International Journal of Mathematical, Engineering and Management Sciences

ISSN: 2455-7749

A Joint Return Policy for a Multi-Item Perishable Inventory Model with Deterministic Demands, Return and All-Units Discount

Dharma Lesmono
Department of Mathematics, Universitas Katolik Parahyangan, Indonesia.

Taufik Limansyah
Department of Mathematics, Universitas Katolik Parahyangan, Indonesia.

Neilshan Loedy
Department of Mathematics, Universitas Katolik Parahyangan, Indonesia.


Received on August 06, 2019
Accepted on January 07, 2020


In this paper, we develop a multi-item perishable inventory model with deterministic demands, return and all-units discount. We consider a situation where a retailer sells several products to the customer and orders the products from one supplier. Demands are assumed to be deterministic following an inventory-dependent demand, and the supplier offers all-units discount to the retailer who has an opportunity to return unsold or deteriorated products to the supplier at a certain cost. In order to minimize the total cost for the retailer, the decision variables are the optimal return time and the optimal ordering quantity. Considering a multi-item case as an extension of the model by Setiawan et al. (2018) is the main contribution of this paper. We also develop an algorithm to find the optimal solution of the model. Numerical examples for three items are given to illustrate the model and a sensitivity analysis is performed to study the effect of the changes in parameter values on the optimal solution. We consider two scenarios, one with all-units discount and one with no discount. Within these two scenarios, we consider conditions of individual or joint return time for these three items. It is found that the individual return time with no discount gives the least total inventory cost in the numerical examples. Also, in general increasing the value of holding cost, deteriorating rate, return cost per unit and backorder cost will increase the total inventory cost in all scenarios.

Keywords- Joint return policy, Perishable inventory model, Multi-item, All-units discount.


Lesmono, D., Limansyah, T., & Loedy, N. (2020). A Joint Return Policy for a Multi-Item Perishable Inventory Model with Deterministic Demands, Return and All-Units Discount. International Journal of Mathematical, Engineering and Management Sciences, 5(3), 416-431.

Conflict of Interest

The authors confirm that there is no conflict of interest to declare for this publication.


Research funding from the Directorate of Research and Community Services, Indonesian Directorate General of Higher Education (DP2M-DIKTI) and valuable comments from the reviewers are highly appreciated.


Baker, R.C., & Urban, T.L. (1988a). Single-period inventory dependent demand models. Omega, 16(6), 605-607.

Baker, R.C., & Urban, T.L. (1988b). A deterministic inventory system with an inventory-level-dependent demand rate. Journal of the Operational Research Society, 39(9), 823-831.

Chang, C.-T. (2004). Inventory models with stock-dependent demand and nonlinear holding costs for deteriorating items. Asia-Pacific Journal of Operational Research, 21(04), 435-446.

Chang, C.-T., Chen, Y-J., Tsai, T-R., & Wu, S-J. (2010). Inventory models with stock- and price-dependent demand for deteriorating items based on limited shelf space. Yugoslav Journal of Operations Research 20(1), 55-69.

Duong, L.N.K., Wood, L.C., & Wang, W.Y.C. (2015). A multi-criteria inventory management system for perishable & substitutable products. Procedia Manufacturing, 2, 66-76.

Dye, C.Y. (2012). A finite horizon deteriorating inventory model with two-phase pricing and time-varying demand and cost under trade credit financing using particle swarm optimization. Swarm and Evolutionary Computation, 5, 37-53.

Filina-Dawidowicz, L., & Postan, M. (2016). Optimal inventory control for perishable items under additional cost for deterioration reduction. LogForum, 12(2), 147-156.

Heizer, J., & Render, B. (2014). Operation management sustainability and supply chain management. 11th ed. Pearson, Boston.

Kaliraman, N.K., Raj, R., Chandra, S., & Chaudhry, H. (2015). An EPQ inventory model for deteriorating items with Weibull deterioration under stock dependent demand. International Journal of Scientific & Technology Research, 4(1), 232-236.

Kavithapriya, R., & Senbagam, K. (2018). An EOQ inventory model for two parameter Weibull deterioration with quadratic time dependent demand and shortages. International Journal of Pure and Applied Mathematics, 119(7), 467-478.

Kumar S., Kumar P., & Saini M. (2012). An order level inventory model for deteriorating items with quadratic demand rate and variable holding cost. Inernational Journal of Scientific Research Engineering and Technology, 1(5), 253-263

Levin, R.I., McLaughlin, C.P., Lamone, R.P., & Kottas, J.F. (1972) Productions/Operations management: contemporary policy for managing operating systems. McGraw-Hill, New York.

Li, R., Lan, H., & Mawhinney, J.R. (2010). A review on deteriorating inventory study. Journal of Service Science and Management, 3(1), 117-129.

Loedy, N., Lesmono, D., & Limansyah, T. (2018, November). An inventory-dependent demand model with deterioration, all-units discount, and return. In Journal of Physics: Conference Series (Vol. 1108, No. 1, p. 012010). IOP Publishing.

Mishra, V.K., Singh, L.S., & Kumar, R. (2013). An inventory model for deteriorating items with time-dependent demand and time-varying holding cost under partial backlogging. Journal of Industrial Engineering International, 9(1), 1-5.

Nagare, M., & Dutta, P. (2012). Continuous review model for perishable products with inventory dependent demand. In Proceedings of the International Multi-Conference of Engineers and Computer Scientists (Vol. 2, pp. 14-16). Hong Kong. ISBN: 978-988-19251-9-0.

Nahmias, S. (1982). Perishable inventory theory: a review. Operations Research, 30(4), 680-708.

Pervin, M., Roy, S.K., & Weber, G.W. (2018). Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration. Annals of Operations Research, 260(1-2), 437-460.

Prasad Patnaik, V.V.S., & Patnaik, D.P. (2015). A numerical study on deterministic inventory model for deteriorating items with selling price dependent demand and variable cycle length. Jordan Journal of Mechanical & Industrial Engineering, 9(3), 223-240.

Rathod, K.D., & Bhathawala, P.H. (2013) Inventory model with inventory-level dependent demand rate, variable holding cost and shortages. International Journal of Scientific and Engineering Research, 4(8), 368-372.

Ray, J. (2017). Deterioration and its uncertainty in inventory systems. Global Journal of Pure and Applied Mathematics, 13(8), 4003-4014.

Russell, R.S., & Taylor III, B.W. (2014). Operations and supply chain management. 8th ed. International Student Version. Wiley, Singapore.

Setiawan, S.W., Lesmono, D., & Limansyah, T. (2018). A perishable inventory model with return. IOP Conference Series: Materials Science and Engineering, 335, 012049. doi:10.1088/1757-899X/335/1/012049.

Sharmila, D., & Uthayakumar, R. (2015). Inventory model for deteriorating items involving fuzzy with shortages and exponential demand. International Journal of Supply and Operations Management, 2(3), 888-904.

Shenoy, D., & Rosas, R. (2018) Problems & solutions in inventory management. Springer, Switzerland.

Silver, E.A., & Peterson, R. (1985) Decision systems for inventory management and production planning, 2nd ed, Wiley, New York.

Singh, T., Mishra, P.J., & Pattanayak, H. (2017). An optimal policy for deteriorating items with time-proportional deterioration rate and constant and time-dependent linear demand rate. Journal of Industrial Engineering International, 13(4), 455-463.

Tersine, R.J. (1994) Principles of inventory and materials management, 4th ed., Prentice Hall, New Jersey.

Uthayakumar, R., & Tharani, S. (2017). An economic production model for deteriorating items and time dependent demand with rework and multiple production setups. Journal of Industrial Engineering International, 13(4), 499-512.

Wolfe, H.B. (1968). A model for control of style merchandise. Industrial Management Review, 9(2), 68-82.

Yadav, R. K., & Vats, A. K. (2014). A deteriorating inventory model for quadratic demand and constant holding cost with partial backlogging and inflation. IOSR Journal of Mathematics, 10(3), 47-52.

Privacy Policy| Terms & Conditions