A hybrid approach using TOPSIS, Differential Evolution, and Tabu Search to find multiple solutions of constrained non-linear integer optimization problems
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13 tia4 o5
9 nto,10 FES11
151617 Received in revised form 19 February 201418 Accepted 25 February 201419
2 62728integer optimization problems. First, the constrained optimization problem is cast into a bi-objective29optimization problem, where the constraints are inserted as another objective function. Next, the novel303132333435363738
48 methods to deal with non-linear integer programming problems49
54 optimization problems in which some of the variables are55 continuous and other are discrete (MINLP) as in [3,11,25,26].56 Many researchers have proposed to use lled functions57 [30,29,28,10,22,18]. One of the main disadvantages of such58 approach is that the addition of a lled function introduces extra59 parameters which need to be tuned and they generally are
68are multimodal and present many solutions . Niching methods69odal optimization70nt niching meth-71icted tournament72stering, cl73niching m74play an important role when incorporated into evolutionar75rithms to maintain multiple solutions within a stable popu76Unfortunately, most of the methods listed above present d77ties for solving multi-modal optimization problems with multiple78local or global optima because is necessary to specify, a priori,79the niching parameters, which are difcult to tune and they typi-80cally are dependent on the problem to be optimized.81An interesting approach recently proposed in , which does82not require specication of any niching parameters uses a PSO with83ring topology. In this case, the local memory of the individual
CorrespQ2 onding author. Tel.: +55 2799475139.E-mail addresses: email@example.com (E.R.F.A. Schneider), krohling.renato@
gmail.com (R.A. Krohling).
Knowledge-Based Systems xxx (2014) xxxxxx
Contents lists availab
KNOSYS 2761 No. of Pages 9, Model 5G
18 March 2014cannot ensure that the optimal solution is always found, thus, itis necessary to use approximate methods. In recent years, methodssuch as the biologically inspired algorithms have gained impor-tance to tackle these types of optimization problems .
There are some approaches to deal with multi-modal non-linear
have been developed to nd solution of multimproblems with multiple optima. The most relevaods include : tness sharing, derating, restrselection, crowding, deterministic crowding, cluparallelization, speciation, among others. Thesehttp://dx.doi.org/10.1016/j.knosys.2014.02.0150950-7051/ 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: E.R.F.A. Schneider, R.A. Krohling, A hybrid approach using TOPSIS, Differential Evolution, and Tabu Search to nd msolutions of constrained non-linear integer optimization problems, Knowl. Based Syst. (2014), http://dx.doi.org/10.1016/j.knosys.2014.02.015earing,ethodsy algo-lation.ifcul-There is a growing interest in methods that solve optimizationproblems effectively and efciently. Depending on the nature ofthe problem, there are methods that ensure that the optimal solu-tion is found, for instance, the Simplex algorithm for linear pro-gramming problems, and the Branch and Bound algorithm forinteger linear programming problems [1,16]. However, the existing
for solving non-linear optimization problems with linear con-straints has been presented. In  an algorithm for solving non-linear integer optimization problems by means of linearization ofthe objective function and constraints has been proposed.
Generally, Evolutionary Algorithms (EA) and Particle SwarmOptimization (PSO) converge to a single nal solution when usedfor optimization purposes. However, many optimization problemsAvailable online xxxx
Keywords:Constrained non-linear integer optimizationDifferential EvolutionTabu SearchTOPSIS
1. Introductionmethod to solve multi-objective optimization problems is developed and applied to solve the reformu-lated problem. The novel method developed to solve multi-objective optimization problems is basedon the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) whereas the multi-objec-tive problem is cast in single-objectives problems. The Differential Evolution (DE) algorithm in its threeversions (standard DE, DEbest and DEGL) are used as optimizer. Since the solutions found by the DE algo-rithms are continuous, a Tabu Searh (TS) is employed to nd integer solutions during the optimizationprocess. Experimental results show the effectiveness of the proposed method.
2014 Elsevier B.V. All rights reserved.
dependent on the problem to be optimized. In  an algorithmArticle history:Received 1 April 2013
This paper presents a novel method to nd multiple solutions of multi-modal constrained non-linearA hybrid approach using TOPSIS, Differento nd multiple solutions of constrained nproblems
Erick R.F.A. Schneider a, Renato A. Krohling b,aGraduate Program in Computer Science, PPGI, UFES Federal University of Esprito SabDepartment of Production Engineering & Graduate Program in Computer Science, PPGI, UEsprito Santo, ES, Brazil
a r t i c l e i n f o a b s t r a c t
journal homepage: wwwl Evolution, and Tabu Searchn-linear integer optimization
Av. Fernando Ferrari, 514, CEP 29075-910 Vitria, Esprito Santo, ES, Brazil Federal University of Esprito Santo, Av. Fernando Ferrari, 514, CEP 29075-910 Vitria,
le at ScienceDirect
sevier .com/ locate /knosysultiple
KNOSYS 2761 No. of Pages 9, Model 5G
18 March 2014particles of the PSO are able to maintain the best positions found sofar, while the particles explore the search space. In the niching PSOwithout niching parameters proposed in  it is shown that largepopulations using PSO with ring topology is capable of forming sta-ble niche and able to nd multiple local and global optima. Thepromising results suggest that this method presents good resultswithout requiring parameters to tune. The need of large popula-tions in niching PSO without niching parameters is its main disad-vantage. Nonetheless, the method developed in , motivated usto extend the approach to Differential Evolution (DE). In fact, DEwith ring topology [5,6] were applied to multimodal problemswith promising results to nd a single solution of optimizationproblems. However, for problems with multiple optima, DE withring topology called DEGL  has not been used yet. The DEGL pro-vides a good balance between exploration and exploitation duringthe search improving the performance of the algorithm.
In this work, inspired by PSO with ring topology, the DEGL isused to explore the search space, but since DEGL has no memory,this motivated us to combine the algorithm with Tabu Search(TS). In this paper, we propose a method which uses only the valueof the objective function and the constraints, without no assump-tions regarding linearity, continuity, and convexity of the objectivefunctions and the constraints. The proposed method consists ofthree stages, where each stage uses the DE (Differential Evolution)algorithm hybridized with the TOPSIS (Technique for Order Prefer-ence by Similarity to Ideal Solution). The three stages are based onTOPSIS  to transform a k-objective optimization problem insingle-objectives optimization problem. In the last stage, a hybrid-ization with the Tabu Search (TS) algorithm to nd integer solu-tions from the real solutions obtained previously is performed.
The remainder of this paper is organized as follows. In Section 2,we present the mathematical formulation of the optimizationproblem. The three algorithms used in our approach are describedin Section 3. In Section 4, we propose a novel hybrid method to ndmultiple solutions of constrained optimization problems. Compu-tational results for several benchmarks are provided in Section 5to show the suitability of the method. In Section 6, we give someconclusions with directions for further works.
2. Mathematical formulation
A usual notation is used here: Z denotes the set of integers, Rthe set of real numbers, and Zn Z n Z. The problem of inter-est in this paper is the constrained non-linear integer optimizationproblem, which, without loss of generality, can be dened as:
min f xs:t: : gix 0; with i 1; . . . ;m
lj xj uj; with j 1; . . . ;nx 2 Zn
where f : Zn ! R is the objective function, gi : Zn ! R are inequal-ity constraints; lj and uj represents the lower and upper bounds ofthe integer search space, respectively.
The reformulated problem can be dened as:
min f x; gxs:t: : lj xj uj with j 1; . . .n
x 2 Zn2
where g(x) = max(0, g1(x), . . . , gm(x)).
3. Algorithms description
2 E.R.F.A. Schneider, R.A. Krohling / KnowTo deal with the problem dened in Section 2, the constraintswere incorporated into the objective function. A method to solve
Please cite this article in press as: E.R.F.A. Schneider, R.A. Krohling, A hybrid appsolutions of constrained non-linear integer optimization problems, Knowl. Basmulti-objective optimization problems is developed, whereas theTOPSIS is used to formulate the multi-objective problem inmono-objective problems. Next, the DE and TOPSIS are used tosolve the resulting mono-objective problems. At the last stage ofthe proposed method, the TS algorithm is executed to nd integersolutions provided by DE algorithm. Next, we describe the threealgorithms used in this novel method.
3.1. The technique for order preference by similarity to ideal solution
TOPSIS  is a technique to nd the best alternative(s) for amulti-criteria decision making (MCDM) problem consisting ofalternatives and criteria (benet or cost). The algorithm selectsthe best alternative within the set of alternatives so that there isa compromise between the criteria. The algorithm calculates thepositive ideal solution (PIS) and the negative ideal solution (NIS)and chooses the alternative that best approximates the PIS and atthe same time increase the distance to the NIS. The PIS is denedas the solution containing the best values of the criteria withinthe set of alternatives, whereas the NIS contains the worst valueswithin the set of alternatives.
First, we dene a matrix A = (xi,j)m2 which contains in each linean alternative. In this paper, an alternative means a possiblesolution. The matrix A is composed of two columns, namely, twocriteria. The rst criterion is the objective function for eachcandidate solution (or alternative), while the second criterion isthe maximum value of the constraint violation. The matrix A is de-scribed as:
A f1 g1
where fi f X!i and gi max0; g1X!
i; . . . ; gmX!
i.The weight vector composed by individual weights wj(j = 1, 2)
to each criteria Cj satisesP2
j1 wj 1: In this paper, the matrixA is normalized for each criterion Cj throughpij xijMAX xj ; with j 1;2 and MAX xj represents the maximum va-lue for each criterion Cj. Thus, the decision matrix B representsthe normalized relative rating of alternatives and is described by:
B pi;jm2The TOPSIS  begins with the calculation of the positive ideal
solutions B+(benets) and negative ideal solutions B(cost) asfollows:
B p1 ; p2 ; . . . ;pn where pj maxi pij; j 2 J1;mini pij; j 2 J2
B p1 ; p2 ; . . . ;pn where pj mini
pij; j 2 J1;maxi
pij; j 2 J2
4where J1 and J2 represent the benet and cost criteria, respectively.
Secondly, compute the Euclidean distances between Bi and B+
(benets) and between Bi and B (cost) as follows:
where dij pj pij; with i
1; . . . ;m: 5
where d p pij; with i
e-Based Systems xxx (2014) xxxxxxj1 ij ij j
1; . . . ;m: 6 197197
roach using TOPSIS, Differential Evolution, and Tabu Search to nd multipleed Syst. (2014), http://dx.doi.org/10.1016/j.knosys.2014.02.015
198 Next, one calculates the relative closeness coefcient ni to each199 alternative Bi in relation to the positive ideal B+ as:200
di di 7
203 Finally, ranking according to the relative closeness coefcient.204 The best alternatives are those that have the greatest value ni205 and should be chosen because they are closer to the positive ideal206
252where gbest is the best individual of the population.253The DEGL algorithm differs from the standard DE due the insert-254ing of a ring neighborhood topology  as shown in Fig. 2. This ring255neighborhood topology was initially proposed in PSO  and ex-256tended to other bio-inspired algorithms. It favors a balance be-257tween exploration and exploitation improving the capabilities of258DE algorithm. The donor vector is calculated by performing a linear259combination of the local donor vector L
!i, created using members
ing iB a
E.R.F.A. Schneider, R.A. Krohling / Knowledge-Based Systems xxx (2014) xxxxxx 3
KNOSYS 2761 No. of Pages 9, Model 5G
18 March 2014solution.The TOPSIS pseudo-code is shown in Fig. 1.In this paper, TOPSIS is used to select the best solution within a
population of candidate solutions. In order to obtain a compromiseamong the set of criteria, which in this case are the value of theobjective function and the maximum constraint violation.
3.2. Differential Evolution
The optimization algorithm Differential Evolution (DE) wasintroduced by Storn and Price . Similar to other EvolutionaryAlgorithms (EAs), DE is based on the idea of evolution of popula-tions of possible candidate solutions, which undergoes the opera-tions of mutation, crossover and selection . The parametervectors are denoted by xi = [xi,1, xi,2, . . . , xi,n]T with components xi,j.The lower and upper bounds of xi,j are lj and uj, respectively. The in-dex i = 1, . . . , NP represents the individuals index in the populationand j = 1, . . . , n is the position in D-dimensional individual. NP andn stand for population size and problem dimension, respectively.
First, the population of candidate solutions is initialized asfollows:
xi;j lj U0;1i;juj lj j 1; . . . ;n; i 1; . . . ;NP 8where U(0,1)i,j is a uniformly distributed random number within[0,1].
The DE algorithm undergoes the following operators:
Mutation. Crossover. Seletion.
Mutation is seen as a change or perturbation with a random ele-ment. In DE-literature, a parent vector from the current generationis called target vector, a mutant vector obtained thro...