A conjunctive multiple-criteria decision-making approach for cloud service supplier selection of manufacturing enterprise

Abstract

In this article, a conjunctive multiple-criteria decision-making approach for cloud service supplier selection of manufacturing enterprise is proposed. According to the two perspectives of technology and technology management, the index framework of cloud service supplier selection is constructed from four dimensions: cloud service performance, supplier capability, supplier service level, and supplier service quality. Then, a conjunctive multiple-criteria decision-making approach is built. Neural network is used to realize the expert importance degree determination; fuzzy analytic hierarchy process considering the expert importance degree is used to determine the subjective weight; and CRITIC (CRiteria Importance Through Inter-criteria Correlation) is used to determine the objective weight. The multi-weight contribution equilibration model is built to integrate the subjective and objective weight. Finally, the improved TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) replacing Euclidean distance with connection distance is used to sequence the suppliers by their weighted index-value matrix. An automotive manufacturing enterprise application is given finally. The proposed approach can provide a scientific and efficient solution for cloud service supplier selection of manufacturing enterprise.

Keywords

Cloud service supplier selection multiple-criteria decision-making expert importance degree integrated weight connection distance

Introduction

As a new network computing service mode, cloud computing has been paid more and more attention in recent years. A large number of computing resources, storage resources, and software resources were integrated by cloud computing through the information technologies such as virtualization and grid computing. Then, a huge virtually shared resource pool was formed to provide the required information service for users.^1–3 Generally, with the integration of information resources such as infrastructure, platform, and software, just like social resources such as water, electricity, and gas, cloud computing has changed the traditional information service mode to provide users with on-demand services. Based on the advanced service concept of cloud computing, medical, financial, energy, e-government, education, scientific research, telecommunications, manufacturing, and other industries were in an information revolution.^4–6 A new development model based on industry characteristics is gradually formed by the depth integration with cloud computing technology.

Among them, manufacturing industry is a typical application field, and the service-oriented networked manufacturing mode named cloud manufacturing is being generated by the integration of cloud computing into manufacturing industry.^7–11 In cloud manufacturing mode, networked manufacturing technology, cloud calculation, networking, high-performance calculation, and other existing technology are integrated to realize the unified and centralized management of manufacturing resources, which can provide in-time, on-demand, safe, and high-quality manufacturing services for the whole life cycle. The two kinds of thought “distributed resources’ concentrated use” and “concentrated resources’ distributed service” can achieve the unified embodiment by cloud manufacturing. Moreover, demonstration, design, production, processing, experiment, simulation, management, and other business processes in life cycle are shown and provided to users in on-demand service form.^9,10 The achievement of this goal cannot be separated from the support of virtualization technology. Through Internet of Things (IOT), Cyber-Physical Systems (CPS), calculation virtualization systems, and other technologies, physical manufacturing resources are translated into logical manufacturing resources, which make the services efficient, agile, reliable, safe, and convenient to users.^8–12

To provide better services to users, the multi-granularity classifying and clustering of resources will further increase the efficiency and performance of resource virtualization, and sharing and allocation in cloud manufacturing.¹³ Cloud manufacturing is different from the existing networked manufacture. On the technology level, it generally has the characteristics of IOT, virtualization, service, coordination, intelligence, and so on. These characteristics can be detailed as service and demand-oriented, transparent and integration, outsourcing, agile, specialization, capacity sharing, green, low carbon, and so on.¹⁴ On the technology management level, it has the characteristics of uncertainty, user-centered, demand-driven, fare-paying, knowledge-based, group innovation-based, virtual organization- supported, intelligent perception, and full coverage of manufacturing resources, dynamic reconstruction of resource supply, intelligent demand, responsibility, and benefit share.^12,15,16

Based on the above analysis, manufacturing enterprises or enterprise union can effectively reduce the information cost by cloud manufacturing mode. As a result, the unified management, sharing, and on-demand use of manufacturing resources can be achieved, and the resource utilization rate can be increased. So, the market reaction capability and core competitiveness are enhanced. To deploy and implement cloud manufacturing mode for enterprise or enterprise union, one of the most important challenges is how to choose a suitable cloud service supplier. This is also a public key problem that medical, financial, energy, e-government, and cloud computing industry face.¹⁴

The rest of this article is organized as follows. In section “Related works,” the related works are reviewed and the weakness of the existing approach is analyzed. Section “Research methodology” builds the index framework of cloud service supplier selection and proposes a conjunctive multi-criteria decision-making (MCDM) approach based on integrated weight and improved TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution). In section “Case study,” the case study is reported. We discuss the advantages of the proposed approach by comparing the sorting results of three kinds of weight (objective weight, subjective weight, and integrated weight) and four sorting methods (improved TOPSIS replacing Euclidean distance with connection distance, traditional TOPSIS, angle measure evaluation TOPSIS, and vertical projection TOPSIS) in section “Discussion.” Finally, conclusions are drawn in section “Conclusion.”

Related works

To solve the cloud service supplier selection problem, Cloud Services Measurement Initiative Consortium (CSMIC) from Carnegie Mellon University’s Silicon Valley Campus designed and released the service measurement index framework in which cloud service can be evaluated from seven aspects: accountability, agility, assurance, financial, performance, security privacy, and usability.¹⁵

The cloud service supplier selection problem can be abstracted as a MCDM problem. Therefore, MCDM approach based on this framework has become an important trend of cloud service supplier evaluation.^17–20 Generally, MCDM approach can be divided into two stages: index weight determination and comprehensive sequencing by weighted index value.

Analytic hierarchy process (AHP) is a common method for subjective index weight determination. It was proposed by TL Saaty²¹ in the early 1970s and is a systematic analysis method based on the combination of qualitative analysis and quantitative analysis. Traditional AHP has the following shortcomings. First, the judgment matrix is obtained by 1–9 levels scale. As the dominant factor in the practical operation, experts’ subjective factor makes the evaluation result easy to be biased. So, it is hard to reach the consistency of judgment matrix. Second, when judgment matrix is not consistent, the adjustment is often with blindness and large calculation.

The common methods used to determinate the objective index weight include entropy weight method,²² standard deviation method,²³ and CRITIC (CRiteria Importance Through Inter-criteria Correlation).²⁴ By CRITIC, the objective weight is comprehensively measured by the contrast intensity and conflict between different indexes, which is more scientific and reasonable.

In the stage of comprehensive sequencing by weighted index value, the technique for order preference by similarity to an ideal solution (TOPSIS) is a classic sequencing method for MCDM.²⁵ Euclidean distances between the evaluation object and two ideal points are used to calculate the closeness in TOPSIS. The evaluation objects on the perpendicular bisector of two ideal points have the same closeness value and cannot be distinguished. Therefore, some improved TOPSIS methods have been proposed. For example, angle measure evaluation method²⁶ only considers the angle closeness between the evaluation object and two ideal points, and ignores the difference in length. When two evaluation objects have the same angle closeness but different length, it will draw the wrong conclusion. Vertical projection method²⁷ also cannot distinguish these evaluation objects when two or more objects have the same projective point on the connection line of two ideal points.

Furthermore, Sun et al. analyzed and summarized the existing cloud service selection methods and pointed out these methods’ shortness which is lack of cloud service automatic identification and update engine, advanced multi-criteria user preference measurement, consideration of attribute relationship, processing of qualitative and fuzzy information, long-term performance prediction, and dynamic service.²⁸

The disadvantages of the researches mentioned above can be divided into two aspects: one is that cloud service is considered more on the technology standard but less on the technology management standard in service measurement index framework, and the other is that the processing of uncertain preference is lacked. Based on the above analysis, we put forward a conjunctive MCDM approach for cloud service supplier selection of manufacturing enterprise. According to the two perspectives of technology and technology management, the index framework of cloud service supplier selection is constructed from four dimensions: cloud service performance (CSP), supplier capability (SC), supplier service level (SSL), and supplier service quality (SSQ). Then, a conjunctive MCDM approach is built. Neural network is used to realize the expert importance degree determination; fuzzy analytic hierarchy process (FAHP) considering the expert importance degree is used to determine the subjective weight; and CRITICis used to determine the objective weight. The multi-weight contribution equilibration model is built to integrate the subjective and objective weight. Finally, the improved TOPSIS replacing Euclidean distance with connection distance is used to sequence the suppliers by their weighted index-value matrix.

Research methodology

Index framework

The index framework is one of the key problems of cloud service supplier selection. Cloud service supplier selection is well abstracted by CSMIC on the technology level, but not on the technology management level.¹⁵ Different suppliers can provide similar or same cloud service, so cloud service supplier selection is different from cloud service selection. Meanwhile, cloud service supplier selection has some inherent relevance to the application industry.

Based on this, considering the cloud manufacturing’s characteristics and other industries, following the principles of systematic, comprehensive, scientific, flexible and operable, review cloud service supplier from two perspectives of technology and technology management, the evaluation criteria of cloud service supplier selection can be defined as four dimensions: CSP, SC, SSL, and SSQ. Among them, CSP embodies the technology perspective which includes the core content of CSMIC’s service measurement index framework, and SC, SSL, and SSQ embody the technology management perspective. Each dimension is further detailed to indexes. The methods of affinity diagram, systematic diagram, Delphi method, and so on, are used to make it hierarchical. After proper simplification, the index framework of cloud service supplier selection is built in Figure 1.

Figure 1.

Index framework of cloud service supplier selection.

Based on the index framework built in section “Related works,” cloud service supplier selection can be abstracted as a MCDM problem with the consideration of the index uncertainty. The proposed conjunctive MCDM approach for cloud service supplier selection is shown in Figure 2.

Figure 2.

Proposed conjunctive MCDM approach for cloud service supplier selection.

Expert importance degree

The experts have different experiences, abilities, and employment positions; so, their distinguishing of importance degree should be taken into consideration.^28,29 In this article, we propose a back propagation (BP) neural network–based method to solve the expert importance degree determination problem in group decision, which has the ability of self-learning and makes the importance degree assignment more scientific and reasonable.³⁰

In the expert importance degree assignment model based on BP neural network, several successful decisions including sorting judgments and actual sorting results are necessary as the sample data. The prior decision results with exact number subjective weight assignment form is hard to obtain, while indicator-sorting form is easy. So, for a successful decision, every sorting judgment is used as one of the inputs and actual sorting result as the output of BP neural network.

There is a group of N′ (N′ is a natural number) samples. In every sample, s (s > 1) experts sort the objects O₁, O₂, …, O_r. In the sample N (N ≤ N′), the sorting judgment for O₁, O₂, …, O_r by judge i (i = 1, 2, …, s) can be expressed as a linear order ${LI}_{i}^{(N)} : O_{i 1}^{(N)}, O_{i 2}^{(N)}, \dots, O_{ir}^{(N)}$ . The symbol ${LI}_{i}^{(N)} (O_{j}^{(N)})$ is used to represent the number of objects behind O_j in LI_i (including itself), which is also named as the score of O_j in LI_i. The expert importance degree vector is $ρ = [ρ_{1}, ρ_{2}, \dots, ρ_{s}]$ $(\sum_{i = 1}^{s} ρ_{i} = 1, 0 \leq ρ_{i} \leq 1)$ . $L I^{(N)} (O_{j}) = \sum_{i = 1}^{s} ρ_{i} \cdot {LI}_{i}^{(N)} (O_{j})$ is called the weighted Borda number of O_j. Finally, we sort O₁, O₂, …, O_r by their weighted Borda numbers and get the sorting result of s experts which is also a linear order $L I^{(N)} : O_{1}^{(N)}, O_{2}^{(N)}, \dots, O_{r}^{(N)}$ . $L I^{*} : O_{1}^{*}, O_{2}^{*}, \dots, O_{r}^{*}$ is the real linear order of the objects O₁, O₂, …, O_r. So, we can build the BP neural network model as shown in Figure 3.

Figure 3.

Structure of BP neural network for determining the expert importance degree.

Structure

Input layer

The neuron number of input layer is s×r. The input matrix is $I N^{(N)} = ({in}_{ij}^{(N)})_{s \times r}$ , where ${in}_{ij}^{(N)}$ is equal to ${LI}_{i}^{(N)} (O_{j})$ .

Hidden layer

The neuron number of hidden layer is r (equal to the object number). For neuron i, its input is $y_{i}^{(N)}$ and the output is $d_{i}^{(N)}$ (i = 1, 2, …, r). The connection weight matrix between the input layer and the hidden layer is composed of s matrix blocks, and the matrix block j (j = 1, 2, …, s) is

\underset{j}{Γ^{(N)}} = {(\underset{j}{γ_{ik}^{(N)}})}_{r \times r} = [\begin{matrix} \underset{j}{γ_{11}^{(N)}} & 0 & \dots & 0 \\ 0 & \underset{j}{γ_{22}^{(N)}} & \dots & 0 \\ ⋮ & ⋮ & ⋮ \\ 0 & 0 & \dots & \underset{j}{γ_{rr}^{(N)}} \end{matrix}]

(1)

Output layer

There is only one neuron in the output layer. The connection weight matrix between the hidden layer and the output layer is (1, 1, …, 1)^T.

Reasoning process

For the hidden layer node j (j = 1, 2, …, r), its input is $γ_{i}^{(N)} = \sum_{i = 1}^{s} ρ_{i}^{(N)} {in}_{ij}^{(N)} = L I^{(N)} (O_{j})$ and the output is $d_{j}^{(N)} = f (y_{j}^{(N)} + θ_{j})$ (the effect function is $f (y) = 1 / (1 + e^{- y})$ ).

For the output layer node, its input is $d^{(N)} = (d_{1}^{(N)}, d_{2}^{(N)}, \dots, d_{r}^{(N)})$ and the output is $b^{(N)} = (b_{1}^{(N)}, b_{2}^{(N)}, \dots, b_{r}^{(N)})$ in which $b_{i}^{(N)}$ is the score of O_i in the linear order $L I^{(N)} : O_{1}^{(N)}, O_{2}^{(N)}, \dots, O_{r}^{(N)}$ sorted by $d_{1}^{(N)}, d_{2}^{(N)}, \dots, d_{r}^{(N)}$ . Its effect is sorted in O₁, O₂, …, O_r order and the scores are given.

Error function

For the sample N, the error is $Erro r^{(N)} = \sum_{j = 1}^{r} {(b_{j}^{(N)} - b_{j}^{*})}^{2}$ where $b_{j}^{(N)}$ is the score of O_j in $L I^{(N)} : O_{1}^{(N)}, O_{2}^{(N)}, \dots, O_{r}^{(N)}$ and $b_{j}^{*}$ is the score of O_j in $L I^{*} : O_{1}^{*}, O_{2}^{*}, \dots, O_{r}^{*}$ .

Learning process

The learning process is shown in Figure 4.

Figure 4.

Learning process of BP neural network for determining the expert importance degree.

Calculation of expert importance degree

After the BP neural network has been trained successfully, we need to normalize the connection weights between the input layer and the hidden layer to obtain the expert importance degree as follows:

Step 1. For the connection weight matrix between the input layer and the hidden layer, calculate the diagonal elements’ average value of its every matrix block. To the matrix block j (j = 1, 2, …, s), that is, $\underset{j}{γ^{(N')}} = (1 / r) \sum_{i = 1}^{r} \underset{j}{γ_{ii}^{(N')}}$ .

Step 2. Normalize the average values $\underset{1}{γ^{(N')}}, \underset{2}{γ^{(N')}}, \dots, \underset{s}{γ^{(N')}}$ and the importance degree of expert j is $ρ_{j} = \underset{j}{γ^{(N')}} / \underset{j}{γ^{(N')}} \sum_{k = 1}^{s} \underset{k}{γ^{(N')}} \sum_{k = 1}^{s} \underset{k}{γ^{(N')}}$ . So, we can obtain the expert importance degree vector as follows

ρ = (ρ_{1}, ρ_{2}, \dots, ρ_{s})^{T}

(2)

Subjective weight

To improve AHP method, FAHP has been proposed.^31–35 In FAHP, 1–3 levels scale is used to construct the judgment matrix; so, experts can easily make the decision that which one has a higher priority between two indexes. Experts’ fuzzy priority judgment matrixes can be directly transformed to fuzzy consistent matrixes, so there is no need for consistency check or adjustment.

The specific steps of FAHP considering the expert importance degree are as follows:

Step 1. Collect s (s > 1) experts’ judgment opinions of the criteria C₁, C₂, …, C_m dominated by the overall evaluation target T in Figure 1. The judgment opinion made by the expert k (k = 1, 2, …, s) can be expressed as a fuzzy priority judgment matrix $F^{k} = (f_{ij}^{k})_{m \times m}$ where

f_{ij}^{k} = {\begin{matrix} 0, j is more important than i \\ 0.5, i is as important as j \\ 1, i is more important than j \end{matrix}

(3)

Step 2. Transform the fuzzy priority judgment matrixes F¹, F², …, F^s to the fuzzy consistent matrixes Q¹, Q², …, Q^s, and $Q^{k} = (q_{ij}^{k})_{m \times m}, q_{ij}^{k} = (q_{i}^{k} - q_{j}^{k}) / 2 m + 0.5, k = 1, 2, \dots, s$ where $q_{i}^{k} = \sum_{j = 1}^{m} f_{ij}^{k}$ .

Step 3. Calculate $λ_{1}, λ_{2}, \dots, λ_{m}$ of Q^k by $| Q^{k} - λ E | = 0$ , k = 1, 2, …, s. The eigenvalue of Q^k is $λ_{\max} = {λ_{1}, λ_{2}, \dots, λ_{m}}$ . The eigenvector $α = (α_{1}, α_{2}, \dots, α_{m})^{T}$ can be calculated by $Q^{k} \cdot α = λ_{\max} \cdot E$ . After normalizing the eigenvector $α$ , we can get the relative weight vector of C₁, C₂, …, C_m dominated by T is $θ^{k} = (θ_{1}^{k}, θ_{2}^{k}, \dots, θ_{m}^{k})^{T}$ where $θ_{i}^{k} = (α_{i}) / (\sum_{i = 1}^{m} α_{i})$ according to the judgment of expert k.

Step 4. Synthesize s experts’ judgment results $θ^{1}, θ^{2}, \dots, θ^{s}$ and obtain the relative weight vector of C₁, C₂, …, C_m is $θ = (θ_{1}, θ_{2}, \dots, θ_{m})^{T}$ where $θ_{i} = \sum_{k = 1}^{s} ρ_{k} θ_{i}^{k}$ .

Step 5. Repeat steps 1–4. The relative weight vectors of the indexes I₁, I₂, …, I₇ dominated by criterion C₁, the indexes I₈, I₉, I₁₀, I₁₁ dominated by criterion C₂, the indexes I₁₂, I₁₃, I₁₄ dominated by criterion C₃, and the indexes I₁₅, I₁₆, I₁₇, I₁₈ dominated by criterion C₄ are $(γ_{1}, γ_{2}, \dots, γ_{7}), (γ_{8}, γ_{9}, γ_{10}, γ_{11}), (γ_{12}, γ_{13}, γ_{14}), and (γ_{15}, γ_{16}, γ_{17}, γ_{18})$ , respectively. So, the subjective weight vector of the indexes I₁, I₂, …, I₁₈ to overall evaluation target T is

\begin{array}{l} φ = (θ_{1} γ_{1}, θ_{1} γ_{2}, . . ., θ_{1} γ_{7}, θ_{2} γ_{8}, θ_{2} γ_{9}, θ_{2} γ_{10}, θ_{2} γ_{11}, \\ θ_{3} γ_{12}, θ_{3} γ_{13}, θ_{3} γ_{14}, θ_{4} γ_{15}, θ_{4} γ_{16}, θ_{4} γ_{17}, θ_{4} γ_{18}) \end{array}

(4)

Objective weight

Under the index I_p (p = 1, 2, …, l, l = 18), the judgment opinion of evaluation objects S₁, S₂, …, S_n made by the expert k (k = 1, 2, …, s) can be expressed as a fuzzy priority judgment matrix $G^{p, k} = (g_{ij}^{p, k})_{n \times n}$ which is similar to F^k (step 1 in section “Subjective weight”). Then, we do the same steps 2–4 in section “Subjective weight,” and the priority value of the suppliers S₁, S₂, …, S_n under the index I_p can be obtained as z_p = [z_1p, z_2p, …, z_np]^T. After repeating the above steps to get the priority value of the evaluation objects under other indexes in Figure 1, we obtain the priority value decision matrix Z = (z_ij)_n×l, and z_p is the column p of Z. The steps of CRITIC are as follows.

The contrast intensity, which shows the value difference degree of all objects under the same index, is expressed in the form of standard deviation. The bigger the standard deviation of an index, the more the information this index has.

The conflict between, which is based on the correlation relationship between different indexes, is expressed in the form of correlation coefficient. The bigger the negative correlation coefficient between two different indexes, the greater the conflict they have. This shows that the information reflected by the two indexes in the evaluation results has a greater difference, and the weights of two indexes should be bigger.

The correlation coefficient between I_i and I_j is

cc r_{ij} = \frac{\sum_{k = 1}^{l} (z_{ki} - \bar{z_{, i}}) (z_{kj} - \bar{z_{, j}})}{\sqrt{\sum_{k = 1}^{l} {(z_{ki} - \bar{z_{, i}})}^{2}} \sqrt{\sum_{k = 1}^{l} {(z_{kj} - \bar{z_{, j}})}^{2}}}

(5)

where $\bar{z_{, i}} and \bar{z_{, j}}$ are the arithmetic average values of I_i and I_j, respectively.

The conflict degree between I_j and other indexes is

R_{j} = \sum_{i = 1, i \neq j}^{l} (1 - cc r_{ij})

(6)

The information volume of I_j can be expressed as $C_{j} = σ_{j} R_{j}$ where $σ_{j}$ is the standard deviation of the index I_j.

Finally, we get the objective weight vector of the indexes I₁, I₂, …, I_l (l = 18) as follows

χ = (χ_{1}, χ_{2}, \dots, χ_{l})

(7)

where the objective weight of the index I_j is $χ_{j} = C_{j} / \sum_{i = 1}^{l} C_{i}$ .

Integrated weight

The sequencing result of evaluation objects depends on their weighted index values.³⁶ Because the subjective weight is equal to the objective weight in comprehensive sequencing process, their contributions to the evaluating result tend to an equilibration. So, we assume that the integrated weight vector is $ω = λ_{1} \cdot φ + λ_{2} \cdot χ$ , and $λ_{1}, λ_{2} (λ_{1}, λ_{2} \geq 0, λ_{1} + λ_{2} = 1)$ are the equilibration coefficients of the subjective and objective weights, respectively.

For the evaluation object i, the subjective weight’s contribution on index I_j is $λ_{1} φ_{j} z_{ij}$ and the objective weight’s contribution on index I_j is $λ_{2} χ_{j} z_{ij}$ (i = 1, 2, …, n, j = 1, 2, …, l), and the multi-weight contribution deviation is $\sum_{j = 1}^{l} {(λ_{1} φ_{j} z_{ij} - λ_{2} χ_{j} z_{ij})}^{2}$ .

All evaluation objects are equal, so the total multi-weight contribution deviation is $\sum_{i = 1}^{n} \sum_{j = 1}^{l} {(λ_{1} φ_{j} z_{ij} - λ_{2} χ_{j} z_{ij})}^{2}$ .

Therefore, the multi-weight contribution equilibration model is built as

\min \sum_{i = 1}^{n} \sum_{j = 1}^{l} {(λ_{1} φ_{j} z_{i j} - λ_{2} χ_{j} z_{i j})}^{2}, s . t . λ_{1}, λ_{2} \geq 0, λ_{1} + λ_{2} = 1

(8)

We can express $λ_{2}$ by $1 - λ_{1}$ , and then set the first-order derivative of the target function in formula (20) to zero, so we get

λ_{1} = \frac{\sum_{i = 1}^{n} \sum_{j = 1}^{l} z_{ij}^{2} χ_{j} (φ_{j} + χ_{j})}{\sum_{i = 1}^{n} \sum_{j = 1}^{l} z_{ij}^{2} {(φ_{j} + χ_{j})}^{2}}, λ_{2} = \frac{\sum_{i = 1}^{n} \sum_{j = 1}^{l} z_{ij}^{2} φ_{j} (φ_{j} + χ_{j})}{\sum_{i = 1}^{n} \sum_{j = 1}^{l} z_{ij}^{2} {(φ_{j} + χ_{j})}^{2}}

(9)

It is known that the minimum of the target function in formula (8) is obtained at the value of $λ_{1}, λ_{2}$ in formula (9) according to the target function’s property. So, we can get the integrated weight vector as

ω = λ_{1} \cdot φ + λ_{2} \cdot χ

(10)

Supplier sequencing

The theory of set pair analysis (SPA), proposed by K Zhao and Xuan³⁷ in 1996, is a systematic analysis method to solve the uncertainty problem with connection degree. A set pair is constructed from two related sets in the uncertainty system; then the sameness, contrariety, and difference analysis will be done on the uncertainty of the set pair; and finally, the connection degree of the set pair can be obtained. SPA method can be used to describe the relationship in certainty–uncertainty system.

According to the priority value decision matrix Z = (z_ij)_n×l and the equilibration weight vector $ω$ , we can get the weighted index-value matrix $H = (h_{ij})_{n \times l} = (ω_{j} z_{ij})_{n \times l}$ .

The positive ideal point and the negative ideal point are as follows

H^{+} = [h_{1}^{+}, h_{2}^{+}, \dots, h_{l}^{+}], H^{-} = [h_{1}^{-}, h_{2}^{-}, \dots, h_{l}^{-}], h_{i}^{+} = \max {h_{1 i}, h_{2 i}, \dots, h_{n i}}, h_{i}^{-} = \min {h_{1 i}, h_{2 i}, \dots, h_{n i}}

(11)

The weighted index value of the supplier k (k = 1, 2, …, n) can be expressed as H^k, that is, the row k of H. Among the element pairs $(h_{k 1}, h_{1}^{+}), (h_{k 2}, h_{2}^{+}), \dots, (h_{kl}, h_{l}^{+})$ comprising the corresponding elements of H^k and H⁺, the number of element pairs with the sameness relationship is $N_{k, +}^{1}$ , the number of element pairs with the contrariety relationship is $N_{k, +}^{2}$ , and the number of element pairs with the difference relationship is $N_{k, +}^{3}$ and $N_{k, +}^{1} + N_{k, +}^{2} + N_{k, +}^{3} = l$ . So, the connection degree between H^k and H⁺ can be expressed as $D (H^{k}, H^{+}) = ((N_{k, +}^{1}) / l) Δ + ((N_{k, +}^{2}) / l) Ω + ((N_{k, +}^{3}) / l) Ψ$ where $Δ, Ω, and Ψ$ are respectively, the symbols of the sameness, contrariety, and difference relationships in turn.

We assume that $D (H^{k}, H^{+}) = Θ_{1} \cdot D (h_{k 1}, h_{1}^{+}) + Θ_{2} \cdot D (h_{k 2}, h_{2}^{+}) + \dots + Θ_{l} \cdot D (h_{kl}, h_{l}^{+})$ where $D (h_{ki}, h_{1}^{+}) = A_{ki}^{+} \cdot Δ + B_{ki}^{+} \cdot Ω + C_{ki}^{+} \cdot Ψ, i = 1, 2, \dots, m$ . If $h_{ki} = h_{i}^{-}$ , $A_{ki}^{+} = C_{ki}^{+} = 0$ and $B_{ki}^{+} = 1$ ; if $h_{k, i} \in (h_{i}^{-}, h_{i}^{+})$ , $A_{ki}^{+} = h_{ki} / h_{i}^{+}$ , $C_{ki}^{+} = 1 - (h_{ki} / h_{i}^{+})$ , and $B_{ki}^{+} = 0$ ; and if $h_{ki} = h_{i}^{+}, A_{ki}^{+} = 1$ and $B_{ki}^{+} = C_{ki}^{+} = 0$ . So, we can get the connection degree between H^k and H⁺ as follows

D (H^{k}, H^{+}) = A_{k}^{+} Δ + B_{k}^{+} Ω + C_{k}^{+} Ψ

(12)

where $A_{k}^{+} = (\sum_{i = 1}^{l} A_{ki}^{+}) / l, B_{k}^{+} = (\sum_{i = 1}^{l} B_{ki}^{+}) / l, C_{k}^{+} = (\sum_{i = 1}^{l} C_{ki}^{+}) / l$ .

According to SPA theory, the connection vector between H^k and H⁺ can be expressed as $V (H^{k}, H^{+}) = (A_{k}^{+}, B_{k}^{+}, C_{k}^{+})^{T}$ and the connection vector between H⁺ and itself is V(H⁺, H⁺) = (1, 0, 0)^T, so the connection distance from H^k to H⁺ is $d (H^{k}, H^{+}) = \sqrt{{(A_{k}^{+} - 1)}^{2} + {(B_{k}^{+})}^{2} + {(C_{k}^{+})}^{2}}$ .

Similarly, we can get the connection distance from H^k to H⁻ as $d (H^{k}, H^{-}) = \sqrt{{(A_{k}^{-} - 1)}^{2} + {(B_{k}^{-})}^{2} + {(C_{k}^{-})}^{2}}$ .

Finally, we can get the relative closeness degree from H^k to the ideal points $H^{+}, H^{-}$ as follows

r c_{k} = \frac{d (H^{k}, H^{-})}{d (H^{k}, H^{+}) + d (H^{k}, H^{-})}

(13)

The relative closeness degree rc_k has the following characteristics:

If H^k = H⁺, rc_k = 1.

If H^k = H⁻, rc_k = 0.

When $d (H^{k}, H^{+}) \to 0$ (i.e. H^k ≠ H⁺ and H^k ≠ H⁻, H^k→H⁺), rc_k→1.

Therefore, using the connection distance from the evaluation object to the ideal points fits well with the basic sequencing principles of TOPSIS, so replacing Euclidean distance with the connection distance is reasonable. We can calculate the relative closeness from each evaluation object to the ideal points successively and obtain the final results by sequencing them in the descending order.

Case study

With the rapid development of the automotive electronics technology and the continuous improvement of the customers’ need for entertainment and safety of automotive, the traditional vertical manufacturing mode is gradually hard to guarantee the whole process of automobile’s design, manufacture, and integration under the certain cost and periodic constraints. Cloud manufacturing mode combines the IT capacity with the traditional automotive manufacturing industry on-demand and low-priced and provides an effective solution to this problem and realizes the rapid integrating and updating of the automobile manufacturing enterprise. Cloud manufacturing platform has greatly facilitated the deep collaboration among different enterprises in the automotive industry chain. More importantly, the automotive manufacturing enterprises can gradually shift from the production-oriented pattern to service-oriented pattern through the cloud manufacturing platform, and the enterprise’s independent innovation and core competitiveness can be continuously improved.

To meet the development requirement, the management team of an automotive manufacturing enterprise has decided to introduce cloud manufacturing mode after careful analysis and discussion. After automobile industry cloud services market researching and electronic integration and update demand-oriented preliminary screening for cloud service suppliers, enterprise management team intends to choose the best one from IBM (S₁), Microsoft (S₂), Huawei (S₃), TEAMSUN (S₄), Intple (S₅), Inspur (S₆), and youyou (S₇). According to the cloud service supplier selection index framework and method, the enterprise decision-making team decides to evaluate the cloud service suppliers with the comprehensive consideration of the opinions of administrative department, finance department, IT department, manufacturing department, and relevant experts and managers.

There are three experts to sort four objects. There are six samples shown in Table 1. In sample 1, four objects are O₁, O₂, O₃, O₄ and their real order is O₂, O₃, O₄, O₁. In sample 2, four objects are O₅, O₆, O₇, O₈ and their real order is O₅, O₆, O₈, O₇. In sample 3, four objects are O₉, O₁₀, O₁₁, O₁₂ and their real order is O₁₀, O₁₂, O₁₁, O₉. In sample 4, four objects are O₁₃, O₁₄, O₁₅, O₁₆ and their real order is O₁₃, O₁₆, O₁₅, O₁₄. In sample 5, four objects are O₁₇, O₁₈, O₁₉, O₂₀ and their real order is O₂₀, O₁₇, O₁₈, O₁₉. In sample 6, four objects are O₂₁, O₂₂, O₂₃, O₂₄ and their real order is O₂₃, O₂₂, O₂₄, O₂₁.

Table 1.

Six samples of BP neural network model.

Sample no.	Input												Expected output
	Judge 1				Judge 2				Judge 3				Expected output
	in ₁₁	in ₁₂	in ₁₃	in ₁₄	in ₂₁	in ₂₂	in ₂₃	in ₂₄	in ₃₁	in ₃₂	in ₃₃	in ₃₄	$b_{1}^{}$*	$b_{2}^{}$*	$b_{3}^{}$*	$b_{4}^{}$*
1	1	4	2	3	4	1	3	2	3	2	1	4	1	4	3	2
2	4	3	1	2	3	1	4	2	4	2	1	3	4	3	1	2
3	1	4	2	3	2	3	1	4	4	1	3	2	1	4	2	3
4	4	2	1	3	2	4	3	1	2	4	1	3	4	1	2	3
5	2	3	1	4	2	3	1	4	3	4	1	2	3	2	1	4
6	1	3	2	4	1	2	3	4	4	3	2	1	1	3	4	2

Limited to the text space, the detailed training process of BP neural network is omitted. Finally, the judge importance degree vector is calculated as follows

ρ = {(0.4285 0.3346 0.2369)}^{T}

(14)

From up to down, we first use FAHP considering the expert importance degree to determine the relative weight vector of C₁ (CSP), C₂ (SC), C₃ (SSL), and C₄ (SSQ) dominated by T in Figure 1. On the issue of influence degree for the total target T, the expert E₁ thought that C₁ > C₃ > C₂ > C₄. So, the fuzzy priority judgment matrix of the expert E₁ is given by

F^{1} = [\begin{matrix} 0.5 & 0 & 0 & 0 \\ 1 & 0.5 & 0 & 1 \\ 1 & 1 & 0.5 & 1 \\ 1 & 0 & 0 & 0.5 \end{matrix}]

(15)

The matrix F¹ can be transformed to the fuzzy consistent matrix as follows

Q^{1} = [\begin{matrix} 0.5 & 0.25 & 0.125 & 0.375 \\ 0.75 & 0.5 & 0.375 & 0.625 \\ 0.875 & 0.625 & 0.5 & 0.75 \\ 0.625 & 0.375 & 0.25 & 0.5 \end{matrix}]

(16)

The eigenvalue of Q¹ is $λ_{\max} = {1.8292, 0.1708, 0, 0} = 1.8292$ , and the eigenvector $α = (- 0.2821, - 0.5435, - 0.6742, - 0 {. 4128)}^{T}$ can be calculated by $Q^{1} \cdot α = λ_{\max} \cdot E$ . After normalizing the eigenvector $α$ , the relative weight vector of C₁, C₂, C₃, C₄ dominated by T is $θ^{1} = (0.1475, 0.2842, 0.3525, 0 {. 2158)}^{T}$ according to the judgment of the expert E₁.

Similarly, we can get

\begin{array}{l} θ^{2} = {(\begin{array}{l} 0.1078, & 0.2534, & 0.4108, & 0.2280 \end{array})}^{T}, \\ θ^{2} = {(\begin{array}{l} 0.2006, & 0.2986, & 0.4037, & 0.0971 \end{array})}^{T} \end{array}

(17)

After synthesizing three experts’ judgment results $θ^{1}, θ^{2}, θ^{3}$ , we can get the relative weight vector of C₁, C₂, C₃, C₄ as $θ = (0.1468, 0.2773, 0.3841, 0 {. 1918)}^{T}$ .

We repeat the above steps; the relative weight vectors of the indexes I₁, I₂, …, I₇ dominated by criterion C₁, the indexes I₈, I₉, I₁₀, I₁₁ dominated by criterion C₂, the indexes I₁₂, I₁₃, I₁₄ dominated by criterion C₃, and the indexes I₁₅, I₁₆, I₁₇, I₁₈ dominated by criterion C₄ are (0.2031, 0.0987, 0.1765, 0.3078, 0.0098, 0.1002, 0.1039), (0.3786, 0.2087, 0.4002, 0.0125), (0.2876, 0.5876, 0.1248), and (0.1245, 0.5002, 0.1018, 0.2735), respectively. So, the subjective weight vector of the indexes I₁, I₂, …, I₁₈ to overall evaluation target T is

φ = {(\begin{matrix} 0.0298 0.0145 0.0259 0.0452 0.0014 0.0147 0.0153 0.1050 0.0579 \\ 0.1110 0.0035 0.1105 0.2257 0.0479 0.0239 0.0959 0.0195 0.0525 \end{matrix})}^{T}

(18)

Service resource’s virtualization management and the judgment opinion of seven suppliers: IBM (S₁), Microsoft (S₂), Huawei (S₃), TEAMSUN (S₄), Intple (S₅), Inspur (S₆), and youyou (S₇) made by experts E₁, E₂, and E₃ under index I₁ are shown in Table 2.

Table 2.

Judgment opinion of seven suppliers: IBM (S₁), Microsoft (S₂), Huawei (S₃), TEAMSUN (S₄), Intple (S₅), Inspur (S₆), and youyou (S₇) made by experts E₁, E₂, and E₃ under index I₁.

The judgment opinion made by experts E₁, E₂, and E₃ can, respectively, be expressed as fuzzy priority judgment matrixes $G^{1, 1} = (g_{ij}^{1, 1})_{7 \times 7}, G^{1, 2} = (g_{ij}^{1, 2})_{7 \times 7}, and G^{1, 3} = (g_{ij}^{1, 3})_{7 \times 7}$

G^{1, 1} = (g_{ij}^{1, 1})_{7 \times 7} = [\begin{matrix} 0.5 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0.5 & 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0.5 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0.5 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.5 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0.5 & 1 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0.5 \end{matrix}]

(19)

G^{1, 2} = (g_{ij}^{1, 2})_{7 \times 7} = [\begin{matrix} 0.5 & 1 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0.5 & 0 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0.5 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0.5 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.5 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0.5 & 0 \\ 0 & 1 & 0 & 1 & 1 & 1 & 0.5 \end{matrix}]

(20)

G^{1, 3} = (g_{ij}^{1, 3})_{7 \times 7} = [\begin{matrix} 0.5 & 0 & 0 & 1 & 1 & 1 & 1 \\ 1 & 0.5 & 0 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0.5 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0.5 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0.5 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0.5 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0.5 \end{matrix}]

(21)

Limited to the text space, the detailed calculation processes are omitted. We repeat the same steps 2–4 in section “Index framework,” and the priority value of the suppliers S₁, S₂, …, S₇ under index I₁ can be obtained as z₁ = [0.1920 0.1631 0.2005 0.0927 0.0809 0.1300 0.1408]^T. After repeating the above steps to get the priority value of the evaluation objects under other indexes, we obtain the index-value decision matrix Z = (z_ij)_7×18

Z = [\begin{matrix} 0.1920 & 0.2080 & 0.0401 & 0.1781 & 0.1654 & 0.0139 & 0.2675 & 0.1224 & 0.1973 & 0.0522 & 0.1823 & 0.0796 & 0.0822 & 0.2399 & 0.3574 & 0.2393 & 0.2110 & 0.2091 \\ 0.1631 & 0.1440 & 0.2467 & 0.1541 & 0.1846 & 0.1211 & 0.0097 & 0.1114 & 0.0490 & 0.1753 & 0.0462 & 0.3037 & 0.1942 & 0.1211 & 0.0497 & 0.0938 & 0.2290 & 0.1363 \\ 0.2005 & 0.0222 & 0.2433 & 0.1866 & 0.1811 & 0.0202 & 0.1235 & 0.1615 & 0.0358 & 0.0595 & 0.0497 & 0.1144 & 0.1368 & 0.1807 & 0.2177 & 0.1593 & 0.1379 & 0.2522 \\ 0.0927 & 0.0634 & 0.1234 & 0.1275 & 0.0956 & 0.0425 & 0.1074 & 0.1773 & 0.1501 & 0.1181 & 0.0858 & 0.0643 & 0.1285 & 0.0241 & 0.1796 & 0.0499 & 0.0257 & 0.0198 \\ 0.0809 & 0.1245 & 0.2034 & 0.0069 & 0.1597 & 0.3600 & 0.2155 & 0.1886 & 0.2890 & 0.1631 & 0.2801 & 0.0821 & 0.2144 & 0.0172 & 0.0046 & 0.1814 & 0.0701 & 0.1121 \\ 0.1300 & 0.2181 & 0.0361 & 0.1651 & 0.0417 & 0.3038 & 0.2238 & 0.0690 & 0.1025 & 0.2079 & 0.0847 & 0.2013 & 0.0668 & 0.1689 & 0.1290 & 0.0792 & 0.2796 & 0.0270 \\ 0.1408 & 0.2197 & 0.1072 & 0.1816 & 0.1720 & 0.1386 & 0.0526 & 0.1699 & 0.1763 & 0.2239 & 0.2713 & 0.1547 & 0.1770 & 0.2480 & 0.0621 & 0.1971 & 0.0466 & 0.2435 \end{matrix}]

(22)

Finally, we get the objective weight vector of the indexes I₁, I₂, …, I_l (l = 18) as

χ = {(\begin{matrix} 0.0478 0.0505 0.0623 0.0553 0.0455 0.0713 0.0619 0.0560 0.0576 \\ 0.0575 0.0530 0.0587 0.0530 0.0498 0.0683 0.0430 0.0609 0.0477 \end{matrix})}^{T}

(23)

The standardized decision matrix is shown in formula (25). We can obtain the equilibration coefficients $λ_{1} = 0.4221 and λ_{2} = 0.5779$ , so the integrated weight vector is

ω = {[\begin{matrix} 0.0402 0.0353 0.0469 0.0510 0.0269 0.0474 0.0422 0.0767 0.0577 \\ 0.0801 0.0321 0.0806 0.1259 0.0490 0.0496 0.0653 0.0434 0.0497 \end{matrix}]}^{T}

(24)

According to the integrated weight vector $ω$ in formula (35) and the index-value decision matrix in formula (25), the weighted index-value decision matrix is H = (h_ij)_7×18.

The positive ideal point H⁺ = [0.0081 0.0078 0.0116 0.0095 0.0050 0.0171 0.0113 0.0145 0.0167 0.0179 0.0090 0.0245 0.0270 0.0122 0.0177 0.0156 0.0121 0.0125] and the negative ideal point H⁻ = [0.0033 0.0008 0.0017 0.0004 0.0011 0.0007 0.0004 0.0053 0.0021 0.0042 0.0015 0.0052 0.0084 0.0008 0.0002 0.0033 0.0011 0.0010].

As is shown in Table 3, the evaluation results of IBM (S₁), Microsoft (S₂), Huawei (S₃), TEAMSUN (S₄), Intple (S₅), Inspur (S₆), and youyou (S₇) are shown in Table 4; so, the sequencing result is TEAMSUN (S₄) > Inspur (S₆) > Microsoft (S₂) > Huawei (S₃) > youyou (S₇) > Intple (S₅) > IBM (S₁). As a result, the automotive manufacturing enterprise selected TEAMSUN (S₄) as the best cloud service supplier.

Table 3.

Evaluation results of T₁, T₂, T₃, T₄, and T₅.

	The connection distance from H^k to H⁺	The connection distance from H^k to H⁻	The connection distance closeness of H^k
IBM (S₁)	11.9318	10.5249	0.4687
Microsoft (S₂)	11.1434	10.7915	0.4920
Huawei (S₃)	11.0883	10.4317	0.4847
TEAMSUN (S₄)	10.1314	12.0245	0.5427
Intple (S₅)	10.8935	9.8665	0.4753
Inspur (S₆)	10.2283	11.4795	0.5288
youyou (S₇)	12.7796	11.7837	0.4797

Table 4.

Sequencing results using three kinds of weight.

Discussion

Three kinds of weights

As shown in Figure 5, the assignments of the proposed integrated weight, objective weight,²⁴ and subjective weight³⁵ are compared. The objective weight, which comes from the actual index values of evaluation objects, is the characterization of the inherent relationships among index values. Instead, the subjective weight, which comes from the human judgment on the effect of each index to the overall evaluation target, is often influenced by the subjective factors of experts.

Figure 5.

Assignments of integrated weight, objective weight,²⁴ and subjective weight.³⁵

Through the comparison of three kinds of weights in Figure 5, the findings are as follows:

Objective weight shows a more even distribution characteristics and this is because the index-value fluctuation of 18 indexes among seven suppliers is not very obvious.

The distribution of subjective weight is more irregular.

Integrated weight is in between.

Using the subjective weight, objective weight, and integrated weight, the sequencing results are shown in Table 4.

As shown in Table 4, the overall trends of the sequencing results using the three kinds of weights are consistent. The supplier TEAMSUN (S₄) has the highest priority, and the suppliers Intple (S₅) and IBM (S₁) have a lower priority than others. Thus, it can be seen that the weighting information loss exists in the individual objective or subjective weight. The integrated weight considers not only the objective difference but also the experts’ subjective preference, so it can reduce the information loss caused by the individual objective or subjective weighting.

Four sequencing methods

The sequencing results using the proposed improved TOPSIS replacing Euclidean distance with connection distance, TOPSIS,²⁵ improved TOPSIS by angle measure evaluation²⁶ and improved TOPSIS by vertical projection²⁷ are compared in Table 5.

Table 5.

Sequencing results using four sequencing methods.

From the comparison shown in Table 5, the findings are as follows:

It is known that the sequencing result of the proposed improved TOPSIS replacing Euclidean distance with connection distance is as same as the result of TOPSIS,²⁵ which confirms the validity of the proposed method. The supplier TEAMSUN (S₄) has the highest priority, and the supplier IBM (S₁) has the lowest priority, so the results of the four sequencing methods show a consistent overall trend.

Improved TOPSIS by angle measure evaluation²⁶ only considers the angle closeness between the evaluation object and two ideal points, and ignores the difference in length. When two evaluation objects have the same angle closeness but different length, it will draw the wrong conclusion. The sequencing result of Microsoft (S₂), youyou (S₇) by angle measure evaluation method is youyou (S₇) >Microsoft (S₂), which is obviously contrary to the sequencing result by other three methods. It can prove its disadvantage.

When two or more objects have the same projective point on the connection line of two ideal points, improved TOPSIS by vertical projection²⁷ cannot distinguish these objects to sequence them. The closeness of Huawei (S₃) and youyou (S₇) obtained by vertical projection method are equal, so they cannot be sequenced.

Therefore, improved TOPSIS by angle measure evaluation²⁶ and improved TOPSIS by vertical projection²⁷ cannot meet the sequencing requirements in some special cases. From the case of this article, we can see that the improved TOPSIS by connection distance can solve the shortages of the improved TOPSIS by angle measure evaluation or vertical projection.

In this section, the application case of proposed approach and other approaches (objective weight,²⁴ subjective weight,³⁵ TOPSIS,²⁵ improved TOPSIS by angle measure,²⁶ and improved TOPSIS by vertical projection²⁷) are compared and analyzed. Considering the expert importance degree, the integration of objective and subjective weight, and the improved TOPSIS replacing Euclidean distance with connection distance, we explore a scientific and rational solution for cloud service supplier selection of manufacturing enterprise. Under the circumstance of cloud manufacturing, enterprise managers can achieve the goal of more scientific, reliable, and reasonable cloud service supplier selection by the propose approach.

Conclusion

With the development of cloud computing technology, manufacturing industry is rapidly promoting its information revolution, and a new mode named cloud manufacturing has emerged now. An important challenge which the enterprise or enterprise alliance must face under the circumstance of cloud computing technology is how to select the best cloud service supplier.

In this article, based on the analysis of existing researches, the index framework of cloud service supplier selection is built from two perspectives of technology and technology management. Considering the uncertain preference of enterprise or enterprise alliance decision-maker, the cloud service supplier selection is abstracted as an uncertain MCDM problem based on the index framework, and then a conjunctive MCDM approach is put forward based on integrated weight and improved TOPSIS.

The novelty of this article may include the following aspects:

By the aid of BP neural network’s self-learning ability, the expert importance degree can be determined more scientifically and reasonably.

The expert importance degree can make the group decision-making method of FAHP gain a beneficial supplement.

The weighting information loss exists in the individual objective or subjective weight, and the integrated weight can reduce this information loss.

The traditional TOPSIS and the commonly used improved TOPSIS by angle measure evaluation or vertical projection all have some shortages in special situations. The case application proves that the improved TOPSIS by connection distance can solve this problem.

For the possible limitations using a three-grade scale $f_{ij}^{k}$ value to express the judgment opinion, we will further focus on the uncertainty of expert judgment, and trapezoidal fuzzy number or triangular fuzzy number will be used to express the expert judgment. Simultaneously, we will study the generalization of the proposed approach in other fields, such as enterprise strategy decision, region industry planning, and medical diagnosis.

Footnotes

Academic Editor: ZW Zhong

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Science Research Project of Ningxia High Education (no. NGY2016141).

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