Continuous Review Perishable Inventory System with OneSupplier, One Retailer andPositive Lead Time

Document Type: Research Paper


1 Department of Applied Mathematics and Statistics,Madurai Kamaraj University,Madurai, India.

2 Department of Applied Mathematics and Statistics, Madurai Kamaraj University, Madurai, India.


We consider a two-echelon single commodity inventory system with one warehouse(supplier) at the higher echelon and one retailer at the lower echelon. The retailer stocks individual items of a commodity and satisfies unit demands which occur according to a Poisson process. The supplier stocks these items in packets and uses these packetsto: (i) satisfy the demands that occur for single packet and that form an independent Poisson process and (ii) replenish the retailer's stock. The supplier implements (s,S)ordering policy to replenish the stock of packets and the lead time is assumed to have exponential distribution. Though the retailer's stock is replenished instantaneously by the supplier if packets are available, and a random stock out period may occur at the retailer node when the supplier has zero stock. It is assumed that because of better stocking facility that is usually available at the warehouse, the items do not perish at the warehouse, but they do perish at the retailer node. It is also assumed that the items have exponential life time distribution at the lower echelon. The joint probability distribution of the inventory levels on both nodes is obtained in the steady state. Various system performance measures are calculated. The long run total expected cost per unit time is derived. These results are illustrated with numerical examples. Some special cases are discussed in detail.


Main Subjects

[1] Andersson J., Melchiors P. (2001), A two-echelon inventory model with lost sales;International
Journal of Production Economics 69(3); 307-315.
[2] Axsäter S. (1990), Simple solution procedures for a class of two-echelon inventory problems;
Operations research38(1); 64-69.
[3] Axsäter S., Marklund J. (2008), Optimal position-based warehouse ordering in divergent two-echelon
inventory systems;Operations research56(4); 976-991.
[4] BeamonB.M. (1998), Supply chain design and analysis;International Journal of Production
Economics55(3); 281-294.
[5] Bykadorov I., Ellero A., Moretti E., Vianello S. (2009), The role of retailer's performance in optimal
wholesale price discount policies;European Journal of Operational Research 194(2); 538-550.
[6] Clark A.J., Scarf H. (1960), Optimal policies for a multi-echelon inventory problem;Management
Science 6(4); 475-490.
[7] Clark A.J. (1972), An informal survey of multi-echelon inventory theory;Naval Research Logistics
Quarterly 19(4); 621-650.
[8] Duc T.T.H., Luong H.T., Kim Y.D. (2008), A measure of bull-whip effect in supply chains with a
mixed autoregressive-moving average demand process;European Journal of Operational Research
187(1); 243-256.

[9] Goyal S.K., Tao C.C. (2009), Optimal ordering and transfer policy for an inventory with stock
dependent demand;European Journal of Operational Research 196(1); 177-185.
[10] Graves S.C. (1985), A multi-echelon inventory model for a repairable item with one-for-one
replenishment;Management Science 31(10), 1247-1256.
[11] Hill R.M., Seifbarghy M., Smith D.K. (2007), A two-echelon inventory model with lost sales;
European Journal of Operational Research 181(2); 753-766.
[12] Kalpakam S.,Arivarignan G. (1988), A continuous review perishable inventory model;Statistics 19(3);
[13] Kogan K.,Herbon A. (2008), A supply chain under limited-time promotion: The effect of customer
sensitivity;European Journal of Operational Research 188(1); 273-292.
[14] Ming T.C., Li D., Yan H. (2008), Mean-variance analysis of a single supplier and retailer supply chain
under a returns policy;European Journal of Operational Research184(1); 356-376.
[15] Olsson R.J.,Hill R.M. (2007), A two-echelon base-stock inventory model with Poisson demand and the
sequential processing of orders at the upper echelon;European Journal of Operational Research
177(1); 310-324.
[16] Powell S.G., Pyke D.F. (1998), Buffering unbalanced assembly systems;IIE Transactions 30(1); 55-
[17] Roundy R.O. (1985),98%-Effective integer-ratio lot-sizing for one-warehouse multi-retailer
systems;Management Science 31(11); 1416-1430.
[18] Simon R.M.(1971), Stationary properties of a two echelon inventory model for low demand items;
Operations Research19; 761-777.
[19] Sivazilian S.D. (1974), A continuous review (s,S) inventory system with arbitrary inter arrival
distribution between unit demands;Operations Research 22; 65-71.
[20] Szmerekovsky J.G., Zhang J. (2009), Pricing and two-tier advertising with one manufacturer and one
retailer;European Journal of Operational Research 192(3); 904-917.
[21] Wong H., Kranenburg B., Houtum G.J. van,Cattrysse D. (2007), Efficient heuristics for two-echelon
spare parts inventory systems with an aggregate mean waiting time constraint per local warehouse;OR
Spectrum 29; 699-722.