The graph games with fuzzy characteristic functions

Abstract

This paper proposes the H-average tree solution based on a graph game with fuzzy characteristic function. This solution is a generalization of the classical average tree solution, and is an imputation of a graph game with fuzzy characteristic function. When the game is cycle-free the H-average tree solution is the unique solution that satisfies component efficiency and component fairness. The interval average tree solution is introduced. Furthermore, we show the relationship between the H-average tree solution and the interval average tree solution. A practical application of the proposed model in outsource production is given.

Keywords

Graph game with fuzzy characteristic function H-average tree solution imputation interval average tree solution

1 Introduction

Due to the impact of competition, politics and other factors, cooperation is not free, but limited. Myerson [1] formulated a transferable utility cooperative game with communication structure, called graph game. Players can cooperate only if they are directly or indirectly connected. Demange [2] interpreted the tree as a hierarchy on the set of players and showed that hierarchies yield stability. In 2008 Talman [3] and Herings et al. [4] proposed the average tree solution of a cycle-free graph game respectively. Later, the solution was generalized to the class of all graph games by Herings et al. [5].

Because of many uncertain factors the real outcome of cooperation only is an approximate evaluation. Mares [6, 7] and Vlach [8] discussed the cooperative games with fuzzy characteristic functions. In their models the allocations of players are fuzzy numbers. However, there is not any imputation for a graph game when its fuzzy characteristic function is fuzzy.

With economic globalization, outsource produc-tion is taken by many companies. In this paper, we study a graph game with fuzzy characteristic function and define the H-average tree solution. The properties of the solution are discussed. Finally we give an example to show the application of the H-average tree solution in outsource production.

2 Preliminaries

2.1 Operations of fuzzy numbers

Let us begin with reminding the most general definition of operations of fuzzy numbers [9]. We denote the collection of all bounded fuzzy numbers by $\tilde{F} R$ . For any λ ∈ [0, 1], we have the level set of $\tilde{A} \in \tilde{F} R : {\tilde{A}}_{λ} = {x \in R : \tilde{A} (x) \geq λ}$ , where $\tilde{A} (x)$ is the membership function of $\tilde{A}$ . The λ-cuts ${\tilde{A}}_{λ}$ can be represented by an interval number ${\tilde{A}}_{λ} = [{\tilde{A}}_{λ}^{-}, {\tilde{A}}_{λ}^{+}]$ .

Definition 2.1. Let $\tilde{A}, \tilde{B} \in \tilde{F} R$ . If there exists $\tilde{C} \in \tilde{F} R$ such that $\tilde{A} = \tilde{B} + \tilde{C}$ , then $\tilde{C}$ is called the Hukuhara difference, represented by $\tilde{A} -_{H} \tilde{B}$ .

Theorem 2.1. Let $\tilde{A}, \tilde{B} \in \tilde{F} R$ . If $\tilde{A} -_{H} \tilde{B}$ exists, then $\begin{matrix} {(\tilde{A} -_{H} \tilde{B})}_{λ} = {\tilde{A}}_{λ} -_{H} {\tilde{B}}_{λ} \\ = [{\tilde{A}}_{λ}^{-} - {\tilde{B}}_{λ}^{-}, {\tilde{A}}_{λ}^{+} - {\tilde{B}}_{λ}^{+}], \forall λ \in [0, 1] . \end{matrix}$

Theorem 2.2. Let $\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D} \in FR$ . If $\tilde{A} -_{H} \tilde{B}$ and $\tilde{C} -_{H} \tilde{D}$ exist, then $(\tilde{A} + \tilde{C}) -_{H} (\tilde{B} + \tilde{D}) = (\tilde{A} -_{H} \tilde{B}) + (\tilde{C} -_{H} \tilde{D}) .$

2.2 Graph games

We consider a transferable utility cooperative game with communication structure, also called a graph game [4]. It is denoted by (N, v, L) where N ={ 1, ⋯, n } is a node set of players, v : 2^N → R a characteristic function such that v (φ) = 0 and L⊆ { { i, j } |i ≠ j, i, j ∈ N } a set of edges. Meanwhile, (N, L) is called an (undirected) graph.

Definition 2.2. A coalition S ∈ 2^N is called a network of (N, v, L) if S is connected on the graph (N, L). Furthermore, a network S forms a component on (N, v, L) if S can not form a large network with any other players of N ∖ S.

For a graph game (N, v, L) we denote C ^L (N) the set of all networks and ${\overset{\land}{C}}^{L} (N)$ the collection of all components.

Let (N, L) be a graph, W ∈ 2^N and L (W) ={{ i, j } ∈ L|i, j ∈ W }. Then (W, L (W)) is called a subgraph of the graph (N, L). For $K \in {\overset{\land}{C}}^{L} (N)$ of a graph game (N, v, L) we get two new components after deleting a link {i, j} on (K, L (K)). Then we denote by K ^h the component containing the player h, where h = i, j. On a graph game (N, v, L) a sequence of different nodes (i ₁, …, i _k) is called a cycle when it satisfies: (1) k ≥ 2; (2) i _k = i ₁; (3) {i _h, i _h+1} ∈ L, h = 1, …, k - 1. A graph game (N, v, L) is cycle-free if it does not contain any cycle.

For a graph (N, L) an n-tuple B ={ B ₁, ⋯ B _n } of n subsets of N is admissible if

i ∈ B _i for all i ∈ N, and B _j = N for some j ∈ N;

K = B _j for all i ∈ N and $K \in {\hat{C}}^{L} (B_{i} ∖ {i})$ , and {i, j } ∈ L for some j ∈ N.

The class of all admissible n-tuples B ={ B ₁, ⋯, B _n } of (N, L) is denoted by B ^L.

We define a graph game with fuzzy characteristic function by $(N, \tilde{v}, L)$ in which the characteristic function $\tilde{v} : P (N) \to \tilde{F} R$ and $\tilde{v} (φ) = 0$ . A graph game with fuzzy characteristic function degenerates to be a crisp game when the domain of the characteristic function is the set of real numbers. So we only give some existing facts on a graph game with fuzzy characteristic function.

Definition 2.3. For a graph game with fuzzy characteristic function $(N, \tilde{v}, L)$ and W ∈ 2^N, a fuzzy vector $\tilde{x} (W) =$ $({\tilde{x}}_{1} (W), \dots, {\tilde{x}}_{n} (W))$ is said to be an imputation about W if it satisfies:

${\tilde{x}}_{i} (W) = 0, \forall i \notin W$ ;

$\sum_{i \in N} {\tilde{x}}_{i} (W) = \tilde{v} (W)$ ;

${\tilde{x}}_{i} (W) \underline{≻}$ $\tilde{v} ({i}), i \in W$ .

Herings [4, 5] has discussed the average tree is unique if the solution satisfies component fairness and component efficiency for a classical graph game. We can get the similar conclusion for a graph game with fuzzy characteristic function.

Theorem 2.3. Let f be a function of a cycle-free graph game with fuzzy characteristic function $(N, \tilde{v}, L)$ and W ∈ 2^N. Then f is the unique solution about W on $(N, \tilde{v}, L)$ if the followings hold:

component fairness

$\begin{matrix} \frac{1}{K^{h}} \sum_{i \in K^{h}} (f_{i} (\tilde{v}, L (W)) - f_{i} (\tilde{v}, L (W) ∖ {h, l})) = \\ \frac{1}{K^{l}} \sum_{i \in K^{l}} (f_{i} (\tilde{v}, L (W)) - f_{i} (\tilde{v}, L (W) ∖ {h, l})) \\ \forall {h, l} \in L (K) and K \in C^{L (W)} (W); \end{matrix}$

component efficiency

$\sum_{i \in K} f_{i} (\tilde{v}, L (W)) = \tilde{v} (K), \forall K \in {\hat{C}}^{L (W)} (W) .$

3 H-average tree solution on a graph game with fuzzy characteristic function

Let $(N, \tilde{v}, L)$ be a graph game with fuzzy characteristic function. ∀λ, β ∈ [0, 1], λ ⩽ β, $\begin{matrix} {\tilde{v}}_{λ}^{-} (S) - {\tilde{v}}_{λ}^{-} (T) ⩽ {\tilde{v}}_{β}^{-} (S) - {\tilde{v}}_{β}^{-} (T) ⩽ \\ {\tilde{v}}_{β}^{+} (S) - {\tilde{v}}_{β}^{+} (T) ⩽ {\tilde{v}}_{λ}^{+} (S) - {\tilde{v}}_{λ}^{+} (T), \\ \forall S, T \in 2^{N}, T \subseteq S . \end{matrix}$ (1)

We denote by ${\tilde{G}}_{H} (N)$ the class of graph games with fuzzy characteristic functions that satisfy Equation (1). By the existence of Hukuhara difference described in [9], we know $\tilde{v} (S) -_{H} \tilde{v} (T)$ exists if and only if Equation (1) holds.

Definition 3.1 For $(N, \tilde{v}, L)$ $\in {\tilde{G}}_{H} (N)$ and W ∈ 2^N, we define the H-average tree solution $\tilde{HAT} (W, \tilde{v}, L (W))$ is the vector by $\begin{matrix} {\tilde{HAT}}_{i} (W, \tilde{v}, L (W)) = \\ {\begin{matrix} \frac{1}{| B^{L (W)} |} \sum_{B \in B^{L (W)}} [\tilde{v} (B_{i}) -_{H} \sum_{K \in {\hat{C}}^{L (W)} (B_{i} ∖ {i})} \tilde{v} (K)], i \in W, \\ 0, i \in N ∖ W . \end{matrix} \end{matrix}$ (2)

When $(N, \tilde{v}, L) \in {\tilde{G}}_{H} (N)$ is cycle-free, we write T ^j (W) a spanning tree induced by the root j ∈ W in the subgraph (W, L (W)) and $K_{i}^{j} (W)$ the set consisting of i ∈ W and all its subordinates in T ^j (W). It is clear that

${\begin{matrix} \frac{1}{| W |} \sum_{j \in W} (\tilde{v} (K_{i}^{j} (W)) - H \sum_{{i^{'} | (i, i^{″}) \in T^{j} (W)}} \tilde{v} (K_{i}^{j} (W))), i \in W \\ 0 & i \in N \ W . \end{matrix}$ (3)

From Equation (2) we can get the average tree solution AT (W, v, L (W)) of the crisp graph game (N, v, L) when the fuzzy characteristic function $\tilde{v}$ degenerates to be the crisp characteristic function v. $\begin{matrix} A T_{i} (W, v, L (W)) = \\ {\begin{matrix} \frac{1}{| B^{L (W)} |} \sum_{B \in B^{L (W)}} [v (B_{i}) - \sum_{K \in {\hat{C}}^{L (W)} (B_{i} ∖ {i})} v (K)], i \in W, \\ 0, i \in N ∖ W . \end{matrix} \end{matrix}$

It is easy to prove that AT (W, v, L (W)) is an imputation about W in (N, v, L). And the solution satisfies component efficiency and component fairness. Thus, the H-average tree solution is an extension of AT (W, v, L (W)). By Theorem 2.1, we get the following.

Lemma 3.1. Let $(N, \tilde{v}, L) \in {\tilde{G}}_{H} (N)$ and W ∈ 2^N, then ${({H \tilde{A} T}_{i} (W, \tilde{v}, L (W)))}_{λ} =$ ${H \tilde{A} T}_{i} (W, {\tilde{v}}_{λ},$ $L (W)) = [{H \tilde{A} T}_{i} (W, {\tilde{v}}_{λ}^{-}, L (W)), {H \tilde{A} T}_{i} (W, {\tilde{v}}_{λ}^{+},$ L (W))], ∀i ∈ N and λ ∈ [0, 1], where $H \tilde{A} T (W, {\tilde{v}}_{λ}^{-}, L (W))$ and $H \tilde{A} T (W, {\tilde{v}}_{λ}^{+}, L (W))$ are respectively the average tree solutions of the crisp graph games $(N, {\tilde{v}}_{λ}^{-}, L)$ and $(N, {\tilde{v}}_{λ}^{+}, L)$ .

Theorem 3.1. Let $(N, \tilde{v}, L)$ and $(N, \tilde{w}, L)$ $\in {\tilde{G}}_{H} (N)$ , W ∈ 2^N, $(N, \tilde{v} + \tilde{w}, L)$ is defined by: $(\tilde{v} + \tilde{w}) (S) = \tilde{v} (S) + \tilde{w} (S), \forall S \in P (N)$ . Then the H-average tree solutions satisfy additivity, i.e. ${H \tilde{A} T}_{i} (W, \tilde{v} + \tilde{w}, L (W)) = {H \tilde{A} T}_{i} (W, \tilde{v}, L (W)) +$ ${H \tilde{A} T}_{i} (W, \tilde{w}, L (W)), \forall i$ ∈N.

Proof. If i ∈ N ∖ W, it is obvious that $H \tilde{A} T_{i} (W, \tilde{v}, L (W))$ satisfies additivity. If i ∈ W, then from Theorem 2.2, $\begin{matrix} H \tilde{A} T (W, \tilde{v} + \tilde{w}, L (W)) \\ = \frac{1}{| B^{L (W)} |} \sum_{B \in B^{L (W)}} [(\tilde{v} + \tilde{w}) (B_{i}) -_{H} \sum_{K \in {\hat{C}}^{L (W)} (B_{i} ∖ {i})} (\tilde{v} + \tilde{w}) (K)] \\ \frac{1}{| B^{L (W)} |} \sum_{B \in B^{L (W)}} {[\tilde{w} (B_{i}) + \tilde{v} (B_{i})] -_{H} \\ \sum_{K \in {\hat{C}}^{L (W)} (B_{i} ∖ {i})} [\tilde{v} (K) + \tilde{w} (K)]} \\ = \frac{1}{| B^{L (W)} |} \sum_{B \in B^{L (W)}} [\tilde{v} (B_{i}) -_{H} \sum_{K \in {\hat{C}}^{L (W)} (B_{i} ∖ {i})} \tilde{v} (K)] + \\ \frac{1}{| B^{L (W)} |} \sum_{B \in B^{L (W)}} [\tilde{w} (B_{i}) -_{H} \sum_{K \in {\hat{C}}^{L (W)} (B_{i} ∖ {i})} \tilde{w} (K)] \end{matrix} = H \hat{A} T (W, \tilde{v}, L (W)) + H \tilde{A} T (W, \tilde{w}, L (W))$ The proof is completed. □ Theorem 3.2. If $(N, \tilde{v}, L)$ $\in {\tilde{G}}_{H} (N)$ is cycle-free and W ∈ 2^N, then the H-average tree solution $H \tilde{A} T (W, \tilde{v}, L (W))$ is the unique solution that satisfies component efficiency and component fairness about the coalition W on $(N, \tilde{v}, L)$ . Proof. Given any λ ∈ [0, 1] and W ∈ 2^N, by Lemma 3.1 and Theorem 2.1, we have $\begin{matrix} {({H \tilde{A} T}_{i} (W, \tilde{v}, L (W)))}_{λ}^{-} = {H \tilde{A} T}_{i} (W, {\tilde{v}}_{λ}^{-}, L (W)) = \\ \frac{1}{| B^{L (W)} |} \sum_{B \in B^{L (W)}} [{\tilde{v}}^{-} (B_{i}) -_{H} \sum_{K \in {\hat{C}}^{L (W)} (B_{i} ∖ {i})} {\tilde{v}}^{-} (K)], \\ {({H \tilde{A} T}_{i} (W, \tilde{v}, L (W)))}_{λ}^{+} = {H \tilde{A} T}_{i} (W, {\tilde{v}}_{λ}^{+}, L (W)) = \\ \frac{1}{| B^{L (W)} |} \sum_{B \in B^{L (W)}} [{\tilde{v}}^{+} (B_{i}) -_{H} \sum_{K \in {\hat{C}}^{L (W)} (B_{i} ∖ {i})} {\tilde{v}}^{+} (K)], \\ \forall i \in N . \end{matrix}$ Moreover, the crisp average tree solution is component efficient, the following is valid: $\begin{matrix} \sum_{i \in K} {({H \tilde{A} T}_{i} (W, \tilde{v}, L (W)))}_{λ}^{-} = {\tilde{v}}_{λ}^{-} (K), \\ \sum_{i \in K} {({H \tilde{A} T}_{i} (W, \tilde{v}, L (W)))}_{λ}^{+} = {\tilde{v}}_{λ}^{+} (K), \\ \forall K \in {\hat{C}}^{L (W)} (W) . \end{matrix}$ Consequently, $\sum_{i \in K} {({H \tilde{A} T}_{i} (W, \tilde{v}, L (W)))}_{λ} = {\tilde{v}}_{λ} (K)$ which means $\sum_{i \in K} ({H \tilde{A} T}_{i} (W, \tilde{v}, L (W))) = \tilde{v} (K), K \in {\hat{C}}^{L (W)} (W) .$ Hence, H-average tree solution $H \tilde{A} T (W, \tilde{v},$ L (W)) is component efficient. It is apparent that $H \tilde{A} T (W, {\tilde{v}}_{λ}^{-}, L (W))$ and $H \tilde{A} T (W, {\tilde{v}}_{λ}^{+}, L (W))$ satisfy component fairness. By Lemma 3.1, we obtain $({H \tilde{A} T}_{i} (W, \tilde{v},$ L (W))) _λ = $[{H \tilde{A} T}_{i} (W, {\tilde{v}}_{λ}^{-}, L (W)), {H \tilde{A} T}_{i} (W,$ ${\tilde{v}}_{λ}^{+}, L (W))], \forall i \in N$ . Then for any $K \in {\hat{C}}^{L (W)} (W)$ and {i, j } ∈ L (K), the following holds: $\begin{matrix} \frac{1}{| K^{h} |} \sum_{i \in K^{h}} {{H \tilde{A} T}_{i} (W, {\tilde{v}}_{λ}, L (W)) - \\ {H \tilde{A} T}_{i} (W ∖ {h, l}, {\tilde{v}}_{λ}, L (W ∖ {h, l})) = \\ \frac{1}{| K^{l} |} \sum_{i \in K^{l}} {{H \tilde{A} T}_{i} (W, {\tilde{v}}_{λ}, L (W)) - \\ {H \tilde{A} T}_{i} (W ∖ {h, l}, {\tilde{v}}_{λ}, L (W ∖ {h, l}))} . \end{matrix}$ Therefore, $H \tilde{A} T (W, \tilde{v}, L (W))$ satisfies component fairness, i.e. $\begin{matrix} \frac{1}{| K^{h} |} \sum_{i \in K^{h}} {{H \tilde{A} T}_{i} (W, \tilde{v}, L (W)) - \\ {H \tilde{A} T}_{i} (W ∖ {h, l}, \tilde{v}, L (W ∖ {h, l})) = \\ \frac{1}{| K^{l} |} \sum_{i \in K^{l}} {{H \tilde{A} T}_{i} (W, \tilde{v}, L (W)) - \\ {H \tilde{A} T}_{i} (W ∖ {h, l}, \tilde{v}, L (W ∖ {h, l}))}, \\ \forall K \in {\hat{C}}^{L (W)} (W) . \end{matrix}$ For the coalition W, $H \tilde{A} T (W, \tilde{v}, L (W))$ is the unique solution that satisfies component efficiency and component fairness from Theorem 2.3. □

4 Properties of H-average tree solution

Theorem 4.1. Let $(N, \tilde{v}, L)$ $\in {\tilde{G}}_{H} (N)$ and W ∈ 2^N, then the H-average tree solution $H \tilde{A} T (W, \tilde{v}, L (W))$ is an imputation about the coalition W on $(N, \tilde{v}, L)$ .

Proof.

By Definition 3.1, ${H \tilde{A} T}_{i} (W, \tilde{v}, L (W)) = 0, \forall i \notin W .$

Because $H \tilde{A} T (W, \tilde{v}, L (W))$ is component efficient, $\begin{matrix} \sum_{i \in N} {H \tilde{A} T}_{i} (W, \tilde{v}, L (W)) = \\ \sum_{i \in W} {H \tilde{A} T}_{i} (W, \tilde{v}, L (W)) \\ = \sum_{K \in {\hat{C}}^{L (W)} (W)} \sum_{i \in K} {H \tilde{A} T}_{i} (W, \tilde{v}, L (W)) \\ = \sum_{K \in {\hat{C}}^{L (W)} (W)} \tilde{v} (K) \\ = \tilde{v} (W) . \end{matrix}$

For any λ ∈ [0, 1] and i ∈ W, from Lemma 3.1, it is obvious that

${({H \tilde{A} T}_{i} (W, \tilde{v}, L (W)))}_{λ}^{-} = {H \tilde{A} T}_{i} (W, {\tilde{v}}_{λ}^{-}, L (W)) .$

Due to ${H \tilde{A} T}_{i} (W, {\tilde{v}}_{λ}^{-}, L (W))$ and ${H \tilde{A} T}_{i} (W,$ ${\tilde{v}}_{λ}^{+}, L (W))$ are respectively imputations of crisp graph games $(N, {\tilde{v}}_{λ}^{-}, L)$ and $(N, {\tilde{v}}_{λ}^{+}, L)$ , we have that

$\begin{matrix} {H \tilde{A} T}_{i} (W, {\tilde{v}}_{λ}^{-}, L (W)) \geq {\tilde{v}}_{λ}^{-} ({i}), \\ {H \tilde{A} T}_{i} (W, {\tilde{v}}_{λ}^{+}, L (W)) \geq {\tilde{v}}_{λ}^{+} ({i}) . \end{matrix}$

Furthermore, ${H \tilde{A} T}_{i} (W, \tilde{v}, L (W)) \underline{≻} \tilde{v} ({i})$ which implies that ${H \tilde{A} T}_{i} (W, \tilde{v}, L (W)) \underline{≻}$ $\tilde{v} ({i})$ . The proof is completed. □

Definition 4.1. For a game with fuzzy characteristic function $(N, \tilde{v}, L)$ , the interval average tree solution $M \tilde{A} T (N, \tilde{v}, L)$ is defined by

$\begin{matrix} {M \tilde{A} T}_{i} (N, \tilde{v}, L) = \\ \frac{1}{| B^{L} |} (\sum_{B \in B^{L}} (\tilde{v} (B_{i}) - \sum_{K \in {\hat{C}}^{L} (B_{i} ∖ {i})} \tilde{v} (K))), i \in N . \end{matrix}$ (4)

Theorem 4.2. Let $(N, \tilde{v}, L)$ $\in {\tilde{G}}_{H} (N)$ , then the relationship between the interval average tree solution $M \tilde{A} T (N, \tilde{v}, L)$ and the H-average tree solution $H \tilde{A} T (W, \tilde{v}, L (W))$ is $M \tilde{A} T (N, \tilde{v}, L) \supseteq$ $H \tilde{A} T (W, \tilde{v}, L (W)), \forall i \in N .$

Proof. From Equation (4), it follows that for any λ ∈ [0, 1] and i ∈ N $\begin{matrix} {(M \tilde{A} T_{i} (N, \tilde{v}, L))}_{λ}^{-} = \frac{1}{| B^{L} |} (\sum_{B \in B^{L}} ({\tilde{v}}_{λ}^{-} (B_{i}) - \\ \sum_{K \in {\hat{C}}^{L} (B_{i} ∖ {i})} {\tilde{v}}_{λ}^{+} (K))) ⩽ \frac{1}{| B^{L} |} (\sum_{B \in B^{L}} ({\tilde{v}}_{λ}^{-} (B_{i}) - \\ \sum_{K \in {\hat{C}}^{L} (B_{i} ∖ {i})} {\tilde{v}}_{λ}^{-} (K))) = {(H \tilde{A} T_{i} (N, \tilde{v}, L))}_{λ}^{-}, \\ {(M \tilde{A} T_{i} (N, \tilde{v}, L))}_{λ}^{+} = \frac{1}{| B^{L} |} (\sum_{B \in B^{L}} ({\tilde{v}}_{λ}^{+} (B_{i}) - \\ \sum_{K \in {\hat{C}}^{L} (B_{i} ∖ {i})} {\tilde{v}}_{λ}^{-} (K))) \geq \frac{1}{| B^{L} |} (\sum_{B \in B^{L}} ({\tilde{v}}_{λ}^{+} (B_{i}) - \\ \sum_{K \in {\hat{C}}^{L} (B_{i} ∖ {i})} {\tilde{v}}_{λ}^{+} (K))) = {(H \tilde{A} T_{i} (N, \tilde{v}, L))}_{λ}^{+} . \end{matrix}$

Consequently, ${(M \tilde{A} T_{i} (N, \tilde{v}, L))}_{λ} \supseteq {(H \tilde{A} T_{i} (N, \tilde{v}, L))}_{λ} .$

Hence, $M \tilde{A} T (N, \tilde{v}, L) \supseteq H \tilde{A} T (W, \tilde{v}, L (W))$ which completes the proof. □

5 Case analysis

Example 5.1. There is an outsource production model that company A plans to cooperate with two service providers, named B and C. Sake of discussion, we write company A, B and C by nodes 1, 2 and 3. Then N ={ 1, 2, 3 }, L ={{ 1, 2 }, { 1, 3 }}. Because of many reasons such an consumer demand, product cost, etc. each company’s share is represented by triangular fuzzy numbers (m, L _m, R _m) _T, not a precise value. The effective output of each product is shown in Table 1.

Table 1

The output and average profit of each coalition’s product

Coalition	Output of coalition’s product (tons)	Average Profit (thousands of dollars)
{1}	15	(2.40, 0.08, 0.08) _T
{1, 3}	22	(3.00, 0.10, 0.15) _T
{1, 2}	25	(2.60, 0.12, 0.10) _T
{1, 2, 3}	30	(3.20, 0.15, 0.15) _T
Otherwise	0	(0, 0, 0) _T

Let $\tilde{v}$ be the fuzzy characteristic function on N. Then $\begin{matrix} \tilde{v} ({1}) = 15 \times {(2.40, 0.08, 0.08)}_{T} = {(36.00, 1.20, 1.20)}_{T}, \\ \tilde{v} ({1, 3}) = 22 \times {(3.10, 0.10, 0.15)}_{T} = {(66.00, 2.20, 3.30)}_{T}, \\ \tilde{v} ({1, 2}) = 25 \times {(2.60, 0.12, 0.10)}_{T} = {(65.00, 3.00, 2.50)}_{T}, \\ \tilde{v} ({1, 2, 3}) = 30 \times {(3.20, 0.15, 0.15)}_{T} = {(96.00, 4.50, 4.50)}_{T} . \end{matrix}$

Now we can employ the H-average tree solution in Equation (3) to estimate each company’s share. $\begin{matrix} {H \tilde{A} T}_{1} (N, \tilde{v}, L) = \frac{1}{3} [(\tilde{v} ({1, 2, 3}) -_{H} \tilde{v} ({2}) -_{H} \tilde{v} ({3})) \\ + (\tilde{v} ({1, 3}) -_{H} \tilde{v} ({3})) + (\tilde{v} ({1, 2}) -_{H} \tilde{v} ({2}))] \\ = {(75.67, 3.23, 3.43)}_{T}, \end{matrix}$ $\begin{matrix} {H \tilde{A} T}_{2} (N, \tilde{v}, L) \\ = \frac{1}{3} [\tilde{v} ({2}) + (\tilde{v} ({1, 2, 3}) -_{H} \tilde{v} ({1, 3})) + \tilde{v} ({2})] \\ = {(10.00, 0.77, 0.40)}_{T}, \end{matrix}$ $\begin{matrix} {H \tilde{A} T}_{3} (N, \tilde{v}, L) \\ = \frac{1}{3} [\tilde{v} ({3}) + \tilde{v} ({3}) + (\tilde{v} ({1, 2, 3}) -_{H} \tilde{v} ({1, 2}))] \\ = {(10.33, 0.50, 0.67)}_{T} . \end{matrix}$

Therefore, $\sum_{i = 1}^{3} \tilde{A} T_{i} (N, \tilde{v}, L) = {(96.00, 4.50, 4.50)}_{T} = \tilde{v} (N) .$

For example, for W = { 1, 2 } ∈ 2^N, we get the H-average tree solution as follows: $\begin{matrix} {H \tilde{A} T}_{1} ({1, 2}, \tilde{v}, L ({1, 2})) \\ = \frac{1}{2} [(\tilde{v} ({1, 2}) -_{H} \tilde{v} ({2})) + \tilde{v} ({1})] \\ = {(50.50, 2.10, 1.85)}_{T}, \end{matrix}$ $\begin{matrix} {H \tilde{A} T}_{2} ({1, 2}, \tilde{v}, L ({1, 2})) \\ = \frac{1}{2} [(\tilde{v} ({1, 2}) -_{H} \tilde{v} ({1})) + \tilde{v} ({2})] \\ = {(14.50, 0.90, 0.65)}_{T}, \\ {H \tilde{A} T}_{3} ({1, 2}, \tilde{v}, L ({1, 2})) = {(0, 0, 0)}_{T} . \end{matrix}$

Given confidence level λ = 0.7, we calculate the allocation of each company is the interval $\begin{matrix} {\tilde{A} T}_{1} (N, {\tilde{v}}_{0.7}, L) = [65.98, 85.96], \\ {\tilde{A} T}_{2} (N, {\tilde{v}}_{0.7}, L) = [9.77, 10.12], \\ {\tilde{A} T}_{3} (N, {\tilde{v}}_{0.7}, L) = [10.18, 10.53] . \end{matrix}$

Footnotes

Acknowledgments

The research work is supported by the National Natural Science Foundation of China (Nos. 71371030, 71871002, 11601348), the National Natural Science Foundation of Hebei Province (No. A2015106040) and the Doctoral Foundation of Shijiazhang College (18BS012).

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