• Previous Article
    Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs
  • JIMO Home
  • This Issue
  • Next Article
    A closed-form solution for outperformance options with stochastic correlation and stochastic volatility
October  2015, 11(4): 1211-1245. doi: 10.3934/jimo.2015.11.1211

On EOQ cost models with arbitrary purchase and transportation costs

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.
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.

[1]

Biswajit Sarkar, Bijoy Kumar Shaw, Taebok Kim, Mitali Sarkar, Dongmin Shin. An integrated inventory model with variable transportation cost, two-stage inspection, and defective items. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1975-1990. doi: 10.3934/jimo.2017027

[2]

Shuhua Zhang, Longzhou Cao, Zuliang Lu. An EOQ inventory model for deteriorating items with controllable deterioration rate under stock-dependent demand rate and non-linear holding cost. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021156

[3]

Vincent Choudri, Mathiyazhgan Venkatachalam, Sethuraman Panayappan. Production inventory model with deteriorating items, two rates of production cost and taking account of time value of money. Journal of Industrial and Management Optimization, 2016, 12 (3) : 1153-1172. doi: 10.3934/jimo.2016.12.1153

[4]

Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. A two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 21-50. doi: 10.3934/naco.2017002

[5]

Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. An integrated inventory model with variable holding cost under two levels of trade-credit policy. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 169-191. doi: 10.3934/naco.2018010

[6]

Shuren Liu, Qiying Hu, Yifan Xu. Optimal inventory control with fixed ordering cost for selling by internet auctions. Journal of Industrial and Management Optimization, 2012, 8 (1) : 19-40. doi: 10.3934/jimo.2012.8.19

[7]

Lars Grüne, Marleen Stieler. Multiobjective model predictive control for stabilizing cost criteria. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3905-3928. doi: 10.3934/dcdsb.2018336

[8]

Adriana Navarro-Ramos, William Olvera-Lopez. A solution for discrete cost sharing problems with non rival consumption. Journal of Dynamics and Games, 2018, 5 (1) : 31-39. doi: 10.3934/jdg.2018004

[9]

Yanyi Xu, Arnab Bisi, Maqbool Dada. New structural properties of inventory models with Polya frequency distributed demand and fixed setup cost. Journal of Industrial and Management Optimization, 2017, 13 (2) : 931-945. doi: 10.3934/jimo.2016054

[10]

Sudip Adak, G. S. Mahapatra. Effect of reliability on varying demand and holding cost on inventory system incorporating probabilistic deterioration. Journal of Industrial and Management Optimization, 2022, 18 (1) : 173-193. doi: 10.3934/jimo.2020148

[11]

R. Enkhbat , N. Tungalag, A. S. Strekalovsky. Pseudoconvexity properties of average cost functions. Numerical Algebra, Control and Optimization, 2015, 5 (3) : 233-236. doi: 10.3934/naco.2015.5.233

[12]

Giuseppe Buttazzo, Serena Guarino Lo Bianco, Fabrizio Oliviero. Optimal location problems with routing cost. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1301-1317. doi: 10.3934/dcds.2014.34.1301

[13]

Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086

[14]

Tai Chiu Edwin Cheng, Bertrand Miao-Tsong Lin, Hsiao-Lan Huang. Talent hold cost minimization in film production. Journal of Industrial and Management Optimization, 2017, 13 (1) : 223-235. doi: 10.3934/jimo.2016013

[15]

Xiaoli Yang, Jin Liang, Bei Hu. Minimization of carbon abatement cost: Modeling, analysis and simulation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2939-2969. doi: 10.3934/dcdsb.2017158

[16]

Fan Sha, Deren Han, Weijun Zhong. Bounds on price of anarchy on linear cost functions. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1165-1173. doi: 10.3934/jimo.2015.11.1165

[17]

Wei Xu, Liying Yu, Gui-Hua Lin, Zhi Guo Feng. Optimal switching signal design with a cost on switching action. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2531-2549. doi: 10.3934/jimo.2019068

[18]

Onur Şimşek, O. Erhun Kundakcioglu. Cost of fairness in agent scheduling for contact centers. Journal of Industrial and Management Optimization, 2022, 18 (2) : 873-896. doi: 10.3934/jimo.2021001

[19]

Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial and Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043

[20]

Biswajit Sarkar, Arunava Majumder, Mitali Sarkar, Bikash Koli Dey, Gargi Roy. Two-echelon supply chain model with manufacturing quality improvement and setup cost reduction. Journal of Industrial and Management Optimization, 2017, 13 (2) : 1085-1104. doi: 10.3934/jimo.2016063

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (145)
  • HTML views (0)
  • Cited by (5)

[Back to Top]