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October  2011, 7(4): 875-890. doi: 10.3934/jimo.2011.7.875

Optimal fleet composition via dynamic programming and golden section search

Ryan Loxton 1, and  Qun Lin 1,

1. 

Department of Mathematics and Statistics, Curtin University, GPO Box U1987 Perth, Western Australia 6845, Australia, Australia

Received  September 2010 Revised  May 2011 Published  August 2011

In this paper, we consider an optimization problem arising in vehicle fleet management. The problem is to construct a heterogeneous vehicle fleet in such a way that cost is minimized subject to a constraint on the overall fleet size. The cost function incorporates fixed and variable costs associated with the fleet, as well as hiring costs that are incurred when vehicle requirements exceed fleet capacity. We first consider the simple case when there is only one type of vehicle. We show that in this case the cost function is convex, and thus the problem can be solved efficiently using the well-known golden section method. We then devise an algorithm, based on dynamic programming and the golden section method, for solving the general problem in which there are multiple vehicle types. We conclude the paper with some simulation results.
Keywords: Dynamic Programming, Convex Optimization, Golden Section Method., Fleet Composition.
Mathematics Subject Classification: Primary: 90B06, 90C25, 90C39; Secondary: 90C1.
Citation: Ryan Loxton, Qun Lin. Optimal fleet composition via dynamic programming and golden section search. Journal of Industrial & Management Optimization, 2011, 7 (4) : 875-890. doi: 10.3934/jimo.2011.7.875
References:
[1]

M. S. Bazaraa, H. D. Sherali and C. M. Shetty, "Nonlinear Programming: Theory and Algorithms," 3rd edition, Wiley-Interscience [John Wiley & Sons], Hoboken, New Jersey, 2006.  Google Scholar

[2]

R. Bellman, "Dynamic Programming," Dover Publications, Inc., Mineola, New York, 2003.  Google Scholar

[3]

J. Couillard, A decision support system for vehicle fleet planning, Decision Support Systems, 9 (1993), 149-159. doi: 10.1016/0167-9236(93)90009-R.  Google Scholar

[4]

G. Ghiani, G. Laporte and R. Musmanno, "Introduction to Logistics Systems Planning and Control," John Wiley, Chichester, 2004. Google Scholar

[5]

A. Hoff, H. Andersson, M. Christiansen, G. Hasle and A. Løkketangen, Industrial aspects and literature survey: Fleet composition and routing, Computers & Operations Research, 37 (2010), 2041-2061. doi: 10.1016/j.cor.2010.03.015.  Google Scholar

[6]

A. Imai and F. Rivera, Strategic fleet size planning for maritime refrigerated containers, Maritime Policy & Management, 28 (2001), 361-374. doi: 10.1080/03088830010020629.  Google Scholar

[7]

D. Kirby, Is your fleet the right size?, Operational Research Quarterly, 10 (1959), 252. doi: 10.1057/jors.1959.25.  Google Scholar

[8]

M.-Y. Lai, C.-S. Liu and X.-J. Tong, A two-stage hybrid meta-heuristic for pickup and delivery vehicle routing problem with time windows, Journal of Industrial & Management Optimization, 6 (2010), 435-451. doi: 10.3934/jimo.2010.6.435.  Google Scholar

[9]

D. G. Luenberger and Y. Ye, "Linear and Nonlinear Programming," 3rd edition, International Series in Operations Research & Management Science, 116, Springer, New York, 2008.  Google Scholar

[10]

H. L. Royden, "Real Analysis," 3rd edition, Macmillan Publishing Company, New York, 1988.  Google Scholar

[11]

I. F. A. Vis, R. B. M. de Koster and M. W. P. Savelsbergh, Minimum vehicle fleet size under time-window constraints at a container terminal, Transportation Science, 39 (2005), 249-260. doi: 10.1287/trsc.1030.0063.  Google Scholar

show all references

References:
[1]

M. S. Bazaraa, H. D. Sherali and C. M. Shetty, "Nonlinear Programming: Theory and Algorithms," 3rd edition, Wiley-Interscience [John Wiley & Sons], Hoboken, New Jersey, 2006.  Google Scholar

[2]

R. Bellman, "Dynamic Programming," Dover Publications, Inc., Mineola, New York, 2003.  Google Scholar

[3]

J. Couillard, A decision support system for vehicle fleet planning, Decision Support Systems, 9 (1993), 149-159. doi: 10.1016/0167-9236(93)90009-R.  Google Scholar

[4]

G. Ghiani, G. Laporte and R. Musmanno, "Introduction to Logistics Systems Planning and Control," John Wiley, Chichester, 2004. Google Scholar

[5]

A. Hoff, H. Andersson, M. Christiansen, G. Hasle and A. Løkketangen, Industrial aspects and literature survey: Fleet composition and routing, Computers & Operations Research, 37 (2010), 2041-2061. doi: 10.1016/j.cor.2010.03.015.  Google Scholar

[6]

A. Imai and F. Rivera, Strategic fleet size planning for maritime refrigerated containers, Maritime Policy & Management, 28 (2001), 361-374. doi: 10.1080/03088830010020629.  Google Scholar

[7]

D. Kirby, Is your fleet the right size?, Operational Research Quarterly, 10 (1959), 252. doi: 10.1057/jors.1959.25.  Google Scholar

[8]

M.-Y. Lai, C.-S. Liu and X.-J. Tong, A two-stage hybrid meta-heuristic for pickup and delivery vehicle routing problem with time windows, Journal of Industrial & Management Optimization, 6 (2010), 435-451. doi: 10.3934/jimo.2010.6.435.  Google Scholar

[9]

D. G. Luenberger and Y. Ye, "Linear and Nonlinear Programming," 3rd edition, International Series in Operations Research & Management Science, 116, Springer, New York, 2008.  Google Scholar

[10]

H. L. Royden, "Real Analysis," 3rd edition, Macmillan Publishing Company, New York, 1988.  Google Scholar

[11]

I. F. A. Vis, R. B. M. de Koster and M. W. P. Savelsbergh, Minimum vehicle fleet size under time-window constraints at a container terminal, Transportation Science, 39 (2005), 249-260. doi: 10.1287/trsc.1030.0063.  Google Scholar

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