Document Type : Research Paper

**Authors**

Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran

**Abstract**

In this paper we introduce the optimal solution for a simple and yet practical inventory policy with the important characteristic which eliminates the uncertainty in demand for suppliers. In this new policy which is different from the classical inventory policies, the time interval between any two consecutive orders is fixed and the quantity of each order is one. Assuming the fixed ordering costs are negligible, lead times are constant, and demand forms a Poisson process, we use queuing theory concepts to derive the long-run average total inventory costs, consisting of holding and shortage costs in terms of the average inventory. We show that the total cost rate has the important property of being entirely free of the lead time. We prove that the average total cost rate is a convex function and thus has a unique solution. We, then derive the relation for the optimal value of the time interval between any two consecutive orders. Finally we present a numerical example to compare the performance of this new policy with the classical one-for-one ordering policy. The provided example intends to re-examine the optimality of (s, S) policy in continuous review inventory models as well to establish the fact that even for the case where demand forms a Poisson process the optimality does not hold.

**Keywords**

[1] Andersson J., Melchiors P. (2001), A two-echelon inventory model with lost sales;

International Journal of Production Economics 69(3); 307-315.

[2] Archibald B.C. (1981), Continuous review (s, S) policies with lost sales; Management Science

27; 1171-1178.

[3] Axsäter S. (1990), Simple solution procedures for a class of two-echelon inventory problems;

Operation Research 38; 64-69.

[4] Axsäter S. (1993), Exact and approximate evaluation of batch-ordering policies for two-level

inventory systems; Operations .Research 41; 777-785.

[5] Axsäter S. (2003), Approximate optimization of a two-level distribution inventory system;

International Journal of Production Economics 81-82; 545-553.

[6] Axsäter S., Forsberg R., Zhang W.F. (1994), Approximating general multi-echelon inventory

systems by Poisson models; International Journal of Production Economics 35; 201-206.

[7] Axsäter S., Zhang W.F. (1999), A joint replenishment policy for multi-echelon inventory

control; International Journal of Production Economics 59; 243-250.

[8] Beckman M. (1961), An inventory model for arbitrary interval and quantity distribution of

demand; Management Science 8; 35-57.

International Journal of Production Economics 69(3); 307-315.

[2] Archibald B.C. (1981), Continuous review (s, S) policies with lost sales; Management Science

27; 1171-1178.

[3] Axsäter S. (1990), Simple solution procedures for a class of two-echelon inventory problems;

Operation Research 38; 64-69.

[4] Axsäter S. (1993), Exact and approximate evaluation of batch-ordering policies for two-level

inventory systems; Operations .Research 41; 777-785.

[5] Axsäter S. (2003), Approximate optimization of a two-level distribution inventory system;

International Journal of Production Economics 81-82; 545-553.

[6] Axsäter S., Forsberg R., Zhang W.F. (1994), Approximating general multi-echelon inventory

systems by Poisson models; International Journal of Production Economics 35; 201-206.

[7] Axsäter S., Zhang W.F. (1999), A joint replenishment policy for multi-echelon inventory

control; International Journal of Production Economics 59; 243-250.

[8] Beckman M. (1961), An inventory model for arbitrary interval and quantity distribution of

demand; Management Science 8; 35-57.

[9] Federgruen A., Zheng Y. S. (1992), An efficient algorithm for computing an optimal (r, Q)

policy for continuous review stochastic inventory systems; Operations Research 40; 808-812.

[10] Forsberg R. (1995), Optimization of order-up-to-S policies for two-level inventory systems

with compound Poisson demand; European Journal of Operational Research 81; 143-153.

[11] Forsberg, R. (1996), Exact evaluation of (R,Q) -policies for two-level inventory systems with

Poisson demand; European Journal of Operational Research; 96; 130-138.

[12] Forsberg R. (1997), Evaluation of (R, Q) policies for two-level inventory systems with

generally distributed customer inter-arrival times; European Journal of Operational

Research 99; 401-411.

[13] Ganeshan R. (1999), Managing supply chain inventories: A multiple retailer, one warehouse,

multiple supplier model; International Journal of Production Economics 59; 341-354.

[14] Ghalebsaz-Jeddi B, Shultes B. C, Haji R. (2004), A multi-production continuous review

inventory system with stochastic demand, backorders, and a budget constraint; European

Journal of Operational Research 158; 456-469.

[15] Graves S. (1985), A multi-echelon inventory model for a repairable item with one–for-one

replenishment; Management Science 31; 1247-1256.

[16] Hadley G., Whitin. T. (1963), Analysis of inventory systems. Prentice Hall. Englewood

Cliffs. NJ.

[17] Haji R., Pirayesh M., Baboli A. (2006), A new replenishment policy in a two-echelon

inventory system with stochastic demand, International Conference on Service System and

Service Management-ICSSSM’06 Proceeding, 256–260 Troyes, France, 25-27 October

[18] Iglehart. D. (1963), Optimality of (s, S) policies in the infinite horizon dynamic inventory

problem; Management Science 9; 259-267.

[19] Johnson L.A., Montgomery D.C. (1974), Operation research in production planning,

scheduling, and inventory control, John Wiley & Sons, New York.

[20] Kruse W. K. (1980), Waiting time in an (S-1, S) inventory system with arbitrary distributed

lead times; Operations Research 28; 348-352.

[21] Kruse W. K. (1981), Waiting time in a continuous review (s, S) inventory system with

constant lead times; Operations Research 29; 202-206.

[22] Lee H.L., Moinzadeh K. (1987a), Two-parameter approximations for multi-echelon

repairable inventory models with batch ordering policy; IIE Transactions 19; 140-149.

[23] Lee H.L., Moinzadeh K. (1987b), Operating characteristics of a two-echelon inventory

system for repairable and consumable items under batch ordering and shipment policy; Naval

Research Logistics Quarterly 34; 356-380.

[24] Love S. (1979), Inventory control, McGraw-Hill Book Company, New York

policy for continuous review stochastic inventory systems; Operations Research 40; 808-812.

[10] Forsberg R. (1995), Optimization of order-up-to-S policies for two-level inventory systems

with compound Poisson demand; European Journal of Operational Research 81; 143-153.

[11] Forsberg, R. (1996), Exact evaluation of (R,Q) -policies for two-level inventory systems with

Poisson demand; European Journal of Operational Research; 96; 130-138.

[12] Forsberg R. (1997), Evaluation of (R, Q) policies for two-level inventory systems with

generally distributed customer inter-arrival times; European Journal of Operational

Research 99; 401-411.

[13] Ganeshan R. (1999), Managing supply chain inventories: A multiple retailer, one warehouse,

multiple supplier model; International Journal of Production Economics 59; 341-354.

[14] Ghalebsaz-Jeddi B, Shultes B. C, Haji R. (2004), A multi-production continuous review

inventory system with stochastic demand, backorders, and a budget constraint; European

Journal of Operational Research 158; 456-469.

[15] Graves S. (1985), A multi-echelon inventory model for a repairable item with one–for-one

replenishment; Management Science 31; 1247-1256.

[16] Hadley G., Whitin. T. (1963), Analysis of inventory systems. Prentice Hall. Englewood

Cliffs. NJ.

[17] Haji R., Pirayesh M., Baboli A. (2006), A new replenishment policy in a two-echelon

inventory system with stochastic demand, International Conference on Service System and

Service Management-ICSSSM’06 Proceeding, 256–260 Troyes, France, 25-27 October

[18] Iglehart. D. (1963), Optimality of (s, S) policies in the infinite horizon dynamic inventory

problem; Management Science 9; 259-267.

[19] Johnson L.A., Montgomery D.C. (1974), Operation research in production planning,

scheduling, and inventory control, John Wiley & Sons, New York.

[20] Kruse W. K. (1980), Waiting time in an (S-1, S) inventory system with arbitrary distributed

lead times; Operations Research 28; 348-352.

[21] Kruse W. K. (1981), Waiting time in a continuous review (s, S) inventory system with

constant lead times; Operations Research 29; 202-206.

[22] Lee H.L., Moinzadeh K. (1987a), Two-parameter approximations for multi-echelon

repairable inventory models with batch ordering policy; IIE Transactions 19; 140-149.

[23] Lee H.L., Moinzadeh K. (1987b), Operating characteristics of a two-echelon inventory

system for repairable and consumable items under batch ordering and shipment policy; Naval

Research Logistics Quarterly 34; 356-380.

[24] Love S. (1979), Inventory control, McGraw-Hill Book Company, New York

[25] Marklund J., (2002), Centralized inventory control in a two-level distribution system with

Poisson demand; Naval Research Logistics 49;798-822.

[26] Matta K.F., Sinha D. (1995), Policy and cost approximations of two-echelon distribution

systems with a procurement cost at the higher echelon; IIE Transactions 27(5); 646-656.

[27] Moinzadeh K. (2001), An improved ordering policy for continuous review inventory systems

with an arbitrary inter-demand time distributions; IIE Transactions 33; 111-118.

[28] Moinzadeh K. (2002), A multi-echelon inventory system with information exchange,

Management Science 48(3); 414-426.

[29] Moinzadeh; K., Lee H.L. (1986), Batch size and stocking levels in multi-echelon repairable

systems; Management Science 32; 1567-1581.

[30] Nahmias S. (1976), On the equivalence of three approximate continuous review inventory

model; Naval Research Logistics Quarterly 23; 31-38.

[31] Nahmias S. (2005), Production and operations analysis; 5th edition, McGraw-Hill/Irwin; New

York.

[32] Nahmias S., Smith S.A. (1994), Optimizing inventory levels in a two-echelon retailer system

with partial lost sales; Management Science 40; 582-596.

[33] Ross S. M. (1993), Introduction to probability models Fifth Edition, Academic press, New

York.

[34] Sahin I. (1979), On the stationary analysis of continuous review (s, S) inventory systems with

constant lead times; Operations Research 27; 717-729.

[35] Sahin I. (1982), On the objective function behavior in (s, S) inventory models; Operations

Research 30; 709-724.

[36] Sahin I. (1990), Regenerative inventory systems Springer-Verlag, New York. NY.

[37] Scarf H. E. (1960), The optimality of (s, S) policies in the dynamic inventory problem. In

Mathematical Methods in the social Sciences. Stanford University Press. Stanford. CA.

[38] Schultz C. R. (1989), Replenishment delays for expensive slow-moving items; Management

Science 35; 1454-1462.

[39] Seo Y., Jung S,. Hahm J. (2002), Optimal reorder decision utilizing centralized stock

information in a two-echelon distribution system; Computers & Operations Research 29;

171-193.

[40] Sherbrooke C. C. (1968), METRIC: a multi-echelon technique for recoverable item control;

Operations Research 16; 122-141.

[41] Sivlazian B. D. (1974), A continuous review (s, S) inventory system with arbitrary interval

distribution between unit demand; Operations Research 22; 65-71

Poisson demand; Naval Research Logistics 49;798-822.

[26] Matta K.F., Sinha D. (1995), Policy and cost approximations of two-echelon distribution

systems with a procurement cost at the higher echelon; IIE Transactions 27(5); 646-656.

[27] Moinzadeh K. (2001), An improved ordering policy for continuous review inventory systems

with an arbitrary inter-demand time distributions; IIE Transactions 33; 111-118.

[28] Moinzadeh K. (2002), A multi-echelon inventory system with information exchange,

Management Science 48(3); 414-426.

[29] Moinzadeh; K., Lee H.L. (1986), Batch size and stocking levels in multi-echelon repairable

systems; Management Science 32; 1567-1581.

[30] Nahmias S. (1976), On the equivalence of three approximate continuous review inventory

model; Naval Research Logistics Quarterly 23; 31-38.

[31] Nahmias S. (2005), Production and operations analysis; 5th edition, McGraw-Hill/Irwin; New

York.

[32] Nahmias S., Smith S.A. (1994), Optimizing inventory levels in a two-echelon retailer system

with partial lost sales; Management Science 40; 582-596.

[33] Ross S. M. (1993), Introduction to probability models Fifth Edition, Academic press, New

York.

[34] Sahin I. (1979), On the stationary analysis of continuous review (s, S) inventory systems with

constant lead times; Operations Research 27; 717-729.

[35] Sahin I. (1982), On the objective function behavior in (s, S) inventory models; Operations

Research 30; 709-724.

[36] Sahin I. (1990), Regenerative inventory systems Springer-Verlag, New York. NY.

[37] Scarf H. E. (1960), The optimality of (s, S) policies in the dynamic inventory problem. In

Mathematical Methods in the social Sciences. Stanford University Press. Stanford. CA.

[38] Schultz C. R. (1989), Replenishment delays for expensive slow-moving items; Management

Science 35; 1454-1462.

[39] Seo Y., Jung S,. Hahm J. (2002), Optimal reorder decision utilizing centralized stock

information in a two-echelon distribution system; Computers & Operations Research 29;

171-193.

[40] Sherbrooke C. C. (1968), METRIC: a multi-echelon technique for recoverable item control;

Operations Research 16; 122-141.

[41] Sivlazian B. D. (1974), A continuous review (s, S) inventory system with arbitrary interval

distribution between unit demand; Operations Research 22; 65-71

[42] Tijms H.C. (1986), Stochastic modeling and analysis-A computational approach. John Wiley

and Sons, New York.

[43] Zheng Y. S. (1992), On properties of stochastic inventory systems; Management Science 38;

87-130.

[44] Zipkin P. (1986a), Stochastic lead times in continuous review inventory systems; Naval

Research Logistics Quarterly 35; 763-774.

[45] Zipkin P. (1986b), Inventory service level measures: convexity and approximations;

Management Science 32; 975-981.

[46] Zipkin P. H. (2000), Foundation of inventory management. McGraw-Hill, New York

and Sons, New York.

[43] Zheng Y. S. (1992), On properties of stochastic inventory systems; Management Science 38;

87-130.

[44] Zipkin P. (1986a), Stochastic lead times in continuous review inventory systems; Naval

Research Logistics Quarterly 35; 763-774.

[45] Zipkin P. (1986b), Inventory service level measures: convexity and approximations;

Management Science 32; 975-981.

[46] Zipkin P. H. (2000), Foundation of inventory management. McGraw-Hill, New York