Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 7358621, 12 pages
https://doi.org/10.1155/2018/7358621

Research Article

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

Neng-gang Xie,¹ Zhong Chen,² Kang Hao Cheong,³ Rui Meng,¹ and Wei Bao¹

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

Correspondence should be addressed to Neng-gang Xie; xienenggang@aliyun.com

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.

Table 1: Computation results of a simple test case.

Ranking

Ranking

0.667
0.333
1
0.333
0.667
2
0.222

0.333
0.667
2
0.667
0.333
1
0.222

The threshold of moment is computed according to (5) and determined to be 0.222. According to Table 1 and step of Algorithm 1 in Section 2.2.1, we will then have and .

The numerical results can be easily obtained by following Algorithm 2 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.

Objective value,
Objective value,

Stackelberg game [13]
Preferred
1.4
2.2
0.8
1.28

Preferred
1.8
2.6
1.28
0.8

Method given in [13]
Preferred
1.399945
2.200030
0.800092
1.280002

Preferred
1.799998
2.600001
1.280001
0.800004

This paper
Preferred
1.409656
2.202957
0.797145
1.264605

Preferred
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 .

The dam volume is defined to be the economic objective; . The maximum principal tensile stress (of the dam) is the safety goal of arch dam’s local area, , and is the principal tensile stress. The strain energy of the arch dam is taken as the overall safety goal; . 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

 Let objective functions be

 And satisfy the following constraints:

 Volume constraints: (arch dam volume does not exceed the allowable value) Geometric constraints: (arch dam thickness does not exceed the allowable value); (upstream overhang degree does not exceed the allowable value); (downstream overhang degree does not exceed the allowable value) Stress constraints: (the maximum principal compressive stress does not exceed the allowable value) Stability constraints: (the maximum central angle does not exceed the allowable value)

where is the strain energy, . represents the displacement matrix of the whole nodes and 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: the foundation rock in upstream and depth directions is calculated at twice the height of the dam. 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 Table 3, provided by a design institution. The constraints are as follows: and ; the upstream overhang degree: ; the downstream overhang degree: ; the largest central angle: ; the chord length and quasi semicentral angle constraints; the allowable principal compressive stress value is given as .

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

(14.000)
(341.617)
(19.035)
(261.046)
(19.000)

780.0

(301.730)
(30.650)
(241.896)
(39.679)

740.0

(41.510)
(268.473)
(42.822)
(223.934)
(55.000)

690.0

(229.704)
(58.165)
(201.973)
(69.945)

640.0

(58.677)
(200.492)
(70.442)
(185.014)
(78.787)

600.0

(176.757)
(75.744)
(171.928)
(80.437)

570.0

(163.024)
(75.929)
(163.842)
(77.997)

550.0
(−37.816)
(70.000)
(154.557)
(73.834)
(154.836)
(74.435)

The impact index, space distance, and space moment are computed, and all design variables are sorted according to the rules set out in Section 2.2.1. According to the partition rules, the strategy subspace of the game player : , the strategy subspace of the game player : , and the strategy subspace of the game player : .

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, when is regarded as preferred target, the game profit of the leader and the game profit of the follower . The strategy space of the leader is and the strategy space of the follower is . The situation is handled in the same way if or 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 Table 4. 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 is the smallest and the corresponding thickness of arch dam is minimal. On comparison, the shape of the preferred 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 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 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

/°

Initial shape
689.26
3.733
11.57
0.254
0.048
96.312

Game shape
Preferred
622.51
4.088
12.57
0.127
0.113
98.912

Preferred
723.65
4.273
10.01
0.213
0.145
99.754

Preferred
697.90
3.278
11.52
0.186
0.123
95.104

Table 5: Parameters of the shape of the arch dam for preferred .

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

25.447
27.953
36.078
465.152
244.917
35.127
45.260

740.00

35.019
42.090
48.274
356.685
205.378
39.768
49.207

690.00

43.695
68.095
65.178
256.498
167.642
45.143
53.242

640.00

50.142
80.723
84.727
198.301
154.219
46.694
52.218

600.00

54.718
84.585
83.448
162.681
253.498
45.356
32.774

570.00

58.324
82.458
81.886
196.758
263.621
31.942
25.069

550.00

60.994
74.118
68.642
203.955
207.586
14.205
13.992

Table 6: Parameters of the shape of the arch dam for preferred .

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

38.282
26.162
26.584
462.596
246.299
35.275
45.099

740.00

49.633
48.349
37.736
356.380
209.717
39.793
48.614

690.00

59.984
45.885
45.812
252.188
167.403
45.628
53.281

640.00

68.188
61.853
61.781
193.024
153.820
47.464
52.290

600.00

74.732
67.038
73.513
169.103
249.697
44.247
33.169

570.00

80.391
66.466
67.368
187.931
254.426
33.135
25.859

550.00

84.794
79.348
82.285
205.829
205.511
14.081
14.128

Table 7: Parameters of the shape of the arch dam for preferred .

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

27.175
36.851
35.675
358.614
226.646
42.381
47.478

740.00

38.333
49.818
36.752
241.491
244.455
50.871
44.233

690.00

54.252
67.894
69.054
371.721
362.338
34.741
31.775

640.00

69.426
84.503
84.759
272.058
331.687
37.715
30.956

600.00

78.917
82.532
82.126
291.854
252.444
29.440
32.883

570.00

83.434
84.265
83.213
289.663
337.388
22.953
20.078

550.00

84.830
84.754
84.666
233.141
163.397
12.487
17.567

(a) Initial shape

(b) Leader-follower game shape (preferred )

(c) Leader-follower game shape (preferred )

(d) Leader-follower game shape (preferred )

Figure 1: Comparison of the upstream element strain energy (MJ).

(a) Initial shape

(b) Leader-follower game shape (preferred )

(c) Leader-follower game shape (preferred )

(d) Leader-follower game shape (preferred )

Figure 2: Comparison of the downstream element strain energy (MJ).

(a) Initial shape

(b) Leader-follower game shape (preferred )

(c) Leader-follower game shape (preferred )

(d) Leader-follower game shape (preferred )

Figure 3: Comparison of the displacement along the river of the downstream surface (cm).

(a) Initial shape

(b) Leader-follower game shape (preferred )

(c) Leader-follower game shape (preferred )

(d) Leader-follower game shape (preferred )

Figure 4: Comparison of contour line of the upstream surface principal tensile stress (MPa).

(a) Initial shape

(b) Leader-follower game shape (preferred )

(c) Leader-follower game shape (preferred )

(d) Leader-follower game shape (preferred )

Figure 5: Comparison of the contour line of downstream surface principal compressive stress (MPa).

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 Table 4), the shape of the preferred (volume) is reduced by 667500 m³ (about 9.68%). The shape of the preferred (stress) is reduced by 1.56 MPa (about 13.48%). The shape of the preferred (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 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.

Conflicts of Interest

The authors declare no conflicts of interest related to the publication of this paper.

Acknowledgments

This project was supported by the National Natural Science Foundation of China (Grant no. 61375068 and no. 51605005) and the Talent Project for Higher Education Promotion Program of Anhui Province.

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