H. Donald Ratliff

Finding the n Most Vital Links in Flow Networks

The n most vital links of a flow network are defined as those n arcs whose simultaneous removal from the network causes the greatest decrease in the throughput capability of the remaining system between a specified pair of nodes. These n arcs are shown to be the n largest capacity arcs in a particular cut. A solution procedure is developed which involves sequentially modifying the network so as to make this cut eventually become the cut with smallest capacity. An algorithm with computational results is presented.

A Cut Approach to the Rectilinear Distance Facility Location Problem

This paper is concerned with the problem of locating n new facilities in the plane when there are m facilities already located. The objective is to minimize the weighted sum of rectilinear distances. Necessary and sufficient conditions for optimality are established. We show that the optimum locations of the new facilities are dependent on the relative orderings of old facilities along the two coordinate axes but not on the distances between them. Based on these results an algorithm is presented that requires the solution of at most m-1 minimum cut problems on networks with at most n + 2 vertices. All of these results are easily extended to the same location problem on a tree graph.

A Network-Flow Model for Racially Balancing Schools

This paper is concerned with developing a procedure for assigning students to public schools optimally, given that a specified racial balance must be attained in each school. The criterion for optimality is to minimize the total number of miles traveled. The problem is formulated as a minimum-cost flow problem in a single-commodity network. A summary of the results achieved by the model for the Gainesville, Florida, school system is discussed.

A Graph-Theoretic Equivalence for Integer Programs

This paper is concerned with the relation between 0-1 integer programs and graphs. An equivalence is established between solving 0-1 integer programs with quadratic or linear objective functions and solving a cut problem on a related graph.

Network Models for Production Scheduling Problems with Convex Cost and Batch Processing

Abstract This paper considers a class of production scheduling problems which can be modeled as network flow problems. The problems are addressed under the assumption that production occurs in batches. We also require that the cost function be convex and separable. The model applies for a number of well known production scheduling problems and will handle multiple products and multiple facilities.

Scheduling Rules for Parallel Processors.

Abstract This paper investigates various criteria under which simple scheduling rules generate optimum schedules for m machine scheduling problems. The problems are considered under a “job-splitting” assumption. Some of the rules which generate optimum schedules with job splitting for the multi-machine problems, do not split jobs for the single machine problems. The relationship between the job splitting and no job splitting problem in these cases, is analogous to the situation when an optimum solution to a linear program has all the variables integer and hence is optimum to the corresponding integer program.