# Food Preservation within Multi-Echelon Supply Chain Considering Single Setup and Multi-Deliveries of Unequal Lot Size

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## Abstract

**:**

## 1. Introduction

- What is the set of solutions to optimize deterioration using multiple deliveries and preservation simultaneously?
- Does the SSMD policy affect investment in preservation?
- Does the investment in preservation have any effect on the number of deliveries/shipments within an SSMD setup?
- How does the preservation effect quantitatively the lifetime/deterioration/freshness of a fresh food product?
- What is the effect of SSMD policy on lot size and replenishment cycle?
- How do the preservation and SSMD policies affect total supply chain profit?

## 2. Literature Review

#### 2.1. Single Setup Multiple Deliveries (SSMD)

#### 2.2. Product Deterioration

#### 2.3. Preservation Policies

Authors | Two-Echelon SCM | Multiple Retailers | Unequal Lot Size | SSMD | Preservation Policy |
---|---|---|---|---|---|

Goyal [13] | ✓ | ✓ | |||

Lu [15] | ✓ | ✓ | |||

Goyal [16] | ✓ | ✓ | ✓ | ✓ | |

Hill [18] | ✓ | ✓ | ✓ | ||

Goyal and Nebebe [20] | ✓ | ✓ | ✓ | ||

Woo et al. [59] | ✓ | ✓ | |||

Yang and Wee [21] | ✓ | ✓ | |||

Khouja [22] | ✓ | ✓ | |||

Wang and Sarker [23] | ✓ | ||||

Siajadi et al. [24] | ✓ | ✓ | ✓ | ||

Chan and Kingsman [25] | ✓ | ✓ | ✓ | ||

Ertogral et al. [60] | ✓ | ✓ | |||

Ben-Daya and Al-Nassar [26] | ✓ | ✓ | ✓ | ||

Darwish and Odah [27] | ✓ | ✓ | ✓ | ||

Hsu et al. [50] | Conventional | ||||

Ben-Daya et al. [11] | ✓ | ✓ | |||

Dye [51] | Conventional | ||||

Sana et al. [30] | ✓ | ✓ | |||

Yang et al. [53] | Conventional | ||||

Yang et al. [31] | ✓ | ✓ | |||

Jia et al. [32] | ✓ | ||||

Dye and Yang [55] | ✓ | ✓ | Conventional | ||

Giri et al. [56] | ✓ | Conventional | |||

Azadi et al. [6] | ✓ | ✓ | ✓ | ||

Sarkar et al. [8] | ✓ | ✓ | ✓ | ||

This paper | ✓ | ✓ | ✓ | ✓ | MDRDRMIP |

## 3. Problem Definition, Notation and Assumptions

#### 3.1. Problem Definition

#### 3.2. Assumptions

- A single manufacturer supplies fresh products to the multi-retailers to constitute a supply chain system.
- The manufacturer supplies the produced items to retailers in multiple deliveries, which is known as a single setup multi-delivery (SSMD) policy. Therefore, the cycle time of the manufacturer is the integer multiple of the retailers’ cycle time. This integer is the number of deliveries/shipments to the retailers per cycle of the manufacturer.
- Shipments/deliveries for retailers are prepared from a production batch while the production is continued [26].
- The cycle time of all the retailers is equal, i.e., the inventory is replenished at all the retailers at the same point of time [11].
- The customer demand at all the retailers is known, constant, and different.
- As the demand at each retailer is different, therefore, this model assumes an unequal lot size for each retailer.
- The ordering cost and cost of inventory carrying are different for each retailer.
- The products under consideration are deteriorating in nature, and deteriorate at a constant rate. Practically, the products start deteriorating after being replenished at the retailer. This study includes this fact by considering no deterioration at the manufacturer.
- The rate of production depends on the demand rate [40], i.e., assuming where the production rate and the demand rate at the manufacturer are.
- The inventory holding cost at the manufacturer is less than the inventory holding cost at the retailers, i.e.
- There are no shortages, i.e., all the customers are satisfied to fulfill their demand.
- The supply chain is vertically integrated, such that the optimal value of profit is obtained as a centralized system.

## 4. Model Formulation and Solution

#### 4.1. Retailers’ Model

#### 4.1.1. Ordering Cost

#### 4.1.2. Purchasing Cost

#### 4.1.3. Inventory Holding Cost

#### 4.1.4. Preservation Cost

#### 4.1.5. Total Cost per Unit Time

#### 4.1.6. Sales Revenue per Unit Time

#### 4.1.7. Retailers’ Profit per Unit Time

#### 4.2. Manufacturer’s Model

_{1}, mT).

#### 4.2.1. Setup Cost

#### 4.2.2. Material Purchasing Cost

#### 4.2.3. Production Cost

#### 4.2.4. Inventory Holding Cost

#### 4.2.5. Total Cost per Unit Time

#### 4.2.6. Sales Revenue per Unit Time

#### 4.2.7. Manufacturer’s Profit per Unit Time

#### 4.2.8. Total Profit per Unit Time of the Supply Chain

## 5. Solution Methodology

#### 5.1. Solution Algorithm

**Step 1**Start with $m=1$ and input appropriate values of other parameters.**Step 2**For the first iterative value of $T$, start with $p=0$ and perform the following steps.- (i)
- Compute the value of $T$ that satisfies Equation (9).
- (ii)
- Using the value of $T$, calculated in step (i), compute the value of $p$ that satisfies Equation (10).
- (iii)
- Using the value of $p$, calculated in Step (ii), repeat Step (i) and Step (ii) for $n$ times, until no further change occurs in the value of ${T}_{i}$ and ${p}_{i}$, where $i$ denotes the $i\u2013\mathrm{th}$ iteration.

**Step 3**For the $i\u2013\mathrm{th}$ iteration, using the pair of variables $\left({T}_{i},{p}_{i},m\right)$, compute $TP\left({T}_{i},{p}_{i},m\right)$ from Equation (8).**Step 4**Set $TP\left({T}_{m}^{*},{p}_{m}^{*},m\right)={\mathrm{max}}_{i=1}^{n}TP\left({T}_{i},{p}_{i},m\right)$, then $\left({T}_{m}^{*},{p}_{m}^{*},m\right)$ is the optimal solution for given value of m.**Step 5**Set $m=m+1$, repeat Step 2 to Step 4 to attain $TP\left({T}_{m}^{*},{p}_{m}^{*},m\right)$.**Step 6**If $TP\left({T}_{m}^{*},{p}_{m}^{*},m\right)\ge TP\left({T}_{m-1}^{*},{p}_{m-1}^{*},m-1\right)$, go to Step 5, otherwise go to Step 7.**Step 7**Set $\left({T}_{m}^{*},{p}_{m}^{*},m\right)=\left({T}_{m-1}^{*},{p}_{m-1}^{*},m-1\right)$, then $\left({T}_{m}^{*},{p}_{m}^{*},m\right)$ is a set of optimal solutions.

#### 5.2. Numerical Experiments

#### 5.2.1. Input Parameters

#### 5.2.2. Results and Discussion

**Example**

**1.**

**Example**

**2.**

#### Comparative Analysis of the Results from Examples 1 and 2

**Example**

**3.**

#### 5.2.3. Sensitivity Analysis

- The magnitude of variation in the value of profit is different from the variation in different cost parameters. The effect of variation in the value of each cost parameter on the value of profit is illustrated in Figure 5c.
- The cost of production and the cost of materials affect the profit the most, while the variation in the value of inventory holding cost of retailer 1 has no effect on the value of the profit. Other cost parameters including manufacturer’s setup cost and inventory holding cost, and retailer’s ordering cost have an insignificant effect on the value of profit.
- Variations in production and material cost have a significant and inverse effect on the value of the retailers’ cycle time. The cycle time varies directly with variations in the retailer’s ordering cost. Retailers’ and manufacturers’ inventory holding costs and setup costs of manufacture have no significant effect on the value of the cycle time.
- The investment in the preservation is affected directly by the variation in the cost parameters. This effect is significant for cost of material and production cost, while other cost parameters do not have a considerable effect on the value of preservation investment.
- Variations in production cost, retailer’s ordering, and inventory holding cost have no effect on the value of the number of deliveries/shipments to the retailers per cycle of the manufacturer. Variations in the cost of material affect directly, while that in the manufacturer’s inventory holding cost and setup cost affect the number of shipments/deliveries inversely.

## 6. Managerial Insights

#### 6.1. Insight 1—Transportation/Delivery Reduction

#### 6.2. Insight 2—Demand Improvement

#### 6.3. Insight 3—Profit Maximization

#### 6.4. Insight 4—Preservation during Transportation

#### 6.5. Insight 5—Environmental Protection

## 7. Conclusions

#### Limitations and Future Research Directions

- This research considered variable ordering cost, inventory holding cost, and selling price due to several demographical, geographical, and setup structure reasons, which can be modeled to extend this research.
- As the demand at each retailer is assumed to be different due to several reasons, those reasons can be considered and modeled to improve the customer demand, e.g., by considering local advertisement-dependent demand for each retailer, as considered by Palanivel and Uthayakumar [64].
- This research assumed a constant size of each replenishment for a single retailer, which can be considered as different for each replenishment, as proposed by Goyal [16].
- The proposed model considered that uniform preservation investment though the lot size is different at each retailer. This model can be extended by relating the amount of preservation investment to the lot size.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Index | |

$i$ | $i=1,2,\dots ,n$$,$$$. |

Variables | |

$m$ | number of shipments/deliveries to retailers per manufacturer’s cycle (number) |

$T$ | retailers’ cycle time (time units) |

$p$ | preservation investment at retailers’ level ($/unit/unit time) |

Retailers’ Parameters | |

${d}_{ri}$ | $\mathrm{customer}\u2019\mathrm{s}\mathrm{demand}\mathrm{at}i-th$ retailer per unit time (units/unit time) |

${D}_{ri}$ | $\mathrm{customer}$ per cycle (units/cycle) |

${N}_{di}$ | $\mathrm{number}$ (units/cycle) |

$P{Q}_{ri}$ | $\mathrm{purchasing}$ (units/cycle) |

${I}_{ri}^{o}$ | $\mathrm{on}$$\mathrm{at}\mathrm{any}\mathrm{time}t,0\le t\le T$ (units) |

${I}_{ri}$ | $\mathrm{total}$ (units/cycle) |

${A}_{ri}$ | $\mathrm{ordering}$ ($/order) |

$P{C}_{ri}$ | $\mathrm{purchasing}$ ($/unit) |

${h}_{ri}$ | $\mathrm{inventory}$ ($/unit/unit time) |

$T{C}_{ri}$ | $\mathrm{total}$ ($/unit time) |

$S{P}_{ri}$ | $\mathrm{selling}$ ($/unit) |

$S{R}_{ri}$ | $\mathrm{sales}$ ($/unit time) |

$T{P}_{ri}$ | $\mathrm{total}$ ($/unit time) |

$T{P}_{r}$ | total profit per unit time of all the retailers ($/unit time) |

Manufacturer’s Parameters | |

${d}_{m}$ | demand per unit time (units/unit time) |

${D}_{m}$ | demand per cycle (units/cycle) |

$P$ | rate of production (units/unit time) |

${I}_{m}^{a}$ | $\mathrm{on}-\mathrm{hand}\mathrm{inventory}\mathrm{at}\mathrm{any}\mathrm{time}t,0\le t\le {t}_{1}$ (units) |

${I}_{m}^{b}$ | $\mathrm{on}-\mathrm{hand}\mathrm{inventory}\mathrm{at}\mathrm{any}\mathrm{time}t,{t}_{1}\le t\le mT$ (units) |

${I}_{m}$ | total inventory carried during one cycle (units/cycle) |

${N}_{p}$ | number of items produced per cycle (units/cycle) |

${C}_{set}$ | setup cost per setup ($/setup) |

${C}_{mt}$ | material cost per unit ($/unit) |

${C}_{p}$ | production cost per unit ($/unit) |

${h}_{m}$ | inventory holding cost per unit per unit time ($/unit/unit time) |

$T{C}_{m}$ | total cost per unit time ($/unit time) |

$S{P}_{m}$ | selling price per unit ($/unit) |

$S{R}_{m}$ | sales revenue per unit time ($/unit time) |

$T{P}_{m}$ | total profit per unit time ($/unit time) |

Other Parameters | |

$TP$ | total profit per unit time of the supply chain as a centralized system ($/unit time) |

$L$ | maximum lifetime of the product (time units) |

$\theta $ | rate of deterioration |

$\alpha $ | degree of vulnerability to deterioration |

$x$ | degree of effectiveness of preservation cost |

$k$ | scaling parameter within production and demand at manufacturer |

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**Figure 2.**(

**a**) Reduced effect of additional investment in preservation technology on lifetime. (

**b**) Reduced effect of additional investment in preservation technology on deterioration.

**Figure 3.**(

**a**) Behavior of inventory level per cycle at $i\u2013\mathrm{th}$ retailer. (

**b**) Behavior of inventory level per manufacturer’s cycle.

**Figure 4.**(

**a**) Total profit per unit time versus investment in preservation technology. (

**b**) Total profit per unit time versus number of replenishments per manufacturer’s cycle. (

**c**) Total profit per unit time versus retailers’ cycle time. (

**d**) Total profit per unit time versus retailers’ cycle time and number of shipments/deliveries to retailers per manufacturer’s cycle. (

**e**) Total profit per unit time versus investment in preservation technology and number of shipments/deliveries to retailers per manufacturer’s cycle. (

**f**) Total profit per unit time versus investment in preservation technology and retailers’ cycle time.

**Figure 5.**(

**a**) Improvement in product’s lifetime, retailers’ cycle time, number of shipments, lot size, and profit with investment in preservation technology. (

**b**) Losses when SSSD policy is adopted instead of SSMD policy. (

**c**) Variation in profit per unit time by varying retailer 1′s ordering cost, inventory holding cost, manufacturer’s setup cost, production cost, inventory holding cost, and material cost.

${A}_{ri}=\$\left(30,25,28,30,27,28,29\right)/\mathrm{order}$ | |||

$S{P}_{ri}=\$\left(200,180,160,190,170,175,185\right)/\mathrm{unit}$ | |||

${d}_{ri}=\left(100,110,105,95,115,102,108\right)\mathrm{units}/\mathrm{month}$ | |||

${h}_{ri}=\$\left(0.4,0.6,0.5,0.45,0.55,0.52,0.48\right)/\mathrm{unit}/\mathrm{month}$ | |||

${C}_{mt}=\$10/\mathrm{unit}$ | ${C}_{p}=\$5/\mathrm{unit}$ | ${h}_{m}=\$0.3/\mathrm{unit}/\mathrm{month}$ | $L=0.5\mathrm{months}$ |

$x=2$ | $\gamma =0.2$ | $k=4$ |

**Table 3.**(

**a**) Optimal solution for Example 1. (

**b**) Optimal replenishment quantities per delivery at each retailer for Example 1. (

**c**) Optimal solution for Example 2. (

**d**) Optimal replenishment quantities per delivery at each retailer for Example 2.

(a) | ||||||

$\begin{array}{l}T{P}^{*}\\ \$118,783/\mathrm{month}\end{array}$ | $\begin{array}{l}{T}^{*}\\ 0.19\mathrm{month}\end{array}$ | $\begin{array}{l}{m}^{*}\\ 8/\mathrm{manufacturer}\u2019\mathrm{s}\mathrm{cycle}\end{array}$ | ||||

(b) | ||||||

$\begin{array}{l}P{Q}_{r1}^{*}\\ 22\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r2}^{*}\\ 24\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r3}^{*}\\ 23\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r4}^{*}\\ 20\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r5}^{*}\\ 25\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r6}^{*}\\ 22\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r7}^{*}\\ 23\mathrm{units}\end{array}$ |

(c) | ||||||

$\begin{array}{l}T{P}^{*}\\ \$119,475/\mathrm{month}\end{array}$ | $\begin{array}{l}{T}^{*}\\ 0.31\mathrm{month}\end{array}$ | $\begin{array}{l}{m}^{*}\\ 5/\mathrm{manufacturer}\u2019\mathrm{s}\mathrm{cycle}\end{array}$ | $\begin{array}{l}{p}^{*}\\ \$0.58/\mathrm{unit}/\mathrm{month}\end{array}$ | |||

(d) | ||||||

$\begin{array}{l}P{Q}_{r1}^{*}\\ 32\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r2}^{*}\\ 35\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r3}^{*}\\ 34\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r4}^{*}\\ 31\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r5}^{*}\\ 37\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r6}^{*}\\ 33\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r7}^{*}\\ 35\mathrm{units}\end{array}$ |

**Table 4.**(

**a**) Comparative analysis of important results with and without preservation technology. (

**b**) Optimal replenishment quantities per delivery at each retailer.

(a) | |||||||

Parameters | Without Preservation | With Preservation | Percent Variation | ||||

Lifetime (month/s) | 0.5 | 1.4 | 180 | ||||

Cycle time (month/s) | 0.19 | 0.31 | 63 | ||||

Number of shipments | 8 | 5 | −37.5 | ||||

Profit/month ($/month) | 118,783 | 119,475 | 0.58 | ||||

(b) | |||||||

$\begin{array}{l}P{Q}_{r1}^{*}\\ 195\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r2}^{*}\\ 214\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r3}^{*}\\ 204\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r4}^{*}\\ 185\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r5}^{*}\\ 224\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r6}^{*}\\ 199\mathrm{units}\end{array}$ | $\begin{array}{l}P{Q}_{r7}^{*}\\ 210\mathrm{units}\end{array}$ |

Percentage Variation in Parameters | Percentage Variation in Optimal Values of Decision Variables and Profit | ||||
---|---|---|---|---|---|

$\mathit{m}$ | $\mathit{p}$ | $\mathit{T}$ | $\mathit{T}\mathit{P}$ | ||

${C}_{mt}$ | −50 | −20 | −31.21 | 14.52 | 3.23 |

−25 | 0 | −15.69 | 5.48 | 1.61 | |

25 | 0 | 14.83 | −7.42 | −1.60 | |

50 | 20 | 29.83 | −11.94 | −3.20 | |

${C}_{p}$ | −50 | 0 | −15.69 | 5.48 | 1.61 |

−25 | 0 | −7.93 | 1.61 | 0.80 | |

25 | 0 | 7.24 | −4.52 | −0.80 | |

50 | 0 | 14.83 | −7.42 | −1.60 | |

${A}_{r1}$ | −50 | 0 | −0.52 | −4.68 | 0.04 |

−25 | 0 | −0.34 | −3.55 | 0.02 | |

25 | 0 | −0.34 | 10.32 | −0.02 | |

50 | 0 | −0.17 | 1.94 | −0.04 | |

${h}_{r1}$ | −50 | 0 | −0.34 | −1.61 | 0.00 |

−25 | 0 | −0.34 | −1.29 | 0.00 | |

25 | 0 | −0.34 | −1.61 | 0.00 | |

50 | 0 | −0.34 | −1.94 | 0.00 | |

${h}_{m}$ | −50 | 40 | −0.69 | −1.29 | 0.06 |

−25 | 20 | −0.52 | −1.61 | 0.03 | |

25 | −20 | −0.17 | −1.61 | −0.03 | |

50 | −40 | −0.17 | −1.61 | −0.05 | |

${C}_{set}$ | −50 | −20 | −0.69 | −1.61 | 0.06 |

−25 | −20 | −0.52 | −1.61 | 0.03 | |

25 | 20 | −0.17 | −1.61 | −0.03 | |

50 | 20 | −0.17 | −1.61 | −0.05 |

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## Share and Cite

**MDPI and ACS Style**

Iqbal, M.W.; Ramzan, M.B.; Malik, A.I.
Food Preservation within Multi-Echelon Supply Chain Considering Single Setup and Multi-Deliveries of Unequal Lot Size. *Sustainability* **2022**, *14*, 6782.
https://doi.org/10.3390/su14116782

**AMA Style**

Iqbal MW, Ramzan MB, Malik AI.
Food Preservation within Multi-Echelon Supply Chain Considering Single Setup and Multi-Deliveries of Unequal Lot Size. *Sustainability*. 2022; 14(11):6782.
https://doi.org/10.3390/su14116782

**Chicago/Turabian Style**

Iqbal, Muhammad Waqas, Muhammad Babar Ramzan, and Asif Iqbal Malik.
2022. "Food Preservation within Multi-Echelon Supply Chain Considering Single Setup and Multi-Deliveries of Unequal Lot Size" *Sustainability* 14, no. 11: 6782.
https://doi.org/10.3390/su14116782