Parametric Ghost Image Processes for Fixed-Charge Problems: A Study of Transportation Networks

Abstract

We present a parametric approach for solving fixed-charge problems first sketched in Glover (1994). Our implementation is specialized to handle the most prominently occurring types of fixed-charge problems, which arise in the area of network applications. The network models treated by our method include the most general members of the network flow class, consisting of generalized networks that accommodate flows with gains and losses. Our new parametric method is evaluated by reference to transportation networks, which are the network structures most extensively examined, and for which the most thorough comparative testing has been performed. The test set of fixed-charge transportation problems used in our study constitutes the most comprehensive randomly generated collection available in the literature. Computational comparisons reveal that our approach performs exceedingly well. On a set of a dozen small problems we obtain ten solutions that match or beat solutions found by CPLEX 9.0 and that beat the solutions found by the previously best heuristic on 11 out of 12 problems. On a more challenging set of 120 larger problems we uniformly obtain solutions superior to those found by CPLEX 9.0 and, in 114 out of 120 instances, superior to those found by the previously best approach. At the same time, our method finds these solutions while on average consuming 100 to 250 times less CPU time than CPLEX 9.0 and a roughly equivalent amount of CPU time as taken by the previously best method.

Fixed Charge Transportation Problems: a new heuristic approach based on Lagrangean relaxation and the solving of core problems

“…It is applied along the subgradient algorithm, and takes elements of other known heuristics such as the dynamic slope scaling procedure (Kim and Pardalos 1999) and the parametric ghost image processes (Glover et al 2005). Let x ∈ X(S) be the solution at the current iteration, let x ij be the maximum value attained by the variable x ij until this iteration, and consider the LTP(S) with objective coefficients for (i, j ) ∈ S given by:…”

“…Other heuristic methods that must be emphasized are the Lagrangean heuristic method of Wright et al (1989Wright et al ( , 1991, the tabu search heuristic procedure (TS) of Sun et al (1998) and the recently proposed parametric ghost image processes (GIP) of Glover et al (2005).…”

“…Until very recently, the TS method of Sun et al (1998) could be considered as the most efficient method to attack FCTP problems. The appearance of the Parametric GIP method of Glover et al (2005) and its superior computational performance makes it currently the most efficient presently. Both methods are based on different algorithmic strategies and use local search intensively.…”

“…Both methods are based on different algorithmic strategies and use local search intensively. The results corresponding to the TS and GIP methods are extracted from Glover et al (2005), and were obtained in a Dell, Latitude Laptop, 1 GHz, with 256K Cache running on Windows 2000 operating system.…”

Fixed-Charge Transportation with Product Blending

“…As a consequence, the FCTPwB is N P-hard since if p ij > 0, ∀(i, j) ∈ A, then the blending constraints (1b) become redundant and the resulting problem is simply the FCTP. By and large, researchers have focused on developing heuristics and exact algorithms for solving the FCTP [1,4,6,10,12,13,19,22]. More generally, the FCTP is a special case of the fixed-charge network flow problem for which substantial polyhedral theory and numerous algorithms have been developed.…”