An extended multiple attribute group decision-making TODIM method based on the neutrosophic numbers

Abstract

Neutrosophic numbers (NNs), which contain determinate part and indeterminate part, are considered as a useful tool to express the indeterminate evaluation information which is common in real situation. The TODIM which is the short of Interactive and Multicriteria Decision Making in Portuguese can take the bounded rationality of decision makers into consideration based on the prospect theory. However, the classical TODIM method can only process the multiple attribute decision making (MADM) problems in which the attribute values are crisp numbers. The purpose of the paper is to give a new method to handle multiple attribute group decision making problems with neutrosophic numbers, which means we combine the TODIM with neutrosophic numbers to deal with MADM problems. In this paper, we firstly introduce the definition, the properties and the operational laws of the neutrosophic numbers, and then the possibility degree function is introduced concisely. Secondly, we represent the idea and steps of TODIM, and then we extend the TODIM method to the multiple attribute group decision making (MAGDM) with neutrosophic numbers. Finally, we use an illustrative example to demonstrate the practicality and effectiveness of the proposed method.

Keywords

Multiple attribute group decision making neutrosophic numbers TODIM

1 Introduction

Multiple attribute decision group making (MAGDM) problems are common in many fields such as politics, economy, military and culture. The attribute values in the decision-making problems are usually uncertain due to the complexity and fuzziness of object things, and they are difficult to be described by the crisp number. The fuzzy set (FS) firstly proposed by Zadeh [1] was used to deal with the fuzzy information. Atanassov [2] overcame the disadvantage of the fuzzy set which only considered the membership degree and neglected the non-membership degree and proposed the intuitionistic fuzzy set (IFS) which have the membership (or called truth-membership) and the non-membership (or called falsity-membership). Then, Atanassov and Gargov [3] presented the interval-valued intuitionistic fuzzy sets (IVIFS) by extending the membership and non-membership of IFS to interval numbers. Liu and Zhang [4] proposed the triangular intuitionistic fuzzy numbers (TIFN) by extending the membership and non-membership of IFS to triangular fuzzy numbers. Wang [5] defined intuitionistic trapezoidal fuzzy numbers (ITFNs) and interval intuitionistic trapezoidal fuzzy numbers (IITFNs), then some decision making methods based on them had been proposed [6, 7]. So, Smarandache [8] further proposed the neutrosophic numbers (NNs) which consisted of determinate part and indeterminate part and even more practical to deal with indeterminate information in real decision problems. Therefore, the neutrosophic numbers (NNs) can be conveyed by the function of N = a + bI where a and bI are the determinate part and the indeterminate part respectively. Clearly, the fewer indeterminate part of the neutrosophic numbers (NNs), more precise the neutrosophic numbers. So, if N is unknown, that is the worst-case (N = bI). If N is a crisp number, that is the best cases (N = a). We can know that it is more suitable to deal with the indeterminate information in decision making problems. Further, Ye [9] proposed a possibility degree for neutrosophic numbers and developed a neutrosophic number ranking methods, and applied it to the multiple-attribute group decision-making. Kong et al. [10] propsoed the cosine similarity measure of neutrosophic numbers, and developed a misfire fault diagnosis method of gasoline engines based on this similarity measure.

So far, there are many methods to deal with MADM or MAGDM problems [11 –20] and the TODIM is a better one because it can consider decision makers’ bounded rationality. Gomes and Lima [21] firstly proposed the TODIM method which is the short of Interactive and Multicriteria Decision Making in Portuguese, and TODIM method considered the bounded rationality of the decision makers based on Prospect Theory. The method applied in some aspects, such as investment assessment, rental evaluation of residential properties [22] and so on. The researches on TODIM method made some achievements. Krohling and de Souza [23] proposed a generalized TODIM method, which was known as fuzzy TODIM (F-TODIM) to handle fuzzy information. Fan et al. [24] proposed an extended TODIM method to deal with the hybrid MADM problems in which the attribute values take the form of crisp numbers, interval numbers and fuzzy numbers. Krohling et al. [25] proposed an extended TODIM method to process the intuitionistic fuzzy information. Wang [26] extended the traditional TODIM method to the environment under multi-valued neutrosophic numbers with the form (T, I, F). Liu [27] and Liu [28] proposed an extended TODIM method to process the 2-dimension uncertain linguistic information or intuitionistic uncertain linguistic information, respectively.

Because NNs are more suitable to deal with the uncertain information and the TODIM is a good decision making method based on Prospect Theory, our aim is to propose an extend TODIM method to handle multiple attribute group decision making problems in which the evaluation information is expressed by neutrosophic numbers.

The remainder of this paper is shown as follows. In Section 2, we briefly introduce the basic concepts and the operational rules and the characteristics of the neutrosophic numbers (NNs), then, the steps of TODIM method are proposed. In Section 3, we introduce the steps of the extend TODIM method in detail. In Section 4, we give an numerical example to demonstrate the function of the proposed method and use the different indeterminate ranges for I to analyze theeffectiveness.

2 Preliminaries

Definition 1. [29 –31] Suppose I ∈ [β^-, β⁺] be an indeterminate part, a neutrosophic number N is given by $N = a + bI$ (1) in which a and b are real numbers, and I is indeterminacy, satisfying I² = I, 0 · I = 0 and I/I = undefined.

We can further understand the definition of NNs through an example, N₁ = 7 + I, in which I ∈ [0.8, 0.95], that means N₁ ∈ [7.8, 7.95], i.e. the determinate part of N₁ is 7, while the indeterminate part of N₁ is I ∈ [0.8, 0.95].

Definition 2. [29 –31] Suppose N₁ = a₁ + b₁I and N₂ = a₂ + b₂I be two neutrosophic numbers, the operational laws of NNs are defined as follows: $(1) N_{1} + N_{2} = a_{1} + a_{2} + (b_{1} + b_{2}) I$ (2) $(2) N_{1} - N_{2} = a_{1} - a_{2} + (b_{1} - b_{2}) I$ (3) $(3) N_{1} \times N_{2} = a_{1} a_{2} + (a_{1} b_{2} + a_{2} b_{1} + b_{1} b_{2}) I$ (4) $(4) N_{1}^{2} = a_{1}^{2} + (2 a_{1} b_{1} + b_{1}^{2}) I$ (5) $(5) λ N_{1} = λ a_{1} + λ b_{1} I$ (6) $(6) N_{1}^{λ} = a_{1}^{λ} + ((a_{1} + b_{1})^{λ} - a_{1}^{λ}) I λ > 0$ (7) $(7) \frac{N_{1}}{N_{2}} = \frac{a_{1} + b_{1} I}{a_{2} + b_{2} I} = \frac{a_{1}}{a_{2}} + \frac{a_{2} b_{1} - a_{1} b_{2}}{a_{2} (a_{2} + b_{2})} I$ (8) $for a_{2} \neq 0 and a_{2} \neq - b_{2}$

$\begin{matrix} (8) \sqrt{N_{1}} \\ = \sqrt{a_{1} + b_{1} I} = {\begin{matrix} \sqrt{a_{1}} - (\sqrt{a_{1}} + \sqrt{a_{1} + b_{1}}) I \\ \sqrt{a_{1}} - (\sqrt{a_{1}} - \sqrt{a_{1} + b_{1}}) I \\ - \sqrt{a_{1}} + (\sqrt{a_{1}} + \sqrt{a_{1} + b_{1}}) I \\ - \sqrt{a_{1}} + (\sqrt{a_{1}} - \sqrt{a_{1} + b_{1}}) I \end{matrix} \end{matrix}$ (9)

Example 1. Suppose N₁ = 1 +3I and N₂ = 2 +5I be two neutrosophic numbers, we can get these results in the following part according to the operational laws of NNs mentioned above. $\begin{matrix} (1) N_{1} + N_{2} & = & a_{1} + a_{2} + (b_{1} + b_{2}) I \\ = & 1 + 2 + (3 + 5) I = 3 + 8 I \end{matrix}$ $\begin{matrix} (2) N_{1} - N_{2} & = & a_{1} - a_{2} + (b_{1} - b_{2}) I \\ = & 1 - 2 + (3 - 5) I = - 1 - 2 I \end{matrix}$ $\begin{matrix} (3) N_{1} \times N_{2} & = & a_{1} a_{2} + (a_{1} b_{2} + a_{2} b_{1} + b_{1} b_{2}) I \\ = & 1 \cdot 2 + (1 \cdot 5 + 2 \cdot 3 + 3 \cdot 5) I = 2 + 26 I \end{matrix}$ $\begin{matrix} (4) N_{1}^{2} & = & a_{1}^{2} + (2 a_{1} b_{1} + b_{1}^{2}) I \\ = & 1^{2} + (2 \cdot 1 \cdot 3 + 3^{2}) I = 1 + 15 I \end{matrix}$ $\begin{matrix} (5) \frac{N_{1}}{N_{2}} & = & \frac{a_{1} + b_{1} I}{a_{2} + b_{2} I} = \frac{a_{1}}{a_{2}} + \frac{a_{2} b_{1} - a_{1} b_{2}}{a_{2} (a_{2} + b_{2})} I \\ = & \frac{1}{2} + \frac{2 \cdot 3 - 1 \cdot 5}{2 (2 + 5)} I = \frac{1}{2} + \frac{1}{14} I \end{matrix}$ $\begin{matrix} \sqrt{N_{1}} & = & \sqrt{a_{1} + b_{1} I} = {\begin{matrix} \sqrt{a_{1}} - (\sqrt{a_{1}} + \sqrt{a_{1} + b_{1}}) I \\ \sqrt{a_{1}} - (\sqrt{a_{1}} - \sqrt{a_{1} + b_{1}}) I \\ - \sqrt{a_{1}} + (\sqrt{a_{1}} + \sqrt{a_{1} + b_{1}}) I \\ - \sqrt{a_{1}} + (\sqrt{a_{1}} - \sqrt{a_{1} + b_{1}}) I \end{matrix} \\ = & {\begin{matrix} \sqrt{1} - (\sqrt{1} + \sqrt{1 + 3}) I \\ \sqrt{1} - (\sqrt{1} - \sqrt{1 + 3}) I \\ - \sqrt{1} + (\sqrt{1} + \sqrt{1 + 3}) I \\ - \sqrt{1} + (\sqrt{1} - \sqrt{1 + 3}) I \end{matrix} = {\begin{matrix} 1 - 3 I \\ 1 + I \\ - 1 + 3 I \\ - 1 - I \end{matrix} \end{matrix}$

Theorem 1. Suppose N₁ = a₁ + b₁I and N₂ = a₂ + b₂I be two neutrosophic number s, and λ, λ₁, λ₂ > 0, then we have $(1) N_{1} \oplus N_{2} = N_{2} \oplus N_{1}$ (10) $(2) N_{1} \otimes N_{2} = N_{2} \otimes N_{1}$ (11) $(3) λ (N_{1} \oplus N_{2}) = λ N_{1} \oplus λ N_{2}$ (12) $(4) λ_{1} N_{1} \oplus λ_{2} N_{1} = (λ_{1} + λ_{2}) N_{1}$ (13) $(5) N_{1}^{λ} \otimes N_{2}^{λ} = (N_{1} \otimes N_{2})^{λ}$ (14) $(6) N_{1}^{λ_{1}} \otimes N_{1}^{λ_{2}} = N_{1}^{λ_{1} + λ_{2}}$ (15)

Proof.

the formula (10) is obviously right according to the operational rule (1) expressed by (2).

the formula (11) is obviously right according to the operational rule (2) expressed by (3).

for the left of the formula (12)

\begin{matrix} λ (N_{1} \oplus N_{2}) & = & λ ((a_{1} + b_{1} I) \oplus (a_{2} + b_{2} I)) \\ = & λ ((a_{1} + a_{2}) + (b_{1} + b_{2}) I) \end{matrix}

for the right of the formula (12)

\begin{matrix} λ N_{1} \oplus λ N_{2} & = & λ (a_{1} + b_{1} I) \oplus λ (a_{2} + b_{2} I) \\ = & (λ a_{1} + λ b_{1} I) \oplus (λ a_{2} + λ b_{2} I) \\ = & (λ a_{1} + λ a_{2}) + (λ b_{1} + λ b_{2}) I \\ = & λ ((a_{1} + a_{2}) + (b_{1} + b_{2}) I) \end{matrix}

So, we can get λ (N₁ ⊕ N₂) = λN₁ ⊕ λN₂, which completes the formula (12). $\begin{matrix} (4) λ_{1} N_{1} \oplus λ_{2} N_{1} & = & λ_{1} (a_{1} + b_{1} I) + λ_{2} (a_{1} + b_{1} I) \\ = & (λ_{1} a_{1} + λ_{2} a_{1}) + (λ_{1} b_{1} + λ_{2} b_{1}) I \\ = & (λ_{1} + λ_{2}) a_{1} + (λ_{1} + λ_{2}) b_{1} I \\ = & (λ_{1} + λ_{2}) N_{1} \end{matrix}$

So, we can get the formula (13) is right.

(5) for the left of the formula (14) $\begin{matrix} N_{1}^{λ} \otimes N_{2}^{λ} \\ = (a_{1}^{λ} + ((a_{1} + b_{1})^{λ} - a_{1}^{λ}) I) \\ \otimes (a_{2}^{λ} + ((a_{2} + b_{2})^{λ} - a_{2}^{λ}) I) \\ = a_{1}^{λ} a_{2}^{λ} + a_{1}^{λ} ((a_{2} + b_{2})^{λ} - a_{2}^{λ}) I \\ + a_{2}^{λ} ((a_{1} + b_{1})^{λ} - a_{1}^{λ}) I \\ + ((a_{2} + b_{2})^{λ} - a_{2}^{λ}) ((a_{1} + b_{1})^{λ} - a_{1}^{λ}) I \\ = a_{1}^{λ} a_{2}^{λ} + (a_{1}^{λ} (a_{2} + b_{2})^{λ} - a_{1}^{λ} a_{2}^{λ}) I \\ + (a_{2}^{λ} (a_{1} + b_{1})^{λ} - a_{2}^{λ} a_{1}^{λ}) I \\ + ((a_{2} + b_{2})^{λ} (a_{1} + b_{1})^{λ} - a_{2}^{λ} (a_{1} + b_{1})^{λ} \\ - a_{1}^{λ} (a_{2} + b_{2})^{λ} + a_{1}^{λ} a_{2}^{λ}) I \\ = (a_{1} a_{2})^{λ} + ((a_{2} + b_{2})^{λ} (a_{1} + b_{1})^{λ} - a_{1}^{λ} a_{2}^{λ}) I \end{matrix}$ for the right of the formula (14) $\begin{matrix} (N_{1} \otimes N_{2})^{λ} & = & {((a_{1} + b_{1}) I \otimes (a_{2} + b_{2}) I)}^{λ} \\ = & {(a_{1} a_{2} + (a_{1} b_{2} + a_{2} b_{1} + b_{1} b_{2}) I)}^{λ} \\ = (a_{1} a_{2})^{λ} \\ + ((a_{1} a_{2} + a_{1} b_{2} + a_{2} b_{1} + b_{1} b_{2})^{λ} - (a_{1} a_{2})^{λ}) I \\ = (a_{1} a_{2})^{λ} + ((a_{1} + b_{1})^{λ} (a_{2} + b_{2})^{λ} - a_{1}^{λ} a_{2}^{λ}) I \end{matrix}$

So, we can get the formula (14) is right. $\begin{matrix} (6) N_{1}^{λ_{1}} \otimes N_{1}^{λ_{2}} \\ = (a_{1}^{λ_{1}} + ((a_{1} + b_{1})^{λ_{1}} - a_{1}^{λ_{1}}) I) \\ \otimes (a_{1}^{λ_{2}} + ((a_{1} + b_{1})^{λ_{2}} - a_{1}^{λ_{2}}) I) \\ = a_{1}^{λ_{1}} a_{1}^{λ_{2}} + (a_{1}^{λ_{1}} ((a_{1} + b_{1})^{λ_{2}} - a_{1}^{λ_{2}}) I \\ + a_{1}^{λ_{2}} ((a_{1} + b_{1})^{λ_{1}} - a_{1}^{λ_{1}}) I \\ + ((a_{1} + b_{1})^{λ_{2}} - a_{1}^{λ_{2}}) ((a_{1} + b_{1})^{λ_{1}} - a_{1}^{λ_{1}}) I) \\ = a_{1}^{λ_{1}} a_{1}^{λ_{2}} + ((a_{1} + b_{1})^{λ_{2}} (a_{1} + b_{1})^{λ_{1}} - a_{1}^{λ_{2}} a_{1}^{λ_{1}}) I \\ = a_{1}^{λ_{1} + λ_{2}} + ((a_{1} + b_{1})^{λ_{1} + λ_{2}} - a_{1}^{λ_{1} + λ_{2}}) I \\ = N_{1}^{λ_{1} + λ_{2}} \end{matrix}$

So, we can get the formula (15) is right.

Definition 3. [9] Suppose N_i = a_i + b_iI be a neutrosophic number in which I ∈ [β^-, β⁺] (i = 1, 2, …, n), a_i, b_i, β^-, β⁺ ∈ R, where R is all real numbers, the neutrosophic number N_i is equivalent to N_i ∈ [a_i + b_iβ^-, a_i + b_iβ⁺], then the possibility degree is $P_{ij} = P (N_{i} \geq N_{j}) = max {1 - max (ɛ, 0), 0}$ (16)

where, $ɛ = \frac{(a_{j} + b_{j} β^{+}) - (a_{i} + b_{i} β^{-})}{(a_{i} + b_{i} β^{+}) - (a_{i} + b_{i} β^{-}) + (a_{j} + b_{j} β^{+}) - (a_{j} + b_{j} β^{-})}$

Thus, the matrix of possibility degrees can be conveyed by P = (P_ij) _n×n, in which P_ij ≥ 0, P_ij + P_ji = 1,and P_ii = 0.5. We can rank N_i (i = 1, 2, …, n) by the value of q_i (i = 1, 2, …, n) and acquire the best choice(s). $q_{i} = \frac{(\sum_{j = 1}^{n} P_{ij} + \frac{n}{2} - 1)}{n (n - 1)}$ (17)

Definition 4. Suppose N_i = a_i + b_iI be a neutrosophic number in which I ∈ [β^-, β⁺] (i = 1, 2, …, n), a_i, b_i, β^-, β⁺ ∈ R, where R is all real numbers, the neutrosophic number N_i is equivalent to N_i = [a_i + b_iβ^-, a_i + b_iβ⁺], so the distance between N_i and N_j can be defined as following: $\begin{matrix} d (N_{i}, N_{j}) \\ = \sqrt{\frac{{[(a_{j} + b_{j} β^{-}) - (a_{i} + b_{i} β^{-})]}^{2} + {[(a_{j} + b_{j} β^{+}) - (a_{i} + b_{i} β^{+})]}^{2}}{2}} \\ = \sqrt{\frac{{((a_{j} - a_{i}) + (b_{j} - b_{i}) β^{-})}^{2} + {((a_{j} - a_{i}) + (b_{j} - b_{i}) β^{+})}^{2}}{2}} \end{matrix}$ (18)

3 An extended TODIM method for MAGDM problems based on the neutrosophic numbers

As we all know, the shortcoming of the classical TODIM method is that it can only deal with the MADM problems where the attribute values are crisp numbers. But in real decision making, it is not easy to acquire the crisp numbers for the alternative under the different attributes because the pressure of time or cost and the complexity of the decision environment. However, the neutrosophic numbers can express the indeterminate information easily and precisely. In this section, we will extend the TODIM method to process the MAGDM problems under the environment of the neutrosophic numbers.

3.1 Description of the MAGDM problems

Consider the multiple attribute group decision making problems under the environment of the neutrosophic numbers as follows.

Suppose A ={ A₁, A₂, …, A_m }, C = {C₁, C₂, …, C_n} be the discrete set of alternatives and the set of attributes respectively. W ={ w₁, w₂, …, w_n } is the weight vector of the attribute C_j (j = 1, 2, …, n), where w_j ≥ 0, j = 1, 2, …, n, $\sum_{j = 1}^{n} w_{j} = 1$ . Let D = {D₁, D₂, …, D_P} be the set of decision makers, and λ = (λ₁, λ₂, …, λ_p) ^T be the weight vector of decision makers D_k (k = 1, 2, …, p), where λ_k ≥ 0, $\sum_{k = 1}^{p} λ_{k} = 1$ . Suppose that ${\tilde{R}}^{k} = [{\tilde{r}}_{ij}^{k}]_{m \times n}$ is the decision matrix, where ${\tilde{r}}_{ij}^{k} = a_{ij}^{k} + b_{ij}^{k} I$ takes the form of the neutrosophic numbers given by the decision maker D_k for alternative A_i with respect to attribute A_j, and $a_{ij}^{k}$ and $b_{ij}^{k}$ are real numbers, and I is indeterminacy. ${\tilde{R}}^{k} = \begin{matrix} A_{1} \\ ⋮ \\ A_{m} \end{matrix} \overset{\begin{matrix} C_{1} \dots C_{n} \end{matrix}}{[\begin{matrix} {\tilde{r}}_{11}^{k} & \dots & {\tilde{r}}_{1 n}^{k} \\ ⋮ & ⋱ & ⋮ \\ {\tilde{r}}_{m 1}^{k} & \dots & {\tilde{r}}_{mn}^{k} \end{matrix}]}$ (19)

3.2 The extended TODIM method for neutrosophic numbers

In the following, we will apply the TODIM to solve the multiple attribute group decision making problems with the neutrosophic numbers.

The method has the following steps:

Step 1: Obtain the relative weight w_cr of criterion C_c to reference criterion C_r, $w_{cr} = \frac{w_{c}}{w_{r}}$ (20)

Where w_r = max {w_c|c = 1, 2, …, n}.

Step 2: Obtain the dominance of each alternative A_i over each alternative A_j under the criterion C_c of the kth decision maker D_k by the Equation (21).

$\begin{matrix} φ_{c}^{k} (A_{i}, A_{j}) \\ = {\begin{matrix} \sqrt{\frac{w_{cr}}{\sum_{c = 1}^{n} w_{cr}} d ({\tilde{x}}_{ic}^{k}, {\tilde{x}}_{jc}^{k})} & if ({\tilde{x}}_{ic}^{k} - {\tilde{x}}_{jc}^{k}) > 0 \\ 0 & if ({\tilde{x}}_{ic}^{k} - {\tilde{x}}_{jc}^{k}) = 0 \\ \frac{- 1}{θ} \sqrt{\frac{(\sum_{c = 1}^{n} w_{cr})}{w_{cr}} d ({\tilde{x}}_{ic}^{k}, {\tilde{x}}_{jc}^{k})} & if ({\tilde{x}}_{ic}^{k} - {\tilde{x}}_{jc}^{k}) < 0 \end{matrix} \end{matrix}$ (21)

Step 3: Obtain the global dominance of alternative A_i over each alternative A_j by the Equation (22). $δ^{k} (A_{i}, A_{j}) = \sum_{c = 1}^{n} φ_{c}^{k} (A_{i}, A_{j})$ (22)

Step 4: Obtain the collective overall dominance of alternative A_i over each alternative A_j by the Equation (23). $δ (A_{i}, A_{j}) = \sum_{k = 1}^{p} λ_{k} δ^{k} (A_{i}, A_{j})$ (23)

Step 5: After normalizing the collective overall dominance of alternative A_i by the Equation (24), acquire the all value of the alternative A_i. $ξ_{i} = \frac{\sum_{j = 1}^{n} δ (A_{i}, A_{j}) - min_{i} (\sum_{j = 1}^{n} δ (A_{i}, A_{j}))}{max_{i} (\sum_{j = 1}^{n} δ (A_{i}, A_{j})) - min_{i} (\sum_{j = 1}^{n} δ (A_{i}, A_{j}))}$ (24)

Step 6: According to the all values of alternatives, rank the alternatives. The better alternative A_i has the larger value of ξ_i.

4 A numerical example

In this section, we give a numerical example to demonstrate the multiple attribute group decision making method based on neutrosophic number (which is cited from [32]). An investment company wants to choose a best investment project. Four possible alternatives can be selected: (1) A₁ is a car company; (2) A₂ is a food company; (3) A₃ is a computer company; (4) A₄ is an arms company. The investment company choose best option according to the following three attributes: (1) C₁ is the risk index; (2) C₂ is the growth index; (3) C₃ is the environmental index. Assume that the weight vector of the attributes is W = (0.35, 0.25, 0.4) ^T. There are three experts {D₁, D₂, D₃} who evaluate the four alternatives in the evaluation process. The weighting vector of three experts is V = (0.37, 0.33, 0.3) ^T, the kth (k = 1,2,3) expert evaluates the four possible alternatives of A_i (i = 1, 2, 3, 4) with respect to the three attributes of C_j (j = 1, 2, 3) by the form of neutrosophic numbers $N_{ij}^{k} = a_{ij}^{k} + b_{ij}^{k} I$ for $a_{ij}^{k}, b_{ij}^{k} \in R$ (k = 1, 2, …, s ; j = 1, 2, …, n ; i = 1, 2, …, m). The evaluation results are shown in Tables 1–3.

4.1 The evaluation steps

(1) Since the all criteria are benefit type and the all evaluation values are described as neutrosophic numbers, we don’t need to normalize the decision matrix.

(2) Calculate the relative weight w_cr.

According to (20), we can get w_cr = (0.875, 0.625, 1) ^T

(3) Calculate the dominance of each alternative A_i over each alternative A_j according to the kth decision maker under the criterion C_c. By (21) (suppose the attenuation coefficient θ is 1), we get the following results listed in Tables 4–12.

(4) Calculate the global dominance of alternative A_i over each alternative A_j according the kth decision maker. By (22), we can get the following results listed in Tables 13–15.

(5) Calculate the collective overall dominance of alternative A_i over each alternative A_j. By (23), we can get the following result listed in Table 16.

(6) Calculate the overall value of the alternative A_i. By (24), we can get the following result listed in Table 17.

(7) Rank all the alternatives A_i (i = 1,2,3,4) in accordance with the value ξ_i, we can get the ranking of investments A₃ ≻ A₁ ≻ A₄ ≻ A₂.

4.2 Analysis on the effect of the different indeterminate ranges for I

In order to analyze the effect of the different indeterminate ranges of I, we can rank the alternatives based on the different indeterminate I, and get the ranking results shown in Table 18.

As we can see from Table 18, when I = 0, I ∈ [0, 0.2] , I ∈ [0, 0.4], I ∈ [0, 0.6], I ∈ [0, 0.8] , I ∈ [0, 1] the ordering of the alternatives is always A₁ ≻ A₃ ≻ A₂ ≻ A₄ and the best alternative is A₁.

4.3 Analysis on the effectiveness of the proposed method

In order to verify the effectiveness of the proposed method, we can compare it with the method proposed Ye [9]. Firstly, we got the same ranking results by these methods based on this example. However, the advantage of the extended TODIM method in this paper is that it can fully consider the decision makers’ bounded rationality based on Prospect Theory.

5 Conclusion

Neutrosophic numbers (NNs), which contain two sections including determinate part and indeterminate part, are considered as a useful tool to express the inconsistent evaluation information which is common in real situation. The TODIM which is the short of Interactive and Multicriteria Decision Making in Portuguese can take the bounded rationality of decision makers into consideration based on the prospect theory. In this paper, we proposed TODIM to deal with the multiple attribute group decision making problems under the environment of neutrosophic numbers. We firstly introduced the definition, the properties, the operational laws and the possibility degree function of the neutrosophic numbers. Secondly, we developed an extended TODIM method to process the multiple attribute group decision making problems with neutrosophic numbers. Lastly, we gave a numerical example to demonstrate the practicability of the proposed method. Especially, we used the different indeterminate ranges for I to analyze the effectiveness. In the future research, we will further research some aggregation operators for neutrosophic numbers, and apply them to multiple attribute group decision making problems.

Footnotes

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Nos. 71471172 and 71271124), the Special Funds of Taishan Scholars Project of Shandong Province the Humanities and Social Sciences Research Project of Ministry of Education of China (No. 13YJC630104), Shandong Provincial Social Science Planning Project (No. 13BGLJ10), the National Soft Science Research Project of China (2014GXQ4D192), the Natural Science Foundation of Shandong Province (No. ZR2014JL046) and Graduate education innovation projects in Shandong Province (SDYY12065). The authors also would like to express appreciations to the anonymous reviewers and Editors for their very helpful comments that improved the paper.

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