Fuzzy Integral Multiple Criteria Decision Making Method Based on Fuzzy Preference Relation on Alternatives

1 Introduction

Selecting the most desirable alternative from all of the possible alternatives evaluated on a family of criteria is the problem in multiple criteria decision making (MCDM)[1]. The methods that have been designed to solve the MCDM problem consist of two steps: 1) Obtaining the decision information; 2) Ranking alternatives by choosing an appropriate aggregation operator.

In general, it is the most popular for additive model to aggregate the DM’s preference[2]. Unfortunately, This model is not able to represent interactions among criteria because it does need an idealistic assumption about preferential independence among criteria. To deal with the interactions among criteria, the Choquet integral[3, 4] was introduced to the MCDM. It is an extension of the weighted sum and able to address interaction between criteria in many situations[5]. However, In order to use Choquet integral, it is necessary to construct both the value functions and the capacity. Generally, The value functions can be given by the decision maker (DM) easily, the construction of the capacity has an exponential complexity[6–8].

Several particular families of capacities[9–12] have been proposed to reduce the number of parameters of capacity evaluation process. λ-measures[9] and k-additive[10] capacities are the very commonly used of all capacities. In particular, the use of 2-additive capacities is realistic because it’s noteworthy that the DM have to express their preference on positive and negative interactions between couples of criteria only.

The approaches to solve MCDM problem with interacting criteria can be divided according to preference information, into two classes. One is that some studies tried to solve MCDM with interacting criteria only from the preference information on criteria like the following: The relative importance and interacting for each pair of criteria[13, 14], the partial weak order and the sign of interaction over criteria[15]. The other is that the preference information on alternatives, including overall values, preference relation and preference orderings. Mori and Murofushi[16] minimized the average quadratic distance between the overall utilities computed by the Choquet integral and the desired overall values given by the DM, then constructed a quadratic program to identify capacities. Beliakov[17] adopted the least absolute deviation and constructed a linear programming model to identify capacities. Marichal and Roubens[18] proposed the approach, which maximize the minimal difference between the overall utilities of alternatives that have been compared by the DM through the preference relation. Lately, Angilella, Greco and Matarazzo[19] proposed a non-additive robust regression model based on the Choquet integral to identify capacities. This is done on the basis of the preference relation over alternatives and the criteria, the intensity of preference over couples of alternatives and the criteria, and the sign of interaction of couples of criteria as well as the pairwise comparisons on the interaction intensities. Angilella, Greco, et al.[20] presented a heuristic method to construct non-linear optimization model based on the Choquet integral for MCDM with interacting criteria, according to the complete preference orderings in subset of alternatives, some relations of importance between criteria and the sign of interaction between criteria given by the DM.

However, these approaches do not handle the problem where the preference information on alternatives is provided in fuzzy terms. In the paper, we are concerned with the MCDM problem with interacting criteria where the fuzzy preference information on alternatives is provided by the DM. Giving two types of models to solve this problem, i.e., the least squares model and the linear programming model, which seek the capacities that holds the minimum squared deviation and the minimum absolute deviation between the final overall values based on the fuzzy preference relation of alternatives and the final overall values computed by means of the Choquet integral, then to select the most desirable alternative. The main characteristic of the proposed methods is that it has more general for MCDM problem with interacting criteria. The reason is that the other kinds of preference information over alternatives (except for overall values) are always converted into fuzzy preference relation[21, 22].

This paper is organized as follows. In Section 2, some basic concepts are introduced. Then, two types of models are constructed in Section 3. Section 4 shows an illustrative example of the proposed methods and some conclusions are included in Section 5.

2 Preliminaries

Let D = {d₁, d₂, …, d_m} denote a set of m alternatives. G = {g₁, g₂, …, g_n} denote a set of n criteria. Each alternative d_i ∈ D is associated with a profile g(d_i) = (g₁(d_i), g₂(d_i), …, g_n(d_i)) ∈ Rⁿ where g_j(d_i) represents the utility of d_i related to the criterion g_j, with g_j(d_i) ∈ Ri = 1, 2, …, m, j = 1, 2, …, n.

The fuzzy preference information on alternatives is described by a binary fuzzy relation P in D, where P is a mapping D × D → [0,1]. P_ik denotes the preference degree of alternative d_i over d_k. We suppose that P is reciprocal, by definition[23, 24]

1. P_ik + P_ki = 1, ∀ i,k;

2. P_ii = – (symbol ′–′ means that the DM dose not give any preference information on d_i), ∀i.

In the context of MCDM analysis, for each subset of criteria T ⊆ G, μ(T) can be considered as the weight or the importance of T[25]. A capacity is said to be additive if μ(S ∪ T) = μ(S) + μ(T) where S ∩ T = ∅. In this situation, the n weight (i.e., μ({g₁}), μ({g₂}), …, μ({g_n})) are sufficient to construct the whole capacity. In general, one has to calculate (2^|G| – 2) values μ (T), ∅ ⊂ T ⊂ G when the capacity is non-additive[26].

If the criteria from G are interacting and their importance is represented by a capacity, a suitable aggregation operator is the Choquet integral[25].

The Möbius representation a(S) is obtained by μ(T) in the following way[19, 26]:

where lower case letters s and t denote the corresponding cardinalities of subsets S and T.

Under Möbius representation, properties (1) and (2) can be formulated as[19, 26]

1. a(∅) = 0, ∑_{T ⊆ G}a(T) = 1;

2. a({g_i}) + ∑_{T ⊆ S}a(T ∪ {g_i}) ≥ 0, ∀ g_i ∈ G and ∀S ⊆ G \ {g_i}.

The Choquet integral can be rewritten in terms of Möbius representation as[20]

We can see that a capacity μ on G is defined by the knowledge of (2^|G| – 2) coefficients. With the aim of reducing the number of coefficients to be elicited, Grabsich proposed the notion of k-additive capacities[10].

2-additive capacities has received more and more attention because of its simplicity and realistic in many real decision problems. Hence, we shall consider 2-additive capacities. According to Möbius representation, a value a({g_i}) for every criterion g_i and a value a({g_i, g_j}) for every couple of criteria {g_i, g_j}.

Taking into account the 2-additive capacities, capacity μ assigns to set S ⊆ G can be expressed in terms of the Möbius representation as follows[19, 26]:

With respect to 2-additive capacities, properties (1a) and (2a) can be restated as follows[19, 26]:

1. a(∅)=0,   Σgi∈Ga({gi})+Σ{gi,gj}⊆Ga({gi,gj})=1,
2. a({gi})≥0,  ∀gi∈G,a({gi})+∑gj∈Ta({gi,gj})≥0,  ∀gi∈G  and  ∀T⊆G∖{gi}, T≠∅.

The representation of the Choquet integral is given by

and the importance of criteria g_i ∈ G, described by the Shapley value can be written as follows[19, 26]:

while the interaction index for a couple of criteria {g_i, g_j} ⊆ G is given by[19, 26]

3 Description of the Methodology

4 An Illustrative Example

The whole methodology will be illustrated by a practical example. We suppose that the DM has to rank the cars of Table 1 according to his/her preference. The cars are evaluated on four criteria: price in Euro (P), acceleration 0–100km/h in seconds (A), maximum speed in km/h (MS), consumption in km/l (C) (adapted from [20]).

The DM’s fuzzy preference relation on seven cars are given in Table 2. Making the evaluations on all criteria is expressed on the same scale. As shown in [24] (Table 3). And explicit preference information with respect to the important and the interaction of the considered criteria are given as follows:

The above constraints can be translated as:

where the indifference thresholds δ_I and δ_Sh are supposed to have been set to 0.05 and 0.01 by the DM[25].

We can construct the least squares programming model. Solving this programming by the mathematical software lingo. It is possible to obtain their Möbius representations and the fuzzy measures as well as the Shapley value. The results are show in Tables 4–6. Finally, we have the overall values of seven cars and get the ranking of seven cars, as shown in Table 7.

According to the linear programming model, we have the optional solution, as shown in Table 8. The fuzzy measures of couples of criteria and the Shapley value are listed in Tables 9–10, the overall values and the ranking of seven cars are in Table 11.

In Table 6 and Table 10, we observe that the Shapley value of price is greater than that of the criterion acceleration, which is greater than that of criterion consumption. It showed that the preference of the DM are presented by the model. At the same time, From Table 7 and Table 11, we can see the ranking obtained by the least squares programming is the same as that obtained by the method of linear programming.

5 Conclusions

In this paper, We gave the least squares programming model and the linear programming model to solve MCDM problem with interacting criteria. Compared with the methods based on the preference information. The proposed approach that considers the fuzzy relation on alternatives given by the DM is relatively reasonable. The ranking of the alternatives combined the fuzzy preference information with the partial evaluations of the alternatives, and reflected the preference information of DM to the maximum extent.