Journal of Industrial and Management Optimization
January 2006 , Volume 2 , Issue 1
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Support set invariancy sensitivity analysis deals with finding the range of the parameter variation where there are optimal solutions with the same positive variables for all parameter values throughout this range. This approach to sensitivity analysis has been studied for Linear Optimization (LO) and Convex Quadratic Optimization (CQO) problems, when they are in standard form. In practice, most problems are in general form, in addition to nonnegative variables and equalities, they include free variables and inequalities. The LO problem in general form can be converted into the standard form, but this transforming changes the meaning of the support set invariancy sensitivity analysis.
In this paper, we consider the primal and dual LO problems in general form and introduce the associated general standard form. It is shown that investigating support set invariancy sensitivity analysis for this general standard form is able to accommodate not only the support set invariancy sensitivity analysis for usual standard form, but also the classic study of sensitivity analysis based on simplex methods as well as the recent point of view of sensitivity analysis based on interior point methods.
We present a periodic review inventory model with multiple delivery modes. We generalize the notion of the base-stock policy for inventory systems with multiple delivery modes. While base-stock policies are optimal for one or two consecutive delivery modes, it is not so otherwise. For multiple consecutive delivery modes, we show that only the fastest two modes have optimal base stocks, and provide simple counterexamples to show that the remaining ones do not in general. We investigate why the base-stock policy is not optimal through detailed analyses of two numerical examples.
Our aim is to provide a short analysis of the generalized variational inequality (GVI) problem from both theoretical and algorithmic points of view. First, we show connections among some well known existence theorems for GVI and for inclusions. Then, we recall the proximal point approach and a splitting algorithm for solving GVI. Finally, we propose a class of differentiable gap functions for GVI, which is a natural extension of a well known class of gap functions for variational inequalities (VI).
A two-dimensional discrete optimal control problem is considered. In this problem it is required that the first component admits the given value and the second component attains the largest value at the last step. The explicit solution of this problem is obtained under some assumptions.
Supervisory control for discrete event systems (DESs) belongs essentially to the logic level for control problems in DESs. Its corresponding control task is hard. In this paper, we study a new optimal control problem in DESs. The performance measure is to maximize the maximal discounted total reward among all possible strings (i.e., paths) of the controlled system. The condition we need for this is only that the performance measure is well defined. We then divide the problem into three sub-cases where the optimal values are respectively finite, positive infinite and negative infinite. We then show the optimality equation in the case with a finite optimal value. Also, we characterize the optimality equation together with its solutions and characterize the structure of the set of all optimal policies. All the results are still true when the performance measure is to maximize the minimal discounted total reward among all possible strings of the controlled system. Finally, we apply these equations and solutions to a resource allocation system. The system may be deadlocked and in order to avoid the deadlock we can either prohibit occurrence of some events or resolve the deadlock. It is shown that from the view of the maximal discounted total cost, it is better to resolve the deadlock if and only if the cost for resolving the deadlock is less than the threshold value.
In this paper we consider one-buyer, one-seller, finite horizon, multi-period inventory models in which the economic order quantity is integrated with the economic production quantity (EOQ-EPQ in short). We introduce the Stackelberg equilibrium framework in which the objective is to maximize the vendor's total benefit subject to the minimum total cost that the buyer is willing to incur. Some existence results, optimality conditions and the optimal replenishment policy under the Stackelberg equilibrium concept are obtained and a numerical algorithm and examples are presented to find the optimal replenishment policy in practice.
An open-loop two-person zero-sum linear quadratic (LQ for short) stochastic differential game is considered. The controls for both players are allowed to appear in both the drift and diffusion of the state equation, the weighting matrices in the payoff/cost functional are not assumed to be definite/non-singular, and the cross-terms between two controls are allowed to appear. A forward-backward stochastic differential equation (FBSDE, for short) and a generalized differential Riccati equation are introduced, whose solvability leads to the existence of the open-loop saddle points for the corresponding two-person zero-sum LQ stochastic differential game, under some additional mild conditions. The main idea is a thorough study of general two-person zero-sum LQ games in Hilbert spaces.
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