A branch and bound algorithm for allocation problems in which constraint coefficients depend upon decision variables

A branch and bound algorithm is developed for a class of allocation problems in which some constraint coefficients depend on the values of certain of the decision variables. Were it not for these dependencies, the problems could be solved by linear programming. The algorithm is developed in terms of a strategic deployment problem in which it is desired to find a least-cost transportation fleet, subject to constraints on men/materiel requirements in the event of certain hypothesized contingencies. Among the transportation vehicles available for selection are aircraft which exhibit the characteristic that the amount of goods deliverable by an aircraft on a particular route in a given time period (called aircraft productivity and measured in kilotons/aircraft/month) depends on the ratio of type 1 to type 2 aircraft used on that particular route. A model is formulated in which these relationships are first approximated by piecewise linear functions. A branch and bound algorithm for solving the resultant nonlinear problem is then presented; the algorithm solves a sequence of linear programming problems. The algorithm is illustrated by a sample problem and comments concerning its practicality are made.     

A maximum utility solution to a vehicle constrained tanker scheduling problem

A modification to the Dantzig and Fulkerson Tanker Scheduling Problem is described. An insufficient number of vehicles and a utility associated with each vehicle delivery are assumed. The new problem is shown to be equivalent to a Transshipment Problem, the solution of which is the same as the maximal utility solution of the modified Tanker Scheduling Problem. An example is given.     

An application of servomechanisms to inventory

In recent years, some attention has been devoted to the application of techniques of control theory to inventory management. In particular, H. Vassian (1955) developed a model for a periodic review inventory system utilizing techniques of discrete variable servomechanisms to analyze the system in a cost-free structure. The resulting model is inherently deterministic, however, and emphasizes the control of inventory fluctuation about a safety level by selecting an appropriate order policy. Such an order policy is defined only up to an arbitrary method of forecasting customer demands. The present paper is a continuation of the model developed by Vassian in which exponential smoothing is used as a specific forecasting technique. Full recognition of the probabilistic nature of demand is taken into account and the requirement of minimizing expected inventory level is imposed. In addition, explicit formulas for the variance in inventory are derived as functions of the smoothing constant and the tradeoff between small variance and rapid system response is noted. Finally, in an attempt to remove the bias inherent in exponential smoothing, a modification of that technique is defined and discussed as an alternate forecasting method.     

A chance-constrained distribution problem

The transportation model with supplies (S_i) and demands (D_i) treated as bounded variables developed by Charnes and Klingman is extended to the case where the S_i and D_i are independently and uniformly distributed random variables. Chance constraints which require that demand at the jth destination will be satisfied with probability at least β_i and that stockout at the ith origin will occur with probability less than α_i are imposed. Conversion of the chance constraints to their linear equivalents results in a transportation problem with one more row and column than the original with some of the new arcs capacitated. The chance-constrained formulation is extended to the transshipment problem.     

A priori error bounds for procurement commodity aggregation in logistics planning models

A complete logistical planning model of a firm or public system should include activities having to do with the procurement of supplies. Not infrequently, however, procurement aspects are difficult to model because of their relatively complex and evanescent nature. This raises the issue of how to build an overall logistics model in spite of such difficulties. This paper offers some suggestions toward this end which enable the procurement side of a model to be simplified via commodity aggregation in a “controlled” way, that is, in such a manner that the modeler can know and control in advance of solving his model how much loss of accuracy will be incurred for the solutions to the (aggregated) overall model.     

An analytical minefield evaluation model without space averages

The paper describes an approach to the evaluation of the effectiveness of a minefield in terms of the number of mines that are detonated by a convoy of sweepers and ships and the corresponding number of vessels that are immobilized. The positions of the mines and the tracks of the vessels are assumed to be known, which means that the evaluation measures are dependent on a large number of disjoint events, each event being the immobilization of particular vessels by particular mines. This may render combinatorial methods computationally infeasible, but by introducing approximations in the assumptions, the difficulty can be overcome, specifically by modelling the arrival of each individual vessel in the neighborhood of a mine by an inhomogeneous Poisson stream for which the arrival rate is nonzero only over a short time interval. The plausibility of the approach is supported by results of a critical-event simulation model.     

Auxiliary procedures for solving long transportation problems

An efficient auxiliary algorithm for solving transportation problems, based on a necessary but not sufficient condition for optimum, is presented.     

On the distribution of the optimal value for a class of stochastic geometric programs

An approach is presented for obtaining the moments and distribution of the optimal value for a class of prototype stochastic geometric programs with log-normally distributed cost coefficients. It is assumed for each set of values taken on by the cost coefficients that the resulting deterministic primal program is superconsistent and soluble. It is also required that the corresponding dual program has a unique optimal point with all positive components. It is indicated how one can apply the results obtained under the above assumptions to stochastic programs whose corresponding deterministic dual programs need not satisfy the above-mentioned uniqueness and positivity requirements.     

Techniques for establishing ergodic and recurrence properties of continuous-valued markov chains

We present techniques for classifying Markov chains with a continuous state space as either ergodic or recurrent. These methods are analogous to those of Foster for countable space chains. The theory is presented in the first half of the paper, while the second half consists of examples illustrating these techniques. The technique for proving ergodicity involves, in practice, three steps: showing that the chain is irreducible in a suitable sense; verifying that the mean hitting times on certain (usually bounded) sets are bounded, by using a “mean drift” criterion analogous to that of Foster; and finally, checking that the chain is such that bounded mean hitting times for these sets does actually imply ergodicity. The examples comprise a number of known and new results: using our techniques we investigate random walks, queues with waiting-time-dependent service times, dams with general and random-release rules, the s-S inventory model, and feedback models.