Expert Systems with Applications 38 (2011) 15370–15376

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Expert Systems with Applications

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A tabu search meta-heuristic approach to the dual response systems problem

Cagdas Hakan Aladag a, Onur Köksoy b,⇑a Department of Statistics, Hacettepe University, Ankara 06800, Turkeyb Department of Mathematical Sciences, The State University of New York, Binghamton, NY 13902-6000, USA

a r t i c l e i n f o a b s t r a c t

Keywords:Robust designPareto optimizationQuality improvementTabu search

0957-4174/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.eswa.2011.06.026

⇑ Corresponding author.E-mail addresses: aladag@hacettepe.edu.tr (C.

binghamton.edu (O. Köksoy).

This paper presents an alternative approach to the dual response systems problem by utilizing a tabusearch algorithm that yields a string of solutions and examine the trade-offs graphically and systemati-cally how the controllable variables simultaneously impact the mean and the standard deviation of acharacteristic of interest relevant to an industrial process. Heuristic-based search techniques may be veryuseful for cases where interactive multi-objective optimization techniques are not available due to lack ofwillingness of decision-makers. A further advantage of tabu search is its simplicity and we show that theentire process only occupies a few lines of codes and generates string of solutions in speedy manner espe-cially for the larger-the-better/smaller-the-better cases of Taguchi’s robust parameter design. The proce-dure is illustrated with an example.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In today’s increasingly competitive marketplace more attentionis being paid to off-line quality control and the idea of robust prod-uct design. Recent advances in quality technology have resultedfrom considering the variation of a quality characteristic as wellas its mean value. Taguchi and Wu (1985) and Taguchi (1986) havebeen a major proponent of this philosophy. The recent push forquality improvement in industry has brought response surfacemethodology (RSM) to the attention of many users (Khuri, 1996).In the 1990s, much attention was given to the optimization of dualresponse systems (DRS) as an important RSM tool for qualityimprovement. In our context, the dual response refers to the meanand the standard deviation of the process.

Taguchi’s robust parameter design (RPD) calls for simultaneousoptimization of the mean and standard deviation responses. TheRPD problem is a special case of the multiple response problems,where two responses, the mean and variance of a fundamental re-sponse/characteristic observed during the experiment. The DRSproblem requires an overall optimization – that is a simultaneoussatisfaction with respect to both the mean and the standard devi-ation of a quality characteristic. Basically one builds two empiricalresponse surface models – one for the mean and one for the stan-dard deviation – and then optimizes one of the responses subjectto an appropriate constraint on the other’s value. The decisionhow to use the dual response approach to achieve the basic goalsof Taguchi’s philosophy depends upon the ultimate purpose of

ll rights reserved.

H. Aladag), koksoy@math.

the experiment. Vining and Myers (1990) adapted to the three ba-sic cases of RPD, i.e., ‘‘larger-the-better’’, ‘‘smaller-the-better’’, and‘‘target-is-best’’. In the larger-the-better/smaller-the-better cases,one seeks the settings of the control parameters that maximize/minimize the mean response while controlling the standard devi-ation at some specified value. In the target-is-best case, one isinterested in minimizing the standard deviation while keepingthe mean response at a specified target value. In each of threecases, a solution is found under an additional constraint on thevector of control variables. Let x = [x1, . . . , xk] be a k � 1 vector ofcontrol variables. If a factorial type design is used for the purposeof experimentation, then a cuboidal region, defined by�1 6 xi 6 1, i = 1, 2, . . . , k (k is the number of control variables),may be a good choice for defining the region of interest. Whenusing a spherical type design (e.g., a central composite design),the additional constraint is defined by x0x 6 q2 where q is thedesign radius.

A major drawback of selecting the most dominant response asthe objective function and then taking the other one as a constraint(i.e., single objective optimization) imposes an unnecessary restric-tion on the value of secondary response especially when dealingwith the larger-the-better/smaller-the-better cases. Keeping thestandard deviation below a specified value may rule out betterconditions during the optimization process, since an acceptable va-lue for the standard deviation response is usually unknown. In fact,process conditions that result in a smaller standard deviation areoften preferable. Recently, Köksoy and Doganaksoy (2003) realizedthat the standard deviation of any performance property could betreated as a new property in its own right as far as Pareto optimizerwas concern (i.e., multi-objective optimization). The interactionamong different conflicting objectives gives rise to a string of

C.H. Aladag, O. Köksoy / Expert Systems with Applications 38 (2011) 15370–15376 15371

solutions, called Pareto optimal solutions. Pareto solutions arethose for which improvement in one objective can only occur withthe worsening of at least one other objective. Thus, instead of a un-ique solution to the problem, the solution to a multi-objectiveproblem is a (possibly infinite) set of Pareto points. Since none ofthese alternative solutions can be identified as better than otherswithout any further examination, the goal in multi-objective opti-mization is to find as many alternative solutions as possible inspeedy manner. Once such set of compromised solutions is found,it usually requires a higher level decision making with other con-siderations to choose one of them for implementation. We believethat such analysis is useful compared to a single optimal solution,and that is required in order to achieve an improved understandingof the problem before searching for a final optimal solution.

Even though we support and follow the main philosophy pro-posed by Köksoy and Doganaksoy (2003), their optimizationmethod, namely the NIMBUS (Nondifferentiable Interactive Multi-objective Bundle-based Optimization System) algorithm, based oninteractive articulation of preference information has the followingdifficulty in applications: the interest devoted to interactive meth-ods can be explained by the fact that assuming the decision makerhas enough time and capabilities for co-operation. The conver-gence is not necessarily fast if the decision maker is not purposeful.The freedom of the decision maker has both positive and negativeaspects. The decision maker can direct the solution process and isfree to change her or his mind during the process. Because of thesubjectivity of the decision makers, different starting points, differ-ent types of questions or interaction styles may lead to different fi-nal solutions.

In order to overcome the difficulties associated with the optimi-zation method of Köksoy and Doganaksoy, we propose a tabusearch algorithm for finding the Pareto solutions for the DRS prob-lem. First we convert the problem into a scalar one by using aweighted linear sum of the objectives and then optimize theweighted objective function. The major advantage of the proposedformulation is that it does not require any constraints on the sec-ondary response. Unlike the NIMBUS method, the proposed tabuapproach does not set any specific assumptions on the behavioror the preference structure of the decision maker. It means thatthe proposed method will still work and generate many alternativesolutions whether or not the decision maker has enough time andcapabilities for co-operation.

The rest of the paper is organized as follows. In the next sectionwe present a revised problem formulation of the DRS problem. Wethen briefly review the fundamental concepts of tabu search. Thisis followed by a numerical example that illustrates the proposedapproach. We conclude the paper with a summary. The proposedtabu search algorithm is described in the Appendix.

2. Revised problem formulation of the DRS problem

Assume that an appropriate 2nd order response surface exper-iment is conducted. Let l̂ and r̂ denote the fitted response surfacesof the process mean and standard deviation, respectively. Assumethat these responses may be modelled by

l̂ ¼ b0 þXk

i¼1

bixi þXk

i¼1

biix2i þ

XXk

i<j

bijxixj ð1Þ

r̂ ¼ a0 þXk

i¼1

aixi þXk

i¼1

aiix2i þ

XXk

i<j

aijxixj ð2Þ

where the b’s and a’s represent the estimated coefficients, and thex’s are the control variables (x e R, where R is a region of interest).Therefore, the DRS optimization problem for the smaller-the-bettercase may be defined as:

Minimizeðw1l̂þw2r̂Þ ð3Þ

x 2 R

where w1 and w2 are pre-specified positive constants which arechosen based on the relative importance of the mean and standarddeviation responses, usually obtained through the advice of expertson the process of interest. The objective is to find the settings of x’sthat would optimize the weighted objective function subject only tothe constraint that defines the region of interest R. As mentionedearlier, we consider two different regions of interest, cuboidal andspherical. For the larger the better case, the mean should be maxi-mized therefore w1 is replaced with (�w1) for solving the DRS prob-lem using the tabu algorithm. The proposed formulation of the DRSproblem is directly applicable for the smaller-the-better and the lar-ger-the-better cases of RPD. The target-is-best case, however, needsa preset constraint on the value of the mean response function and,thus, the weighted objective optimization is not necessary. In some-what limited manner by changing the weight values in Eq. (3) onecan still find string of solutions for the target-is-best case, however,this case can be more directly addressed by a single objective opti-mization (e.g., Del Castillo and Montgomery (1993) or Copeland andNelson (1996)).

3. A brief overview of tabu search

Tabu search is a local search-based metaheuristic method thathas been successfully applied to a wide class of hard optimizationproblems. Appropriate subject areas include bioengineering, fi-nance, manufacturing, scheduling, and political districting. It wasfirst presented by Glover (1986) and also sketched by Hansen in1986.

Tabu search uses a short-term memory structure called a ‘tabulist’. A potential solution is marked as ‘‘tabu’’ so that the algorithmdoes not visit that possibility repeatedly. Tabu search starts withan initial solution. Algorithms based on tabu search perform aneighborhood search (i.e., a local search) starting from a currentsolution to its best neighbor (the one with the best objective valueamong all examined candidates). Tabu search modifies the neigh-borhood structure of each solution as the search progresses. Allthe neighbors of a current solution are examined and the bestnon forbidden move is selected. Note that this move may decreasethe quality of the solution, but necessary in order to increase thelikelihood of escaping from so-called local optimum ‘‘traps’’. A tabulist stores all the previously exploited moves or solutions which arenow forbidden. The search continues until some stopping criterionhas been satisfied. To avoid cycling during the search process, thereverses of the last certain number of moves, formed as a tabu list,are prohibited or announced as tabu restricted for certain numberof iterations (i.e., the tabu duration). To prevent a too rigorousparameter settings of the tabu restriction, some aspiration criteriaare usually introduced which allow overriding the tabu restrictionand thereby to guide the search toward a promising region. Inten-sification and diversification strategies with tabu search are alsoapplied to emphasize and broaden the search in the solution space,respectively. More discussion can be found in articles by Chao(2002), Vilcot and Billaut (2008), and Caserta and Uribe (2009).

The basic components of the tabu algorithm are outlined below:

1. Configuration: Coding of a solution.2. Move: Selected feasible direction of the search.3. Set of candidate moves: Feasible directions of the search.4. Tabu restrictions: The length of the tabu list.5. Aspiration criteria: Overriding the tabu restriction.6. Stopping condition: Terminating the search.

15372 C.H. Aladag, O. Köksoy / Expert Systems with Applications 38 (2011) 15370–15376

Fig. 1 by Güngör and Ünler (2008) shows the flowchart diagramof basic tabu search procedure.

4. Numerical example

To demonstrate the feasibility of applying the proposed meth-od, the following example of Vining and Myers (1990) (adaptedfrom Box and Draper (1987, p.247) will be presented in this sec-tion. The experimental data has appeared repeatedly in the litera-ture on DRS. The purpose of the experiment was to analyze theeffect of the speed (x1), pressure (x2), and distance (x3) variableson a printing machine’s ability to apply colored inks to package la-bels (y). The experiment was conducted using a 33 factorial designwith three runs at each design point. The fitted response surfacefunctions of mean and standard deviation using ordinary leastsquares from Vining and Myers (1990) are

l̂ ¼ 327:6þ 177x1 þ 109:4x2 þ 131:5x3 þ 32:0x21 � 22:4x2

2

� 29:1x23 þ 66x1x2 þ 75:5x1x3 þ 43:6x2x3 ð4Þ

and

r̂ ¼ 34:9þ 11:5x1 þ 15:3x2 þ 29:2x3 þ 4:2x21 � 1:3x2

2

þ 16:8x23 þ 7:7x1x2 þ 5:1x1x3 þ 14:1x2x3 ð5Þ

In applying the proposed approach to the printing example, we firstconsider the larger-the-better case since our interest is to find

Fig. 1. The flowchart diagram of

operating conditions that produce a large response (i.e., betterprinting machine quality) accompanied by the minimal process var-iability. Furthermore, we consider both cuboidal and spherical re-gion constrains. Since the design is factorial, it can be argued thatcuboidal region is more appropriate. We also, however, considerthe spherical region for the sake of completeness, since this regionwas used by other researchers who studied the same example. Theproposed formulation of the DRS problem is directly applicable forthe smaller-the-better and larger-the-better cases of RPD. However,a solution for the target-is-best case can also be found by introduc-ing a modified version of an objective function with a penalty term.

4.1. ‘Larger-the-better’ case

4.1.1. Cuboidal regionTable 1 displays the results which obtained by using the cuboi-

dal region �1 6 xi 6 1, i = 1, 2, 3 (see the Appendix for a descriptionof the proposed tabu algorithm). Here, the solutions are producedby setting w1 = 1 and randomly increasing w2 between 1 and 9. Wegive high priority to r̂ by increasing the value of w2. Hence, findingthe minimum value of r̂ gains more importance in the optimiza-tion of larger-the-better case since the proposed search minimizesthe given composite objective function. On the other hand, it is alsopossible to employ another interval for the coefficient w2. In Table1, we present string of solutions (i.e., Pareto optimal solutions) thathighlight the trade-offs between the mean and the standard

basic tabu search procedure.

Table 1Results for the ‘‘Larger-the-Better’’ case (cuboidal R.,�16 xi6 1, i = 1, 2, 3).

Alternatives l̂ r̂ x1 x2 x3

1 911.02 137.47 0.999991 0.999995 0.9996222 893.55 130.26 0.999953 0.999985 0.9101343 863.53 118.89 0.999956 0.999970 0.7614774 843.40 111.92 0.999978 0.999906 0.6651825 824.20 105.75 0.999661 0.999981 0.5761306 810.38 101.57 0.999351 0.999805 0.5136077 797.97 97.93 0.999832 0.999206 0.4574648 728.14 80.29 0.999978 0.999926 0.1567079 611.61 59.27 0.999961 0.998814 �0.30015610 537.06 49.60 0.999953 0.410388 �0.29029011 508.80 46.16 0.999930 0.167035 �0.25464612 449.54 39.29 0.992127 �0.366098 �0.09387813 433.30 37.24 0.999526 �0.434604 �0.11873914 401.55 33.53 0.999933 �0.625331 �0.09101115 361.11 28.84 0.999988 �0.840517 �0.070960

Fig. 2. Pareto optimal solutions for the ‘‘Larger-the-Better’’ case.

C.H. Aladag, O. Köksoy / Expert Systems with Applications 38 (2011) 15370–15376 15373

deviation responses. For example, the interpretation of the fifthalternative is as follows: under the cuboidal region, 105.75 is thesmallest standard deviation value that can be attained if the meanresponse function is held fixed at 824.20 (or, 824.20 is the largestmean value that can be attained if the standard deviation is heldfixed at 105.75). The other solutions can be interpreted similarly.The first alternative produces the highest value of the mean,l̂ ¼ 911:02, with a standard deviation of r̂ ¼ 137:46. If a smallervalue of standard deviation is required, one needs to sacrifice somevalue for the mean function and move in the direction of thefifteenth alternative. One cannot improve any criterion withoutdeteriorating a value of at least one other criterion. The best com-promising solutions perhaps might be the middle alternatives.

4.1.2. Spherical regionTable 2 displays the results based on a spherical region x0x 6 1.

The first alternative gives the highest mean, l̂ ¼ 638:11, while thestandard deviation is 74.90. However, this may not be the bestcompromise solution since the value of the standard deviationseems high. Again, one needs to sacrifice some value for the meanresponse for getting an improved result for the standard deviation.

Fig. 2 graphically displays the Pareto optimal solutions withq2 = 1, 2, 3. As the bound on x’s is relaxed, it is possible to achievea smaller standard deviation (larger mean) for a set value of mean(standard deviation). Our present findings show excellent agree-ment with those obtained by Köksoy and Doganaksoy (2003).

4.2. ‘Target-is-best’ case

In the proposed tabu search, the mean and the standard devia-tion responses can be individually made desirably close to their

Table 2Results for the ‘‘Larger-the-Better’’ case (spherical R., x0x 6 1).

Alternatives Weights for r̂ l̂ r̂

1 1 638.11 742 2 631.53 703 3 616.84 654 4 602.94 615 5 583.10 576 6 560.48 537 7 535.12 498 8 499.05 459 9 450.53 3910 10 386.19 3211 11 318.08 2612 12 276.70 22

respective desired/implied target values. A target mean of 500and target standard deviation of 60 were used as desired target val-ues by past authors who studied the same problem. We use thesame desired target values to illustrate the performance of the pro-posed tabu search. For the sake of simple illustration here we onlyconsider the cuboidal region of interest.

In order to reach the desired goals, the objective function for thelarger-the-better case needs to be modified by adding a so-calledpenalty term. For instance, when the target standard deviationr = 60 is considered, the modified objective function is constructedas follows: The penalty term (p) is calculated by

p ¼ j60� r̂j

where r̂ is calculated from a fitted model given in Equation (5).Thus, the modified objective function is constructed by adding thisvalue to the objective function as follows:

f 1ðxÞ ¼ ð�l̂þ r̂Þ þ c � p ð6Þ

where c is a constant multiplied by the penalty term p. We take thisconstant as 100. Here l̂ and r̂ represent the fitted response surfacesgiven in (4) and (5), respectively. Table 3 summarizes the solutionsobtained from the proposed algorithm. The proposed penalty-basedformulation and its solutions are very close to desired target r = 60.Also, the decision maker gains some flexibility to pick the right va-lue of control parameters x1, x2, and x3 based on her/his sacrificeinterest level for the mean response at ‘almost fixed’ level of stan-dard deviation.

For the mean target l = 500, we again construct a modifiedobjective function by adding a new penalty term as follows:

x1 x2 x3

.90 0.80601 0.42302 0.41401

.76 0.85181 0.41828 0.31525

.60 0.89608 0.39718 0.19706

.99 0.94228 0.30282 0.14283

.80 0.95467 0.29501 0.03924

.83 0.97416 0.22302 �0.03494

.97 0.98053 0.16102 �0.11239

.20 0.98447 0.01814 �0.17461

.55 0.95877 �0.19371 �0.20794

.80 0.86605 �0.45352 �0.21042

.28 0.70237 �0.68141 �0.20580

.66 0.57058 �0.79200 �0.21720

Table 3Results for the ‘‘Target-is-best’’ case for the target standard deviation r = 60.

Alternatives l̂ r̂ x1 x2 x3

1 613.192479 59.999176 0.970429 0.963129 �0.2464582 610.333828 60.000939 0.947088 0.944606 �0.2233913 603.761625 59.999938 0.899762 0.884410 �0.1678074 616.475036 60.000564 0.997888 0.998561 �0.2796325 608.677983 60.000286 0.934203 0.933758 �0.2105476 608.880479 60.000040 0.938056 0.911181 �0.2033037 605.511891 59.999977 0.903405 0.987123 �0.2134068 605.889663 60.000012 0.911959 0.928052 �0.1940879 612.781697 59.999928 0.967814 0.949561 �0.23893910 611.795153 60.000127 0.960072 0.939356 �0.229534

Table 4Results for the ‘‘Target-is-best’’ case for the target mean l = 500.

Alternatives l̂ r̂ x1 x2 x3

1 499.998492 47.327856 0.851057 0.026738 �0.0294632 500.000295 47.469489 0.841615 0.034666 �0.0249543 500.000069 47.452121 0.845704 0.023673 �0.0206874 499.999682 47.630160 0.843448 0.002527 �0.0003715 499.999627 45.499895 1.000000 �0.125374 �0.0700376 500.000139 46.319765 0.942671 �0.100711 �0.0270157 500.001395 46.144159 0.974450 �0.171611 0.0000468 500.000374 45.472879 0.953593 0.248976 �0.3003429 500.000026 45.505631 0.987875 �0.076847 �0.09793010 499.998700 45.429854 0.978789 �0.002952 �0.147872

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f 2ðxÞ ¼ ð�l̂þ r̂Þ þ 100 � j500� l̂j ð7Þ

where l̂ and r̂ are the fitted models given in (4) and (5), respec-tively. Table 4 displays the results from the proposed method. Onceagain the proposed method generates solutions at ‘almost fixed‘ le-vel of mean response and some flexibility for the standard deviationin the range between 45.42–47.63. According to a decision-maker’ssacrifice level of preference for the standard deviation, there wouldbe some flexibility in deciding the values of control parameters x1,x2, and x3.

Overall, for the target mean of 500 and the target standard devi-ation of 60, the results from the proposed approach are in closeagreement with those obtained by previous researchers (e.g., DelCastillo and Montgomery, 1993 or Kim and Lin, 1998).

5. Summary

This paper presents a tabu search metaheuristic algorithm forfinding the Pareto optimal solutions of the dual response systemsproblem in industrial applications. The proposed formulation doesnot require any constraints on the secondary response (i.e., theprocess standard deviation). Unlike the other multi-objective alter-natives, tabu approach does not set any specific assumptions onthe behavior or the preference structure of the decision maker.Tabu search algorithms are especially appealing due to the factthat the entire process only occupies a few lines of codes and gen-erates string of solutions in speedy manner. This makes the searchsimpler and also computationally attractive than the other heuris-tic algorithms. We believe that tabu search algorithms provide agood alternative to the methods of Pareto optimization in the re-cent literature.

Appendix A. The proposed tabu search algorithm

In order to solve the DRS problem, we developed a tabu searchalgorithm based on its core frame. In addition, intensification anddiversification strategies are employed in order to make the

algorithm more efficient. The constructed algorithm is describedbelow:

A.1. Coding of a solution x

Based on our numerical example we have three variables foreach of the presented solution. A vector including these variablesrepresents a solution as follows:

x ¼ ½x1; x2; x3�

A.2. Objective function f(x)

The developed tabu search algorithm tries to minimize the fol-lowing objective function

f ðxÞ ¼ �w1l̂þw2r̂ ð8Þ

where l̂ and r̂ represent the fitted surfaces given in (4) and (5),respectively. Also, w1 and w2 represent pre-specified positive con-stants which are chosen based on the relative importance of themean and standard deviation responses, usually obtained throughthe advice of experts on the process of interest.

A.3. The solution space R

The coded program using the developed tabu search can findsolutions under four different constraints given below. Therefore,a solution x is a feasible solution if x e R.

(i) �1 6 x1, x2, x3 6 1(ii) x2

1 þ x22 þ x2

3 � 1(iii) x2

1 þ x22 þ x2

3 � 2(iv) x2

1 þ x22 þ x2

3 � 3

A.4. Initial solution xi

The control variables can take values in between �1 and 1 forthe first constraint (i). Three randomly generated values of thosevariables in this interval initiate the proposed algorithm for thefirst constraint. Thus, the initial solution xi can be represented asfollows:

xi ¼ ½x1; x2; x3�; for x1; x2; x3 2 ½�1; 1�

When the developed tabu search operates for the constraints (ii),(iii) and (iv), the search starts from the initial solutionxi = [0, 0, 0]. This is called a design center.

A.5. Neighborhood N(x)

We construct the neighborhood of a solution N(x) by decreasingor increasing the values of variables of a given solution. In otherwords, to generate a neighbor solution xt from a solution x, we em-ploy the moves based on decreasing or increasing the values ofvariables. In order to do this, a parameter D is used. A value of avariable can be changed by adding or subtracting D so neighborscan be produced. The value of D is shifting during the search pro-cedure according to the intensification and diversification strate-gies which will be defined later. For instance, as these definedmoves are performed, some neighbors xt

k of a solutionx = [x1, x2, x3] can be generated as follows:

xt1 ¼ ½x1; x2; ðx3 þ DÞ�

xt2 ¼ ½ðx1 � DÞ; x2; x3�

xt3 ¼ ½ðx1 þ DÞ; ðx2 � DÞ; x3�

xt4 ¼ ½ðx1 þ DÞ; ðx2 þ DÞ; ðx3 � DÞ�

C.H. Aladag, O. Köksoy / Expert Systems with Applications 38 (2011) 15370–15376 15375

A.6. Candidate list V(x)

For a given solution x, it is computationally too expensive to ex-plore its whole neighborhood N(x). Therefore, a candidate liststrategy may be used for the proposed tabu search algorithm.When the next current solution xnext is being determined, the can-didate list of the current solution V(xc) is firstly be constructed,xnext e V(xc), and then is selected as a next solution.

The employed candidate strategy operates as follows: Threecases arise for a variable in a solution such as decreasing, increas-ing or not changing. In the printing study example we have 33 = 27combinations for three main variables. If the values of all variablesdo not change, the obtained neighbor solution will remain thesame as the current one. Therefore, this move is useless and 26neighbors of a current solution are generated by doing remain 26moves. In other words, the size of the candidate list is 26. It isclearly seen that V(x) � N(x) for any solution x. It is very hard toexamine the whole neighborhood of a solution so using this candi-date list strategy is helpful. After producing 26 neighbors of thecurrent solution, the one which has the best objective function va-lue is chosen as a next solution. For instance, when D = 0.1, theproduced candidate list V(x) of a given solution x = [x1 = 0, x2 = 0,x3 = 0] are given in Table 5.

A.7. Tabu list

An examined current solution is defined as tabu and kept in thetabu list (T) for a sufficient number of iterations. We consider atabu list of size eight. Once the best neighbor solution is deter-mined, the tabu status of a solution is checked by using the tabulist. If a candidate solution belongs to this tabu list then it willnot be selected again as a new current solution in the nextiteration.

A.8. Aspiration criterion

As an aspiration criterion, a global aspiration by objective is used.If a tabu solution xt has an objective function value less than anobjective function value of the best solution (i.e xbest) obtained sofar, this tabu solution xt is selected as the next current solution.In other words, if f(xt) < f(xbest), and then xt will be selected asthe next current solution in spite of its tabu status.

A.9. Selection of a solution

For the printing study example the number of candidate solu-tions (xt

i 2 VðxcÞ) produced from the current solution xc is 26.These solutions ðxt

i Þ are sorted by the objective function value.Then the best feasible objective value solution (i.e.,x0best 2 R) is cho-sen if it is not in the tabu list (i.e., x0best R T). If it is in the tabu list(x0best 2 T) but satisfying the aspiration criterion f ðx0bestÞ < f ðxbestÞthen it is accepted; otherwise, another candidate solution (x00) with

Table 5The produced candidate list V(x) for the given solution x.

No x1 x2 x3 No x1 x2 x3 No x1 x2 x3

1 0.1 0.1 0.1 10 �0.1 0.1 0.1 19 0 0.1 0.12 0.1 0.1 0 11 �0.1 0.1 0 20 0 0.1 03 0.1 0.1 �0.1 12 �0.1 0.1 �0.1 21 0 0.1 �0.14 0.1 0 0.1 13 �0.1 0 0.1 22 0 0 0.15 0.1 0 0 14 �0.1 0 0 23 0 0 �0.16 0.1 0 �0.1 15 �0.1 0 �0.1 24 0 �0.1 0.17 0.1 �0.1 0.1 16 �0.1 �0.1 0.1 25 0 �0.1 08 0.1 �0.1 0 17 �0.1 �0.1 0 26 0 �0.1 �0.19 0.1 �0.1 �0.1 18 �0.1 �0.1 �0.1

the best objective function value among the remaining candidatesis examined in a similar manner.

A.10. Intensification and diversification strategies

We generate intensification and diversification effects by shift-ing the value of D, which is the step size of the search. Based on theobtained solutions during the search, the search space can beexamined more efficiently by decreasing or increasing the valueof D. After obtaining the good solutions in the search region, D isdecreased in order to examine the region more carefully. If this re-gion is not a promising one then it can be skipped by increasing thevalue of D. Thus, new regions may be examined if necessary. Alter-nation of intensification and diversification phases of the proposedalgorithm allows finding the global optimum with a good accuracy.

A.11. Stopping criterion

When the maximum iteration bound is reached by the algo-rithm the search will be terminated.

A.12. The developed algorithm

Let ‘‘NG(Solution, D)’’ be the neighborhood generating functionwith two input fields. Let ‘‘OF(Solution)’’ be a function which calcu-lates an objective function value from a given solution. Let‘‘TS(Solution)’’ be a procedure which adds the current solution tothe tabu list so that its tabu status can be checked during the selec-tion procedure of the best neighborhood. Finally, SS() is a proce-dure that generates a random vector solution on a givenrestricted interval [�1, 1]. We present the developed tabu searchalgorithm with the following pseudo-code:

Best_Solution = SS()tolerance = 0.5eks = 0for k := 1 to Bound_1 dobeginSolution = Best_Solutionfor j := 1 to Bound_2 dobegin

Current_ Solution = NG(Solution,(0.1 + eks)k � j)TS(Current_ Solution)if OF(Current_ Solution) < OF(Best_Solution) thenbegin

if OF(Current_ Solution) – OF(Best_Solution) > tolerance thenif (eks > 0) then eks = eks � 0.01

elseeks = eks + 0.01

Best_Solution = Current_ Solutionendendend

As seen from the algorithm, the maximum iteration bound isBound_1 � Bound_2. Based on our numerical example, the valuesof Bound_1 and Bound_2 are 5 and 30, respectively. The variabletolerance is fixed at 0.5. The initial value for the variable eks is 0and this value is subject to change during the search. The greaterthe value of eks, the greater the value of D is obtained. D is calcu-lated by the formula (0.1 + eks)k � j) so that the different neighborscan also be obtained for the different step sizes. An intensificationeffect is generated by the first loop with an index k. After thesecond loop with an index j, the first loop starts again with the bestpreviously found solution. Since k is increased by 1, D is todecrease and the search will focus on the region around the best

15376 C.H. Aladag, O. Köksoy / Expert Systems with Applications 38 (2011) 15370–15376

previous solution. In the second loop when the current solution isbetter than the best solution found so far, it means that if the dif-ference between these solutions is greater than a given tolerancevalue, D is decreased so that the search can focus on this region;otherwise, D is increased to make the search pass through this re-gion as quickly as possible. Thus, the search can go towards differ-ent solutions in different regions and a diversification effect isgenerated.

A.13. Restrictions

The NG function needs to be modified based on a user-selectedconstraint which defines the search region. Candidate solutions areexamined in terms of their objective function values as well astheir tabu status. In addition to these, we also check the feasibilityof a given solution by satisfying the given constraint. Therefore, thebest chosen neighbor at the each selection step will be a solutionthat satisfies the given constraint.

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