Convex envelope results and strong formulations for a class of mixed-integer programs

In this article we present a novel technique for deriving the convex envelope of certain nonconvex fixed-charge functions of the type that arise in several related applications that have been considered in the literature. One common attribute of these problems is that they involve choosing levels for the undertaking of several activities. Two or more activities share a common resource, and a fixed charge is incurred when any of these activities is undertaken at a positive level. We consider nonconvex programming formulations for these problems in which the fixed charges are expressed in the form of concave functions. With the use of the developed convex envelope results, we show that the convex envelope relaxations of the nonconvex formulations lead to the linear programming relaxations of the strong IP/MIP formulations of these problems. Moreover, our technique for deriving convex envelopes offers a useful construct that could be exploited in other related contexts as well. © 1996 John Wiley & Sons, Inc.     

The bottleneck transportation problem

The bottleneck transportation problem can be stated as follows: A set of supplies and a set of demands are specified such that the total supply is equal to the total demand. There is a transportation time associated between each supply point and each demand point. It is required to find a feasible distribution (of the supplies) which minimizes the maximum transportaton time associated between a supply point and a demand point such that the distribution between the two points is positive. In addition, one may wish to find from among all optimal solutions to the bottleneck transportation problem, a solution which minimizes the total distribution that requires the maximum time Two algorithms are given for solving the above problems. One of them is a primal approach in the sense that improving fcasible solutions are obtained at each iteration. The other is a “threshold” algorithm which is found to be far superior computationally.     

Economic impact and the notion of compensated procurement

The paper consists essentially of two parts. In the first part a linear economic impact model is presented whose structure is based on subcontracting flows. The structural coefficients are defined in terms of flows per area. The model is derived from two identities that are analogous to the income and expenditure identities of national income accounting. The parameters are prime contracts and when one or several of the prime contracts are changed, the model determines the impacts of such changes on the various regions that have been selected. The impacts can be combined with regional multipliers to derive changes in regional income and regional employment. Fragmentary data for this kind of model have been collected on a one-time basis by DOD in 1965 and some results based on the data are presented. The second part of the paper is concerned with normative economics. A scheme is suggested, called compensated procurement, that outlines how the Department of Defense might employ the impact model in a macroeconomic setting. The basic idea is that a stabilization fund be established to finance an array of potential projects which are contracted for to balance sudden shifts in defense demand. Only short-run stabilization is advocated.     

Parametric analysis of linear programs with upper bounded variables

Linear programming problems with upper bounded variables can be solved by regular simplex method by considering upper bounding constraints as explicit constraints of the problem. However, more efficient methods exist which consider these upper bound constraints implicitly. When parametric analysis for problems with upper bounds is to be carried out, one can use the regular parameter analysis by considering the upper bound constraints explicitly. This paper develops formulas for parametric analysis where upper bound constraints are used implicitly, thus reducing the size of the basic matrix.     

Discounted production scheduling and employment smoothing

We consider the problem of minimizing the sum of production, employment smoothing, and inventory costs over a finite number of time periods where demands are known. The fundamental difference between our model and that treated in [1] is that here we permit the smoothing cost to be nonstationary, thereby admitting a model with discounting. We show that the values of the instrumental variables are nondecreasing in time when demands are nondecreasing. We also derive some asymptotic properties of optimal policies.     

Bayes sequential design of stock levels

Bayesian determination of optimal stock levels is studied for the case of Poisson distribution of the demand variable, and prior gamma distribution of the expected demand. Bayes sequential procedure is derived, assuming that stock level can be adjusted at the beginning of each period so that a shortage can be immediately replenished and an overstock can be corrected. The Bayes sequential procedure is more difficult to obtain if this assumption is removed. A dynamic programming method for obtaining the general Bayes sequential procedure is outlined. Finally, an empiric Bayes estimation procedure of the optimal Bayesian stock level is presented.     

Application of the fundamental theorem of games to an example concerning antiballistic missile defense

This paper considers a two sided resource allocation game in which both players initially have fixed resources which may be distributed over various targets. Their effectiveness depends on the manner of distribution and also on the strategy of the opponent, a natural payoff function for such a situation being used. The complete solution to the game is derived and a numerical example given.     

An improved branch and bound procedure for n × m flow shop problems

This paper considers the classical nXm flow shop sequencing problem. An improved branch and bound procedure is proposed. Computational experience shows that the proposed procedure is more efficient compared to the existing optimizing procedures.     

An algorithm for a class of loading problems

The loading problem we consinder is to assign a set of discrete objects, each having a weight, to a set of boxes, each of which has a capacity limit, in such a way that every object is assigned to a box and the number of boxes used is minimized. A characterization of the assignments is offered and used to develop a set of rules for generating nonredundant assignments. The rules are incorporated into an implicit enumeration algorithm. The algorithm is tested against a very good heuristic. Computational experience shows that the algorithm is highly efficient, solving problems of up to 3600 0-1 variables in a CPU second.