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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.

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.

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. [

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 [

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. [

For

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

We define

The leader can achieve its lower bound

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

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

A spatial game approach has been proposed in [

(1) Optimize

(2) Every

(3)

(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

(b) If one design variable

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

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.

(1) Obtain the strategy space

(2) Generate

(3) For any

(4) For any

(5) Define the strategy combination

(6) If

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:

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 (

The numerical results can be easily obtained by following Algorithm

Comparisons of the analytical and numerical results.

| | Objective value, | Objective value, | ||
---|---|---|---|---|---|

Stackelberg game [ | Preferred | 1.4 | 2.2 | 0.8 | 1.28 |

Preferred | 1.8 | 2.6 | 1.28 | 0.8 | |

Method given in [ | 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 |

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 [

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 [

The dam volume

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:

Geometric constraints:

Stress constraints:

Stability constraints:

where

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 [^{4} 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 × 10^{4} MPa, Poisson’s ratio is 0.167, and its density is 2.4 t/m^{3}. 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

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 | | | | | | |

| | | | | ||

780.0 | | | | | ||

| | | | |||

740.0 | | | | | | |

| | | | | | |

690.0 | | | | | ||

| | | | |||

640.0 | | | | | | |

| | | | | | |

600.0 | | | | | ||

| | | | |||

570.0 | | | | | ||

| | | | |||

550.0 | | | | | | |

| | | | | |

The impact index, space distance, and space moment are computed, and all design variables are sorted according to the rules set out in Section

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

The main parameters of the initial shape and the shape featured in leader-follower game are shown in Table

The main technical parameters of the initial shape and leader-follower game shape.

Volume/10^{4} m^{3} | 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 |

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 |

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 |

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 |

Comparison of the upstream element strain energy (MJ).

Initial shape

Leader-follower game shape (preferred

Leader-follower game shape (preferred

Leader-follower game shape (preferred

Comparison of the downstream element strain energy (MJ).

Initial shape

Leader-follower game shape (preferred

Leader-follower game shape (preferred

Leader-follower game shape (preferred

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

Initial shape

Leader-follower game shape (preferred

Leader-follower game shape (preferred

Leader-follower game shape (preferred

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

Initial shape

Leader-follower game shape (preferred

Leader-follower game shape (preferred

Leader-follower game shape (preferred

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

Initial shape

Leader-follower game shape (preferred

Leader-follower game shape (preferred

Leader-follower game shape (preferred

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 ^{3} (about 9.68%). The shape of the preferred

(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.

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

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.