The fixed charge problem

Hirsch, Warren M.; Dantzig, George B.

doi:10.1002/nav.3800150306

Cited by 221 publications

(68 citation statements)

References 4 publications

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“…Hirsch and Dantzig (1968) established that the feasible region of FCTP is a bounded convex set with a concave objective function. An optimal solution occurs at an extreme point of the constraint set, and for a non-degenerate problem with all positive fixed costs, every extreme point of the feasible region is a local minimum.…”

Section: Literature Reviewmentioning

confidence: 99%

Glowworm swarm optimisation algorithm for nonlinear fixed charge transportation problem in a single stage supply chain network

Manimaran¹,

Selladurai²

2014

IJLEG

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Abstract:In this paper, Glowworm swarm optimisation (GSO) is proposed to solve nonlinear fixed charge transportation problem (NLFCTP) in a single stage supply chain network. In a fixed charge transportation problem, fixed cost is incurred for every route, along with the variable cost that is proportional to the amount shipped. In some cases, the variable cost is associated with quadratic variables in which the cost function will be nonlinear. This problem is more difficult to solve due to presence of the fixed costs that result in discontinuities in the objective function. The objective of this paper is to determine the least cost transportation plan that minimises the total variable and fixed costs while satisfying the supply and demand requirements of each plant and customer. The performance of GSO is compared in terms of total distribution cost with a spanning tree-based genetic algorithm. The comparison reveals that GSO provides better solutions.Keywords: distribution problem; glowworm swarm optimisation; GSO; fixed cost; NP-hard; supply chain.Reference to this paper should be made as follows: Manimaran, P. and Selladurai, V. (2014) 'Glowworm swarm optimisation algorithm for nonlinear fixed charge transportation problem in a single stage supply chain network', Int.

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Section: Literature Reviewmentioning

confidence: 99%

Glowworm swarm optimisation algorithm for nonlinear fixed charge transportation problem in a single stage supply chain network

Manimaran¹,

Selladurai²

2014

IJLEG

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“…The observation of Hirsch and Dantzig (1968), which showed that optimal solutions to the fixed-charge problem occur at extreme points, opened a fertile area for developing a class of exact methods. These and other types of exact methods include cutting plane approaches (Rousseau, 1973), vertex ranking strategies (Murty, 1968;McKeown, 1975), and a number of branch and bound approaches with penalty based search tree pruning mechanisms and capacity improvement techniques (Gray, 1971;Kennington, 1976;Kennington and Unger, 1976;Fisk and McKeown, 1979;McKeown and Sinha, 1980;Barr, Glover and Klingman, 1981;McKeown, 1981;Cabot and Erenguc, 1984;and 1986;Schaffer, 1989;McKeown and Ragsdale, 1990;Palekar, Karwan and Zionts, 1990;Lamar and Wallace, 1997;Bell, Lamar and Wallace, 1999;Glover, Amini, and Kochenberger, 2003;and Ortega and Wolsey, 2003).…”

Section: Fc-mip Minimize X O [Fc] = Cx + Fzmentioning

confidence: 99%

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

Glover

2005

J Heuristics

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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.

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“…Wright and Haehling von Lanzenauer (1989) develop a Lagrangian heuristic for the FCTP which is based on relaxing the variable upper bound constraints Xij < Uijyij. It can be proved that an optimal Solution to the FCTP is an extreme point of the convex region defined by constraints (4b)-(4d) (see Hirsch and Dantzig (1968)). Basic feasible solutions to the system (4b)- (4d) define spanning trees of the transportation network.…”

Section: Fixed-charge Transportation Problemmentioning

confidence: 99%

Combinatorial Optimisation Problems of the Assignment Type and a Partitioning Approach

Klose

Drexl

2002

Lecture Notes in Economics and Mathematical Systems

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Abstract Assignment type problems consist in optimally assigning or allocating a given set of "activities" to a given set of "resources". Optimisation problems of the assignment type have numerous applications in production planning and logistics. A populär approach to solve such problems or to compute lower b ounds on the optimal Solution value (in case of a minimisation problem) is to employ column generation. By means of co nsidering subsets of "activities" which can feasibly be assigned to a Single resource, the problem is reformulated as some kind of set-partitioning problem. Column generation is then used in order to solve the linear relaxation of the reformulation. The lower bound obtainable from this approach may, however, be improved by partitioning the set of resources into subsets and by considering subsets of activities which can feasibly be assigned to subsets of resources. This paper outlines the application of this partitioning method to a number of important combinatorial optimisation problems of the assignment type. Terms of use: Documents in EconStor may

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The fixed charge problem

Cited by 221 publications

References 4 publications

Glowworm swarm optimisation algorithm for nonlinear fixed charge transportation problem in a single stage supply chain network

Glowworm swarm optimisation algorithm for nonlinear fixed charge transportation problem in a single stage supply chain network

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

Combinatorial Optimisation Problems of the Assignment Type and a Partitioning Approach

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