A Machine-Job Scheduling Model

This paper makes use of a device proposed by Fulkerson and Ford for maximal multi-commodity network flows (Ford, L. R. Jr., D. R. Fulkerson. 1958. A suggested computation for maximal multi-commodity network flows. The RAND Corporation, Paper P-1114, March 27, 1958. Also Management Sci. 5(1) 97-101.). A machine-job scheduling model is formulated that avoids the usual in-process inventories and thereby has fewer equations. The activities are the possible sequence for each job of machine steps and interspersed delays; this, of course, greatly multiplies the number of activities. However, a compact network representation of the set of possible activities is developed which permits generating just the activity to enter the basis on each iteration without explicitly generating the others. This is done by using one of the efficient procedures that now exist for computing the shortest route through a network (Dantzig, George B. 1960. On the shortest route through a network. Management Sci. 6(2); Moore, E. F. The shortest path through a maze. Unpublished mimeographed report, 16 pages; Bellman, Richard. 1958. On a routing problem. JORSA, XVI(1) 87-90.), (Ford, L. R. Jr. 1956. Network flow theory. The RAND Corporation, Paper P-923).

A Control Problem of Bellman

The control problem discussed in this paper is a variant of one considered by Bellman in a seminar at the RAND Corporation. A solution was presented to the seminar by the author in October 1952 based on the idea of placing a "loose" string between end points and "pulling tight." Recently, Arthur Veinott has greatly extended the class of problems which admit a "string" solution. It appeared of value that the author publish his original notes on Bellman's problem. The problem will be considered here in a discrete version. The reader should have no difficulties developing its continuous analogue.

The ORSA New Orleans Address on Compact City

The failure of urban planners, politicians, and bureaucrats to cope with urban crises and come up with solutions that work, stems from a fundamental inability to deal with complexity. The urban environment is complex, inter-related, highly dynamic. Nothing less than a total-system approach using the tools of mathematical and computer simulation models can hope to attack the problems of urban over growth. It is important to develop analytic, computer and simulation type models of the total urban system and of detailed subsystems in order to compare various arrangements such as the current megalopolis with its sprawl, and new satelite towns and cities built to make more effective use of vertical space and the round-the-clock use of facilities. Cities that make effective use of the vertical dimension and time, represent a viable alternative and should be given serious consideration.

Linear Programming Under Uncertainty

No abstract available.

Recent Advances in Linear Programming

As interest grows rapidly in industry on the potentialities of mathematical programming techniques, it appears worthwhile to have a paper devoted to some of the more promising developments which may speed up the transition from interest to use. Three topics have been selected (in three sections that follow) which have recently come into prominence: uncertainty, combinatorial problems, and large scale systems. The reader will find in the course of their discussions that a survey--though perhaps not a systematic survey--has been made of current techniques in the linear programming field.