An Interval-Valued Pythagorean Fuzzy AHP and COPRAS Hybrid Methods for the Supplier Selection Problem

Abstract

Companies must be able to identify their suppliers appropriately and effectively in order to survive in the competitive market conditions. In order to fulfill and surpass the expectations of the consumers and clients, companies need to interact with the relevant suppliers. It is a tough manner for companies to select the best supplier from a large number of relevant alternatives. The selection process of the appropriate supplier involves multiple interacting and competing factors. Generally, the selection process and its results cause a waste of time and money. For this purpose, MCDM methodologies are utilized to manage this complex process efficiently. MCDMs allows for consistent and accurate decision-making as well as the selection of the most appropriate supplier. MCDM is one the most preferred tool to select the best alternative under the conflicting and competitive criteria when the evaluations are made in crisp numbers. Therefore, MCDM methods are preferred in various applications in academia and real life. However, the evaluations could not be always possible with crisp numbers, especially in vague environments or evaluations needs qualitative data. This study is one of the first to combine the AHP and COPRAS supplier selection methods with interval-valued Pythagorean fuzzy (IPF) logic. The effectiveness of these IPF-AHP and IPF-COPRAS evaluations for the supplier selection problem is compared and examined. The experimental results of case scenarios show that IPF is an effective way to apply in decision-making applications. In addition, sensitivity analysis is conducted to evaluate the proposed methodologies. According to sensitivity analysis, the IPF-AHP and IPF-COPRAS be able to illustrate the effects of small changings in criteria weights. Therefore, companies can use the IPF-AHP and IPF-COPRAS to assist their decision-makers in identifying and selecting the best suppliers.

1 Introduction

Businesses have had to make tough choices to enhance organizational structures, reduce expenses, and produce high-quality items in a competitive market. Evaluating various elements, exactly comparing alternatives, and making consistent and successful judgments have become critical and difficult for enterprises. This is one of the first studies to combine AHP and COPRAS supplier selection techniques with interval-valued Pythagorean fuzzy (IPF) logic. Comparing and analyzing the effectiveness of these IPF-AHP and IPF-COPRAS evaluations for the supplier selection problem. The efficient use of a company’s limited resources such as human, financial, and intellectual properties is contingent on making the optimal choice from the alternatives available. MCDM is a structured process that helps companies select the best supplier by considering multiple criteria. It offers benefits such as improved decision-making, increased transparency, better risk management, increased efficiency, and better supplier relationships. MCDM helps companies make informed decisions, reduce time and resources required for supplier selection, and build better relationships with suppliers by identifying and managing risks associated with supplier selection. In this direction, procedures such as MCDM are utilized to select the most appropriate solution for the objective while considering competing criteria.

In supplier selection, there may be criteria that contradict each other, such as when a company wants to select a supplier who offers the lowest price while also ensuring that the supplier has a good reputation for quality. In this case, price and quality are diametrically opposed criteria. MCDM techniques provide a structured approach to supplier selection decision-making, taking into account multiple criteria and ensuring that all are taken into account. Companies can prioritize the most important criteria and determine their appropriate weighting in the decision-making process by breaking them down into smaller, manageable components. MCDM techniques are mathematical models that aid decision-making when multiple contradicting criteria are applied to analyze feasible solutions [1]. They facilitate accurate decision-making in fields where identifying the best alternative is challenging [2]. MCDM-based methods assist the selection of the optimal alternative, which is determined by examining the weights associated with each criterion.

Selecting a supplier is one of the most crucial business decisions. Quantitative and qualitative factors play a role in the strategic importance of supplier selection for numerous businesses. Since a poor supplier selection could reduce supply chain efficacy and result in a loss of competitive advantage, it is important to carefully select suppliers. Thus, selecting the most qualified candidate from a pool of candidates is a difficult multi-criteria decision-making process.

Businesses must choose appropriate suppliers to maintain production and meet client needs. Supply chains have challenges in obtaining commodities from the right source at the right time and at the lowest price. Decision-makers must choose the right supplier to manage the supply chain from production to consumption. A reliable supplier also helps organizations meet their manufacturing goals. The right supplier improves production flexibility and quality. Thus, consumer satisfaction, purchasing costs, and the company’s competitiveness can improve. Supplier-supplied raw materials and production capacity determine enterprise product quality [3]. As a result, organizations have prioritized evaluating various suppliers and choosing the best one based on predetermined criteria [4].

Selecting the wrong supplier can cost the organization money, time, clients, and reputation. Thus, providers should be selected using scientific and required criteria. Strategic decisions that meet the goal grow the company.

The primary objective of supplier selection is to select a supplier that is compatible with the organization and provides the greatest value [3]. In this instance, MCDM techniques are used to determine the most suitable alternative for achieving the goal, taking into account competing criteria. This study investigated the MCDM Problem of selecting the best supplier. Before analyzing potential suppliers, the necessary criteria were established.

Weights for these criteria were determined using the AHP approach. Following that, each alternative supplier was evaluated quantitatively and qualitatively. The COPRAS approach was used to determine the best acceptable supplier alternative throughout the option evaluation process. The crisp numbers may not reflect the decision maker's judgments accurately. For example, “Very Strongly Important (VSI)” is shown by 7 on a linguistic scale from integer numbers between 1 and 9. However, VSI could be defined more accurately using a triangular fuzzy number that assigns VSI around 7 such as (6.5, 7.0, 7.5).

This definition better illustrates decision-makers’ judgments. To manage the lack of knowledge and uncertain data regarding decision-making, most MCDMs uses fuzzy logic models such as type-2 fuzzy, intuitionistic fuzzy, Pythagorean fuzzy, and neutrosophic fuzzy. The first fuzzy set applications represented membership functions as system complexity. After the first fuzzy set applications, fuzzy logic is extended to type-n fuzzy ideas (Zadeh, 1975). Atanossov (1999) introduces intuitionistic type-2 fuzzy (IFS2) sets with membership and non-membership functions. Yager (2013) extended IFT2 via Pythagorean Fuzzy Sets (PFS). PFS extends membership and non-membership functions to help decision-makers handle uncertainty better than fuzzy sets [5]. Thus, this study compares PFS-based MCDM algorithms for imprecise information. This study develops an interval-valued Pythagorean Fuzzy AHP (IPF-AHP) and IPF-COPRAS to pick the best supplier among multiple alternatives under conflicting criteria.

The remaining sections of the paper are organized as follows: In Sect. 2, a literature review of the different versions of AHP and COPRAS, Pythagorean fuzzy sets, and supplier selection is provided. Section 3 explains fuzzy sets with Pythagorean coefficients. The IPF-AHP and IPF-COPRAS procedures are described in Sect. 4. In Sect. 5, applications of the IPF-AHP and IPF-COPRAS are presented with a sensitivity analysis. Section 6 concludes this paper with its conclusions and recommendations for further research.

2 Literature Review

Numerous studies on supplier selection criteria, supplier selection, and evaluation have been published in the literature. Alkahtani et al. [6] developed an MCDM tool that requires answering two questions in order to select the best supplier for a business. To begin, the queries “What criteria should be used to evaluate each supplier?” and “How should the best supplier be selected?” were addressed. A literature review was undertaken on these two challenges, and many viable approaches were provided. Madic et al. [7] examined the COPRAS technique for supplier selection again. A construction and agricultural tools manufacturer employed this method. Results were also compared to previous studies. Rouyendegh et al. [8] investigated green supplier selection (GSS) for sustainability. They choose the finest green provider using IFTOPSIS. Wang Chen et al. [9] recommended fuzzy MCDM for green supplier selection and evaluation. They presented an economic and environmental approach. A case study established the practicality and importance of the approach. Percin [10] adapted MCDM to the cyclical supplier selection (CSS) problem of a cement company. He proposed a CSS strategy employing interval-valued intuitionistic fuzzy sets (IVIFS) with AHP and COPRAS. In addition, AHP and COPRAS were used in different application areas such as evaluating the website quality of banks by defining weights of evaluation criteria related to the quality of bank websites [11]. Several MCDM techniques are used to define sustainable supplier selection, such as FUCOM [12, 13]; therefore, the MCDM methodology for supplier selection has been published in the literature. The table below provides a summary of current research. Analyzing the studies reveals that the approaches are used independently or combination with one another.

In the literature, there are many studies in which AHP and COPRAS have been integrated to into supply chain management in terms of vendor, supplier, or location decisions. For example, Erdebilli et al. (2023) suggest integrating AHP-COPRAS for vendor selection in SCM [14]. In that study, firstly, the vendor selection criteria are established, and then the relative relevance of the various criteria is evaluated using the AHP. The next step is to assess the potential providers and choose the best vendor using the COPRAS approach [15]. However, in real-time, there may not always be precise information available to evaluate the criteria and alternatives. For this reason, researchers use fuzzy logic in the MCDM technique to handle uncertainty and imprecision in decision-making [16]. Fuzzy logic allows decision-makers to express judgments or preferences in linguistic terms rather than precise numerical values, which are often more realistic and practical. Fuzzy MCDM methods enable more realistic results in solving decision-making problems. Efficient energy use is crucial for economic development, but excessive fossil fuel use harms the environment [17]. Renewable energy emits low greenhouse gases, leading countries to increase usage. Sectoral specific and asymmetric foreign exchange volatility effects affect crude oil, coal, electricity, and petroleum products. A three-dimensional hexagonal structure of nano-inclusions demonstrated better wear resistance and a reduced friction coefficient in polymer films [18, 19]. Therefore, this proposed study provides a decision-making model for supplier selection problems through integrated AHP-COPRAS, which is extended by Pythagorean fuzzy logic. PFL relaxes the requirement that the sum of squares representing an element's membership degree and non-membership degree cannot be larger than 1. As a result, modeling uncertainty and ambiguity in decision-making processes is now more flexible as shown in Table 1. Therefore, a new approach is applied to select the most appropriate supplier for the entire supply chain.

Table 1 Literature review of supplier selection with MCDM methods

The following criteria are presented in the literature review:

C1: Cost/price	C12: Environmental
C2: Quality	C13: Geographical location
C3: Lead/delivery time	C14: Sustainability
C4: Technology	C15: Performance
C5: Service	C16: Reputation
C6: Flexibility	C17: Cooperation
C7: Distance	C18: Green design
C8: Variety	C19: Green manufacturing system
C9: Technical competence/capability	C20: Management system
C10: Economic	C21: Other criteria
C11: Social

3 Preliminaries and Methodology

In this section, the AHP and COPRAS procedures, which are both MCDM approaches, were used to evaluate the supplier selection alternatives available to a company. MCDM methodologies offer a structured approach to decision-making, considering all relevant factors, identifying important criteria, weighing them appropriately, identifying trade-offs, and reducing personal biases. However, MCDM can be complex, time-consuming, require significant data, be sensitive to criteria and weights, and may not always produce clear or unambiguous results. Brunelli [46] and Kulakowski [47] describe the phases of the generic AHP method. Alinezhad and Kahlili [15] present the phases and applications of the COPRAS method. In addition, approach-specific details have been provided first and foremost. This information was used to submit an application to identify the most qualified service provider. The interval-valued Pythagorean fuzzy AHP (IPF-AHP) and interval-valued Pythagorean fuzzy (IPF-COPRAS) techniques are combined with the following interval-valued Pythagorean fuzzy sets prerequisites:

3.1 Preliminaries of Interval-Valued Pythagorean Fuzzy Sets

Yager (2013) defines PFS as the sum of membership degrees, ${\mu }_{\widetilde{p}}\left(x\right)$, and non-membership degrees, ${v}_{\widetilde{p}}\left(x\right)$, of a function $\widetilde{p}(x)$ might be greater than 1; however, the sum of squares of the ${\mu }_{\widetilde{P}}(x)$ and ${v}_{\widetilde{p}}(x)$ could be less than or equal to 1 [5].

Definition 1

A PFS, $\widetilde{p}$, is an object that has the form (Yager 2013):

$$\widetilde{{\widetilde{p}}_{1}\oplus {\widetilde{p}}_{2}}\widetilde{=} \left\{ \langle x, {\mu }_{\widetilde{P}}\left(x\right), { v}_{\widetilde{P}}\left(x\right)\rangle ;x\in X\right\},$$

where ${\mu }_{\widetilde{P}}\left(x\right):\to [\mathrm{0,1}]$ and ${v}_{\widetilde{P}}\left(x\right):\to [\mathrm{0,1}]$, $x\in X$ and $\forall x\in X$ holds that

$$0\le {\mu }_{\widetilde{P}}\left(x\right)+{v}_{\widetilde{P}}\left(x\right)\le 1.$$

The degree of hesitancy condition is defined as

$${\pi }_{\widetilde{P}}{\left(x\right)}^{2}=1-{\mu }_{\widetilde{P}}{\left(x\right)}^{2}+{v}_{\widetilde{P}}{\left(x\right)}^{2}.$$

Definition 2

Let $\widetilde{{p}_{1}}=\langle {\mu }_{1}, {v}_{1}\rangle$ and ${\widetilde{p}}_{2}=\langle {\mu }_{2}, {v}_{2}\rangle$ be two PFNs and summation and multiplication of two PFN are

$${\widetilde{p}}_{1}\oplus {\widetilde{p}}_{2}=\left(\sqrt{{\mu }_{1}^{2}+{\mu }_{2}^{2}-{\mu }_{1}^{2}{\mu }_{2}^{2}},{v}_{1}{v}_{2}\right),$$

$${\widetilde{p}}_{1}\oplus {\widetilde{p}}_{2}=\left({\mu }_{1}{\mu }_{2}, \sqrt{{v}_{1}^{2}+{v}_{2}^{2}-{v}_{1}^{2}{v}_{2}^{2}},\right).$$

Definition 3

Let $\widetilde{{P}_{1}}=\langle {\mu }_{1}, {v}_{1}\rangle$ be a PFNs and $\uplambda > 0$ then operations could be defined as

$${\uplambda \widetilde{p}}_{1}=\left(\sqrt{1-{\left(1-{\mu }_{1}^{2}\right)}^{\uplambda }},{v}_{1}^{\uplambda }\right),$$

$${\widetilde{p}}_{1}^{\uplambda }=\left({\mu }_{1}^{2},\sqrt{{{1-(1-v}_{1}^{2})}^{\uplambda }}\right).$$

This study is operating in interval-valued fuzzy space; therefore, the hesitation degree should be extended for the lower and upper points of $\widetilde{P}$ as follows:

Definition 4

Let $\widetilde{P}=\langle \left[{\upmu }_{L},{\upmu }_{U}\right],\left[{v}_{L},{v}_{U}\right]\rangle$ be an interval-valued PFN and hesitancy degree of lower and upper points of $\widetilde{P}$, ${\uppi }_{L}\mathrm{and}{\uppi }_{U}$, respectively, which are calculated as follows:

$${\pi }_{l}^{2}=1-\left({\mu }_{U}^{2}+{v}_{U}^{2}\right),$$

$${\pi }_{U}^{2}=1-\left({\mu }_{L}^{2}+{v}_{L}^{2}\right).$$

The decision-makers evaluate the alternatives and criteria using the linguistic scale. Table 2 displays the linguistic scale proposed by Karasan et al. (2019) for IPFNs.

Table 2 Linguistic scale for performance weighting for IPFV

In this study, decision-makers use the linguistic terms in Table 2 to select the best supplier from a group of PFS-evaluated options. The mathematical explanation of the supplier selection problem is defined as a set of decision-makers, $DM=\{D{M}_{1}\dots D{M}_{k}\}$, evaluate a set of alternatives, $A=\{{A}_{1}\dots {A}_{n}\}$, under the set of criteria, $C=\{{C}_{1}\dots {C}_{m}\}$.The opinion of the kth Decision-maker, ${o}_{ij}^{k}$, regarding the ith alternative under the jth criteria is defined as ${o}_{ij}^{k}=\langle \left[{\mu }_{{L}_{ij}}^{k},{\mu }_{{U}_{ij}}^{k}\right], \left[{v}_{{L}_{ij}}^{k},{v}_{{U}_{ij}}^{k}\right]\rangle$ and weight vector of decision-makers is defined as ${w}_{DM}=\{{w}_{D{M}_{1}}\dots {w}_{D{M}_{k}}\}$ based on the IPFV. The membership degree of ${A}_{i}$ under ${C}_{j}$ given by $D{M}_{k}$ is represented as $\left[{\mu }_{{L}_{ij}}^{k},{\mu }_{{U}_{ij}}^{k}\right]$. The membership degree of ${A}_{i}$ under ${C}_{j}$ given by $D{M}_{k}$ is represented as $\left[{v}_{{L}_{ij}}^{k},{v}_{{U}_{ij}}^{k}\right]$.

3.2 Proposed IPF-AHP Method

The steps of the proposed IPF-AHP are derived from Karasan (2019) and Ayyildiz and Taskin Gumus (2021) as follows:

Step 1. Create an IPF Decision matrix for decision-makers’ opinions.

Step 2. Applying Eqs. 1 and 2, compute the difference matrix between the lower and upper points of membership and non-membership:

$${d}_{{L}_{ij}}= {\mu }_{{L}_{ij}}^{2}- {v}_{{L}_{ij}}^{2},$$

(1)

$${d}_{{U}_{ij}}= {\mu }_{{U}_{ij}}^{2}- {v}_{{U}_{ij}}^{2}.$$

(2)

Step 3. Construct the interval multiplicative matrix by applying Eqs. 3 and 4:

$${S}_{{L}_{ij}}=\sqrt{{1000}^{{d}_{{L}_{ij}}}},$$

(3)

$${S}_{{U}_{ij}}= \sqrt{{1000}^{{d}_{{U}_{ij}}}}.$$

(4)

Step 4. Calculate the indeterminacy value of ${o}_{ij}$ using Eq. 5:

$${h}_{ij}=1-\left({\mu }_{{U}_{ij}}^{2}-{\mu }_{{L}_{ij}}^{2}\right)-\left({v}_{{U}_{ij}}^{2}-{v}_{{L}_{ij}}^{2}\right).$$

(5)

Step 5. Construct the unnormalized weights matrix by applying Eq. 6:

$${\tau }_{ij}=\left(\frac{{S}_{{L}_{ij}}+{S}_{{U}_{ij}}}{2}\right) {h}_{ij}.$$

(6)

Step 6. Calculate normalized weight for each criterion using Eq. 7:

$${w}_{j}^{c}=\left(\frac{{\sum }_{i=1}^{n}{\tau }_{ij}}{{\sum }_{i=1}^{n}{\sum }_{j}^{m}{\tau }_{ij}}\right).$$

(7)

Step 7. Apply Steps 1–6 for each alternative under each criterion and calculate normalized weight using Eq. 8:

$${w}_{i}^{a}=\left(\frac{{\sum }_{j=1}^{m}{\tau }_{ij}}{{\sum }_{i=1}^{n}{\sum }_{j}^{m}{\tau }_{ij}}\right).$$

(8)

Step 8. Calculate priority weights for each alternative using Eq. 9:

$$p\left({A}_{i}\right)= \sum_{j=1}^{m}{w}_{i}^{A}{w}_{j}^{C}, \forall i.$$

(9)

Step 9. Prioritize the alternatives in descending order of value $p\left({A}_{i}\right)$.

3.3 Proposed IPF-COPRAS Method

Step 1. Create an IPF Decision matrix for decision-makers’ opinions.

Step 2. Calculate criteria weights using Eq. 10:

$${w}_{j}=\frac{\left({\mu }_{{U}_{j}}^{2}+{\mu }_{{L}_{j}}^{2}\right)\left(2+\sqrt{1- {\mu }_{{L}_{j}}^{2}- {v}_{{L}_{j}}^{2}} + \sqrt{1- {\mu }_{{U}_{j}}^{2}- {v}_{{U}_{j}}^{2}}\right)}{\sum_{j=1}^{m}\left(\left({\mu }_{{U}_{j}}^{2}+{\mu }_{{L}_{j}}^{2}\right)\left(2+\sqrt{1- {\mu }_{{L}_{j}}^{2}- {v}_{{L}_{j}}^{2}} + \sqrt{1- {\mu }_{{U}_{j}}^{2}- {v}_{{U}_{j}}^{2}}\right)\right)}.$$

(10)

Step 3. Applying Eqs. 1 and 2, respectively, calculate the difference matrix between the lower and upper points of the membership and non-membership.

Step 4. Construct the interval multiplicative matrix by applying Eqs. 3 and 4, respectively.

Step 5. Determine the indeterminacy value of ${o}_{ij}$ using Eq. 5.

Step 6. Construct the unnormalized weights matrix by applying Eq. 6.

Step 7. Calculate normalized weight for each criterion using Eq. 7.

Step 8. Calculate weighted normalized matrix based on the criteria weights using Eq. 11:

$${D}_{ij}={w}_{i}^{a}{w}_{j}.$$

(11)

Step 9. Calculate beneficiary and non-beneficiary indexes ${S}_{i}^{+}$ and ${S}_{i}^{-}$ by applying Eqs. 12 and 13, respectively:

$${S}_{i}^{+}=\sum_{j=1}^{k}{D}_{ij}, i=1, \dots , k\, \mathrm{beneficary\, criteria},$$

(12)

$${S}_{i}^{-}=\sum_{j=k}^{m}{D}_{ij}, i=k+1, \dots , m \,\mathrm{non}-\mathrm{beneficary\,criteria}.$$

(13)

Step 10. Calculate the COPRAS index for the relative significance of alternatives using Eq. 14:

$${Q}_{i}={S}_{i}^{+}+\frac{\sum_{i=1}^{n}{S}_{i}^{-}}{{S}_{i}^{-}\sum_{i=1}^{n}\frac{1}{{S}_{i}^{-}}}.$$

(14)

Step 11. Calculate the maximum relative significance values and performance index using Eqs. 15 and 16, respectively:

$${Q}_{\mathrm{max}}=\mathrm{max} \left\{{Q}_{1},\dots ,{Q}_{n}\right\},$$

(15)

$$p\left({A}_{i}\right)=\frac{{Q}_{i}}{{Q}_{max}} 100\%.$$

(16)

Step 12. Rank the alternatives in descending order of importance $p\left({A}_{i}\right)$.

4 Case Study with Applications and Results

This research was conducted for a military-focused research organization in Ankara, Turkey. Choose the correct source because military businesses manufacture delicate components. Thus, using the literature analysis and expert opinions, the application was designed according to the most important criteria.

Four criteria and five options are being explored to choose a provider. C1–C4 were cost, quality, delivery time, and service performance. In this regard, an interval-valued Pythagorean Fuzzy AHP (IPF-AHP) and IPF-COPRAS are developed and used to pick the best provider among 5 supplier alternatives (A1,…, A5) under conflicting criteria. According to Table 2, the decision-maker evaluates the alternatives using linguistic terms. For example, if the decision-maker assumes that A₁ is high important than A₅ under the same criterion, then the decision-maker must assign the HI from Table 2 to make evaluations accurately.

4.1 Application of IPF-AHP Method

Step 0. Create the hierarchical structure as shown in Fig. 1.

Fig. 1

Structural hierarchy of the problem

Step 1. Decision-maker uses Table 3 linguistic phrases to analyze criteria. Table 4 shows the IPFN-based pairwise comparison matrix.

Table 3 Evaluation of criteria in linguistic variable

Table 4 Evaluation of criteria in IPFV

Step 2. The differences matrix between $D={\left({d}_{ij}\right)}_{mxm}$ the lower and upper points of the membership and non-membership by applying Eqs. 1 and 2 as shown in Table 5

Table 5 Differences matrix between upper and lower values of μ and v

Step 3. The interval multiplicative matrix is constructed by applying Eqs. 3 and 4 as represented in Table 6.

Table 6 Interval multiplicative matrix

Step 4. The indeterminacy value matrix is created using Eq. 5 as shown in Table 7.

Table 7 Indeterminacy values

Step 5. The unnormalized weights matrix is created by applying the Eq. 6 as shown in Table 8.

Table 8 Unnormalized weights

Step 6. The normalized weight for each criterion is calculated using Eq. 7 as shown in Table 9.

Table 9 Normalized weights

Step 7. Steps 1–6 are applied for each alternative under each criterion. The calculation of how to obtain the weights is shown with respect to goal. Therefore, due to page and word limitations computations of Tables 10, 11, 12 and 13 are not shown in the manuscript. Tables 10, 11, 12 and 13 show the comparison matrixes and final weights of alternatives under each criterion. Normalized weights of each alternative under each criterion are calculated using Eq. 8

Table 10 Evaluation of alternatives respected to C₁

Table 11 Evaluation of alternatives respected to C₂

Table 12 Evaluation of alternatives respected to C₃

Table 13 Evaluation of alternatives respected to C₄

Step 8. Table 14 shows how Eq. 9 calculates alternate priority weights.

Table 14 Normalized alternatives’ weights under each criterion

Step 9. Alternatives rank in descending order of importance $p\left({A}_{i}\right)$ as shown in Table 15.

Table 15 The rank of the alternatives

According to the results of the IPF-AHP methodology given in Table 15, the alternatives are ranked as A₅, A₄, A₂, A₁, and A₃. Therefore, the best alternative for supplier selection is found as A₅, and the worst alternative was A₃.

4.2 The COPRAS Approach Applied Into Practice

The IPF-AHP methodology was applied in order to derive the weights that should be assigned to the various viable options for the provider selection process. After that, the COPRAS approach was applied in order to calculate the weights of the various options.

Step 1. Decision-maker uses Table 16 linguistic phrases to analyze criteria. Table 17 shows the IPFN-based pairwise comparison matrix.

Table 16 Evaluation of alternatives for each criterion in linguistic variable

Table 17 Evaluation of criteria in IPFV

Step 2. The weights of each criterion are calculated using Eq. 11 as shown in Table 18.

Table 18 The weights of each criterion

Step 3. The differences matrix between $D={\left({d}_{ij}\right)}_{mxm}$ the lower and upper points of the membership and non-membership by applying Eqs. 1 and 2 as shown in Table 19.

Table 19 Differences matrix between upper and lower values of μ and v

Step 4. The interval multiplicative matrix is constructed by applying Eqs. 13 and 14 as represented in Table 20.

Table 20 Interval multiplicative matrix

Step 5. The indeterminacy value matrix is created using Eq. 15 as shown in Table 21.

Table 21 Indeterminacy values

Step 6. The unnormalized weights matrix is created by applying the Eq. 16 as shown in Table 22.

Table 22 The unnormalized weights

Step 7. The normalized weights are calculated for each criterion using Eq. 17 as shown in Table 23.

Table 23 The normalized weights

Step 8. The weighted normalized weights based on the criteria weights are determined using Eq. 11 as shown in Table 24.

Table 24 The normalized weights

Step 9. The beneficiary and non-beneficiary indexes ${S}_{i}^{+}$ and ${S}_{i}^{-}$ are calculated by applying Eqs. 12 and 13, respectively, as shown in Table 25. C₁ and C₂ are defined as beneficiary criteria. On the other hand, C₃ and C₄ are accepted as non-beneficiary criteria.

Table 25 ${S}_{i}^{+}$, ${S}_{i}^{-}$, ${Q}_{i}$, and $p\left({A}_{i}\right)$ values with ranking of alternatives

Step 10. The COPRAS index for the relative significance of alternatives is computed using Eq. 14 as shown in Table 25.

Step 11. The maximum relative significance values and performance index are calculated using Eqs. 15 and 16 as shown in Table 25.

Step 12. Alternatives are ranked in descending order of importance $p\left({A}_{i}\right)$.

After finding the weight matrix, ${S}_{i}^{+}$ and ${S}_{i}^{-}$ values have been calculated for each alternative. ${S}_{i}^{+}$ is equal to the sum of the weighted normalized values of C1 and C2 among the alternatives. The ${S}_{i}^{-}$ value was derived from the aggregate of the weighted normalized values of the delivery time and service performance, which was determined to be the minimum among the alternatives. The option with a performance index of 100, represented by$p\left({A}_{5}\right)$, is the finest option. The order of preference was determined by sorting the performance index values from greatest to least. The greatest alternative according to Table 25 was the A₅ with a performance index value of 100%, while the worst alternative was the A₁ with a performance index value of 56.53%. The alternatives are ranked in descending importance order as follows: A₅, A₄, A₂, A₃, and A₁ respectively.

4.3 Sensitivity Analysis

A sensitivity analysis is carried out so that the degree to which the results are affected by changes in the parameters representing the various weight scenarios may be determined. In order to discover how the weights of the key criteria affect the ranks of the alternative choices, a sensitivity analysis must first be carried out. In order to accomplish this goal, we have created four unique scenarios by adjusting the relative importance of the major criteria. The following is a definition of the criteria that were obtained:

• Case-1 All criteria has equal importance on each other

• Case-2 C4 has higher importance than the other criteria

• Case-3 C3 has higher importance than the other criteria

• Case-4 C2 has higher importance than the other criteria

The results of the case scenarios are compared with the expert evaluations (Current Case) to show effectiveness of the proposed methodology. The various ranks derived from such case scenarios are used to examine the effects of the weighted criteria. The various weight cases that are employed in the sensitivity analysis for IPF-AHP and IPF-TOPSIS are shown in Tables 26 and 27.

Table 26 Weights of criteria for different cases for IPF-AHP

Table 27 Weights of criteria for different cases for IPF-COPRAS

As shown in Tables 26 and 27, criteria weights changed gradually. The results of sensitivity analysis are presented in Figs. 2 and 3 for IPF-AHP and IPF-COPRAS. In the current case of IPF-AHP, the best alternative is as A₅ that is followed by A₄, A₂, A₁, and A₃, respectively. On the other hand, the current case of IPF-AHP shows the rank of alternatives as A₅, A₄, A₂, A₃, and A₁, respectively. After the changes on each criterion, the weights of alternatives are changed. As shown in Fig. 3, Case-3 in which the C₂ has higher importance than the other criteria implies that rank of the alternatives is A₂, A₅, A₄, A₃, and A₁ in descending order. Similar results are appeared in the each cases of IPF-AHP and IPF-COPRAS.

Fig. 2

Sensitivity analysis for IPF-AHP

Fig. 3

Sensitivity analysis and results for IPF-COPRAS

According to the sensitivity analysis and results are shown in Figs. 2 and 3, the proposed IPF-AHP and IPF-COPRAS methods are robust and reliable. Therefore, sensitivity analysis shows that the ranking among A₅ could be accepted as robust to changes in most importance levels. On the other hand, it could be thought that the rankings among A₁, A₂, A₃, and A₄ are highly sensitive to changes in the different levels of criteria weights.

In this research, in order to identify and select the best-suited supplier among all five possible choices under four criteria, consistent and effective assessments are made utilizing the IPF-AHP and IPF-COPRAS methodologies. As a managerial implication of the results, it is recommended to put more attention on the most essential risk factors to select the most appropriate supplier. Table 9 implies that the most essential criterion is ‘‘Quality.’’ However, the fact that the quality conditions are extremely unpredictable and tough to foresee and avoid is well established. Therefore, the decision-makers could focus on the other variables which are controllable by managers to decide on the appropriate provider. In addition, from the management point of view, it is recommended to improve alternative diversity in order to increase flexibility in the decision-making process. On the other hand, from the practical consequences, choosing the best suitable supplier under the fuzzy environment could the research one step ahead. In a fuzzy environment, evaluating criteria or alternatives is difficult to quantify. The IPF-AHP and IPF-COPRAS capture a board frame to represent fuzzy judgments in order to minimize failure in supplier selection, which could result in increased costs and undesirable consequences. The IPF-AHP and IPF-COPRAS approaches are demonstrated to be a beneficial way to handle the fuzzy multiple attribute group decision-making problems more flexibly and fully, according to the sensitivity analysis results of the suggested methodology. Consequently, fuzzy logic and MCDM present unique study topics with a range of various managerial and practical implications.

5 Conclusion

In today’s tough business environment, companies must make the best decisions to succeed. The best supplier choice affects the entire supply chain, thus firms must make this decision carefully. Supplier selection techniques help companies choose providers that meet quality standards [48]. Businesses have to select appropriate providers to meet consumer demands. Thus, MCDM is utilized to make consistent, effective decisions and choose the best provider. The literature research and expert feedback helped this study’s application meet the most important criteria. Four criteria and five options were chosen for the most suitable provider. Supplier selection was solved using cost, quality, delivery time, and service performance.

MCDM and AHP were used to create a hierarchical model based on selection criteria. The expanded hierarchical model’s findings help decision-makers choose providers by considering predetermined criteria. This instance used AHP to rank numerous criteria. After that, potential suppliers were assessed using the COPRAS method. Comparing options with a ratio showed how good or bad they were. Finally, after ranking the possibilities, the best provider was chosen. Based on the value of the performance index assigned to each alternative, the best supplier among the alternatives was found using the COPRAS technique. The options were identified as A₅, A₄, A₂, A₃, and A₁, respectively, based on the results collected from the test. As a result, A₁ was determined to be the most viable solution for supplier selection. Aside from that, the choices were rated based on both approaches, and the validity of the methodologies was compared. Eventually, a sensitivity analysis was undertaken to assess whether or not the reordering of the alternatives was a direct consequence of the adjustments that were made.

As further research directions, it could be grateful to investigate different methods of decision-making, and it would also be beneficial to broaden the scope of the study to include a larger number of participants in terms of the number of experts. In addition, additional research could investigate the influence that exogenous factors, such as the state of the economy or political unrest, have on the decisions made regarding which suppliers to work with. It is also essential to take into account the long-term repercussions of the decisions made regarding the selection of suppliers, including how those decisions will affect the overall performance and sustainability of the supply chain. In general, the findings of this study emphasize how critical it is to base decisions regarding supplier selection on methodologically sound research and in-depth analysis, and they provide insightful information that is useful to both researchers and practitioners.