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Abstract

The supplier selection problem has been largely explored in the extant literature and attracted considerable attention of academics and purchasing managers. Practical supplier selection problem is usually revolved around multicriteria and a committee of experts. However, even using the exact values of the input data, certain experts may generate uncertain evaluation results on a supplier, because the exact weights with respect to each criterion are extremely difficult to reach a group consensus. In this article, first the interval data to describe all experts’ evaluation on all suppliers are formulated and then a stochastic multicriteria acceptability analysis (SMAA-2) is applied to provide a full rank of all candidate suppliers. SMAA-2 method is considered as an effective instrument to deal with stochastic decision-making problems. The rank acceptability indices and holistic rank indices are obtained to support the supplier selection. A numerical example drawn from the previous paper is recalculated to show the effectiveness of our approach.

Keywords

Supplier selection stochastic multicriteria acceptability analysis rank multiple criteria decision-making

Introduction

In today’s highly complex and competitive business world, how to choose and collaborate with the right suppliers has become an important management responsibility. The cost of supply acquisition commonly represents a large part of the aggregate costs. Supplier selection is the procedure through which the purchasers identify, evaluate, and contract with supplier and has underpinning effects on purchasers’ cost reduction and performance.¹ This problem has received considerable attention in both decision analysis and supply chain management literature and is becoming a fertile research topic for operations research and management science disciplines. Ho et al.² exhaustively reviewed the individual and integrated decision-making approaches from 2000 to 2008 to aid the supplier selection problem. Chai et al.³ complementarily provided a systematic literature review of the decision-making techniques assisting supplier selection from 2008 to 2012, which classifies the mentioned techniques into three categories: multiple criteria decision-making (MCDM) techniques, mathematical programming (MP) techniques, and artificial intelligence (AI) techniques. Wetzstein et al.⁴ conducted a structured review of supplier selection literature from 1990 to 2015 and showed that Analytic Hierarchy Process (AHP) and Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) are the most popular methods in MCDM approaches.

The contemporary supply chain management requires decision-maker to build and maintain a strategic partnership with few but reliable suppliers,² which effectively reduces the materials purchasing costs and improve the competitive advantages.^5–8 Therefore, besides the conventional price factor, promising supplier selection policy should also depend on a broad spectrum of qualitative and quantitative criteria such as quality, delivery, flexibility, and lead time.⁹ Dickson¹⁰ identified 23 criteria to be considered during the process of the purchasing manager determines supplier selection. Although the large body of research on multicriteria supplier selection in the literature is helpful to effectively guide purchasing manager to choose appropriate suppliers, it is crucial to understand the impact of interval values on supplier evaluation and selection.

In the previous literature, supplier selection can be seen as a decision-making process under predefined decision criteria. Therefore, the supplier selection problem examined in this article is described as follows. A set of $I$ candidate suppliers are evaluated according to $J$ criteria, with the involvement of a group of $K$ experts. Each expert $k, k = 1, 2, . . ., K$ is represented by a specific preference on the criteria importance. It is common that individual preferences may lead to a high level of variability, as observed in the decision-making by a committee of experts and collective choice problem.¹¹ In the presence of deterministic values for each supplier associated with each criterion, each expert knows the lower and upper bounds about the evaluation results for each supplier. Therefore, individual experts may produce interval evaluation values to measure the performance of each supplier, such that an interval supplier selection matrix (ISSM) is formulated to support supplier evaluation and selection. Different experts may generate different intervals for certain suppliers, due to their different preferences among the evaluation criteria. The interval formulation is motivated from the observation that in the domain of MCDM, different weight determination schemes may generate different weights even for the identical problem, and it is extremely hard to achieve a group consensus about the precise weights.¹²

The main aim of this study is to develop a sophisticated technique for solving the aforementioned ISSM and then provide a comprehensive rank of candidate suppliers. To the best of our knowledge, the existing literature has left this interesting and important topic largely unexplored. This article bridges this gap by first building the ISSM and then applying stochastic multicriteria acceptability analysis (SMAA-2) to provide a holistic rank of candidate suppliers. Such an investigation sheds much-needed light on potential incentives and directions for academic, managerial, and policy-related implications.

As initially proposed by Lahdelma et al.,¹³ SMAA is a family of methods proposed to support MCDM with many experts in scenarios in which little or no weight message is known, and the values associated with criteria are imprecise. SMAA does not require the experts to precisely or implicitly provide the input data and develops three useful and meaningful indices including acceptability index for each alternative measuring the variety of input data that give each alternative the best ranking position, central weight describing the preferences of an expert supporting an alternative, and confidence factor representing the reliability of the analysis. Lahdelma and Salminen¹² improved the SMAA in terms of taking into account all ranks and presented a holistic SMAA-2 investigation to determine good compromise alternatives. Lahdelma et al.¹⁴ provided an SMAA-O model to untangle the decision problems with ordinal criteria data. Durbach¹⁵ proposed an SMAA using achievement functions (SMAA-A) for discrete-choice problems by studying what combinations of aspirations are necessary to make each alternative the preferred one. Lahdelma and Salminen¹⁶ provided cross-confidence factors in terms of computing alternatives’ confidence factors based on others’ central weights. Lahdelma and Salminen¹⁷ integrated SMAA-2 and data envelopment analysis (DEA) to assess multicriteria alternatives. Lahdelma and Salminen¹⁸ presented an SMAA-P method by combining SMAA with the piecewise linear difference functions of prospect theory. Lahdelma et al.^19,20 provided and compared simulation and multivariate Gaussian distribution models to investigate the dependency information and uncertainty arisen in MCDM. Tervonen and Lahdelma²¹ developed efficient methods to perform the computations through Monte Carlo simulation, conducted the complexity analysis, and assessed the accuracy of the proposed algorithms. Corrente et al.²² combined SMAA and Preference Ranking Organisation Method for Enrichment Evaluations (PROMETHEE) mechanisms to study the parameters compatible with preference knowledge offered by the decision-maker. Angilella et al.²³ and Angilella et al.²⁴ integrated SMAA with the Choquet integral preference model to derive robust ordinal regression and robust recommendations, respectively. Durbach and Calder²⁵ explored the circumstance in which decision-makers are unable or unwilling to precisely evaluate trade-off message in SMAA.

Besides the method development of SMAA, there exist substantial application papers in the literature: facility location,¹⁴ forest planning,²⁶ elevator planning,²⁷ descriptive multiattribute choice model,²⁸ estimation of a satisficing model of choice,²⁹ DEA cross-efficiency aggregation,³⁰ performance assessment of mutual funds,³¹ project portfolio optimization,³² multicriteria ABC inventory classification,³³ and constructing composite indicators.³⁴

The main contribution of this article is summarized as follows. First, an ISSM to describe the supplier selection problem is formulated, in which each expert has specific but uncertain evaluation results on a set of candidate suppliers. Therefore, the supplier selection problem with interval values is deemed as a stochastic optimization problem. Second, SMAA-2 is introduced, along with the concepts of rank acceptability index, central weight vector, and confidence factor. Third, SMAA-2 to the supplier selection problem with interval data is applied, and a holistic rank of candidate suppliers is proposed. Even though the classical supplier selection problem has been sufficiently investigated in the literature, such investigation in this study is completely new and of both academic and practical significances and values.

The remainder of this study is structured as follows. Section “Problem formulation” provides the problem description. Section “Stochastic multicriteria acceptability analysis” presents SMAA-2 and some related important indices. Section “Numerical example” uses SMAA-2 to solve the supplier selection with interval inputs. Section “Conclusion” concludes the article and proposes meaningful directions for future research.

Problem formulation

The supplier selection problem studied in this article is modeled as follows, and the parameters used in this article are summarized in Table 1.

Table 1.

Notation of parameters.

Parameters	Notation
$S_{i}$	Evaluation score of supplier $i$
$x_{ij}$	Normalized values of all experts for criteria $j$ of supplier $i$ , $x_{ij} \in [0, 1]$
$w_{ij}$	Weight of criterion $j$ associated with supplier $i$ , $\sum_{j = 1}^{J} w_{ij} = 1, w_{ij} \geq 0$
$w_{ij}^{k}$	Weight of criterion $j$ associated with supplier $i$ provided by expert $k$ , $\sum_{j = 1}^{J} w_{ij}^{k} = 1, w_{i 1}^{k} \geq w_{i 2}^{k} \geq \dots \geq w_{iJ}^{k} \geq 0$
${US}_{i}^{k}$	Maximum evaluation of expert $k$ for supplier $i$
${LS}_{i}^{k}$	Minimum evaluation of expert $k$ for supplier $i$
$b_{i}^{r}$	Rank acceptability index
$a_{i}^{n}$	$n$ best ranks (nbr) acceptabilities
$p_{i}^{n}$	$n$ best ranks (nbr) confidence factor
$α^{r}$	Metaweights to construct holistic acceptability indices, $1 = α^{1} \geq α^{2} \geq \dots \geq α^{I} \geq 0$
$a_{i}^{h}$	Holistic acceptability index

A set of $I$ candidate suppliers are evaluated according to $J$ criteria, with the involvement of a committee of $K$ experts. All criteria are assumed to be benefit types. With regard to the cost-type criteria, the transformation of negativity or reciprocal may be taken. Therefore, the basic framework of the multicriteria supplier selection problem is depicted by a decision matrix $G_{IJ} = [x_{ij}]_{IJ}$

$G_{IJ} = [\begin{matrix} x_{11} & x_{12} & \dots & x_{1 J} \\ x_{21} & x_{22} & \dots & x_{2 J} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ x_{I 1} & x_{I 2} & \dots & x_{IJ} \end{matrix}]$ (1)

where $x_{ij}, x_{ij} \in [0, 1], i = 1, 2, . . ., I, j = 1, 2, . . ., J$ are exact values for all experts and have been normalized to eliminate the effect of magnitude of data. The evaluation score of a supplier is computed by the weighted sum of criteria measures with respect to the mentioned supplier, that is

$S_{i} = \sum_{j = 1}^{J} x_{ij} w_{ij}, i = 1, 2, \dots, I$ (2)

where $w_{ij}$ are the weights of criterion $j$ associated with supplier $i$ , and $\sum_{j = 1}^{J} w_{ij} = 1, w_{ij} \geq 0$ .

Each expert $k, k = 1, 2, . . ., K$ is identified by a specific preference on the sequence of criteria. Without loss of generality, this work assumes that for typical expert $k, k = 1, 2, . . ., K$ , the criteria are presented in a descending sequence of importance, that is, $w_{i 1}^{k} \geq w_{i 2}^{k} \geq \dots \geq w_{iJ}^{k}$ . This sequence definitely changes across different experts. Therefore, certain expert $k, k = 1, 2, \dots, K$ may formulate the following mathematical model to aggregate the most favorable performance for each supplier $i$

$\begin{matrix} {US}_{i}^{k} = max \sum_{j = 1}^{J} x_{ij} w_{ij}^{k} \\ s . t . w_{i 1}^{k} \geq w_{i 2}^{k} \geq \dots \geq w_{iJ}^{k}, i = 1, 2, \dots, I \end{matrix}$ (3)

$\sum_{j = 1}^{J} w_{ij}^{k} = 1, w_{ij}^{k} \geq 0, i = 1, 2, \dots, I, k = 1, 2, \dots, K$

Theorem 1

The optimal score of supplier $i$ derived from the mathematical model (equation (3)) is $max_{j = 1, 2, \dots, J} {1 / j (\sum_{t = 1}^{j} x_{it})}$ .³⁵

Proof

After denoting $ν_{ij}^{k} = w_{ij}^{k} - w_{i (j + 1)}^{k} \geq 0, j = 1, 2, \dots, J - 1, ν_{iJ}^{k} = w_{iJ}^{k} \geq 0$ , the following is obtained

$\begin{matrix} \sum_{j = 1}^{J} w_{ij}^{k} = (w_{i 1}^{k} - w_{i 2}^{k}) + 2 (w_{i 2}^{k} - w_{i 3}^{k}) + \dots + (J - 1) \\ (w_{i (J - 1)}^{k} - w_{iJ}^{k}) + J (w_{iJ}^{k}) \\ = ν_{i 1}^{k} + 2 ν_{i 2}^{k} + \dots + J ν_{iJ}^{k} \\ = \sum_{j = 1}^{J} j ν_{ij}^{k} \\ = 1 \end{matrix}$ (4)

Incorporating $φ_{ij}^{k} = \sum_{i = 1}^{j} x_{it}$ , the following is obtained

$\begin{array}{l} \sum_{j = 1}^{J} x_{i j} w_{i j}^{k} = x_{i 1} w_{i 1}^{k} + x_{i 2} w_{i 2}^{k} + \dots + x_{i J} w_{i J}^{k} \\ = [(w_{i 1}^{k} - w_{i 2}^{k}) x_{i 1}] + [(w_{i 2}^{k} - w_{i 3}^{k}) (x_{i 1} + x_{i 2})] \\ + \dots + [(w_{i (J - 1)}^{k} - w_{i J}^{k}) (x_{i 1} + x_{i 2} + \dots + x_{i (J - 1)})] \\ + [w_{i J}^{k} (x_{i 1} + x_{i 2} + \dots + x_{i J})] \\ = ν_{i 1}^{k} φ_{i 1}^{k} + ν_{i 2}^{k} φ_{i 2}^{k} + \dots + ν_{i J}^{k} φ_{i J}^{k} \\ = \sum_{j = 1}^{J} ν_{i j}^{k} φ_{i j}^{k} \end{array}$ (5)

Therefore, the mathematical model (equation (3)) is equivalent to the following formulation

$\begin{matrix} {US}_{i}^{k} = max \sum_{j = 1}^{J} ν_{ij}^{k} φ_{ij}^{k} \\ s . t . \sum_{j = 1}^{J} j ν_{ij}^{k} = 1 \end{matrix}$ (6)

$ν_{ij}^{k} \geq 0, j = 1, 2, \dots, J$

The dual of equation (6) is

$\begin{matrix} min z_{i}^{k} \\ s . t . z_{i}^{k} \geq \frac{1}{j} φ_{ij}^{k} \end{matrix}$ (7)

The optimal solution to equation (7) is realized at the point that

$z_{i}^{k} = max_{j = 1, 2, \dots, J} {\frac{1}{j} φ_{ij}^{k}}$

which is the optimal objective value of equation (3) in terms of

${US}_{i}^{k} = max_{j = 1, 2, \dots, J} {\frac{1}{j} \sum_{t = 1}^{j} x_{it}}$

This is the most favorable evaluation values determined by expert $k$ for supplier $i$ , with the given input of decision matrix (equation (1)). Given the determined sequence of criteria provided by typical expert, model (equation (3)) is simple-to-understand and easy-to-apply and can be conveniently solved in the absence of the elicitation of the exact values of weights.

Similarly, it is also necessary to consider the least favorable evaluation scores by expert $k$ for supplier $i$ . Therefore, an analogous mathematical model is presented as follows

$\begin{matrix} {LS}_{i}^{k} = min \sum_{j = 1}^{J} x_{ij} w_{ij}^{k} \\ s . t . w_{i 1}^{k} \geq w_{i 2}^{k} \geq \dots \geq w_{iJ}^{k}, i = 1, 2, \dots, I \end{matrix}$ (8)

$\sum_{j = 1}^{J} w_{ij}^{k} = 1, w_{ij}^{k} \geq 0, i = 1, 2, \dots, I, k = 1, 2, \dots, K$

Theorem 2

The optimal score of supplier $i$ derived from the mathematical model (equation (8)) is

$min_{j = 1, 2, \dots, J} {\frac{1}{j} \sum_{t = 1}^{j} x_{it}}$

On the strength of the obtained least and most favorable evaluation scores for supplier $i$ by expert $k$ , ISSM $Ω_{IK} = ([{LS}_{i}^{k}, {US}_{i}^{k} {])}_{IK}$ that describes the uncertain judgment of each expert on each supplier is formulated. Reasonable evaluation of supplier $i$ by expert $k$ should lie in $[{LS}_{i}^{k}, {US}_{i}^{k}]$ , where ${US}_{i}^{k}$ and ${LS}_{i}^{k}$ represent the upper and lower evaluation of expert k, respectively

$Ω_{IK} = [\begin{matrix} [{LS}_{1}^{1}, {US}_{1}^{1}] & [{LS}_{1}^{2}, {US}_{1}^{2}] & \dots & [{LS}_{1}^{K}, {US}_{1}^{K}] \\ [{LS}_{2}^{1}, {US}_{2}^{1}] & [{LS}_{2}^{2}, {US}_{2}^{2}] & \dots & [{LS}_{2}^{K}, {US}_{2}^{K}] \\ ⋮ & ⋮ & ⋮ & ⋮ \\ [{LS}_{I}^{1}, {US}_{I}^{1}] & [{LS}_{I}^{2}, {US}_{I}^{2}] & \dots & [{LS}_{I}^{K}, {US}_{I}^{K}] \end{matrix}]$ (9)

In line with Yang et al.,³⁰ the derived ISSM can be viewed as a stochastic MCDM problem. The following section briefly describes the SMAA-2 method developed by Lahdelma and Salminen,¹² which effectively solves these series of stochastic MCDM problems by providing a holistic rank of all alternatives.

Stochastic multicriteria acceptability analysis

SMAA covers a family of approaches that assist MCDM with imprecise, uncertain, or partially missing information. The logic of SMAA is exploring the weight space to describe the preferences that guarantee a specified ranking position for a certain alternative or make each alternative the most preferred one. Lahdelma and Salminen¹² initiated the adventure on this topic and develop rank acceptability index, central weight vector, and confidence factor for all alternatives. In terms of taking into account all ranks in the analysis, Lahdelma et al.¹³ made an extension of the SMAA and provide more holistic SMAA-2 analysis to graphically determine good compromise alternatives.

Preliminaries

According to the ISSM introduced in section “Problem formulation,” this work considers that a committee of $K$ experts has a set of $I$ suppliers to be evaluated and selected. Neither expert-specific evaluation values nor weights are exactly available. This article assumes that the individual preferences of decision-maker across all experts’ evaluations could be represented by a real-value utility function $g (i, w), i = {1, 2, \dots, I}$ , in which the weight vector $w$ quantifies subjective preferences of decision-maker across experts’ judgments. Moreover, the uncertain evaluation values from experts on suppliers are denoted by stochastic variables $ξ_{ik}$ with assumed or estimated density function $f (ξ)$ in the space $X \subseteq ℜ^{I \times K}$ . In addition, the uncertain weight vector is defined by a weight distribution with density function $f (w)$ in a set of weights described as

$W = {w \subseteq ℜ^{K} : \sum_{k = 1}^{K} w_{k} = 1, w_{k} \geq 0}$ (10)

The total absence of weight vector knowledge is defined in “Bayesian” manner by a uniform weight distribution in $W$ , that is

$f (w) = \frac{1}{Vol (W)} = \frac{(K - 1)!}{\sqrt{K}}$

Based on the above descriptions, the utility function is thereby employed to map the stochastic experts’ weight distributions and evaluation values into utility distributions $g (ξ_{i}, w)$ .

This study defines a ranking function representing the rank of each supplier as an integer from the best ranking position (=1) to the worst ranking position (=I) as follows

$rank (ξ_{i}, w) = 1 + \sum_{l} ρ (g (ξ_{l}, w) > g (ξ_{i}, w))$ (11)

in which $ρ (true) = 1$ and $ρ (false) = 0$ .

The SMAA-2 analysis is totally relied on analyzing the sets of favorable rank weights $W_{i}^{r} (ξ)$ known as

$W_{i}^{r} (ξ) = {w \in W : rank (ξ_{i}, w) = r}$ (12)

in which a weight $w \in W_{i}^{r} (ξ)$ guarantees that alternative $ξ_{i}$ obtains rank $r$ .

Useful indices

This section introduces various useful indices proposed by SMAA-2 method. The first index is the rank acceptability index $b_{i}^{r}$ , which is defined as the expected volume of the set of favorable rank weights.⁸ Specifically, $b_{i}^{r}$ evaluates the variety of possible valuations that guarantee alternative $ξ_{i}$ rank $r$ , which is computed by

$b_{i}^{r} = \int_{X} f (ξ) \int_{W_{i}^{r} (ξ)} f (w) dwd ξ$ (13)

Evidently, $b_{i}^{r}$ is distributed across the interval $[0, 1]$ , while $b_{i}^{r} = 0$ indicates that the alternative $ξ_{i}$ never reaches rank $r$ , and $b_{i}^{r} = 1$ represents that the alternative $ξ_{i}$ always obtains rank $r$ , neglecting the effect of the choice of weights. Furthermore, the rank acceptability index can be directly employed in MCDM. With regard to the large-scale problems, an iterative process is developed as follows, wherein the analysis of $n$ best ranks (nbr) acceptabilities is conducted at each interaction $n$

$a_{i}^{n} = \sum_{r = 1}^{n} b_{i}^{r}$ (14)

The nbr-acceptability $a_{i}^{n}$ represents a measure of the different individual preferences that guarantee alternative $ξ_{i}$ any of the $n$ best rank. The present analysis will not terminate until one or more alternatives realize a sufficient majority of the weights.

The weight space of the $n$ best rank with respect to an alternative could be described by the central nbr weight vector $w_{i}^{n}$ as follows

$w_{i}^{n} = \frac{\int_{X} f (ξ) \sum_{r = 1}^{n} \int_{W_{i}^{r} (ξ)} f (w) wdwd ξ}{a_{i}^{n}}$ (15)

In light of the known weight distribution, $w_{i}^{n}$ is the best single vector description for the individual preferences of a decision-maker who determines an alternative any rank from 1 to $n$ .

The third index is the nbr confidence factor $p_{i}^{n}$ , which is described as the probability that the alternative achieves any rank from 1 to $n$ when the central nbr weight vector is calculated by

$p_{i}^{n} = \int_{ξ : rank (ξ_{i}, w_{i}^{n})} f (ξ) d ξ$ (16)

Detailed information about the above indices has been presented in the study by Lahdelma and Salminen.¹² A manual to implement SMAA in real-life is proposed by Tervonen and Lahdelma.²¹

Holistic evaluation of rank acceptabilities

Based on the above rank acceptabilities, the next step is to propose a comprehensive approach that integrates the rank acceptabilities into holistic acceptability indices associated with all alternatives as follows

$a_{i}^{h} = \sum_{r = 1}^{I} α^{r} b_{i}^{r}$ (17)

in which $α^{r}$ are defined as metaweights to construct holistic acceptability indices and meet $1 = α^{1} \geq α^{2} \geq \dots \geq α^{I} \geq 0$ .

The elicitation of so-called metaweights is critical for the lexicographic decision problem, which naturally assign the largest value to $α^{1}$ , and the least value to $α^{I}$ . With regard to assigning weights to ranks, Barron and Barrett³⁶ developed three different mechanisms, that are, rank-sum (RS) approach, that is

$α^{r} (RS) = \frac{2 (I + 1 - r)}{I (I + 1)}, r = 1, 2, \dots, I$

reciprocal of the rank (RR) approach, that is

$α^{r} (R R) = \frac{1 / r}{\sum_{r = 1}^{I} 1 / r} r = 1, 2, \dots, I$

and rank-order centroid (ROC) approach, that is

$α^{r} (ROC) = \frac{1}{I} \sum_{r = 1}^{I} \frac{1}{r}, r = 1, 2, \dots, I$

This study employs ROC to decide $α^{r}, r = 1, 2, \dots, I$ , since they are more straightforward, accurate, and efficacious and offer a more appropriate implementation instrument.³⁶

In summary, the working process of SMAA-2 is first producing the rank acceptability index and then giving rise to the holistic acceptability index using the metaweights. The holistic evaluation of rank acceptability indices generates an overall measure of the acceptability of all alternatives. This is helpful to effectively rank and sort alternatives.

Numerical example

To apply SMAA-2 to solve supplier selection problem (Table 2), data from the multiple criteria supplier selection problem studied by Xia and Wu³⁷ are drawn. Three criteria, namely, price, quality, and service, are evaluated by means of the three-point scale, that is 1, 2, and 3, which indicate “low,”“middle,” and “high” for price criterion, and “good,”“middle,” and “poor” for quality and service criteria. The problem is to select 5 out of 14 candidate suppliers, with the involvement of a committee of six experts. Each expert has a specific preference on the criteria importance, that is, $price ≻ quality ≻ service$ , $price ≻ service ≻ quality$ , $quality ≻ price ≻ service$ , $quality ≻ service ≻ price$ , $service ≻ price ≻ quality$ , and $service ≻ quality ≻ price$ , which are denoted by notations “1,”“2,”“3,”“4,”“5,” and “6,” respectively.

Table 2.

Data for supplier selection.

Supplier	Price	Quality	Service	Price(norm)	Quality(norm)	Service(norm)
1	2	1	1	0.0600	0.0370	0.0400
2	3	1	1	0.0400	0.0370	0.0400
3	1	2	2	0.1200	0.0741	0.0800
4	2	2	2	0.0600	0.0741	0.0800
5	3	2	1	0.0400	0.0741	0.0400
6	1	2	3	0.1200	0.0741	0.1200
7	1	3	1	0.1200	0.1111	0.0400
8	1	1	3	0.1200	0.0370	0.1200
9	2	2	1	0.0600	0.0741	0.0400
10	2	2	3	0.0600	0.0741	0.1200
11	3	3	1	0.0400	0.1111	0.0400
12	3	2	2	0.0400	0.0741	0.0800
13	2	3	1	0.0600	0.1111	0.0400
14	2	1	3	0.0600	0.0370	0.1200

The ISSM $Ω_{IK} = ([{LS}_{i}^{k}, {US}_{i}^{k} {])}_{IK}$ is obtained by formulations (3) and (8), in which the interval evaluations on all suppliers by all experts are reported in Table 3.

Table 3.

Interval supplier selection matrix.

Supplier	Expert
	1	2	3	4	5	6
1	$[0.0457, 0.0600]$	$[0.0457, 0.0600]$	$[0.0370, 0.0485]$	$[0.0370, 0.0457]$	$[0.0400, 0.0500]$	$[0.0385, 0.0457]$
2	$[0.0385, 0.0400]$	$[0.0390, 0.0400]$	$[0.0370, 0.0390]$	$[0.0370, 0.0390]$	$[0.0390, 0.0400]$	$[0.0385, 0.0400]$
3	$[0.0914, 0.1200]$	$[0.0914, 0.1200]$	$[0.0741, 0.0970]$	$[0.0741, 0.0914]$	$[0.0800, 0.1000]$	$[0.0770, 0.0914]$
4	$[0.0600, 0.0714]$	$[0.0600, 0.0714]$	$[0.0670, 0.0741]$	$[0.0714, 0.0770]$	$[0.0700, 0.0800]$	$[0.0714, 0.0800]$
5	$[0.0400, 0.0570]$	$[0.0400, 0.0514]$	$[0.0514, 0.0741]$	$[0.0514, 0.0741]$	$[0.0400, 0.0514]$	$[0.0400, 0.0570]$
6	$[0.0970, 0.1200]$	$[0.1047, 0.1200]$	$[0.0741, 0.1047]$	$[0.0741, 0.1047]$	$[0.1047, 0.1200]$	$[0.0970, 0.1200]$
7	$[0.0904, 0.1200]$	$[0.0800, 0.1200]$	$[0.0904, 0.1156]$	$[0.0756, 0.1111]$	$[0.0400, 0.0904]$	$[0.0400, 0.0904]$
8	$[0.0785, 0.1200]$	$[0.0923, 0.1200]$	$[0.0370, 0.0923]$	$[0.0370, 0.0923]$	$[0.0923, 0.1200]$	$[0.0785, 0.1200]$
9	$[0.0580, 0.0670]$	$[0.0500, 0.0600]$	$[0.0580, 0.0741]$	$[0.0570, 0.0741]$	$[0.0400, 0.0580]$	$[0.0400, 0.0580]$
10	$[0.0600, 0.0847]$	$[0.0600, 0.0900]$	$[0.0637, 0.0847]$	$[0.0741, 0.0970]$	$[0.0847, 0.1200]$	$[0.0847, 0.1200]$
11	$[0.0400, 0.0756]$	$[0.0400, 0.0637]$	$[0.0637, 0.1111]$	$[0.0637, 0.1111]$	$[0.0400, 0.0637]$	$[0.0400, 0.0756]$
12	$[0.0400, 0.0647]$	$[0.0400, 0.0647]$	$[0.0570, 0.0741]$	$[0.0647, 0.0770]$	$[0.0600, 0.0800]$	$[0.0647, 0.0800]$
13	$[0.0600, 0.0856]$	$[0.0500, 0.0704]$	$[0.0704, 0.1111]$	$[0.0704, 0.1111]$	$[0.0400, 0.0704]$	$[0.0400, 0.0756]$
14	$[0.0485, 0.0723]$	$[0.0600, 0.0900]$	$[0.0370, 0.0723]$	$[0.0370, 0.0785]$	$[0.0723, 0.1200]$	$[0.0723, 0.1200]$

Furthermore, the metaweights to formulate the holistic acceptability indices are

$\begin{matrix} α^{14} = (1.00, 0.69, 0.54, 0.44, 0.36, 0.30, 0.25, \\ 0.20, 0.16, 0.13, 0.10, 0.07, 0.05, 0.02) \end{matrix}$ (18)

The aforementioned SMAA-2 model could be readily solved using the open-source software developed by Tervonen.³⁸

Normal distribution

This article assumes that the interval data $[{LS}_{i}^{k}, {US}_{i}^{k}]$ satisfy the normal distribution, and their mean and variance are represented by

$μ_{i}^{k} = \frac{{LS}_{i}^{k} + {US}_{i}^{k}}{2}$

and

$(σ^{2})_{i}^{k} = \frac{{US}_{i}^{k} - {LS}_{i}^{k}}{6}$

respectively. The results about the rank acceptability indices and the holistic acceptability indices derived according to SMAA-2 are shown in Table 4 and graphically reported in Figure 1.

Table 4.

Holistic acceptability indices and rank acceptability indices (normal distribution).

Supplier	$b^{1}$	$b^{2}$	$b^{3}$	$b^{4}$	$b^{5}$	$b^{6}$	$b^{7}$	$b^{8}$	$b^{9}$	$b^{10}$	$b^{11}$	$b^{12}$	$b^{13}$	$b^{14}$	$a^{h}$
1	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.01	0.35	0.64	0.0307
2	0.01	0.01	0.00	0.01	0.02	0.03	0.03	0.02	0.04	0.04	0.08	0.11	0.27	0.33	0.0960
3	0.00	0.25	0.46	0.23	0.05	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.5392
4	0.00	0.00	0.00	0.00	0.00	0.09	0.47	0.38	0.07	0.00	0.00	0.00	0.00	0.00	0.2311
5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.04	0.55	0.37	0.03	0.0607
6	0.91	0.08	0.01	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.9708
7	0.07	0.24	0.16	0.16	0.21	0.12	0.03	0.01	0.00	0.00	0.00	0.00	0.00	0.00	0.5128
8	0.03	0.24	0.18	0.26	0.17	0.06	0.05	0.02	0.01	0.00	0.00	0.00	0.00	0.00	0.5035
9	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.03	0.10	0.59	0.27	0.01	0.00	0.0962
10	0.02	0.17	0.17	0.26	0.31	0.09	0.01	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.4833
11	0.00	0.00	0.00	0.01	0.03	0.08	0.13	0.12	0.22	0.29	0.10	0.01	0.00	0.00	0.1798
12	0.00	0.00	0.00	0.00	0.00	0.00	0.01	0.13	0.31	0.39	0.14	0.03	0.00	0.00	0.1464
13	0.01	0.00	0.01	0.05	0.09	0.28	0.14	0.20	0.16	0.06	0.01	0.00	0.00	0.00	0.2630
14	0.00	0.01	0.01	0.02	0.12	0.25	0.13	0.12	0.16	0.11	0.05	0.03	0.00	0.00	0.2426

Figure 1.

Rank acceptability indices (normal distribution).

Based on the holistic acceptability indices in Table 4, a full and comprehensive rank of all suppliers is obtained: $6 ≻ 3 ≻ 7 ≻ 8 ≻ 10 ≻ 13 ≻ 14 ≻ 4 ≻ 11 ≻ 12 ≻ 9 ≻ 2 ≻ 5 ≻ 1$ .

The selected suppliers are suppliers 6, 3, 7, 8, and 10. More specifically, the most favorable supplier is supplier 6 whose holistic rank index is 97.08% and first rank support is 91% of the possibility, whereas the least favorable supplier is supplier 1 whose holistic rank index is 3.07% and last rank support is 64% of the possibility.

Uniform distribution

This study alternatively assumes that the interval data satisfy the uniform distribution. With such assumptions, the holistic acceptability indices and the rank acceptability indices are reported in Table 5 and Figure 2, respectively.

Table 5.

Holistic acceptability indices and rank acceptability indices (uniform distribution).

Supplier	$b^{1}$	$b^{2}$	$b^{3}$	$b^{4}$	$b^{5}$	$b^{6}$	$b^{7}$	$b^{8}$	$b^{9}$	$b^{10}$	$b^{11}$	$b^{12}$	$b^{13}$	$b^{14}$	$a^{h}$
1	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.03	0.55	0.41	0.0362
2	0.01	0.00	0.00	0.00	0.01	0.01	0.01	0.01	0.02	0.02	0.04	0.07	0.23	0.56	0.0587
3	0.02	0.23	0.36	0.26	0.11	0.02	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.5321
4	0.00	0.00	0.00	0.00	0.01	0.16	0.43	0.32	0.08	0.01	0.00	0.00	0.00	0.00	0.2365
5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.03	0.14	0.61	0.20	0.02	0.0708
6	0.82	0.13	0.04	0.01	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.9359
7	0.10	0.19	0.17	0.16	0.17	0.12	0.04	0.03	0.02	0.01	0.00	0.00	0.00	0.00	0.5103
8	0.03	0.23	0.19	0.20	0.15	0.08	0.05	0.03	0.02	0.01	0.01	0.01	0.00	0.00	0.4813
9	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.01	0.06	0.17	0.55	0.19	0.01	0.00	0.1025
10	0.02	0.17	0.18	0.24	0.28	0.09	0.02	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.4717
11	0.00	0.01	0.01	0.02	0.05	0.10	0.12	0.14	0.19	0.24	0.09	0.02	0.00	0.00	0.1995
12	0.00	0.00	0.00	0.00	0.00	0.01	0.05	0.18	0.34	0.31	0.10	0.02	0.00	0.00	0.1592
13	0.01	0.01	0.03	0.05	0.11	0.22	0.16	0.16	0.14	0.10	0.01	0.00	0.00	0.00	0.2687
14	0.00	0.01	0.03	0.05	0.12	0.20	0.13	0.13	0.13	0.10	0.06	0.04	0.01	0.00	0.2495

Figure 2.

Rank acceptability indices (uniform distribution).

It is observed that the sequence of candidate suppliers using SMAA-2 under uniform distribution is $6 ≻ 3 ≻ 7 ≻ 8 ≻ 10 ≻ 13 ≻ 14 ≻ 4 ≻ 11 ≻ 12 ≻ 9 ≻ 5 ≻ 2 ≻ 1$ , and the selected suppliers are suppliers 6, 3, 7, 8, and 10 as well. This sequence is mildly different from that derived from normal distribution case. The only difference lies in the rank positions of suppliers 2 and 5. In detail, the holistic rank index and first rank support possibility of the most favorable supplier 6 is 93.59% and 82%, respectively, both of which are lower than that of normal distribution case. Meanwhile, the holistic rank index and last rank support possibility of the least favorable supplier 1 are 3.62% and 41%, respectively. In summary, SMAA-2 under both the normal distribution and uniform distribution assumptions may produce complete ranks with sufficient discrimination power among all alternatives, in the case of that each expert has uncertain evaluations across all suppliers.

Conclusion

Multicriteria supplier selection problem with the involvement of a group of experts has been widely explored in decision science and supply chain management literature. Given the exact input data, different experts may generate uncertain evaluation results for all suppliers. However, the extant literature has left this topic largely undiscovered. This article is initially engaged in this effort by first formulating the interval values to be optimized and then innovatively applying the SMAA-2 method to obtain an overall rank for all the candidate suppliers. The interval data are assumed to be either normally or uniformly distributed in this study, and a metaweight scheme to derive holistic rank indices is elicited from the previous literature. A numerical example from the existing work is reexamined to show the effectiveness of our approach.

This article not only provides the decision-maker with more methodological options, but also enriches the theory and method of supplier selection problem. Future research should consider the determination of the uncertain sets for decision-making and investigate more practical distributions over the uncertainties. Furthermore, this suggested methodology could be applied in a real context such as green supply chain management.

Footnotes

Handling Editor: Guogqing Zhang

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 research was funded by the Major Program of the National Social Science Foundation of China (no. 18ZDA104).

ORCID iDs

Qian Zhang

Kin Keung Lai

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Multicriteria supplier selection using acceptability analysis

Abstract

Keywords

Introduction

Problem formulation

Theorem 1

Proof

Theorem 2

Stochastic multicriteria acceptability analysis

Preliminaries

Useful indices

Holistic evaluation of rank acceptabilities

Numerical example

Normal distribution

Uniform distribution

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References