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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  August 2018

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, doi: 10.3934/jimo.2018126
References:
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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.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

[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).

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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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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