Multiple-attribute decision-making approach for an energy-efficient facility layout design

ORIGINAL ARTICLE

Multiple-attribute decision-making approachfor an energy-efficient facility layout design

Lei Yang & Jochen Deuse & Pingyu Jiang

Received: 27 February 2012 /Accepted: 2 July 2012 /Published online: 20 July 2012# Springer-Verlag London Limited 2012

Abstract Due to the trends of energy shortage and energyprice rise, energy efficiency, which was always ignored overthe past decades, becomes a worldwide hot issue and also asignificant challenge for most factories. Therefore, it is neces-sary to incorporate energy-relevant criterion as a key criterionwith traditional criteria in the layout planning phase. As a multi-attribute decision-making (MADM) problem, the evaluationand selection of facility layout alternatives are often difficultand time consuming since the criteria generally have differentunits and conflicting features. In this article, aMADM approachwhich incorporates the advantages of rough set theory, analytichierarchy process (AHP), and technique for order preference bysimilarity to ideal solution (TOPSIS) is proposed to solve thefacility layout design problem with considering both traditionallayout criteria and energy relevant criteria. At first, rough settheory is integrated with AHP to determine the weights for eachcriterion of alternatives. Then, TOPSIS is applied to get the finalalternative ranking. Besides, sensitivity analysis for both deci-sion weights and production rates is performed, and a compar-ison among different decision-making approaches for the sameproblem is also studied to demonstrate the rationality of the finaldecision. Finally, a practical expanding case is studied to vali-date the proposed approach.

Keywords Decision making . Energy efficiency . Facilitylayout design . Sensitivity analysis

1 Introduction

Nowadays, many manufacturing companies share the com-mon goals towards cost effectiveness, energy efficiency, andsustainability. In particular, in the age of energy shortageand energy price rise, energy efficiency should be consid-ered as an essential factor in early planning phase with thepurpose of obtaining more benefit and becoming more com-petitive in the market. Besides, developing an energy-efficient facility planning is not only a problem about costreduction but also a great contribution to the environmentalprotection [1].

Facility layout design has an essential impact on theperformance of the whole manufacturing system, and it isalways considered as a key for manufacturing systems toimprove their productivity. The traditional facility layoutproblem (FLP) generally focuses on quantitative criteriasuch as shape ratio, material handling cost, adjacency score,and space demand, as well as the qualitative criteria such asflexibility and quality. However, due to the trends of energyshortage and energy price rise, energy relevant criteriashould be incorporated with the traditional criteria in thefacility layout planning phase.

2 Literature review

Over the past decades, in the purpose of solving FLP,most studies have investigated the allocation activities.Many researches try to simplify practical FLP intomathematical programming models or simulation modelsto optimize the objectives of the FLP [2]. Unfortunately,such optimization problem always belongs to the classof NP-hard, which means large amount of computingtime consumption and difficulty of finding an exactsolution [3]. Furthermore, such optimization approaches

L. Yang (*) : J. DeuseDepartment of Mechanical Engineering, TU Dortmund University,Leonhard-Euler-Straße 5,44227 Dortmund, Germanye-mail: [email protected]

P. JiangState key Laboratory for Manufacturing Systems Engineering,Xi’an Jiaotong University,No.99.Yanxiang Road,710054 Xi’an, China

Int J Adv Manuf Technol (2013) 66:795–807DOI 10.1007/s00170-012-4367-x

are difficult to solve the problem with qualitative crite-ria. In addition, most layout design problems have manyoptimization objectives with different units andconflicting features. Aiming at acquiring the best layoutwith considering all objectives, many multiobjective op-timization approaches have been developed which final-ly obtain an optimal solution set instead of a singleoptimal solution. Ye et al. [4] use genetic algorithmwhich applied a random weight approach to combinethe normalized value of two objectives for solving thelayout problem with considering the material handlingcosts and nonmaterial relation requirements. Aiello et al.[5] develop a genetic search algorithm which determinesPareto optimal solutions to solve the multiobjectiveoptimization problem with the objectives of minimizingmaterial handling costs while maximizing the satisfac-tion of distance requests, closeness rating, and the satisfactionof aspect ratio requests. Ye et al. [6] employ a GA-TS algo-rithm based on weighted sum method for maximizing close-ness rating and minimizing material handling cost. Under thiscondition, the layout designers should choose a best solutionaccording to the practical situation and their preference. How-ever, layout decision making is a multiattribute decision-making (MADM) problem; hence, the evaluation of FLPalternatives is always difficult and time consuming becauseof its inherent multiple attribute feature.

In order to avoid relayout that will cause extra costsand waste of production time, many researchersconcerned the layout evaluation by using MADM tech-niques. They investigate the characteristics of a finitenumber of layout alternatives, the information on theperformance among alternatives of an attribute, and thepreferences across all involved attributes to choose theoptimum layout alternative that can satisfy all the rele-vant attributes [3]. Cambron et al. [7] use variouscomputer-aided layout approaches to obtain several lay-out alternatives and employ analytic hierarchy process(AHP) to evaluate them by considering material han-dling cost, safety level, aesthetic appeal, noise level, andthe level of satisfaction for special requirements. AHP isable to provide weights for qualitative layout evaluationcriteria. However, many quantitative criteria are difficultto be distinguished with its nine-point scale. Yang et al.[8] use AHP to collect qualitative data, and the dataenvelopment analysis is then employed to take bothquantitative data (distance, adjacency, and shape ratio)and qualitative data (such as flexibility, accessibility,and maintenance) to solve plant layout design problem.Kuo et al. [9] take the same criteria into considerationand then apply gray relational analysis to solve themultiple attribute layout decision-making problems.However, energy relevant criteria are always ignored inthe decision-making process.

For the MADM problem that has different attributeswith different units and feature, it is always difficult todetermine the relative weights among the involved cri-teria. AHP is always used to solve this problem. Yangand Deuse (2012) use AHP to obtain the weights ofcriteria and then apply PROMETEE to choose the bestlayout alternative from layout candidates [10]. Önüt etal. (2008) develop a hybrid fuzzy MCDM approachbased on integrated fuzzy AHP and fuzzy techniquefor order preference by similarity to ideal solution(TOPSIS) to solve CNC machining center selectionproblem for a company in Istanbul [11]. However,AHP strongly depends on human judgment that maycause assessment bias, and the results of AHP are easilyinfluenced by different person and situation. Althoughfuzzy AHP can classify human judgment in fuzzy way,the most widely used fuzzy AHP method—extent anal-ysis method—cannot evaluate the true weights by usinga fuzzy comparison matrix and has led to quite anumber of misapplications in practice [12]. Therefore,more researchers focus on rough set theory with thepurpose of getting weights of criteria. Wen and Chen(2010) employ rough set theory and AHP to get thecriteria weights then used TOPSIS to make the riskassessment [13]. Sen et al. apply rough set theory toobtain the criteria weights and then developed an inter-active multiobjective decision-making method to solve adistribution center location problem [14].

After getting the relative weights, it is necessary to findan efficient method to get the final ranking among finitealternatives. Although AHP can help decision maker notonly to calculate the ratio of the decision maker’s inconsis-tency but also to make best decision considering with bothtangible and nontangible aspects of a decision problem,large amount of pairwise comparisons leads to impracticalusage of AHP, especially under fuzzy conditions [11]. TOP-SIS is effective in handling with the tangible attributes andlarge number of alternatives, but it requires an efficientapproach to provide the relative weights of different deci-sion criteria [15]. Unlike other decision-making methods,for example, PROMETHEE which needs much experienceto determine the preference functions and thresholds, as wellas AHP which is very difficult to make the pairwise com-parisons precisely, TOPSIS calculates the Euclidean dis-tance from chosen alternatives to the ideal solutions. Itsmathematical simplicity is preferred by many decisionmakers.

In addition, due to the fluctuation of economic and politicsituation and the change of seasons, the criteria weights andthe production rates inevitably fluctuate correspondingly.These fluctuations lead to the change of final rankingresults. Önüt et al. [11] and Azadeh et al. [16] use a sensi-tivity analysis method that exchanges the weights of criteria

796 Int J Adv Manuf Technol (2013) 66:795–807

with each other. However, only limited scenarios can bestudied by this method. Dağdeviren [17] employ softwareDecision Lab 2000 to analyze the influence of weightchange on the final ranking results. But this analysis is basedon a hypothesis: when the analysis is studied on a givenweight of a decision criterion, the ratios among otherweights are remained constant. Nevertheless, in industrialapplication, the criteria weights are fluctuated randomly.Besides, the sensitivity analysis for production rates fluctu-ation is always ignored in facility layout decision-makingprocess. As for the sensitivity analysis considering the bothfluctuations of criteria weights and production rates, thereare few researches in this field.

As mentioned above, a hybrid approach, which considersboth traditional and energy relevant criteria, is developed in thisarticle. It integrates rough set theory and AHP to get the criteriaweights, and then based on the results of rough set–AHP ap-proach, TOPSIS is applied to obtain the final layout alternativeranking. The sensitivity analysis of criteria weights is studied tofind the influence of criteria weight variation on rankingresults. Moreover, the sensitivity analysis of production ratesis performed with considering the possibility of differentproduction rate conditions. Finally, the results obtained bydifferent decision-making approaches are compared.

The remaining parts of this paper are organized as follows:Section 3 describes the proposed approach briefly. InSection 4, a case study is applied to validate this approach.Finally, the main conclusions and future researches are sum-marized in Section 5.

3 The proposed methodology

In this section, a hybrid approach based on rough set–AHPand TOPSIS is proposed for the facility layout decision-making problem. In addition, the sensitivity analysis ofcriteria weights and production rates is studied. Figure 1shows its flowchart. This approach mainly includes thefollowing steps:

Step 1: Data collection. Data collection should considerthe features of products, transport routing, quanti-ties, restrictions, and so on. Those collected dataare used to validate the input data at the designstage.

Step 2: Alternatives generation. In this step, the layoutalternatives are obtained from a simulation model.In addition, the quantitative performances of thegenerated layout alternatives are also obtained.

Step 3: Rough set—AHP calculation. AHP developed bySaaty (1980) provides a method to decompose thecomplex problem into a hierarchy of subproblemswhich can be evaluated and handled more easilyand rationally [18]. Moreover, AHP makes it pos-sible to quantify the experiences of experts and tointegrate those quantified experiences to thedecision-making process. Especially, when thestructure of the object is complex and the data ismissing, such quantified experiences are extremelyvaluable for the decision makers.

Collecting data

Generating layoutalternatives

Ranking layoutalternatives via TOPSIS

Getting relative importancevia rough set theory

Findingcriteria weights via AHP

Creating theevaluation matrix

Sensitivity analysisof weights

Sensitivity analysisof production rates

Are objectivesachieved?

Decision making:select the best layout

N

Y

Step 1:Data collection

Step 2:Alternativesgeneration

Step 3:Rough set-AHP

Step 4:TOPSIS

calculation

Step 5:Results analysis

Step 6:Decision making

Comparing the resultsby different methods

Fig. 1 The flow chart ofproposed approach

Int J Adv Manuf Technol (2013) 66:795–807 797

However, AHP is strongly connected to humanjudgment that is easily influenced by differentperson and different situation. It will cause eval-uator’s assessment bias that makes the judgmentmatrix inconsistent. Therefore, in this paper, roughset theory is integrated with AHP to solve theevaluation bias problem in AHP. Rough settheory is used to get the relative significancesamong decision criteria based on data gottenfrom previous project, and then based on therelative significances, AHP is applied to get thecriteria weights.

Rough set theory was firstly developed byPawlak (1982) [19]. It is a well-known mathe-matical tool to express imprecise, incomplete,and uncertain information or knowledge that isobtained from human experience. Besides, it isa suitable method to deal with qualitative in-formation which is difficult to be analyzed byusing standard statistical methods [20]. It canalso be used to find the hidden patterns indata, make the data and attribute reduction,evaluate the significance of data, and find theminimal decision rules based on obtained data.In addition, its straightforward interpretation ofresults and no requirement for priori knowledgebrings a wide application prospect.

Some important definitions of rough set the-ory are described as follows [21, 22]:

Definition 1 Assume that S 0 (U,A,V,f) is an informa-tion system, where U is nonempty finitesets of object; A 0 C∪D is an attributesset where subset C denotes the conditionattribute set and subset D indicates thedecision attribute set and they satisfyC∩D 0 Ф; V 0 ∪a∈A, Va, Va is the rangeof a; and f: U×A → V is an informationfunction and it gives values to each ob-ject for each attribute, 8x 2 U , and a∈A,f(x,a)∈Va.

Definition 2 For a nonempty subset B of attributes A(B � A), a indiscernibility relation IND(B) is defined and can be denoted as

INDðBÞ¼ ðx:yÞjðx:yÞ 2 U � U ; 8b 2 BðbðxÞ ¼ bðyÞÞf g:

ð1Þ

It is clear that the indiscernibility rela-tion expresses an equivalence relation.The family of all the equivalence classesof the relation IND(B) can be expressed

as U jINDðBÞ ¼ X1;X2; . . . ;Xnð Þ , whereXi indicates different equivalence classes.

Definition 3 Entropy H(P) of knowledge P (attributesset) can be calculated as

HðpÞ ¼ �Xni¼1

pðXiÞ log pðXiÞ: ð2Þwhere p Xið Þ ¼ Xij j Uj j= is the probabilityof Xi when P is in the partition X 0 {X1,X2,…, Xn} of U.

Definition 4 Conditional entropy H QjPð Þ thatexpresses how knowledge Q�U jINDðQÞð Þ ¼ Y1; Y2; . . . ; Ymf g is rel-ative to knowledge P U jINDðPÞð Þ ¼X1;X2; . . . ;Xnf g can be described as

HðQjPÞ¼ �

Xni¼1

pðXiÞXmj¼1

pðYjjXiÞ log pðYjjXiÞ

ð3Þwhere p YjjXi

� �; i ¼ 1; 2; . . . ; n;ð j ¼

1; 2; . . . ; mÞ denotes conditionalprobability.

Definition 5 The attribute significance SGA(a, E, D)of attribute a can be calculated as

SGAða;E;DÞ ¼ HðDjEÞ � HðDjE [ aÞð4Þ

where E is a subset of A.The higher value of SGF(a,E,D)

means more important is attribute a fordecision attribute set D.

Based on the significance values of allattribute in condition attribute set C, thepairwise comparison judgment matrix ofdecision criteria can be made. Then, withthe help of AHP, the weights of eachdecision criterion can be obtained. Moredetail of the AHP can be found in litera-ture [10, 17, 18].

Step 4: TOPSIS calculation. After assigning the weights ofdecision criteria from the rough set–AHP, the lay-out alternatives are evaluated and ranked by theTOPSIS method.

TOPSIS calculates the distance from both thepositive and the negative ideal solutions to eachalternative to evaluate the alternatives. The alter-native which has the shortest distance to the posi-tive ideal solution and the farthest distance to thenegative ideal solution is considered as the bestalternative.


This method mainly consists of the followingsteps [12, 23, 24]:

1. Establishing the evaluation matrix D based on thevalues of each layout alternative obtained by thesimulation model as follows:

D ¼

x11 x12 ::: x1j ::: x1nx21 x22 ::: x2j ::: x2n

::: :::xi1 xi2 ::: xij ::: xin

::: ::: . ..

xm1 xm2 ::: xmj ::: xmn

0BBBBBBB@

1CCCCCCCA

ð5Þ

where xij is the performance of jth criterion for ithlayout alternative, with i01, 2,…, m and j01, 2,…,n.

2. Normalizing the evaluation matrix D:

R ¼ ½rij� i ¼ 1; 2; :::;m; j ¼ 1; 2; :::; n ð6Þ

rij ¼ xijffiffiffiffiffiffiffiffiffiffiffiPni¼1

xij

s i ¼ 1; 2; :::;m; j ¼ 1; 2; :::; n ð7Þ

3. Constituting the weighted and normalized decisionmatrix V by using the weights gotten from roughset–AHP:

V ¼

w1r11 w2r12 ::: wjr1j ::: wnr1nw1r21 w2r22 ::: wjr2j ::: wnr2n

::: :::w1rj1 w2rj2 ::: wjrjj ::: wnrjn

::: ::: . ..

w1rm1 w2rm2 ::: wjrmj ::: wnrmn

0BBBBBBB@

1CCCCCCCAð8Þ

where wj stands for the weight of the jth criterion.4. Choosing both the positive ideal solution (V+) and

negative ideal solution (V−) with the following rep-resentations:

Vþ ¼ fðmaxi

vijjj 2 JÞ; ðmini

vijjj 2 J 0 Þji

¼ 1; 2; :::;m; j ¼ 1; 2; :::; ng ð9Þ

Vþ ¼ fðmaxi

vijjj 2 JÞ; ðmini

vijjj 2 J 0 Þji

¼ 1; 2; :::;m; j ¼ 1; 2; :::; ng ð10Þ

where J represents the benefit criteria and J′ indicates thecost criteria.

5. Calculating the distances from each alternativeto the positive ideal and negative ideal solutions.The distance from positive ideal solution isexpressed as

diþ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXnj¼1

ðvij � VjþÞ

vuut i ¼ 1; 2; :::;m; j ¼ 1; 2; :::; n:

ð11Þ

The distance from negative ideal solution isexpressed as

di� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXnj¼1

ðvij � Vj�Þ

vuut i ¼ 1; 2; :::;m; j ¼ 1; 2; :::; n;

ð12Þ

6. Measuring the relative closeness to the ideal solu-tion (Ci

*) defined as follows:

Ci* ¼ di

�

diþ þ di

� i ¼ 1; 2; :::; n; 0 � Ci* � 1:

ð13Þ

7. Ranking the layout alternative based on values ofCi

*. The alternative which has higher closenessmeans the better ranking.

Step 5: Analysis of results. In this step, sensitivity analysesfor both criteria weights and production rates arestudied. Besides, comparisons among differentdecision-making approaches for the same problemare also performed.

1. Sensitivity analysis for criteria weights. Theweights for criteria are significant for the finalranking, and they are usually obtained based onthe subjective experiences which can be easilyinfluenced by different situation and persons.Therefore, it is valuable to analyze the influenceof varying criteria weights on the final ranking.With the help of this sensitivity analysis, themost sensitive weight can be found. As a result,the final layout alternative ranking can be mademore rationally.

Assume that the optimal layout alternative iss*, and T*

þ and w*are the corresponding relativecloseness to the ideal solution and weight vec-tor, respectively. S is the set of all layout alter-natives. Suppose that ri (i01, 2,…, m) is theminimum variation of weights which makesT*þ < Ti

þ , which means layout alternative si is


better than optimal layout alternative s*. Then,ri can be calculated as

ri ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXnj¼1

ðkj � kj*Þ2

vuut ð14Þ

where kj indicates the weight of the jth criterion,k*j expresses the weight of the jth criterion for

the optimal layout alternative s*, and n is thenumber of criteria.

If the layout alternative si has the minimum riin m layout alternatives, it means that the layouts* is most sensitive to the layout alternative si.Usually, the value of ri and the correspondingweight vector can be calculated by the followingnonlinear programming with n variables [25]:

min riðw;w*Þ ð15Þ

s:t:Tþi > T*þ; ð16Þ

Xnj¼1

kj ¼ 1; kj � 0 ð17Þ

where w is the corresponding weight vector for ri, Tiþ

denotes the corresponding relative closeness obtained fromTOPSIS, kj indicates the weight of the jth criterion in theweight vector w.

Based on the variation of each weight, we can know thatunder the restriction of having smallest weight variation,varying the criterion with greatest variation is most efficientto make layout alternative si better than s*. Therefore, thiscriterion should be treated carefully.

2. Sensitivity analysis for production rates. Besides,the sensitivity analysis for fluctuation of productionrates is studied. In practice, due to the change ofeconomic situation or company’s strategy, the pro-duction rates always vary. Consequently, both theperformance of layouts and the ranking of layoutalternatives will change. Therefore, analyzing theinfluence of production rates variation on the layoutperformance can help layout designer to make theirdecision more rationally.

Suppose that the production rate is ŋ and then theperformance of ith layout alternative at productionrate ŋ is

PiðηÞ ¼ T*i;η ð18Þ

where T*i;η denotes the relative closeness to the ideal

solution of the ith layout alternative at production rate ŋ.

Based on the value of Pi(ŋ), a ranking results matrixP can be established:

P ¼

p11 p12 ::: p1j ::: p1tp21 p22 ::: p2j ::: p2t

::: :::pi1 pi2 ::: pij ::: pit

::: ::: . ..

pm1 pm2 ::: pmj ::: pmt

0BBBBBBB@

1CCCCCCCA

ð19Þ

where pij indicates the relative closeness to theideal solution of the ith layout alternative at jthproduction rate, t denotes the number of differentproduction rates, and m is the number of layoutalternatives.

Then, the ranking results matrix P is normalized:

G ¼ ½gij� i ¼ 1; 2; :::;m; j ¼ 1; 2; :::; t ð20Þ

gij ¼ pijffiffiffiffiffiffiffiffiffiffiffiPni¼1

pij

s i ¼ 1; 2; . . . ;m; j ¼ 1; 2; . . . ; t ð21Þ

Afterwards, assume that the possibility of under jth pro-duction rate condition is λj. Therefore, the normalized ma-trix considering the possibility under each production ratecondition can be expressed as

FR ¼

l1g11 l2g12 ::: ljg1j ::: ltg1tl1g21 l2g22 ::: ljg2j ::: ltg2t

::: :::ligi1 l2gi2 ::: ligij ::: ltgit

::: ::: . ..

l1gm1 l2gm1 ::: ljgmj ::: ltgmt

0BBBBBBB@

1CCCCCCCA

ð22Þ

Finally, the synthetic ranking value of the ith layoutalternative for all production rates FRi is obtained:

FRi ¼Xt

i¼1

hij i ¼ 1; 2; :::;m; j ¼ 1; 2; :::; t ð23Þ

Based on the value of FRi, the layout alternative rankingresults with considering the possibility of all productionrates are obtained. The layout alternative with high valuemeans the better ranking.

3. Comparing the results by different methods. In thisstep, the same problem is analyzed by differentdecision-making methods. The comparison of


ranking results by using different decision-makingmethods can help the layout designer to validate thecorrectness of the proposed model.

Step 6: Decision making. In this step, all objectives arechecked by decision makers, and if all objectivesare satisfied, the final decision is made and thebest layout is chosen based on the above obtainedresults. Otherwise, the procedure returns to step 2to generate new layout alternatives.

4 Case study

In this section, an expanding case study based on [26]is used to validate the proposed hybrid approach ofrough set–AHP and TOPSIS. In a paint department,there are six ovens which have a great amount ofenergy consumption. In addition, workplaces for fillerapplication, basecoat application, and clearcoat applica-tion are included.

Nowadays, due to the energy policy and the trend ofenergy price rise, energy consumption reduction isregarded as a key objective in the process of facilitylayout design because of its long-term effect on expensereduction and clime protection. On the other hand,several traditional layout criteria such as transport perfor-mance, investment, and space requirement will be taken intoaccount.

4.1 Data collection

The paint department in this studied case has two pro-duction lines. The original layout design is displayed inFig. 2. The production route of the production line 1 is1→4→2→5→3→6, and the production route of line 2is 7→10→8→11→9→12, regardless of the product type.

4.2 Alternatives generation

New layout alternatives are generated by a Pareto-based multi-objective optimization approach that simultaneously considerstransport performance, energy loss, and space requirement [27].All new generated layouts apply energy recovery network toreduce energy consumption, and ten optimized layouts arechosen from the Pareto optimal solutions because of theirbalancing performances on all involved criteria. In addition,the original layout design is also accepted as a layout alternative.

4.3 Rough set–AHP

The hybrid approach rough set–AHP is applied to find theweights of criteria determined by decision makers. Firstly, theproblem is analyzed, and the decision criteria determined bydecision makers are as follows: space requirement (SR), trans-port performance (TP), investment for energy recovery net-work (Inv.), distance request (DR), and energy saving (ES).The decision hierarchy structure is shown in Fig. 3.

The space requirement is equal to the needed minimumrectangle area, and it can be evaluated as

SR ¼ ðXbr � XtlÞðYtl � YbrÞ ð24Þwhere Xbr is the maximum x-coordinate value of bottomright corner in all facilities, Xtl denotes the minimum x-coordinate value of top left corner in all facilities, Ytl indi-cates the maximum y-coordinate value of top left corner inall facilities, and Ybr stands for the minimum y-coordinatevalue of bottom right corner in all facilities.

The transport performance is the sum of the material flowmultiplying the corresponding rectangle distance betweenthe output point of previous facility and input point of thenext facility which can be formulated as

TP ¼Xni¼1

Xnj¼1

fijdij ð25Þ

where fij is the material transport flow from ith facilityto jth facility, dij denotes the rectilinear distance be-tween the output point of ith facility and the input pointof jth facility.

The distance request is measured by multiplying distancerating and distance between facilities. This requirement isrelated to the satisfaction of environmental issues like noise,vibration, pollution, or risks of fire or explosion. It can bedescribed as

DR ¼Xi

Xj

sijdij ð26Þ

where sij indicates the distance rating of facility i and jand dij denotes the distance between the centers offacility i and j.

Oven 2 (5)

Oven 3 (6)Oven 1 (4)

Filler ApplicationBasecoat Application

Clearcoat Application

Filler ApplicationBasecoat Application

Clearcoat Application

Oven 4 (10) Oven 6 (12)

Oven 5 (11)Compressedair machine

Door Door

1 32

7 8 9

13

Fig. 2 The original layout design


Energy saving is used to measure the amount of energyconsumption reduction of optimized layout alternativescompared to the original layout design. It can be calculatedby

ES ¼ ER� EL ð27Þwhere ER indicates the amount of recovered energywith the help of energy recovery network and ELdenotes the amount of energy loss because of heat lossand pressure drop in pipes, which is brought by energy recov-ery network.

Although energy recovery network brings long-termbenefit of reducing energy consumption, it brings alsoextra investment. Therefore, the investment for energyrecovery network is accepted as a decision criterion tocalculate the sum costs for heat exchange network including

heat transfers, corresponding pipes, labor costs, and soon.

Inv: ¼ Cht þ aL ð28Þ

where Cht denotes the costs for heat transfer devices. Inaddition, the material costs of pipes and labor costs for instal-lation depend on the length of pipes, and L indicates the lengthof pipes and α is the cost coefficient.

After forming the problem hierarchy, a decision table isbuilt in Table 1. Rows show the distinct alternatives, andcolumns indicate value of different alternatives on differentdecision attributes. A three-grade value is used to evaluate theperformance of criteria. In Table 1, 19 different combinationsof criteria rates are listed before evaluation process accordingto the expert advice according to previous layout design, inwhich “1” means good, “2” corresponds to “medium,” and

Selecting best Layout

Space requirement

(SR)

Investment(Inv.)

TransportPerformance

(TP)

Energy Saving (ES)

Original layout Layout 2Layout 1

Distance Request

(DR)

Goal layer

Criteria layer

Alternative layer

Layout 3 Layout 10...

Fig. 3 Hierarchy structure

Table 1 Decision tableU SR (a) Inv. (b) TP (c) DR (d) ES (e) Decision (D)

1 2 1 3 1 2 0

2 1 1 2 2 1 1

3 1 1 2 2 3 0

4 2 1 1 1 2 1

5 1 1 1 3 1 1

6 2 1 2 1 2 0

7 1 2 2 2 2 0

8 1 2 3 2 1 1

9 2 2 2 2 1 1

10 1 1 3 2 1 1

11 1 2 2 2 1 1

12 1 3 3 2 1 0

13 2 1 2 2 2 0

14 2 2 3 2 1 0

15 3 1 1 1 2 0

16 1 1 3 2 2 0

17 2 3 2 2 1 0

18 2 1 1 2 2 1

19 1 1 3 1 2 1


“3” indicates “poor” by trichotomy. Besides, in thedecision column, 1 means it is valuable to build the layoutand “0” means not to establish the layout.

From the decision table, the relative significances of thementioned five criteria can be obtained by the followingprocess:

U IND a; b; c; d; eð Þj ¼ 1f g; 2f g; 3f g; 4f g; 5f g; 6f g; 7f g; 8f g; 9f g; 10f g; 11f g; 12f g; 13f g; 14f g; 15f g; 16f g; 17f g; 18f g; 19f gf g;

U INDðDÞj ¼ 2; 4; 5; 8; 9; 10; 11; 18; 19f g; 1; 3; 6; 7; 12; 13; 14; 15; 16; 17f gf g ¼ D1;D2f g;

U IND b; c; d; eð Þj ¼ 1; 19f g; 4; 15f g; 8; 14f g; 9; 11f g; 2f g; 3f g; 5f g; 6f g; 7f g; 10f g; 12f g; 13f g; 16f g; 17f g; 18f gf g¼ X1; X2; X3; X4; X5; X6; X7; X8; X9; X10; X11; X12; X13; X14; X15f g;

P X1ð Þ ¼ 2 19= ; P X2ð Þ ¼ 2=19; P X3ð Þ ¼ 2 19= ; P X4ð Þ¼ 2 19= ; P X5ð Þ ¼ P X6ð Þ ¼ P X7ð Þ ¼ P X8ð Þ ¼ P X9ð Þ¼ P X10ð Þ ¼ P X11ð Þ ¼ P X12ð Þ ¼ P X13ð Þ ¼ P X14ð Þ¼ P X15ð Þ ¼ 1 19= ;

P D1 X1jð Þ ¼ 1 2= ; P D2 X1jð Þ ¼ 1 2= ;

P D1 X2jð Þ ¼ 1 2= ; P D2 X2jð Þ ¼ 1=2;

P D1 X3jð Þ ¼ 1=2; P D2 X3jð Þ ¼ 1 2= ;

P D1 X4jð Þ ¼ 1; P D2 X4jð Þ ¼ 0;

P D1 X5jð Þ ¼ P D2 X5jð Þ ¼ 0;





P D1 X10jð Þ ¼ P D2 X10jð Þ ¼ 0;

P D1 X11jð Þ ¼ P D2 X11jð Þ ¼ 0;

P D1 X12jð Þ ¼ P D2 X12jð Þ ¼ 0;

P D1 X13jð Þ ¼ P D13 X5jð Þ ¼ 0;

P D1 X14jð Þ ¼ P D2 X14jð Þ ¼ 0;

P D1 X15jð Þ ¼ P D2 X15jð Þ ¼ 0:

SGF a; b; c; d; ef g; fDgð Þ ¼ H Df g fb; c; d; egjð Þ� H Df g fa; b; c; d; egjð Þ

¼ �2 19= * 1 2= * log 1 2=ð Þð Þ þ 1 2= * log 1 2=ð Þ*3� 2 19= * 1* logð1Þð Þ

¼ 0:0951

Therefore, the relative significance of attribute a is0.0951. Similarly, the significance of attribute b, c, d, ande can be calculated and they are 0.0757, 0.1267, 0.0317,0.0951, respectively.

Then, AHP is used to get the weights of decision criteria.The preference of alternative i–j can be calculated as Wi/Wj

where Wi and Wj denote the significant value of attribute iand j. Therefore, the judgment matrix can be described asthe following according to the significant values obtained byrough set theory:

J ¼

1 1:256 0:751 3 10:796 1 0:597 2:388 0:7961:332 1:674 1 4 1:3320:333 0:419 0:25 1 0:3331 1:256 0:75 3 1

266664

377775 ð29Þ


Finally, the following criteria weights are obtained:0.224, 0.178, 0.299, 0.075, and 0.224, respectively. In ad-dition, the consistency index which is calculated as 0 whichmeans pairwise comparison matrix established according torough set theory is complete consistency.

4.4 TOPSIS calculation

After finding the weights of criteria by using rough set–AHP, TOPSIS method is employed to rank the layout alter-natives. In this step, each layout alternative is evaluated withrespect to all involved decision criteria by the simulationmodel. The evaluation matrix is established as shown inTable 2.

By using Eq. 7, the performance values of all layoutalternatives are normalized. The weighted normalized deci-sion matrix is given in Table 3. It is calculated by multiply-ing the normalized matrix and the criteria weights obtainedby rough set–AHP. Afterwards, the positive ideal and

negative ideal solutions, which respectively mean the bestvalues and the worst values of layout alternatives, areobtained as shown in Table 4.

The distances from each layout alternative to the positiveideal and negative ideal solutions are calculated by Eqs. 11and 12. Then, Eq. 13 is applied to get the relative closenessfrom each layout alternative to the ideal solution. Finally, allthe layout alternatives are ranked. The results are summa-rized in Table 5.

4.5 Analysis of results

Based on the results in Table 5, the ranking of the layoutalternatives is obtained. Layout 2 with minimum transportperformance and minimum space requirement has the high-est value of Ti

*. Therefore, it is regarded as the best layout.Although layout 10 and layout 8 do not have the bestperformance in any decision criterion, their balancing per-formances are preferred by decision makers. Hence, theytake the second and third place, respectively. Besides, al-though the original design does not need extra investmentand layout 3 requires least energy as well as space, theirpoor performances in other criteria make them as the worsttwo layout alternatives.

In the case discussed above, only layout 2, layout 4,layout 8, and layout 10 are studied in the sensitivity analysisof weights due to their good performances. k1, k2, k3, k4, andk5 denote the decision criterion SR, Inv., TP, DR, and ES,

Table 3 Weighted normalized decision matrix

SR Inv. TP DR ES

Original design 0.077 0 0.134 0.022 0

Layout 1 0.063 0.058 0.085 0.024 0.070

Layout 2 0.061 0.041 0.058 0.022 0.070

Layout 3 0.061 0.075 0.134 0.020 0.074

Layout 4 0.076 0.052 0.069 0.024 0.070

Layout 5 0.067 0.054 0.080 0.023 0.071

Layout 6 0.067 0.055 0.077 0.024 0.071

Layout 7 0.068 0.061 0.092 0.023 0.071

Layout 8 0.071 0.051 0.066 0.023 0.071

Layout 9 0.064 0.063 0.097 0.022 0.071

Layout 10 0.067 0.046 0.062 0.022 0.070

Table 4 Positive ideal solutions and negative ideal solutions

SR Inv. TP DR ES

V+ 0.061 0 0.058 0.024 0.074

V- 0.077 0.075 0.134 0.020 0

Table 2 The evaluation matrixfor TOPSIS calculation Criteria SR Inv. TP DR ES

Unit m2 103 € 106 kg m/year m 105 kwh/year

Max/min Min Min Min Max Max

Weights 0.224 0.178 0.299 0.075 0.224

Original design 1,242 0 406 3,612 0

Layout 1 1,012 210 259 3,872 8.412

Layout 2 972 150 176 3,530 8.385

Layout 3 972 270 406 3,322 8.887

Layout 4 1,217 190 208 3,892 8.471

Layout 5 1,069 195 241 3,777 8.521

Layout 6 1,069 200 234 3,836 8.554

Layout 7 1,090 220 278 3,666 8.578

Layout 8 1,139 185 199 3,738 8.496

Layout 9 1,026 230 293 3,513 8.576

Layout 10 1,071 165 189 3,630 8.453


respectively. Layout 2, layout 4, layout 8, and layout 10were indicated as s2, s4, s8, and s10, respectively. Because s2is the optimal solution obtained by the proposed approach, itis considered as s*, and its corresponding weight vector(0.224, 0.178, 0.299, 0.075, 0.224) is denoted as w*. Duringvariation of weights, if the minimum weight variation riwhich make si where i0{4, 8, 10} better than s2 is found,the corresponding weight vector is expressed as wi,2.

By using nonlinear programming developed with Matlab,the minimum weight variation r400.362 that makes s4 betterthan s2 is found out. The corresponding weight vector w4,2 is(0.089, 0.014, 0.223, 0.330, 0.347). Similarly, r8 is 0.317and w8,2 is (0.117, 0.011, 0.246, 0.289, 0.336); r10 is 0.274and w10,2 is (0.142, 0.014, 0.272, 0.252, 0.320). The weightvariation graph is shown in Fig. 4.

From the above analysis, it is clear that s2 is moresensitive to s10 since s10 can get better ranking than s2 withthe smallest weight variation. In addition, from Fig. 4, it canbe seen that k2 and k4 have larger influence than othercriteria on the final ranking for s10 because varying themis more efficient to make s10 better than s2 under the restric-tion of having smallest weight variation. Therefore, duringthe decision-making process, the weight setting of w2 andw4 should be made more carefully.

In the sensitivity analysis of production rate, it is assumedthat the production rate fluctuates ±0.05, ±0.1, ±0.15, ±0.2 ,and ±0.25 because of the change of seasons, economicsituation, etc. As the result, values of the transport perfor-mance and energy saving of each layout alternatives arechanged correspondingly. Here, the weight vectors obtainedfrom rough set–AHP are used, and the possibility for eachproduction rate is assumed to be the same. The rankings ofeach layout alternative under each production rate conditionare shown in Fig. 5, and the synthetic ranking values of eachlayout alternative for all production rates are shown inTable 7.

From Fig. 5, it is clear that, during the variation ofproduction rate from 0.75 to 1.25, layout 2 is always thebest solution; meanwhile, layout 10, layout 8, and layout 4always take the second, third, and fourth place, respectively.Layout 5 and layout 6 change their rankings at productionrate 0.85. In addition, the ranking of layout 1 becomesworse from production rate 1.1 to 1.2, and its ranking goesfrom seventh place to ninth place. Layout 9 and layout 7 getbetter rankings at production rates 1.15 and 1.2, respective-ly. Due to the poor performances of layout 3 and the originallayout, they are always the worst two alternatives during thevariation of production rate. From Table 6, we can knowthat layout 2 is the best layout considering all productionrate conditions.

Finally, in the scope of the application, the same decision-making problem has also been studied by AHP, PROME-THEE. The results are given in Table 7.

From Table 7, it can be seen that due to the bestperformance of layout 2, it is always the best layoutalternative in all alternatives. Layout 10 and layout8 are apparently the second and third choice in all threeranking methods, respectively. Because of the poor per-formances of layout 3 and the original design, they arethe worst two layout alternatives during the studies by all threemethods. Other alternatives vary their rankings by using dif-ferent methods.

As above mentioned, layout 2 is finally selected as thebest layout because of its good performances in all criteriaas well as its stabilities during varying criteria weights andchanging production rate.

5 Conclusions

The layout design problem with the essential impact on theperformance of manufacturing system is a strategic issuethat should be treated carefully. Usually, layout designersmake several layout alternatives by using different methodsor from different aspects. However, due to the multipleattribute nature of layout design problem, it is always diffi-cult to make an optimal decision.Fig. 4 Weights variation graph

Table 5 Overall scores and ranking of layout alternatives

dþi d�i C*i Ranking

Original design 0.107 0.075 0.410 11

Layout 1 0.064 0.088 0.578 7

Layout 2 0.042 0.109 0.724 1

Layout 3 0.106 0.076 0.416 10

Layout 4 0.056 0.099 0.639 4

Layout 5 0.058 0.092 0.613 6

Layout 6 0.059 0.094 0.614 5

Layout 7 0.070 0.085 0.547 8

Layout 8 0.053 0.101 0.657 3

Layout 9 0.074 0.082 0.525 9

Layout 10 0.046 0.105 0.694 2


In this paper, a decision-making approach for facilitylayout design is proposed. Due to the energy shortage,energy relevant criterion is introduced as an important factorand integrated with other traditional layout criteria in theprocess of layout decision making. The aim of this study isto find the most effective layout solution for the paintdepartment. The proposed approach is useful for managersto improve company’s strategic planning or to be used in thefurther applications of layout planning.

By using rough set–AHP, rational weights for decisioncriteria are easily made. Based on this, the TOPSIS is used

to obtain the final results that represent not only the rankingof different alternatives but also the degree of superiorityamong the studied layout alternatives. Furthermore, sensi-tivity analysis of varying criteria weights and fluctuatingproduction rate is studied. The most sensitive layout alter-native, the greatest influencing criterion, and the changes offinal ranking during the variation of production rate arefound. These results are valuable for other applications aswell as the further strategic planning. Finally, the results’comparison is implemented by using three differentdecision-making methods for the same problem, and theresults show that the best layout and the worst layout

Table 6 Ranking results of layout alternatives for all production rates

Layout alternative Ranking value for allproduction rates FRi

Ranking

Original design 0.414 10

Layout 1 0.577 7

Layout 2 0.722 1

Layout 3 0.413 11

Layout 4 0.635 4

Layout 5 0.610 6

Layout 6 0.612 5

Layout 7 0.543 9

Layout 8 0.654 3

Layout 9 0.550 8

Layout 10 0.691 2

Table 7 Results obtained from different ranking methods

Ranking AHP PROMETHEE Current approach

1 Layout 2 Layout 2 Layout 2










11 Original design Original design Original design

OriginalLayout 1Layout 2Layout 3Layout 4

Layout 5Layout 6Layout 7Layout 8Layout 9Layout 10

11

10

9

8

7

6

5

4

3

2

1

0.75 0.8 0.85 0.9 0.95 1.0 1.05 1.1 1.15 1.2 1.25

Ran

king

Production rate

Fig. 5 The ranking changes by fluctuating production rate


obtained by using different methods are same. Therefore,the final decision is made, and the best layout is chosen. It iswell known that, due to the nature of MADM, the optimalsolution may not exist. However, the effective systematicdecision-making approach can help designers to reduce therisk of a poor layout design.

In our future research, some qualitative criteria (e.g.,quality and flexibility), which cannot be measured precisely,will be integrated in the decision-making process.

References

1. Yang L, Deuse J, Droste M (2011) Energy efficiency at energyintensive factory—a facility planning approach. In: IEEE the 18thinternational conference on industrial engineering and engineeringmanagement, pp: 699-703

2. Ertay T, Ruan D, Tuzkaya UR (2006) Integrating data envelop-ment analysis and analytic hierarchy for the facility layout designin manufacturing systems. Inform Sci 176:237–262

3. Yang T, Hang CC (2007) Multiple-attribute decision making meth-ods for plant layout design problem. Robot Comput Integr Manuf23:126–137

4. Ye MJ, Zhou GG (2005) The application of genetic algorithm inthe bi-criteria layout problem with aisles. Syst Eng Theory Pract10:101–107, In Chinese

5. Aiello G, Enea M, Galante G (2006) A multi-objective approach tofacility layout problem by genetic search algorithm and Electremethod. Robot Comput Integr Manuf 22:447–455

6. Ye MJ, Zhou GG (2007) A local genetic approach to multi-objective, facility layout problems with fixed aisles. Int J ProdRes 45:5243–5264

7. Cambron KE, Evans GW (1991) Layout design using the analytichierarchy process. Comput Ind Eng 20:211–229

8. Yang T, Kuo CW (2003) A hierarchical AHP/DEA methodologyfor the facilities layout design problem. Eur J Oper Res 147:128–136

9. Kuo Y, Yang T, Huang GW (2008) The use of grey relationalanalysis in solving multiple attribute decision-making problems.Comput Ind Eng 55:80–93

10. Yang, L., Deuse, J. (2012) Multiple-attribute decision making foran energy efficient facility layout design. In: 45th CIRP conferenceon manufacturing systems, 16-18 May 2012, Athens

11. Önüt S, Kara SS, Efendigil T (2008) A hybrid fuzzy MCDMapproach to machine tool selection. J Intell Manuf 19:443–453

12. Wang YM, Luo Y, Hua ZS (2008) On the extent analysis methodfor fuzzy AHP and its application. Eur J Oper Res 186(2008):735–747

13. Wen, G. F., Chen, L. W (2010) Construction project bidding riskassessment model based on rough set-TOPSIS. In: 2010 WASEinternational conference on information engineering: 296-300

14. Sen L, Felix TS, Chung SH (2011) A study of distribution centerlocation based on the rough sets and interactive multi-objectivefuzzy decision theory. Robot Comput Integr Manuf 27:426–433

15. Rao RV, Davim JP (2008) A decision making framework modelfor material selection using a combined multiple attribute decision-making method. Int J Adv Manuf Technol 35:751–760

16. Azadeh A, Nazari-Shirkouhi S (2011) A unique fuzzy multi-criteria decision making: computer simulation approach for produc-tive operators’ assignment in cellular manufacturing systems withuncertainty and vagueness. Int J Adv Manuf Technol 56:329–343

17. Dağdeviren M (2008) Decision making in equipment selection: anintegrated approach with AHP and PROMETHEE. J Intell Manuf19:397–406

18. Saaty TL (1980) The analytic hierarchy process. McGraw-Hill,New York

19. Pawlak Z (1982) Rough sets. Int J Inf Comput Sci 11:345–35620. Zou Z, Tseng TL, Sohn H, Song G, Gutierrez R (2011) A rough set

based approach to distributor selection in supply chain management.Expert Syst Appl 38:106–115

21. Aydogan EK (2011) Performance measurement model for Turkishaviation firms using the rough-AHP and TOPSIS methods underfuzzy environment. Expert Syst Appl 38:3992–3998

22. Liu S, Chan FTS, Chung SH (2011) A study of distribution centerlocation based on the rough sets and interactive multi-objectivefuzzy decision theory. Robot Comput Integr Manuf 27:426–433

23. Wang Y, Zhen H (2008) A TOPSIS based robust optimizationmethodology for multivariable quality characteristics. In IEEEinternational conference on service operations and logistics andinformatics: 2558-2561

24. Chang CW (2010) Collaborative decision making algorithm forselection of optimal wire saw in photovoltaic wafer manufacture. JIntell Manuf. doi:10.1007/s10845-010-0391-6

25. Wang CY, Shi JC, Niu XS, Jia SJ, Liu SJ, Mu L (2011) Sensitivityanalysis on the weights in power system restoration decision-making. In: 2011 4th international conference on electric utilityderegulation and restructuring and power technologies, pp 653-656

26. Geldermann J, Treitz M, Schollenberger H, Ludwig J, Rentz O(2007) Integrated process design for the inter-company plant lay-out planning of dynamic mass flow networks. UniversitätsverlagKarlsruhe, Kalrsruhe

27. Yang L, Deuse J, Jiang P (2012) Multi-objective optimization offacility planning for energy intensive companies. J Intell Manuf.doi:10.1007/s10845-012-0637-6


http://dx.doi.org/10.1007/s10845-010-0391-6

http://dx.doi.org/10.1007/s10845-012-0637-6

Documents

Multiple-attribute decision-making approach for an energy-efficient facility layout design