S. Thomas McCormick

Sequencing in an assembly line with blocking to minimize cycle time

We consider an assembly line with m stations in series having finite capacity buffers. Blocking occurs when buffers are full. There are M different types of products to be assembled, each with its own processing requirements. There is a production target set for each type. The problem is to operate the line to maximize throughput. We propose heuristic approaches to this problem based on an equivalent maximum flow problem and on critical path techniques.

Finding disjoint paths with different path-costs: Complexity and algorithms

Consider a network G = (V,E) with distinguished vertices s and t, and with k different costs on every edge. We consider the problem of finding k disjoint paths from s to t such that the total cost of the paths is minimized, where the j th edge-cost is associated with the j th path. The problem has several variants: The paths may be vertex-disjoint or arcdisjoint and the network may be directed or undirected. We show that all four versions of the problem are strongly NP-complete even for k = 2. We describe polynomial time heuristics for the problem and a polynomial time algorithm for the acyclic directed case.

Submodular Function Minimization

Abstract This chapter describes the submodular function minimization problem (SFM); why it is important; techniques for solving it; algorithms by Cunningham, by Schrijver as modified by Fleischer and Iwata, by Iwata, Fleischer and Fujishige, and by Iwata for solving it; and extensions of SFM to more general families of subsets.

Polynomial Methods for Separable Convex Optimization in Unimodular Linear Spaces with Applications

We consider the problem of minimizing a separable convex objective function over the linear space given by a system Mx=0 with M a totally unimodular matrix. In particular, this generalizes the usual minimum linear cost circulation and cocirculation problems in a network and the problems of determining the Euclidean distance from a point to the perfect bipartite matching polytope and the feasible flows polyhedron. We first show that the idea of minimum mean cycle canceling originally worked out for linear cost circulations by Goldberg and Tarjan [J. Assoc. Comput. Mach., 36 (1989), pp. 873--886.] and extended to some other problems [T. R. Ervolina and S. T. McCormick, Discrete Appl. Math., 46 (1993), pp. 133--165], [A. Frank and A. V. Karzanov, Technical Report RR 895-M, Laboratoire ARTEMIS IMAG, Universite Joseph Fourier, Grenoble, France, 1992], [T. Ibaraki, A. V. Karzanov, and H. Nagamochi, private communication, 1993], [M. Hadjiat, Technical Report, Groupe Intelligence Artificielle, Faculte des Sciences de Luminy, Marseille, France, 1994] can be generalized to give a combinatorial method with geometric convergence for our problem. We also generalize the computationally more efficient cancel-and-tighten method. We then consider objective functions that are piecewise linear, pure and piecewise quadratic, or piecewise mixed linear and quadratic, and we show how both methods can be implemented to find exact solutions in polynomial time (strongly polynomial in the piecewise linear case). These implementations are then further specialized for finding circulations and cocirculations in a network. We finish by showing how to extend our methods to find optimal integer solutions, to linear spaces of larger fractionality, and to the case when the objective functions are given by approximate oracles.

Incentives for Transshipment in a Supply Chain with Decentralized Retailers

We examine transshipment incentives in a decentralized supply chain where a monopolist distributes a product through independent retailers. A key insight is that the transshipment price determines whether the firms benefit from, or are hurt by, transshipment. In particular, we show that the manufacturer prefers to set the transshipment price as high as possible, whereas retailers prefer a lower transshipment price. Given the important role of the transshipment price in determining the benefits that each firm gets from transshipment, it is useful to consider transshipment in the case where retailers are under joint ownership (a “chain store”) and the transshipment price does not play a role. This comparison yields two surprising results. First, if decentralized retailers control the transshipment price, they will choose a relatively low transshipment price as a way to mitigate the manufacturers ability to extract profits by increasing wholesale prices; therefore, the manufacturer may prefer dealing with the chain store, which does not have a transshipment price, rather than with decentralized retailers. Similarly, the decentralized retailers can use a low transshipment price to achieve higher total profits than a chain store.

Scheduling n Independent Jobs on m Uniform Machines with both Flowtime and Makespan Objectives: A Parametric Analysis

We consider the problem of scheduling n jobs without precedence constraints on m uniform machines (i.e., the machines are identical except for speed), with preemptions allowed at no cost. We are interested in generating the entire tradeoff curve of schedules which are Pareto-optimal (undominated) for the flowtime and makespan objectives. To achieve this, we first develop an O(mn) algorithm that produces a schedule with minimum flowtime, subject to a fixed makespan deadline. This algorithm alternates between the Shortest Processing Time on Fastest Machine (SPT-FM) rule and the Longest Remaining Processing Time on Fastest Machine (LRPT-FM) rule. We then investigate how the behavior of the algorithm changes as the deadline is varied parametrically. Our knowledge of the structure of optimal schedules allows us to characterize breakpoints on the (piecewise linear) tradeoff curve, and then to compute all of the O(mn) breakpoints in O(m 3 n) time. Our analysis yields various useful sensitivity results as a by-product. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

The point-to-point delivery and connection problems: complexity and algorithms

Abstract We consider the computational complexity of point-to-point delivery problems. These problems involve shipping one item from each one of p sources to p destinations might be prematched to sources (the fixed destination case), or a sources item might go to any destination (the nonfixed destination case). The networks can be directed or undirected. Up to K items at once can share a truck on an arc, and costs are linear in the number of trucks used. We also consider the closely related point-to-point connection problems, which are to find a minimum cost arc subset connecting sources with destinations. We find that all variations of both problems are strongly NP-hard for all K≤2, but that there are polynomial algorithms in some cases if p is fixed, or if the underlying network is a grid with sources on one side, destinations on the other.

A Polynomial Algorithm for Multiprocessor Scheduling with Two Job Lengths

The following multiprocessor scheduling problem was motivated by scheduling maintenance periods for aircraft. Each maintenance period is a job, and the maintenance facilities are machines. In this context, there are very few different types of maintenances performed, so it is natural to consider the problem with only a small, fixed number C of different types of jobs. Each job type has a processing time, and each machine is available for the same length of time. A machine can handle at most one job at a time, all jobs are released at time zero, there are no due dates or precedence constraints, and preemption is not allowed. The question is whether it is possible to finish all jobs. We call this problem the Multiprocessor Scheduling Problem with C job lengths MSPC. Scheduling problems such as MSPC where we can partition the jobs into relatively few types such that all jobs of a certain type are identical are often called high-multiplicity problems. High-multiplicity problems are interesting because their input is very compact: The input to MSPC consists of only 2C + 2 numbers. For the case C = 2 we present a polynomial-time algorithm. We show that this algorithm guarantees a schedule that uses the minimum possible number of different one-machine schedules, namely three. Further, we extend this algorithm to the case of machine-dependent deadlines uniform parallel machines, to a multi-parametric case that contains the case of unrelated parallel machines, and to some related covering problems. Finally, we give some counterexamples showing why our results do not extend to the case C > 2.

Two strongly polynomial cut cancelling algorithms for minimum cost network flow

Abstract We present two new strongly polynomial algorithms for the minimum cost network flow problem (MCNF). They are dual algorithms based on cancelling positive augmenting cuts, which are the duals of negative augmenting cycles. The first cancels maximum mean cuts, which are cuts whose increase in the dual objective function per arc is maximum. The second, Dual Cancel and Tighten, employs a more flexible cut selection rule that allows it to be more efficient. These algorithms are duals to the Minimum Mean Cycle Cancelling and (Primal) Cancel and Tighten algorithms of Goldberg and Tarjan. These algorithms do not use explicit scaling to achieve polynomiality.

Strongly polynomial and fully combinatorial algorithms for bisubmodular function minimization

Bisubmodular functions are a natural “directed”, or “signed”, extension of submodular functions with several applications. Recently Fujishige and Iwata showed how to extend the Iwata, Fleischer, and Fujishige (IFF) algorithm for submodular function minimization (SFM) to bisubmodular function minimization (BSFM). However, they were able to extend only the weakly polynomial version of IFF to BSFM. Here we investigate the difficulty that prevented them from also extending the strongly polynomial version of IFF to BSFM, and we show a way around the difficulty. This new method gives a somewhat simpler strongly polynomial SFM algorithm, as well as the first combinatorial strongly polynomial algorithm for BSFM. This further leads to extending Iwata’s fully combinatorial version of IFF to BSFM.