Derek R. Atkins

Bicriteria linear fractional programming

As a step toward the investigation of the multicriteria linear fractional program, this paper provides a thorough analysis of the bicriteria case. It is shown that the set of efficient points is a finite union of linearly constrained sets and the efficient frontier is the image of a finite number of connected line segments of efficient points. A simple algorithm using only one-dimensional parametric linear programming techniques is developed to evaluate the efficient frontier.

An interactive algorithm for multicriteria programming

Abstract An interactive algorithm for general multicriteria programming is proposed. The decision maker merely selects his most preferred point amongst presented alternatives. The algorithm ensures that the alternatives generated in each iteration are evenly distributed over the desired neighbourhood of the efficient frontier. The algorithm was motivated by the need for solutions to the multiple linear fractional problem and is shown to have a particularly simple and easy to implement structure in this case. A discussion is included of why this problem is important and of how it arises in practice, particularly in financial problems.

Newsvendors Under Simultaneous Price and Inventory Competition

This paper extends the theory of N competitive newsvendors to the case where competition occurs simultaneously in price and inventory. The basic research questions are whether the Nash equilibrium exists in this game, whether it is unique, and how the resulting inventories and prices are affected by competition. Using a novel method, we show the quasiconcavity of the competitive newsvendors problem and establish the existence of the pure-strategy Nash equilibrium. Through a contraction mapping approach, we develop sufficient conditions for the Nash equilibrium to be unique. We then analyze the properties of the equilibrium and compare it with the optimal solution for the (noncompeting) price-sensitive newsvendor. We prove that at a symmetric equilibrium, retail prices and safety stocks strictly increase with the proportion of a newsvendors unsatisfied customers that switch to a competitor, but strictly decrease with the intensity of price competition. Total inventories, on the other hand, increase with the intensity of price competition. Furthermore, the competitive equilibrium never has lower safety stocks and higher retail prices (a situation that definitely hurts the customers) than the solution for noncompetitive newsvendors.

Proper Efficiency in Nonconvex Multicriteria Programming

Proper efficient solutions of nonconvex vector maximum problems can be generated by solving a parametric family of ordinary nonlinear programs. This parametric scheme follows from the characterization of proper efficiency by an extended form of the generalized Tchebycheff norm.

98%-Effective Lot Sizing for Series Inventory Systems with Backlogging

For deterministic series inventory systems with backlogging, we show that the continuous relaxation of the best integer frequency policies is a lower bound on all feasible policies. The problem with backlogging can be reduced, in On time, to an equivalent problem without backlogging. Therefore, based on the result of R. O. Roundy Roundy, R. O. 1983. 94%-Effective lot-sizing in multistage assembly systems. Technical Report 674, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York., the best integer frequency policy provides a solution which is within 2% of the optimal.

A lower bound on a class of coordinated inventory/production problems

A new lower bound is given for a class of coordinated multi-product inventory problems. The bound has considerable generality and so far shows promise to be good. This promise includes duplicating two recent excellent results from the literature.

Lot sizing policies for finite production rate assembly systems

An assembly production system with n facilities has a constant external demand occurring at the end facility. Production rates at each facility are finite and nonincreasing along any path in the assembly network. Associated with each facility are a setup cost and positive echelon holding cost rate. The formulation of the lot sizing problem is developed in terms of integer-ratio lot size policies. This formulation provides a unification of the integer-split policies formulation of L. B. Schwarz and L. Schrage and the integer-multiple policies formulation of J. P. Moily, allowing either assumption to be operative at any point in the system. A relaxed solution to this unified formulation provides a lower bound to the cost of any feasible policy. The derivation of this Lower Bound Theorem is novel and relies on the notion of path holding costs, which is a generalization of echelon holding costs. An optimal power-of-two lot size policy is found by an On5 algorithm and its cost is within 2% of the optimum in the worst case.

On the effect of demand randomness on a price/quantity setting firm

Replenishment and pricing decisions are of great importance to firms. Traditionally replenishment and pricing strategies are determined by separate units of a firm, the former by production and the latter by marketing. In determining these two strategies, firms frequently have to face two challenges. One stems from a lack of coordination between production and marketing, which leads to revenue loss or excess inventory. The other is caused by lack of information when making these two decisions. That is, firms usually have to make decisions when market information is poor. We consider the joint effects of coordination and information when market demand becomes more variable in the sense of a specific mean preserving transformation. We have three main findings. First, when these two strategies are coordinated by the Headquarters (HQ) of the firm, the HQ uses pricing and inventory as two instruments to manage uncertainty. How exactly increasing demand variability affects the decisions depends on the type of demand uncertainty faced. In particular, for additive demand uncertainty, both price and service level decrease in demand variability, whereas for multiplicative demand uncertainty, they both increase in demand variability. Second, the value of information increases with the level of demand variability, suggesting that it is more beneficial to have information when demand is more variable. The impact of demand variability on the value of coordination, however, is indeterminate. Furthermore, perfect information has more value than perfect coordination if and only if demand variability is high. Third, coordinating these two decisions reduces the value of obtaining the demand information and similarly coordination is more valuable when the demand information is unavailable than otherwise.

A survey of lower bounding methodologies for production/inventory models

A review is undertaken of lower bounding methodologies for lot-sizing production/inventory models. This is done for both dynamic discrete time demand and continuous review, constant demand situations. The view is taken that lower bounds are an essential ingredient in the evaluation of heuristics when optima are unknown or too computationally complex to comfortably evaluate. Several directions of future research are suggested.

A simple lower bound to the dynamic assembly problem

Abstract We consider the multi-stage assembly problem where many components are assembled to make a final product. Demand is dynamic but known over a finite horizon. Optimal solutions are available but only with considerable computations. To properly evaluate heuristic solutions, good lower bounds with relatively low computational requirements are needed. We describe a very simple bound and illustrate that despite its simplicity it compares reasonably with more sophisticated bounds. The bound readily admits of many improvements and generalisations.