A multimodel approach to stochastic nash games

A decentralized filtering and control scheme is presented for obtaining low-order Nash equilibrium strategies for decision makers using different models of the same large-scale system. Multiparameter singular perturbations are employed to capture the multimodel nature of the fast dynamic subsystems interconnected through slow dynamic variables. The small parameters are appropriately scaled so that the variables in both time scales are well defined. The decision makers have decentralized information structures and are constrained to use only finite dimensional compensators of a specified dimension. It is shown that the proposed scheme is in fact the asymptotic limit of the exact solution as the small parameters go to zero.     

Team decision theory and information structures in optimal control problems--Part I

Information structures of organizations are studied and applied to problems of dynamic team decisions. For a causal system it is shown that there is a partially ordered precedence relation existing among the decision makers. The team decision problem with linear information structure and quadratic payoff function is dealt with. The primitive random variables are assumed to be jointly Gaussian. The optimal solutions for the teams in which precedents' information is available for the followers are obtained. It is shown that the well-known linear-quadratic-Gaussian stochastic control problem and static team decision problem are special cases of the structure considered.     

Interactive fuzzy random cooperative two-level linear programming through level sets based probability maximization

In this paper, assuming cooperative behavior of the decision makers, two-level linear programming problems under fuzzy random environments are considered. To deal with the formulated fuzzy random two-level linear programming problems, α-level sets of fuzzy random variables are introduced and an α-stochastic two-level linear programming problem is defined for guaranteeing the degree of realization of the problem. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced and the α-stochastic two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through probability maximization, the transformed stochastic two-level programming problem can be reduced to a deterministic one. Interactive fuzzy programming to derive a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method.     

Interactive fuzzy programming for fuzzy random two-level linear programming problems through probability maximization with possibility

This paper considers two-level linear programming problems involving fuzzy random variables under cooperative behavior of the decision makers. Through the introduction of fuzzy goals together with possibility measures, the formulated fuzzy random two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. By adopting probability maximization, the transformed stochastic two-level programming problem can be reduced to a deterministic one. Interactive fuzzy programming to derive a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented. An illustrative numerical example demonstrates the feasibility and efficiency of the proposed method.     

On optimal distributed decision architectures in a hypothesistesting environment

The authors consider the distributed detection problem, in which a set of decision makers (DMs) receive observations of the environment and transmit finite-values messages to other DMs according to prespecified communication protocols. A designated primary DM makes the final decision on one out of two alternative hypothesis. All DMs make decisions, in order to maximize a measure of organizational performance. The problem is to find an architecture for the organization which remains optimal for a variety of operating conditions. It is shown that even for very small organizations this problem is quite complex because the optimal architecture depends on variables external to the team, so that global conclusions on optimal organizational structures cannot be drawn. Suboptimal solutions are considered, and bounds on their performance are obtained     

Control strategies for decision makers using different models of the same system

Situations in which strategies of various decision makers are designed using different models of the same system are a characteristic of large-scale system practice. For interconnected systems with slow and fast dynamics we develop a design algorithm which takes into account such multimodel situations. Conditions for the validity of this approximate design are formulated and illustrated by a power system example.     

Min-max sliding-mode control for multimodel linear time varying systems

An original linear time-varying system with unmatched disturbances and uncertainties is replaced by a finite set of dynamic models such that each one describes a particular uncertain case including exact realizations of possible dynamic equations as well as external bounded disturbances. Such a tradeoff between an original uncertain linear time varying dynamic system and a corresponding higher order multimodel system with a complete knowledge leads to a linear multi-model system with known bounded disturbances. Each model from a given finite set is characterized by a quadratic performance index. The developed min-max sliding-mode control strategy gives an optimal robust sliding-surface design algorithm, which is reduced to a solution of an equivalent linear quadratic problem that corresponds to the weighted performance indices with weights from a finite dimensional simplex. An illustrative numerical example is presented.     

The dynamic linear exponential Gaussian team problem

The dynamic team problem for a linear system with Gaussian noise, exponential of a quadratic performance index, and one-step delayed sharing information pattern is considered. It is shown, via dynamic programming, that the multistage problem can be decomposed into a series of static team problems. Moreover, the optimal policy of theith team member at timekis an affine function of both the one-step predicted Kalman filter estimate and theith team member's observation at timek. Efficient algorithms are available for determining the gains of this affine controller. This model and solution are applied to an approximate resource allocation problem associated with a defense network, and a numerical example is discussed.     

Interactive fuzzy programming for random fuzzy two-level programming problems through possibility-based fractile model

This paper focuses on interactive decision making methods for random fuzzy two-level linear programming problems. Considering the probabilities that the decision makers’ objective function values are smaller than or equal to target variables, fuzzy goals of the decision makers are introduced. Using the fractile model to optimize the target variables under the condition that the degrees of possibility with respect to the attained probabilities are greater than or equal to certain permissible levels, the original random fuzzy two-level programming problems are reduced to deterministic ones. Interactive fuzzy nonlinear programming to obtain a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented. An illustrative numerical example demonstrates the feasibility and efficiency of the proposed method.     

Symmetric team problems and multi access wire communication

In this paper we deal with the problem of sharing one communication wire among a (possibly large) number of communication stations. The fact that all communication stations are considered identical and that they share one objective of using the communication wire as efficiently as possible leads to the concept of symmetric team problems. For symmetric team problems we define a symmetric solution by the restriction that all decision makers must have identical decision rules. In the first section of this paper the concepts of symmetric team problems and symmetric solutions are developed and motivated. A theorem is given that relates symmetric solutions to randomized decision rules. An example is given which illustrates that the concept of symmetric solutions explains some everyday phenomena. In the second section the access problem in multi-access wire communication is considered as a symmetric team problem. It is shown that the symmetric solution, which corresponds to randomized access rules, tends to give as good performance as the unrestricted solution when the number of stations becomes large (asymptotic optimality). The solutions are also determined numerically, giving quantitative information on the asymptotic behavior.     

A team algorithm for robust stability analysis and control design of uncertain time‐varying linear systems using piecewise quadratic Lyapunov functions

A team algorithm based on piecewise quadratic simultaneous Lyapunov functions for robust stability analysis and control design of uncertain time‐varying linear systems is introduced. The objective is to use robust stability criteria that are less conservative than the usual quadratic stability criterion. The use of piecewise quadratic Lyapunov functions leads to a non‐convex optimization problem, which is decomposed into a convex subproblem in a selected subset of decision variables, and a lower‐dimensional non‐convex subproblem in the remaining decision variables. A team algorithm that combines genetic algorithms (GA) for the non‐convex subproblem and interior‐point methods for the solution of linear matrix inequalities (LMI), which form the convex subproblem, is proposed. Numerical examples are given, showing the advantages of the proposed method. Copyright © 2001 John Wiley & Sons, Ltd.     

An approach to decentralized control of large scale systems using aggregation methods

A two-level decentralized control structure is formulated for large scale interconnected subsystems controlled byNdecision makers. Chained aggregation is used to decompose the overall team problem with a decentralized information structure into (N + 1) subproblems: one low order team problem with a centralized information structure andNdecentralized optimal control problems. Accordingly, the control of each decision maker is decomposed into three components: a decoupling control which induces aggregation, a local control which controls the subsystem dynamics, and an aggregate control which controls the dynamics of the interconnection variables. The robustness of this composite control with respect to perturbations in the system dynamics and the cost functional is established.     

A distributed M-ary hypothesis testing problem withcorrelated observations

The distributed M-ary hypothesis testing problem of detecting one of M₀ events by an (N+1)-person hierarchical team, when the observations are correlated, is examined. In this problem, each of the N subordinate decision makers (DMs) transmits one of a prespecified set of messages based on their data to a primary decision maker who, in turn, combines the messages with his or her own data to make the final team decision. The necessary conditions for the optimal decision rules of the DMs are derived. A nonlinear Gauss-Seidel iterative algorithm is developed for the person-by-person optimal decision rules, and its monotonic convergence to the person-by-person optimum is established. A fast approximation algorithm is proposed for computing certain conditional probabilities arising in the person-by-person optimal decision rules. The algorithms are illustrated with several examples, and implications for distributed organizational designs are pointed out     

An α-Fuzzy Goal Approximate Algorithm for Solving Fuzzy Multiple Objective Linear Programming Problems

Multiple conflicting objectives in many decision making problems can be well described by multiple objective linear programming (MOLP) models. This paper deals with the vague and imprecise information in a multiple objective problem by fuzzy numbers to represent parameters of an MOLP model. This so-called fuzzy MOLP (or FMOLP) model will reflect some uncertainty in the problem solution process since most decision makers often have imprecise goals for their decision objectives. This study proposes an approximate algorithm based on a fuzzy goal optimization under the satisfactory degree α to handle both fuzzy and imprecise issues. The concept of a general fuzzy number is used in the proposed algorithm for an FMOLP problem with fuzzy parameters. As a result, this algorithm will allow decision makers to provide fuzzy goals in any form of membership functions.     

Dynamic selling of quality-graded products under demand uncertainties

This paper examines dynamic selling (DS) problems under demand uncertainties. Quality-graded products with fully downward substitutable demands are considered. Downward demand substitution indicates that demands for lower quality grade products can be fulfilled by either designated or higher quality grade products. In this dynamic selling problem, decision makers need to choose an optimal selling policy in each decision epoch. The objective is to identify an optimal policy for the dynamic selling of quality-graded inventory.DS problems are formulated as a discrete-time Markov decision process (MDP) model. In the MDP model, demand type and inventory levels are state variables. The objective is to maximize expected profits. In such a multi-dimensional dynamic decision problem, computational complexity is a chief concern. This study proves the structure of optimal policies that significantly reduce computational complexity. Performance of optimal dynamic selling policies is evaluated in detailed numerical studies.     

Mini-max integral sliding-mode control for multimodel linear uncertain systems

An original linear time-varying system with matched and unmatched disturbances and uncertainties is replaced by a finite set of dynamic models such that each one describes a particular uncertain case including exact realizations of possible dynamic equations as well as external unmatched bounded disturbances. Such a tradeoff between an original uncertain linear time varying dynamic system and a corresponding higher order multimodel system containing only matched uncertainties leads to a linear multi-model system with known unmatched bounded disturbances and unknown matched disturbances as well. Each model from a given finite set is characterized by a quadratic performance index. The developed minimax integral sliding mode control strategy gives an optimal minimax linear quadratic (LQ)-control with additional integral sliding mode term. The design of this controller is reduced to a solution of an equivalent mini-max LQ problem that corresponds to the weighted performance indices with weights from a finite dimensional simplex. The additional integral sliding mode controller part completely dismisses the influence of matched uncertainties from the initial time instant. Two numerical examples illustrate this study.     

Analysis of the environmental efficiency in China based on the DEA cross-efficiency approach under different policy objectives

The existing studies on environmental efficiency evaluation generally have the problem of efficiency overestimation. To solve this problem, a new data envelopment analysis (DEA) cross-efficiency approach with undesirable outputs is developed to evaluate environmental efficiency from the perspectives of both self-evaluation and peer evaluation. Then, three new evaluation strategies, namely, economic development strategy, environmental protection strategy, and win–win strategy, are proposed to reflect the needs of decision makers under different policy objectives. The proposed cross-efficiency approach with different evaluation strategies not only realizes the cross evaluation of environmental efficiency, but also guarantees the relative uniqueness of the optimal solution on the basis of the preferences of decision makers. Combining the metafrontier DEA approach and DEA window analysis, a new cross-efficiency analytical framework is constructed to gradually analyse the influences of policy objectives, technology heterogeneity, and dynamic correlation on the environmental efficiency. Subsequently, the environmental efficiency of China's economic development during 2006–2015 is in-depth analysed on the basis of the proposed analytical framework, and some interesting conclusions, and some useful suggestions are obtained.     

Nonlinear robust control and minimax team problems

The problem of robust stabilization of a linear system leads to the classical ℋ︁_∞ control problem. The same analysis applied to a nonlinear system leads to the problem of ensuring via output feedback that a nonlinear operator be Lipshitz continuous, with a prescribed Lipshitz modulus. We show that, in the same way as the ℋ︁_∞ control problem is equivalent to a minimax control problem, the Lipshitz modulus control problem can be approached via a minimax team decision problem. This motivates us to re-visit a class of the so-called ‘static’ team decision problems for nonlinear dynamical control systems. Because of the ‘static’ character, signaling plays no role in that case, which is important for the equivalence with the Lipshitz modulus control problem. We show that under some conditions, a certainty equivalence principle applies that yields a practical solution to the team problem at hand. To reach that conclusion we must first investigate a ‘partial team’ problem where one of the team members has all the information. Copyright © 1999 John Wiley & Sons, Ltd.     

Scenario relaxation algorithm for finite scenario-based min–max regret and min–max relative regret robust optimization

Most practical decision-making problems are compounded in difficulty by the degree of uncertainty and ambiguity surrounding the key model parameters. Decision makers may be confronted with problems in which no sufficient historical information is available to make estimates of the probability distributions for uncertain parameter values. In these situations, decision makers are not able to search for the long-term decision setting with the best long-run average performance. Instead, decision makers are searching for the robust long-term decision setting that performs relatively well across all possible realizations of uncertainty without attempting to assign an assumed probability distribution to any ambiguous parameter. In this paper, we propose an iterative algorithm for solving min–max regret and min–max relative regret robust optimization problems for two-stage decision-making under uncertainty (ambiguity) where the structure of the first-stage problem is a mixed integer (binary) linear programming model and the structure of the second-stage problem is a linear programming model. The algorithm guarantees termination at an optimal robust solution, if one exists. A number of applications of the proposed algorithm are demonstrated. All results illustrate good performance of the proposed algorithm.