Picture Fuzzy ARAS Method for Freight Distribution Concept Selection

Abstract

Companies can perform their freight distribution in three different ways. The first concept, the in-house concept, represents the use of a company’s own resources and knowledge to organize transportation from the production to retailers or from the warehouse to customers. The opposite concept is to outsource distribution activities by hiring third-party logistics providers. The third concept represents a combination of the previous two. Although the arguments in favor of outsourcing can be found in the literature, an appropriate selection of a freight distribution concept is specific for each company and depends on many evaluation criteria and their symmetrical roles. This paper presents a methodology that can be used by companies that need to choose their freight distribution concept. An advanced extension of the Additive Ratio ASsessment (ARAS) method is developed to solve the freight distribution concept selection problem. To illustrate the implementation of the proposed methodology, a tire manufacturing company from the Czech Republic is taken as a case study. However, the proposed picture fuzzy ARAS method is general and can be used by any other company. To validate the novel picture fuzzy ARAS method, a comparative analysis with the nine existing state-of-the-art picture fuzzy multi-criteria decision-making methods is provided.

1. Introduction

The study of the methods of physical connection between producers of goods and customers has been an intriguing topic since ancient times. Carrying out freight distribution activities in a proper way is of crucial importance for companies. The internal benefits of freight distribution are improved business operations, cost reduction, brand building, etc. On the other hand, its external benefits affect a broader economy, society, and environment. Some companies perform distribution activities by themselves, while others engage third-party logistics (3PL) providers. Determining how to select a freight distribution concept (FDC) is a matter of importance. The term FDC can refer to the choice of transport mode in distribution activities [1]. In this paper, FDC refers to the organizational issue, i.e., whether a company should organize the distribution on its own or outsource.

This paper aims to propose a methodology that can support FDC decision-making. The proposed methodology will suggest one of the following alternatives: investing in a company’s own transport fleet, employing a 3PL provider, or using a combined approach. It should be kept in mind that there is no unique solution that fits all companies, i.e., the preferable solution is based on the characteristics of the company considered.

Logistics companies operate in a changeable and uncertain environment [2,3,4,5,6]. They are increasingly confronted with complexity in the situations where decision-making is needed. Besides, FDC selection is a complex multidisciplinary problem. It requires numerous factors to be taken into consideration. The knowledge from various fields such as operational research, forecasting, statistics, and economics should be used to solve this problem. Selecting an appropriate FDC is a multi-criteria decision-making (MCDM) problem that demands expert opinions. Its goal is to find the best freight distribution alternative that satisfies some predetermined criteria and sub-criteria. However, the freight distribution concept selection problem includes some imprecise, uncertain, and inconsistent information. Typically, the experts are not completely familiar with features of all criteria and freight distribution alternatives due to their distinct education, field of expertise, position, and experience. Therefore, they may abstain or refuse to provide evaluations for the criteria and/or alternatives. Unfortunately, the available MCDM methods for outsourcing logistics do not take into account such abstention or refusal.

Picture fuzzy sets (PFSs) are advanced fuzzy sets that are characterized by degrees of positive, neutral, negative, and refusal membership [7,8]. PFSs allow experts to handle vague information with ease [9]. PFS-based MCDM models are suitable in cases when expert opinions include more answers; i.e., yes, abstain, no, and refusal [8]. Voting can be a good example of such cases. As a result, PFSs are realistic and accurate for describing uncertain information in the FDC selection problem.

The Additive Ratio ASsessment (ARAS) method is a new MADM technique, introduced by Zavadskas and Turskis [10] and Zavadskas et al. [11]. The ARAS method helps decision-makers to assess the performance of alternatives as well as the ratio of each alternative to the ideal alternative [12]. It aims to select the best alternative based on several criteria, and the final ranking of alternatives is made by determining the utility degree of each alternative [13]. Its numerous applications are the result of its straightforward, direct, and easy-to-follow steps, which yield reasonable and acceptable ranking results [14]. Its concept has a profound logic [12]. Unfortunately, there has been no work extending the ARAS method with PFSs to solve MADM problems in the literature.

This study presents an advanced extension of the ARAS method to tackle decision-making issues in real-life situations. For the first time, a picture fuzzy set based MCDM approach is used for solving transportation problems. The criteria and sub-criteria for solving the freight distribution concept selection problem are distinguished. The proposed method is applied to the freight distribution concept selection problem for a tire manufacturing company in the Czech Republic. The experts’ assessments for the criteria, sub-criteria, and freight distribution alternatives are represented by PFSs so that their attitudes can be expressed precisely and information loss can be mitigated. The comparative analysis with the nine existing state-of-the-art picture fuzzy MCDM methods is provided to validate the novel picture fuzzy ARAS method. Finally, the Spearman’s rank correlation coefficients between the developed picture fuzzy ARAS method and the available PFS-based MCDM methods are calculated to check the consistency.

The paper is organized in the following manner: Section 2 presents a review of related research. Section 3 details the used methodology and the development of the picture fuzzy ARAS method. A case study is described in Section 4. Section 5 provides the case study results and discussions. Section 6 offers the conclusions of the work.

2. Literature Review

The literature review is organized into two subsections to provide better insights into the concepts underlying this research and more clearly address the contributions of this study. The first subsection explores the existing research methods used to solve the corresponding issues in the field of outsourcing logistics. The second subsection investigates the available applications of the ARAS method.

2.1. Research Methods for Analyzing the Freight Distribution Concept

Logistics outsourcing refers to a business activity where a company employs an external partner to carry out logistics services. This phenomenon, as confirmed in the literature [15,16], has been attracting growing attention over the past few decades, and this trend is continuing. As organizations place more focus on their core activities, they examine the possibilities of contracting with 3PL providers in order to reduce costs, save time, and improve the quality of products and services. Logistics outsourcing mainly includes transport, packaging, warehousing, and documentation manipulation. According to Rao and Young [17], the most dominant factors that influence organizations’ decisions to outsource all or some of their functions to third parties are the cost/service trade-offs and quality of information systems. Bhatnagar et al. [18] emphasized that various cost-saving opportunities, increased supply chain flexibility, and improved logistics efficiency could be realized by logistics outsourcing with professional help from third-party logistics (3PL) service providers. However, there are also arguments against outsourcing. Kremic et al. [19] noticed that certain hidden costs could outweigh any short-term financial benefits from outsourcing. Besides, Handley [20] emphasized that when companies made outsourcing decisions, they would inevitably encounter the consequence of internal capability erosion, changing their internal structure. Therefore, a decision to outsource is not so simple; it should be based on the consideration of many influencing factors, making it suitable to be considered as a multi-criteria decision-making problem. To accurately describe this MCDM problem, it is very useful to analyze the examples from the literature that contribute to solving the considered task.

Considering the literature published in the last decade, a good theoretical basis was set by Bajec and Jakomin [21]. They proposed a make-or-buy decision process methodology through the following stages: planning, evaluation, internal costs, and performance analysis. Pirannejad et al. [22] used the ANP method for solving public problems in terms of outsourcing governmental logistic functions. Liou and Chuang [23] considered the outsourcing provider selection problem by using the DEMATEL, ANP, and VIKOR methods. Cheng and Lee [24] used the ANP method in solving the problem of outsourcing reverse logistics of high-tech manufacturing firms. In their study, the ANP was used to calculate the most appropriate 3PL alternative as well as to find the relative importance of reverse logistics service requirements.

Aktas et al. [25] proposed a descriptive research model to estimate the outsourcing perception of the companies operating in different business areas in Turkey. Based on the statistical analyses, considering 299 companies in Finland, Solakivi et al. [26] found that cost savings together with flexibility and customer service were the major motives for outsourcing. The outsourcing provider selection problem was considered in a real case of a Taiwanese company [27]. The DEMATEL and ANP approach was used to solve the problem. Rezaeisaray et al. [28] researched the pipe manufacturing industry and proposed a combination of MCDM methods to solve the outsourcing-supplier selection problem. Chen et al. [29] carried out a study concerning the evaluation and selection of the best outsourcing service country in East and Southeast Asia by using an AHP approach. Kahraman et al. [30] applied the fuzzy AHP-TOPSIS approach to evaluate outsourcing manufacturers. Arif and Jawab [31] considered the impact of the outsourcing strategy on logistics performances. Pedregosa et al. [32] investigated the determinants of success in transport services outsourcing in Europe by applying the partial least squares simultaneous equation models (PLS-SEM).

Hwang and Kim [33] compared the effects of in-house and outsourced logistics services on supply chain agility and organizational performance using structural equation modeling. Wan et al. [34] implemented a fuzzy-set qualitative comparative analysis (fsQCA) to examine the drivers of outsourcing decisions in China. Bucovetchi et al. [35] proposed several key performance indicators to assess whether logistics or outsourcing is the best option for business performance. Zarbakhshnia et al. [36] integrated a fuzzy AHP method and grey multiobjective optimization by ratio analysis (MOORA-G) for evaluating reverse logistics 3PL providers. Kiani et al. [37] utilized a fuzzy MCDM approach for prioritizing outsourceable activities in universities. They combined the AHP, SAW, TOPSIS, and VIKOR methods. Vazifehdan and Darestani [38] evaluated the outsourcing components in the petrochemical industry. Sayed et al. [39] applied an inductive case study approach to explore the impact of outsourcing versus in-house concepts. Mokrini and Aouam [40] evaluated risks of healthcare logistics outsourcing by using the combination of fuzzy AHP, TOPSIS, and PROMETHEE methods. Tavana et al. [41] considered the reverse logistics outsourcing problem by using integrated intuitionistic fuzzy AHP and SWOT methods. Wu et al. [42] extended the VIKOR method under the linguistic environment to investigate the supplier selection problem in the nuclear power industry. A structured overview of the considered literature is shown in

Table 1. Summary of the available models for the freight distribution concept (FDC) selection problem.

Authors (Publication Year)	Research Focus	Methods
Bajec and Jakomin [21]	Make-or-buy decision process for outsourcing	Theoretical study—technological consideration
Pirannejad et al. [22]	Defining outsourcing priorities of public organizations	ANP method
Liou and Chuang [23]	Outsourcing provider selection problem	DEMATEL, ANP, VIKOR
Cheng and Lee [24]	Outsourcing reverse logistics of high-technology manufacturers	ANP method
Aktas et al. [25]	Motives for outsourcing logistics activities	Statistical analysis
Solakivi et al. [26]	Motives for outsourcing logistics activities	Statistical analysis
Hsu et al. [27]	Outsourcing provider selection problem	DEMATEL, ANP, and GRT
Rezaeisaray et al. [28]	Outsourcing provider selection problem	DEMATEL, FANP, and DEA
Chen et al. [29]	Evaluation and selection of the best outsourcing service country in East and Southeast Asia	AHP
Kahraman et al. [30]	Evaluation of outsource manufacturers	Fuzzy-AHP and TOPSIS
Arif and Jawab [31]	The impact of the outsourcing strategy on logistics performance	Theoretical study—technological consideration
Pedregosa et al. [32]	Determinants of success in transport services outsourcing	Partial least squares simultaneous equation models (PLS-SEM)
Hwang and Kim [33]	Effects of in-house and outsourced logistics services	Statistical analysis
Wan et al. [34]	Drivers of outsourcing decisions	Fuzzy-set qualitative comparative analysis (fsQCA)
Bucovetchi et al. [35]	Key performance indicators—in-house or outsourcing	Statistical analysis
Zarbakhshnia et al. [36]	Outsourcing sustainable reverse logistics providers	Fuzzy-AHP and MOORA-G
Kiani et al. [37]	Prioritizing outsourceable activities in universities	Fuzzy MCDM (Fuzzy-AHP, Fuzzy-SAW, Fuzzy-TOPSIS, Fuzzy-VIKOR)
Vazifehdan & Darestani [38]	Evaluation of the drivers of outsourcing for green logistics	Fuzzy-ANP, QFD, and SIR
Sayed et al. [39]	Comparing in-house and outsourcing implementation	Statistical analysis
Mokrini and Aouam [40]	Risk evaluation in healthcare logistics outsourcing	Fuzzy-AHP, Fuzzy-TOPSIS, Fuzzy-PROMETHEE
Tavana et al. [41]	Outsourcing reverse logistics activities	Fuzzy-AHP and SWOT
Wu et al. [42]	Supplier selection in nuclear power industry	Extended VIKOR method
Our study	Freight distribution concept selection—in-house or outsourcing	Picture fuzzy ARAS method

.

2.2. Applications of the ARAS Method

Although the ARAS method is a new approach in the MCDM literature, it has been applied in various areas (

Table 2. Summary of the available applications of the additive ratio assessment (ARAS) method.

Author(s)	Research Focus	Application Type	Multi-Criteria Group Decision-Making	Parameter Type
				Criterion	Alternative
Tupenaite et al. [43]	Human environment renovation	Real-life	No	Deterministic
Turskis and Zavadskas [44]	Logistics center location	Illustrative example	Yes	Deterministic	Fuzzy
Turskis and Zavadskas [45]	Supplier selection	Illustrative example	No	Interval
Zavadskas and Turskis [10]	Microclimate in offices	Illustrative example	Yes	Deterministic
Zavadskas et al. [11]	Foundation installment	Illustrative example	No	Deterministic
Keršulienė and Turskis [46]	Personnel selection	Illustrative example	Yes	Deterministic	Fuzzy
Baležentis et al. [47]	Economic sector comparison	Real-life	No	Fuzzy
Dadelo et al. [48]	Personnel selection	Illustrative example	Yes	Deterministic
Zavadskas et al. [49]	Construction technology	Real-life	Yes	Deterministic
Zavadskas et al. [50]	Personnel selection	Illustrative example	Yes	Deterministic
Turskis et al. [51]	Built heritage	Real-life	Yes	Deterministic	Interval
Keršulienė and Turskis [52]	Personnel selection	Illustrative example	Yes	Deterministic	Fuzzy
Kutut et al. [53]	Historic building preservation	Real-life	Yes	Deterministic
Zamani et al. [54]	Brand extension strategy selection	Real-life	Yes	Deterministic	Fuzzy
Medineckiene et al. [55]	Sustainable building certification	Illustrative example	No	Deterministic
Stanujkic [56]	Website evaluation	Illustrative example	Yes	Fuzzy	Interval fuzzy
Zavadskas et al. [57]	Seaport location	Real-life	Yes	Deterministic	Fuzzy
Liao et al. [58]	Green supplier selection	Illustrative example	Yes	Deterministic	Fuzzy
Nguyen et al. [59]	Conveyor selection	Illustrative example	Yes	Fuzzy
Štreimikienė et al. [60]	Electricity generation technology	Real-life	Yes	Deterministic
Rostamzadeh et al. [12]	Supply chain performance measurement	Illustrative example	Yes	Fuzzy
Büyüközkan and Göçer [61]	Digital supply chain supplier selection	Real-life	Yes	Interval intuitionistic fuzzy
Dahooie et al. [62]	Personnel selection	Illustrative example	Yes	Deterministic	Interval
Dahooie et al. [63]	Oil and gas well drilling	Real-life	Yes	Deterministic	Interval fuzzy
Radović et al. [64]	Transportation company	Real-life	Yes	Deterministic	Rough interval
Bahrami et al. [65]	Mineral potential mapping	Real-life	Yes	Deterministic
Dahooie et al. [66]	Financial performances	Real-life	No	Deterministic	Fuzzy
Iordache et al. [67]	Hydrogen storage site selection	Real-life	Yes	Interval type-2 hesitant fuzzy
Fu [68]	Catering supplier selection	Real-life	Yes	Deterministic
Naicker and Thopil [69]	Renewable energy systems	Real-life	Yes	Deterministic
Pehlivan and Gürsoy [70]	Life satisfaction	Real-life	Yes	Fuzzy
Turskis et al. [71]	Structural solutions for buildings	Real-life	Yes	Deterministic	Fuzzy
Ghenai et al. [14]	Renewable energy systems	Illustrative example	Yes	Deterministic
Our study	Freight distribution concept selection	Real-life	Yes	Picture fuzzy

).

Tupenaite et al. [43] utilized the ARAS method to evaluate alternatives for built and human environment renovation. Turskis and Zavadskas [44] developed the fuzzy ARAS method for locating logistics centers. The AHP method was used to determine crisp criteria weights. Turskis and Zavadskas [45] presented the grey ARAS method to solve the supplier selection problem. Zavadskas et al. [11] applied the ARAS method to find the most appropriate and safe foundation installment alternative.

Keršulienė and Turskis [46] combined the stepwise weight assessment ratio analysis (SWARA) and the fuzzy ARAS methods to overcome difficulties in the personnel selection process. Baležentis et al. [47] utilized the fuzzy ARAS method to compare the efficiency of Lithuanian economic sectors. Dadelo et al. [48] used the ARAS method for solving the personnel selection problem. Zavadskas et al. [49] adopted the ARAS method to generate a decision on the most suitable construction technology for installing pile-columns. Zavadskas et al. [50] applied the AHP-ARAS approach to assess project managers in construction processes.

Turskis et al. [51] coupled the AHP and grey ARAS methods to rank built heritage projects. Keršulienė and Turskis [52] combined the AHP and the fuzzy ARAS methods with the fuzzy weighted-product model to assess chief accounting officers. Kutut et al. [53] used the AHP-ARAS approach to prioritize cultural heritage buildings. Zamani et al. [54] integrated the ANP and fuzzy ARAS methods to solve the brand-extension strategy selection problem in the dairy food industry. Medineckiene et al. [55] applied the AHP-ARAS approach for solving the sustainable building certification problem. Stanujkic [56] proposed an interval-valued fuzzy set based ARAS method. Zavadskas et al. [57] coupled the AHP and the fuzzy ARAS methods to rank seaport locations. Liao et al. [58] integrated the AHP-ARAS approach and the multisegment goal programming for solving the green supplier selection problem. Nguyen et al. [59] used the fuzzy AHP-ARAS approach to solve the conveyor selection problem. Štreimikienė et al. [60] applied the AHP-ARAS approach for assessing electricity generation technologies in Lithuania. Rostamzadeh et al. [12] used the fuzzy ARAS method for evaluating supply chain management performance.

Büyüközkan and Göçer [61] developed an interval-valued intuitionistic fuzzy set based AHP-ARAS approach to support the supplier selection process in a digital supply chain. Dahooie et al. [62] combined the SWARA and the grey ARAS methods for choosing the best information technology expert. Dahooie et al. [63] used the interval-valued fuzzy ARAS method to evaluate oil and gas well drilling projects. The fuzzy Delphi and the SWARA methods were used to identify and determine criteria and weights, respectively. Radović et al. [64] suggested a rough ARAS method for evaluating performance indicators of transportation companies.

Bahrami et al. [65] utilized the best-worst method and the ARAS method to calculate weights of criteria and rank mineral deposits, respectively. Dahooie et al. [66] integrated the fuzzy c-means clustering algorithm and the ARAS method to evaluate the financial performances of manufacturing companies. Iordache et al. [67] developed an interval type-2 hesitant fuzzy set based ARAS method to tackle the underground hydrogen storage site selection problem. Fu [68] coupled the AHP-ARAS approach and multichoice goal programming to rank catering suppliers in the airline industry. Naicker and Thopil [69] used the AHP-ARAS approach to highlight renewable technology options. Pehlivan and Gürsoy [70] utilized the fuzzy ARAS method to assess life satisfaction levels. Turskis et al. [71] integrated the AHP and the fuzzy ARAS methods with the fuzzy multiplicative utility function for analyzing structural elements of buildings. Ghenai et al. [14] applied the SWARA-ARAS approach to rank renewable energy systems.

Our assessment of the literature has revealed some noticeable gaps:

• Uncertainty is the key factor influencing the selection of a freight distribution concept. However, uncertainty analysis is mainly ignored in the available studies.
• The available MCDM methods for outsourcing logistics do not take into account the neutral/refusal information.
• The available system analysis methods for selection of FDC are inadequate in situations when decision-makers’ opinions involve more answers, such as yes, abstain, no, and refusal.
• Advanced methodological approaches for solving outsourcing logistics problems, which can capture a higher degree of uncertainty and take into account numerous conflicting criteria, are missing.
• No previous work has elucidated the criteria for the selection of a freight distribution concept.
• Deterministic numbers or type-1 fuzzy sets have been used in the majority of the previous studies for solving outsourcing logistics problems.
• A picture fuzzy set based MCDM approach for solving transportation problems has not been applied in previous research.
• The ARAS method has not been extended before using picture fuzzy sets.

Hence, to fill these gaps with the help of the proposed picture fuzzy ARAS method, this paper solves the FDC selection problem in the Czech scenario.

3. Methodology

In this section, definitions of picture fuzzy sets and the developed picture fuzzy ARAS method are provided.

3.1. Picture Fuzzy Sets

Definition 1

[7,8]. Let PFS A on a universe X be an object in the form of

$$A=\left\{<x,{\mu }_A\left(x\right),{\eta }_A\left(x\right),{\upsilon }_A\left(x\right)>│xϵX\right\},$$

(1)

where ${\mu }_A\left(x\right),{\eta }_A\left(x\right),{\upsilon }_A\left(x\right)\in \left[0, 1\right];$ ${\mu }_A\left(x\right), {\eta }_A\left(x\right)$, and ${\upsilon }_A\left(x\right)$ represent the degrees of positive, neutral, and negative membership of x in A, respectively. The following condition needs to be satisfied:

$$0\le {\mu }_A\left(x\right)+{\eta }_A\left(x\right)+{\upsilon }_A\left(x\right)\le 1,\forall xϵX.$$

(2)

The word “picture” in PFS relates to generality, as this set is the direct extension of fuzzy sets and intuitionistic fuzzy sets (IFSs). In the case where ${\eta }_A\left(x\right)=0,$ the PFS returns to the IFS set. When both ${\eta }_A\left(x\right)={\upsilon }_A\left(x\right)=0,$ the PFS returns to the fuzzy set. The integration of the degree of neutral membership ${\eta }_A\left(x\right)$ measures the information of objects more accurately and increases the quality and accuracy of achieved results. In PFS theory, decision-makers are divided into four groups: vote for (its ratio is denoted as µ), abstain (its ratio is denoted as η), vote against (its ratio is denoted as υ), and refusal (its ratio is denoted as ξ) [72].

The degree of refusal membership of x in the PFS A can be calculated as follows:

$${\xi }_A\left(x\right)=1-\left({\mu }_A\left(x\right)+{\eta }_A\left(x\right)+{\upsilon }_A\left(x\right)\right), \forall xϵX.$$

(3)

In particular, if X has only one element, then$A=\left\{<x,{\mu }_A\left(x\right),{\eta }_A\left(x\right),{\upsilon }_A\left(x\right) > │xϵX\right\}$is called a picture fuzzy number (PFN), in which${\mu }_A,{\eta }_A,{\upsilon }_A\in \left[0, 1\right]$and$0\le {\mu }_A+{\eta }_A+{\upsilon }_A\le 1$. For convenience, a PFN is denoted by$A=< {\mu }_A,{\eta }_A, {\upsilon }_A>.$

Definition 2

[7,8]. The complement of a PFS $A=\left\{<x,{\mu }_A\left(x\right),{\eta }_A\left(x\right),{\upsilon }_A\left(x\right)>│xϵX\right\}$ on a universe X is represented as

$$A^c=\left\{<x,{\upsilon }_A\left(x\right),{\eta }_A\left(x\right),{\mu }_A\left(x\right)>│xϵX\right\}.$$

(4)

Definition 3

[68,69]. Let $A=<{\mu }_A,{\eta }_A, {\upsilon }_A>$, $A_1=<{\mu }_{A_1}, {\eta }_{A_1},{\upsilon }_{A_1}>,$ and $A_2=<{\mu }_{A_2}, {\eta }_{A_2},{\upsilon }_{A_2}>$ be three PFNs, and λ > 0. Their operations are defined as follows:

$$A_1\oplus A_2=<1-(1-{\mu }_{A_1})(1-{\mu }_{A_2}),{\eta }_{A_1}{\eta }_{A_2}, ({\eta }_{A_1}+v_{A_1})({\eta }_{A_2}+v_{A_2})-{\eta }_{A_1}{\eta }_{A_2}>$$

(5)

$$A_1\otimes A_2=<\left({\mu }_{A_1}+{\eta }_{A_1}\right)\left({\mu }_{A_2}+{\eta }_{A_2}\right)-{\eta }_{A_1}{\eta }_{A_2}, {\eta }_{A_1}{\eta }_{A_2}, 1-\left(1-{\upsilon }_{A_1}\right)\left(1-{\upsilon }_{A_2}\right)>,$$

(6)

$$\lambda ·A=<1-{(1-{\mu }_A)}^{\lambda },{({\eta }_A)}^{\lambda }, {\left({\eta }_A{+}_A\right)}^{\lambda }-{({\eta }_A)}^{\lambda }>,$$

(7)

$$A^{\lambda }=<{\left({\mu }_A+{\eta }_A\right)}^{\lambda }-{({\eta }_A)}^{\lambda }, {({\eta }_A)}^{\lambda },1-{\left(1-{\upsilon }_A\right)}^{\lambda }>.$$

(8)

Definition 4

[72,73]. Let $A_i=< {\mu }_{A_i}, {\eta }_{A_i},{\upsilon }_{A_i}>$ (i = 1, ..., m) be a collection of PFNs and λ = (λ₁, ..., λ_m)^T be the weight vector with λ_i∈[0, 1] and ${\displaystyle \sum }_{i=1}^m{\lambda }_i=1$. The picture fuzzy weighted geometric average (PFWGA) operator is defined as follows:

$$PFWGA\left(A_1,\dots ,A_m\right)={\otimes }_{i=1}^m{(A_i)}^{{\lambda }_i}=<{{\displaystyle \prod }}_{i=1}^m{\left({\mu }_{A_i}+{\eta }_{A_i}\right)}^{{\lambda }_i}-{{\displaystyle \prod }}_{i=1}^m{\left({\eta }_{A_i}\right)}^{{\lambda }_i}, {{\displaystyle \prod }}_{i=1}^m{\left({\eta }_{A_i}\right)}^{{\lambda }_i}, 1-{{\displaystyle \prod }}_{i=1}^m{\left(1-{\upsilon }_{A_i}\right)}^{{\lambda }_i}>.$$

(9)

Definition 5

[74,75]. Let $A=<{\mu }_A,{\eta }_A, {\upsilon }_A>$ be a PFN. A two-step defuzzification method to obtain a crisp value of the PFN A is given as follows:

Step 1. Distribute the neutral degree to the positive and negative degrees as follows:

$${\mu }_A^{’}={\mu }_A+\frac{{\eta }_A}{2},$$

(10)

$${\upsilon }_A^{’}={\upsilon }_A+\frac{{\eta }_A}{2}.$$

(11)

Step 2. Calculate the defuzzification value y by

$$y={\mu }_A^{’}+\xi \frac{1+{\mu }_A^{’}-{\upsilon }_A^{’}}{2}.$$

(12)

3.2. Picture Fuzzy ARAS Method

Let A = {A₁, …, A_m} (m ≥ 2) be a finite set of alternatives which experts have to choose from, C = {C₁, …, C_n} (n ≥ 2) be a finite set of criteria with which performances of the alternatives can be measured, $C_{s_j}=\left\{C_{J1},\dots ,C_{j{n}_j}\right\}$ be a finite set of sub-criteria with respect to the j-th criterion C_j, and D = {D₁, …, D_k} (k ≥ 2) be a set of invited experts. The steps of the developed picture fuzzy ARAS method are given as follows:

Step 1. Construct the linguistic criteria weight matrices ${\Psi }^e={\left[{\psi }_j^e\right]}_{n\times 1}$:

$${\Psi }^e=\begin{array}{c}{C}_1\\ ⁞\\ C_n\end{array}\left[\begin{array}{c}{\psi }_1^e\\ ⁞\\ {\psi }_n^e\end{array}\right], e=1,\dots ,k,$$

(13)

where ${\psi }_j^e$ is the linguistic importance evaluation given by the expert D_e (e = 1, ..., k) towards the criterion C_j (j = 1, ..., n). Generally, importance evaluations can be yes, abstain, no, and refusal. For instance, if an expert has a positive, neutral, or negative attitude towards criterion importance, then they will answer yes, abstain, or no in a questionnaire, respectively. Therefore, experts are divided into four groups.

Step 2. Determine the picture fuzzy criteria weight matrix $V={\left[v_j\right]}_{n\times 1}$:

$$V=\begin{array}{c}{C}_1\\ ⁞\\ C_n\end{array}\left[\begin{array}{c}{v}_1\\ ⁞\\ v_n\end{array}\right],$$

(14)

where $v_j=<{\mu }_j,{\eta }_j,{\upsilon }_j>$ is a PFN that represents an importance evaluation of the criterion C_j given by the experts. The four types of voting results are fully in accordance with the four components of a PFN. Criteria importance evaluations given by the experts can be expressed as PFNs by calculating the proportion of each item in the voting results.

Step 3. Calculate the weight of each criterion as follows:

$$w_j=\frac{{\mu }_j+\frac{{\eta }_j}{2}+\frac{{\xi }_j}{2}\left(1+{\mu }_j-{\upsilon }_j\right)}{{{\displaystyle \sum }}_{l=1}^n\left[{\mu }_l+\frac{{\eta }_l}{2}+\frac{{\xi }_l}{2}\left(1+{\mu }_l-{\upsilon }_l\right)\right]}, j=1,\dots ,n,$$

(15)

where white papers are divided into half, i.e., one half for the experts who vote for and the other half for the experts who vote against; ${\xi }_j=1-{\mu }_j-{\eta }_j-{\upsilon }_j$ (j = 1, ..., n) is the ratio of experts who refuse to provide importance evaluation towards the criterion C_j; and w_j∈[0, 1] (j = 1, ..., n), ${\displaystyle \sum }_{j=1}^n{w}_j=$1.

Step 4. Construct the linguistic sub-criteria weight matrices ${\Phi }^e={\left[{\varphi }_{j{s}_j}^e\right]}_{n\times 1}$:

$${\Phi }_j^e=\begin{array}{c}{C}_{j1}\\ ⁞\\ C_{j{n}_j}\end{array}\left[\begin{array}{c}{\varphi }_{j1}^e\\ ⁞\\ {\varphi }_{j{n}_j}^e\end{array}\right], j=1,\dots ,n; e=1,\dots ,k,$$

(16)

where ${\varphi }_{j{s}_j}^e$ is a linguistic importance evaluation given by the expert D_e towards the sub-criterion $C_{s_j}$ (s_j = 1, ..., n_j) of the criterion C_j. Generally, if an expert has a positive, neutral, or negative attitude towards the importance of some sub-criterion, then they will answer yes, abstain, or no in a questionnaire, respectively.

Step 5. Determine the picture fuzzy sub-criteria weight matrices $O_j={\left[o_{j{s}_j}\right]}_{n_j\times 1}$:

$$O_j=\begin{array}{c}{C}_{j1}\\ ⁞\\ C_{j{n}_j}\end{array}\left[\begin{array}{c}{o}_{j1}\\ ⁞\\ o_{j{n}_j}\end{array}\right], j=1,\dots ,n,$$

(17)

where $o_{j{s}_j}=<{\mu }_{j{s}_j},{\eta }_{j{s}_j},{\upsilon }_{j{s}_j}>$ is a PFN which represents an importance evaluation of the sub-criterion $C_{s_j}$ of the criterion C_j given by the experts. It is calculated as the proportion of each item in the voting results.

Step 6. Calculate the weight of each sub-criterion as follows:

$${\delta }_{j{s}_j}=\frac{{\mu }_{j{s}_j}+\frac{{\eta }_{j{s}_j}}{2}+\frac{{\xi }_{j{s}_j}}{2}\left(1+{\mu }_{j{s}_j}-{\upsilon }_{j{s}_j}\right)}{{{\displaystyle \sum }}_{s_l=1}^{n_j}\left[{\mu }_{j{s}_l}+\frac{{\eta }_{j{s}_l}}{2}+\frac{{\xi }_{j{s}_l}}{2}\left(1+{\mu }_{j{s}_l}-{\upsilon }_{j{s}_l}\right)\right]}, j=1, \dots , n; s_j=1,\dots ,n_j,$$

(18)

where ${\delta }_s{}_jϵ\left[0,1\right]$ (s_j = 1, ..., n_j) and ${\displaystyle \sum }_{s_j=1}^{n_j}{\delta }_s{}_j=1$.

Step 7. Construct the linguistic evaluation matrices ${\Gamma }_{ij }={\left[{\gamma }_{ij{s}_j}^e\right]}_{n_j\times k}$:

$${\Gamma }_{ij}=\begin{array}{c}{C}_{j1}\\ ⋮\\ C_{j{n}_j}\end{array}\stackrel{D_1 \cdots D_k}{\left[\begin{array}{ccc}{\gamma }_{ij1}^1& \cdots & {\gamma }_{ij1}^k\\ ⋮& \ddots & ⋮\\ {\gamma }_{ij{n}_j}^1& \cdots & {\gamma }_{ij{n}_j}^k\end{array}\right]}, i=1,\dots ,m;j=1,\dots ,n,$$

(19)

where ${\gamma }_{ij{s}_j}^e$ is a linguistic evaluation given by the expert D_e towards the alternative A_i (i = 1, ..., m) with respect to the sub-criterion $C_{s_j}$ of the criterion C_j. Generally, if an expert has a positive, neutral, or negative attitude towards an alternative with respect to some sub-criterion, then they will answer yes, abstain, or no in a questionnaire, respectively.

Step 8. Determine the picture fuzzy evaluation matrices $Z_j={\left[z_{ij{s}_j}\right]}_{m\times n_j}$:

$$Z_j=\begin{array}{c}{A}_1\\ ⁞\\ A_m\end{array}\stackrel{C_{j1 \cdots C_{j{n}_j}}}{\left[\begin{array}{ccc}{z}_{1j1}& \cdots & z_{1j{n}_j}\\ ⋮& \ddots & ⋮\\ z_{mj1}& \cdots & z_{mj{n}_j}\end{array}\right]}, j=1,\dots ,n,$$

(20)

where $z_{ij{s}_j}=<{\mu }_{ij{s}_j},{\eta }_{ij{s}_j},{\upsilon }_{ij{s}_j}>$ is a PFN which represents an evaluation of the alternative A_i with respect to the sub-criterion $C_{s_j}$ of the criterion C_j given by the experts. It is calculated as the proportion of each item in the voting results.

Step 9. Determine the picture fuzzy normalized evaluation matrices $R_j={\left[r_{ij{s}_j}\right]}_{m\times n_j}$ as follows:

$$r_{ijsj}=\{\begin{array}{c}{Z}_{ijsj}=<{\mu }_{ijsj},{\eta }_{ijsj},v_{ijsj}>\mathrm{if} C_{sj}\mathrm{is} \mathrm{a} \mathrm{benefit} \mathrm{sub}-\mathrm{criterion}\\ {\left(Z_{ijsj}\right)}^c=<v_{ijsj},{\eta }_{ijsj},{\mu }_{ijsj}>\mathrm{if} C_{sj}\mathrm{is} \mathrm{a} \mathrm{cost} \mathrm{sub}-\mathrm{criterion}\end{array}, i=1,\dots ,m; j=1,\dots n; s_j=1,\dots ,n_j,$$

(21)

where $r_{ij{s}_j}$ denotes a normalized evaluation of the alternative A_i with respect to the sub-criterion $C_{s_j}$ of the criterion C_j given by the experts. Only experts’ evaluations with respect to cost sub-criteria are transformed by utilizing the complement operation.

Step 10. Determine the picture fuzzy decision matrix $Q={\left[q_{ij}\right]}_{m\times n}$:

$$Q=\begin{array}{c}{A}_1\\ ⋮\\ A_m\end{array}\stackrel{C_1 \cdots C_n}{\left[\begin{array}{ccc}{q}_{11}& \cdots & q_{1n}\\ ⋮& \ddots & ⋮\\ q_{m1}& \cdots & q_{mn}\end{array}\right]},$$

(22)

where $q_{ij}=<{\mu }_{q_{ij}}, {\eta }_{q_{ij}}, {\upsilon }_{q_{ij}}>$ is a PFN which represents aggregated normalized evaluation of the alternative A_i with respect to the criterion C_j. It is calculated as follows:

$$\begin{gathered} q_{ij}=<{\mu }_{q_{ij}}, {\eta }_{q_{ij}}, {\upsilon }_{q_{ij}}\ge PFWGA\left(r_{ij1},\dots ,r_{ij{n}_j}\right)={\otimes }_{S_j=1}^{n_j}{\left(r_{ij{s}_j}\right)}^{{\delta }_{s_j}} \\ =<{{\displaystyle \prod }}_{s_j=1}^{n_j}{\left({\mu }_{ij{s}_j}\right)}^{{\delta }_{s_j}}, 1-{{\displaystyle \prod }}_{s_j=1}^{n_j}{\left(1-{\eta }_{ij{s}_j}\right)}^{{\delta }_{s_j}}, 1-{{\displaystyle \prod }}_{s_j=1}^{n_j}{\left(1-{\upsilon }_{ij{s}_j}\right)}^{{\delta }_{s_j}}>, i=1,\dots ,m;j=1,\dots n, \end{gathered}$$

(23)

where ${\delta }_j={\left({\delta }_1,\dots ,{\delta }_{n_j}\right)}^T$(j = 1, ..., n) is the weight vector of the sub-criteria of the criterion C_j with ${\delta }_s{}_jϵ\left[0,1\right]$ and ${\displaystyle \sum }_{s_j=1}^{n_j}{\delta }_s{}_j=1$.

Step 11. Determine the picture fuzzy weighted decision matrix $G={\left[g_{ij}\right]}_{m\times n}$ as follows:

$$g_{ij}=w_j·q_{ij}=<1-{\left(1-{\mu }_{ij}\right)}^{w_j},{\left({\eta }_{ij}\right)}^{w_j},{({\eta }_{ij}+{\upsilon }_{ij})}^{w_j}-{\left({\eta }_{ij}\right)}^{w_j}>, i=1,\dots ,m;j=1,\dots n,$$

(24)

where $g_{ij}=<{\mu }_{g_{ij}},{\eta }_{g_{ij}},{\upsilon }_{g_{ij}}>$ is a PFN which represents an aggregated normalized weighted evaluation of the alternative A_i with respect to the criterion C_j and $w={\left(w_1\dots w_n\right)}^T$ is the weight vector of the criteria, with $w_jϵ\left[0,1\right]$ and ${\displaystyle \sum }_{j=1}^n{w}_j=1$.

Step 12. Determine the ideal alternative $A_0=\left\{g_{01},\dots ,g_{0n}\right\}$ as follows:

$$g_{0j}=<{\mu }_{g_{0j}},{\eta }_{g_{0j}},{\upsilon }_{g_{0j}}>=<ma{x}_i{\mu }_{g_{ij}},mi{n}_i{\eta }_{g_{ij}},mi{n}_i{\upsilon }_{g_{ij}}>, j=1,\dots ,n.$$

(25)

Step 13. Calculate the picture fuzzy optimality function as follows:

$$S_i=<{\mu }_{s_i},{\eta }_{s_i},{\upsilon }_{s_i}>={\oplus }_{j=1}^n{g}_{ij}=<1-{{\displaystyle \prod }}_{j=1}^n\left(1-{\mu }_{g_{ij}}\right), {{\displaystyle \prod }}_{j=1}^n{\eta }_{g_{ij}}, {{\displaystyle \prod }}_{j=1}^n\left({\eta }_{g_{ij}}+{\upsilon }_{g_{ij}}\right)-{{\displaystyle \prod }}_{j=1}^n{\eta }_{g_{ij}}>, i=0, \dots , m.$$

(26)

Step 14. Calculate the defuzzified value of the optimality function as follows:

$$E_i={\mu }_{S_i}+\frac{{\eta }_{s_i}}{2}+\frac{{\xi }_{s_i}}{2}\left(1+{\mu }_{S_i}-{\upsilon }_{s_i}\right), i=0,\dots ,m,$$

(27)

where neutral degrees are distributed equally to positive and negative degrees.

Step 15. Calculate the utility degree of each alternative as follows:

$$B_i=\frac{E_i}{E_0}, i=1,\dots ,m.$$

(28)

Step 16. Rank the alternatives according to the decreasing values of their utility degree. The highest utility degree is the most desirable alternative.

The flowchart of the proposed picture fuzzy ARAS method is presented in Figure 1.

4. Case Study

The developed PFS-based ARAS method is applied to the freight distribution concept selection problem for a tire manufacturing company in the Czech Republic.

The proposed methodology was tested in the case of a tire manufacturing company from the Czech Republic. The company name is not mentioned according to the agreement with its management; however, it is one of the major players in the Czech tire market. The main activities of the company are manufacturing, wholesaling, and retailing of tires. It produces a wide range of tires, mostly for trucks and passenger cars. In its warehouse, there are more than 90,000 pieces of tires available to its customers at each moment. Some of the major benefits of the company are reflected through the high level of service, professional staff, online ordering system, and delivery of goods within 24 h across the Czech Republic. The manufacturing company considers the possibilities of tire distribution. According to a discussion with the logisticians working in the company, the authors of this paper defined the criteria important for the selection of FDC that would be of the greatest interest of the company and society.

Five experts from the considered tire manufacturing company participated in this case study. They hold managerial positions to directly influence the selection of FDC. Interviews were carried out with them to collect linguistic importance evaluations considering criteria, sub-criteria, and alternatives.

The hierarchical structure of the freight distribution concept selection problem is presented in Figure 2. As can be noticed in Figure 2, there are 23 sub-criteria under the four criteria.

The economical criterion (C₁) has seven sub-criteria:

• Vehicle procurement cost (C₁₁). This emerges when the company procures its transport fleet. The cost can be considered as a relatively high burden, especially for start-ups.
• Vehicle maintenance cost (C₁₂). This cost is expressed in terms of parts consumption and maintenance. It is the amount spent for servicing during transport fleets’ life cycle.
• Time to achieve the equilibrium point of investment (C₁₃). The period after which the invested funds start to bring benefits to the company.
• Financial performance (C₁₄). Indication of the company’s endurance. Sound financial performance ensures the stability of services.
• Human capital cost (C₁₅). This cost is related to employees, such as costs for drivers, dispatchers, training, etc. The core aim of outsourcing logistics is to decrease this cost.
• Workspace cost (C₁₆). This includes the costs of garages, parking places, warehouses, and other auxiliary facilities.
• Number of kilometers driven (C₁₇). If the company expects that its distribution system would require a relatively high number of kilometers to cover, then it should invest in its own fleet of vehicles.

The environmental criterion (C₂) has five sub-criteria:

• Air pollution (C₂₁). This represents the percentage of air pollution by a certain transport fleet. Emissions could vary in proportion to the alternative that is selected.
• Noise pollution (C₂₂). This has a negative impact on both natural ecosystems and urban populations. It causes discomfort, complaints, sleep disorders, etc.
• Effect on public health (C₂₃). This represents the occurrence of injuries, threats to health and life, fires, explosions, and other hazards. It is important to apply technical/technological and organizational solutions that minimize the effect on public health.
• Energy consumption (C₂₄). The pollution from energy consumption is not only restricted to carbon emissions; other types of air pollution, from smog to acid rain, have harmful effects.
• Vehicle utilization (C₂₅). This is determined mainly by delivery requirements and exploitation of the backhauling and transport fleet management. If each company were to have its own transport fleet, it could result in poor vehicle utilization.

The social criterion (C₃) has five sub-criteria:

• Social reputation (C₃₁). This is based on the social appraisal in terms of prestige in society. Adhering to ethical business practices such as supplying quality products on time and acting according to what is agreed secures a high social reputation.
• Brand building (C₃₂). Successful brand building is essential to introduce new products and services. It can be considered as a catalyst for the development of a modern company.
• Enterprise culture construction (C₃₃). This improves competitiveness by providing a guarantee, feedback, and long-term effect mechanisms.
• Job opportunities (C₃₄). This is the number and quality of jobs created.
• Local community influence (C₃₅). This encompasses service infrastructure, public services, community projects, etc.

The technical criterion (C₄) has six sub-criteria:

• Quality of service (C₄₁). This is the most important factor in obtaining customer loyalty. It is measured by the standard of customer service satisfaction.
• Flexibility (C₄₂). This is the ability to react faster to turbulences in the market.
• Workforce availability (C₄₃). The total number of workers can also influence a decision about purchasing a fleet of vehicles.
• Land availability (C₄₄). The availability of enough land is a vital infrastructure prerequisite.
• Vehicle reliability index (C₄₅). This is the number of reliable vehicles in relation to the total fleet of vehicles.
• Parking space utilization index (C₄₆). This is the ratio of the required number of vehicles to the number of available parking spaces.

5. Results and Discussion

Step 1. Four criteria are evaluated by five invited experts. Linguistic importance evaluations for the criteria are presented in

Table 3. Linguistic importance evaluations for the criteria.

Criterion	Expert
	D1	D2	D3	D4	D5
C1	Yes	Abstain	Yes	Yes	Yes
C2	Yes	Yes	Abstain	No	Abstain
C3	No	No	No	Abstain	Abstain
C4	Abstain	Abstain	Abstain	Abstain	No

. Five linguistic criteria weight matrices are determined by using Equation (13).

Step 2.

Table 4. The picture fuzzy criteria weight matrix and defuzzified values.

Criterion	Degree of Positive Membership	Degree Of Neutral Membership	Degree of Negative Membership	Weight
C1	0.8	0.2	0	0.4286
C2	0.4	0.4	0.2	0.2857
C3	0	0.4	0.6	0.0952
C4	0	0.8	0.2	0.1905

presents the picture fuzzy criteria weight matrix. It is determined based on five linguistic criteria weight matrices with the help of Equation (14). The linguistic importance evaluations of four criteria given by experts (Table 3) are expressed as PFNs by computing the proportion of each item in the voting results.

Step 3. The criteria weights are provided in Table 4. They are computed by using Equation (15).

Step 4. Linguistic importance evaluations given by five invited experts towards sub-criteria are provided in

Table 5. Linguistic importance evaluations for the sub-criteria.

Criterion	Sub-Criterion	Expert
		D1	D2	D3	D4	D5
C1	C11	Yes	Yes	Yes	Abstain	Yes
	C12	Abstain	Abstain	No	Refusal	Yes
	C13	Abstain	Refusal	Abstain	Yes	Refusal
	C14	Yes	Yes	Yes	Yes	Abstain
	C15	Abstain	No	Abstain	Refusal	No
	C16	Abstain	Abstain	Refusal	Abstain	No
	C17	Yes	Yes	Abstain	Abstain	Refusal
C2	C21	Yes	Abstain	No	Yes	Yes
	C22	No	Refusal	Abstain	Abstain	No
	C23	Abstain	No	Refusal	Abstain	Refusal
	C24	Abstain	Abstain	Abstain	Refusal	No
	C25	Yes	Yes	Yes	Yes	Yes
C3	C31	Refusal	Abstain	Yes	Abstain	Yes
	C32	Abstain	Yes	Abstain	Abstain	Refusal
	C33	Abstain	Abstain	Abstain	Refusal	No
	C34	Yes	Abstain	No	Yes	Yes
	C35	Refusal	No	Refusal	Abstain	Yes
C4	C41	Yes	Yes	Yes	Yes	Yes
	C42	Yes	Yes	Yes	Abstain	Yes
	C43	Abstain	Abstain	Refusal	No	Yes
	C44	Refusal	No	Refusal	No	Abstain
	C45	Abstain	Yes	No	Abstain	No
	C46	Abstain	Refusal	No	Refusal	No

. The linguistic sub-criteria weight matrices are constructed with the help of Equation (16).

Step 5. The picture fuzzy sub-criteria weight matrices are given in

Table 6. The picture fuzzy sub-criteria weight matrices and defuzzified values.

Criterion	Sub-Criterion	Degree of Positive Membership	Degree of Neutral Membership	Degree of Negative Membership	Weight
C1	C11	0.8	0.2	0	0.2083
	C12	0.2	0.4	0.2	0.1157
	C13	0.2	0.4	0	0.1481
	C14	0.8	0.2	0	0.2083
	C15	0	0.4	0.4	0.0602
	C16	0	0.6	0.2	0.0880
	C17	0.4	0.4	0	0.1713
C2	C21	0.6	0.2	0.2	0.2593
	C22	0	0.4	0.4	0.0963
	C23	0	0.4	0.2	0.1333
	C24	0	0.6	0.2	0.1407
	C25	1	0	0	0.3704
C3	C31	0.4	0.4	0	0.2517
	C32	0.2	0.6	0	0.2109
	C33	0	0.6	0.2	0.1293
	C34	0.6	0.2	0.2	0.2381
	C35	0.2	0.2	0.2	0.1701
C4	C41	1	0	0	0.3086
	C42	0.8	0.2	0	0.2778
	C43	0.2	0.4	0.2	0.1543
	C44	0	0.2	0.4	0.0679
	C45	0.2	0.4	0.4	0.1235
	C46	0	0.2	0.4	0.0679

. PFNs which represent importance evaluation of the sub-criteria are computed as the proportion of each item in the voting results.

Step 6. The sub-criteria weights are found with the help of Equation (18). The obtained values are provided in Table 6.

Step 7. Linguistic evaluations given by the invited experts towards three alternatives for solving the freight distribution concept selection problem are presented in

Table 7. Linguistic evaluations for the alternatives.

Alternative	Expert	Sub-Criterion
		C11	C12	C13	C14	C15	C16	C17	C21	C22	C23	C24	C25	C31	C32	C33	C34	C35	C41	C42	C43	C44	C45	C46
A1	D1	Y	Y	A	Y	A	N	Y	N	N	A	N	N	A	Y	A	Y	Y	Y	A	Y	N	A	A
	D2	Y	A	Y	R	Y	A	A	R	N	A	N	N	Y	Y	A	A	N	A	Y	Y	N	Y	A
	D3	A	Y	Y	N	A	N	A	A	Y	Y	A	A	A	R	N	Y	A	Y	Y	A	N	N	Y
	D4	Y	Y	Y	A	A	N	R	Y	A	R	N	N	Y	A	Y	Y	Y	Y	A	R	A	A	N
	D5	Y	Y	Y	A	Y	A	Y	A	R	A	N	N	A	A	Y	Y	A	Y	Y	Y	N	A	A
A2	D1	N	N	A	A	N	N	R	A	N	A	Y	Y	A	Y	Y	A	A	A	A	A	A	N	A
	D2	A	N	A	A	N	A	N	R	Y	N	A	A	Y	N	Y	A	N	A	Y	A	Y	A	R
	D3	N	A	N	A	N	N	A	N	R	A	Y	Y	A	N	A	N	R	Y	Y	N	A	Y	A
	D4	N	N	N	A	A	A	N	A	A	R	Y	Y	R	A	Y	Y	A	Y	N	A	N	A	A
	D5	N	N	N	Y	A	A	N	N	A	R	Y	Y	A	Y	N	Y	Y	N	Y	R	A	A	A
A3	D1	Y	A	Y	Y	Y	A	A	N	N	N	Y	A	N	N	A	N	N	Y	A	A	A	Y	A
	D2	A	R	N	N	A	R	Y	A	Y	A	Y	Y	A	N	A	R	N	A	Y	A	Y	A	N
	D3	A	A	A	N	N	A	A	N	A	R	Y	Y	N	N	N	A	R	Y	Y	N	N	A	A
	D4	N	A	Y	Y	A	N	Y	R	A	A	A	N	N	R	Y	N	A	Y	A	A	A	Y	A
	D5	A	Y	Y	A	Y	Y	Y	A	A	Y	A	A	N	A	R	N	N	A	Y	N	A	A	A

Yes: Y; Abstain: A; No: N; Refusal: R.

. Twelve linguistic evaluation matrices are obtained by using Equation (19).

Step 8.

Table 8. The picture fuzzy evaluation matrices.

Criterion	Sub-Criterion	Alternative
		A1	A2	A3
C1	C11	<0.8, 0.2, 0>	<0, 0.2, 0.8>	<0.2, 0.6, 0.2>
	C12	<0.8, 0.2, 0>	<0, 0.2, 0.8>	<0.2, 0.6, 0>
	C13	<0.8, 0.2, 0>	<0, 0.4, 0.6>	<0.6, 0.2, 0.2>
	C14	<0.2, 0.4, 0.2>	<0.2, 0.8, 0>	<0.4, 0.2, 0.4>
	C15	<0.4, 0.6, 0>	<0, 0.4, 0.6>	<0.4, 0.4, 0.2>
	C16	<0, 0.4, 0.6>	<0, 0.6, 0.4>	<0.2, 0.4, 0.2>
	C17	<0.4, 0.4, 0>	<0, 0.2, 0.6>	<0.6, 0.4, 0>
C2	C21	<0.2, 0.4, 0.2>	<0, 0.4, 0.4>	<0, 0.4, 0.4>
	C22	<0.2, 0.2, 0.4>	<0.2, 0.4, 0.2>	<0.2, 0.6, 0.2>
	C23	<0.2, 0.6, 0>	<0, 0.4, 0.2>	<0.2, 0.4, 0.2>
	C24	<0, 0.2, 0.8>	<0.8, 0.2, 0>	<0.6, 0.4, 0>
	C25	<0, 0.2, 0.8>	<0.8, 0.2, 0>	<0.4, 0.4, 0.2>
C3	C31	<0.4, 0.6, 0>	<0.2, 0.6, 0>	<0, 0.2, 0.8>
	C32	<0.4, 0.4, 0>	<0.4, 0.2, 0.4>	<0, 0.2, 0.6>
	C33	<0.4, 0.4, 0.2>	<0.6, 0.2, 0.2>	<0.2, 0.4, 0.2>
	C34	<0.8, 0.2, 0>	<0.4, 0.4, 0.2>	<0, 0.2, 0.6>
	C35	<0.4, 0.4, 0.2>	<0.2, 0.4, 0.2>	<0, 0.2, 0.6>
C4	C41	<0.8, 0.2, 0>	<0.4, 0.4, 0.2>	<0.6, 0.4, 0>
	C42	<0.6, 0.4, 0>	<0.6, 0.2, 0.2>	<0.6, 0.4, 0>
	C43	<0.6, 0.2, 0>	<0, 0.6, 0.2>	<0, 0.6, 0.4>
	C44	<0, 0.2, 0.8>	<0.2, 0.6, 0.2>	<0.2, 0.6, 0.2>
	C45	<0.2, 0.6, 0.2>	<0.2, 0.6, 0.2>	<0.4, 0.6, 0>
	C46	<0.2, 0.6, 0.2>	<0, 0.8, 0>	<0, 0.8, 0.2>

presents the picture fuzzy evaluation matrices. The picture fuzzy evaluations of three different freight distribution concepts are computed as the proportion of each item in the voting results.

Step 9. Twelve picture fuzzy normalized evaluation matrices are presented in

Table 9. The picture fuzzy normalized evaluation matrices.

Criterion	Sub-Criterion	Alternative
		A1	A2	A3
C1	C11	<0, 0.2, 0.8>	<0.8, 0.2, 0>	<0.2, 0.6, 0.2>
	C12	<0, 0.2, 0.8>	<0.8, 0.2, 0>	<0, 0.6, 0.2>
	C13	<0, 0.2, 0.8>	<0.6, 0.4, 0>	<0.2, 0.2, 0.6>
	C14	<0.2, 0.4, 0.2>	<0.2, 0.8, 0>	<0.4, 0.2, 0.4>
	C15	<0, 0.6, 0.4>	<0.6, 0.4, 0>	<0.2, 0.4, 0.4>
	C16	<0.6, 0.4, 0>	<0.4, 0.6, 0>	<0.2, 0.4, 0.2>
	C17	<0.4, 0.4, 0>	<0, 0.2, 0.6>	<0.6, 0.4, 0>
C2	C21	<0.2, 0.4, 0.2>	<0.4, 0.4, 0>	<0.4, 0.4, 0>
	C22	<0.4, 0.2, 0.2>	<0.2, 0.4, 0.2>	<0.2, 0.6, 0.2>
	C23	<0, 0.6, 0.2>	<0.2, 0.4, 0>	<0.2, 0.4, 0.2>
	C24	<0.8, 0.2, 0>	<0, 0.2, 0.8>	<0, 0.4, 0.6>
	C25	<0, 0.2, 0.8>	<0.8, 0.2, 0>	<0.4, 0.4, 0.2>
C3	C31	<0.4, 0.6, 0>	<0.2, 0.6, 0>	<0, 0.2, 0.8>
	C32	<0.4, 0.4, 0>	<0.4, 0.2, 0.4>	<0, 0.2, 0.6>
	C33	<0.4, 0.4, 0.2>	<0.6, 0.2, 0.2>	<0.2, 0.4, 0.2>
	C34	<0.8, 0.2, 0>	<0.4, 0.4, 0.2>	<0, 0.2, 0.6>
	C35	<0.4, 0.4, 0.2>	<0.2, 0.4, 0.2>	<0, 0.2, 0.6>
C4	C41	<0.8, 0.2, 0>	<0.4, 0.4, 0.2>	<0.6, 0.4, 0>
	C42	<0.6, 0.4, 0>	<0.6, 0.2, 0.2>	<0.6, 0.4, 0>
	C43	<0.6, 0.2, 0>	<0, 0.6, 0.2>	<0, 0.6, 0.4>
	C44	<0, 0.2, 0.8>	<0.2, 0.6, 0.2>	<0.2, 0.6, 0.2>
	C45	<0.2, 0.6, 0.2>	<0.2, 0.6, 0.2>	<0.4, 0.6, 0>
	C46	<0.2, 0.6, 0.2>	<0, 0.8, 0>	<0, 0.8, 0.2>

. Only experts’ evaluations with respect to cost sub-criteria are transformed by utilizing the complement operation (Equation (21)). Cost sub-criteria are vehicle procurement cost (C₁₁), vehicle maintenance cost (C₁₂), time to achieve the equilibrium point of investment (C₁₃), human capital cost (C₁₅), workspace cost (C₁₆), air pollution (C₂₁), noise pollution (C₂₂), effect on public health (C₂₃), and energy consumption (C₂₄).

Step 10. The picture fuzzy decision matrix is provided in

Table 10. The picture fuzzy decision matrix.

Alternative	Criterion
	C1	C2	C3	C4
A1	<0.097, 0.296, 0.567>	<0.152, 0.277, 0.506>	<0.517, 0.376, 0.065>	<0.531, 0.299, 0.14>
A2	<0.419, 0.340, 0.145>	<0.389, 0.281, 0.220>	<0.367, 0.350, 0.204>	<0.367, 0.398, 0.188>
A3	<0.298, 0.356, 0.306>	<0.282, 0.416, 0.23>	<0.012, 0.219, 0.633>	<0.414, 0.482, 0.103>

. Aggregated normalized evaluations of three different freight distribution concepts with respect to each criterion are determined based on Table 6 and Table 9 with the help of Equation (23).

Step 11. The picture fuzzy weighted decision matrix is presented in

Table 11. The picture fuzzy weighted decision matrix.

Alternative	Criterion
	C1	C2	C3	C4
A1	<0.043, 0.593, 0.345>	<0.046, 0.693, 0.240>	<0.067, 0.911, 0.014>	<0.134, 0.795, 0.06>
A2	<0.208, 0.630, 0.104>	<0.131, 0.696, 0.125>	<0.043, 0.905, 0.040>	<0.083, 0.839, 0.064>
A3	<0.141, 0.642, 0.196>	<0.090, 0.778, 0.105>	<0.001, 0.865, 0.119>	<0.097, 0.870, 0.033>

. Its values are determined based on the criteria weights (Table 4) and the aggregated normalized evaluations of alternatives (Table 10) with the help of Equation (24).

Step 12. The ideal alternative is: A₀ = {<0.208, 0.593, 0.104>, <0.131, 0.693, 0.105>, <0.067, 0.865, 0.014>, <0.134, 0.795, 0.033>}. It is determined by using Equation (25).

Step 13. The picture fuzzy optimality functions of the ideal alternative and three different freight distribution concepts are calculated with the help of Equation (26). The values can be found in

Table 12. Optimality function, utility degree, and alternative ranking.

Alternative	Optimality Function		Utility Degree	Rank
	Picture Fuzzy Value	Crisp Value
A0	<0.444, 0.283, 0.112>	0.685	−	−
A1	<0.262, 0.298, 0.395>	0.431	0.629	3
A2	<0.396, 0.333, 0.18>	0.617	0.901	1
A3	<0.295, 0.376, 0.28>	0.507	0.740	2

.

Step 14. The picture fuzzy optimality functions are defuzzified by using Equation (27). The obtained crisp values are provided in Table 12.

Step 15. The utility degrees of three different freight distribution concepts can be found in Table 12. These values are calculated with the help of Equation (28).

Step 16. Six alternative locations are ranked according to the decreasing values of their utility degree (Table 12). The ordering is A₂ > A₃ > A₁. Hence, according to the proposed picture fuzzy ARAS method, “freight distribution provided by 3PL” is the best freight distribution concept for the tire manufacturing company.

For validating the developed picture fuzzy ARAS method, the freight distribution concept selection problem in the Czech scenario is solved with all available PFS-based MCDM methods.

Table 13. The comparison of different PFS-based MCDM methods.

Method		Alternative
		A1	A2	A3
Picture Fuzzy ARAS (our study)	Score	0.629	0.901	0.740
	Rank	3	1	2
Picture fuzzy TOPSIS (Torun and Gördebil [76])	Score	0.296	0.775	0.548
	Rank	3	1	2
Picture fuzzy EDAS (Liang et al. [69]; Zhang et al. [77])	Score	0.160	1.0	0.570
	Rank	3	1	2
Picture fuzzy TODIM (Wei [78]; Wang et al. [79])	Score	0.616	1.0	0.0
	Rank	2	1	3
Picture fuzzy VIKOR (Wang et al. [80])	Score	1.0	0.015	0.064
	Rank	3	1	2
Picture fuzzy MABAC (Wang et al. [81])	Score	-0.928	0.344	0.013
	Rank	3	1	2
Picture fuzzy cross-entropy (Wei [82])	Score	0.1	0.009	0.061
	Rank	3	1	2
Picture fuzzy projection (Wei et al. [83])	Score	0.250	0.306	0.287
	Rank	3	1	2
Picture fuzzy grey relational projection (Ju et al. [84])	Score	0.30	0.660	0.541
	Rank	3	1	2
Picture fuzzy grey relational analysis (Liu et al. [9])	Score	0.580	0.855	0.613
	Rank	3	1	2

shows the score and ranking results of the considered PFS-based MCDM methods. The proposed method and eight existing state-of-the-art picture fuzzy MCDM methods have the ideal agreement between themselves since they generate the same ordering of three alternatives for the freight distribution in the Czech scenario; i.e., A₂ > A₃ > A₁.

The Spearman’s rank correlation coefficient is used to check the ranking similarity between the presented PFS-based MCDM method and all available state-of-the-art picture fuzzy MCDM methods. The relationships between the proposed PFS-based ARAS method and picture fuzzy TOPSIS, EDAS, VIKOR, MABAC, cross-entropy, projection, grey relational projection, and grey relational analysis methods are perfect. On the other hand, the correlation coefficient between the proposed PFS-based ARAS method and the PFS-based TODIM method is equal to 0.5. The provided PFS-based ARAS method has 94.4% of ranks matched. Hence, the results of the proposed method are highly consistent with the results of available state-of-the-art picture fuzzy MCDM methods.

In the provided real-life case study, the best alternative for the tire manufacturing company is the freight distribution by a 3PL provider (A₂). In reality, by engaging the 3PL provider, the investigated company can greatly save on costs and time, allowing it to focus on core activities. On the other hand, the FDC by its own transportation fleet (A₁) could pay off only for large-scale companies that generate significant transportations flows. Freight distribution by combining a company’s own transport fleet with 3PL services (A₃) can also present a viable solution for mid-size companies to cut costs and save time. Additionally, companies that collaborate with 3PL providers can learn from them, since they are specialized in distribution activities.

6. Conclusions

The implementation of fuzzy logic in combination with MCDM methods is becoming more and more popular and represents a powerful tool for solving problems related to outsourcing logistics. This paper presents the novel picture fuzzy ARAS method. It is highly reliable and accurate. This is confirmed by its comparison with the existing state-of-the-art picture-fuzzy-based MCDM methods.

The insight into the FDC selection in a tire manufacturing company in the Czech Republic is provided to fully illustrate the real-life applicability of the formulated method. The picture fuzzy ARAS method shows effectiveness in dealing with the highly complex FDC selection problem. The obtained results indicate that the best possible solution is to engage the 3PL provider to carry out the outsourcing activities. It should be highlighted that the managers from the investigated tire manufacturing company agreed to this FDC proposal.

Managers dealing with uncertainty could use the proposed advanced decision-making method to solve the FDC selection problem. The introduced picture-fuzzy-based method is especially useful for a start-up company, where the FDC selection is of crucial importance. The right selection of the FDC can greatly benefit the company, especially in organizing the whole business process, and increase profit and customer satisfaction. This method may be considered as a powerful tool in solving complex decision-making problems.

This study can be seen as an important initiator of future papers in the field. Besides, future studies could apply the developed picture fuzzy ARAS method to solve various MCDM problems.