# An Imperfect Production Model for Breakable Multi-Item with Dynamic Demand and Learning Effect on Rework over Random Planning Horizon

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## Abstract

**:**

## 1. Introduction

- Step 1
- Formulate the imperfect production-inventory model for first ${N}^{j}$ fully accommodated cycles considering production, screening, reworking, holding, and shortage costs.
- Step 2
- Determine the expression of the expected total cost for the first ${N}^{j}$ fully accommodated cycles concerning the random time horizon.
- Step 3
- Calculate the expression of the total cost for the last cycle under the following mutually exclusive and disjoint cases:
- –
- Case I: The random time horizon ends before the production period concludes.
- –
- Case II: The random time horizon ends in the between of the end of the production period and exhaust of inventory period.
- –
- Case III: The random time horizon ends during the shortage period.
- –
- Case IV: The random time horizon ends during the shortage period when the production restarts to cover the shortages.

- Step 4
- Compute the expression of the projected total cost for the last period which joins the four mutually exclusive and disjoint events described in Step-3.
- Step 5
- Minimize the expected total cost including the expressions determined in Step-2 and Step-4.

## 2. Notation and Assumptions

#### 2.1. Notation

${q}^{j}\left(t\right)$ | : | Inventory level at time t for perfect quality items in each cycle except last cycle |

${q}_{L}^{j}\left(t\right)$ | : | Inventory quantity in the last cycle at time t for perfect quality items |

${S}^{j}\left(t\right)$ | : | Shortage level at time t for perfect quality items in each cycle except last cycle |

${S}_{L}^{j}\left(t\right)$ | : | Shortage level in the last cycle at time t for perfect quality items |

${S}^{j}$ | : | Maximum shortage level |

${P}^{j}$ | : | Production rate |

${D}^{j}$ | : | Customers’ demand rate |

${t}_{p}^{j}$ | : | Time at which production stopped in each cycle |

${t}_{s}^{j}$ | : | Time at which inventory exhausted in each cycle |

${t}_{r}^{j}$ | : | Time at which production restarts in shortage period in each cycle |

${T}^{j}$ | : | Length of each cycle |

${\gamma}^{j}$ | : | Portion of the demand that is not backlogged |

$1-{\gamma}^{j}$ | : | Portion of the demand that is backlogged |

${\alpha}^{j}$ | : | Achieved learning parameter to increase the rework rate |

${\beta}^{j}$ | : | Achieved learning parameter for production and screening costs |

${\theta}^{j}$ | : | Rate of breakability of the produced items |

${\delta}^{j}$ | : | Percentage of rework for breakable items related to the learning effect |

${c}_{p}^{j}$ | : | Production cost per unit in the first cycle, ${c}_{p}^{j}{e}^{(i-1){\beta}^{j}}$ is the unit productioncost in $ith$ cycle |

${c}_{sr}^{j}$ | : | Screening cost per unit in first cycle, ${c}_{sr}^{j}{e}^{(i-1){\beta}^{j}}$ is screening cost per unit in $ith$ cycle |

${r}_{c}^{j}$ | : | Reworking cost per unit |

${h}_{c}^{j}$ | : | Holding cost per unit per unit time for perfect quality items |

${c}_{sh}^{j}$ | : | Shortage cost per unit per unit time for perfect quality items |

${s}^{j}$ | : | Selling price per unit perfect quality items |

R | : | Difference between inflation and the time value of money |

${N}^{j}$ | : | Number of entirely accommodated cycles |

M | : | Total number of items |

H | : | Random time horizon length |

#### 2.2. Assumptions

- (i)
- Non-perfect quality of multiple items is produced (breakable items) in the production system. A portion of breakable items is reworked to get a nearly perfect item. The perfect quality products are immediately set for sale.
- (ii)
- Demand rate (${D}^{j}$) of $jth$ item ($j=1,2,\dots ,M$) is dependent on the displayed inventory level and selling price, which is of the form:$$\begin{array}{c}\hfill {D}^{j}({q}^{j},{s}^{j})=\left\{\begin{array}{cc}{d}_{0}^{j}+{d}_{1}^{j}{q}^{j}\left(t\right)-{d}_{2}^{j}{s}^{j},\hfill & {q}^{j}\left(t\right)\ge 0\hfill \\ {d}_{0}^{j}-{d}_{1}^{j}{S}^{j}\left(t\right),\hfill & {q}^{j}\left(t\right)\le 0\hfill \end{array}\right.\end{array}$$
- (iii)
- The time horizon H is not necessarily fixed and known. It has some uncertainty, therefore, it is finite and randomly distributed. Here, it is assumed that H follows an exponential distribution with the following probability density function (p.d.f)$$\begin{array}{c}\hfill f\left(h\right)=\left\{\begin{array}{cc}\lambda {e}^{-\lambda h}\phantom{\rule{0.166667em}{0ex}}\hfill & h\ge 0\hfill \\ 0\phantom{\rule{0.166667em}{0ex}}\hfill & otherwise\phantom{\rule{93.89418pt}{0ex}}\hfill \end{array}\right.\end{array}$$
- (iv)
- First ${N}^{j}$ cycles are completely contained in the time horizon and it finishes during $({N}^{j}+1)$th cycle.
- (v)
- Shortages are allowed in each cycle. After occurring shortage, some customers will wait in during the stockout period. So, it is considered the partial backlogging during the stockout period of unsatisfied market demand.
- (vi)
- The opening and terminal stock levels in each period are zero in each cycle.
- (vii)
- Due to fluctuation of the economy, the cost or price of every commodity is changed. For this reason, inflation and the time value of capital are considered.
- (viii)
- The learning experience of inspection process increases the rework rate of defective units and the amount of finished product. Learning effect influences the decision-maker to reduce more screening costs and rework costs to the next cycle.

## 3. Mathematical Formulation of the Production-Inventory Model

#### 3.1. Formulation for the $ith$ ($1\le i\le {N}^{j}$) Cycle for the $jth$ Item

#### 3.2. Formulation for the Last Cycle for the $jth$ Item

#### 3.2.1. Case-I (${N}^{j}{T}^{j}\le h\le {N}^{j}{T}^{j}+{t}_{p}^{j}$)

#### 3.2.2. Case-II (${N}^{j}{T}^{j}+{t}_{p}^{j}\le h\le {N}^{j}{T}^{j}+{t}_{s}^{j}$)

#### 3.2.3. Case-III (${N}^{j}{T}^{j}+{t}_{s}^{j}\le h\le {N}^{j}{T}^{j}+{t}_{r}^{j}$)

#### 3.2.4. Case-IV (${N}^{j}{T}^{j}+{t}_{r}^{j}\le h\le ({N}^{j}+1){T}^{j}$)

#### 3.3. Objective Function of the Production-Inventory Model

## 4. Numerical Analysis

#### Sensitivity Analysis

## 5. Practical Implications

## 6. Conclusions

- (i)
- To avoid the loss due to presence of defective items, the manufacturers should adopt the rework policy.
- (ii)
- To decrease the total cost of the manufacturing system and to increase the rework rate of imperfect items, the learning effect plays an important role. So, keeping in mind this effect, the manufacturer should consider a learning effect policy.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Author(s) (Year) | Model | Learning Effect | Inflation | Backlogging | Demand Rate Depends on | Multi-Item |
---|---|---|---|---|---|---|

Mandal and Phaujdar (1989) | Production | No | No | Yes | Stock level | No |

Datta and Pal (1990) | Purchase | No | No | No | Stock level | No |

Lau and Lau (1998) | Purchase | No | No | Yes | Random | No |

Abad (2001) | Purchase | No | No | Yes | Price and time | No |

Hayek and Salameh (2001) | Production | No | No | Yes | Constant | No |

Ben-Daya (2002) | Production | No | No | No | Time | No |

Chiu (2003) | Production | No | No | Yes | Constant | No |

Sheu and Chens (2004) | Production | No | No | No | Constant | No |

Dye and Ouyang (2005) | Purchase | No | No | Yes | Stock level | No |

Wu et al. (2006) | Purchase | No | No | Yes | Stock level | No |

Lee and Dye (2012) | Purchase | No | No | Yes | Stock level | No |

Avinadav et al. (2013) | Purchase | No | No | No | Price and time | No |

Taleizadeh et al. (2013) | Purchase | Yes | No | Yes | Constant | No |

Manna et al. (2016) | Production | No | Yes | No | Price and advertisement | Yes |

Manna et al. (2017a) | Production | Yes | Yes | Yes | Stock level | Yes |

Bhunia et al. (2018) | Purchase | No | No | No | Stock level | No |

Pervin et al. (2019) | Supply chain | No | No | No | Price and stock level | Yes |

Banu et al. (2021) | Supply chain | No | No | No | Stock level | No |

Present paper | Production | Yes, effect on | Yes | Yes | Price and stock level | Yes |

reworked rate |

Production | Screening | Rework Cost | Holding | Shortage | Selling | |
---|---|---|---|---|---|---|

Cost (${\mathit{c}}_{\mathit{p}}^{\mathit{j}}$) | Cost (${\mathit{c}}_{\mathbf{sr}}^{\mathit{j}}$) | Cost (${\mathit{r}}_{\mathit{c}}^{\mathit{j}}$) | Cost (${\mathit{h}}_{\mathit{c}}^{\mathit{j}}$) | Cost (${\mathit{c}}_{\mathbf{sh}}^{\mathit{j}}$) | Price (${\mathit{s}}^{\mathit{j}}$) | |

item-1 | $\$12$ | $1.15 | $ 6 | $4.5 | $ 14 | $ 43 |

item-2 | $\$10$ | $1.20 | $ 5 | $4.5 | $ 11 | $ 38 |

${\mathit{d}}_{0}^{\mathit{j}}$ | ${\mathit{d}}_{1}^{\mathit{j}}$ | ${\mathit{d}}_{2}^{\mathit{j}}$ | ${\mathit{\theta}}^{\mathit{j}}$ | ${\mathit{\delta}}^{\mathit{j}}$ | ${\mathit{\alpha}}^{\mathit{j}}$ | ${\mathit{\beta}}^{\mathit{j}}$ | ${\mathit{\gamma}}^{\mathit{j}}$ | |
---|---|---|---|---|---|---|---|---|

item-1 | 12 | 0.010 | 0.038 | $0.18$ | $0.54$ | $0.20$ | 0.29 | 0.70 |

item-2 | 14 | 0.011 | 0.040 | $0.25$ | $0.59$ | $0.18$ | 0.25 | 0.75 |

Item | ${\mathit{P}}^{\mathit{j}}$ | ${\mathit{t}}_{\mathit{p}}^{\mathit{j}}$ | ${\mathit{t}}_{\mathit{s}}^{\mathit{j}}$ | ${\mathit{t}}_{\mathit{r}}^{\mathit{j}}$ | ${\mathit{T}}^{\mathit{j}}$ | ETC |
---|---|---|---|---|---|---|

item-1 | 11.139 | 5.21 | 7.04 | 8.17 | 9.83 | 1774.941 |

item-2 | 17.683 | 5.78 | 7.26 | 8.25 | 10.29 |

**Table 5.**Results for distinct values of defective rate (${\theta}^{j}$) and rework rate (${\delta}^{j}$) and their comparison.

Item | ${\mathit{\theta}}^{\mathit{j}}$ | ${\mathit{\delta}}^{\mathit{j}}$ | ${\mathit{P}}^{\mathit{j}}$ | Reworking Cost | Holding Cost | ETC |
---|---|---|---|---|---|---|

item-1 | 0.15 | 0.54 | 10.18 | 10.37 | 184.78 | 1727.29 |

item-2 | 0.21 | 0.59 | 16.75 | 11.24 | ||

item-1 | 0.15 | 0.64 | 10.65 | 12.26 | 200.32 | 1737.23 |

item-2 | 0.21 | 0.69 | 16.91 | 13.17 | ||

item-1 | 0.18 | 0.44 | 11.27 | 8.87 | 176.20 | 1764.52 |

item-2 | 0.25 | 0.49 | 17.86 | 9.74 | ||

item-1 | 0.18 | 0.54 | 11.14 | 10.86 | 190.30 | 1774.94 |

item-2 | 0.25 | 0.59 | 17.68 | 11.74 | ||

item-1 | 0.18 | 0.64 | 11.01 | 12.83 | 204.46 | 1785.45 |

item-2 | 0.25 | 0.69 | 17.86 | 13.76 | ||

item-1 | 0.22 | 0.54 | 11.67 | 11.44 | 193.62 | 1832.32 |

item-2 | 0.29 | 0.59 | 18.72 | 12.21 | ||

item-1 | 0.22 | 0.64 | 11.51 | 13.52 | 208.34 | 1843.26 |

item-2 | 0.29 | 0.69 | 18.92 | 14.31 |

**Table 6.**Comparison results for different values of learning effect parameters ${\alpha}^{j}$ and ${\beta}^{j}$.

Item | ${\mathit{\alpha}}^{\mathit{j}}$ | ${\mathit{\beta}}^{\mathit{j}}$ | ${\mathit{\theta}}^{\mathit{j}}$ | ${\mathit{P}}^{\mathit{j}}$ | Reworking Cost | Production & Screening Cost | ETC |
---|---|---|---|---|---|---|---|

item-1 | 0.16 | 0.25 | 0.14 | 10.68 | 10.03 | 405.95 | 1726.87 |

item-2 | 0.14 | 0.21 | 0.22 | 16.92 | 10.75 | 568.71 | |

item-1 | 0.16 | 0.25 | 0.18 | 11.16 | 10.54 | 484.37 | 1770.26 |

item-2 | 0.14 | 0.21 | 0.25 | 17.62 | 11.02 | 592.34 | |

item-1 | 0.16 | 0.29 | 0.18 | 11.16 | 10.54 | 424.37 | 1770.23 |

item-2 | 0.14 | 0.25 | 0.25 | 17.62 | 11.02 | 592.33 | |

item-1 | 0.20 | 0.29 | 0.18 | 11.14 | 10.86 | 423.53 | 1774.97 |

item-2 | 0.18 | 0.25 | 0.25 | 17.68 | 11.74 | 594.49 | |

item-1 | 0.24 | 0.25 | 0.14 | 10.65 | 10.37 | 405.08 | 1735.61 |

item-2 | 0.22 | 0.21 | 0.22 | 17.03 | 12.03 | 572.47 | |

item-1 | 0.24 | 0.33 | 0.14 | 10.65 | 10.37 | 405.07 | 1735.59 |

item-2 | 0.22 | 0.30 | 0.22 | 17.03 | 12.03 | 572.46 | |

item-1 | 0.24 | 0.33 | 0.22 | 11.63 | 11.90 | 442.31 | 1848.50 |

item-2 | 0.22 | 0.30 | 0.30 | 19.04 | 13.30 | 641.99 |

Item | R | ${\mathit{P}}^{\mathit{j}}$ | Reworking Cost | Production & Screening Cost | Holding Cost | ETC |
---|---|---|---|---|---|---|

item-1 | 0.35 | 11.24 | 09.56 | 375.88 | 154.18 | 1447.43 |

item-2 | 17.67 | 10.21 | 519.84 | |||

item-1 | 0.30 | 11.14 | 10.86 | 423.53 | 190.30 | 1774.94 |

item-2 | 17.68 | 11.74 | 594.50 | |||

item-1 | 0.25 | 11.03 | 12.68 | 488.17 | 240.58 | 2257.23 |

item-2 | 17.69 | 13.91 | 697.01 |

Item | $\mathit{\lambda}$ | ${\mathit{P}}^{\mathit{j}}$ | Reworking Cost | Production & Screening Cost | Holding Cost | ETC |
---|---|---|---|---|---|---|

item-1 | 0.0005 | 11.14 | 10.88 | 424.13 | 146.92 | 1682.69 |

item-2 | 17.68 | 11.76 | 595.37 | |||

item-1 | 0.0010 | 11.14 | 10.86 | 423.53 | 190.30 | 1774.94 |

item-2 | 17.68 | 11.74 | 594.50 | |||

item-1 | 0.0015 | 11.14 | 10.84 | 422.94 | 233.32 | 1866.45 |

item-2 | 17.68 | 11.72 | 593.62 |

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**MDPI and ACS Style**

Manna, A.K.; Cárdenas-Barrón, L.E.; Das, B.; Shaikh, A.A.; Céspedes-Mota, A.; Treviño-Garza, G. An Imperfect Production Model for Breakable Multi-Item with Dynamic Demand and Learning Effect on Rework over Random Planning Horizon. *J. Risk Financial Manag.* **2021**, *14*, 574.
https://doi.org/10.3390/jrfm14120574

**AMA Style**

Manna AK, Cárdenas-Barrón LE, Das B, Shaikh AA, Céspedes-Mota A, Treviño-Garza G. An Imperfect Production Model for Breakable Multi-Item with Dynamic Demand and Learning Effect on Rework over Random Planning Horizon. *Journal of Risk and Financial Management*. 2021; 14(12):574.
https://doi.org/10.3390/jrfm14120574

**Chicago/Turabian Style**

Manna, Amalesh Kumar, Leopoldo Eduardo Cárdenas-Barrón, Barun Das, Ali Akbar Shaikh, Armando Céspedes-Mota, and Gerardo Treviño-Garza. 2021. "An Imperfect Production Model for Breakable Multi-Item with Dynamic Demand and Learning Effect on Rework over Random Planning Horizon" *Journal of Risk and Financial Management* 14, no. 12: 574.
https://doi.org/10.3390/jrfm14120574