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October  2019, 15(4): 1857-1879. doi: 10.3934/jimo.2018126

An economic order quantity for deteriorating items with allowable rework of deteriorated products

Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran

* Corresponding author

Received  December 2017 Revised  April 2018 Published  July 2019

This paper presents an inventory model for deteriorating items with variable demand when shortage is permitted and quantity discount in purchase cost, and rework on deteriorating products are also allowed. The main idea of this research is to study the effects of the discount and the rework on the inventory costs. In this paper, it is assumed that for a certain quantity of purchased items, the seller would offer a discount and the manager would have the choice to either accept the discount or dismiss. On the other hand, there is also a similar decision-making scenario, where the manager makes a decision to reduce the total costs by using the rework and reducing the shortage periods or reducing the total costs by ignoring the rework cost and increasing the shortage periods. The implementation of the mathematical model is illustrated with a numerical example and sensitivity analysis describes the effects of the parameters on the total costs. The results show that the rework will decrease the total costs of the inventory system, significantly.

Citation: Mahdi Karimi, Seyed Jafar Sadjadi, Alireza Ghasemi Bijaghini. An economic order quantity for deteriorating items with allowable rework of deteriorated products. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1857-1879. doi: 10.3934/jimo.2018126
References:
[1]

S. Aggarwal and C. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, Journal of the Operational Research Society, 46 (1995), 658-662.   Google Scholar

[2]

A. K. BhuniaA. A. Shaikh and R. Gupta, A study on two-warehouse partially backlogged deteriorating inventory models under inflation via particle swarm optimisation, International Journal of Systems Science, 46 (2015), 1036-1050.  doi: 10.1080/00207721.2013.807385.  Google Scholar

[3]

A. K. BhuniaA. A. Shaikh and L. Sahoo, A two-warehouse inventory model for deteriorating item under permissible delay in payment via particle swarm optimisation, International Journal of Logistics Systems and Management, 24 (2016), 45-69.  doi: 10.1504/IJLSM.2016.075662.  Google Scholar

[4]

A. K. BhuniaA. A. ShaikhG. Sharma and S. Pareek, A two storage inventory model for deteriorating items with variable demand and partial backlogging, Journal of Industrial and Production Engineering, 32 (2015), 263-272.  doi: 10.1080/21681015.2015.1046508.  Google Scholar

[5]

M. Bounkhel, Nonlinear receding horizon control of production inventory systems with deteriorating items, Yugoslav Journal of Operations Research, 18 (2018), 37-45.  doi: 10.2298/YJOR0801037B.  Google Scholar

[6]

C. K. ChanW. H. WongA. Langevin and Y. Lee, An integrated production-inventory model for deteriorating items with consideration of optimal production rate and deterioration during delivery, International Journal of Production Economics, 189 (2017), 1-13.  doi: 10.1016/j.ijpe.2017.04.001.  Google Scholar

[7]

C.-T. Chang, Inventory models with stock-and pricedependent demand for deteriorating items based on limited shelf space, Yugoslav Journal of Operations Research, 20 (2016). Google Scholar

[8]

C.-T. ChangL.-Y. Ouyang and J.-T. Teng, An EOQ model for deteriorating items under supplier credits linked to ordering quantity, Applied Mathematical Modelling, 27 (2003), 983-996.  doi: 10.1016/S0307-904X(03)00131-8.  Google Scholar

[9]

H. J. Chang, A partial backlogging inventory model for non-instantaneous deteriorating items with stock-dependent consumption rate under inflation, Yugoslav Journal of Operations Research, 20 (2016). Google Scholar

[10]

S.-C. Chen and J.-T. Teng, Inventory and credit decisions for time-varying deteriorating items with up-stream and down-stream trade credit financing by discounted cash flow analysis, European Journal of Operational Research, 243 (2015), 566-575.  doi: 10.1016/j.ejor.2014.12.007.  Google Scholar

[11]

Z. Chen and B. R. Sarker, Integrated production-inventory and pricing decisions for a single-manufacturer multi-retailer system of deteriorating items under JIT delivery policy, The International Journal of Advanced Manufacturing Technology, 89 (2017), 2099-2117.   Google Scholar

[12]

K. D. ChoudhuryB. KarmakarM. Das and T. K. Datta, An inventory model for deteriorating items with stock-dependent demand time-varying holding cost and shortages, Opsearch, 52 (2015), 55-74.  doi: 10.1007/s12597-013-0166-x.  Google Scholar

[13]

V. ChoudriM. Venkatachalam and S. Panayappan, Production inventory model with deteriorating items, two rates of production cost and taking account of time value of money, Journal of Industrial and Management Optimization, 12 (2016), 1153-1172.  doi: 10.3934/jimo.2016.12.1153.  Google Scholar

[14]

R. R. ChowdhuryS. Ghosh and K. Chaudhuri, An inventory model for deteriorating items with stock and price sensitive demand, International Journal of Applied and Computational Mathematics, 1 (2015), 187-201.  doi: 10.1007/s40819-014-0011-9.  Google Scholar

[15]

P. Ghare and G. Schrader, An inventory model for deteriorating item for exponentially deteriorating items, Journal of Industrial Engineering, 14 (1963), 238-243.   Google Scholar

[16]

Y. Ghiami and T. Williams, A two-echelon production-inventory model for deteriorating items with multiple buyers, International Journal of Production Economics, 159 (2015), 233-240.  doi: 10.1016/j.ijpe.2014.09.017.  Google Scholar

[17]

M. GhoreishiG.-W. Weber and A. Mirzazadeh, An inventory model for non-instantaneous deteriorating items with partial backlogging, permissible delay in payments, inflation-and selling price-dependent demand and customer returns, Annals of Operations Research, 226 (2015), 221-238.  doi: 10.1007/s10479-014-1739-7.  Google Scholar

[18]

S. K. GhoshT. Sarkar and K. Chaudhuri, A multi-item inventory model for deteriorating items in limited storage space with stock-dependent demand, American Journal of Mathematical and Management Sciences, 34 (2015), 147-161.  doi: 10.1080/01966324.2014.980870.  Google Scholar

[19]

S. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory, European Journal of Operational Research, 134 (2001), 1-16.  doi: 10.1016/S0377-2217(00)00248-4.  Google Scholar

[20]

K.-L. Hou, An inventory model for deteriorating items with stock-dependent consumption rate and shortages under inflation and time discounting, European Journal of Operational Research, 168 (2006), 463-474.  doi: 10.1016/j.ejor.2004.05.011.  Google Scholar

[21]

T.-P. Hsieh and C.-Y. Dye, Optimal dynamic pricing for deteriorating items with reference price effects when inventories stimulate demand, European Journal of Operational Research, 262 (2017), 136-150.  doi: 10.1016/j.ejor.2017.03.038.  Google Scholar

[22]

Y.-F. Huang, Optimal retailer's ordering policies in the EOQ model under trade credit financing, Journal of the operational Research Society, 54 (2003), 1011-1015.  doi: 10.1057/palgrave.jors.2601588.  Google Scholar

[23]

C. K. JaggiK. Aggarwal and S. K. Goel, Optimal order policy for deteriorating items with inflation induced demand, International Journal of Production Economics, 10 (2006), 707-714.  doi: 10.1016/j.ijpe.2006.01.004.  Google Scholar

[24]

JaggiPareekGoel and Nidhi, An inventory model for deteriorating items with ramp type demand under fuzzy environment, International Journal of Logistics Systems and Management, 22 (2015), 436-463.  doi: 10.1504/IJLSM.2015.072748.  Google Scholar

[25]

A. JamalB. Sarker and S. Wang, An ordering policy for deteriorating items with allowable shortage and permissible delay in payment, Journal of the operational Research Society, 48 (1997), 826-833.   Google Scholar

[26]

D. K. JanaB. Das and M. Maiti, Multi-item partial backlogging inventory models over random planning horizon in random fuzzy environment, Applied Soft Computing, 21 (2014), 12-27.   Google Scholar

[27]

N. Kumar and S. Kumar, Effect of learning and salvage worth on an inventory model for deteriorating items with inventory-dependent demand rate and partial backlogging with capability constraints, Uncertain Supply Chain Management, 4 (2016), 123-136.  doi: 10.5267/j.uscm.2015.11.002.  Google Scholar

[28]

S. Kumar and U. Rajput, Fuzzy inventory model for deteriorating items with time dependent demand and partial backlogging, Applied Mathematics, 6 (2015), Article ID 54567, 13 pages. doi: 10.4236/am.2015.63047.  Google Scholar

[29]

S. KumarA. K. Singh and M. K. Patel, Optimization of Weibull deteriorating items inventory model under the effect of price and time dependent demand with partial backlogging, Sadhana, 41 (2016), 977-984.   Google Scholar

[30]

Y. LiS. Zhang and J. Han, Dynamic pricing and periodic ordering for a stochastic inventory system with deteriorating items, Automatica, 76 (2017), 200-213.  doi: 10.1016/j.automatica.2016.11.003.  Google Scholar

[31]

J.-J. Liao, A note on an EOQ model for deteriorating items under supplier credit linked to ordering quantity, Applied Mathematical Modelling, 31 (2007), 1690-1699.  doi: 10.1016/j.apm.2006.05.003.  Google Scholar

[32]

G. C. Mahata, An EPQ-based inventory model for exponentially deteriorating items under retailer partial trade credit policy in supply chain, Expert Systems with Applications, 39 (2012), 3537-3550.  doi: 10.1016/j.eswa.2011.09.044.  Google Scholar

[33]

R. MaihamiB. Karimi and S. M. Ghomi, Pricing and Inventory Control in a Supply Chain of Deteriorating Items: A Non-cooperative Strategy with Probabilistic Parameters, International Journal of Applied and Computational Mathematics, 3 (2017), 2477-2499.  doi: 10.1007/s40819-016-0250-z.  Google Scholar

[34]

W. A. Mandal and S. Islam, Fuzzy inventory model for weibull deteriorating items, with time depended demand, shortages, and partially backlogging, Pak. J. Stat. Oper. Res., 12 (2016), 101-109.  doi: 10.18187/pjsor.v12i1.1153.  Google Scholar

[35]

V. K. Mishra, Inventory Model of Deteriorating Items with Revenue Sharing on Preservation Technology Investment under Price Sensitive Stock Dependent Demand, International Journal of Mathematical Modelling and Computations, 21 (2016), 37-48. Google Scholar

[36]

D. J. MohantyR. S. Kumar and A. Goswami, A two-warehouse inventory model for non-instantaneous deteriorating items over stochastic planning horizon, Journal of Industrial and Production Engineering, 33 (2016), 516-532.  doi: 10.1080/21681015.2016.1176964.  Google Scholar

[37]

B. Mukherjee and A. Bansal, An Approach for Developing an Optimum Quantity Discount Policy of Deteriorating Items Inventory Transportation System, International Journal of Innovative Technology and Research, 5 (2017), 5811-5816. Google Scholar

[38]

B. Naik and R. Patel, Deteriorating items inventory model with different deterioration rates for imperfect quality items and shortages, International Journal of Computational and Applied Mathematics, 12 (2017), 273-284.   Google Scholar

[39]

S. PalG. Mahapatra and G. Samanta, A production inventory model for deteriorating item with ramp type demand allowing inflation and shortages under fuzziness, Economic Modelling, 46 (2015), 334-345.  doi: 10.1016/j.econmod.2014.12.031.  Google Scholar

[40]

X. Pan and S. Li, Optimal control of a stochastic production-inventory system under deteriorating items and environmental constraints, International Journal of Production Research, 53 (2015), 607-628.  doi: 10.1080/00207543.2014.961201.  Google Scholar

[41]

S. Papachristos and K. Skouri, An optimal replenishment policy for deteriorating items with time-varying demand and partial-exponential type-backlogging, Operations Research Letters, 27 (2000), 175-184.  doi: 10.1016/S0167-6377(00)00044-4.  Google Scholar

[42]

P. Pathma Raja, A study on factors affecting quality deterioration in housing construction industry in Klang Valley, in Open University Malaysia. Google Scholar

[43]

M. PervinG. C. Mahata and S. K. Roy, An inventory model with declining demand market for deteriorating items under a trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016), 243-251.  doi: 10.1080/17509653.2015.1081082.  Google Scholar

[44]

M. PervinS. K. Roy and G.-W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460.  doi: 10.1007/s10479-016-2355-5.  Google Scholar

[45]

M. PervinS. K. Roy and G.-W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control and Optimization, 7 (2017), 21-50.  doi: 10.3934/naco.2017002.  Google Scholar

[46]

R. S. Rajan and R. Uthayakumar, Optimal pricing and replenishment policies for instantaneous deteriorating items with backlogging and trade credit under inflation, Journal of Industrial Engineering International, 13 (2017), 428-443.   Google Scholar

[47]

A. Roy, Fuzzy inventory model for deteriorating items with price dependent demand, International Journal of Management Science and Engineering Management, 10 (2015), 237-241.  doi: 10.1080/17509653.2014.959086.  Google Scholar

[48]

G. Samanta, A production inventory model with deteriorating items and shortages, Yugoslav Journal of Operations Research, 14 (2016). Google Scholar

[49]

P. Samanta, J. Das and S. Indrajitsingha, Fuzzy Inventory Model for Two Parameter Weibull Deteriorating Items, (2017). Google Scholar

[50]

B. Sarkar, A production-inventory model with probabilistic deterioration in two-echelon supply chain management, Applied Mathematical Modelling, 37 (2017), 3138-3151.  doi: 10.1016/j.apm.2012.07.026.  Google Scholar

[51]

N. H. Shah and L. E. Cárdenas-Barrón, Retailer's decision for ordering and credit policies for deteriorating items when a supplier offers order-linked credit period or cash discount, Applied Mathematics and Computation, 259 (2015), 569-578.  doi: 10.1016/j.amc.2015.03.010.  Google Scholar

[52]

H. ShavandiH. Mahlooji and N. E. Nosratian, A constrained multi-product pricing and inventory control problem, Applied Soft Computing, 12 (2012), 2454-2461.   Google Scholar

[53]

S. W. ShinnH. Hwang and S. P. Sung, Joint price and lot size determination under conditions of permissible delay in payments and quantity discounts for freight cost, European Journal of Operational Research, 91 (1996), 528-542.   Google Scholar

[54]

M. Srichandan, Inventory Model of Deteriorating Items for Linear Holding Cost with Time Dependent Demand, (2015). Google Scholar

[55]

A. A. Taleizadeh, An economic order quantity model for deteriorating item in a purchasing system with multiple prepayments, Applied Mathematical Modelling, 38 (2014), 5357-5366.  doi: 10.1016/j.apm.2014.02.014.  Google Scholar

[56]

A. A. Taleizadeh and M. Nematollahi, An inventory control problem for deteriorating items with back-ordering and financial considerations, Applied Mathematical Modelling, 38 (2014), 93-109.  doi: 10.1016/j.apm.2013.05.065.  Google Scholar

[57]

A. A. TaleizadehM. Noori-daryan and L. E. Cárdenas-Barrón, Joint optimization of price, replenishment frequency, replenishment cycle and production rate in vendor managed inventory system with deteriorating items, International Journal of Production Economics, 159 (2015), 285-295.  doi: 10.1016/j.ijpe.2014.09.009.  Google Scholar

[58]

R. TatA. A. Taleizadeh and M. Esmaeili, Developing economic order quantity model for non-instantaneous deteriorating items in vendor-managed inventory (VMI) system, International Journal of Systems Science, 46 (2015), 1257-1268.  doi: 10.1080/00207721.2013.815827.  Google Scholar

[59]

S. TayalS. Singh and R. Sharma, An inventory model for deteriorating items with seasonal products and an option of an alternative market, Uncertain Supply Chain Management, 3 (2015), 69-86.  doi: 10.5267/j.uscm.2014.8.003.  Google Scholar

[60]

J.-T. TengL. E. Cárdenas-BarrónH.-J. ChangJ. Wu and Y. Hu, Inventory lot-size policies for deteriorating items with expiration dates and advance payments, Applied Mathematical Modelling, 40 (2016), 8605-8616.  doi: 10.1016/j.apm.2016.05.022.  Google Scholar

[61]

J.-T. Teng and C.-T. Chang, Optimal manufacturer's replenishment policies in the EPQ model under two levels of trade credit policy, European Journal of Operational Research, 195 (2009), 358-363.  doi: 10.1016/j.ejor.2008.02.001.  Google Scholar

[62]

J.-T. TengL.-Y. Ouyang and L.-H. Chen, A comparison between two pricing and lot-sizing models with partial backlogging and deteriorated items, International Journal of Production Economics, 105 (2007), 190-203.  doi: 10.1016/j.ijpe.2006.03.003.  Google Scholar

[63]

J. T. TengH. Yang and L. Ouyang, On an EOQ model for deteriorating items with time-varying demand and partial backlogging, Journal of the Operational Research Society, 54 (2003), 432-436.   Google Scholar

[64]

J. T. TengM. S. Chern and H. L. Yang, An optimal recursive method for various inventory replenishment models with increasing demand and shortages, Naval Research Logistics, 44 (1997), 791-806.  doi: 10.1002/(SICI)1520-6750(199712)44:8<791::AID-NAV6>3.0.CO;2-5.  Google Scholar

[65]

S. TiwariL. E. Cárdenas-BarrónA. Khanna and C. K. Jaggi, Impact of trade credit and inflation on retailer's ordering policies for non-instantaneous deteriorating items in a two-warehouse environment, International Journal of Production Economics, 176 (2016), 154-169.  doi: 10.1016/j.ijpe.2016.03.016.  Google Scholar

[66]

S. TiwariC. K. JaggiA. K. BhuniaA. A. Shaikh and M. Goh, Two-warehouse inventory model for non-instantaneous deteriorating items with stock-dependent demand and inflation using particle swarm optimization, Annals of Operations Research, 254 (2017), 401-423.  doi: 10.1007/s10479-017-2492-5.  Google Scholar

[67]

W.-C. WangJ.-T. Teng and K.-R. Lou, Seller's optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime, European Journal of Operational Research, 232 (2014), 315-321.  doi: 10.1016/j.ejor.2013.06.027.  Google Scholar

[68]

X. XuQ. Bai and M. Chen, A comparison of different dispatching policies in two-warehouse inventory systems for deteriorating items over a finite time horizon, Applied Mathematical Modelling, 41 (2017), 359-374.  doi: 10.1016/j.apm.2016.08.024.  Google Scholar

[69]

H.-L. Yang, Two-warehouse partial backlogging inventory models for deteriorating items under inflation, International Journal of Production Economics, 103 (2006), 362-370.  doi: 10.1016/j.ijpe.2005.09.003.  Google Scholar

[70]

P.-C. Yang, Pricing strategy for deteriorating items using quantity discount when demand is price sensitive, European Journal of Operational Research, 157 (2004), 389-397.  doi: 10.1016/S0377-2217(03)00241-8.  Google Scholar

[71]

Q. ZhangH. DongJ. Luo and A. Segerstedt, Supply chain coordination with trade credit and quantity discount incorporating default risk, International Journal of Production Economics, 153 (2014), 352-360.  doi: 10.1016/j.ijpe.2014.03.019.  Google Scholar

show all references

References:
[1]

S. Aggarwal and C. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, Journal of the Operational Research Society, 46 (1995), 658-662.   Google Scholar

[2]

A. K. BhuniaA. A. Shaikh and R. Gupta, A study on two-warehouse partially backlogged deteriorating inventory models under inflation via particle swarm optimisation, International Journal of Systems Science, 46 (2015), 1036-1050.  doi: 10.1080/00207721.2013.807385.  Google Scholar

[3]

A. K. BhuniaA. A. Shaikh and L. Sahoo, A two-warehouse inventory model for deteriorating item under permissible delay in payment via particle swarm optimisation, International Journal of Logistics Systems and Management, 24 (2016), 45-69.  doi: 10.1504/IJLSM.2016.075662.  Google Scholar

[4]

A. K. BhuniaA. A. ShaikhG. Sharma and S. Pareek, A two storage inventory model for deteriorating items with variable demand and partial backlogging, Journal of Industrial and Production Engineering, 32 (2015), 263-272.  doi: 10.1080/21681015.2015.1046508.  Google Scholar

[5]

M. Bounkhel, Nonlinear receding horizon control of production inventory systems with deteriorating items, Yugoslav Journal of Operations Research, 18 (2018), 37-45.  doi: 10.2298/YJOR0801037B.  Google Scholar

[6]

C. K. ChanW. H. WongA. Langevin and Y. Lee, An integrated production-inventory model for deteriorating items with consideration of optimal production rate and deterioration during delivery, International Journal of Production Economics, 189 (2017), 1-13.  doi: 10.1016/j.ijpe.2017.04.001.  Google Scholar

[7]

C.-T. Chang, Inventory models with stock-and pricedependent demand for deteriorating items based on limited shelf space, Yugoslav Journal of Operations Research, 20 (2016). Google Scholar

[8]

C.-T. ChangL.-Y. Ouyang and J.-T. Teng, An EOQ model for deteriorating items under supplier credits linked to ordering quantity, Applied Mathematical Modelling, 27 (2003), 983-996.  doi: 10.1016/S0307-904X(03)00131-8.  Google Scholar

[9]

H. J. Chang, A partial backlogging inventory model for non-instantaneous deteriorating items with stock-dependent consumption rate under inflation, Yugoslav Journal of Operations Research, 20 (2016). Google Scholar

[10]

S.-C. Chen and J.-T. Teng, Inventory and credit decisions for time-varying deteriorating items with up-stream and down-stream trade credit financing by discounted cash flow analysis, European Journal of Operational Research, 243 (2015), 566-575.  doi: 10.1016/j.ejor.2014.12.007.  Google Scholar

[11]

Z. Chen and B. R. Sarker, Integrated production-inventory and pricing decisions for a single-manufacturer multi-retailer system of deteriorating items under JIT delivery policy, The International Journal of Advanced Manufacturing Technology, 89 (2017), 2099-2117.   Google Scholar

[12]

K. D. ChoudhuryB. KarmakarM. Das and T. K. Datta, An inventory model for deteriorating items with stock-dependent demand time-varying holding cost and shortages, Opsearch, 52 (2015), 55-74.  doi: 10.1007/s12597-013-0166-x.  Google Scholar

[13]

V. ChoudriM. Venkatachalam and S. Panayappan, Production inventory model with deteriorating items, two rates of production cost and taking account of time value of money, Journal of Industrial and Management Optimization, 12 (2016), 1153-1172.  doi: 10.3934/jimo.2016.12.1153.  Google Scholar

[14]

R. R. ChowdhuryS. Ghosh and K. Chaudhuri, An inventory model for deteriorating items with stock and price sensitive demand, International Journal of Applied and Computational Mathematics, 1 (2015), 187-201.  doi: 10.1007/s40819-014-0011-9.  Google Scholar

[15]

P. Ghare and G. Schrader, An inventory model for deteriorating item for exponentially deteriorating items, Journal of Industrial Engineering, 14 (1963), 238-243.   Google Scholar

[16]

Y. Ghiami and T. Williams, A two-echelon production-inventory model for deteriorating items with multiple buyers, International Journal of Production Economics, 159 (2015), 233-240.  doi: 10.1016/j.ijpe.2014.09.017.  Google Scholar

[17]

M. GhoreishiG.-W. Weber and A. Mirzazadeh, An inventory model for non-instantaneous deteriorating items with partial backlogging, permissible delay in payments, inflation-and selling price-dependent demand and customer returns, Annals of Operations Research, 226 (2015), 221-238.  doi: 10.1007/s10479-014-1739-7.  Google Scholar

[18]

S. K. GhoshT. Sarkar and K. Chaudhuri, A multi-item inventory model for deteriorating items in limited storage space with stock-dependent demand, American Journal of Mathematical and Management Sciences, 34 (2015), 147-161.  doi: 10.1080/01966324.2014.980870.  Google Scholar

[19]

S. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory, European Journal of Operational Research, 134 (2001), 1-16.  doi: 10.1016/S0377-2217(00)00248-4.  Google Scholar

[20]

K.-L. Hou, An inventory model for deteriorating items with stock-dependent consumption rate and shortages under inflation and time discounting, European Journal of Operational Research, 168 (2006), 463-474.  doi: 10.1016/j.ejor.2004.05.011.  Google Scholar

[21]

T.-P. Hsieh and C.-Y. Dye, Optimal dynamic pricing for deteriorating items with reference price effects when inventories stimulate demand, European Journal of Operational Research, 262 (2017), 136-150.  doi: 10.1016/j.ejor.2017.03.038.  Google Scholar

[22]

Y.-F. Huang, Optimal retailer's ordering policies in the EOQ model under trade credit financing, Journal of the operational Research Society, 54 (2003), 1011-1015.  doi: 10.1057/palgrave.jors.2601588.  Google Scholar

[23]

C. K. JaggiK. Aggarwal and S. K. Goel, Optimal order policy for deteriorating items with inflation induced demand, International Journal of Production Economics, 10 (2006), 707-714.  doi: 10.1016/j.ijpe.2006.01.004.  Google Scholar

[24]

JaggiPareekGoel and Nidhi, An inventory model for deteriorating items with ramp type demand under fuzzy environment, International Journal of Logistics Systems and Management, 22 (2015), 436-463.  doi: 10.1504/IJLSM.2015.072748.  Google Scholar

[25]

A. JamalB. Sarker and S. Wang, An ordering policy for deteriorating items with allowable shortage and permissible delay in payment, Journal of the operational Research Society, 48 (1997), 826-833.   Google Scholar

[26]

D. K. JanaB. Das and M. Maiti, Multi-item partial backlogging inventory models over random planning horizon in random fuzzy environment, Applied Soft Computing, 21 (2014), 12-27.   Google Scholar

[27]

N. Kumar and S. Kumar, Effect of learning and salvage worth on an inventory model for deteriorating items with inventory-dependent demand rate and partial backlogging with capability constraints, Uncertain Supply Chain Management, 4 (2016), 123-136.  doi: 10.5267/j.uscm.2015.11.002.  Google Scholar

[28]

S. Kumar and U. Rajput, Fuzzy inventory model for deteriorating items with time dependent demand and partial backlogging, Applied Mathematics, 6 (2015), Article ID 54567, 13 pages. doi: 10.4236/am.2015.63047.  Google Scholar

[29]

S. KumarA. K. Singh and M. K. Patel, Optimization of Weibull deteriorating items inventory model under the effect of price and time dependent demand with partial backlogging, Sadhana, 41 (2016), 977-984.   Google Scholar

[30]

Y. LiS. Zhang and J. Han, Dynamic pricing and periodic ordering for a stochastic inventory system with deteriorating items, Automatica, 76 (2017), 200-213.  doi: 10.1016/j.automatica.2016.11.003.  Google Scholar

[31]

J.-J. Liao, A note on an EOQ model for deteriorating items under supplier credit linked to ordering quantity, Applied Mathematical Modelling, 31 (2007), 1690-1699.  doi: 10.1016/j.apm.2006.05.003.  Google Scholar

[32]

G. C. Mahata, An EPQ-based inventory model for exponentially deteriorating items under retailer partial trade credit policy in supply chain, Expert Systems with Applications, 39 (2012), 3537-3550.  doi: 10.1016/j.eswa.2011.09.044.  Google Scholar

[33]

R. MaihamiB. Karimi and S. M. Ghomi, Pricing and Inventory Control in a Supply Chain of Deteriorating Items: A Non-cooperative Strategy with Probabilistic Parameters, International Journal of Applied and Computational Mathematics, 3 (2017), 2477-2499.  doi: 10.1007/s40819-016-0250-z.  Google Scholar

[34]

W. A. Mandal and S. Islam, Fuzzy inventory model for weibull deteriorating items, with time depended demand, shortages, and partially backlogging, Pak. J. Stat. Oper. Res., 12 (2016), 101-109.  doi: 10.18187/pjsor.v12i1.1153.  Google Scholar

[35]

V. K. Mishra, Inventory Model of Deteriorating Items with Revenue Sharing on Preservation Technology Investment under Price Sensitive Stock Dependent Demand, International Journal of Mathematical Modelling and Computations, 21 (2016), 37-48. Google Scholar

[36]

D. J. MohantyR. S. Kumar and A. Goswami, A two-warehouse inventory model for non-instantaneous deteriorating items over stochastic planning horizon, Journal of Industrial and Production Engineering, 33 (2016), 516-532.  doi: 10.1080/21681015.2016.1176964.  Google Scholar

[37]

B. Mukherjee and A. Bansal, An Approach for Developing an Optimum Quantity Discount Policy of Deteriorating Items Inventory Transportation System, International Journal of Innovative Technology and Research, 5 (2017), 5811-5816. Google Scholar

[38]

B. Naik and R. Patel, Deteriorating items inventory model with different deterioration rates for imperfect quality items and shortages, International Journal of Computational and Applied Mathematics, 12 (2017), 273-284.   Google Scholar

[39]

S. PalG. Mahapatra and G. Samanta, A production inventory model for deteriorating item with ramp type demand allowing inflation and shortages under fuzziness, Economic Modelling, 46 (2015), 334-345.  doi: 10.1016/j.econmod.2014.12.031.  Google Scholar

[40]

X. Pan and S. Li, Optimal control of a stochastic production-inventory system under deteriorating items and environmental constraints, International Journal of Production Research, 53 (2015), 607-628.  doi: 10.1080/00207543.2014.961201.  Google Scholar

[41]

S. Papachristos and K. Skouri, An optimal replenishment policy for deteriorating items with time-varying demand and partial-exponential type-backlogging, Operations Research Letters, 27 (2000), 175-184.  doi: 10.1016/S0167-6377(00)00044-4.  Google Scholar

[42]

P. Pathma Raja, A study on factors affecting quality deterioration in housing construction industry in Klang Valley, in Open University Malaysia. Google Scholar

[43]

M. PervinG. C. Mahata and S. K. Roy, An inventory model with declining demand market for deteriorating items under a trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016), 243-251.  doi: 10.1080/17509653.2015.1081082.  Google Scholar

[44]

M. PervinS. K. Roy and G.-W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460.  doi: 10.1007/s10479-016-2355-5.  Google Scholar

[45]

M. PervinS. K. Roy and G.-W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control and Optimization, 7 (2017), 21-50.  doi: 10.3934/naco.2017002.  Google Scholar

[46]

R. S. Rajan and R. Uthayakumar, Optimal pricing and replenishment policies for instantaneous deteriorating items with backlogging and trade credit under inflation, Journal of Industrial Engineering International, 13 (2017), 428-443.   Google Scholar

[47]

A. Roy, Fuzzy inventory model for deteriorating items with price dependent demand, International Journal of Management Science and Engineering Management, 10 (2015), 237-241.  doi: 10.1080/17509653.2014.959086.  Google Scholar

[48]

G. Samanta, A production inventory model with deteriorating items and shortages, Yugoslav Journal of Operations Research, 14 (2016). Google Scholar

[49]

P. Samanta, J. Das and S. Indrajitsingha, Fuzzy Inventory Model for Two Parameter Weibull Deteriorating Items, (2017). Google Scholar

[50]

B. Sarkar, A production-inventory model with probabilistic deterioration in two-echelon supply chain management, Applied Mathematical Modelling, 37 (2017), 3138-3151.  doi: 10.1016/j.apm.2012.07.026.  Google Scholar

[51]

N. H. Shah and L. E. Cárdenas-Barrón, Retailer's decision for ordering and credit policies for deteriorating items when a supplier offers order-linked credit period or cash discount, Applied Mathematics and Computation, 259 (2015), 569-578.  doi: 10.1016/j.amc.2015.03.010.  Google Scholar

[52]

H. ShavandiH. Mahlooji and N. E. Nosratian, A constrained multi-product pricing and inventory control problem, Applied Soft Computing, 12 (2012), 2454-2461.   Google Scholar

[53]

S. W. ShinnH. Hwang and S. P. Sung, Joint price and lot size determination under conditions of permissible delay in payments and quantity discounts for freight cost, European Journal of Operational Research, 91 (1996), 528-542.   Google Scholar

[54]

M. Srichandan, Inventory Model of Deteriorating Items for Linear Holding Cost with Time Dependent Demand, (2015). Google Scholar

[55]

A. A. Taleizadeh, An economic order quantity model for deteriorating item in a purchasing system with multiple prepayments, Applied Mathematical Modelling, 38 (2014), 5357-5366.  doi: 10.1016/j.apm.2014.02.014.  Google Scholar

[56]

A. A. Taleizadeh and M. Nematollahi, An inventory control problem for deteriorating items with back-ordering and financial considerations, Applied Mathematical Modelling, 38 (2014), 93-109.  doi: 10.1016/j.apm.2013.05.065.  Google Scholar

[57]

A. A. TaleizadehM. Noori-daryan and L. E. Cárdenas-Barrón, Joint optimization of price, replenishment frequency, replenishment cycle and production rate in vendor managed inventory system with deteriorating items, International Journal of Production Economics, 159 (2015), 285-295.  doi: 10.1016/j.ijpe.2014.09.009.  Google Scholar

[58]

R. TatA. A. Taleizadeh and M. Esmaeili, Developing economic order quantity model for non-instantaneous deteriorating items in vendor-managed inventory (VMI) system, International Journal of Systems Science, 46 (2015), 1257-1268.  doi: 10.1080/00207721.2013.815827.  Google Scholar

[59]

S. TayalS. Singh and R. Sharma, An inventory model for deteriorating items with seasonal products and an option of an alternative market, Uncertain Supply Chain Management, 3 (2015), 69-86.  doi: 10.5267/j.uscm.2014.8.003.  Google Scholar

[60]

J.-T. TengL. E. Cárdenas-BarrónH.-J. ChangJ. Wu and Y. Hu, Inventory lot-size policies for deteriorating items with expiration dates and advance payments, Applied Mathematical Modelling, 40 (2016), 8605-8616.  doi: 10.1016/j.apm.2016.05.022.  Google Scholar

[61]

J.-T. Teng and C.-T. Chang, Optimal manufacturer's replenishment policies in the EPQ model under two levels of trade credit policy, European Journal of Operational Research, 195 (2009), 358-363.  doi: 10.1016/j.ejor.2008.02.001.  Google Scholar

[62]

J.-T. TengL.-Y. Ouyang and L.-H. Chen, A comparison between two pricing and lot-sizing models with partial backlogging and deteriorated items, International Journal of Production Economics, 105 (2007), 190-203.  doi: 10.1016/j.ijpe.2006.03.003.  Google Scholar

[63]

J. T. TengH. Yang and L. Ouyang, On an EOQ model for deteriorating items with time-varying demand and partial backlogging, Journal of the Operational Research Society, 54 (2003), 432-436.   Google Scholar

[64]

J. T. TengM. S. Chern and H. L. Yang, An optimal recursive method for various inventory replenishment models with increasing demand and shortages, Naval Research Logistics, 44 (1997), 791-806.  doi: 10.1002/(SICI)1520-6750(199712)44:8<791::AID-NAV6>3.0.CO;2-5.  Google Scholar

[65]

S. TiwariL. E. Cárdenas-BarrónA. Khanna and C. K. Jaggi, Impact of trade credit and inflation on retailer's ordering policies for non-instantaneous deteriorating items in a two-warehouse environment, International Journal of Production Economics, 176 (2016), 154-169.  doi: 10.1016/j.ijpe.2016.03.016.  Google Scholar

[66]

S. TiwariC. K. JaggiA. K. BhuniaA. A. Shaikh and M. Goh, Two-warehouse inventory model for non-instantaneous deteriorating items with stock-dependent demand and inflation using particle swarm optimization, Annals of Operations Research, 254 (2017), 401-423.  doi: 10.1007/s10479-017-2492-5.  Google Scholar

[67]

W.-C. WangJ.-T. Teng and K.-R. Lou, Seller's optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime, European Journal of Operational Research, 232 (2014), 315-321.  doi: 10.1016/j.ejor.2013.06.027.  Google Scholar

[68]

X. XuQ. Bai and M. Chen, A comparison of different dispatching policies in two-warehouse inventory systems for deteriorating items over a finite time horizon, Applied Mathematical Modelling, 41 (2017), 359-374.  doi: 10.1016/j.apm.2016.08.024.  Google Scholar

[69]

H.-L. Yang, Two-warehouse partial backlogging inventory models for deteriorating items under inflation, International Journal of Production Economics, 103 (2006), 362-370.  doi: 10.1016/j.ijpe.2005.09.003.  Google Scholar

[70]

P.-C. Yang, Pricing strategy for deteriorating items using quantity discount when demand is price sensitive, European Journal of Operational Research, 157 (2004), 389-397.  doi: 10.1016/S0377-2217(03)00241-8.  Google Scholar

[71]

Q. ZhangH. DongJ. Luo and A. Segerstedt, Supply chain coordination with trade credit and quantity discount incorporating default risk, International Journal of Production Economics, 153 (2014), 352-360.  doi: 10.1016/j.ijpe.2014.03.019.  Google Scholar

Figure 2.  Inventory level $(I)$ vs. Time.
Figure 1.  Purchase cost vs. Ordering quantity$(Q_i)$
Figure 3.  Sensitivity analysis for backlogging rate
Figure 4.  Sensitivity analysis for deterioration rate
Figure 5.  Sensitivity analysis for rework percentage
Figure 6.  Sensitivity analysis for discount threshold
Table 1. A.  Review of previous works
Number 1 2 3 4 5 6 7 8 9 10
Paper [49] [29] [12] [46] [48] [5] [9] [7] [37] [54]
Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Type Production $\surd$ $\surd$ $\surd$
Three $\surd$
Levels Two $\surd$
one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Product Yes
Inflation Discount $\surd$ $\surd$
and Inflation $\surd$ $\surd$
Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Time Infinity $\surd$ $\surd$
Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Ware Two
-houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Chain Cyclic
Lost Sales
Shortage Part. Backlog $\surd$ $\surd$ $\surd$
Com. Backlog $\surd$ $\surd$ $\surd$
Not Allowed $\surd$ $\surd$ $\surd$ $\surd$
Lead Psitive
Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Deter. Variable $\surd$ $\surd$ $\surd$ $\surd$
rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Fuzzy
Demand Prob.
Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Det. Fixed $\surd$ $\surd$
Number 11 12 13 14 15 16 17 18 19 20
Paper [28] [2] [6] [10] [11] [14] [16] [17] [18] [21]
Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Type Production $\surd$ $\surd$ $\surd$
Three $\surd$ $\surd$ $\surd$
Levels Two $\surd$
one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Product Yes $\surd$
Inflation Discount $\surd$
and Inflation $\surd$
Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Time Infinity $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Ware Two $\surd$
-houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Chain Cyclic
Number 1 2 3 4 5 6 7 8 9 10
Paper [49] [29] [12] [46] [48] [5] [9] [7] [37] [54]
Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Type Production $\surd$ $\surd$ $\surd$
Three $\surd$
Levels Two $\surd$
one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Product Yes
Inflation Discount $\surd$ $\surd$
and Inflation $\surd$ $\surd$
Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Time Infinity $\surd$ $\surd$
Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Ware Two
-houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Chain Cyclic
Lost Sales
Shortage Part. Backlog $\surd$ $\surd$ $\surd$
Com. Backlog $\surd$ $\surd$ $\surd$
Not Allowed $\surd$ $\surd$ $\surd$ $\surd$
Lead Psitive
Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Deter. Variable $\surd$ $\surd$ $\surd$ $\surd$
rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Fuzzy
Demand Prob.
Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Det. Fixed $\surd$ $\surd$
Number 11 12 13 14 15 16 17 18 19 20
Paper [28] [2] [6] [10] [11] [14] [16] [17] [18] [21]
Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Type Production $\surd$ $\surd$ $\surd$
Three $\surd$ $\surd$ $\surd$
Levels Two $\surd$
one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Product Yes $\surd$
Inflation Discount $\surd$
and Inflation $\surd$
Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Time Infinity $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Ware Two $\surd$
-houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Chain Cyclic
Table 1. B.  Review of previous works
Number 11 12 13 14 15 16 17 18 19 20
Paper [28] [2] [6] [10] [11] [14] [16] [17] [18] [21]
Lost Sales $\surd$
Shortage Part. Backlog $\surd$ $\surd$
Com. Backlog $\surd$
Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Lead Psitive $\surd$
Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Deter. Variable $\surd$ $\surd$
rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Fuzzy $\surd$
Demand Prob.
Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Det. Fixed $\surd$ $\surd$
Number 21 22 23 24 25 26 27 28 29 30
Paper [24] [30] [33] [36] [38] [39] [40] [51] [13] [57]
Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Type Production $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Three $\surd$
Levels Two $\surd$ $\surd$ $\surd$
one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Product Yes
Inflation Discount $\surd$
and Inflation $\surd$ $\surd$
Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Time Infinity $\surd$ $\surd$
Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Ware Two $\surd$
-houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Chain Cyclic
Lost Sales
Shortage Part. Backlog $\surd$ $\surd$ $\surd$ $\surd$
Com. Backlog $\surd$
Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Lead Psitive $\surd$
Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Deter. Variable $\surd$ $\surd$ $\surd$
rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Fuzzy
Demand Prob. $\surd$
Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Det. Fixed $\surd$ $\surd$
Number 31 32 33 34 35 36 37 38 39 40
Paper [58] [60] [65] [66] [59] [27] [68] [3] [20] [23]
Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Type Production $\surd$ $\surd$
Three
Levels Two $\surd$ $\surd$ $\surd$
one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Number 11 12 13 14 15 16 17 18 19 20
Paper [28] [2] [6] [10] [11] [14] [16] [17] [18] [21]
Lost Sales $\surd$
Shortage Part. Backlog $\surd$ $\surd$
Com. Backlog $\surd$
Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Lead Psitive $\surd$
Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Deter. Variable $\surd$ $\surd$
rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Fuzzy $\surd$
Demand Prob.
Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Det. Fixed $\surd$ $\surd$
Number 21 22 23 24 25 26 27 28 29 30
Paper [24] [30] [33] [36] [38] [39] [40] [51] [13] [57]
Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Type Production $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Three $\surd$
Levels Two $\surd$ $\surd$ $\surd$
one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Product Yes
Inflation Discount $\surd$
and Inflation $\surd$ $\surd$
Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Time Infinity $\surd$ $\surd$
Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Ware Two $\surd$
-houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Chain Cyclic
Lost Sales
Shortage Part. Backlog $\surd$ $\surd$ $\surd$ $\surd$
Com. Backlog $\surd$
Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Lead Psitive $\surd$
Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Deter. Variable $\surd$ $\surd$ $\surd$
rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Fuzzy
Demand Prob. $\surd$
Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Det. Fixed $\surd$ $\surd$
Number 31 32 33 34 35 36 37 38 39 40
Paper [58] [60] [65] [66] [59] [27] [68] [3] [20] [23]
Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Type Production $\surd$ $\surd$
Three
Levels Two $\surd$ $\surd$ $\surd$
one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Table 1. C.  Review of previous works
Number 31 32 33 34 35 36 37 38 39 40
Paper [58] [60] [65] [66] [59] [27] [68] [3] [20] [23]
Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Product Yes
Inflation Discount $\surd$
and Inflation $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$
Time Infinity $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Ware Two $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
-houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Chain Cyclic
Lost Sales
Shortage Part. Backlog $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Com. Backlog $\surd$ $\surd$ $\surd$
Not Allowed $\surd$
Lead Psitive $\surd$ $\surd$
Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Deter. Variable $\surd$ $\surd$
rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Fuzzy
Demand Prob.
Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Det. Fixed $\surd$ $\surd$ $\surd$ $\surd$
Number 41 42 43 44 45 46 47 48 49 50
Paper [32] [34] [35] [47] [56] [63] [69] [55] [31] [8]
Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Type Production $\surd$
Three
Levels Two $\surd$ $\surd$ $\surd$ $\surd$
one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Product Yes
Inflation Discount
and Inflation $\surd$ $\surd$
Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Time Infinity $\surd$
Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Ware Two $\surd$
-houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Chain Cyclic
Lost Sales
Shortage Part. Backlog $\surd$ $\surd$ $\surd$
Com. Backlog $\surd$ $\surd$ $\surd$ $\surd$
Not Allowed $\surd$ $\surd$ $\surd$
Lead Psitive
Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Number 31 32 33 34 35 36 37 38 39 40
Paper [58] [60] [65] [66] [59] [27] [68] [3] [20] [23]
Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Product Yes
Inflation Discount $\surd$
and Inflation $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$
Time Infinity $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Ware Two $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
-houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Chain Cyclic
Lost Sales
Shortage Part. Backlog $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Com. Backlog $\surd$ $\surd$ $\surd$
Not Allowed $\surd$
Lead Psitive $\surd$ $\surd$
Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Deter. Variable $\surd$ $\surd$
rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Fuzzy
Demand Prob.
Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Det. Fixed $\surd$ $\surd$ $\surd$ $\surd$
Number 41 42 43 44 45 46 47 48 49 50
Paper [32] [34] [35] [47] [56] [63] [69] [55] [31] [8]
Model Ordering $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Type Production $\surd$
Three
Levels Two $\surd$ $\surd$ $\surd$ $\surd$
one $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Multi No $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Product Yes
Inflation Discount
and Inflation $\surd$ $\surd$
Discount Not Allowed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Time Infinity $\surd$
Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Ware Two $\surd$
-houses One $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Chain Cyclic
Lost Sales
Shortage Part. Backlog $\surd$ $\surd$ $\surd$
Com. Backlog $\surd$ $\surd$ $\surd$ $\surd$
Not Allowed $\surd$ $\surd$ $\surd$
Lead Psitive
Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Table 1. D.  Review of previous works
Number 41 42 43 44 45 46 47 48 49 50
Paper [32] [34] [35] [47] [56] [63] [69] [55] [31] [8]
Deter. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Fuzzy
Demand Prob.
Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Det. Fixed $\surd$ $\surd$
Number 51 52 53 54 Percentage(rounded) This Paper
Paper [43] [45] [44] [41] $\%$ $^*$
Model Ordering $\surd$ $\surd$ $\surd$ $73\%$ $\surd$
Type Production $\surd$ $27\%$
Three $10\%$
Levels Two $\surd$ $26\%$
one $\surd$ $\surd$ $\surd$ $64\%$ $\surd$
Multi No $\surd$ $\surd$ $\surd$ $\surd$ $98\%$ $\surd$
Product Yes $2\%$
Inflation Discount $\surd$ $16\%$ $\surd$
and Inflation $\surd$ $24\%$
Discount Not Allowed $\surd$ $\surd$ $\surd$ $68\%$
Time Infinity $27\%$
Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $73\%$ $\surd$
Ware Two $15\%$
-houses One $\surd$ $\surd$ $\surd$ $\surd$ $85\%$ $\surd$
Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $100\%$
Chain Cyclic $0\%$ $\surd$
Lost Sales $2\%$
Shortage Part. Backlog $\surd$ $\surd$ $\surd$ $39\%$ $\surd$
Com. Backlog $22\%$
Not Allowed $\surd$ $37\%$
Lead Psitive $8\%$
Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $92\%$ $\surd$
Deter. Variable $\surd$ $32\%$
rate Fixed $\surd$ $\surd$ $\surd$ $68\%$ $\surd$
Fuzzy $2\%$
Demand Prob. $\surd$ $4\%$
Det. Variable $\surd$ $\surd$ $\surd$ $72\%$ $\surd$
Det. Fixed $22\%$
Number 41 42 43 44 45 46 47 48 49 50
Paper [32] [34] [35] [47] [56] [63] [69] [55] [31] [8]
Deter. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
rate Fixed $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Fuzzy
Demand Prob.
Det. Variable $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Det. Fixed $\surd$ $\surd$
Number 51 52 53 54 Percentage(rounded) This Paper
Paper [43] [45] [44] [41] $\%$ $^*$
Model Ordering $\surd$ $\surd$ $\surd$ $73\%$ $\surd$
Type Production $\surd$ $27\%$
Three $10\%$
Levels Two $\surd$ $26\%$
one $\surd$ $\surd$ $\surd$ $64\%$ $\surd$
Multi No $\surd$ $\surd$ $\surd$ $\surd$ $98\%$ $\surd$
Product Yes $2\%$
Inflation Discount $\surd$ $16\%$ $\surd$
and Inflation $\surd$ $24\%$
Discount Not Allowed $\surd$ $\surd$ $\surd$ $68\%$
Time Infinity $27\%$
Horizon Finitie $\surd$ $\surd$ $\surd$ $\surd$ $73\%$ $\surd$
Ware Two $15\%$
-houses One $\surd$ $\surd$ $\surd$ $\surd$ $85\%$ $\surd$
Suply Open $\surd$ $\surd$ $\surd$ $\surd$ $100\%$
Chain Cyclic $0\%$ $\surd$
Lost Sales $2\%$
Shortage Part. Backlog $\surd$ $\surd$ $\surd$ $39\%$ $\surd$
Com. Backlog $22\%$
Not Allowed $\surd$ $37\%$
Lead Psitive $8\%$
Time Zero $\surd$ $\surd$ $\surd$ $\surd$ $92\%$ $\surd$
Deter. Variable $\surd$ $32\%$
rate Fixed $\surd$ $\surd$ $\surd$ $68\%$ $\surd$
Fuzzy $2\%$
Demand Prob. $\surd$ $4\%$
Det. Variable $\surd$ $\surd$ $\surd$ $72\%$ $\surd$
Det. Fixed $22\%$
Table 2.  Optimal solution of Numerical Example
$n$ $TC$ $i$ $t_i$ $tw_i$ $s_i$
6 12785.79 1 0 0.0310 0.0310
7 12279.73 2 0.0310 0.0314 0.0410
8 7978.37 3 0.0410 0.0410 0.0520
9 14197.62 4 0.0521 0.0522 0.0814
10 14886.37 5 0.0816 0.0817 0.194
11 21056.36 6 0.1965 0.1972 0.2075
7 0.2075 0.2086 0.2335
$n^*$=8 8 0.2345 0.2346 1.3667
$TC^*$=7978.37 9 - 1.3681 2
$n$ $TC$ $i$ $t_i$ $tw_i$ $s_i$
6 12785.79 1 0 0.0310 0.0310
7 12279.73 2 0.0310 0.0314 0.0410
8 7978.37 3 0.0410 0.0410 0.0520
9 14197.62 4 0.0521 0.0522 0.0814
10 14886.37 5 0.0816 0.0817 0.194
11 21056.36 6 0.1965 0.1972 0.2075
7 0.2075 0.2086 0.2335
$n^*$=8 8 0.2345 0.2346 1.3667
$TC^*$=7978.37 9 - 1.3681 2
Table 3.  Sensitivity analysis for backlogging rate
$\beta$ 0(Complete backlogging) 10 20 40 $\infty$(No shortage)
$TC$ 7874.34 7894.87 7978.37 8078.46 8213.75
$\beta$ 0(Complete backlogging) 10 20 40 $\infty$(No shortage)
$TC$ 7874.34 7894.87 7978.37 8078.46 8213.75
Table 4.  Sensitivity analysis for deterioration rate
$\rho$ 0.01 0.016 0.02 0.024 0.03
$TC$ 7975.35 7977.15 7978.37 7979.56 7981.37
$\rho$ 0.01 0.016 0.02 0.024 0.03
$TC$ 7975.35 7977.15 7978.37 7979.56 7981.37
Table 5.  Sensitivity analysis for rework percentage
$\eta$ 0.1 0.16 0.2 0.24 0.3
$TC$ 7978.63 7978.47 7978.37 7978.25 7978.09
$\eta$ 0.1 0.16 0.2 0.24 0.3
$TC$ 7978.63 7978.47 7978.37 7978.25 7978.09
Table 6.  Sensitivity analysis for discount threshold
$M$ 50 200 400 600 $\infty$
$TC$ 12917 7978.37 10546 13848 11180
$M$ 50 200 400 600 $\infty$
$TC$ 12917 7978.37 10546 13848 11180
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