A mathematical programming approach to identification and optimization of a class of unknown systems

There exists a class of decision problems for which: (1) models of input-output response functions are not available in a closed-form, functional representation; (2) informational costs associated with learning about the response function are significant. For these problems, combining identification with optimization using mathematical programming is potentially attractive. Three approaches to the identification-optimization problem are proposed: an outer-linearized approximation using relaxation (OLR); an inner-linearized approximation using restriction (ILR); and a sequential combination of inner- and outer-linearized subproblems (SIO). Algorithms based on each approach are developed and computational experience reported.     

A mixed optimization technique for the generalized machine replacement problem

A mixed optimization technique for optimal machine replacement is presented which allows much more flexibility than previous models. Optimal purchase, maintenance and sale of a given machine between any two given points in time is treated as a subproblem, which one may choose to solve via control theory, dynamic programming, or practical engineering considerations. (A control theory formulation is used in the paper as an illustration.) These subproblem solutions are then incorporated into a Wagner-Whitin formulation for solution of the full problem. The technique is particularly useful for problems with such asymmetries as an existing initial machine or uneven technological change. A simple numerical example is solved in the Appendix.     

On constraint qualifications in nonlinear programming

In this paper we examine the relationship between two constraint qualifications developed by Abadie and Arrow, Hurwicz, and Uzawa. A third constraint qualification is discussed and shown to be weaker than either of those mentioned above.     

Optimal spares for stochastically failing equipment

A mathematical model is formulated for determining the number of spare components to purchase when components stochastically fail according to a known life distribution function and there is a cost incurred when a component is replaced. Bounds are determined for the optimal inventory which indicate that the inclusion of the replacement cost lowers the optimal inventory. Since these bounds are no easier to calculate than the optimal spares level, the theory is specialized to components with exponentially distributed time to failure. Procedures are given for calculating the optimal spares level, and numerical examples are provided.     

Optimal interdiction policy for a flow network

This paper analyzes the problem faced by a field commander who, confronted by an enemy on N battlefields, must determine an interdiction policy for the enemy's logistics system which minimizes the amount of war material flowing through this system per unit time. The resource utilized to achieve this interdiction is subject to constraint. It can be shown that this problem is equivalent to determining the set of arcs Z* to remove subject to constraint from a directed graph G such that the resulting maximal flow is minimized. A branch and bound algorithm for the solution to this problem is described, and a numerical example is provided.     

Application of the GLM technique to a production planning problem

Generalized Lagrange Multipliers (GLM) are used to develop an algorithm for a type of multiproduct single period production planning problem which involves discontinuities of the fixed charge variety. Several properties of the GLM technique are developed for this class of problems and from these properties an algorithm is obtained. The problem of resolving the gaps which are exposed by the GLM procedure is considered, and an example involving a quadratic cost function is explored in detail.     

On a sequential rule for estimating the location parameter of an exponential distribution

Let us assume that observations are obtained at random and sequentially from a population with density function In this paper we consider a sequential rule for estimating μ when σ is unknown corresponding to the following class of cost functions In this paper we consider a sequential rule for estimating μ when σ is unknown corresponding to the following class of cost functions Where δ(X_I,…,X_N) is a suitable estimator of μ based on the random sample (X₁,…, X_N), N is a stopping variable, and A and p are given constants. To study the performance of the rule it is compared with corresponding “optimum fixed sample procedures” with known σ by comparing expected sample sizes and expected costs. It is shown that the rule is “asymptotically efficient” when absolute loss (p=-1) is used whereas the one based on squared error (p = 2) is not. A table is provided to show that in small samples similar conclusions are also true.     

Applications of a generalized combinatorial problem of smirnov

In this paper applications of results obtained by these authors for a generalized version of a problem proposed by Smirnov, are considered. The areas of application explored are system interface, queueing, transportation flow, and sequential analysis. The included table should be invaluable to the reader in applying these results. Finally the relationship between the limiting and exact expressions relating to this table is also explored.     

A branch-bound algorithm for the capacitated facilities location problem

This paper discusses a mixed integer programming method for solving the Facilities Location Problem with capacities on the facilities. The algorithm uses a Decomposition technique to solve the dual of the associated continuous problem in each branch-bound iteration. The method was designed to produce the global optimum solution for problems with up to 100 facilities and 1,000 customers. Computational experience and a complete example are also presented in the appendix.