Document Type : Article

**Authors**

^{1}
School of Management, JiuJiang University, 551 Qianjin East Road, 332005, JiuJiang City, JiangXi Province, China

^{2}
School of Economics, JiuJiang University, 551 Qianjin East Road, 332005, JiuJiang City, JiangXi Province, China

**Abstract**

In this paper, two inventory models with starting shortages and without shortages for perishable products in supply chain are proposed. The demand for perishable products is dependent on price and stock. Supply chain is composed of one manufacturer, one distribution center, and one retailer. The objective of these two models is to maximize the average profit per unit time by determining the optimal replenishment cycle, frequency, and quantity. The property of optimal solutions for two cases of two models is discussed to verify the existence of optimal solutions. Algorithms for searching optimal solutions are presented. In order to investigate the effect of parameters on optimal solutions and obtain some management insights, computational experiments with sensitivity analyses are carried out. Finally, conclusions and future researches are provided.

**Keywords**

References:

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24. Coelho, L.C. and Laporte, G. "Optimal joint replenishment, delivery and inventory management policies for perishable products", Comput. Oper. Res., 47, pp. 42-52 (2014).

25. Liu, H., Zhang, J., Zhou, C., et al. "Optimal purchase and inventory retrieval policies for perishable seasonal agricultural products", Omega-Int. J. Manage. Sci., 79, pp. 133-145 (2018).

26. Kaasgari, M.A., Imani, D.M., and Mahmoodjanloo, M. "Optimizing a vendor managed inventory (VMI) supply chain for perishable products by considering discount: Two calibrated meta-heuristic algorithms", Comput. Ind. Eng., 103, pp. 227-241 (2017).

27. Jaggi, C.K., Cardenas-Barron, L.E., Tiwari, S., et al. "Two-warehouse inventory model for deteriorating items with imperfect quality under the conditions of permissible delay in payments", Sci. Iran., 24(1), pp. 390-412 (2017).

28. Teimoury, E. and Kazemi, S.M.M. "An integrated pricing and inventory model for deteriorating products in a two stage supply chain under replacement and shortage", Sci. Iran., 24(1), pp. 342-354 (2017).

29. Sarkar, B. and Sarkar, S. "An improved inventory model with partial backlogging, time varying deterioration and stock-dependent demand", Econ. Model., 30, pp. 924-932 (2013).

30. Panda, S., Saha, S., and Goyal, S.K. "Dilemma of rented warehouse and shelf for inventory systems with displayed stock level dependent demand", Econ. Model., 32, pp. 452-462 (2013).

31. Ghiami, Y., Williams, T., and Wu, Y. "A two-echelon inventory model for a deteriorating item with stock dependent demand, partial backlogging and capacity constraints", Eur. J. Oper. Res., 231(3), pp. 587-597 (2013).

32. Yang, C.T. "An inventory model with both stockdependent demand rate and stock-dependent holding cost rate", Int. J. Prod. Econ., 155, pp. 214-221 (2014).

33. Bhunia, A.K., Shaikh, A.A., Dhaka, V., et al. "An application of genetic algorithm and PSO in an inventory model for single deteriorating item with variable demand dependent on marketing strategy and displayed stock level", Sci. Iran., 25(3), pp. 1641-1655 (2018).

34. Lim, S. "A note on a robust inventory model with stock-dependent demand", J. Oper. Res. Soc., 70(5), pp. 851-866 (2019).

35. Maihami, R. and Kamalabadi, I.N. "Joint pricing and inventory control for non-instantaneous deteriorating items with partial backlogging and time and price dependent demand", Int. J. Prod. Econ., 136(1), pp. 116-122 (2012).

36. Banerjee, S. and Agrawal, S. "Inventory model for deteriorating items with freshness and price dependent demand: optimal discounting and ordering policies", Appl. Math. Model., 52, pp. 53-64 (2017).

37. Jadidi, O., Jaber, M.Y., and Zolfaghari, S. "Joint pricing and inventory problem with price dependent stochastic demand and price discounts", Comput. Ind.Eng., 114, pp. 45-53 (2017).

38. San-Jose, L.A., Sicilia, J., and Alcaide-Lepez-de-Pablo, D. "An inventory system with demand dependent on both time and price assuming backlogged shortages", Eur. J. Oper. Res., 270(3), pp. 889-897 (2018).

39. Johari, M., Hosseini-Motlagh, S.M., Nematollahi, M., et al. "Bi-level credit period coordination for periodic review inventory system with price-credit dependent demand under time value of money", Transp. Res. Pt. e-Logist. Transp. Rev., 114, pp. 270-291 (2018).

40. Mishra, U., Wu, J.Z., and Tseng, M.L. "Effects of a hybrid-price-stock dependent demand on the optimal solutions of a deteriorating inventory system and trade credit policy on re-manufactured product", J. Clean Prod., 241, pp. 1-15 (2019).

41. Chen, L., Chen, X., Keblis, M.F., et al. "Optimal pricing and replenishment policy for deteriorating inventory under stock-level-dependent, time-varying and price-dependent demand", Comput. Ind. Eng., 135, pp. 1294-1299 (2019).

42. Pervin, M., Roy, S.K., and Weber, G.W. "Multi-item deteriorating two-echelon inventory model with priceand stock-dependent demand: A trade-credit policy", J. Ind. Manag. Optim., 15(3), pp. 1345-1373 (2019).

2. Gorji, M.H., Setak, M., and Karimi, H. "Optimizing inventory decisions in a two-level supply chain with order quantity constraints", Appl. Math. Model., 38(3),pp. 814-827 (2014).

3. Chu, Y., You, F., Wassick, J.M., et al. "Simulationbased optimization framework for multi-echelon inventory systems under uncertainty", Comput. Chem. Eng., 73, pp. 1-16 (2015).

4. Panda, D., Maiti, M.K., and Maiti, M. "Two warehouse inventory models for single vendor multiple retailers with price and stock dependent demand", Appl. Math. Model., 34(11), pp. 3571-3585 (2010).

5. Rad, M.A., Khoshalhan, F., and Glock, C.H. "Optimizing inventory and sales decisions in a two-stage supply chain with imperfect production and backorders", Comput. Ind. Eng., 74, pp. 219-227 (2014).

6. Sadeghi, J., Mousavi, S. M., Niaki, S.T. A., et al. "Optimizing a bi-objective inventory model of a three-echelon supply chain using a tuned hybrid bat algorithm", Transp. Res. Pt. e-Logist. Transp. Rev., 70, pp. 274-292 (2014).

7. Alhaj, M.A., Svetinovic, D., and Diabat, A. "A carbon-sensitive two-echelon-inventory supply chain model with stochastic demand", Resour. Conserv. Recycl., 108, pp. 82-87 (2016).

8. Mousavi, S.M., Alikar, N., Niaki, S.T.A., et al. "Optimizing a location allocation-inventory problem in a two-echelon supply chain network: A modified fruit y optimization algorithm", Comput. Ind. Eng., 87, pp. 543-560 (2015).

9. Sarakhsi, M.K., Ghomi, S.F., and Karimi, B. "Joint economic lot-sizing problem for a two-stage supply chain with price-sensitive demand", Sci. Iran., 23(3), pp. 1474-1487 (2016).

10. Taleizadeh, A.A., Satarian, F., and Jamili, A. "Optimal multi-discount selling prices schedule for deteriorating product", Sci. Iran., 22(6), pp. 2595-2603 (2015).

11. Alfares, H.K. and Ghaithan, A.M. "Inventory and pricing model with price-dependent demand, timevaryng holding cost, and quantity discounts", Comput.Ind. Eng., 94, pp. 170-177 (2016).

12. Azadeh, A., Elahi, S., Farahani, M.H., et al. "A genetic algorithm-Taguchi based approach to inventory routing problem of a single perishable product with transshipment", Comput. Ind. Eng., 104, pp. 124-133 (2017).

13. Vahdani, B., Niaki, S.T.A., and Aslanzade, S. "Production-inventory-routing coordination with capacity and time window constraints for perishable products: Heuristic and meta-heuristic algorithms", J. Clean Prod., 161, pp. 598-618 (2017).

14. Cheng, M., Zhang, B., and Wang, G. "Optimal policy for deteriorating items with trapezoidal type demand and partial backlogging", Appl. Math. Model, 35(7), pp. 3552-3560 (2011).

15. Guerrero, W.J., Yeung, T.G., and Gueret, C. "Jointoptimization of inventory policies on a multi-product multi-echelon pharmaceutical system with batching and ordering constraints", Eur. J. Oper. Res., 231(1), pp. 98-108 (2013).

16. Tsai, S.C. and Liu, C.H. "A simulation-based decision support system for a multi-echelon inventory problem with service level constraints", Comput. Oper. Res., 53, pp. 118-127 (2015).

17. Zhao, Y. and Zhao, X. "On human decision behavior in multi-echelon inventory management", Int. J. Prod. Econ., 161, pp. 116-128 (2015).

18. Fichtinger, J. and Yates, N. "A joint network design and multi-echelon inventory optimisation approach for supply chain segmentation", Int. J. Prod. Econ., 209, pp. 103-111 (2017).

19. Ross, A., Khajehnezhad, M., Otieno, W., et al. "Integrated location-inventory modelling under forward and reverse product ows in the used merchandise retail sector: A multi-echelon formulation", Eur. J. Oper. Res., 259(2), pp. 664-676 (2017).

20. Wang, Z., Cui, B., Feng, Q., et al. "An agent-based approach for resources' joint planning in a multi-echelon inventory system considering lateral transshipment", Comput. Ind. Eng., 138, pp. 1-12 (2019).

21. Shen, H., Tian, T., and Zhu, H. "A two-echelon inventory system with a minimum order quantity requirement", Sustainability, 11(18), pp. 1-22 (2019).

22. Duan, Q. and Liao, T.W. "A new age-based replenishment policy for supply chain inventory optimization of highly perishable products", Int. J. Prod. Econ., 145(2), pp. 658-671 (2013).

23. Haijema, R. "Optimal ordering, issuance and disposal policies for inventory management of perishable products", Int. J. Prod. Econ., 157, pp. 158-169 (2014).

24. Coelho, L.C. and Laporte, G. "Optimal joint replenishment, delivery and inventory management policies for perishable products", Comput. Oper. Res., 47, pp. 42-52 (2014).

25. Liu, H., Zhang, J., Zhou, C., et al. "Optimal purchase and inventory retrieval policies for perishable seasonal agricultural products", Omega-Int. J. Manage. Sci., 79, pp. 133-145 (2018).

26. Kaasgari, M.A., Imani, D.M., and Mahmoodjanloo, M. "Optimizing a vendor managed inventory (VMI) supply chain for perishable products by considering discount: Two calibrated meta-heuristic algorithms", Comput. Ind. Eng., 103, pp. 227-241 (2017).

27. Jaggi, C.K., Cardenas-Barron, L.E., Tiwari, S., et al. "Two-warehouse inventory model for deteriorating items with imperfect quality under the conditions of permissible delay in payments", Sci. Iran., 24(1), pp. 390-412 (2017).

28. Teimoury, E. and Kazemi, S.M.M. "An integrated pricing and inventory model for deteriorating products in a two stage supply chain under replacement and shortage", Sci. Iran., 24(1), pp. 342-354 (2017).

29. Sarkar, B. and Sarkar, S. "An improved inventory model with partial backlogging, time varying deterioration and stock-dependent demand", Econ. Model., 30, pp. 924-932 (2013).

30. Panda, S., Saha, S., and Goyal, S.K. "Dilemma of rented warehouse and shelf for inventory systems with displayed stock level dependent demand", Econ. Model., 32, pp. 452-462 (2013).

31. Ghiami, Y., Williams, T., and Wu, Y. "A two-echelon inventory model for a deteriorating item with stock dependent demand, partial backlogging and capacity constraints", Eur. J. Oper. Res., 231(3), pp. 587-597 (2013).

32. Yang, C.T. "An inventory model with both stockdependent demand rate and stock-dependent holding cost rate", Int. J. Prod. Econ., 155, pp. 214-221 (2014).

33. Bhunia, A.K., Shaikh, A.A., Dhaka, V., et al. "An application of genetic algorithm and PSO in an inventory model for single deteriorating item with variable demand dependent on marketing strategy and displayed stock level", Sci. Iran., 25(3), pp. 1641-1655 (2018).

34. Lim, S. "A note on a robust inventory model with stock-dependent demand", J. Oper. Res. Soc., 70(5), pp. 851-866 (2019).

35. Maihami, R. and Kamalabadi, I.N. "Joint pricing and inventory control for non-instantaneous deteriorating items with partial backlogging and time and price dependent demand", Int. J. Prod. Econ., 136(1), pp. 116-122 (2012).

36. Banerjee, S. and Agrawal, S. "Inventory model for deteriorating items with freshness and price dependent demand: optimal discounting and ordering policies", Appl. Math. Model., 52, pp. 53-64 (2017).

37. Jadidi, O., Jaber, M.Y., and Zolfaghari, S. "Joint pricing and inventory problem with price dependent stochastic demand and price discounts", Comput. Ind.Eng., 114, pp. 45-53 (2017).

38. San-Jose, L.A., Sicilia, J., and Alcaide-Lepez-de-Pablo, D. "An inventory system with demand dependent on both time and price assuming backlogged shortages", Eur. J. Oper. Res., 270(3), pp. 889-897 (2018).

39. Johari, M., Hosseini-Motlagh, S.M., Nematollahi, M., et al. "Bi-level credit period coordination for periodic review inventory system with price-credit dependent demand under time value of money", Transp. Res. Pt. e-Logist. Transp. Rev., 114, pp. 270-291 (2018).

40. Mishra, U., Wu, J.Z., and Tseng, M.L. "Effects of a hybrid-price-stock dependent demand on the optimal solutions of a deteriorating inventory system and trade credit policy on re-manufactured product", J. Clean Prod., 241, pp. 1-15 (2019).

41. Chen, L., Chen, X., Keblis, M.F., et al. "Optimal pricing and replenishment policy for deteriorating inventory under stock-level-dependent, time-varying and price-dependent demand", Comput. Ind. Eng., 135, pp. 1294-1299 (2019).

42. Pervin, M., Roy, S.K., and Weber, G.W. "Multi-item deteriorating two-echelon inventory model with priceand stock-dependent demand: A trade-credit policy", J. Ind. Manag. Optim., 15(3), pp. 1345-1373 (2019).

Transactions on Industrial Engineering (E)

January and February 2022Pages 320-342