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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 and 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.

[2]

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

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[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.

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.

[2]

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

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[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.

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