The purpose of this paper is to describe the development, implementation, and availability of a computer program for generating a variety of feasible network problems together with a set of benchmarked problems derived from it. The code "NETGEN" can generate capacitated and uncapacitated transportation and minimum cost flow network problems, and assignment problems. In addition to generating structurally different classes of network problems the code permits the user to vary structural characteristics within a class. Problems benchmarked on several codes currently available are provided in this paper since NETGEN will also allow other researchers to generate identical problems. In particular, the latter part of the paper contains the solution time and objective function value of 40 assignment, transportation, and network problems varying in size from 200 nodes to 8,000 nodes and from 1,300 arcs to 35,000 arcs.
This paper examines different algorithms for calculating the shortest path from one node to all other nodes in a network. More specifically, we seek to advance the state‐of‐the‐art of computer implementation technology for such algorithms and the problems they solve by exmining the effect of innovative computer science list structures and labeling techniques on algorithmic performance.
The study shows that the procedures examined indeed exert a powerful influence on solution efficiency, with the identity of the best dependent upon the topology of the network and the range of the arc distance coefficients. The study further discloses, for the problems tested, that the lable‐setting shortest path algorithm previously documented as the most efficient is dominated for all problem structures examined by the new methods.
This paper develops a new polynomially bounded shortest path algorithm, called the partitioning shortest path (PSP) algorithm, for finding the shortest path from one node to all other nodes in a network containing no cycles with negative lengths. This new algorithm includes as variants the label setting algorithm, many of the label correcting algorithms, and the apparently computationally superior threshold algorithm.
This paper presents an in-depth computational comparison of the basic solution algorithms for solving transportation problems. The comparison is performed using "state of the art" computer codes for the dual simplex transportation method, the out-of-kilter method, and the primal simplex transportation method (often referred to as the Row-Column Sum Method or M O D I method). In addition, these codes are compared against a state of the art large scale LP code, O P H E L I E/LP. The study discloses that the most efficient solution procedure arises by coupling a primal transportation algorithm (embodying recently developed methods for accelerating the determination of basis trees and dual evaluators) with a version of the Row Minimum start rule and a "modified row first negative evaluator" rule. The resulting method has been found to be at least 100 times faster than OPHELIE, and 9 times faster than a streamlined version of the SHARE out-of-kilter code. The method's median solution time for solving 1000 \times 1000 transportation problems on a CDC 6600 computer is 17 seconds with a range of 14 to 22 seconds. Some of the unique characteristics of this study are (1) all of the fundamental solution techniques are tested on the same machine and the same problems, (2) a broad spectrum of problem sizes are examined, varying from 10 \times 10 to 1000 \times 1000; (3) a broad profile of nondense problems are examined ranging from 100 percent to 1 percent dense; and (4) additional tests using the best of the codes have been made on three other machines (IBM 360/65, UNIVAC 1108, and CDC 6400), providing surprising insights into conclusions based on comparing times on different machines and compilers.
The augmented predecessor indexing method provides an efficient way to update the basis and cost evaluators in adjacent extreme point solution methods for transportation problems. The method extends the predecessor indexing method, which was designed to exploit an ancestry relation in the basis tree, by showing how to exploit a more elaborate “family relation,” commonly known as a binary tree representation in computer list processing. This representation, sometimes called the “triple-label method,” was first suggested for application to network optimization problems by Ellis Johnson, who examined its use in the context of maximal flow problems. The augmented predecessor indexing method shows how to take special advantage of this representation in the context of the transportation problem, elaborating Johnson's work by providing an efficient method for characterizing successive basis trees with minimal relabeling. Moreover, we show how this procedure can be applied to update the transportation cost evaluators simultaneously without calculating current node potentials (as ordinarily required in the “stepping-stone” and related basis exchange algorithms), thus greatly speeding the arithmetic calculations.
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