Two normalized projection models and application to group decision-making

Abstract

Projection can measure not only the distance but also the angle between two decision objects. It has become one of important tools for complex decisions. This research finds that the existing projection formulae are unreasonable in real vector and interval vector settings. To solve this problem, this paper develops two new normalized projection measures. And new projection measures are applied to group decision-making with hybrid decision information. The applicability and feasibility are shown by an experimental analysis. The results show that the projection measures improved in this paper are superior to the existing ones.

Keywords

Normalized projection measure real vector interval vector group decision-making hybrid information

1 Introduction

Early concept of projection occurs in physics for calculating work. If a force is applied to a particle moving along a path, it often needs to know the magnitude of the force in the direction of motion. It is applied to finding the projection of one vector onto another [26]. Projection is now used for measure the “closeness” or “similarity” between two vectors in decision science. It can consider not only the distance but also the angle between two decision objects [41].

Projection methods have attracted great attention from researchers [52, 54] and successfully applied to many multi-attribute decision-making (MADM) problems [16 , 55]. For example, Xu and Da [40] and Xu [37] established two projection models in uncertain MADM context. Zheng et al. [55] and Fu et al. [6] proposed grey relational projection methods and applied them to MADM. Tsao and Chen [28] proposed a projection-based compromising method for MADM with interval-valued intuitionistic fuzzy information. Ji et al. [15] gave a projection-based TODIM method under multi-valued neutrosophic environments and application to personnel selection. Xu and Hu [41], Wang et al. [31], Xu and Cai [39] and Wei [32] explored projection model-based approaches to MADM with intuitionistic fuzzy information.

The above-mentioned methods show that projection is a very important and useful tool to deal with some decision problems. However, this research finds that the existing projection formulae are unreasonable in real and interval vector settings (see Examples 1–4 below). In order to improve them, the main contributions of this work are as follows.

This paper intendeds to contribute two new normalized projection measures toliterature.

The new projection measures will be applied to group decision-making (GDM) with hybrid decision information.

The rest of the paper is structured as follows. Section 2 reviews the related work. Section 3 briefly reviews some basic concepts, primary questions and research motivation. Section 4 presents two new normalized projection measures in real and interval vector settings, and applies them to GDM. Section 5 gives an experimental analysis to illustrate thepracticability, feasibility, effectiveness and advantages of introduced method. And Section 6 draws conclusions and future research.

2 Related work

This section introduces the related work, which includes the GDM methods and applications of projection measures to GDM problems.

GDM [3 , 49] is a type of decision-making problems [10 , 56] in which multiple decision makers (DMs) acting collectively, analyze problems, evaluate alternatives, and select a solution from a collection of alternatives [2]. The GDM methods have been widely developed. For example, Yue [42] developed a geometric approach for ranking interval-valued intuitionistic fuzzy numbers with an application to GDM. Merigó et al. [23] proposed some aggregation operators in economic growth analysis and entrepreneurial GDM. Gupta et al. [9] established an intuitionistic fuzzy GDM model with an application to plant location selection based on a new extended VIKOR method. Li et al. [19] developed some personalized individual semantics in computing with words for supporting linguistic GDM. Naamani-Dery et al. [24] introduced a reducing preference elicitation in a GDM setting. Ureña et al. [29] addressed a new framework in R to support fuzzy GDM processes. Gupta and Mohanty [8] explored an algorithmic approach to GDM problems under fuzzy and dynamic environment. Massanet et al. [22] suggested a model based on subjective linguistic preference relations for GDM problems. Maio et al. [20] focused a framework for context-aware heterogeneous GDM in business processes. Kumar and Claudio [18] introduced implications of estimating confidence intervals on group fuzzy decision making scores. Xu et al. [36] modeled the priority weights from incomplete hesitant fuzzy preference relations in GDM. Dan et al. [5] explored an empirical evaluation of a process to increase consensus in group architectural decision making. Wu et al. [34] described the selection of maritime safety control options for NUC ships using a hybrid GDM approach. He et al. [11] introduced an intuitionistic fuzzy power Bonferroni mean and application to GDM problem. He et al. [12] modeled an interval-valued hesitant fuzzy weighted power Bonferroni mean and application to GDM problem. He et al. [13] explored hesitant fuzzy power Bonferroni mean operators, and applied them to a GDM problem.

The projection measures have been applied to some GDM problems. Yue [43] proposed a model for evaluating software quality based on projection measure. Wei [33] proposed a projection method for GDM in two-tuple linguistic setting. Ju and Wang [16] proposed a projection method for GDM with incomplete weight information in linguistic setting. Wei [33] established a projection-based GDM method in two-tuple linguistic setting. Zeng et al. [52], Yue and Jia [51] and Yue [46, 47] developed some GDM methods based on projection measurement with intuitionistic fuzzy information. Xu and Liu [35] and Yue [45] established the projection methods in uncertain GDM context. Recently, Yue and Jia [50] proposed a direct projection-based GDM methodology with crisp values and interval data.

However, this research finds that the projection-based decision methods are lacking in GDM problems with hybrid decision information. To fill this research gap, this paper intends to develop two normalized projection models and apply them to a GDM problem with hybrid decision information.

3 Preliminaries

In this section, some basic concepts are briefly reviewed, including some primary questions and research motivations.

3.1 Projection measure between two real vectors

Definition 1. [37] Let α = (α₁, α₂, ⋯ , α_t) and β = (β₁, β₂, ⋯ , β_t) be two real vectors, then ${Proj}_{β} (α) = \frac{α \cdot β}{| β |}$ (1) is called a projection of vector α onto vector β, where the |β| is the module of vector β, and the α · β is the inner product between α and β.

The Proj_β (α) is a measure that the α is close to the β [6, 55]. Generally, the larger the value Proj_β (α) is, the closer the α is to the β. However, this is not always the case.

Example 1. If take α = (2, 1) and β = (1, 0), then Proj_β (α) =2 and Proj_β (β) =1. According to Equation (1), it can be found that α · β = 2 and |β|² = 1. So Proj_β (α) =2/1 =2 and Proj_β (β) =1 . In this case, we have Proj_β (β) < Proj_β (α) . However, it is obvious that the β is much closer to itself than to the α since α ≠ β.

To improve the Equation (1), Xu and Liu [35] gave the following definition.

Definition 2. Let α = (α₁, α₂, ⋯ , α_t) and β = (β₁, β₂, ⋯ , β_t) be two real vectors, then the relative projection of vector α onto vector β is defined as: ${RProj}_{β} (α) = \frac{α \cdot β}{| β |^{2}} .$ (2)

The closer the RProj_β (α) is to 1, the closer the vector α is to the β.

Let us reconsider the α = (2, 1) and β = (1, 0) in Example 1. According to Equation (2), we have RProj_β (α) =2/1 =2 and RProj_β (β) =1/1 =1.Hereby, it shows that RProj_β (β) is closer to 1 than RProj_β (α). Therefore, the β here is much closer to itself than to the α.

However, the RProj_β (α) is not a normalization measure. And for this, Yue and Jia [50] gave the following definition.

Definition 3. Let α = (α₁, α₂, ⋯ , α_t) and β = (β₁, β₂, ⋯ , β_t) be two real vectors, then the normalized projection of vector α onto vector β is defined as: ${NP}_{β} (α) = \frac{{RProj}_{β} (α)}{{RProj}_{β} (α) + | 1 - {RProj}_{β} (α) |},$ (3) where the RProj_β (α) is shown in Equation (2).

It is noted that the closer or similar the vector α is to the β, the closer the αβ/|β|² is to 1, then the closer the NProj_β (α) is to 1.

It is obvious that if α and β are nonnegative vectors, then 0 ≤ NP_β (α) ≤1. And if α = β, then NP_β (α) =1. In general, if α ≠ β, it should be NP_β (α) <1. However, a counterexamples can be seen from the following example.

Example 2. Let us take α = (1, 1) and β = (1, 0). It is noted that NP_β (α) = NP_β (β) =1. That is, the closeness of α to β is the same as the closeness of β to itself. This is a contradiction sinceα ≠ β.

3.2 Projection measurement between two interval vectors

Xu [38] and Zhang et al. [53] described the definition of interval as follows:

Definition 4. Let a = [a^l, a^u] = {x|0 ≤ a^l ≤ x ≤ a^u}, then a is called a nonnegative interval. Especially, if a^l = a^u, then a is reduced to a nonnegative real number.

Unless otherwise stated, all intervals are considered to be nonnegative in this paper.

Definition 5. Let $\tilde{α} = ([α_{1}^{l}, α_{1}^{u}], [α_{2}^{l}, α_{2}^{u}], \dots, [α_{t}^{l}, α_{t}^{u}])$ and $\tilde{β} = ([β_{1}^{l}, β_{1}^{u}], [β_{2}^{l}, β_{2}^{u}], \dots, [β_{t}^{l}, β_{t}^{u}])$ be two interval vectors, then ${Proj}_{\tilde{β}} (\tilde{α}) = \frac{\tilde{α} \cdot \tilde{β}}{| \tilde{β} |}$ (4) is called a projection [37] of vector $\tilde{α}$ onto vector $\tilde{β}$ , where $| \tilde{β} | = \sqrt{\sum_{k = 1}^{t} ((β_{k}^{l})^{2} + (β_{k}^{u})^{2})}$ and $\tilde{α} \cdot \tilde{β} = \sum_{k = 1}^{t} (α_{k}^{l} β_{k}^{l} + α_{k}^{u} β_{k}^{u})$ .

In general, the larger the value ${Proj}_{\tilde{β}} (\tilde{α})$ is, the closer the $\tilde{α}$ is to the $\tilde{β}$ . However, there are some hidden flaws, as the following example shows.

Example 3. Suppose that $\tilde{α} = ([0, 4], [0, 5])$ and $\tilde{β} = ([0, 3], [0, 4])$ . Since the $\tilde{α} \cdot \tilde{β} = 12 + 20 = 32$ and $\tilde{β} \cdot \tilde{β} = | \tilde{β} |^{2} = 25$ , according to Equation (4), we have that ${Proj}_{\tilde{β}} (\tilde{α}) = 32 / 5 > 5$ and ${Proj}_{\tilde{β}} (\tilde{β}) = 25 / 5 = 5$ . In this case, we have ${Proj}_{\tilde{β}} (\tilde{β}) < {Proj}_{\tilde{β}} (\tilde{α})$ . As mentioned in Example 1, since $\tilde{α} \neq \tilde{β}$ , the $\tilde{β}$ should be much closer to itself than to the $\tilde{α}$ .

Similar to Definition 2, Xu and Liu [35] gave the following definition.

Definition 6. Let $\tilde{α} = ([α_{1}^{l}, α_{1}^{u}], [α_{2}^{l}, α_{2}^{u}], \dots, [α_{t}^{l}, α_{t}^{u}])$ and $\tilde{β} = ([β_{1}^{l}, β_{1}^{u}], [β_{2}^{l}, β_{2}^{u}], \dots, [β_{t}^{l}, β_{t}^{u}])$ be two interval vectors, then the relative projection of vector $\tilde{α}$ onto vector $\tilde{β}$ is defined as: ${RProj}_{\tilde{β}} (\tilde{α}) = \frac{\tilde{α} \cdot \tilde{β}}{| \tilde{β} |^{2}} .$ (5)

The closer the ${RProj}_{\tilde{β}} (\tilde{α})$ is to 1, the closer the vector $\tilde{α}$ is to the $\tilde{β}$ .

Let us reconsider the $\tilde{α} = ([0, 4], [0, 5])$ and $\tilde{β} = ([0, 3], [0, 4])$ in Example 3. Since the $\tilde{α} \cdot \tilde{β} = 32$ and $\tilde{β} \cdot \tilde{β} = | \tilde{β} |^{2} = 25$ , according to Equation (5), we have that ${RProj}_{\tilde{β}} (\tilde{α}) = 32 / 25 > 1$ and ${RProj}_{\tilde{β}} (\tilde{β}) = 25 / 25 = 1$ . In this case, it is noted that the $\tilde{β}$ is much closer to itself than to the $\tilde{α}$ .

Similar to real-normalized projection in Equation (3), Yue and Jia [50] gave the following definition.

Definition 7. Let $\tilde{α} = ([α_{1}^{l}, α_{1}^{u}], [α_{2}^{l}, α_{2}^{u}], \dots, [α_{t}^{l}, α_{t}^{u}])$ and $\tilde{β} = ([β_{1}^{l}, β_{1}^{u}], [β_{2}^{l}, β_{2}^{u}], \dots, [β_{t}^{l}, β_{t}^{u}])$ be two interval vectors, then the normalized projection of vector $\tilde{α}$ onto vector $\tilde{β}$ is defined as: ${NP}_{\tilde{β}} (\tilde{α}) = \frac{{RProj}_{\tilde{β}} (\tilde{α})}{{RProj}_{\tilde{β}} (\tilde{α}) + | 1 - {RProj}_{\tilde{β}} (\tilde{α}) |},$ (6) where the ${RProj}_{\tilde{β}} (\tilde{α})$ is shown in Equation (5).

It is obvious that if $\tilde{α}$ and $\tilde{β}$ are nonnegative vectors, then the Equation (6) satisfies $0 \leq {NP}_{\tilde{β}} (\tilde{α}) \leq 1$ .

Similar to Equation (3), if $\tilde{α} = \tilde{β}$ , then ${NP}_{\tilde{β}} (\tilde{α}) = 1$ according to Equation (6). In general, if $\tilde{α} \neq \tilde{β}$ , it should be ${NP}_{\tilde{β}} (\tilde{α}) < 1$ . However, a counterexamples is shown in the following example.

Example 4. Let us take $\tilde{α} = ([0, 1], [0, 1])$ and $\tilde{β} = ([0, 1], [0, 0])$ . Since the $\tilde{α} \cdot \tilde{β} = 1$ and $| \tilde{β} |^{2} = 1$ , according to Equation (6), we have that ${NP}_{\tilde{β}} (\tilde{α}) = {NP}_{\tilde{β}} (\tilde{β}) = 1$ . In this case, it fails to see that the $\tilde{β}$ is much closer to itself than to the $\tilde{α}$ .

4 Presented method and application to group decision-making

In this section, two new normalized projection formulae will be presented and applied them toGDM.

4.1 Presented projection methods

This section follows the Equation (3), a new normalized projection measure between two real vectors will be developed.

Definition 8. Let α = (α₁, α₂, ⋯ , α_t) and β = (β₁, β₂, ⋯ , β_t) be two real vectors, then the normalized projection of vector α onto vector β is improved as follows: ${NProj}_{β} (α) = \frac{α \cdot β}{α \cdot β + | | α |^{2} - | β |^{2} |} .$ (7)

For the α = (1, 1) and β = (1, 0) in Example 2, if Equation (7) is used to calculate the normalized projection of α onto β, we have α · β = 1, |α|² = 2 and β · β = |β|² = 1. It follows that NProj_β (α) =1/(1 + 1) =1/2, NProj_β (β) =1/(1 + 0) =1. This is exactly what we would expect.

A particular case is shown in the following example.

Example 5. Let α = (-1, 0) and β = (1, 0), then α · β = -1, |α|² = |β|² = 1 and β · β = |β|² = 1. We have NProj_β (α) = NProj_β (β) =1.

It is obvious that if α and β are nonnegative vectors, then Equation (7) satisfies the following properties:

0 ≤ NProj_β (α) ≤1;

NProj_α (α) = NProj_β (β) =1;

NProj_β (α) <1 if α ≠ β;

NProj_α (β) <1 if α ≠ β;

NProj_β (α) →1 as α → β.

The limitation of nonnegative vectors is based on above Example 5.

Similar to Equation (7), the normalized projection between two interval vectors is proposedbelow.

Definition 9. Let $\tilde{α} = ([α_{1}^{l}, α_{1}^{u}], [α_{2}^{l}, α_{2}^{u}], \dots, [α_{t}^{l}, α_{t}^{u}])$ and $\tilde{β} = ([β_{1}^{l}, β_{1}^{u}], [β_{2}^{l}, β_{2}^{u}], \dots, [β_{t}^{l}, β_{t}^{u}])$ be two interval vectors, then the normalized projection of vector $\tilde{α}$ onto vector $\tilde{β}$ is defined as follows: ${NProj}_{\tilde{β}} (\tilde{α}) = \frac{\tilde{α} \cdot \tilde{β}}{\tilde{α} \cdot \tilde{β} + | | \tilde{α} |^{2} - | \tilde{β} |^{2} |} .$ (8)

For the $\tilde{α} = ([0, 1], [0, 1])$ and $\tilde{β} = ([0, 1], [0, 0])$ in Example 4, if Equation (8) is used to calculate the normalized projection of $\tilde{α}$ on $\tilde{β}$ , we have $\tilde{α} \cdot \tilde{β} = 1, | \tilde{α} |^{2} = 2$ and $| \tilde{β} |^{2} = 1$ . It follows that that ${NProj}_{\tilde{β}} (\tilde{α}) = 1 / (1 + 1) = 1 / 2, {NProj}_{\tilde{β}} (\tilde{β}) = 1 / (1 + 0) = 1$ . This is also what we would expect.

A particular case is shown in the following example.

Example 6. Let $\tilde{α} = ([- 1, 0], [- 1, 0])$ and $\tilde{β} = ([1, 0], [1, 0])$ , then $\tilde{α} \cdot \tilde{β} = - 2, | \tilde{α} |^{2} = | \tilde{β} |^{2} = 2$ and $\tilde{β} \cdot \tilde{β} = | \tilde{β} |^{2} = 2$ . It means that ${NProj}_{\tilde{β}} (\tilde{α}) = {NProj}_{\tilde{β}} (\tilde{β}) = 1$ .

The Equation (8) also satisfies the five properties in Equation (7), and the limitation of nonnegative vectors is based on above Example 6.

4.2 Approach to group decision-making

This section devotes to a projection-based GDM methodology with real numbers and intervaldata.

For convenience, some symbols used in current model are introduced as follows.

A set of m feasible alternatives written A = {A_i|i ∈ M} and M = {1, 2, ⋯ , m}.

A set of attributes written U = {u_j|j ∈ N} and N = {1, 2, ⋯ , n}.

A set of attribute weights written w = {w_j|j ∈ N}, with 0 ≤ w_j ≤ 1 and $\sum_{j = 1}^{n} w_{j} = 1$ .

A set of DMs written D = {d_k|k ∈ T} and T = {1, 2, ⋯ , t}.

In a GDM problem, if the attribute values allow to be characterized by real numbers or by intervals, the following definition is introduced.

Definition 10. [50] Let X = (x_kj) _t×n be a decision matrix and U = {u₁, u₂, …, u_n} be the set of attributes. If C and I are two subsets of U such that C ∪ I = U and C ∩ I = φ, then X is called a hybrid decision, where if u_j ∈ C, then the attribute values x_kj are characterized by real numbers; if u_j ∈ I, then the attribute values x_kj are characterized by interval data.

Definition 11. [50] Suppose that $X_{1} = (x_{kj}^{1})_{t \times n}$ and $X_{2} = (x_{kj}^{2})_{t \times n}$ are two hybrid decisions, the U, C and I are the same as in Definition 10, and X₁ and X₂ are measured based on the same attribute sets U, C and I.More specifically, if u_j ∈ C in X₁ if and only if the u_j ∈ C in X₂, and the u_j ∈ I in X₁ if and only if the u_j ∈ I in X₂, then X₁ and X₂ are called hybrid decisions with the same type. Sometimes, the X₁ and X₂ are called the same type for short.

Example 7. Suppose that $\begin{array}{l} X_{1} = {(x_{k j}^{1})}_{3 \times 3} = \begin{matrix} \begin{matrix} u_{1} & u_{2} \end{matrix} \\ \begin{matrix} e_{1} \\ e_{2} \\ e_{3} \end{matrix} (\begin{matrix} 3 & [41, 46] & 9 \\ 6 & [33, 86] & 7 \\ 9 & [55, 88] & 1 \end{matrix}) \end{matrix}, \\ X_{2} = {(x_{k j}^{2})}_{3 \times 3} = \begin{matrix} \begin{matrix} u_{1} & u_{2} \end{matrix} \\ \begin{matrix} e_{1} \\ e_{2} \\ e_{3} \end{matrix} (\begin{matrix} 2 & [68, 77] & 3 \\ 5 & [70, 78] & 7 \\ 10 & [80, 83] & 8 \end{matrix}) \end{matrix}, \end{array}$ then the X₁ and X₂ are of the same type.

Let $X_{i} = (x_{kj}^{i})_{t \times n}, i \in M$ (9) be group decision matrix of the ith (i ∈ M) alternative, in which the attribute values $x_{kj}^{i}$ are provided by kth DM.

Suppose that the X₁, X₂, …, X_m in Equation (9) are hybrid group decisions with the same type, and w = (w₁, w₂, …, w_n) is the weight vector of attributes, which is determined by all DMs, then $Y_{i} = (y_{kj}^{i})_{t \times n}, i \in M$ (10) is the weighted group decision of X_i, where, if the $y_{kj}^{i}$ in Equation (9) is a real number, then $y_{kj}^{i} = w_{j} x_{kj}^{i}$ ; if the $x_{kj}^{i}$ in Equation (9) is an interval $[x_{kj}^{il}, x_{kj}^{iu}]$ , then $y_{kj}^{i} = [w_{j} x_{kj}^{il}, w_{j} x_{kj}^{iu}]$ .

For all group decisions Y_i (i ∈ M), the positive and negative ideal decisions, Y₊ and Y_-, are determined by: $Y_{+} = (y_{kj}^{+})_{t \times n}, Y_{-} = (y_{kj}^{-})_{t \times n},$ (11) where if the $y_{kj}^{i}$ in Equation (10) is a real number, and u_j is a benefit attribute, then the $y_{kj}^{+} = max_{i \in M} {y_{kj}^{i}}$ and $y_{kj}^{-} = min_{i \in M} {y_{kj}^{i}}$ ; if the $y_{kj}^{i}$ in Equation (10) is a real number, and u_j is a cost attribute, then the $y_{kj}^{+} = min_{i \in M} {y_{kj}^{i}}$ and $y_{kj}^{-} = max_{i \in M} {y_{kj}^{i}}$ . If $y_{kj}^{i}$ in Equation (10) is an interval $[y_{kj}^{il}, y_{kj}^{iu}]$ , and u_j is a benefit attribute, then $y_{kj}^{+} = [y_{kj}^{+ l}, y_{kj}^{+ u}] = [max_{i \in M} {y_{kj}^{il}}, max_{i \in M} {y_{kj}^{iu}}]$ and $y_{kj}^{-} = [y_{kj}^{- l}, y_{kj}^{- u}] = [min_{i \in M} {y_{kj}^{il}}, min_{i \in M} {y_{kj}^{iu}}]$ ; if $y_{kj}^{i}$ in Equation (10) is an interval $[y_{kj}^{il}, y_{kj}^{iu}]$ , and u_j is a cost attribute, then $y_{kj}^{+} = [y_{kj}^{+ l}, y_{kj}^{+ u}] = [min_{i \in M} {y_{kj}^{il}}, min_{i \in M} {y_{kj}^{iu}}]$ and $y_{kj}^{-} = [y_{kj}^{- l}, y_{kj}^{- u}] = [max_{i \in M} {y_{kj}^{il}}, max_{i \in M} {y_{kj}^{iu}}]$ .

According to the Equations (7) and (8), the normalized projection of each alternative decision Y_i onto the positive ideal decision Y₊ is calculated by: ${NProj}_{Y_{+}} (Y_{i}) = \frac{Y_{i} \cdot Y_{+}}{Y_{i} \cdot Y_{+} + | | Y_{i} |^{2} - | Y_{+} |^{2} |}, i \in M,$ (12) where the $Y_{i} \cdot Y_{+} = \sum_{k = 1}^{t} (\sum_{u_{j} \in C, j = 1}^{n} y_{kj}^{i} y_{kj}^{+} + \sum_{u_{j} \in I, j = 1}^{n} (y_{kj}^{il} y_{kj}^{+ l} + y_{kj}^{iu} y_{kj}^{+ u}))$ and $| Y_{+} |^{2} = \sum_{k = 1}^{t} (\sum_{u_{j} \in C, j = 1}^{n} (y_{kj}^{+})^{2} + \sum_{u_{j} \in I, j = 1}^{n} ((y_{kj}^{+ l})^{2} + (y_{kj}^{+ u})^{2}))$ , and C and I are the same as in Definition 10.

Example 8. Suppose that $Y_{1} = (y_{kj}^{1})_{2 \times 2} = (\begin{matrix} y_{11}^{1} & [y_{12}^{1 l}, y_{12}^{1 u}] \\ y_{21}^{1} & [y_{22}^{1 l}, y_{22}^{1 u}] \end{matrix})$ and $Y_{2} = (y_{kj}^{2})_{2 \times 2} = (\begin{matrix} y_{11}^{2} & [y_{12}^{2 l}, y_{12}^{2 u}] \\ y_{21}^{2} & [y_{22}^{2 l}, y_{22}^{2 u}] \end{matrix})$ are two weighted group decisions, where the $y_{kj}^{i} (k \in T, j \in N, i \in M)$ are real numbers, and the $[y_{kj}^{il}, y_{kj}^{iu}] (k \in T, j \in N, i \in M)$ are intervals. Then the $Y_{1} \cdot Y_{2} = (y_{11}^{1} y_{11}^{2} + y_{21}^{1} y_{21}^{2}) + (y_{12}^{1 l} y_{12}^{2 l} + y_{12}^{1 u} y_{12}^{2 u} + y_{22}^{1 l} y_{22}^{2 l} + y_{22}^{1 u} y_{22}^{2 u}); | Y_{2} |^{2} = ((y_{11}^{2})^{2} + (y_{21}^{2})^{2}) + ((y_{12}^{2 l})^{2} + (y_{12}^{2 u})^{2} + (y_{22}^{2 l})^{2} + (y_{22}^{2 u})^{2}) .$

Similarly, the normalized projection of each group decision Y_i onto the negative ideal decision Y_- is calculated by: ${NProj}_{Y_{-}} (Y_{i}) = \frac{Y_{i} \cdot Y_{-}}{Y_{i} \cdot Y_{-} + | | Y_{i} |^{2} - | Y_{-} |^{2} |}, i \in M,$ (13) where the $Y_{i} \cdot Y_{-} = \sum_{k = 1}^{t} (\sum_{u_{j} \in C, j = 1}^{n} y_{kj}^{i} y_{kj}^{-} + \sum_{u_{j} \in I, j = 1}^{n} (y_{kj}^{il} y_{kj}^{- l} + y_{kj}^{iu} y_{kj}^{- u})),$ and the $| Y_{-} |^{2} = \sum_{k = 1}^{t} (\sum_{u_{j} \in C, j = 1}^{n} (y_{kj}^{-})^{2} + \sum_{u_{j} \in I, j = 1}^{n} ((y_{kj}^{- l})^{2} + (y_{kj}^{- u})^{2}))$ , and C and I are the same as in Definition 10.

The smaller the projection NProj_{Y
_-} (Y_i), the better the alternative A_i is.

The relative closeness of each group decision Y_i with respect to ideal decisions Y₊ and Y_- is calculated based on their normalized projection measurements as follows: ${NPRC}_{i} = \frac{{NProj}_{Y_{+}} (Y_{i})}{{NProj}_{Y_{+}} (Y_{i}) + {NProj}_{Y_{-}} (Y_{i})}, i \in M .$ (14)

All alternatives are ranked in accordance with the order of their relative closeness. The larger RC_i means the better alternative A_i.

4.3 Presented algorithm

This subsection establishes a GDM algorithm based on the normalized projection measurements with hybrid information. The procedure involves in the following steps and major programs.

Step 1. Establish group decisions.

The group decisions $X_{i} = (x_{kj}^{i})_{t \times n}$ are established by Equation (9), where the X₁, X₂, …, X_m are the same type.

Step 2. Construct the weighted group decisions.

For a given weight vector w = (w₁, w₂, …, w_n) of attributes, the weighted group decisions are constructed by Equation (10).

The weighted group decisions can be implemented by a MATLAB program: $\begin{matrix} for k = 1 : t \\ Y (:, :, k) = X (:, :, k) * diag ([w]); \\ end \end{matrix}$

Step 3. Determine the ideal decision.

The ideal decision Y₊ and Y_- are determined by Equation (11).

The ideal decision can be implemented by a MATLAB program: $\begin{matrix} Y_pos & = & max (Y, [], 3); \\ Y_neg & = & min (Y, [], 3); \end{matrix}$

Step 4. Calculate the normalized projections.

The normalized projection measurements are calculated by Equations (12) and (13).

The normalized projection measurements can be implemented by a MATLAB program: $\begin{matrix} for & k = 1 : t \\ NP_pos (k) = sum (sum (Y (:, :, k) . * \\ Y_p os)) / (sum (sum (Y (:, :, k) . * \\ Y_p os + abs ((Y (:, :, k) .^{2} - Y_pos .^{2}))))); \\ NP_n eg (k) = sum (sum (Y (:, :, k) . * \\ Y_neg)) / (sum (sum (Y (:, :, k) . * \\ Y_neg + abs (Y (:, :, k) .^{2} - Y_neg .^{2})))); \\ end \end{matrix}$

Step 5. Calculate the relative closeness for each alternative.

For each alternative, the relative closeness is based on the normalized projection measurements, which is shown in Equation (14).

The relative closeness can be implemented by a MATLAB program: $NPRC = NP_pos . / (NP_neg + NP_pos);$

Step 6. Rank the preference order of alternatives.

The alternatives are ranked in descending order in accordance with the order of the relative closeness.

5 Experimental analysis

An experimental analysis turns to show the practicability, feasibility, effectiveness and advantages of the proposed model in this paper.

5.1 Illustrative example

In this subsection, the illustrative example is based on the literature [50], by which can provide a valid comparison with the proposed method in this paper.

A manufacturing company is located in Shenzhen, China. This company produces agricultural machines and has an important worldwide market share. The manufacturing company is desiring to select some appropriate partners in order to increase its customer base. After pre-evaluation, four potential partners A₁, A₂, A₃ and A₄ as alternatives have remained for further evaluation.

To evaluate the four partners, the following five DMs are represented on the committee:

the production manager,

the quality manager,

the material manager,

the planning manager, and

the purchasing manager.

Four attributes are considered by DMs as follows:

trust,

market share,

financial service, and

quality stability.

The procedure for selecting partner(s) involves the following steps.

First, by the Step 1, the potential partners are evaluated by DMs in hundred-mark system (0-100), in which the evaluation scores in u₁ and u₃ are characterized by real numbers and the evaluation scores in u₂ and u₄ are characterized by interval data. The evaluation scores are shown as X₁, X₂, X₃ and X₄ in Table 1.

Table 1
Evaluation decisions of four partners

Decision DM u ₁ u ₂ u ₃ u ₄

X ₁ d ₁ 77.0 [85.0, 91.0] 92.0 [58.0, 66.0>]

d ₂ 93.0 [73.0, 88.0] 76.0 [76.0, 86.0]

d ₃ 79.0 [79.0, 85.0] 83.0 [72.0, 82.0]

d ₄ 78.0 [80.0, 86.0] 81.0 [65.0, 67.0]

d ₅ 76.0 [61.0, 67.0] 71.0 [80.0, 83.0]

X ₂ d ₁ 87.0 [70.0, 80.0] 89.0 [72.0, 86.0]

d ₂ 90.0 [77.0, 81.0] 98.0 [71.0, 81.0]

d ₃ 70.0 [80.0, 86.0] 90.0 [59.0, 63.0]

d ₄ 79.0 [55.0, 62.0] 91.0 [70.0, 98.0]

d ₅ 67.0 [68.0, 85.0] 40.0 [82.0, 85.0]

X ₃ d ₁ 75.0 [75.0, 86.0] 83.0 [76.0, 86.0]

d ₂ 89.0 [79.0, 87.0] 91.0 [75.0, 79.0]

d ₃ 82.0 [70.0, 72.0] 72.0 [84.0, 89.0]

d ₄ 93.0 [80.0, 87.0] 84.0 [61.0, 83.0]

d ₅ 77.0 [68.0, 82.0] 89.0 [58.0, 80.0]

X ₄ d ₁ 77.0 [65.0, 86.0] 78.0 [76.0, 80.0]

d ₂ 69.0 [70.0, 97.0] 85.0 [86.0, 87.0]

d ₃ 78.0 [82.0, 90.0] 72.0 [84.0, 92.0]

d ₄ 83.0 [73.0, 85.0] 69.0 [58.0, 95.0]

d ₅ 68.0 [58.0, 61.0] 76.0 [73.0, 82.0]

Decision	DM	u ₁	u ₂	u ₃	u ₄
X ₁	d ₁	77.0	[85.0, 91.0]	92.0	[58.0, 66.0>]
	d ₂	93.0	[73.0, 88.0]	76.0	[76.0, 86.0]
	d ₃	79.0	[79.0, 85.0]	83.0	[72.0, 82.0]
	d ₄	78.0	[80.0, 86.0]	81.0	[65.0, 67.0]
	d ₅	76.0	[61.0, 67.0]	71.0	[80.0, 83.0]
X ₂	d ₁	87.0	[70.0, 80.0]	89.0	[72.0, 86.0]
	d ₂	90.0	[77.0, 81.0]	98.0	[71.0, 81.0]
	d ₃	70.0	[80.0, 86.0]	90.0	[59.0, 63.0]
	d ₄	79.0	[55.0, 62.0]	91.0	[70.0, 98.0]
	d ₅	67.0	[68.0, 85.0]	40.0	[82.0, 85.0]
X ₃	d ₁	75.0	[75.0, 86.0]	83.0	[76.0, 86.0]
	d ₂	89.0	[79.0, 87.0]	91.0	[75.0, 79.0]
	d ₃	82.0	[70.0, 72.0]	72.0	[84.0, 89.0]
	d ₄	93.0	[80.0, 87.0]	84.0	[61.0, 83.0]
	d ₅	77.0	[68.0, 82.0]	89.0	[58.0, 80.0]
X ₄	d ₁	77.0	[65.0, 86.0]	78.0	[76.0, 80.0]
	d ₂	69.0	[70.0, 97.0]	85.0	[86.0, 87.0]
	d ₃	78.0	[82.0, 90.0]	72.0	[84.0, 92.0]
	d ₄	83.0	[73.0, 85.0]	69.0	[58.0, 95.0]
	d ₅	68.0	[58.0, 61.0]	76.0	[73.0, 82.0]

Through discussion and negotiation of DMs, the attribute weights are determined as w = (w₁, w₂, w₃, w₄) = (0.3, 0.2, 0.3, 0.2). By Step 2, the weighted group decisions, Y₁, Y₂, Y₃ and Y₄, are calculated and shown in Table 2.

Table 2

Weighted decisions of four partners

Decision	DM	u ₁	u ₂	u ₃	u ₄
Y ₁	d ₁	23.1	[17.0, 18.2]	27.6	[11.6, 13.2]
	d ₂	27.9	[14.6, 17.6]	22.8	[15.2, 17.2]
	d ₃	23.7	[15.8, 17.0]	24.9	[14.4, 16.4]
	d ₄	23.4	[16.0, 17.2]	24.3	[13.0, 13.4]
	d ₅	22.8	[12.2, 13.4]	21.3	[16.0, 16.6]
Y ₂	d ₁	26.1	[14.0, 16.0]	26.7	[14.4, 17.2]
	d ₂	27.0	[15.4, 16.2]	29.4	[14.2, 16.2]
	d ₃	21.0	[16.0, 17.2]	27.0	[11.8, 12.6]
	d ₄	23.7	[11.0, 12.4]	27.3	[14.0, 19.6]
	d ₅	20.1	[13.6, 17.0]	12.0	[16.4, 17.0]
Y ₃	d ₁	22.5	[15.0, 17.2]	24.9	[15.2, 17.2]
	d ₂	26.7	[15.8, 17.4]	27.3	[15.0, 15.8]
	d ₃	24.6	[14.0, 14.4]	21.6	[16.8, 17.8]
	d ₄	27.9	[16.0, 17.4]	25.2	[12.2, 16.6]
	d ₅	23.1	[13.6, 16.4]	26.7	[11.6, 16.0]
Y ₄	d ₁	23.1	[13.0, 17.2]	23.4	[15.2, 16.0]
	d ₂	20.7	[14.0, 19.4]	25.5	[17.2, 17.4]
	d ₃	23.4	[16.4, 18.0]	21.6	[16.8, 18.4]
	d ₄	24.9	[14.6, 17.0]	20.7	[11.6, 19.0]
	d ₅	20.4	[11.6, 12.2]	22.8	[14.6, 16.4]

The ideal decision is determined by Step 3, which is shown in Table 3.

Table 3

Ideal decisions of four partners

Decision	DM	u ₁	u ₂	u ₃	u ₄
Y ₊	d ₁	26.1	[17.0, 18.2]	27.6	[15.2, 17.2]
	d ₂	27.9	[15.8, 19.4]	29.4	[17.2, 17.4]
	d ₃	24.6	[16.4, 18.0]	27.0	[16.8, 18.4]
	d ₄	27.9	[16.0, 17.4]	27.3	[14.0, 19.6]
	d ₅	23.1	[13.6, 17.0]	26.7	[16.4, 17.0]
Y _-	d ₁	22.5	[13.0, 16.0]	23.4	[11.6, 13.2]
	d ₂	20.7	[14.0, 16.2]	22.8	[14.2, 15.8]
	d ₃	21.0	[14.0, 14.4]	21.6	[11.8, 12.6]
	d ₄	23.4	[11.0, 12.4]	20.7	[11.6, 13.4]
	d ₅	20.1	[11.6, 12.2]	12.0	[11.6, 16.0]

The normalized projections of each Y_i (i = 1, 2, 3) on the ideal decisions Y₊ and Y_-, NProj_{Y
₊} (Y_i) and NProj_{Y
_-} (Y_i), are calculated by Step 4, which are shown in Table 4.

Table 4

Projections, relative closeness and rankings of four partners

Partner	NProj_{Y ₊} (Y_i)	Ranking	NProj_{Y _-} (Y_i)	Ranking	NPRC _i	Ranking
A ₁	0.8352	3	0.7793	3	0.5173	3
A ₂	0.8359	2	0.7779	2	0.5180	2
A ₃	0.8744	1	0.7468	1	0.5394	1
A ₄	0.8102	4	0.8021	4	0.5025	4

By Step 5, the relative closeness of each weighted decision with respect to the ideal decisions, RC_i, are calculated; and the four partners are ranked by Step 6, which are also shown in Table 4.

Table 4 shows that the preference order of potential suppliers is as follows: $A_{3} ≻ A_{2} ≻ A_{1} ≻ A_{4},$ that is to say, the A₃ is the best or most suitable supplier for this manufacturing company, followed by A₂, A₁ and A₄.

5.2 Comparison with other measures

This subsection shows an experimental analysis to illustrate effectiveness and advantages of introduced method.

If the Equation (7) is replaced by the Equation (3), and the Equation (8) is replaced by the Equation (6), then the normalized projections in Equation (12) will be transformed to the following form:

$\begin{matrix} {NP}_{Y_{+}} (Y_{i}) & = & \frac{{RProj}_{Y_{+}} (Y_{i})}{{RProj}_{Y_{+}} (Y_{i}) + | 1 - {RProj}_{Y_{+}} (Y_{i}) |}, \\ i \in M . \end{matrix}$ (15)

The normalized projections in Equation (13) will be transformed to the following form:

$\begin{matrix} {NP}_{Y_{-}} (Y_{i}) & = & \frac{{RProj}_{Y_{-}} (Y_{i})}{{RProj}_{Y_{-}} (Y_{i}) + | 1 - {RProj}_{Y_{-}} (Y_{i}) |}, \\ i \in M . \end{matrix}$ (16)

The relative closeness in Equation (14) will be transformed to the following form: ${NRC}_{i} = \frac{{NP}_{Y_{+}} (Y_{i})}{{NP}_{Y_{+}} (Y_{i}) + {NP}_{Y_{-}} (Y_{i})}, i \in M .$ (17)

The larger the value NRC_i, the better the alternative A_i is.

For the above-mentioned example, if the normalized projections in Equations (12) and (13) are replaced by the Equations (15) and (16), the relative closeness in Equation (14) is replaced by the Equation (17), then the projections, relative closeness and ranking of four partners are shown in Table 5.

Table 5

Projections, relative closeness and rankings of four partners based on another normalized projection measure

Partner	NP_{Y ₊} (Y_i)	Ranking	NP_{Y _-} (Y_i)	Ranking	NRC _i	Ranking
A ₁	0.9025	2	0.8897	3	0.5036	2
A ₂	0.8965	3	0.8883	2	0.5023	3
A ₃	0.9283	1	0.8754	1	0.5147	1
A ₄	0.8861	4	0.9049	4	0.4948	4

Table 5 shows that the ranking based another normalized projection measure is as follows: $A_{3} ≻ A_{1} ≻ A_{2} ≻ A_{4} .$

This order is different than the ranking based on the normalized projection measure in Equation (14). In this case, we do not know which ranking is actual.

Next, the ranking in Table 4 is further compared with the general projection measures in Equations (1) and (4).

If the Equations (12) and (13) are replaced by the general projection measure, then they will change to the following formulae: ${Proj}_{Y_{+}} (Y_{i}) = \frac{Y_{i} \cdot Y_{+}}{| Y_{+} |}, i \in M .$ (18) ${Proj}_{Y_{-}} (Y_{i}) = \frac{Y_{i} \cdot Y_{-}}{| Y_{-} |}, i \in M .$ (19)

And the Equation (14) is changed to the following form: ${PRC}_{i} = \frac{{Proj}_{Y_{+}} (Y_{i})}{{Proj}_{Y_{+}} (Y_{i}) + {Proj}_{Y_{-}} (Y_{i})}, i \in M .$ (20)

The larger the value PRC_i, the better the alternative A_i is.

The projections calculated by Equations (18) and (19), relative closeness calculated by Equation (20), and ranking of four partners are shown in Table 6.

Table 6

Projections, relative closeness and rankings of four partners based on the general projection measure

Partner	Proj_{Y ₊} (Y_i)	Ranking	Proj_{Y _-} (Y_i)	Ranking	PRC _i	Ranking
A ₁	0.9025	2	1.1415	2	0.4415	3
A ₂	0.8965	3	1.1438	3	0.4394	2
A ₃	0.9283	1	1.1659	4	0.4433	1
A ₄	0.8861	4	1.1174	1	0.4423	4

Table 6 shows that the ranking based on the general projection measure is also different than the ranking based on the normalized projection measure in Equation (14).

Now, three different measures lead to different results. In this case, we do not know which measure is better. For this reason, we need to go back and to review three different measures.

First, we review the normalized projection measure. It is noted that the NProj_{Y
₊} (·) in Equation (12) is also a measure, which can measure alternatives. By Definitions 8 and 9, the larger the value NProj_{Y
₊} (Y_i), the better the alternative A_i is. Similarly, the NProj_{Y
_-} (·) in Equation (13) is also a measure to measure alternatives. The smaller the value NProj_{Y
_-} (Y_i), the better the alternative A_i is. The rankings based on NProj_{Y
₊} (·) and NProj_{Y
_-} (·) are also shown in Table 4.

Table 4 shows that three rankings are consistent. So it proves that the normalized projection is a robust measure.

Similarly, the NP_{Y
₊} (·) in Equation (15) is also a measure. The larger the value NP_{Y
₊} (Y_i), the better the alternative A_i is. Also the NP_{Y
_-} (·) in Equation (16) is also a measure. The smaller the value NP_{Y
_-} (Y_i), the better the alternative A_i is. Two rankings of partners based on the Equations (15) and (16) are also shown in Table 5.

Table 5 shows that three rankings are not consistent. So it proves that the normalized projection in Equations (7) and (8) are superior to previous normalized projection measure in Equations (3) and (6).

Finally, the general projection measure is discussed. Similar to Equation (15), the Proj_{Y
₊} (·) in Equation (18) is also a measure. The larger the value Proj_{Y
₊} (Y_i), the better the alternative A_i is. And the Proj_{Y
_-} (·) in Equation (19) is also a measure. The smaller the value Proj_{Y
_-} (Y_i), the better the alternative A_i is. Their rankings are also shown in Table 6.

Table 6 shows that the rankings based on Proj_{Y
₊} (·) and Proj_{Y
_-} (·) are not consistent with PRC_i. So the general projection measure is inferior to the normalized projection measure.

The above-mentioned nine measures led to four rankings. Now, the question is which ranking is ideal. To find an ideal ranking of alternatives, a natural idea is that the more choices a ranking is, the higher robustness this ranking is, and the greater credibility this ranking will be [50]. For this reason, these rankings are listed in Table 7.

Table 7

Statistics of rankings based on nine measures

Measure	A₃ ≻ A₂ ≻ A₁ ≻ A₄	A₂ ≻ A₃ ≻ A₁ ≻ A₄	A₃ ≻ A₄ ≻ A₁ ≻ A₂	A₂ ≻ A₃ ≻ A₄ ≻ A₁
NProj_{Y ₊} (·)	✓
NProj_{Y _-} (·)	✓
NPRC _(·)		✓
NP_{Y ₊} (·)	✓
NP_{Y _-} (·)		✓
NRC _(·)		✓
Proj_{Y ₊} (·)				✓
Proj_{Y _-} (·)			✓
PRC _(·)	✓

Table 7 shows that the ranking A₃ ≻ A₂ ≻ A₁ ≻ A₄ appears four times, the ranking A₂ ≻ A₃ ≻ A₁ ≻ A₄ appears three times, and other rankings appear one time. So the A₃ ≻ A₂ ≻ A₁ ≻ A₄ is an ideal ranking. And it is consistent with the normalized projection measurement. This result illustrates once again the superiority of normalized projection measure provided in this paper.

In a word, the normalized projection measure is superior to two relevant measures because (1) it has improved their flaws; (2) above experimental analysis has shown that it is a robust measure; and (3) above experimental analysis has shown that its result is reliable.

5.3 A dynamic comparison with other measures

It is noted that just a set of data is used in above experimental analysis. If the validity of these results are not enough to be convinced, this subsection further shows a dynamic comparison with above two measures.

For the evaluation matrix X₁ in Table 1, this comparison focuses on the $x_{12}^{1} = [x_{12}^{1 l}, x_{12}^{1 u}] = [85.0, 91.0]$ . To show a dynamic comparison, let $x_{12}^{1} = [α, 6 + α]$ , where the α ∈ [0, 94] is a parameter. Other values are the same as in Table 1. If let α increase from 0 to 94, and employ the normalized projection in Equation (12) to rank the alternatives A₁, A₂, A₃, A₄, then observe the changes of their rankings. The curves of rankings of A₁, A₂, A₃ and A₄ are shown in Fig. 1.

Fig.1

Rankings of four alternatives A₁, A₂, A₃, A₄ based on NProj_Y
₊ (Y_i) in Equation (12).

Similarly, it is noted the changes of rankings of four alternatives based on Equation (13). For the convenience of comparison, the Equation (13) is transformed as follows: ${NP}_{-} (Y_{i}) = 1 - {NProj}_{Y_{-}} (Y_{i}), i \in M .$ (21)

It is obvious that the larger the value ${NP}_{i}^{-}$ , the better the alternative A_i is. The curves of rankings of A₁, A₂, A₃ and A₄ based on Equation (21) are shown in Fig. 2.

Fig.2

Rankings of four alternatives A₁, A₂, A₃, A₄ based on NP_- (Y_i) in Equation (21).

The rankings of four alternatives A₁, A₂, A₃, A₄ based on relative closeness in Equation (14) are shown in Fig. 3.

Fig.3

Rankings of four alternatives A₁, A₂, A₃, A₄ based on NPRC_i in Equation (14).

It is noted that (1) the changes of rankings of four alternatives are pretty much the same way in Figs. 1–3; (2) the changes of rankings are stable; and (3) the A₁ is monotone increase as α increase from 0 to 94 except Fig. 2 in an interval. So the proposed normalized projection measures in this paper is a robust measure.

Next the changes of rankings based on Equations (15)-(17) are shown. For the convenience of comparison, the Equation (16) is transformed asfollows: ${NP}_{i}^{-} = 1 - {NP}_{Y_{-}} (Y_{i}), i \in M .$ (22)

The changes of alternatives’ rankings based on Equations (15), (22) and (17) are shown in Figs. 4–6 respectively.

Fig.4

Rankings of four alternatives A₁, A₂, A₃, A₄ based on NP_Y
₊ (Y_i) in Equation (15).

Fig.5

Rankings of four alternatives A₁, A₂, A₃, A₄ based on ${NP}_{i}^{-}$ in Equation (22).

Fig.6

Rankings of four alternatives A₁, A₂, A₃, A₄ based on NRC_i in Equation (17).

Figures 4–6 show that (1) the rankings of four alternatives are very different, especially in Fig. 5; (2) the changes of rankings are also very different, especially in Fig. 5. So the proposed normalized projection measures in this paper is superior to another normalized projection measures in Equations (15)–(17).

The following changes of rankings are based on Equations (18)–(20). For the convenience of comparison, the Equation (19) is transformed asfollows: $P_{i}^{-} = 1.5 - {Proj}_{Y_{-}} (Y_{i}), i \in M .$ (23)

Similarly, the changes of alternatives’ rankings based on Equations (18), (23) and (20) are shown in Figs. 7–9 respectively.

Fig.7

Rankings of four alternatives A₁, A₂, A₃, A₄ based on Proj_Y
₊ (Y_i) in Equation (18).

Fig.8

Rankings of four alternatives A₁, A₂, A₃, A₄ based on $P_{i}^{-}$ in Equation (23).

Fig.9

Rankings of four alternatives A₁, A₂, A₃, A₄ based on PRC_i in Equation (20).

It is noted that (1) the rankings of four alternatives are very different in Figs. 7–9; (2) the changes of rankings are also very different. Especially, the A₃ is ranked first in Figs. 7 and 9; the A₄ is ranked first in Fig. 8. They are inconsistent; (3) the A₁ is not monotone increase as α increase from 0 to 94 except Fig. 7.

So the proposed normalized projection measures in this paper is superior to general projection measure based on above comparisons.

6 Conclusion

This paper has developed two normalized projection measures in real and interval vector settings. The developed method has been successfully applied to GDM problems. The improved projection measures in this paper are more perfect than the classical projection measures and another normalized projection measures because they have made up for their defects. And the improved projection methods are straightforward, and can be implemented easily on a computer.

Without a doubt, this model has its limitations. First, this normalized projection measures are bidirectional, i.e., NProj_β (α) = NProj_α (β) for all α and β. This is a limitation, although its experimental results are satisfactory. Second, some properties of normalized projection measures and particular cases of this method are not enough, although it is limited by the available space. Third, the most significant contribution of this study can be explained primarily in both theoretical and practical perspectives: i.e., managerial, marketing, and educational implications. Just an illustrative example is not enough.

As future work this research plans to apply the normalized projection method to other fields [21], such as the supplier selection [17], the strategic alliance partner selection [14], the robot selection [7], and green supplier development program selection [1].

Acknowledgments

The author would like to thank the editors and the anonymous reviewers for their insightful and constructive comments and suggestions that have led to this improved version of the paper. This work was partially supported by the Education and Teaching Reform Program of Guangdong Ocean University (XJG201644).

References

Awasthi

and Kannan

, Green supplier development program selection using NGT and VIKOR under fuzzy environment, Computers & Industrial Engineering 91 (2016), 100–108.

Cabrerizo

F.J.

, Herrera-Viedma

and Pedrycz

, A method based on PSO and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts, European Journal of Operational Research 230(3) (2013), 624–633.

Chen

T.-Y.

, An interval-valued intuitionistic fuzzy LINMAP method with inclusion comparison possibilities and hybrid averaging operations for multiple criteria group decision making, Knowledge-Based Systems 45 (2013), 134–146.

Chen

T.-Y.

, The inclusion-based TOPSIS method with interval-valued intuitionistic fuzzy sets for multiple criteria group decision making, Applied Soft Computing 26 (2015), 57–73.

Dan

, Galster

, Lytra

, Avgeriou

, Zdun

, Fouche

M.A.

, Boer

R.D.

and Solms

, Empirical evaluation of a process to increase consensus in group architectural decision making, Information & Software Technology 72(C) (2016), 31–47.

, Gao

, Liu

, Han

and Chen

, GRAP: Grey risk assessment based on projection in ad hoc networks, Journal of Parallel and Distributed Computing 71(9) (2011), 1249–1260.

Ghorabaee

M.K.

, Developing an MCDM method for robot selection with interval type-2 fuzzy sets, Robotics and Computer-Integrated Manufacturing 37 (2015), 221–232.

Gupta

and Mohanty

B.K.

, An algorithmic approach to group decision making problems under fuzzy and dynamic environment, Expert Systems with Applications 55 (2016), 118–132.

Gupta

, Mehlawat

M.K.

and Grover

, Intuitionistic fuzzy multi-attribute group decision-making with an application to plant location selection based on a new extended VIKOR method, Information Sciences (2016), 370–371 184–203.

10.

and He

, Extensions of atanassov’s intuitionistic fuzzy interaction bonferroni means and their application to multiple attribute decision making, IEEE Transactions on Fuzzy Systems 24(3) (2016), 558–573.

11.

, He

, Deng

and Zhou

, IFPBMs and their application to multiple attribute group decision making, Journal of the Operational Research Society 67(1) (2015), 127–147.

12.

, He

, Shi

and Meng

, Multiple attribute group decision making based on IVHFPBMs and a new ranking method for interval-valued hesitant fuzzy information, Computers & Industrial Engineering 99(1) (2016), 63–77.

13.

, He

, Wang

and Chen

, Hesitant fuzzy power bonferroni means and their application to multiple attribute decision making, IEEE Transactions on Fuzzy Systems 23 (2015), 1655–1668.

14.

Igoulalene

, Benyoucef

and Tiwari

M.K.

, Novel fuzzy hybrid multi-criteria group decision making approaches for the strategic supplier selection problem, Expert Systems with Applications 42(7) (2015), 3342–3356.

15.

, Zhang

H.Y.

and Wang

J.Q.

, A projection-based TODIM method under multi-valued neutrosophic environments and its application in personnel selection, Neural Computing & Applications (2016), in press, DOI: 10.1007/s00521-016-2436-z

16.

and Wang

, Projection method for multiple criteria group decision making with incomplete weight information in linguistic setting, Applied Mathematical Modelling 37(20) (2013), 9031–9040.

17.

Kar

A.K.

, A hybrid group decision support system for supplier selection using analytic hierarchy process, fuzzy set theory and neural network, Journal of Computational Science 6 (2015), 23–33.

18.

Kumar

F.P.

and Claudio

, Implications of estimating confidence intervals on group fuzzy decision making scores, Expert Systems with Applications 65 (2016), 152–163.

19.

C.C.

, Dong

, Herrera

, Herrera-Viedma

and Martínez

, Personalized individual semantics in computing with words for supporting linguistic group decision making. An application on consensus reaching, Information Fusion 33 (2017), 29–40.

20.

Maio

C.D.

, Fenza

, Loia

, Orciuoli

and Herrera-Viedma

, A framework for context-aware heterogeneous group decision making in business processes, Knowledge-Based Systems 102 (2016), 39–50.

21.

Mardani

, Jusoh

and Zavadskas

E.K.

, Fuzzy multiple criteria decision-making techniques and applications–Two decades review from 1994 to 2014, Expert Systems with Applications 42(8) (2015), 4126–4148.

22.

Massanet

, Riera

J.V.

, Torrens

and Herrera-Viedma

, A model based on subjective linguistic preference relations for group decision making problems, Information Sciences (2016), 355–356: 249–264.

23.

Merigó

J.M.

, Peris-Ortíz

, Navarro-García

and Rueda-Armengot

, Aggregation operators in economic growth analysis and entrepreneurial group decision-making, Applied Soft Computing 47 (2016), 141–150.

24.

Naamani-Dery

, Kalech

, Rokach

and Shapira

, Reducing preference elicitation in group decision making, Expert Systems with Applications 61 (2016), 246–261.

25.

Peng

J.J.

, Wang

J.Q.

and Wu

X.H.

, Novel multi-criteria decision-making approaches based on hesitant fuzzy sets and prospect theory, International Journal of Information Technology & Decision Making 15(3) (2016), 1–23.

26.

Stewart

, Thomas’ Calculus Early Transcendentals. McMaster University and University of Toronto, 2010.

27.

Tian

Z.P.

, Wang

J.Q.

and Zhang

H.Y.

, A likelihoodbased qualitative flexible approach with hesitant fuzzy linguistic information, Cognitive Computation 8(4) (2016), 670–683.

28.

Tsao

C.-Y.

and Chen

T.-Y.

, A projection-based compromising method for multiple criteria decision analysis with intervalvalued intuitionistic fuzzy information, Applied Soft Computing 45 (2016), 207–223.

29.

Ureña

, Cabrerizo

F.J.

, Morente-Molinera

J.A.

and Herrera-Viedma

, GDM-R: A new framework in R to support fuzzy group decision making processes,357:, Information Sciences (2016), 161–181.

30.

Wang

and Wang

, A multi-criteria decision-making method based on triangular intuitionistic fuzzy preference information, Intelligent Automation & Soft Computing 22(3) (2016), 473–482.

31.

Wang

J.-Q.

, Li

K.-J.

and Zhang

H.-Y.

, Interval-valued intuitionistic fuzzy multi-criteria decision-making approach based on prospect score function, Knowledge-Based Systems 27 (2012), 119–125.

32.

Wei

, Decision-making based on projection for intuitionistic fuzzy multiple attributes, Chinese Journal of Management 6(9) (2009), 1154–1156.

33.

Wei

, Project method for multiple attribute group decision making in two-tuple linguistic setting, Operations Research and Management Science 18(5) (2009), 59–63.

34.

, Yan

, Wang

and Soares

C.G.

, Selection of maritime safety control options for NUC ships using a hybrid group decision-making approach, Safety Science 88 (2016), 108–122.

35.

and Liu

, An approach to group decision making based on interval multiplicative and fuzzy preference relations by using projection, Applied Mathematical Modelling 37(6) (2013), 3929–3943.

36.

, Chen

, Rodríguez

R.M.

, Herrera

and Wang

, Deriving the priority weights from incomplete hesitant fuzzy preference relations in group decision making, Knowledge-Based Systems 99 (2016), 71–78.

37.

, On method for uncertain multiple attribute decision making problems with uncertain multiplicative preference information on alternatives, Fuzzy Optimization and Decision Making 4(2) (2005), 131–139.

38.

, Dependent uncertain ordered weighted aggregation operators, Information Fusion 9(2) (2008), 310–316.

39.

and Cai

, Projection model-based approaches to intuitionistic fuzzy multi-attribute decision making, In Intuitionistic Fuzzy Information Aggregation, 2012, pp. 249–258. Springer.

40.

and Da

, Projection method for uncertain multiattribute decision making with preference information on alternatives, International Journal of Information Technology & Decision Making 3(03) (2004), 429–434.

41.

and Hu

, Projection models for intuitionistic fuzzy multiple attribute decision making, International Journal of Information Technology & Decision Making 9(2) (2010), 267–280.

42.

Yue

, A geometric approach for ranking interval-valued intuitionistic fuzzy numbers with an application to group decisionmaking, Computers & Industrial Engineering 102 (2016), 233–245.

43.

Yue

, A model for evaluating software quality based on symbol information, Journal of Guangdong Ocean University 36(1) (2016), 85–92.

44.

Yue

, An extended TOPSIS for determining weights of decision makers with interval numbers, Knowledge-Based Systems 24(1) (2011), 146–153.

45.

Yue

, Application of the projection method to determine weights of decision makers for group decision making, Scientia Iranica 19(3) (2012), 872–878.

46.

Yue

, Approach to group decision making based on determining the weights of experts by using projection method, Applied Mathematical Modelling 36(7) (2012), 2900–2910.

47.

Yue

, An intuitionistic fuzzy projection-based approach for partner selection, Applied Mathematical Modelling 37(23) (2013), 9538–9551.

48.

Yue

, Aggregating crisp values into intuitionistic fuzzy number for group decision making, Applied Mathematical Modelling 38(11-12) (2014), 2969–2982.

49.

Yue

, TOPSIS-based group decision-making methodology in intuitionistic fuzzy setting, Information Sciences 277 (2014), 141–153.

50.

Yue

and Jia

, A direct projection-based group decision-making methodology with crisp values and interval data, Soft Computing (2015), in press. DOI: 10.1007/s00500-015-1953-5

51.

Yue

and Jia

, A group decision making model with hybrid intuitionistic fuzzy information, Computers & Industrial Engineering 87 (2015), 202–212.

52.

Zeng

, Baležentis

, Chen

and Luo

, A projection method for multiple attribute group decision making with intuitionistic fuzzy information, Informatica 24(3) (2013), 485–503.

53.

Zhang

, Wu

and Olson

, The method of grey related analysis to multiple attribute decision making problems with interval numbers, Mathematical and Computer Modelling 42(9-10) (2005), 991–998.

54.

Zhang

, Jin

and Liu

, A grey relational projection method for multi-attribute decision making based on intuitionistic trapezoidal fuzzy number, Applied Mathematical Modelling 37(5) (2013), 3467–3477.

55.

Zheng

, Jing

and Huang

, Application of improved grey relational projection method to evaluate sustainable building envelope performance, Applied Energy 87(2) (2010), 710–720.

56.

Zhou

, Wang

and Zhang

, Multi-criteria decision-making approaches based on distance measures for linguistic hesitant fuzzy sets, Journal of the operational research Society (2016), in press. doi: 10.1057/jors.2016.41