems in MOPLIB [38] which stands for multiobjective problem library.Problem 48 is such that the constraint matrix is sparse while the objective matrix is dense. All the components of the RHS vector are ones except for 200 at the end as the largest entry. Problem 49 has a dense constraint matrix with an identity matrix of ordern as its objective matrix where n is the number of variables in the problem.All the components of the RHS vectors are zeros except for the first entry which is one (1) as the only nonzero component it is a degenerate problem. Problems 50 to 53 have dense objective matrices with identity matrices of order n as their constraint matrices where n is also the number of variables in the respective problem. All the components of the RHS vectors are ones. Note that problem 54 is also highly degenerate. Its structure is such that the constraint and objective matrices are sparse while all the components of the RHS vector are zeros except for a one (1) as the only nonzero entry. Finally, problems 55 and 56 have sparse constraints and objective matrices with dense RHS vectors.Results for Algorithms1 were obtained using a MATLAB implementation of the algorithm provided by Rudloff et al. [30]. We modified and extended Algorithm 1 of Evans and Steuer [3] into Algorithm 2 or EMSA, the Extended Multiobjective Simplex Algorithm introduced here. We have implemented it in # in the same way as in [30] and experimented with it on a set of test problems. We also used a MATLAB implementation of Algorithm 3 (BOA), known as Bensolve-1.2 [34]. The current version, Bensolve-2.0 of Löhne and Weißing [41] is implemented in the C programming language. We employed Bensolve-1.2 of Löhne [34] which is implemented in MATLAB to test the algorithms with the same tools and for a meaningful comparison. Note that the current version returns the same number of nondominated points as Bensolve-1.2 but has improved running time as noted in [42]. All algorithms were executed on an Intel Core i5–2500 CPU at 3.30 GHz with 16.0 GB RAM. In all tests, n is the number of variables, m the number of constraints, and q the number of objectives. Algorithm 1 is MSA of Evans and Steuer [3], Algorithm 2 its extended version, and Algorithm 3 is BOA as presented in [15]. We recorded the number of efficient solutions (NES) returned by MSA, the number of nondominated points (NNP) returned by EMSA, and the NNP returned by BOA for each problem.As can be seen in Table1, the NNP returned by EMSA is the same as that returned by BOA for most of the problems considered. This is due to the fact that EMSA is designed to avoid returning redundant nondominated points. This feature is also reported in [8] that BOA avoids redundant calculations of points that would be of little or no use to the DM. This makes EMSA compare favourably in terms of the NNP it returns. Though, we noticed a few differences in the NNP returned for some of the problems considered. These differences occur when some of the nondominated extreme points computed by BOA are repeated. We also noticed a significant reduction in the NNP returned by EMSA compared to the NES returned by MSA. This is due to the fact that MSA returns different efficient extreme points that yield the same nondominated points.It was also observed that the simplex-type algorithms could not produce results for problems 39, 40, 45, 46, 48, and 56 despite the long running time allowed (3 days); they were aborted. As we noted in [1], the fact that some problems were aborted after 3 days of running time does not necessarily mean that the algorithms cannot solve these problems; if allowed to run further, they could potentially return a huge number of efficient extreme points or nondominated points, or run out of memory which would indicate that the total number of efficient extreme points or nondominated points has exceeded the MATLAB storage capacity of the machine used.It was also found that for these problems where EMSA could not return a solution after running for three (3) days, these problems may have a huge number of efficient solutions or nondominated points as can be seen from the nondominated extreme points computed by BOA for these problems (see problems 53 and 56) which shows the sensitivity of EMSA to the number of nondominated extreme points in a given problem. EMSA may also find it difficult to return a solution for the degenerate problems (problems 49 and 54) as already mentioned in Section 6.2 due to cycling; that is, one may remain at the same vertex of the feasible region for many iterations or return the same efficient extreme points in more than one iteration. Though this is not the case when the algorithm is computing the nondominated points, as the nondominated points are sorted after computation. Thus, leading to a nondominated set that is devoid of redundant ones.


## 10. Conclusion

We have reviewed the relevant literature on MSA and BOA [8]. We have also extended the MSA of Evans and Steuer [3] to compute the entire set of all nondominated extreme points and illustrated the algorithms on a small MOLP instance. We then proceeded to compare the total number of nondominated extreme points computed by BOA and EMSA. It was observed that the total number of nondominated extreme points computed by EMSA is the same as that returned by BOA for most of the problems considered.



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*Source: 1857030-2021-07-05.xml*