#### Abstract

A quadratic programming model is constructed for solving the fuzzy cooperative games with coalition values expressed by triangular fuzzy numbers, which will be abbreviated to TFN-typed cooperative games from now on. Based on the concept of -cut set and the representation theorem for the fuzzy set, the least square distance solution for solving TFN-typed cooperative games is proposed. The least square distance solution successfully avoids the subtraction operation of TFNs, which may inevitably lead to the amplification of uncertainty and the distortion of decision information. A calculating example related to the profit distribution of logistics coalition is illustrated to show the advantages, validity, and applicability of the proposed method. Besides, the least square distance solution for solving TFN-typed cooperative games satisfies many important properties of cooperative games, such as uniqueness, additivity, symmetry, and uniqueness.

#### 1. Introduction

There exist many problems and phenomena related to fuzzy cooperative games in our daily lives. An increasing number of researchers turn their attention to the theory and application of fuzzy cooperative games. Generally speaking, we usually divide the fuzzy cooperative games into three different types as follows: cooperative games with fuzzy coalition values [1–3], cooperative games with fuzzy coalitions [4–6], and cooperative games with both fuzzy coalitions and fuzzy coalition values [7, 8], respectively. As for the three types of fuzzy cooperative games mentioned above, the cooperative games with fuzzy coalition values gradually become a research hotspot in recent years. Some researchers extend the common solutions of crisp cooperative games to fuzzy cooperative games and propose some corresponding solution methods and solution concepts for solving fuzzy cooperative games, such as fuzzy Shapley value [9–11], fuzzy set-valued solution [12], fuzzy least square prenucleolus and B-nucleolus [13–15], fuzzy bargaining sets [16, 17], and fuzzy equalizer and lexicographical solution [18].

However, many existing models and methods for solving fuzzy cooperative games inevitably use the subtraction operation of fuzzy numbers. As is known to all, some operations of fuzzy numbers especially the subtraction operation may easily result in the amplification of uncertainty and the distortion of decision information.

In this paper, we are absorbed in developing a new and intuitionistic method for the TFN-typed cooperative games, which can successfully avoid the subtraction operation of TFNs. The rest of the paper is arranged as follows. Section 2 briefly introduces some key concepts for constructing the quadratic programming model, such as the TFN, the -cut set, and the representation theorem for the fuzzy set. Section 3 puts forward the least square distance solution for TFN-typed cooperative games based on the square distance and -cut sets of TFNs. In order to show the superiority, advantages, and applicability of the proposed method, a calculating example about the profit distribution of logistics coalition is illustrated. Besides, the interval Shapley-like value proposed in [9] is used to redetermine the optimal allocation strategy and the results from the two methods are compared with each other. Conclusion is made in Section 5.

#### 2. Preliminaries

##### 2.1. The Definition of TFNs

The membership function of a random TFN [19] can be shown as follows: and, in (1), denotes the lower bound of , denotes the mean of , and denotes the upper bound of , respectively. It is not difficult to find that the TFN will reduce to a real number if the three values , , and are equal. In other words, an arbitrary real number can be changed to the form of the corresponding TFN, which is shown as .

##### 2.2. The -Cut Set of the TFN and the Representation Theorem for the Fuzzy Set

As is known to all, the definition of the -cut set of an arbitrary TFN can be shown as . Therefore, the -cut set of an arbitrary TFN at any confidence level can be obtained, which is actually an interval expressed by .

According to (1), we haveandThat is to say,It is obvious thatandThrough the further analysis based on the interval operations [20], we havewhich means as long as we know the 1-cut set and 0-cut set of an arbitrary TFN , then we can conveniently calculate its -cut set at any confidence level.

According to the representation theorem for the fuzzy set [21], a random TFN can be denoted as the following form:

#### 3. The Least Square Distance Solution for TFN-Typed Cooperative Games Based on the Square Distance and -Cut Sets

##### 3.1. TFN-Typed Cooperative Games

In this section, we will demonstrate the mathematical representation of the TFN-typed cooperative games. A TFN-typed cooperative game in coalitional form can be shown as an ordered pair . is the set of all the players conducting cooperation. is the characteristic function of the coalition (), which is a TFN expressed as . For any coalition (), denotes the minimal profit, denotes the mean profit, and denotes the maximal profit the coalition can realize if all of the players in form a coalition and cooperate with each other. What needs illustration is that the memberships of the possible coalition values (i.e., profits) are usually different from one another. Particularly, is an empty coalition, which means nobody is willing to join the cooperation, so . For the sake of convenient description, () is abbreviated to and () is abbreviated to , respectively, in this paper.

##### 3.2. Quadratic Programming Model for Solving the Least Square Distance Solution

As mentioned before, at any confidence level, the -cut set of the characteristic function of the coalition is an interval, which can be shown as follows:

For an arbitrary TFN-typed cooperative game, conclusion can be drawn that every player () should obtain a TFN-typed payoff, which is described by because the characteristic function of the coalition () is a TFN. Hence, denotes a payoff vector, where the payoff of every player () is expressed by a TFN. As is known to all, if a payoff vector satisfies the effectiveness of attribution (i.e., ), the payoff vector is regarded as a preimputation or an efficient payoff. Similarly, if a payoff vector simultaneously satisfies the effectiveness of attribution (i.e., ) and the individual rationality (i.e., for all ), the payoff vector is regarded as an imputation. Otherwise, the payoff vector may be unsatisfactory.

According to the definition and properties of the -cut set, the -cut set of the TFN-typed payoff of the player () can be written as follows:

denotes the sum of -cut sets of the TFN-typed payoffs of all players in the coalition (). Therefore, we havewhich is also an interval.

In order to construct the quadratic programming model for solving the optimal attribution strategy of players, we use the square distance to measure the difference between and based on the least square method, which is shown as follows:

To some extent, the square distance between and can be regarded as a measure of the dissatisfaction of coalition () once the payoff vector is advised to be the final attribution strategy.

For the sake of concise description, is replaced by and is replaced by , respectively. Hence, the square distance between and can be rewritten as and the sum of the dissatisfaction of coalition can be shown as

Generally speaking, denotes the mean value of and denotes the mean value of , respectively. Conclusions can be drawn thatand

Based on the principle of fairness, i.e., each player () would like to minimize its loss [20, 21], we construct the following quadratic programming model:

Because ofandthe quadratic programming model (17) can be rewritten as follows:

According to the Lagrange multiplier method, the Lagrange function of the quadratic programming model (20) can be obtained as follows:

Take the solution process of () as an example. Let partial derivatives of with regard to the variables and be equal to 0, respectively. So, and

The following result can be obtained through mathematical derivation that

Based on (22) and (24), we have

It is obvious to see thatTherefore,As a result,Based on (23) and (28), we can obtainHence,where denotes the number of players in coalition .

Combining with (28) and (30), we can obtain the following analysis formula of :

In the similar way, we can finally obtain the following analysis formula of :

Until now, we have obtained the optimal solution of the quadratic programming model (17) (i.e., (20)). According to the representation theorem for the fuzzy set, the TFN-typed imputation of the player () can be expressed as

#### 4. A Numerical Example and Computational Result Analysis

In Section 3, we construct a quadratic programming model for solving the least square distance solution of the TFN-typed cooperative games. The method proposed in this paper can be applied to many fields, which may relate to cooperation and the distribution of profits such as supply chain management, logistics coalition, environmental collaboration, and strategic cooperation. We have elaborated with detail the process of the least square distance solution of the TFN-typed cooperative games and what is following is a calculating example about the profit distribution of logistics coalition to examine its practicability, rationality, and superiority.

*Example 1. *Considering a logistics coalition composed of three logistics enterprises, which are called player 1, player 2, and player 3, respectively, each logistics enterprise can operate alone with small profit. However, once all of them form a coalition and work together, the operational hazard will reduce, the market share will increase, and the divisible profit will rise. Owing to the fuzzy uncertainty in the freight transport market, the cooperative profits cannot be forecast accurately and just the value ranges, the profit, and the corresponding membership degrees can be estimated. As a result, we use the TFN to denote the characteristic function (i.e., profit) of the coalition (). Through the elaborated market research, we can know the coalition profits in different situations as follows: , , , , , , and .

##### 4.1. Computational Results Obtained by the Proposed Method

According to (31), for logistics enterprise (i.e., player) 1, we haveFor logistics enterprise (i.e., player) 2, we haveFor logistics enterprise (i.e., player) 3, we have

In the similar way, according to (32), for logistics enterprise (i.e., player) 1, we haveFor logistics enterprise (i.e., player) 2, we haveFor logistics enterprise (i.e., player) 3, we have

According to (33), we can calculate the optimal distribution strategy of the three logistics enterprises when they form a cooperative coalition and work together, which are shown as follows:andrespectively.

##### 4.2. Results Analysis and Comparison

By watching the distribution results carefully, we see that the sum of the lower bounds of the three logistics enterprises’ distribution can be shown as , which is equal to the lower bound of the characteristic function of the grand coalition . The similar conclusions apply to logistics enterprise 2 and logistics enterprise 3. All of the profits created by the cooperative coalition have been absolutely distributed to the three logistics enterprises. In other words, the optimal distribution strategy based on the least square distance solution satisfies the property of efficiency. What is more, each logistics enterprise obtains more profit than they operate business alone. Therefore, all of the three logistics enterprises are willing to work together to make more profits.

In order to show more intuitively the superiority of the least square distance solution proposed in this paper, we redetermine the allocation strategy according to the interval Shapley-like value [9].

Based on (9), the -cut sets of the TFN-typed profits of coalition () (i.e., ) in Example 1 can be written as the following intervals: and

Therefore, the -cut sets of the TFN-typed Shapley-like values () can be shown as follows:and

According to the representation theorem for the fuzzy set, we can obtain the following results: