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April  2016, 12(2): 449-470. doi: 10.3934/jimo.2016.12.449

## Location and capacity design of congested intermediate facilities in networks

 1 School of Management, South-Central University for Nationalities, Wuhan, 430074, China 2 Department of Automation, School of Power and Mechanical Engineering, Wuhan University, Wuhan, 430072, China

Received  August 2014 Revised  February 2015 Published  June 2015

This article deals with the problem of making simultaneous decisions on the location, capacity and demand flow assignment for intermediate facilities in a network. Two nonlinear mixed-integer program (NMIP) models for continuous and discrete capacity decisions are proposed, respectively. The objective is to minimize the total costs, including fixed location cost, transportation cost, congestion cost and capacity cost. Congestion at intermediate facilities is modeled as the ratio of total flow to surplus capacity by viewing each facility as an M/M/1 queuing system. To solve NMIP with continuous capacity decision, we apply the Lagrangean algorithm that has been proposed to solve the classic inventory-location model. For the NMIP with discrete capacity decision, we propose another Lagrangean algorithm where the problem is decomposed into $|K|$ subproblems that can be solved to optimality. The measures of allocation heuristic, capacity increase and capacity adjustment are taken to construct feasible solutions. Computational results indicate that the heuristics for the two models are both efficient and effective.
Citation: Dandan Hu, Zhi-Wei Liu. Location and capacity design of congested intermediate facilities in networks. Journal of Industrial & Management Optimization, 2016, 12 (2) : 449-470. doi: 10.3934/jimo.2016.12.449
##### References:
 [1] R. Aboolian, O. Berman and D. Krass, Profit maximizing distributed service system design with congestion and elastic demand,, Transportation Science, 46 (2012), 247. doi: 10.1287/trsc.1110.0392. [2] S.R. Agnihothri, S. Narasimhan and H. Pirkul, An assignment problem with queueing time cost,, Naval Research Logistics, 37 (1990), 231. doi: 10.1002/1520-6750(199004)37:2<231::AID-NAV3220370204>3.0.CO;2-N. [3] M. Armony, E. Plambeck and S. Seshadri, Sensitivity of optimal capacity to customer impatience in an unobservable m/m/s queue (why you shouldn't shout at the dmv),, Manufacturing & Service Operations Management, 11 (2009), 19. doi: 10.1287/msom.1070.0194. [4] O. Berman and Z. Drezner, Location of congested capacitated facilities with distance-sensitive demand,, IIE Transactions, 38 (2006), 213. doi: 10.1080/07408170500288190. [5] O. Berman and Z. Drezner, The multiple server location problem,, Journal of the Operational Research Society, 58 (2006), 91. doi: 10.1057/palgrave.jors.2602126. [6] M. L. Brandeau and S. S. Chiu, A center location problem with congestion,, Annals of operations research, 40 (1992), 17. doi: 10.1007/BF02060468. [7] M. L. F. Cheong, R. Bhatnagar and S. C. Graves, Logistics network design with supplier consolidation hubs and multiple shipment options,, Journal of Industrial and Management Optimization, 3 (2007), 51. doi: 10.3934/jimo.2007.3.51. [8] S. M. Choi, X. Huang and W. K. Ching, Minimizing equilibrium expected sojourn time via performance-based mixed threshold demand allocation in a multiple-server queueing environment,, Journal of Industrial and Management Optimization, 8 (2012), 299. doi: 10.3934/jimo.2012.8.299. [9] M. S. Daskin, C. R. Coullard and Z.-J. M. Shen, A maximum expected covering location model: formulation, properties and heuristic solution,, Transportation Science, 17 (1983), 48. doi: 10.1287/trsc.17.1.48. [10] M. S. Daskin, C. R. Coullard and Z.-J. M. Shen, An inventory-location model: Formulation, solution algorithm and computational results,, Annals of Operations Research, 110 (2002), 83. doi: 10.1023/A:1020763400324. [11] M. S. Daskin, Network and Discrete Location: Models, Algorithms, and Applications,, John Wiley & Sons, (2011). doi: 10.1002/9781118032343. [12] S. Elhedhli and H. Wu, A lagrangean heuristic for hub-and-spoke system design with capacity selection and congestion,, INFORMS Journal on Computing, 22 (2010), 282. doi: 10.1287/ijoc.1090.0335. [13] A. F. Gabor and J. Van Ommeren, An approximation algorithm for a facility location problem with stochastic demands and inventories,, Operations research letters, 34 (2006), 257. doi: 10.1016/j.orl.2005.04.009. [14] R. Hassin and M. Haviv, To Queue or not to Queue: Equilibrium Behavior in Queueing Systems,, Kluwer Academic Publishers, (2002). doi: 10.1007/978-1-4615-0359-0. [15] D. Hu, C. Yang and J. Yang, Budget constrained flow interception location model for congested systems,, Journal of Systems Engineering and Electronics, 20 (2009), 1255. [16] S. Huang, R. Batta and R. Nagi, Distribution network design: Selection and sizing of congested connections,, Naval Research Logistics, 52 (2005), 701. doi: 10.1002/nav.20106. [17] V. Marianov and D. Serra, Probabilistic, maximal covering location-allocation models for congested systems,, Journal of Regional Science, 38 (1998), 401. [18] S. H. R. Pasandideh, S. T. A. Niaki and V. Hajipour, A multi-objective facility location model with batch arrivals: two parameter-tuned meta-heuristic algorithms,, Journal of Intelligent Manufacturing, 24 (2013), 331. [19] S. H. A. Rahmati, A. Ahmadi, M. Sharifi and A. Chambari, A multi-objective model for Facility Location-allocation Problem with immobile servers within queuing framework,, Computers and Industrial Engineering, 74 (2014), 1. doi: 10.1016/j.cie.2014.04.018. [20] H. Shavandi and H. Mahlooji, A fuzzy queuing location model with a genetic algorithm for congested systems,, Applied mathematics and computation, 181 (2006), 440. doi: 10.1016/j.amc.2005.12.058. [21] Q. Wang, R. Batta and C. M. Rump, Algorithms for a facility location problem with stochastic customer demand and immobile servers,, Annals of Operations Research, 111 (2002), 17. doi: 10.1023/A:1020961732667. [22] Q. Wang, R. Batta and C. M. Rump, Facility location models for immobile servers with stochastic demand,, Naval Research Logistics, 51 (2004), 137. doi: 10.1002/nav.10110. [23] L. Zhang and G. Rushton, Optimizing the size and locations of facilities in competitive multi-site service systems,, Computers & Operations Research, 35 (2008), 327. doi: 10.1016/j.cor.2006.03.002.

show all references

##### References:
 [1] R. Aboolian, O. Berman and D. Krass, Profit maximizing distributed service system design with congestion and elastic demand,, Transportation Science, 46 (2012), 247. doi: 10.1287/trsc.1110.0392. [2] S.R. Agnihothri, S. Narasimhan and H. Pirkul, An assignment problem with queueing time cost,, Naval Research Logistics, 37 (1990), 231. doi: 10.1002/1520-6750(199004)37:2<231::AID-NAV3220370204>3.0.CO;2-N. [3] M. Armony, E. Plambeck and S. Seshadri, Sensitivity of optimal capacity to customer impatience in an unobservable m/m/s queue (why you shouldn't shout at the dmv),, Manufacturing & Service Operations Management, 11 (2009), 19. doi: 10.1287/msom.1070.0194. [4] O. Berman and Z. Drezner, Location of congested capacitated facilities with distance-sensitive demand,, IIE Transactions, 38 (2006), 213. doi: 10.1080/07408170500288190. [5] O. Berman and Z. Drezner, The multiple server location problem,, Journal of the Operational Research Society, 58 (2006), 91. doi: 10.1057/palgrave.jors.2602126. [6] M. L. Brandeau and S. S. Chiu, A center location problem with congestion,, Annals of operations research, 40 (1992), 17. doi: 10.1007/BF02060468. [7] M. L. F. Cheong, R. Bhatnagar and S. C. Graves, Logistics network design with supplier consolidation hubs and multiple shipment options,, Journal of Industrial and Management Optimization, 3 (2007), 51. doi: 10.3934/jimo.2007.3.51. [8] S. M. Choi, X. Huang and W. K. Ching, Minimizing equilibrium expected sojourn time via performance-based mixed threshold demand allocation in a multiple-server queueing environment,, Journal of Industrial and Management Optimization, 8 (2012), 299. doi: 10.3934/jimo.2012.8.299. [9] M. S. Daskin, C. R. Coullard and Z.-J. M. Shen, A maximum expected covering location model: formulation, properties and heuristic solution,, Transportation Science, 17 (1983), 48. doi: 10.1287/trsc.17.1.48. [10] M. S. Daskin, C. R. Coullard and Z.-J. M. Shen, An inventory-location model: Formulation, solution algorithm and computational results,, Annals of Operations Research, 110 (2002), 83. doi: 10.1023/A:1020763400324. [11] M. S. Daskin, Network and Discrete Location: Models, Algorithms, and Applications,, John Wiley & Sons, (2011). doi: 10.1002/9781118032343. [12] S. Elhedhli and H. Wu, A lagrangean heuristic for hub-and-spoke system design with capacity selection and congestion,, INFORMS Journal on Computing, 22 (2010), 282. doi: 10.1287/ijoc.1090.0335. [13] A. F. Gabor and J. Van Ommeren, An approximation algorithm for a facility location problem with stochastic demands and inventories,, Operations research letters, 34 (2006), 257. doi: 10.1016/j.orl.2005.04.009. [14] R. Hassin and M. Haviv, To Queue or not to Queue: Equilibrium Behavior in Queueing Systems,, Kluwer Academic Publishers, (2002). doi: 10.1007/978-1-4615-0359-0. [15] D. Hu, C. Yang and J. Yang, Budget constrained flow interception location model for congested systems,, Journal of Systems Engineering and Electronics, 20 (2009), 1255. [16] S. Huang, R. Batta and R. Nagi, Distribution network design: Selection and sizing of congested connections,, Naval Research Logistics, 52 (2005), 701. doi: 10.1002/nav.20106. [17] V. Marianov and D. Serra, Probabilistic, maximal covering location-allocation models for congested systems,, Journal of Regional Science, 38 (1998), 401. [18] S. H. R. Pasandideh, S. T. A. Niaki and V. Hajipour, A multi-objective facility location model with batch arrivals: two parameter-tuned meta-heuristic algorithms,, Journal of Intelligent Manufacturing, 24 (2013), 331. [19] S. H. A. Rahmati, A. Ahmadi, M. Sharifi and A. Chambari, A multi-objective model for Facility Location-allocation Problem with immobile servers within queuing framework,, Computers and Industrial Engineering, 74 (2014), 1. doi: 10.1016/j.cie.2014.04.018. [20] H. Shavandi and H. Mahlooji, A fuzzy queuing location model with a genetic algorithm for congested systems,, Applied mathematics and computation, 181 (2006), 440. doi: 10.1016/j.amc.2005.12.058. [21] Q. Wang, R. Batta and C. M. Rump, Algorithms for a facility location problem with stochastic customer demand and immobile servers,, Annals of Operations Research, 111 (2002), 17. doi: 10.1023/A:1020961732667. [22] Q. Wang, R. Batta and C. M. Rump, Facility location models for immobile servers with stochastic demand,, Naval Research Logistics, 51 (2004), 137. doi: 10.1002/nav.10110. [23] L. Zhang and G. Rushton, Optimizing the size and locations of facilities in competitive multi-site service systems,, Computers & Operations Research, 35 (2008), 327. doi: 10.1016/j.cor.2006.03.002.
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