A T-spherical fuzzy ELECTRE approach for multiple criteria assessment problem from a comparative perspective of score functions

Abstract

The theory involving T-spherical fuzziness provides an exceptionally good tool to efficiently manipulate the impreciseness, equivocation, and vagueness inherent in multiple criteria assessment and decision-making processes. By exploiting the notions of score functions and distance measures for complex T-spherical fuzzy information, this paper aims to propound an innovational T-spherical fuzzy ELECTRE (ELimination Et Choice Translating REality) approach to handling intricate and convoluted evaluation problems. Several newly-created score functions are employed from the comparative perspective to constitute a core procedure concerning concordance and discordance determination in the current T-spherical fuzzy ELECTRE method. By the agency of a realistic application, this paper appraises the usefulness and efficacy of available score functions in the advanced ELECTRE mechanism under T-spherical fuzzy uncertainties. This paper incorporates two forms of Minkowski distance measures into the core procedure; moreover, the effectuality of the advocated measure in differentiating T-spherical fuzzy information is validated. The effectiveness outcomes of the evolved method have been investigated through the medium of an investment decision regarding potential company options for extending the business scope. The real-world application also explores the comparative advantages of distinct score functions in tackling multiple criteria decision-making tasks. Finally, this paper puts forward a conclusion and future research directions.

Keywords

T-spherical fuzziness multiple criteria assessment decision-making score function T-spherical fuzzy elimination and choice translating reality (ELECTRE)

1 Introduction

Uncertainties frequently arise in down-to-earth decision-making processes because of operational complexity in expressing subjective judgments and intrinsic equivocation in human consciousness [13 , 40]. To inquire into imprecise, indistinct, and ambiguous contents in uncertain circumstances, the current theories of fuzzy mathematical models have been generally recognized and exploited for problem solving and intelligent decision support [17 , 43]. Notably, a variety of extensions regarding ordinary fuzzy sets have been propounded throughout most of the decision making under uncertainty, but have been particularly active in the decades following the introduction of non-standard fuzzy models such as the mathematical configurations involving intuitionistic fuzziness [5], Pythagorean fuzziness [41], picture fuzziness [9], q-rung orthopair fuzziness [42], Fermatean fuzziness [37], spherical fuzziness [19], and T-spherical fuzziness [28].

In relation to the foregoing non-standard fuzzy models, the intuitionistic fuzzy (IF), Pythagorean fuzzy (PyF), Fermatean fuzzy (FF), and q-rung orthopair fuzzy (q-ROF) configurations symbolize uncertain information via delineating a membership value and a nonmembership value, wherein the ranges of the two values are between 0 and 1. Moreover, the notion of q-ROF sets is a generalized version of the remainder [3 , 42]. On the flip side, the picture fuzzy (PF), spherical fuzzy (SF), and T-spherical fuzzy (T-SF) configurations externalize uncertain information via explicating a membership value, an abstinence value, and a nonmembership value, wherein the ranges of the three functions are between 0 and 1. It is worthwhile to notice that the notion of T-SF sets can be a generalized version of the aforesaid fuzzy models [16 , 45]. By way of explanation, for a positive integer t, the sum of three functions to the t-th power belongs to [0, 1] in the T-SF configuration [24, 28]. A T-SF set becomes a PF set and a SF set in cases where t = 1, 2, respectively. Furthermore, if the value of an abstinence function is equal to 0, then the T-SF configuration is mathematically equivalent to the q-ROF configuration. As a consequence of this, the theory of T-SF sets plays a leading role because of its generality of several non-standard fuzzy models [12 , 39].

Since the creation of the notion of T-SF sets, there have been significant developments in the multiple criteria decision-making discipline leaning on soft computing and intelligent informatics. As an illustration, Garg et al. [12] exploited a T-SF power aggregation operator-based method with the aim of treating multiple criteria assessment tasks. Guleria and Bajaj [15] utilized new measures of correlation coefficients in T-SF settings and built a multiple criteria evaluation approach. Ju et al. [22] brought forward a T-SF interactive decision-making method supported by interaction aggregation operators in T-SF contexts. Liu et al. [24] addressed multiple criteria evaluation problems through the agency of complex T-SF 2-tuple linguistic Muirhead mean operations. Mahmood et al. [26] amalgamated interval-valued T-SF sets and soft sets to advance average aggregation operators in an effort to prioritize multiple available alternatives. Based on T-SF theory, Mahmood et al. [27] unfolded a generalized multiple objective optimization model supported by ratio analyses plus a full multiplicative form (MULTIMOORA) for decision aiding. Munir et al. [31] took advantage of interactive geometric aggregation operators and associated immediate probability to construct a T-SF decision-making method. Ölü and Karaaslan [33] amalgamated the type-2 hesitant fuzzy model into a T-SF set and launched a correlation coefficient-based decision-making approach via such a hybrid fuzzy model.

Although there are numerous advanced studies in the literature exploiting the T-SF theory for decision-making related problems, these studies have not developed a T-SF elimination and choice translating reality (ELECTRE) methodology to process exceptionally equivocation and indeterminateness in decision situations. The ELECTRE brings forward a family of outranking decision-aiding methods [21, 29], such as the evolutions of ELECTRE I−IV, IS, and TRI [2,25, 2,25]. The particularity possessed by ELECTRE involves the composition of outranking relations and the investigation procedure for choosing, ranking, or sorting in most cases [2,25,29,35 , 2,25,29,35]. Nonetheless, when the state of uncertainty is high, it would be advantageous to utilize T-spherical fuzziness for properly structuring the multiple criteria assessment problem and explicitly appraising available alternatives in terms of multiple criteria. However, the ELECTRE-related literature under T-SF uncertainties is limited. To deal with this research gap, this paper attempts to advance a T-SF version of the ELECTRE approach and furnish an advantageous means that instructs decision makers to the most appropriate alternative for enhancing decision quality. On the flip-side, another challenge confronting the ELECTRE-based techniques in T-SF settings is how to select an appropriate score function to handle the T-spherical fuzziness from a comparative perspective. Nevertheless, few studies discussed the usefulness and effectuality of various T-SF score functions in the ELECTRE mechanism. These considerations constitute a forceful motivation for conducting this study.

The main purpose of this research is to propound an innovational T-SF ELECTRE methodology for handling intricate and convoluted evaluation problems through the agency of the concepts of score functions and distance measures for complex T-SF information. From a viable viewpoint of different T-SF score functions, this paper configures a multiple criteria evaluation problem within T-SF environments. To doing this, grades of satisfaction, neutral satisfaction, and dissatisfaction are exploited to ascertain the positive, neutral, and negative membership grades, respectively, embedded in T-SF evaluation values. This paper exploits some newly-evolved T-SF score functions to constitute a core procedure regarding concordance and discordance determination in the proposed T-SF ELECTRE methodology.

To be specific, by virtue of various types of T-SF score functions, this paper identifies two collections of concordance and discordance with respect to a contrasting effect. This paper puts forward a score function-dependent concordance index using weighted differences of score functions. Over and above that, two forms of Minkowski distance measures are presented in T-SF settings, consisting of a current three-term representation and a new four-term representation. Additionally, this paper defines a Minkowski distance-dependent discordance index using weighted Minkowski distances. This paper provides two prioritization procedures in the T-SF ELECTRE approach. In the proposed T-SF ELECTRE I prioritization procedure, the concordance Boolean matrix, discordance Boolean matrix, and prioritization Boolean matrix are constructed to delineate a domination graph for partial-priority rankings of available alternatives. In the evolved T-SF ELECTRE II prioritization procedure, the score function-dependent leaving flows and Minkowski distance-dependent entering flows are elucidated to derive the notion of superiority indices for complete-priority rankings among alternatives.

To validate the practicality of the advanced T-SF ELECTRE approach in uncertain circumstances, this paper explores a business investment decision that was originated from Munir et al. [30] for exhibiting the efficaciousness of the solution outcomes yielded by the developed techniques. Moreover, this paper appraises the practicability and efficacy of various score functions in the current T-SF ELECTRE procedure supported by the model application in real decisions. Based on a comparative perspective of score functions, a comprehensive analysis is implemented to give a demonstration of the reasonability and merits of the initiated methodology.

This article is organized in this manner. First, Section 2 introduces preparatory notions in connection with T-SF sets. This section also presents different formulations of score function for T-SF information. Section 3 exploits score functions and two forms of Minkowski distance measures to carry forward a T-SF ELECTRE approach to conducting multiple criteria assessment tasks. Section 4 explored an investment decision issue about potential company options in anticipation of validating the usefulness of the evolved method. Section 5 carries out some systematic comparisons to reveal the favorable features of the current techniques. Section 6 comes to conclusions and suggests promising exploration directions.

2 Preliminaries on T-SF sets

This section provides a preliminary explanation for T-SF sets. Some formulations of score functions for T-SF information are explicated as well.

This paper refers to Mahmood et al. [28] to define basic mathematical notations relevant to T-SF sets. First, designate a finite nonempty set U to represent a universe of discourse; moreover, let Z+ indicate a collection of positive integers. A T-SF set T is constituted by a collection of T-SF numbers. Let t(u) denote a T-SF number of an element u ∈ U belonging to T. They are delineated along these lines: $T = {〈 u, t (u) 〉 | u \in U},$ (1) $t (u) = (μ_{T} (u), η_{T} (u), ν_{T} (u)) .$ (2)

The T-SF set T is symbolized in terms of three functions known as the grade of positive membership μ_T (u) : U → [0, 1], the grade of neutral membership (i.e., abstinence) η_T (u) : U → [0, 1], and the grade of negative membership ν_T (u) : U → [0, 1] of each element u ∈ U associated with T. Herein, for each positive integer t ∈ Z⁺, the three functions must fulfill the following restriction: $0 ⩽ {(μ_{T} (u))}^{t} + {(η_{T} (u))}^{t} + {(ν_{T} (u))}^{t} ⩽ 1 .$ (3)

Let r_T (u) : U → [0, 1] manifest the grade of refusal membership corresponding to a T-SF number t(u). It is identified using the succeeding manner: $r_{T} (u) = \sqrt[t]{1 - {(μ_{T} (u))}^{t} - {(η_{T} (u))}^{t} - {(ν_{T} (u))}^{t}} .$ (4)

A T-SF set T reduces to a PF set [9] and a SF set [19] when the t value is taken as 1 and 2, respectively. Considering the situation where η_T (u) =0, a T-SF set T becomes a q-ROF set [42]; moreover, T reduces to an IF set [5], a PyF set [41], and a FF set [37] when the t value is taken as 1, 2, and 3, respectively. On the grounds of this, the T-SF theory can provide a comprehensive framework of the foregoing non-standard fuzzy models.

Let S(t(u)) denote a score function of a T-SF number t(u). The higher the S(t(u)) value is, the greater the T-SF number t(u) would be. It is noted that formulations of the score function S(t(u)) differ according to miscellaneous aspects of scholars’ view. Referring to newly devoted literature, this paper presents several worthwhile definitions of S(t(u)) below: $S^{I} (t (u)) = {(μ_{T} (u))}^{t} - {(ν_{T} (u))}^{t},$ (5) $S^{II} (t (u)) = {(μ_{T} (u) - η_{T} (u))}^{t} - {(ν_{T} (u) - η_{T} (u))}^{t},$ (6) $\begin{matrix} S^{III} (t (u)) = {(μ_{T} (u))}^{t} - {(η_{T} (u))}^{t} - {(ν_{T} (u))}^{t} \\ + (\frac{e^{{(μ_{T} (u))}^{t} - {(η_{T} (u))}^{t} - {(ν_{T} (u))}^{t}}}{e^{{(μ_{T} (u))}^{t} - {(η_{T} (u))}^{t} - {(ν_{T} (u))}^{t}} + 1} - \frac{1}{2}) {(r_{T} (u))}^{t}, \end{matrix}$ (7)

$S^{IV} (t (u)) = {(μ_{T} (u) - \frac{η_{T} (u)}{2})}^{t} - {(ν_{T} (u) - \frac{η_{T} (u)}{2})}^{t},$ (8)

$S^{V} (t (u)) = {(μ_{T} (u))}^{t} - {(η_{T} (u))}^{t} - {(ν_{T} (u))}^{t},$ (9) $S^{VI} (t (u)) = {(μ_{T} (u))}^{t} - {(ν_{T} (u))}^{t} - {(\frac{η_{T} (u)}{2})}^{t},$ (10) $S^{VII} (t (u)) = \frac{1}{2} [1 + {(μ_{T} (u))}^{t} - {(η_{T} (u))}^{t} - {(ν_{T} (u))}^{t}],$ (11) $S^{VIII} (t (u)) = \frac{1}{3} [2 + {(μ_{T} (u))}^{t} - {(η_{T} (u))}^{t} - {(ν_{T} (u))}^{t}],$ (12) where -1 ⩽ S^I (t (u)) , S^II (t (u)) , ⋯ , S^VI (t (u)) ⩽1, 0 ⩽ S^VII (t (u)) ⩽1, and 1/3 ⩽ S^VIII (t (u)) ⩽1.

The score function S^I(t(u)) presented in Garg et al. [11] is focused on a long-established definition. Because S^I(t(u)) does not involve the grade of neutral membership (i.e., abstinence), several scholars propounded distinct score functions to address such issues. The score functions S^II(t(u)), S^III(t(u)),Λ, S^VIII(t(u)) are formulated in which neutral membership is involved. The score function S^II(t(u)) was originated from the formulation based on SF sets. To be specific, Gündoğdu and Kahraman [18 –20] delineated the score value by way of (μ_T (u) -η_T (u)) ² minus (ν_T (u) - η_T (u)) ² for SF information. Motivated by Gündoğdu and Kahraman [18 –20], this paper displays an extended score function S^II(t(u)) through the utility of (μ_T (u)- η_T (u)) ^t minus (ν_T (u) - η_T (u)) ^t. Zeng et al. [44] exploited a curve function form e^x/(e^x + 1) to propose a new score function S^III(t(u)). Note that the refusal membership r_T (u) is involved in S^III(t(u)).

Moreover, influenced by Gündoğdu and Kahraman [19], the score function S^IV(t(u)) was advanced by Donyatalab et al. [10], where their formulation was originally explicated in SF contexts. The score function S^V(t(u)) was adopted in Garg et al. [12], Liu et al. [24], and Munir et al. [31]. For S^V(t(u)), three membership grades, i.e., positive, neutral, and negative memberships, are utilized for calculating score values. Gul and Yucesan [14] exhibited a score function in the context based on interval-valued T-SF sets. By transforming interval-valued T-SF information into T-SF information, the formulation in Gul and Yucesan [14] gives rise to the score function S^VI(t(u)) that can be suitably employed in T-SF circumstances. The score function S^VII(t(u)) was applied by Ju et al. [22]. Finally, the score function S^VIII(t(u)) was presented in Akram et al. [1] and Barukab et al. [6].

3 A T-SF ELECTRE approach

This section describes the multiple criteria assessment problem, presents two forms of Minkowski distance measures in T-SF settings, and propounds a new T-SF ELECTRE approach for decision support.

Let us structure a multiple criteria evaluation task that focuses on m available alternatives and n considered criteria, wherein m, n ⩾ 2. Specifically, A and C are used to denote a collection of available alternatives {a₁, a₂, ⋯ , a_m} and a collection of considered criteria {c₁, c₂, ⋯ , c_n}, respectively. Relating to a considered criterion c_j ∈ C, let w_j ∈ [0, 1] represent an importance weight with a restriction that $\sum_{j = 1}^{n} w_{j} = 1$ . The collection of all w_j is expressed as a weight vector W = (w₁, w₂, ⋯ , w_n).

Consider a T-SF uncertain environment. Each available alternative’s performance evaluation and overall contribution to a decision objective can be measured through the medium of T-SF set theory. Let μ _ij , η _ij , and ν _ij represent the grade of satisfaction (i.e., positive membership), grade of neutral satisfaction (i.e., neutral membership), and grade of dissatisfaction (i.e., negative membership), respectively, of an alternative a_i ∈ A pertaining to c_j ∈ C. Moreover, the T-SF evaluative rating t_ij = (μ_ij, η_ij, ν_ij). These grades μ_ij, η_ij, ν_ij ∈ [0, 1] can be appraised in accordance with the decision maker’s expertise, experience, and judgment; moreover, they must satisfy 0 ⩽ (μ_ij) ^t + (η_ij) ^t + (ν_ij) ^t ⩽ 1 towards a specific positive integer t ∈ Z⁺. The T-SF characteristic T_i in connection with each alternative a_i is manifested on this wise:

$T_{i} = {〈 c_{j}, t_{ij} 〉 | c_{j} \in C} = {〈 c_{j}, (μ_{ij}, η_{ij}, ν_{ij}) 〉 | c_{j} \in C},$ (13)

where the grade of refusal membership corresponding to t_ij is derived by $r_{ij} = \sqrt[t]{1 - (μ_{ij})^{t} - (η_{ij})^{t} - (ν_{ij})^{t}}$ .

The notion of distance measures provides a proper model in ascertaining the separation between T-SF information. This paper exploits two forms of Minkowski distance measures to efficaciously manipulate T-SF information. Through the utility of positive, neutral, and negative memberships, Ju et al. [22] put forward a Minkowski distance measure between two T-SF numbers. Given two T-SF evaluative ratings t_ij = (μ_ij, η_ij, ν_ij) and t_i′j = (μ_i′j, η_i′j, ν_i′j), the notations $Δ_{μ}^{t} = (μ_{ij})^{t} - (μ_{i^{'} j})^{t}$ , $Δ_{η}^{t} = (η_{ij})^{t} - (η_{i^{'} j})^{t}$ , and $Δ_{ν}^{t} = (ν_{ij})^{t} - (ν_{i^{'} j})^{t}$ are used for simplicity purposes. Let ρ ∈ Z⁺ represent a distance parameter. The Minkowski distance of order ρ between t_ij and t_i′j using a three-term representation is explicated as follows: $D_{ρ}^{3 - term} (t_{ij}, t_{i^{'} j}) = {({| Δ_{μ}^{t} |}^{ρ} + {| Δ_{η}^{t} |}^{ρ} + {| Δ_{ν}^{t} |}^{ρ})}^{\frac{1}{ρ}} .$ (14)

The range of the three-term Minkowski distance is in the real interval [0, 2]. As is well known, the Minkowski distance is a generalized distance measurement. More precisely, when ρ = 1, 2 and ρ→ ∞, the $D_{ρ}^{3 - term} (t_{ij}, t_{i^{'} j})$ becomes the Manhattan, Euclidean, and Chebyshev distances, respectively, as shown: $D_{1}^{3 - term} (t_{ij}, t_{i^{'} j}) = | Δ_{μ}^{t} | + | Δ_{η}^{t} | + | Δ_{ν}^{t} |,$ (15) $D_{2}^{3 - term} (t_{ij}, t_{i^{'} j}) = \sqrt{{| Δ_{μ}^{t} |}^{2} + {| Δ_{η}^{t} |}^{2} + {| Δ_{ν}^{t} |}^{2}},$ (16) $lim_{ρ \to \infty} D_{ρ}^{3 - term} (t_{ij}, t_{i^{'} j}) = max {| Δ_{μ}^{t} |, | Δ_{η}^{t} |, | Δ_{ν}^{t} |} .$ (17)

The second type of representation of the Minkowski distance measure in T-SF contexts takes into account the positive, neutral, negative, and refusal memberships. This paper calls it four term representation of the Minkowski distance for T-SF information. Letting $Δ_{r}^{t} = (r_{ij})^{t} - (r_{i^{'} j})^{t}$ for two T-SF evaluative ratings t_ij and t_i′j, their Minkowski distance of order ρ (ρ ∈ Z⁺) in a four-term representation is explicated in this fashion:

$D_{ρ}^{4 - term} (t_{ij}, t_{i^{'} j}) = {({| Δ_{μ}^{t} |}^{ρ} + {| Δ_{η}^{t} |}^{ρ} + {| Δ_{ν}^{t} |}^{ρ} + {| Δ_{r}^{t} |}^{ρ})}^{\frac{1}{ρ}} .$ (18)

Analogously, the range of the four-term Minkowski distance is in [0, 2]. Based on $D_{ρ}^{4 - term} (t_{ij}, t_{i^{'} j})$ , the Manhattan, Euclidean, and Chebyshev distances are stated precisely in this manner:

$D_{1}^{4 - term} (t_{ij}, t_{i^{'} j}) = | Δ_{μ}^{t} | + | Δ_{η}^{t} | + | Δ_{ν}^{t} | + | Δ_{r}^{t} |,$ (19) $D_{2}^{4 - term} (t_{ij}, t_{i^{'} j}) = \sqrt{{| Δ_{μ}^{t} |}^{2} + {| Δ_{η}^{t} |}^{2} + {| Δ_{ν}^{t} |}^{2} + {| Δ_{r}^{t} |}^{2}},$ (20) $lim_{ρ \to \infty} D_{ρ}^{4 - term} (t_{ij}, t_{i^{'} j}) = max {| Δ_{μ}^{t} |, | Δ_{η}^{t} |, | Δ_{ν}^{t} |, | Δ_{r}^{t} |} .$ (21)

Both forms of the Minkowski distances, i.e., the three- and four-term representations, are valid and acceptable from the mathematical perspective. Nonetheless, from the viewpoint of decision analysis, the outcomes yielded by the two forms $D_{ρ}^{3 - term} (t_{ij}, t_{i^{'} j})$ and $D_{ρ}^{4 - term} (t_{ij}, t_{i^{'} j})$ might differ. This issue will be examined in the comparative analysis of practical applications in details.

The employment of score functions can facilitate comparing complicated T-SF uncertain information. In an effort to identifying a concordance collection and a discordance collection in the evolved T-SF PF ELECTRE procedure, this paper utilizes the aforesaid score functions such as S^I(t_ij), S^II(t_ij),Λ, S^VIII(t_ij) related to each T-SF evaluative rating t_ij for determining the score function-based concordance and discordance collections connected with all couples of available alternatives. Consider that the positive and negative memberships expound the grades of satisfaction and dissatisfaction, respectively. In this regard, higher score functions imply a stronger preference, whereas lower score functions indicate a weaker preference. Given two alternatives a_i and a_i’ belonging to the collection A, the concordance collection $Θ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ and the discordance collection $Θ_{D}^{ϑ} (a_{i}, a_{i^{'}})$ by virtue of the score functions S^ϑ (t_ij) and S^ϑ (t_i′j) (ϑ ∈ {I, II, ⋯ , VIII}) are explicated by: $Θ_{C}^{ϑ} (a_{i}, a_{i^{'}}) = {c_{j} | S^{ϑ} (t_{ij}) ⩾ S^{ϑ} (t_{i^{'} j}) for c_{j} \in C},$ (22) $Θ_{D}^{ϑ} (a_{i}, a_{i^{'}}) = {c_{j} | S^{ϑ} (t_{ij}) < S^{ϑ} (t_{i^{'} j}) for c_{j} \in C} .$ (23)

It is discernible to separate the collection C into two sub-collections $Θ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ and $Θ_{D}^{ϑ} (a_{i}, a_{i^{'}})$ in conjunction with a couple of a_i and a_i′; moreover, they are disjoint. That is, $Θ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ and $Θ_{D}^{ϑ} (a_{i}, a_{i^{'}})$ are an absolute complement of each other. The score function-based concordance and discordance sets fulfill the following properties:

$Θ_{C}^{ϑ} (a_{i}, a_{i^{'}}) \cap Θ_{D}^{ϑ} (a_{i}, a_{i^{'}}) = \emptyset$ .

$Θ_{C}^{ϑ} (a_{i}, a_{i^{'}}) \cup Θ_{D}^{ϑ} (a_{i}, a_{i^{'}}) = C$ .

$Θ_{C}^{ϑ} (a_{i}, a_{i^{'}}) = C ∖ Θ_{D}^{ϑ} (a_{i}, a_{i^{'}})$ .

$Θ_{C}^{ϑ} (a_{i}, a_{i^{'}}) = Θ_{D}^{ϑ} (a_{i^{'}}, a_{i})$ if S^ϑ (t_ij) ≠ S^ϑ (t_i′j).

The foregoing fourth property can be validated in the following manner. In the face of no tied score functions, i.e., S^ϑ (t_ij) ≠ S^ϑ (t_i′j), the concordance collection $Θ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ in Equation (22) can be written as {c_j|S^ϑ (t_ij) > S^ϑ (t_i′j) for c_j ∈ C}. By using Equation (23), the discordance collection $Θ_{D}^{ϑ} (a_{i^{'}}, a_{i}) = {c_{j} | S^{ϑ} (t_{i^{'} j}) < S^{ϑ} (t_{ij}) for c_{j} \in C}$ . Therefore, from these outcomes one can infer that $Θ_{C}^{ϑ} (a_{i}, a_{i^{'}}) = Θ_{D}^{ϑ} (a_{i^{'}}, a_{i})$ if S^ϑ (t_ij) ≠ S^ϑ (t_i′j).

Concerning each couple of a_i and a_i’ (a_i, a_i′ ∈ A and i ≠ i′) and each ϑ ∈ {I, II, ⋯ , VIII}, this paper conceives the score function-dependent concordance index $Φ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ on this wise: $Φ_{C}^{ϑ} (a_{i}, a_{i^{'}}) = \frac{\sum_{c_{j} \in Θ_{C}^{ϑ} (a_{i}, a_{i^{'}})} w_{j} (S^{ϑ} (t_{ij}) - S^{ϑ} (t_{i^{'} j}))}{\sum_{j = 1}^{n} w_{j} (S^{ϑ} (t_{ij}) - S^{ϑ} (t_{i^{'} j}))} .$ (24)

Notably, even though the ranges of the score functions S^I(t_ij), S^II(t_ij),Λ, S^VIII(t_ij) are more or less different, i.e., S^I (t_ij) , S^II (t_ij) , ⋯ , S^VI (t_ij) ∈ [-1, 1], S^VII (t_ij)∈ [0, 1], and S^VIII (t_ij) ∈ [1/3, 1], the score function-dependent concordance index $Φ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ satisfies some common beneficial properties for each ϑ ∈ {I, II, ⋯ , VIII} in case that S^ϑ (t_ij) ≠ S^ϑ (t_i′j):

$Φ_{C}^{ϑ} (a_{i}, a_{i^{'}}) + Φ_{C}^{ϑ} (a_{i^{'}}, a_{i}) = 1$ .

$\sum_{i^{'} = 1, i^{'} \neq i}^{m} \sum_{i = 1, i \neq i^{'}}^{m} Φ_{C}^{ϑ} (a_{i}, a_{i^{'}}) = m (m - 1) / 2$ .

${\bar{Φ}}_{C}^{ϑ} = 0.5$ , where ${\bar{Φ}}_{C}^{ϑ}$ is the average of all $Φ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ values.

The foregoing characteristics can be easily corroborated. In the first place, we are aware of $Θ_{C}^{ϑ} (a_{i}, a_{i^{'}}) = Θ_{D}^{ϑ} (a_{i^{'}}, a_{i})$ because S^ϑ (t_ij) ≠ S^ϑ (t_i′j), which means directly that $Θ_{C}^{ϑ} (a_{i^{'}}, a_{i}) = Θ_{D}^{ϑ} (a_{i}, a_{i^{'}})$ . According to $Θ_{C}^{ϑ} (a_{i}, a_{i^{'}}) = C ∖ Θ_{D}^{ϑ} (a_{i}, a_{i^{'}})$ , one realizes that $Θ_{D}^{ϑ} (a_{i}, a_{i^{'}}) = C ∖ Θ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ ; thus, $Θ_{C}^{ϑ} (a_{i^{'}}, a_{i}) = C ∖ Θ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ . The later consequence is obtained: $\begin{matrix} Φ_{C}^{ϑ} (a_{i}, a_{i^{'}}) + Φ_{C}^{ϑ} (a_{i^{'}}, a_{i}) \\ = \frac{\sum_{c_{j} \in Θ_{C}^{ϑ} (a_{i}, a_{i^{'}})} w_{j} (S^{ϑ} (t_{ij}) - S^{ϑ} (t_{i^{'} j}))}{\sum_{j = 1}^{n} w_{j} (S^{ϑ} (t_{ij}) - S^{ϑ} (t_{i^{'} j}))} \\ + \frac{\sum_{c_{j} \in Θ_{C}^{ϑ} (a_{i^{'}}, a_{i})} w_{j} (S^{ϑ} (t_{i^{'} j}) - S^{ϑ} (t_{ij}))}{\sum_{j = 1}^{n} w_{j} (S^{ϑ} (t_{i^{'} j}) - S^{ϑ} (t_{ij}))} \\ = \frac{\sum_{c_{j} \in Θ_{C}^{ϑ} (a_{i}, a_{i^{'}})} w_{j} (S^{ϑ} (t_{ij}) - S^{ϑ} (t_{i^{'} j}))}{\sum_{c_{j} \in C} w_{j} (S^{ϑ} (t_{ij}) - S^{ϑ} (t_{i^{'} j}))} \\ + \frac{\sum_{c_{j} \in C ∖ Θ_{C}^{ϑ} (a_{i}, a_{i^{'}})} w_{j} (S^{ϑ} (t_{ij}) - S^{ϑ} (t_{i^{'} j}))}{\sum_{c_{j} \in C} w_{j} (S^{ϑ} (t_{ij}) - S^{ϑ} (t_{i^{'} j}))} \\ = \frac{\sum_{c_{j} \in C} w_{j} (S^{ϑ} (t_{ij}) - S^{ϑ} (t_{i^{'} j}))}{\sum_{c_{j} \in C} w_{j} (S^{ϑ} (t_{ij}) - S^{ϑ} (t_{i^{'} j}))} = 1 . \end{matrix}$

Next, based on $Φ_{C}^{ϑ} (a_{i}, a_{i^{'}}) + Φ_{C}^{ϑ} (a_{i^{'}}, a_{i}) = 1$ , one can infer straightforwardly that: $\begin{matrix} \sum_{i^{'} = 1, i^{'} \neq i}^{m} \sum_{i = 1, i \neq i^{'}}^{m} Φ_{C}^{ϑ} (a_{i}, a_{i^{'}}) \\ = \sum_{i = 1, i \neq i^{'}}^{m} (Φ_{C}^{ϑ} (a_{i}, a_{i^{'}}) + Φ_{C}^{ϑ} (a_{i^{'}}, a_{i})) \\ = \sum_{i = 1, i \neq i^{'}}^{m} (1) = \frac{m (m - 1)}{2} . \end{matrix}$

Finally, the average ${\bar{Φ}}_{C}^{ϑ}$ of all $Φ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ values is calculated in this manner:

${\bar{Φ}}_{C}^{ϑ} = \frac{\sum_{i^{'} = 1, i^{'} \neq i}^{m} \sum_{i = 1, i \neq i^{'}}^{m} Φ_{C}^{ϑ} (a_{i}, a_{i^{'}})}{m (m - 1)} = \frac{\frac{m (m - 1)}{2}}{m (m - 1)} = 0.5 .$

For each couple of a_i and a_i’ (a_i, a_i′ ∈ A and i ≠ i′), this paper exploits the $D_{ρ}^{ι - term} (t_{ij}, t_{i^{'} j})$ (ι = 3, 4 for the three- and four-term representations, respectively) to formulate the Minkowski distance-dependent discordance index $Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}})$ as follows: $Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}) = \frac{\sum_{c_{j} \in Θ_{D}^{ϑ} (a_{i}, a_{i^{'}})} w_{j} D_{ρ}^{ι - term} (t_{ij}, t_{i^{'} j})}{\sum_{j = 1}^{n} w_{j} D_{ρ}^{ι - term} (t_{ij}, t_{i^{'} j})},$ (25) where ϑ ∈ {I, II, ⋯ , VIII}, ι ∈ {3, 4}, and ρ ∈ Z⁺. It is noted that for all ϑ, ι, and ρ, the index $Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}})$ fulfills the succeeding helpful features:

$0 ⩽ Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}) ⩽ 1$ .

$Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}) + Φ_{D}^{ϑ, ι, ρ} (a_{i^{'}}, a_{i}) = 1$ .

$\sum_{i^{'} = 1, i^{'} \neq i}^{m} \sum_{i = 1, i \neq i^{'}}^{m} Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}) = m (m - 1) / 2$ .

${\bar{Φ}}_{D}^{ϑ, ι, ρ} = 0.5$ , where ${\bar{Φ}}_{D}^{ϑ, ι, ρ}$ is the average of all $Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}})$ outcomes.

First, the boundary condition is obvious because of the normalization prerequisite, i.e., $\sum_{j = 1}^{n} w_{j} = 1$ . For the second property, one recognizes that $Θ_{D}^{ϑ} (a_{i^{'}}, a_{i}) = Θ_{C}^{ϑ} (a_{i}, a_{i^{'}}) = C ∖ Θ_{D}^{ϑ} (a_{i}, a_{i^{'}})$ . Because the Minkowski distance $D_{ρ}^{ι - term} (t_{ij}, t_{i^{'} j})$ satisfies the symmetry axiom, it can be realized that $D_{ρ}^{ι - term} (t_{i^{'} j}, t_{ij}) = D_{ρ}^{ι - term} (t_{ij}, t_{i^{'} j})$ . As a consequence, one obtains: $\begin{matrix} Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}) + Φ_{D}^{ϑ, ι, ρ} (a_{i^{'}}, a_{i}) \\ = \frac{\sum_{c_{j} \in Θ_{D}^{ϑ} (a_{i}, a_{i^{'}})} w_{j} D_{ρ}^{ι - term} (t_{ij}, t_{i^{'} j})}{\sum_{c_{j} \in C} w_{j} D_{ρ}^{ι - term} (t_{ij}, t_{i^{'} j})} \\ + \frac{\sum_{c_{j} \in C ∖ Θ_{D}^{ϑ} (a_{i}, a_{i^{'}})} w_{j} D_{ρ}^{ι - term} (t_{ij}, t_{i^{'} j})}{\sum_{c_{j} \in C} w_{j} D_{ρ}^{ι - term} (t_{ij}, t_{i^{'} j})} \\ = \frac{\sum_{c_{j} \in C} w_{j} D_{ρ}^{ι - term} (t_{ij}, t_{i^{'} j})}{\sum_{c_{j} \in C} w_{j} D_{ρ}^{ι - term} (t_{ij}, t_{i^{'} j})} = 1 . \end{matrix}$

According to $Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}) + Φ_{D}^{ϑ, ι, ρ} (a_{i^{'}}, a_{i}) = 1$ , one can yield that $\sum_{i^{'} = 1, i^{'} \neq i}^{m} \sum_{i = 1, i \neq i^{'}}^{m} Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}) = \sum_{i = 1, i \neq i^{'}}^{m} (Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}) + Φ_{D}^{ϑ, ι, ρ} (a_{i^{'}}, a_{i})) = \sum_{i = 1, i \neq i^{'}}^{m} (1) = m (m - 1) / 2$ . Thus, the third property is valid. The average ${\bar{Φ}}_{D}^{ϑ, ι, ρ}$ of all $Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}})$ values is derived by $\sum_{i^{'} = 1, i^{'} \neq i}^{m} \sum_{i = 1, i \neq i^{'}}^{m} Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}) / m (m - 1) = [m (m - 1) / 2] / m (m - 1) = 0.5$ . Therefore, the fourth property is fulfilled.

By exploiting the measurements of score function-dependent concordance indices and Minkowski distance-dependent discordance indices, this paper establishes an efficacious T-SF ELECTRE approach to support decision-making activities under complicated uncertainty. More specifically, this paper provides the T-SF ELECTRE I and II prioritization procedures for rendering partial- and complete-priority rankings among available alternatives.

To draw up the T-SF ELECTRE I prioritization procedure, this paper first compares the score function-dependent concordance index $Φ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ and the average ${\bar{Φ}}_{C}^{ϑ}$ to elucidate the concordance entry $B_{C}^{ϑ} (a_{i}, a_{i^{'}})$ and to construct the concordance Boolean matrix $B_{C}^{ϑ}$ , as shown: $B_{C}^{ϑ} (a_{i}, a_{i^{'}}) = {\begin{matrix} 1 & if Φ_{C}^{ϑ} (a_{i}, a_{i^{'}}) ⩾ {\bar{Φ}}_{C}^{ϑ}, \\ 0 & if Φ_{C}^{ϑ} (a_{i}, a_{i^{'}}) < {\bar{Φ}}_{C}^{ϑ}, \end{matrix}$ (26) $B_{C}^{ϑ} = [\begin{matrix} - & B_{C}^{ϑ} (a_{1}, a_{2}) & \dots & B_{C}^{ϑ} (a_{1}, a_{m}) \\ B_{C}^{ϑ} (a_{2}, a_{1}) & - & \dots & B_{C}^{ϑ} (a_{2}, a_{m}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ B_{C}^{ϑ} (a_{m}, a_{1}) & B_{C}^{ϑ} (a_{m}, a_{2}) & \dots & - \end{matrix}] .$ (27)

Next, this paper contrasts the Minkowski distance-dependent discordance index $Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}})$ and the average ${\bar{Φ}}_{D}^{ϑ, ι, ρ}$ to put forward the discordance entry $B_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}})$ and construct the discordance Boolean matrix $B_{D}^{ϑ, ι, ρ}$ , in this manner: $B_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}) = {\begin{matrix} 1 & if Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}) ⩽ {\bar{Φ}}_{D}^{ϑ, ι, ρ}, \\ 0 & if Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}) > {\bar{Φ}}_{D}^{ϑ, ι, ρ}, \end{matrix}$ (28)

$B_{D}^{ϑ, ι, ρ} = [\begin{matrix} - \\ B_{D}^{ϑ, ι, ρ} (a_{2}, a_{1}) \\ ⋮ \\ B_{D}^{ϑ, ι, ρ} (a_{m}, a_{1}) \end{matrix} \begin{matrix} B_{D}^{ϑ, ι, ρ} (a_{1}, a_{2}) \\ - \\ ⋮ \\ B_{D}^{ϑ, ι, ρ} (a_{m}, a_{2}) \end{matrix} \begin{matrix} \dots \\ \dots \\ ⋱ \\ \dots \end{matrix} \begin{matrix} B_{D}^{ϑ, ι, ρ} (a_{1}, a_{m}) \\ B_{D}^{ϑ, ι, ρ} (a_{2}, a_{m}) \\ ⋮ \\ - \end{matrix}] .$ (29)

Furthermore, this paper defines the prioritization entry $B_{P}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}})$ and construct the prioritization Boolean matrix $B_{P}^{ϑ, ι, ρ}$ , using this fashion: $B_{P}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}) = B_{C}^{ϑ} (a_{i}, a_{i^{'}}) \cdot B_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}),$ (30)

$B_{P}^{ϑ, ι, ρ} = [\begin{matrix} - \\ B_{P}^{ϑ, ι, ρ} (a_{2}, a_{1}) \\ ⋮ \\ B_{P}^{ϑ, ι, ρ} (a_{m}, a_{1}) \end{matrix} \begin{matrix} B_{P}^{ϑ, ι, ρ} (a_{1}, a_{2}) \\ - \\ ⋮ \\ B_{P}^{ϑ, ι, ρ} (a_{m}, a_{2}) \end{matrix} \begin{matrix} \dots \\ \dots \\ ⋱ \\ \dots \end{matrix} \begin{matrix} B_{P}^{ϑ, ι, ρ} (a_{1}, a_{m}) \\ B_{P}^{ϑ, ι, ρ} (a_{2}, a_{m}) \\ ⋮ \\ - \end{matrix}] .$ (31)

When $B_{P}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}) = 1$ , it indicates that the alternative a_i is preferred to a_i’ with respect to both dimensions of the score function-dependent concordance index and the Minkowski distance-dependent discordance index. When $B_{P}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}}) = 0$ , it implies that either a_i is less preferred to a_i’ or a_i is indifferent or incomparable to a_i′. On the basis of the yielded outcomes in $B_{P}^{ϑ, ι, ρ}$ , a domination graph is then portrayed for manifesting a partial-priority ranking of available alternatives. This stipulates the T-SF ELECTRE I prioritization procedure.

To formulate the T-SF ELECTRE II prioritization procedure, this paper establishes a score function-dependent leaving flow and a Minkowski distance-dependent entering flow to create a superiority index for ranking alternatives. One should notice that the $Φ_{C}^{ϑ}$ values are not bounded in the unit interval due to different ranges of the score function S^ϑ (t_ij). To improve data comparability, this paper exploits a normalization approach for adjusting the score function-dependent concordance indices ranged on different scales to a notionally common scale. Consider that feature scaling, also known as unity-based normalization, can regulate the $Φ_{C}^{ϑ}$ values into the unit interval [0,1]. Let $Φ_{C}^{' ϑ} (a_{i}, a_{i^{'}})$ represent the normalized score function-dependent concordance index for each couple of a_i and a_i′. It is delineated by way of a well-established min-max feature scaling approach: $Φ_{C}^{' ϑ} (a_{i}, a_{i^{'}}) = \frac{Φ_{C}^{ϑ} (a_{i}, a_{i^{'}}) - {min}_{l i, i^{'} = 1 i \neq i^{'}}^{m} Φ_{C}^{ϑ} (a_{i}, a_{i^{'}})}{{max}_{l i, i^{'} = 1 i \neq i^{'}}^{m} Φ_{C}^{ϑ} (a_{i}, a_{i^{'}}) - {min}_{l i, i^{'} = 1 i \neq i^{'}}^{m} Φ_{C}^{ϑ} (a_{i}, a_{i^{'}})} .$ (32)

Herein, the value range of the normalized concordance index would be $0 ⩽ Φ_{C}^{' ϑ} (a_{i}, a_{i^{'}}) ⩽ 1$ . By virtue of Equation (32), the original concordance index $Φ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ can be converted to the range [0,1] for facilitating data comparability.

By applying the notions of the net concordance/discordance values developed by Chen [7], this paper defines the score function-dependent leaving flow $Γ_{C}^{ϑ} (a_{i})$ of alternative a_i on this wise: $Γ_{C}^{ϑ} (a_{i}) = \frac{\sum_{i^{'} = 1, i^{'} \neq i}^{m} Φ_{C}^{' ϑ} (a_{i}, a_{i^{'}})}{m - 1} .$ (33)

The Minkowski distance-dependent entering flow $Γ_{D}^{ϑ, ι, ρ} (a_{i})$ of a_i is determined in a similar way: $Γ_{D}^{ϑ, ι, ρ} (a_{i}) = \frac{\sum_{i^{'} = 1, i^{'} \neq i}^{m} Φ_{D}^{ϑ, ι, ρ} (a_{i^{'}}, a_{i})}{m - 1} .$ (34)

By combining the leaving and entering flows, the superiority index $Γ_{P}^{ϑ, ι, ρ} (a_{i})$ of a_i is identified by: $Γ_{P}^{ϑ, ι, ρ} (a_{i}) = Γ_{C}^{ϑ} (a_{i}) \cdot Γ_{D}^{ϑ, ι, ρ} (a_{i}) .$ (35)

Moreover, these two flows and the superiority index fulfill the following essential properties:

$0 ⩽ Γ_{C}^{ϑ} (a_{i}) ⩽ 1$ .

$0 ⩽ Γ_{D}^{ϑ, ι, ρ} (a_{i}) ⩽ 1$ .

$0 ⩽ Γ_{P}^{ϑ, ι, ρ} (a_{i}) ⩽ 1$ .

It is worthwhile to mention that a higher level of $Γ_{P}^{ϑ, ι, ρ} (a_{i})$ represents a superior alternative a _i. The complete-priority ranking towards available alternatives is then generated depending on the descending order of the superiority indices, which stipulates the T-SF ELECTRE II prioritization procedure.

Figure 1 reveals a systematic framework for exploiting the proposed T-SF ELECTRE approach to tackle multiple criteria assessment issues under T-SF uncertainty. As shown in this figure, the implementation procedure consists of configuration of a multiple criteria evaluation problem (as shown in Steps I.1−I.3 and Steps II.1−II.3), identification of concordance and discordance collections (as shown in Steps I.4−I.6 and Steps II.4−II.6), ascertainment of concordance and discordance indices (as shown in Steps I.7−I.9 and Steps II.7−II.9), and T-SF ELECTRE I (as shown in Steps I.10−I.12) and II (as shown in Steps II.10−II.12) prioritization procedures.

Fig. 1

Framework of T-SF ELECTRE I and II approaches.

The evolved T-SF ELECTRE I approach is delineated along these implemental steps:

Step I.1: Formulate a multiple criteria evaluation problem that comprises m available alternatives (i.e., a₁, a₂, ⋯ , a_m) and n considered criteria (i.e., c₁, c₂, ⋯ , c_n). Build the collections A and C.

Step I.2: Investigate the importance weight w_j with a normalization restriction and erect a collection for the weight vector W.

Step I.3: Collect the grades of satisfaction μ _ij , neutral satisfaction η _ij , and dissatisfaction ν _ij to establish the T-SF evaluative rating t_ij.

Step I.4: Set up the T-SF characteristic T_i using Equation (13). Derive the grade of refusal membership r_ij for each t_ij in T_i using Equation (4).

Step I.5: Calculate the score functions S^I(t_ij), S^II(t_ij),Λ, S^VIII(t_ij) associated with each t_ij using Equations (5)−(12), respectively.

Step I.6: Constitute the concordance collection $Θ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ and the discordance collection $Θ_{D}^{ϑ} (a_{i}, a_{i^{'}})$ by means of Equations (22) and (23), respectively, for a_i, a_i′ ∈ A and ϑ ∈ {I, II, ⋯ , VIII}.

Step I.7: Combine the weight vector W to derive the score function-dependent concordance index $Φ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ using Equation (24).

Step I.8: Designate a distance parameter ρ ∈ Z⁺, and compute the three-term Minkowski distance $D_{ρ}^{3 - term} (t_{ij}, t_{i^{'} j})$ using Equation (14) or the four-term Minkowski distance $D_{ρ}^{4 - term} (t_{ij}, t_{i^{'} j})$ using Equation (18) for each couple of t_ij and t_i′j.

Step I.9: Incorporate the weight vector W to generate the Minkowski distance-dependent discordance index $Φ_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}})$ via Equation (25) for a_i, a_i′ ∈ A and ι ∈ {3, 4}.

Step I.10: Generate the concordance entry $B_{C}^{ϑ} (a_{i}, a_{i^{'}})$ and the concordance Boolean matrix $B_{C}^{ϑ}$ via Equations (26) and (27), respectively.

Step I.11: Receive the discordance entry $B_{D}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}})$ and the discordance Boolean matrix $B_{D}^{ϑ, ι, ρ}$ via Equations (28) and (29), respectively.

Step I.12: Obtain the prioritization entry $B_{P}^{ϑ, ι, ρ} (a_{i}, a_{i^{'}})$ and the prioritization Boolean matrix $B_{P}^{ϑ, ι, ρ}$ using Equations (30) and (31), respectively. Draw a domination graph for yielding a partial-priority ranking.

The advanced T-SF ELECTRE II approach is delineated along the following implemental steps:

Steps II.1−II.9: See Steps I.1−I.9.

Step II.10: Calculate the normalized score function-dependent concordance index $Φ_{C}^{' ϑ} (a_{i}, a_{i^{'}})$ by means of Equation (32).

Step II.11: Attain the score function-dependent leaving flow $Γ_{C}^{ϑ} (a_{i})$ and the Minkowski distance-dependent entering flow $Γ_{D}^{ϑ, ι, ρ} (a_{i})$ by means of Equations (33) and (34), respectively.

Step II.12: Gain the superiority index $Γ_{P}^{ϑ, ι, ρ} (a_{i})$ using Equation (35) to render a complete-priority ranking of alternatives depending on a descending order of $Γ_{P}^{ϑ, ι, ρ} (a_{i})$ .

4 Numerical application

This section intends to investigate a multiple criteria assessment problem concerning company investment by exploiting the evolved T-SF ELECTRE I and II methodology and prioritization procedures.

The company investment decision issue was originated from Munir et al. [30]. By way of explanation, a company attempts to expand its business scope. The company’s board of directors decided to invest its funds in three potential business options based on six considered criteria. Figure 2 provides a concise description of the issue to be addressed in the company investment decision.

Fig. 2

Description of the company investment problem.

Let us demonstrate the implementation process of the T-SF ELECTRE I approach. As indicated in Fig. 2, the collection of available alternatives and the collection of considered criteria were formulated as shown: A = {a₁, a₂, a₃} and C = {c₁, c₂, ⋯ , c₅}. The weight vector was built by W = (w₁, w₂, ⋯ , w₅), in which the value of the importance weight w_j is also shown in Fig. 2. Moreover, according to Munir et al. [30], the T-SF evaluative rating t_ij of each business company a_i with respect to criterion c_j is displayed in this figure. Herein, the t value associated with each T-SF evaluative rating is taken as 3, in conformity with the specification in Munir et al. [30]. In previous descriptions it was noted that Steps I.1−I.3 have been completed.

In Step I.4, this study constructed the T-SF characteristic T_i for each a_i and computed the grade of refusal membership r_ij relating to each t_ij. Consider T₁ and r₁₂ for instance. On has T₁ = {〈c_j, t_1j〉|c_j ∈ C} = ác₁, (0.5, 0.3, 0.4)ñ, ác₂, (0.9, 0.4, 0.5)ñ, ác₃, (0.7, 0.5, 0.2)ñ, ác₄, (0.8, 0.5, 0.5)ñ, ác₅, (0.2, 0.2, 0.8)ñ; moreover, r₁₂ = [1 - (0.9) ³ - (0.4) ³ - (0.5) ³] ^1/-3 = 0.4344.

In Step I.5, this study computed the score function S^ϑ(t_ij) for each t_ij contained in the T-SF characteristic T_i, where ϑ ∈ {I, II, ⋯ , VIII}. The determination results are exhibited in Table 1.

Table 1

Resulting outcome of the score function S^ϑ(t_ij)

S^ϑ(t_ij)	c ₁	c ₂	c ₃	c ₄	c ₅
Results of S^ϑ(t_1j) in relation to the T-SF characteristic T₁
S^I(t_1j)	0.0610	0.6040	0.3350	0.3870	–0.5040
S^II(t_1j)	0.0070	0.1240	0.0350	0.0270	–0.2160
S^III(t_1j)	0.5626	0.6035	0.5829	0.4339	–0.2547
S^IV(t_1j)	0.0273	0.3160	0.0913	0.1508	–0.3420
S^V(t_1j)	0.0340	0.5400	0.2100	0.2620	–0.5120
S^VI(t_1j)	0.0576	0.5960	0.3194	0.3714	–0.5050
S^VII(t_1j)	0.5170	0.7700	0.6050	0.6310	0.2440
S^VIII(t_1j)	0.6780	0.8467	0.7367	0.7540	0.4960
Results of S^ϑ(t_2j) in relation to the T-SF characteristic T₂
S^I(t_2j)	–0.3350	0.0560	0.6040	–0.1890	–0.2180
S^II(t_2j)	–0.0350	0.0260	0.3160	–0.0630	–0.0560
S^III(t_2j)	–0.0638	0.6842	0.7042	0.2685	0.0632
S^IV(t_2j)	–0.1250	0.0395	0.4480	–0.1170	–0.1235
S^V(t_2j)	–0.3990	0.0550	0.5960	–0.1970	–0.2450
S^VI(t_2j)	–0.3430	0.0559	0.6030	–0.1900	–0.2214
S^VII(t_2j)	0.3005	0.5275	0.7980	0.4015	0.3775
S^VIII(t_2j)	0.5337	0.6850	0.8653	0.6010	0.5850
Results of S^ϑ(t_3j) in relation to the T-SF characteristic T₃
S^I(t_3j)	0.1520	–0.3160	0.2790	0.1170	–0.0610
S^II(t_3j)	0.0560	–0.0160	0.1170	0.0630	–0.0070
S^III(t_3j)	0.6409	–0.1568	0.6945	0.7152	0.4191
S^IV(t_3j)	0.0980	–0.0910	0.1890	0.0878	–0.0273
S^V(t_3j)	0.1440	–0.4410	0.2710	0.1160	–0.0880
S^VI(t_3j)	0.1510	–0.3316	0.2780	0.1169	–0.0644
S^VII(t_3j)	0.5720	0.2795	0.6355	0.5580	0.4560
S^VIII(t_3j)	0.7147	0.5197	0.7570	0.7053	0.6373

To give an instance, by exploiting Equations (5)−(12), the eight score functions relating to t₁₂ = (0.9, 0.4, 0.5) (with r₁₂ = 0.4344) were derived in this fashion:

$\begin{matrix} S^{I} (t_{12}) = (0.9)^{3} - (0.5)^{3} = 0.6040, \\ S^{II} (t_{12}) = (0.9 - 0.4)^{3} - (0.5 - 0.4)^{3} = 0.1240, \\ S^{III} (t_{12}) = (0.9)^{3} - (0.4)^{3} - (0.5)^{3} \\ + (\frac{e^{(0.9)^{3} - (0.4)^{3} - (0.5)^{3}}}{e^{(0.9)^{3} - (0.4)^{3} - (0.5)^{3}} + 1} - \frac{1}{2}) (0.4344)^{3} = 0.6035, \\ S^{IV} (t_{12}) = (0.9 - 0.4 / 2)^{3} - (0.5 - 0.4 / 2)^{3} = 0.3160, \\ S^{V} (t_{12}) = (0.9)^{3} - (0.4)^{3} - (0.5)^{3} = 0.5400, \\ S^{VI} (t_{12}) = (0.9)^{3} - (0.5)^{3} - (0.4 / 2)^{3} = 0.5960, \\ S^{VII} (t_{12}) = [1 + (0.9)^{3} - (0.4)^{3} - (0.5)^{3}] / 2 = 0.7700, \\ S^{VIII} (t_{12}) = [2 + (0.9)^{3} - (0.4)^{3} - (0.5)^{3}] / 3 = 0.8467 . \end{matrix}$

In Step I.6, for each couple of a_i and a_i′, the concordance collection $Θ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ and the discordance collection $Θ_{D}^{ϑ} (a_{i}, a_{i^{'}})$ were established by virtue of Equations (22) and (23), respectively, for all ϑ∈ {I, II, ⋯ , VIII}. The obtained concordance and discordance collections are manifested in Table 2. The important point to notice is $Θ_{C}^{ϑ} (a_{i}, a_{i^{'}}) = Θ_{D}^{ϑ} (a_{i^{'}}, a_{i})$ due to S^ϑ (t_ij) ≠ S^ϑ (t_i′j) in all pairs of t_ij and t_i′j.

Table 2

Resulting outcome of the collections $Θ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ and $Θ_{D}^{ϑ} (a_{i}, a_{i^{'}})$

ϑ	$Θ_{C}^{ϑ} (a_{1}, a_{2})$ (i.e., $Θ_{D}^{ϑ} (a_{2}, a_{1})$ )	$Θ_{C}^{ϑ} (a_{1}, a_{3})$ (i.e., $Θ_{D}^{ϑ} (a_{3}, a_{1})$ )	$Θ_{C}^{ϑ} (a_{2}, a_{3})$ (i.e., $Θ_{D}^{ϑ} (a_{3}, a_{2})$ )
I	c₁, c₂, c₄	c₂, c₃, c₄	c₂, c₃
II	c₁, c₂, c₄	c ₂	c₂, c₃
III	c₁, c₄	c ₂	c₂, c₃
IV	c₁, c₂, c₄	c₂, c₄	c₂, c₃
V	c₁, c₂, c₄	c₂, c₄	c₂, c₃
VI	c₁, c₂, c₄	c₂, c₃, c₄	c₂, c₃
VII	c₁, c₂, c₄	c₂, c₄	c₂, c₃
VIII	c₁, c₂, c₄	c₂, c₄	c₂, c₃
	$Θ_{C}^{ϑ} (a_{2}, a_{1})$ (i.e., $Θ_{D}^{ϑ} (a_{1}, a_{2})$ )	$Θ_{C}^{ϑ} (a_{3}, a_{1})$ (i.e., $Θ_{D}^{ϑ} (a_{1}, a_{3})$ )	$Θ_{C}^{ϑ} (a_{3}, a_{2})$ (i.e., $Θ_{D}^{ϑ} (a_{2}, a_{3})$ )
I	c₃, c₅	c₁, c₅	c₁, c₄, c₅
II	c₃, c₅	c₁, c₃, c₄, c₅	c₁, c₄, c₅
III	c₂, c₃, c₅	c₁, c₃, c₄, c₅	c₁, c₄, c₅
IV	c₃, c₅	c₁, c₃, c₅	c₁, c₄, c₅
V	c₃, c₅	c₁, c₃, c₅	c₁, c₄, c₅
VI	c₃, c₅	c₁, c₅	c₁, c₄, c₅
VII	c₃, c₅	c₁, c₃, c₅	c₁, c₄, c₅
VIII	c₃, c₅	c₁, c₃, c₅	c₁, c₄, c₅

Taking (a₁, a₂) and ϑ = I for example, it was observed that S^I (t₁₁) ⩾ S^I (t₂₁) (i.e., 0.0610≥−0.3350), S^I (t₁₂) ⩾ S^I (t₂₂) (i.e., 0.6040≥0.0560), and S^I (t₁₄)⩾ S^I (t₂₄) (i.e., 0.3870≥−0.1890). Moreover, it was acquired that S^I (t₁₃) < S^I (t₂₃) (i.e., 0.3350 <0.6040) and S^I (t₁₅) < S^I (t₂₅) (i.e., −0.5040< −0.2180). Thus, one has $Θ_{C}^{I} (a_{1}, a_{2}) = {c_{1}, c_{2}, c_{4}}$ and $Θ_{D}^{I} (a_{1}, a_{2}) =$ {c₃, c₅}. Over and above that, it was inferred that $Θ_{D}^{I} (a_{2}, a_{1}) = {c_{1}, c_{2}, c_{4}}$ and $Θ_{C}^{I} (a_{2}, a_{1}) = {c_{3}, c_{5}}$ .

In Step I.7, this study combined the weight vector W = (0.25, 0.20, 0.15, 0.18, 0.22) and the difference between S^ϑ (t_ij) and S^ϑ (t_i′j) to generate the score function-dependent concordance index $Φ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ . The resulting outcomes are indicated in Table 3. To give an example, the $Φ_{C}^{I} (a_{1}, a_{2})$ was calculated using Equation (24) in this manner:

$\begin{matrix} Φ_{C}^{I} (a_{1}, a_{2}) = \frac{\sum_{c_{j} \in {c_{1}, c_{2}, c_{4}}} w_{j} (S^{I} (t_{1 j}) - S^{I} (t_{2 j}))}{\sum_{j = 1}^{5} w_{j} (S^{I} (t_{1 j}) - S^{I} (t_{2 j}))} \\ = [0.25 (0.0610 - (- 0.3350)) + 0.20 (0.6040 - 0.0560) \\ + 0.18 (0.3870 - (- 0.1890))] / [0.25 (0.0610 - (- 0.3350)) \\ + 0.20 (0.6040 - 0.0560) + 0.15 (0.3350 - 0.6040) \\ + 0.18 (0.3870 - (- 0.1890)) + 0.22 (- 0.5040 \\ - (- 0.2180))] = 1.4941 . \end{matrix}$

Table 3

Resulting outcome of the concordance index $Φ_{C}^{ϑ} (a_{i}, a_{i^{'}})$

ϑ	(a₁, a₂)	(a₁, a₃)	(a₂, a₁)	(a₂, a₃)	(a₃, a₁)	(a₃, a₂)
I	1.4941	1.9952	–0.4941	–1.3959	–0.9952	2.3959
II	–1.4911	–0.5713	2.4911	–2.1297	1.5713	3.1297
III	2.2703	–1.8291	–1.2703	–1.0269	2.8291	2.0269
IV	3.5412	–10.4732	–2.5412	–1.3301	11.4732	2.3301
V	1.6812	2.4039	–0.6812	–1.8804	–1.4039	2.8804
VI	1.5137	2.0258	–0.5137	–1.4507	–1.0258	2.4507
VII	1.6812	2.4039	–0.6812	–1.8804	–1.4039	2.8804
VIII	1.6812	2.4039	–0.6812	–1.8804	–1.4039	2.8804

In Step I.8, this paper utilized the Manhattan metric to compute the three- and four-term Minkowski distances; accordingly, the distance parameter was set as ρ = 1. First, the three-term Manhattan distance $D_{1}^{3 - term} (t_{ij}, t_{i^{'} j})$ was acquired by use of Equation (15), and the results are displayed in the top half of Table 4. Applying Equation (19), the four-term Manhattan distance $D_{1}^{4 - term} (t_{ij}, t_{i^{'} j})$ was calculated, as shown in the bottom half of Table 4. Consider the four-term representation as an illustration. The Manhattan distance between t₁₂ = (0.9, 0.4, 0.5) (with r₁₂ = 0.4344) and t₃₂ = (0.3, 0.5, 0.7) (with r₃₂ = 0.7963) was computed on this wise:

Table 4

Results of the Manhattan distances $D_{1}^{3 - term} (t_{ij}, t_{i^{'} j})$ and $D_{1}^{4 - term} (t_{ij}, t_{i^{'} j})$

(t_ij, t_i’_j) & (t_i’_j, t_ij)	c ₁	c ₂	c ₃	c ₄	c ₅
Manhattan distance via the three-term representation
(t_1j, t_2j) & (t_2j, t_1j)	0.4330	0.8450	0.6200	0.6930	0.3050
(t_1j, t_3j) & (t_3j, t_1j)	0.1100	0.9810	0.1730	0.6280	0.4620
(t_2j, t_3j) & (t_3j, t_2j)	0.5430	0.4960	0.4470	0.3130	0.2790
Manhattan distance via the four-term representation
(t_1j, t_2j) & (t_2j, t_1j)	0.6320	1.6900	1.0060	1.2040	0.3380
(t_1j, t_3j) & (t_3j, t_1j)	0.1820	1.4040	0.2340	1.2560	0.7740
(t_2j, t_3j) & (t_3j, t_2j)	0.6700	0.9180	0.8940	0.4300	0.5580

$\begin{matrix} D_{1}^{4 - term} (t_{12}, t_{32}) = | Δ_{μ}^{3} | + | Δ_{η}^{3} | + | Δ_{ν}^{3} | + | Δ_{r}^{3} | \\ = | (0.9)^{3} - (0.3)^{3} | + | (0.4)^{3} - (0.5)^{3} | + | (0.5)^{3} - (0.7)^{3} | \\ + | (0.4344)^{3} - (0.7963)^{3} | = 1.4040 . \end{matrix}$

In Step I.9, the determination outcomes of the Manhattan distance-dependent discordance index $Φ_{D}^{ϑ, ι, 1} (a_{i}, a_{i^{'}})$ are depicted in Table 5, where ι∈ {3, 4}. Letting ϑ = III, ι = 4, and ρ = 1, for instance, the $Φ_{D}^{III, 4, 1} (a_{1}, a_{2})$ was derived using Equation (25) as shown:

$\begin{matrix} Φ_{D}^{III, 4, 1} (a_{1}, a_{2}) = \frac{\sum_{c_{j} \in {c_{2}, c_{3}, c_{5}}} w_{j} D_{1}^{4 - term} (t_{1 j}, t_{2 j})}{\sum_{j = 1}^{5} w_{j} D_{1}^{4 - term} (t_{1 j}, t_{2 j})} \\ = (0.20 \times 1.6900 + 0.15 \times 1.0060 + 0.22 \times 0.3380) / \\ (0.25 \times 0.6320 + 0.20 \times 1.6900 + 0.15 \times 1.0060 \\ + 0.18 \times 1.2040 + 0.22 \times 0.3380) = 0.6005 . \end{matrix}$

Table 5

Resulting outcome of the discordance index $Φ_{D}^{ϑ, ι, 1} (a_{i}, a_{i^{'}})$

ϑ	(a₁, a₂)	(a₁, a₃)	(a₂, a₁)	(a₂, a₃)	(a₃, a₁)	(a₃, a₂)
Manhattan distance-dependent discordance index $Φ_{D}^{ϑ, 3, 1} (a_{i}, a_{i^{'}})$
I	0.2848	0.2781	0.7152	0.6039	0.7219	0.3961
II	0.2848	0.5775	0.7152	0.6039	0.4225	0.3961
III	0.5855	0.5775	0.4145	0.6039	0.4225	0.3961
IV	0.2848	0.3340	0.7152	0.6039	0.6660	0.3961
V	0.2848	0.3340	0.7152	0.6039	0.6660	0.3961
VI	0.2848	0.2781	0.7152	0.6039	0.7219	0.3961
VII	0.2848	0.3340	0.7152	0.6039	0.6660	0.3961
VIII	0.2848	0.3340	0.7152	0.6039	0.6660	0.3961
Manhattan distance-dependent discordance index $Φ_{D}^{ϑ, 4, 1} (a_{i}, a_{i^{'}})$
I	0.2402	0.2848	0.7598	0.5364	0.7152	0.4636
II	0.2402	0.6294	0.7598	0.5364	0.3706	0.4636
III	0.6005	0.6294	0.3995	0.5364	0.3706	0.4636
IV	0.2402	0.3311	0.7598	0.5364	0.6689	0.4636
V	0.2402	0.3311	0.7598	0.5364	0.6689	0.4636
VI	0.2402	0.2848	0.7598	0.5364	0.7152	0.4636
VII	0.2402	0.3311	0.7598	0.5364	0.6689	0.4636
VIII	0.2402	0.3311	0.7598	0.5364	0.6689	0.4636

In Step I.10, it is known that the average ${\bar{Φ}}_{C}^{ϑ} = 0.5$ . By exploiting Equations (26) and (27), the concordance Boolean matrix $B_{C}^{ϑ}$ for each ϑ ∈ {I, II, ⋯ , VIII} was rendered as shown: $\begin{matrix} B_{C}^{I} = B_{C}^{V} = B_{C}^{VI} = B_{C}^{VII} = B_{C}^{VIII} = [\begin{matrix} - & 1 & 1 \\ 0 & - & 0 \\ 0 & 1 & - \end{matrix}], \\ B_{C}^{II} = [\begin{matrix} - & 0 & 0 \\ 1 & - & 0 \\ 1 & 1 & - \end{matrix}], B_{C}^{III} = B_{C}^{IV} = [\begin{matrix} - & 1 & 0 \\ 0 & - & 0 \\ 1 & 1 & - \end{matrix}] . \end{matrix},$

In Step I.11, it is recognized that the average ${\bar{Φ}}_{D}^{ϑ, ι, ρ} = 0.5$ . In this illustration, the same discordance Boolean matrices were acquired based on the three- and four-term Manhattan distances. Applying Equations (28) and (29), the discordance Boolean matrix $B_{D}^{ϑ, ι, 1}$ for ϑ ∈ {I, II, ⋯ , VIII} and ι ∈ {3, 4} was generated along these lines:

$\begin{matrix} B_{D}^{I, ι, 1} = B_{D}^{IV, ι, 1} = B_{D}^{V, ι, 1} = \dots = B_{D}^{VIII, ι, 1} = [\begin{matrix} - & 1 & 1 \\ 0 & - & 0 \\ 0 & 1 & - \end{matrix}], \\ B_{D}^{II, ι, 1} = [\begin{matrix} - & 1 & 0 \\ 0 & - & 0 \\ 1 & 1 & - \end{matrix}], B_{D}^{III, ι, 1} = [\begin{matrix} - & 0 & 0 \\ 1 & - & 0 \\ 1 & 1 & - \end{matrix}] . \end{matrix},$

In Step I.12, this study exploited Equations (30) and (31) to yield the prioritization Boolean matrix $B_{P}^{ϑ, ι, 1}$ for ϑ ∈ {I, II, ⋯ , VIII} and ι ∈ {3, 4} as follows:

$\begin{matrix} B_{P}^{I, ι, 1} = B_{P}^{V, ι, 1} = B_{P}^{VI, ι, 1} = B_{P}^{VII, ι, 1} = B_{P}^{VIII, ι, 1} = [\begin{matrix} - & 1 & 1 \\ 0 & - & 0 \\ 0 & 1 & - \end{matrix}], \\ B_{P}^{II, ι, 1} = B_{P}^{III, ι, 1} = [\begin{matrix} - & 0 & 0 \\ 0 & - & 0 \\ 1 & 1 & - \end{matrix}], B_{P}^{IV, ι, 1} = [\begin{matrix} - & 1 & 0 \\ 0 & - & 0 \\ 0 & 1 & - \end{matrix}] . \end{matrix}$

The partial-priority rankings of the three potential business options were demonstrated by means of the domination graphs in Fig. 3. Herein, the domination relationships using the T-SF ELECTRE I approach were portrayed using blue arrows in cases where ϑ ∈ {I, V, VI, VII, VIII} and ι ∈ {3, 4}. That is, $a_{1} ≻_{P - I}^{ϑ, ι, 1} a_{2}$ , $a_{1} ≻_{P - I}^{ϑ, ι, 1} a_{3}$ , and $a_{3} ≻_{P - I}^{ϑ, ι, 1} a_{2}$ . Moreover, the domination relationships were delineated using pink arrows in cases where ϑ ∈ {II, III} and ι ∈ {3, 4} (i.e., $a_{3} ≻_{P - I}^{ϑ, ι, 1} a_{1}$ and $a_{3} ≻_{P - I}^{ϑ, ι, 1} a_{2}$ ), as well as brown arrows in cases where ϑ ∈ {IV} and ι ∈ {3, 4} (i.e., $a_{1} ≻_{P - I}^{ϑ, ι, 1} a_{2}$ and $a_{3} ≻_{P - I}^{ϑ, ι, 1} a_{2}$ ). One can notice that the ranking consequence yielded by Munir et al. [30] was a₁ ≻ a₃ ≻ a₂. It can be observed that the domination relationships generated by the proposed T-SF ELECTRE I procedure are almost concordant with Munir et al.’s solution result, excluding $a_{3} ≻_{P - I}^{ϑ, ι, 1} a_{1}$ for ϑ ∈ {II, III} and ι ∈ {3, 4}.

Fig. 3

Domination graphs for the company investment problem.

Next, this paper demonstrates the execution process of the T-SF ELECTRE II methodology. Recall that Steps II.1−II.9 are identical to Steps I.1−I.9. In Step II.10, this study exploited Equation (32) to determine the normalized score function-dependent concordance index $Φ_{C}^{' ϑ} (a_{i}, a_{i^{'}})$ , as revealed in Table 6. Taking $Φ_{C}^{' I} (a_{1}, a_{2})$ as an example, one obtains:

$\begin{matrix} Φ_{C}^{' I} (a_{1}, a_{2}) = \frac{Φ_{C}^{I} (a_{1}, a_{2}) - {min}_{l i, i^{'} = 1 i \neq i^{'}}^{3} Φ_{C}^{I} (a_{i}, a_{i^{'}})}{{max}_{l i, i^{'} = 1 i \neq i^{'}}^{3} Φ_{C}^{I} (a_{i}, a_{i^{'}}) - {min}_{l i, i^{'} = 1 i \neq i^{'}}^{3} Φ_{C}^{I} (a_{i}, a_{i^{'}})} \\ = (1.4941 - min {1.4941, 1.9952, - 0.4941, - 1.3959, \\ - 0.9952, 2.3959}) / (max {1.4941, 1.9952, - 0.4941, \\ - 1.3959, - 0.9952, 2.3959} - min {1.4941, 1.9952, \\ - 0.4941, - 1.3959, - 0.9952, 2.3959}) = 0.7622 . \end{matrix}$

Table 6

Outcome of the normalized concordance index $Φ_{C}^{' ϑ} (a_{i}, a_{i^{'}})$

ϑ	(a₁, a₂)	(a₁, a₃)	(a₂, a₁)	(a₂, a₃)	(a₃, a₁)	(a₃, a₂)
I	0.7622	0.8943	0.2378	0.0000	0.1057	1.0000
II	0.1214	0.2963	0.8786	0.0000	0.7037	1.0000
III	0.8800	0.0000	0.1200	0.1722	1.0000	0.8278
IV	0.6386	0.0000	0.3614	0.4166	1.0000	0.5834
V	0.7481	0.8999	0.2519	0.0000	0.1001	1.0000
VI	0.7598	0.8911	0.2402	0.0000	0.1089	1.0000
VII	0.7481	0.8999	0.2519	0.0000	0.1001	1.0000
VIII	0.7481	0.8999	0.2519	0.0000	0.1001	1.0000

In Step II.11, this study employed Equation (33) to derive the score function-dependent leaving flow $Γ_{C}^{ϑ} (a_{i})$ , and the computation outcomes are indicated in the top part of Table 7. By the agency of Equation (34), the Manhattan distance-dependent entering flows $Γ_{D}^{ϑ, 3, 1} (a_{i})$ and $Γ_{D}^{ϑ, 4, 1} (a_{i})$ for the three- and four-term representations, respectively, were determined, as shown in the middle parts of Table 7. By way of illustration, $Γ_{C}^{I} (a_{1}) =$ (0.7622 + 0.8943)/(3−1) = 0.8282, $Γ_{D}^{I, 3, 1} (a_{1})$ = (0.7152 + 0.7219)/(3−1) = 0.7185, and $Γ_{D}^{I, 4, 1} (a_{1}) =$ (0.7598 + 0.7152)/(3−1) = 0.7375.

Table 7

Outcome of leaving flows, entering flows, and superiority indices

a_i	I	II	III	IV	V	VI	VII	VIII
Score function-dependent leaving flow $Γ_{C}^{ϑ} (a_{i})$
a ₁	0.8282	0.2089	0.4400	0.3193	0.8240	0.8255	0.8240	0.8240
a ₂	0.1189	0.4393	0.1461	0.3890	0.1259	0.1201	0.1259	0.1259
a ₃	0.5528	0.8518	0.9139	0.7917	0.5500	0.5545	0.5500	0.5500
Manhattan distance-dependent entering flow $Γ_{D}^{ϑ, 3, 1} (a_{i})$
a ₁	0.7185	0.5689	0.4185	0.6906	0.6906	0.7185	0.6906	0.6906
a ₂	0.3405	0.3405	0.4908	0.3405	0.3405	0.3405	0.3405	0.3405
a ₃	0.4410	0.5907	0.5907	0.4690	0.4690	0.4410	0.4690	0.4690
Manhattan distance-dependent entering flow $Γ_{D}^{ϑ, 4, 1} (a_{i})$
a ₁	0.7375	0.5652	0.3850	0.7144	0.7144	0.7375	0.7144	0.7144
a ₂	0.3519	0.3519	0.5320	0.3519	0.3519	0.3519	0.3519	0.3519
a ₃	0.4106	0.5829	0.5829	0.4338	0.4338	0.4106	0.4338	0.4338
Outcome of the superiority index $Γ_{P}^{ϑ, 3, 1} (a_{i})$
a ₁	0.5951	0.1188	0.1842	0.2205	0.5690	0.5931	0.5690	0.5690
a ₂	0.0405	0.1496	0.0717	0.1324	0.0429	0.0409	0.0429	0.0429
a ₃	0.2438	0.5032	0.5398	0.3713	0.2579	0.2445	0.2579	0.2579
Outcome of the superiority index $Γ_{P}^{ϑ, 4, 1} (a_{i})$
a ₁	0.6109	0.1181	0.1694	0.2281	0.5887	0.6088	0.5887	0.5887
a ₂	0.0418	0.1546	0.0777	0.1369	0.0443	0.0423	0.0443	0.0443
a ₃	0.2270	0.4966	0.5327	0.3434	0.2386	0.2277	0.2386	0.2386

In Step II.12, this study exploited Equation (35) to determine the superiority index $Γ_{P}^{ϑ, 3, 1} (a_{i})$ and $Γ_{P}^{ϑ, 4, 1} (a_{i})$ , and the outcomes are displayed in the bottom parts of Table 7. For giving an illustration, $Γ_{P}^{I, 3, 1} (a_{1}) =$ 0.8282 × 0.7185 = 0.5951, and $Γ_{P}^{I, 4, 1} (a_{1}) =$ 0.8282 × 0.7375 = 06109. Supported by a descending order of $Γ_{P}^{ϑ, 3, 1} (a_{i})$ or $Γ_{P}^{ϑ, 4, 1} (a_{i})$ , the T-SF ELECTRE II approach yielded the complete-priority rankings of three alternatives were yielded in this fashion: $a_{1} ≻_{P - II}^{ϑ, ι, 1} a_{3} ≻_{P - II}^{ϑ, ι, 1} a_{2}$ for ϑ ∈ {I, V, VI, VII, VIII} and ι ∈ {3, 4}, $a_{3} ≻_{P - II}^{ϑ, ι, 1} a_{1} ≻_{P - II}^{ϑ, ι, 1} a_{2}$ for ϑ ∈ {III, IV} and ι ∈ {3, 4}, and $a_{3} ≻_{P - II}^{ϑ, ι, 1} a_{2} ≻_{P - II}^{ϑ, ι, 1} a_{1}$ for ϑ ∈ {II} and ι ∈ {3, 4}. As demonstrated in the acquired consequence, the exploitation of the score functions S^I(t_ij), S^V(t_ij), S^VI(t_ij), S^VII(t_ij), and S^VIII(t_ij) produced the same ranking as the solution result by Munir et al. [30]. The application outcomes with the selection problem of a business company for investment have demonstrated the utility and practicability of the initiated T-SF ELECTRE approach in real-world decision circumstances.

5 Comparative analysis

This section intends to implement a systematic comparison to exhibit some favorable features of the evolved methods and techniques.

The first comparative analysis concentrates on the determination outcomes of the score function-dependent concordance index $Φ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ and its normalized version $Φ_{C}^{' ϑ} (a_{i}, a_{i^{'}})$ for alternatives a_i, a_i′ ∈ A and ϑ ∈ {I, II, ⋯ , VIII}, as manifested in Fig. 4. To be specific, the consequences of $Φ_{C}^{ϑ} (a_{i}, a_{i^{'}})$ and $Φ_{C}^{' ϑ} (a_{i}, a_{i^{'}})$ are contrasted in Fig. 4(a) and 4(b), respectively. There was a large discrepancy among the resulting concordance indices concerning the couple of a₁ and a₃ (i.e., pairs (a₁, a₃) and (a₃, a₁)), especially in situations of employing the score function S^II(t_ij), S^III(t_ij), and S^IV(t_ij). In particular, from Fig. 4(a), a significant difference was observed between a₁ and a₃, namely, $Φ_{C}^{IV} (a_{1}, a_{3}) = - 10.4732$ and $Φ_{C}^{IV} (a_{3}, a_{1}) = 11.4732$ . As a whole, the pair (a₃, a₂) enjoyed the highest concordance indices. From Fig. 4(b), except for $Φ_{C}^{' III} (a_{3}, a_{2}) = 0.8278$ and $Φ_{C}^{' IV} (a_{3}, a_{2}) = 0.5834$ , one receives $Φ_{C}^{' ϑ} (a_{3}, a_{2}) = 1$ when ϑ = I, II, V, VI, VII, VIII. Accordingly, the reversed pair (a₂, a₃) enjoyed the lowest concordance indices. Overall, the employment of the score function S^I(t_ij), S^V(t_ij), S^VI(t_ij), S^VII(t_ij), and S^VIII(t_ij) produced more reliable and reasonable outcomes.

Fig. 4

Juxtaposition of the (normalized) score function-dependent concordance indices.

The second comparative analysis centralizes the comparisons of the Manhattan distance-dependent discordance index $Φ_{D}^{ϑ, ι, 1} (a_{i}, a_{i^{'}})$ for a_i, a_i′ ∈ A and ϑ ∈ {I, II, ⋯ , VIII}, as exhibited in Fig. 5. The outcomes of $Φ_{D}^{ϑ, 3, 1} (a_{i}, a_{i^{'}})$ and $Φ_{D}^{ϑ, 4, 1} (a_{i}, a_{i^{'}})$ are juxtaposed in Fig. 5(a) and 5(b), respectively. In comparison, the pairs (a₂, a₁), (a₃, a₁), and (a₂, a₃) received the top three discordance indices based on the three- and four-term Manhattan distance measures. Thus, it seems unlikely that a₂ performs better than a₁, a₃ better than a₁, and a₂ better than a₃. Such findings were in agreement with the priority ranking a₁ ≻ a₃ ≻ a₂ generated by Munir et al. [30]. The overall effects of the three- and four-term representations on the rendered discordance indices were roughly consistent in the comparison outcomes.

Fig. 5

Juxtaposition of the Manhattan distance-dependent discordance indices.

The third comparative analysis focuses on the superiority indices and their complete-priority rankings rendered by the T-SF ELECTRE II prioritization procedure under various settings of the distance parameter ρ in the three- and four-term Minkowski distance measures. For conducting comparisons, this study designated the ρ values as 1, 2, ...,10 for all ι ∈ {3, 4} and ϑ ∈ {I, II, ⋯ , VIII}. The comparison outcomes of the superiority indices $Γ_{P}^{ϑ, 3, ρ} (a_{i})$ and $Γ_{P}^{ϑ, 4, ρ} (a_{i})$ are displayed in Figs. 6 and 7, respectively. On account of the three-term representation, the juxtaposition consequences of the $Γ_{P}^{ϑ, 3, ρ} (a_{i})$ values based on the score functions S^I(t_ij), S^II(t_ij),Λ, S^VIII(t_ij) are portrayed in Fig. 6(a)−6(h), respectively. These graphs reveal the contrasts of the superiority indices in distinct settings of ρ = 1, 2, ..., 10. Specifically, the use of S^I(t_ij), S^V(t_ij), S^VI(t_ij), S^VII(t_ij), and S^VIII(t_ij) generated the complete-priority ranking $a_{1} ≻_{P - II}^{ϑ, 3, ρ} a_{3} ≻_{P - II}^{ϑ, 3, ρ} a_{2}$ for all ρ ∈ {1, 2, ⋯ , 10}. The employment of S^II(t_ij) brought about the ranking $a_{3} ≻_{P - II}^{II, 3, ρ} a_{2} ≻_{P - II}^{II, 3, ρ} a_{1}$ for all ρ values; moreover, the use of S^III(t_ij) and S^IV(t_ij) gave rise to the ranking $a_{3} ≻_{P - II}^{ϑ, 3, ρ} a_{1} ≻_{P - II}^{ϑ, 3, ρ} a_{2}$ for each ρ.

By the agency of the four-term Minkowski distance measure, the contrasts of the superiority index $Γ_{P}^{ϑ, 4, ρ} (a_{i})$ in varied settings of ρ = 1, 2, ⋯ , 10 in accordance with S^I(t_ij), S^II(t_ij),Λ, S^VIII(t_ij) are sketched in Fig. 7(a)−7(h), respectively. The exploitation of S^I(t_ij), S^V(t_ij), S^VI(t_ij), S^VII(t_ij), and S^VIII(t_ij) yielded the complete-priority ranking $a_{1} ≻_{P - II}^{ϑ, 4, ρ} a_{3} ≻_{P - II}^{ϑ, 4, ρ} a_{2}$ for all ρ ∈ {1, 2, ⋯ , 10}. The use of S^II(t_ij) rendered the ranking $a_{3} ≻_{P - II}^{II, 4, ρ} a_{2} ≻_{P - II}^{II, 4, ρ} a_{1}$ for all ρ values; moreover, the use of S^III(t_ij) and S^IV(t_ij) produced the ranking $a_{3} ≻_{P - II}^{ϑ, 4, ρ} a_{1} ≻_{P - II}^{ϑ, 4, ρ} a_{2}$ for each ρ. Obviously, the comparison outcomes of the graphs in Fig. 7 were consistent with those in Fig. 6, which leads to a consensus consequence by virtue of the three- and four-term Minkowski distance measures. Because the acquired rankings $a_{1} ≻_{P - II}^{ϑ, 3, ρ} a_{3} ≻_{P - II}^{ϑ, 3, ρ} a_{2}$ and $a_{1} ≻_{P - II}^{ϑ, 4, ρ} a_{3}$ $≻_{P - II}^{ϑ, 4, ρ} a_{2}$ were in agreement with the ranking result by Munir et al. [30], the score functions S^I(t_ij), S^V(t_ij), S^VI(t_ij), S^VII(t_ij), and S^VIII(t_ij) are more reliable and appropriate to be incorporated into the evolved T-SF ELECTRE I and II approaches.

Fig. 6

Juxtaposition of the superiority index $Γ_{P}^{ϑ, 3, ρ} (a_{i})$ .

Fig. 7

Juxtaposition of the superiority index $Γ_{P}^{ϑ, 4, ρ} (a_{i})$ .

6 Conclusions

This paper has established an inventive T-SF ELECTRE methodology to manipulate complicated multiple criteria evaluation problems. Some beneficial notions have been advanced to facilitate the construction of the T-SF ELECTRE I and II prioritization procedures. First, the concordance collection and the discordance collection have been identified through the medium of various types of T-SF score functions. As a further matter, this paper has utilized the weighted differences of score functions to create the score function-dependent concordance indices. Next, this paper has exploited the three- and four-term Minkowski distance measures to elucidate the Minkowski distance-dependent discordance indices in T-SF contexts. On the grounds of the concordance Boolean matrix, discordance Boolean matrix, and prioritization Boolean matrix, this paper has propounded the T-SF ELECTRE I prioritization procedure for generating partial-priority rankings of available alternatives. Over and above that, this paper has presented the score function-dependent leaving flows and Minkowski distance-dependent entering flows to expound the superiority indices. This paper has constructed the T-SF ELECTRE II prioritization procedure for yielding complete-priority rankings of alternatives. The application concerning a business investment decision has been implemented; moreover, the practicality and usefulness of the initiated methodology have been examined on grounds of the investigation of resulting outcomes. More importantly, the juxtaposition of application outcomes under distinct designations has revealed the comparative advantages of different score functions in tackling multiple criteria decision-making tasks.

The principal limitation on the employment of the current T-SF ELECTRE methodology as a multiple criteria assessment approach is its specification of score functions. This limitation restricts the manipulation of T-SF information. More precisely, the conversion of T-SF evaluative ratings into crisp numbers via score functions would eliminate the fuzziness. The exploration of a T-SF data-processing task using other measurements can enhance the T-SF PF ELECTRE procedure to prevent this limitation.

This paper provides some promising exploration directions for reference. First, it is valuable to utilize the evolved T-SF ELECTRE methodology in the various real-world fields, such as material selection, supplier selection, logistic outsourcing, facility location planning, airport location selection, and evaluation of industrial robots, because of its ease of use and comprehensibility. Second, the use of the T-SF score functions S^I(t_ij), S^V(t_ij), S^VI(t_ij), S^VII(t_ij), and S^VIII(t_ij) was demonstrated as one of the most prominent features of the current T-SF ELECTRE methods, which can be incorporated into other decision-making techniques. Third, the T-SF score functions S^II(t_ij), S^III(t_ij), and S^IV(t_ij) are potentially worthwhile in forming other outranking decision models in T-SF contexts, even though the effectiveness about application effects is unobvious in the current approach. Other newly available outranking decision models might potentially create the enrichment of the three score functions in practice.

Footnotes

Acknowledgments

The authors acknowledge the assistance of the respected editor and the anonymous referees for their insightful and constructive comments, which helped to improve the overall quality of the paper. The authors are grateful for grant funding support from the Ministry of Science and Technology, Taiwan (MOST 110-2410-H-182-005) and Chang Gung Memorial Hospital, Linkou, Taiwan (BMRP 574) during the completion of this study.

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