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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[19]

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

[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.

[21]

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

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[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.

[35]

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

[36]

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

[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.

[38]

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

[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.

[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.

[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.

[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.

[43]

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

[44]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[19]

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

[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.

[21]

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

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[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.

[35]

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

[36]

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

[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.

[38]

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

[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.

[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.

[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.

[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.

[43]

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

[44]

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

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