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A Dynamic Programming Approach to the Design of Composite Aircraft Wings  [PDF]
Prashant K. Tarun, Herbert W. Corley
American Journal of Operations Research (AJOR) , 2022, DOI: 10.4236/ajor.2022.125011
Abstract: A light and reliable aircraft has been the major goal of aircraft designers. It is imperative to design the aircraft wing skins as efficiently as possible since the wing skins comprise more than fifty percent of the structural weight of the aircraft wing. The aircraft wing skin consists of many different types of material and thickness configurations at various locations. Selecting a thickness for each location is perhaps the most significant design task. In this paper, we formulate discrete mathematical programming models to determine the optimal thicknesses for three different criteria: maximize reliability, minimize weight, and achieve a trade-off between maximizing reliability and minimizing weight. These three model formulations are generalized discrete resource-allocation problems, which lend themselves well to the dynamic programming approach. Consequently, we use the dynamic programming method to solve these model formulations. To illustrate our approach, an example is solved in which dynamic programming yields a minimum weight design as well as a trade-off curve for weight versus reliability for an aircraft wing with thirty locations (or panels) and fourteen thickness choices for each location.
Decomposition of Mathematical Programming Models for Aircraft Wing Design Facilitating the Use of Dynamic Programming Approach  [PDF]
Prashant K. Tarun, Herbert W. Corley
American Journal of Operations Research (AJOR) , 2023, DOI: 10.4236/ajor.2023.135007
Abstract: Aircraft designers strive to achieve optimal weight-reliability tradeoffs while designing an aircraft. Since aircraft wing skins account for more than fifty percent of their structural weight, aircraft wings must be designed with utmost care and attention in terms of material types and thickness configurations. In particular, the selection of thickness at each location of the aircraft wing skin is the most consequential task for aircraft designers. To accomplish this, we present discrete mathematical programming models to obtain optimal thicknesses either to minimize weight or to maximize reliability. We present theoretical results for the decomposition of these discrete mathematical programming models to reduce computer memory requirements and facilitate the use of dynamic programming for design purposes. In particular, a decomposed version of the weight minimization problem is solved for an aircraft wing with thirty locations (or panels) and fourteen thickness choices for each location to yield an optimal minimum weight design.
Normative Utility Models for Pareto Scalar Equilibria in n-Person, Semi-Cooperative Games in Strategic Form  [PDF]
H. W. Corley
Theoretical Economics Letters (TEL) , 2017, DOI: 10.4236/tel.2017.76113
Abstract: Semi-cooperative games in strategic form are considered in which either a negotiation among the n players determines their actions or else an arbitrator specifies them. Methods are presented for selecting such action profiles by using multiple-objective optimization techniques. In particular, a scalar equilibrium (SE) is an action profile for the n players that maximize a utility function over the acceptable joint actions. Thus the selection of “solutions” to the game involves the selection of an acceptable utility function. In a greedy SE, the goal is to assign individual actions giving each player the largest payoff jointly possible. In a compromise SE, the goal is to make individual player payoffs equitable, while a satisficing SE achieves a target payoff level while weighting each player for possible additional payoff. These SEs are formally defined and shown to be Pareto optimal over the acceptable joint actions of the players. The advantage of these SEs is that they involve only pure strategies that are easily computed. Examples are given, including some well-known coordination games, and the worst-case time complexity for obtaining these SEs is shown to be linear in the number of individual payoffs in the payoff matrix. Finally, the SEs of this paper are checked against some standard game-theoretic bargaining axioms.
A Regret-Based Algorithm for Computing All Pure Nash Equilibria for Noncooperative Games in Normal Form  [PDF]
H. W. Corley
Theoretical Economics Letters (TEL) , 2020, DOI: 10.4236/tel.2020.106076
Abstract: The concept of a pure Nash equilibrium (NE) for a noncooperative game is simpler than that of a mixed NE, which always exists. However, pure NEs probably have more practical significance even though such a game may not have a pure NE. An efficient algorithm is presented here to determine whether an n-person game in normal form has a pure NE and, if so, to obtain all NEs. This algorithm uses the notion of regret, and the payoff matrix (PM) is transformed into a regret matrix (RM)—a loss matrix with an intuitive interpretation. The RM has the property that an action profile of the PM is a pure NE if and only if (0,· · ·,0) is the corresponding element of the RM. The computational complexity of the algorithm is O(N) in the number of individual utilities N in the PM, and so it is substantially faster than a total enumeration.
Forming Coalitions in Normal-Form Games  [PDF]
Emma Dwobeng, Herbert Corley
Theoretical Economics Letters (TEL) , 2022, DOI: 10.4236/tel.2022.125080
Abstract: For a given n-person normal-form game G, we consider all possible sets of mutually exclusive and collective exhaustive coalitions of the n players. For each such set of coalitions, we define a coalitional semi-cooperative game Γ of G as one in which 1) the coalitions are taken as the players of this new game, 2) each coalition tries to maximize the sum of its individual players’ payoffs, and 3) the players within a coalition cooperate to do so. The purpose of this paper is to determine an optimal set of coalitions for G for some relevant notion of optimality. To do so, for the payoff matrix of each possible Γ of G, we determine all Greedy Scalar Equilibria (GSEs), where a GSE is an analog of the Nash equilibrium but always exists in pure strategies. For each of these GSEs, we divide the total payoff for each coalition among its members in the same proportions as its members average over the entire payoff matrix of G. Doing so gives n modified individual player payoffs associated with each GSE of all the Γs. For each of these GSEs, we then compute the geometric mean of its n modified payoffs. A set of coalitions associated with a GSE is deemed optimal for G if the corresponding geometric mean is a maximum among all the GSEs for all the Γs. An optimal set of coalitions thus incorporates the selfishness of the coalitions via the GSE, while the geometric mean of the redistribution of the players’ payoffs models the cooperation of and the fairness for the individual players.
The Mixed Berge Equilibrium in Extensive Form Games  [PDF]
Ahmad Nahhas, H. W. Corley
Theoretical Economics Letters (TEL) , 2017, DOI: 10.4236/tel.2017.77152
Abstract: In this paper we apply the concept of a mixed Berge equilibrium to finite n-person games in extensive form. We study the mixed Berge equilibrium in both perfect and imperfect information finite games. In addition, we define the notion of a subgame perfect mixed Berge equilibrium and show that for a 2-person game, there always exists a subgame perfect Berge equilibrium. Thus there exists a mixed Berge equilibrium for any 2-person game in extensive form. For games with 3 or more players, however, a mixed Berge equilibrium and a subgame perfect mixed Berge equilibrium may not exist. In summary, this paper extends extensive form games to include players acting altruistically.
An Alternative Interpretation of Mixed Strategies in n-Person Normal Form Games via Resource Allocation  [PDF]
Ahmad Nahhas, H. W. Corley
Theoretical Economics Letters (TEL) , 2018, DOI: 10.4236/tel.2018.810122
Abstract: In this paper, we give an interpretation of mixed strategies in normal form games via resource allocation games, where all players utilize the same resource. We define a game in normal form such that each player allocates to each of his pure strategies a fraction of the maximum resource he has available. However, he does not necessarily allocate all of the resource at his disposal. The payoff functions in the resource allocation games vary with how each player allocates his resource. We prove that a Nash equilibrium always exists in mixed strategies for n-person resource allocation games. On the other hand, we show that a mixed Berge equilibrium may not exist in such games.
A Scalar Compromise Equilibrium for N-Person Prescriptive Games  [PDF]
H. W. Corley, Surachai Charoensri, Narakorn Engsuwan
Natural Science (NS) , 2014, DOI: 10.4236/ns.2014.613098
Abstract:

A scalar equilibrium (SE) is defined for n-person prescriptive games in normal form. When a decision criterion (notion of rationality) is either agreed upon by the players or prescribed by an external arbiter, the resulting decision process is modeled by a suitable scalar transformation (utility function). Each n-tuple of von Neumann-Morgenstern utilities is transformed into a nonnegative scalar value between 0 and 1. Any n-tuple yielding a largest scalar value determines an SE, which is always a pure strategy profile. SEs can be computed much faster than Nash equilibria, for example; and the decision criterion need not be based on the players’ selfishness. To illustrate the SE, we define a compromise equilibrium, establish its Pareto optimality, and present examples comparing it to other solution concepts.

Relating Optimization Problems to Systems of Inequalities and Equalities  [PDF]
H. W. Corley, E. O. Dwobeng
American Journal of Operations Research (AJOR) , 2020, DOI: 10.4236/ajor.2020.106016
Abstract: In quantitative decision analysis, an analyst applies mathematical models to make decisions. Frequently these models involve an optimization problem to determine the values of the decision variables, a system S of possibly non- linear inequalities and equalities to restrict these variables, or both. In this note, we relate a general nonlinear programming problem to such a system S in such a way as to provide a solution of either by solving the other—with certain limitations. We first start with S and generalize phase 1 of the two-phase simplex method to either solve S or establish that a solution does not exist. A conclusion is reached by trying to solve S by minimizing a sum of artificial variables subject to the system S as constraints. Using examples, we illustrate how this approach can give the core of a cooperative game and an equilibrium for a noncooperative game, as well as solve both linear and nonlinear goal programming problems. Similarly, we start with a general nonlinear programming problem and present an algorithm to solve it as a series of systems S by generalizing the “sliding objective function method” for
Constraint Optimal Selection Techniques (COSTs) for Linear Programming  [PDF]
Goh Saito, H. W. Corley, Jay M. Rosenberger
American Journal of Operations Research (AJOR) , 2013, DOI: 10.4236/ajor.2013.31004
Abstract:

We describe a new active-set, cutting-plane Constraint Optimal Selection Technique (COST) for solving general linear programming problems. We describe strategies to bound the initial problem and simultaneously add multiple constraints. We give an interpretation of the new COST’s selection rule, which considers both the depth of constraints as well as their angles from the objective function. We provide computational comparisons of the COST with existing linear programming algorithms, including other COSTs in the literature, for some large-scale problems. Finally, we discuss conclusions and future research.

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