Retail store performance measurement using a novel interval-valued Pythagorean fuzzy WASPAS method

Abstract

Retail store selection is an important decision for both customers and retailers because it is directly linked to customer satisfaction and profitability of retailers. Because of the competitive market conditions, retailers severely try to find out what they should do to be preferred by potential customers, and consequently to grow their sales. In this context, the criteria influencing customers’ store selection decision have to be analyzed. Although the relationship between customer preferences and retail store attributes has been widely studied through exploratory studies, a comprehensive framework using a multi-criteria decision making method under uncertainty to provide an overall assessment for retail stores has not yet been proposed. Pythagorean fuzzy sets are quite capable of representing uncertainty and vagueness in a decision making process by providing a larger domain to experts in expressing their opinions. Therefore, in this study, a novel interval-valued Pythagorean fuzzy WASPAS method is developed to evaluate the performance of retail stores. The obtained results are compared with crisp WASPAS and interval-valued intuitionistic WASPAS methods and it is revealed that the proposed method provides reliable and informative outputs.

Keywords

WASPAS Pythagorean fuzzy sets retail store selection multi-criteria decision making intuitionistic fuzzy sets

1 Introduction

Retailers are the most effective components of the distribution channels with their proximity to customers [1]. Retail, which is one of the most dynamic sectors of the Turkish economy with its employment contribution and turnover volume, has raised the target to 180 billion USD for the year 2018. The sector has reached a total turnover of 135 billion USD in 2015 [2]. Retail trade in Turkey has increased over the years with shopping center investments and increasing competition [3]. Increasing competition has forced retailers to seek new ways of attracting the customers. Retail store selection is an important decision for both customers and retailers because it is directly linked to customer satisfaction, and consequently profitability of retailers [1].

With the increasing number of retail outlets in the market, customers have more options to select the place where they want to make their purchases [4]. Because of the competitive market conditions, retailers are severely looking at what they need to do to be preferred by the potential customers, and consequently to increase their sales and profits. In this context, the factors influencing customers’ store selection decision have to be examined. However, store selection decision is a complex problem and involves various criteria that should be considered in the evaluation process, even some of them are conflicting. Therefore, to make such a decision, utilization of multi-criteria decision making methods is required.

Weighted Aggregated Sum Product Assessment (WASPAS) is a relatively new method, but it has been widely employed in the literature since it was introduced in 2012. WASPAS is preferred over Weighted Sum Model (WSM) and Weighted Product Model (WPM), because it provides a more accurate and comprehensive performance yielding a more reliable composite solution than its components [5]. Extensions of WASPAS with fuzzy sets such as single-valued neutrosophic sets, interval valued intuitionistic fuzzy sets, and interval type-2 fuzzy sets have been also commonly studied [6 –12]. The reason why fuzzy sets are needed is to handle impreciseness and vagueness of the linguistic terms in the evaluation processes. In the literature, new types of fuzzy sets have been derived from ordinary fuzzy sets in order to define membership functions in a more detailed manner. One of these extensions is Pythagorean fuzzy sets (PFS). Pythagorean fuzzy sets provide more freedom to experts in expressing their opinions because experts may assign membership and non-membership degrees whose sum is greater than 1 unlike the intuitionistic fuzzy sets [13]. PFS have been also used for the solutions of multi-attribute decision making problems [14, 15].

Consumer loyalty and the impact of the store attributes on customer preferences have been widely studied in the literature. Mostly empirical studies exist, and factor analysis, inferential analysis, and statistical analysis are the most common methods employed throughout the literature. Yoo et al. [16] investigated the influence of store characteristics, product assortment, product value, salesperson’s service, after-sale service, atmosphere, facilities, and location, on customer attitudes. It was shown that store characteristics affect in-store emotions of customers. Yalch and Spangenberg [17] examined the influence of music in retail stores and found out that shopping time increases with the exposure to unfamiliar music. Michon et al. [18] examined the impact of ambient odors on customers’ emotions and perceptions of the retail store under different levels of retail density. It was revealed that ambient odors positively affect customer perceptions only under the medium retail density condition. Summers and Hebert [19] investigated the impact of display lighting on customer behavior by examining time of display, number of items touched and number of items picked up. It was indicated that improving in-store lighting can be useful to attract customer attention. Moreover, Babin et al. [20] indicated that store color is important to understand patronage behavior, but the relationships between color, patronage and purchase intentions mostly depend on other factors. Sharma and Stafford [21] investigated the impact of store atmospherics on the level of persuasion by salespeople, and it was revealed that store atmospherics have a positive impact on customers’ persuasion and their positive perceptions of salespeople.

Skinner [22] studied customer motives in supermarket selection. Factor analysis indicated that a pleasant shopping experience, prices, social influence, near other services, meats, advertising, conveniences, and location are the customer motives. Wel et al. [4] examined the important determinants of retail store selection such as store personnel and physical characteristics of the store, advertising, store convenience & merchandise selection, peer influence, store location, product variety and quality, and services. It was revealed that customer store choice is affected by several factors and their decisions change with respect to the types of goods purchased [4]. The significance of determining key retail store attributes has been recognized in the consumer behavior as well as the retailing literature [23, 24]. There are many studies showing the positive relationship between attitude and behavior; in other words, retailers might increase patronage by creating more positive attitude toward their stores [24 –26]. Jones [27] found the factors that have an impact on in-store experiences of customers as sales people, store environment, selection, retail prices. Backstrom and Johansson [28] utilized personnel, service elements, selection, price, design, display, layout, atmospherics as the retailer factors that affect in-store experiences of customers. Hsu and Chen [29] proposed a selection model for bedding chain retail store franchisee by using Delphi and fuzzy analytic hierarchy process (AHP). The selection criteria used in this study are personal background, financial situation, business ability, area/location, traffic, and population.

Reynolds-McIlnay et al. [30] examined how product-environment brightness contrast and product disarray influence customer choices. It was shown that as the contrast between products’ brightness levels and the retail environment increases, products become more preferred. Gauri et al. [31] examined the effect of promotional discounts on store performance, and it was found that promotions generate store traffic, and consequently might increase sales and profit. Santini et al. [32] also investigated the impact of discount sales promotion in consumer buying intent, and it was revealed that other actions such as advertising might be better than discount actions to encourage the purchase of a product whose perception is of financial risk.

Swoboda et al. [33] analyzed the significance of retail brand equity and store accessibility in store loyalty. It was stated that strong brand is more important than a conveniently accessible location for store loyalty. Burnaz and Topcu [34] developed an ANP based approach for retail location selection by using the following evaluation criteria: store, other retailers, shopping centre/business district, accessibility, and environmental. Wu et al. [35] examined customer flow in retail store (traffic, visited areas, customers’ paths, dwell time or loyalty) to learn the preferences of customers, to provide customer needs to retailers, marketing, and attentions of products, and consequently to increase trading volume. Nikhashemi et al. [36] employed structural equation modelling to examine the effect of customers’ perceived value of retail store, and demonstrated that store attributes directly affect customers’ perceived value and their loyalty with the store brand. Moreover, the store attributes also indirectly influence store brand loyalty through customers’ perceived value of the brand [36]. Wang et al. [37] assessed the competitiveness of chain retail enterprises based on AHP and fuzzy comprehensive evaluation by using the following criteria: brand influence ability, store management ability, profitability, solvency, and store development ability.

Zhang and Peng [38] investigated how to improve customer loyalty by employing AHP. Customer loyalty is examined under four main categories as cognitive loyalty, affective loyalty, loyalty intention and behavioral loyalty [38]. Wenyong and Jing [39] evaluated service quality of brand retailers by utilizing exploratory factor analysis, AHP, and fuzzy comprehensive appraisal. The main factors of the evaluation process is identified as prestige, reliability, responsiveness, recovery, and tangibles. It was revealed that the sub-factors related to the brand reputation, reasonableness of price, and availability of service personnel are the most significant criteria [39]. Deb and Lomo-David [40] assessed retail service quality using AHP. Reliability, problem solving, personal interaction, physical aspects, and policy are the main evaluation criteria employed in this study. Policy factor which involves transaction safety, error free transaction, high quality merchandise, and convenient operating hours for target segments is the most important criterion, followed by the physical aspects [40]. Gopalan et al. [41] evaluated retail service quality by adopting the same main criteria with fuzzy AHP. High quality merchandise and the store’s willingness to handle returns and exchanges were determined to be the most important sub-criteria influencing the overall service quality [41].

The relationship between customer preferences and retail store attributes have been widely studied through experimental studies; however, to the best of our knowledge, a framework using MCDM to provide an overall assessment for retail stores has not been introduced yet. The novelty of this paper is twofold. The first one is the development of Pythagorean fuzzy WASPAS method which presents a wider domain for assigning membership and non-membership degrees. The second one is the application of the proposed model to a new area which is the retail store performance measurement problem.

The rest of this paper is structured as follows: the methods and approaches utilized in this study are presented in Section 2. Application of the proposed method, sensitivity analysis, and comparative analysis are given in Section 3. Finally, concluding remarks are provided in Section 4.

2 Methodology

In this section, after the preliminaries for Pythagorean fuzzy sets and crisp WASPAS method are provided, we propose a novel interval-valued Pythagorean fuzzy WASPAS (IVPF – WASPAS) method.

2.1 Preliminaries

Pythagorean fuzzy sets are proposed as an extension to intuitionistic fuzzy sets with the purpose of not enforcing decision makers to assign membership and non-membership degrees whose sum must be at most 1. In other words, the sum of membership and non-membership degrees can exceed 1 in Pythagorean fuzzy sets, but the sum of their squares can not. It is expressed in Definition 1 as follows:

Definition 1. Let X be a fixed set. A Pythagorean fuzzy set $\tilde{P}$ is an object in the form [42]: $\tilde{P} ≅ {〈 x, μ \tilde{p} (x), ν \tilde{p} (x); x ϵ X 〉}$ (1) where $μ_{\tilde{P}} (x)$ and $ν_{\tilde{P}} (x)$ define the degree of membership and non-membership of the element xϵX to P, respectively. For every xϵX, Equation (2) is satisfied. $0 ⩽ μ_{\tilde{A}} (x)^{2} + ν_{\tilde{A}} (x)^{2} ⩽ 1$ (2)

Definition 2. Let, be two Pythagorean fuzzy numbers (PFNs), and λ > 0, then the operations on these two PFNs are defined as follows [43]: $\tilde{A} \oplus \tilde{B} = (\sqrt{μ_{1}^{2} + μ_{2}^{2} - μ_{1}^{2} μ_{2}^{2}}, ν_{1} ν_{2})$ (3) $\tilde{A} \otimes \tilde{B} = (μ_{1} μ_{2}, \sqrt{ν_{1}^{2} + ν_{2}^{2} - ν_{1}^{2} ν_{2}^{2}})$ (4) $λ \tilde{A} = (\sqrt{1 - {(1 - μ^{2})}^{λ}}, ν^{λ})$ (5) ${\tilde{A}}^{λ} = (μ^{λ}, \sqrt{1 - {(1 - ν^{2})}^{λ}})$ (6)

Definition 3. Let an interval valued Pythagorean fuzzy set be defined as follows [44, 45]: $\tilde{P} = {〈 x, μ \tilde{p} (x), ν \tilde{p} (x) 〉; x ϵ X}$ (7) where $μ_{\tilde{P}} (x) = [μ_{\tilde{P}}^{L} (x), μ_{\tilde{P}}^{U} (x)] \subset [0, 1]$ and $ν_{\tilde{P}} (x) = [ν_{\tilde{P}}^{L} (x), ν_{\tilde{P}}^{U} (x)] \subset [0, 1]$ . For $\tilde{P}$ , Equation (8) must be satisfied: $0 ⩽ μ_{\tilde{P}}^{U} (x)^{2} + ν_{\tilde{P}}^{U} (x)^{2} ⩽ 1$ (8) Then, the degree of indeterminacy, $π_{\tilde{P}} (x) = [π_{\tilde{P}}^{L} (x), π_{\tilde{P}}^{U} (x)]$ , is obtained as follows [44]: $π_{\tilde{P}}^{L} (x) = \sqrt{1 - μ_{\tilde{P}}^{U} (x)^{2} - ν_{\tilde{P}}^{U} (x)^{2}}$ (9) $π_{\tilde{P}}^{U} (x) = \sqrt{1 - μ_{\tilde{P}}^{L} (x)^{2} - ν_{\tilde{P}}^{L} (x)^{2}}$ (10)

2.2 Crisp WASPAS

Weighted Aggregated Sum Product Assessment (WASPAS), a multi-criteria decision making method, is proposed by Zavadskas [46] by facilitating equal contribution of Weighted Sum Model (WSM) and Weighted Product Model (WPM) for an overall assessment. In the WASPAS method, initially, decision matrix given below is formed based on expert opinions. $[\begin{matrix} x_{11} & x_{12} & . . . & x_{1 n} \\ x_{21} & x_{22} & . . . & x_{2 n} \\ . . . & . . . & . . . & . . . \\ x_{m 1} & x_{m 2} & . . . & x_{mn} \end{matrix}]$

In the matrix, x_ij is the performance of i^th alternative with respect to j^th th criterion whereas m and n represent the number of alternatives and the number of evaluation criteria, respectively. Normalization of x_ij values is performed using Equations (11) and (12), and normalized decision matrix, r_ij, is formed. $r_{ij} = \frac{x_{ij}}{\max_{i} x_{ij}}$ (11) $r_{ij} = \frac{\min_{i} x_{ij}}{x_{ij}}$ (12) Then, the weighted normalized decision matrix for WSM and the weighted normalized decision matrix for WPM are calculated by using Equations (13) and (14). Finally, a more generalized equation, Equation (15), is utilized to obtain the total relative importance of alternatives [46 –48]. $Q_{i}^{(1)} = \sum_{j = 1}^{n} r_{ij} w_{j}$ (13) $Q_{i}^{(2)} = \prod_{j = 1}^{n} r_{ij}^{w_{j}}$ (14) $Q_{i} = λ Q_{i}^{(1)} + (1 - λ) Q_{i}^{(2)}, λ ϵ [0, 1]$ (15)

2.3 Steps of the Proposed IVPF-WASPAS

The proposed approach is a combination of WASPAS method and Pythagorean fuzzy sets. Experts are asked to evaluate alternatives with respect to certain criteria using linguistic terms. In decision making problems, it is often required to aggregate the expert judgments in a single matrix [6] before the mathematical operations suggested by the proposed method are performed.

Step 1. In the proposed approach, expert judgments are obtained in linguistic form and converted to interval valued Pythagorean fuzzy numbers by using the scales given in Table 1. These values aggregated through the interval-valued Pythagorean fuzzy weighted average (IVPFWA) operator given in Equation (16) [44].

IVPFWA = ([\sum_{i = 1}^{n} w_{i} μ_{i}^{L}, \sum_{i = 1}^{n} w_{i} μ_{i}^{U}],

(16)

[\sum_{i = 1}^{n} w_{i} ν_{i}^{L}, \sum_{i = 1}^{n} w_{i} ν_{i}^{U}])

where n represents the number of experts.

Table 1

An IVPFN scale to rate alternatives on criteria

Linguistic Term	μ _L	μ _U	ν _L	ν _U
Very very good (VVG)	0.75	0.90	0.03	0.18
Very good (VG)	0.66	0.81	0.12	0.27
Good (G)	0.57	0.72	0.21	0.36
Medium good (MG)	0.48	0.63	0.30	0.45
Fair (F)	0.39	0.54	0.39	0.54
Medium bad (MB)	0.30	0.45	0.48	0.63
Bad (B)	0.21	0.36	0.57	0.72
Very bad (VB)	0.12	0.27	0.66	0.81
Very very bad (VVB)	0.03	0.18	0.75	0.90

Step 2. In order to normalize the values of the aggregated decision matrix as in Equations (18) and (19), we initially need to obtain $\max_{i} p_{ij}$ using the deffuzification formula given in Equation (17). This defuzzification formula is derived based on the defuzzification of interval valued intuitionistic fuzzy numbers [49]. After taking the maximum of crisp numbers obtained from IVPFNs, Equation (20) is used to calculate the multiplication of a crisp number and an IVPFN. $p = \frac{μ_{L} + μ_{U} + \sqrt{1 - ν_{L}^{2}} + \sqrt{1 - ν_{U}^{2}}}{4}$ (17) $+ \frac{μ_{L} μ_{U} - \sqrt{\sqrt{1 - ν_{L}^{2}} \sqrt{1 - ν_{U}^{2}}}}{4}$ ${\tilde{r}}_{ij} = \frac{{\tilde{x}}_{ij}}{\max_{i} p_{ij}}$ (18) ${\tilde{r}}_{ij} = \frac{\min_{i} p_{ij}}{{\tilde{x}}_{ij}}$ (19) $\begin{matrix} λ \tilde{p} = ([\sqrt{1 - (1 - μ_{L}^{2})^{λ}}, \sqrt{1 - (1 - μ_{U}^{2})^{λ}}], \\ [ν_{L}^{λ}, ν_{U}^{λ}]) \end{matrix}$ (20)

Step 3. Criteria weights are obtained via linguistic terms given in Table 2 by using the corresponding IVPFNs. The aggregation of these IVPFNs is performed through Equation (16).

Table 2

Linguistic terms for rating the importance of criteria

Linguistic Term	μ _L	μ _U	ν _L	ν _U
Very important (VI)	0.70	0.90	0.06	0.26
Important (I)	0.54	0.74	0.22	0.42
Medium (M)	0.38	0.58	0.38	0.58
Unimportant (U)	0.22	0.42	0.54	0.74
Very unimportant (VU)	0.06	0.26	0.70	0.90

Step 4. The Pythagorean fuzzy weighted sum values, ${\tilde{Q}}_{i}^{1}$ , given in Equation (21), are calculated from the normalized decision matrix by using the operators of IVPFNs given in Equations (22) and (23) [44]. ${\tilde{Q_{i}}}^{(1)} = \sum_{j = 1}^{n} {\tilde{r}}_{ij} \tilde{w_{j}}$ (21) $\tilde{p_{1}} \otimes \tilde{p_{2}} = ([μ_{1}^{L} μ_{2}^{L}, μ_{1}^{U} μ_{2}^{U}],$ (22) $\begin{matrix} [\sqrt{(ν_{1}^{L})^{2} + (ν_{2}^{L})^{2} - (ν_{1}^{L})^{2} (ν_{2}^{L})^{2}}, \\ \sqrt{(ν_{1}^{U})^{2} + (ν_{2}^{U})^{2} - (ν_{1}^{U})^{2} (ν_{2}^{U})^{2}}]) \end{matrix}$ $\tilde{p_{1}} \oplus \tilde{p_{2}} = ([\sqrt{(μ_{1}^{L})^{2} + (μ_{2}^{L})^{2} - (μ_{1}^{L})^{2} (μ_{2}^{L})^{2}},$ (23) $\begin{matrix} \sqrt{(μ_{1}^{U})^{2} + (μ_{2}^{U})^{2} - (μ_{1}^{U})^{2} (μ_{2}^{U})^{2}}], \\ [ν_{1}^{L} ν_{2}^{L}, ν_{1}^{U} ν_{2}^{U}]) \end{matrix}$

Step 5. In order to obtain the Pythagorean fuzzy weighted product values, ${\tilde{Q}}_{i}^{2}$ , given in Equation (24), criteria weights are defuzzified using Equation (17). Then, Equations (22) and (25) [44] are used to complete the computation. ${\tilde{Q_{i}}}^{(2)} = \prod_{j = 1}^{n} {\tilde{r}}_{ij}^{w_{j}}$ (24) $\begin{matrix} {\tilde{p}}^{λ} = ([μ_{L}^{λ}, μ_{U}^{λ}], [\sqrt{1 - (1 - ν_{L}^{2})^{λ}}, \\ \sqrt{1 - (1 - ν_{U}^{2})^{λ}}]) \end{matrix}$ (25)

Step 6. In order to obtain the total relative importance of the alternatives as in Equation (26), Pythagorean fuzzy weighted sum values and Pythagorean fuzzy weighted product values are combined using Equations (20) and (23) [44]. $\tilde{Q_{i}} = λ {\tilde{Q_{i}}}^{(1)} + (1 - λ) {\tilde{Q_{i}}}^{(2)}, λ ϵ [0, 1]$ (26)

Step 7. The obtained relative importance scores are defuzzified using Equation (17) and the alternative with the highest score is determined to be the best alternative.

3 Application

In this section, the proposed method is applied to select the best retail store. After the evaluation process is carried out through the proposed method, sensitivity analysis is performed. Then, the comparison of the proposed method with crisp and intuitionistic fuzzy WASPAS is presented.

3.1 Application of the proposed method

There are various criteria affecting customers’ retail store preferences. As a result of a comprehensive literature analysis, the main criteria that influence customers’ retail store preferences have been determined as follows: prices, queues at billing counter, retail atmosphere, sales personnel, location, product variety, retail layout, home delivery, product/price displays, and placement of merchandise. The notation for these criteria that will be used thereafter is presented in Table 3. In this section, five alternative retail stores are evaluated with respect to the identified criteria by using the proposed method.

Table 3
Criteria used in the assessment

Criteria
C1	Prices
C2	Queues at billing counter
C3	Atmosphere
C4	Sales personnel
C5	Location
C6	Product variety
C7	Retail layout
C8	Home delivery
C9	Product/price displays
C10	Placement of merchandise

Table 4 shows the aggregated decision matrix obtained by Equation (16). The maximum IVPFNs of each criterion with respect to the alternatives are provided in Table 5. The normalized IVPFNs are computed in Table 6 by using the maximum IVPNs given in Table 5. The aggregated Pythagorean fuzzy criteria weights calculated by using Equation (16) are shown in Table 7. The results of IVPF-WASPAS method are presented in Table 8. With respect to the defuzzified values, AL4 is the best alternative, followed by AL3, AL5, AL2, and AL1, respectively.

Table 4

Aggregated decision matrix in the form of IVPFNs

Alternatives	AL1				AL2				AL3				AL4				AL5
Criteria	μ _L	μ _U	ν _L	ν _U	μ _L	μ _U	ν _L	ν _U	μ _L	μ _U	ν _L	ν _U	μ _L	μ _U	ν _L	ν _U	μ _L	μ _U	ν _L	ν _U
C1	0.03	0.18	0.75	0.9	0.21	0.3	0.63	0.78	0.48	0.63	0.3	0.45	0.63	0.78	0.15	0.3	0.45	0.6	0.33	0.48
C2	0.57	0.72	0.21	0.36	0.66	0.81	0.12	0.27	0.48	0.63	0.3	0.45	0.36	0.51	0.42	0.57	0.42	0.57	0.36	0.51
C3	0.33	0.48	0.45	0.6	0.24	0.39	0.54	0.69	0.33	0.48	0.45	0.6	0.6	0.75	0.18	0.33	0.42	0.57	0.36	0.51
C4	0.12	0.27	0.66	0.81	0.33	0.48	0.45	0.6	0.66	0.81	0.12	0.27	0.51	0.66	0.27	0.42	0.3	0.45	0.48	0.63
C5	0.03	0.18	0.75	0.9	0.06	0.21	0.72	0.87	0.15	0.3	0.63	0.78	0.54	0.69	0.24	0.39	0.33	0.48	0.45	0.6
C6	0.51	0.66	0.27	0.42	0.48	0.63	0.3	0.45	0.33	0.48	0.45	0.6	0.39	0.54	0.39	0.54	0.18	0.33	0.6	0.75
C7	0.6	0.75	0.18	0.33	0.45	0.6	0.33	0.48	0.63	0.78	0.15	0.3	0.36	0.51	0.42	0.57	0.51	0.66	0.27	0.42
C8	0.09	0.24	0.69	0.84	0.24	0.39	0.54	0.69	0.33	0.48	0.45	0.6	0.48	0.63	0.3	0.45	0.57	0.72	0.21	0.36
C9	0.39	0.54	0.39	0.54	0.33	0.48	0.45	0.6	0.63	0.78	0.15	0.3	0.48	0.63	0.3	0.45	0.36	0.51	0.42	0.57
C10	0.18	0.33	0.6	0.75	0.39	0.54	0.39	0.54	0.24	0.39	0.54	0.69	0.6	0.75	0.18	0.33	0.57	0.72	0.21	0.36

Table 5

Maximum IVPFNs with respect to criteria

Criteria	Defuzzified values of IVPFNs					Max
	AL1	AL2	AL3	AL4	AL5
C1	0.194	0.320	0.584	0.718	0.558	0.718
C2	0.664	0.746	0.584	0.480	0.532	0.746
C3	0.454	0.377	0.454	0.691	0.532	0.691
C4	0.274	0.454	0.746	0.611	0.428	0.746
C5	0.194	0.221	0.300	0.637	0.454	0.637
C6	0.611	0.584	0.454	0.506	0.326	0.611
C7	0.691	0.558	0.718	0.480	0.611	0.718
C8	0.248	0.377	0.454	0.584	0.664	0.664
C9	0.506	0.454	0.718	0.584	0.480	0.718
C10	0.326	0.506	0.377	0.691	0.664	0.691

Table 6

Normalized decision matrix in the form of IVPF numbers

Alternatives	AL1				AL2				AL3				AL4				AL5
Criteria	μ _L	μ _U	ν _L	ν _U	μ _L	μ _U	ν _L	ν _U	μ _L	μ _U	ν _L	ν _U	μ _L	μ _U	ν _L	ν _U	μ _L	μ _U	ν _L	ν _U
C1	0.04	0.21	0.67	0.86	0.25	0.35	0.53	0.71	0.55	0.71	0.19	0.33	0.71	0.85	0.07	0.19	0.52	0.68	0.21	0.36
C2	0.64	0.79	0.12	0.25	0.73	0.87	0.06	0.17	0.54	0.70	0.20	0.34	0.41	0.58	0.31	0.47	0.48	0.64	0.25	0.41
C3	0.39	0.56	0.31	0.48	0.29	0.46	0.41	0.58	0.39	0.56	0.31	0.48	0.69	0.84	0.08	0.20	0.49	0.66	0.23	0.38
C4	0.14	0.31	0.57	0.75	0.38	0.54	0.34	0.50	0.73	0.87	0.06	0.17	0.58	0.73	0.17	0.31	0.34	0.51	0.37	0.54
C5	0.04	0.22	0.64	0.85	0.08	0.26	0.60	0.80	0.19	0.37	0.48	0.68	0.65	0.80	0.11	0.23	0.41	0.58	0.29	0.45
C6	0.62	0.78	0.12	0.24	0.59	0.75	0.14	0.27	0.41	0.59	0.27	0.43	0.49	0.66	0.21	0.36	0.23	0.41	0.43	0.62
C7	0.68	0.83	0.09	0.21	0.52	0.68	0.21	0.36	0.71	0.85	0.07	0.19	0.42	0.59	0.30	0.46	0.59	0.74	0.16	0.30
C8	0.11	0.29	0.57	0.77	0.29	0.47	0.40	0.57	0.40	0.57	0.30	0.46	0.57	0.73	0.16	0.30	0.67	0.82	0.10	0.21
C9	0.45	0.62	0.27	0.42	0.39	0.55	0.33	0.49	0.71	0.85	0.07	0.19	0.55	0.71	0.19	0.33	0.42	0.59	0.30	0.46
C10	0.22	0.39	0.48	0.66	0.46	0.63	0.26	0.41	0.29	0.46	0.41	0.58	0.69	0.84	0.08	0.20	0.66	0.81	0.10	0.23

Table 7

Criteria weights in the form of IVPFNs

Criteria	μ _L	μ _U	ν _L	ν _U
C1	0.70	0.90	0.06	0.26
C2	0.65	0.85	0.11	0.31
C3	0.17	0.37	0.59	0.79
C4	0.22	0.42	0.54	0.74
C5	0.59	0.79	0.17	0.37
C6	0.54	0.74	0.22	0.42
C7	0.33	0.53	0.43	0.63
C8	0.43	0.63	0.33	0.53
C9	0.49	0.69	0.27	0.47
C10	0.27	0.47	0.49	0.69

Table 8

Results of IVPF WASPAS

Alternetives	WASPAS Resuls				Defuzzified value	Rank
	μ _L	μ _U	ν _L	ν _U
AL1	0.45	0.77	0.02	0.14	0.643	5
AL2	0.52	0.81	0.01	0.10	0.687	4
AL3	0.58	0.87	0.00	0.06	0.739	2
AL4	0.67	0.92	0.00	0.04	0.800	1
AL5	0.56	0.85	0.01	0.06	0.723	3

3.2 Sensitivity analysis

Sensitivity analysis is a measure of effectiveness with respect to changes in the inputs [50]. In this sub-section, a multi-parameter sensitivity analysis is conducted by changing the λ value and linguistic terms assigned by the decision makers at the same time. We applied the sensitivity analysis in order to observe the effects of the linguistic term changes in the decision matrices on the final ranking of the alternatives. Some of the linguistic terms in each of the decision matrices have been subjected to only slight changes to see whether the decision is sensitive to this kind of changes. Fig. 1 shows the actual result whereas Fig. 2 indicates the results of the sensitivity analysis.

Fig. 1

Rank of alternatives with respect to different λ values.

The sensitivity graphs given in Fig. 2 show that AL4 is always the best alternative whatever our slight linguistic changes are. This means that AL4 is insensitive to the slight changes in linguistic terms and λ values. It can be revealed that the changes in the alternative ranking occur with extremely small and high λ values. AL1 is always the worst alternative unless the λ value is extremely small or high. In Fig. 2d, it is seen that AL2 which usually is in the fourth rank takes the second rank if λ value becomes larger than 0.3. Similarly, in Fig. 2k, AL2 moves to the second rank when λ value becomes larger than 0.8. These results indicate that AL2 is the most sensitive alternative to the changes in linguistic terms and λ values.

Fig. 2

Results of the sensitivity analysis.

As a result of the sensitivity analysis, it is demonstrated that decisions made by using the proposed IVPF-WASPAS method are robust and reliable.

3.3 Comparison with crisp and intuitionistic Fuzzy WASPAS methods

In this sub-section, the results of IVPF-WASPAS are compared with the results obtained from crisp WASPAS [46], interval-valued intuitionistic fuzzy (IVIF) WASPAS [6] and interval-valued Pythagorean fuzzy weighted geometric (IVPFWG) aggregation operator [51]. Due to space constraints, the steps of these method are not provided in this study. The results obtained from these methods are provided in Table 9. According to Table 9, the rank of the alternatives does not change with respect to method. In the classical WASPAS method, the uncertainty in human judgments is neglected. Hence, the difference between alternatives is observed more sharply. After incorporating the fuzzy sets theory with WASPAS method, these differences significantly reduce. It is revealed that the difference between AL4 and AL3 is much less than the crisp. This is because the vagueness in the evaluation process is taken into account.

Table 9
Comparison of the results with Crisp WASPAS, IVIF WASPAS

Alternatives	Crisp WASPAS		IVIF WASPAS		IVPF WASPAS		IVPFWG operator
	Overall Score	Rank	Overall Score	Rank	Overall Score	Rank	Rank
A1	0.454	5	0.642	5	0.643	5	5
A2	0.555	4	0.691	4	0.687	4	4
A3	0.697	2	0.740	2	0.739	2	2
A4	0.827	1	0.803	1	0.800	1	1
A5	0.686	3	0.718	3	0.723	3	3

4 Conclusions

Retailers are the most effective components of the distribution channels with their proximity to customers. With the increasing number of retail outlets in the market, customers have more options to select the place where they want to make their purchases. However, store selection decision is a complex problem and involves various criteria that should be considered in the evaluation process, even some of them are conflicting. Therefore, to make such a decision, utilization of multi-criteria decision making methods is required.

In the literature, mostly empirical studies exist. Factor analysis, inferential analysis, and statistical analysis are the most common methods used to examine the impact of the store attributes on customer preferences. However, a comprehensive approach under uncertainty to provide an overall assessment for retail stores has not yet been proposed. Therefore, in this study, a novel Pythagorean fuzzy WASPAS method is proposed to assess the performance of retail stores. Pythagorean fuzzy sets provided a wider range for the decision makers to define membership and non-membership degrees. It has also been observed that fuzzy WASPAS methods provide better results by involving more information on the uncertainty of decision parameters. More specifically, it has been seen that the overall scores of the first two alternatives become closer when the analysis is made with fuzzy sets. Sensitivity analyses have shown that the given decisions through the proposed method are robust against changes in the λ value and the slight changes in the linguistic terms. The comparative analyses of IVPF-WASPAS with crisp WASPAS and IVIF-WASPAS were performed and it has been shown that the proposed method provides reliable and consistent outcomes.

For further research, we suggest other types of fuzzy numbers such as trapezoidal, triangular or left-right fuzzy numbers to be employed in the analysis in order to compare with our results. Besides, other multi-criteria decision making methods integrated with fuzzy sets such as intuitionistic fuzzy AHP, hesitant fuzzy TOPSIS or Type-2 fuzzy VIKOR can be utilized for comparative purposes.

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Retail store performance measurement using a novel interval-valued Pythagorean fuzzy WASPAS method

Abstract

Keywords

1 Introduction

2 Methodology

2.1 Preliminaries

3.1 Application of the proposed method

Table 3 Criteria used in the assessment

Table 9 Comparison of the results with Crisp WASPAS, IVIF WASPAS

References

Table 3
Criteria used in the assessment

Table 9
Comparison of the results with Crisp WASPAS, IVIF WASPAS