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October  2015, 11(4): 1211-1245. doi: 10.3934/jimo.2015.11.1211

On EOQ cost models with arbitrary purchase and transportation costs

Ş. İlker Birbil 1, , Kerem Bülbül 1, , J. B. G. Frenk 2, and  H. M. Mulder 3,

1. 

Sabanci University, Manufacturing Systems and Industrial Engineering, Orhanli-Tuzla, 34956 Istanbul
2. 

Manufacturing Systems and Industrial Engineering, Sabancı University, Istanbul, Turkey
3. 

Erasmus University Rotterdam, Postbus 1738, 3000 DR Rotterdam, Netherlands

Received  July 2013 Revised  September 2014 Published  March 2015

We analyze an economic order quantity cost model with unit out-of-pocket holding costs, unit opportunity costs of holding, fixed ordering costs, and general purchase-transportation costs. We identify the set of purchase-transportation cost functions for which this model is easy to solve and related to solving a one-dimensional convex minimization problem. For the remaining purchase-transportation cost functions, when this problem becomes a global optimization problem, we propose a Lipschitz optimization procedure. In particular, we give an easy procedure which determines an upper bound on the optimal cycle length. Then, using this bound, we apply a well-known technique from global optimization. Also for the class of transportation functions related to full truckload (FTL) and less-than-truckload (LTL) shipments and the well-known carload discount schedule, we specialize these results and give fast and easy algorithms to calculate the optimal lot size and the corresponding optimal order-up-to-level.
Keywords: Inventory, exact solution., transportation cost function, EOQ cost model, purchasing cost function.
Mathematics Subject Classification: Primary: 90B05, 90B06; Secondary: 90C26, 46N1.
Citation: Ş. İlker Birbil, Kerem Bülbül, J. B. G. Frenk, H. M. Mulder. On EOQ cost models with arbitrary purchase and transportation costs. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1211-1245. doi: 10.3934/jimo.2015.11.1211
References:
[1]

P. L. Abad and V. Aggarwal, Incorporating transport cost in the lot size and pricing decisions with downward sloping demand, International Journal of Production Economics, 95 (2005), 297-305. doi: 10.1016/j.ijpe.2003.12.008.  Google Scholar

[2]

F. J. Arcelus and J. E. Rowcroft, Inventory policies with freight and incremental quantity discounts, International Journal of Systems Science, 22 (1991), 2025-2037. doi: 10.1080/00207729108910771.  Google Scholar

[3]

J. B. Aubin, Optima and Equilibra (An introduction to nonlinear analysis), vol. 140 of Graduate Texts in Mathematics, Springer Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02959-6.  Google Scholar

[4]

D. C. Aucamp, Nonlinear freight costs in the EOQ problem, European Journal of Operational Research, 9 (1982), 61-63. doi: 10.1016/0377-2217(82)90011-X.  Google Scholar

[5]

W. J. Baumol and H. D. Vinod, An inventory theoretic model of freight transport demand, Management Science, 16 (1970), 413-421. doi: 10.1287/mnsc.16.7.413.  Google Scholar

[6]

Z. P. Bayındır, Ş.İ. Birbil and J. Frenk, The joint replenishment problem with variable production costs, European Journal of Operational Research, 175 (2006), 622-640. doi: 10.1016/j.ejor.2005.06.005.  Google Scholar

[7]

M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, Third edition. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006. doi: 10.1002/0471787779.  Google Scholar

[8]

C. R. Bector, Programming problems with convex fractional functions, Operations Research, 16 (1968), 383-391. doi: 10.1287/opre.16.2.383.  Google Scholar

[9]

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511804441.  Google Scholar

[10]

T. H. Burwell, D. S. Dave, K. E. Fitzpatrick and M. R. Roy, Economic lot size model for price-dependent demand under quantity and freight discounts, International Journal of Production Economics, 48 (1997), 141-155. doi: 10.1016/S0925-5273(96)00085-0.  Google Scholar

[11]

J. R. Carter and B. G. Ferrin, Transportation costs and inventory management: Why transportation costs matter, Production and Inventory Management Journal, 37 (1996), 58-62. Google Scholar

[12]

J. R. Carter, B. G. Ferrin and C. R. Carter, The effect of less-than-truckload rates on the purchase order lot size decision, Transportation Journal, 34 (1995), 35-44. Google Scholar

[13]

J. R. Carter, B. G. Ferrin and C. R. Carter, On extending Russell and Krajewski's algorithm for economic purchase quantities, Decision Sciences, 26 (1995), 819-829. doi: 10.1111/j.1540-5915.1995.tb01577.x.  Google Scholar

[14]

C. Das, A generalized discount structure and some dominance rules for selecting price-break EOQ, European Journal of Operational Research, 34 (1988), 27-38. doi: 10.1016/0377-2217(88)90452-3.  Google Scholar

[15]

M. Drake and K. Marley, Handbook of EOQ Inventory Problems (Stochastic and Deterministic Models and Applications), chapter Century of the EOQ, 3-22, Springer, New York, 2014, Editor: Tsan-Ming Choi. Google Scholar

[16]

J. B. G. Frenk, M. Kaya and B. Pourghannad, chapter Generalizing the Ordering Cost and Holding-Backlog Cost Rate Functions in EOQ-Type Inventory Models, Handbook of EOQ Inventory Problems (Stochastic and Deterministic Models and Applications), Springer, New York, (2014), 79-119, Editor: Tsan-Ming Choi. Google Scholar

[17]

G. Hadley and T. Whitin, Analysis of Inventory Systems, Prentice Hall, Englewood Cliffs, 1963. Google Scholar

[18]

F. Harris, How many parts to make at once, Factory, The Magazine of Management, 10 (1913), 135-136. doi: 10.1287/opre.38.6.947.  Google Scholar

[19]

R. Horst, P. M. Pardalos and N. V. Thoai, Introduction to Global Optimization, Kluwer Academic Publishers, Dordrecht, 1995.  Google Scholar

[20]

H. Hwang, D. H. Moon and S. W. Shinn, An EOQ model with quantity discounts for both purchasing price and freight cost, Computers and Operations Research, 17 (1990), 73-78. doi: 10.1016/0305-0548(90)90029-7.  Google Scholar

[21]

K. Iwaniec, An inventory model with full load ordering, Management Science, 25 (1979), 374-384. doi: 10.1287/mnsc.25.4.374.  Google Scholar

[22]

J. V. Jucker and M. J. Rosenblatt, Single-period inventory models with demand uncertainty and quantity discounts: Behavioral implications and a new solution procedure, Naval Research Logistics Quarterly, 32 (1985), 537-550. doi: 10.1002/nav.3800320402.  Google Scholar

[23]

T. W. Knowles and P. Pantumsinchai, All-units discounts for standard container sizes, Decision Sciences, 19 (1988), 848-857. doi: 10.1111/j.1540-5915.1988.tb00307.x.  Google Scholar

[24]

D. Konur and A. Toptal, Analysis and applications of replenishment problems under stepwise transportation costs and generalized wholesale prices, International Journal of Production Economics, 140 (2012), 521-529. doi: 10.1016/j.ijpe.2012.07.003.  Google Scholar

[25]

P. D. Larson, The economic transportation quantity, Transportation Journal, 28 (1988), 43-48. Google Scholar

[26]

C. Lee, The economic order quantity for freight discount costs, IIE Transactions, 18 (1986), 318-320. doi: 10.1080/07408178608974710.  Google Scholar

[27]

S. A. Lippman, Optimal inventory policy with multiple set-up costs, Management Science, 16 (1969), 118-138. doi: 10.1287/mnsc.16.1.118.  Google Scholar

[28]

S. A. Lippman, Economic order quantities and multiple set-up costs, Management Science, 18 (1971), 39-47. doi: 10.1287/mnsc.18.1.39.  Google Scholar

[29]

A. Mendoza and J. A. Ventura, Incorporating quantity discounts to the EOQ model with transportation costs, International Journal of Production Economics, 113 (2008), 754-765. doi: 10.1016/j.ijpe.2007.10.010.  Google Scholar

[30]

J. A. Muckstadt and A. Sapra, Principles of Inventory Management: When You Are Down to Four Order More, Springer, New York, 2010. doi: 10.1007/978-0-387-68948-7.  Google Scholar

[31]

S. Nahmias, Production and Operations Analysis (Third Edition), Irwin/McGraw-Hill, New York, 1997. Google Scholar

[32]

E. Porteus, Handbooks in Operations Research and Management Science, Volume 2, Stochastic Models, chapter Stochastic Inventory Theory, 605-652, North-Holland, Amsterdam, 1990, Editors: Heyman, D.P and Sobel, M.J.  Google Scholar

[33]

B. Q. Rieksts and J. A. Ventura, Two-stage inventory models with a bi-modal transportation cost, Computers & Operations Research, 37 (2010), 20-31. doi: 10.1016/j.cor.2009.02.026.  Google Scholar

[34]

B. Q. Rieskts and J. A. Ventura, Optimal inventory policies with two modes of freight transportation, European Journal of Operational Research, 186 (2008), 576-585. doi: 10.1016/j.ejor.2007.01.042.  Google Scholar

[35]

A. Roberts and E. Varberg, Convex Functions, Academic Press, New York, 1973.  Google Scholar

[36]

R. T. Rockafellar, Convex Analysis, Princeton University Press, New Jersey, 1997.  Google Scholar

[37]

R. M. Russell and L. J. Krajewski, Optimal purchase and transportation cost lot sizing for a single item, Decision Sciences, 22 (1991), 940-952. doi: 10.1111/j.1540-5915.1991.tb00373.x.  Google Scholar

[38]

E. A. Silver, D. F. Pyke and R. Peterson, Inventory Management and Production Planning and Scheduling, John Wiley and Sons, 1998. Google Scholar

[39]

S. R. Swenseth and M. R. Godfrey, Incorporating transportation costs into inventory replenishment decisions, International Journal of Production Economics, 77 (2002), 113-130. doi: 10.1016/S0925-5273(01)00230-4.  Google Scholar

[40]

R. J. Tersine and S. Barman, Economic inventory/transport lot sizing with quantity and freight rate discounts, Decision Sciences, 22 (1991), 1171-1179. doi: 10.1111/j.1540-5915.1991.tb01914.x.  Google Scholar

[41]

A. Toptal, Replenishment decisions under an all-units discount schedule and stepwise freight costs, European Journal of Operational Research, 198 (2009), 504-510. doi: 10.1016/j.ejor.2008.09.037.  Google Scholar

[42]

A. Toptal and S. Bingöl, Transportation pricing of a truckload carrier, European Journal of Operational Research, 214 (2011), 559-567. doi: 10.1016/j.ejor.2011.05.005.  Google Scholar

[43]

A. F. Veinott Jr, The status of mathematical inventory theory, Management Science, 12 (1966), 745-777. doi: 10.1287/mnsc.12.11.745.  Google Scholar

[44]

P. H. Zipkin, Foundations of Inventory Management, McGraw-Hill, New York, 2000. Google Scholar

show all references

References:
[1]

P. L. Abad and V. Aggarwal, Incorporating transport cost in the lot size and pricing decisions with downward sloping demand, International Journal of Production Economics, 95 (2005), 297-305. doi: 10.1016/j.ijpe.2003.12.008.  Google Scholar

[2]

F. J. Arcelus and J. E. Rowcroft, Inventory policies with freight and incremental quantity discounts, International Journal of Systems Science, 22 (1991), 2025-2037. doi: 10.1080/00207729108910771.  Google Scholar

[3]

J. B. Aubin, Optima and Equilibra (An introduction to nonlinear analysis), vol. 140 of Graduate Texts in Mathematics, Springer Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02959-6.  Google Scholar

[4]

D. C. Aucamp, Nonlinear freight costs in the EOQ problem, European Journal of Operational Research, 9 (1982), 61-63. doi: 10.1016/0377-2217(82)90011-X.  Google Scholar

[5]

W. J. Baumol and H. D. Vinod, An inventory theoretic model of freight transport demand, Management Science, 16 (1970), 413-421. doi: 10.1287/mnsc.16.7.413.  Google Scholar

[6]

Z. P. Bayındır, Ş.İ. Birbil and J. Frenk, The joint replenishment problem with variable production costs, European Journal of Operational Research, 175 (2006), 622-640. doi: 10.1016/j.ejor.2005.06.005.  Google Scholar

[7]

M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, Third edition. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006. doi: 10.1002/0471787779.  Google Scholar

[8]

C. R. Bector, Programming problems with convex fractional functions, Operations Research, 16 (1968), 383-391. doi: 10.1287/opre.16.2.383.  Google Scholar

[9]

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511804441.  Google Scholar

[10]

T. H. Burwell, D. S. Dave, K. E. Fitzpatrick and M. R. Roy, Economic lot size model for price-dependent demand under quantity and freight discounts, International Journal of Production Economics, 48 (1997), 141-155. doi: 10.1016/S0925-5273(96)00085-0.  Google Scholar

[11]

J. R. Carter and B. G. Ferrin, Transportation costs and inventory management: Why transportation costs matter, Production and Inventory Management Journal, 37 (1996), 58-62. Google Scholar

[12]

J. R. Carter, B. G. Ferrin and C. R. Carter, The effect of less-than-truckload rates on the purchase order lot size decision, Transportation Journal, 34 (1995), 35-44. Google Scholar

[13]

J. R. Carter, B. G. Ferrin and C. R. Carter, On extending Russell and Krajewski's algorithm for economic purchase quantities, Decision Sciences, 26 (1995), 819-829. doi: 10.1111/j.1540-5915.1995.tb01577.x.  Google Scholar

[14]

C. Das, A generalized discount structure and some dominance rules for selecting price-break EOQ, European Journal of Operational Research, 34 (1988), 27-38. doi: 10.1016/0377-2217(88)90452-3.  Google Scholar

[15]

M. Drake and K. Marley, Handbook of EOQ Inventory Problems (Stochastic and Deterministic Models and Applications), chapter Century of the EOQ, 3-22, Springer, New York, 2014, Editor: Tsan-Ming Choi. Google Scholar

[16]

J. B. G. Frenk, M. Kaya and B. Pourghannad, chapter Generalizing the Ordering Cost and Holding-Backlog Cost Rate Functions in EOQ-Type Inventory Models, Handbook of EOQ Inventory Problems (Stochastic and Deterministic Models and Applications), Springer, New York, (2014), 79-119, Editor: Tsan-Ming Choi. Google Scholar

[17]

G. Hadley and T. Whitin, Analysis of Inventory Systems, Prentice Hall, Englewood Cliffs, 1963. Google Scholar

[18]

F. Harris, How many parts to make at once, Factory, The Magazine of Management, 10 (1913), 135-136. doi: 10.1287/opre.38.6.947.  Google Scholar

[19]

R. Horst, P. M. Pardalos and N. V. Thoai, Introduction to Global Optimization, Kluwer Academic Publishers, Dordrecht, 1995.  Google Scholar

[20]

H. Hwang, D. H. Moon and S. W. Shinn, An EOQ model with quantity discounts for both purchasing price and freight cost, Computers and Operations Research, 17 (1990), 73-78. doi: 10.1016/0305-0548(90)90029-7.  Google Scholar

[21]

K. Iwaniec, An inventory model with full load ordering, Management Science, 25 (1979), 374-384. doi: 10.1287/mnsc.25.4.374.  Google Scholar

[22]

J. V. Jucker and M. J. Rosenblatt, Single-period inventory models with demand uncertainty and quantity discounts: Behavioral implications and a new solution procedure, Naval Research Logistics Quarterly, 32 (1985), 537-550. doi: 10.1002/nav.3800320402.  Google Scholar

[23]

T. W. Knowles and P. Pantumsinchai, All-units discounts for standard container sizes, Decision Sciences, 19 (1988), 848-857. doi: 10.1111/j.1540-5915.1988.tb00307.x.  Google Scholar

[24]

D. Konur and A. Toptal, Analysis and applications of replenishment problems under stepwise transportation costs and generalized wholesale prices, International Journal of Production Economics, 140 (2012), 521-529. doi: 10.1016/j.ijpe.2012.07.003.  Google Scholar

[25]

P. D. Larson, The economic transportation quantity, Transportation Journal, 28 (1988), 43-48. Google Scholar

[26]

C. Lee, The economic order quantity for freight discount costs, IIE Transactions, 18 (1986), 318-320. doi: 10.1080/07408178608974710.  Google Scholar

[27]

S. A. Lippman, Optimal inventory policy with multiple set-up costs, Management Science, 16 (1969), 118-138. doi: 10.1287/mnsc.16.1.118.  Google Scholar

[28]

S. A. Lippman, Economic order quantities and multiple set-up costs, Management Science, 18 (1971), 39-47. doi: 10.1287/mnsc.18.1.39.  Google Scholar

[29]

A. Mendoza and J. A. Ventura, Incorporating quantity discounts to the EOQ model with transportation costs, International Journal of Production Economics, 113 (2008), 754-765. doi: 10.1016/j.ijpe.2007.10.010.  Google Scholar

[30]

J. A. Muckstadt and A. Sapra, Principles of Inventory Management: When You Are Down to Four Order More, Springer, New York, 2010. doi: 10.1007/978-0-387-68948-7.  Google Scholar

[31]

S. Nahmias, Production and Operations Analysis (Third Edition), Irwin/McGraw-Hill, New York, 1997. Google Scholar

[32]

E. Porteus, Handbooks in Operations Research and Management Science, Volume 2, Stochastic Models, chapter Stochastic Inventory Theory, 605-652, North-Holland, Amsterdam, 1990, Editors: Heyman, D.P and Sobel, M.J.  Google Scholar

[33]

B. Q. Rieksts and J. A. Ventura, Two-stage inventory models with a bi-modal transportation cost, Computers & Operations Research, 37 (2010), 20-31. doi: 10.1016/j.cor.2009.02.026.  Google Scholar

[34]

B. Q. Rieskts and J. A. Ventura, Optimal inventory policies with two modes of freight transportation, European Journal of Operational Research, 186 (2008), 576-585. doi: 10.1016/j.ejor.2007.01.042.  Google Scholar

[35]

A. Roberts and E. Varberg, Convex Functions, Academic Press, New York, 1973.  Google Scholar

[36]

R. T. Rockafellar, Convex Analysis, Princeton University Press, New Jersey, 1997.  Google Scholar

[37]

R. M. Russell and L. J. Krajewski, Optimal purchase and transportation cost lot sizing for a single item, Decision Sciences, 22 (1991), 940-952. doi: 10.1111/j.1540-5915.1991.tb00373.x.  Google Scholar

[38]

E. A. Silver, D. F. Pyke and R. Peterson, Inventory Management and Production Planning and Scheduling, John Wiley and Sons, 1998. Google Scholar

[39]

S. R. Swenseth and M. R. Godfrey, Incorporating transportation costs into inventory replenishment decisions, International Journal of Production Economics, 77 (2002), 113-130. doi: 10.1016/S0925-5273(01)00230-4.  Google Scholar

[40]

R. J. Tersine and S. Barman, Economic inventory/transport lot sizing with quantity and freight rate discounts, Decision Sciences, 22 (1991), 1171-1179. doi: 10.1111/j.1540-5915.1991.tb01914.x.  Google Scholar

[41]

A. Toptal, Replenishment decisions under an all-units discount schedule and stepwise freight costs, European Journal of Operational Research, 198 (2009), 504-510. doi: 10.1016/j.ejor.2008.09.037.  Google Scholar

[42]

A. Toptal and S. Bingöl, Transportation pricing of a truckload carrier, European Journal of Operational Research, 214 (2011), 559-567. doi: 10.1016/j.ejor.2011.05.005.  Google Scholar

[43]

A. F. Veinott Jr, The status of mathematical inventory theory, Management Science, 12 (1966), 745-777. doi: 10.1287/mnsc.12.11.745.  Google Scholar

[44]

P. H. Zipkin, Foundations of Inventory Management, McGraw-Hill, New York, 2000. Google Scholar

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