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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product", Sci. Iran., 22(6), pp. 2595{2603 (2015).

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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).

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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).

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inventory system with a minimum order quantity

requirement", Sustainability, 11(18), pp. 1{22 (2019).

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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 stockdependent

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).

338 Z. Dai et al./Scientia Iranica, Transactions E: Industrial Engineering 29 (2022) 320{342

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. E ects 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 Z. Dai et al./Scientia Iranica, Transactions E: Industrial Engineering 29 (2022) 320{342 337 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 modi ed fruit 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 stockdependent

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).

338 Z. Dai et al./Scientia Iranica, Transactions E: Industrial Engineering 29 (2022) 320{342

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. E ects 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