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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:
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##### References:
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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. Choudhury, B. Karmakar, M. 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. Choudri, M. 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. Chowdhury, S. 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. Ghoreishi, G.-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. Ghosh, T. 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. Jaggi, K. 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] Jaggi, Pareek, Goel 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. Jamal, B. 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. Jana, B. 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. Kumar, A. 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. Li, S. 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. 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Inventory level $(I)$ vs. Time.
Purchase cost vs. Ordering quantity$(Q_i)$
Sensitivity analysis for backlogging rate
Sensitivity analysis for deterioration rate
Sensitivity analysis for rework percentage
Sensitivity analysis for discount threshold
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
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 $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
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 $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
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 $22\%$
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
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
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
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
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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