Optimal service and arrival rates in Jackson queueing networks

In this paper we present an algorithm for solving a class of queueing network design problems. Specifically, we focus on determining both service and arrival rates in an open Jackson network of queueing stations. This class of problems has been widely studied and used in a variety of applications, but not well solved due to the difficulty of the resulting optimization problems. As an example, consider the classic application in computer network design which involves determining the minimum cost line capacities and flow assignments while satisfying a queueing performance measure such as an upper limit on transmission delay. Other application areas requiring the selection of both service and arrival rates in a network of queues include the design of communication, manufacturing, and health care systems. These applications yield optimization problems that are difficult to solve because typically they are nonconvex, which means they may have many locally optimal solutions that are not necessarily globally optimal. Therefore, to obtain a globally optimal solution, we develop an efficient branch and bound algorithm that takes advantage of the problem structure. Computational testing on randomly generated problems and actual problems from a health care organization indicate that the algorithm is able to solve realistic sized problems in reasonable computing time on a laptop computer. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 1–17, 2000     

Solving the generalized knapsack problem with variable coefficients

In this article we present methods based on Lagrangian duality and decomposition techniques for the generalized knapsack problem with variable coefficients. The Lagrangian dual is solved with subgradient optimization or interval bisection. We also describe a heuristic that yields primal feasible solutions. Combining the Lagrangian relaxation with a primal (Benders) subproblem yields the subproblem phase in cross decomposition. By using averages in this procedure, we get the new mean-value cross-decomposition method. Finally, we describe how to insert this into a globally convergent generalized Benders decomposition framework, in the case that there is a duality gap. Encouraging computational results for the optimal generating unit commitment problem are presented. © 1996 John Wiley & Sons, Inc.     

A penalty for concave minimization derived from the tuy cutting plane

A wide variety of optimization problems have been approached with branch-and-bound methodology, most notably integer programming and continuous nonconvex programming. Penalty calculations provide a means to reduce the number of subproblems solved during the branch-and-bound search. We develop a new penalty based on the Tuy cutting plane for the nonconvex problem of globally minimizing a concave function over linear constraints and continuous variables. Computational testing with a branch-and-bound algorithm for concave minimization indicates that, for the problems solved, the penalty reduces solution time by a factor ranging from 1.2 to 7.2. © 1994 John Wiley & Sons, Inc.     

An algorithm and new penalties for concave integer minimization over a polyhedron

We present a branch-and-bound algorithm for globally minimizing a concave function over linear constraints and integer variables. Concave cost functions and integer variables arise in many applications, such as production planning, engineering design, and capacity expansion. To reduce the number of subproblems solved during the branch-and-bound search, we also develop a framework for computing new and existing penalties. Computational testing indicates that penalties based on the Tuy cutting plane provide large decreases in solution time for some problems. A combination of Driebeek-Tomlin and Tuy penalties can provide further decreases in solution time. © 1994 John Wiley & Sons, Inc.     

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.     

Automated air load planning

We describe the development of a heuristic algorithm for determining efficient 2-dimensional packings in cargo aircraft where cargo placement constraints are critically important in determining the feasibility of packing locations. We review the performance of a new algorithm versus some traditional ones for aircraft loading. The algorithm is also tested in a more generalized setting where there exist no additional constraints on items, to suggest applicability in other environments. The new algorithm has been used worldwide in the Automated Air Load Planning System (AALPS) for cargo aircraft loading, with much success. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 751–768, 1998