Implementation and computational comparisons of primal, dual and primal‐dual computer codes for minimum cost network flow problems

Abstract

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:

Relaxation Methods for Minimum Cost Ordinary and Generalized Network Flow Problems

“…This conclusion agrees with empirical observations of others (e.g. [16]) as well as our own (see Section 7).…”

“…(16)] that the rate of change of the dual cost w at t along v is C(v,t). Since w is piecewise linear the actual change of w along the direction v is linear in the stepsize y up to the point where y becomes large enough so that the pair [w(t+yv), t+yv] meets a new face of the graph of w. This value of y is the one for which a new arc incident to S becomes balanced and it equals the scalar 6 of (20) Q.E.D.…”

“…the direction v corresponds to decreasing the price of each node i in S by an amount proportional to ui, while keeping the prices of all other nodes unchanged. From (5), (16) and (17) it is seen that the directional derivative of the dual cost at t along a vector v is' again given by…”

A primal simplex variant for the maximum‐flow problem

“…Many different pivot strategies were developed and tested during this study. This includes simple modifiL tions of the pivot strategies that have proven successful for more general network flow problems [20,21,36), as well as new strategies developed from insights into the special structure of the maximum flow 25 network problem. The level of complexity of these pivot strategies range from the simple sequential examination of forward stars to a complex candidate list with a steepest descent evaluation criteria.…”

Model Extension and Computation In Goal-Arc Network Approaches for Eeo Planning

“…It has been formulated in network form via the Goal-Arc formalisms that were introduced in [5]. This provides access to the highly efficient network codes such as PNET [8] or D. Klingman's CAPNET for computation and implementation of the very large models that are likely to be involved in dealing with all of the RNS categories in an integrated manpower-EEO planning system. This is a subject for further research.…”