Mathematical Problems in Engineering

Volume 2018, Article ID 7358621, 12 pages

https://doi.org/10.1155/2018/7358621

## A Multiobjective Game Approach with a Preferred Target Based on a Leader-Follower Decision Pattern

^{1}School of Mechanical Engineering, Anhui University of Technology, Ma’anshan, Anhui Province, China^{2}Zhongtian Construction Group Co., Hangzhou, Zhejiang Province, China^{3}Engineering Cluster, Singapore Institute of Technology, 10 Dover Drive, Singapore 138683

Correspondence should be addressed to Neng-gang Xie; moc.nuyila@gnaggneneix

Received 5 September 2017; Revised 26 January 2018; Accepted 7 February 2018; Published 20 March 2018

Academic Editor: Vladimir Turetsky

Copyright © 2018 Neng-gang Xie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose a multiobjective leader-follower game based on the Stackelberg model, where the designer’s preferred target is taken into account. Here, the preferred target is regarded as a leader and the other targets are regarded as followers. A partition method of strategy subspace is also given. Finally, a real-life example of the multiobjective optimization design of a Chinese arch dam named “Baihetan” is presented to demonstrate the effectiveness of our proposed method.

#### 1. Introduction

Given the similarity between the multiobjective design and game theory, game theory has been applied to solve many multiobjective design problems in recent years. The crux of the multiobjective method is to establish the mapping relationships between the elements of the multiobjective optimization model and the elements of the game. The multiobjective optimization model includes design objectives, objective functions, design variables, and constraints, while the game elements contain game players, benefit functions, strategy subspaces, and constraints. Two mapping relationships are listed as follows.

*(1) The Space Mapping*. Design variable space is divided into the parallel strategy subspaces owned by each game player. It satisfies ; .

*(2) The Bionic Mapping*. There are design objectives, which are modeled as game players with a certain intelligence. The design objectives are given the behaviors of the game players (such as competition, cooperation, and adaptive behaviors). The interdependent game pattern of each design goal is formed, and the quantitative mapping relationships between the benefit functions of the game and the objective functions are constructed according to the qualitative description characteristics of different behavioral patterns. The constraints of the multiobjective optimization model also correspond to the constraints of the model of the game.

The conventional multiobjective optimization method usually adopts the method of “combination” for the objective function, while the game approach uses “partition” to deal with the problem of the design variables. At present, the space partition approaches of design variables include the fuzzy clustering method proposed by Wang et al. [1], a sensitivity analysis method proposed by Hu and Rao [2], and a step adjustment method by Clarich et al. [3].

It is well known that the key issue of the multiobjective optimization model is to take into account the interests of each target and to reflect their status. For the multiobjective game, the particular embodiment is to give each design objective the appropriate behavior patterns and construct game patterns between design objectives. Regarding the behavior patterns of the design objectives, there are mainly two types, one is the “egoistic” type with the competitive behavior, and the other is the “you have me, I have you” type of cooperative behavior. Furthermore, there exist four kinds of game patterns (pure competitive game pattern [4–9], pure cooperative game pattern [9–11], hybrid game pattern [3], and evolution game pattern [12]). In the pure competitive game pattern, all the game players (design objectives) obtain game profits through a competitive behavior. In the pure cooperative game pattern, all the game players gain game profits through a cooperative behavior. In the hybrid game pattern, some game players achieve game profits through a competitive behavior, but some game players obtain game profits through a cooperative behavior. In the evolutionary game pattern, the behavior of all game players is automatically adjusted based on evolution rules.

In engineering practice, there exists a kind of multiobjective problems where the designers have a preferred target, and their target status may be unequal. There are many processing methods based on the conventional multiobjective optimization, for instance, the weighted sum method by adjusting the target weights to reflect the preferred target and the hierarchical sequence method by adjusting the target optimization order according to the target's preference. Using game theory to solve the multiobjective optimization design problems with the preference for the target, Wang et al. [13] proposed Stackelberg genetic algorithms based on the Stackelberg equilibrium to solve problems in Aerospace Engineering involving high lift multiairfoil systems. In this paper, we present a method of a “leader-follower” game pattern for a target preference according to Stackelberg model. In Section 2, we will discuss the theories related to the Stackelberg model. In Section 3, a simple numerical example will be presented. The results will be compared with those found in [13]. In Section 4, a real-life example of the multiobjective optimization design of an arch dam is given.

#### 2. A Multiobjective Method Based on a Leader-Follower Game Pattern

##### 2.1. A Leader-Follower Decision and a Stackelberg Model

For* a leader-follower decision* problem, each decision-maker is in a different level. Each decision-maker has its own objective function. The higher level decision-maker is endowed with a more important objective function, so the final decision is often a coordinated result which the decision makers at all levels seek. Under this scheme, the goal of the top decision-maker can be optimized, and the goal of the lower level decision-maker can be optimized in the subordinate position.

The decision making problem with a leader-follower hierarchical structure was first proposed by a German economist V. Stackelberg in 1952 when studying market economy problems. Therefore, the leader-follower decision problem is also called* Stackelberg game*. A typical application of the Stackelberg model exists in the oligopoly market in economics. We assume two producers in the oligopoly market where one producer is a leader, and the other one is the follower. The game aims to minimize their cost function. The leader gives priority to the decision. The follower must make its own decision following the decision made by the leader.

We define as the game profit (cost function) of the leader and as the game profit of the follower. is defined as the strategy space of the leader and as the strategy space of the follower. Let and be an arbitrary strategy. If there exists , then is called the Stackelberg strategy of the leader.

The leader can achieve its lower bound , where is the reaction function of the follower in relation to the leader.

The presence of leader and follower in the Stackelberg oligopoly game means that the satisfaction of the game players will be different. The leader can obtain a higher satisfaction level than the follower. Hence, for the multiobjective optimization with a preferred target, the preferred target is regarded as the leader, and the other targets are taken as the follower. If there are oligopoly markets, then followers are in the subordinate position. For a real-life example of the multiobjective optimization design of an arch dam, we have three objectives where one preferred target will be the leader with the other two targets as the subordinate position. The corresponding game profit of the followers is a weighted combination of these two objectives.

The decision mechanism of the Stackelberg model is as follows: the leader first announces the strategy of making its objective function optimal, which will influence the constraint set and the objective function of the follower’s optimal decision. Then, the follower selects the strategy to make its objective function reach the optimum under this premise. Since the choice of the follower affects the constraint and the objective function of the leader’s optimal decision. The leader can further adjust its strategic variables . The process will repeat until the leader's objective function, , is optimal.

##### 2.2. Solution Steps

###### 2.2.1. Exploration Method of the Strategy Space

A spatial game approach has been proposed in [14], which can sort the items and provide a way to simplify the knapsack problem. For the multiobjective optimization problem based on the game approach, design variables are first required to be divided into multiple strategy subspaces owned by game players. Here, we establish related concepts from [14] (such as the space distance and sorting methods) to propose a novel exploration method involving game player's strategy space [15]. The computation steps are as follows.

*Algorithm 1. *(1) Optimize single objective, and then obtain optimal solution , where .

(2) Every is divided into fragments with a step length in its feasible space. The effect of on the objective is first computed as follows:The normalization gives an impact index , which is defined below:(3) is defined as the space distance from to as follows:Also, is defined as the moment of to all objective functions, which represents the full influence degree of on all objective functions: is defined as the threshold of moment:(4) All design variables assigned to each objective function (each game player) are sorted according to the descending order of .

(5) Each game player first chooses a design variable being the first ranking and then the second ranking and so on until the accumulative moment of the selected variables is greater than or equal to the threshold of moment, .

As an illustration, the partition rules of design variables are performed as follows:

(a) If one chosen design variable has a ranking different from the game players, then this design variable is assigned to one game player with a relatively higher ranking;

(b) If one design variable has the same highest ranking among the multiple game players, the ownership of this design variable is determined by the impact index. In particular, if is the greatest, then is assigned to .

(c) According to the partition rules, all the design variables are assigned to the corresponding game players (objective functions).

###### 2.2.2. Game Algorithm

According to the above decision mechanism based on the Stackelberg model, the steps of the multiobjective game based on the leader-follower game pattern can be described as follows.

*Algorithm 2. *(1) Obtain the strategy space attached to each player according to the method in Section 2.2.1 and then form (the strategy space of the leader) and (the strategy space of the follower).

(2) Generate sets of the initial feasible strategies . Here, is the initial feasible strategy of the leader and is the initial feasible strategy of the follower.

(3) For any , optimize in , and obtain , which is the “follower strategy” of the follower obeying the leader.

(4) For any , optimize in , and obtain , which is the “response strategy” of the leader to the follower.

(5) Define the strategy combination and construct the fitness function . is the strategy combination with a minimum fitness function value.

(6) If ( refers to the number of iterations; is a decimal parameter given in advance; when , ), then a solution is produced, and the algorithm terminates. If it is not satisfied, then the leader’s strategy of the generation is generated according to (6) given below. Return to step of Algorithm 2 to perform a loop.where and denote the component of the strategy of the leader in a and generation, respectively. represents the number of design variables owned by the leader. and are referred to as the variance variable and , , , , and are the independent standard normal random variables. is also a normal random variable.

#### 3. Test Example

A simple test case will be used to illustrate the process of the multiobjective optimization proposed in this paper. We have the following equations:

Optimize single objective: ; ; ; ; ; . Impact index , space distance , and moment are computed, and all design variables are sorted according to Section 2.2.1. The related results are shown in Table 1.