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

- 178 Downloads
- 12 Citations

## 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.

## Keywords

fixed-charge problems networks generalized networks ghost image processes tabu search## Preview

Unable to display preview. Download preview PDF.

## References

- Adlakha, V. and K. Kowalski. (2003). “A Simple Heuristic for Solving Small Fixed-Charge Transportation Problems.”
*Omega*31, 205–211.CrossRefGoogle Scholar - Balas, E. and M.W. Padberg. (1976). “Set Partitioning: A Survey.”
*SIAM Review*18, 710–760.CrossRefGoogle Scholar - Balinski, M.L. (1961). “Fixed Cost Transportation Problem.”
*Naval Research Logistics Quarterly*8, 41–54.Google Scholar - Barr, R.S., F. Glover, and D. Klingman. D. (1981). “A New Optimization Method for Large Scale Fixed Charge Transportation Problems.”
*Operations Research*29, 443–463.Google Scholar - Bell, G.J., B.W Lamar, and C.A. Wallace. (1999). “Capacity Improvement, Penalties, and the Fixed Charge Transportation Problem.”
*Naval Research Logistics Quarterly*46, 341–355.CrossRefGoogle Scholar - Cabot, A.V. and S.S Erenguc. (1984). “Some Branch and Bound Procedures for Fixed Charge Transportation Problems,”
*Naval Research Logistics Quarterly*31, 145–154.Google Scholar - Cabot, A.V. and S.S. Erenguc. (1986). “Improved Penalties for Fixed Cost Transportation Problems.”
*Naval Research Logistics Quarterly*31, 856–869.Google Scholar - Cooper, L. (1975). “The Fixed Charge Problem-1: A New Heuristic.”
*Computers and Mathematics with applications*1(1), 89–96.CrossRefGoogle Scholar - Cooper, L. and C. Drebes. (1967). “An Approximation Algorithm for the Fixed Charge Problem.”
*Naval Research Logistics Quarterly*14, 89–96.Google Scholar - Cornuejols, G., L. Fisher, and G.L. Nemhauser. (1977). “Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms.”
*Management Science*23, 789–810.Google Scholar - Crainic, T.G., A. Frangioni, and B. Gendron. (2001). “Bundle-Based Relaxation Methods for Multicommodity Capacitated Fixed Charge Network Design.”
*Discrete Applied Mathematics*112, 73–99.CrossRefGoogle Scholar - Crainic, T.G., B. Gendron, and G. Hernu. 2004. A slope scaling/Lagrangian perturbation heuristic with long-term memory for multicommodity capacitated fixed-charge network design.
*Journal of Heuristics*10(5), 525–545.CrossRefGoogle Scholar - Denzler, D.R. (1964). “An Approximate Algorithm for the Fixed Charge Problem.”
*Naval Research Logistics Quarterly*16, 411–416.Google Scholar - Diaby, M. (1991). “Successive Linear Approximation Procedure for Generalized Fixed-Charge Transportation Problem.”
*Journal of Operation Research Society*42, 991–1001.Google Scholar - Dongarra, J.J. (2002). “Performance of Various Computers Using Standard Linear Equations Software.” Research working paper CS-89–85, University of Tennessee, Knoxville, TN 37996.Google Scholar
- Dwyer, P.S. (1966). “Use of Completely Reduced Numbers in Solving Transportation Problems with Fixed Charge.”
*Naval Research Logistics Quarterly*13(3), 289–313.Google Scholar - Fisk, J. and P.G. McKeown. (1979). “The Pure Fixed Charge Transportation Problem.”
*Naval Research Logistics Quarterly*26, 631–641.Google Scholar - Gendron, B., J.Y. Potvin, and P. Sorian. (2003a). “A Tabu Search with Slope Scaling for the Multicommodity Capacitated Location Problem with Balancing Requirements.”
*Annals of Operations Research*122, 193–217.Google Scholar - Gendron, B., J.Y. Potvin, and P. Sorian. (2003b). “A Parallel Hybrid Heuristic for the Multicommodity Capacitated Location Problem with Balancing Requirements.
*Parallel Computing*29, 592–606.Google Scholar - Glover, F. (1994). “Optimization by Ghost Image Processes in Neural Networks.”
*Computers and Operations Research*21(8) 801–822.CrossRefMathSciNetGoogle Scholar - Glover, F. 1996. GN2PC: An MS DOS Based Network Optimizing System.Google Scholar
- Glover, F., M.M. Amini, and G. Kochenberger. (2003). “Discrete Optimization via Netform Representations and a New Dynamic Branch and Bound Method.” Working research paper, University of Colorado-Boulder, CO.Google Scholar
- Glover, F., D. Klingman, and N. Phillips. (1992).
*Network Models in Optimization and their Applications in Practice*. New York:Wiley.Google Scholar - Glover, F. and M. Laguna. (1997).
*Tabu Search*. Kluwer Academic Publishers. Hingham, MA.Google Scholar - Gottlieb, J. and C. Eckert. (2002). “A Comparison of Two Representations for the Fixed Charge Transportation Problem.” In J.J. Merelo et al. (eds.),
*Parallel Problem Solving From Nature*—*PP3NVII, Lecture Notes in Computer Science*Vol. 2439. Springer Publishing Company, pp. 77–87.Google Scholar - Gray, P. (1971). “Exact Solution of the Fixed Charge Transportation Problem.”
*Operations Research*19, 1529–1538.Google Scholar - Hirsch, W.M. and G.B. Dantzig. (1968). “The Fixed Charge Problem.”
*Naval Research Logistics Quarterly*15, 413–424.Google Scholar - Kennington, J.L. (1976). “The Fixed Charge Transportation Problem: A Computational Study with a Branch and Bound Code.”
*AIIE Transaction*8, 241–247.Google Scholar - Kennington, J.L. and E. Unger. (1976). “A New Branch and Bound Algorithm for the Fixed Charge Transportation Problem.”
*Management Science*22, 1116–1126.Google Scholar - Khang, D.B. and O. Fujiwara. (1991). “Approximation Solution of Capacitated Fixed-Charge Minimum Cost Network Flow Problems.”
*Networks*21, 689–704.Google Scholar - Kim, D. and P.M. Pardalos. (1999). “Ä Solution Approach to the Fixed-Charge Network Flow Problem Using a Dynamic Slope Scaling Procedure.”
*Operations Research Letters*24, 195–203.CrossRefMathSciNetGoogle Scholar - Kim, D. and P.M. Pardalos. (2000). “Dynamic Slope Scaling and Trust Interval Techniques for Solving Concave Piecewise Linear Network Flow Problems.”
*Networks*35, 216–222.CrossRefGoogle Scholar - Klingman, D., A. Napier, and J. Stutz. (1974). “NETGEN: A Program for Generating Large Scale Capacitated Assignment, Transportation, and Minimum Cost Network Problems.”
*Management Science*5, 814–821.Google Scholar - Kuhn, H.W. and W.J Baumol. (1961). “An Approximate Algorithm for the Fixed Charge Transportation Problem.”
*Naval Research Logistics Quarterly*9(1), 1–16.Google Scholar - Lamar, B.W. and C.A. Wallace. (1997). “Revised Modified Penalties for Fixed Charge Transportation Problems.”
*Management Science*43, 1431–1436.Google Scholar - Luna, H.P.L., N. Ziviani, and R.M.B. Cabral. (1987). “The Telephonic Switching Center Network Problem: Formalization and Computational Experience.”
*Discrete and Applied Mathematics*18, 199–210.CrossRefGoogle Scholar - McKeown, P.G. (1975). A Vertex Ranking Procedure for Solving the Linear Fixed Charge Problem.”
*Operations Research*23, 1183–1191.Google Scholar - McKeown, P.G. (1981). “A Branch-and-Bound Algorithm for Solving Fixed Charge Problems.”
*Naval Research Logistics Quarterly*28, 607–617.Google Scholar - McKeown, P.G. and C.T. Ragsdale. (1990). “A Computational Study of Using Preprocessing and Stronger Formulations to Solve General Fixed Charge Problem.”
*Operations Research*17, 9–16.Google Scholar - McKeown, P.G. and P. Sinha. (1980). “An Easy Solution to Special Class of Fixed Charge Problems.”
*Naval Research Logistics Quarterly*2, 621–624.Google Scholar - Mirzain, A. (1985). “Lagrangian Relaxation for the Star-Star Concentrator Location Problem: Approximation Algorithm and Bounds.”
*Networks*15, 1–20.Google Scholar - Murty, K.G. (1968). “Solving the Fixed Charge Problem by Ranking Extreme Points.”
*Operations Research*16, 268–279.Google Scholar - Nozick, L. and M. Turnquist. (1998a). “Two-Echelon Inventory Allocation and Distribution Center Location Analysis.” In
*Proceedings of Tristan III*(Transportation Science Section of INFORMS), San Juan, Puerto Rico.Google Scholar - Nozick, L. and M. Turnquist. (1998b). “Integrating Inventory Impacts into a Fixed Charge Model for Locating Distribution Centers.”
*Transportation Research Part E 31 E*(3), 173–186.Google Scholar - Ortega, F. and L.A. Wolsey. (2003). “A Branch-and-Cut Algorithm for the Single-Commodity, Uncapacitated, Fixed-Charge Network Flow Problem.”
*Networks*41, 143–158.CrossRefGoogle Scholar - Palekar, U.S., M.K. Karwan, and S. Zionts. (1990). “A Branch and Bound Method for the Fixed Charge Transportation Problem.”
*Management Science*36, 1092–1105.MathSciNetGoogle Scholar - Rothfarb, B., H. Frank, D.M. Rosembaun, and K. Steiglitz. (1970). “Optimal Design of Offshore Natural-Gas Pipeline Systems.”
*Operations Research*18, 992–1020.Google Scholar - Rousseau, J.M. (1973). “A Cutting Plane Method for the Fixed Cost Problem.” Doctoral dissertation Massachusetts Institute of Technology. Cambridge, MA.Google Scholar
- Schaffer, J.E. (1989). “Use of Penalties in the Branch-and-Bound Procedure for the Fixed Charge Transportation Problem.”
*European Journal of Operations Research*43, 305–312.CrossRefGoogle Scholar - Shetty, U.S. (1990). “A Relaxation Decomposition Algorithm for the Fixed Charge Network Problem.”
*Naval Research Logistics Quarterly*32(2), 327–340.Google Scholar - Steinberg, D.I. (1970). “The Fixed Charge Problem.”
*Naval Research Logistics Quarterly*7(2), 217–236.Google Scholar - Steinberg, D.I. (1977). “Designing a Heuristic for the Fixed Charge Transportation Problem.” Reprints in Mathematics and the Mathematical Sciences. Southern Illinois University at Edwardsville, Edwardsville. ILGoogle Scholar
- Sun, M., J.E. Aronson., P.G. McKeown, and D. Drinka. (1998). “A Tabu Search Heuristic Procedure for the Fixed Charge Transportation Problem.”
*European Journal of Operational Research*106, 441–456.CrossRefGoogle Scholar - Sun, M. and P.G. McKeown. (1993). Tabu Search Applied to the General Fixed Charge Problems.”
*Annals of Operations Research*41(1–4), 405–420.CrossRefGoogle Scholar - Walker, W.E. (1976). “A Heuristic Adjacent Extreme Point Algorithm for the Fixed Charge Problem.”
*Management Science*22(3), 587–596.Google Scholar - Woodruff, D. (1995). “Ghost Image Processing for Minimum Covariance Determinant Estimators.”
*ORSA Journal on Computing*7, 468–473.Google Scholar - Wright, D. and C. Haehling von Lanzenauer. (1989). “Solving the Fixed Charge Problem with Lagrangian Relaxation and Cost Allocation Heuristics.”
*European Journal of Operations Research*42, 304–312.CrossRefGoogle Scholar - Wright, D. and C. Haehling von Lanzenauer. (1991). “COLE: A New Heuristic Approach for Fixed Charge Problem Computational Results.”
*European Journal of Operations Research*52, 235–246.CrossRefGoogle Scholar