Three-way multi-attribute decision making under incomplete mixed environments using probabilistic similarity

Abstract

In recent years, many scholars have explored a variety of methods integrating three-way decision (3WD) and multi-attribute decision making (MADM), which enables the classification and priority ranking of alternatives possible and fully reflects the effectiveness and advantages of 3WD in solving MADM problems. However, few of these methods can effectively deal with the MADM problems with incomplete mixed information that are frequently encountered in real-world situations. This study proposes a three-way MADM method for an incomplete mixed information system (IMIS), where the objective determination of conditional probabilities and utility functions in IMIS without decision label is the pivotal issue. To overcome this issue, we define a probabilistic similarity measure for incomplete mixed information. The probabilistic similarity measure is used to replace the distance measure of classical TOPSIS for estimating the conditional probabilities objectively. The probabilistic similarity class is introduced with arithmetic average method to design a conversion mechanism and obtain the objective relative utility functions of incomplete evaluation values. We then construct a novel 3WD model in IMIS and combine it with two customized ranking principles, to solve the incomplete mixed MADM problems from the perspective of classification and ranking in a more thoughtful and interpretable manner. Our study provides a new perspective for the research on MADM in incomplete mixed information environment. Several examples and experimental comparisons verify the effectiveness and stability of the proposed method. The experiments demonstrate that our method can meet more decision-making requirements and is more accurate and rational in some decision-making scenarios than several prior similar methods.

Introduction

Decision making behavior is ubiquitous in politics, economy, military, and people’s daily life. Multiple evaluation indicators/attributes of alternatives in decision-making environment and multi-attribute decision making (MADM) have emerged because of the complexity of socioeconomic system [45]. MADM is a decision-making problem of ranking alternatives and selecting an optimal/satisfactory alternative in accordance with clear needs. In present, MADM has formed a relatively mature research system. Outranking-based MADM methods, including the well-known PROMETHEE [1], TOPSIS [42], VIKOR [5], [24], a series of ELECTREs [3], the alternative queuing method [7], and the gained and lost dominance score method [31]emerged in recent years. Some MADM methods considered the preference of decision makers (DMs) [2], [30]. These methods obtain decision results by constantly adjusting the parameters with a human–computer interaction mode. Fuzzy MADM has emerged and has been studied extensively to adapt to various increasingly complex and uncertain decision-making environments. Some fuzzy MADM methods on interval number [10], [28], intuitive fuzzy number [4], [13], [21], [34], Pythagorean fuzzy number [17], hesitant fuzzy number [29], and linguistic term set [[5], [6]] have been proposed to discuss the indecisive and hesitate attitude of DMs. Some scholars have studied fuzzy MADM methods that can handle complex linguistic information, such as probabilistic hesitant fuzzy set [14] and double hierarchy hesitant fuzzy linguistic term set [6]. However, most existing MADM methods can only be regarded as “either-or” decision mode because they only form two options of “select” or “do not select” and ignore the decision situation of “waiting for selection,” which may lead to greater decision cost or smaller decision benefit. Considering that most MADM methods cannot divide the alternatives into several parts, the selection of some alternatives can only be realized through the subjective judgment of DMs, which is inapplicable to the decision-making scenario where the information is uncertain, and the DMs’ experience is insufficient. Hence, objective classifying ability should be added for the existing MADM research system based on its ranking ability. In this regard, three-way decision (3WD) method offers an effective solution.

3WD [37], [38], [39] is an extension of classical, probabilistic, and decision-theoretic rough set (DTRS). The core idea of 3WD is to divide a nonempty finite universal set into three disjoint regions (positive region, boundary region, and negative region) based on the expected loss minimization in Bayesian decision procedure. Three different strategies (acceptance, rejection, and noncommitment [delayed acceptance/rejection]) are adopted on the objects in the three regions in accordance with the realistic requirements and semantic environment. The noncommitment idea of 3WD is more consistent with people’s decision habits in uncertain decision environment, so 3WD has attracted much attention and made remarkable achievements in theory and application. In terms of theoretical research, many 3WD models combined with concept analysis and concept lattices, clustering, game-theory, cost-sensitive, and group decision making in addition to the well-known multi-granularity 3WD, sequential 3WD, 3WD spaces, three-way attribute reduction [8], [11], [16], [18], [38], [39]. Related studies, such as 3WD combined with optimization models [19], [20], regret theory [30], [32], and utility theory [44], [45] have been reported. In the research branch for different data types, many scholars have utilized 3WD to analyze discrete, continuous, fuzzy information systems [39], and incomplete information systems (IISs) [18], [36], [40], [44] based on equivalence relationship, neighborhood/similarity relationship, fuzzy relationship, and tolerance relationship, and has conducted many discussions around the acquisition of the thresholds. In terms of application research, 3WD has been widely applied to recommendation system, medical diagnosis, government decision support [38], [39], conflict analysis [22], text sentiment classification [35], and threat assessment [4], [13], [25].

3WD can compensate for the limitation that the existing MADM method cannot classify the alternatives. The idea of classification and the concept of boundary region can make the decision results of MADM richer and more interpretable. In recent years, some three-way MADM (3WMADM) models integrating the ranking advantages of MADM and the classification advantages of 3WD have emerged; these models are applicable to solve MADM problems in uncertain and fuzzy environment. The first category is the 3WMADM models in single-valued fuzzy decision environment. Jia et al. [9] proposed a new concept of relative loss functions on evaluation values and ranked alternatives in line with the numeric values of thresholds $\alpha$, $\beta$, and $\gamma$. However, a semantic explanation is lacking for the inconsistent ranking results under the consistent conditional probabilities in their discussion. Peng et al. [25] proposed a multi-attribute 3WD method for the threat assessment of aerial targets by combining the outranking relationship of ELECTRE-I method and the relative loss functions defined in [9]. Ye et al. [41] presented a decision-making approach based on 3WD in fuzzy continuous information system by using a reflexive fuzzy $\alpha$-neighborhood operator. However, the addition of the constraint condition of obtaining neighborhood sets leads to a relatively higher complexity of the algorithm. Zhan et al. [43] combined ELECTRE-1 with membership degrees to obtain objective conditional probabilities and presented optimistic, pessimistic and compromise 3WMADM models according to the DMs’ subjective attitude toward decision risk. The second category is the 3WMADM models in complex fuzzy decision environment. Jiang et al. [10] proposed an evaluation-based multi-attribute 3WD model under an interval-valued environment by using the optimization models in [20]. Liang et al. [17] introduced a 3WD model for Pythagorean fuzzy information and applied it to project selection. They obtained attribute weights by using the maximum deviation method and computed conditional probabilities by using an ideal TOPSIS. Wang et al. [29] studied 3WMADM under hesitant fuzzy environment by combining ELECTRE-I and a hesitant fuzzy integration operator, and performed the example analysis of infectious disease diagnosis, product investment, and teacher recruitment. Subsequently, they considered the psychological behavior of DMs and introduced regret theory into the construction of 3WD model in hesitant fuzzy MADM environment [30]. Deng et al. presented a MADM model for multiscale information system based on regret theory and 3WD framework [2], which reflects the risk attitude and psychological behavior of DMs. These regret-theoretic 3WMADM models make the decision results more credible and more in line with DM’s expectations. The third category is the 3WMADM models under incomplete decision environment. Zhan et al. [44] created a utility-based 3WD model for MADM problem with incomplete discrete information based on the θ-level similarity relation and the utility values of objects, which was illustrated by an example of project investment. Ye et al. [40] proposed an incomplete 3WMADM method based on the hesitation degrees of evaluation values and the improved fuzzy neighborhood operators. This method contains three decision strategies and is applied to a real hepatitis diagnosis selected from the University of California Irvine (UCI) database.

The risk measurement functions and conditional probabilities are the essential components of a 3WMADM model, which have a direct influence on decision results. Objective calculation rather than subjective setting of the two elements is conducive to form more accurate and reasonable decision results, because the lack of knowledge and experience of DMs and the uncertain decision-making environment may lead to greater decision deviation and decision error. However, neither of the conditional probabilities in [9], [25], [44] and the loss/utility functions in [29], [43] have been calculated synchronously and objectively. Besides, the other 3WMADM methods only aimed at complete decision environment and cannot deal with incomplete MADM problems, except for [40], [44]. In a word, although studies have laid the foundation for the further exploration to MADM in this work, the objective calculation of the key elements and more complex multi-attribute decision environment can be further explored.

In real-world situations, the decision information in many MADM problems is complex, and may consist of multi-source heterogeneous mixed data of discrete type (e.g., categorical values) and continuous type (e.g., numerical values). For example, CV data include categorical values, such as nationality, gender, educational level, specialty, and marital status, and continuous values, such as age, height, weight, and salary expectation. Project investment data consist of categorical values, such as types of projects, risk level, management level, and production capability, and continuous values, such as quantity and profit. Game data include discrete values, such as lethality and stealth capability, and continuous values such as velocity, distance, and angle. Incomplete data with missing attribute values are extremely common in real life due to data privacy, intentional data hiding, data acquisition ability and other limitations. For convenience, we call this complex MADM environment containing discrete and continuous values accompanied by missing values “incomplete mixed information system (IMIS).” The MADM problems in IMIS generally exist in real life and are a topic worth studying with practical importance.

Our study has three motivations based on the above systematic literature review and the practical requirement in daily life. (a) The classification and ranking of alternatives should be realized at the same time to meet more decision-making requirements. The classification function of 3WD can be added the conventional MADM that can only rank alternatives. (b) In some existing 3WMADM methods, the objective obtainment of conditional probability and risk functions is not realized synchronously. Most of these methods only aim at a relatively single decision-making environment and cannot process incomplete mixed information directly. These limitations will be compensated in this study. (c) The MADM problems under incomplete mixed environment with discrete and continuous values accompanied by missing values are frequently encountered in practical applications. However, few prior 3WMADM methods can solve such MADM problems effectively. On this basis, we explore a novel 3WMADM method for incomplete mixed MADM problems, and then realize the classification and ranking of alternatives through a series of objective calculations. For this purpose, we need to address three pivotal issues based on the existing relevant studies, which are as follows:

The estimation of the conditional probabilities in IMIS. The conditional probabilities cannot be calculated by using the decision class-based classical conditional probability formulas, such as in [37], because no decision label in most MADM information systems. The membership degree of each object cannot be estimated because of the existence of unknown values. Consequently, the conditional probabilities cannot be calculated on the basis of membership degrees, such as in [43], either. To combat these problems, we first define a new probabilistic similarity measure for incomplete mixed information, and then integrate it into TOPSIS to estimate the relative closeness between an object and the ideal solution. In this way, the similarity-based relative closeness is regarded as the objective conditional probability of an object.

• The determination of the risk measurement functions for different decision behavior. The existing risk measurement functions include the loss functions [9], [17], [20], [29], [40], [41], [43] and the utility functions [44], [45]. Among them, the risk measurement functions in [17], [29], [43], [45] were subjectively set in a direct or indirect manner, whereas the risk measurement functions in [9], [40], [41] were objectively generated from the evaluation values. This study intends to choose the risk measurement functions from the application perspective of MADM. On the one hand, the MADM methods originated from management field usually rank and select alternatives in accordance with the highest comprehensive quality or aggregate utility of alternatives [34], [42]. On the other hand, we intend to apply the proposed method to the threat assessment under incomplete mixed game environment, which needs to classify and rank the opponent agents according to their comprehensive threat degrees. Therefore, we describe the risk measurement by the relative utility functions for the more practical semantic interpretation and more objective decision results.

• The obtainment of the objective relative utility functions, especially that of the objects with unknown values. For the known values, the relative utility functions can be obtained by fusing the concepts of the relative loss functions defined in [9] and the utility matrix defined in [45]. For the unknown value, Ye et al. [40] first utilized the average value of all objects to fill in the missing values, and then obtained loss functions of unknown values. In this study, we use arithmetic average method to fill in the missing values, but the difference is that we intend to narrow the filling range by using the proposed probabilistic similarity class, so as to obtain more accurate relative utility functions of unknown values.

We can establish a 3WMADM model for IMIS (IMIS-3WMADM) by solving the above key issues of determining conditional probabilities and relative utility functions in IMIS. The core concepts of the established model are refined in Fig. 1.

The rest of this paper is organized as follows. Section 2 reviews the works related to 3WD. Section 3 establishes a novel 3WD model for IMIS system, including the definition of probabilistic similarity measure for incomplete mixed information, the determination of the attribute weights, the conditional probabilities and the relative utility functions. Section 4 utilizes the IMIS-3WMADM based on the classification rules and ranking principles to solve the MADM problems in incomplete mixed decision environment. Meanwhile, the decision-making process and corresponding algorithm are given to implement classification and ranking of objects. Section 5 cites the application instances of hepatitis diagnosis and threat assessment, and analyzes the influence of relevant parameters on decision results, as well as proves the effectiveness and superiority of the proposed model by some comparative experiments and discussions. Section 6 gives the conclusion and outlies direction for future work.