Approximate decision rules for continuous review inventory systems

Constrained multi-item inventory models have long presented signifcant computational problems. This article presents a general algorithm to obtain simultaneous solutions for order quantities and safety stocks for each line item in an inventory, while satisfying constraints on average inventory investment and reordering workload. Computational experience is presented that demonstrates the algorithm's efficiency in handling large-scale applications. Decision rules for several customer service objectives are developed, with a discussion of the characteristics of the inventory systems in which each objective would be most appropriate. The decision rules are approximations, based on the assumptions commonly used in practice.     

Nonadjacent extreme point methods for solving linear programs

In this article we present some advanced basis or block-pivoting, relaxation, and feasible direction methods for solving linear programming problems. Preliminary computational results appear to indicate that the former two types of simplex-based procedures may hold promise for solving linear programming problems, unlike the third type of scheme which is shown to be computationally unattractive.     

Evaluation of costs of ordering policies in large machine repair problems

This article examines a version of the machine repair problem where failures may be irreparable. Since the number of machines in the system keeps decreasing, we impose a fixed state-dependent ordering policy of the type often encountered in inventory models. Although the system is Markovian, the number of states becomes very large. The emphasis of the article, therefore, is on deriving computationally tractable formulas for the steady-state probabilities, the long-run average cost per unit time, and the vector of expected discounted costs. When the state space is so large that exact computations may be infeasible, we propose approximations which are relatively quick and simple to compute and which yield very accurate results for the test problems examined.     

An efficient cyclic coordinate method for optimizing penalty functions

In this study, a simple and efficient cyclic coordinate search procedure is used to optimize penalty functions. Since the contours of the penalty function are very ill-behaved, an accurate line search is very difficult to achieve. Due to accumulated errors in line search, this makes “simple-minded” search directions just as good as more sophisticated directions, and actually better since they require a smaller effort per iteration. Of course this is only true if the search procedure is able to “ride” along steep ridges, and meanwhile move an appreciable distance towards the optimal, if at all possible. Computational results on the cyclic coordinate method seems to support this point of view, and shows robustness, reliability, and efficiency of the method.     

Adjacent vertices on transportation polytopes

To solve linear fixed charge problems with Murty's vertex ranking algorithm, one uses a simplex algorithm and a procedure to determine the vertices adjacent to a given vertex. In solving fixed charge transportation problems, the simplex algorithm simplifies to the stepping-stone algorithm. To find adjacent vertices on transportation polytopes, we present a procedure which is a simplification of a more general procedure for arbitrary polytopes.     

An operator theory of parametric programming for the generalized transportation problem-III-weight operators

This paper investigates the effect on the optimum solution of a capacitated generalized transportation problem when any coefficient of any row constraint is continuously varied as a linear function of a single parameter. The entire analysis is divided into three parts. Results are derived relative to the cases when the coefficient under consideration is associated, to a cell where the optimal solution in that cell attains its lower bound or its upper bound. The discussion relative to the case when the coefficient under consideration is associated to a cell in the optimal basis is given in two parts. The first part deals with the primal changes of the optimal solution while the second part is concerned with the dual changes. It is shown that the optimal cost varies in a nonlinear fashion when the coefficient changes linearly in certain cases. The discussion in this paper is limited to basis-preserving operators for which the changes in the data are such that the optimum bases are preserved. Relevant algorithms and illustrations are provided throughout the paper.     

Denumerable state markovian sequential control processes: On randomizations of optimal policies

We consider a denumerable state Markovian sequential control process. It is well known that when we consider the expected total discounted income as a criterion, there exists a nonrandomized stationary policy that is optimal. It is also well known that when we consider the expected average income as a criterion, an optimal nonrandomized stationary policy exists when a certain system of equations has a solution. The problem considered here is: if there exist two optimal nonrandomized stationary policies, will a randomization of these two policies be optimal? It is shown that in the discounted case the answer is always yes, but in the average income case, the answer is yes only under certain additional conditions.     

An operator theory of parametric programming for the generalized transportation problem: II Rim,cost and bound operators

This paper investigates the effect on the optimum solution of a capacitated generalized transportation problem when certain data of the problem are continuously varied as a linear function of a single parameter. First the rim conditions, then the cost coefficients, and finally the cell upper bounds are varied parametrically and the effect on the optimal solution, the associated change in costs and the dual changes are derived. Finally the effect of simultaneous changes in both cost coefficients and rim conditions are investigated. Bound operators that effect changes in upper bounds are shown to be equivalent to rim operators. The discussion in this paper is limited to basis preserving operators for which the changes in the data are such that the optimum bases are preserved.     

On the assignment and sequencing of operations on a crew-served project

This paper is concerned with assigning and sequencing a set of activities for some or all members of a crew of operators so that the completion time of all such operations is minimized. It is assumed that each of the operators in the crew possesses, initially, certain tasks that only he can perform. A branch-and-bound scheme is proposed to treat the problem, and suitable computational experience is provided.