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

Authors: Neng-gang Xie; Zhong Chen; Kang Hao Cheong; Rui Meng; Wei Bao
Journal: Mathematical Problems in Engineering
 (2018)

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## 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.


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## Body

## 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 X=x1,x2,…,xn is divided into the parallel strategy subspaces S1=xi,…,xj,…,Sm=xk,…,xl owned by each game player. It satisfies S1∪⋯∪Sm=X; Sa∩Sb=0(a,b=1,…,m;a≠b).(2) The Bionic Mapping. There are m design objectives, which are modeled as m 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

Fora 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 calledStackelberg 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 defineFS as the game profit (cost function) of the leader and FW as the game profit of the follower. SS is defined as the strategy space of the leader and SW as the strategy space of the follower. Let sS∈SS and sW∈SW be an arbitrary strategy. If there exists supsW∈R(sS∗)⁡FS(sS∗,sW)≤supsW∈R(sS)⁡FS(sS,sW)∀sS∈SS, then sS∗∈SS is called the Stackelberg strategy of the leader.The leader can achieve its lower boundFS∗=infsS∈SS⁡{supsW∈R(sS)⁡FS(sS,sW)}, where R(sS)=sW∗∣FW(sS,sW∗)≤FW(sS,sW),∀sW∈SW 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 arem(m>2) oligopoly markets, then m-1 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 FW 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 strategysW 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 sS. The process will repeat until the leader's objective function, FS, 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) Optimizem single objective, and then obtain optimal solution f1X1∗,f2X2∗,…,fm(Xm∗), where Xi∗=x1i∗,x2i∗,…,xni∗(i=1,2,…,m).
(2) Everyxj is divided into T fragments with a step length Δxj in its feasible space. The effect of xj on the objective fi is first computed as follows:(1)Θj,i=∑t=1Tfix1i∗,…,xj-1i∗,xjt,xj+1i∗,…,xni∗-fix1i∗,…,xj-1i∗,xjt-1,xj+1i∗,…,xni∗T·Δxj.The normalization gives an impact index Δ(j,i), which is defined below:(2)Δj,i=Θj,i∑l=1nΘl,ij=1,2,…,n;i=1,2,…,m.
(3)d(j,i) is defined as the space distance from xj to fi as follows:(3)dj,i=1/Δj,i∑h=1m1/Δj,hj=1,2,…,n;i=1,2,…,m.Also, Mo(j) is defined as the moment of xj to all objective functions, which represents the full influence degree of xj on all objective functions:(4)Moj=1∑h=1m1/Δj,hj=1,2,…,n.λ is defined as the threshold of moment:(5)λ=∑j=1nMoj2.
(4) All design variables assigned to each objective function (each game player) are sorted according to the descending order ofd(j,i).
(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,λ.
As an illustration, the partition rules of design variables are performed as follows:
(a) If one chosen design variablexj(j=1,2,…,n) 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 variablexj(j=1,2,…,n) 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 Δ(j,i) is the greatest, then xj is assigned to fi.
(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 spaceS1,…,Sm attached to each player according to the method in Section 2.2.1 and then form SS (the strategy space of the leader) and SW (the strategy space of the follower).
(2) GenerateL sets of the initial feasible strategies si(0)=sSi0,sWi0(i=1,…,L). Here, sSi0(i=1,…,L) is the initial feasible strategy of the leader and sWi0(i=1,…,L) is the initial feasible strategy of the follower.
(3) For anysSi0(i=1,…,L), optimize FW in SW, and obtain sWiR(i=1,…,L), which is the “follower strategy” of the follower obeying the leader.
(4) For anysWiR(i=1,…,L), optimize FS in SS, and obtain sSiR(i=1,…,L), which is the “response strategy” of the leader to the follower.
(5) Define the strategy combinationsi(1)=sSiR∪sWiR and construct the fitness function SSiR-SSi(0). smin(1) is the strategy combination with a minimum fitness function value.
(6) IfFS(smin(k))-FS(smin(k-1))/FS(smin(k))≤ε(k=1,2,…) (k refers to the number of iterations; ε is a decimal parameter given in advance; when k=0, FS(smin(0))=mini=1,2,…,L⁡FS(si(0))), then a solution is produced, and the algorithm terminates. If it is not satisfied, then the leader’s strategy of the kth generation sSik(i=1,2,…,L) is generated according to (6) given below. Return to step (3) of Algorithm 2 to perform a loop.(6)ρijk=ρijk-1exp⁡θ′·N0,1+θ·Nj0,1j=1,2,…,H;i=1,2,…,Lxijk=xijk-1+N0,ρijkj=1,2,…,H;i=1,2,…,L,where xij(k) and xij(k-1) denote the jth component of the ith strategy of the leader in a kth and k-1th generation, respectively. H represents the number of design variables owned by the leader. ρij(k) and ρij(k-1) are referred to as the variance variable and ρij(0)=1(j=1,2,…,H;i=1,2,…,L), θ′=1/2L, θ=1/2L, N(0,1), and Nj(0,1) are the independent standard normal random variables. N(0,ρij(k)) is also a normal random variable.


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

Fora 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 calledStackelberg 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 defineFS as the game profit (cost function) of the leader and FW as the game profit of the follower. SS is defined as the strategy space of the leader and SW as the strategy space of the follower. Let sS∈SS and sW∈SW be an arbitrary strategy. If there exists supsW∈R(sS∗)⁡FS(sS∗,sW)≤supsW∈R(sS)⁡FS(sS,sW)∀sS∈SS, then sS∗∈SS is called the Stackelberg strategy of the leader.The leader can achieve its lower boundFS∗=infsS∈SS⁡{supsW∈R(sS)⁡FS(sS,sW)}, where R(sS)=sW∗∣FW(sS,sW∗)≤FW(sS,sW),∀sW∈SW 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 arem(m>2) oligopoly markets, then m-1 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 FW 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 strategysW 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 sS. The process will repeat until the leader's objective function, FS, 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) Optimizem single objective, and then obtain optimal solution f1X1∗,f2X2∗,…,fm(Xm∗), where Xi∗=x1i∗,x2i∗,…,xni∗(i=1,2,…,m).
(2) Everyxj is divided into T fragments with a step length Δxj in its feasible space. The effect of xj on the objective fi is first computed as follows:(1)Θj,i=∑t=1Tfix1i∗,…,xj-1i∗,xjt,xj+1i∗,…,xni∗-fix1i∗,…,xj-1i∗,xjt-1,xj+1i∗,…,xni∗T·Δxj.The normalization gives an impact index Δ(j,i), which is defined below:(2)Δj,i=Θj,i∑l=1nΘl,ij=1,2,…,n;i=1,2,…,m.
(3)d(j,i) is defined as the space distance from xj to fi as follows:(3)dj,i=1/Δj,i∑h=1m1/Δj,hj=1,2,…,n;i=1,2,…,m.Also, Mo(j) is defined as the moment of xj to all objective functions, which represents the full influence degree of xj on all objective functions:(4)Moj=1∑h=1m1/Δj,hj=1,2,…,n.λ is defined as the threshold of moment:(5)λ=∑j=1nMoj2.
(4) All design variables assigned to each objective function (each game player) are sorted according to the descending order ofd(j,i).
(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,λ.
As an illustration, the partition rules of design variables are performed as follows:
(a) If one chosen design variablexj(j=1,2,…,n) 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 variablexj(j=1,2,…,n) 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 Δ(j,i) is the greatest, then xj is assigned to fi.
(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 spaceS1,…,Sm attached to each player according to the method in Section 2.2.1 and then form SS (the strategy space of the leader) and SW (the strategy space of the follower).
(2) GenerateL sets of the initial feasible strategies si(0)=sSi0,sWi0(i=1,…,L). Here, sSi0(i=1,…,L) is the initial feasible strategy of the leader and sWi0(i=1,…,L) is the initial feasible strategy of the follower.
(3) For anysSi0(i=1,…,L), optimize FW in SW, and obtain sWiR(i=1,…,L), which is the “follower strategy” of the follower obeying the leader.
(4) For anysWiR(i=1,…,L), optimize FS in SS, and obtain sSiR(i=1,…,L), which is the “response strategy” of the leader to the follower.
(5) Define the strategy combinationsi(1)=sSiR∪sWiR and construct the fitness function SSiR-SSi(0). smin(1) is the strategy combination with a minimum fitness function value.
(6) IfFS(smin(k))-FS(smin(k-1))/FS(smin(k))≤ε(k=1,2,…) (k refers to the number of iterations; ε is a decimal parameter given in advance; when k=0, FS(smin(0))=mini=1,2,…,L⁡FS(si(0))), then a solution is produced, and the algorithm terminates. If it is not satisfied, then the leader’s strategy of the kth generation sSik(i=1,2,…,L) is generated according to (6) given below. Return to step (3) of Algorithm 2 to perform a loop.(6)ρijk=ρijk-1exp⁡θ′·N0,1+θ·Nj0,1j=1,2,…,H;i=1,2,…,Lxijk=xijk-1+N0,ρijkj=1,2,…,H;i=1,2,…,L,where xij(k) and xij(k-1) denote the jth component of the ith strategy of the leader in a kth and k-1th generation, respectively. H represents the number of design variables owned by the leader. ρij(k) and ρij(k-1) are referred to as the variance variable and ρij(0)=1(j=1,2,…,H;i=1,2,…,L), θ′=1/2L, θ=1/2L, N(0,1), and Nj(0,1) are the independent standard normal random variables. N(0,ρij(k)) is also a normal random variable.


## 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) Optimizem single objective, and then obtain optimal solution f1X1∗,f2X2∗,…,fm(Xm∗), where Xi∗=x1i∗,x2i∗,…,xni∗(i=1,2,…,m).
(2) Everyxj is divided into T fragments with a step length Δxj in its feasible space. The effect of xj on the objective fi is first computed as follows:(1)Θj,i=∑t=1Tfix1i∗,…,xj-1i∗,xjt,xj+1i∗,…,xni∗-fix1i∗,…,xj-1i∗,xjt-1,xj+1i∗,…,xni∗T·Δxj.The normalization gives an impact index Δ(j,i), which is defined below:(2)Δj,i=Θj,i∑l=1nΘl,ij=1,2,…,n;i=1,2,…,m.
(3)d(j,i) is defined as the space distance from xj to fi as follows:(3)dj,i=1/Δj,i∑h=1m1/Δj,hj=1,2,…,n;i=1,2,…,m.Also, Mo(j) is defined as the moment of xj to all objective functions, which represents the full influence degree of xj on all objective functions:(4)Moj=1∑h=1m1/Δj,hj=1,2,…,n.λ is defined as the threshold of moment:(5)λ=∑j=1nMoj2.
(4) All design variables assigned to each objective function (each game player) are sorted according to the descending order ofd(j,i).
(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,λ.
As an illustration, the partition rules of design variables are performed as follows:
(a) If one chosen design variablexj(j=1,2,…,n) 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 variablexj(j=1,2,…,n) 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 Δ(j,i) is the greatest, then xj is assigned to fi.
(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 spaceS1,…,Sm attached to each player according to the method in Section 2.2.1 and then form SS (the strategy space of the leader) and SW (the strategy space of the follower).
(2) GenerateL sets of the initial feasible strategies si(0)=sSi0,sWi0(i=1,…,L). Here, sSi0(i=1,…,L) is the initial feasible strategy of the leader and sWi0(i=1,…,L) is the initial feasible strategy of the follower.
(3) For anysSi0(i=1,…,L), optimize FW in SW, and obtain sWiR(i=1,…,L), which is the “follower strategy” of the follower obeying the leader.
(4) For anysWiR(i=1,…,L), optimize FS in SS, and obtain sSiR(i=1,…,L), which is the “response strategy” of the leader to the follower.
(5) Define the strategy combinationsi(1)=sSiR∪sWiR and construct the fitness function SSiR-SSi(0). smin(1) is the strategy combination with a minimum fitness function value.
(6) IfFS(smin(k))-FS(smin(k-1))/FS(smin(k))≤ε(k=1,2,…) (k refers to the number of iterations; ε is a decimal parameter given in advance; when k=0, FS(smin(0))=mini=1,2,…,L⁡FS(si(0))), then a solution is produced, and the algorithm terminates. If it is not satisfied, then the leader’s strategy of the kth generation sSik(i=1,2,…,L) is generated according to (6) given below. Return to step (3) of Algorithm 2 to perform a loop.(6)ρijk=ρijk-1exp⁡θ′·N0,1+θ·Nj0,1j=1,2,…,H;i=1,2,…,Lxijk=xijk-1+N0,ρijkj=1,2,…,H;i=1,2,…,L,where xij(k) and xij(k-1) denote the jth component of the ith strategy of the leader in a kth and k-1th generation, respectively. H represents the number of design variables owned by the leader. ρij(k) and ρij(k-1) are referred to as the variance variable and ρij(0)=1(j=1,2,…,H;i=1,2,…,L), θ′=1/2L, θ=1/2L, N(0,1), and Nj(0,1) are the independent standard normal random variables. N(0,ρij(k)) 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:(7)minf1=x-12+x-y2minf2=y-32+x-y2-5≤x,y≤5.Optimize single objective:f1∗=0.0; x11∗=1.0; x21∗=1.0; f2∗=0.0; x12∗=3.0; x22∗=3.0. Impact index Δ(j,i), space distance d(j,i), and moment Mo(j) are computed, and all design variables are sorted according to Section 2.2.1. The related results are shown in Table 1.Table 1
Computation results of a simple test case.
x j f 1 f 2 M o ( j ) Δ j , 1 d j , 1 Ranking Δ j , 2 d j , 2 Ranking x 1 0.667 0.333 1 0.333 0.667 2 0.222 x 2 0.333 0.667 2 0.667 0.333 1 0.222The threshold of moment is computed according to (5) and determined to be 0.222. According to Table 1 and step (5) of Algorithm 1 in Section 2.2.1, we will then have S1=x1 and S2=x2.The numerical results can be easily obtained by following Algorithm2 in Section 2.2.2. These results are then compared with those obtained in [13], as well as the results obtained by the Stackelberg game given in [13]. Table 2 shows that our proposed method is better than the method in [13] for this simple test (because both the leader’s objective value and the follower’s objective value are better than those in [13]).Table 2
Comparisons of the analytical and numerical results.
x 1 x 2 Objective value,f1 Objective value,f2 Stackelberg game [13] Preferredf1 1.4 2.2 0.8 1.28 Preferredf2 1.8 2.6 1.28 0.8 Method given in [13] Preferredf1 1.399945 2.200030 0.800092 1.280002 Preferredf2 1.799998 2.600001 1.280001 0.800004 This paper Preferredf1 1.409656 2.202957 0.797145 1.264605 Preferredf2 1.800240 2.597472 1.275963 0.797608


## 4. Case Study: A Triobjective Shape Optimization Design for an Arch Dam

The shape of an arch dam will have a large influence on the volume, stress, and displacement of the dam. Hence, the shape of an arch dam determines the economic aspects and its safe operation. To ensure the safety and economy viability of a high arch dam to the static load case, many have carried out studies on the multiobjective optimization design of an arch dam [15–18]. These studies focus on the construction of the geometric model, selection of design variables, defining the objective functions, constraint conditions, structural analysis, and optimization method.


### 4.1. Model Formulation

In our model, the continuous geometry is modeled such that the shape of the horizontal arch and shape of the arch crown section of an arch dam are described separately like in [15]. Here, the shape of the horizontal arch is determined by the curves of the upper and lower planes of the arch, which can be easily defined by the equations of the arch axis and the arch thickness. The shape of the crown beam section is determined by the curves of the upper and lower planes of the arch crown beam and can be defined by the curve equations of the upstream surface of the arch crown beam and the thickness equations of the arch crown beam. According to established geometric model of the arch dam, the design variables of the shape are determined to be X=x1,x2,…,xn.The dam volumeV is defined to be the economic objective; f1(X)=V. The maximum principal tensile stress (of the dam) is the safety goal of arch dam’s local area, f2(X)=maxV⁡(σ1), and σ1 is the principal tensile stress. The strain energy U of the arch dam is taken as the overall safety goal; f3(X)=U. There are also other constraints like the geometric constraints, stress constraints, stability constraints, and volume constraints. The finite-element method was primarily used for the structure analysis.A triobjective shape optimization model for an arch dam is given as follows:Seeking the design variables(8)X=x1x2⋯xn.Let objective functions be(9)FX=VmaxVσ1U⟶min.And satisfy the following constraints:
Volume constraints:Vmin≤V≤Vmax (arch dam volume does not exceed the allowable value)Geometric constraints:Tmin≤T≤Tmax (arch dam thickness T does not exceed the allowable value); Ku≤Ku (upstream overhang degree Ku does not exceed the allowable value); Kd≤Kd (downstream overhang degree Kd does not exceed the allowable value)Stress constraints:minV⁡σ3≥σ3 (the maximum principal compressive stress σ3 does not exceed the allowable value)Stability constraints:φmax≤φmax (the maximum central angle φmax does not exceed the allowable value)whereU is the strain energy, U=1/2δTKδ. δ represents the displacement matrix of the whole nodes and K represents the whole stiffness matrix.Here, a Chinese arch dam named “Baihetan” (ready for construction) is used as a way of example to illustrate the effectiveness of our proposed method. Furthermore, a static analysis of the arch dam-water-foundation rock system is under a gravity load and hydrostatic pressure. The foundation rock is assumed to be massless [19–21]. The calculations related to the foundation rock using finite-element analysis are as follows: (1) the foundation rock in upstream and depth directions is calculated at twice the height of the dam. (2) The downstream side and the left and right sides of the foundation rock are calculated to be two times higher than the dam. The foundation rock is simulated by the linear elastic material, where the elastic modulus is 2.0 × 104 MPa and Poisson’s ratio is 0.2. The parabolic-hyperbolic shape of the arch dam is first considered, and the elevation of dam bottom is 550.0 m with a height of 277.0 m. The upstream normal storage level is 820.0 m, the static elastic modulus is 2.10 × 104 MPa, Poisson’s ratio is 0.167, and its density is 2.4 t/m3. The finite-element method is used for the structural analysis, where a hexahedron shape with 20 nodes and a pentahedron shape with 15 nodes are used. A two-layer element is arranged along the direction of the dam thickness, and a five-layer element is arranged along the direction of the dam height.The upstream curves of the arch dam and the thickness of the crown cantilever are simulated using cubic curves, and the other geometry characteristics are simulated using the Lagrange interpolation equation. There are 39 design variables. The design variables distribution, upper and lower limit, and the initial shape parameters are shown in Table3, provided by a design institution. The constraints are as follows: (1)Vmin=600×104m3 and Vmax=750×104m3; (2) the upstream overhang degree: Ku≤0.3; the downstream overhang degree: Kd≤0.25; (3) the largest central angle: φmax<100∘; (4) the chord length and quasi semicentral angle constraints; (5) the allowable principal compressive stress value is given as σ3=-15.0MPa.Table 3
Distribution of design variables (including initial shape parameters and upper and lower limit).
Elevation/m Arch crown curvature Arch dam’s left bank Arch dam’s right bank Upstream coordinates/m Thickness/m Curvature radius of arch ring axis/m Thickness/m Curvature radius of arch ring axis/m Thickness/m 827.0  x 7 (14.000) x 15 (341.617) x 23 (19.035) x 31 (261.046) x 39 (19.000) ( 10.0,20.0 ) ( 255.5,521.2 ) ( 10.0,25.0 ) ( 183.2,373.8 ) ( 10.0,25.0 ) 780.0   x 14 (301.730) x 22 (30.650) x 30 (241.896) x 38 (39.679) ( 229.1,467.4 ) ( 25.0,40.0 ) ( 173.1,353.0 ) ( 25.0,40.0 ) 740.0 x 3   ( - 32.563 ) x 6 (41.510) x 13 (268.473) x 21 (42.822) x 29 (223.934) x 37 (55.000) ( - 102.0,0.0 ) ( 35.0,50.0 ) ( 207.9,424.0 ) ( 35.0,60.0 ) ( 166.6,339.9 ) ( 35.0,60.0 ) 690.0   x 12 (229.704) x 20 (58.165) x 28 (201.973) x 36 (69.945) ( 180.5,446.5 ) ( 45.0,70.0 ) ( 157.2,388.7 ) ( 45.0,70.0 ) 640.0 x 2   ( - 48.000 ) x 5 (58.677) x 11 (200.492) x 19 (70.442) x 27 (185.014) x 35 (78.787) ( - 102.0,0.0 ) ( 50.0,70.0 ) ( 147.3,364.4 ) ( 60.0,85.0 ) ( 139.3,344.6 ) ( 60.0,85.0 ) 600.0   x 10 (176.757) x 18 (75.744) x 26 (171.928) x 34 (80.437) ( 138.2,353.2 ) ( 60.0,85.0 ) ( 136.9,350.0 ) ( 60.0,85.0 ) 570.0   x 9 (163.024) x 17 (75.929) x 25 (163.842) x 33 (77.997) ( 146.2,337.0 ) ( 60.0,85.0 ) ( 147.0,338.8 ) ( 60.0,85.0 ) 550.0 x 1 (−37.816) x 4 (70.000) x 8 (154.557) x 16 (73.834) x 24 (154.836) x 32 (74.435) ( - 102.0,0.0 ) ( 60.0,85.0 ) ( 110.7,292.8 ) ( 60.0,85.0 ) ( 110.9,293.4 ) ( 60.0,85.0 )The impact index, space distance, and space moment are computed, and all design variables are sorted according to the rules set out in Section2.2.1. According to the partition rules, the strategy subspace of the game player (f1): S1=x16~x23,x32~x39, the strategy subspace of the game player (f2): S2=x8~x15,x24~x31, and the strategy subspace of the game player (f3): S3=x1~x7.When leader-follower game method is used to solve optimization problems with three objective functions, the preferred target is regarded as the leader, and the other two targets are taken as the follower. For example, whenf1 is regarded as preferred target, the game profit of the leader FS=f1 and the game profit of the follower FW=0.5f2/f2∗+0.5f3/f3∗. The strategy space of the leader is SS=S1 and the strategy space of the follower is SW=S2∪S3. The situation is handled in the same way if f2 or f3 is regarded as the preferred target.


### 4.2. Numerical Results

The main parameters of the initial shape and the shape featured in leader-follower game are shown in Table4. There are three kinds of leader-follower game shape parameters, and their respective results are presented in Tables 5, 6, and 7. Results in the tables show the shape of the preferred f1 is the smallest and the corresponding thickness of arch dam is minimal. On comparison, the shape of the preferred f2 appears to be more curvy with its crown cantilever section towards the downstream. Such features have the potential to reduce the value of the maximum principal tensile stress. The shape of the preferred f3 appears to be more upright, with its crown cantilever section towards the upstream, which can reduce the dam deformation and decrease deformation energy. In Figure 1, the upstream element strain energy is compared with various preferred targets. The downstream element strain energy is compared with various preferred targets in Figure 2. Figures 1 and 2 show that the distribution of the element deformation energy is suitably reasonable when f3 is selected as the preferred target. Here, the deformation energy at the bottom of the dam is large due to the relatively thicker dam’s bottom and the deformation energy at the top of the dam is small due to the relatively thinner dam’s top. The displacement along the river of the downstream surface is compared with different preferred targets in Figure 3. The contour line of the upstream surface principal tensile stress is compared with various preferred targets in Figure 4. The comparison of the contour line of downstream area principal compressive stress with various preferred targets is shown in Figure 5. Figures 3, 4, and 5 indicate that the distributions of the displacement and the stress are identical, but the numerical size is different.Table 4
The main technical parameters of the initial shape and leader-follower game shape.
Volume/104 m3 Strain energy/GJ The maximum principal tensile stress/MPa K u K d φ m a x/° Initial shape 689.26 3.733 11.57 0.254 0.048 96.312 Game shape Preferredf1 622.51 4.088 12.57 0.127 0.113 98.912 Preferredf2 723.65 4.273 10.01 0.213 0.145 99.754 Preferredf3 697.90 3.278 11.52 0.186 0.123 95.104Table 5
Parameters of the shape of the arch dam for preferredf1.
Elevation/m Crown cantilever/m The thickness of the arch abutment/m Arch crown curvature radius of Arch axis/m Semicenter angle/° Upstream coordinates Thickness Left bank Right Bank Left bank Right bank Left bank Right bank 827.00 0.000 10.074 24.536 14.105 481.883 194.787 37.136 53.339 780.00 - 21.494 25.447 27.953 36.078 465.152 244.917 35.127 45.260 740.00 - 36.667 35.019 42.090 48.274 356.685 205.378 39.768 49.207 690.00 - 51.100 43.695 68.095 65.178 256.498 167.642 45.143 53.242 640.00 - 59.927 50.142 80.723 84.727 198.301 154.219 46.694 52.218 600.00 - 62.543 54.718 84.585 83.448 162.681 253.498 45.356 32.774 570.00 - 61.708 58.324 82.458 81.886 196.758 263.621 31.942 25.069 550.00 - 59.746 60.994 74.118 68.642 203.955 207.586 14.205 13.992Table 6
Parameters of the shape of the arch dam for preferredf2.
Elevation/m Crown cantilever/m The thickness of the arch abutment/m Arch crown curvature radius of arch axis/m Semicenter angle/° Upstream coordinates Thickness Left bank Right bank Left bank Right bank Left bank Right bank 827.00 0.000 19.760 21.005 22.878 470.074 191.740 37.822 53.771 780.00 - 27.791 38.282 26.162 26.584 462.596 246.299 35.275 45.099 740.00 - 49.119 49.633 48.349 37.736 356.380 209.717 39.793 48.614 690.00 - 70.978 59.984 45.885 45.812 252.188 167.403 45.628 53.281 640.00 - 85.466 68.188 61.853 61.781 193.024 153.820 47.464 52.290 600.00 - 90.281 74.732 67.038 73.513 169.103 249.697 44.247 33.169 570.00 - 89.206 80.391 66.466 67.368 187.931 254.426 33.135 25.859 550.00 - 85.996 84.794 79.348 82.285 205.829 205.511 14.081 14.128Table 7
Parameters of the shape of the arch dam for preferredf3.
Elevation/m Crown cantilever/m The thickness of the arch abutment/m Arch crown curvature radius of arch axis/m Semicenter angle/° Upstream coordinates Thickness Left bank Right Bank Left bank Right bank Left bank Right bank 827.00 0.000 18.203 24.485 20.347 464.445 194.050 38.157 53.443 780.00 - 9.442 27.175 36.851 35.675 358.614 226.646 42.381 47.478 740.00 - 14.089 38.333 49.818 36.752 241.491 244.455 50.871 44.233 690.00 - 15.928 54.252 67.894 69.054 371.721 362.338 34.741 31.775 640.00 - 13.826 69.426 84.503 84.759 272.058 331.687 37.715 30.956 600.00 - 9.645 78.917 82.532 82.126 291.854 252.444 29.440 32.883 570.00 - 5.220 83.434 84.265 83.213 289.663 337.388 22.953 20.078 550.00 - 1.717 84.830 84.754 84.666 233.141 163.397 12.487 17.567Figure 1
Comparison of the upstream element strain energy (MJ).
(a)
Initial shape (b)
Leader-follower game shape (preferredf1) (c)
Leader-follower game shape (preferredf2) (d)
Leader-follower game shape (preferredf3)Figure 2
Comparison of the downstream element strain energy (MJ).
(a)
Initial shape (b)
Leader-follower game shape (preferredf1) (c)
Leader-follower game shape (preferredf2) (d)
Leader-follower game shape (preferredf3)Figure 3
Comparison of the displacement along the river of the downstream surface (cm).
(a)
Initial shape (b)
Leader-follower game shape (preferredf1) (c)
Leader-follower game shape (preferredf2) (d)
Leader-follower game shape (preferredf3)Figure 4
Comparison of contour line of the upstream surface principal tensile stress (MPa).
(a)
Initial shape (b)
Leader-follower game shape (preferredf1) (c)
Leader-follower game shape (preferredf2) (d)
Leader-follower game shape (preferredf3)Figure 5
Comparison of the contour line of downstream surface principal compressive stress (MPa).
(a)
Initial shape (b)
Leader-follower game shape (preferredf1) (c)
Leader-follower game shape (preferredf2) (d)
Leader-follower game shape (preferredf3)Our numerical results have demonstrated that all three shapes based on the leader-follower game can achieve their preferred targets. Compared to the initial shape (see Table4), the shape of the preferred f1 (volume) is reduced by 667500 m3 (about 9.68%). The shape of the preferred f2 (stress) is reduced by 1.56 MPa (about 13.48%). The shape of the preferred f3 (energy) is reduced by 0.455 GJ (about 12.19%). These results have highlighted the status of the preferred targets and demonstrated the effectiveness of our proposed method. Besides, the maximum center angle of the arch dam is related to the stability of the arch abutment and the dam shoulder. The smaller maximum center angle yields a better stability of the arch abutment and the dam shoulder. The shape of the preferred f3 appears to be more robust (Table 4 shows that the maximum center angle is 95.104°, which is the smallest among the four shapes), and its multiobjective functions have been further improved as well.


## 4.1. Model Formulation

In our model, the continuous geometry is modeled such that the shape of the horizontal arch and shape of the arch crown section of an arch dam are described separately like in [15]. Here, the shape of the horizontal arch is determined by the curves of the upper and lower planes of the arch, which can be easily defined by the equations of the arch axis and the arch thickness. The shape of the crown beam section is determined by the curves of the upper and lower planes of the arch crown beam and can be defined by the curve equations of the upstream surface of the arch crown beam and the thickness equations of the arch crown beam. According to established geometric model of the arch dam, the design variables of the shape are determined to be X=x1,x2,…,xn.The dam volumeV is defined to be the economic objective; f1(X)=V. The maximum principal tensile stress (of the dam) is the safety goal of arch dam’s local area, f2(X)=maxV⁡(σ1), and σ1 is the principal tensile stress. The strain energy U of the arch dam is taken as the overall safety goal; f3(X)=U. There are also other constraints like the geometric constraints, stress constraints, stability constraints, and volume constraints. The finite-element method was primarily used for the structure analysis.A triobjective shape optimization model for an arch dam is given as follows:Seeking the design variables(8)X=x1x2⋯xn.Let objective functions be(9)FX=VmaxVσ1U⟶min.And satisfy the following constraints:
Volume constraints:Vmin≤V≤Vmax (arch dam volume does not exceed the allowable value)Geometric constraints:Tmin≤T≤Tmax (arch dam thickness T does not exceed the allowable value); Ku≤Ku (upstream overhang degree Ku does not exceed the allowable value); Kd≤Kd (downstream overhang degree Kd does not exceed the allowable value)Stress constraints:minV⁡σ3≥σ3 (the maximum principal compressive stress σ3 does not exceed the allowable value)Stability constraints:φmax≤φmax (the maximum central angle φmax does not exceed the allowable value)whereU is the strain energy, U=1/2δTKδ. δ represents the displacement matrix of the whole nodes and K represents the whole stiffness matrix.Here, a Chinese arch dam named “Baihetan” (ready for construction) is used as a way of example to illustrate the effectiveness of our proposed method. Furthermore, a static analysis of the arch dam-water-foundation rock system is under a gravity load and hydrostatic pressure. The foundation rock is assumed to be massless [19–21]. The calculations related to the foundation rock using finite-element analysis are as follows: (1) the foundation rock in upstream and depth directions is calculated at twice the height of the dam. (2) The downstream side and the left and right sides of the foundation rock are calculated to be two times higher than the dam. The foundation rock is simulated by the linear elastic material, where the elastic modulus is 2.0 × 104 MPa and Poisson’s ratio is 0.2. The parabolic-hyperbolic shape of the arch dam is first considered, and the elevation of dam bottom is 550.0 m with a height of 277.0 m. The upstream normal storage level is 820.0 m, the static elastic modulus is 2.10 × 104 MPa, Poisson’s ratio is 0.167, and its density is 2.4 t/m3. The finite-element method is used for the structural analysis, where a hexahedron shape with 20 nodes and a pentahedron shape with 15 nodes are used. A two-layer element is arranged along the direction of the dam thickness, and a five-layer element is arranged along the direction of the dam height.The upstream curves of the arch dam and the thickness of the crown cantilever are simulated using cubic curves, and the other geometry characteristics are simulated using the Lagrange interpolation equation. There are 39 design variables. The design variables distribution, upper and lower limit, and the initial shape parameters are shown in Table3, provided by a design institution. The constraints are as follows: (1)Vmin=600×104m3 and Vmax=750×104m3; (2) the upstream overhang degree: Ku≤0.3; the downstream overhang degree: Kd≤0.25; (3) the largest central angle: φmax<100∘; (4) the chord length and quasi semicentral angle constraints; (5) the allowable principal compressive stress value is given as σ3=-15.0MPa.Table 3
Distribution of design variables (including initial shape parameters and upper and lower limit).
Elevation/m Arch crown curvature Arch dam’s left bank Arch dam’s right bank Upstream coordinates/m Thickness/m Curvature radius of arch ring axis/m Thickness/m Curvature radius of arch ring axis/m Thickness/m 827.0  x 7 (14.000) x 15 (341.617) x 23 (19.035) x 31 (261.046) x 39 (19.000) ( 10.0,20.0 ) ( 255.5,521.2 ) ( 10.0,25.0 ) ( 183.2,373.8 ) ( 10.0,25.0 ) 780.0   x 14 (301.730) x 22 (30.650) x 30 (241.896) x 38 (39.679) ( 229.1,467.4 ) ( 25.0,40.0 ) ( 173.1,353.0 ) ( 25.0,40.0 ) 740.0 x 3   ( - 32.563 ) x 6 (41.510) x 13 (268.473) x 21 (42.822) x 29 (223.934) x 37 (55.000) ( - 102.0,0.0 ) ( 35.0,50.0 ) ( 207.9,424.0 ) ( 35.0,60.0 ) ( 166.6,339.9 ) ( 35.0,60.0 ) 690.0   x 12 (229.704) x 20 (58.165) x 28 (201.973) x 36 (69.945) ( 180.5,446.5 ) ( 45.0,70.0 ) ( 157.2,388.7 ) ( 45.0,70.0 ) 640.0 x 2   ( - 48.000 ) x 5 (58.677) x 11 (200.492) x 19 (70.442) x 27 (185.014) x 35 (78.787) ( - 102.0,0.0 ) ( 50.0,70.0 ) ( 147.3,364.4 ) ( 60.0,85.0 ) ( 139.3,344.6 ) ( 60.0,85.0 ) 600.0   x 10 (176.757) x 18 (75.744) x 26 (171.928) x 34 (80.437) ( 138.2,353.2 ) ( 60.0,85.0 ) ( 136.9,350.0 ) ( 60.0,85.0 ) 570.0   x 9 (163.024) x 17 (75.929) x 25 (163.842) x 33 (77.997) ( 146.2,337.0 ) ( 60.0,85.0 ) ( 147.0,338.8 ) ( 60.0,85.0 ) 550.0 x 1 (−37.816) x 4 (70.000) x 8 (154.557) x 16 (73.834) x 24 (154.836) x 32 (74.435) ( - 102.0,0.0 ) ( 60.0,85.0 ) ( 110.7,292.8 ) ( 60.0,85.0 ) ( 110.9,293.4 ) ( 60.0,85.0 )The impact index, space distance, and space moment are computed, and all design variables are sorted according to the rules set out in Section2.2.1. According to the partition rules, the strategy subspace of the game player (f1): S1=x16~x23,x32~x39, the strategy subspace of the game player (f2): S2=x8~x15,x24~x31, and the strategy subspace of the game player (f3): S3=x1~x7.When leader-follower game method is used to solve optimization problems with three objective functions, the preferred target is regarded as the leader, and the other two targets are taken as the follower. For example, whenf1 is regarded as preferred target, the game profit of the leader FS=f1 and the game profit of the follower FW=0.5f2/f2∗+0.5f3/f3∗. The strategy space of the leader is SS=S1 and the strategy space of the follower is SW=S2∪S3. The situation is handled in the same way if f2 or f3 is regarded as the preferred target.


## 4.2. Numerical Results

The main parameters of the initial shape and the shape featured in leader-follower game are shown in Table4. There are three kinds of leader-follower game shape parameters, and their respective results are presented in Tables 5, 6, and 7. Results in the tables show the shape of the preferred f1 is the smallest and the corresponding thickness of arch dam is minimal. On comparison, the shape of the preferred f2 appears to be more curvy with its crown cantilever section towards the downstream. Such features have the potential to reduce the value of the maximum principal tensile stress. The shape of the preferred f3 appears to be more upright, with its crown cantilever section towards the upstream, which can reduce the dam deformation and decrease deformation energy. In Figure 1, the upstream element strain energy is compared with various preferred targets. The downstream element strain energy is compared with various preferred targets in Figure 2. Figures 1 and 2 show that the distribution of the element deformation energy is suitably reasonable when f3 is selected as the preferred target. Here, the deformation energy at the bottom of the dam is large due to the relatively thicker dam’s bottom and the deformation energy at the top of the dam is small due to the relatively thinner dam’s top. The displacement along the river of the downstream surface is compared with different preferred targets in Figure 3. The contour line of the upstream surface principal tensile stress is compared with various preferred targets in Figure 4. The comparison of the contour line of downstream area principal compressive stress with various preferred targets is shown in Figure 5. Figures 3, 4, and 5 indicate that the distributions of the displacement and the stress are identical, but the numerical size is different.Table 4
The main technical parameters of the initial shape and leader-follower game shape.
Volume/104 m3 Strain energy/GJ The maximum principal tensile stress/MPa K u K d φ m a x/° Initial shape 689.26 3.733 11.57 0.254 0.048 96.312 Game shape Preferredf1 622.51 4.088 12.57 0.127 0.113 98.912 Preferredf2 723.65 4.273 10.01 0.213 0.145 99.754 Preferredf3 697.90 3.278 11.52 0.186 0.123 95.104Table 5
Parameters of the shape of the arch dam for preferredf1.
Elevation/m Crown cantilever/m The thickness of the arch abutment/m Arch crown curvature radius of Arch axis/m Semicenter angle/° Upstream coordinates Thickness Left bank Right Bank Left bank Right bank Left bank Right bank 827.00 0.000 10.074 24.536 14.105 481.883 194.787 37.136 53.339 780.00 - 21.494 25.447 27.953 36.078 465.152 244.917 35.127 45.260 740.00 - 36.667 35.019 42.090 48.274 356.685 205.378 39.768 49.207 690.00 - 51.100 43.695 68.095 65.178 256.498 167.642 45.143 53.242 640.00 - 59.927 50.142 80.723 84.727 198.301 154.219 46.694 52.218 600.00 - 62.543 54.718 84.585 83.448 162.681 253.498 45.356 32.774 570.00 - 61.708 58.324 82.458 81.886 196.758 263.621 31.942 25.069 550.00 - 59.746 60.994 74.118 68.642 203.955 207.586 14.205 13.992Table 6
Parameters of the shape of the arch dam for preferredf2.
Elevation/m Crown cantilever/m The thickness of the arch abutment/m Arch crown curvature radius of arch axis/m Semicenter angle/° Upstream coordinates Thickness Left bank Right bank Left bank Right bank Left bank Right bank 827.00 0.000 19.760 21.005 22.878 470.074 191.740 37.822 53.771 780.00 - 27.791 38.282 26.162 26.584 462.596 246.299 35.275 45.099 740.00 - 49.119 49.633 48.349 37.736 356.380 209.717 39.793 48.614 690.00 - 70.978 59.984 45.885 45.812 252.188 167.403 45.628 53.281 640.00 - 85.466 68.188 61.853 61.781 193.024 153.820 47.464 52.290 600.00 - 90.281 74.732 67.038 73.513 169.103 249.697 44.247 33.169 570.00 - 89.206 80.391 66.466 67.368 187.931 254.426 33.135 25.859 550.00 - 85.996 84.794 79.348 82.285 205.829 205.511 14.081 14.128Table 7
Parameters of the shape of the arch dam for preferredf3.
Elevation/m Crown cantilever/m The thickness of the arch abutment/m Arch crown curvature radius of arch axis/m Semicenter angle/° Upstream coordinates Thickness Left bank Right Bank Left bank Right bank Left bank Right bank 827.00 0.000 18.203 24.485 20.347 464.445 194.050 38.157 53.443 780.00 - 9.442 27.175 36.851 35.675 358.614 226.646 42.381 47.478 740.00 - 14.089 38.333 49.818 36.752 241.491 244.455 50.871 44.233 690.00 - 15.928 54.252 67.894 69.054 371.721 362.338 34.741 31.775 640.00 - 13.826 69.426 84.503 84.759 272.058 331.687 37.715 30.956 600.00 - 9.645 78.917 82.532 82.126 291.854 252.444 29.440 32.883 570.00 - 5.220 83.434 84.265 83.213 289.663 337.388 22.953 20.078 550.00 - 1.717 84.830 84.754 84.666 233.141 163.397 12.487 17.567Figure 1
Comparison of the upstream element strain energy (MJ).
(a)
Initial shape (b)
Leader-follower game shape (preferredf1) (c)
Leader-follower game shape (preferredf2) (d)
Leader-follower game shape (preferredf3)Figure 2
Comparison of the downstream element strain energy (MJ).
(a)
Initial shape (b)
Leader-follower game shape (preferredf1) (c)
Leader-follower game shape (preferredf2) (d)
Leader-follower game shape (preferredf3)Figure 3
Comparison of the displacement along the river of the downstream surface (cm).
(a)
Initial shape (b)
Leader-follower game shape (preferredf1) (c)
Leader-follower game shape (preferredf2) (d)
Leader-follower game shape (preferredf3)Figure 4
Comparison of contour line of the upstream surface principal tensile stress (MPa).
(a)
Initial shape (b)
Leader-follower game shape (preferredf1) (c)
Leader-follower game shape (preferredf2) (d)
Leader-follower game shape (preferredf3)Figure 5
Comparison of the contour line of downstream surface principal compressive stress (MPa).
(a)
Initial shape (b)
Leader-follower game shape (preferredf1) (c)
Leader-follower game shape (preferredf2) (d)
Leader-follower game shape (preferredf3)Our numerical results have demonstrated that all three shapes based on the leader-follower game can achieve their preferred targets. Compared to the initial shape (see Table4), the shape of the preferred f1 (volume) is reduced by 667500 m3 (about 9.68%). The shape of the preferred f2 (stress) is reduced by 1.56 MPa (about 13.48%). The shape of the preferred f3 (energy) is reduced by 0.455 GJ (about 12.19%). These results have highlighted the status of the preferred targets and demonstrated the effectiveness of our proposed method. Besides, the maximum center angle of the arch dam is related to the stability of the arch abutment and the dam shoulder. The smaller maximum center angle yields a better stability of the arch abutment and the dam shoulder. The shape of the preferred f3 appears to be more robust (Table 4 shows that the maximum center angle is 95.104°, which is the smallest among the four shapes), and its multiobjective functions have been further improved as well.


## 5. Conclusion

(1) In this paper, we have developed an original and novel Stackelberg model in solving a multiobjective design problem with the preferred target. In our model, we utilize a method for computing the strategy subspace. A multiobjective game based on the leader-follower game pattern has also been established. Our optimization example of an arch dam with three targets shows that our approach is efficient and can implement preferred objective (among other objectives) in a leader-follower game-theoretic manner.(2) From the process of the leader-follower game we notice that the leader player gives priority to a decision making. After the leader player makes the decision, the follower player selects its own strategy variables according to the decision of the leader player to optimize its goal. Therefore, the leader player restricts the achievement of the subordinate goal, and the decision of the follower player will also influence the goal of the leader player. The optimization problem of the follower player is actually a constraint to the optimization problem of the leader player. The strategy set of the leader and the follower forms a nonseparable whole and a relation that restricts each of them.



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*Source: 7358621-2018-03-20.xml*