A fuzzy-based decision making procedure for machine selection problem

Abstract

The selection of appropriate machines is one of the most critical decisions in the design and development of an efficient production environment. It is the fact that improper machine selection can result in quality, flexibility, productivity, etc., problems and negatively affect the overall performance and productivity of a manufacturing system. On the other hand, selecting the best machine among many alternatives is a multi-criteria decision making (MCDM) problem. In this paper, a fuzzy-based MCDM approach is used. For this aim, the fuzzy analytic network process (FANP) is used to determine weights of the criteria and preference ranking organization method for enrichment evaluations (PROMETHEE) is used to obtain final ranking of alternative machines. The proposed approach is applied for the selection of a CNC router machine (RM) to be purchased in an international company. In the problem addressed, there are four main criteria, namely cost, quality, flexibility and performance with the corresponding fourteen sub-criteria. The results for the case study indicate the best machine among six potential alternatives and provide different managerial insights for the decision makers.

Keywords

Fuzzy analytic network process machine selection multi-criteria decision making PROMETHEE

1 Introduction

Multi-criteria decision making (MCDM) comprises a finite set of alternatives, among which the decision makers (DMs) have to select, evaluate or rank according to the weights of a finite set of criteria. Many kinds of MCDM methods have been offered by different researchers, operational researchers and decision theorists for years such as analytic hierarchy process (AHP), technique for order performance by similarity to ideal solution (TOPSIS), VIKOR method (VlseKriterijumska Optimizacija I Kompromisno Resenje in Serbian, meaning Multi-criteria Optimization and Compromise Solution), etc. defining how a DM may decide about preferences in choosing situation among the multiple attribute alternatives. Some methods apply to the assessment of qualitative criteria assessment while others can be shown as suitable for quantitative criteria. These methods also involve subjective assessments, resulting with imprecise data in qualitative manner. Due to the availability and uncertainty of information in decision process as well as the vagueness of human feeling and recognition, it is difficult to make an exact evaluation and convey the feeling and recognition of objects for decision makers. Fuzzy set theory can play a significant role in this kind of decision situation [58]. Hence DMs generally fail to make a good numerical prediction for criteria; evaluation is expressed in linguistic terms. By the help of linguistic models verbal expressions can be turned into numerical ones. Thereby, when dealing quantitatively with imprecision in the expression of the importance of each criterion, some multi-criteria methods based on fuzzy relations are used [18].

For the design and development of an efficient production environment you need to think about the type of machines carefully. It is the fact that improper machine selection can result in quality, flexibility, productivity, etc., problems and it affects all performance and how productive the manufacturing system is in negative way. The outputs of manufacturing system mostly depend on what kinds of properly selected and implemented machines are used [4]. In addition to this, companies need to decide carefully due to competitive market conditions in consequence of globalization, limited resources, etc. If resources such as money, time, workforce, etc. due to wrong decisions are wasted in any way it affects the costs of companies in the matter of increasing which is also reflected to the customer. Companies where machining process is very important for the product need to select machine tool well since it is very critical for these companies. The importance of machine selection cannot be overlooked. However, with the wide range of machine available today, determination of the best machine alternative for a given production scenario is not an easy task [12].

A finite number of alternatives which need ranking according to many different and conflicting criteria are possible for the problem of machine selection. Then this problem can be called as a MCDM problem. Several methods exist for MCDM in the literature [52, 59]. Almost every evaluation technique has its strong points or defects and issues about the suitability for different situations [29]. These methods can explain both financial and non-financial impacts which are the advantage of them. The most popular ones in these methods are scoring models [33], AHP [36], and grey relational analysis (GRA) [19, 47], ANP [57], axiomatic design (AD) [22], utility models [32], TOPSIS [50], ELECTRE (ELimination Et Choix Traduisant la REalit $\overset{´}{e}$ ) [54] and PROMETHEE [29]. It is essential to develop all elements related to the situation of MCDM in detail before selecting an appropriate MCDM method to solve the problem under consideration [10, 29]. The MCDM method choice decisions should wait until the analyst and the decision makers understand the problem, the feasible alternatives, different outcomes, conflicts between the criteria and level of the data uncertainty [29].

In this paper, FANP-PROMETHEE integrated approach for selection of the most suitable machine is introduced and the implementation process is explained with a real case application. The FANP method is used to analyze the structure of the machine selection problem and determine the weights of criteria, and PROMETHEE method is used for final ranking. The rest of the paper is organized as follows: In Section 2, a general overview of the existing MCDM literature on machine selection is summarized. Section 3 briefly describes the FANP and PROMETHEE methodologies and the proposed FANP-PROMETHEE integrated approach for machine selection is presented and the stages of the proposed approach are explained. How the proposed approach is used on a real case application and scenarios analyzes are detailed in Section 4. In Section 5, conclusions and future research directions are discussed.

2 Literature review

Although machine selection plays an important role in the design of an effective manufacturing system, the publications on this subject are limited [23]. An integer programming model and a heuristic algorithm are developed to solve the problem of multiple time periods [13]. Lagrangean relaxation is used to generate lower bounds for the integer programming model to evaluate the quality of the heuristic solution. Machine selection from fixed number of available machines is also considered by Atmani and Lashkari [3] who developed a model for machine tool selection and operation allocation in flexible manufacturing systems (FMS). It is also proposed a fuzzy multi-attribute decision making model to assist the decision-maker to deal with the machine selection problem for a FMS [53]. A Monte Carlo simulation model is presented for designing and selecting integrated circuit (IC) inspection systems and equipment choices [16] while some researchers consider the cell formation and the machine selection problems for the design of a new cellular manufacturing system using a heuristic algorithm at the same year [5]. A model that links machine alternatives to manufacturing strategy for machine tool selection is presented by Yurdakul [56]. In their study, AHP and analytic network process (ANP) are applied for calculation of the contributions of machine tool alternatives to the manufacturing strategy of a manufacturing organization. A FANP approach is proposed for machine tool selection problem [4] while AHP and ANP are proposed for the same problem by Paramasivam et al. [40]. Özgen et al. [39] combine the modified DELPHI method, AHP and PROMETHEE approaches with fuzzy sets theory to consider machine selection problem of a fixing products manufacturer from Turkey, Istanbul. Peng and Xiao [41] combine the PROMETHEE method with ANP to select the best material for a given application, where ANP is used to identify weights, and PROMETHEE to rank alternatives. Taha and Rostam [49] present a decision support system to select the best alternative machine using a hybrid approach of fuzzy AHP and PROMETHEE. A MATLAB-based fuzzy AHP is used to determine the weights of the criteria and the PROMETHEE method is applied for the final ranking. Liu et al. [26] propose a general framework for evaluating and selecting the best material for a given application. A novel hybrid MCDM model combining DEMATEL-based ANP and modified VIKOR is used to solve the material selection problems of multiple dimensions and criteria that are interdependent. A hybrid approach of the fuzzy ANP and COPRAS-G (COmplex PRoportional ASsessment of alternatives with Grey relations) for fuzzy multi-attribute decision-making in evaluating machine tools with consideration of the interactions of the attributes is presented by Nguyen et al. [34]. The fuzzy ANP is used to handle the imprecise, vague and uncertain information to determine the weights of the attributes and COPRAS-G is employed to present the preference ratio of the alternatives in interval values with respect to each attribute and calculate the weighted priorities of the machine alternatives. The VIKOR compromise ranking method combined with generalized fuzzy number has explored for determining a CNC machine tool by Sahu et al. [46]. CNC machine tool performances are measured in term of subjective attributes such as productivity, precision and accuracy. Izadikhah [20] describes a fuzzy goal programming method for evaluation and selection of machine tool alternatives for a manufacturing company in Iran. Their approach applied triangular numbers into traditional goal programming method, and we investigated deriving fuzzy weights of criteria from the pair-wise comparison matrix with fuzzy elements.

Multi-criteria decision making methods such as AHP used for machine selection problems in the literature make the evaluations in a hierarchical structure based on the assumption that elements between layers and among elements of layers are independent of each other. But there are some shortcomings in the conventional AHP [4 , 51]; (1) It is generally applied in nearly crisp decision makings, (2) It creates and deals with a very unbalanced scale of judgment, (3) It does not take into account the uncertainty associated with the mapping of one’s judgment to a number, (4) Its ranking is rather imprecise, (5) The method is greatly affected by the DMs’ subjective judgment, selection and preferences. In this study, criteria are weighted by FANP which can also incorporate dependence and feedback within a set of elements (inner dependence) and among different sets of elements (outer dependence). Alternatives can be also compared by using FANP. But sometimes DMs should perform many pairwise comparisons and then we see this situation as impractical the usage of FANP process in some cases. With the aim of dealing with this problem, PROMETHEE is used to rank alternative machines. Definition of different preference functions for the criteria is an important factor which affects the correctness of the decision made at the same time. PROMETHEE method has some advantages according to the other ranking methods in the literature: (1) Different preference functions can be defined for criteria (2) PROMETHEE-I and PROMETHEE-II allow both partial and total ranking of the alternatives [11 , 55].

As mentioned in the first section, fuzzy ANP and PROMETHEE combination is proposed to evaluate machine alternatives in this study. It is known that the situation that criteria may affect criteria or alternatives and alternatives may affect criteria or alternatives requires building a network to calculate criteria weights and select the alternative more precisely. ANP allows for complex interrelationships among decision levels and factors and to build a network [45]. It is the reason that why ANP and fuzzy ANP (due to uncertainty in decision making) is preferred to determine factors weights in the proposed method. In the PROMETHEE method, six different functions can be chosen by decision makers to evaluate alternatives’ performance. This feature, compared to other methods, is a major concern affecting the PROMETHEE method to be able to model more accurately the way human decide and reach more realistic results for the decision problem [11, 21]. In this study, to solve the machine selection problem, the PROMETHEE method is selected due to its capacity to approximate the way that human mind expresses and synthesizes preferences when facing multiple contradictory decision perspectives.

In contrast to existing studies, this paper breaks away from the literature with following scientific contributions: (i) considering four main criteria, namely cost (C), quality (Q), flexibility (F), performance (P) with the corresponding fourteen sub-criteria; (ii) providing a MCDM model presenting a hybrid approach of the fuzzy ANP and PROMETHEE to select the most suitable machine tool and (iii) comparing proposed hybrid approach with TOPSIS method.

3 Proposed integrated approach with uncertainties

In this section, after describing the FANP and ELECTREE methods, proposed integrated approach is introduced.

3.1 ANP and FANP methods

The AHP allows for complex interrelationships among decision levels and attributes [45]. The ANP feedback approach replaces hierarchies (Fig. 1a) with networks (Fig. 1b) in which the relationships between levels cannot be easily represented as higher or lower, dominant or subordinate, direct or indirect [28].

In ANP, the modelling process can be divided to three steps, which are described as follows [30 , 57]:

Step 1: The pairwise comparisons and relative weight estimation

Before performing the pairwise comparisons, all criteria and clusters compared are linked to each other. The pairwise comparisons are made depending on the scale shown on Table 1. In the pairwise comparison matrix, the score of a_ij represents the relative importance of the component on row (i) over the component on column (j), i.e. a_ij = w_i/w_j. The reciprocal value of the expression (1/a_ij) is used when the component j is more important than the component i. The comparison matrix A is defined as

$\begin{matrix} A & = & [\begin{matrix} w_{1} / w_{1} & w_{1} / w_{2} & \dots & w_{1} / w_{n} \\ w_{2} / w_{1} & w_{2} / w_{2} & \dots & w_{2} / w_{n} \\ ⋮ & ⋮ & ⋮ \\ w_{n} / w_{1} & w_{n} / w_{2} & \dots & w_{n} / w_{n} \end{matrix}] \\ = & [\begin{matrix} 1 & a_{12} & \dots & a_{1 n} \\ 1 / a_{12} & 1 & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋮ \\ 1 / a_{1 n} & 1 / a_{2 n} & \dots & 1 \end{matrix}] \end{matrix}$ (1)

Then, a local priority vector (eigenvector) w is computed as an estimate of the relative importance accompanied by the elements being compared by solving the following equation: $Aw = λ_{max} w$ (2) where λ_max is the largest eigenvalue of matrix A.

Step 2: Formation of the initial supermatrix

The obtained vectors are further normalized to represent the local weight vector. Supermatix is formed, local weight vectors are entered in the appropriate columns of the matrix of influence among the elements, to obtain global priorities. The supermatrix representation of a network with three levels is given as follows (Fig. 1b): $W = \begin{matrix} Goal (G) \\ Criteria (C) \\ Alternatives (A) \end{matrix} \begin{matrix} G C A \\ (\begin{matrix} 0 & 0 & 0 \\ W_{21} & W_{22} & 0 \\ 0 & W_{32} & I \end{matrix}) \end{matrix}$ (3) where W₂1 is a vector that represents the impact of the goal on the criteria, W₂2 is a vector that represents impact of the interdependences among criteria, W₃2 is also a vector that represents the impact of criteria on each of alternatives, and I is the identity matrix. Any zero value in the super-matrix can be replaced by a matrix if there is an interrelationship of elements within a cluster or between to clusters.

Step 3: Formation of the weighted super-matrix

An eigenvector is obtained from the pair-wise comparison matrix of the row clusters with respect to the column cluster, which in turn yields an eigenvector for each column cluster. The first entry of the respective eigenvector for each column cluster is multiplied by all the elements in the first cluster of that column, the second by all the elements in the second cluster of that column and so on. In this way, the cluster in each column of the super-matrix is weighted, and the result, known as the weighted super-matrix, is stochastic. DM cannot always explain his judgments about certain attributes, quality, performance, etc., with discrete scales. For these reasons fuzzy scales are defined. In the application, triangle fuzzy numbers have been used by DM to state their preferences to compare attributes.

In the proposed methodology, pair-wise comparison matrices are formed with the help of triangle fuzzy numbers shown in Fig. 2, the FANP has been used to determine weights of machine selection criteria. The FANP is good at accommodating the interrelationships among the functional activities. For obtaining the composite weights that overcome the existing relationships the concept of supermatrices is used.

Pairwise comparison matrices are structured by using triangle fuzzy numbers (l, m, u). The fuzzy matrix can be given as follows: $\begin{matrix} \tilde{A} \\ [- 2 pt] & = (\begin{matrix} (a_{11}^{l}, a_{11}^{m}, a_{11}^{u}) & (a_{12}^{l}, a_{12}^{m}, a_{12}^{u}) & . . . & (a_{1 n}^{l}, a_{1 n}^{m}, a_{1 n}^{u}) \\ (a_{21}^{l}, a_{21}^{m}, a_{21}^{u}) & (a_{22}^{l}, a_{22}^{m}, a_{22}^{u}) & . . . & (a_{2 n}^{l}, a_{2 n}^{m}, a_{2 n}^{u}) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ (a_{m 1}^{l}, a_{m 1}^{m}, a_{m 1}^{u}) & (a_{m 1}^{l}, a_{m 1}^{m}, a_{m 1}^{u}) & . . . & (a_{mn}^{l}, a_{mn}^{m}, a_{mn}^{u}) \end{matrix}) \end{matrix}$ (4)

The a_mn represents the of comparison m (row) with component n (column). The pair-wise comparison matrix (Ã) is assumed as reciprocal. $\begin{matrix} \tilde{A} \\ = (\begin{matrix} (1, 1, 1) & (a_{12}^{l}, a_{12}^{m}, a_{12}^{u}) & . . . & (a_{1 n}^{l}, a_{1 n}^{m}, a_{1 n}^{u}) \\ (\frac{1}{a_{12}^{u}}, \frac{1}{a_{12}^{m}}, \frac{1}{a_{12}^{l}}) & (1, 1, 1) & . . . & (a_{2 n}^{l}, a_{2 n}^{m}, a_{2 n}^{u}) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ (\frac{1}{a_{1 n}^{u}}, \frac{1}{a_{1 n}^{m}}, \frac{1}{a_{1 n}^{l}}) & (\frac{1}{a_{2 n}^{u}}, \frac{1}{a_{2 n}^{m}}, \frac{1}{a_{2 n}^{l}}) & \dot{.} . & (1, 1, 1) \end{matrix}) \end{matrix}$ (5)

In this study, logarithmic least squares method is used for getting estimates for fuzzy priorities ${\tilde{w}}_{i}$ . The logarithmic least squares method for calculating triangular fuzzy weights can be given as follows [42]: $\tilde{W} = (W_{k}^{l}, W_{k}^{m}, W_{k}^{u}) k = 1, 2, 3, \dots, n$ (6) where $W_{k}^{s} = \frac{{(\prod_{j = 1}^{n} a_{kj}^{s})}^{1 / n}}{\sum_{i = 1}^{n} {(\prod_{j = 1}^{n} a_{ij}^{m})}^{1 / n}}, s \in {l, m, u}$ (7)

Then, crisp values for fuzzy weights are calculated as follows [13, 43]: $Crisp value = \frac{l + 2 m + u}{4}$ (8)

ANP can be used to calculate the relative importance of the criteria and outrank the alternatives. In our proposed model, FANP will be used only to calculate the triangular fuzzy weights for the relative importance of the criteria and the interdependence priorities of the criteria. Equation (9) will be used to support PROMETHEE for outranking the alternatives. $W = (\begin{matrix} 0 & 0 \\ W_{21} & W_{22} \end{matrix})$ (9)

3.2 The PROMETHEE method

The PROMETHEE is a multi-criteria decision-making method introduced in 1985 and developed in 1986 [7, 8]. It is a quite simple ranking method in conception and application compared with other methods used for multi-criteria analysis. Outranking principle is used to rank the alternatives which are combined with the ease of use and decreased complexity by this method. According to a number of criteria it performs a pairwise comparison of alternatives to put them in order. It is well adapted to problems where a finite number of alternatives are to be ranked according to several, sometimes conflicting criteria [1]. The evaluation table is the starting point of the PROMETHEE method. In this table, the alternatives are evaluated on the different criteria.

The implementation of PROMETHEE requires two additional types of information, namely: (1) Information on the relative importance that is the weights of the criteria considered. (2) Information on the DM’s preference function, which he/she uses when comparing the contribution of the alternatives in terms of each separate criterion. The weights coefficients can be determined according to various methods [29, 35].

The PROMETHEE method is appropriate to treat the multi-criteria problem of the following type: $max {f_{1} (a), f_{2} (a), \dots, f_{n} (a) | a \in A},$ (10) where A is a finite set of possible alternatives, and f_j denotes n criteria to be maximized. For each alternative, f_j (a) is an evaluation of this alternative. When we compare two alternatives a, b ∈ A, we must be able to express the result of these comparisons in terms of preference. We, therefore, consider a preference function P. The preference function translates the difference between the evaluations of two alternatives (a and b) in terms of a particular criterion, into a preference degree ranging from 0 to 1. Let $P_{j} (a, b) = G_{j} [f_{j} (a) - f_{j} (b)]$ (11) $0 \leq P_{j (a, b)} \leq 1$ (12) be the preference function associated to the criterion, f_j (i) where G_j is a non-decreasing function of the observed deviation (d) between f_j (a) and f_j (b). In order to facilitate the selection of specific preference function, six basic types of this preference function are proposed to decision maker (usual function, U-shape function, V-shape function, level function, linear function and Gaussian function) in each case no more than two parameters (threshold, q, p or s) have to fixed [7, 9, 55].

Indifference threshold q: the largest deviation to consider as negligible on that criterion. It is a small value with respect to the scale of measurement. Preference threshold p: the smallest deviation to consider decisive in the preference of one alternative over another. It is a large value with respect to the scale of measurement. Gaussian threshold s: it is only used with the Gaussian preference function. It is usually fixed as an intermediate value between indifference and a preference threshold. PROMETHEE permits the computation of the following quantities for each alternative aand b: $π (a, b) = \frac{\sum_{j = 1}^{n} w_{j} P_{j} (a, b)}{\sum_{j = 1}^{n} w_{j}}$ (13) $φ^{+} (a) = \sum_{x \in A} π (x, a)$ (14) $φ^{-} (a) = \sum_{x \in A} π (a, x)$ (15) $φ (a) = φ^{+} (a) - φ^{-} (a)$ (16)

For each alternative a, belonging to the set A of alternatives, π (a, b) is an overall preference index of a over b. The leaving flow φ⁺ (a) shown in Fig. 3 is the measure of the outranking character of a (how a dominates all the other alternatives of A). Symmetrically, the entering flow φ^- (a) shown in Fig. 3 gives the outranked character of a (how a is dominated by all the other alternatives of A). φ (a) represents a value function, whereby a higher value reflects a higher attractiveness of alternative a and is called net flow.

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

The PROMETHEE-I partial ranking provides a ranking of alternatives. In PROMETHEE-I, alternative a is preferred to alternative b, aPb, if alternative a has a greater leaving flow than that of alternative b and a smaller entering flow than the entering flow of alternative b: $\begin{matrix} aPb if : \\ φ^{+} (a) > φ^{+} (b) and φ^{-} (a) < φ^{-} (b) or \end{matrix}$

$\begin{matrix} φ^{+} (a) > φ^{+} (b) and φ^{-} (a) = φ^{-} (b); or \\ φ^{+} (a) = φ^{+} (b) and φ^{-} (a) < φ^{-} (b) . \end{matrix}$ (17)

PROMETHEE-I evaluation allows indifference and incomparability situations. Therefore, sometimes partial rankings can be obtained. In the indifference situation (aIb), two alternatives a and b has the same leaving and entering flows: $aIb if : φ^{+} (a) = φ^{+} (b) and φ^{-} (a) = φ^{-} (b)$ (18)

Two alternatives are considered incomparable, aRb, if alternative a is better than alternative b in terms of leaving flow, while the entering flows indicate the reverse: $\begin{matrix} aRb if : \\ φ^{+} (a) > φ^{+} (b) and φ^{-} (a) > φ^{-} (b) or \\ φ^{+} (a) < φ^{+} (b) and φ^{-} (a) < φ^{-} (b) \end{matrix}$ (19)

PROMETHEE-II provides a complete ranking of the alternatives from the best to the worst one. Here, the net flow (φ) is used to rank the alternatives. The alternative with the higher net flow is assumed to be superior. Since PROMETHEE-I does not provide a complete ranking, resulting ranking cannot be compared with the ranking provided by PROMETHEE-II. PROMETHEE-I ensure creation of indifferent and incomparable alternatives. In some ranking problems, PROMETHEE-I can give a complete ranking depending on the evaluation matrix values and, this ranking cannot be different from the one achieved with PROMETHEE-II.

The geometrical analysis for interactive aid (GAIA) plane displays the relative position of the alternatives graphically, in terms of contributions to the various criteria [7, 9]. Principal components analysis is applied to the matrix of “normed flows”, defined for alternative a and criterion j by: $φ_{j} (a) = \frac{1}{n - 1} \sum_{b \neq a} [P_{j} (a, b) - P_{j} (b, a)]$ (20) where n is the number of alternatives, and this is used to generate a two-dimensional plot in which the alternatives and criteria are represented in the same plan [6].

There are some studies in the literature which consider the PROMETHEE. The PROMETHEE method is used in the ranking of alternative energy exploitation projects [17]. In addition to these studies, other researchers used PROMETHEE method for information systems outsourcing [55], the selection of the best compromise management scheme for end of life vehicles [29], the evaluation of alternative energy exploitation projects [24] and the outsourcing management [2]. SOM and PROMETHEE techniques are used for earthquake-induced landslide hazard monitoring and assessment in Central Taiwan [25]. A preventive maintenance decision model based on PROMETHEE-II integrated with Bayesian approach is developed [11].

3.3 The proposed approach

The integrated approach -combination of FANP and PROMETHEE methods- for the machine selection problem consists of 3 basic stages: (1) research, (2) FANP computations, (3) PROMETHEE computations and ranking. In the first stage, criteria and sub-criteria used for evaluating machines are derived first from the literature and experts’ opinions. In the criteria determination phase, related literature, especially, some papers are utilized [4 , 51]. Alternative machines are determined and the hierarchical structure is formed. In the last step of the first stage, the decision model is approved by experts.

Following determining the hierarchical structure and the approval of decision model, evaluation of criteria with FANP is realized in the second stage. In this phase, pairwise comparison matrices are formed to determine the criteria weights. The experts make individual evaluations to compare the criteria linguistically. Then the linguistic evaluations are converted to triangular fuzzy numbers (TFN). Computing the geometric mean of the values obtained from individual evaluations, a final pairwise comparison matrix on which there is a consensus is found. The local weights of the criteria and sub-criteria by using pairwise comparison matrices and the inner dependence matrix of each criterion with respect to the other criteria are determined. This inner dependence matrix is multiplied with the local weights of the criteria to compute the interdependent weights of the criteria. Global sub-criteria weights are computed by multiplying local weight of the sub-criteria with the interdependent weights of the criterion to which it belongs. At the last step of this phase, after calculated weights of the criteria are approved by decision making team, using Equation (8), and the crisp values for fuzzy weights are calculated.

After the determination the relative weights of qualitative criteria, these weights are used as coefficients of PROMETHEE model in the third stage. Preference functions and parameters to be used for PROMETHEE computations are determined by the decision making team. After the approval of the functions -partial ranking with PROMETHEE-I- complete ranking with PROMETHEE-II and GAIA plane are determined by using Decision Lab software. In the last step of the proposed procedure, the best machine is selected according to the ranking results. Schematic representation of the proposed approach is presented in Fig. 4.

4 A real case application

Thanks to technological advances, increasing taxes and customer demand the manufacturing industry in Turkey has gone through a number of significant changes in the last years. The changes have forced companies to increase efficiency, decrease cost, change their old machines, etc. The proposed machine selection model was applied in a manufacturing company, located in Ankara, Turkey. The company wants to purchase a few router machines (RM) to increase its efficiency and reduce cost at long terms by replacing its old machines. The high technology machines make significant improvements in the manufacturing processes of the firms and the correct decisions made at this stage brings the companies competitive advantage. Therefore, selecting the most proper router machine has a great importance for the company. But it is hard to choose the most suitable one among alternative RMs which dominate each other in different characteristics.

4.1 Research phase

In the application, firstly the experts, who will take a part in machine selection process, are determined. With a preliminary work, experts determine six possible RMs suitable for the needs of the company. In the problem addressed, there are four main criteria, namely cost (C), quality (Q), flexibility (F), performance (P) with the corresponding fourteen sub-criteria shown in Table 2. For example, cost depends on four sub-criteria such as capital (CC), operation (OC), maintenance cost (MC) and depreciation (D) while quality depends on three sub-criteria namely scrap & rework (SR), reliability (R) and product conformance (PC). Interdependencies among main criteria are provided in Fig. 5 and decision hierarchy structured with the determined alternative machines.

As it seen in Fig. 5, there are four levels in the decision hierarchy structured for machine selection problem. In the first level of the hierarchy, there is the overall goal of the decision process which is determined as “the selection of the best machine”. The criteria are on the second level, the sub-criteria are on the third level and alternative RMs are on the fourth level of the hierarchy. The criteria of second level are connected to the goal with a single directional arrow. The arrows shown in Fig. 5 represent the interdependence among the criteria. The interdependencies among criteria which are in this level are taken into account and by this way the effects of the criteria on each other are analyzed. The arrow from flexibility to quality means flexibility effects quality; cost is affected by flexibility, quality and performance. Sub-criteria related to the criteria are in the third level of the model and the criteria determined before are also in this level.

4.2 FANP phase

After forming the decision hierarchy for machine selection problem, the weights of criteria to be used in evaluation process are assigned by using FANP method. In this phase, the experts are given the task of forming individual pairwise comparison matrix by using the scale given in Table 1. Geometric means of these values are found to obtain the pairwise compassion matrix on which there is a consensus. The details of this phase are explained below.

Step 1: Local weights of the criteria and sub-criteria are calculated. Pairwise comparison matrices are formed by the experts. For example, the cost criteria are compared with quality criteria using the question “Which is considered more important by the experts selecting the machine, and how much more important is it with respect to satisfaction with the machine? All the evaluation matrices are produced in the same manner. Pairwise comparison matrices for criteria and sub-criteria of cost are given in Tables 3 and 4 together with the calculated local weights.

Step 2: In this step, interdependent weights of the criteria are calculated and the dependencies among the criteria are considered. Dependence among the criteria is determined by analyzing the impact of each criterion on every other criterion using pairwise comparisons. The dependencies between criteria are shown at Fig. 5. These dependencies are determined by an expert team on the basis of a group study and following statements are obtained: a) “flexibility criteria” affect “quality criteria”, b) Cost and performance affect each other, etc. Based on the dependencies presented above, expert team defined dependence among all criteria via pairwise comparison matrices. For this purpose, four pairwise comparison matrices are formed for C, Q, F and P criteria. The results of relative importance weights of these matrices are calculated. These weights are listed in Table 5. The value “(0, 0, 0)” presented in Table 5 means that there is no dependence between two criteria and the numerical values show the degree of relative impact between two criteria. For example, the C’s degree of relative impact for F is (0.30, 0.33, 0.38). $\begin{matrix} w_{criteria} & = & [\begin{matrix} C \\ Q \\ F \\ P \end{matrix}] = [\begin{matrix} (0, 0, 0) (0.13, 0.14, 0.16) (0.30, 0.33, 0.38) (0.11, 0.12, 0.13) \\ (0.45, 0.59, 0.70) (0, 0, 0) (0, 0, 0) (0.58, 0.69, 0.79) \\ (0.13, 0.14, 0.18) (0.17, 0.19, 0.23) (0, 0, 0) (0.16, 0.19, 0.23) \\ (0.21, 0.27, 0.34) (0.55, 0.67, 0.72) (0.57, 0.67, 0.73) (0, 0, 0) \end{matrix}] \times [\begin{matrix} (0.10, 0.12, 0, 15) \\ (0.33, 0.40, 0.54) \\ (0.12, 0.14, 0.18) \\ (0.25, 0.33, 0.42) \end{matrix}] \\ w_{criteria} & = & [\begin{matrix} C \\ Q \\ F \\ P \end{matrix}] = [\begin{matrix} (0.10, 0.14, 0.21) \\ (0.19, 0.30, 0.43) \\ (0.11, 0.16, 0.25) \\ (0.27, 0.39, 0.57) \end{matrix}] \end{matrix}$

Relative importance of the criteria on the basis of interdependence can be calculated by using the data given in Tables 3–5 as above. According to the results, P is the most important considering criteria relating to machine preference. Significant differences are observed in the results obtained for the criteria priorities when interdependent priorities of the criteria (w_criteria) and dependencies are not taken into account. For example, the result for quality changes from (0.33, 0.40, 0.54) to (0.19, 0.30, 0.43).

Step 3: Using interdependent weights of the criteria and local weights sub-criteria, global weights for the sub-criteria are calculated in this step. Global sub-criteria weights are computed by multiplying local weight of the sub-criterion with the interdependent weight of the criteria to which it belongs. Computed global weights and crisp values for sub-criteria are shown in Table 6. The results reveal that the three most significant sub-criteria are SCA, R, PC, and then their weights are 0.153, 0.144, and 0.137, respectively.

Meanwhile, the three least important sub-criteria are: D, CT and SR.

4.3 PROMETHEE phase

Step 4: After the FANP evaluations, PROMETHEE calculations are realized. Alternative machines are evaluated based on the evaluation criteria and the evaluation matrix is formed. As a first step of PROMETHEE, a specific preference function (PF) for each criterion with its thresholds is defined.

Preference functions and threshold values have been defined by experts. CC, OC, MC, D, CT and MR are quantitative sub-criteria. The others are qualitative and experts have set the other values 1 through 10 by taking into consideration the features of alternative machines. The preference functions, thresholds and evaluations of the six alternatives according to criteria are provided in Table 7.

Step 5: After evaluation matrix and preference functions are determined, alternative machines are evaluated by using Decision Lab software. Alternatives’ positive flows (φ⁺), negative flows (φ^-) and net flows (φ) are calculated as shown in Table 8. Then, partial ranking of alternatives are found via PROMETHEE-I as shown in Fig. 6. PROMETHEE-I used positive and negative flow values to find the partial ranking. Based on this partial ranking, RM-2 determined as the worst alternative; RM-5 is preferred to RM-4, RM-1 and RM-2. According the results that obtained with PROMETHEE-I, RM-3, RM-5 and RM-6 cannot be compared. The reason of this situation for RM3 and RM6 is that, RM3 is superior to RM6 in the positive flow comparison, but, RM6 is superior to RM3 in the negative flow comparison. To compare these two alternatives, the net flow should be calculated in PROMETHEE-II. PROMETHEE-I did not provide information about the best alternative. Net flow values given in the fourth column of Table 8, are used in PROMETHEE-II complete ranking to identify the best alternative. At the last phase of calculations, the complete ranking of alternatives is determined as shown in Fig. 7. RM-5 is the best, RM-2 is the worst alternative under the desired requirements of machine properties, main and sub-criteria; and then the other alternatives are ranked in the order of RM-3, RM-6, RM-4, and RM-1. The ranking, the whole criteria have equal weights, for alternative machines are also shown in Table 8.

Step 6: The decision problem can be represented in the GAIA plane as in Fig. 8 where alternative machines are represented by points and criteria by vectors. In this way, conflicting criteria may appear clearly. Criteria vectors expressing similar preferences 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 alternative machines differentiation.

4.4 Results and discussion

This plane is the result of principal component analysis (PCA), projecting the 14-dimensional space of criteria onto a two-dimensional plane, i.e. the 14 original variables are transformed to the two new variables that are obtained by two linear combinations of the original variables. By applying the PCA related criteria are handled by these combinations and double counting never occurs [1]. As it is shown in the Fig. 8, the Delta-parameter is 94.19% ; this means only 5.81% of the total information gets lost by the projection.

We observe that SCA and FMP have a high differentiation power and expresses independent preferences, different from those expressed by most of all other criteria. A cluster of conflicting criteria (OC and VFP expressing opposite preferences) are clearly represented. It is also possible to appreciate clearly the quality of the alternative machines with respect to the different criteria. RM-3 is particularly good on FMP and U. RM-5 and RM-4 are good on SCA and PC. Vector pi (decision axis) represents the direction of the compromise derived from the assignment; the decision maker is invited to appreciate the alternative machines located in that direction [55]. It can be seen from Fig. 8 that pi vector is in the direction of criterion FMP and PC and the closest alternatives to the pi vector are RM-3, RM-6 and RM-5. This result is consistent with the complete ranking of PROMETHEE-II. According to the fuzzy ANP and PROMETHEE computations, it is decided to prefer RM-5, RM-3, RM-6, RM-4, RM-1, RM-2, respectively. Figure 9 illustrates the selected machine (RM-5) among the alternatives.

In addition to PROMETHEE, TOPSIS method is also applied to rank the alternatives to make a comparison. Table 9 indicates the ranking order obtained by TOPSIS. The readers can find the detail formulation of applied TOPSIS method in the study of Kabak et al. [21].

The similarity of rankings is assessed on the basis of Spearman’s rank correlation coefficient, defined as [31]: $r_{s} = 1 - \frac{6 \sum_{i = 1}^{n} d_{i}^{2}}{n (n^{2} - 1)}$ (21) where n denotes the number of alternatives, d_i is the difference between the ranks of alternatives in the pair of rankings compared and r_s is the Spearman rank correlation coefficient. Using the values in Table 10, the r_s value is 0.83 that indicates a stronger association between the two sets of ranking. We should point out that our interest does not lie in studying the computational properties of the model, or investigating which MCDM method is better, but rather in shedding light on the effect of the changes in applied method on the final rank. Although similar outcomes can be observed from TOPSIS, company owners accept the rank of PROMTHEE method.

Besides the comparison of PROMETHEE and TOPSIS methods, an extra analyze is done by changing the current weights of each criterion to see effects of each criterion on machine ranking. To do so, each criterion is decreased and increased step by step. Table 11 shows the results of changing the criteria weights. According to the Table 11, increasing the weights of CC, MC, D, R, PC, S, SCA and MR has no effect on machine ranking. However, increasing the weights of the rest of the criteria (OC, SR, FMP, VFP, CT and U) creates a new order which is given in Table 11. Increased weights column indicates the new weights of that criterion which changes the machine rank.

In addition to increasing the weights, the weights of criteria are also decreased. Decreasing the weights of OC, FMP, VFP and SCA changes the machine ranking which is shown in decreased weights ranking column. For instance, Fig. 10 shows the sensitivity analyze of SCA criterion. It illustrates the new machine rank and also updated weights of other criteria. As it seen from Fig. 10, decreasing the weight of SCA from 0.153 to 0.09, the ranking of machines is updated as RM-3, RM-5, RM-6, RM-4, RM-1 and RM-2. This analyzes shows that the most dominants criteria are operation cost (OC), flexibility in mass production (FMP) and variety and flexibility of products (VFP) among other criteria. In the ranking frame, it is clear that while RM-5 and RM-3 are mostly at the top of the list; RM-1 and RM-2 are at the end of the list in all alternative ranks.

5 Conclusion and future research

Selecting the most appropriate machine from a number of available machines in the market is a challenging task. Cost, quality, flexibility and performance of the company depend on machine properties. In this paper, a decision approach is provided for machine selection problem and FANP-PROMETHEE based methodology is proposed. FANP is used to assign weights to the criteria to be used in machine selection, while PROMETHEE is employed to determine the priorities of the alternatives. In particular, firstly, a DM team works together to determine the attributes and a set of potential alternatives. Secondly, fuzzy ANP is used to determine the interaction and weights of the attributes. Then, the ranking of alternatives is evaluated based on PROMETHEE method. Lastly, a comparison with another MCDM method and a sensitivity analysis is conducted to verify the ranking and further support the decision when selecting the final solution. In general, the proposed methodology is very flexible in the sense that it can be applied to other types of selection problems, e.g., selection of vehicle, equipment, the material handling system, robots, project, appliances, etc.

Main contributions of the study to the literature are (i) considering four main criteria with the corresponding fourteen sub-criteria; (ii) providing a MCDM model presenting a hybrid approach of the fuzzy ANP and PROMETHEE and (iii) comparing proposed hybrid approach with TOPSIS method.

The proposed methodology has some advantages comparing the previously proposed methodologies. The vagueness embedded in this decision making area may easily incorporated into the decision making process with this methodology. FANP method is used to reflect the weights of qualitative criteria and integrate the various expectations from different evaluators into evaluating the machines. The FANP-PROMETHEE integrated method is well suited to deal with multi-criteria decisions that involve both qualitative and quantitative criteria. PROMETHEE method takes into account the preference function of each criterion, determined by the decision-makers. By this way, each criterion is evaluated on a different basis and it is possible to make better decisions. PROMETHEE-I identifies the alternatives which cannot be compared and the alternatives which are indifferent, by making a partial ranking, while PROMETHEE-II provides a complete ranking for alternatives. The GAIA plane is a useful analytical tool that some remarks can be detected from the alternatives and criteria sets.

It is acknowledged that there are some limitations and shortcomings of the research. However, these shortcomings should be handled for future studies. First, PROMETHEE method which is used to rank the alternatives in this study is applied as crisp. PROMETHEE method should be considered as fuzzy to reflect the uncertainties in ranking phase. Second, four-main criteria with corresponding fourteen sub-criterions are considered to select a machine in this study. In addition to mentioned criteria, social and environmental impacts of machines should be considered. Finally, no data base or expert systems are used in this study. Therefore, expert systems to reduce the number of judgments in the pairwise comparison matrix should be embedded for future studies.

Footnotes

Acknowledgments

The authors express their gratitude to the three anonymous reviewers and area editor for their valuable comments on the paper.

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