J Intell Manuf (2008) 19:397406DOI 10.1007/s10845-008-0091-7

Decision making in equipment selection: an integrated approachwith AHP and PROMETHEE

Metin Dagdeviren

Received: 3 September 2007 / Accepted: 7 January 2008 / Published online: 31 January 2008 Springer Science+Business Media, LLC 2008

Abstract Multi-attribute equipment selection is a veryimportant issue for an effective manufacturing system, sincethe improper equipment selection might cause manyproblems affecting productivity, precision, flexibility andquality of the products negatively. On the other hand, select-ing the best equipment among many alternatives is a multi-criteria decision making (MCDM) problem. In this study,an integrated approach which employs analytic hierarchyprocess (AHP) and preference ranking organization methodfor enrichment evaluations (PROMETHEE) together, is pro-posed for the equipment selection problem. The AHP is usedto analyze the structure of the equipment selection problemand to determine weights of the criteria, and PROMETHEEmethod is used to obtain final ranking, and to make a sensitiv-ity analysis by changing the weights. Proposed approach isapplied to a problem of selecting milling machines to be pur-chased in an international company. Company managementfound the application and results satisfactory and implement-able in their equipment selection decisions.

Keywords Equipment selection Multi criteriadecision-making AHP PROMETHEE GAIA Sensivity analysis

Introduction

A proper equipment selection is a very important activity formanufacturing systems due to the fact that improper equip-ment selection can negatively affect the overall performanceand productivity of a manufacturing system. The outputs of

M. Dagdeviren (B)Department of Industrial Engineering, Gazi University,Maltepe, Ankara 06570, Turkeye-mail: metindag@gazi.edu.tr

manufacturing system (i.e., the rate, quality and cost) mostlydepend on what kinds of properly selected and implementedequipments are used (Ayag and Ozdemir 2006). In additionto this, equipment selection has a major effect on the com-panies global competitiveness. Using proper equipment canenhance the production process, provide effective utilizationof manpower, increase production, and improve system flexi-bility. The importance of equipment selection cannot be over-looked. However, with the wide range of equipment availabletoday, determination of the best equipment alternative for agiven production scenario is not an easy task (Chan et al.2001).

Selecting the new equipment is a time-consuming and dif-ficult process, requiring advanced knowledge and experiencedeeply. So, the process can be a hard task for engineers andmanagers, and also for equipment manufacturer or vendor, tocarry out. For a proper and effective evaluation, the decision-maker may need a large amount of data to be analyzed andmany factors to be considered (Ayag and Ozdemir 2006).

Although equipment selection plays an important role inthe design of an effective manufacturing system, the publica-tions on this subject are limited (Kulak et al. 2005). The stud-ies performed could be classified into two groups as equip-ment selection and machine selection. One of the recent stud-ies is by Standing et al. (2001) which uses multi-attributeutility theory to quantify the contribution of various struc-tural and infrastructural factors for an equipment selectiondecision. Tabucanon et al. (1994) developed a decision sup-port system for multi-criteria machine selection problem forflexible manufacturing systems (FMS), and used the AHPtechnique for the selection process. Chen (1999) developsan integer programming model and a heuristic algorithmto solve the problem of multiple time periods. Lagrangeanrelaxation is used to generate lower bounds for the integerprogramming model to evaluate the quality of the heuristic

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solution. Machine selection from fixed number of availablemachines is also considered by Atmani and Lashkari (1998),who developed a model for machine tool selection and oper-ation allocation in FMS. Wang et al. (2000) proposed a fuzzymulti-attribute decision making model to assist the decision-maker to deal with the machine selection problem for a FMS.Dellurgio et al. (1997) presents a Monte Carlo simulationmodel for designing and selecting integrated circuit (IC)inspection systems and equipment choices. Beaulieu et al.(1997) consider the cell formation and the machine selectionproblems for the design of a new cellular manufacturing sys-tem using a heuristic algorithm. Kulak et al. (2005) developsunweighted and weighted multi-attribute axiomatic design(AD) approaches for fuzzy multi-attribute equipment selec-tion. These approaches include both crisp and fuzzy criteria.Arslan et al. (2002), Lin and Yang (1996) proposed AHP formachine selection problem. Yurdakul (2004) applied AHPand ANP (Analytic Network Process) for calculation of thecontributions of machine tool alternatives to the manufac-turing strategy of a manufacturing organization. Ayag andOzdemir (2006) proposed a fuzzy AHP approach for machinetool selection problem. In addition, the articles for an equip-ment replacement decisions are presented by Oeltjenbruns etal. (1995) and Sullivan et al. (2002).

In the equipment selection problem, there are a finite num-ber of alternatives which have to be ranked considering manydifferent and conflicting criteria. Accordingly, this problemis considered as a multi-criteria decision-making (MCDM)problem. Several methods exist for MCDM (Zeleny 1982;Vincke 1992). There are no better or worse techniques, butsome techniques better suit to particular decision problemsthan others do (Mergias et al. 2007). The advantage of thesemethods is that they can account for both financial andnon-financial impacts. Among these methods, the most pop-ular ones are scoring models (Nelson 1986), analytic hier-archy process (AHP) (Ong et al. 2001), analytic networkprocess (ANP) (Yksel and Dagdeviren 2007), axiomaticdesign (AD) (Kulak and Kahraman 2005), utility models(Munoz and Sheng 1995), TOPSIS (Tong et al. 2003), ELEC-TRE (Wang and Triantaphyllou 2006) and PROMETHEE(Mergias et al. 2007). It is essential to develop all elementsrelated to the situation of MCDM in detail before selectingan appropriate MCDM method to solve the problem underconsideration (Bufardi et al. 2004; Mergias et al. 2007). TheMCDM method choice decision should wait until the analystand the decision makers understand the problem, the feasiblealternatives, different outcomes, conflicts between the crite-ria and level of the data uncertainty (Mergias et al. 2007).

Multi-criteria decision making (MCDM) methods such asAHP and Fuzzy AHP, used for equipment selection problemsin the literature, make the evaluations using the same eval-uation scale and preference functions on the criteria basis.Definition of different preference functions for the criteria is

an important factor which affects the correctness of the deci-sion made. Unlike other ranking methods in the literature,different preference functions can be defined for criteria inPROMETHEE method. In this paper, to solve the equipmentselection problem, the PROMETHEE method was selecteddue to its simplicity and capacity to approximate the waythat human mind expresses and synthesizes preferences whenfacing multiple contradictory decision perspectives. Thismethod has some strength in comparison with existing meth-ods, such as: the PROMETHEE I does not aggregate goodscores on some criteria and bad scores on other criteria, it hasless pairwise comparisons and it does not have the artificiallimitation of the use of the standard scale for evaluation. Deci-sion Lab (1999) is software which supports this method andit also makes sensitivity analysis for the results. This methodprovides a visual and powerful tool called Geometrical Ana-lytic for Interactive Aid (GAIA) plane to identify conflictsamong criteria and to group the alternatives (Albadvi et al.2007).

In this paper, AHPPROMETHEE integrated approachfor selection of the most suitable equipment will be intro-duced and the implementation process will be explained witha real world example. We shall use the AHP method to ana-lyze the structure of the equipment selection problem anddetermine the weights of criteria, and use PROMETHEEmethod for final ranking. In the application, the criteria,which have the greatest effect on the equipment selectionare determined via a sensitivity analysis, moreover how theranking changes if the criteria did not have weights will beanalyzed. This paper is divided into five sections.In Sect. Introduction, the studied problem is introduced.Section Principles of AHP and PROMETHEE methodsbriefly describes the two proposed methodologies. In Sect.Proposed AHP-PROMETHEE integrated approach, pro-posed AHPPROMETHEE integrated approach for equip-ment selection is presented and the stages of the proposedapproach and steps are determined in detail. How the pro-posed approach is used on a real world example is explainedin Sect. A numerical application of proposed approach.In Sect.Conclusions and future research, conclusions andfuture research areas are discussed.

Principles of AHP and PROMETHEE methods

The AHP method

AHP, developed by Saaty (1980), addresses how to determinethe relative importance of a set of activities in a multi-criteriadecision problem. The process makes it possible to incorpo-rate judgments on intangible qualitative criteria alongsidetangible quantitative criteria (Badri 2001). The AHP methodis based on three principles: first, structure of the model;

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second, comparative judgment of the alternatives and the cri-teria; third, synthesis of the priorities.

The first step, a complex decision problem is structured asa hierarchy. AHP initially breaks down a complex MCDMproblem into a hierarchy of interrelated decision elements(criteria, decision alternatives). With the AHP, the objectives,criteria and alternatives are arranged in a hierarchical struc-ture similar to a family tree. A hierarchy has at least threelevels: overall goal of the problem at the top, multiple criteriathat define alternatives in the middle, and decision alterna-tives at the bottom (Albayrak and Erensal 2004).

The second step is the comparison of the alternatives andthe criteria. Once the problem has been decomposed andthe hierarchy is constructed, prioritization procedure startsin order to determine the relative importance of the crite-ria within each level. The pairwise judgment starts from thesecond level and finishes in the lowest level, alternatives. Ineach level the criteria are compared pairwise according totheir levels of influence and based on the specified criteriain the higher level (Albayrak and Erensal 2004). In AHPmultiple pairwise comparisons are based on a standardizedcomparison scale of nine levels (Table 1).

Let C = {C j | j = 1, 2, . . . , n} be the set of criteria. Theresult of the pairwise comparison on n criteria can be sum-marized in an (n n) evaluation matrix A in which everyelement ai j (i, j = 1, 2, . . . , n) is the quotient of weights ofthe criteria, as shown in Eq. 1:

A =

a11 a12 . . . a1na21 a22 . . . a2n...

.... . .

...

an1 an2 . . . ann

, aii = 1, a ji = 1/ai j , ai j = 0.

(1)

At the last step, the mathematical process commences tonormalize and find the relative weights for each matrix. Therelative weights are given by the right eigenvector (w) cor-responding to the largest eigenvalue (max), as

Aw = maxw. (2)

Table 1 Nine-point intensity of importance scale and its description

Definition Intensity of importance

Equally important 1Moderately more important 3Strongly more important 5Very strongly more important 7Extremely more important 9Intermediate values 2,4,6,8

If the pairwise comparisons are completely consistent, thematrix A has rank 1 and max = n. In this case, weights canbe obtained by normalizing any of the rows or columns of A(Wang and Yang 2007).

It should be noted that the quality of the output of the AHPis strictly related to the consistency of the pairwise compar-ison judgments. The consistency is defined by the relationbetween the entries of A: ai j a jk = aik . The consistencyindex (CI) isCI = (max n)/(n 1). (3)

The final consistency ratio (CR), using which one can con-clude whether the evaluations are sufficiently consistent, iscalculated as the ratio of the CI and the random index (RI),as indicated in Eq. 4.

CR = CI/RI (4)The number 0.1 is the accepted upper limit for CR. If

the final consistency ratio exceeds this value, the evaluationprocedure has to be repeated to improve consistency. Themeasurement of consistency can be used to evaluate the con-sistency of decision makers as well as the consistency of allthe hierarchy (Wang and Yang 2007).

In the literature, AHP, has been widely used in solvingmany complicated decision-making problems. Recently,Tiwari and Banerjee (2001) used the AHP for the selection ofa casting process. Albayrak and Erensal (2004) applied theAHP to improve human performance. Gngr and Arkan(2007) also used this method to improve quality-based invest-ment. In spite of its popularity and simplicity in concept,the AHP is criticized for its inability to adequately handlethe inherent uncertainty and imprecision associated with themapping of the decision-makers perception to crisp val-ues. In this case, many researchers have suggested use offuzzy numbers in pairwise comparison process (Bozdag et al.2003; Chan et al. 2007; Dagdeviren and Yksel 2008). Thesecond point why AHP method has been criticized is thatdecision problems are structured in a hierarchical manner.Some decision-making problems cannot be structured hier-archically because they involve the interaction and depen-dence of higher level elements on lower level elements (Saaty1996). Saaty suggested the use of ANP to solve the prob-lem of dependence among alternatives or criteria and manyresearchers used ANP in their study (Niemira and Saaty2004; Mohanty et al. 2005; Ulutas 2005; Chung et al. 2005;Agarwal et al. 2006; Yksel and Dagdeviren 2007).

The PROMETHEE method

The PROMETHEE (Preference Ranking OrganizationMETHod for Enrichment Evaluation) is a multi-criteria deci-sion-making method developed by Brans et al. (Brans andVincke 1985; Brans et al. 1986). It is a quite simple ranking

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method in conception and application compared with othermethods used for multi-criteria analysis. It is well adaptedto problems where a finite number of alternatives are to beranked according to several, sometimes conflicting criteria(Albadvi et al. 2007). The evaluation table is the startingpoint of the PROMETHEE method. In this table, the alter-natives are evaluated on the different criteria.

The implementation of PROMETHEE requires two addi-tional types of information, namely: (1) Information on therelative importance that is the weights of the criteria con-sidered. (2) Information on the decision-makers preferencefunction, which he/she uses when comparing the contributionof the alternatives in terms of each separate criterion.

The weights coefficients can be determined according tovarious methods (Nijkamp et al. 1990; Mergias et al. 2007).AHP method is used to determine the criteria weights in thisstudy.

The PROMETHEE method is appropriate to treat themulti-criteria problem of the following type:

max{ f1(a), f2(a), . . . , fn(a)|a A}, (5)where A is a finite set of possible alternatives, and f j denotesn criteria to be maximized. For each alternative, f j (a) is anevaluation of this alternative. When we compare two alterna-tives a, b A, we must be able to express the result of thesecomparisons in terms of preference. We, therefore, considera preference function P . The preference function translatesthe difference between the evaluations of two alternatives(a and b) in terms of a particular criterion, into a preferencedegree ranging from 0 to 1. Let

Pj (a,b) = G j [ f j (a) f j (b)], (6)

0 Pj (a,b) 1, (7)be the preference function associated to the criterion, f j (i)where G j is a non-decreasing function of the observed devi-ation (d) between f j (a) and f j (b). In order to facilitatethe selection of specific preference function, six basic typesof this preference function are proposed to decision makerby Brans and Vincke (1985) (usual function, U-shape func-tion, V-shape function, level function, linear function andGaussian function) in each case no more than two parame-ters (threshold, q, p or s) have to fixed (Brans and Mareschall1994; Wang and Yang 2007).

Indifference threshold q: the largest deviation to consideras negligible on that criterion. It is a small value with respectto the scale of measurement. Preference threshold p: thesmallest deviation to consider decisive in the preference ofone alternative over another. It is a large value with respectto the scale of measurement. Gaussian threshold s: it is onlyused with the Gaussian preference function. It is usually fixedas an intermediate value between indifference and a prefer-ence threshold.

PROMETHEE permits the computation of the followingquantities for each alternative a and b:

(a, b) =

nj=1

w j Pj (a, b)

nj=1

w j, (8)

+(a) =xA

(x, a), (9)

(a) =xA

(a, x), (10)

(a) = +(a) (a). (11)For each alternative a, belonging to the set A of alterna-

tives, (a, b) is an overall preference index of a over b. Theleaving flow +(a) is the measure of the outranking char-acter of a (how a dominates all the other alternatives of A).Symmetrically, the entering flow (a) gives the outrankedcharacter of a (how a is dominated by all the other alter-natives of A). (a) represents a value function, whereby ahigher value reflects a higher attractiveness of alternative aand is called net flow.

The three main PROMETHEE tools can be used to ana-lyze the evaluation problem: (1) the PROMETHEE I partialranking, (2) the PROMETHEE II complete ranking and (3)the geometrical analysis for interactive aid (GAIA).

The PROMETHEE I partial ranking provides a ranking ofalternatives. In PROMETHEE I, alternative a is preferred toalternative b, aPb, if alternative a has a greater leaving flowthan that of alternative b and a smaller entering flow than theentering flow of alternative b:

aPb if : +(a) > +(b) and (a) < (b); or+(a) > +(b) and (a) = (b); or+(a) = +(b) and (a) < (b). (12)

PROMETHEE I evaluation allows indifference and inco-mparability situations. Therefore, sometimes partial rankingscan be obtained. In the indifference situation (aIb), two alter-natives a and b has the same leaving and entering flows:

aIb if : +(a) = +(b) and (a) = (b). (13)Two alternatives are considered incomparable, aRb, if

alternative a is better than alternative b in terms of leavingflow, while the entering flows indicate the reverse:

aRb if : +(a) > +(b) and (a) > (b); or+(a) < +(b) and (a) < (b). (14)

PROMETHEE II provides a complete ranking of the alter-natives from the best to the worst one. Here, the net flow () is

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used to rank the alternatives. The alternative with the highernet flow is assumed to be superior. Since PROMETHEEI does not provide a complete ranking, resulting rankingcan not be compared with the ranking provided by PROM-ETHEE II. PROMETHEE I ensure creation of indifferentand incomparable alternatives. In some ranking problems,PROMETHEE I can give a complete ranking depending onthe evaluation matrix values and, this ranking can not be dif-ferent from the one achieved with PROMETHEE II.

The geometrical analysis for interactive aid (GAIA) planedisplays the relative position of the alternatives graphically,in terms of contributions to the various criteria (Brans andVincke 1985; Brans and Mareschall 1994). Principal com-ponents analysis is applied to the matrix of normed flows,defined for alternative a and criterion j by:

j (a) = 1n 1

b =a

[Pj (a, b) Pj (b, a)], (15)

where n is the number of alternatives, and this is used to gen-erate a two-dimensional plot in which the alternatives andcriteria are represented in the same plan (Belton and Stewart2002).

There are some studies in the literature which considerthe PROMETHEE. Goumas and Lygerou (2000) used thePROMETHEE method in the ranking of alternative energyexploitation projects. Albadvi et al. (2007) developed a deci-sion making model for stock trading with PROMETHEE. Inaddition to these studies Wang and Yang (2007), for infor-mation systems outsourcing; Mergias et al. (2007), for theselection of the best compromise management scheme forend of life vehicles; Araz et al. (2007) in the outsourcingmanagement, used PROMETHEE method.

Proposed AHP-PROMETHEE integrated approach

The integrated approach, composed of AHP and PROM-ETHEE methods, for the equipment selection problemconsists of 4 basic stages: (1) Data gathering, (2) AHP com-putations, (3) PROMETHEE computations, (4) Decisionmaking. In the first stage, alternative equipments and thecriteria which will be used in their evaluation are determinedand the decision hierarchy is formed. In the last step of thefirst stage, the decision hierarchy is approved by decisionmaking team.

After the approval of decision hierarchy, criteria used inequipment selection are assigned weights using AHP in thesecond stage. In this phase, pairwise comparison matrices areformed to determine the criteria weights. The experts fromdecision making team make individual evaluations usingthe scale, provided in Table 1, to determine the values ofthe elements of pairwise comparison matrices. Computing

the geometric mean of the values obtained from individualevaluations, a final pairwise comparison matrix on whichthere is a consensus is found. The weights of the criteriaare calculated based on this final comparison matrix. In thelast step of this phase, calculated weights of the criteria areapproved by decision making team.

Equipment priorities are found by using PROMETHEEcomputations in the third stage. Firstly, preference functionsand parameters are determined by the decision making team.After the approval of the functions, partial ranking withPROMETHEE I and complete ranking with PROMETHEEII and GAIA plane are determined. Decision Lab software isused in this process.

In the last step of the proposed procedure, the best equip-ment is selected according to the rankings and GAIA planeobtained by PROMETHEE I and II. Schematic representa-tion of the proposed approach is presented in Fig. 1.

Forming decision making teamStep 1:

Determining alternative equipmentsStep 2:

Determining the criteria to be used inevaluationStep 3:

STAGE 1: Data GatheringStructuring decision hierarchyStep 4:

Approvedecision

hierarchy?Step 5: N

Y

Step 6: Assigning criteria weigths via AHP

STAGE 2: AHP calculationsApprovecriteria

weights?Step 7: N

Y

Determining the preference functionsand parameters for the criteriaStep 8:

Approvepreferencefunctions?

Step 9: N

STAGE 3: PROMETHEEY calculations

Partial ranking via PROMETHEE IStep 10:

Complete ranking via PROMETHEE IIStep 11:

Step 12: Determining GAIA plane

Step 13: STAGE 4: Decision makingSelecting the best equipment

Fig. 1 Schematic representation of the process proposed for equip-ment selection

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A numerical application of proposed approach

The proposed approach is applied in a manufacturingcompany, located in Ankara, Turkey. The company wants topurchase a few milling machines (MM) to reduce the work-in-process inventory and to replace its old equipment. Thehigh technology equipments make significant improvementsin the manufacturing processes of the firms and the correctdecisions made at this stage brings the companies competi-tive advantage. Therefore, selecting the most proper millingmachines is of great importance for the company. But it ishard to choose the most suitable one among the MMs whichdominate each other in different characteristics.

Data gathering

In the application, firstly the decision making team, whichwill take a part in equipment selection process, is formed.With a preliminary work, this decision making team deter-mined five possible milling machines suitable for the needs ofthe company. The six criteria, namely price, weight, power,spindle, diameter and stroke, which will be taken into accountin the selection process, are determined. Decision hierarchystructured with the determined alternative equipments andcriteria is provided in Fig. 2.

There are three levels in the decision hierarchy structuredfor equipment selection problem. In the first level of the hier-archy, there is the overall goal of the decision process whichis determined as the selection of the best equipment. Thecriteria are on the second level and alternative MMs are onthe third level of the hierarchy.

AHP calculations

After forming the decision hierarchy for equipment selec-tion problem, the criteria to be used in evaluation processare assigned weights by using AHP method. In this phase,the experts in the decision making team are given the taskof forming individual pairwise comparison matrix by usingthe scale given in Table 1. Geometric means of these valuesare found to obtain the pairwise compassion matrix on whichthere is a consensus (Table 2).

Selection of the best equipment

Price Weight Power Spindle Diameter Stroke

MM-3 MM-4 MM-5MM-1 MM-2

Fig. 2 The decision hierarchy of equipment selection

Table 2 Pairwise comparison matrix for criteria and weights

Criteria Price Weight Power Spindle Diameter Stroke

Price 1.00 0.52 0.62 0.41 0.55 0.59Weight 1.93 1.00 2.6 1.21 1.3 2.39Power 1.61 0.38 1.00 0.40 0.41 1.64Spindle 2.44 0.83 2.47 1.00 0.40 0.46Diameter 1.80 0.77 2.40 2.49 1.00 2.40Stroke 1.69 0.42 0.61 2.17 0.42 1.00

Table 3 Results obtained from AHP computations

Criteria Weights (w) max, CI, RI CR

Price 0.090 max = 6.201 0.032Weight 0.244Power 0.113 CI = 0.040Spindle 0.266Diameter 0.186 RI = 1.24Stroke 0.101

The results obtained from the computations based on thepairwise comparison matrix, provided in Table 2, are pre-sented in Table 3.

The spindle, weight and diameter are determined as thethree most important criteria in the equipment selection pro-cess by AHP. Consistency ratio of the pairwise comparisonmatrix is calculated as 0.032

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Table 5 Preference functionsCriteria PF Thresholds

q p s

Price Level 600 800 Weight Gaussian 4Power Level 800 1,200 Spindle Level 20,000 23,000 Diameter Gaussian 6Stroke V-shape 50

Before using the PROMETHEE method to rank thealternative milling machines, for each criterion, a specificpreference function (PF), with its thresholds is defined. Pref-erence functions and threshold values have been defined bythe decision-making team established at the beginning of theapplication. Decision-making team has set these values bytaking into consideration the features of alternative MMsand purchasing policy of the firm. The preference functionsand thresholds defined are provided in Table 5.

After evaluation matrix and preference functions are deter-mined, alternative MMs are evaluated by using Decision Labsoftware. The positive flow (+), negative flow () and netflow () values obtained from this evaluation are given inTable 6.

By using the flow values in Table 6, firstly the partialranking is determined via PROMETHEE I (Fig. 3). PROM-ETHEE I used positive and negative flow values to find thepartial ranking.

MM-3 is determined as the worst alternative according tothe PROMETHEE I partial ranking MM-5, MM-2, MM-4and MM-1 alternatives are preferred to MM-3 alternative,and MM-4 alternative is preferred to MM-1 alternative. Onthe other hand MM-5, MM-4 and MM-2 (also MM-5 andMM-1, MM-2 and MM-1) are incomparable alternatives.PROMETHEE I did not provide information about the bestalternative.

Net flow values given in the last column of Table 6, areused in PROMETHEE II complete ranking to identify thebest alternative (Fig. 4).

Table 6 PROMETHEE flowsAlternatives +

MM-1 0.0199 0.0139 0.0061MM-2 0.0553 0.0480 0.0073MM-3 0.0192 0.0810 0.0618MM-4 0.0298 0.0130 0.0168MM-5 0.0478 0.0163 0.0315

Fig. 3 PROMETHEE I partial ranking

Fig. 4 PROMETHEE II complete ranking

MM-5 is selected as the best alternative based on the infor-mation PROMETHEE II provides, and the other alternativesare ranked in the order of MM-4, MM-2, MM-1, MM-3.

The decision problem can be represented in the GAIAplane (see Fig. 5) where alternative machines are representedby points and criteria by vectors. In this way, conflicting cri-teria may appear clearly. Criteria vectors expressing similarpreferences on the data are oriented in the same direction,while conflicting criteria are pointing in opposite directions.The length of each vector is a measure of its power in alter-native machines differentiation.

This plane is the result of principal component analysis(PCA), projecting the 6-dimensional space of criteria onto atwo-dimensional plane, i.e., the 6 original variables are trans-formed to the two new variables that are obtained by twolinear combinations of the original variables. By applyingthe PCA related criteria are handled by these combinationsand double counting never occurs (Albadvi et al. 2007). Asit is shown in the Fig. 5, the Delta-parameter is 95.70%; thismeans only 4.3% of the total information gets lost by theprojection.

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Fig. 5 GAIA plane for equipment selection

We observe that Price (Criterion 1) has a high differen-tiation power and expresses independent preferences, dif-ferent from those expressed by most of all other criteria.A cluster of conflicting criteria (Criterion 1 and criterion 3expressing opposite preferences) are clearly represented. Itis also possible to appreciate clearly the quality of the alter-native machines with respect to the different criteria. MM-2(Action 2) is particularly good on Criterion 3. MM-3 (Action3) is good on Criterion 24.

Vector pi (decision axis) represents the direction of thecompromise derived from the assignment; the decision makeris invited to appreciate the alternative machines located inthat direction (Wang and Yang 2007). It can be seen fromFig. 5 that pi vector is in the direction of criterion 5- criterion6 and the closest alternatives to the pi vector are MM-4 andMM-5. This result is consistent with the complete ranking ofPROMETHEE II.

Decision-making

According to the AHP and PROMETHEE computations, itis decided to purchase MM-5 and MM-4. How the variationin the criteria weights after the decision will affect the rank-ing is analyzed. Sensitivity analysis has been performed withDecision Lab Software and the resulting stability intervalsvalues are given in Table 7. Table 7 gives for each criterion thelimits within weights values which can vary without chang-ing the PROMETHEE II complete ranking. From the resultof sensitivity analysis, it is clear that Price (Criterion 1) and

Table 7 Stability intervals

Criteria Weight Interval

Min Max

Price 0.090 0.0645 0.0947Weight 0.244 0.1194 0.2569Power 0.113 0.1083 0.1512Spindle 0.266 0.0000 InfinityDiameter 0.186 0.0000 0.6009Stroke 0.101 0.0943 0.2283

Table 8 Weighted and unweighted rankings

Alternatives Weighted Weighted Unweighted Unweightedranking ranking

MM-1 0.0061 4 0.0035 4MM-2 0.0073 3 0.0175 3MM-3 0.0618 5 0.0855 5MM-4 0.0168 2 0.0328 1MM-5 0.0315 1 0.0317 2

Power (Criterion 3) have the greatest impacts on the completeranking.

The case in which criteria weights are not considered, i.e.,the criteria have equal priorities, is analyzed and the net flowvalues obtained in this condition are presented in Table 8 withtheir comparisons with previous values.

The best alternative has changed according to the unweigh-ted ranking results. The first alternatives have inter-changed,while the other ranking remained the same. The change in thebest alternative when criteria weights are taken into accounthas shown that criteria weights found consistently constitutean important phase in decision making process.

In the scope of the application, finally, the same problemhas been analyzed with AHP, TOPSIS and ELECTRE meth-ods and, obtained results are given in Table 9 in a comparablemanner.

When the results obtained for the same problem by usingother ranking methods are compared, it is seen that the best

Table 9 Ranks obtained with different ranking methods

Current study AHP TOPSIS ELECTRE

MM-5 MM-5 MM-5 MM-5MM-4 MM-1 MM-3 MM-1MM-2 MM-4 MM-1 MM-4MM-1 MM-3 MM-4 MM-3MM-3 MM-2 MM-2 MM-2

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alternative does not change (MM-5) while the ranking ofthe alternatives changed. None of other ranking methodshas been able to produce the result obtained by integratedAHPPROMETHEE approach. This situation has showedthat defining of different preference functions for the crite-ria considered in decision-making process is an importantdifference effective on the decision to be made.

Conclusions and future research

In this paper, a decision approach is provided for equipmentselection problem. This selection problem is based on thecomparisons of equipment alternatives according to identi-fied criteria. PROMETHEE and AHP compound decision-making methods have been used in proposed approach. Withits above-mentioned structure, the proposed approach dif-fers from the present equipment selection literature. AHP isused to assign weights to the criteria to be used in equip-ment selection, while PROMETHEE is employed to deter-mine the priorities of the alternatives. The weights obtainedfrom AHP are included in decision making process by usingthem in PROMETHEE computations and the alternative pri-orities are determined based on these weights. By this way,weighting of the criteria considered during decision-makingand evaluation of these criteria via preference functions areperformed simultaneously. Additionally, in the application, itis shown that calculation of the criteria weights is importantin PROMETHEE method and they could change the ranking.

The proposed decision approach can help decision-makersto choose and analyze factors and attributes easily. In addi-tion, the strengths of this approach over the existing methodscan be explained as follows. PROMETHEE method takesinto account the preference function of each criterion, deter-mined by the decision-makers. By this way, each criterion isevaluated on a different basis and it is possible to make betterdecisions. PROMETHEE I identifies the alternatives whichcannot be compared and the alternatives which are indif-ferent, by making a partial ranking, while PROMETHEEII provides a complete ranking for alternatives. The GAIAplane is a useful analytical tool that some remarks can bedetected from the alternatives and criteria sets. For exam-ple, in MM evaluation, MM-3 and MM-5 are the same onsome criteria and MM-1 and MM-4 are the same on othercriteria. In addition, differentiation power of the criteria, sim-ilar criteria, independent criteria, and opposite criteria can bedetermined from the GAIA analysis. By utilizing the PROM-ETHEE method to make sensitivity analysis of the result, themost effective criteria in decision making are determined.These opportunities are not available in present methods suchas AHP, fuzzy AHP, ELECTRE and TOPSIS. The proposedmodel has only been implemented on an equipment selectionproblem in the company, however, company management has

found the proposed model satisfactory and implementable inothers equipment selection decisions.

Since the parameters related with the criteria included inthe model in equipment selection in this study have beenable to be determined as crisp parameters, classical AHPand PROMETHEE have been used. The proposed integratedapproach might be incapable if the evaluation matrix for thealternatives (Table 4) cannot be formed with crisp values.Some criteria could have a qualitative structure or have anuncertain structure which cannot be measured precisely. Insuch cases, fuzzy numbers can be used to obtain the evalu-ation matrix. This will improve the proposed method and isone of the directions in our future research.

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Decision making in equipment selection: an integrated approachwith AHP and PROMETHEEAbstractReferences

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