Omega 42 (2014) 25–32

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An extension of MACBETH method for a fuzzy environment to analyzealternatives in reverse logistics for automobile tire wastes

Diala Dhouib n

Research unit Logistics, Industrial and Quality Management, Superior Institute of Industrial Management, Sfax, Tunisia

a r t i c l e i n f o

Article history:

Received 1 May 2012

Accepted 9 February 2013

Processed by B. Levrubbers, carbon blacks, steel cord and other organic and inorganic minor components. The most

important problem in the scrap tire recycling program is the type of product recovery option because

Available online 26 February 2013

Keywords:

Reverse Logistics

Waste tire

Decision making model

MACBETH methodology

2-tuple model

83/$ - see front matter & 2013 Elsevier Ltd. A

x.doi.org/10.1016/j.omega.2013.02.003

þ216 22 58 54 46; fax: þ216 74 403 278.

ail addresses: dialadhouib@yahoo.fr, sondaihe

a b s t r a c t

Waste tire related environmental problems and its recycling alternatives have been a major issue

nowadays because of their complex combination of very different materials, which include several

there are few specific data available. Multi-criteria decision analysis (MCDA) was used to assess options

in reverse logistics for waste tire. MCDA is a widely used decision methodology that considers

conflicting systems of criteria. However, many real-world decision problems involve ambiguity and

imprecise information. In this study, the analysis has been undertaken using an extended version of

MACBETH methodology to take into account the imprecise and linguistic assessments provided by a

decision-maker by integrating the 2-tuple model dealing with non-homogeneous information data.

The proposed fuzzy MACBETH method has been applied to a real case related to the automobile tire

waste to elucidate its details.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

With growing concern in environmental protection in recentyears, a problem of tire wastes disposal has increased especiallybecause they are virtually non-degradable and take up landfillspaces [1]. Discarded tires are useful in many recycling alternativeslike recovering energy from waste tires by incineration/combustion[2], co-combustion with coal or other fuels [3], pyrolysis [4] andgasification [5]. In practice, waste tires can be used by the civilengineering applications or by marine applications, as a wave break-ing material, ship/dock protective bumpers, or even to constructartificial reefs [6] in the ocean farming industry. Supplemental fuel forthe cement kilns [7], roadway pavement material [8], refuse derivedfuel [9] are different shapes of recycled waste tires. Waste tire canalso contribute to build and construct materials [10], produce road-way guard rails, as engineered protective cushions or bumpers.

Thus, due to the diversity of recycling alternatives regardingwaste tire, producers are looking for efficient ways to integratereverse logistics into their supply chains to recover economicvalue from discarded tires. This paper aims to structure theproblem related to the selection of an alternative for the reverselogistics option for waste tire and links the criteria with differentalternatives. In the same context, Gomes et al., [23] have applied amulticriteria decision aiding hybrid algorithm (THOR) as a multi-criteria decision support system that helps social agents to

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evaluate different disposal alternatives for plastic waste. Patiet al., [24] have developed a mixed integer goal programming(MIGP) model to assist in proper management of the paperrecycling logistics system. Morais, Almeida [25] have focused ona group decision making procedure based on the analysis ofindividual rankings with the aim of choosing an appropriatealternative for a water resources problem.

This paper presents a multicriteria decision making model fordecisions in reverse logistics practices based on MACBETH (Measur-ing Attractiveness by a Categorical Based Evaluation Technique)methodology. MACBETH methodology is an interactive approachthat uses semantic judgments about the differences in attractivenessof several stimuli to help a decision maker quantify the relativeattractiveness of each ([13]). The proposed method is designed forranking problem. Our fuzzy MACBETH approach considers thefuzziness in the decision data. Therefore, the model relies not onlyon the quality of the process data but also on the impreciseassessment modeling by applying the 2-tuple model [16].

This paper is organized as follows: In Section 2, we review someof the basic definitions of the MACBETH method and the 2-tupplefuzzy linguistic model. In Section 3, we introduce our method.In Section 4, we present a real case to elucidate the details of theproposed method and in Section 5, we interpret our results. InSection 6, we present our conclusions and future research directions.

2. Preliminary definitions

In this section, some basic definitions of MACBETH methodand the 2-tuple fuzzy linguistic model are reviewed.

D. Dhouib / Omega 42 (2014) 25–3226

Definition 1. MACBETH (Measuring Attractiveness by a Catego-rical Based Evaluation Technique) is an interactive approach thatallows a decision maker or a decision-advising group to evaluatealternatives by simply making qualitative comparisons regardingtheir differences of attractiveness in multiple criteria. Thus, whatdistinguishes MACBETH from the other multicriteria models isthat it needs only qualitative judgments about the difference ofattractiveness between two elements at a time, in order togenerate numerical scores for the options in each criterion andto weight the criteria.

M-MACBETH software is offered in ([12]). It verifies automa-tically the consistency of the judgements expressed by thedecision-maker and suggests to resolve inconsistencies if theyarise. Using the functionalities offered by the software, criteriaweights are provided from the decision-maker’s semantic judge-ments. By taking all the criteria into consideration, the valuescores of the options are, then, aggregated additively to generatethe overall value scores that reflect their attractiveness. M-MACBETH software also gives extensive analysis of the sensitivityand robustness of the model’s results.

Definition 2. The 2-tuple fuzzy linguistic representation model isone computational model in decision making that easily com-bines linguistic and numerical information. It deals with qualita-tive aspects that are presented in qualitative terms by means oflinguistic variables, i.e., variables whose values are words orsentences in a natural or artificial language instead of numbers.Each linguistic value is characterized by a syntactic value or labeland a semantic value or meaning. The label is a word or sentencebelonging to a linguistic term set and the meaning is a fuzzysubset in a universe of discourse. For example, a set of seventerms S, could be given as follows:

S¼{s0¼none, s1¼ very low, s2¼ low, s3¼medium, s4¼high,s5¼very high, s6¼perfect}. The semantics of the terms is given byfuzzy numbers defined in the [0, 1] interval. A way to characterizea fuzzy number is the use of a representation which is based onparameters of its membership function [18]. For example, we mayassign the following semantics to the set of seven terms viatriangular fuzzy numbers:

N¼ none¼ (0, 0, 0.17), VL¼ very low¼ (0, 0.17, 0.33), L¼low¼ (0.17, 0.33, 0.5), M¼ medium¼ (0.33, 0.5, 0.67), H¼ high¼(0.5, 0.67, 0.83), VH¼ very high¼ (0.67, 0.83, 1), P¼ perfect¼(0.83, 1, 1).

This assignment is depicted in Fig. 1.The 2-tuple fuzzy linguistic representation model takes as a

basis the symbolic aggregation model [20]. In addition, it definesthe concept of Symbolic Translation. The latter is used torepresent the linguistic information by means of a pair of valuescalled linguistic 2-tuple, (s, a), where s is a linguistic term and a isa numeric value representing the symbolic translation.

Definition 3. Let S¼{so,y,sg} be a linguistic term set andB(V)¼{(so,o0),y,sg,og)} be a fuzzy set that represents a numer-ical value V A[0, 1] over the linguistic term set S¼{so,y,sg}.

N VL L M H VH P

0 0.17 0.33 0.5 0.67 0.83 1

N VL L M H VH P

0 0.17 0.33 0.5 0.67 0.83 1

Fig. 1. A set seven terms with their semantics [19].

We obtain a numerical value b which is the result of asymbolic aggregation operation from the fuzzy set, assessed inthe interval [0, g] by means of the function w [16]

w : FðST Þ- 0,g½ �,

wðFðST ÞÞ ¼ w ðsj,gjÞ,j¼ 0,. . .,gn o� �

¼

Pgj ¼ 0 jgjPgj ¼ 0 gj

¼ b ð1Þ

Definition 4. Herrera and Martinez [21] developed a linguisticrepresentation model which represents the linguistic informationby means of 2-tuples (sk,a) where skAS is a linguistic term andaA[�0.5, 0.5) is called a symbolic translation.

The 2-tuple expresses the equivalent information to b and it isobtained with the following function where round (.) is the usualround operation, sk has the closest index label to ‘‘b ’’ and ‘‘a’’ isthe value of the symbolic translation

D : ½0,g�-S� ½�0:5,0:5Þ

DðbÞ ¼ ðsk,aÞ, withsk k¼ roundðbÞa¼ b�k aA ½�0:5,0:5Þ

(ð2Þ

Example. Let ST¼{s0, s1, s2, s3, s4, s5, s6} be a linguistic term andB(V)¼{(so,0), (s1,0), (s2,0), (s3,0.75), (s4,0.25), (s5,0), (s6,0),} be thecorresponding fuzzy set. By applying the function w, b¼3.25 is itsresult. Therefore, the representation of this quantity of informationwith the 2-tuples becomes D (3.25)¼(s3, 0.25), as illustrated in Fig. 2.

3. Fuzzy MACBETH method

Multi-criteria decision analysis (MCDA) methods have becomeincreasingly popular in decision-making for reverse logistic fieldbecause of the multi-dimensionality of the strategy goal and thecomplexity of environment policy [11]. MCDA includes variousmethods such as general utility analysis and outranking meth-odologies. This paper presents a multicriteria decision makingmodel for policy decisions in reverse logistics using MACBETH(Measuring Attractiveness by a Categorical Based EvaluationTecHnique). The latter is an interactive approach that requiresonly qualitative judgements about differences to help a decisionmaker or a decision-advising group quantify the relative attrac-tiveness of options. However, it is often difficult for a decision-maker (DM) to assign precise judgments. The merit of using afuzzy approach such as the 2-tuple model is its contemplation ofambiguity and the imprecision in the decision making process.Similarly, integrating the 2-tuple model is very suitable forscreening the qualitative deviations required by MACBETHmethod. In this section, we extend MACBETH method under afuzzy environment for developing reverse manufacturing options.

Fig. 3 presents stepwise procedure to implement our methodol-ogy. The procedure is composed into four steps explained hereafter

3.1. Step 1. identification of a MCDA problem with qualitative and

quantitative assessments

Most decision-making problems could be described by meansof the following sets:

�

A set of n actions called A¼{A1, A2,y, An}; � A set of m criteria called C¼{C1, C2,y, Cm};

3.250 3 4 521 6

0.25

Fig. 2. Example of symbolic translation computing.

Step1. Identification of a MCDA problem with qualitative and quantitative assessments

Step2. Transforming assessments into 2-tuples

Step3. Construction of linguistic pairwise comparison matrices

Step4. Ranking alternatives by applying M-MACBETH software

Fig. 3. Stepwise procedure for the fuzzy MACBETH approach.

D. Dhouib / Omega 42 (2014) 25–32 27

�

A set of performance ratings of Aj (j¼1, 2,y, n) on criteria Ci

(i¼1, 2,y, m) called X¼ {xij/, i¼1, 2,y, m, j¼1, 2,y, n}.

3.2. Step 2. transforming assessments into 2-tuples

First of all, the linguistic 2-tuple representation model hasbeen used as a good choice to manage non-homogeneous infor-mation. It unifies the information into fuzzy sets over a specificlinguistic domain, called Basic Linguistic Term Set (BLTS), ST. For asuitable choice of the BLTS, ST, we refer the reader to [19].

Whatever the nature of the input values: numerical, intervalvalued or linguistic, each one should be expressed by means of afuzzy set on the BLTS, F(ST) [17]. The process is carried out in thefollowing manner:

(a)

transforming numerical values in [�UBi, þUBi] into F(ST),with UBi is the numerical values upper bound for eachcriterion ki. Let di (Aj,Ak) be a deviation that denotes thedifference between the evaluations of Aj and Ak on eachquantitative criterion ki.

diðAj,AkÞ ¼ Aj�Ak8iA ½1,m�;Aj,AkAA; j,kA ½1,n� ð3Þ

Accordingly,

UBi ¼MaxAj ,Ak

9diðAj,AkÞ98iA ½1,m�;Aj,AkAA; j,kA ½1,n� ð4Þ

(b)

transforming linguistic terms into F(ST), (c) transforming interval valued into F(ST).

The resulting homogeneous information should be convertedinto a numerical value b by using the function w (see Eq. (1)).

After that, b is translated into linguistic 2-tuples (Sk, a) byapplying the function D already presented in Eq. (2). Conse-quently, for each criterion Ci, we obtain a 2-tuple (Sk, a) accordingto each alternative Aj. The unification of input information byapplying the 2-tuple model is an opportunity to combine impre-cise data without any loss of information. However, for an elicitedpurpose, the fuzzy MACBETH method ignores decision makingproblems with multiple experts as group decision making pro-blems. For managing non-homogeneous information in groupdecision making, we refer the reader to [19].

3.3. Step 3. construction of linguistic pairwise comparison matrices

For each criterion, alternatives are first ranked in a decreasingway according to the resulting linguistic terms. If two alternativeshave the same linguistic terms, they are ranked according to the

smallest distance in the absolute value of a. Pairwise comparisonmatrices contain linguistic judgments. For two alternatives Ae andAf with respectively their 2-tuples (Sk, ae), (Sh, af) and by applyingthe BLTS, ST¼{s0, s1,y,s6} for the input information, formally, alinguistic deviation def is computed as follows:

If 9k�h9¼0, then def¼noIf 9k�h9¼1, then def¼very weakIf 9k�h9¼2, then def¼weakIf 9k�h9¼3, then def¼moderateIf 9k�h9¼4, then def¼strongIf 9k�h9¼5, then def¼very strongIf 9k�h9¼6, then def¼extreme

Instead of MACBETH method, linguistic pairwise comparisonmatrices are not given directly by the DM but they are developedfor each criterion via an objective way and by taking into accountfuzziness that could have occurred in a decision-making process.We have proposed to compute pairwise comparison using thisscale with keeping the same qualitative categories of difference inattractiveness as used in the original version of MACBETH method:is there no difference (indifference), or is the difference very weak,weak, moderate, strong, very strong, or extreme? ([15]). However,this scale is not fixed and could be adapted to the related contextand the needs of the decision maker. It should be the same as inthe chosen BLTS, ST over which input values are converted intofuzzy sets.

The key distinction from MACBETH approach is that fuzzyMACBETH doesn’t require qualitative judgements about differ-ences of attractiveness from the decision maker. Indeed, cognitiveuneasiness may be expressed by the decision maker when tryingto provide his judgements. Therefore, since MACBETH approachhas tested the consistency of the rating scale in such a way thatdifferences among numerical ratings should reflect differences ofattractiveness ([15]), no inconsistencies could be arising using thefuzzy logic approach.

3.4. Step 4. ranking alternatives by applying M-MACBETH software

After providing linguistic pairwise comparison matrices, M-MACBETH software is applied to weight the model’s criteria byranking them in terms of their overall attractiveness to obtain theweighting matrix of judgements. As observed, the fuzzy MAC-BETH approach followed the same process as adopted in MAC-BETH method for building alternatives weightings. The proposedapproach does not seek to develop weightings for options in orderto avoid the risk of a cumbersome model. Indeed, it is not verydifficult for the decision-maker to rank criteria according to hispreferences. To improve the approach proposed in this paper,future work could involve assigning weights using the 2-tuplemodel or using other MCDA tools to evaluate the level of priorityfor particular criteria or developing a consensus on weightings bymultiple experts.

Finally, M-MACBETH builds a weights scale from the weight-ing matrix of judgements to create a weighting scale of the set ofalternatives. Once the model has been built, the model’s resultsare provided in a concise results table called ‘‘Table of scores’’which gives the alternatives ranking.

Generally speaking, a broad methodological approach toMCDA has been taken in this analysis, based on a modifiedprocedure of MACBETH method. The fuzzy MACBETH approachis a generic model, that is, the led process to compute linguisticdeviations could constitute an autonomous model to unify per-formance ratings. Furthermore, it is a flexible model that can beapplied to any multicriteria evaluation problem with a mix of

D. Dhouib / Omega 42 (2014) 25–3228

qualitative and quantitative assessment. The developed model isnot only to produce a ranking of alternatives but also to rationa-lize the problem and provide a framework for communicationamong decision-advisory group.

4. Real-case study

This study is based on data collected from the Promotion Agencyof Industry and Innovation in Tunisia and the model has been builtin direct interaction with a civil engineer, an expert from theNational Agency of Environment Protection of Sfax and an expertfrom the National Waste Management Agency from Tunis.

The fuzzy MACBETH model that is presented in this researchhas been evaluated in an actual automobile tire wastes manu-facturing company, which has been interested in the implemen-tation of the reverse logistics practices. The company wants asystematic way to determine the best possible option for con-ducting the reverse logistics alternatives. After a review of theliterature and a discussion with experts in the field of automobiletire wastes, the important selection criteria for the assessment ofdifferent options have been identified. The main criteria held inthis case are described in Table 1 and represent:

�

TabCrit

M

P

Jo

E

TabInp

M

P

Jo

E

Market factor (N1): it refers to the State intervention in thedemand for automobile tire wastes. In Tunisia, the State seeksto enhance exposure to the waste tires recycling techniques bytilting the investment toward the long-term growth potentialof each project related to environmental problems. The ‘‘Mar-ket Factor’’ criterion measures the degree of the State inter-vention in the demand for tire wastes by the product recoveryoptions to support investors to meet their longer term productappreciation needs. These interventions can be categorized intwo modes: indirect and direct intervention. The former is anexternal restriction by law, regulations, and policy; and thelatter is a direct involvement by state shareholding, assign-ment of officials, and control of capital by state-owned banks.In the case of indirect intervention, the market factor criteriontakes positive values. It is equal to 100% if there is nointervention and equal to 0% if there is a total intervention.However, in the case of direct intervention, negative values aregiven to the market factor criterion such as the value for themarket factor of Play-S in Table 2.

� Profit (N2): in modeling recycling processes like scrap tire

recycling, many uncertainties could be identified such as the

le 1eria for reverse manufacturing alternative selection.

Unit

arket factor Percentage

rofit One thousand tunisian dinars

bs created Number of jobs per capita consumption

nvironmental impact Percentage of the contribution to the environment

le 2ut data of the decision maker.

Walls-S

arket factor (%) 20

rofit (MD) 166,564

bs created (number of jobs per capita consumption) 4.4

nvironmental impact Very high

period of time to which the waste collection is related, theamount of total waste generated and the composition ofwaste. In the case study, an economic evaluation is appliedto get a quick estimation of recycling cost and revenue.This evaluation considers uncertainties in the estimation out-puts which include collecting cost, recycling cost, waste dis-posal cost and the revenue is simply based on normal profit(i.e., break-even) from the considered alternative

Total benefit¼ Revenue�CCollecting�CRecycling�CDisposal

Collecting cost is estimated based on collection time, cost forpaying the collectors and transportation cost. The latter isbased on the transportation distance and the distance isassumed an average transportation distance in the currentcollection system.Recycling cost include recycling equipment cost, facility cost,labor and overhead cost.Disposal cost represents the cost of hazardous materials whichare assumed to be fully taken out at the recycling operation. Itdepends on the amount of total waste generated and thecomposition of waste.

� Jobs created (N3): it represents the number of jobs per capita

consumption.

� Environmental impact (N4): in Tunisia, major rapid environ-

mental changes are becoming gradually controversial and arepresenting human society with unprecedented challengesnowadays. Since assessed environmental impact has been anew culture for managers, Industrialists and citizens, no clearparameters are included in the evaluation process and sovarious scales of environmental problems are considered froma societal perspective. Thus, from a policy standpoint, localgovernment encourages recycling industry to model recyclingprocesses as an assessment tool for the environmental impactor environmental benefit of the processes that act as a helpfultool in the decision making process.

In the current automobile tire wastes manufacturing company,environmental impact remains dependent on subjective or value-laden judgement to identify potential contribution to the envir-onment protection.

The scale for this criterion is: ‘‘very low, low, medium, high,very high’’ and it is proposed by the three decision-advisors.The alternatives were classified in a comparative way and twoalternatives could have the same assessment, affecting the envir-onment at the same proportion.

Score type Optimization criterion

Quantitative Maximize

Quantitative Maximize

Quantitative Maximize

protection Qualitative Maximize

Play-S Civil-Eng Cement-k Reuse

�56 30 100 12

173,277 154,858 161,858 188,889

4.4 4.7 4.2 3.1

Very low Low High Medium

D. Dhouib / Omega 42 (2014) 25–32 29

The four steps of the developed model that are presented inSection 3 have been evaluated in the real case as shown hereafter:

(1) In this model, we analyze five distinct alternatives thatinclude Walls with soundproofing (Walls-SP), Playground surfaces(Play-S), Civil engineering (Civil-Eng), Cement kilns (Cement-K)and Reuse (Reuse). These alternatives represent all recoveryoptions that could occur in a Tunisian scrap tire recycling programand they have been identified by the three decision-advisors: thecivil engineer, the expert from the National Agency of Environ-ment Protection of Sfax and the expert from the National WasteManagement Agency from Tunis.

The list of possible strategies for recycling the tire wastes andthe corresponding criteria are shown in Table 2. The caseexperience helps us to understand better the advantages anddisadvantages of the methodology from a practical point of view.

(2) The input information as presented in Table 2 is expressedby means of fuzzy sets on the BLTS, ST¼{s0, s1,y,s6}, with 7 labelsand with the following associated semantics:

s0¼(0, 0, 0.17),s1¼(0, 0.17, 0.33),s2¼(0.17, 0.33, 0.5),s3¼(0.33, 0.5, 0.67),s4¼(0.5, 0.67, 0.83),s5¼(0.67, 0.83, 1),s6¼(0.83, 1, 1).

As mentioned before, for a homogeneous goal, all input values:numerical, interval valued or linguistic should be expressed bymeans of a fuzzy set on the BLTS, ST. ST must maintain the abilityof discrimination to express the performance values, therefore,we should choose as BLTS, a term set with a granularity largerthan a person who is able to discriminate. Different psychologicalstudies about this can be found in the literature [16]. In our case,we have a scale with 5 items for the environmental impactcriterion. So, we shall choose a term set with at least 5 items.According to Miller’s [22] observation, human beings can reason-ably manage to bear in mind seven or so items. Cardinality valuesof classical scales used in the linguistic models are odd, such us7 and 9. Thus, we have used a scale with 7 items.

Herrera and Martinez [16] demonstrated that among theconditions to be imposed on the linguistic term set S, to avoidany loss of information during the transformation processes, themembership functions of its terms are triangular. That is why,triangular membership functions are selected for our analysis.

The resulted fuzzy sets are transformed over the BLTS, ST¼

{s0, s1,y,s6}, into b by applying the function w defined in Eq. (3).

Table 3

Calculation of b.

Walls-S Play-S Civil-Eng Cement-k Reuse

Market factor 3.6 1.32 3.9 6 3.36

Profit 5.64 5.752 5.46 5.57 6

Jobs created 5.85 5.8 6 5.68 4.97

Environmental impact 5.45 0.54 1.5 4.5 3

Table 4Linguistic 2-tuples.

Walls-S Play-S

Market factor (S4, �0.4) (S1, 0.32)

Profit (S6, �0.36) (S6, �0.248)

Jobs created (S6, �0.2) (S6, �0.2)

Environmental impact (S5, 0.45) (S1, 0.46)

The resulted b are presented in Table 3. Then, the use of thefunction D to b leads to linguistic 2-tuples shown in Table 4.

(3) For each criterion, after ranking alternatives, linguisticjudgement matrices are given in Table 5.

Example. Let us take the example of b computing of thecriterion Market factor for the alternative Walls-SP. W¼20 be anumerical value to be transformed in [�100, 100] into F(ST)where �100 and 100 are the numerical values upper bound forthe Market factor criterion and F(ST) is the fuzzy set on the BLTS.

Here, we shall compute seven fuzzy sets of W:

s0¼(�100, �100, �66.66)s1¼(�100, �66.66, �33.33)s2¼(�66.66, �33.33, 0)s3¼(�33.33, 0, 33.33)s4¼(0, 33.33, 66.66)s5¼(33.33, 66.66, 100)s6¼(66.66, 100, 100)B(20)¼{( s0, 0), (s1, 0), (s2, 0), (s3, 0.4), (s4, 0.6), (s5, 0), (s6, 0)}

The graphical representation is shown in Fig. 4.The value b is obtained by the function w defined in (3).

b¼ wðfðs0,0Þ,ðs1,0Þ,ðs2,0Þ,ðs3,0:4Þ,ðs4,0:6Þ,ðs5,0Þ,ðs6,0ÞgÞ ¼ 3:6

The 2-tuple which represents the information of 20 in eachlinguistic term set si, is the following: D (3.6)¼(s4, �0.4).The same procedure is applied to obtain the other 2-tuples inTable 4.

For the Market factor criterion, in the pairwise comparisonmatrices, actions are ranked in an increasing way according to thesmallest distance in absolute value of a. Therefore, we obtain thefollowing ranking:

Cement kilns-Civil engineering-Walls with soundproofing

-Reuse-Playground surfaces

Then, for example, for the comparison of the 2-tuples ofCement kilns and Civil engineering which are respectively (s6, 0)and (s4, �0.1), a linguistic deviation is computed: 96�49¼2. Thisleads to the linguistic judgement: ‘‘weak’’ according to the scaledeveloped in Section 3.

Then, the decision maker must provide his preferences amongthe criteria. He is just involved in the decision process to rankcriteria according to their overall attractiveness. The resultingweighting matrix of judgments is given in Table 6.

(4) Finally, M-MACBETH software is applied to weight themodel’s criteria and to provide scores of alternatives as shown inTable 7. Thus, the resulting ranking of the strategies is generatedand the best option can be chosen. The issued ranking isdetermined as follows:

Cement kilns-Walls with soundproofing

-Playground surfaces-Civil engineering-Reuse

The recommended strategy for the company is Cement kilns.

Civil-Eng Cement-k Reuse

(S4, �0.1) (S6, 0) (S3, 0.36)

(S5, 0.46) (S6, �0.43) (S6, 0)

(S6, 0) (S6, �0.32) (S5, �0.03)

(S1, 0.5) (S4, 0.5) (S3, 0)

Table 5Linguistic judgement matrices for each criterion.

(a) Market factor Cement-k Civil-Eng Walls-S Reuse Play-S

Cement-K – Weak Weak Moderate Very strong

Civil-Eng – No Very weak Moderate

Walls-S – Very weak Moderate

Reuse – Weak

Play-S –

(b) Profit Reuse Play-S Walls-S Cement-K Civil-EngReuse – No No No Very weak

Play-S – No No Very weak

Walls-S – No Very weak

Cement-K – Very weak

Civil-Eng –

(c) Jobs created Civil -Eng Play-S Walls-S Cement-K ReuseCivil-Eng – No No No Very weak

Play-S – No No Very weak

Walls-S – No Very weak

Cement-K – Very weak

Reuse –

(d) Environmental impact Walls-S Cement-k Reuse Play-S Civil-EngWalls-S – Very weak Weak Strong Strong

Cement-K – Very weak Moderate Moderate

Reuse – Weak Weak

Play-S – No

Civil-Eng –

Fig. 4. Matching between the numerical value 20 and linguistic values.

Table 6Weighting matrix of judgments.

Market

factor

Profit Jobs

created

Environmental

impact

Market factor – Very

weak

Weak Moderate

Profit – Very weak Weak

Jobs created – Very weak

Environmental

impact

–

Table 7Results of the fuzzy MACBETH model for the automobile tire wastes.

Overall N1 N2 N3 N4

Walls-SP 84.00 60.00 100.00 100.00 100.00

Play-S 57.50 0.00 100.00 100.00 75.00

Civil-Eng 49.00 60.00 0.00 100.00 50.00

Cement-K 90.00 100.00 100.00 100.00 0.00

Reuse 46.00 40.00 100.00 0.00 0.00

[All upper] 100.00 100.00 100.00 100.00 100.00

[All lower] 0.00 0.00 0.00 0.00 0.00

Weights: 0.4000 0.3000 0.2000 0.1000

Table 8Results of the MACBETH model for the automobile tire wastes.

Overall N1 N2 N3 N4

Walls-SP 43.49 23.08 45.45 64.29 88.89

Play-S 57.56 53.85 63.64 64.29 11.11

Civil-Eng 40.85 38.46 9.09 92.86 22.22

Cement-K 57.64 92.31 27.27 42.86 66.67

Reuse 37.05 7.69 90.91 7.14 44.44

[All upper] 100.00 100.00 100.00 100.00 100.00

[All lower] 0.00 0.00 0.00 0.00 0.00

Weights: 0.3809 0.3333 0.2381 0.0477

D. Dhouib / Omega 42 (2014) 25–3230

5. Results and discussion

MACBETH method has been applied to the same real casestudy for reverse manufacturing alternative selection. Resultsprovided by M-MACBETH Microsoft (shown in Table 8) suggestthe following ranking:

Cement kilns-Playground surfaces-Walls with soundproofing

-Civil engineering-Reuse

The comparison between the two approaches shows that therank is different only in terms of the two alternatives: Playground

surfaces and Walls with soundproofing. This is due to the greatdifference between the two alternatives according to the envir-onmental impact criterion. Indeed, decision-makers argue thatthis last criterion captures much fuzziness intensity.

Sensitivity analysis of the criteria has been carried out in twodifferent analyses. Fig. 5 present the effects of a change in eachcriterion weight on the options’ overall scores. Each option’s linein the graph shows the variation of the option’s overall scorewhen the criterion weight varies from 0 to 100%. The vertical linerepresents the current weight of the criterion.

Fig. 5. Sensitivity analysis on weight for each criterion. (a) Sensitivity analysis on weight for the Market factor criterion. (b) Sensitivity analysis on weight for the Profit

criterion. (c) Sensitivity analysis on weight for the Jobs created criterion. (d) Sensitivity analysis on weight for the Environmental impact criterion.

Fig. 6. Intersection between options for the market factor criterion.

D. Dhouib / Omega 42 (2014) 25–32 31

While modifying the weight of the Market factor from 0 to 100%,the overall scores of Walls-SP, Play-S and Reuse decrease and thoseof Civil-Eng and Cement-K increase (Fig. 5(a)). For the Profitcriterion, note that when its weight increases, the overall scores ofall alternatives increase except the Civil-Eng one (Fig. 5(b)). In thecase of Job created criterion, the overall scores of all alternativesincrease except the Reuse one (Fig. 5(c)). However, for the Environ-mental impact criterion, the overall scores of Walls-SP and Play-Sincrease, but those of Cement-K and Reuse decrease (Fig. 5(d)).

These modifications do not question the study of sensitivitythat we have led; it is simply explained by the fact that thedecision maker can change his perception regarding these weights.The engineers showed some discomfort with the decreasing of theCement-K overall score that resulted for the Environmental impactcriterion. Given this statement, it was proposed to carefully revisethe weighting judgements, but no change was introduced.

Fig. 6 allows finding, for the market factor criterion as anexample; the weight associated with the intersection of any twoof the options’ lines i.e. the weight necessary to swap their rank inoverall attractiveness. This figure shows that as long as the weightplaced upon N1 is less than 29.4, Walls-SP will score the highestoverall; however, if the weight placed upon N1 is greater than29.4, Cement-K will receive the model’s highest overall score.Similarly, the remaining intersections have been investigated.

If the lines of two options do not intersect, one is always moreattractive than the other regardless of the weight of the criterion.We notice that there are no intersections neither between Play-Sand Walls-SP, nor Walls-SP and Reuse, nor Civil-Eng and Cement-K, nor Cement-K and Reuse.

A different type of sensitivity analysis consisted in exploringthe extent to which the final ranking of the 5 alternatives would

be affected if uncertainty in the estimation of their consequenceswas introduced in the model. The sensitivity analyses haveconfirmed that the constructed model has been consideredadequate in form and contents to resolve the reverse manufactur-ing alternative selection problem.

6. Conclusion

This paper shows how MACBETH, which is a multi-criteriaapproach that can be used for decision-aiding, can be applied toreverse manufacturing alternative selection problem, regarding a

D. Dhouib / Omega 42 (2014) 25–3232

mix of qualitative and quantitative assessments. This probleminvolves conflicting systems of criteria, ambiguity and impreciseinformation. MACBETH is a well-established MCDA method thathas a history of successful real-world applications ([14]). In thispaper, we have proposed a methodological and computationalenhancement of MACBETH decision aid method for processingfuzzy preferences without loss of information by integrating the2-tuple model. The proposed approach for decision-aiding underfuzzy environments is flexible and could be applied easily to othermanagerial decision-making problems.

The contribution of this paper is fivefold: (1) Providing thedecision makers with simple and wide application of MACBETHmulticriteria method without reducing any of its beneficial proper-ties; (2) Integrating new information types into the decision-makingprocess without loss of information and involving conflicting systemsof criteria, ambiguity and imprecise information; (3) Providing theinput information to M-MACBETH software in an objective way byusing the 2-tuple representation model instead of subjective attrac-tiveness; (4) Identifying the best reverse manufacturing alternativefor tire waste automobile in order to demonstrate the approachapplicability and user-friendliness, as well as to response to a needclearly identified by industry; (5) Performing a sensitivity analysis toelucidate the robustness of the proposed approach.

Future work could include: (i) Comparing fuzzy MACBETHapproach with other MCDA methods in order to validate orcorroborate results; (ii) Developing weightings for different cri-teria; (iii) Improving the environmental impact analysis using anappropriate assessment method; (iv) Varying alternatives andcriteria and (v) Undertaking further sensitivity analysis.

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