Consumer Behavior

This paper presents an allocation model for a perishable product, distributed from a regional center to a given set of locations with random demands. We consider the combined problem of allocating the available inventory at the center while deciding how these deliveries should be performed. Two types of delivery patterns are analyzed: the first pattern assumes that all demand points receive individual deliveries; the second pattern subsumes the frequently occuring case in which deliveries in multistop routes traveled by a fleet of vehicles. Computational experience is reported.

In many resource allocation problems, the objective is to allocate discrete resource units to a set of activities so as to maximize a concave objective function subject to upper bounds on the total amounts allotted to certain groups of activities. If the constraints determine a polymatroid and the objective is linear, it is well known that the greedy procedure results in an optimal solution. In this paper we extend this result to objectives that are "weakly concave," a property generalizing separable concavity.

In the classical maximal flow problem, the objective is to maximize the supply to a single sink in a capacitated network. In this paper we consider general capacitated networks with multiple sinks: the objective is to optimize a general "concave" preference relation on the set of feasible supply vectors. We show that an optimal solution can be obtained by a marginal allocation procedure. An efficient implementation results in an adaptation of the augmenting path algorithm. We also discuss an application of the procedure for an investment company that deals in oil and gas ventures.

Special algorithms have been developed to compute an optimal (s,S) policy for an inventory model with discrete demand and under standard assumptions (stationary data, a well-behaved one-period cost function, full backlogging and the average cost criterion). We present here an iterative algorithm for continuous demand distributions which avoids any form of prior discretization. The method can be viewed as a modified form of policy iteration applied to a Markov decision process with continuous state space. For phase-type distributions, the calculations can be done in closed form.

This paper establishes a simple existence proof for a solution to the optimality equations arising in finite undiscounted Markov Renewal Programs, by applying Brouwer's fixed point theorem to the so-called reduced value-iteration operator. Because of its simplicity, our approach lends itself to new existence results for more general models.

This paper presents a successive approximation method for solving systems of nested functional equations which arise, e.g., when considering Markov renewal programs in which policies that are maximal gain or optimal under more selective discount — and average overtaking optimality criteria are to be found. In particular, a successive approximation method is given to find the optimal bias vector and bias-optimal policies. Applications with respect to a number of additional stochastic control models are pointed out.

In many practical applications of multi-item inventory systems significant economies of scale can be exploited when coordinating replenishment orders for groups of items. This paper considers a continuous review multi-item inventory system with compound Poisson demand processes; excess demands are backlogged and each replenishment requires a lead time. There is a major setup cost associated with any replenishment of the family of items, and a minor (item dependent) setup cost when including a particular item in this replenishment. Moreover there are holding and penalty costs.

Consider a central depot (or plant) which supplies several locations experiencing random demands. Orders are placed (or production is initiated) periodically by the depot. The order arrives after a fixed lead time, and is then allocated among the several locations. (The depot itself does not hold inventory.) The allocations are finally received at the demand points after another lag. Unfilled demand at each location is backordered. Linear costs are incurred at each location for holding inventory and for backorders. Also, costs are assessed for orders placed by the depot.

Clark and Scarf [1960] characterize optimal policies in a two-echelon, two-location inventory model. We extend their result to the infinite-horizon case (for both discounted and average costs). The computations required are far easier than for the finite horizon problem. Further simplification is achieved for normal demands. We also consider the more interesting case of multiple locations at the lower echelon. We show that, under certain conditions, this problem can be closely approximated by a model with one such location.