John G. Klincewicz

Solving a freight transport problem using facility location techniques

Use of consolidation terminals to transport products from various sources to various destinations can take advantage of economies of scale in transportation costs. Instead of making direct shipments, each source can ship in bulk to one or more consolidation terminals. There, shipments can be broken down, and material bound for the same destination can be combined. We consider such a freight transport problem FTP. For each source-destination pair, it must be decided whether to ship the product directly or via a consolidation terminal. Shipping costs are piecewise linear concave functions of the volume shipped. Shipping via a terminal can also incur a linear inventory holding cost. We seek a minimum cost pattern of direct and indirect i.e., via a terminal shipments. This is a type of concave cost multiproduct network flow problem. We can solve this problem optimally if either the source-to-terminal or the terminal-to-destination shipping costs are linear. In this case, FTP decomposes into a set of concave cost facility location problems CFLP. In more general cases, heuristic methods that solve sequences of linear problems can be used. Some computational results are presented.

A Newton method for convex separable network flow problems

Previous feasible direction algorithms for convex, separable network flow problems that have appeared in the literature have all been linearly convergent. In this paper we propose an approximate implementation of the quadratically convergent Newton algorithm. The key to this implementation is a conjugate direction method that determines second-order dual multiplier estimates at each iteration. This method exploits the network structure in a way that allows certain crucial matrix-vector products to be computed in a simple pass through the arcs. Since this novel approach requires no matrix storage to compute these products, the algorithm can be used for large-scale problems. Some computational experience with this new algorithm is reported.

A Dual-Based Algorithm for Multiproduct Uncapacitated Facility Location

The multiproduct uncapacitated facility location problem (MUFLP) is a generalization of the classic uncapacitated facility location problem (UFLP). In MUFLP, different products are required by the customers. In addition to the fixed cost for opening a facility, there is an added fixed cost for handling a particular product. Assignment costs are incurred for satisfying a customers requirement for each of the separate products. The objective is to minimize the total fixed costs plus assignment costs subject to satisfying all customer requirements. We propose a new dual-based algorithm for MUFLP that extends previous work for UFLP that has proved so successful. Dual ascent and dual adjustment procedures generate a good feasible solution to the dual of the linear programming relaxation of MUFLP. A feasible primal solution to MUFLP can then be constructed based on this dual solution. Like the analogous procedures for UFLP, these procedures can be used either as a stand-alone heuristic or, else, they can be incorporated into a branch-and-bound algorithm. Computational experience is given for both randomly generated and nonrandom problems with quite favorable results.

A large-scale multilocation capacity planning model

Abstract In this paper, a large-scale multilocation capacity planning model is described. The model chooses a multiperiod schedule of openings, expansions, and closings of facilities, and assigns demand locations to these facilities. Although generic in nature, this model was developed to plan the evolution of material logistics systems over time. In order to have a truly practical tool, numerous features are considered including existing configuration, arbitrary demand patterns, concave operating costs, single-source assignments, demand location reassignment costs, and others. Such capacity planning models are highly combinatorial in nature, and are solved, in general, by heuristics. Our solution method has three major modules. First, an initial solution is generated by solving successively single-period problems using network optimization techniques complemented by other heuristics. Next, opening and closing decisions are adjusted and improved. Finally, demand location assignment decisions throughout the planning horizon are modified. The heuristic was tested on many problems of various sizes; computational experience is described.

Optimal timing decisions for the introduction of new technologies

Abstract High technology industries, such as the communications industry, are characterized by frequent development of new technologies. These new technologies are often available before the capacities of existing facilities that use an old technology are exhausted. Whenever a new technology facility is introduced, a fixed set-up cost is generally incurred; however, the annual operating costs are often reduced. The optimal timing of the introduction of new facilities is therefore of interest. In this paper, we examine such timing decisions. The study was motivated by an application involving electronic plug-in units that enhance the operation of communication facilities. First, we develop optimal timing decisions for linearly growing demand. The analysis is then extended to nonlinear demand. For linear demand, one of two decisions is optimal: Either introduce the new technology immediately, or as late as possible. However, for nonlinear demand, these decisions may be nonoptimal.

THE AIRLINE EXCEPTION SCHEDULING PROBLEM

As part of its schedule planning task, a domestic airline must assign fleets (aircraft types) to legs (non-stop flight segments). Initially, for planning purposes, this fleet assignment is done for a daily, repetitive schedule, called a “skeleton” schedule, in which the same set of legs is assumed to fly daily. In practice, however, there are often significant changes in passenger demand patterns on weekends (and other days as well), warranting changes in scheduled legs and fleet assignments. Thus, the airline will include, in its actual schedule, certain variations or changes in the nominal “skeleton” schedule on particular days of the week. We refer to such changes as exceptions. A typical domestic airline may vary the fleet assignment of over 10% of its legs on the weekend alone. Thus, handling exceptions is an important airline scheduling function. We describe an approach, centered around formulating and solving a network flow problem, for efficiently scheduling such exceptions to the usual daily schedule. In addition, another procedure was developed to identify and suggest possible profitable exceptions. It implements graph theoretic methods to detect “cycles” of legs whose fleet assignment is unprofitable on a particular day. Some computational experience is discussed.

On cycling in the network simplex method

There are well-known examples of cycling in the linear programming simplex method having basis size two and requiring only six pivots. We prove that any example having basis size two for the network simplex method requires at least ten pivots. We also present an example that achieves this lower bound. In addition, we show that an attractive variant of Cunninghams noncyling method does admit cycling.

Dealing with degeneracy in reduced gradient algorithms

Many algorithms for linearly constrained optimization problems proceed by solving a sequence of subproblems. In these subproblems, the number of variables is implicitly reduced by using the linear constraints to express certain ‘basic’ variables in terms of other variables. Difficulties may arise, however, if degeneracy is present; that is, if one or more basic variables are at lower or upper bounds. In this situation, arbitrarily small movements along a feasible search direction in the reduced problem may result in infeasibilities for basic variables in the original problem. For such cases, the search direction is typically discarded, a new reduced problem is formed and a new search direction is computed. Such a process may be extremely costly, particularly in large-scale optimization where degeneracy is likely and good search directions can be expensive to compute. This paper is concerned with a practical method for ensuring that directions that are computed in the reduced space are actually feasible in the original problem. It is based on a generalization of the ‘maximal basis’ result first introduced by Dembo and Klincewicz for large nonlinear network optimization problems.