Selection of suppliers using FBWM-FGRA hybrid approach in hospital food supply chain

Abstract

In the current study, a new approach to assess and select food suppliers in hospitals is presented using integrated group evaluation method of fuzzy best- worst method (FBWM) and fuzzy gray relational analysis (FGRA). Evaluation criteria are selected by experts and weighed by the fuzzy best-worst method. After that, suppliers are rated using FGRA method. The proposed approach was implemented with seven criteria in one of the Iranian hospitals, and the results showed that quality, delivery time and trust criteria had the highest and skilled manpower and lack of surplus production criteria had the lowest score. Using FGRA, existing suppliers were ranked and the appropriate supplier was identified. In order to evaluate the reliability of the results, sensitivity analysis was performed on the criteria changes. The results showed that the supplier’s selection is greatly influenced by the criteria estimation values by the experts.

Keywords

Food supply chain supplier evaluation and selection fuzzy best-worst method fuzzy gray relational analysis health and medical centers

1 Introduction

In recent decades, the increasing competition among organizations around the world has given rise to the philosophy of supply chain management, which is responsible for integrating the levels of the chain and coordinating logistics, information and financial flows. These activities are carried out with the aim of responding to the demand of final customers to increase the competitiveness of the supply chain [1]. The food supply chain is one of the most important chains requires planning at the macro and micro levels for the supply of raw materials [2]. The nutrition department is one of the most important departments of hospitals to ensure the recovery of patients and their satisfaction is strongly influenced by the performance and activity of this department. Moreover, optimal nutrition has a therapeutic effect on patients [3]. Therefore, the nutrition of patients and staff is effective in their assessment of the hospital or hospital department where they work or are hospitalized [4]. As a result, the food supply chain of hospitals is very important. However, in the investigations of the hospital’s nutrition department, factors like lack of proper control over food distribution, high amount of waste, increase in costs and low quality of food, have led to more dissatisfaction of employees and patients [5]. One of the effective ways to increase the quality of hospital food is the proper selection of suppliers, which leads to the satisfaction of employees and patients [6]. Several studies have been conducted to evaluate and select the appropriate suppliers of food supply chains [7, 8].

Besides, several decision-making methods, including those limited to cover single and simultaneously multi-objective methods have been developed in order to select suitable suppliers [9]. The selection of suppliers is not efficient based on a single criterion such as price, and different criteria should be considered. In addition, each supplier meets some of the buyer’s criteria. Therefore, choosing a suitable supplier requires a structured and systematic approach, otherwise such an important decision is likely to fail [9].

Due to its high flexibility and applicability of the obtained results, the use of hybrid methods and a combination of more than one single methods has recently gained special attention [10]. However, because of the relative complexity of hybrid methods more than 80% of published models are based on individual methods [11]. Additionally, there is practically a degree of uncertainty in the decision-making process due to the existence of subjective judgments in the evaluation of quantitative and qualitative criteria. Also, models based on these subjective decision preferences are not always accurate because these judgments require a high level of knowledge, expertise, and experience [12].

In consensus achieving models for new type of group decision making problems based on ordinal classification, criteria weights and classification features are not provided, but heterogeneous and imprecise preference information is provided [13]. This paper proposes a consensus-building model for multi-criteria group sorting problems based on a threshold-based value-based sorting method of minimum adjustment view [14]. That research proposes a framework based on trust consensus of two-way interaction for decision making of large social network groups [15].

A bargaining game was presented to develop a feedback mechanism for group decision making of dynamic social networks. A feedback mechanism based on the bargaining game driven by the trust relationship was proposed [16]. Studies show that feedback mechanisms based on overlapping communities are superior to that with non-overlapping communities [17].

The best-worst method (BWM) was proposed by Rezaei (2015) to solve multi-criteria decision-making (MCDM) problems. In an MCDM problem, a number of alternatives are evaluated according to a number of criteria to select the best alternative(s) [18].

Therefore, the decision-making methods used in this research were used in a group and in an uncertain environment. In the present study, for the first time, the combination of FBWM and FGRA has been used to rank and select hospital food suppliers.

The COVID-19 pandemic has many implications for food supply chains. Experience in the early stages of the pandemic suggests that supply chain management may be vulnerable to short-term disruptions caused by demand shocks. Once the initial demand shock subsides, supply chains respond with a delay, and the COVID-19 pandemic has no longer lasting effects on the nature of food supply chains [19]. One of the researches conducted during the epidemic of COVID-19 was the selection of suppliers of face masks and face shields in the supply chain of health and treatment centers in order to prevent the shortage of these items. Taking into account technical and sustainability measures, health care centers should choose the best supplier for their required products. The dynamism and uncertainty of the pandemic are other factors that increase the complexity of the supplier selection problem. A new decision-making approach using attractiveness measurement through classification-based evaluation technique (MACBETH) and a new hybrid distance-based evaluation method was developed to address the supplier selection problem [20]. Based on the results of Macbeth’s method, job creation and occupational health and safety systems are the two best criteria. Considering that this research was conducted before the corona virus pandemic, the impact of this disease has not been considered in the selection of suppliers. However, the proposed approach of the current research can be extended to the parameters and criteria in Covid-19 environment.

Despite the high importance of hospital food supply, the review of related literature shows that no study has specially been done on the suppliers of this supply chain so far. Although, the FGRA method and the FBWM method have been used separately in the evaluation and selection of supply chain suppliers, the combination of these two methods (FBWM-FGRA) has not been used. Therefore, the current research aims to identify and rank the most important criteria for the evaluation and selection of food suppliers in hospitals using the basic approaches of the supply chain (lean, agile, resilient and green) including various traditional, social and environmental criteria with regard to the decisive role of suppliers in improving the food chain [21]. The innovations of the current research are:

Identifying and ranking the most important criteria for more accurate evaluating and selecting food suppliers in hospitals.

Using a hybrid FBWM-FGRA method to evaluate and select suppliers in the food supply chain of hospitals.

Determining the criteria sensitive to the ranking of suppliers in the proposed new approach of the current research

In the following, a part of the related literature of the topic has been reviewed due to the novelty of the research approach. Then, to analyze the data and results and identify and prioritize the most appropriate criteria, as well as evaluate and select the best suppliers, and perform the sensitivity of the results the research method is presented. The Final section deals with the conclusion and analysis of the results. Also, some suggestions are provided for future researches.

2 Literature review

Supplier selection is an important issue in supply chain management area [22, 23]. The optimal supplier selection requires the selection of appropriate criteria and the method of weighing them. Numerous studies have weighed the criteria. These studies are related to definite numbers or linguistic criteria such as gray and fuzzy numbers [24–26]. The number of pairwise comparisons is an important factor in the volume of calculations of management decision methods [27].

The FBWM is one of the new decision-making methods. This method minimizes the number of required pairwise comparisons and thus reduces the computational rate and improves the decision-making speed and compatibility of the results simultaneously [28]. This approach was first proposed by Rezaei (2015) to reduce pairwise comparisons of criteria in existing multi-criteria decision-making methods [29]. With the aim of showing similarity or difference in development trends between an alternative and a reference alternative (ideal), Grey Relational Analysis (GRA), examines and measures the correlations of the desired alternative and other alternatives in a system [30]. Since human judgments are usually ambiguous, Guo and Zhao (2017) developed the best-worst method in an uncertain environment to reduce ambiguity in decision-making [30].

Another widely used group decision-making method is FGRA [31]. This method uses ambiguous information in the form of distance values to rank decision alternatives [32]. FGRA is an accurate and reliable approach for prioritizing the order of gray relations between independent and dependent decision alternatives [33]. Selecting a supplier in an effective manner ensures that different quantitative and qualitative criteria are taken into account [34]. Since more environmental and social factors can be expressed by mathematical and numerical methods, with the growth of sustainable supply chain management, it is expected that the use of qualitative methods to cover these factors will increase [35]. Ryder and Fearne (2003) considered criteria such as quality, service, and price as criteria of flexibility in production, and trust as the basis of the organization’s key strategy [36]. Long (2007) used five criteria including timely delivery, production diversity, quality, price and geographical distance to evaluate and select a supplier in an agricultural company [37]. In another study conducted by Villanyi and Pakurar (2007) on the selection of fruit and vegetable suppliers in Hungary, various criteria such as timing and number of deliveries, quality, flexibility, non-hazardous transport, complaint management, time required between receipt and delivery and price were considered [38]. A summary of Hu et al. (2010) from researches conducted between 2001 and 2008 showed that the criteria included ability to meet and provide service, technology used, research and development, timely delivery, flexibility, quality, price (cost), validity, Security and the work environment, and the ability to take risks, are most important [39].

3 Research methodology

3.1 Conceptual framework of research

In the present study, for the first time, a combination of the two FBWM and the FGRA has been used to evaluate and select the appropriate supplier in the food supply chain in the hospital. In this approach, a number of experts in the field of hospital processes and supply chain are selected. Also, a list of criteria will be prepared from written scientific sources and have been distributed among experts three times by Delphi method to select the most appropriate criteria. Then, using the FBWM, the criteria are weighted and ranked. The results obtained using the FGRA lead to the selection of the best supplier. The conceptual research model is shown in Fig. 1.

Fig. 1

Conceptual research model.

The description of the proposed approach of the current research is as follows:

Identify the criteria for evaluating and selecting suppliers from different sources. Expert’s selection: Considering that the implementation of some multi-criteria decision-making methods involves the existence of specialized knowledge and skills from different fields of study, their implementation requires the existence of more than one expert [16]. In this case, group decision-making on the selection of ideal alternatives is made from the available alternatives [40].

Selection of evaluation criteria and selection of suppliers by experts and by Delphi method.

Determine the weight of the criteria by the FBDM. Choosing the best alternative by the FGRA.

Sensitivity analysis of the results by changing the values of the criteria. Determine the different scenarios and the ranking of suppliers in each of the scenarios.

3.2 Fuzzy Best Worst Method (FBWM)

In the best-worst method, first the best and worst criteria are determined by the expert (experts), and then the pairwise comparison matrix between each of these two indicators is formed compared with other indicators. In the following, the problem is transformed into a non-linear programming problem, which determines the weight of the criteria by applying the minimization function to the maximum absolute value of the difference in weights. Therefore, by improving the compatibility of comparisons, the number of paired comparisons is also significantly reduced so results with higher reliability are presented in this method. The steps to implement the best-worst method are as follows:

Select and finalize decision-making criteria by reviewing the literature and expert(s) opinion $\begin{matrix} c_{1} & c_{2} & \dots & c_{n} \end{matrix}$ , pairwise comparing and organizing the matrix in equation 1. $\begin{matrix} \begin{matrix} c_{1} & c_{2} & \dots & c_{n} \end{matrix} \\ \tilde{A} = \begin{matrix} c_{1} \\ c_{2} \\ ⋮ \\ c_{n} \end{matrix} [\begin{matrix} {\tilde{a}}_{11} & {\tilde{a}}_{12} & \dots & {\tilde{a}}_{1 n} \\ {\tilde{a}}_{21} & {\tilde{a}}_{22} & \dots & {\tilde{a}}_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{a}}_{n 1} & {\tilde{a}}_{n 2} & \dots & {\tilde{a}}_{nn} \end{matrix}] \end{matrix}$ (1)

Determine the best (with the highest priority) and the worst (with the least priority) criterion among the criteria.

Prioritizing the importance of the best criterion compared to other criteria based on the linguistic spectrum of Table 1. The result of this step is best to others (BO) vector.

Compare the priority and importance of other criteria to the worst one. The result of this is others to worst (OW) vector.

Calculate the optimal weights of the criteria ( $w_{1}^{*}, w_{2}^{*}, . . ., w_{n}^{*}$ ) by solving the nonlinear programming problem in the equation 2.

Table 1

Linguistic spectrum and their equivalent triangular fuzzy numbers

Linguistic scale	Triangular fuzzy number	Symbol
Equally important	(1,1,1)	EI
Weakly important	(2/3,1,3/2)	WI
Fairly important	(3/2,2,5/2)	FI
Very important	(5/2,3,7/2)	VI
Absolutely important	(7/2,4,9/2)	AI

$\begin{matrix} min - max_{j} {| \frac{{\tilde{w}}_{B}}{{\tilde{w}}_{j}} - {\tilde{a}}_{Bj} |, | \frac{{\tilde{w}}_{j}}{{\tilde{w}}_{W}} - {\tilde{a}}_{jW} |} \\ s . t . {\begin{matrix} \sum_{j = 1}^{n} R ({\tilde{w}}_{j}) = 1 \\ l_{j}^{w} ⩽ m_{j}^{w} ⩽ u_{j}^{w} \\ l_{j}^{w} ⩾ 0 \\ j = 1, 2, . . ., n \end{matrix} \end{matrix}$ (2)

The above problem can be written as equation 3. $\begin{matrix} min \tilde{ξ} \\ s . t . {\begin{matrix} | \frac{{\tilde{w}}_{B}}{{\tilde{w}}_{j}} - {\tilde{a}}_{Bj} | ⩽ \tilde{ξ} \\ | \frac{{\tilde{w}}_{j}}{{\tilde{w}}_{W}} - {\tilde{a}}_{jW} | ⩽ \tilde{ξ} \\ \sum_{j = 1}^{n} R ({\tilde{w}}_{j}) = 1 \\ l_{j}^{w} ⩽ m_{j}^{w} ⩽ u_{j}^{w} \\ l_{j}^{w} ⩾ 0 \\ j = 1, 2, . . ., n \end{matrix} \end{matrix}$ (3)

All the model parameters are assumed to be triangular fuzzy numbers as:

${\tilde{w}}_{B} = (l_{B}^{w}, m_{B}^{w}, u_{B}^{w})$ , ${\tilde{w}}_{j} = (l_{j}^{w}, m_{j}^{w}, u_{j}^{w})$ , ${\tilde{w}}_{W} = (l_{W}^{w}, m_{W}^{w}, u_{W}^{w})$ , ${\tilde{a}}_{Bj} = (l_{Bj}, m_{Bj}, u_{Bj})$ , ${\tilde{a}}_{jW} = (l_{jW}, m_{jW}, u_{jW})$ $\tilde{ξ} = (l^{ξ}, m^{ξ}, u^{ξ})$ .

Assuming $ξ^{*} = (k^{*}, k^{*}, k^{*}), k^{*} \leq l^{ξ}$ , the above model is converted to the final model of equation 4.

$\begin{matrix} min {\tilde{ξ}}^{*} \\ s . t . {\begin{matrix} | \frac{(l_{B}^{w}, m_{B}^{w}, u_{B}^{w})}{(l_{j}^{w}, m_{j}^{w}, u_{j}^{w})} - (l_{Bj}, m_{Bj}, u_{Bj}) | ⩽ (k^{*}, k^{*}, k^{*}) \\ | \frac{(l_{j}^{w}, m_{j}^{w}, u_{j}^{w})}{(l_{W}^{w}, m_{W}^{w}, u_{W}^{w})} - (l_{jW}, m_{jW}, u_{jW}) | ⩽ (k^{*}, k^{*}, k^{*}) \\ \sum_{j = 1}^{n} R ({\tilde{w}}_{j}) = 1 \\ l_{j}^{w} ⩽ m_{j}^{w} ⩽ u_{j}^{w} \\ l_{j}^{w} ⩾ 0 \\ j = 1, 2, . . ., n \end{matrix} \end{matrix}$ (4)

The optimal weights of the criteria $({\tilde{w}}_{1}^{*}, {\tilde{w}}_{2}^{*}, . . ., {\tilde{w}}_{n}^{*})$ are obtained by solving the mathematical programming model 4.

3.3 Gray Relational Analysis Method (GRAM)

This method was first developed by Dong (1982). GRA examines multi-criteria decision-making problems considering the combination of all functional values of the criteria in a unique value for all available alternatives. It simplifies the initial problem into a single-criterion decision-making problem. As a result, multiple criteria alternatives can be compared effectively after applying the process of GRA.

The more consistency of change process between an alternative and optimal alternative is, the strong relationship between them is, otherwise the relationship between them will be low [16]. By adopting this approach, GRA can be used to measure the relationship between reference and comparative sets.

The steps to implement GRA include: creating gray relationships, defining referenced sequences, calculating gray relational coefficient, and assigning gray relational score. First, the decision matrix ( $\tilde{X}$ ) is converted to the normal decision matrix ( $\tilde{R}$ ). By considering ${\tilde{r}}_{ij} = (l_{ij}, m_{ij}, u_{ij})$ , the normal performance rank is calculated as equation 5. ${\tilde{r}}_{ij} = (\frac{l_{ij}}{u_{ij}^{+}}, \frac{m_{ij}}{u_{ij}^{+}}, \frac{u_{ij}}{u_{ij}^{+}}), i = 1, . . . m, j = 1, . . ., n$ (5) Where:

$u_{j}^{+} = max_{i} {u_{ij}} \forall i i = 1, . . ., m Δ$ (6)

The reference set is determined by Equation 7.

${\tilde{R}}_{0} = [{\tilde{r}}_{01}, {\tilde{r}}_{02}, . . ., {\tilde{r}}_{0 n} = max ({\tilde{r}}_{ij})] j = 1, . . ., n$ (7)

Then the distance matrix is formed. The distance between the reference value and any comparison value (δ_ij) is calculated by the equation 8.

$d (\tilde{A}, \tilde{B}) = \sqrt{\frac{1}{3} [{(l_{1} - l_{2})}^{2} + {(m_{1} - m_{2})}^{2} + {(u_{1} - u_{2})}^{2}]}$ (8)

In the next step, the Gray Relational Coefficient (ξ_ij) is obtained by equation 9 [41].

$ζ_{ij} = \frac{δ_{min} + ρ δ_{max}}{δ_{ij} + ρ δ_{max}}, δ_{max} = max (δ_{ij}), δ_{min} = min (δ_{ij})$ (9)

ρ is called the solution factor and ρ ∈ [0, 1]. To difuzzify the weight of the j-th criterion ( ${\tilde{w}}_{j}$ ), the definition (5) (equation (9)) is used. Then, the gray relational grade (γ_i) is achieved from equation 10. $γ_{i} = \sum_{j = 1}^{n} w_{j} ζ_{ij}, i = 1, . . ., m$ (10)

Where ${\tilde{w}}_{j}$ is the weight of j-th criterion and the sum of the weights is one. Therefore, all options are ranked according to their gray relational grade, and the option with a higher value of gray relational grade will be more preferable.

4 Research findings

The statistical population of this study was the food supply department of the hospitals of West Azerbaijan province and the case study was Shahid Taleghani Hospital of Urmia city. Considering various health and environmental requirement and due to the various measures and special standards in the selection of food suppliers, this hospital has been selected. Therefore, examining its food supply chain is associated with more favorable conditions.

Lingo software has been used to implement the FBWM and FGRA. Based on above mentioned issues, the present study is cross-sectional in terms of research time (September 2018 to March 2019); applied in terms of research results, combined in terms of the process of conducting research; descriptive (case study) in terms of the purpose of the research, and inductive in terms of the logic of conducting the research.

Step 1) Identify criteria for evaluating and selecting suppliers and selecting experts

In this study, the views of experts in the field of food supply chain in hospitals have been used to collect data, prepare and identify criteria for determining hospital food supply (first expert: procurement officer (E1), second expert: nutritionist (E2) and third expert: university professor (E3)).

The characteristics of the experts are as described in Table 2

Table 2
Supplier selection criteria

Symbol Criteria Description

C1 Price Reasonable price while maintaining acceptable quality

C2 Quality Satisfying customers, easy delivery, complete and in accordance with agreed quality standards

C3 Delivery time Commitment to agreed delivery time and minimization of delay time

C4 No surplus production Delivery of the exact amount of the order, eliminate waste, efficient use of energy, waste and disposal

C5 Trust Commitment to contracts

C6 Skilled manpower Multi-skill workforce

C7 Accountability How to handle criticism and complaints and commitment to meet expectations, documented responses

Symbol	Criteria	Description
C1	Price	Reasonable price while maintaining acceptable quality
C2	Quality	Satisfying customers, easy delivery, complete and in accordance with agreed quality standards
C3	Delivery time	Commitment to agreed delivery time and minimization of delay time
C4	No surplus production	Delivery of the exact amount of the order, eliminate waste, efficient use of energy, waste and disposal
C5	Trust	Commitment to contracts
C6	Skilled manpower	Multi-skill workforce
C7	Accountability	How to handle criticism and complaints and commitment to meet expectations, documented responses

Experts Code	Job	Field of Study	Work experience
E1	Procurement officer	Master’s degree in procurement management	14
E2	Nutritionist	PhD in nutrition, clinical and diet therapy	21
E3	University professor	PhD in occupational health	19

After reviewing conducted studies, a list of criteria for selecting suppliers was identified and provided to the experts.

Step 2) Selection of evaluation criteria and selection of suppliers by experts

In order to implement the methodology in the conceptual framework of food supply of hospitals, the localization questionnaire was provided to experts and they were asked to determine the degree of importance of these criteria. Table 2 shows the desired criteria of the study for hospital food supplier’s selection.

Step 3) Determine the weight of the criteria by the FBWM

The experts were asked to determine the degree of importance of criteria based on linguistic variables. Then, among the above criteria, the best and worst criteria were selected by each of the experts. In the next step, BO and OW matrices were formed as described in Table 3.

Table 3

Comparison matrix of the best and worst criteria compared to other criteria from the perspective of each expert

Best -Worst	Expert	Criteria	C1	C2	C3	C4	C5	C6	C7
Best	E1	C3	FI	WI	EI	AI	WI	FI	VI
	E2	C2	WI	EI	FI	AI	WI	FI	VI
	E3	C5	WI	FI	WI	VI	EI	FI	WI
Worst	E1	C6	WI	VI	FI	WI	FI	EI	FI
	E2	C4	VI	AI	VI	EI	FI	WI	FI
	E3	C6	WI	VI	FI	WI	FI	EI	FI

The linguistic variables in Table 3 are then transformed into triangular fuzzy numbers. For example, the BO and OW vectors for expert 1 are as follows: $\begin{matrix} {\tilde{A}}_{BO 1} = [(3 / 2, 2, 5 / 2), (2 / 3, 1, 3 / 2), (1, 1, 1), (7 / 2, \\ 4, 9 / 2), (2 / 3, 1, 3 / 2), (3 / 2, 2, 5 / 2), (5 / 2, 3, 7 / 2)] \end{matrix}$ $\begin{matrix} {\tilde{A}}_{OW 1} = [(2 / 3, 1, 3 / 2), (5 / 2, 3, 7 / 2), (3 / 2, 2, 5 / 2), \\ (2 / 3, 1, 3 / 2), (3 / 2, 2, 5 / 2), (1, 1, 1), (3 / 2, 2, 5 / 2)] \end{matrix}$

Then, the FBWM mathematical model is formed. As an example, model calculations based on first expert opinion are presented below.

Min ξ

l₃ - 1.5u₁ ⩽ ξu₁	l₃ - 1.5u₁ ⩾ - ξu₁	m₃ - 2m₁ ⩽ ξu₁
m₃ - 2m₁ ⩾ - ξu₁	u₃ - 2.5l₁ ⩽ ξl₁	u₃ - 2.5l₁ ⩾ - ξl₁
l₃ - 0.67u₂ ⩽ ξu₂	l₃ - 0.67u₂ ⩾ - ξu₂	m₃ - 1m₂ ⩽ ξm₂
m₃ - 1m₂ ⩾ - ξm₂	u₃ - 1.5l₂ ⩽ ξl₂	u₃ - 1.5l₂ ⩾ - ξl₂
l₃ - 3.5u₄ ⩽ ξu₄	l₃ - 3.5u₄ ⩾ - ξu₄	m₃ - 4m₄ ⩽ ξm₄
m₃ - 4m₄ ⩾ - ξm₄	u₃ - 4.5l₄ ⩽ ξm₄	u₃ - 4.5l₄ ⩾ - ξm₄
l₃ - 0.67u₅ ⩽ ξu₅	l₃ - 0.67u₅ ⩾ - ξu₅	m₃ - 1m₅ ⩽ ξm₅
m₃ - 1m₅ ⩾ - ξm₅	u₃ - 1.5l₅ ⩽ ξl₅	u₃ - 1.5l₅ ⩾ - ξl₅
l₃ - 1.5u₆ ⩽ ξu₆	l₃ - 1.5u₆ ⩾ - ξu₆	m₃ - 2m₆ ⩽ ξm₆
m₃ - 2m₆ ⩾ - ξm₆	u₃ - 2.5l₆ ⩽ ξl₆	u₃ - 2.5l₆ ⩾ - ξl₆
l₃ - 2.5u₇ ⩽ ξu₇	l₃ - 2.5u₇ ⩾ - ξu₇	m₃ - 3u₇ ⩽ ξm₇
m₃ - 3u₇ ⩾ - ξm₇	u₃ - 3.5l₇ ⩽ ξl₇	u₃ - 3.5l₇ ⩾ - ξl₇
l₁ - 0.67u₆ ⩽ ξu₆	l₁ - 0.67u₆ ⩾ - ξu₆	m₁ - 1m₆ ⩽ ξm₆
m₁ - 1m₆ ⩾ - ξm₆	u₁ - 1.5l₆ ⩽ ξl₆	u₁ - 1.5l₆ ⩾ - ξl₆
l₂ - 2.5u₆ ⩽ ξu₆	l₂ - 2.5u₆ ⩾ - ξu₆	m₂ - 3m₆ ⩽ ξm₆ (11)
m₂ - 3m₆ ⩾ - ξm₆	u₂ - 3.5l₆ ⩽ ξl₆	u₂ - 3.5l₆ ⩾ - ξl₆
l₄ - 0.67u₆ ⩽ ξu₆	l₄ - 0.67u₆ ⩾ - ξu₆	m₄ - 1m₆ ⩽ ξm₆
m₄ - 1m₆ ⩾ - ξm₆	u₄ - 1.5l₆ ⩽ ξl₆	u₄ - 1.5l₆ ⩾ - ξl₆
l₅ - 1.5u₆ ⩽ ξu₆	l₅ - 1.5u₆ ⩾ - ξu₆	m₅ - 2m₆ ⩽ ξm₆
m₅ - 2m₆ ⩾ - ξm₆	u₅ - 2.5l₆ ⩽ ξl₆	u₅ - 2.5l₆ ⩾ - ξl₆
l₇ - 1.5u₆ ⩽ ξu₆	l₇ - 1.5u₆ ⩾ - ξu₆	m₇ - 2m₆ ⩽ ξm₆
m₇ - 2m₆ ⩾ - ξm₆	u₇ - 2.5l₆ ⩽ ξl₆	u₇ - 2.5l₆ ⩾ - ξl₆
$\begin{matrix} \frac{l_{1} + 4 m_{1} + u_{1}}{6} + \frac{l_{2} + 4 m_{2} + u_{2}}{6} + \frac{l_{3} + 4 m_{3} + u_{3}}{6} + \frac{l_{4} + 4 m_{4} + u_{4}}{6} + \frac{l_{5} + 4 m_{5} + u_{5}}{6} \\ + \frac{l_{6} + 4 m_{6} + u_{6}}{6} + \frac{l_{7} + 4 m_{7} + u_{7}}{6} = 1 \end{matrix}$
l₁ ⩽ m₁ l₂ ⩽ m₂ l₃ ⩽ m₃ l₄ ⩽ m₄ l₅ ⩽ m₅ l₆ ⩽ m₆ l₇ ⩽ m₇
m₁ ⩽ u₁ m₂ ⩽ u₂ m₃ ⩽ u₃ m₄ ⩽ u₄ m₅ ⩽ u₅ m₆ ⩽ u₆ m₇ ⩽ u₇
l₁ > 0 l₂ > 0 l₃ > 0 l₄ > 0 l₅ > 0 l₆ > 0 l₇ > 0 ξ ⩾ 0

These calculations are similarly applicable to other experts. The objective function is conversion of initial min-max model to the above nonlinear programming model. The above model is solved with Lingo software 17 and the final optimal weights are as follows: $\begin{matrix} \begin{matrix} ξ^{*} = (0 . 763, 0 . 763, 0 . 763) {\tilde{w}}^{*}_{C 1} \\ = (l_{1}, m_{1}, u_{1}) = (0.109, 0.118, 0.118) \\ {\tilde{w}}^{*}_{C 3} = (l_{3}, m_{3}, u_{3}) = (0.185, 0.185, 0.200) \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} {\tilde{w}}^{*}_{C 2} = (l_{2}, m_{2}, u_{2}) = (0.173, 0.252, 0.252) \\ {\tilde{w}}^{*}_{C 4} = (l_{2}, m_{2}, u_{2}) = (0.054, 0.057, 0.068) \\ {\tilde{w}}^{*}_{C 5} = (l_{3}, m_{3}, u_{3}) = (0.185, 0.185, 0.185) \\ {\tilde{w}}^{*}_{C 6} = (l_{2}, m_{2}, u_{2}) = (0.061, 0.067, 0.099) \\ {\tilde{w}}^{*}_{C 7} = (l_{3}, m_{3}, u_{3}) = (0.073, 0.083, 0.0106) \end{matrix} \end{matrix}$

The consistency index (CI) will be equal to: 0 . 763/8.04 = 0.095 < 0.1 which shows the high rate of consistency [28]. Similarly, the final optimal weights of the seven factors for each expert are presented in Table 4.

Table 4

Final fuzzy weights of decision-making criteria in the form of triangular fuzzy numbers

Criteria	E1	E2	E3	Final optimal weight (w_j)
C1	(0.109, 0.118, 0.118)	(0.146, 0.146, 0.180)	(0.100, 0.100, 0.133)	(0.118, 0.121, 0.144)
C2	(0.173, 0.252, 0.252)	(0.222, 0.222, 0.284)	(0.100, 0.100, 0.150)	(0.165, 0.191, 0.229)
C3	(0.185, 0.185, 0.200)	(0.146, 0.146, 0.173)	(0.167, 0.167, 0.200)	(0.166, 0.166, 0.191)
C4	(0.054, 0.057, 0.068)	(0.059, 0.060, 0.076)	(0.100, 0.100, 0.133)	(0.071, 0.072, 0.092)
C5	(0.185, 0.185, 0.185)	(0.142, 0.060, 0.076)	(0.150, 0.150, 0.200)	(0.159, 0.132, 0.154)
C6	(0.061, 0.067, 0.099)	(0.093, 0.093, 0.121)	(0.067, 0.067, 0.100)	(0.074, 0.076, 0.107)
C7	(0.073, 0.083, 0.0106)	(0.090, 0.094, 0.114)	(0.167, 0.167, 0.200)	(0.110, 0.115, 0.108)
Optimal value (ξ^*)	0 . 763	0 . 559	0 . 499	–

According to Table 4 the final weights of the decision criteria are expressed in the form of triangular fuzzy numbers as follows: $\begin{matrix} \begin{matrix} {\tilde{w}}^{*}_{C 1} = (0.118, 0.121, 0.144) \\ {\tilde{w}}^{*}_{C 2} = (0.166, 0.166, 0.191) \\ {\tilde{w}}^{*}_{C 3} = (0.165, 0.191, 0.229) \\ {\tilde{w}}^{*}_{C 4} = (0.071, 0.072, 0.092) \\ {\tilde{w}}^{*}_{C 5} = (0.159, 0.132, 0.154) \\ {\tilde{w}}^{*}_{C 6} = (0.074, 0.076, 0.107) \\ {\tilde{w}}^{*}_{C 7} = (0.110, 0.115, 0.108) \end{matrix} \end{matrix}$

The final values of weights are obtained using the definition of equation (8) as follows: $\begin{matrix} {w^{*}}_{C 1} = 0.125, {w^{*}}_{C 2} = 0.193, {w^{*}}_{C 3} = 0.170 \\ {w^{*}}_{C 4} = 0.075 {w^{*}}_{C 5} = 0.140 {w^{*}}_{C 6} = 0.081 \\ , {w^{*}}_{C 7} = 0.113 \end{matrix}$

The results show that quality C2 had the highest weight and delivery time C3 is ranked second. The next ranks were dedicated to trust C5, price C1, accountability C7, skilled manpower C6, and no surplus production C4, respectively. $\begin{matrix} {w^{*}}_{C 2} > {w^{*}}_{C 3} > {w^{*}}_{C 5} > \\ {w^{*}}_{C 1} > {w^{*}}_{C 7} > {w^{*}}_{C 6} > {w^{*}}_{C 4} \end{matrix}$

Step 4) Choosing the best alternative by the FGARM

In this section, the proposed approach FGRA is used to rate the existing suppliers. The linguistic spectrum of Table 5 is used to evaluate the 5 suppliers based on the seven research criteria.

Table 5

Linguistic spectrum variables and equivalent triangular fuzzy numbers for ranking suppliers

Linguistic variable (Importance)	Fuzzy Triangular Number
Very low (VL)	(0,1,2)
Low (L)	(1,2,3)
Medium low (ML)	(2,3.5,5)
Medium (M)	(4,5,6)
Medium High (MH)	(5,6.5,8)
High (H)	(7,8,9)
Very High (VH)	(8,9,10)

In this step, existing suppliers are assessed on the basis of the seven criteria by experts that the obtained evaluation matrix can be observed in Table 6.

Table 6

Supplier’s assessment matrix taking into account existing criteria

Supplier	Expert	Criteria
		C1	C2	C3	C4	C5	C6	C7
S1	E1	M	VL	M	ML	ML	ML	VL
	E2	ML	VH	MH	MH	M	L	VL
	E3	L	H	VL	MH	VH	M	L
S2	E1	M	MH	H	VH	ML	H	M
	E2	MH	M	MH	H	L	H	M
	E3	H	ML	VL	L	H	ML	ML
S3	E1	M	ML	VH	L	M	ML	M
	E2	H	M	H	L	M	ML	L
	E3	VH	H	H	M	M	L	VL
S4	E1	M	VH	ML	MH	VL	M	M
	E2	H	M	ML	VL	M	H	M
	E3	H	H	ML	L	MH	M	M
S5	E1	VH	VH	ML	L	L	M	ML
	E2	MH	M	MH	M	M	MH	MH
	E3	ML	ML	VH	ML	L	ML	M

According to Table 6, the average of experts’ opinions for all suppliers is calculated according to Table 8. For example, Average(C1:S1)=((4,5,6)+(2,3.5,5)+(1,2,3)) / 3)=(2.333,3.500,4.667).

For example, the proposed FGRA is applied to existing suppliers using the evaluation matrix of Table 7 considering the first two criteria of price (C1) and quality (C2).

Table 7

Supplier evaluation matrix by converting linguistic variables into equivalent fuzzy numbers

Supplier	Expert	C1	C2	C3	C4	C5	C6	C7
S1	E1	(4,5,6)	(0,1,2)	(4,5,6)	(2,3.5,5)	(2,3.5,5)	(2,3.5,5)	(0,1,2)
	E2	(2,3.5,5)	(8,9,10)	(5,6.5,8)	(5,6.5,8)	(4,5,6)	(1,2,3)	(0,1,2)
	E3	(1,2,3)	(7,8,9)	(0,1,2)	(5,6.5,8)	(8,9,10)	(4,5,6)	(1,2,3)
S2	E1	(4,5,6)	(5,6.5,8)	(7,8,9)	(8,9,10)	(2,3.5,5)	(7,8,9)	(4,5,6)
	E2	(5,6.5,8)	(4,5,6)	(5,6.5,8)	(7,8,9)	(1,2,3)	(7,8,9)	(4,5,6)
	E3	(7,8,9)	(2,3.5,5)	(0,1,2)	(1,2,3)	(7,8,9)	(2,3.5,5)	(2,3.5,5)
S3	E1	(4,5,6)	(2,3.5,5)	(8,9,10)	(1,2,3)	(4,5,6)	(2,3.5,5)	(4,5,6)
	E2	(7,8,9)	(4,5,6)	(7,8,9)	(1,2,3)	(4,5,6)	(2,3.5,5)	(1,2,3)
	E3	(8,9,10)	(7,8,9)	(7,8,9)	(4,5,6)	(4,5,6)	(1,2,3)	(0,1,2)
S4	E1	(4,5,6)	(8,9,10)	(2,3.5,5)	(5,6.5,8)	(0,1,2)	(4,5,6)	(4,5,6)
	E2	(7,8,9)	(4,5,6)	(2,3.5,5)	(0,1,2)	(4,5,6)	(7,8,9)	(4,5,6)
	E3	(7,8,9)	(7,8,9)	(2,3.5,5)	(1,2,3)	(5,6.5,8)	(4,5,6)	(4,5,6)
S5	E1	(8,9,10)	(8,9,10)	(2,3.5,5)	(1,2,3)	(1,2,3)	(4,5,6)	(2,3.5,5)
	E2	(5,6.5,8)	(4,5,6)	(5,6.5,8)	(4,5,6)	(4,5,6)	(5,6.5,8)	(5,6.5,8)
	E3	(2,3.5,5)	(2,3.5,5)	(8,9,10)	(2,3.5,5)	(1,2,3)	(2,3.5,5)	(4,5,6)

Table 8

Average matrix of experts’ opinions

Criteria	Suppliers
S1	S2	S3	S4	S5
C1	(2.333, 3.500, 4.667)	(5.333, 6.500, 7.667)	(6.333, 7.333, 8.333)	(6.000, 7.000, 8.000)	(4.667, 6.000, 7.333)
C2	(5.000, 6.000, 7.000)	(3.667, 5.000, 6.333)	(4.333, 5.500, 6.667)	(6.333, 7.333, 8.333)	(4.667, 5.833, 7.000)
C3	(3.000, 4.167, 5.333)	(4.000, 5.167, 6.333)	(7.333, 8.333, 9.333)	(2.000, 3.500, 5.000)	(5.000, 6.333, 7.667)
C4	(4.000, 5.500, 7.000)	(5.333, 6.333, 7.333)	(2.000, 3.000, 4.000)	(2.000, 3.167, 4.333)	(2.333, 3.500, 4.667)
C5	(4.667, 5.833, 7.000)	(3.333, 4.500, 5.667)	(4.000, 5.000, 6.000)	(3.000, 4.167, 5.333)	(2.000, 3.000, 4.000)
C6	(2.333, 3.500, 4.667)	(5.333, 6.500, 7.667)	(1.667, 3.000, 4.333)	(5.000, 6.000, 7.000)	(3.667, 5.000, 6.333)
C7	(0.333, 1.333, 2.333)	(3.333, 4.500, 5.667)	(1.667, 2.667, 3.667)	(4.000, 5.000, 6.000)	(3.667, 5.000, 6.333)

$\begin{matrix} u_{j}^{+} = max_{i} {8.333, 8.333} \end{matrix}$

Then the normal matrix of Table 7 is obtained as follows: $\begin{matrix} \tilde{R} = [\begin{matrix} 0.280 & 0.420 & 0.560 & 0.600 & 0.720 & 0.840 \\ 0.640 & 0.780 & 0.920 & 0.440 & 0.600 & 0.760 \\ 0.760 & 0.880 & 1.000 & 0.520 & 0.660 & 0.800 \\ 0.720 & 0.840 & 0.960 & 0.760 & 0.880 & 1.000 \\ 0.560 & 0.720 & 0.880 & 0.560 & 0.700 & 0.840 \\ 0.760 & 0.880 & 1.000 & 0.760 & 0.880 & 1.000 \end{matrix}] \end{matrix}$

Considering the above normal matrix and equation 6, we have the reference set as: $\begin{matrix} \tilde{R} = [(0.760, 0.880, 1.000), (0.760, 0.880, 1.000)] \end{matrix}$

Then, the distance between the reference value and the desired value is obtained from the Equation 7: $\begin{matrix} δ_{11} = \sqrt{\frac{1}{3} [{(0.760 - 0.280)}^{2} + {(0.880 - 0.420)}^{2} + {(1.000 - 0.560)}^{2}]} = 0.460 \end{matrix}$

Similarly, the distance matrix (δ_ij) is obtained: $\begin{matrix} δ_{ij} = [\begin{matrix} 0.460 & 0.160 & 0.447 & 0.127 & 0.000 & 0.392 & 0.598 \\ 0.101 & 0.282 & 0.340 & 0.000 & 0.190 & 0.000 & 0.098 \\ 0.000 & 0.220 & 0.000 & 0.455 & 0.121 & 0.457 & 0.387 \\ 0.040 & 0.000 & 0.520 & 0.432 & 0.238 & 0.068 & 0.031 \\ 0.163 & 0.180 & 0.216 & 0.387 & 0.405 & 0.197 & 0.030 \end{matrix}] \end{matrix}$

The maximum and minimum values of the distance matrix are respectively δ_max = 0.598 and δ_min = 0.000. Using equation 8, gray relational coefficients matrix (ξ_ij) are obtained as: $\begin{matrix} ξ_{ij} = [\begin{matrix} 0.394 & 0.652 & 0.401 & 0.703 & 1.000 & 0.433 & 0.334 \\ 0.748 & 0.515 & 0.468 & 1.000 & 0.611 & 1.000 & 0.754 \\ 0.999 & 0.576 & 1.000 & 0.397 & 0.713 & 0.395 & 0.436 \\ 0.883 & 0.999 & 0.365 & 0.409 & 0.557 & 0.816 & 0.908 \\ 0.647 & 0.624 & 0.581 & 0.436 & 0.425 & 0.603 & 0.908 \end{matrix}] \end{matrix}$

Then, according to equation 10 and with regard to the final weights of seven criteria the gray relational grade (γ_i) of the first supplier will be achieved: $\begin{matrix} γ_{1} = (0.394 * 0.125) + (0.652 * 0.193) \\ + (0.401 * 0.170) + (0.703 * 0.075) \\ + (1.000 * 0.140) + (0.433 * 0.081) \\ + (0.334 * 0.113) = 0.509 \end{matrix}$

Similarly, gray relational grades for all suppliers are obtained as follows: $\begin{matrix} γ_{1} = 0.509, γ_{2} = 0.599, γ_{3} = 0.617, γ_{4} = 0.643, \\ γ_{5} = 0.544 \end{matrix}$

According to the results, the final ranking of suppliers is:

S4 > S3 > S2 > S5 > S1. Finally, the S4 supplier is known as the most preferable supplier.

Step 5) Sensitivity analysis

Sensitivity analysis is a practice by which the amount and distribution of input data is determined with the greatest impact on the output of the model. In this section, sensitivity analysis is presented by changing the weight of 7 criteria and the results of the suppliers’ rating. For this purpose, the base scenario (BSC), including the criteria weight values from the FBWM of current research, is given in the first line of Table 8. Six new scenarios (SC1 to SC6) are also designed by the researcher with minor and controlled changes in the values of the criteria, which are shown in the second to seventh lines of Table 9.

Table 9

Weight of supplier selection criteria according to different scenarios

Scenario	Criteria
	C1	C2	C3	C4	C5	C6	C7
BSC	.1246	.1932	.1702	.0754	.1399	.0805	.1128
SC1	.2846	.1662	.1432	.0484	.1129	.0505	.0858
SC2	.0976	.1662	.3502	.0454	.1129	.0505	.0858
SC3	.0976	.1662	.1432	.0484	.2999	.0505	.0858
SC4	.0976	.1662	.1432	.0484	.1129	.0505	.2728
SC5	.0976	.1662	.1432	.0484	.1129	.2405	.0858
SC6	.0976	.1662	.1432	.2554	.1129	.0505	.0858

These changes are in such a way that in each scenario only one criterion is increased and other criteria remain unchanged. In the first scenario, C2 criterion (time) has been increased from 0.976 to.2846. In the second scenario, the C3 criterion (delivery time) has increased from 0.1432 to 0.3502. In the third scenario, C5 criterion (confidence) has increased from 0.1129 to 0.2999. In the fourth scenario, C7 (responsiveness) is increased from 0.828 to 0.2728. In the fifth scenario, C6 (skilled manpower) increased from 0.0505 to 0.2405. Finally, in the sixth scenario, C4 (lack of surplus production) increased from 0.484 to 0.2554. Meanwhile, the criterion C2 (quality) did not change in any scenario.

Numbers in Table 10 show a change in the priority of suppliers’ rankings in scenarios 1 to 6 due to minor changes in the weight of the criteria.

Table 10

Priority score of suppliers considering different scenarios

Supplier	Scenarios
	BSC	SC 1	SC 2	SC 3	SC 4	SC 5	SC 6
S1	.5068	.4766	.4754	.5899	.4653	.4839	.5387
S2	.5988	.6011	.5443	.5755	.6023	.6482	.6558
S3	.6166	.6815	.6911	.6279	.5761	.5686	.5644
S4	.6420	.6739	.5706	.6129	.6784	.6613	.5798
S5	.5433	.5503	.5385	.5086	.5990	.5421	.5081

According to Table 10, the final ranking of suppliers in each scenario will be as shown in Table 11.

Table 11

Suppliers’ ranking by considering different scenarios

Supplier	Scenarios
	BSC	SC 1	SC 2	SC 3	SC 4	SC 5	SC 6
S1	5	5	5	3	5	5	4
S2	3	3	3	4	2	2	1
S3	2	1	1	1	4	3	3
S4	1	2	2	2	1	1	2
S5	4	4	4	5	3	4	5

As Table 11 shows, a small change in the weight of the criteria leads to a change in the ranking of suppliers, which indicates the sensitivity of the results to changes in independent variables. For instance, in Scenario 1, a small change in the weight of the first criterion (price) has led to a change in the rankings of suppliers 3 and 4.

The results showed that the first, second, and fifth scenarios caused the least turnover in suppliers’ positions. Therefore, the objective function is not highly sensitive to change in the criteria of existing scenarios. The fourth scenario relating to the responsiveness criterion, led to a change in the rating of 3 suppliers has a moderate sensitivity. Finally, the third and sixth scenarios made the most change in the suppliers’ ranking. As a result, the objective function showed the highest sensitivity to the criteria of trust and lack of surplus production.

The hospital is free to choose its entire demand from one or more suppliers with high priorities. In the normal case and the zero scenarios (BSC) for example, the fourth, third and second suppliers are prioritized respectively. Either one or a combination of suppliers is selected to meet the demand depending on the policy of the hospital managers. Also, it is possible to combine suppliers in each of the scenarios.

5 Summary and conclusion

Appropriate Supplier(s) selection is not an easy and single criterion (such as cost) based task and a variety of criteria must be considered for this purpose. In addition, the more complexity of this choice stems from the fact that each supplier meets some of the buyer’s criteria. Therefore, choosing from them requires a structured and systematic approach. The development of food production networks, as well as the diversity of producers, the variety of supply methods and the existence of uncertainties such as: uncertainty in the supply of main production inputs, market demand and prices, has made the food supply chain more complicated. Due to the high importance of hospital food supply, significant changes in the food industry and the key role of suppliers in improving this chain, this study seeks to identify and rank the most important criteria for evaluating and selecting food suppliers in hospitals.

The aim of this study was to introduce and use the fuzzy best-worst method as an approach to weigh the criteria for selecting hospital food suppliers in uncertainty conditions considering qualitative judgments and to use fuzzy gray relational analysis method for suppliers’ rating. First, among the criteria extracted from scientific sources, 7 criteria including: price, quality, delivery time, no surplus production, trust, skilled manpower, and accountability were selected by 5 experts. In the second step, by performing the fuzzy best-worst method, 7 criteria were evaluated and the weight of each of these criteria was obtained. Based on the results, the order of weight of the criteria for the hospital food supplier’s selection was determined respectively, quality, delivery time, trust, price, accountability, skilled manpower, and no surplus production in descending order. Then, by implementing the fuzzy gray relational analysis and based on the obtained weights, the 5 existing suppliers were ranked and the supplier No. 4 was ranked first. Suppliers No. 3, 2, 5, and 1 got the other ranks successively. In order to evaluate the effect of changes on the value of each criterion, sensitivity analysis was performed. The results showed that the proposed approach is influenced by the change in criteria. In other words, the expert’s opinion has a decisive effect on the ranking of suppliers. All scenarios changed the ranking of suppliers. Scenarios 3 and 6 had the greatest impact on ranking and changing the rankings of all suppliers. Also, most of the shifts are related to suppliers in the first and second priority. As a result, the experts’ selection and opinions are highly sensitive and must be done more carefully. Compared to researches in the food industry field, determining role of criteria such as quality, delivery time, trust, and price in selecting suppliers was confirmed. Also, compared to similar studies such as [42, 43] which used fuzzy hierarchical analysis methods and fuzzy TOPSIS, it is possible to facilitate the implementation of the method and faster access to the final answer.

The proposed approach of the present study can be used to decide on the hospital food supplier’s selection. For further research, it is recommended that the proposed approach be used in the decision-making of industrial and service organizations. Considering competitive and cooperation situations, and a combination of them can include a variety of scenarios of different types of coalitions at different levels of the supply chain. Also, other weighting techniques can be used to weigh the criteria on similar issues. For further research, the results of this study can be compared with other decision-making methods such as TOPSIS 1 , MOORA 2 , VIKOR 3 , ANP 4 , so forth. It is suggested that different value functions be used in group decision making to evaluate and rate suppliers.

6 Conclusion

A combination of fuzzy best-worst and gray relational analysis has been used to increase the efficiency of supplier selection in the proposed approach of the current research. Taking into account qualitative judgments, in order to evaluate the selection criteria of hospital food suppliers in uncertainty conditions, the best-worst fuzzy method was used. Also, fuzzy gray relational analysis method has been used to rank suppliers. By implementing the best-worst fuzzy method, the weight of the selection criteria of hospital food suppliers was determined as quality, delivery time, trust, price, responsiveness, skilled manpower and lack of surplus production in descending order. The sensitivity analysis showed that the results are affected by criteria change. In other words, the experts’ opinion has a decisive effect on the ranking of suppliers. As a result, the preferences and opinions of experts are sensitive and should be done more carefully. Furthermore, compared to the previous researches, the determining role of quality, delivery time, trust and price criteria was confirmed in the supplier selection field of food industry. Moreover, it is possible to facilitate the implementation of the proposed method and access the final answer faster compared to the conventional methods. By selecting new experts and extracting related criteria, this approach can be extended to other supply chains.

Footnotes

Acknowledgments

The authors would like to thank the editors and anonymous reviewers for their valuable and constructive comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article.

Additional information

The Technique for Order of Preference by Similarity to Ideal Solution

Multi-Objective Optimization on the Basis of Ratio Analysis

Vlsekriterijumska Optimizacija I Kompromisno Resenje

Analytic Network Process

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