Choquet based TOPSIS and TODIM for dynamic and heterogeneous decision making with criteria interaction
Introduction
In the past few decades, the complexity of decision making problems has grown substantially and different aspects must be considered simultaneously. With the growing complexity of the problems, an increasing effort to develop more resourceful methods to support decision makers in their activities is being made.
The TOPSIS is one of the most known MCDM methods and it is very simple and intuitive. Generally speaking, the idea behind TOPSIS is that the best alternative should be as close as possible to the best possible solution and as far as possible from the worst possible solution. The standard formulation of TOPSIS was only able to deal with crisp numbers. This is a very prohibitive limitation. To require that the experts provide their evaluations as crisp numbers, which have only one parameter, demands a precision that they may not have. Other types of data, such as interval number, fuzzy numbers and intuitionistic fuzzy numbers, are very important tools to help the decision makers to express their knowledge more appropriately. As the data types get more general, the number of parameters to be defined increases. For example, an interval number has two parameters, a trapezoidal fuzzy number has four parameters and an intuitionistic trapezoidal fuzzy number has 10 parameters. Obviously, an excessive number of parameter may be problematic. However, it is worth noting that this increase in the number of parameter is, as a matter of fact, an attempt to aid the decision makers to express their knowledge by providing space for uncertainty and ambiguity. Naturally, due to the seriousness of this limitation, the TOPSIS has been extensively extended. There are many variants of TOPSIS to deal with interval number [11], fuzzy information [5], intuitionistic fuzzy data [4], interval-valued intuitionistic fuzzy data [67], probability distributions [39], [61] and hesitant fuzzy [65].
On the other hand, Kahneman and Tversky [24] studied how individuals make decisions in situations involving risks and detected strong bias in some situations, such as loss aversion. In this same work, the Prospect Theory to describe such behavior was proposed. Later, Gomes and Lima [14] proposed one of the first MCDM methods, named TODIM, to present a value function with the same shape of the Prospect function which has been successfully applied in real world problems [16]. However, similarly to TOPSIS, in its standard formulation the TODIM method is only able to process crisp numbers. Naturally, several variants of TODIM have been proposed to deal with fuzzy data [33], heterogeneous data types [13] and hesitant fuzzy [72].
There are still other issues related to MCDM problems. It is becoming increasingly common to have groups of individuals participating in the process of making decisions. The methods to support decision making need to adapt in order to process the input from all stakeholders. Several variants of TOPSIS [36] and TODIM [37] are able to process group decision making. Although one could argue that as long as there is a variant of the method for a single decision maker, one could still solve the group decision making problem by using aggregation operators [64], however, there are some concerns involved that, in fact, also affect most of the variants for group decision making. One major concern is that all decision makers agree with a fixed criteria set and use the same type of data.
There may be situations where it is convenient to work with different types of data in different criteria. For example, in a supplier selection problem while evaluating the price of the product the company may naturally use a crisp number, since the price is precisely provided, but while evaluating the delivery time of the product the company may want to use a random variable. This situation can even be intensified when there are groups of decision makers involved, where different decision makers may want to use different types of data to evaluate an alternative according to a specific criterion. This requires the MCDM methods to be able to process heterogeneous data types. However, although there are many adaptation of both methods, TOPSIS and TODIM, for dealing with several types of data, little effort was made to adapt them for a computation with heterogeneous data. Peng et al. [45] proposed a TOPSIS version for group decision making that deals with crisp numbers, interval values and linguistic variables with different granularity. The TOPSIS was also extended to deal with crisp, interval, fuzzy numbers and Gaussian random variables in [57]. A TOPSIS based method which is able to process different data types was proposed by Li et al. [36]. On the TODIM side, Fan et al. [13] proposed a variant of TODIM to deal with crisp number, interval number and fuzzy data. Of course, there are also other approaches to deal with heterogeneous data types [12].
Although some efforts have been made to process heterogeneous data types, all the above cited methods have some serious shortcomings. The methods in [13], [45], [57] and Espinilla et al. [12] all require transformations of the data types. So, while they accept heterogeneous input, the processing is made with a homogeneous data type. As argued in [41], transformation of the information types may cause some problems.
Another shortcoming of the above mentioned methods is that not all of them can process group decision making and the ones that can require an agreement to be reached on the criteria set. Different decision makers may have different backgrounds and may consider that different criteria are relevant to the problem. Constraining the decision makers to consider a unique criteria set reduces drastically the usefulness of group decision making. Additionally, none of the methods, except in [57], are able to process random variable and even the method in [57] is only able to process Gaussian distributions by transforming them into interval numbers. Random variables are a key type of data since they are a very consolidate method to model uncertainty caused by stochasticity. In what may be considered an even more serious limitation, all the mentioned methods assume a static environment.
Assuming a static environment may lead to an unrealistic modeling. The performance of the alternatives may depend on several factors that can change or are not completely predictable beforehand. For instance, investment profiles can behave differently with the variation of the interest rate. The price of a seasonal product can vary depending on the period of the year. The weather conditions can affect the performance of different strategies in cases of accidents in the sea. Not only that, but also the alternatives can be affected differently by such underlying changes. Interesting works have been done considering the case of heterogeneous preferences relations and dynamic environment [47]. Methods that use preference relations are different of methods based on ratings, such as the TOPSIS and TODIM. For more about heterogeneous preferences relations the interested reader is referred to Ureña et al. [55].
Another interesting issue in MCDM problems is criteria interaction. When individual decision makers define their set of criteria, two or more criteria can present redundant or complementary information. Actually, the hypothesis of criteria independence is rarely verified in practice [20]. Two very important tools to deal with criteria interaction are the fuzzy measure [53] and the Choquet integral [10]. The fuzzy measure relaxes the additivity requirement of the classical measure to the monotonicity requirement. The Choquet integral is based on the fuzzy measure and generalizes a wide range of aggregation operators [17, and references therein]. Naturally, the Choquet integral spread in the MCDM field [19], [46] and several variants of TOPSIS and TODIM that incorporate the Choquet integral have been proposed [4], [15], [71]. However, to apply the Choquet integral it is necessary a fuzzy measure and, with the exponential growth of the number of parameters in a fuzzy measure, this can become a difficult computational problem to solve. Several methods have been proposed for identification of a fuzzy measure [21], [30], [59] but as it will be pointed out in Section 3.5 they have some shortcomings when applied in the MCDM context.
The paper is organized as follows. In Section 2, a quick review of the necessary concepts is included. In Section 3 we introduce the GMC-RTOPSIS and the GMC-RTODIM, a new approach for fuzzy measure identification which is more suitable for MCDM problems and two examples. Section 4 includes two case studies illustrating complex problems in imperfect settings. Finally, Section 5 presents some conclusions and directions for future research.
Section snippets
Basic concepts and definitions
In this section we present the definitions of the data types that will be used in this paper, which are: crisp, interval, fuzzy and intuitionistic fuzzy numbers. Here, the crisp numbers are denoted by lower case letters (a), interval numbers are denoted by bold lower case letters (a), fuzzy sets are denoted by lower case letters with tilde () and intuitionistic fuzzy sets are denoted by capital letters with tilde (). Also, we present some basic definitions on fuzzy measures and Choquet
Group Modular Choquet (GMC) approaches
In this section, we present the Group Modular Choquet Random TOPSIS (GMC-RTOPSIS) and the Group Modular Choquet Random TODIM (GMC-RTODIM) that handle several practical problems of multi-criteria decision making. The methods are able to process groups of decision makers, different types of information, which here are crisp numbers, interval numbers, fuzzy numbers, intuitionistic fuzzy numbers and random variables. In addition, the methods are able to process underlying factors that are not a
Case Study 1
This problem evaluates the air quality of the city of Guangzhou/China in November of 2006, 2007 and 2008. The air is monitored by sixteen stations across the Pearl River Delta region. From the 16 stations, three were selected to provide the data. The stations measure the concentration of respirable suspended particulate (PM10), sulphur dioxide (SO2) and nitrogen dioxide (NO2). In this situation, the stations would be considered as the decision makers, the months of November of 2006, 2007 and
Conclusion
In this work we introduced new methods for group decision making based on TOPSIS and TODIM, named GMC-RTOPSIS and GMC-RTODIM, respectively. These methods allow each decision maker to act independently if desired and it is able to process several types of information, to name crisp, interval number, fuzzy number, intuitionistic fuzzy number and probability distributions. The methods are able to deal with interacting criteria by incorporating the Choquet integral in the aggregation step. As
Acknowledgment
R. Lourenzutti would like to thank the Brazilian agency CAPES for the financial support and R. A. Krohling would like to thank the financial support of the Brazilian agency CNPq under grant nr. 309161/2015-0.
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On leave - Department of Electrical and Computer Engineering, University of Alberta Edmonton, Alberta T6G 2V4, Canada.