Extension of Fuzzy ELECTRE I for Evaluating Demand Forecasting Methods in Sustainable Manufacturing

Abstract

The selection of a demand forecasting method is critical for companies aiming to avoid manufacturing overproduction or shortages in pursuit of sustainable development. Various qualitative and quantitative criteria with different weights must be considered during the evaluation of a forecasting method. The qualitative criteria and criteria weights are usually assessed in linguistic terms. Aggregating these various criteria and linguistic weights for evaluating and selecting demand forecasting methods in sustainable manufacturing is a major challenge. This paper proposes an extension of fuzzy elimination and choice translating reality (ELECTRE) I to resolve this problem. In the proposed method, fuzzy weighted ratings are defuzzified with the signed distance to develop a crisp ELECTRE I model. Moreover, an extension to ELECTRE I is developed by suggesting an extended modified discordance matrix and a closeness coefficient for ranking alternatives. The proposed extension can overcome the problem of information loss, which can lead to incorrect ranking results when using the Hadamard product to combine concordance and modified discordance matrices. A comparison is conducted to show the advantage of the proposed extension. Finally, a numerical example is used to demonstrate the feasibility of the proposed method. Furthermore, a numerical comparison is made to display the advantage of the proposed method.

1. Introduction

Manufacturing is a key driver of growth and global development and is a major contributor to the creation of prosperity and employment, especially in growing economies. However, industrial activities have a substantial environmental burden [1]. The key considerations in sustainable manufacturing (green manufacturing) are the efficient use of resources through the enhancement of resource productivity [2] and minimization of waste [3]. Therefore, to avoid overproduction or shortages during production, accurately forecasting demand is necessary to ensure the production of an adequate number of intermediate parts and final products. Manufacturing systems based on advanced forecasting subsystems play a key role in supply chains and can facilitate environmental protection and long-term sustainable development [3].

A forecasting method is a method of predicting the solution to a task to enable users to accurately predict outcomes [4]. A poor manufacturing forecast could cause a buildup of product stock, leading to increasing part ordering and holding costs [5]. Therefore, demand forecasting is a crucial element of the planning process for companies with the goal of sustainable development. More than 200 forecasting methods are described in the economic literature, and these methods can be classified on the basis of the following criteria: type of information, forecast time span, forecast object, and forecast goal [4]. Dweiri et al. [6] used the analytic hierarchy process (AHP) to select a production planning forecasting method in a supply chain. Dahooie et al. [7] provided a hybrid method of the fuzzy MULTIMOORA approach for multi-criteria decision making and the objective weighting method (CCSD) to select a forecasting method for technology. Various qualitative criteria, such as ease of use and data validity, and quantitative criteria, including implementation cost and forecast accuracy, must be considered when evaluating demand forecasting methods, as these criteria may differ in importance. Evaluating demand forecasting methods is thus a multiple criteria decision-making (MCDM) problem. Therefore, how to aggregate various criteria and their weights to select the most suitable demand forecasting method is a key challenge in forecasting research. To overcome this challenge, this paper proposes an extension of the fuzzy ELECTRE method for selecting the best demand forecasting method.

The ELECTRE method (Roy [8]) is an MCDM method based on outranking relations. An advantage of the ELECTRE method is that it achieves a more realistic decision-making process by including both the criteria weights and the preferences of the decisionmaker in the selection process (Singh and Kaushik [9]). The ELECTRE method is one of the most effective decision-making techniques; the method outputs a reduced set of suitable alternatives by using outranking relations to remove options outranked by other options (Akram [10]). Several versions of the ELECTRE method have been proposed, namely ELECTRE I, II, III, IV, IS, and TRI. Among these versions, ELECTRE I involves a choice problem and attempts to select a small group of favorable alternatives to facilitate the ultimate selection of a single alternative (Zandi and Roghanian [11]). However, because some decision makers provide their opinions using linguistic terms, the performance ratings and criteria weights in the ELECTRE method cannot be measured precisely (Hatami-Marbini and Tavana [12]). Scholars have investigated combining ELECTRE I with fuzzy set theory for addressing the imprecise or vague nature of linguistic assessments. Shojaie et al. [13] used fuzzy ELECTRE to evaluate green health suppliers, and they conducted a case study on Tehran Chemie Pharmaceutical Company. Akram et al. [14] introduced bipolar fuzzy TOPSIS and bipolar fuzzy ELECTTRE I to determine a disease that explains a patient’s symptoms. Moreover, Massami and Manyasi [15] used fuzzy ELECTRE to determine the importance of various criteria and subcriteria for evaluating the performance of sailors.

To the best of our knowledge, fuzzy ELECTRE I’s application for selecting demand forecasting methods has never been studied before. To fill this gap, this paper proposes an extension of fuzzy ELECTRE I for selecting the most suitable demand forecasting method. In the proposed method, the membership function of the fuzzy weighted rating of each alternative for each qualitative criterion is defuzzified by applying the signed distance (Yao and Wu [16]) to form a crisp ELECTRE I model. Defuzzification formulas can be precisely derived to improve the model for assisting in decision making. In addition, the proposed ELECTRE I model uses a closeness coefficient, derived on the basis of an expanded modified discordance matrix, to rank alternatives. The proposed closeness coefficient can resolve the problem of information loss, which can otherwise lead to incorrect ranking results when the Hadamard product is used to combine the concordance matrix and modified discordance matrix. A comparison with some other methods will be used to present the advantage of the proposed extension based on the closeness coefficient. Moreover, a numerical example will be provided to display the feasibility of the proposed method. In addition, a numerical comparison with other methods will be conducted to display the advantage of the proposed method. The rest of this paper is organized as follows. Section 2 presents a review of the literature. Section 3 introduces the basic concepts of fuzzy set theory. Section 4 describes the model establishment process, with Section 4.1 presenting a comparison of the developed extension with other approaches to demonstrate the advantages of the proposed closeness coefficient using an expanded modified discordance matrix. Section 5 provides a numerical example to demonstrate the feasibility of the proposed method, with Section 5.1 presenting a numerical comparison and analysis to display the advantage of the proposed method. Finally, Section 6 addresses the conclusion.

2. Literature Review

2.1. Sustainable Manufacturing

Since the 1980s, the core goal of sustainable manufacturing has been waste reduction, and the aim of cleaner manufacturing is to increase available resources and reduce energy usage in manufacturing (Seliger et al. [17]). Recently, sustainable manufacturing has been defined as “the ability to smartly use natural resources for manufacturing, by creating products and solutions that, thanks to new technology, regulatory measures and coherent social behaviours, are able to satisfy economical, environmental, and social objectives, thus preserving the environment while continuing to improve the quality of human life” (Garetti and Taisch [18]). The pursuit of sustainability affects operations and manufacturing activities in which input materials and energy are converted into commercial products (Haapala et al. [1]). Moreover, materials and equipment that are adaptable to various situations are required for flexible manufacturing, which is responsive to variations in material flows, and flexible manufacturing can enhance sustainability while maintaining competitiveness (Rosen and Kishawy [19]). Owing to the increased complexity and performance expectations in supply chains for high-tech products, forecasting product demands is now key for efficiently managing operations (Dweiri et al. [6]). Therefore, an accurate demand forecasting method that can avoid overproduction or shortages and facilitate sustainable manufacturing should be the cornerstone of a sustainable supply chain. Forecasting methods in sustainable manufacturing have drawn the attention of numerous scholars in various fields. Hart et al. [20] introduced effective manufacturing systems for supply chains based on demand forecasting. Rivera-Castro et al. [21] presented diagonal feeding, a useful technique for forecasting build-to-order lean manufacturing supply chains. A review of the literature regarding sustainable manufacturing strategies can be found in the article by Garetti and Taisch [18].

2.2. Demand Forecasting Method Selection

Demand forecasting is critical for industrial firms because many decision-making processes require accurate forecasts for the selection of appropriate strategies for sales budgeting, production planning, new product launches, and other business activities (Choudhury [5]). Moreover, accurate demand forecasts are necessary to create an effective master plan that can facilitate all managerial processes involved in internal and external material flows, enabling comprehensive supply chain management (Hart et al. [20]). Forecasting methods can be classified as quantitative methods, including the simple moving average method and the exponential smoothing method, and qualitative methods, including the Delphi method and the nominal group technique. Various categories of forecasting methods are available to businesses. Therefore, selecting an appropriate method is key. Qualitative forecasting methods are based on experts’ opinions and may thus be marred by several biases. In contrast, quantitative methods analyze previously acquired data, which may not be applicable if the business environment changes substantially (Dweiri et al. [6]). Combining qualitative and quantitative methods could achieve substantially improved results compared with those produced with a single approach (Dweiri et al. [6]). Therefore, various quantitative criteria, such as accuracy and maintenance cost, and qualitative criteria, including method adaptability and data validity, are necessary for selecting the optimal demand forecasting method. The weights of these criteria are also necessary for this selection process. Accordingly, evaluating demand forecasting methods is an MCDM problem. Selecting the best forecasting method is a critical task for many manufacturers, and the inappropriate selection of a forecasting method can result in reduced sales and market share. To overcome this challenge, this paper proposes an extension of fuzzy ELECTRE for selecting the most suitable method. The selection of forecasting methods has been investigated by researchers in various fields. For example, Acar and Gardner [22] used tradeoff curves, considering total costs and customer service, to select the optimal forecasting method. Intepe et al. [23] used TOPSIS, an intuitionistic fuzzy environment method, to select the optimal forecasting method. Dahooie et al. [7] developed a hybrid method of the fuzzy MULTIMOORA approach for MCDM and the objective weighting method (CCSD) for selecting a forecasting method for technology. Furthermore, Taghiyeh et al. [24] proposed a new forecasting method selection scheme by considering intermediate classifications. Meira et al. [25] used a prediction interval to select and enhance the predictive power of forecasting models. Hanifi et al. [26] studied the literature regarding wind power forecasting with physical, statistical, and hybrid methods. Kuznietsova et al. [27] introduced a data technique for evaluating and forecasting roaming cell services in Ukraine.

2.3. Fuzzy ELECTRE Method

The ELECTRE method was developed in 1965 and is suitable for selecting the best action from a given set of actions. The ELECTRE family is one of the most powerful MCDM techniques based on outranking relations. An introduction to ELECTRE and ELECTRE TRI can be seen in the work of Roy and Bouyssou [28]. Fundamentally, the ELECTRE method eliminates options that are worse than other options by a specified degree (Akram et al. [10]). Because ELECTRE allows combining qualitative and quantitative information, it is considered a flexible method that requires less complicated information (Tolga [29]). Moreover, the ELECTRE method enables rating the alternatives for each criterion independently without aggregating the score of the alternatives for all criteria (Çalı and Balaman [30]). Among the methods in the ELECTRE family, ELECTRE I is one of the most widely used versions. The ELECTRE I method is applied to selection problems (Adeel et al. [31]), and its complexity can be easily increased through combination with other methods (Govindan and Jepsen [32]). Furthermore, when considering a choice problem in which a is preferred to options b and c, analyzing the preference between b and c becomes irrelevant; these two actions can remain completely unmatched without degrading the decision procedure, and therefore, the basic idea of this series of methods is to emphasize the analysis of dominance relations (Basilio, et al. [33]). This is also the reason why ELECTRE I is used in this study instead of other existing methods. However, the ELECTRE method lacks precise measurements for producing criteria weights and performance ratings (Hatami-Marbini and Tavana [12]) because exact (or crisp) numbers are often inadequate for describing real-life situations. Fuzzy set theory (Zadeh [34]) is an ideal solution for overcoming this problem in that it resembles human reasoning in its use of approximate information and uncertainty to generate decisions (Belbag et al. [35]). The core advantages of the fuzzy ELECTRE method can be summarized as being highly applicable and non-compensatory when criteria are described in the ordinal scale (Chhipi-Shrestha et al. [36]). Therefore, this paper combines the ELECTRE I method and fuzzy set theory to select the best demand forecasting method. Fuzzy ELECTRE has been investigated by researchers in various fields. Belbag et al. [35] used fuzzy ELECTRE to rank four smart phone brands on the basis of a survey of 250 students. Akram et al. [37] indicated that the Pythagorean fuzzy set model can effectively capture the vagueness in human evaluations and thus proposed a Pythagorean fuzzy ELECTRE I method. Ayyildiz et al. [38] proposed integrating the AHP and ELECTRE methods using interval type2 trapezoidal fuzzy ELECTRE to evaluate individual credit. Chen [39] developed an extension of the ELECTRE method by using novel Chebyshev distance measures as Pythagorean membership grades and applied it to bridgesuperstructure construction methods for validating feasibility and applicability. Wang and Chen [40] used a T-spherical fuzzy ELECTRE approach to select potential companies for extending the scope of a business. Some recent fuzzy MCDM works can be seen in the work of Badi et al. [41], Martin and Edalatpanah [42], Puška and Stojanović [43], and Su et al. [44]. However, fuzzy ELECTRE I has yet to be applied to select demand forecasting methods for sustainable manufacturing. To fill this gap, the study proposes an extension of fuzzy ELECTRE I for selecting the most suitable demand forecasting method.

In fuzzy ELECTRE I, the division of two fuzzy numbers is needed to produce a discordance matrix. However, the membership function produced by this division has not been precisely defined. Thus, a proper defuzzification method is necessary to produce the discordance matrix. Numerous ranking and defuzzification methods have been investigated. Peddi [45] proposed a defuzzification method for ranking fuzzy numbers based on centroids and maximizing and minimizing sets. The literature on defuzzification methods has a long history which can be seen in the works of Kataria [46], Kumar [47], and Talon and Curt [48]. Recently published works are described in the articles by Arman et al. [49] and Menïz [50]. Each method has advantages and disadvantages. In this paper, the signed distance (Yao and Wu [16]) method is used because it is simple and can be applied to both negative and positive fuzzy numbers. Moreover, this paper derives defuzzification formulas based on signed distance (Yao and Wu [16]) to derive an ELECTRE I model that can assist in decisionmaking. Zhang et al. [51] used the ELECTRE method to determine the ranking order of substrate nodes for resolving a virtual network embedding problem. They obtained the modified weighted summation matrix for ranking alternatives by using the Hadamard product to combine the concordance and modified discordance matrices. Despite the merits of the method proposed by Zhang et al. [51], it may have information loss that could lead to the production of an incorrect ranking order. Nghiem and Chu [52] suggested ranking sustainable conceptual designs by using a total net dominance value based on Nijkamp and Van Delft’s [53] net concordance dominance value and net modified discordance dominance value in order to avoid information loss in the method of Zhang et al. [51]. Moreover, Ke and Chen [54] suggested an ELECTRE method for selecting e-services. The Hadamard product of the concordance matrix and modified discordance matrix was used to obtain the modified total matrix for ranking alternatives. Despite the merits of their method, it also can produce an incorrect ranking due to information loss resulting from zero values in the modified discordance matrix when the Hadamard product is used. To resolve this problem from Ke and Chen [54], Nghiem and Chu [55] proposed subtracting discordance values from concordance values to obtain the total dominance matrix and produce the Boolean matrix to obtain ranking results, and they further applied it to develop a BWM-based fuzzy ELECTRE I method and evaluate lean facility layout designs. Nevertheless, the two suggested methods still exhibit the problem of information loss when the Hadamard product is used. To resolve this problem, the present study adopts a closeness coefficient based on an extended modified discordance matrix. Herein, the proposed extension is compared with the methods of Ke and Chen [54] and Zhang et al. [51] to demonstrate its advantages. Finally, a numerical example is used to show the feasibility of the proposed method. Furthermore, a numerical comparison is conducted with some other methods to display the advantages of the proposed fuzzy ELECTRE I method.

3. Fuzzy Set Theory

3.1. Fuzzy Sets

A fuzzy set $\tilde{A}$ can be denoted as $\tilde{A}=\left\{\left(x,f_{\tilde{A}}\left(x\right)\right)\left|x\in X\right.\right\}$, where $X$ is the universe of discourse. The fuzzy set $\tilde{A}$ in the universe of discourse $X$ is characterized by a membership function $f_{\tilde{A}}\left(x\right)$, $\forall x\in X$, $f_{\tilde{A}}\left(x\right)\in \left[0,1\right]$ (Kaufmann and Gupta [56]). The larger $f_{\tilde{A}}\left(x\right)$, the stronger is the grade of membership for $x$ in $\tilde{A}$. A fuzzy number is a fuzzy set.

The definition of a fuzzy number by Dubois and Prade [57] is described as follows. A fuzzy number $\tilde{A}$ is described as any fuzzy subset of the real line R with a membership function $f_{\tilde{A}}$ possessing the following properties: $f_{\tilde{A}}$ is a continuous mapping from R to [0, 1], $f_{\tilde{A}}\left(x\right)=0 \mathrm{for} \mathrm{all} x\in \left(-\infty ,\left.a\right]\right.$, $f_{\tilde{A}}$ is strictly increasing in the left membership function on $\left[a,b\right]$ and is strictly decreasing in the right membership function on $\left[c,d\right]$, $f_{\tilde{A}}\left(x\right)=1 \mathrm{for} \mathrm{all} x\in \left[b,c\right]$, and $f_{\tilde{A}}\left(x\right)=0 \mathrm{for} \mathrm{all} x\in \left[d,\left.\infty \right)\right.$, where a, b, c, and d are real numbers. We may let $a=-\infty$, a = b, b = c, c = d, or d = $+\infty$. Unless elsewhere defined, $\tilde{A}$ is assumed to be convex, normalized, and bounded (i.e., $-\infty <a$, $d<\infty$). $\tilde{A}$ can be indicated as $\left[a,b,c,d\right]$, $a\le b\le c\le d$. Let $f_{\tilde{A}}^L\left(x\right), a\le x\le b$ and $f_{\tilde{A}}^R\left(x\right), c\le x\le d$ represent the left and the right membership functions of $\tilde{A}$, respectively.

3.2. Triangular Fuzzy Numbers

Herein, triangular fuzzy numbers are used because they are intuitive and easy to calculate. A triangular fuzzy number is denoted as $\tilde{A}=\left(a^l,a^{\lambda },a^u\right)$, with the membership function $f_{\tilde{A}}\left(x\right)$ presented by Laarhoven and Pedrycz [58], and $\tilde{A}$ can be denoted as $\left[a^l,a^{\lambda },a^u\right]$ if $f_{\tilde{A}}\left(x\right)$ is nonlinear:

$$f_{\tilde{A}}(x)=\left\{\begin{array}{c}\begin{array}{cc}\frac{x-a^l}{a^{\lambda }-a^l},& \hfill a^l\le x\le a^{\lambda }\hfill \end{array},\\ \begin{array}{cc}\frac{x-a^u}{a^{\lambda }-a^u},& \hfill a^{\lambda }\le x\le a^u\hfill \end{array},\\ \hfill 0\begin{array}{c},\end{array}\begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}otherwise,\hfill \end{array}\right.$$

(1)

3.3. Arithmetic Operations on Fuzzy Numbers

Assume two fuzzy numbers ${\tilde{A}}_1$ and ${\tilde{A}}_2$. With the $\alpha$-cut $\left(0\le \alpha \le 1\right)$, the closed interval can be defined as ${\tilde{A}}_1^{\alpha }=\left[A_1^{L\left(\alpha \right)},A_1^{R\left(\alpha \right)}\right]$, ${\tilde{A}}_2^{\alpha }=\left[A_2^{L\left(\alpha \right)},A_2^{R\left(\alpha \right)}\right]$. Some main operations of two fuzzy numbers can be determined as follows (Kaufmann and Gupta [56]):

$${\left({\tilde{A}}_1\oplus {\tilde{A}}_2\right)}^{\alpha }=\left[A_1^{L\left(\alpha \right)}+A_2^{L\left(\alpha \right)},A_1^{R\left(\alpha \right)}+A_2^{R\left(\alpha \right)}\right]$$

(2)

$${\left({\tilde{A}}_1\ominus {\tilde{A}}_2\right)}^{\alpha }=\left[A_1^{L\left(\alpha \right)}-A_2^{R\left(\alpha \right)},A_1^{R\left(\alpha \right)}-A_2^{L\left(\alpha \right)}\right]$$

(3)

$${\left({\tilde{A}}_1\otimes {\tilde{A}}_2\right)}^{\alpha }=\left[A_1^{L\left(\alpha \right)}\times A_2^{L\left(\alpha \right)},A_1^{R\left(\alpha \right)}\times A_2^{R\left(\alpha \right)}\right], {\tilde{A}}_1,{\tilde{A}}_1\in R^{+}$$

(4)

$$z\otimes {\tilde{A}}_1^{\alpha }=\left[z\times A_1^{L\left(\alpha \right)},z\times A_1^{R\left(\alpha \right)}\right], z\in R^{+}$$

(5)

3.4. Ranking Fuzzy Numbers by Signed Distance

Fuzzy number ranking is one of the steps in a fuzzy MCDM model. Scholars have investigated various ranking methods, including Chen and Hong [59], Yu et al. [60], Nayagam et al. [61], De et al. [62], Aguilera et al. [63], and Hop [64]. Yao and Wu [16] proposed the use of the signed distance to rank fuzzy numbers. This method has been used by many researchers and can be applied to both negative and positive fuzzy numbers. The present study applies the signed distance because of its intuitive nature. The signed distance calculates the distance of the middle point of the left and right end point from the y axis. If a fuzzy number with the middle point is further away from the y-axis, then the fuzzy number receives the larger value. The signed distance of two fuzzy numbers ${\tilde{A}}_1$ and ${\tilde{A}}_2$ is calculated as follows:

$$d\left({\tilde{A}}_1,{\tilde{A}}_2\right)=d\left({\tilde{A}}_1,0_1\right)-d\left({\tilde{A}}_2,0_2\right)=\frac{1}{2}{\displaystyle \underset{0}{\overset{1}{\int }}\left[{\tilde{A}}_1^L+{\tilde{A}}_1^R-{\tilde{A}}_2^L-{\tilde{A}}_2^R\right]}d\alpha$$

(6)

3.5. Linguistic Values

A linguistic variable is one whose values are not numbers but are instead words or linguistic terms. The concept of a linguistic variable is useful in complex situations (Zadeh [65]); linguistic variables can be used to describe the degrees of a criterion if crisp data are inefficient for modeling a real situation in MCDM (Wang and Lee [66]). For example, the ratings of alternative versus qualitative criteria constitute a linguistic variable whose values can be defined as very low (VL), low (L), medium (M), high (H), and very high (VH). These linguistic variables can be represented by triangular fuzzy numbers such as the following: VL = (0, 0.1, 0.3), L = (0.1, 0.3, 0.5), M = (0.3, 0.5, 0.7), H = (0.5, 0.7, 0.9), and VH = (0.7, 0.9, 1). These fuzzy numbers have the characteristic of larger being better, and thus qualitative criteria are regarded as benefit criteria.

4. Model Establishment

Assume that a committee of k experts (i.e., $E_t, t=1~k$) is responsible for the evaluation of m alternatives (i.e., $O_i,i=1~m$) under n criteria ($C_j,j=1~n$). The criteria can be categorized as quantitative and qualitative. Quantitative criteria can be further classified as benefit (B), for which a larger value is better, and cost (C), for which a smaller value is better.

• Step 1. Develop a decision matrix

Assume that ${\tilde{p}}_{ijt}=\left(p_{ijt}^l,p_{ijt}^{\lambda },p_{ijt}^u\right)$, ${\tilde{p}}_{ijt}\in R^{+},i=1~m,j=1~n$, and $t=1~k$ are the ratings of $O_i$ with respect to $C_j$ criterion given by the expert $E_t$. Further assume that the ratings of alternative versus qualitative criteria are provided by the experts, employing linguistic terms with equivalent fuzzy numbers as displayed in Section 3.5, which can be aggregated by Equation (7):

$${\tilde{p}}_{ij}=\left(\frac{1}{k}\right)\times ({\tilde{p}}_{ij1}\oplus \dots \oplus {\tilde{p}}_{ijt}\oplus \dots \oplus {\tilde{p}}_{ijk})$$

(7)

• Step 2. Normalization of values under quantitative criteria

The normalized decision matrix is necessary to ensure that all the performance ratings of alternative versus quantitative criteria have a homogeneous and comparable scale. Assume that ${\tilde{r}}_{ij}=\left(r_{ij}^l,r_{ij}^{\lambda },r_{ij}^u\right)$ is the normalized value of ${\tilde{p}}_{ij}$. The formulas for normalizing the values under quantitative criteria are presented in Equations (8) and (9):

$${\tilde{r}}_{ij}=\left(r_{ij}^l,r_{ij}^{\lambda },r_{ij}^u\right)=\left(\frac{p_{ij}^l}{b_j^{+}},\frac{p_{ij}^{\lambda }}{b_j^{+}},\frac{p_{ij}^u}{b_j^{+}}\right),j\in B$$

(8)

$${\tilde{r}}_{ij}=\left(r_{ij}^l,r_{ij}^{\lambda },r_{ij}^u\right)=\left(\frac{a_j^{-}}{p_{ij}^u},\frac{a_j^{-}}{p_{ij}^{\lambda }},\frac{a_j^{-}}{p_{ij}^l}\right),j\in C \mathrm{where} b_j^{+}=\underset{i}{\max }\left(p_{ij}^u\right),a_j^{-}=\underset{i}{\min }\left(p_{ij}^l\right)$$

(9)

• Step 3. Determine the criteria weights

Suppose that ${\tilde{w}}_{jt}=\left(w_{jt}^l,w_{jt}^{\lambda },w_{jt}^u\right)$ denotes the weight of criterion $C_j$ given by the expert $E_t$. The average value of the criteria weight is produced by the committee of experts with Equation (10):

$${\tilde{w}}_j=\left(\frac{1}{k}\right)\times \left({\tilde{w}}_{j1}+\dots +{\tilde{w}}_{jt}+\dots +{\tilde{w}}_{jk}\right)$$

(10)

where

$$w_j^l={\displaystyle {\sum }_{t=1}^k\frac{w_{jt}^l}{k}},w_j^{\lambda }={\displaystyle {\sum }_{t=1}^k\frac{w_{jt}^{\lambda }}{k}},w_j^u={\displaystyle {\sum }_{t=1}^k\frac{w_{jt}^u}{k}}$$

• Step 4. Weighted normalization matrix

The weighted normalized value, ${\tilde{v}}_{ij}=\left[v_{ij}^l,v_{ij}^{\lambda },v_{ij}^u\right]$, in the decision matrix is obtained through the following equation:

$${\tilde{v}}_{ij}=\left(r_{ij}^l,r_{ij}^{\lambda },r_{ij}^u\right)\otimes \left(w_j^l,w_j^{\lambda },w_j^u\right)$$

(11)

Suppose that the $\alpha -\mathrm{cut}$ of ${\tilde{v}}_{ij}$ is denoted as $v_{ij}^{\alpha }=\left[v_{ij}^L,v_{ij}^R\right]$. Formulas for $v_{ij}^{\alpha }$ can be developed with Equations (2)–(5) as shown in the following equation (Kaufmann and Gupta [56]):

$$v_{ij}^{\alpha }=\left[r_{ij}^L\cdot w_j^L, r_{ij}^R\cdot w_j^R\right]=\left[{\alpha }^2\cdot G_{ij1}+\alpha \cdot H_{ij1}+L_{ij1}, {\alpha }^2\cdot G_{ij2}+\alpha \cdot H_{ij2}+L_{ij2}\right]$$

(12)

where

$$G_{ij1}=\left(r_{ij}^{\lambda }-r_{ij}^l\right)\cdot \left(w_j^{\lambda }-w_j^l\right), H_{ij1}=r_{ij}^l\cdot \left(w_j^{\lambda }-w_j^l\right)+w_j^l\cdot \left(r_{ij}^{\lambda }-r_{ij}^l\right), L_{ij1}=r_{ij}^l\cdot w_j^l$$

$$G_{ij2}=\left(r_{ij}^{\lambda }-r_{ij}^u\right)\cdot \left(w_j^{\lambda }-w_j^u\right), H_{ij2}=r_{ij}^u\cdot \left(w_j^{\lambda }-w_j^u\right)+w_j^u\cdot \left(r_{ij}^{\lambda }-r_{ij}^u\right), L_{ij2}=r_{ij}^u\cdot w_j^u$$

• Step 5. Defuzzification

The signed distance of two fuzzy numbers, ${\tilde{v}}_{ij} \mathrm{and} {\tilde{v}}_{sj}$, is obtained by the Yao and Wu method [16] as shown in Equation (13):

$$\begin{array}{cc}d\left({\tilde{v}}_{ij},{\tilde{v}}_{sj}\right)& =\frac{1}{2}{\displaystyle \underset{0}{\overset{1}{\int }}\left(v_{ij}^L+v_{ij}^R-v_{sj}^L-v_{sj}^R\right)}d\alpha \\ & =\frac{1}{2}{\displaystyle \underset{0}{\overset{1}{\int }}\left({\alpha }^2\left(G_{ij1}+G_{ij2}\right)+\alpha \left(H_{ij1}+H_{ij2}\right)+L_{ij1}+L_{ij2}-{\alpha }^2\left(G_{sj1}+G_{sj2}\right)-\alpha \left(H_{sj1}+H_{sj2}\right)-L_{sj1}-L_{sj2}\right)}d\alpha \\ & \hspace{1em}=\frac{G_{ij1}+G_{ij2}-G_{sj1}-G_{sj2}}{6}+\frac{H_{ij1}+H_{ij2}-H_{sj1}-H_{sj2}}{4}+\frac{L_{ij1}+L_{ij2}-L_{sj1}-L_{sj2}}{2}\end{array}$$

(13)

• Step 6. Identify the concordance and discordance sets

According to Yao and Wu [16], ${\tilde{v}}_{ij}>{\tilde{v}}_{sj}$ if $d\left({\tilde{v}}_{ij},{\tilde{v}}_{sj}\right)>0$; ${\tilde{v}}_{ij}<{\tilde{v}}_{sj}$ if $d\left({\tilde{v}}_{ij},{\tilde{v}}_{sj}\right)<0$; and ${\tilde{v}}_{ij}={\tilde{v}}_{sj}$ if $d\left({\tilde{v}}_{ij},{\tilde{v}}_{sj}\right)=0$. The concordance and discordance sets can be determined as follows:

$$C_{is}=\left\{j,{\tilde{v}}_{ij}\ge {\tilde{v}}_{sj}\right\}$$

(14)

$$D_{is}=\left\{j,{\tilde{v}}_{ij}<{\tilde{v}}_{sj}\right\}$$

(15)

• Step 7. Produce concordance and discordance matrices

First, the fuzzy weight ${\tilde{w}}_j$ is defuzzified by the signed distance as shown in Equation (16) and normalized to obtain the crisp weight $w_j$ as shown in Equation (17):

$$w_j^{\prime }=d\left({\tilde{w}}_j,0\right)=\frac{1}{2}{\displaystyle \underset{0}{\overset{1}{\int }}(w_j^L}+w_j^R)d\alpha =\frac{1}{4}\left(2w_j^{\lambda }+w_j^l+w_j^u\right)$$

(16)

$$w_j=\frac{w_j^{\prime }}{{\displaystyle \sum _{j=1}^n{w}_j^{\prime }}} \mathrm{where} {\displaystyle \sum _{j=1}^n{w}_j}=1$$

(17)

The concordance matrix is produced by aggregating the criteria weights in the concordance set. The formula for the concordance matrix Con can be obtained using Equation (18):

$$Con={\left[c_{is}\right]}_{m\times m}, c_{is}=\frac{{\displaystyle \sum _{j\in C_{is}}{w}_j}}{{\displaystyle \sum _{j=1}^n{w}_j}}$$

(18)

The discordance matrix is produced by Equation (19), in which $d\left({\tilde{v}}_{ij},{\tilde{v}}_{sj}\right)$ is obtained through the signed distance (Yao and Wu [16]):

$$D={\left[d_{is}\right]}_{m\times m}, d_{is}=\frac{\underset{j\in D_{is}}{\max }\left\{\left|d\left({\tilde{v}}_{ij},{\tilde{v}}_{sj}\right)\right|\right\}}{\underset{j\in J}{\max }\left\{\left|d\left({\tilde{v}}_{ij},{\tilde{v}}_{sj}\right)\right|\right\}}, J=\left\{1,2,\dots \dots ,n\right\}$$

(19)

• Step 8. An extended modified discordance matrix

Ke and Chen [54] introduced the Hadamard product of $c_{is}$ and $d_{is}^{\prime }$, where $d_{is}^{\prime }=1-d_{is}$, to obtain a modified total matrix for ranking alternatives. Despite the merits of this method, it can result in information loss because if a value of $d_{is}^{\prime }$ is zero, then the corresponding value in the modified total matrix will be zero owing to the nature of multiplication, which can influence the ranking order. Zhang et al. [51] used the Hadamard product to obtain a modified weighted summation matrix to produce net dominating values for ranking alternatives. Despite the merits, their method could also result in information loss that influences the ranking order. To resolve this problem, we propose an extended modified discordance matrix as follows:

$$D^{\prime }={\left[d_{is}^{\prime }\right]}_{m\times m}+{\left[1_{is,}{}_{i\ne s}\right]}_{m\times m}, \mathrm{where} d_{is}^{\prime }=1-d_{is}, i,s=1~m$$

(20)

Because “1” is the maximum value in the discordance matrix according to Equation (19), adding “1” to the corresponding $d_{is}^{\prime }$ can avoid a zero value in the modified discordance matrix and avoid losing information when calculating the Hadamard product. By contrast, the concordance matrix ${\left[c_{is}\right]}_{m\times m}$ has a zero, and no information is lost because its corresponding value in the modified discordance matrix will also be zero according to Equations (19) and (20).

• Step 9. Closeness coefficient for ranking alternatives

The closeness coefficient proposed by Hwang and Yoon [67] is used to rank alternatives based on the Hadamard product, which is a new application. The extended modified total matrix is obtained as presented in Equation (21). The closeness coefficient index $c{c}_i$ can be derived as shown in Equation (22):

$$G={\left[e_{is}\right]}_{m\times m}, \mathrm{where} e_{is}=c_{ij}\circ \left(d_{is}^{\prime }+1_{is}\right) i,s=1~m$$

(21)

$$c{c}_i=\frac{{\displaystyle \sum _{s=1\wedge s\ne i}^m{e}_{is}}}{{\displaystyle \sum _{s=1\wedge s\ne i}^m{e}_{is}}+{\displaystyle \sum _{i=1\wedge i\ne s}^m{e}_{si}}},i,s=1~m$$

(22)

According to the concept of net dominance value presented by Nijkamp and Van Delft [53], if ${\displaystyle \sum _{s=1\wedge s\ne i}^m{e}_{is}}$ is larger, or if ${\displaystyle \sum _{i=1\wedge i\ne s}^m{e}_{si}}$ is smaller, then the ranking order of the corresponding alternative is higher. Therefore, a larger $c{c}_i$ value indicates that the corresponding alternative has a higher ranking order. Accordingly, the closeness coefficient index used in this study is effective for ranking alternatives.

4.1. Comparison with Similar Methods

The proposed extension is compared with the methods presented by Ke and Chen [54] and Zhang et al. [51] to demonstrate its advantages. Assume that three alternatives ($A_1,A_2,A_3$) under four benefit criteria ($C_1,C_2,C_3,C_4$) must be evaluated by decisionmakers who have determined the performance ratings of the alternatives, $p_{ij},i=1~3,j=1~4$, under four criteria and the criteria weights $w_j, j=1~4$ as listed in

Table 1. The performance ratings of alternatives versus criteria.

	C1	C2	C3	C4
A1	6	5.5	6.5	8
A2	7	6.5	4.5	8
A3	6	2	4	3
Weight	0.450	0.185	0.255	0.110

. The normalization matrix can be obtained using $r_{ij}=\frac{p_{ij}}{\sqrt{{\displaystyle \sum _{i=1}^3{p}_{ij}^2}}}$, as presented in

Table 2. Normalization matrix.

	C1	C2	C3	C4
A1	0.545	0.629	0.734	0.683
A2	0.636	0.743	0.508	0.683
A3	0.545	0.229	0.451	0.256

. The weighted normalized matrix can be produced using $v_{ij}=r_{ij}\times w_j, i=1~3,j=1~4$, as displayed in

Table 3. Weighted normalization matrix.

	C1	C2	C3	C4
A1	0.245	0.116	0.187	0.075
A2	0.286	0.137	0.130	0.075
A3	0.245	0.042	0.115	0.028

. The concordance matrix can be obtained using $c_{is}=\frac{{\displaystyle \sum _{j\in C_{is}}{w}_j}}{{\displaystyle \sum _{j=1}^n{w}_j}}$, $C_{is}$= $\left\{j,v_{ij}\ge v_{sj}\right\}$, as shown in

Table 4. Concordance matrix.

	C1	C2	C3
A1	-	0.365	1.000
A2	0.745	-	1.000
A3	0.450	0.000	-

. The discordance matrix can be obtained using $d_{is}=\frac{\underset{j\in D_{is}}{\max }\left\{\left|v_{ij}-v_{sj}\right|\right\}}{\underset{j\in J}{\max }\left\{\left|v_{ij}-v_{sj}\right|\right\}},$ $J=\left\{1,2,\dots \dots ,n\right\},$ $D_{is}$= $\left\{j,v_{ij}<v_{hj}\right\}$, as presented in

Table 5. Discordance matrix.

	A1	A2	A3
A1	-	0.711	0.000
A2	1.000	-	0.000
A3	1.000	1.000	-

, and the modified discordance matrix can then be obtained as indicated in

Table 6. Modified discordance matrix.

	A1	A2	A3
A1	-	0.289	1.000
A2	0.000	-	1.000
A3	0.000	0.000	-

. Through the use of the Hadamard product, the modified total matrix can be obtained using $f_{is}=c_{is}\times d_{is}^{\prime }$, as shown in

Table 7. The modified total matrix.

	A1	A2	A3
A1	-	0.106	1.000
A2	0.000	-	1.000
A3	0.000	0.000	-

. According to Ke and Chen [54], the Boolean matrix $Q={\left[q_{is}\right]}_{m\times m},\left\{\begin{array}{c}{q}_{is}=1,f_{is}\ge \overline{f}\\ q_{is}=1,f_{is}<\overline{f}\end{array}\right.$ can be obtained on the basis of Table 7, as presented in

Table 8. Boolean matrix of Ke and Chen [54].

	A1	A2	A3
A1	-	1.000	1.000
A2	0.000	-	1.000
A3	0.000	0.000	-

, in which the threshold $\overline{f}$ is set between the smallest value $f_1$ and the next smallest value $f_2$, $f_s=\max \left\{f_{is}\left|i=1~m\right.\right\},s=1~m$. The net dominating values in the method of Zhang et al. [51] with $H_i={\displaystyle \sum _{s=1\wedge s\ne i}^m{f}_{is}}-{\displaystyle \sum _{i=1\wedge i\ne s}^m{f}_{is}}$ can be obtained on the basis of Table 7.

The extended modified discordance matrix $d_{is}^{\prime }$ is obtained using Equation (20), as presented in

Table 9. Extended modified discordance matrix.

	A1	A2	A3
A1	-	1.289	2.000
A2	1.000	-	2.000
A3	1.000	1.000	-

. The extended modified total matrix can be obtained using Equation (21), as listed in

Table 10. The extended modified total matrix.

	A1	A2	A3
A1	-	0.471	2.000
A2	0.745	-	2.000
A3	0.450	0.000	-

. The closeness coefficients can then be obtained using Equation (22), as presented in

Table 11. Ranking values.

	Zhang et al.’s Values [51]	Closeness Coefficients
A1	1.106	0.674
A2	0.894	0.854
A3	−2.000	0.101

. According to the values in Table 3, $v_{21}(0.286)>v_{11}(0.245)$, $v_{22}(0.137)>v_{12}(0.116)$, $v_{24}(0.075)=v_{14}(0.075)$, and $v_{23}(0.130)<v_{13}(0.187)$. Thus, the alternative $A_2$ is clearly preferable to $A_1$. However, the Boolean matrix derived by the method of Ke and Chen [54] in Table 8 produced the ranking order $A_1>A_2>A_3$ (

Table 12. Ranking order.

	Ranking
Ke and Chen [54]	$A_1>A_2>A_3$
Zhang et al. [51]	$A_1>A_2>A_3$
Proposed method	$A_2>A_1>A_3$

). The values in Table 11 derived by the method of Zhang et al. [51] also produced the ranking order $A_1>A_2>A_3$ (Table 12). These ranking results contradict the values presented in Table 3. However, the proposed extension using the closeness coefficient (Table 11) correctly obtained the ranking order $A_2>A_1>A_3$ (Table 12), which is consistent with the values in Table 3. Therefore, the proposed extension can overcome the limitations of the methods of Ke and Chen [54] and Zhang et al. [51]. Moreover, if the concordance matrix has a zero value, such as $c_{32}=0$ in Table 4, then the corresponding value in the modified discordance matrix also becomes zero, such as $d_{32}=0$ in Table 6, because if Equation (18) produces a zero value, then Equation (19) produces a value of one, resulting in a zero value in the modified discordance matrix. Thus, no information is lost using the Hadamard product if there is (or are) a zero value (or zero values) in the concordance matrix.

5. Numerical Example

A hypothetical numerical example is conducted to demonstrate feasibility and effectiveness of the proposed method. Assume that an industrial company intends to select a suitable demand product forecasting method to establish a production plan and achieve its sustainable manufacturing goals. A committee of four decisionmakers is formed to evaluate four forecasting techniques $A_i$, $i=1~4$. Furthermore, assume that eleven qualitative and quantitative criteria in

Table 13. List of criteria.

Symbol	Criteria	Quantitative	Qualitative
C1	Data availability (B)		x
C2	Data validity (B)		x
C3	Technology development predictability (B)		x
C4	Technology similarity (B)		x
C5	Method adaptability (B)		x
C6	Ease of operation (B)		x
C7	Implementation cost (C, UDS)	x
C8	Maintenance cost (C, USD)	x
C9	Accuracy (C, %)	x
C10	Timeliness in providing forecasts (C, months)	x
C11	Ease of interpretation (B)		x

are selected by decision makers based on their professional perceptions, experience, and group discussion. Herein, qualitative criteria are beneficial. The solution is found using the following steps.

Step 1. The decisionmakers give the performance ratings of four options versus the criteria with Equation (7) based on their subjective professional perceptions, which are displayed in

Table 14. Performance ratings of alternatives versus criteria.

	C1			C2			C3			C4			C5			C6
	xl	xγ	xu	xl	xγ	xu	xl	xγ	xu	xl	xγ	xu	xl	xγ	xu	xl	xγ	xu
A1	0	0.1	0.3	0	0.1	0.3	0.5	0.7	0.9	0.3	0.5	0.7	0.5	0.7	0.9	0.7	0.9	1
A2	0.1	0.3	0.5	0.3	0.5	0.7	0.5	0.7	0.9	0.5	0.7	0.9	0.3	0.5	0.7	0.7	0.9	1
A3	0.3	0.5	0.7	0.3	0.5	0.7	0.5	0.7	0.9	0.3	0.5	0.7	0.3	0.5	0.7	0.5	0.7	0.9
A4	0.7	0.9	1	0.7	0.9	1	0.5	0.7	0.9	0.5	0.7	0.9	0.7	0.9	1	0	0.1	0.3
	C7			C8			C9			C10			C11
	xl	xγ	xu	xl	xγ	xu	xl	xγ	xu	xl	xγ	xu	xl	xγ	xu
A1	348	348	348	50	50	50	35	35	35	1	1	1	0.7	0.9	1
A2	340	340	340	60	60	60	20	20	20	2	2	2	0.3	0.5	0.7
A3	350	350	350	60	60	60	23	23	23	3	3	3	0.5	0.7	0.9
A4	360	360	360	50	50	50	10	10	10	2	2	2	0.5	0.7	0.9

. The linguistic terms converted into triangular fuzzy numbers in Section 3.5 are employed to provide performance ratings for options versus the qualitative criteria, which are also presented in Table 14.

Step 2. The ratings of options versus quantitative criteria are normalized with Equations (8) and (9) as shown in

Table 15. Normalization matrix.

	C1			C2			C3			C4
	xl	xγ	xu	xl	xγ	xu	xl	xγ	xu	xl	xγ	xu
A1	0.000	0.100	0.300	0.000	0.100	0.300	0.500	0.700	0.900	0.300	0.500	0.700
A2	0.300	0.500	0.700	0.300	0.500	0.700	0.100	0.300	0.500	0.500	0.700	0.900
A3	0.300	0.500	0.700	0.300	0.500	0.700	0.500	0.700	0.900	0.300	0.500	0.700
A4	0.700	0.900	1.000	0.700	0.900	1.000	0.300	0.500	0.700	0.500	0.700	0.900
Weight	0.300	0.500	0.700	0.300	0.500	0.700	0.000	0.100	0.300	0.000	0.100	0.300
	C5			C6			C7			C8
	xl	xγ	xu	xl	xγ	xu	xl	xγ	xu	xl	xγ	xu
A1	0.500	0.700	0.900	0.700	0.900	1.000	0.977	0.977	0.977	1.000	1.000	1.000
A2	0.300	0.500	0.700	0.700	0.900	1.000	1.000	1.000	1.000	0.833	0.833	0.833
A3	0.300	0.500	0.700	0.500	0.700	0.900	0.971	0.971	0.971	0.833	0.833	0.833
A4	0.700	0.900	1.000	0.000	0.100	0.300	0.944	0.944	0.944	1.000	1.000	1.000
Weight	0.000	0.100	0.300	0.300	0.500	0.700	0.500	0.700	0.900	0.300	0.500	0.700
	C9			C10			C11
	xl	xγ	xu	xl	xγ	xu	xl	xγ	xu
A1	0.286	0.286			1.000	1.000	0.700	0.900	1.000
A2	0.500	0.500	0.500	0.500	0.500	0.500	0.300	0.500	0.700
A3	0.435	0.435	0.435	0.333	0.333	0.333	0.500	0.700	0.900
A4	1.000	1.000	1.000	0.500	0.500	0.500	0.500	0.700	0.900
Weight	0.700	0.900	1.000	0.300	0.500	0.700	0.300	0.500	0.700

.

Step 3. Suppose that the criteria weights are assigned by decisionmakers and the average weights can be obtained with Equation (10), as also displayed in Table 15.

Step 4. The weighted normalization values are calculated using Equations (11) and (12), as presented in

Table 16. The weighted normalization matrix.

	C1						C2
	Gij1	Hij1	Lij1	Gij2	Hij2	Lij2	Gij1	Hij1	Lij1	Gij2	Hij2	Lij2
A1	0.020	0.030	0.000	0.040	−0.200	0.210	0.020	0.030	0.000	0.040	−0.200	0.210
A2	0.040	0.120	0.090	0.040	−0.280	0.490	0.040	0.120	0.090	0.040	−0.280	0.490
A3	0.040	0.120	0.090	0.040	−0.280	0.490	0.040	0.120	0.090	0.040	−0.280	0.490
A4	0.040	0.200	0.210	0.020	−0.270	0.700	0.040	0.200	0.210	0.020	−0.270	0.700
	C3						C4
	Gij1	Hij1	Lij1	Gij2	Hij2	Lij2	Gij1	Hij1	Lij1	Gij2	Hij2	Lij2
A1	0.020	0.050	0.000	0.040	−0.240	0.270	0.020	0.030	0.000	0.040	−0.200	0.210
A2	0.020	0.010	0.000	0.040	−0.160	0.150	0.020	0.050	0.000	0.040	−0.240	0.270
A3	0.020	0.050	0.000	0.040	−0.240	0.270	0.020	0.030	0.000	0.040	−0.200	0.210
A4	0.020	0.030	0.000	0.040	−0.200	0.210	0.020	0.050	0.000	0.040	−0.240	0.270
	C5						C6
	Gij1	Hij1	Lij1	Gij2	Hij2	Lij2	Gij1	Hij1	Lij1	Gij2	Hij2	Lij2
A1	0.02	0.05	0.00	0.04	−0.24	0.27	0.04	0.20	0.21	0.02	−0.27	0.70
A2	0.02	0.03	0.00	0.04	−0.20	0.21	0.04	0.20	0.21	0.02	−0.27	0.70
A3	0.02	0.03	0.00	0.04	−0.20	0.21	0.04	0.16	0.15	0.04	−0.32	0.63
A4	0.02	0.07	0.00	0.02	−0.23	0.30	0.02	0.03	0.00	0.04	−0.20	0.21
	C7						C8
	Gij1	Hij1	Lij1	Gij2	Hij2	Lij2	Gij1	Hij1	Lij1	Gij2	Hij2	Lij2
A1	0.00	0.20	0.49	0.00	−0.20	0.88	0.00	0.20	0.30	0.00	−0.20	0.70
A2	0.00	0.20	0.50	0.00	−0.20	0.90	0.00	0.17	0.25	0.00	−0.17	0.58
A3	0.00	0.19	0.49	0.00	−0.19	0.87	0.00	0.17	0.25	0.00	−0.17	0.58
A4	0.00	0.19	0.47	0.00	−0.19	0.85	0.00	0.20	0.30	0.00	−0.20	0.70
	C9						C10
	Gij1	Hij1	Lij1	Gij2	Hij2	Lij2	Gij1	Hij1	Lij1	Gij2	Hij2	Lij2
A1	0.00	0.06	0.20	0.00	−0.03	0.29	0.00	0.20	0.30	0.00	−0.20	0.70
A2	0.00	0.10	0.35	0.00	−0.05	0.50	0.00	0.10	0.15	0.00	−0.10	0.35
A3	0.00	0.09	0.30	0.00	−0.04	0.43	0.00	0.07	0.10	0.00	−0.07	0.23
A4	0.00	0.20	0.70	0.00	−0.10	1.00	0.00	0.10	0.15	0.00	−0.10	0.35
	C11
	Gij1	Hij1	Lij1	Gij2	Hij2	Lij2
A1	0.04	0.20	0.21	0.02	−0.27	0.70
A2	0.04	0.12	0.09	0.04	−0.28	0.49
A3	0.04	0.16	0.15	0.04	−0.32	0.63
A4	0.04	0.16	0.15	0.04	−0.32	0.63

.

Step 5. The defuzzified values can be obtained with Equation (13), as also displayed in

Table 17. Defuzzification.

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11
A12	−0.191	−0.191	0.050	−0.025	0.025	0.000	−0.016	0.083	−0.188	0.250	0.184
A13	−0.191	−0.191	0.000	0.000	0.025	0.084	0.004	0.083	−0.130	0.333	0.084
A14	−0.375	−0.375	0.025	−0.025	−0.019	0.375	0.023	0.000	−0.625	0.250	0.084
A21	0.191	0.191	−0.050	0.025	−0.025	0.000	0.016	−0.083	0.188	−0.250	−0.184
A23	0.000	0.000	−0.050	0.025	0.000	0.084	0.020	0.000	0.057	0.083	−0.100
A24	−0.184	−0.184	−0.025	0.000	−0.044	0.375	0.039	−0.083	−0.438	0.000	−0.100
A31	0.191	0.191	0.000	0.000	−0.025	−0.084	−0.004	−0.083	0.130	−0.333	−0.084
A32	0.000	0.000	0.050	−0.025	0.000	−0.084	−0.020	0.000	−0.057	−0.083	0.100
A34	−0.184	−0.184	0.025	−0.025	−0.044	0.291	0.019	−0.083	−0.495	−0.083	0.000
A41	0.375	0.375	−0.025	0.025	0.019	−0.375	−0.023	0.000	0.625	−0.250	−0.084
A42	0.184	0.184	0.025	0.000	0.044	−0.375	−0.039	0.083	0.438	0.000	0.100
A43	0.184	0.184	−0.025	0.025	0.044	−0.291	−0.019	0.083	0.495	0.083	0.000
Weight	0.101	0.101	0.025	0.025	0.025	0.101	0.141	0.101	0.177	0.101	0.101

.

Step 6. The concordance and discordance sets can be determined by Equations (14) and (15) as presented in

Table 18. Concordance and discordance sets.

Concordance = 1, Discordance = 0
	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11
A12	0	0	1	0	1	1	0	1	0	1	1
A13	0	0	1	1	1	1	1	1	0	1	1
A14	0	0	1	0	0	1	1	1	0	1	1
A21	1	1	0	1	0	1	1	0	1	0	0
A23	1	1	0	1	1	1	1	1	1	1	0
A24	0	0	0	1	0	1	1	0	0	1	0
A31	1	1	1	1	0	0	0	0	1	0	0
A32	1	1	1	0	1	0	0	1	0	0	1
A34	0	0	1	0	0	1	1	0	0	0	1
A41	1	1	0	1	1	0	0	1	1	0	0
A42	1	1	1	1	1	0	0	1	1	1	1
A43	1	1	0	1	1	0	0	1	1	1	1

.

Step 7. The crisp weights of the criteria can be obtained using Equations (16) and (17), as also displayed in Table 17. The concordance matrix can be obtained with Equation (18), as shown in

Table 19. Concordance matrix.

	A1	A2	A3	A4
A1	-	0.455	0.621	0.571
A2	0.646	-	0.000	0.369
A3	0.429	0.455	-	0.369
A4	0.530	0.758	0.268	-

, and the discordance matrix can be obtained with Equation (19), as displayed in

Table 20. Discordance matrix.

	A1	A2	A3	A4
A1	-	0.763	0.573	1.000
A2	1.000	-	1.000	1.000
A3	1.000	0.842	-	1.000
A4	0.600	0.857	0.588	-

.

Step 8. The modified discordance matrix can be easily produced, as shown in

Table 21. Modified discordance matrix.

	A1	A2	A3	A4
A1	-	0.237	0.428	0.000
A2	0.000	-	0.000	0.000
A3	0.000	0.158	-	0.000
A4	0.400	0.143	0.412	-

. The extended modified discordance matrix can be obtained with Equation (20), as shown in

Table 22. Extended modified discordance matrix.

	A1	A2	A3	A4
A1	-	1.237	1.428	1.000
A2	1.000	-	1.000	1.000
A3	1.000	1.1583	-	1.000
A4	1.400	1.1429	1.412	-

.

Step 9. The extended modified total matrix can be obtained with Equation (21), as shown in

Table 23. Extended modified total matrix.

	A1	A2	A3	A4
A1	-	0.562	0.887	0.571
A2	0.6465	-	0.000	0.369
A3	0.4293	0.527	-	0.369
A4	0.7424	0.866	0.378	-

. The closeness coefficient $c{c}_i$ can then be obtained via Equation (22), as shown in

Table 24. Final ranking results.

	Closeness Coefficients	Ranking
A1	0.526	2
A2	0.342	4
A3	0.512	3
A4	0.603	1

. According to the closeness coefficients in Table 24, the ranking order is $A_4>A_1>A_3>A_2$ because $0.603>0.526>0.512>0.342$. Therefore, the forecasting method $A_4$ should be selected as the best solution.

5.1. Numerical Comparison

The proposed method was compared with those of Ke and Chen [54], Zhang et al. [51], Nghiem and Chu [52], and Nghiem and Chu [55] by using the numerical example presented in Section 5. According to Ke and Chen [54] and Zhang et al. [51], the modified total matrix (

Table 25. Modified total matrix.

	A1	A2	A3	A4
A1	-	0.108	0.266	0.000
A2	0.000	-	0.000	0.000
A3	0.000	0.072	-	0.000
A4	0.212	0.108	0.110	-

) can be derived using the Hadamard product based on Table 19 and Table 21. The Boolean matrix derived using the method of Ke and Chen [54] is presented in

Table 26. Boolean matrix of Ke and Chen [54].

	A1	A2	A3	A4
A1	-	0	1	0
A2	0	-	0	0
A3	0	0	-	0
A4	1	0	1	-

. The net dominating values are derived by the method of Zhang et al. [51]. The total dominance matrix based on subtracting discordance values from concordance values under the method of Nghiem and Chu [55] can be obtained through the equation $G={\left[e_{is}\right]}_{m\times m}$, $e_{is}=c_{is}-d_{is}$, as presented in

Table 27. Total dominance matrix.

	A1	A2	A3	A4
A1	-	−0.309	0.049	−0.429
A2	−0.354	-	−1	−0.631
A3	−0.571	−0.387	-	−0.631
A4	−0.070	−0.100	−0.320	-

. The Boolean matrix (

Table 28. Boolean matrix of Nghiem and Chu [55].

	A1	A2	A3	A4
A1	-	1	1	0
A2	1	-	0	0
A3	0	1	-	0
A4	1	1	1	-

) can be derived with the method of Nghiem and Chu [55], based on Table 27. In the method of Nghiem and Chu [52], the net concordance and net modified discordance values can be derived using $c_i={\displaystyle \sum _{s=1\wedge s\ne i}^m{c}_{is}}-{\displaystyle \sum _{i=1\wedge i\ne s}^m{c}_{is}}$ and $d_i^{\prime }={\displaystyle \sum _{s=1\wedge s\ne i}^m{d}_{is}^{\prime }}-{\displaystyle \sum _{i=1\wedge i\ne s}^m{d}_{is}^{\prime }}$, respectively, as shown in

Table 29. Net concordance and net modified discordance matrix.

	Net Concordance	Net Modified Discordance
A1	0.040	0.264
A2	−0.652	−0.538
A3	0.364	−0.681
A4	0.247	0.955

. Their net total dominance values can be obtained using $U_i=c_i\oplus d_i^{\prime },i=1~m$, as listed in

Table 30. Final ranking results.

	Ke and Chen [54]	Zhang et al. [51]		Nghiem and Chu [55]	Nghiem and Chu [52]
	Ranking	Values	Ranking	Ranking	Net Total Dominance	Ranking
A1	2	0.161	2	2	0.305	2
A2	3	−0.288	3	4	−1.189	4
A3	3	−0.304	4	3	−0.317	3
A4	1	0.431	1	1	1.202	1

.

According to the Boolean matrix in Table 26, the method of Ke and Chen [54] produced the following ranking order: $A_4>A_1>A_3=A_2$ (Table 30). According to the net dominating values in Table 30, the method of Zhang et al. [51] yielded the following ranking order: $A_4>A_1>A_2>A_3$ (Table 30). Their methods are inconsistent with the ranking order $A_4>A_1>A_3>A_2$, presented in Table 24, which was obtained by the proposed method. The methods of Ke and Chen [54] and Zhang et al. [51] can lose information when using the Hadamard product to generate the modified total matrix, leading to incorrect ranking results. The proposed method can resolve the problem of information loss in the methods of Ke and Chen [54] and Zhang et al. [51], where the Hadamard product is used. The method of Nghiem and Chu [52] uses the total net dominance value (Table 30), and it yielded a ranking order of $A_4>A_1>A_3>A_2$ (Table 30), which is consistent with that derived by the proposed method, as shown in Table 24. In addition, the method of Nghiem and Chu [55] uses the total dominance matrix method based on subtracting discordance values from concordance values to produce the Boolean matrix and it obtained the ranking order of $A_4>A_1>A_3>A_2$ (Table 30), which is also consistent with that derived by the proposed method, as shown in Table 24. The comparison with the methods of Nghiem and Chu [52,55] further indicates the effectiveness of the prosed method. However, in the methods of Nghiem and Chu [52,55], the problem of information loss remains if the Hadamard product is used. The proposed method in this study overcomes this problem.

6. Conclusions

Demand forecasting method selection plays a key role in sustainable development for manufacturing companies; selecting a suitable forecasting method can help companies avoid overproduction or shortages. Therefore, evaluating demand forecasting methods has attracted the attention of numerous scholars and practitioners. More than 200 forecasting methods have been investigated in the economic literature. Thus, companies must compare forecasting methods and select the most suitable one to improve their production procedures.

Evaluating demand forecasting methods is a fuzzy MCDM problem because qualitative criteria, quantitative criteria, including benefit and cost, and criteria weights must be considered. To the best of our knowledge, this study is the first to apply fuzzy ELECTRE I to the evaluation and selection of demand forecasting methods. Specifically, this study proposes an extension of fuzzy ELECTRE I for selecting the most suitable demand forecasting method. In the proposed method, the fuzzy weighted ratings are defuzzified by applying the signed distance (Yao and Wu [16]) to form a crisp ELECTRE I model. The defuzzification formulas can help yield a complete model that can assist in decision making. Moreover, the proposed extension uses a closeness coefficient based on the extended modified discordance matrix to rank alternatives. The use of this extension avoids the problem of information loss when calculating the Hadamard product. Herein, a comparison with the methods of Ke and Chen [54] and Zhang et al. [51] demonstrated the advantages of the proposed extension. Moreover, a numerical example was used to present the feasibility of the proposed fuzzy ELECTRE I method, and a numerical comparison of the proposed method with other methods was conducted to reveal the advantages of the proposed method.

In future studies, the proposed method could be applied to a case study, could be used to resolve other problems under a fuzzy MCDM environment, and could be further investigated using other types of fuzzy numbers, such as interval type2 fuzzy numbers and intuitionistic fuzzy numbers, to expand its applicability. However, there are several factors that can affect the final results, such as the weight derivation method, defuzzification method, normalization method, or number of criteria, etc. These factors could be further investigated in further research.