# A Green Approach—Cost Optimization for a Manufacturing Supply Chain with MFIFO Warehouse Dispatching Policy and Inspection Policy

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## Abstract

**:**

## 1. Introduction and Literature Review

- (a)
- LIFO (last in first out) policy—This policy works with the concept that RW inventories should be used first as it reduces extra rent, which leads to minimum cost because it is an extra cost burden on the supply chain. Chakraborty et al. (2018) [25] considered LIFO policy in their work.
- (b)
- FIFO (first in first out)—This policy works with the concept that OW inventories should be used first as it reduces the deterioration rate, because the inventory that comes first is manufactured first so it will become deteriorated first, so if it is used first, it reduces the deterioration cost. There is one more concept behind this policy that some RWs provide better storage policies compared to OW which reduces deterioration. So, in RW inventories can be kept fresh for a longer period and it minimizes deterioration cost and indirectly maximizes profit instead of greater cost of RW. In the FIFO policy, inventories that is kept first, must be utilized first. There are some RWs that charge less with a good quality of storage facilities that reduce deterioration so that it is more profitable for firms to use OW inventories. Some works on this policy are Niu and Xie (2008) [26], Yu (2019) [27], and Ghiami and Patrick (2020) [28]. Sometimes RW provides good preserving environmental conditions and provide permissible delay in payments as well. Many organizations work with this policy to make sure better originality of the stored inventory.
- (c)
- MFIFO (mix first in first out) policy—This works with the concept of the FIFO policy but with some modifications. In this policy, deteriorated inventories are replaced with RW so RW inventories are also simultaneously utilized. In the present paper, we use this policy. Xu et al. (2017) [12] worked on this policy with a constant demand rate.
- (d)
- MLIFO (mix last in first out) dispatching policy—This policy works upon the LIFO policy but there is some modification in the LIFO policy. In this policy RW inventories are used first. Inventories from RW are sent to OW and then to the market. Some of the manufactures manufacture inventories in RW and then place them in OW. Xu et al. (2017) [12] worked with this policy on deteriorating inventories. They also considered all the above policies and compared mathematical modeling in each case.
- (e)
- (f)
- AIFO (allocation-in-fraction-out)—In this policy, OW inventories are utilized first but some fraction of inventories are replaced from RW so under this policy, both warehouses are used simultaneously and inventories are finished at the same time. Other than LIFO and FIFO, Alamri and Syntetos (2018) [31] used the AIFO policy.

## 2. Assumption

**Assumption**

**1.**

- 1.
- The average deterioration rate of the inventory is the linear function of the deterioration rate of each item in each period, and it is dependent on the function of the number of inspections. The mathematical rate of deterioration is calculated as follows for OW and RW:${\theta}_{1}={\theta}^{\prime}+\frac{g}{n+1};{\theta}_{2}={\theta}^{\u2033}+\frac{g}{n+1},$respectively.
- 2.
- Demand is considered constant, which means demand remains constant in the whole cycle.
- 3.
- Inspection policy considers removing deteriorated products.
- 4.
- MFIFO dispatching policy assumes using OW and RW inventories.
- 5.
- Carbon tax policy is considered for producing each unit of carbon emission.
- 6.
- Lead time is constant.

## 3. Mathematical Modelling

#### 3.1. Fixed Manufacturing Cost

#### 3.2. Holding Cost of Desirable Items in OW and RW in ith Cycle

#### 3.3. Deterioration Cost in OW and RW in ith Cycle

#### 3.4. Cost of Inspection

#### 3.5. Emission Cost

#### 3.6. Total Cost

**Lemma**

**1.**

_{ij}(i = 1, 2, …, N, j = 1, 2, 3, 4) that satisfy${t}_{i2}-{t}_{i1}={t}_{i-1,2}-{t}_{i-1,1};{t}_{i3}-{t}_{i2}={t}_{i-1,3}-{t}_{i-1,2};{t}_{i4}-{t}_{i3}={t}_{i-1,4}-{t}_{i-1,3};{t}_{i+1,1}-{t}_{i4}={t}_{i,1}-{t}_{i-1,4}$.

**Proof.**

**Theorem**

**1.**

**Proof.**

## 4. Numerical Illustration

## 5. Sensitivity Analysis and Observations

- By the mathematical calculation in Table 1 it is observed that as the values of ${t}_{i2}-{t}_{i1},{t}_{i3}-{t}_{i2},{t}_{i4}-{t}_{i3},{t}_{i+1,1}-{t}_{i4}\mathrm{and}N$ increase, the total cost also increases. This implies that for minimizing the total cost, cycle length and sub cycles should be smaller. If the cycle is smaller then inventories need to be utilized in a limited time to minimize deterioration. Therefore, smaller cycles and sub cycles minimize the total cost and management becomes easy.
- As the value of $n$ increases, total cost decreases. This shows that inspection during supply chain for deteriorating inventories is a profitable step. Inspection during supply chain reduces the deterioration rate and reduces deterioration cost. In the same way, this policy reduces carbon emission that also reduces carbon tax; therefore this policy helps to minimize the cost. Along with this, it helps to make a supply chain a green supply chain. It is observed in Table 1 that if we increase $n$, we can minimize the total cost. But it is also observed that when we put $n=3,4,5$ total cost decreases, but as we put n = 6 or more the total cost becomes negative; therefore, it is observed that n should be between 3 and 5 for minimum cost.
- In Table 2, we studied the behaviour of cost model with respect to the deterioration rate. It is observed that as ${\theta}^{\prime}\mathrm{and}{\theta}^{\u2033}$ increases, total cost also increases but when ${\theta}^{\prime}<{\theta}^{\u2033}$ total cost is lower compared to when ${\theta}^{\prime}>{\theta}^{\u2033}$. Therefore, it is concluded that the deterioration rate should be lower to minimize cost. The same way as we studied the behaviour of total cost with ${h}_{w}$ and ${h}_{r}$, it is observed that as ${h}_{w}$ increases total cost increases, but when ${h}_{r}$ increases total cost decreases.
- Table 3 is constructed by using the result of Lemma 1 which is ${t}_{i+1,1}-{t}_{i4}=\frac{H}{N}-\left({t}_{i2}-{t}_{i1}+{t}_{i3}-{t}_{i2}+{t}_{i4}-{t}_{i3}\right)$; this result gives the value of $\left({t}_{i+1,1}-{t}_{i4}\right)$ length of fourth sub cycle.
- Figure 2, Figure 3, Figure 4 and Figure 5 shows the behaviour of total cost with respect to different decision variables. Figure 2 represents the behaviour of total cost with respect to ${t}_{i2}-{t}_{i1}\mathrm{and}n$. It is observed that the graphical representation also justified our results of Table 1 that the total cost is increasing with increase in ${t}_{i2}-{t}_{i1}$ (same way with each sub cycle) and decreasing with increases in $n$. Similarly Figure 3 represents the behaviour of total cost with respect to $N\mathrm{and}n$ and it is observed that as $n$ increases total cost decreases, and as $N$ increases the total cost also increases. This supports our results in Table 1. Figure 4 is a graphical representation of total cost with respect to ${t}_{i2}-{t}_{i1}$ and ${t}_{i3}-{t}_{i2}$ and it is concluded that with the increase in sub cycle length, total cost increases. Hence, by all the graphical representations and numerical calculations in Table 1 and Table 2, it is observed that total cost increases as ${t}_{i2}-{t}_{i1},{t}_{i3}-{t}_{i2},{t}_{i4}-{t}_{i3}\mathrm{and}N$ increases; therefore to minimize the cost, ${t}_{i2}-{t}_{i1},{t}_{i3}-{t}_{i2},{t}_{i4}-{t}_{i3}\mathrm{and}N$ should be minimized and the number of inspection $n$ should be maximized.
- Figure 5 represents the behavior of the total cost with respect to the deterioration rate. The graph with blue colour represents the cost when ${\theta}^{\prime}>{\theta}^{\u2033}$ meaning ${\theta}^{\u2033}$ is constant and ${\theta}^{\prime}$ is increasing whereas the graph with red colour represents the total cost when ${\theta}^{\prime}<{\theta}^{\u2033}$ meaning ${\theta}^{\prime}$ is constant and ${\theta}^{\u2033}$ is increasing. This graph also concludes that when ${\theta}^{\prime}>{\theta}^{\u2033}$, TC is lower compared to when ${\theta}^{\prime}>{\theta}^{\u2033}$.

## 6. Conclusions

#### Application, Limitations and Further Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notation

Symbol | Description |

D | Constant demand |

P | Number of manufactured items |

${\theta}^{\prime}$ | The fix rate of deterioration in OW |

${\theta}_{1}$ | The effective rate of deterioration in OW |

${\theta}^{\u2033}$ | The fix rate of deterioration in RW |

${\theta}_{2}$ | The effective rate of deterioration in RW |

${h}_{w}$ | Stocking charges of inventory in OW |

${h}_{r}$ | Stocking charges of inventory in RW |

$H{C}_{ow}$ | Total Stocking charges of inventory in OW |

$H{C}_{or}$ | Total Stocking charges of inventory in RW |

$W$ | Maximum limit of OW |

${C}_{1}$ | Fixed manufacturing cost |

${D}_{c}$ | Deterioration cost/unit deteriorating inventory |

${I}_{ow}$ | Inventories at time $t$ on OW |

${I}_{or}$ | Inventories at time $t$ on OR |

${I}_{ow}^{i}$ | Total inventory level in OW in ith manufacturing cycle |

${I}_{or}^{i}$ | Total inventory level in RW in ith manufacturing cycle |

${{I}^{\prime}}_{ow}$ | Total deteriorating inventory level in OW in ith manufacturing cycle |

${I}_{or}^{\prime}$ | Total deteriorating inventory level in RW in ith manufacturing cycle |

${E}^{r}$ | Emission cost |

${C}_{ins}$ | Cost of inspection per cycle |

TC | Total cost for one cycle |

$r$ | Hiring cost of inspector |

$g$ | Coefficient representing the effect of deteriorated product on desirable product |

$\chi $ | Tax/unit emission production |

$\nu $ | Emission/unit due to disposal of a deteriorating product |

Decision variables | |

${t}_{i1}$ | Initial period of the ith manufacturing cycle along with ${t}_{11}$ = 0 and ${t}_{n+1,1}=H;i=1,2,......,N$ |

${t}_{i2}$ | At this time OW inventory level reaches to its maximum limit W in ith manufacturing cycle where $i=1,2,......,N$ |

${t}_{i3}$ | At this time OR inventory level reaches to its maximum inventory limit and manufacturing work stops in ith manufacturing cycle where $i=1,2,......,N$ |

${t}_{i4}$ | At this time RW inventory level reaches to zero in ith manufacturing cycle where $i=1,2,......,N$ |

N | Number of cycles |

$n$ | Number of inspections during cycle |

## Appendix A

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${\mathit{t}}_{\mathit{i}2}-{\mathit{t}}_{\mathit{i}1}$ | ${\mathit{t}}_{\mathit{i}3}-{\mathit{t}}_{\mathit{i}2}$ | ${\mathit{t}}_{\mathit{i}4}-{\mathit{t}}_{\mathit{i}3}$ | $\mathit{N}$ | n | TC |
---|---|---|---|---|---|

0.1760 | 0.005958 | 0.350 | 10 | 3 | 1087.83 |

0.1770 | 0.005958 | 0.350 | 10 | 3 | 1103.94 |

0.1780 | 0.005958 | 0.350 | 10 | 3 | 1120.11 |

0.1760 | 0.005959 | 0.350 | 10 | 3 | 1071.83 |

0.1760 | 0.005960 | 0.350 | 11 | 3 | 1071.89 |

0.1760 | 0.005961 | 0.350 | 12 | 3 | 1071.94 |

0.1760 | 0.005958 | 0.351 | 13 | 3 | 1088.05 |

0.1760 | 0.005958 | 0.352 | 10 | 3 | 1104.35 |

0.1760 | 0.005958 | 0.353 | 10 | 3 | 1120.69 |

0.1760 | 0.005958 | 0.350 | 10 | 4 | 639.538 |

0.1760 | 0.005958 | 0.350 | 10 | 5 | 289.579 |

0.1760 | 0.005958 | 0.350 | 15 | 3 | 1711.12 |

0.1760 | 0.005958 | 0.350 | 16 | 3 | 2397.05 |

0.1760 | 0.005958 | 0.350 | 17 | 3 | 3109.65 |

**Table 2.**Behaviour of total cost with the changes in parameters such as ${\theta}^{\prime}$, ${\theta}^{\u2033}$, $N$, $n$, ${h}_{w}$, ${h}_{r}$, $TC$.

${\mathit{\theta}}^{\prime}$ | ${\mathit{\theta}}^{\u2033}$ | N | n | ${\mathit{h}}_{\mathit{w}}$ | ${\mathit{h}}_{\mathit{r}}$ | TC |
---|---|---|---|---|---|---|

0.051 | 0.050 | 10 | 2 | 5 | 8 | 1072.27 |

0.052 | 0.050 | 10 | 2 | 5 | 8 | 1072.77 |

0.053 | 0.050 | 10 | 3 | 5 | 8 | 1073.26 |

0.050 | 0.052 | 10 | 4 | 5 | 8 | 1073.74 |

0.050 | 0.053 | 11 | 2 | 5 | 8 | 1074.71 |

0.050 | 0.054 | 10 | 2 | 5 | 8 | 1075.69 |

0.050 | 0.054 | 11 | 2 | 5.1 | 8 | 1088.84 |

0.050 | 0.054 | 10 | 3 | 5.2 | 8 | 1102.00 |

0.050 | 0.054 | 10 | 2 | 5.3 | 8 | 1115.15 |

0.050 | 0.050 | 10 | 2 | 5 | 8.1 | 1057.92 |

0.050 | 0.050 | 10 | 2 | 5 | 8.2 | 1044.07 |

0.050 | 0.050 | 10 | 2 | 5 | 8.3 | 1030.21 |

0.050 | 0.050 | 10 | 2 | 5 | 8.4 | 1016.35 |

0.050 | 0.050 | 10 | 2 | 5.4 | 8.5 | 1055.12 |

**Table 3.**Changes in ${t}_{i+1,1}-{t}_{i4}$ with the changes in $N$, ${t}_{i2}-{t}_{i1}$, ${t}_{i3}-{t}_{i2}$, ${t}_{i4}-{t}_{i3}$.

N | ${\mathit{t}}_{\mathit{i}2}-{\mathit{t}}_{\mathit{i}1}$ | ${\mathit{t}}_{\mathit{i}3}-{\mathit{t}}_{\mathit{i}2}$ | ${\mathit{t}}_{\mathit{i}4}-{\mathit{t}}_{\mathit{i}3}$ | ${\mathit{t}}_{\mathit{i}+1,1}-{\mathit{t}}_{\mathit{i}4}$ |
---|---|---|---|---|

15 | 0.1750 | 0.005958 | 0.350 | 0.13570 |

16 | 0.1750 | 0.005958 | 0.350 | 0.09404 |

15 | 0.1760 | 0.005958 | 0.350 | 0.182327 |

15 | 0.1770 | 0.005958 | 0.350 | 0.181327 |

14 | 0.1780 | 0.005958 | 0.350 | 0.180327 |

14 | 0.1750 | 0.005959 | 0.350 | 0.183326 |

14 | 0.1750 | 0.005960 | 0.350 | 0.183325 |

14 | 0.1750 | 0.005958 | 0.351 | 0.182327 |

14 | 0.1750 | 0.005958 | 0.352 | 0.182326 |

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## Share and Cite

**MDPI and ACS Style**

Shekhawat, S.; Alessa, N.; Rathore, H.; Sharma, K.
A Green Approach—Cost Optimization for a Manufacturing Supply Chain with MFIFO Warehouse Dispatching Policy and Inspection Policy. *Sustainability* **2022**, *14*, 14664.
https://doi.org/10.3390/su142114664

**AMA Style**

Shekhawat S, Alessa N, Rathore H, Sharma K.
A Green Approach—Cost Optimization for a Manufacturing Supply Chain with MFIFO Warehouse Dispatching Policy and Inspection Policy. *Sustainability*. 2022; 14(21):14664.
https://doi.org/10.3390/su142114664

**Chicago/Turabian Style**

Shekhawat, Santosh, Nazek Alessa, Himanshu Rathore, and Kalpna Sharma.
2022. "A Green Approach—Cost Optimization for a Manufacturing Supply Chain with MFIFO Warehouse Dispatching Policy and Inspection Policy" *Sustainability* 14, no. 21: 14664.
https://doi.org/10.3390/su142114664