1. INTRODUCTION

Multiple criteria group decision-making (MCGDM) intends to prioritize a set of candidate alternatives based on a set of evaluative criteria regarding subjective preference and judgments of multiple decision-makers. The linear programming technique for multidimensional analysis of preference (LINMAP) initiated by Srinivasan and Shocker [1] provides an efficient decision-making tool to address MCGDM problems with preference information over alternatives and unknown weights of criteria [2–4]. When decision-makers provide preference relations between pairs of alternatives in the MCGDM process, the LINMAP methods can be effectively applied to resolve the objective weights of criteria and specify the best compromise choice by ranking alternatives. According to preference relations through pairwise comparisons over alternatives, a linear programming model that desires to accomplish the minimum inconsistency is established in the LINMAP procedure to determine the optimal weights of criteria objectively and generate the priority ranking results of alternatives [5–7].

However, the classical LINMAP methods cannot be directly employed to handle the decision-making problems involving fuzzy information because of the uncertainty contained in performance information or evaluation values of candidate alternatives in terms of criteria [8–10]. Accordingly, numerous studies have extended the classical LINMAP for conducting multiple criteria decision analysis in a variety of different fuzzy circumstances. For example, Wan and Li [11] emanated from LINMAP to construct an intuitionistic fuzzy programming method for handling heterogeneous MCGDM containing intuitionistic fuzzy truth degrees. Furthermore, Wan and Li [8] considered the hesitancy degrees about pairwise comparisons as interval-valued intuitionistic fuzzy sets to establish a fuzzy LINMAP-based method for conducting a heterogeneous decision analysis. Zhang et al. [12] utilized the LINMAP and Shapley values to develop an interval-valued intuitionistic fuzzy mathematical programming model to manage uncertain MCGDM problems. Moreover, Zhang et al. [13] established a mathematical programming-based approach to heterogeneous MCGDM involving aspirations and incomplete preference information. Zhang and Xu [10] applied the LINMAP structure to present an interval programming method for handling MCGDM problems with hesitant fuzzy ratings of alternatives and pairwise judgments between alternatives using interval numbers. Wan et al. [14] developed a hesitant fuzzy mathematical programming model to deal with hybrid MCGDM involving hesitant fuzzy truth degrees and incomplete criteria weight information. Xu et al. [15] combined hesitant fuzzy linguistic term sets with LINMAP to provide a hesitant fuzzy linguistic LINMAP method. Liu et al. [5] explored a double-hierarchy hesitant fuzzy linguistic mathematical programming technique for MCGDM. To multiple criteria decision analysis, Gou et al. [6] proposed a hesitant fuzzy linguistic possibility degree-based linear assignment method and made some comparisons with a hesitant fuzzy LINMAP. Song et al. [7] solved a decision-making problem with multi-stage uncertain risk on the grounds of interval grey numbers and an extended LINMAP method. Liao et al. [16] established a linear programming model to address decision-making issues, where the decision-maker's preferences over alternatives in connection with criteria are described as probabilistic linguistic term sets. With the assistance of LINMAP, Qin et al. [17] constructed some linear programming models to solve an MCGDM problem in interval type-2 fuzzy contexts. Haghighi et al. [3] developed a soft computing model involving interval type-2 fuzzy information by the agency of a linear assignment method and LINMAP.

Vagueness and impreciseness are unavoidable uncertainties in human evaluation processes [18]. The concept of Pythagorean fuzzy (PF) sets, initiated by Yager [19–21] and Yager and Abbasov [22], is a powerful tool in handling real-world uncertainty because PF sets slacken the prerequisite in which the sum of membership and nonmembership degrees is less than or equal to one with the square sum is less than or equal to one [23–25]. Accordingly, PF sets allow decision-makers to portray uncertain assessment data agilely and conveniently during the MCGDM process.

The LINMAP methodology has been successfully extended to diverse uncertain representations for conducting multiple criteria decision analysis or handling MCGDM issues under a variety of distinct fuzzy environments. Nonetheless, most of the current LINMAP-based techniques are not applicable to the decision environments under complex uncertainty of Pythagorean fuzziness. In this regard, Wan et al. [9] and Xue et al. [26] put forward new LINMAP-based techniques in PF uncertain circumstances. Wan et al. [9] utilized the information entropy to derive individual subjective criterion weight vectors for multiple decision-makers and synthesized them into a collective one via a cross-entropy optimization model. Subsequently, Wan et al. [9] constructed a PF mathematical programming model for addressing MCGDM problems based on the PF truth degree. Xue et al. [26] defined new PF entropy measures and exploited a PF LINMAP method for solving MCGDM problems. These two studies have made significant contributions to enrich the theory of PF sets and have been applied to diverse fields of green supplier selection of a smelting equipment [9] and the selection of railway project investment [26]. In contrast to abovementioned information entropy-based techniques, this paper adopts the perspective of PF representation to construct the core concept in the PF LINMAP framework. Different from Wan et al. [9] and Xue et al. [26], this paper intends to employ the concept of PF scalar functions from fuzzy rule bases and develop a novel PF LINMAP-based compromising approach to deal with MCGDM problems within PF decision environments.

It is worthwhile to mention that the concept of PF scalar functions presented by Yager [19,20] and Yager and Abbasov [22] can be utilized to facilitate comparisons among complicated PF information. Yager [20] and Yager and Abbasov [22] explored the relationship between complex numbers and Pythagorean membership grades; then they demonstrated that Pythagorean membership grades are regarded as a subclass of complex numbers called Π−i numbers. Moreover, they introduced a mapping, i.e., a PF scalar function, from the Π−i numbers to the unit intervals to enhance the usage of PF sets in the decision-making field. Liu and Zhang [27] further demonstrated several useful properties of PF scalar functions and obtained some partial orders on Π−i numbers based on such functions. Zeng et al. [28] introduced the following four commonly used approaches to comparing and ranking Pythagorean membership grades in PF sets: the approach via score functions [29], the approach via score functions and accuracy functions [30], the approach via closeness indices [31], and the approach via PF scalar functions [20]. Zeng et al. [28] investigated several comparative examples and indicated that the approach by virtue of PF scalar functions is more helpful than the other existing comparison approaches. Consistent with Zeng et al.'s findings, Li and Zeng [32] also validated the effectiveness and superiority of PF scalar functions in comparing complex PF information. Moreover, Li and Zeng [32] and Zeng et al. [28] mentioned that the PF scalar function can thoroughly consider the influence of the fundamental characteristics of PF information. On the grounds of the PF scalar function in terms of certain anchor points of reference, Chen [33] introduced the measurement of PF precedence indices and precedence-based preference functions and further established a new PF preference ranking organization method for enrichment evaluations (PROMETHEE). Chen demonstrated the usefulness and effectiveness of the PF scalar function because its based PF precedence index can simplify the data-processing procedure and avoid the loss of high-order uncertain information. The previous studies indicated the merits of PF scalar functions; it would be desirable to apply the PF scalar function-based method to address intricate decision-making problems. Nevertheless, the concept of PF scalar functions has not employed in the current LINMAP-based methods within PF decision environments. Considering the benefits of the PF scalar function in measuring the magnitudes of Pythagorean membership grades as well as the usefulness of comparisons of PF information, this paper utilizes the PF scalar functions to present PF scalar function-based dominance measures that form the PF LINMAP basis for building a strong foundation of valuable concepts related to PF dominance relations, individual consistency and inconsistency levels, and individual fit measurements.

The primary purpose of this paper is to exploit an innovative PF LINMAP-based compromising approach for coping with MCGDM problems within PF environment. More fundamentally, this paper fully utilizes the essential characteristics of PF sets and provides a PF scalar function-based approach for determining PF dominance relations and building a core structure of the PF LINMAP procedure. First, the idea of PF scalar functions is fully employed to construct two types of PF scalar function-based dominance measures with respective to displaced and fixed reference points. Next, Types I and II comprehensive dominance measures are established for the sake of determining PF dominance relations. Some essential and important properties of the proposed measures are investigated to validate the practical usefulness of the proposed measures in synthetic comparisons for PF information. This paper utilizes the PF scalar function-based approach to specify the overall dominance relations and contrasts the obtained results with the decision-makers' paired preference relations. Based on the outcomes, this paper identifies the individual levels of rank consistency and rank inconsistency for acquiring individual fit measurements, i.e., individual goodness of fit and individual poorness of fit. Based on Types I and II dominance measures, two bi-objective mathematical programming models are formulated with the intention of maximal total group comprehensive dominance measure and minimal the group poorness of fit. To improve the computation efficiency, the bi-objective models are transformed into single-objective parametric linear programming models for simplicity. Two algorithmic procedures of the developed PF LINMAP models are presented to address MCGDM problems in PF contexts. The optimal collective weights of criteria and the individual degrees of violation yielded by the proposed models can facilitate generating the overall priority ranking orders and identifying the best compromise alternative. To examine the feasibility and effectiveness of the developed PF LINMAP approaches in practice, this paper investigates an MCGDM problem involving the railway project investment and conducts some sensitivity analyses and comparative studies.

The significant contributions of this paper are highlighted as four aspects: (i) incorporation of PF scalar functions into the core procedure for enriching the LINMAP-based methodology; (ii) exploitation of useful concepts such as PF scalar function-based (comprehensive) dominance measures and fit measurements; (iii) development of an innovative PF LINMAP-based compromising approach for manipulating subjective preferences over alternatives; and (iv) construction of applicable and effective parametric optimization models to assist multiple decision-makers in tackling MCGDM problems.

The remainder of this article is organized as follows: Section 2 concisely presents several essential concepts related to PF sets, Pythagorean membership grades, and PF scalar functions. Section 3 formulates the MCGDM problem in the uncertain context based on PF sets. Section 4 exploits novel PF scalar function-based dominance measures and explores their useful properties. Next, this section provides comprehensive dominance measures for the sake of acquiring overall dominance of alternatives. Section 5 develops some useful concepts, such as individual levels of rank consistency and inconsistency, individual fit measurements, and group comprehensive dominance measures. Based on the proposed concepts, this section employs the PF scalar function-based approach to originate a novel PF LINMAP methodology via effective parametric optimization models. Section 6 presents a practical decision-making case concerning a railway investment issue to show the implementation procedure of the developed approaches. Section 7 conducts a sensitivity analysis and makes some comparative discussions to explore the influences of relevant parameters and to evaluate the usefulness and strengths of the developed models and techniques. Lastly, Section 8 delivers the conclusions.

2. PRELIMINARY

This section describes preliminary definitions and concepts of PF sets, such as Pythagorean membership grades and PF scalar functions, all of which are necessary for the subsequent study.

Yager [19–21] and Yager and Abbasov [22] initiated a Pythagorean membership grade that belongs to a class of nonstandard membership grades. Furthermore, Li and Zeng [32] and Zeng et al. [28] revealed that a Pythagorean membership grade can be characterized by five parameters comprising the degrees of membership, nonmembership, and indeterminacy, the strength of commitment about membership, and the direction of commitment. Nonetheless, Chen [34] indicated that the degree of indeterminacy and the strength of commitment are dual concepts. Due to the duality of the two parameters, Chen [34,35] suggested the employment of the four parameters consisting of membership, nonmembership, strength, and direction for PF characterization. Zhang and Xu [29] presented a useful mathematical representation of PF sets. Motivated by Zhang and Xu's expressions [29], Chen [34,35] provided an effective definition and contributed a new operationally mathematical representation of PF sets involving Pythagorean membership grades.

3. MCGDM PROBLEM UNDER PF UNCERTAINTY

This section intends to formulate an MCGDM problem under complex uncertainty of Pythagorean fizziness.

Consider an MCGDM problem based on PF sets. Let Z={z1,z2,⋯,zm} represent a discrete set of m m≥2 candidate alternatives. Let C={c1,c2,⋯,cn} be a finite set of n n≥2 evaluative criteria with the weight vector w=(w1,w2,⋯,wn), such that the weight wj∈[0,1] for all j∈{1,2,⋯,n} and ∑j=1nwj=1. Because all of the n criteria belong to salient attributes during an MCGDM process, nonnegative boundary conditions should be imposed into the feasible ranges of the weights. Specifically, this paper designates that wj≥εj for all cj∈C, where εj is a sufficiently small nonnegative value. Note that the weight vector w is unknown and required to be resolved, in which the weight wj satisfies ∑j=1nwj=1 and wj≥εj (εj∈ [0, 1]) for all j∈{1,2,⋯,n}. Let E={e1,e2,⋯,eK} represent the set of decision-makers participated in the MCGDM process.

Let μij k and νij k indicates the degrees of satisfaction and dissatisfaction, respectively, of the decision-maker ek∈E for the alternative zi∈Z in connection with a criterion cj∈C. They are subject to 0≤μij k2+νij k2≤1. The PF evaluative rating of z_i in terms of c_j provided by the decision-maker e_k is represented as a Pythagorean membership grade pij k, in which μij k=rij k⋅cosθij k, νijk=rijk⋅sin(θijk), rij k=μij k2+νij k20.5, and dijk=(π−2⋅θijk)/π for θij k∈[0,π/2]; pij k is expressed as follows:

where i∈{1,2,⋯,m}, j∈{1,2,⋯,n}, and k∈{1,2,⋯,K}. Moreover, the degree of indeterminacy associated with each pij k is τij k=1−μij k2−νij k20.5.

Let P^k denote the decision-maker e_k's PF decision matrix in an MCGDM problem within the PF environment; P^k is concisely expressed as follows:

It is worth noting that the PF evaluative ratings can be conveniently acquired through the medium of applicable linguistic rating scales. Consider a nine-point rating scale as an example. Table 1 depicts some practical and beneficial approaches to evaluate the alternatives by use of nine-point linguistic scales. To be specific, Gündoǧdu [36] introduced a useful nine-point linguistic scale for facilitating the construction of the evaluation matrix based on PF sets and spherical fuzzy sets. Mete [37], Oz et al. [38], and Pérez-Domínguez et al. [39] employed an identical nine-point scale for expressing linguistic evaluations in PF settings. Rani et al. [40] introduced a nine-point linguistic scale for assisting decision-makers' judgments concerning the performance evaluations of alternatives in terms of criteria. Seker and Aydin [41] provided nine-point linguistic terms and their corresponding interval-valued PF numbers that can be used for the evaluation of alternatives with criteria. By applying a middle-point approach, these interval-valued PF numbers can be converted into Pythagorean membership grades, and the obtained results are shown in Table 1.

Decision-makers can express the linguistic evaluation values of alternatives toward each criterion. Afterward, these linguistic terms are transformed into PF evaluative ratings by the agency of the linguistic rating system in Table 1. Accordingly, the PF decision matrices from decision-makers can be determined using an appropriate linguistic rating system (e.g., the linguistic terms given in Table 1).

The decision-maker e_k provides individual preference relations between the alternatives in the set Z with the preference set Ω^k in which Ω^k denotes a set of ordered pairs (ϕ,φ) and is defined as follows:

for k∈{1,2,⋯,K}, where the preference relation zϕ≻_zφ signifies that the alternative z_ϕ is noninferior to the alternative z_φ. Namely, either z_ϕ is preferred to z_φ or z_ϕ and z_φ are evenly preferred for the decision-maker e_k.

Theoretically, the preference set Ω^k contains at most m(m−1)/2 paired comparison judgments. Nonetheless, the decision-makers frequently express incomplete preference judgments toward two alternatives in real-world situations. That is, only partial preference relations are filled in by the decision-makers' subjective judgments. For instance, there are five candidate alternatives in Z, i.e., Z={z1,z2,⋯,z5}. Suppose that the decision-maker e_k provides his/her confirmed preference relations z3≻_z1, z2≻_z4, and z3≻_z4, i.e., Ω^k = {(3, 1), (2, 4), (3, 4)}. In this case, there are only three paired comparison judgments in Ω^k. Another issue may be faced with preference conflicts regarding some decision-makers' ordered pairs. For example, the preference relation (3, 5) (i.e., z3≻_z5) in the set Ω¹ is filled in by the decision-maker e₁'s subjective judgments, while the inconsistent relation (5, 3) (i.e., z5≻_z3) in the set Ω² is filled in by the decision-maker e₂'s subjective judgments. In this regard, this paper would like to propose an effective procedure in the PF LINMAP method for the sake of overcoming the difficulty about incomplete and inconsistent information.

To deal with partial preference relations based on human subjective judgments, this paper devises a scheme that comes as close as possible to meet most of the decision-makers' paired preference relations in developing the PF LINMAP methods. In particular, this paper defines a useful synthetic index, named a comprehensive dominance measure, to evaluate the overall dominance of competing alternatives. The obtained overall dominance relations are contrasted with paired preference relations in the preference sets Ω^k for all k∈{1,2,⋯,K}. No errors are attributed to the consistency between paired preference relations and overall dominance relations; whereas errors are to be minimized through the subsequent PF LINMAP models. Moreover, this paper presents new consistency and inconsistency measurements for handling incomplete and inconsistent information among the sets Ω^k in the proposed PF LINMAP procedure.

4. DOMINANCE MEASURE VIA PF SCALAR FUNCTIONS

This section introduces different points of reference for PF information and establishes two types of PF scalar function-based dominance measures for facilitating intracriterion comparisons of PF evaluative ratings.

In general, different points of reference might have distinct influences on the contrast of various PF evaluative ratings, which would result in the change of the decision-makers' preference intensities. In this respect, this paper considers two types of reference points to define the PF scalar function-based dominance measures. First, the displaced reference points (i.e., the positive-ideal and negative-ideal reference points) under anchored judgments are utilized to construct the Type I dominance measure. Second, this paper regard the largest Pythagorean membership grade (1, 0; 1, 1) and the smallest grade (0, 1; 1, 0) as the fixed reference points for presenting the Type II dominance measure.