Research Article

An Improved Method for Solving Multiobjective Integer LinearFractional Programming Problem

Meriem Ait Mehdi, Mohamed El-Amine Chergui, and Moncef Abbas

Department of Operational Research, Faculty of Mathematics, Houari Boumediene University of Sciences and Technology,P.O. Box 32, El-Alia Bab Ezzouar, 16111 Algiers, Algeria

Correspondence should be addressed to Meriem Ait Mehdi; myriam aitmehdi@yahoo.fr

Received 10 February 2014; Revised 28 May 2014; Accepted 6 June 2014; Published 1 July 2014

Academic Editor: Shelton Peiris

Copyright © 2014 Meriem Ait Mehdi et al. his is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We describe an improvement of Chergui and Moulaı’s method (2008) that generates the whole eicient set of a multiobjectiveinteger linear fractional program based on the branch and cut concept. he general step of this method consists in optimizing(maximizing without loss of generality) one of the fractional objective functions over a subset of the original continuous feasibleset; then if necessary, a branching process is carried out until obtaining an integer feasible solution. At this stage, an eicient cut isbuilt from the criteria’s growth directions in order to discard a part of the feasible domain containing only noneicient solutions.Our contribution concerns irstly the optimization process where a linear program that we deine later will be solved at each steprather than a fractional linear program. Secondly, local ideal and nadir points will be used as bounds to prune some branchesleading to noneicient solutions. he computational experiments show that the new method outperforms the old one in all thetreated instances.

1. Introduction

In this paper we focus our interest on multiobjective integerlinear fractional programming (MOILFP) where several lin-ear fractional objectives (i.e., ratio of two linear functions) areto be optimized simultaneously subject to a set of linear con-straints and nonnegative integer variables. he motivationbehind this choice comes in particular from the fact that inour knowledge a very few number of papers treating this typeof problem were published in the literature [1–3] contraryto continuous case which has received much more attentionfrom researchers (see, e.g., [4–13]). For the interested reader,Stancu-Minasian [14] has presented a comprehensive bibliog-raphy with 491 entries in addition to his book [15] containingthe state of the art in the theory and practice of fractionalprogramming.

Abbas and Moulaı [1] and Gupta and Malhotra [3] havepresented each a technique to generate the eicient set of aMOILFP based on the same principle: solving a sequence ofinteger linear fractional programs (ILFPs) until the stopping

criterion they propose is met. he irst ILFP solved is deinedby optimizing one of the objective functions subject to theentire feasible set; then a cutting plane is recursively addedto eliminate the current optimal solution and eventuallythe solutions lying on an adjacent edge. he choice of thislatter makes the main diference between these two methods.Unfortunately, they have both the same inconvenience thatis scanning almost the whole search space which is veryexpensive in time and memory space.

In [2], Chergui and Moulaı proposed a branch and cutapproach to solve MOILFP problems. At each step, a newnode is added to the search tree and the correspondingproblem deined by optimizing one of the fractional objectivefunctions over a subset of the original continuous feasibleset is solved. Two types of nodes are distinguished: thoserelative to the branching process and are created for thesearch of integer feasible solutions and those relative to theaddition of what the authors called an eicient cut. hepurpose of adding such a cut to a given problem is toremove from its feasible domain the optimal solution and

Hindawi Publishing CorporationAdvances in Decision SciencesVolume 2014, Article ID 306456, 7 pageshttp://dx.doi.org/10.1155/2014/306456

2 Advances in Decision Sciences

eventually a set of noneicient solutions. A node is fathomedif the corresponding problem is infeasible or if the set ofthe criteria’s growth directions determined to construct theeicient cut is empty. he procedure terminates when all thecreated nodes have been examined. he major drawback ofthis method lies in the fact that the integer feasible solutionsfound during the resolution are not all eicient (the eiciencyof a solution is guaranteed only at the end of the procedure;otherwise, it is only potentially eicient). he larger thesize of the problem, the greater the number of noneicientsolutions generated and thus the slower the convergence ofthe algorithm. To overcome this, we will introduce two newnode fathoming rules based on the calculation of local idealand nadir points that will avoid exploring other branchesleading to unnecessary solutions. Also, we will give theformulation of the linear program that we propose to solveas subproblem at each node rather than a fractional linearprogram.

he rest of the paper is organized as follows: in Section 2,we give some deinitions and some theoretical results relatedto our work. In Section 3, we describe in detail the improvedmethod. We present a numerical example in Section 4followed by a discussion of the results obtained from thecomputational experiments in Section 5. We conclude inSection 6.

2. Definitions and Previous Results

Amultiobjective integer linear fractional program (MOILFP)can be written as follows:

(MOILFP)

{{{{{{{{{{{{{{{{{{{{{{{

max�1 (�) = �1� + �1�1� + �1

...

max�� (�) = ��� + ����� + ��

s.t. � ∈ � = {� ∈ R� | �� = �, � ≥ 0} ,

� integer,(1)

where ��,�� are (1 × �) vectors and��,�� are scalars for � = 1, �;� is a (� × �) real matrix and � ∈ R�.

We assume that � is a nonempty compact polyhedronand all denominators are positive everywhere in�.

he solution to the problem (MOILFP) is to ind allsolutions that are eicient in the sense of the followingdeinition.

An integer solution � ∈ � is called eicient if theredoes not exist another integer solution � ∈ � such that

��(�) ≥ ��(�) for all � = 1, �with at least one strict inequality.he resulting criterion vector (�1(�), . . . , ��(�))� is said to benondominated.he set of eicient solutions of (MOILFP)willbe denoted by Ef throughout the paper.

he vectors �� = (��1, . . . , ���)� with ��� =max�∈�,� integer��(�) and�� = (��1, . . . , ���)� with ��� =min�∈Ef��(�) are called, respectively, ideal and nadir pointsof the problem (MOILFP).

A payof table (suggested by Benayoun et al. [16]) is asquare matrix of order � where its ith row represents valuesof all objective functions calculated at a point where the ithobjective obtained its maximum value. An estimate of thenadir point is obtained by inding the worst objective valuesin each column.hismethod gives accurate information onlyin the biobjective case; otherwise, it may be an over or anunderestimation.

We give in the following the results that justify themethoddescribed in [2]. We irst need to introduce some notations:

�� : the optimal integer solution of the linear fractionalprogram (��) : max{�1(�) | � ∈ ��, �� ⊆ �};note that instead of�1, one can similarly consider theproblem (��) with any of the remaining objectives ��,� ∈ {2, . . . , �}. Moreover, in the new method that wewill state later, the fractional objective function of (��)will be replaced by a linear one;

�� (resp., ��): the set of indices of basic variables

(resp., nonbasic variables) of ��;��: the reduced gradient vector of the ith objective. Itis deined by

�� = ���� − ����, (2)

where ��, ��, ��, and �� are updated values obtained from theoptimal simplex tableau of (��);

�� : the set deined by {� ∈ �� | ∃ � ∈ {1, 2, . . . , �}; ��� >0} ∪ {� ∈ �� | ��� = 0, ∀� ∈ {1, 2, . . . , �}};��+1: the set deined by {� ∈ �� | ∑�∈�� �� ≥ 1}.

heorem 1. Suppose that�� = 0 at the current integer solution��. If � is an integer eicient solution in domain �� \ {��} then� ∈ ��+1.For proof, see [2].

Deinition 2. An eicient cut is a constraint added to themodel and does not eliminate any feasible eicient integersolutions.

Corollary 3. Suppose that �� = 0 at the current integer solu-

tion ��; then the following constraint is an eicient cut:

∑�∈��

�� ≥ 1. (3)

See [2] for proof.

Proposition 4. If�� = 0 at the current integer solution �� then�� \ {��} is an explored domain.

he proof is also in [2].

Advances in Decision Sciences 3

Table 1: Comparison between Chergui and Moulaı’s method and its improved version on this example.

Old method New method

Number of eicient sol. 7 7

Number of integer sol. generated/number feasiblesol.

32/42 17/42

Number of created nodes 223 50

Number of simplex iterations 610 148

Table 2: he eicient set of the problem.

� �(�)(0, 0, 0, 0, 0, 0, 1)� (8347 ,

5699)�

(0, 0, 0, 1, 0, 0, 1)� (177103 ,121150)

�

(0, 0, 1, 1, 0, 0, 0)� (3023 ,8170)�

(0, 1, 0, 1, 0, 0, 1)� (227143 ,167168)

�

(0, 1, 0, 1, 0, 0, 0)� (3425 ,6659)�

(0, 2, 0, 0, 0, 0, 1)� (183127 ,148135)

�

(0, 0, 0, 2, 0, 0, 0)� (4529 , 1)�

Table 3: Results obtained by Seshan and Tikekar’s method.

�1 �2 �max�1 83

475699

8347

max�2 3023

8170

8170

3. Description of the Method

In this section, we describe some improvements that we willbring to Chergui and Moulaı’s method [2] but irst we recallbriely its principle and its limitations.

Let us consider a multiobjective integer linear fractionalprogram in the form of (MOILFP). he method of CherguiandMoulaımanages a search tree consisting of nodes. At eachnode (say �) the following linear fractional program is solvedusing Cambini andMartein’s method [17] or the dual simplexmethod:

(��) : max {�1 (�) | � ∈ ��} , (4)

where�� ⊆ � and�0 = �.(i) If this problem is infeasible, then the node � is

fathomed.

(ii) On the contrary case, if the obtained optimal solutionis not integer, a branching process is performed. Oth-erwise, the set�� deined above is determined. If it is

empty then the node � is fathomed; else a new node iscreated and the corresponding problem is obtained byadding to (��) the eicient cut constructed from��.

he procedure ends when all the created nodes have beenexamined.

As it can be seen, the node fathoming rules are reducedto the two following cases: the problem (��) is infeasible and�� is empty which happens very rarely (see Table 7) duringthe search process. Since the performance of this methoddepends, among other things, on its ability to avoid exploringnodes of the search tree leading to noneicient solutions,we propose to add at each node � the two new fathomingconditions given below.

(i) he ideal point, say ���, of the multiobjective lin-ear fractional program (MOF��) : max{�1(�), . . . ,��(�) | � ∈ ��} is dominated by at least one of thepotentially nondominated points already found.

(ii) here exists � ∈ {1, . . . , �} such that ���� < ���, where��� is the ith component of the original problem’snadir point.

Note that the use of the last rule ismore appropriate in thebiobjective case since the nadir point can be easily computedusing the payof table.

On the other hand, as we need to generate a feasiblesolution at each step of the method and not necessarilythe solution which optimizes one of the fractional objectivefunctions, we suggest to replace for each node � the linearfractional program by the following linear program (we keepthe same notation for the new problem):

(��) : max {�1� − ��1� | � ∈ ��} , (5)

where � is the parameter of Seshan and Tikekar’s method[18] taken at the optimum when the irst fractional objectiveis optimized under the integer feasible set. he idea behindthis choice is to start with a solution close to an eicient one(in our case, that with the maximum value of �1). However,any linear objective function can be used in (��) since thepurpose is to obtain a feasible solution; the only disadvantageis that the starting solution will be any vertex of the feasibleset which may increase the iteration number.

Handling linear programs will not only reduce the num-ber of simplex iterations performed but also facilitate the useof the dual simplex algorithm whenever a constraint is addedto a node problem.

4 Advances in Decision Sciences

Table 4: Reduced gradient vectors corresponding to Node 2.

�2 �1 �2 �3 �4 �5 �6 �11�1 −6724 −970 −2056 −230 −1138 −1016 −708�2 1274 3546 5284 3579 634 −4795 −665

By taking into account the suggested changes and byadopting the same notations as above, we obtain the fol-lowing algorithm that generates the entire eicient set Ef of(MOILFP).he Algorithm

Step 0 (initialization). Denote by � the list of untreatedproblems. Set Ef to the empty set and � to the linear program(�0). Compute the nadir point �� of (MOILFP) (for thebiobjective case, the payof table is used and the individualoptimization of both fractional objectives is performed bySeshan and Tikekar’s method [18]).Step 1 (problem selection and resolution). If� is empty, returnthe eicient set Ef and terminate. Otherwise, select the linearprogram with the greatest index � in �. Solve the problem (��)using the simplex or the dual simplex method; remove (��)from � and go to Step 2.

Step 2 (Fathoming). If (��) is infeasible, go to Step 1.

If not, let �� be its optimal solution,�� (resp.,��) the set ofindices of basic variables (resp., nonbasic variables) of �� and�� the corresponding criterion vector.

If�� is not dominated, go to Step 3. Else, compute the ideal

point ��� of (�����) as follows.he individual optimization of each fractional objective �

under�� is done by adding three rows to the optimal tableau

of (��); the irst and second rows correspond, respectively, tothe numerator and denominator of the fractional function� written in the basis �� and the last one to the reduced

gradient vector of the ith objective �� (see (2)). he ordinarysimplex pivot operations are then applied to the new table’srows except the last one which is modiied using (2) until the

optimal condition is met; that is, ��� ≤ 0, ∀� ∈ ��.If ��� is dominated by the criterion vector of at least one

solution of Ef or if there exists � ∈ {1, . . . , �} such that ���� <��� then fathom the corresponding node and go to Step 1.On the contrary case, go to Step 3.

Step 3 (branching). If �� is integer, go to Step 4; else select themost fractional variable ��� = ��. Create and add to � two

problems identical to (��) with the additional constraint �� ≥⌊��⌋ + 1 in the irst one and �� ≤ ⌊��⌋ in the second; go toStep 1.

Step 4 (cut generating). Update Ef: If there does not exist � ∈Ef such that the corresponding criterion vector dominates�� then add �� to Ef and remove all solutions for which thecriterion vector is dominated by ��.

Determine the set��. If�� is empty, go to Step 1. In casethat�� = ��, generate the incident edge �� with the greatestnumber of integer solutions and update Ef; set �� = �� \{�}. Add to � the problem obtained by extending (��) withthe eicient cut of (3) and go to Step 1.

4. Numerical Example

Consider the following MOILFP problem:

(MOILFP)

{{{{{{{{{{{{{{{{{{{{{{{{{

max �1 = 30�1 + 50�2 + 94�3 + 94�4 + 27�5 + 19�6 + 91�7 − 898�1 + 40�2 + 78�3 + 56�4 + 29�5 + 23�6 + 43�7 + 4

max �2 = 14�1 + 46�2 + 76�3 + 65�4 + 46�5 + 7�6 + 35�7 + 212�1 + 18�2 + 40�3 + 51�4 + 70�5 + 98�6 + 50�7 + 49

s.t. 21�1 + 3�2 + 8�3 − 9�4 − �5 + 26�6 + 34�7 ≤ 4133�1 + 28�2 + 29�3 + 14�4 + 9�5 − 7�6 − 9�7 ≤ 4928�1 − �2 + 40�3 + 38�4 + 28�5 + 43�6 + 35�7 ≤ 106�1, �7 ∈ N.

(6)

Following the steps described above to solve (MOILFP), weind the eicient set given in Table 2.

During the construction of the search tree, 50 nodes werecreated: the root node, 36 nodes of the branching process,and 13 of the cutting process; 19 nodes among them werefathomed. Only 17 solutions were generated from a total of42 feasible solutions (the entire feasible set).

In Table 1, we compare the inal results obtained for thepresent example using Chergui and Moulaı’s method [2] andits improved version.

In what follows, we give the calculation details for somenodes of the problem’s search tree.

Node 0. We irst calculate the payof table and thus thenadir point by simply inding the individual optima of bothobjectives with Seshan and Tikekar’s method. he resultsobtained are shown in Table 3.

he nadir point is�� = (30/23, 56/99).We then deine the linear program (�0) as max{�1� −

(83/47)�1� | � ∈ �0}, where �1 = (30, 50, 94, 94, 27, 19, 91),�1 = (98, 40, 78, 56, 29, 23, 43), � = (�1, . . . , �7)�, and �0

Advances in Decision Sciences 5

Table 5: Reduced gradient vectors corresponding to Node 29.

�29 �1 �2 �6 �8 �18 �19 �20�1 −52290 −5734 −47474 −1758 −15500 −54530 −18874�2 −57196 −2128 −89977 −2846 −14004 −98349 −30121

is the set of � ∈ R7 satisfying the three constraints of the

(MOILFP) problem.Recall that the objective function of (�0) taken here �1�−

(83/47)�1� can be replaced by any linear objective function.Set � = {�0} and Ef = 0. he resolution of (�0) gives

the optimal solution �0 = (0, 0, 0, 0, 0, 0, 41/34)� which is notinteger. Since only �7 is fractional this is the variable that webranch on. We obtain the two following subproblems:

Subproblem 1 : (�1) { (�0)�7 ≥ 2Subproblem 2 : (�2) { (�0)�7 ≤ 1,

(7)

where � = {�1, �2}.Node 2. Select from � the problem (�2). he resolutionof this problem yields the optimal integer solution �2 =(0, 0, 0, 0, 0, 0, 1)� with the corresponding criterion vector

�2 = (83/47, 56/99)�. hen Ef = {�2}. he set of nonbasicvariables �2 = {1, 2, 3, 4, 5, 6, 11} and the reduced gradientvectors are given in Table 4.

We have �2 = {1, 2, 3, 4, 5} =�2. We create a newproblem (�3) by adding to (�2) the constraint �1 + �2 + �3 +�4 + �5 ≥ 1. Consider � = {�1, �3}.Node 18. he potentially nondominated solutions found so

far are (83/47, 56/99)�, (177/103, 121/150)�, and (43/30,43/50)�.

he resolution of (�18) gives the optimal solution

(1, 0, 0, 0, 0, 0, 0)� with the criterion vector (11/51, 35/51)�which is dominated then Ef remains the same.

We have ��181 = 121/281 < ��1 = 30/23; we then fathomthe Node 18.

Node 29. he potentially nondominated solutions found

so far are (83/47, 56/99)�, (177/103, 121/150)�, (43/30,43/50)�, and (30/23, 81/70)�.

he resolution of (�29) gives the optimal

solution (0, 0, 1, 0, 1, 0, 1)� with the criterion vector(102/77, 178/209)�. his latter is dominated by (43/30,43/50)�; therefore, Ef remains the same. Here, the localideal point ��29 is not dominated so we proceed to theconstruction of the set�29: see Table 5.

Since we have�29 = 0, we fathom the Node 29.

Node 43. he potentially nondominated solutions found

so far are (83/47, 56/99)�, (177/103, 121/150)�, (30/23,81/70)�, (227/143, 167/168)�, and (34/25, 66/59)�.

he resolution of (�43) gives the optimal solution

(0, 1, 0, 1, 1, 0, 1)� with the criterion vector (127/86,

213/238)� which is dominated; then Ef remains thesame.

We have that ��43 = (127/86, 309/331)� is dominated by

(227/143, 167/168)�; we then fathom the Node 43.

5. Computational Results

he method described in Section 3 and the one presented in[2] (referred in Table 6 asNewAlgorithm andOldAlgorithm,respectively) were implemented in a Matlab 7.0 environ-ment and tested on randomly generated MOILFP problems.he data are uncorrelated integers uniformly distributedin the interval [1, 99] for the numerator and denominatorcoeicients, [−10, 50] for the numerator constant, [1, 50]for the denominator constant, and [−10, 50] for constraintscoeicients. For each constraint, the right-hand side valuewas set to �% of the sum of its coeicients. For each instance(�,�, �) (� is the number of variables, � the number ofconstraints, and � the number of objectives), a series of10 problems were solved. Computational experiments werecarried out on a computer with 2,53GHz Core i3 Processorand 3GB ofmemory. Table 6 summarizes the obtained resultswhere mean and maximum number of created nodes (TreeSize), number of eicient solutions (|Ef|), and execution timein seconds are reported.he xmark in the table refers tomorethan 2 hours of execution time in the CPU Time column anda number of created nodes of order of millions in Tree Sizecolumn.

Due to the important amount of information to be pro-vided, we have not presented the number of nodes saturatedcorresponding to each of the four fathoming rules used inour method. However, we can give in Table 7, as an example,what we have found for the 10 problems treated of the irsttype instance that is with 20 variables, 5 constraints, and 2objectives.

As expected, the results obtained by the new method aremuch better compared to those obtained by the original onein all the treated instances. Indeed, the oldmethod takesmorethan two hours of execution time for a number of variablesexceeding twenty whereas the new one does not reach halfan hour for sixty variables (972,01 sec. on average). One canalso notice the inluence of the use of nadir and ideal pointsin the search tree size; for example, in the biobjective case,the number of nodes of the search tree corresponding to aMOILFP problem with twenty variables is of order of tensof thousands on average for the old method; this number ofnodes is attained only for sixty variables andmore for the newmethod.We should also notice that themethodwe propose ismore suited for the biobjective case; this is because, irst, thecomputation of the nadir point can be done quickly using thepayof table and, second, the less calculation performed (tworeoptimizations) to obtain the ideal point at each stage.

6Advan

cesin

Decisio

nScien

ces

Table 6: he results obtained by both new and old methods on randomMOILFPs.

� � � �%New algorithm Old algorithm

CPU Time (sec) Tree Size CPU Time (sec) Tree size |Ef|Mean Max Mean Max Mean Max Mean Max Mean Max

20 5 2 25 6,59 23,19 835,20 2803 264,37 334,41 38220,80 47461 13,10 1820 10 2 25 1,99 4,13 279,30 482 103,94 175,14 10798,30 18766 7,40 1230 5 2 16,67 44,15 181,59 3050,30 12703 x x x x 18,60 4230 10 2 16,67 18,78 63,32 1201,90 3151 x x x x 11,90 2540 10 2 12,5 69,12 154,84 2717,60 5642 x x x x 15,10 2360 10 2 10 972,01 3109,24 18568,70 54107 x x x x 20,20 3930 5 3 16,67 439,07 851,79 14023,30 25618 x x x x 80,30 16440 10 3 12,5 693,44 1597,12 15013 37819 x x x x 78,40 21330 10 4 16,67 448,76 850,53 11202,30 19734 x x x x 189,30 335

Advances in Decision Sciences 7

Table 7: he results of the irst type instance concerning the number of saturated nodes using each of the 4 fathoming rules.

Problem numberNumber of saturated nodes using

�� Nadir point Ideal point Infeasibility Total

1 0 16 65 37 118

2 0 38 5 33 76

3 4 13 234 131 382

4 2 34 512 612 1160

5 3 32 362 320 717

6 3 4 115 68 190

7 0 59 120 105 284

8 2 22 139 67 230

9 0 3 47 40 90

10 3 35 55 56 149

6. Conclusion

In this paper, we have presented an improvement of Cherguiand Moulaı’s method [2] which generates the whole eicientset of MOILFP problems. One of the main changes thatwe have made was the addition of two fathoming rulesusing both ideal and nadir points resulting in a remarkablecomputational savings. Also, we have proposed to linearizeone of the fractional objective functions to solve linear pro-grams as subproblems rather than fractional linear programswhich facilitates the use of the dual simplex algorithm andreduces the number of simplex iterations performed. Weshould inally point out that this method does not require anynonlinear optimization and its tree structure can be exploitedfor construction of a parallel algorithm to handle large scaleproblems.

Conflict of Interests

he authors declare that there is no conlict of interestsregarding the publication of this paper.

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