Service supplier selection under fuzzy and stochastic uncertain environments

Abstract

Owing to the heterogeneity and inherent uncertainty of services, the selection of service suppliers is a complicated multi-attribute group decision-making (MAGDM) problem in which fuzzy criteria and stochastic criteria coexist. During the past few decades, many real-world supplier selection problems have been resolved using MAGDM methods. Nevertheless, extant research on supplier selection considers either fuzzy criteria or stochastic criteria, and hence most of these methods cannot address the complex and unstructured nature of contemporary service supplier selection problems. In this study, a novel technique for order preference by similarity to the ideal solution (TOPSIS) approach, integrating both fuzzy criteria and stochastic criteria, is developed; in this approach, the interval-valued intuitionistic fuzzy (IVIF) cross-entropy for fuzzy criteria and the Euclidean distance for stochastic criteria are used to acquire the rankings of alternatives. Moreover, a sensitivity analysis is conducted for a case study of hoisting service supplier selection, and a comparative analysis with other existing methods is performed to confirm the effectiveness and efficiency of the proposed approach.

Keywords

Service supplier selection MAGDM fuzzy criteria stochastic criteria TOPSIS

1 Introduction

Contemporary supply management aims to maintain long-term partnerships with suppliers and use fewer but more reliable suppliers. Therefore, issues concerning outstanding supplier selection are important in supply chain management [1]. Since the rapid advancement of manufacturing technologies and the emergence of customised requirements. Services are regarded as the add-in values for physical products. Products and services are expected to be integrated as a whole to fulfil customer value requirements. Supplier selection concerning services has scarcely been studied, and hence it is important to examine this aspect.

Compared with the traditional supplier-selection process in deterministic environments, the selection of service suppliers in hybrid uncertain environments can be considered a complicated multi-attribute group decision-making (MAGDM) problem [2]. The service supplier selection process is characterised using different types of qualitative criteria, including fuzzy and stochastic criteria. For fuzzy criteria such as service reliability and service quality, it is difficult for decision makers (DMs) to provide exact numerical evaluation values because of subjectiveness. Instead, DMs often describe a criterion using linguistic terms, such as “low/good”, “medium”, and “high/bad”. In addition, serviceability is essential for the selection of service suppliers and is related to the ability of the employees of the service suppliers. In other words, service is heterogeneous and cannot be produced repeatedly as a product. Considering the selection of rescue service suppliers as an example, criteria such as arrival time and service cost are of stochastic nature, and their probability distributions tend to be beyond acquisition owing to several unknown factors such as transportation conditions, weather, and personnel equipment readiness. For this purpose, the values of the stochastic criteria can be characterised in the form of discrete stochastic variables by using the statistical results for several subject judgments from DMs [3]. Then, it is easy to define the probability distribution of the values of the stochastic criteria.

To the best of our knowledge, few studies that concurrently consider fuzzy criteria and stochastic criteria have been reported, except for an evaluation of product service system concepts based on the information axiom in a fuzzy-stochastic environment. The criteria were classified into four types: crisp deterministic criteria, crisp random criteria, fuzzy deterministic criteria, and fuzzy random criteria. According to the characteristics of the concept evaluation problem for product and service systems, the information content of a fuzzy random criterion is described as an integral with fuzzy boundaries and is then calculated using a new algorithm based on fuzzy random simulation methods and an expected information content model [4, 5]. However, the information axiom cannot be applied to solve the service supplier selection problem. The method proposed by them can deal with only a hybrid situation in which the fuzziness and randomness occur in the same criterion, whereas the fuzzy criteria and stochastic criteria are independent in our research. Considering the coexistence of fuzzy criteria and stochastic criteria, it is important to develop a new approach to resolve the service supplier selection problem in fuzzy and stochastic uncertain environments.

Considering the MAGDM method based on the interval-valued intuitionistic fuzzy set (IVIFS) and the stochastic dominance (SD) rule, an integrated TOPSIS methodology incorporating fuzzy criteria and stochastic criteria simultaneously is proposed in this study for service supplier selection. In the proposal, the evaluation criteria are first categorised into fuzzy criteria and stochastic criteria, and interval-valued intuitionistic fuzzy numbers (IVIFNs) and stochastic variables (defined by SD) are employed to describe the evaluation data of the fuzzy and stochastic criteria, respectively. Then, the positive ideal solution (PIS) and negative ideal solution (NIS) under the TOPSIS framework are defined, and the ranking order of the service suppliers is obtained by virtue of the comprehensive closeness degree (CCD), which is calculated using the IVIF cross-entropy and Euclidean distance.

The remainder of this paper is organised as follows. In Section 2, a literature review is provided. In Section 3, some basic concepts of IVIFS and SD rules are introduced. In Section 4, the framework of the proposed approach is described. In Section 5, the proposed approach is discussed. In Section 6, a case study for the selection of a hoisting service supplier is presented to demonstrate the effectiveness and validity of the proposed method. In Section 7, the conclusions and future work are presented.

2 Related works

There is limited research addressing the service supplier selection problem, which is a specific MAGDM problem involving fuzzy criteria and stochastic criteria, as mentioned in the introduction. Hence, the literature related to the MAGDM problem in fuzzy environments, stochastic environments, and fuzzy and stochastic environments (simultaneously) are reviewed.

According to the nature of service supplier selection, the application of MAGDM techniques incorporating fuzzy criteria is widespread. Many researchers have used fuzzy sets theory [6, 7] to solve the supplier selection problem. A fuzzy analytic hierarchy process (AHP) method was developed [8] for the supplier selection of a white goods manufacturer in Turkey. An integrated multi-criteria optimisation and compromise solution (VIKOR) method was presented [9] with subjective and objective weights to select the most suitable supplier for an information system project. An integrated fuzzy technique was proposed [10] for order preference by similarity to an ideal solution (TOPSIS) and a multi-choice goal programming approach for supply chain management. Moreover, owing to the complexity and uncertainty of objective things and the ambiguity of human thinking, many attributes seem to be more suitable to be described using distributed linguistic representations [11, 13] Pythagorean fuzzy sets [12, 14], fuzzy numbers [15], intuitionistic fuzzy sets (IFSs) [16, 18], and IVIFSs [19, 20], while dealing with uncertain and imprecise judgments of DMs in many MAGDM problems. The studies on supplier selection that incorporate considerations of fuzzy criteria are summarised in Table 1.

Table 1
Research on supplier selection in fuzzy environments [15 , 21–31]

Fuzzification method Research Method

Fuzzy set Ertay, Kahveci, and Tabanl (2011) FAHP

Pitchipoo, Venkumar, Rajakarunakaran (2013) FAHP and GRA

Lima-Junior and Carpinetti (2016) TOPSIS

Trapezoidal fuzzy set Shemshadi, Shirazi, Toreihi, and Tarokh (2011) VIKOR

Yildiz, Ayyildiz, Taskin Gumus, and Ozkan (2020) Type-2 AHP

Pythagorean fuzzy set Villa Silva, Perez Dominguez et al (2019) Dimensional analysis

Yu, Shao, Wang, and Zhang (2019) TOPSIS

IFS Wood (2016) TOPSIS

Phochanikorn and Tan (2019) ANP and VIKOR

Sarwar, Akram, and Liu (2021) ELECTRE-II

IVIFS Boran, Genc, Kurt, and Akay (2009) TOPSIS

Ren, Chen, Fei, and Li (2016) TOPSIS

Zhao, You, Liu, and Wu (2017) VIKOR

Fuzzification method	Research	Method
Fuzzy set	Ertay, Kahveci, and Tabanl (2011)	FAHP
	Pitchipoo, Venkumar, Rajakarunakaran (2013)	FAHP and GRA
	Lima-Junior and Carpinetti (2016)	TOPSIS
Trapezoidal fuzzy set	Shemshadi, Shirazi, Toreihi, and Tarokh (2011)	VIKOR
	Yildiz, Ayyildiz, Taskin Gumus, and Ozkan (2020)	Type-2 AHP
Pythagorean fuzzy set	Villa Silva, Perez Dominguez et al (2019)	Dimensional analysis
	Yu, Shao, Wang, and Zhang (2019)	TOPSIS
IFS	Wood (2016)	TOPSIS
	Phochanikorn and Tan (2019)	ANP and VIKOR
	Sarwar, Akram, and Liu (2021)	ELECTRE-II
IVIFS	Boran, Genc, Kurt, and Akay (2009)	TOPSIS
	Ren, Chen, Fei, and Li (2016)	TOPSIS
	Zhao, You, Liu, and Wu (2017)	VIKOR

In conclusion, multi-attribute decision support frameworks based on the principle of pairwise comparison (FAHP, ELECTRE) and ideal points (TOPSIS, GRA, and VIKOR) have already been well developed. The trapezoidal fuzzy set can only describe the membership degree without the non-membership degree, whereas the (non-) membership degree and hesitation degree can be revealed by the Pythagorean fuzzy set, IFS and IVIFS. Compared with other fuzzy sets, the IVIFS can not only describe membership, non-membership, and hesitancy degree intervals, but also handle the uncertainty and fuzziness in group decision-making problems more effectively. Therefore, several approaches have been developed to solve the supplier selection problem that incorporates fuzzy criteria using the IVIFS.

Research on stochastic multi-attribute group decision-making (SMAGDM) can mainly be divided into stochastic multi-criteria acceptability analysis (SMAA)-based methods and SD-based methods. SMAA is a multi-criteria decision support technique for multiple DMs based on the exploration of the weight space [32]. In SMAA, the DMs do not need to express their preferences explicitly or implicitly. Instead, the technique analyses the type of evaluations that would make each alternative the preferred one. The theoretical discussions and real-life applications of the SMAA are summarised in Table 2. The assumption of a specific utility function is the primary limitation for the widespread use of the SMAA. The most famous rules are the SD rules [33], which have been derived from investment decision making, because the SD rules do not require a strong assumption of a utility function [34]. Some SD-based methods have been developed to resolve SMAGDM problems, as listed in Table 2. Besides these two types of research on SMAGDM, there are also some other methodologies. A stochastic formulation of AHP was developed [35] using an approach based on Bayesian categorical data models. A dynamic stochastic decision-making method was proposed [36] based on discrete-time sequences. These studies quantified the stochastic criteria as specific distribution values without implicitly addressing the fuzzy MAGDM problem.

Table 2

Research on SMAGDM using SMAA and SD rules [37, 46]

Research on SMAA	Application
Menou, Benallou, Lahdelma, and Salminen (2010)	Selection of location of a centralised air cargo hub
Corrente, Figueira, and Greco (2014)	Car evaluation
Angilella, Corrente, and Greco (2015)	Scale-construction problem
Angilella, Corrente, Greco, and Słowiński (2016)	University ranking
Zhang, Lai, and Yen (2019)	Supplier selection
Research on SD rules	Application
Batur and Choobineh (2012)	Simulated system selection
Tan, Ip, and Chen (2014)	Investment evaluation
Wu (2015)	Shortest path model
Chang, Jiménez-Martín et al (2015)	Financial risk management
Guo, Hu, Wang, and Zhao (2016)	Risk comparison

However, the theories and methods used in previous studies require some improvements. For instance, approaches based on SD rules can hardly determine the dominance relation between two distinct alternatives to obtain the ranking order [47]. Some studies used TOPSIS to transform the criteria values into crisp or interval values, but this transformation tends to cause information loss.

Motivated by the above discussion, our research is focused on the following aspects.

The co-occurrence phenomenon of fuzzy criteria and stochastic criteria in service supplier selection is revealed and elaborated.

A novel integrated TOPSIS framework is proposed to establish service supplier selection, which is modelled as a hybrid uncertain MAGDM problem incorporating fuzzy criteria and stochastic criteria simultaneously.

3 Preliminaries

Definition 1. Interval-valued intuitionistic fuzzy sets (IVIFS) [19]

An IVIFS in a finite set of the universe of discourse X ={ x₁, x₂, …, x_m } is given by $IVF = {〈 x_{i}, [\underline{μ} (x_{i}), \bar{μ} (x_{i})], [\underline{ν} (x_{i}), \bar{ν} (x_{i})] 〉 | x_{i} \in X}$ (1) -3ptwhere $[\underline{μ} (x_{i}), \bar{μ} (x_{i})]$ and $[\underline{ν} (x_{i}), \bar{ν} (x_{i})]$ denote the interval membership degree and non-membership degree, respectively, of element x_i belonging to IVF. Additionally, $0 \leq \underline{μ} (x_{i}) \leq \bar{μ} (x_{i}) \leq 1$ , $0 \leq \underline{ν} (x_{i}) \leq \bar{ν} (x_{i}) \leq 1$ , and $0 \leq \bar{μ} (x_{i}) + \bar{ν} (x_{i}) \leq 1$ . The IVF can be simplified to $IVF = 〈 [\underline{μ} (x_{i}), \bar{μ} (x_{i})], [\underline{ν} (x_{i}), \bar{ν} (x_{i})] 〉$ (2)

The hesitation degree π_IVF (x_i) of x_i to IVF is given by $\begin{matrix} π_{IVF} (x_{i}) = [\underline{π_{IVF}} (x_{i}), \bar{π_{IVF}} (x_{i})] = \\ [1 - \bar{μ} (x_{i}) - \bar{ν} (x_{i}), 1 - \underline{μ} (x_{i}) - \underline{ν} (x_{i})] \end{matrix}$ (3)

Further, Equation (2) can be transformed to $IVF = {[\underline{μ} (x_{i}), \bar{μ} (x_{i})], [1 - \bar{ν} (x_{i}), 1 - \underline{ν} (x_{i})]}$ (4)

Definition 2. Stochastic dominance rules [33]

The SD rules include first-order stochastic dominance (SD₁), second-order stochastic dominance (SD₂), and third-order stochastic dominance (SD₃). These SD rules are described based on their probability distributions.

Suppose λ₁ and λ₂ are two discrete stochastic variables, and E₁ (x) and E₂ (x) are the cumulative distribution functions of λ₁ and λ₂, respectively. Then, for a, b ∈ X⁺ with a ≤ b, the SD rules are

E₁ (x) SD₁ E₂ (x) if and only if E₁ (x) - E₂ (x) ≤ 0 and E₁ (x) ≠ E₂ (x) for all a ≤ x ≤ b.

E₁ (x) SD₂ E₂ (x) if and only if $\int_{a}^{x} (E_{1} (y) - E_{2} (y)) dy \leq 0$ and E₁ (x) ≠ E₂ (x) for all a ≤ x ≤ b.

E₁ (x)SD₃ E₂ (x) if and only if $\int_{a}^{x} \int_{a}^{z} (E_{1} (y) - E_{2} (y)) dydz \leq 0$ and E₁ (x) ≠ E₂ (x) for all a ≤ x ≤ b.

Definition 3. PIS and NIS

The PIS and NIS are vectors that include a series of fuzzy variables and stochastic variables in a hybrid uncertain MAGDM problem. Each component of the PIS and NIS represents a positive ideal variate and a negative ideal variate, respectively. For the fuzzy criteria, each variate is a fuzzy number. For the stochastic criteria, each variate is defined by a set of discrete values described by an associated probability distribution according to the SD rules.

Suppose θ₁, θ₂, …, θ_k, …, θ_n (n≥2) are discrete stochastic variables. The probability distribution P (θ⁺) of θ⁺ (PIS) is defined as follows. $\begin{matrix} - 1.5 pt & P (θ^{+}) \\ - 1.5 pt & {\begin{matrix} P_{1} (θ^{+}) = 0, & θ^{+} = x_{1} \\ ⋮ & ⋮ \\ P_{h - 1} (θ^{+}) = 0, & θ^{+} = x_{h - 1} \\ P_{h} (θ^{+}) = 1 - \sum_{u = h + 1}^{n} P_{u} (θ^{+}), & θ^{+} = x_{h} \\ P_{h + 1} (θ^{+}) = \max_{v} {P_{h + 1} (θ_{v}) | v = 1, 2, \dots, k}, & θ^{+} = x_{h + 1} \\ ⋮ & ⋮ \\ P_{n} (θ^{+}) = \max_{v} {P_{n} (θ_{v}) | v = 1, 2, \dots, k}, & θ^{+} = x_{n} \end{matrix} \end{matrix}$ (5) subject to $\sum_{u = 1}^{n} P_{u} (θ^{+}) = 1$ .

The subscript h is the smallest value satisfying P_u (θ⁺) = 0 for u ∈ [1, h–1], $P_{h} (θ^{+}) = 1 - \sum_{u = h + 1}^{n} P_{u} (θ^{+})$ , and $P_{u} (θ^{+}) = \max_{v} {P_{u} (θ_{v}) | v = 1, 2, \dots, k}$ for u ∈ [h + 1, n]. If $\sum_{u = q}^{n} \max_{v} {P_{u} (θ_{v}) | v = 1, 2, \dots, k} > 1$ and $\sum_{u = q + 1}^{n} \max_{v} {P_{u} (θ_{v}) | v = 1, 2, \dots, k} < 1$ for q ∈{1, 2, …, n - 1}, then h = q and $P_{q} (θ^{+}) = 1 - \sum_{u = q + 1}^{n} P_{u} (θ^{+})$ .

Suppose θ₁, θ₂, …, θ_k, …, θ_n (n≥2) are stochastic variables. The probability distribution P (θ^-) belonging to θ^- (NIS) is defined as follows. $\begin{matrix} - 1.5 pt & P (θ^{-}) \\ - 1.5 pt & {\begin{matrix} P_{1} (θ^{-}) = \max_{v} {P_{1} (θ_{v}) | v = 1, 2, \dots, k}, & θ^{-} = x_{1} \\ ⋮ & ⋮ \\ P_{i - 1} (θ^{-}) = \max_{v} {P_{i - 1} (θ_{v}) | v = 1, 2, \dots, k}, & θ^{-} = x_{i - 1} \\ P_{i} (θ^{-}) = 1 - \sum_{u = 1}^{i - 1} P_{u} (θ^{-}), & θ^{-} = x_{i} \\ P_{i + 1} (θ^{-}) = 0, & θ^{-} = x_{i + 1} \\ ⋮ & ⋮ \\ P_{n} (θ^{-}) = 0, & θ^{-} = x_{n} \end{matrix} \end{matrix}$ (6) subject to $\sum_{u = 1}^{n} P_{u} (θ^{-}) = 1$ .

The subscript i is the smallest value satisfying $P_{i} (θ^{-}) = \max_{v} {P_{u} (θ_{v}) | v = 1, 2, \dots, k}$ for u ∈ [1, i–1], $P_{i} (θ^{-}) = 1 - \sum_{u = 1}^{i - 1} P_{v} (θ^{-})$ , and P_i+1 (θ^-) =0 for u ∈ [i + 1, n]. If $\sum_{u = 1}^{q - 1} \max_{v} {P_{u} (θ_{v}) | v = 1, 2, \dots, k} < 1$ and $\sum_{u = 1}^{q} \max_{v} {P_{u} (θ_{v}) | v = 1, 2, \dots, k} > 1$ for q ∈ {2, …, n}, then i = q and $P_{q} (θ^{-}) = 1 - \sum_{u = 1}^{q - 1} P_{u} (θ^{-})$ .

Definition 4. Distance between discrete stochastic variables [48]

Suppose λ₁ and λ₂ are two discrete stochastic variables with probability distributions P (λ₁) and P (λ₂), respectively. The distance between λ₁ and λ₂, denoted as d (λ₁, λ₂), is defined as follows. $d (λ_{1} {, λ}_{2}) = \sqrt{\frac{1}{2} [P (λ_{1}) - P (λ_{2})] M {[P (λ_{1}) - P (λ_{2})]}^{T}}$ (7) where $\begin{matrix} P (λ_{1}) - P (λ_{2}) = (P_{1} (λ_{1}) - P_{1} (λ_{2}), P_{2} (λ_{1}) \\ - P_{2} (λ_{2}), \dots, P_{n} (λ_{1}) - P_{n} (λ_{2})) \end{matrix}$ (8)

Additionally, each element of the symmetric positive definite matrix M = (m_ij) _n×n is defined as $m_{ij} {= x}_{n}^{2} - {(x_{i} {- x}_{j})}^{2}, i, j = 1, 2, \dots, n$ (9) where x_i represents the i^th possible evaluation value of a stochastic criterion.

Definition 5. Cross-entropy between two discrete IVIFNs [49]

Suppose A, B are two IVIFNs in the universe X ={ x₁, x₂, …, x_{n
_f} }. The divergence measure between A and B is defined by $D (A, B) = \sum_{i = 1}^{n_{f}} (\begin{matrix} \frac{α_{A}^{i}}{4} \ln \frac{{2 α}_{A}^{i}}{α_{A}^{i} {+ α}_{B}^{i}} + \frac{β_{A}^{i}}{4} \ln \frac{{2 β}_{A}^{i}}{β_{A}^{i} {+ β}_{B}^{i}} \end{matrix}),$ (10) where $α_{A}^{i}$ substitutes ${\underline{μ}}_{A} (x_{i}) + {\bar{μ}}_{A} (x_{i}) + 2 - ν -_{A} (x_{i}) - {\bar{ν}}_{A} (x_{i})$ , and $β_{A}^{i}$ substitutes ${\underline{ν}}_{A} (x_{i}) + {\bar{ν}}_{A} (x_{i}) + 2 - μ -_{A} (x_{i}) - {\bar{μ}}_{A} (x_{i})$ for simplicity, similar to other notations. Then, D (A, B) is converted to a symmetric form as follows. $D^{*} (A, B) = D (A, B) + D (B, A)$ (11)

D^* (A, B) can also be called the cross-entropy of A and B, which indicates that the greater the increment of D^* (A, B), the more the information obtained from one observation for discriminating A from B. It can be inferred that D^* (A, B) ≥ 0, and D^* (A, B) = 0 if μ ⁱ _A =μ ⁱ _B , vⁱ_A = vⁱ_B for ∀ x_i. For the rationale, characteristics, and analysis of this measure, the source [49] may be referred.

4 Problem description and solution procedures

4.1 Problem description

As discussed in Section 1, the selection of service suppliers is a hybrid uncertain MAGDM problem with fuzzy and stochastic criteria.

The expert set is given by D = {d₁, d₂, ... , d_h} with respect to their weight B = {b₁, b₂, ... , b_h}, where d_l represents the l^th DM and b_l is his weight (l = 1, …, h). The alternative set is given by A = {a₁, a₂, ... , a_m}, where m ≥ 2, and a_i (i = 1, …, m) denotes an alternative of service supplier. The evaluation criteria set and their weight are separately given by C = {c₁, ... , c_{j
_f}, ... , c_{n
_f}, c_{n_f+1}, ... , c_{n_f+j_s}, c_{n_f+n_s}} and W = {w₁, ... , w_{j
_f}, ... , w_{n
_f}, w_{n_f+1}, ... , w_{n_f+j_s}, ... , w_{n_f+n_s}}, where j_f and j_s are the subscripts of the fuzzy criteria and stochastic criteria, respectively; n_f and n_s are the numbers of fuzzy and stochastic criteria; and n_f + n_s = n.

4.2 Solution procedures

For the selection of service suppliers in a hybrid uncertain environment, it is necessary to determine the weighted evaluation criteria, transform the initial linguistic evaluation judgments into calculative values, and establish an evaluation model. The logical flow of the proposed methodology is illustrated in Fig. 1. This problem can be solved using the following three phases.

Classify the criteria into two types: fuzzy criteria and stochastic criteria.

Collect the linguistic judgments of each fuzzy criterion on each alternative and the assessment value in a specified constant scale in the form of the scores of each stochastic criterion on each alternative. Convert the linguistic judgments into interval-valued intuitionistic fuzzy numbers (IVIFNs) and obtain the probability distribution for the stochastic component in the PIS F⁺ and NIS F^-.

For the fuzzy criteria, aggregate the DM judgments by incorporating b_l (l = 1, 2, ... , h) and w_{j
_f} (j_f = 1, 2, ... , n_f), and identify the fuzzy component in the PIS F⁺ and NIS F^- based on the TOPSIS. Then, calculate the degrees of divergence between each alternative and the PIS/NIS using IVIF cross-entropy for the fuzzy criteria. Then, calculate the degrees of divergence between each alternative and PIS/NIS using the Euclidean distance for the stochastic criteria. Calculate the relative closeness between each alternative and S + /S^–. Rank all the alternatives according to their CCD.

Fig. 1

Framework of the proposed TOPSIS approach.

5 Methodology

5.1 Identification of criteria

The evaluation criteria for the selection of service suppliers include both fuzzy and stochastic criteria. It is necessary to consider a situation with both fuzzy and stochastic criteria. Generally, fuzzy criteria result from cognitive obscure factors, and stochastic criteria are mainly affected by some unstable factors.

5.2 Quantification of criteria

A group of DMs is selected to provide their judgments on each alternative against each fuzzy criterion in the form of linguistic terms. The linguistic judgments are then converted into IVIFNs. To simplify the expression of linguistic judgments and variables, this study considers the rating of the criteria by the DMs as linguistic variables described by satisfaction degrees and introduces the homologous IVIFNs, as listed in Table 3.

Table 3
Relationship between linguistic variables and the corresponding IVIFNs

Linguistic variables IVIFNs

Very high (VH) [0.90, 0.95], [0.02, 0.05]

High (H) [0.70, 0.75], [0.20, 0.25]

Fair (F) [0.50, 0.55], [0.40, 0.45]

Low (L) [0.20, 0.25], [0.70, 0.75]

Very low (VL) [0.02, 0.05], [0.90, 0.95]

Linguistic variables	IVIFNs
Very high (VH)	[0.90, 0.95], [0.02, 0.05]
High (H)	[0.70, 0.75], [0.20, 0.25]
Fair (F)	[0.50, 0.55], [0.40, 0.45]
Low (L)	[0.20, 0.25], [0.70, 0.75]
Very low (VL)	[0.02, 0.05], [0.90, 0.95]

In addition, the stochastic criteria are given as a set of discrete values on the same scale as the associated probability distribution, according to definition 3.

5.3 Aggregation of fuzzy criteria

Consider that $p_{{ij}_{f}}^{l} = 〈 [{\underline{μ}}_{{ij}_{f}}^{l} {, \bar{μ}}_{{ij}_{f}}^{l}], [{\underline{ν}}_{{ij}_{f}}^{l} {, \bar{ν}}_{{ij}_{f}}^{l}] 〉$ denotes the assessment value of d_l on c_{j
_f} of a_i, where $0 \leq {\underline{μ}}_{{ij}_{f}}^{l} \leq {\bar{μ}}_{{ij}_{f}}^{l} \leq 1$ , $0 \leq {\underline{ν}}_{{ij}_{f}}^{l} \leq {\bar{ν}}_{{ij}_{f}}^{l} \leq 1$ , $0 \leq {\bar{μ}}_{{ij}_{f}}^{l} {+ \bar{ν}}_{{ij}_{f}}^{l} \leq 1$ , 1≤l≤h. Then, $\begin{matrix} y_{{ij}_{f}} = \sum_{l = 1}^{h} b_{l} p_{{ij}_{f}}^{l} = 〈 [\sum_{l = 1}^{h} b_{l} {\underline{μ}}_{{ij}_{f}}^{l}, \sum_{l = 1}^{h} b_{l} {\bar{μ}}_{{ij}_{f}}^{l}], \\ [\sum_{l = 1}^{h} b_{l} {\underline{ν}}_{{ij}_{f}}^{l}, \sum_{l = 1}^{h} b_{l} {\bar{ν}}_{{ij}_{f}}^{l}] 〉 \end{matrix}$ (12) where y_{ij
_f} is the aggregated assessment value denoting the j_f ^th criterion for the i^th alternative, 1≤j_f≤n_f, 1≤i≤m. Consequently, the aggregated decision matrix is obtained as follows. $\begin{matrix} \begin{matrix} c_{1} & c_{2} & \dots & c_{n_{f}} \end{matrix} \\ Y = \begin{matrix} a_{1} \\ a_{2} \\ ⋮ \\ a_{m} \end{matrix} [\begin{matrix} y_{11} & y_{12} & \dots & y_{{1 n}_{f}} \\ y_{21} & y_{22} & \dots & y_{{2 n}_{f}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ y_{m 1} & y_{m 2} & \dots & y_{{mn}_{f}} \end{matrix}] \end{matrix}$ (13)

5.4 Ranking of alternatives

TOPSIS is an effective ideal-point-based technique for dealing with MAGDM problems. The best alternative should be closest to the PIS and farthest from the NIS.

5.4.1 Relative closeness of fuzzy criteria

As the similarity and distance measure of IVIFS have been applied to MAGDM, the entropy of IVIFS can be naturally applied to such fields due to their close relationships, as discussed in [49]. For the fuzzy criteria, the IVIF cross-entropy is employed to measure the relative closeness degree between the IVIFNs and PIS/NIS.

According to Equation (12), IVIFS Y_i denotes the performance of a fuzzy criterion for the i^th alternative. $\begin{matrix} Y_{i} = {y_{i 1} {, y}_{i 2} {, \dots, y}_{{in}_{f}}} = \\ {\begin{matrix} 〈 [{\underline{y}}_{i 11} {, \bar{y}}_{i 11}], [{\underline{y}}_{i 12} {, \bar{y}}_{i 12}] 〉, 〈 [{\underline{y}}_{i 21} {, \bar{y}}_{i 21}], [{\underline{y}}_{i 22} {, \bar{y}}_{i 22}] 〉, \\ \dots, 〈 [{\underline{y}}_{{in}_{f} 1} {, \bar{y}}_{{in}_{f} 1}], [{\underline{y}}_{{in}_{f} 2} {, \bar{y}}_{{in}_{f} 2}] 〉 \end{matrix}} \end{matrix}$ (14) Let $F^{+} = {y_{1}^{+} {, y}_{2}^{+} {, \dots, y}_{n_{f}}^{+}}$ be the fuzzy components of the PIS, where $y_{n_{f}}^{+}$ is the positive ideal variate of the (n_f) ^th fuzzy criterion. Similarly, suppose that $F^{-} = {y_{1}^{-} {, y}_{2}^{-} {, \dots, y}_{n_{f}}^{-}}$ are the fuzzy components of the NIS, where $y_{n_{f}}^{-}$ is the negative ideal variate of the (n_f) ^th fuzzy criterion.

The fuzzy components of the PIS are $\begin{matrix} y_{j_{f}}^{+} = 〈 \max_{i = 1, 2, \dots, m} [{\underline{y}}_{{ij}_{f} 1} {, \bar{y}}_{{ij}_{f} 1}], \\ \min_{i = 1, 2, \dots, m} [{\underline{y}}_{{ij}_{f} 2} {, \bar{y}}_{{ij}_{f} 2}] 〉 = 〈 [{\underline{y}}_{i j_{f} 1}^{+} {, \bar{y}}_{i j_{f} 1}^{+}], \\ [{\underline{y}}_{{ij}_{f} 2}^{+} {, \bar{y}}_{i j_{f} 2}^{+}] 〉 \end{matrix}$ (15)

Similarly, the fuzzy components of the NIS are $\begin{matrix} y_{j_{f}}^{-} = 〈 \min_{i = 1, 2, \dots, m} [{\underline{y}}_{{ij}_{f} 1} {, \bar{y}}_{{ij}_{f} 1}], \\ \max_{i = 1, 2, \dots, m} [{\underline{y}}_{{ij}_{f} 2} {, \bar{y}}_{{ij}_{f} 2}] 〉 = 〈 [{\underline{y}}_{{ij}_{f} 1}^{-} {, \bar{y}}_{{ij}_{f} 1}^{-}], \\ [{\underline{y}}_{i j_{f} 2}^{-} {, \bar{y}}_{i j_{f} 2}^{-}] 〉 \end{matrix}$ (16)

Hence, the IVIF cross-entropy $D (y_{{ij}_{f}} {, y}_{j_{f}}^{-})$ or $D (y_{j_{f}}^{-} {, y}_{{ij}_{f}})$ between y_{ij
_f} and $y_{j_{f}}^{-}$ is calculated according to Equation (10).

Then, the symmetric relative closeness degree is $D^{*} (y_{{ij}_{f}} {, y}_{j_{f}}^{-}) = D (y_{{ij}_{f}} {, y}_{j_{f}}^{-}) + D (y_{j_{f}}^{-} {, y}_{{ij}_{f}})$ (17)

Herein, the term $D^{*} (y_{{ij}_{f}} {, y}_{j_{f}}^{-})$ describes the relative closeness degree between y_{ij
_f} and $y_{j_{f}}^{-}$ .

Analogous to the above process, the relative closeness degree $D^{*} (y_{{ij}_{f}} {, y}_{j_{f}}^{+})$ between y_{ij
_f} and $y_{j_{f}}^{+}$ can be obtained.

5.4.2 Relative closeness of stochastic criteria

Let $S^{+} = {s_{n_{f} + 1}^{+} {, s}_{n_{f} + 2}^{+} {, \dots, s}_{n_{f} {+ n}_{s}}^{+}}$ be the stochastic components of the PIS, where $s_{n_{f} + 1}^{+}$ is the positive ideal variate of the (n_f +1) ^th stochastic criterion. Similarly, suppose that $S^{-} = {s_{n_{f} + 1}^{-} {, s}_{n_{f} + 2}^{-} {, \dots, s}_{n_{f} {+ n}_{s}}^{-}}$ are the stochastic components of the NIS, where $s_{n_{f} + 1}^{-}$ is the negative ideal variate of the (n_f +1) ^th stochastic criteria.

The distances $d (s_{{ij}_{s}}^{'} {, s}_{j_{s}}^{+})$ between $s_{{ij}_{s}}^{'}$ and $s_{j_{s}}^{+}$ and $d (s_{{ij}_{s}}^{'} {, s}_{j_{s}}^{-})$ between $s_{{ij}_{s}}^{'}$ and $s_{j_{s}}^{-}$ can be calculated using Equations. (7) and (8), where $s_{{ij}_{s}}^{'}$ is the normalised value after aggregating all the evaluations of the DMs, and $s_{{ij}_{s}}^{'}$ , $s_{j_{s}}^{+}$ , and $s_{j_{s}}^{-}$ are all stochastic variables [48]. $\begin{matrix} d (s_{{ij}_{s}}^{'} {, s}_{j_{s}}^{+}) = \\ \sqrt{\frac{1}{2} [P (s_{{ij}_{s}}^{'}) - P (s_{j_{s}}^{+})] M {[P (s_{{ij}_{s}}^{'}) - P (s_{j_{s}}^{+})]}^{T}} \end{matrix}$ (18) $\begin{matrix} d (s_{{ij}_{s}}^{'} {, s}_{j_{s}}^{-}) = \\ \sqrt{\frac{1}{2} [P (s_{{ij}_{s}}^{'}) - P (s_{j_{s}}^{-})] M {[P (s_{{ij}_{s}}^{'}) - P (s_{j_{s}}^{-})]}^{T}} \end{matrix}$ (19) where i = 1, 2, …, m; j_s = n_f + 1, n_f + 2, …, n_f + n_s.

$P (s_{{ij}_{s}}^{'}) - P (s_{j_{s}}^{+}) = (P_{1} (s_{{ij}_{s}}^{'}) {- P}_{1} (s_{j_{s}}^{+}), P_{2} (s_{{ij}_{s}}^{'}) - P_{2} (s_{j_{s}}^{+}) {, \dots, P}_{n_{s}} (s_{{ij}_{s}}^{'}) {- P}_{n_{s}} (s_{j_{s}}^{+}))$ and $P (s_{{ij}_{s}}^{'}) - P (s_{j_{s}}^{-}) = (P_{1} (s_{{ij}_{s}}^{'}) - P_{1} (s_{j_{s}}^{-}) {, P}_{2} (s_{{ij}_{s}}^{'}) - P_{2} (s_{j_{s}}^{-}), . . ., P_{n_{s}} (s_{{ij}_{s}}^{'}) - P_{n_{s}} (s_{j_{s}}^{-})$ ).

5.4.3 Ranking of alternatives

The comprehensive distance between the alternatives and the ideal points $d_{i}^{+}$ and $d_{i}^{-}$ is defined as follows. $\begin{matrix} d_{i}^{+} = \sum_{j_{f} = 1}^{n_{f}} w_{j_{f}} D^{*} (y_{{ij}_{f}} {, y}_{j_{f}}^{+}) + \sum_{j_{s} {= n}_{f} + 1}^{n_{f} {+ n}_{s}} \\ w_{j_{s}} d (s_{{ij}_{s}}^{'} {, s}_{j_{s}}^{+}), i = 1, 2, \dots, m \end{matrix}$ (20) $\begin{matrix} d_{i}^{-} = \sum_{j_{f} = 1}^{n_{f}} w_{j_{f}} D^{*} (y_{{ij}_{f}} {, y}_{j_{f}}^{-}) + \sum_{j_{s} {= n}_{f} + 1}^{n_{f} {+ n}_{s}} \\ w_{j_{s}} d (s_{{ij}_{s}}^{'} {, s}_{j_{s}}^{-}), i = 1, 2, \dots, m \end{matrix}$ (21) where w_{j
_f} denotes the weight of c_{j
_f}, andw_{j
_s} denotes the weight of c_{j
_s}.

Based on the principle of weighted sum, the CCD of a_i, considering both fuzzy and stochastic criteria, is ${CCD}_{i} = \frac{d_{i}^{-}}{d_{i}^{+} {+ d}_{i}^{-}}, i = 1, 2, \dots, m$ (22)

After the CCD of each alternative is determined, the alternatives are ranked according to the descending order of CCD_i.

6 Case study

A growing number of enterprises with large-scale projects tend to buy engineering machine-leasing services instead of purchasing engineering machines. The major advantages of this approach are that it can reduce fixed investments, improve resource utilisation, and receive professional services. This section presents the hoisting service supplier selection for the curved dome of a nuclear power station in G Company to demonstrate the applicability and efficiency of the proposed approach.

Many nuclear power stations are being constructed in China, and the hoisting of the nuclear curved dome is the most important process in their construction. Figure 2 shows the hoisting process of the curved dome. There are four suppliers leasing hoisting services, denoted as a₁, a₂, a₃, and a₄; the corresponding technical properties of alternative the hoisting supports are explained in Fig. 3.

Fig. 2

Dome with part of the pipe preinstalled is being hoisted.

Fig. 3

Alternative hoisting supports identified for the case study. (a) Hoisting supports for a₁, maximum rated lifting weight of main boom (t): 1600, maximum rated lifting weight of luffing arm (t): 610. (b) Hoisting supports for a₂; maximum rated lifting weight of main boom (t): 1350, maximum rated lifting weight of luffing arm (t): 515. (c) Hoisting supports for a₃; maximum rated lifting weight of main boom (t): 1250, maximum rated lifting weight of luffing arm (t): 477. (d) Hoisting supports for a₄; maximum rated lifting weight of main boom (t): 800, maximum rated lifting weight of luffing arm (t): 650.

Table 4

Original evaluation data for the fuzzy criteria of a₁

d _l	b _l	Fuzzy criteria
		c ₁	c ₂	c ₃
d ₁	0.25	< [0.50, 0.55], [0.40, 0.45]>	< [0.70, 0.75], [0.20, 0.25]>	< [0.70, 0.75], [0.20, 0.25]>
d ₂	0.15	< [0.70, 0.75], [0.20, 0.25]>	< [0.50, 0.55], [0.40, 0.45]>	< [0.70, 0.75], [0.20, 0.25]>
d ₃	0.25	< [0.70, 0.75], [0.20, 0.25]>	< [0.70, 0.75], [0.20, 0.25]>	< [0.20, 0.25], [0.70, 0.75]>
d ₄	0.15	< [0.90, 0.95], [0.02, 0.05]>	< [0.20, 0.25], [0.70, 0.75]>	< [0.70, 0.75], [0.20, 0.25]>
d ₅	0.20	< [0.90, 0.95], [0.02, 0.05]>	< [0.70, 0.75], [0.20, 0.25]>	< [0.50, 0.55], [0.40, 0.45]>

6.1 Classification of evaluation criteria

Suppose that the service prices offered by the four service suppliers are the same in this example, so that we can focus on the qualitative criteria. Based on the survey, some evaluation criteria were determined. In addition, the evaluation criteria were classified into fuzzy criteria and stochastic criteria based on the real-world environment. The results are hoisting stability (c₁, fuzzy), environment fitness (c₂, fuzzy), hoisting precision (c₃, fuzzy), success probability (c₄, stochastic), operation time (c₅, stochastic), and service punctuality (c₆, stochastic).

Hoisting stability is the overall stability of the hoisting equipment, as there may be mechanical deformation of the crane boom during operation.

Environment fitness refers to the ability of the equipment to achieve all predetermined operations and functions without being damaged under the action of comprehensive environmental factors.

Hoisting precision refers to the precision in controlling the position change of the curved dome hoisted by the machine.

Success probability refers to the potential ability to successfully complete the hoisting process.

Operation time refers to the total time taken to complete the hoisting process, which affects the follow-up construction of the nuclear island project.

Service punctuality refers to the ability to deliver the service equipment and teams to the construction site on time.

The weights of these criteria, calculated by a fuzzy AHP [50], are w₁ = 0.11, w₂ = 0.28, w₃ = 0.21, w₄ = 0.14, w₅ = 0.15, and w₆ = 0.11.

6.2 Ranking of alternatives

Five DMs are invited to evaluate the hoisting service suppliers, and the relative weight of each DM is assigned as b₁ = 0.25, b₂ = 0.15, b₃ = 0.25, b₄ = 0.15, and b₅ = 0.20. The linguistic judgments of the fuzzy criteria are presented in Table 3. The stochastic criteria are evaluated by a score scale ranging from “1” to “5”, where “1” is the worst case and “5” is the best case. The ranking procedure for the alternatives is as follows.

The original evaluation data for the fuzzy criteria of a₁ are listed in Table 4. The original evaluation data for the stochastic criteria are presented in Table 5. The IVIFS fusion and fuzzy components of the ideal points F+ and F^-, calculated using Equations (12), (15), and (16), are listed in Table 6. The probability distributions of $s_{j_{s}}^{+}$ and $s_{j_{s}}^{-}$ are listed in Table 7.

Table 5
Original evaluation data for the stochastic criteria

c ₄ c ₅ c ₆

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

a ₁ 2/5 2/5 1/5 2/5 3/5 2/5 2/5 1/5

a ₂ 2/5 1/5 2/5 1/5 3/5 1/5 1/5 1/5 3/5

a ₃ 1/5 3/5 1/5 3/5 2/5 1/5 2/5 1/5 1/5

a ₄ 2/5 1/5 2/5 1/5 2/5 2/5 2/5 1/5 2/5

	c ₄	c ₅	c ₆
a ₁		2/5		2/5	1/5	2/5			3/5		2/5		2/5	1/5
a ₂	2/5		1/5		2/5	1/5	3/5	1/5		1/5		1/5		3/5
a ₃	1/5		3/5	1/5		3/5			2/5	1/5		2/5	1/5	1/5
a ₄	2/5	1/5		2/5		1/5	2/5		2/5		2/5	1/5		2/5

Table 6

Aggregation of IVIFNs for fuzzy criteria

Fuzzy criteria
	c ₁	c ₂	c ₃
a ₁	< [0.7200, 0.7700], [0.1870, 0.2300]>	< [0.5950, 0.6450], [0.3050, 0.3550]>	< [0.5350, 0.5850], [0.3650, 0.4150]>
a ₂	< [0.6700, 0.7200], [0.2080, 0.2500]>	< [0.5000, 0.5500], [0.4030, 0.4500]>	< [0.5900, 0.6400], [0.3100, 0.3600]>
a ₃	< [0.5750, 0.6250], [0.2980, 0.3450]>	< [0.5350, 0.5850], [0.3650, 0.4150]>	< [0.5000, 0.5500], [0.4000, 0.4500]>
a ₄	< [0.6300, 0.6800], [0.2430, 0.2900]>	< [0.4600, 0.5100], [0.4400, 0.4900]>	< [0.4850, 0.5350], [0.4180, 0.4650]>
F ^–	< [0.5750, 0.6250], [0.2980, 0.3450]>	< [0.4600, 0.5100], [0.4400, 0.4900]>	< [0.4850, 0.5350], [0.4180, 0.4650]>
F ⁺	< [0.7200, 0.7700], [0.1870, 0.2300]>	< [0.5950, 0.6450], [0.3050, 0.3550]>	< [0.5900, 0.6400], [0.3100, 0.3600]>

Table 7

Probability distributions of $s_{j_{s}}^{+}$ and $s_{j_{s}}^{-}$

	Score
	1	2	3	4	5
$s_{4}^{+}$			1/5	2/5	2/5
$s_{5}^{+}$				2/5	3/5
$s_{6}^{+}$				2/5	3/5
$s_{4}^{-}$	2/5	2/5	1/5
$s_{5}^{-}$	3/5	2/5
$s_{6}^{-}$	2/5	2/5	1/5

Table 8

Divergence or distance between each alternative and the ideal points

D*(y_i, F+)	a ₁	0.0000	0.0000	0.0032
	a ₂	0.0017	0.0095	0.0000
	a ₃	0.0201	0.0038	0.0084
	a ₄	0.0070	0.0187	0.0116
D(y_i*, F^-)	a ₁	0.0201	0.0187	0.0026
	a ₂	0.0102	0.0015	0.0116
	a ₃	0.0000	0.0057	0.0002
	a ₄	0.0034	0.0000	0.0000
$d (s_{{ij}_{s}}^{'} {, s}_{j_{s}}^{+})$	a ₁	0.6400	1.4400	1.4400
	a ₂	1.4400	3.2400	0.6400
	a ₃	1.2200	4.0000	1.9600
	a ₄	3.2400	4.0000	4.8400
$d (s_{{ij}_{s}}^{'} {, s}_{j_{s}}^{-})$	a ₁	2.5600	4.0000	2.5600
	a ₂	1.4400	1.9600	4.0000
	a ₃	1.0000	1.4400	1.9600
	a ₄	0.3600	1.4400	0.3600

The degrees of divergence between the fuzzy components of each alternative and the ideal points $D^{*} (y_{{ij}_{f}} {, y}_{j_{f}}^{+})$ , and $D^{*} (y_{{ij}_{f}} {, y}_{j_{f}}^{-})$ are calculated using Equation (17). The comprehensive distance between the stochastic components of the alternatives and the ideal points $d (s_{{ij}_{s}}^{'} {, s}_{j_{s}}^{+})$ and $d (s_{{ij}_{s}}^{'} {, s}_{j_{s}}^{-})$ are computed using Equations (18)–(21); the values are listed in Table 8. Finally, the CCD between a_i and the NIS, considering the fuzzy and stochastic relative closeness, is calculated using Equation (22), which ranks the alternatives as a₁ ≻ a₂ ≻ a₃ ≻ a₄. The computational results are listed in Table 9.

Table 9

Comprehensive closeness degree of each alternative

	Alternatives
	a ₁	a ₂	a ₃	a ₄
d_i⁺	0.4647	0.7609	0.9914	1.5944
d_i ^–	1.2480	0.9396	0.5732	0.3064
CCD_i	0.7287	0.5526	0.3664	0.1612

6.3 Sensitivity analysis

Criterion weights play an important role when the proposal is applied, in other words, the ranking results of the alternatives can vary with the criterion weights. Therefore, a sensitivity analysis of this approach was conducted by adjusting the original index weight.

Suppose that the initial weight of the j^th criterion c_j is ω_j, and the disturbed one is $ω_{j}^{'} = λ ω_{j}$ , where 0 ≤ λ ≤ 1. Then, the other weights will also be slightly disturbed according to the following condition. $\begin{matrix} ω_{k}^{'} = ω_{k} + (ω_{j} - ω_{j}^{'}) [ω_{k} / (1 - ω j)], \\ k = 1, 2, \dots n, and k \neq j \end{matrix}$ (23)

Suppose that the initial value of λ is 1, and the step size of each change is ν; then, 10 rounds of tests are established considering $\begin{matrix} v = ω_{j} / 10 \\ λ = 1 - vt (t = 1, 2, \dots, 10) \end{matrix}$ (24)

Based on the above discussion, to ensure the rationality of the sensitivity analysis, c₁, c₂, c₅, and c₆ are selected as c_j, as their weights are representative, and then a total of 4×10 tests are conducted. The results of the sensitivity analysis are shown in Appendix 1 and illustrated in Fig. 4.

Fig. 4

Results of sensitivity analysis.

As shown in Fig. 4, the order of ranking the alternatives does not change, and the optimal scheme is still a₁, which indicates that the robustness of the proposed approach is acceptable. In addition, the fluctuation in the results of 20–40 trials is evident, which means that the influence of the stochastic criteria on the results is more powerful than that of the fuzzy criteria. Thus, the proposed approach can effectively reveal the effects of stochastic criteria in the MAGDM problem.

6.4 Comparison of the methods and discussion

Owing to the limitations of the existing methods discussed above, the processing methods for the fuzzy and stochastic criteria are separate. Without loss of generality, a PROMETHEE-II method based on stochastic dominance degree (SDD) [51] and a fuzzy TOPSIS method [52] are adopted in this case, to observe the difference in corresponding decision-making outcomes.

6.4.1 Part I. Stochastic criteria

During the evaluations, we assume that the alternatives a₁, a₂, a₃, and a₄ are random variables with cumulative distribution functions E₁ (x), E₂ (x), E₃ (x), and E₄ (x), respectively. By definition 2, the SD relations for pairwise comparisons of alternatives are identified. Then, three SD relation matrices R₄ = [r_iq]_4 ×4, R₅ = [r_iq]_4 ×4, and R₆ = [r_iq]_4 ×4 with respect to c₄, c₅, and c₆ are obtained as follows. $\begin{matrix} R_{4} = [\begin{matrix} φ & {SD}_{2} & {SD}_{2} & {SD}_{1} \\ φ & φ & φ & {SD}_{1} \\ φ & {SD}_{3} & φ & {SD}_{2} \\ φ & φ & φ & φ \end{matrix}], \\ R_{5} = [\begin{matrix} φ & {SD}_{2} & {SD}_{1} & {SD}_{2} \\ φ & φ & {SD}_{2} & {SD}_{2} \\ φ & φ & φ & φ \\ φ & φ & {SD}_{2} & φ \end{matrix}], \\ R_{6} = [\begin{matrix} φ & φ & {SD}_{2} & {SD}_{1} \\ {SD}_{2} & φ & {SD}_{1} & {SD}_{1} \\ φ & φ & φ & {SD}_{1} \\ φ & φ & φ & φ \end{matrix}] . \end{matrix}$ where r_iq = φ implies that there is no SD relation between a pair of alternatives a_i and a_q. To estimate the degrees of SD relations, the SDD of E_i (x) SD_hE_q (x), h∈{1, 2, 3}, i, q∈{4, 5, 6}, is given by $\begin{matrix} ψ (E_{i} (x) {SD}_{x} E_{q} (x)) = \\ {\begin{matrix} \int_{a}^{b} (E_{i} (x) - E_{q} (x)) dx / \int_{a}^{b} E_{q} (x) xd, h = 1, 2 \\ \int_{a}^{b} \int_{a}^{z} (E_{i} (y) - E_{q} (y)) dydz / \int_{a}^{b} \int_{a}^{z} E_{q} (y) dydz, h = 3 \end{matrix} \end{matrix}$ (25)

Based on R₄, R₅, and R₆, three SDD matrices D₄ = [ψ_iq]_4 ×4, D₅ = [ψ_iq]_4 ×4, and D₆ = [ψ_iq]_4 ×4 (where ψ_iq denotes ψ (E_i (x) SD_hE_q (x)) for simplicity) are built according to Equation (25) as follows. $\begin{matrix} D_{4} = [\begin{matrix} 0 & 0.250 & 0.250 & 0.357 \\ 0 & 0 & 0 & 0.143 \\ 0 & 0.333 & 0 & 0.143 \\ 0 & 0 & 0 & 0 \end{matrix}], \\ D_{5} = [\begin{matrix} 0 & 0 . 167 & 0 . 333 & 0 . 231 \\ 0 & 0 & 0 . 200 & 0 . 077 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 . 133 & 0 \end{matrix}], \\ D_{6} = [\begin{matrix} 0 & 0 & 0 . 200 & 0 . 467 \\ 0 . 125 & 0 & 0 . 300 & 0 . 533 \\ 0 & 0 & 0 & 0 . 333 \\ 0 & 0 & 0 & 0 \end{matrix}] . \end{matrix}$

With the weighted summation operator, matrices D₄, D₅, and D₆ are aggregated into an overall SDD matrix Q as given below. $Q = [\begin{matrix} 0 & 0 . 150 & 0 . 268 & 0 . 340 \\ 0 . 034 & 0 & 0 . 158 & 0 . 226 \\ 0 & 0 . 117 & 0 & 0 . 142 \\ 0 & 0 & 0 . 050 & 0 \end{matrix}] .$

Let $Φ_{i}^{+}$ be the degree that alternative a_i dominates the other alternatives, and $Φ_{i}^{-}$ be the degree that a_i is dominated by the others. Then, the net dominant degree Φ_i, is calculated using Equation (26), which ranks the alternatives as a₁ ≻ a₂ ≻ a₃ ≻ a₄. The computational results are listed in Table 10. $Φ_{i} = Φ_{i}^{+} - Φ_{i}^{-}, i = 1, 2, \dots, m .$ (26)

Table 10

Ranking result based on stochastic criteria

	a ₁	a ₂	a ₃	a ₄
$Φ_{i}^{+}$	0.757	0.417	0.258	0.050
$Φ_{i}^{-}$	0.034	0.267	0.475	0.707
Φ _i	0.723	0.151	–0.217	–0.657

6.4.2 Part II. Fuzzy criteria

Similar to the processing of fuzzy criteria in this study, the normalised Euclidean distance is used to measure the relative closeness degree between the IVIFNs and PIS/NIS instead. Finally, we obtain a ranking order based on the fuzzy criteria; the result is listed in Table 11.

Table 11
Ranking result based on fuzzy criteria

a ₁ a ₂ a ₃ a ₄

d_i⁺ 0.045 0.086 0.139 0.154

d_i ^– 0.160 0.120 0.063 0.045

CCD_i 0.780 0.582 0.310 0.226

	a ₁	a ₂	a ₃	a ₄
d_i⁺	0.045	0.086	0.139	0.154
d_i ^–	0.160	0.120	0.063	0.045
CCD_i	0.780	0.582	0.310	0.226

The best alternative should be farthest from the NIS and closest from the PIS; thus, the ranking result is obtained as a₁ ≻ a₂ ≻ a₃ ≻ a₄ .

In conjunction with the ranking results of Part I and Part II, the best supplier obtained is still a₁, and the order of rankings of the suppliers is identical to that of the proposed method.

By comparison, it can be seen that the final solutions of the two methods are the same, which indicates that the proposed method is feasible. However, once there is a discrepancy between the ranking results of Part I and Part II, it is difficult to judge which alternative is better by the traditional separate discussion of fuzzy and random criteria. In other words, the proposed method is more applicable.

7 Conclusions and future works

The selection of service suppliers is described as a hybrid uncertain MAGDM problem, where fuzzy and stochastic criteria coexist. Although the information axiom, where the fuzziness and randomness occur in a same criterion, has been used for evaluation of product service system in fuzzy-stochastic environment, but differently, the fuzzy and stochastic criteria are independent in this study. Consequently, a novel TOPSIS framework integrating the IVIFS theory and SD rules is proposed to address this issue. The paper is concluded as follows.

The co-occurrence phenomenon of fuzziness and randomness in service supplier selection problem is revealed and elaborated.

A novel TOPSIS framework integrating the IVIFS theory and SD rules is proposed to solve this type of problem. For fuzzy criteria, the IVIFS is utilised to describe the evaluations of DMs and the IVIF cross-entropy is employed to measure the degree of divergence between the IVIFS-formed variables. Similarly, for stochastic criteria, the SD rules and the Euclidean distance are adopted respectively.

A case study, that is a selection of hoisting service suppliers for nuclear curved domes, is given to show how the proposed method work, sensitivity analysis as well as the comparison with the existing methods are discussed to demonstrate its effectivity and advantages.

However, there are still some issues that need to be considered in future work. Practically, once some pair of stochastic criteria and fuzzy criteria are correlated, then the information fusion would be more complex and the proposed method will fail. In addition, the probability distribution of stochastic criteria for a specific service supplier should be derived from engineering statistics.

Funding

This project was supported by the National Natural Science Foundation, China (No. 51505480, 51875345, 72001203). The authors would like to thank the anonymous referees for their valuable comments and suggestions.

References

Chai

, Liu

J.N.K.

and Ngai

E.W.T.

, Application of decision-making techniques in supplier selection: A systematic review of literature, Expert Systems with Applications 40(10) (2013), 3872–3885.

Geng

and Liu

, A hybrid service supplier selection approach based on variable precision rough set and VIKOR for developing product service system, International Journal of Computer Integrated Manufacturing 28(10) (2015), 1063–1076.

Fan

Z-P.

, Liu

and Feng

, A method for stochastic multiple criteria decision making based on pairwise comons of alternatives with random evaluations, European Journal of Operational Research 207(2) (2010), 906–915 paris.

Chen

, Chu

, Sun

and Li

, A new product service system concept evaluation approach based on Information Axiom in a fuzzy-stochastic environment, International Journal of Computer Integrated Manufacturing 28(11) (2015), 1123–1141.

Chen

, Chu

, Sun

, Li

and Su

, An Information Axiom based decision making approach under hybrid uncertain environments, Information Sciences 312 (2015), 25–39.

Suprasongsin

, Yenradee

and Huynh

V.-N.

, A weight-consistent model for fuzzy supplier selection and order allocation problem, Annals of Operations Research 293(2) (2019), 587–605.

Tong

, Pu

, Chen

and Yi

, Sustainable maintenance supplier performance evaluation based on an extend fuzzy PROMETHEE II approach in petrochemical industry, Journal of Cleaner Production (2020), 273.

Kahraman

, Ulukan

and Cebeci

, Multi-criteria supplier selection using fuzzy AHP, Logistics Information Management 16(6) (2003), 382–394.

Chen

L.Y.

and Wang

T.-C.

, Optimizing partners’ choice in IS/IT outsourcing projects: The strategic decision of fuzzy VIKOR, International Journal of Production Economics 120(1) (2009), 233–242.

10.

Liao

C.-N.

and Kao

H.-P.

, An integrated fuzzy TOPSIS and MCGP approach to supplier selection in supply chain management, Expert Systems with Applications 38(9) (2011), 10803–10811.

11.

, Zhang

, Kou

, Zhang

, Chao

, Li

C.-C.

, et al., Distributed linguistic representations in decision making: Tomy, key elements and applications, and challenges in data science and explainable artificial intelligence, Information Fusion 65 (2021), 165–178 axon.

12.

Akram

, Luqman

and Kahraman

, Hesitant Pythagorean fuzzy ELECTRE-II method for multi-criteria decision-making problems, Applied Soft Computing (2021), 108.

13.

, Zhang

, Kou

, Zhang

, Chao

, Li

C.-C.

14.

Feng

, Zheng

, Sun

and Akram

, Novel score functions of generalized orthopair fuzzy membership grades with application to multiple attribute decision making, Granular Computing (2021).

15.

Ertay

, Kahveci

and Tabanli

R.M.

, An integrated multi-criteria group decision-making approach to efficient supplier selection and clustering using fuzzy preference relations, International Journal of Computer Integrated Manufacturing 24(12) (2011), 1152–1167.

16.

Ren

H.-P.

, Chen

H.-H.

, Fei

and Li

D.-F.

, A MAGDM Method Considering the Amount and Reliability Information of Interval-Valued Intuitionistic Fuzzy Sets, International Journal of Fuzzy Systems 19(3) (2016), 715–725.

17.

Akram

, Peng

and Sattar

, A new decision-making model using complex intuitionistic fuzzy Hamacher aggregation operators, Soft Computing 25(10) (2021), 7059–7086.

18.

Atanassov

, Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems 20(1) (1986), 87–89.

19.

Atanassov

and Gargov

, Interval Valued Intuitionistic FuzzySets, Fuzzy Sets and Systems 31(3) (1989), 343–349.

20.

Krishankumar

, Ravichandran

K.S.

and Ramprakash

, A Scientific Decision Framework for Supplier Selection under Interval Valued Intuitionistic Fuzzy Environment, Mathematical Problems in Engineering 2017 (2017), 1–18.

21.

Pitchipoo

, Venkumar

and Rajakarunakaran

, Fuzzy hybrid decision model for supplier evaluation and selection, International Journal of Production Research 51(13) (2013), 3903–3919.

22.

Lima-Junior

F.R.

and Carpinetti

L.C.R.

, Combining SCOR^®model and fuzzy TOPSIS for supplier evaluation and management, International Journal of Production Economics 174 (2016), 128–141.

23.

Sarwar

, Akram

and Liu

, An integrated rough ELECTRE II approach for risk evaluation and effects analysis in automatic manufacturing process, Artificial Intelligence Review. (2021).

24.

Shemshadi

, Shirazi

, Toreihi

and Tarokh

M.J.

, A fuzzy VIKOR method for supplier selection based on entropy measure for objective weighting, Expert Systems with Applications 38(10) (2011), 12160–12167.

25.

Yildiz

, Ayyildiz

, Taskin Gumus

and Ozkan

, A Framework to Prioritize the Public Expectations from Water Treatment Plants based on Trapezoidal Type-2 Fuzzy Ahp Method, Environ Manage (2020).

26.

Wood

D.A.

, Supplier selection for development of petroleum industry facilities, applying multi-criteria decision making techniques including fuzzy and intuitionistic fuzzy TOPSIS with flexible entropy weighting, Journal of Natural Gas Science and Engineering 28 (2016), 594–612.

27.

Phochanikorn

and Tan

, A New Extension to a Multi-Criteria Decision-Making Model for Sustainable Supplier Selection under an Intuitionistic Fuzzy Environment, Sustainability 11(19) (2019).

28.

Villa Silva

A.J.

, Perez Dominguez

L.A.

, Martinez Gomez

, Alvarado-Iniesta

and Perez

I.J.C.

, Olguin Dimensional Analysis under Pythagorean Fuzzy Approach for Supplier Selection, Symmetry-Basel 11(3) (2019).

29.

, Shao

, Wang

and Zhang

, A group decision making sustainable supplier selection approach using extended TOPSIS under interval-valued Pythagorean fuzzy environment, Expert Systems with Applications 121 (2019), 1–17.

30.

Boran

F.E.

, Genc

, Kurt

and Akay

, A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method, Expert Systems with Applications 36(8) (2009), 11363–11368.

31.

Zhao

, You

X.-Y.

, Liu

H.-C.

and Wu

S.-M.

, An Extended VIKOR Method Using Intuitionistic Fuzzy Sets and Combination Weights for Supplier Selection, Symmetry-Basel 9(9) (2017).

32.

Lahdelma

, Hokkanen

and Salminen

, SMAA —Stochastic multiobjective acceptability analysis, European Journal of Operational Research 106(1) (1998), 137–143.

33.

Martel

J.M.

and Zaras

, Stochastic Dominance in Multi -Criteria Analysis under Risk, Theory and Decision 39(1) (1995), 31–49.

34.

Fan

Z.-P.

, Liu

and Feng

, A method for stochastic multiple criteria decision making based on pairwise comons of alternatives with random evaluations, European Journal of Operational Research 207(2) (2010), 906–9015 paris.

35.

Hahn

E.D.

, Link function selection in stochastic multicriteria decision making models, European Journal of Operational Research 172(1) (2006), 86–100.

36.

Sun

and Xu

X.-f.

, A dynamic stochastic decision-making methodbased on discrete time sequences, Knowledge-Based Systems 105 (2016), 23–28.

37.

Menou

, Benallou

, Lahdelma

and Salminen

, Decision support for centralizing cargo at a Moroccan airport hub using stochastic multicriteria acceptability analysis, European Journal of Operational Research 204(3) (2010), 621–629.

38.

Batur

and Choobineh

, Stochastic dominance-based comon for system selection, European Journal of Operational Research 220(3) (2012), 661–672 paris.

39.

Corrente

, Figueira

J.R.

and Greco

, The SMAA-PROMETHEE method, European Journal of Operational Research 239(2) (2014), 514–522.

40.

Angilella

, Corrente

and Greco

, Stochastic multiobjective acceptability analysis for the Choquet integral preference model and the scale construction problem, European Journal of Operational Research 240(1) (2015), 172–182.

41.

Angilella

, Corrente

, Greco

and Słowiński

, RobustOrdinal Regression and Stochastic Multiobjective AcceptabilityAnalysis in multiple criteria hierarchy process for the Choquetintegral preference model, Omega 63 (2016), 154–169.

42.

Zhang

, Lai

K.K.

and Yen

, Multicriteria supplier selection using acceptability analysis, Advances in Mechanical Engineering 11(10) (2019).

43.

Tan

, Ip

W.H.

and Chen

, Stochastic multiple criteria decision making with aspiration level based on prospect stochastic dominance, Knowledge-Based Systems 70 (2014), 231–241.

44.

, Study on mean-standard deviation shortest path problem in stochastic and time-dependent networks: A stochastic dominance based approach, Transportation Research Part B: Methodological 80 (2015), 275–290.

45.

Chang

C.-L.

, Jiménez-Martín

J.-Á.

, Maasoumi

and Pérez-Amaral

, A stochastic dominance approach to financial riskmanagement strategies, Journal of Econometrics 187(2) (2015), 472–485.

46.

Guo

, Hu

, Wang

and Zhao

, Comparing risks with reference points: A stochastic dominance approach, Insurance: Mathematics and Economics 70 (2016), 105–116.

47.

Leshno

and Levy

, Preferred by “All” and Preferred by “Most” Decision Makers: Almost Stochastic Dominance, Management Science 48(8) (2002), 1074–1085.

48.

Jiang

Y.-P.

, Liang

H.-M.

and Sun

, A method for discrete stochastic MADM problems based on the ideal and nadir solutions, Computers & Industrial Engineering 87 (2015), 114–125.

49.

Zhang

Q.S.

, Jiang

, Jia

and Luo

, Some Information Measures for Interval-Valued Intuitionistic Fuzzy Sets, Information Sciences 180(24) (2010), 5130–5145.

50.

Chan

F.T.S.

, Kumar

, Tiwari

M.K.

, Lau

H.C.W.

and Choy

K.L.

, Global supplier selection: a fuzzy-AHP approach, International Journal of Production Research 46(14) (2008), 3825–3857.

51.

Zhang

, Fan

Z.-P.

and Liu

, A method based on stochastic dominance degrees for stochastic multiple criteria decision making, Computers & Industrial Engineering 58(4) (2010), 544–552.

52.

Boran

F.E.

, Genç

, Kurt

and Akay

, A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method, Expert Systems with Applications 36(8) (2009), 11363–11368.

Service supplier selection under fuzzy and stochastic uncertain environments

Abstract

Keywords

1 Introduction

2 Related works

4.1 Problem description

4.2 Solution procedures

5.1 Identification of criteria

5.2 Quantification of criteria

Table 3 Relationship between linguistic variables and the corresponding IVIFNs Linguistic variables IVIFNs Very high (VH) [0.90, 0.95], [0.02, 0.05] High (H) [0.70, 0.75], [0.20, 0.25] Fair (F) [0.50, 0.55], [0.40, 0.45] Low (L) [0.20, 0.25], [0.70, 0.75] Very low (VL) [0.02, 0.05], [0.90, 0.95]

5.4.1 Relative closeness of fuzzy criteria

6.2 Ranking of alternatives

Table 5 Original evaluation data for the stochastic criteria c 4 c 5 c 6 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 a 1 2/5 2/5 1/5 2/5 3/5 2/5 2/5 1/5 a 2 2/5 1/5 2/5 1/5 3/5 1/5 1/5 1/5 3/5 a 3 1/5 3/5 1/5 3/5 2/5 1/5 2/5 1/5 1/5 a 4 2/5 1/5 2/5 1/5 2/5 2/5 2/5 1/5 2/5

6.4.1 Part I. Stochastic criteria

Table 11 Ranking result based on fuzzy criteria a 1 a 2 a 3 a 4 d i + 0.045 0.086 0.139 0.154 d i – 0.160 0.120 0.063 0.045 CCD i 0.780 0.582 0.310 0.226

Funding

References

Table 5
Original evaluation data for the stochastic criteria

c ₄ c ₅ c ₆

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

a ₁ 2/5 2/5 1/5 2/5 3/5 2/5 2/5 1/5

a ₂ 2/5 1/5 2/5 1/5 3/5 1/5 1/5 1/5 3/5

a ₃ 1/5 3/5 1/5 3/5 2/5 1/5 2/5 1/5 1/5

a ₄ 2/5 1/5 2/5 1/5 2/5 2/5 2/5 1/5 2/5

Table 11
Ranking result based on fuzzy criteria

a ₁ a ₂ a ₃ a ₄

d_i⁺ 0.045 0.086 0.139 0.154

d_i ^– 0.160 0.120 0.063 0.045

CCD_i 0.780 0.582 0.310 0.226