Optimal retail assortments for substitutable items purchased in sets

We consider the problem of optimizing assortments in a multi‐item retail inventory system. In addition to the usual holding and stockout costs, there is a fixed cost for including any item in the assortment. Customers' preferences for items include both probabilistic substitution patterns and the desire to purchase sets of complementary items. We develop a demand model to capture this behavior, and derive tractable approximations that allow us to formulate the optimization problem as a 0–1 mixed integer linear program. Numerical examples are solved to illustrate key insights into how both complementary and substitute items affect the optimal assortment and the expected profit. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 793–822, 2003.     

Large deviations of the winsorized mean of an exponential distribution

An asymptotic representation for large deviation probabilities of the Winsorized mean of a sequence of independent, identically distributed exponential random variables is derived. The Winsorized mean, a linear combination of exponential order statistics, is first transformed into a weighted sum of exponential random variables, and then a large deviation theorem for weighted sums can be applied. The representation obtained is then compared with results already known for the mean and the median, the two extreme cases of the Winsorized mean.     

Covariance of weibull order statistics

Let X₁ < X₂ <… < X_n denote an ordered sample of size n from a Weibull population with cdf F(x) = 1 - exp (?x^p), x > 0. Formulae for computing Cov (X_i, X_j) are well known, but they are difficult to use in practice. A simple approximation to Cov(X_i, X_j) is presented here, and its accuracy is discussed.     

Variations on a cutting plane method for solving concave minimization problems with linear constraints

A cutting plane method for solving concave minimization problems with linear constraints has been advanced by Tui. The principle behind this cutting plane has been applied to integer programming by Balas, Young, Glover, and others under the name of convexity cuts. This paper relates the question of finiteness of Tui's method to the so-called generalized lattice point problem of mathematical programming and gives a sufficient condition for terminating Tui's method. The paper then presents several branch-and-bound algorithms for solving concave minimization problems with linear constraints with the Tui cut as the basis for the algorithm. Finally, some computational experience is reported for the fixed-charge transportation problem.     

Transportation problems with some xij negative and transshipment problems

The general solution process of the Hitchcock transportation problem resulting from the application of the method of reduced matrices may give solutions with some negative x_ij values. This paper is devoted to a review of the reduced matrices method, an examination of suitable interpretation of sets of x_ij which include some negative values, and ways of interpreting these values in useful modifications of the Hitchcock problem. Such modifications include a) the reshipment problem, b) the overshipment problem, and c) the transshipment problem. Techniques are developed for determining and eliminating c_ij which are not optimal. These techniques and results are useful in solving the problems indicated above. The natural applicability of the simple and general method of reduced matrices is emphasized.     

Three ways of obtaining the average cost expression in a problem related to joint replenishment inventory control

This paper does not present a new result, rather it is meant to illustrate the choice of modelling procedures available to an analyst in a typical inventory control problem. The same “average cost per unit time” expression is developed by three quite different procedures. This variety of approaches, as well as the recounting of the author's chronological efforts to solve the problem, should be of interest to the reader. The specific inventory problem studied is one where the controller of an item is faced with random opportunities for replenishment at a reduced setup cost; the problem is an integral component of the broader problem of inventory control of a group of items whose replenishments are coordinated to reduce the costs of production, procurement, and/or transportation.     

An explicit general solution in linear fractional programming

A complete analysis and explicit solution is presented for the problem of linear fractional programming with interval programming constraints whose matrix is of full row rank. The analysis proceeds by simple transformation to canonical form, exploitation of the Farkas-Minkowki lemma and the duality relationships which emerge from the Charnes-Cooper linear programming equivalent for general linear fractional programming. The formulations as well as the proofs and the transformations provided by our general linear fractional programming theory are here employed to provide a substantial simplification for this class of cases. The augmentation developing the explicit solution is presented, for clarity, in an algorithmic format.     

Sequential bid selection by stochastic approximation

Suppose that a contractor is faced with a sequence of “minimum bid wins contract” competitions. Assuming that a contractor knows his cost to fulfill the contract at each competition and that competitors are merely informed whether or not they have won, bids may be selected sequentially via a tailored stochastic approximation procedure. The efficacy of this approach in certain bidding environments is investigated.     

Models for multi-item continuous review inventory policies subject to constraints

Models are formulated for determining continuous review (Q, r) policies for a multiitem inventory subject to constraints. The objective function is the minimization of total time-weighted shortages. The constraints apply to inventory investment and reorder workload. The formulations are thus independent of the normal ordering, holding, and shortage costs. Two models are presented, each representing a convex programming problem. Lagrangian techniques are employed with the first, simplified model in which only the reorder points are optimized. In the second model both the reorder points and the reorder quantities are optimized utilizing penalty function methods. An example problem is solved for each model. The final section deals with the implementation of these models in very large inventory systems.