An intuitionistic fuzzy multi-attribute group decision making method with incomplete weight information based on improved VIKOR

Abstract

With respect to the decision making problems where the information of the attribute weights is incomplete under the intuitionistic fuzzy environment, a decision making method based on the improved VIKOR is developed. Firstly considering the advantage of the intuitionistic fuzzy set in expressing the decision makers’ preference information, the intuitionistic fuzzy evaluation matrices are established. In order to derive the attribute weights from the incomplete attribute information of weights, a linear programming model is proposed. Then the traditional VIKOR method is improved by replacing the distance measure with projection model. Finally an illustrative example is given to prove the practicality and effectiveness of the method proposed.

Keywords

Improved VIKOR method projection model intuitionistic fuzzy number decision making linear programming model

1 Introduction

Multiple attribute decision making (MADM) [1] is one of the complex processes in management to find the appropriate alternative among a set of feasible alternatives. Since the increasing complexity of society and economy, decision makers (DMs) have difficulties in expressing their preference information with crisp value. Zadeh [2] introduced the fuzzy set to deal with the uncertain or vague information. Atanassov [3] generalized the fuzzy set to intuitionistic fuzzy set (IFS), which includes the membership degree, non-membership degree and hesitancy degree. IFS can cope with the imprecise information flexibly. Thus the IFS has attracted more and more attentions from experts and been applied in various fields especially in decision making. Many classical methods such as TOPSIS [4, 5], AHP [6, 7], ELECTRE [8, 9], and PROMETHEE [10, 11] are extended to accommodate the intuitionistic fuzzy environment.

VIKOR [12] is also an effective decision making method developed by Opricovic. It determines a compromise solution, providing a maximum utility for the majority and a minimum regret for the opponent. Many experts have utilized it to deal with different decision making problems. Hamidreza et al. [13] used VIKOR in the selection of subcontractor on the construction management. Vinodh et al. [14] applied VIKOR in the selection of fit concept in modern manufacturing environment. Sarrafha et al. [15] used VIKOR to evaluate the supply chain networks. VIKOR method is also extended by different means. Opricovic et al. [16], Sayadi et al. [17], Vahdani et al. [18], and Wan et al. [19] extended VIKOR to solve the decision making problems with the information of interval number, Triangular fuzzy number, Interval fuzzy number and Triangular intuitionistic fuzzy number respectively. Sanayei et al. [20] employed VIKOR in the selection of suppliers where information is expressed in trapezoidal and triangular fuzzy numbers. Shemshadi et al. [21] extended the VIKOR method with the mechanism of extracting and deploying objective weights based on Shannon entropy concept. You et al. [22] proposed an extended VIKOR method for group multi-attribute supplier selection with interval 2-tuple linguistic information. Opricovic and Tzeng [23] developed fuzzy VIKOR with incomplete information for analyzing land-use strategies for reducing the economic and social costs with potential natural hazards. In addition, many studies combined VIKOR with other decision making methods to solve complex multi-attribute decision making problems. Jin et al. [24] combined VIKOR with dynamic intuitionistic fuzzy weighted geometric operator to solve dynamic intuitionistic fuzzy decision making problem. Chatterjee et al. [25] integrated VIKOR and ELECTRE in industrial robots selection problems. Mohammadi et al. [26] integrated VIKOR and ANP to select the project manager. Chang et al. [27] integrated VIKOR, FAHP, GRA and TOPSIS method in deciding the optimal business models. Safari et al. [28] identified and evaluated enterprise architecture risks by using FMEA and fuzzy VIKOR method. Malekian and Azarnivand [29] integrated Shannon’s Entropy conception and VIKOR to evaluate the flood risk. Mardani et al. [30] proposed a new integrated fuzzy hybrid decision making approach with Fuzzy-AHP, TOPSIS and VIKOR method and applied in quality management fields. Ashtiani et al. [31] developed a trust model based on the integration of fuzzy AHP and fuzzy VIKOR. Opricovic and Tzeng [32] made a comparative analysis of VIKOR and TOPSIS method and found that the TOPSIS method was suitable for the risk aversion type decisions, whereas the VIKOR method was suitable for the profit maximization decisions.

In many MADM problems, the attribute weights information is often incomplete due to the decision making time pressure, the lack of knowledge or data, and limited expertise. Presently, the VIKOR method is seldom extended to solve these problems. Meanwhile, the Euclidean distance or Hamming distance is commonly used in traditional VIKOR method. Compared with the distance measure, the projection model considers not only the distance but also the module and included angle between objects [33]. Therefore, an improved VIKOR method can be proposed by replacing the distance measure with projection model. In order to do so, the reminder of the paper is organized as follows. In the Section 2, some basic concepts related to intuitionistic fuzzy set, projection model, and classical VIKOR method are introduced. Section 3 proposes the improved VIKOR method with intuitionistic fuzzy information as well as linear programming model. In Section 4, an illustrative example is given to verify the effectiveness of the method. Finally, the conclusions are made in Section 5.

2 Preliminaries

Firstly, we shall briefly introduce some basic definitions and concepts of intuitionistic fuzzy set (IFS), the projection model of intuitionistic fuzzy set and classical VIKOR method.

Definition 1. [3] Let X be a universal set, an IFS A in X is given by $A = {(x, μ_{A} (x), ν_{A} (x)) | x \in X}$ (1) where μ_A (x) represents the membership degree, ν_A (x) represents the non-membership degree, π_A = 1 - μ_A - ν_A is called the hesitancy degree.

For convenience, Xu [34] called α = (μ_α, ν_α) an intuitionistic fuzzy number (IFN), where μ_α, ν_α, π_α ∈ [0, 1].

Definition 2. [33] Let α = (μ_α, ν_α) and β = (μ_β, ν_β) be two IFNs, then $| α | = \sqrt{μ_{α}^{2} + ν_{α}^{2} + π_{α}^{2}}$ (2) is called the module of α, and $α \cdot β = μ_{α} μ_{β} + v_{α} v_{β} + π_{α} π_{β}$ (3) is called the inner production between α and β, $cos (α, β) = \frac{α \cdot β}{| α | | β |}$ (4) is called the cosine of the included angle between α and β, $\begin{matrix} Pr j_{β} (α) = \frac{α \cdot β}{| β |} \\ = \frac{μ_{α} μ_{β} + ν_{α} ν_{β} + π_{α} π_{β}}{| β |} \end{matrix}$ (5) is called the projection of the α on β. In general, the greater the value Pr j_β (α) is, the closer the α is to β.

Definition 3. [33] Let A = {α₁, α₂,. . . , α_n} and B = {β₁, β₂,. . . , β_n} be two IFSs, then $\begin{matrix} Pr j_{B} A = \frac{A \cdot B}{| B |} \\ = \frac{1}{| B |} \sum_{i = 1}^{n} (μ_{α_{i}} μ_{β_{i}} + ν_{α_{i}} ν_{β_{i}} + π_{α_{i}} π_{β_{i}}) \end{matrix}$ (6) is called the projection of the A on B. The greater the value Pr j_B (A) is, the closer the A is to B.

The VIKOR method is a compromised ranking approach for multiple attribute decision making problems. It determines a compromise solution, providing a maximum utility for the majority and a minimum regret for the opponent [12].

Assuming that the various n alternatives are denoted as a₁, a₂,. . . , a_n, the various m criteria are denoted as c₁, c₂,. . . , c_m, f_ij denotes the rating of alternative a_i, i ∈ {1, 2,. . . , n} on the criterion c_j, j ∈ {1, 2,. . . , m}. The compromise ranking algorithm VIKOR has the following steps:

Step1: Determine the positive ideal solution and negative ideal solution. Let $f_{j}^{*}$ be the positive solution, $f_{j}^{-}$ be the negative positive ideal solution.

Step 2: Calculate the values S_i and R_i by the relations $S_{i} = \sum_{j = 1}^{n} w_{j} \frac{d (f_{j}^{*} - f_{ij})}{d (f_{j}^{*} - f_{j}^{-})}$ (7) $R_{i} = max_{j} {w_{j} \frac{d (f_{j}^{*} - f_{ij})}{d (f_{j}^{*} - f_{j}^{-})}}$ (8) where w_j are the weights of the criteria, expressing the relative importance.

Step 3: Calculate the values Q_i by the relation $Q_{i} = v \frac{(S_{i} - S^{*})}{(S^{-} - S^{*})} + (1 - v) \frac{(R_{i} - R^{*})}{(R^{-} - R^{*})}$ (9) where

$S^{*} = \min_{i} S_{i}$ , $S^{-} = \max_{i} S_{i}$ ,

$R^{*} = \min_{i} R_{i}$ , $R^{-} = \max_{i} R_{i}$ ,

and v ∈ [0, 1] is introduced as the weight of the strategy of “the majority of the criteria”, generally v = 0.5.

Step 4: Rank the alternatives, sorting by the values S, R and Q in increasing order. The results are three ranking lists.

Step 5: Define the compromise solution. a⁽¹⁾ is the best alternative ranked according to the value of Q_i, which can be compromise solution if the following two conditions can be satisfied.

①: “Acceptable advantage”:

$Q (a^{(2)}) - Q (a^{(1)}) ⩾ \frac{1}{(m - 1)}$ , where A⁽²⁾ is the alternative in the second position according to the rank of Q_i. ②: “Acceptable stability in decision making”:

a⁽¹⁾ must be the best alternative according to the rank of S_i or/and R_i.

If one of the two conditions above is not satisfied, a set of compromise solution can be proposed.

(1) The compromise solution is a⁽¹⁾, a⁽²⁾, if condition ② is not satisfied.

(2) The compromise solution is a⁽¹⁾, a⁽²⁾, ... ,a^(J), if condition ① is not satisfied. a^(J) is determined by the relation $Q (a^{(J)}) - Q (a^{(1)}) < \frac{1}{(m - 1)}$ formaximum J.

3 The improved VIKOR method for MADM in intuitionistic fuzzy environment

The multi-attribute group decision making problem has some common elements: alternatives, attributes, the weights of attributes, and multiple decision makers.

(1) A_i: index for alternatives, i ∈ M = {1, 2,. . . , m};

(2) C_j: index for attributes, j ∈ N = {1, 2,. . . , n};

(3) P_k: index for decision makers, k ∈ T = {1, 2,. . . , t};

(4) w_j: index for weight of attribute C_j.

A multi-attribute group decision making problem with k decision makers, m alternatives and n attributes can be contained in the following matrix: $R^{(i)} = (r_{kj}^{(i)})_{t \times n} = (\begin{matrix} r_{11}^{(i)} & r_{12}^{(i)} & \dots & r_{1 n}^{(i)} \\ r_{21}^{(i)} & r_{22}^{(i)} & \dots & r_{2 n}^{(i)} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ r_{t 1}^{(i)} & r_{t 2}^{(i)} & \dots & r_{tn}^{(i)} \end{matrix})$ where R⁽ⁱ⁾ is the intuitionistic fuzzy decision matrix of alternative A_i, $r_{kj}^{(i)}$ represents the attribute value of A_i on C_j given by P_k.

The decision making method based on the improved VIKOR method for multi-attribute group decision making in intuitionistic fuzzy environment involves the following steps:

Step 1: Determine the positive ideal matrix and negative ideal matrix.

(1) Positive ideal matrix: $R^{*} = (r_{kj}^{*})_{t \times n} = (\begin{matrix} r_{11}^{*} & r_{12}^{*} & \dots & r_{1 n}^{*} \\ r_{21}^{*} & r_{22}^{*} & \dots & r_{2 n}^{*} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ r_{t 1}^{*} & r_{t 2}^{*} & \dots & r_{tn}^{*} \end{matrix})$ where $r_{kj}^{*}$ can be determined according to the following relations.

If C_j is the benefit attribute (the bigger the attribute values the better), then $r_{kj}^{*} = (μ_{kj}^{*}, v_{kj}^{*}) = (\max_{i \in m} {μ_{kj}^{(i)}}, min_{i \in m} {v_{kj}^{(i)}})$ (10)

If C_j is the cost attribute (the smaller the attribute values the better), $r_{kj}^{*} = (μ_{kj}^{*}, v_{kj}^{*}) = (\min_{i \in m} {μ_{kj}^{(i)}}, max_{i \in m} {v_{kj}^{(i)}})$ (11)

(2) Negative ideal matrix: $R^{-} = (r_{kj}^{-})_{t \times n} = (\begin{matrix} r_{11}^{-} & r_{12}^{-} & \dots & r_{1 n}^{-} \\ r_{21}^{-} & r_{22}^{-} & \dots & r_{2 n}^{-} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ r_{t 1}^{-} & r_{t 2}^{-} & \dots & r_{tn}^{-} \end{matrix})$ where $r_{kj}^{-}$ can be determined according to the following relations.

If C_j is the benefit attribute (the bigger the attribute values the better), then $r_{kj}^{-} = (μ_{kj}^{-}, v_{kj}^{-}) = (\min_{i \in m} {μ_{kj}^{(i)}}, max_{i \in m} {v_{kj}^{(i)}})$ (12)

If C_j is the cost attribute (the smaller the attribute values the better), $r_{kj}^{-} = (μ_{kj}^{-}, v_{kj}^{-}) = (\max_{i \in m} {μ_{kj}^{(i)}}, min_{i \in m} {v_{kj}^{(i)}})$ (13)

Step 2: Determine the weights of attributes.

In the process of decision making, the information on attribute weights is often incompletely due to the limitation of time or knowledge. Let H be the set of the known weight information, which can be constructed by the following forms [35].

(1) A weak ranking: {w_i ⩾ w_j};

(2) A strict ranking: {w_i - w_j ⩾ α_i (α_i > 0)};

(3) A ranking of differences: {w_i - w_j ⩾ w_k - w_l} , j ≠ k ≠ l;

(4) A ranking with multiples: {w_i ⩾ βw_j} , 0 ⩽ β_i ⩽ 1;

(5) An interval form: {0 ⩽ γ_i ⩽ w_i ⩽ γ_i + ξ_i ⩽ 1};

According to the evaluated information and the known information on attribute weights, the multiple objective optimization models can be established based on projection method.

Let $Pr j_{R^{*}} R^{(i)} = \sum_{j = 1}^{n} w_{j} {Pr}_{j_{r_{j}^{*}}} r_{j}$ be projection of the A_i on A^*. The greater the value Pr j_R*R⁽ⁱ⁾ is, the closer A_i is to A^*. The optimal weights of attribute can make Pr j_R*R⁽ⁱ⁾ larger as soon as possible. The linear programming model can be developed like that

Model 1 $\begin{matrix} max (\sum_{j = 1}^{n} Pr j_{r_{j} *} r_{j}^{(1)}, \sum_{j = 1}^{n} Pr j_{r_{j} *} r_{j}^{(2)}, . . ., \sum_{j = 1}^{n} Pr j_{r_{j} *} r_{j}^{(m)}) \\ s . t . \begin{matrix} \end{matrix} w \in H, w_{j} ⩾ 0, j = 1, 2, . . ., n \end{matrix}$ (14)

Since each alternative is non-inferior, there exists no preference relation on all alternatives. Then the above multiple objective optimization model can be aggregated into the following single objective optimization model:

Model 2 $max \sum_{i = 1}^{m} \sum_{j = 1}^{n} w_{j} Pr j_{r_{j} *} r_{j}^{(i)} s . t . w \in H, w_{j} ⩾ 0, j = 1, 2, . . ., n$ (15)

From the model 2, the optimal weights of attribute can be derived.

Step 3: Calculate the values of S_i, R_i, Q_i for each alternative A_i by the relations. $\begin{matrix} S_{i} = \sum_{j = 1}^{n} w_{j} [Pr j_{r_{j} *} r_{j}^{-} / Pr j_{r_{j} *} r_{j}^{(i)}] = \\ \sum_{j = 1}^{n} w_{j} [\frac{r_{j}^{*} \cdot r_{j}^{-}}{| r_{j}^{*} |} / \frac{r_{j}^{*} \cdot r_{j}^{(i)}}{| r_{j}^{*} |}] = \sum_{j = 1}^{n} w_{j} \frac{r_{j}^{*} \cdot r_{j}^{-}}{r_{j}^{*} \cdot r_{j}^{(i)}} \end{matrix}$ (16) $\begin{matrix} R_{i} = max_{j} {w_{j} [Pr j_{r_{j}^{*}} r_{j}^{-} / Pr j_{r_{j}^{*}} r_{j}^{(i)}]} = \\ max_{j} {w_{j} [\frac{r_{j}^{*} \cdot r_{j}^{-}}{| r_{j}^{*} |} / \frac{r_{j}^{*} \cdot r_{j}^{(i)}}{| r_{j}^{*} |}]} = max_{j} {w_{j} \frac{r_{j}^{*} \cdot r_{j}^{-}}{r_{j}^{*} \cdot r_{j}^{(i)}}} \end{matrix}$ (17) where S_i represents the maximum group utility of the alternative A_i, R_i the minimum individual regret of A_i, w_j the attribute weight of C_j.

Compute the values of Q_i by the relation. $Q_{i} = v \frac{(S_{i} - S^{*})}{(S^{-} - S^{*})} + (1 - v) \frac{(R_{i} - R^{*})}{(R^{-} - R^{*})}$ (18) where $\begin{matrix} S^{*} & = & \min_{i} S_{i}, S^{-} = \max_{i} S_{i}, \\ R^{*} & = & \min_{i} R_{i}, R^{-} = \max_{i} R_{i} . \end{matrix}$ and v is introduced as the weight of the strategy of the majority of criteria, and v ∈ [0, 1], generally v = 0.5.

Step 4: Rank the alternatives according to the value of S_i, R_i, Q_i in increasing order respectively. The smaller the value of the alternative is the better.

Step 5: Define the compromise solution. A⁽¹⁾ is the best alternative ranked according to the value of Q_i, which can be the compromise solution if the following two conditions can be satisfied.

Acceptable advantage:

$Q (A^{(2)}) - Q (A^{(1)}) ⩾ \frac{1}{(m - 1)}$ , where A⁽²⁾ is the alternative in the second position according to the rank of Q_i.

Acceptable stability in decision making:

A⁽¹⁾ must be the best alternative according to the rank of S_i or/and R_i.

If one of the two conditions above is not satisfied, a set of compromise solution can be proposed.

(1) The compromise solution is A⁽¹⁾,A⁽²⁾, if condition ② is not satisfied.

(2) The compromise solution is A⁽¹⁾A⁽²⁾, ... ,A^(J), if condition ① is not satisfied. Where A^(J) is determined by the relation $Q (A^{(J)}) - Q (A^{(1)}) < \frac{1}{(m - 1)}$ formaximum J.

4 An illustrative example

With the increasingly fierce market competition, more and more firms have realized the importance of supply chain strategy. Supply chain logistics outsourcing attracts more and more attention. More and more manufacturers outsource their logistics business to 3PL. GREE is a famous air conditioner manufacturer in China. It wants to select the appropriate third-party logistics service provider to improve the supply chain efficiency. After the initial selection, four alternatives are selected, which are A = {A₁, A₂, A₃, A₄}. The firm invites four experts e = {e₁, e₂, e₃, e₄} to evaluate the four alternatives according to the criteria in Table 1.

Table 1
Criteria used in selection of suppliers

Criteria Type

Quality of service (G₁) Benefit

response capability (G₂) Benefit

Technology (G₃) Benefit

Flexibility (G₄) Benefit

Finance (G₅) Benefit

cost (G₆) Cost

Criteria	Type
Quality of service (G₁)	Benefit
response capability (G₂)	Benefit
Technology (G₃)	Benefit
Flexibility (G₄)	Benefit
Finance (G₅)	Benefit
cost (G₆)	Cost

The evaluation information can be listed in Table 2.

Table 2

Ratings of the alternatives

	G ₁	G ₂	G ₃	G ₄	G ₅	G ₆
A ₁	e ₁	(0.6,0.2)	(0.7,0.3)	(0.6,0.1)	(0.6,0.2)	(0.7,0.2)	(0.3,0.7)
	e	(0.7,0.2)	(0.7,0.3)	(0.6,0.4)	(0.5,0.2)	(0.6,0.3)	(0.2,0.7)
	e	(0.7,0.2)	(0.6,0.2)	(0.6,0.2)	(0.7,0.1)	(0.8,0.1)	(0.2,0.4)
	e ₄	(0.7,0.1)	(0.6,0.2	(0.6,0.2)	(0.8,0.1)	(0.6,0.1)	(0.2,0.5)
A ₂	e ₁	(0.6,0.3)	(0.6,0.2)	(0.6,0.1)	(0.6,0.3)	(0.7,0.1)	(0.4,0.5)
	e ₂	(0.6,0.2)	(0.6,0.3)	(0.7,0.2)	(0.5,0.1)	(0.6,0.1)	(0.3,0.6)
	e ₃	(0.6,0.2)	(0.7,0.3)	(0.6,0.2)	(0.6,0.1)	(0.7,0.2)	(0.3,0.7)
	e ₄	(0.6,0.4)	(0.6,0.3)	(0.8,0.1)	(0.7,0.1)	(0.6,0.2)	(0.3,0.4)
A ₃	e ₁	(0.5,0.2)	(0.6,0.2)	(0.7,0.1)	(0.5,0.2)	(0.7,0.2)	(0.3,0.7)
	e ₂	(0.7,0.2)	(0.7,0.3)	(0.5,0.1)	(0.8,0.1)	(0.6,0.3)	(0.2,0.8)
	e ₃	(0.8,0.1)	(0.5,0.4)	(0.8,0.2)	(0.7,0.2)	(0.6,0.1)	(0.4,0.6)
	e ₄	(0.7,0.1)	(0.6,0.2)	(0.5,0.3)	(0.8,0.1)	(0.6,0.2)	(0.2,0.3)
A ₄	e ₁	(0.7,0.3)	(0.5,0.3)	(0.6,0.3)	(0.6,0.1)	(0.6,0.3)	(0.3,0.4)
	e ₂	(0.8,0.2)	(0.7,0.2)	(0.6,0.2)	(0.8,0.1)	(0.7,0.2)	(0.4,0.4)
	e ₃	(0.6,0.3)	(0.5,0.3)	(0.7,0.2)	(0.7,0.1)	(0.8,0.1)	(0.3,0.5)
	e ₄	(0.6,0.4)	(0.7,0.2)	(0.8,0.1)	(0.6,0.1)	(0.5,0.4)	(0.4,0.5)

Now the proposed improved VIKOR method can be utilized to select the best supplier.

Step1: Determine the positive ideal matrix and negative ideal matrix.

The positive ideal matrix and the negative ideal matrix are given in Table 3.

Table 3

The positive ideal matrix and negative ideal matrix

	G ₁	G ₂	G ₃	G ₄	G ₅	G ₆
R ⁺	e ₁	(0.7,0.2)	(0.7,0.2)	(0.7,0.1)	(0.6,0.1)	(0.7,0.1)	(0.3,0.7)
	e ₂	(0.8,0.2)	(0.7,0.2)	(0.7,0.1)	(0.8,0.1)	(0.7,0.1)	(0.2,0.8)
	e ₃	(0.8,0.1)	(0.7,0.2)	(0.8,0.2)	(0.7,0.1)	(0.8,0.1)	(0.2,0.7)
	e ₄	(0.7,0.1)	(0.7,0.2)	(0.8,0.1)	(0.8,0.1)	(0.6,0.1)	(0.2,0.5)
R ^-	e ₁	(0.5,0.3)	(0.5,0.3)	(0.6,0.3)	(0.5,0.3)	(0.6,0.3)	(0.4,0.4)
	e ₂	(0.6,0.2)	(0.6,0.3)	(0.5,0.4)	(0.5,0.2)	(0.6,0.3)	(0.4,0.4)
	e ₃	(0.6,0.3)	(0.5,0.4)	(0.6,0.2)	(0.6,0.2)	(0.6,0.2)	(0.4,0.4)
	e ₄	(0.6,0.4)	(0.6,0.3)	(0.5,0.3)	(0.6,0.1)	(0.5,0.4)	(0.4,0.3)

Step 2: Determine the weights of attributes.

The information on the criteria weight given by four experts can be listed as follows. $\begin{matrix} w_{1} ⩾ 0.15; \\ 0.2 ⩽ w_{2} ⩽ 0.3; \\ w_{3} ⩾ w_{4}; \\ w_{3} + w_{4} ⩾ 0.3; \\ w_{4} ⩾ 0.1; \\ w_{5} ⩾ 0.1 \\ w_{6} ⩾ 0.2; \end{matrix}$

Utilize the model 2 to establish the optimal programming model to get the weight vector of criteria.

Step 3: Calculate the values of S_i, R_i, Q_i for each alternative A_i.

When v = 0.5, the values of S_i, R_i, Q_i for each alternative A_i can be calculated by using Equation.(16–18). The result can be listed in Table 4.

Table 4

The values of S_i, R_i, Q_i for each alternative

	S	R	Q
A ₁	0.8660	0.1796	0.1581
A ₂	0.8898	0.1940	0.9124
A ₃	0.8512	0.1897	0.3517
A ₄	0.8980	0.1923	0.9410
Ranking order	A₃ ≻ A₁ ≻ A₂ ≻ A₄	A₁ ≻ A₃ ≻ A₄ ≻ A₂	A₁ ≻ A₃ ≻ A₂ ≻ A₄

Step 4: Rank the alternatives according to the value of S_i, R_i, Q_i in increasing order respectively. The smaller the value of the alternative is thebetter.

According to the value of Q, the best alternative is A₁. However, since $Q (A^{(2)}) - Q (A^{(1)}) < \frac{1}{(4 - 1)}$ , thus the decision result is the set of compromise solution {A₁, A₃}.

Due to that the value of parameter v is various in the interval from 0 to 1, the ranking order of the potential third-party logistics service providers may be also various. The profound analysis can be made by figuring out the relation between Q_i and parameter v. The result can be listed in Table 5.

Table 5

The values of S_i, R_i, Q_i for each alternative

	S	R	Q
A ₁	0.8660	0.1796	0.3162v
A ₂	0.8898	0.1940	1-0.1752v
A ₃	0.8512	0.1897	0.7014(1-v)
A ₄	0.8980	0.1923	0.8819 + 0.1181v
Ranking order	A₃ ≻ A₁ ≻ A₂ ≻ A₄	A₁ ≻ A₃ ≻ A₄ ≻ A₂	A₁ ≻ A₃ ≻ A₄ ≻ A₂
			(0 ⩽ v < 0.4027)
			A₁ ≻ A₃ ≻ A₄ ∼ A₂
			(v = 0.4027)
			A₁ ≻ A₃ ≻ A₂ ≻ A₄
			(0.4027 < v < 0.6893)
			A₁ ∼ A₃ ≻ A₂ ≻ A₄
			(v = 0.6893)
			A₃ ≻ A₁ ≻ A₂ ≻ A₄
			(0.6893 < v < 1)

According to the value of Q, we can see that when 0 ⩽ v < 0.6893 the best alternative is A₁, when 0.6893 < v ⩽ 1, the best alternative is A₃_. However, as Fig. 1 shows when 0 ⩽ v ⩽ 0.3416, the best alternative is A₁_, when 0.3416 < v ⩽ 1, $Q (A^{(2)}) - Q (A^{(1)}) < \frac{1}{(4 - 1)}$ the decision result is the set of compromise solution {A₁, A₃}.

Fig.1

The relation between Q_i and parameter v.

Thus, if the decision makers pay more attention to the minimum individual regret of the alternative, A₁ is the appropriate third-party logistics service provider. If decision makers put emphasis on both the maximum group utility and the minimum individual regret or only the maximum group utility, A₁, A₃ are the appropriate third-party logistics service providers.

In order to make a comparison between the improved VIKOR method and classical VIKOR method, we apply the method into the problem (adopted form Jin et al. [24]). On the condition that the decision result is not influenced, Jin et al. [24] simplified the work of the calculation to only calculate the distance between the positive ideal solutions with alternatives. Thus we calculate the projection of the alternatives on positive ideal solution. The values of S_i, R_i, Q_i for each alternative are listed in Table 6.

Table 6

The values of S_i, R_i, Q_i for each alternative

	S	R	Q
X ₁	1.2589	0.4552	0.04v
X ₂	1.6600	0.6811	0.9161 + 0.0839v
X ₃	1.6419	0.7018	1-0..0433v
X ₄	1.2422	0.5182	0.2555(1-v)
X ₅	1.5069	0.5262	0.3457v+0.2879
Ranking order	x₄ ≻ x₁ ≻ x₅ ≻≻ x₃ ≻ x₂	x₁ ≻ x₄ ≻ x₅ ≻ x₂ ≻ x₃	x₁ ≻ x₄ ≻ x₅ ≻ x₂ ≻ x₃
			(0 ⩽ v < 0.6596)
			x₁ ≻ x₄ ≻ x₅ ≻ x₂ ∼ x₃
			(v = 0.6596)
			x₁ ≻ x₄ ≻ x₅ ≻ x₃ ≻ x₂
			(0.6596 < v < 0.8646)
			x₁ ∼ x₄ ≻ x₅ ≻ x₃ ≻ x₂
			(v = 0.8646)
			x₄ ≻ x₁ ≻ x₅ ≻ x₃ ≻ x₂
			(0.8646 < v ⩽ 1)

The decision results of different methods are compared in Table 7.

Table 7

The comparison between the results by different method

Method	Ranking order	Parameter v
The improved VIKOR	x₁ ≻ x₄ ≻ x₅ ≻ x₂ ≻ x₃	0 ⩽ v < 0.6596
	x₁ ≻ x₄ ≻ x₅ ≻ x₂ ∼ x₃	v = 0.6596
	x₁ ≻ x₄ ≻ x₅ ≻ x₃ ≻ x₂	0.6596 < v < 0.8646
	x₁ ∼ x₄ ≻ x₅ ≻ x₃ ≻ x₂	v = 0.8646
	x₄ ≻ x₁ ≻ x₅ ≻ x₃ ≻ x₂	0.8646 < v ⩽ 1
Jin et al. [24]	x₁ ≻ x₄ ≻ x₅ ≻ x₂ ≻ x₃	0 ⩽ v < 0.0516
	x₁ ∼ x₄ ≻ x₅ ≻ x₂ ≻ x₃	v = 0.0516
	x₄ ≻ x₁ ≻ x₅ ≻ x₂ ≻ x₃	0.0516 < v < 0.6056
	x₄ ≻ x₁ ≻ x₅ ≻ x₂ ∼ x₃	v = 0.6056
	x₄ ≻ x₁ ≻ x₅ ≻≻ x₃ ≻ x₂	0.6056 < v ⩽ 1

In the classical VIKOR method the decision result is the set of compromise solution {x₁, x₄} when v ∈ [0, 1]. In the improved VIKOR method the decision result is x₁ when 0 ⩽ v ⩽ 0.0186, the decision result is the set of compromise solution {x₁, x₄} when 0.0186 < v ⩽ 1. We can see some difference in the decision results by two methods. From the view of ranking order, in the classical method the alternative x₄ has more advantages, while in the improved method the alternative x₁ is better. Some reasons can be listed as follows.

Firstly the variances of the values of S_i, R_i on the alternatives are calculated. The variances of the value S_i of the alternatives by improved VIKOR method and classical VIKOR method are 0.0408 and 0.0098. The variances of the values R_i of the alternatives by two methods are 0.0118 and 0.0018. As we know the variances can reflect the deviations of a collection of arguments from their mean. The improved VIKOR method amplifies the separations among alternatives. Secondly the improved VIKOR method highlights the valued of R_i (i.e. minimum individual regret). In the improved VIKOR R_i of x₁ and x₄ are 0.4552 and 0.5182, but in the Table 5(b) of Jin et al. [24] the values are 0.0451 and 0.0456. Thus when decision makers pay more attention to the minimum individual regret of alternatives, the alternatives x₁ has more advantages in the improvedVIKOR.

5 Conclusions

In order to solve the MADM problem in which the information of the attribute weights is incomplete under the intuitionistic fuzzy environment, a decision making method based on the improved VIKOR method is proposed. Compared the distance measure, the projection model consider not only distance but also module and angle between objects. Thus the classical VIKOR method can be improved by replacing the distance measure with the projection model. Also based on the property of projection measure, a linear optimization model is established to derive the attribute weight form the limited information. To verify the effectiveness of the method proposed, the improved VIKOR method is used to solve the supply chain logistics services outsourcing decision making problems. Finally, a comparative analysis shows that compared with the classical VIKOR method, the improved VIKOR method can distinguish the alternatives more effectively and apply in more complex decision making fields.

Footnotes

Acknowledgments

This paper is supported by Research Project of Philosophy and Social Science of Zhejiang Province (No. 18NDJC283YB), the Key Research Center of Philosophy and Social Science of Zhejiang Province – Modern Port Service Industry and Creative Culture Research Center (No. 16JDGH067).

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An intuitionistic fuzzy multi-attribute group decision making method with incomplete weight information based on improved VIKOR

Abstract

Keywords

1 Introduction

2 Preliminaries

Table 1 Criteria used in selection of suppliers Criteria Type Quality of service (G1) Benefit response capability (G2) Benefit Technology (G3) Benefit Flexibility (G4) Benefit Finance (G5) Benefit cost (G6) Cost

Footnotes

Acknowledgments

References

Table 1
Criteria used in selection of suppliers

Criteria Type

Quality of service (G₁) Benefit

response capability (G₂) Benefit

Technology (G₃) Benefit

Flexibility (G₄) Benefit

Finance (G₅) Benefit

cost (G₆) Cost