Multiple criteria based decision making using modified VIKOR

Abstract

Recent year development in each field and area progresses rapidly. For one particular product or service, people have multiple alternative. However, to choose best alternative is most difficult task that most of person facing. VIKOR method is most used approach for choosing best alternative based on different criteria but legging in precision. In this paper, we propose modifier VIKOR with Fuzzy logic integration approach that deal with multiple criteria while making decision as well as integratestriangular fuzzy numbers (TFN) for computing criteria values and weight importance in the processing selecting best alternative. Firstly, positive best points and negative best points were demarcated in a triangular fuzzy number evaluation matrix. Secondly, we analyze the requirements and properties of the VIKOR-based triangular fuzzy number operation rule. Thirdly, an extension of VIKOR method was proposed for comprehensive evaluation of schemes to reduce the information loss arising from defuzzification of triangular fuzzy numbers; Finally, an example demonstrate the efficiency and viability of the method. In conclusion, the proposed method solves the difficulties that subtractive computation between two triple numbers violate the arithmetic regulation, and propose a reasonable procedure according to decision making problems with triple numbers.

Keywords

Vlse kriterijumska optimizacija I kompromisno resenje(VIKOR)technique for order preference by similarity to ideal solution (TOPSIS) triangular fuzzy number (TFN)positive best points and negative best points

1 Introduction

In 1998, Opricovic analyzed the defection of the famous TOPSIS deeply, and hold that the optimal solutions obtained by the TOPSIS might not be the one closest to the ideal point, and put forward a kind of multiple criteria based decision making method which is named as VIKOR. The VIKOR method is superior to TOPSIS in terms of the stability and reliability of ranking the alternatives in which the alternative schemes are ranked by the closeness of their evaluations to the ideal alternative [1–2].

In recent years, the VIKOR method had been applied in many areas. Jinqiu Hu et al. [3] applied the fuzzy VIKOR method into comprehensive orders to explain an identification approach of artificial errors in the hybrid structure for shale fracturing gas operation. Considering the growing number of airports worldwide, Payam S. et al. [4] proposed an assessment and grading model by integrating the Taguchi method and VIKOR method with the best and worst method. To improve the performance of the classic FMEA, Zhang-peng T. et al. [5], obtained the weights of risk factors by the fuzzy optimal difference method and obtained the priority of risks in failure mode with the fuzzy -VIKOR integration method. In combination with the VIKOR method, Behnam K. et al. [6] proposed the best material structure for the repair of concretes and facilitated the model and selection of appropriate repair schemes by civil engineers. R. Rajesh [7] put forward an approach that combined the grey clustering algorithm to calculate the tractability of the supply chain by using the VIKOR method which can allow managers to initially classify barriers by grey clustering.

It is well-known that multi-criteria decision making under incomplete information is very common due to vagueness of objective things and fuzziness of decision makers’ thoughts. TFN is a common fuzzy number value, is applied in practices. Xu Zeshui [8] studied the multiple attribute based decision-making issue in which the values of decision alternatives under attribute values were triangular fuzzy numbers. Hu Lifang [9] investigated the multiple criteria based decision-making issue where both criteria values and criteria weights were triangular fuzzy numbers. Lan Rong, Fan Jiulun [10] analyzed the dual effect of attribute weights on decision-making and offered a decision-making approach based on the TOPSIS method. With regard to the multi-attribute decision-making problem with given decision makers’ preference on schemes, attribute values were given in the form of triangular fuzzy number and attribute weight information could not be fully confirmed, Gong Yanbing [11] proposed a decision-making approach based on the fuzzy proportion and fuzzy deviation where the attribute values of the alternatives were triangular intuitionistic fuzzy numbers and the weights were crisp numbers, Zhang Hongxia, Li Yu [12] studied the triangular intuitionistic fuzzy number-based VIKOR method, in which decision alternatives were ranked in the stable condition of acceptable advantages and decision-making process to get a compromise solution. The optimal alternatives will be proposed by using the above approaches, however, the compromise solutions might be more pragmatic in reality.

According to the VIKOR method with triangular fuzzy numbers, many literatures studied the comparison and computation of triangular fuzzy numbers. Serafim, Yeonjoo Kim, Eun-Sung [13, 14] focused on the triangular fuzzy number-based VIKOR method, and ignored that comparison cannot be directly made among triangular fuzzy numbers. Tolga Kaya, CengizKahraman [15] proposed an integrated VIKOR-AHP method without differentiating endpoint values in processing fuzzy numbers. On the basis of the multi-criteria decision-making problem where both the criteria values and weights were triangular intuitionistic fuzzy numbers, Kavita Devi [16] proposed an extension of VIKOR method, it was likely to cause information loss during the defuzzification.

In conclusion, the above research papers with triangular fuzzy number and the VIKOR method conducted defuzzification in the process of evaluation, might distort evaluation information. It is necessary to analyze how to deal with the triple information effectively and improve the efficiency of fuzzy decision making. Based on the multi-criteria decision-making problem where criteria weights and values were triangular fuzzy numbers, this paper proposed an extension of VIKOR method that reduces the information loss from defuzzification and illustrated its application value by examples.

2 Preliminaries

Definition 1 (Fuzzy-Set): Let A be a universal nonempty set, which consider universal discourse A = {a₁, a₂, … a_n. A fuzzy set F is defined under universal discourse of nonempty set A ={ a₁, a₂, … a_n } with an ordered pair set represented as: { (a₁, μ_F (a₁)) , (a₂, μ_F (a₂))… (a_n, μ_F (a_n)) } and is characterized by degree of membership function μ_F (A) that maps every element a in A to real number in the interval 0,1. The two most widely used fuzzy numbers are the triangular and trapezoidal fuzzy numbers. In this study we employ the Triangular Fuzzy Number (TFN).

Definition 2 [8]: The triangular fuzzy number (TFN) as shown in Fig. 1 is represented as $F_{A} = (f_{A 1}^{s}, f_{A 2}^{m}, f_{A 3}^{b})$ where, $f_{1}^{s}$ denotes the smallest promising value, $f_{2}^{m}$ is the most possible value, and $f_{3}^{b}$ is the biggest promising value and corresponding degree of membership (d_m) can be defined using following function:

Fig. 1

Triangular Membership Function (TFN).

$μ_{F} (d_{m}) = {\begin{matrix} 0, & d_{m} < f_{A 1}^{s} \\ \frac{d_{m} - f_{A 1}^{s}}{f_{A 2}^{s} - f_{A 1}^{s}}, & f_{A 1}^{s} ⩽ d_{m} ⩽ f_{A 2}^{m} \\ \frac{d_{m} - f_{2}^{m}}{f_{A 3}^{b} - f_{A 2}^{m}}, & f_{A 2}^{m} ⩽ d_{m} ⩽ f_{A 3}^{m} \\ 0, & d_{m} > f_{A - 3}^{b} \end{matrix}$ (1)

Each TFN is linear representation of membership function degree (d_m) on its left side and right side shown in following equation:

$\begin{matrix} d_{m} = d_{m}^{lt (y)}, d_{m}^{rt (y)} = \\ (s + (m - s) y, b + (m - b) y), y \in {0, 1} \end{matrix}$ (2)

The defuzzification result is calculated by: $O (F) = \frac{\int d_{m} μ_{F} (d_{m})}{\int μ_{F} (d_{m})} = \frac{f_{A 1}^{s} + {kf}_{A 2}^{m} + f_{A 3}^{b}}{k + 2}$ (3)

Definition 2:

Given that $F_{A} = (f_{A 1}^{s}, f_{A 2}^{m}, f_{A 3}^{b})$ and $F_{B} = (f_{B 1}^{s}, f_{B 2}^{m}, f_{B 3}^{b})$ which are two triangular fuzzy numbers and all the values are greater than 0, the operations of triangular fuzzy number could be depicted as follows:

Summation: $F_{A} + F_{B} = [f_{A 1}^{s} + f_{B 1}^{s}, f_{A 2}^{m} + f_{B 2}^{m}, f_{A 3}^{b} + f_{B 3}^{b}]$

Subtraction: $F_{A} - F_{B} = [f_{A 1}^{s} - f_{B 1}^{s}, f_{A 2}^{m} - f_{B 2}^{m}, f_{A 3}^{b} - f_{B 3}^{b}]$

Multiplication: $F_{A} \times F_{B} = [f_{A 1}^{s} \times f_{B 1}^{s}, f_{A 2}^{m} \times f_{B 2}^{m}, f_{A 3}^{b} \times f_{B 3}^{b}]$

Division: $F_{A} \div F_{B} = [f_{A 1}^{s} \div f_{B 1}^{s}, f_{A 2}^{m} \div f_{B 2}^{m}, f_{A 3}^{b} \div f_{B 3}^{b}]$

Inverse: $1 / F_{A} = [1 / f_{A 1}^{s}, 1 / f_{A 2}^{m}, 1 / f_{A 3}^{b}]$

$ω F_{A} = [ω f_{A 1}^{s}, ω f_{A 2}^{m}, ω f_{A 3}^{b}]$

In relation to a fuzzy multi-criteria decision-making problem, suppose that the decision alternative is as a_ii ∈ (1, …, m), the decision criteria is as c_j, j ∈ (1, …, n) and whose weight is %w_j = j ∈ (1, K, n) in which $% w_{j} = % w_{j}^{L}, % w_{j}^{M}, % w_{j}^{R}$ the criteria values of a_i under a_j evaluated by the decision maker is the triangular fuzzy number %x_ij and $% x_{ij} = (x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{R})$ .

Definition 3:

In an evaluation matrix where the criteria value is a triangular fuzzy number, we suppose $\begin{matrix} max_{p ⩽ i ⩽ q} % x_{ij} = (\max_{p ⩽ i ⩽ q} x_{ij}^{L}, \max_{p ⩽ i ⩽ q} x_{ij}^{M}, \max_{p ⩽ i ⩽ q} x_{ij}^{R}), and \\ \min_{p ⩽ i ⩽ q} % x_{ij} = (\min_{p ⩽ i ⩽ q} x_{ij}^{L}, \min_{p ⩽ i ⩽ q} x_{ij}^{M}, \min_{p ⩽ i ⩽ q} x_{ij}^{R}) then \end{matrix}$

The positive ideal scheme is

$B^{+} = [B_{1}^{+}, K, B_{n}^{+}] = [max_{p ⩽ i ⩽ q} % x_{i 1}, K, max_{p ⩽ i ⩽ q} % x_{in}]$ and

negative ideal scheme is $B^{-} = [B_{1}^{-}, K, B_{n}^{-}] = [min_{p ⩽ i ⩽ q} % x_{i 1}, K, min_{p ⩽ i ⩽ q} % x_{in}]$

In any multi-criteria decision-making matrix, the positive and negative ideal solutions of each criterion are respectively the maximum and minimum criteria values of each alternative. When applying the VIKOR method for a comprehensive evaluation, the positive-best solution (PBS) and negative-best solution (NBS) are required to be elected for computing group utilities and individual regret. In such a case, all criteria are changed to be benefit-oriented criteria.

3 VIKOR-based triangular fuzzy number operation rule analysis

In the VIKOR method, the positive and negative ideal points will be extreme values if the evaluation value is a crisp number. The comparison procedures will be carried out by the operational laws (seen in definition 2).

With the positive and negative ideal solutions of triple numbers obtained according to definition 3, the computational results should satisfy the following (4) and (5) according to Definition 1. $\max_{p ⩽ i ⩽ q} x_{ij}^{L} - x_{ij}^{L} ⩽ \max_{p ⩽ i ⩽ q} x_{ij}^{M} - x_{ij}^{M} ⩽ \max_{p ⩽ i ⩽ q} x_{ij}^{R} - x_{ij}^{R}$ (4) $\min_{p ⩽ i ⩽ q} x_{ij}^{L} - x_{ij}^{L} ⩽ \min_{p ⩽ i ⩽ q} x_{ij}^{M} - x_{ij}^{M} ⩽ \min_{p ⩽ i ⩽ q} x_{ij}^{R} - x_{ij}^{R}$ (5)

Meanwhile, $\max_{p ⩽ i ⩽ q} x_{ij}^{L} - \min_{p ⩽ i ⩽ q} x_{ij}^{L} ⩽ \max_{p ⩽ i ⩽ q} x_{ij}^{M} - \min_{p ⩽ i ⩽ q} x_{ij}^{M} ⩽ \max_{p ⩽ i ⩽ q} x_{ij}^{R} - \min_{p ⩽ i ⩽ q} x_{ij}^{R}$ .

Theorem 1.

the subtractive regulations between positive and negative ideal solutions may not meet the characteristics of triangular fuzzy numbers.

Brief proof: According to the evaluation values $(x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{R})$ of any alternative under any criteria, $x_{ij}^{L} ⩽ x_{ij}^{M} ⩽ x_{ij}^{R}$ , we can obtain $- x_{ij}^{L} ≻ - x_{ij}^{M} ⩾ - x_{ij}^{R}$ .

Due to the unknown difference in which the three points $x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{R}$ , we could not obtain $\max_{p ⩽ i ⩽ q} x_{ij}^{L} - x_{ij}^{L} ⩽ \max_{p ⩽ i ⩽ q} x_{ij}^{M} - x_{ij}^{M} ⩽ \max_{p ⩽ i ⩽ q} x_{ij}^{R} - x_{ij}^{R}$ although $\max_{p ⩽ i ⩽ q} x_{ij}^{L} ⩽ \max_{p ⩽ i ⩽ q} x_{ij}^{M} ⩽ \max_{p ⩽ i ⩽ q} x_{ij}^{R}$ .

Similarly, we may not obtain the inequalities as $x_{ij}^{L} - \min_{p ⩽ i ⩽ q} x_{ij}^{L} ⩽ x_{ij}^{M} - \min_{p ⩽ i ⩽ q} x_{ij}^{M} ⩽ x_{ij}^{R} - \min_{p ⩽ i ⩽ q} x_{ij}^{R}$ since $\min_{p ⩽ i ⩽ q} x_{ij}^{L} ⩽ \min_{p ⩽ i ⩽ q} x_{ij}^{M} ⩽ \min_{p ⩽ i ⩽ q} x_{ij}^{R}$ with unknowndifferences of $x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{R}$ .

Similarly, $\max_{p ⩽ i ⩽ q} x_{ij}^{L} - \min_{p ⩽ i ⩽ q} x_{ij}^{L} ⩽ \max_{p ⩽ i ⩽ q} x_{ij}^{M} - \min_{p ⩽ i ⩽ q} x_{ij}^{M} ⩽ \max_{p ⩽ i ⩽ q} x_{ij}^{R} - \min_{p ⩽ i ⩽ q} x_{ij}^{R}$ . is also incorrect.

If each difference is similar, the above rules are absolutely satisfied.

The above analysis emphasizes that it is necessary to defuzzify the triangular fuzzy numbers, otherwise, we cannot compute group utilities and individual regret, then the compromise solutions cannot be calculated by applying the VIKOR method.

Corollary 1.

In the triangular fuzzy number evaluation matrix, if the differences among the left, middle and right values of any triangular fuzzy number are equal, positive and negative ideal solutions obtained according to definition 3 will satisfy the VIKOR-based triangular fuzzy number operation rule.

Brief proof: when each triangular fuzzy number satisfies $x_{ij}^{M} - x_{ij}^{L} = x_{ij}^{R} - x_{ij}^{M}$ , then: $\max_{p ⩽ i ⩽ q} x_{ij}^{M} - \max_{p ⩽ i ⩽ q} x_{ij}^{L} = x_{ij}^{M} - x_{ij}^{L}$ (6) $\max_{p ⩽ i ⩽ q} x_{ij}^{R} - \max_{p ⩽ i ⩽ q} x_{ij}^{M} = x_{ij}^{R} - x_{ij}^{M}$ (7)

$\max_{p ⩽ i ⩽ q} x_{ij}^{L} ⩾ x_{ij}^{L}$ and $\max_{p ⩽ i ⩽ q} x_{ij}^{M} ⩾ x_{ij}^{M}$ because $\max_{p ⩽ i ⩽ q} x_{ij}^{R} ⩾ x_{ij}^{R}$ . Positive and negative ideal solutions are undoubtedly the triangular fuzzy numbers among them. In such a case, the differences among the left, middle and right values are equal and degraded into clear numbers [17].

Based on the Theorem 1 and Corollary 1, it is necessary to analyze the differences between the left and medium and right points, and judge whether the computational results satisfy the characteristic of triple fuzzy numbers. if it cannot satisfy the operational laws, appropriate defuzzification procedure will conducted here to reduce information losses and improve the feasibility of the decision results [18].

4 Modified VIKOR –fuzzy integrated for multiple criteria based decision making

To reduce the information loss in the course of processing triangular fuzzy numbers, this paper designed the following multi-criteria decision-making steps as shown in Fig. 2.

Fig. 2

Workflow of Modified VIKOR –Fuzzy Integrated for Multiple Criteria based Decision Making.

Step 1: Problem Identification

In this step, purposes of the decision-making method were recognized and the problem space was identified and defined.

Step 2: Significant Criteria Identification

In this step, the significant criteria are identified and described in order to calculate defined problem.

Step 3: Decision Maker Group Formation with Identification of Criteria, Alternative and Weight Importance Alternative and

In relation to a fuzzy multi-criteria decision-making problem, suppose that the decision alternative is as a_i, i ∈ (1, …, m), the decision criteria is as c_j, j ∈ (1, …, n) and whose weight is %w_j = j ∈ (1, K, n) in which $% w_{j} = % w_{j}^{L}, % w_{j}^{M}, % w_{j}^{R}$ , the criteria values of a_i under c_j evaluated by the decision maker FDM_d, whered = 1, 2, … . D.

Step 4: Input transformation into Fuzzy Based Linguistic Variable

In this step, all crisp input including decision alternative a_i, i ∈ (1, …, m) and decision criteria c_j, j∈ (1, …, n) ⩽ are converted into fuzzy linguistic term using Equation (1). Fuzzy linguistic term for decision alternative a_i and decision criteria c_i are defined as: $μ_{A} (a_{i}) = {VeryLow (VL), Low (L), Medium (M),$

High (H) , VeryHigh (VH) and $\begin{matrix} μ_{C} (c_{i}) = {VeryPoor (VP), Poor (P), Fair (F), Good (G), \\ VeryGood (VG), Excellent (E)} \end{matrix}$

Step 5: Fuzzy Aggregation Rating for Decision Alternatives and Weight

In this Step, fuzzy aggregation rate (% x_ij) of each decision alternative in respect to each decision criteria is computed using following equation: $% x_{ij} = (x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{R})$ (8)

Where, $x_{ij}^{L} = \min {x_{ij}^{L}}_{k}, x_{ij}^{M} = \sum_{k}^{K} x_{ij}^{M}, x_{ij}^{R} = \max {x_{ij}^{R}}_{k}$ and aggregation rate of fuzzy weight (% w_j) is computed using following equation $% w_{j} = % w_{j}^{L}, % w_{j}^{M}, % w_{j}^{R}$ (9)

Where, $% w_{j 1}^{L} = \min {% w_{jk 1}^{L}}_{k}, % w_{jk 2}^{M} = \sum_{k}^{K} % w_{jk 2}^{M}, % w_{j 1}^{R} = \max {% w_{jk 3}^{R}}_{k}$

Step 6: Fuzzy Performance Rating Matrix for Decision Making

In this step, Fuzzy Performance Rating Matrix for Decision Making [FDM] _m×n is defined: $\begin{matrix} \begin{matrix} C_{1} & \dots & C_{n} \end{matrix} \\ {[FDM]}_{m \times n} = \begin{matrix} A_{1} \\ ⋮ \\ A_{m} \end{matrix} [\begin{matrix} x_{11} & \dots & x_{1 n} \\ ⋮ & ⋱ & ⋮ \\ x_{m 1} & \dots & x_{mn} \end{matrix}] \end{matrix}$ (10)

Step 7: Fuzzy Best and Worst Value Computation

The Fuzzy best B⁺ and worst B^- value computed using following equation: $B_{j}^{+} = {\begin{matrix} [B_{1}^{+}, K, B_{n}^{+}] = [max_{p ⩽ i ⩽ q} % x_{i 1}, K, max_{p ⩽ i ⩽ q} % x_{in}], \\ for benefit criteria \\ [B_{1}^{-}, K, B_{n}^{-}] = [min_{p ⩽ i ⩽ q} % x_{i 1}, K, min_{p ⩽ i ⩽ q} % x_{in}], \\ for cost criteria \end{matrix}$ (11) $B_{j}^{-} = {\begin{matrix} [B_{1}^{-}, K, B_{n}^{-}] = [min_{p ⩽ i ⩽ q} % x_{i 1}, K, min_{p ⩽ i ⩽ q} % x_{in}], \\ for benefit criteria \\ [B_{1}^{+}, K, B_{n}^{+}] = [max_{p ⩽ i ⩽ q} % x_{i 1}, K, max_{p ⩽ i ⩽ q} % x_{in}], \\ for cost criteria \end{matrix}$ (12)

Where, $B_{j}^{+}$ represent positive-ideal solution and $B_{j}^{-}$ represent negative-ideal solution for j^th decision criteria

Step 8: Fuzzy Difference normalization

According to the computation formula of group utility and individual regret, we compute Fuzzy Difference normalization FD_ij using following equation: ${FD}_{ij} = (B_{j}^{+} - % x_{ij}) / (B_{j}^{+} - B_{j}^{-})$ (13)

If $B_{j}^{+} - B_{j}^{-}$ under a criterion meets the condition where the left endpoints, median values and right endpoints of the triangular fuzzy number gradually increase, we substitute $B_{j}^{+} - B_{j}^{-}$ value into FD_ij; otherwise, it is necessary to defuzzify $B_{j}^{+}$ and $B_{j}^{-}$ respectively, and put their defuzzification values into FD_ij.

If $B_{j}^{+} - % x_{ij}$ value under any criteria meets the condition in Corollary 1, we calculate the value by substituting $B_{j}^{+} - % x_{ij}$ into FD_ij. we will divide the problems into two types:

when the differences are not equal, we can calculate the value by substituting $B_{j}^{+} - % x_{ij}$ into FD_ij if

$\max_{p ⩽ i ⩽ q} x_{ij}^{M} - \max_{p ⩽ i ⩽ q} x_{ij}^{L} ⩾ x_{ij}^{M} - x_{ij}^{L} and \max_{p ⩽ i ⩽ q} x_{ij}^{R} - \max_{p ⩽ i ⩽ q} x_{ij}^{M} ⩾ x_{ij}^{R} - x_{ij}^{M}$ , we, otherwise, defuzzify $B_{j}^{+}$ and %x_ij, and substitute the resulting value into FD_ij.

Step 9: Compute Utility and regret Metrics

In this step, we compute Utility and regret Metrics S_i, R_i, Q_i using following equation: $\begin{matrix} Q_{i} = v \times \frac{S_{i} - min_{i} S_{i}}{max_{i} S_{i} - min_{i} S_{i}} + (1 - v) \times \frac{R_{i} - min_{i} R_{i}}{max_{i} R_{i} - min_{i} R_{i}}, \\ S_{i} = \sum_{j = 1}^{n} {FA}_{ij}, R_{i} = max_{j} F_{A_{ij}}, \\ {whereF}_{A_{ij}} = % w_{j} \times {FD}_{ij} \end{matrix}$ (14) where v is a coefficient of decision-making mechanism subject to the majority of criteria strategies [19]. When v > 0.5, a decision is made according to the opinions of the majority; when v = 0.5, a decision is made according to the agreement; when v < 0.5, a decision is made according to the refusal. As R_i is a triangular fuzzy number, a comparison could be made among F_{A
_ij} values of all schemes under all criteria.

Step 10: Defuzzifying utility and regret metric

In fuzzy system, process of defuzzification perform the conversion of fuzzy number into crisp real values [14]. The defuzzification of Utility and regret Metrics S_i, R_i, Q_i is computed using equation (3).

Step 11: Ranking decision alternatives

In this step, alternative are ranked by arranging S_i, R_i, Q_i in descending manner. Q_i is a measure of separation, smaller the value of Q_i, better the alternative is.

Step 12: Compromise solution

According to the ranking scheme of Q_i, assume a₁ and a₂ are respectively ranked first and second.

Condition 1: Advantage in acceptability. Q₂ - Q₁ ⩾ 1/(m - 1), where m is number of alternatives.

Condition 2: Acceptable stability in the decision-making process. a₁ ranks first in S_i or R_i.

When both condition 1 and 2 are met, a₁ is the scheme ranked first. If either of the condition is not met, for example, if condition 2 is not met, both a₁ and a₂ are compromise solutions; if condition 1 is not met, the maximum M can be obtained according to Q_M - Q₁ < 1/(m - 1), and a₁, …, a_M are close to the ideal solution.

5 Example analysis

Aventure capital enterpriseis committed to findingsome proper device-related VC projects [20]. There are 4 alternative alternatives under criteria of environmental impact c₁, expected return c₂, growth c₃ and social benefit c₄. Different from the first criterion which is cost-oriented, others criteria are benefit oriented. The decision-making matrix is shown in Table 1.

Table 1
Multi-criteria Decision-making Matrix

c ₁ c ₂ c ₃ c ₄

a ₁ [4.3,4.6,4.7] [88,90,92] [0.6,0.7,0.8] [4,5,7]

a ₂ [2.8,2.9,3.1] [91,92,94] [0.4,0.5,0.6] [3,4,5]

a ₃ [4.0,4.1,4.2] [90,92,94] [0.2,0.3,0.4] [5,6,7]

a ₄ [3.5,3.7,3.8] [80,82,83] [0.4,0.5,0.7] [3,4,6]

	c ₁	c ₂	c ₃	c ₄
a ₁	[4.3,4.6,4.7]	[88,90,92]	[0.6,0.7,0.8]	[4,5,7]
a ₂	[2.8,2.9,3.1]	[91,92,94]	[0.4,0.5,0.6]	[3,4,5]
a ₃	[4.0,4.1,4.2]	[90,92,94]	[0.2,0.3,0.4]	[5,6,7]
a ₄	[3.5,3.7,3.8]	[80,82,83]	[0.4,0.5,0.7]	[3,4,6]

Step 1: Normalize the above decision-making matrix, as shown in Table 2.

Table 2

Normalize Multi-criteria Decision-making Matrix

	c ₁	c ₂	c ₃	c ₄
a ₁	[0.489,0.543,0.651]	[0.936,0.968,1.000]	[0.750,1.000,1.000]	[0.571,0.833,1.000]
a ₂	[0.742,0.862,1.000]	[0.968,0.989,1.000]	[0.500,0.714,1.000]	[0.429,0.667,1.000]
a ₃	[0.548,0.610,0.700]	[0.957,0.989,1.000]	[0.250,0.429,0.667]	[0.714,1.000,1.000]
a ₄	[0.605,0.676,0.800]	[0.851,0.882,0.912]	[0.500,0.714,1.000]	[0.429,0.667,1.000]

Step 2: Make a comparison among triangular fuzzy numbers. Based on definition 3: $\begin{matrix} B^{+} = [[0.742, 0.862, 1.000], [0.968, 0.989, 1.000], \\ [0.750, 1.000, 1.000], [0.714, 1.000, 1.000]] \\ B^{-} = [[0.489, 0.543, 0.651], [0.851, 0.882, 0.912], \\ [0.250, 0.429, 0.667], [0.429, 0.667, 1.000]] \end{matrix}$

Among the four criteria, the maximum value minus the minimum value was: $\begin{matrix} B^{+} - B^{-} = [[0.253, 0.319, 0.349], [0.117, 0.108, 0.088], \\ [0.500, 0.571, 0.333], [0.286, 0.333, 0.000]] \end{matrix}$

Only the difference under criterion 1 met the condition of the triangular fuzzy number, while the differences after defuzzification were taken for the other 3 criteria, being 0.30097, 1.26871 and 0.42076 respectively.

Since the values in Table 2 did not satisfy the condition in Corollary 1, we should conduct defuzzification procedure if $B_{j}^{+} - % x_{ij}$ value with the properties of triangular fuzzy number, as shown in Table 3.

Table 3

Compliance $B_{j}^{+} - % x_{ij}$ with the Condition of Triangular Fuzzy Number

	c ₁	c ₂	c ₃	c ₄
a ₁	[0.253,0.319,0.349]	0.041	[0.000,0.000,0.000]	0.242
a ₂	[0.000,0.000,0.000]	[0.000,0.000,0.000]	0.372	0.422
a ₃	[0.194,0.252,0.300]	0.012	1.269	[0.000,0.000,0.000]
a ₄	[0.137,0.186,0.200]	0.301	0.372	0.422

Note: For those which are not triangular fuzzy numbers, insert the result obtained from defuzzification into Table 3.

Step 3: Given that the weights of four criteria offered by the decision maker were [0.2,0.3,0.4], [0.3,0.4,0.5], [0.5,0.6,0.7] and [0.4,0.5,0.6], then the results from normalization were [0.091,0.167, 0.286], [0.136,0.222, 0.357], [0.227,0.333,0.500] and [0.182,0.278,0.429] respectively.

F_{A
_ij} = % w_j × FD_ij, where F_{A
_ij} was calculated for each criteria value, is shown in Table 4.

Table 4

F_{A
_ij} Values

	c ₁	c ₂	c ₃	c ₄
a ₁	[0.066,0.166,0.286]	[0.019,0.030,0.049]	0	[0.105,0.160,0.246]
a ₂	0	0	[0.067,0.098,0.147]	[0.182,0.278,0.429]
a ₃	[0.051,0.132,0.286]	[0.005,0.009,0.014]	[0.227,0.333,0.500]	0
a ₄	[0.036,0.0.97,0.226]	[0.136,0.222,0.357]	[0.067,0.098,0.147]	[0.182,0.278,0.429]

Step 4: Set v = 0.5, calculate the S_i,R_i,Q_i values, as shown in Table 5.

Table 5

%S_i, % R_i, % Q_i Values in VIKOR

	%S_i	%R_i	%Q_i
a ₁	[0.189,0.356,0.581]	[0.066,0.166,0.286]	[0.000,0.000,0.000]
a ₂	[0.248,0.375,0.575]	[0.182,0.278,0.429]	[0.074,0.360,0.468]
a ₃	[0.0.66,0.166,0.286]	[0.227,0.333,0.500]	[0.349,0.694,0.694]
a ₄	[0.0.66,0.166,0.286]	[0.182,0.278,0.429]	[0.249,0.835,0.943]

Step 5: Calculate the results obtained from defuzzification, which satisfied both condition 1 and condition 2, and the compromise solution a₁ was obtained.

Through initial defuzzification of triangular fuzzy numbers, the S_i,R_i,Q_i values finally obtained were listed in Table 6.

Table 6

Comparison between Two Results from Defuzzification

	Results from Defuzzification in Table 5			Results from Initial Defuzzification
	S _i	R _i	Q _i	S _i	R _i	Q _i
a ₁	1.254	0.583	0.000	0.359	0.170	0.000
a ₂	1.321	0.980	0.932	0.374	0.277	0.321
a ₃	1.763	1.162	1.736	0.480	0.330	0.682
a ₄	2.564	0.980	2.044	0.692	0.277	0.832

Obviously, there were great differences in S_i,R_i,Q_i between the two methods, which mainly arose from the information loss caused by defuzzification. Solutions also changed along with the schemes a₁, a₂ obtained by initial defuzzification.

6 Conclusion

With the advancement and constant development many industries and business grow and expand gradually. With this progression people have multiple alternative solution for their problem whether it be multiple website for particular service, personal selection during employment process, or product selection for any requirement. However, the most difficult task is to choose best alternative solution from multiple alternative. With this aim we propose Multiple Criteria based Decision Making Using modified VIKOR - Fuzzy Integration Approach for in order to enhance the decision making capability and based on various decision criteria, best possible alternative is being selected.

Defuzzification of triangular fuzzy numbers will lead to information losses which violate the operational rule of the triangular fuzzy number. This paper analyzed the main processes where information losses arose from defuzzification based on the VIKOR method. In order to show the superiority of the proposed method in the paper which include three k values (k = 1,2,3), we compare it with the others.

The defuzzification results when k = 1 can be depicted as 0.018,0.000,0.004 and 0.104 for a₁,a₂,a₃,a₄ under c₂, the values are 0.000,0.179, 0.468 and 0.179 under c₃, 0.103, 0.206, 0.000 and 0.206 under c₄.

The defuzzification results when k = 2 can be depicted as 0.019,0.000,0.003 and 0.105 for a₁,a₂,a₃,a₄ under c₂, the values are 0.000,0.206, 0.494 and 0.206 under c₃, 0.119, 0.238, 0.000 and 0.238 under c₄.

The defuzzification results when k = 3 can be depicted as 0.019,0.000,0.002 and 0.105 for a₁,a₂,a₃,a₄ under c₂, the values are 0.000,0.222, 0.509 and 0.222 under c₃, 0.129, 0.257, 0.000 and 0.257 under c₄.

It is obvious that the defuzzification values are more than the above three different defuzzification parameters.

In order to obtain compromise solutions, we calculated S_i,R_i and Q_i under three conditions which can be as the following tables.

Similarly, the computation results can be as Table 8 when k = 3.

Table 7

The calculation table according to the parameter k

	Results from Defuzzification (k = 1)			Results from Defuzzification (k = 2)
	S _i	R _i	Q _i	S _i	R _i	Q _i
a ₁	0.896	0.475	0.000	0.786	0.145	0.000
a ₂	1.002	0.873	0.876	0.996	0.116	0.773
a ₃	1.234	1.003	1.453	1.115	0.265	0.923
a ₄	2.143	0.932	1.996	2.021	0.811	1.562

Table 8

The third calculation table

	Results from Defuzzification (k = 3)
	S _i	R _i	Q _i
a ₁	0.567	0.324	0.000
a ₂	0.923	0.734	0.685
a ₃	1.111	0.856	1.237
a ₄	2.003	0.889	1.556

The above three tables show that the differences between the first alternative and the second alternative will decrease, and it will be adverse to selecting appropriate compromise solutions. The differences of the proposed method is larger than the results with three parameters, and it shows the smaller information losses.

However, effective sensitivity analyses are yet to be carried out on the processes that cause losses and the number of losses, and further studies are required in the future.

Funding information: Key Funded Topic of Educational Science for the 13th Five-Year Plan in Jiangsu Province (B-a/2016/01/27); Topic of Higher Education Association of Jiangsu Province for the 13th Five-Year Plan (16YB113).

Author information: Li Ning, (1979-), Male, born in nanjing, jiangsu province, PhD candidate, school of economics and management, nanjing university of science and technology, research direction: management science and engineering.

Corresponding author: Feng Junwen (1959-), Male, born in taiyuan, Shanxi Province, professor, doctoral supervisor, school of economics and management, nanjing university of science and technology, research direction: management science and engineering.

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Multiple criteria based decision making using modified VIKOR - fuzzy integration approach

Abstract

Keywords

1 Introduction

2 Preliminaries

Definition 2:

Definition 3:

Theorem 1.

Corollary 1.

Table 1 Multi-criteria Decision-making Matrix c 1 c 2 c 3 c 4 a 1 [4.3,4.6,4.7] [88,90,92] [0.6,0.7,0.8] [4,5,7] a 2 [2.8,2.9,3.1] [91,92,94] [0.4,0.5,0.6] [3,4,5] a 3 [4.0,4.1,4.2] [90,92,94] [0.2,0.3,0.4] [5,6,7] a 4 [3.5,3.7,3.8] [80,82,83] [0.4,0.5,0.7] [3,4,6]

References

Table 1
Multi-criteria Decision-making Matrix

c ₁ c ₂ c ₃ c ₄

a ₁ [4.3,4.6,4.7] [88,90,92] [0.6,0.7,0.8] [4,5,7]

a ₂ [2.8,2.9,3.1] [91,92,94] [0.4,0.5,0.6] [3,4,5]

a ₃ [4.0,4.1,4.2] [90,92,94] [0.2,0.3,0.4] [5,6,7]

a ₄ [3.5,3.7,3.8] [80,82,83] [0.4,0.5,0.7] [3,4,6]