Abstract

In an open environment, the demands of users are diverse and dynamic because users can participate in product design from beginning to end. Owing to this, the disorderly and unplanned participation of users will greatly increase the complexity of multi-attribute decision-making (MADM) in the product design process. In order to ensure the smooth development of the open design process, the decision support model and method need to repeatedly provide decision makers (DMs) with necessary decision support information in a relatively short period of time, which can realize the evaluation of the scheme and improve the utilization efficiency of community resources. However, the process of eliciting preference information is complex, exhausting, inefficient, and time-consuming in existing methods, which will result in a poor decision-making. With the purpose of optimizing the eliciting process in MADM, a rule-based decision support method is proposed in this paper, where the process of eliciting preference information and decision-making are synchronized and guided by pre-extracted decision rules. The rules are deduced from comparison relations on attributes and their outcomes through the combination of variable precision rough set approach (VPRS) and stochastic multi-objective acceptability analysis (SMAA). With the concept of attribute reduction and approximation accuracy in rough set theory, the extracted rules could eliminate redundant attributes and assign the relative priority of preference information. Based on the extracted rules, the multi-attribute decision-making process could be carried out step by step in an orderly manner. In each step, DMs only need to provide partial preference information by non-quantitative statements according to extracted rules. Once the decision result is reliable enough, the eliciting and decision-making process can be terminated promptly. In order to validate the proposed approach, experiments of decision rule extraction are implemented, and the results show that the proposed approach is effective both in the weak rule extraction and the strong rule extraction.

1. Introduction

Multi-attribute decision-making (MADM), also referred to as multi-objective decision-making, is an important component of modern decision science [1]. It refers to the problem of evaluating and prioritizing a finite set of alternatives based on a collection of attributes. The theory and methods of MADM have been extensively applied to the fields of society, economy, management, military, engineering design, and so on. Generally, the process of solving MADM problems can be divided into two steps. One step is to collect decision information, thus determining attribute values and weights. The other step is to synthesize the collected decision information through a proper aggregation operator (e.g., the simple averaging operator and the ordered weighted averaging operator). The ranking of alternatives can be obtained according to their synthesized values.

The complexity of MADM is mainly due to the difficulty of assessing the relative importance of different attributes. In MADM problems, attributes are assigned different weight values to indicate their relative importance. Depending on the information provided, the methods of determining attribute weights are divided into subjective methods and objective methods [2]. In subjective methods, attribute weights are derived from value judgments about attributes given by decision makers (DMs), including eigenvector-based method [3–6], weighted averaging or least square method [7–10], AHP or ANP-based method [11, 12], and D-TOPSIS method [13]. And the objective methods determine attribute weights according to objective information (e.g., the information in the decision matrix), such as principal component analysis method [14], entropy method [15, 16], and multi-objective programming model-based method [17, 18]. Either way, the process of determining attribute weights is not easy. Especially in subjective methods, attribute weights reflect the group preference of DMs. Therefore, the first thing to address is to collect and aggregate DMs’ preference information through proper means. In the study of most subjective methods, the process of eliciting preference information highly depends on frequent and deep involvement of DMs. For example, the Delphi method uses multiple rounds of surveys to collect and aggregate preference information through a series of data collection and analysis techniques interspersed with feedback [19]. In each round, DMs (experts) need to do a series of questionnaires focusing on problems, opportunities, solutions, or forecasts until an agreement is reached. In AHP and ANP-based methods, a complete pairwise comparison matrix is the prerequisite. Many research studies point out that it is hard to ensure the consistency of the pairwise comparison matrix, especially when the number of attributes is large [20, 21]. In addition, for various reasons, the comparison relation between attributes could not be given by DMs precisely and completely. Predictably, the process of eliciting preference information based on these methods would be very time-consuming and exhausting when the number of attributes and DMs is large. In order to the uncertainty and fuzziness of MADM, Ning et al. [22] proposed a novel probabilistic dual hesitant fuzzy-enabled MADM technique. Wang et al. [23] proposed a multi-objective particle swarm optimization-enabled approach to optimize park-level integrated energy systems with multi-attribute decision analysis frameworks. Li et al. [24] proposed a data-driven approach to solve the problem of personalized individual semantics under the background of MADM, and the experiment showed that the proposed approach can obtain personalized numerical scales of linguistic terms for a decision maker. In order to solve the problem of the AC transmission systems appropriate placement, Chinda et al. [25] proposed a novel MADM approach combined with a particle mobility honey bee algorithm, which can maximize efficiency of the energy planning. Rahman et al. [26] proposed a MADM approach combined with AND OR operations, which can effectively deal with the uncertainties of possibility neutrosophic hypersoft set. Zhang et al. [27] proposed a bisimulation-based generalized fuzzy variable precision rough set model to solve the problem of the decision-making, which can effectively tackle complicated problems including the attribute and the relational data. Ye et al. [28] proposed a fuzzy rough sets-enabled decision-making approach, which can effectively tackle the uncertainty and imprecision problem in MADM. Jiang et al. [29] proposed a rough sets-enabled risk decision-making approach, which not only consider the decision risk but also offer instructions to select the appropriate semantic explanation. Sarwar [30] proposed a rough D-TOPSIS method to manipulate the subjectiveness and vagueness of MADM, which unites the competency to analyze changeable and ambiguous information without additional assumptions. Fei et al. [31] proposed a rough D-TOPSIS method to solve the problem of the human resources selection, and an effectual method was applied by authors to describe the changeable data and information called D-numbers. Lin et al. [32] proposed a rough ELECTRE-II method to deal with the selection problem of the edge nodes, and an entropy measure is devised to measure the uncertainty degree. Akram et al. [33] proposed a rough ELECTRE-II method with the specific structure, through which the diverse opinions of decision experts can be well handled. Kuncova et al. [34] proposed a fuzzy rough PROMETHEE method which can evaluate the order of 14 regions of the Czech Republic in regard to economic indices. Liu et al. [35] proposed a fuzzy rough PROMETHEE method to accomplish the process for optimal alternative selection.

However, existing methods prefer to solve MADM problems through a “top-down” way, in which organizers collect preference information based on the essential characteristics of the problem. For example, in plant location selection (PLS) problem, alternatives are usually evaluated based on environmental index, economic index, social index, and geological index [36, 37]. However, in some cases, alternatives may be similar in some attributes. The relative importance of such attributes, which are called redundant attributes, have no effect on the final ranking result. The existing method cannot eliminate redundant attributes from the perspective of the attribute value. Therefore, it is necessary to clarify which attribute should be involved and which weight should be given priority.

In order to fill in the above research gaps, a rule-based decision support method for MADM (R-MADM) is proposed in this paper to help optimize the process of eliciting preference information in MADM. The contribution and motivation of this paper is summarized as follows:(1)In R-MADM, the process of eliciting preference information and the process of decision-making are synchronized and guided by pre-extracted decision rules. In contrast to existing methods, R-MADM is a “bottom-up” method. The MADM problem could be solved through two stages based on the proposed method, namely, decision rule extraction and decision rule application.(2)In the first stage, outcomes (ranking results) corresponding to different comparison relations are obtained using stochastic multi-objective acceptability analysis (SMAA). Decision rules are then extracted based on SMAA analysis results through the variable precision rough set (VPRS) approach.(3)In the second stage, according to the decision rules, the preference eliciting process and the decision-making process could be carried out step by step. DMs only need to provide partial preference information according to extracted rules in each step. Based on the SMAA analysis result, the measurement of the reliability of decision-making is given. Once the decision result is reliable enough, the process of eliciting preference information and the process of decision-making can be terminated. Therefore, the efficiency of MADM can be improved.

The remainder of this paper is organized as follows. Relevant theoretical foundations about VPRS and SMAA are introduced in Section 2. In Section 3, the proposed R-MADM method is illustrated in detail. A numerical example is used to illustrate the implementation of the proposed method in Section 4. Section 5 provides a conclusion and discusses the limitations of the proposed method as well as research directions for in-depth studies in the future.

2. Theoretical Foundations

2.1. Rough Set Theory

Rough set theory (RST) proposed by Pawlak provides a useful tool for extracting knowledge (decision rules) from inexact, uncertain, or vague information. RST has been widely used in the area of machine learning [38], knowledge management [39], expert system [40], decision support system [41], pattern recognition [42], yield prediction [43], and so on. The basic idea of RST is using available information to perform complete approximation (classification) of the given objects without any preliminary assumptions. In RST, the approximation of an object is related to its attribute information, and an information system is often expressed as a quadruple aswhere is referred to as the universe consisting of a finite set of objects, is a non-empty and finite set of attributes, is the union of attribute domains, i.e., for any , and is an information function which associates every object in with a unique attribute value.

Generally, the information system (IS) in RST is represented in the form of a decision table (knowledge table), in which rows and columns correspond to objects and their attributes, respectively. The attribute set is divided into the condition attribute set and the decision attribute set , i.e., . The value of the decision attribute in is related to the value of the condition attribute in . The objects in can be partitioned into a group of disjoint subsets based on the indiscernibility relation , and it is defined as

Each indiscernibility relation partitions the universe into a family of disjoint subsets called equivalent classes, which is expressed as

Given , can be approximated by its lower approximation and upper approximation . The lower approximation is the union of equivalent classes that belong to the subset with certainty, i.e.,

And the upper approximation is the union of equivalent classes that partly belong to the subset , i.e.,

The pair is called the rough set of with respect to . Based on the lower and upper approximation, the universe could be partitioned into three regions.(1) is referred to as the positive region of with respect to , and it is the lower approximation of , i.e., (2) is referred to as the boundary region of with respect to , and it is the subtraction of the upper approximation and the lower approximation of , i.e., (3) is referred to as the negative region of with respect to , and it is the subtraction of the universe and the upper approximation of , i.e.,

2.1.1. Variable Precision Rough Set Approach

The classical rough set approach only can be used to model fully correct classification problem. This is primarily due to the fact that the classical rough set approach is based on a deterministic approach which deliberately ignores the available probabilistic information in its formalism. The classification with a controlled misclassification error, which is referred to as partial classification, is outside the realm of this approach. This limitation severely reduces the applicability of RST to problems which are more probabilistic than deterministic in nature. To solve this problem, the variable-precision rough set (VPRS) approach is proposed by taking partial classification into account. In VPRS, the standard set inclusion relation used in the classical rough set approach is replaced by the majority inclusion relation. In the majority inclusion relation, the classification error of classifying objects of a set X into another set Y is measured bywhere denotes set cardinality. For a given admissible classification error , the majority inclusion relation in VPRS can be expressed as

It can be observed that the standard set inclusion relation is a special case of the majority inclusion relation when . Based on the majority inclusion relation, Ziarko gives the definitions of -lower and -upper approximations of a set with respect to in VPRS, which are defined as

In VPRS, the -lower approximation of constitutes its -positive region , i.e.,

And the -boundary region is given by

And the -negative region of is the complement of its -upper approximation, i.e.,

2.1.2. Dominance-Based Variable Precision Rough Set Approach

In both the classical rough set approach and VPRS approach, attribute information is assumed to be nominal, which cannot be arranged in any particular order. However, in some real-life applications, the information has obvious ordinal characteristic where better condition attribute values usually bring out better decision attribute values. The dominance-based variable precision rough set (DB-VPRS) approach improves the VPRS approach by replacing the indiscernibility relation with dominance relation, thus taking the preference-orders into consideration [44]. The dominance relation with respect to attribute is represented as , which is also denoted by , and means that dominates (is better than) in terms of attribute . For the benefit index, means that the attribute value of is greater than the attribute value of with respect to attribute , i.e.,

And for the cost index, means that the attribute value of is smaller than the attribute value of with respect to attribute , i.e.,

Based on the dominance relation, the dominance set of is defined by

And the dominated set of is defined by

Let , be decision classes. Each decision class is defined as , where is the decision attribute value. It is assumed that the total order of decision attribute values is ; and then, the order relation of decision classes can be written as . The upward and downward union of decision classes are defined aswhere if then ; and if then .

For a given confidence level in DB-VPRS, the lower and upper approximation of and are defined as

In DB-VPRS, the positive region of with respect to is defined b.

Similarly, the positive region of with respect to is defined as

Based on its duality characteristic, the negative regions of and are defined by

The boundary regions of and can be obtained by

2.1.3. Attribute Reduction and Decision Rule Extraction in VPRS

Attribute reduction for the analysis of important attributes and decision rule extraction for the approximation of the decision attribute by means of the condition attribute are major topics in RST.(1)Attribute dependency and approximation quality. For a given information system , let , , , , and the equivalent classes and are assumed to be known. Then, for a given confidence level , the dependency of on is measured by If , the partition is completely dependent on ; If 0 , is partly dependent on ; If , is completely independent on .(2)Relative importance of attribute. According to the definition given by (28), the relative importance of an attribute can be measured by The bigger the is, the more important is the attribute. If , the attribute is regarded as a redundant attribute.(3)-attribute reduct. For a given confidence level , an -attribute reduct is the minimal subset of condition attributes in preserving the dependency level with the decision attribute , satisfies the following two criteria.(a);(b)No attributes can be further eliminated from without affecting the requirement (a). There are usually more than one rough set reduct The common part of available rough set reduct is called the core of rough set, i.e.,(4)Decision rule extraction.

The approximation relation between condition attributes in and the decision attribute could be translated into a decision rule in the form of

2.2. Stochastic Multi-Objective Acceptability Analysis

Stochastic multi-objective acceptability analysis (SMAA) is a multi-criteria decision support technique for decision-making problems with uncertainty [45, 46]. In SMAA, inaccurate or uncertain input data are represented in the form of probability distribution. It is usually assumed that all feasible combinations of criteria weights and criteria measurements are equally likely to be selected. In SMAA, alternatives are evaluated according to statistically descriptive measurements which are calculated by inverse analysis of uncertainty space.

It is supposed that a set of alternatives with attributes are under evaluation. The value of attribute for alterative is represented by . The weight of attribute is represented by . The function in (32) is usually used to calculate the utility value of an alternative, and it is expressed by a convex combination of the attribute weight vector and the attribute value vector in feasible weight space . Here, attribute values are mapped into the range by a partial utility function .

If attribute weights are available, a multi-attribute decision-making problem can be addressed by calculating utility values and choosing the alternative with the largest overall utility. However, for various reasons, it is often difficult to obtain precise measurements for some attributes. Thus, in SMAA, is replaced with a stochastic variable . The joint probability density functions for attribute values and attribute weights are represented by and , respectively. When attribute weights and values are uncertain, utility value distributions of alternatives can be obtained by computing all possible combinations of uncertain parameters, and it is usually realized through Monte Carlo simulation technique in SMAA.

The core idea of SMAA is to classify alternatives to those which should be taken into consideration and to those which should be eliminated. The standard SMAA method only considers the information about the alternative with the first rank lacking holistic evaluation on all alternatives. SMAA-2 extends the discussion on all ranking results and gives several descriptive measures: rank acceptability index and three best rank-type measures.

2.2.1. Rank Acceptability Index

In SMAA-2, the ranking result for alternative could be calculated bywhere , , and . For a given order , its favorable ranking weights is defined by

Ranking acceptability index describes the share of combinations of uncertain parameter which could make alternative be ranked th. It is computed as a multidimensional integral over criteria measurements and favorable weight space as

Rank acceptability index ranges from zero to one. Alternatives with zero-value ranking acceptability index are never ranked at the certain order, and alternatives with one-value ranking acceptability index could obtain the certain rank with any feasible weights combinations.

2.2.2. K-Best Rank Indices

K-best rank indices in SMAA-2 include k-best rank (k-br) acceptability , k-br central weight vector , and k-br confidence factor , which are defined as

K-best rank indices extended the meaning of the descriptive measures in standard SMAA method by taking more alternatives into consideration. The k-br acceptability is used as a performance indicator of the alternatives ranked in top k. It can help aggregate some alternatives with a good performance into a small group and to eliminate others. The k-br weight vector describes preference information of the k-best alternatives. And the k-br confidence factor gives the credibility for the alternatives to be judged into the k-best ones with k-br central weight vector.

3. A Rule-Based Decision Support Method for Multi-Attribute Decision-Making

In the majority of extant MADM methods, the problem solving process often contains two separate sub-processes: the process of eliciting preference information from DMs and the process of decision-making. Generally, the process of decision-making cannot start if the process of eliciting preference information is not complete. In addition, during the process of eliciting preference information, DMs usually cannot receive feedback of the consequence of their choices. This affects the efficiency and reliability of decision-making. In this paper, a rule-based decision support method is proposed to help optimize the problem solving process in MADM, which is referred to as R-MADM. The proposed method overcomes the defects mentioned above, in which the process of eliciting preference information and the process of decision-making are synchronized (see Figure 1). Once the decision result is reliable, the process of eliciting preference information and the decision-making process could be terminated promptly. In R-MADM, a MADM problem is solved through two stages, namely, decision rule extraction and decision rule application.

Figure 1 

The rule-based decision-making process proposed by this paper.

3.1. Decision Rule Extraction

As shown in Figure 1, the aim of the decision rule extraction stage is to provide rules for subsequent decision-making process. The decision rule extraction stage is implemented through two steps. The first step is to construct a preference information system (IS) which can reflect the relationship between the preference structure of DM and the corresponding alternative ranking result. The second step is to extract decision rules from the constructed IS using variable precision rough set (VPRS) approach.

3.1.1. Construct Preference Information

In this paper, the preference structure of decision maker (DM) is roughly represented by a combination of pairwise comparison relations on attributes called preference combination. If there are kinds of pairwise comparison relations between two attributes, then for a given MADM problem consisting of m alternatives and n attributes, there will be a total of possible combinations. Let represent the pairwise comparison relation of attribute and attribute , then each preference combination can be represented by a tuple as satisfying . Here, three kinds of pairwise comparison relations are defined as follows:(a)If , it means that attribute is at least as important as attribute , i.e., , (b)If , it means that attribute is at most as important as attribute , i.e, , (c)If , it means that the pairwise comparison relation on attribute and attribute is not clear, which is expressed as .

In addition, the defined relations should satisfy the following properties.

(2) The defined pairwise comparison relations are reflexive and transitive, i.e.,

Consequently, for a given MADM problem consisting of m alternatives and n attributes, there are at most preference combinations. Take as an example, in R-MADM, all preference combinations could be depicted by a multi-level decision tree (see Figure 2). Each branch is a possible preference combination consisting of pairwise comparison relations on attributes. Each preference combination is represented by at the end of the branch. Nevertheless, not all combinations are rational in practice. Some preference combinations contain implicit preference information in which one pairwise comparison relations on attributes can be derived from others according to property (2). This kind of preference combination is marked with a green box in Figure 2. Some preference combinations contain inconsistent preference information in which pairwise comparison relations on attributes conflict with each other, and they are marked with red boxes in Figure 2. The preference combinations containing conflict preference information should be filtered.

Figure 2 

Preference combinations in the form of a decision tree (the number of attributes is three).

In fact, each preference combination gives a group of no-conflicting inequality constrains of attribute weights. Under different constrain combinations, ranking results of alternatives can be obtained using stochastic multi-objective acceptability analysis (SMAA). Based on SMAA analysis results and corresponding preference combinations, an information system (IS) containing preference information is constructed, which is represented in the form of a decision table (see Table 1). In Table 1, pairwise comparison relations contained in preference combinations are condition attributes, while ranking results reflected by SMAA indices are decision attributes.

3.1.2. Extract Decision Rules from the Constructed Information System

Due to the stochastic nature of SMAA indices, the information system represented by Table 1 contains a great number of incomplete information. In this method, the variable precision rough set (VPRS) approach is used to extract decision rules from such incomplete information system.

Considering that, in MADM problem, both attribute weights and values are usually uncertain. For a given MADM problem consisting of m alternatives and n attributes, let be the set of possible combinations of attribute weights and values, each combination can be represented by a tuple , and it expressed as

Here, is a possible combination of n attribute values for alternative , is a possible combination of n attribute weights satisfying . Each combination corresponds to a ranking result of m alternatives. Without loss of generality, the simple additive weighting method is used for reasoning the ranking result in this paper. In the proposed method, the ranking result is obtained by evaluating the overall utility values of alternatives. And, the overall utility value for alternative is a convex combination of and , and it is expressed as

(1) Weak Rule Extraction Based on the Combination of VPRS and SMAA. Let all possible combinations of attribute values and weights be the universe , pairwise comparison relations contained in preference combinations be the condition attribute set , and ranking results reflected by SMAA indices be the decision attribute set . The indiscernibility relation with respect to condition attribute set is defined by

The objects in satisfying same pairwise comparison relations are indiscernible and can form an equivalent class with respect to as

In this method, for a MADM problem consisting of m alternatives, there are m decision attributes defined. Each decision attribute describes whether alternative could be ranked in top according to the attribute values and weights given by . Consequently, the indiscernibility relation with respect to a decision attribute is defined aswhere is a Boolean variable, if the bracketed expression is true then , otherwise .

Let and , based on the majority relation in VPRS, the classification error of classifying objects in X into Y is measured by . Here, is the number of objects in satisfying same pairwise comparison relations defined by . It can be calculated bywhere , is the value of pairwise comparison relation between attribute and attribute given by .