# A Sustainable Economic Recycle Quantity Model for Imperfect Production System with Shortages

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## Abstract

**:**

## 1. Introduction and Literature Review

- (i)
- The proposed model is developed focusing on the recycling of defective items collected from regular production after the proper screening with 100% recovery of raw materials used for production in the corresponding item category.
- (ii)
- An inventory model is developed for three-level piecewise constant demand, which varies under three different production time parts, from production run-time to production off-time with positive stock and production off-time with shortages.

## 2. Assumptions and Notation

#### 2.1. Assumptions

- (i)
- The production rate is finite and fixed.
- (ii)
- The defective rate is known, and constant and defective items are produced randomly alongside the perfect product.
- (iii)
- The demand rate of the perfect product is a piecewise constant function (motivated from Bai and Varanasi (1996)):The demand during production-run and production off-times and stock-out period is as follows:$$\{\begin{array}{c}D;\mathrm{during}\mathrm{production}-\mathrm{run}\mathrm{time}\\ xD;\mathrm{during}\mathrm{production}-\mathrm{off}\mathrm{time}\mathrm{with}\mathrm{positive}\mathrm{stock}\\ yD;\mathrm{during}\mathrm{production}-\mathrm{off}\mathrm{time}\mathrm{with}\mathrm{shortages}\end{array}$$
- (iv)
- The sum of the demand rate and defective rate is less than the rate of production.
- (v)
- Lead-time is negligible, and the number of shortages is acceptable while the model is considered for a single items production system.
- (vi)
- Defective items are recyclable with 100% recovery of raw material usable for the production of same product in the next cycle time, and the value of the recycled materials is higher than that of purchased materials.
- (vii)
- The holding cost per defective item is same as that of fresh items, while the recycle cost per defective item is equal irrespective of its number.

#### 2.2. Notation

## 3. Model Formulation

_{1}. Replenishment ends at point A with a maximum number of ${Q}_{d}$ (non-defective) units. Therefore,

_{1}at point K, and the inventory decreases at another demand rate of xD units per unit time, expressed in Figure 2 as line AC during t

_{2}. At the end of time period t

_{2}, the inventory reaches point C with zero on-hand stock. In this case, if the demand rate remains the same as “D”, then the inventory decreases as indicated by the dashed line AB. Therefore,

_{2}, thestock is 0 and then decreases at the demand rate yD units per unit time. During time t

_{3}, shortages reach a total of ${Q}_{s}$ units.

_{s}at point E, production starts. The shortage amount Q

_{s}is served at the rate of (P − D − d) units with the current demand, which is fulfilled at the rate of D during timet

_{4}. After this period, the stock becomes zero again at the point F. Therefore,

- Case 1:
- EPQ model for defective items with three levels of piecewise constant demand under shortages
- Case 2:
- ERQ model for defective items with three levels of piecewise constant demand under shortages

#### 3.1. Case 1 (EPQ Model for Defective Items with Three Levels of Piecewise Constant Demand under Shortages)

- (1)
- Average setup cost or fixed cost (FC):$$\mathrm{FC}\text{}=\text{}\frac{1}{T}{C}_{o}=\frac{\left(2+x+y\right)dD}{4m\left(P-d\right)W}\times {C}_{o}$$
- (2)
- Average production cost (PC):$$\mathrm{PC}\text{}=\text{}\frac{1}{T}Q{C}_{p}=\frac{\left(2+x+y\right)PD}{4m\left(P-d\right)}{C}_{P}$$
- (3)
- Average raw material cost (RMC):Each production lot size is a quantity for which raw materials are purchased for exact production in each time cycle. Therefore,$$\mathrm{RMC}\text{}=\text{}\frac{1}{T}Q{C}_{R}=\frac{\left(2+x+y\right)PD}{4m\left(P-d\right)}{C}_{R}$$
- (4)
- Average holding cost (HC):Defective items have no holding cost in this case. Therefore,Total inventory $=\frac{1}{2}{Q}_{d}\left({t}_{1}+{t}_{2}\right)$And average inventory $=\frac{1}{2}{Q}_{d}\left(\frac{{t}_{1}+{t}_{2}}{T}\right)=\frac{d\left(2+x+y\right)\left\{P-\left(1-x\right)D-d\right\}}{4mx\left(P-D-d\right)\left(P-d\right)}\xb7\frac{{Q}_{d}{}^{2}}{2W}$Therefore,$$\mathrm{HC}\text{}=\text{}\frac{d\left(2+x+y\right)\left\{P-\left(1-x\right)D-d\right\}}{8xm\left(P-d\right)\left(P-D-d\right)W}{\left\{\left(P-D-d\right)\frac{W}{d}-{Q}_{s}\right\}}^{2}{C}_{h}$$
- (5)
- Average shortage cost (SC):$$\mathrm{SC}\text{}=\text{}\frac{{Q}_{s}}{2}\times {C}_{s}\times \frac{{t}_{3}+{t}_{4}}{T}\text{}=\text{}\frac{\left(2+x+y\right)d\left\{P-\left(1-y\right)D-d\right\}}{4my\left(P-D-d\right)\left(P-d\right)}\left(\frac{{Q}_{s}{}^{2}}{2W}\right)\times {C}_{s}$$Hereafter, the total cost function isTCD = FC + PC + RMC + HC + SC$$TCD=\frac{\left(2+x+y\right)}{4m\left(P-d\right)}\left[\begin{array}{c}\frac{dD}{W}{C}_{o}+PD{C}_{P}+PD{C}_{R}+\frac{d\left\{P-\left(1-y\right)D-d\right\}}{2y\left(P-D-d\right)W}{Q}_{s}{}^{2}{C}_{s}\\ +\frac{d\left\{P-\left(1-x\right)D-d\right\}}{2x\left(P-D-d\right)W}{\left\{\left(P-D-d\right)\frac{W}{d}-\text{}{Q}_{s}\right\}}^{2}{C}_{h}\end{array}\right]$$

#### 3.2. Case 2 (ERQ Model for Defective Items with Three Levels of Piecewise Constant Demand under Shortages)

- (1)
- Average production cost (PC) is:$$\mathrm{PC}\text{}=\text{}\frac{1}{T}Q{C}_{p}=\frac{\left(2+x+y\right)PD}{4m\left(P-d\right)}{C}_{P}$$
- (2)
- Average raw material cost (RMC): Each production lot size is $Q$, but raw materials for $\left(Q-W\right)$ units of items in each time cycle are purchased. Therefore,$$\mathrm{RMC}\text{}=\text{}\frac{1}{T}(Q-W){C}_{R}=\frac{\left(2+x+y\right)D}{4m}{C}_{R}$$
- (3)
- Average holding cost (HC):Holding cost of defective items is included during time $\left({t}_{1}+{t}_{4}\right)$ before the items are sent for recycling. Therefore,Total inventory$$=\frac{1}{2}{Q}_{d}\times {t}_{1}+\frac{1}{2}{Q}_{d}\times {t}_{2}+\frac{1}{2}d\left({t}_{1}+{t}_{4}\right)\times \left({t}_{1}+{t}_{4}\right)$$$$=\frac{1}{2}{Q}_{d}\left({t}_{1}+{t}_{2}\right)+\frac{1}{2}d{\left({t}_{1}+{t}_{4}\right)}^{2}$$Average inventory$$=\frac{1}{2}{Q}_{d}\left(\frac{{t}_{1}+{t}_{2}}{T}\right)+\frac{1}{2}d\frac{{\left({t}_{1}+{t}_{4}\right)}^{2}}{T}$$$$=\frac{\left(2+x+y\right)}{4m\left(P-d\right)}\xb7\left[\frac{d\left\{P-\left(1-x\right)D-d\right\}}{2x\left(P-D-d\right)W}{\left\{\left(P-D-d\right)\frac{W}{d}-{Q}_{s}\right\}}^{2}+\frac{DW}{2}\right]$$Hence,$$\mathrm{HC}\text{}=\text{}\frac{\left(2+x+y\right)}{4m\left(P-d\right)}\left[\frac{d\left\{P-\left(1-x\right)D-d\right\}}{2x\left(P-D-d\right)W}{\left\{\left(P-D-d\right)\frac{W}{d}-{Q}_{s}\right\}}^{2}+\frac{DW}{2}\right]{C}_{h}$$

- (4)
- Average shortage cost (SC):$$\mathrm{SC}\text{}=\text{}\frac{{Q}_{s}}{2}\times {C}_{s}\times \frac{{t}_{3}+{t}_{4}}{T}\text{}=\text{}\frac{\left(2+x+y\right)d\left\{P-\left(1-y\right)D-d\right\}}{4my\left(P-D-d\right)\left(P-d\right)}\frac{{Q}_{s}{}^{2}}{2W}\times {C}_{s}$$
- (5)
- Average recycle cost (RC):$$\mathrm{RC}\text{}=\text{}\frac{1}{T}W{C}_{r}=\frac{\left(2+x+y\right)dD}{4m\left(P-d\right)}{C}_{r}$$Hence, the total cost function isTCR = FC + PC + RMC + HC + SC + RC$$\mathrm{TCR}\text{}=\text{}\frac{\left(2+x+y\right)}{4m\left(P-d\right)}\left[\frac{dD}{W}{C}_{o}+PD{C}_{P}+\left(P-d\right)D{C}_{R}\text{}+\frac{d\left\{P-\left(1-x\right)D-d\right\}}{2x\left(P-D-d\right)W}{\left\{\left(P-D-d\right)\frac{W}{d}-{Q}_{s}\right\}}^{2}{C}_{h}+\frac{DW}{2}{C}_{h}+\frac{d\left\{P-\left(1-y\right)D-d\right\}}{y\left(P-D-d\right)}\frac{{Q}_{s}{}^{2}}{2W}\times {C}_{s}+\text{}dD{C}_{r}\right]$$

## 4. Theoretical Derivations

#### 4.1. Case 1 (EPQ Model for Defective Items with Three Levels of Piecewise Constant Demand under Shortages)

**Theorem**

**1.**

^{*}and${Q}_{s}{}^{*}.$

**Proof.**

_{s}, and therefore the first order partial derivatives of TCD with regard to W and ${Q}_{s}$ are

_{s}simultaneously, and TCD is minimum at the unique optimal values ${W}^{*}$ and ${Q}_{s}{}^{*}$. Solving the necessary conditions $\frac{\partial \left(TCD\right)}{\partial W}=0$ and $\frac{\partial \left(TCD\right)}{\partial {Q}_{s}}=0$ from Equations (14) and (15) the following results are obtained.

#### Special Cases

- (i)
- If d = 0, then W
^{*}= $0.$

- (ii)
- If x = y = 1 and d = 0, then we get$$Q{*}^{\text{}}=\sqrt{\frac{2D{C}_{O}}{{C}_{h}}}\sqrt{\frac{P}{\left(P-D\right)}}\xb7\sqrt{\frac{{C}_{s}+{C}_{h}}{{C}_{s}}}$$

- (iii)
- If x = y, then,$$W{*}^{\text{}}=\text{}d\sqrt{\frac{2D{C}_{O}}{{C}_{h}}}\sqrt{\frac{x\left({C}_{s}+{C}_{h}\right)}{\left(P-D-d\right)\left\{P-\left(1-x\right)D-d\right\}{C}_{s}}}$$
^{*}= $P\sqrt{\frac{2D{C}_{O}}{{C}_{h}}}\sqrt{\frac{x\left({C}_{s}+{C}_{h}\right)}{\left(P-D-d\right)\left\{P-\left(1-x\right)D-d\right\}{C}_{s}}}$

- (iv)
- If x = y = 1, then,$$\mathrm{W}{*}^{\text{}}=\text{}d\sqrt{\frac{2D{C}_{O}}{{C}_{h}}}\sqrt{\frac{\left({C}_{s}+{C}_{h}\right)}{\left(P-D-d\right)\left(P-d\right){C}_{s}}}$$
^{*}= $P\sqrt{\frac{2D{C}_{O}}{{C}_{h}}}\sqrt{\frac{\left({C}_{s}+{C}_{h}\right)}{\left(P-D-d\right)\left(P-d\right){C}_{s}}}$

#### 4.2. Case 2 (ERQ Model for Defective Items with Three Levels of Piecewise Constant Demand under Shortages)

**Theorem**

**2.**

^{*}and${Q}_{s}{}^{*}.$

**Proof.**

_{s}, and the relation between TCR and TCD is obtained as

_{s}concurrently and then TCR is minimum at the unique optimal solution ${W}^{*}$ and ${Q}_{s}{}^{*}$.

#### Special Cases

- (i)
- If d = 0, then W* = $0.$

- (ii)
- If x = y = 1 and d = 0, then,$$Q{*}^{\text{}}=\sqrt{\frac{2D{C}_{O}}{{C}_{h}}}\sqrt{\frac{P}{\left(P-D\right)}}\cdot \sqrt{\frac{{C}_{s}+{C}_{h}}{{C}_{s}}}$$

- (iii)
- If x = y then,$$W{*}^{\text{}}=\text{}d\sqrt{\frac{2D{C}_{O}}{{C}_{h}}}\sqrt{\frac{x\left({C}_{s}+{C}_{h}\right)}{\left(P-D-d\right)\left\{P-\left(1-x\right)D-d\right\}{C}_{s}+xdD\left({C}_{s}+{C}_{h}\right)}}$$$$Q{*}^{\text{}}=\text{}P\sqrt{\frac{2D{C}_{O}}{{C}_{h}}}\sqrt{\frac{x\left({C}_{s}+{C}_{h}\right)}{\left(P-D-d\right)\left\{P-\left(1-x\right)D-d\right\}{C}_{s}+xdD\left({C}_{s}+{C}_{h}\right)}}$$

- (iv)
- If x = y = 1, then,$$W{*}^{\text{}}=\text{}d\sqrt{\frac{2D{C}_{O}}{{C}_{h}}}\sqrt{\frac{\left({C}_{s}+{C}_{h}\right)}{\left(P-D-d\right)\left(P-d\right){C}_{s}+dD\left({C}_{s}+{C}_{h}\right)}}$$$$Q{*}^{\text{}}=\text{}P\sqrt{\frac{2D{C}_{O}}{{C}_{h}}}\sqrt{\frac{\left({C}_{s}+{C}_{h}\right)}{\left(P-D-d\right)\left(P-d\right){C}_{s}+dD\left({C}_{s}+{C}_{h}\right)}}$$

#### 4.3. Value of “m”

## 5. Numerical Illustrations and Case Study

_{o}= 1000, C

_{p}= 50, C

_{h}= 10, C

_{R}= 50, C

_{r}= 5, C

_{s}= 3, x = 0.75, and y = 0.5.

#### 5.1. For Case 1

^{*}= 136, Q

^{*}= 6822, Q

_{s}

^{*}= 414, Q

_{d}

^{*}= 131, and ${T}^{*}$= 1.5877

_{s}< 430, Figure 4 depicts the convexity of the total cost function TCD with respect to W and Q

_{s}.

_{s}= 414 and 120 < W < 150, Figure 5 depicts the convexity of the total cost function TCD vs. W.

_{s}< 350, the following Figure 6 shows that TCD is convex with respect to Q

_{s}, and the total inventory cost reaches minimum at “Q

_{s}= 414”.

#### 5.2. For Case 2

^{*}= 98, Q

^{*}= 4910, Q

_{s}

^{*}= 298, Q

_{d}

^{*}= 94, ${T}^{*}$ = 1.1426

_{s}< 350, the following Figure 7 depicts the convexity of the total cost function TCR for W and Q

_{s}.

_{s}= 300 and 80 < W < 120, the following figure (Figure 8) depicts the convexity of the total cost function with respect to “W”.

_{s}< 350, Figure 9 shows that TCR is convex with respect to Q

_{s}and the total cost of the inventory is minimum at “Q

_{s}= 300”.

## 6. Sensitivity Analysis

_{s}is notably reduced.

_{s}notably rises.

#### Cost–Benefitof ERQ Model

## 7. Managerial Insights

- ➢
- Recycling larger or smaller quantities of defective items than the ERQ will increase the total inventory cost. Therefore, inventory managers must evaluate the ERQ of the item before recycling.
- ➢
- When the company decides to recycle, production lot sizes should be set up in favor of the ERQ, and this strategy will provide greater facility to the manager to decline the total cost.
- ➢
- If raw material costs increase significantly, recycling will benefit the company because it will diminish the load on raw-materials and use the wastage items.
- ➢
- The total cost is more sensitive to demand increases during shortage time than it is to increases during production time with positive stock.
- ➢
- If the defective rate is high, companies will receive better cost–benefit from recycling because it produces more defective items, which are usually treated as rejected, but the recycling process converts them to useable, which provides huge revenue for the managers.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Notation | Unit | Description |
---|---|---|

P | Constant | Production rate |

D | Constant | Rate of demand of non-defective items during production run-time |

d | Constant | Production rate defective items |

C_{o} | $/Cycle | Setup cost |

C_{h} | $/Units | Inventory holding cost |

C_{p} | $/Units | Per unit production cost |

C_{s} | $/Units | Shortage cost |

C_{R} | $/Units | Raw material cost per unit item |

C_{r} | $/Units | Recycle cost per unit item of defective items |

TCD | $/Unit time | Total cost of inventory (for case 1) |

TCR | $/Unit time | Total cost of inventory (for case 2) |

t_{1} | Time units | Production run-time with positive stock |

t_{2} | Time units | Production off-time with positive stock |

t_{3} | Time units | Production off-time with negative stock |

t_{4} | Time units | Production run-time with negative stock |

x | Constant | Ratio of demand rates of production off-time with positive stock and production run-time |

y | Constant | Ratio of demand rates of production off-time with negative stock and production run-time |

m | Constant | Shape parameter of the accurate average demand |

Decision variables | ||

W | Units | Quantity of defective items produced per cycle time |

Q | Units | Volume of lot |

Q_{s} | Units | Maximum shortage level |

Q_{d} | Units | Maximum on-hand stock of non-defective items |

T | Time units | Production cycle time. Therefore, T = t_{1} + t_{2} + t_{3} + t_{4} |

**Table 2.**Sensitivity of “x” to different decision variables and costs. The static input values of the parameters are: P = 5000, D = 4500, d = 100, C

_{o}= 1000, C

_{p}= 50, C

_{h}= 10, C

_{R}= 50, C

_{r}= 5, C

_{s}= 3, y = 1.

x | 0.5 | 0.75 | 1.75 | 5 | Observations |
---|---|---|---|---|---|

W^{*} | 98 | 98 | 99 | 99 | Increase |

Q^{*} | 4894 | 4909 | 4929 | 4940 | Increase |

Q_{s}^{*} | 301 | 299 | 295 | 293 | Decrease |

Q_{d}^{*} | 90 | 94 | 99 | 102 | Increase |

${t}_{1}{}^{*}$ | 0.225 | 0.235 | 0.248 | 0.254 | Increase |

${t}_{2}{}^{*}$ | 0.0410 | 0.027 | 0.012 | 0.004 | Decrease |

${t}_{3}{}^{*}$ | 0.133 | 0.132 | 0.131 | 0.130 | Decrease |

${t}_{4}{}^{*}$ | 0.752 | 0.746 | 0.737 | 0.733 | Decrease |

T^{*} | 1.152 | 1.142 | 1.129 | 1.122 | Decrease |

FC | 867 | 875 | 885 | 890 | Increase |

RMC | 208,019 | 210,562 | 213,820 | 215,565 | Increase |

PC | 212,264 | 214,859 | 218,184 | 219,965 | Increase |

HC | 519 | 530 | 545 | 552 | Increase |

SC | 347 | 344 | 340 | 338 | Increase |

RC | 424 | 430 | 436 | 440 | Increase |

TCR | 422,442 | 427,602 | 434,210 | 437,755 | Increase |

**Table 3.**Sensitivity of “y” to different decision variables and costs. The static input values of the parameters are: P = 5000, D = 4500, d = 100, C

_{o}= 1000, C

_{p}= 50, C

_{h}= 10, C

_{R}= 50, C

_{r}= 5, C

_{s}= 3, x = 1.

y | 0.5 | 0.8 | 1.50 | 2 | 5 | Observations |
---|---|---|---|---|---|---|

W^{*} | 98.36 | 99.36 | 100.4 | 100.84 | 101.2 | Increase |

Q^{*} | 4918 | 4968 | 5020 | 5042 | 5058 | Increase |

Q_{s}^{*} | 297 | 305 | 309 | 313 | 316 | Increase |

Q_{d}^{*} | 96 | 91.72 | 90.72 | 89.5 | 88 | Decrease |

t_{1}^{*} | 0.24098 | 0.2293 | 0.22682 | 0.22383 | 0.22157 | Decrease |

t_{2}^{*} | 0.02142 | 0.02547 | 0.02016 | 0.01989 | 0.01969 | Decrease |

t_{3}^{*} | 0.13202 | 0.084992 | 0.04605 | 0.02789 | 0.01404 | Decrease |

t_{4}^{*} | 0.74265 | 0.7643 | 0.77721 | 0.78455 | 0.79016 | Increase |

^{T*} | 1.13709 | 1.10406 | 1.07025 | 1.05619 | 1.04548 | Decrease |

FC | 879 | 905 | 934 | 947 | 956 | Increase |

RMC | 211,938 | 220,500 | 229,841 | 233,914 | 237,093 | Increase |

PC | 216,263 | 225,000 | 234,532 | 238,688 | 241,931 | Increase |

HC | 536 | 552 | 575 | 584 | 591 | Increase |

SC | 342 | 352 | 359 | 362 | 364 | Increase |

RC | 432 | 450 | 469 | 477 | 483 | Increase |

TCR | 430,392 | 447,762 | 466,711 | 474,972 | 481,421 | Increase |

**Table 4.**Sensitivity of defective rate to cost–benefit. The static input values of the parameters are: P = 5000, D = 4500, C

_{o}= 1000, C

_{p}= 50, C

_{h}= 10, C

_{R}= 50, C

_{r}= 5, C

_{s}= 3, x = 1.5, y = 1.5.

d | 100 | 110 | 120 | 130 | 140 |
---|---|---|---|---|---|

PCB | 0.797% | 0.878% | 0.96% | 1.042% | 1.125% |

Remarks | Cost–benefit increases with the increase in “d”. |

**Table 5.**Sensitivity of “recycle cost” to cost–benefit. The static input values of the parameters are: P = 5000, D = 4500, d = 100, C

_{o}= 1000, C

_{p}= 50, C

_{h}= 10, C

_{R}= 50, C

_{s}= 3, x = 1.5, y = 1.5.

C_{r} | 5 | 10 | 15 | 20 | 25 |
---|---|---|---|---|---|

PCB | 0.797% | 0.697% | 0.59% | 0.497% | 0.398% |

Remarks | Cost–benefit decreases with increase in recycle cost. |

**Table 6.**Sensitivity of “holding cost” to cost–benefit. The static input values of the parameters are: P = 5000, D = 4500, d = 100, C

_{o}= 1000, C

_{p}= 50, C

_{R}= 50, C

_{r}= 5, C

_{s}= 3, x = 1.5, y = 1.5.

C_{h} | 10 | 20 | 30 | 40 | 50 |
---|---|---|---|---|---|

PCB | 0.797% | 0.737% | 0.689% | 0.648% | 0.613% |

Remarks | Cost–benefit decreases with increase in holding cost. |

**Table 7.**Sensitivity of “x” to cost–benefit. The static input values of the parameters are: P = 5000, D = 4500, d = 100, C

_{o}= 1000, C

_{p}= 50, C

_{h}= 10, C

_{R}= 50, C

_{r}= 5, C

_{s}= 3, y = 1.5.

x | 0.5 | 1 | 1.5 | 2 | 2.5 |
---|---|---|---|---|---|

PCB | 0.7976% | 0.7972% | 0.7970% | 0.7969% | 0.7969% |

Remarks | “x” is not sensitive to cost–benefit. |

**Table 8.**Sensitivity of “y” to cost–benefit. The static input values of the parameters are: P = 5000, D = 4500, d = 100, C

_{o}= 1000, C

_{p}= 50, C

_{h}= 10, C

_{R}= 50, C

_{r}= 5, C

_{s}= 3, x = 1.5.

y | 0.5 | 1 | 1.5 | 2 | 2.5 |
---|---|---|---|---|---|

Percentage of cost–benefit | 0.7990% | 0.7975% | 0.7970% | 0.7968% | 0.7966% |

Remarks | “y” is not sensitive to cost–benefit. |

**Table 9.**Sensitivity of raw material cost to cost–benefit. The static input values of the parameters are: P = 5000, D = 4500, d = 100, C

_{o}= 1000, C

_{p}= 50, C

_{h}= 10, C

_{r}= 5, C

_{s}= 3, x = 1.5, y = 1.5.

${\mathit{C}}_{\mathit{R}}$ | 50 | 55 | 60 | 65 | 70 |
---|---|---|---|---|---|

PCB | 0.797% | 0.854% | 0.906% | 0.954% | 0.997% |

Remarks | Cost–benefit increases with the increase in raw material cost. |

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## Share and Cite

**MDPI and ACS Style**

AlArjani, A.; Miah, M.M.; Uddin, M.S.; Mashud, A.H.M.; Wee, H.-M.; Sana, S.S.; Srivastava, H.M. A Sustainable Economic Recycle Quantity Model for Imperfect Production System with Shortages. *J. Risk Financial Manag.* **2021**, *14*, 173.
https://doi.org/10.3390/jrfm14040173

**AMA Style**

AlArjani A, Miah MM, Uddin MS, Mashud AHM, Wee H-M, Sana SS, Srivastava HM. A Sustainable Economic Recycle Quantity Model for Imperfect Production System with Shortages. *Journal of Risk and Financial Management*. 2021; 14(4):173.
https://doi.org/10.3390/jrfm14040173

**Chicago/Turabian Style**

AlArjani, Ali, Md. Maniruzzaman Miah, Md. Sharif Uddin, Abu Hashan Md. Mashud, Hui-Ming Wee, Shib Sankar Sana, and Hari Mohan Srivastava. 2021. "A Sustainable Economic Recycle Quantity Model for Imperfect Production System with Shortages" *Journal of Risk and Financial Management* 14, no. 4: 173.
https://doi.org/10.3390/jrfm14040173