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 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.

ABSTRACT. The critical importance of integrating production, distribution, and inventory (PDI) operations has long been recognized by top management of many companies. Now. using (he laiest advances in Managemenl Science modeling and solution technology, an integruied computer-based PDI system has saved approximately $18 million dollars during its first three years of implemenlalion for a major national firm. Agrico Chemical Company. According to ihe Vice-Presideni of Agrico Supply and Dislribulion. an addilional $25 million savings is anticipated over the next two years.Broughi about by close cooperation between company officiais and an outside staff of Management Science consultants, the PDI system has been used extensively to evaluate the benefit/cost impact of alternative capital investments in both shon-term and long-term planning decisions. The development of the system underscores the value of recent Management Science innovations that have made it possible to analyze interacting influences too numerous and complex to be analyzed adequately only a few years ago.Advanced network methodology incorporated inio Ihe PDI system required onl> one one-hundredth of the computer lime and cost of meihodologies previously used. The power and flexibility of the new Management Science tools have also brought about increased communication and understanding of key company operations. This increased communication and understanding stems from ihe inherent '"pictorial" nature of network-based models, which facilitates interpretation of these models and policy recommendations based upon their solution. IN VENTORY/PRODUCTION; NETWORKS/GRAPHS INTERFACES November 1979 21 OverviewAgrico Chemical Company, with annual sales exceeding half a billion dollars, is one of the nation's largest chemical fertilizer companies. A subsidiary of The Williams Companies. Agrico mines, manufactures, and markets eight principal chemical products domestically and internationally. The company's success story, based on aggressive and forward-looking management, is typical of others in which a relatively small firm has been transformed into a leader in its field in less than a decade.In the mid-1970's Agrico encountered unexpected difficulties. The seasonal demand characteristic of the chemical fertilizer industry was creating a chain of intricate and far reaching effects that could not be responded to adequately. As a result, the company's profit margins were being seriously eroded by steeply escalating distribution costs. It became apparent that a multitude of interdependent factors made it impossible to find a remedy through customary metbods, such as studying cost figures and charts. Tbe web of interacting ihfluences wbich spanned tbe company's principal activities -production, distribution, and inventory -required an integrated computer-based planning system to uncover the appropriate decisions.In 1976. David Wilson. Vice-President of Agrico Supply and Distribution, in coordination with Herb Beattie, Vice-President of The Williams Companies Info...

The perfomnance i sThe covputational characteristics of several The study disc2oses the advantages, in both computationOver t h e years a g r e a t deal of code development and comput a t i o n a l t e s t i n g [1,2,4,5,7,9,12,13,17,19,20,21,22,22,23,25,27,281 has been performed on transportation and m i n i m u m c o s t flow network problems. Primarily t h i s development and t e s t i n g has been confined t o transportation codes, and most of the mor'e recently developed codes (since 1960) a r e based on primal-dual algorithms of the out-of-kilter genre [1,2,4,7,9,17,23,25,281. Surprisingly, t h e l i t e r a t u r e does not seem t o report any code development o r computational t e s t i n g of a simplex based m i n i m u m c o s t flow network code. N o doubt, t h i s is due i n p a r t t o the conclusion of Scope of the Computational Analysis Networks, 4:

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This paper develops a primal simplex procedure to solve transshipment problems with an arbitrary additional constraint. The procedure incorporates efficient methods for pricing-out the basis, determining certain key vector representations, and implementing the change of basis. These methods exploit the near triangularity of the basis in a manner that takes advantage of computational schemes and list structures used to solve the pure transshipment problem. We have implemented these results in a computer code, I/O PNETS-I. Computational results (necessarily limited) confirm that this code is significantly faster than APEX-III on some large problems. We have also developed a fast method for determining near optimal integer solutions. Computational results show that the near optimum integer solution value is usually within 0.5% of the value of the optimum continuous solution value.

This paper presents extensive computational experience with a special-purpose primal simplex code using the augmented threaded index method for solving capacitated and uncapacitated transshipment and transportation problems. This code is distinguished from other codes for solving such problems in that not all of the data resides in central memory simultaneously; thus, it is referred to as an in-core, out-of-core code. The major advantages of such a code over an in-core code are (1) it can solve problems that the latter can not solve because of central memory requirements; (2) even for problems that the latter can solve, it requires less central memory, which is critical for fast job processing on multiprogrammed computer systems; and (3) the code can also be used as an in-core code. The design of such codes presents numerous computational difficulties in selecting the best starting and pivot procedures in order to minimize central as well as peripheral processing time. We present computational experience with various pivot, start and capacity buffer procedures, as well as different buffer sizes. Computational results are also provided for different types of network problems, including assignment, transportation, and minimum cost flow problems. These computational results are compared with in-core, out-of-kilter, negative cycle, and primal simplex network codes for problem sizes that these codes could solve.