Bi- and Multi Level Game Theoretic Approaches in Mechanical Design

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1 University o Wisconsin Milwaukee UWM Digital Commons Theses and Dissertations August 03 Bi- and Multi Level Game Theoretic Approaches in Mechanical Design Ehsan Ghotbi University o Wisconsin-Milwaukee Follow this and additional works at: Part o the Economics Commons, Mathematics Commons, and the Mechanical Engineering Commons Recommended Citation Ghotbi, Ehsan, "Bi- and Multi Level Game Theoretic Approaches in Mechanical Design" (03). Theses and Dissertations. Paper 50. This Dissertation is brought to you or ree and open access by UWM Digital Commons. It has been accepted or inclusion in Theses and Dissertations by an authorized administrator o UWM Digital Commons. For more inormation, please contact kristinw@uwm.edu.

2 BI- AND MULTI LEVEL GAME THEORETIC APPROACHES IN MECHANICAL DESIGN By Ehsan Ghotbi A Dissertation Submitted in Partial Fulillment o the Requirements or the Degree o Doctor o Philosophy in Engineering at The University o Wisconsin Milwaukee August 03

3 ABSTRACT BI- AND MULTI LEVEL GAME THEORETIC APPROACHES IN MECHANICAL DESIGN by Ehsan Ghotbi The University o Wisconsin-Milwaukee, 03 Under the Supervision o Proessor Anoop Dhingra This dissertation presents a game theoretic approach to solve bi and multi-level optimization problems arising in mechanical design. Toward this end, Stackelberg (leader-ollower), Nash, as well as cooperative game ormulations are considered. To solve these problems numerically, a sensitivity based approach is developed in this dissertation. Although game theoretic methods have been used by several authors or solving multi-objective problems, numerical methods and the applications o etensive games to engineering design problems are very limited. This dissertation tries to ill this gap by developing the possible scenarios or multi-objective problems and develops new numerical approaches or solving them. This dissertation addresses three main problems. The irst problem addresses the ormulation and solution o an optimization problem with two objective unctions using the Stackelberg approach. A computational procedure utilizing sensitivity o ollower s solution to leader s choices is presented to solve the bi-level optimization problem numerically. Two mechanical design problems including lywheel design and design o ii

4 high speed our-bar mechanism are modeled based on Stackelberg game. The partitioning o variables between the leader and ollower problem is discussed, and a variable partitioning metric is introduced to compare various variable partitions. The second problem this dissertation ocuses on is modeling the multi-objective optimization problem (MOP) as a Nash game. A computational procedure utilizing sensitivity based approach is also presented to ind Nash solution o the MOP numerically. Some test problems including mathematical problems and mechanical design problems are discussed to validate the results. In a Nash game, the players o the game are at the same level unlike the Stackelberg ormulation in which the players are at dierent levels o importance. The third problem this dissertation addresses deals with hierarchical modeling o multi-level optimization problems and modeling o decentralized bi-level multi-objective problems. Generalizations o the basic Stackelberg model to consider cases with multiple leaders and/or multiple ollowers are missing rom the literature. Three mathematical problems are solved to show the application o the algorithm developed in this research or solving hierarchical as well as decentralized problems. iii

5 ACKNOWLEDGMENTS I would like to thank my advisor, Proessor Anoop Dhingra or his continuous guidance, patience, support and riendship throughout my dissertation work. I appreciate my dissertation committee members: Dr Ron Perez, Dr Ilya Avdeev, Dr Matthew McGinty and Dr Wilkistar Otieno or their support, insights and suggestions. I epress my sincere thanks to Proessor Hans Volkmer or his help with a mathematical proo o my algorithm. I would like to thank my good riends in Design Optimization group. I also thank the University o Wisconsin-Milwaukee or providing me inancial support to do my research. I never will orget the good memories that I had here. I also would like to thank my ather and mother or their inluence that shaped me the person I am today. Finally, my special thanks to my dear wie, Zeinab Salari Far. Without her support and patience this success was not possible. iv

6 Dedicated to my dear wie Zeinab and my Parents who always encouraged me and supported me to continue my education. I should mention my little cute son, Parsa. He has given me energy to work on my research. v

7 TABLE OF CONTENTS. Introduction.... Multiple Objective Optimization Problems..... Methods with a Priori Articulation o Preerences..... Methods with a Posteriori Articulation o Preerences Game Theory Approaches in Design Non-Cooperative Games in Design Etensive Games in Design Cooperative Games in Design....3 Summary....4 Dissertation Organization Basic Concepts in Multi-Objective Optimization Techniques or Solving Multi-Objective Optimization Problems Deinitions and Terminology Modeling Multi-Objective Problems Using Game Theory Game Theoretic Models in Design Rational Reaction Set or Stackelberg and Nash Solutions Optimum Sensitivity Derivatives Sensitivity Based Algorithm or Obtaining Stackelberg Solutions Sensitivity Based Algorithm or Obtaining Nash Solutions Convergence Proo Summary... 4 vi

8 4. Generating RRS Using DOE-RSM and Sensitivity Based Approaches Introduction DOE-RSM Method Numerical Eamples Bilevel Problem with Three Followers Design o a Pressure Vessel Two-Bar Truss Problem Conclusions Application o Stackelberg Games in Mechanical Design Optimum Design o Flywheels Design Problem Formulation Thickness Function Mass and Kinetic Energy Stress Analysis Manuacturing Objective Function The Optimization Problem Partitioning the variables Numerical Results Optimum Design o High-Speed 4-bar Mechanisms Introduction Mechanism Design Problem Formulation The Optimization Problem Partitioning the variables vii

9 5..5 Numerical Results Summary Game Based Approaches in Hierarchical and Decentralized Systems Introduction Decentralized Bi-level Model Decentralized Bi-level Model Eample Eample Eample Hierarchical Model Hierarchical Model Eample Conclusions Stackelberg Game Non-Cooperative (Nash) Game Hierarchical and Decentralized Systems Scope or Future Work Reerences... 3 viii

10 LIST OF FIGURES Figure. Flowchart or Possible Cases or Solving Multi-Objective Problems with Game Approach... Figure. Ideal Point... Figure 3. Computational Procedure or Obtaining Stackelberg Solution Using Sensitivity Method Figure 3. Computational Procedure or Obtaining Nash Solution Using Sensitivity Method Figure 4. Thin-Walled Pressure Vessel Figure 4. Nash Solution Length vs Radius or Pressure Vessel Problem Figure 4.3 Nash Solution Thickness vs Radius or Pressure Vessel Problem Figure 4.4 Stress Constraint o Player VOL Figure 4.5 Stress Constraint o Player VOL Figure 4.6 Two-Bar Truss Problem Figure 4.7 The Analytical and RSM Approimation RRS or Figure 4.8 The Leader Objective Function Applying RSM Method Figure 4.9 A 3 Full Factorial Design (7 points) Figure 4.0 Three One-Third Fraction o the 3 Design Figure 4. Central Composite Design or 3 Design Variables at Levels Figure 5. General Shape o the Flywheel Figure 5. Proile Shape o Flywheel or Follower and Leader Problem Figure 5.3 Flywheel Proile or Single Objective and Stackelberg Solutions Figure 5.4 Flywheel Proile or Cases, 3, i

11 Figure 5.5 Flywheel Proile or Cases 4, 5, Figure 5.6 The Path Generating Four Bar Mechanism Figure 5.7 Free Body Diagrams o Four Bar Mechanism Figure 5.8 Input Torque Variation Over the Whole Cycle Figure 5.9 Transmission Angle Deviation rom Ideal Value Over a Whole Cycle Figure 5.0 Desired versus Generated Path Figure 5. Deviation o Transmission Angle rom Ideal Value Figure 5. Desired Versus Generated Path Figure 5.3 Input Torque Variation over the Whole Cycle Figure 6. Decentralized Systems Figure 6. Hierarchical System with Three Levels.... 3

12 LIST OF TABLES Table 4- Comparison o Results or Eample (rom Liu 998) Table 4- Pressure Vessel Problem Parameters... 6 Table 4-3 Eperimental Design to Obtain RRS or the Follower (WGT)... 6 Table 4-4 Number o Optimization Problems Solved or Eample Table 5- Optimum Solutions or Single Objective Optimizations and the Stackelberg Solution Table 5- Optimum Solutions or Dierent Variable Partitions Table 5-3 Objective Values or Single Objective Optimization Table 5-4 Stackelberg Solution or Bi-Level Problem Table 5-5 Stackelberg Solutions or Bi-Level Problem Table 5-6 Optimum Values or Single Objective Optimizations Table 5-7 Stackelberg Solution or Bi-level Problem Table 6- The Best and Worst Values o Objective Functions Table 6- Optimum Solution or Cooperative-Stackelberg Scenario Table 6-3 Optimum Solution or Nash-Stackelberg Scenario Table 6-4 The Best and Worst Values o Objective Functions Table 6-5 Optimum Solution or Cooperative-Stackelberg Scenario Table 6-6 Hierarchical Model Solution i

13 Chapter. INTRODUCTION The topic o multiple objective optimization problems comes rom the ield o multiple criteria decision making. Multiple criteria decision making deals with methods and algorithms to analytically model and solve problems with multiple objective unctions. Multi-objective optimization (MOO) problems requiring a simultaneous consideration o two or more conlicting objective unctions requently arise in design. This dissertation addresses solutions to multi-objective problems arising in the contet o mechanical design.. Multiple Objective Optimization Problems Multiple criteria decision making has two aspects, namely, multi-attribute decision analysis and multiple objective optimization. Multiattribute decision analysis is applicable to problems in which the decision maker is dealing with a small number o alternatives in an uncertain environment. This aspect helps in resolving public policy problems such as nuclear power plant location, location o an airport, location o a waste processing acility, etc. This aspect has been covered in detail by Keeney and Raia (993). The second aspect o multiple criteria decision making deals with the application o optimization techniques in solving these problems. Techniques or solving multiple criteria (objective) optimization have been developed since early 970s. Solutions to multi-objective problems where all objective unctions are simultaneously minimized generally do not eist. Thereore, optimization techniques generally look or the best compromise solution amongst all objectives. Since modeling the decision maker s preerences is a primary goal o multi-objective optimization, to

14 ind the best compromise solution, there should be some procedure to obtain preerence inormation rom the decision maker along with selection o a suitable optimization scheme. Hwang and Masud (979) classiied the optimization techniques into three groups according to the timing o requesting the preerence inormation: (i) Articulation o the decision maker s preerences prior to optimization, (ii) Progressive articulation o preerences (during or in sequence with optimization), and (iii) A posteriori articulation o preerences (ater optimization problem has been solved). Marler and Arora (004) did a comprehensive survey on the multi-objective optimization methods available on literature. They divided the methods based on how the decision makers articulate their preerences including priori articulation, posteriori articulation and no articulation o preerences... Methods with a Priori Articulation o Preerences In these methods, preerences are dictated by the decision maker beore the optimization problem is solved. The dierence between the methods is based on the dierent utility unctions they may use. Some o the methods which are based on an apriori articulation o preerences are discussed below: Weighted Sum Method: The weighting method is a conventional approach to solve multi-objective optimization problems. In this method, a weight is assigned to each objective unction and the summation o weighted objective unctions is considered as the overall objective unction. Steuer (989) related the weights to the preerence o decision maker. Many works have been done to select the weights. Saaty (977) provided an eigenvalue method to determine the weights. This method involves the pairwise comparison between the

15 3 objective unctions. This provides a comparison matri with eigenvalues which are the weights. Yoon and Hwang (995) developed the ranking method to select the weights. In this method, the objective unctions are ranked by importance. The least important objective unction gets a weight o one and the integer weights with increments are assigned to objective unctions that are more important. There are some constraints in applying weighted sum method. For eample, Messac et al. (000) proved that it is impossible or this method to obtain points on non-conve portions o the Pareto optimal rontier set. Also, Papalambros and Wilde (988) stated that this approach can mislead concerning the nature o optimum design. Leicographic Method: In the Leicographic method, the objective unctions are ranked in order o importance by the decision maker. The optimization problem o objective unction deemed most important is solved and the optimum solution is obtained. The second most important objective unction can be optimized by considering that the optimum value o the previous objective unction should not be changed. This procedure is repeated until all objective unctions have been considered. Rentmeesters et al. (996) showed that the optimum solution o leicographic method does not satisy the constraint qualiication o Kuhn-Tucker optimality conditions. The authors developed other optimality conditions or the leicographic approach. Goal Programming Method: The basic idea in goal programming is to establish a goal level or each objective unction. The overall objective is to minimize the deviation o each objective unction rom its own goal level. Charnes and Cooper (96), Lee (97) and Ignizio (980)

16 4 developed the goal programming method. Lee and Olson (999) reviewed the applications o goal programming method. Although the method has the wide range o applications, there is no guarantee that the solution obtained with this method is a Pareto optimal solution. Weighted goal programming method, in which weights are assigned to the deviation o each objective unction rom its goal, was developed by Charnes and Cooper (977). Bounded Objective Function Method: In this approach, only the most important objective unction is minimized and the other objective unctions are considered as constraints. The lower and upper bounds are set or the other objective unctions. Haimes et al. (97) developed -constraint method in which only the upper limits are considered. Miettinen (999) showed that i eists a solution to -constraint, then the solution is a weakly Pareto optimal solution. I the solution is unique, then it is Pareto optimal. Chankong and Haimes (983) proved that i the problem is conve and objective unctions are strictly conve, then the solution is unique. Ehrgott and Ryan (00) improved -constraint by allowing the objective unctions, which are in constraints, to be violated and penalizing any violation in the objective unction. There are some other methods such as weighted min-ma, physical programming and weighted product method in the literature which are based on a priori articulation o preerences. Utility Theory: An approach to solving multiple objective optimization problems is correlating the objective unctions with value unctions; these unctions are comparable, and

17 5 combining these value unctions yields a problem with single objective unction. Figure o Merit (FOM) is an approach to evaluate the multiple objective unctions. A more analytic approach or the evaluation o attributes (objective unctions) is Utility analysis developed by Von Neumann (947), Savage (954) and Keeney and Raia (993). Thurston (99) compared FOM approach by utility analysis. Thurston (994) applied Utility unction in optimization o a design problem. The author deined an overall utility unction or the design problem o single utility unction or each objective unction. For each single utility unction one single scaling constant has been deined which shows the relative merits o the utility unctions. These scaling constants can be obtained by tools such as the Analytic Hierarchy Process (AHP) developed by Saaty (988) or uzzy analysis developed by Zadeh (975). To construct the utility unction or each objective unction, Thurston (994) used the lottery questions to assess a set o points on each single utility unction. The best it o these points shows the orm o the utility unction... Methods with a Posteriori Articulation o Preerences The methods using posteriori articulation o preerences irst look or a set o Pareto optimal solutions and then according to the decision maker preerence, the best compromise solution will be selected rom the Pareto optimal set. The advantage o this method is that the solution set is independent o the decision maker s preerences. These methods are constructed with the target o obtaining Pareto points and then selecting the optimal solution amongst these Pareto optimal points. Algorithms using posteriori articulation o preerences to solve MOLP s can be divided into two categories: () Algorithms inding all eicient etreme points. () Algorithm inding just eicient points. Steuer (976) showed that all algorithms are in

18 6 the irst category consist o three phases. Phase one and two ind an initial etreme and an initial eicient point respectively. Phase three searches or all eicient etreme points. The algorithms in this category dier in their approaches in phase three. Steuer (975) developed two computer codes, ADBASE and ADEX to obtain all eicient etreme points. Most works in the area o posteriori articulation o preerences have been done in category two. Some o the methods with a Posteriori Articulation o Preerences are discussed in below: Normal Boundary Intersection (NBI): Das and Dennis (998) developed NBI method. The weighted sum method has a shortcoming o not being able to ind Pareto optimal points in non-conve problems. But NBI approach uses a scalarization method to produce Pareto optimal set or non-conve problems. However, the method may also produce non-pareto optimal points. It means that it does not provide a suicient condition or the Pareto optimality o the solutions. Das and Dennis (998) applied NBI to a three-bar truss design problem with ive objective unctions and our design variables. Normal Constraint (NC): Messac et al. (003) improved NBI method to eliminate non-pareto optimal solutions rom the optimal solution set. In normal constraint method, irst it determines the ideal point and its components or each objective unction. A plane passing through the ideal points is called the utopia hyper plane. The objective unctions are normalized based on the ideal solution. NC method uses the normalized unction value to tackle with disparate unction scales. This part is dierent than NBI method.

19 7 The other approaches available in the literature are Evolutionary algorithms, Genetic algorithms and Directed search domain. The weighted sum method, goal programming method and leicographic method are some o the common approaches in the literature to solve multiple objective optimization problems. In all these methods, the optimum solution is dependent to the preerences o the decision maker. For eample, in weighted sum method, by changing the weights o the objective unctions, the optimum solution may change. Also, there is no guarantee that the optimum solution o these methods is a Pareto optimal solution. Game theory method is not sensitive to preerences o the decision maker and also it can provide the Pareto optimal solution, or cooperative game. The other methods are attempting to change the multi-objective unction problem to a single objective problem and solve it, but game theoretic models consider each objective unction individually. This makes the game theory an interesting topic to do research. This thesis studies game theoretic models which can be applied in mechanical design. In net section, game theory as a tool or solving multi objective problems is reviewed.. Game Theory Approaches in Design In game theory, the multi-objective optimization problem is treated as a game where each player corresponds to an objective unction being optimized. The notion o designers as players in a game has been demonstrated by several authors (Vincent, 983; Rao, 987; Lewis and Mistree, 997; Badhrinath and Rao, 996; Hernandez and Mistree, 000; Shiau and Michalek, 009). The players control a subset o design variables and seek to optimize their individual payo unctions.

20 8 The objective unctions o players are oten conlicting and the designers may not have the capability o inding a compromise solution. In this situation, game theory can be an appropriate tool to model interactions between designers. There are three types o games that can be uses in the contet o design: cooperative game, non-cooperative (Nash) game, and an etensive game. In a cooperative game, the players have knowledge o the strategies chosen by other players and collaborate with each other to ind a Paretooptimal solution. I a cooperation or coalition among the players is not possible, the players make decision by making assumptions about unknown strategies selected by other players. In etensive games, the players make decisions sequentially. The etensive games can be non-cooperative game but it is considered separately in this research. In the net three sections, these three types o games will be discussed in some detail... Non-Cooperative Games in Design In a non-cooperative game, each player has a set o variables under his control and optimizes his objective unction individually. The player does not care how his selection aects the payo unctions o other players. The players bargain with each other to obtain an equilibrium solution, i one eists. In the literature, this solution is called Nash equilibrium solution (Mcginty (0)). Vincent (983) irst proposed the use o a non-cooperative game in design. Two designers play in a non-cooperative game and end up to the solution. Vincent showed that the Nash solution is usually not on the Pareto optimal set. Rao (987) also discussed the Nash game with two designers as players. The case in which there is more than one intersection or rational reaction sets has been studied by Rao. Rao and Hati (980) etended the idea o two-designer game to deine a

21 9 Nash equilibrium solution to n-player non-cooperative game. Finding an intersection o rational reaction sets or the all players is diicult, so the Nash solution is usually empty. Several approaches have been proposed over the years or the computation o Nash solutions in game-theoretic ormulations. These include methods based on Nikaido Isoda unction (Contreras et al. 004), rational reaction set with DOE-RSM approach (Lewis and Mistree 998) and monotonicity analysis (Rao et al. 997). Recently, Deutsch et al. (0) modeled the interaction between an inspection agency and multiple inspectees as a non-cooperative game and obtained all possible Nash equilibria. Their model employs a n-person player game where there is one player (inspection agency) on one side and multiple players (the inspectees) on other side o the game. Eplicit closed-orm solutions were presented to compute all Nash equilibria. For some problems arising in mechanical design such as the pressure vessel problem considered in Rao et al. (997), closed orm epressions or Nash equilibria can be obtained using the principles o montonicity analysis (Papalambros and Wilde, 000). However, in general, numerical techniques are needed to ind the solution. A design o eperiments based approach (Montgomery 005) coupled with response surace methodology (Myers and Montgomery 00) has been proposed by Lewis and Mistree (998), Marston (000), and Hernandez and Mistree (000). This approach has been used by the authors to obtained Nash solutions or non-cooperative games as well as Stackelberg games. Lewis and Mistree (00) discussed modeling interactions o multiple decision makers. They used statistical techniques such as design o eperiments and second-order response surace or numerical approach.

22 0.. Etensive Games in Design Etensive design games reer to situations in which the designers make the decisions sequentially. Etensive games with two players have been used in engineering design and are called Stackelberg games. There are two groups o players in this game. One is called Leader which dominates the other group called ollower. The leader makes its decision irst and according to its decision, the ollower optimizes its objective unction. Rao and Badhrinath (997) modeled the conlicts between designer s and manuacturer s objective unctions using a Stackelberg game. They construct parametric solution o rational reaction o ollower and substitute this solution in the leader s problem to ind its optimum solution. In both design eamples presented in the paper, the Stackelberg s solution that they obtained was Pareto optimal, although the Stackelberg s solution in general is not Pareto optimal. Lewis and Mistree (997) showed application o the Stackelberg game in the design o a Boeing 77, while Hernandez (000) showed the application in design o absorption chillers. Lewis and Mistree (998) compared the solution o Stackelberg game with cooperative game and Nash solution (non-cooperative game) in design o a pressure vessel and a passenger aircrat. Shiau and Michalek (009) developed an engineering optimization method by considering competitor pricing reactions to the new product design. Nash and Stackelberg conditions are imposed on three product design cases or price equilibrium. One critical point in solving a bi-level problem as a Stackelberg game is obtaining the rational reaction set (RRS) o the ollower. For simple problems, RRS can be obtained by solving the optimization problem o ollower parametrically. It gives an eplicit equation or RRS. Rao and Badhrinath (996) and J.R.Rao and coauthors (997)

23 applied this approach. The other way to construct RRS o ollower is using response surace methodology (RSM) which gives an approimation o RRS. Lewis and Mistree (997), Lewis (998) and Hernandez (000) applied RSM in their problems or solving Stackelberg game...3 Cooperative Games in Design A cooperative game means that all designers or some designers (which orm a coalition) cooperate. In this game, the players have knowledge o the strategies chosen by other players and collaborate with each other to ind a Pareto-optimal solution. In Nash and Stackelberg game, the players do not cooperate. It is not unusual that players improve their non-cooperative solution by cooperating. This approach has been discussed by Vincent (983), S.S.Rao (987), Rao and Badrinath (996) and Marston (000). Also, a model or such a game in the contet o imprecise and uzzy inormation was presented by Dhingra and Rao (995). This cooperative uzzy game theoretic model was used to solve a our bar mechanism design problem. The solution o cooperative games is Pareto optimal. I Player and cooperate, then there may be two approaches to get the cooperative solution. The irst approach deals with obtaining the Pareto optimal rontier set. All points, which are in this set, are Pareto optimal in point o view o player and. There are several techniques to get Pareto optimal rontier set or players. These include the NSGA-II method developed by Deb (00) based on genetic algorithms. The TPM is a population-based stochastic approach or inding Pareto optimal rontier set. Das and Dennis (998) developed NBI method. Shukla and Deb (007) compared these dierent

24 approaches and discussed the limitations o each method. NBI and TPM have some diiculties when the Pareto optimal set is discontinuous or non-uniormly spaced. A problem with these approaches is that a single solution still needs to be selected rom the pareto-optimal set or implementation. These methods do not yield a single solution rom the pareto-optimal set termed the cooperative solution. The other approach or obtaining the cooperative solution is by deining a bargaining unction. In bargaining unction, the players will collaborate to maimize the dierence o their objective unctions rom the worst value that they can get in the game. In the literature, this solution is called Nash bargaining solution (Mcginty (0)). In this research, whenever it talks about Nash it means Nash equilibrium game (Non-cooperative game)..3 Summary Although game theoretic methods have been used by several authors or solving multi-objective problems, applications o etensive games to engineering design problems are limited. The limited applications o Stackelberg games to design problems are based on using response surace methodologies to construct rational reaction sets (RRS). This research presents an alternate approach or obtaining Nash and Stackelberg solutions that utilize sensitivity based ormulation. The sensitivity o optimum solution to problem parameters has been eplored by Sobieski et al. (98) and Hou et al. (004). This idea is adapted herein to construct the RRS or Nash and Stackelberg solutions. Generalizations o the basic leader-ollower model to consider cases with multiple leaders and/or multiple ollowers are also missing rom the literature. This thesis is an attempt to address these identiied shortcomings in the eisting literature.

25 3.4 Dissertation Organization This dissertation has been divided into ive main chapters. Chapter discusses terminology associated with solving MOO problems. Chapter 3 discusses game theoretic mathematical models or solving bi-level optimization problems using Stackelberg game and Nash game approaches. A sensitivity based approach is developed to numerically solve optimization problem modeled as a Stackelberg game. Also, an algorithm is developed to solve the optimization problem modeled as a Nash game. A convergence proo o the proposed algorithm is also presented. Chapter 4 develops the sensitivity based approach to numerically solve the multi-objective optimization problems modeled as a Nash game. It also considers a bilevel problem with one leader and three ollowers where the ollowers have a Nash game among themselves and the interaction between the ollowers and the leader is a Stackelberg game. When solving a bi-level optimization problem using as a Stackelberg game, it is necessary to capture the sensitivity o leader s solution to ollower s variables. Previous work in this area has used design o eperiment techniques (DOE) to get the rational reaction set or the ollower. This chapter provides an introduction to design o eperiments (DOE) and response surace method (RSM). Two eamples are presented to demonstrate the beneit o using the proposed sensitivity based over the DOE-RSM method. Chapter 5 presents two mechanical design problems as an application o the technique which has been presented in chapter 3. The irst problem is the lywheel design optimization problem which has been modeled by a bi-level optimization problem. The

26 4 variable partitioning between the leader and the ollower is an issue in this problem and is discussed in detail. A criterion is proposed to identiy the best variable partitioning. The design o high speed our-bar mechanism is the second design optimization problem discussed in this chapter. The dynamic and kinematic perormances o the mechanism are considered simultaneously. The problem is modeled and solved as a multi-level design optimization problem as a Stackelberg game. Chapter 6 addresses generalization o the basic Stackelberg model (one leader-one ollower problem) to both hierarchical as well as decentralized problems. Towards this end, problems with one leader and several ollowers are considered where the ollowers could be arranged in a hierarchical or decentralized manner. Finally, problems with several ollowers and several leaders are also studied in this research. For decentralized approach with multiple objective unctions in leader and the ollower two dierent scenarios are studied. Two numerical eamples are solved or these two scenarios. Finally, Chapter 7 summarizes the main inding o this research.

27 5 Equation Chapter Section CHAPTER. BASIC CONCEPTS IN MULTI-OBJECTIVE OPTIMIZATION Multi-objective optimization, also known as multi criteria optimization, deals with a simultaneous consideration o two or more conlicting objective unctions in a design problem. When the problem has one objective unction, the optimum solution is easy to obtain. It involves optimizing the objective unction subject to the constraints present in the problem, but when the problem has more than one objective unction, the solution approach is not as in simple as in the single objective unction case. This dissertation deals with multi-objective, multi-level design optimization problems and develops new computational approaches or solving such problems.. Techniques or Solving Multi-Objective Optimization Problems There are several approaches or solving multi-objective optimization problems. These include the weighted sum method, scalarization techniques, methods to ind Pareto optimal rontier, game theory methods, etc. Some o these methods were eplained in chapter. The method that is considered in this research is using game theory to solve multi-objective optimization problems. In the game theory approach, each player corresponds to an objective unction. The players compete/collaborate with each other to improve their respective payo (objective unction value). There are three main types o games: () Non-cooperative game. () Cooperative game. (3) Sequential game (Leader- Follower). Figure. shows theses types o games and the techniques which eist in the literature or solving the problems. For eample, it can be seen that there are two approaches to get the cooperative solution including Pareto optimal rontier set and

28 6 maimizing a multiplication unction. There are our techniques discussed in the literature, NSGA-II, TPM, NBI and Naïve and slow, to get Pareto optimal rontier set. In cooperative games, the players have knowledge o the other player s moves and they work (cooperate) together to ind the best possible solution. Some times because o process or inormation barriers, coalition among the players is not possible. So the players can not cooperate. The non-cooperative (Nash) solution is a solution or this case. Besides the cooperative and non-cooperative models, the players can also make their decision sequentially. This sequential interaction may be advantageous when the inluence o one player on another is strongly uni-directional. Leader-Follower (Stackelberg) game can be used when one or more objective unctions (Leader) make their decision irst. Once the leader makes its decision, the ollower makes its decision. There is an assumption that the ollower will behave rationally. This thesis ocuses more on solving multi-objective optimization problems using the Stackelberg game approach. This is because in certain types o design problems, the decisions are made in a sequential manner. There are some deinitions needed to better understand the concepts discussed in subsequent chapters. These deinitions and associated terminology are given in the net section.. Deinitions and Terminology The general orm or a multi-objective optimization problem can be stated as selecting values or each o n decision variables, (,,..., n ), in order to optimize p objective o unctions, ( ), ( ),..., p( ) subject to constraints. By assuming all objective unctions are to be minimized, the problem can be stated mathematically by:

29 7 min F( ) [ ( ), ( ),..., ( )] subject to X p (.) where n X g ( ) 0, h ( ) 0, j,,..., m, k,,..., q j k where gj ( ) are m inequality constraints and hk ( ) are q equality constraints and X is the set o easible solutions or problem in Eq. (.). A solution, s X, which minimizes each o the objective unctions simultaneously is called a Superior solution. Since at least two o the p objective unctions are conlicting, a superior solution to problem shown in Eq. (.) rarely eists. The deinition o Superior solution mathematically is given below. Superior Solution: A solution s to problem shown in Eq. (.) is said to be superior i and only i s s X and ( ) ( ) or i,..., p or all X. i i The outcome associated with a superior solution is the ideal. The deinition o ideal is as ollows. Ideal: The ideal or problem deined in Eq. (.) is a point in the outcome space, I I I F (,..., p ), such that or i,..., p is the optimum objective unction value or I i the problem: Min I ( ) subject to X. i Suppose there are two objective unctions, then Fig. shows the Ideal point. * Z and Z are the optimum value o objective unctions, respectively when they are * considered separately. Point * Z is the Ideal point which minimizes, simultaneously.

30 8 It was mentioned beore that the Ideal point rarely eists. It is clear rom Fig.. that the Ideal point is not in the easible space. Pareto Solution (Eicient Solution): A Pareto solution easible solution, P X P to problem in Eq. (.) is a, or which there does not eist any other easible solution, P P X, such that ( ) ( ) or all i,..., pand ( ) ( ) or at least one i,..., p. i i Oten, the optimum solutions may not be Pareto optimal solution but they satisy other criteria which are making them signiicant or practical applications. For eample, weakly Pareto optimal criteria can be deined as ollows: Weakly Pareto Solution: A point, * i X, is weakly Pareto optimal i and only i there i does not eist another easible solution, X, such that * i( ) i( ) or all i,..., p. Typically, there will be many Pareto solutions to a multi-objective problem. To determine what solution should be selected requires urther inormation rom the decision maker concerning his preerences. One way to present this inormation is the use o a value unction over the multiple objectives o the problem. Value Function: A unction, which associates a real number ( F ( )) to each X, is said to be a value unction which represents a particular decision maker s preerence provided that: ) F( ) F( ) i and only i ( F( )) ( F( )) or, X ; ) F i and only i ( F( )) ( F( )) or ( ) F( ), X

31 9 where F( ) F( ) denotes that decision maker is indierent between outcomes F( ) and F( ). over outcomes ( ). F ( ) F( ) denotes that the decision maker preers outcomes Given the value unction, problem deined in Eq. (.) can be changed to the ollowing problem: ( ) Ma ( F( )) subject to X (.) Solving problem (.) means inding the solution which maimizes the value unction over all easible solutions. Such a solution is called a best compromise solution. Problem shown in Eq. (.) has changed the multi-objective problem to a single objective problem. It means that i a value unction can be deined, there would not be any need or multi-objective optimization techniques. But, a value unction is diicult to obtain or a multi-objective problem since it requires the preerence structure o decision maker to be deined, which is not easily possible. The value unction is a kind o utility unction discussed in chapter. Bargaining unction: This is a unction providing the cooperative solution or the players who collaborate with each other in the game. The epression or this unction is below: Ma Z ( i wi ) (.3) i where i is the objective unction o the players and wi is the worst value or objective unction i. Suppose player can control and. I player solves its problem, then the optimum values will be * and *. By plugging in these values in player s objective

32 0 unction, then player s payo (, ) w will be the worst value or (, ). * * Similarly, the worst value or (, ) can be obtained. The bargaining unction in Eq. (.3) maimizes the distance o each player s payo rom the worst value. Maimizing Eq. (.3) gives the cooperative solution or players and. It can be shown that solutions which maimize Eq. (.3) are Pareto-optimal.

33 Multi objective unctions problem Non Cooperative Game (Nash) Cooperative Game Sequential Game (Stackelberg) (Leader-Follower) Using Optimal Sensitivity Analytically ind RRS Applying DOE techniques to use RSM to ind RRS ma z ( u v ) i i i Pareto Optimal rontier Set NSGA-II TPM NBI Naïve and Slow Couple o Leaders Couple o Followers One Leader- Couple o Followers in Decentralized System One Leader- Couple o Followers in Hierarchy System One Leader-One Follower Construct an overall bargaining unction or one level and Nash game or the other level Construct an overall bargaining unction or each level Using One Leader-One ollower s techniques Applying DOE to use RSM to get RRS Using Optimal Sensitivity Putting ollower Kuhn-Tucker conditions as constraints in the leader s problem Figure. Flowchart or Possible Cases or Solving Multi-Objective Problems with Game Approach

34 Figure. Ideal Point

35 3 Equation Chapter 3 Section 3 CHAPTER 3 3. MODELING MULTI-OBJECTIVE PROBLEMS USING GAME THEORY This chapter discusses mathematical models or solving multi-objective and multi-level design optimization problems using game theory. For multi-level design optimization, the Stackelberg game has been discussed and or multi-objective problem where all objective unctions are in the same level, the problem has been modeled as a Nash game. A sensitivity based approach is developed to numerically solve the Stackelberg and Nash game ormulations. 3. Game Theoretic Models in Design Consider two players, A and B, who can select strategies and where X R n n and X R. Here X and X are the set o all possible strategies each player can select. U is deined as the set o strategies which are easible or the two players. The objective (cost or loss) unctions (, ) and (, ) account or the cost o players and, respectively. The game theory models deal with inding the optimum strategy (, ) which corresponds to the decision protocol o the speciic game model. The goal o each model is to minimize the objective (loss) unction or each player.

36 4 An optimum strategy pair (, ) is said to be stable i neither o the players have * * an incentive to revise their strategy. When the optimum strategy has this property, it is deined to be a Pareto solution or that problem. The minimum values that the objective unctions and can reach within the easible set (, ) is deined as L in (, ), (, ) U L in (, ) (, ) U (3.) It is epected that there is no solution (, ) that simultaneously satisies ** ** (, ) L and ** ** (, ) L. The shadow minimum is deined as L : ( L, L). ** ** The various models and corresponding solutions or the two players A and B can be classiied into our categories: () Conservative solution () Nash solution (3) Cooperative or Pareto solution (4) Stackelberg solution. The conservative solutions are used when two players do not cooperate. Player one, assumes that player two decides on the strategy which is least advantageous or player one s objective unction. Then player one selects rom the easible set, which corresponds to the minimum value or. A similar approach is used by player two to ind its conservative solution. The strategy (, ) that satisies the above description is called the conservative solution. The mathematical orm o the conservative model is deined as

37 5 ( ) sup (, ), * X ( ) sup (, ) * X (3.) where sup corresponds to the value or which gives the largest based on the selection o and sup corresponds to the value or which gives the largest based on the selection o. T in ( ), * X T in ( ) * X (3.3) where in ( ) * corresponds to the value or which gives the smallest and in ( ) * corresponds to the value or which gives the smallest. The conservative strategies or the two players are T, T. Player A knows that he can not get a value worse than T and will reject any strategy or a given value or which (, ) T. N N The Nash or non-cooperative solution (, ), has the property: (, ) min (, ) N N N X and

38 6 (, ) min (, ) N N N X (3.4) Finding the Nash solution is oten diicult since it is a ied point on a nonlinear map as shown below, (, ) X ( ) X ( ) (3.5) N N N N N N where X ( ) : { X : (, ) min (, )} N N N X (3.6) X ( ) : { X : (, ) min (, )} N N N X (3.7) N N where X ( ) and X ( ) are called rational reaction sets or players and respectively. The term rational reaction set is discussed in the net section. P P The Cooperative or Pareto solution (, ) is epected to yield a better result than the solution related to non cooperative solution. It is likely that the players can P P improve on the Nash solution by cooperating with each other. A pair (, ) is a Pareto solution i there is no other pair (, ) such that, (, ) (, ) P P and (, ) (, ) P P (3.8)

39 7 The set o Pareto solution is usually large and it requires some other selection criteria within the Pareto solutions. C C The core solution (, ) is the same as Pareto solution with these two additions, (, ) T and ( C C, ) T (3.9) C C where T and T have been deined in Eq. (3.3). The Stackelberg game is a special case o a bi-level game where one player dominates the other player. Suppose player A is the leader (or dominant) and player B is the ollower. Player A knows the optimum strategy (solution) o Player B. When player A chooses a strategy (its design variables) player B can see the choices made by Player A. Player B solves its problem and inds the optimum solution with respect to player B. Player A can now adjust its strategy based on choices made by player B. as ollows, The model o the Stackelberg solution when player A is the leader can be written minimize : (, ) (, ) U (3.0) subject to : X ( ). N On the other hand, when B is the leader, the problem is: minimize : (, ) (, ) U (3.) subject to : X ( ). N

40 8 N N where X ( ), X ( ) are given by Eqs. (3.6) and (3.7). 3.. Rational Reaction Set or Stackelberg and Nash Solutions Consider two players, and, with objective unctions (, ) and (, ). n They select strategies and rom a set o possible strategies where X R and X R n respectively. Here X and X are the set o all possible strategies each player can select. The game theory approach deals with inding the optimum strategy (, ) which results in highest possible payo or each player. Three game theoretic models that have been used in the contet o engineering design include non-cooperative (Nash) game, cooperative game, and the Stackelberg game in which one player dominates other player(s). In the Stackelberg method, the leader and the ollower have dierent objective unctions and each player has control over speciic variables. The leader chooses optimum values or its variables by solving its problem, then the ollower observes those values and solves its problem and inds optimum values or its variables. From an implementation view point, the bi-level optimization problem is solved by using backward induction. It begins with ollower s problem. Assuming the value o leader s decision variables are ied, the ollower s objective unction is optimized. By varying ollower s variables, the optimum values o ollower s variables as a unction o leader s variables are obtained. Then these unctions are substituted in the leader s problem and the leader optimizes its objective unction to obtain optimum values o leader s variables. The Stackelberg game can be used to model the behavior o decision makers (players) when they operate in a hierarchical manner. Let, l be a set o objective

41 9 unctions or ollower and leader respectively. The ollower and leader s problems are given by Eqs. (3.) and (3.3) respectively: min (, ) by varying l (3.) min (, ) by varying l l l l (3.3) where subscripts and l correspond to ollower and leader objective unction and variables respectively. The ollower can determine its set o optimum solution(s) based on the choices made by the leader. This solution set is called rational reaction set (RRS) or the ollower. The RRS or the ollower is deined as ollows: R R, min, l l l X (3.4) R where is the optimum solution o the ollower (player ) which varies depending l on the strategy l chosen by the leader (player ). It implies that the optimum values o ollower s variables are given as a unction o leader s variables (Eq.(3.5)): R l (3.5) This RRS o the ollower is substituted in Eq. (3.3) to solve the leader s problem and ind optimum values o the leader s variables. Net, by substituting these optimum values in Eq. (3.5), the optimum values o ollower variables can be obtained. The Nash game is a non-cooperative game where each player determines its set o optimum solutions based on the choices made by other player(s). This set o solutions or

42 30 each player is the rational reaction set (RRS). The RRS or players and are deined as ollows: N N, min, X, N N min, X (3.6) (3.7) N where is the optimum solution o player which varies depending on the strategy N N chosen by player. The unction would be RRS or player. Similarly, is the RRS o player. The intersection o these two sets, i it eists, is the Nash solution N N or the non-cooperative game. Thereore,, is a Nash solution i N N N N, (3.8) When the Stackelberg and Nash problems are solved numerically, it is very R N N diicult to obtain eplicit epressions or, and l. The numerical approach which eists in the literature or generating the RRS is based on design o eperiments (DOE) combined with response surace methodology (RSM). The RSM utilizes DOE (design o eperiments) techniques to construct various eperiments or the players that one is interested in inding the RRS. Then a response surace is itted to the eperiment outcomes to ind an approimation to the RRS. This thesis presetns a new method based on sensitivity inormation to approimate the RRS or the players. The proposed method uses Taylor series to