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

• * Corresponding author
• 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.

Mathematics Subject Classification: Primary: 90B05, 90C30; Secondary: 90C25.

 Citation: • • 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           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           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           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           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           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           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           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           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     $\%$ $^*$ 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

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

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

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

Table 6.  Sensitivity analysis for discount threshold

 $M$ 50 200 400 600 $\infty$ $TC$ 12917 7978.37 10546 13848 11180
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