Coordinated replenishments from multiple suppliers with price discounts

In this study we present an integer programming model for determining an optimal inbound consolidation strategy for a purchasing manager who receives items from several suppliers. The model considers multiple suppliers with limited capacity, transportation economies, and quantity discounts. We propose an integrated branch and bound procedure for solving the model. This procedure, applied to a Lagrangean dual at every node of the search tree, combines the subgradient method with a primal heuristic that interact to change the Lagrangean multipliers and tighten the upper and lower bounds. An enhancement to the branch and bound procedure is developed using surrogate constraints, which is found to be beneficial for solving large problems. We report computational results for a variety of problems, with as many as 70,200 variables and 3665 constraints. Computational testing indicates that our procedure is significantly faster than the general purpose integer programming code OSL. A regression analysis is performed to determine the most significant parameters of our model. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 579–598, 1998     

A specially structured nonlinear integer resource allocation problem

We present an algorithm for solving a specially structured nonlinear integer resource allocation problem. This problem was motivated by a capacity planning study done at a large Health Maintenance Organization in Texas. Specifically, we focus on a class of nonlinear resource allocation problems that involve the minimization of a convex function over one general convex constraint, a set of block diagonal convex constraints, and bounds on the integer variables. The continuous variable problem is also considered. The continuous problem is solved by taking advantage of the structure of the Karush‐Kuhn‐Tucker (KKT) conditions. This method for solving the continuous problem is then incorporated in a branch and bound algorithm to solve the integer problem. Various reoptimization results, multiplier bounding results, and heuristics are used to improve the efficiency of the algorithms. We show how the algorithms can be extended to obtain a globally optimal solution to the nonconvex version of the problem. We further show that the methods can be applied to problems in production planning and financial optimization. Extensive computational testing of the algorithms is reported for a variety of applications on continuous problems with up to 1,000,000 variables and integer problems with up to 1000 variables. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 770–792, 2003.