# Designing a New Location-Allocation and Routing Model with Simultaneous Pick-Up and Delivery in a Closed-Loop Supply Chain Network under Uncertainty

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## Abstract

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## 1. Introduction

## 2. Literature Review

#### 2.1. Closed-Loop Supply Chain

_{2}emissions from industrial operations [20].

_{2}emissions [25]. CLSC models were supplied by Zheng et al., and these models accounted for three inverted channel designs in addition to the two competitive sales channels [26].

#### 2.2. Pick-Up and Multi-Depot Vehicle Routing

_{2}emissions is one of the primary focuses of their model. They regarded hybrid automobiles as a possible solution to satisfy the needs of their consumers, given the uncertainty. They looked at the issue from the perspective of many depots and multiple vehicles [41].

#### 2.3. Location-Allocation Problem

#### 2.4. Uncertainty in Supply Chain

- ∗
- Considering location-routing and allocation simultaneously in the CLSC;
- ∗
- Considering the queuing system in the distribution of products;
- ∗
- Using the robust-fuzzy-probabilistic method to control uncertainty parameters;
- ∗
- Designing an initial solution to solve the model.

## 3. Modelling Process and Methods

$Fix{I}_{i}$ | Supplier selection cost i |

$Fix{J}_{j}$ | Production center establishment cost j |

$Fix{L}_{l}$ | Distribution/collection center Establishment cost l |

$Fix{K}_{k}$ | Warehouse establishment cost k |

$Fix{V}_{v}$ | Fixed cost of using the vehicle v |

${D}_{cpts}$ | Product delivery amount p to customer c in period t under scenario s |

${R}_{cpts}$ | Product pick up amount p from customer c in period t under scenario s |

$Cap{V}_{v}$ | Vehicle capacity v |

$Cap{L}_{lp}$ | Maximum distribution center capacity l of product distribution p |

$Cap{I}_{ir}$ | Maximum supplier capacity i of raw material supply r |

$Cap{K}_{kr}$ | Maximum storage capacity k of raw material storage r |

$Cap{J}_{jp}$ | Maximum production center capacity p from the production of product p |

$Di{s}_{n{n}^{\prime}}$ | Distance between nodes n and n’ |

$T{r}_{n{n}^{\prime}vs}$ | Transportation cost between nodes n and n’ with mode of transport v under scenario s |

$T{r}_{ikvs}$ | Transportation cost between node i and k with mode of transport v under scenario s |

$T{r}_{kjvs}$ | Transportation cost between nodes k and j with mode of transport v under scenario s |

$T{r}_{jlvs}$ | Transportation cost between nodes j and l with mode of transport v under scenario s |

$T{r}_{ljvs}$ | Transportation cost between nodes l and j with mode of transport v under scenario s |

$T{r}_{lovs}$ | Transportation cost between node l and o with mode of transport v under scenario s |

${T}_{n{n}^{\prime}}$ | Transportation time between nodes n and n’ |

${S}_{c}$ | Unloading and loading time of the vehicle in the node c |

${C}_{lp}$ | Distribution cost per unit of product p by the distribution center l |

$[A{S}_{c},B{S}_{c}]$ | Soft time window for pick up delivery of customer products c |

$\mathsf{\alpha}$ | Penalty cost of soft time window |

$\mathrm{H}$ | Product-dependent greenhouse gas emissions |

${p}_{s}$ | Probability of occurrence of scenario s |

${\omega}_{rp}$ | Number of raw materials r used per unit of product p |

${r}_{p}^{c}$ | Average fraction of product recycling p |

${\vartheta}_{l}$ | Number of employees in the distribution center l |

$C{T}_{l}$ | Waiting time cost to serve in the distribution center l |

${\mu}_{l}$ | Distribution center service rate l (exponential distribution) |

${B}_{l}$ | Upper limit of queue length for service in the distribution center l |

${\theta}_{l}$ | High limit probability for excessive service queue length at the distribution center l |

${V}_{lpvts}$ | Product amount total p that can be distributed from the distribution center l by vehicle v under scenario s in period t |

${Z}_{l}$ | If distribution center l is established/selected, 1 and otherwise 0. |

${Z}_{lcvts}$ | If the distribution center l is assigned to customer c and the vehicle v is assigned under scenario s in period t, 1 and otherwise 0. |

${X}_{n{n}^{\prime}vts}$ | If node n’ is visited after node n by vehicle v under scenario s in period t, 1 and otherwise 0. l, c ∈ L∪C. |

${U}_{cvts}$ | Auxiliary variable for sub-tour deletion limit |

$T{c}_{lcvts}$ | Vehicle arrives time v at customer c and exits the distribution center l under scenario s in period t |

$L{c}_{lcpvts}$ | Product amount p in the vehicle load v in the customer node c and out of the distribution center l under scenario s in period t |

$T{e}_{cvts}$ | Time exceeds the vehicle time window v in customer node c under scenario s in period t |

$T{w}_{lvts}$ | Maximum vehicle visit time vehicle v out of distribution center l under scenario s in time period t |

${Q}_{rikvts}^{s}$ | Raw material amount r transferred from the supplier i to the warehouse (silo) k with the transportation mode v in period t under scenario s |

${Q}_{rkjvts}^{h}$ | Raw material amount r transferred from the warehouse (silo) k to the production center j with the transportation mode v in time period t under scenario s |

${Q}_{pjlvts}^{f}$ | Product amount p transferred from the production center j to the distribution collection center l with the transportation mode v in period t under scenario s |

${Q}_{rljvts}^{c}$ | Raw material amount r transferred from the distribution/collection center l to the production center j with the transportation mode v in period t under scenario s |

${Q}_{rlovts}^{d}$ | Raw material amount/scrap product r transferred from distribution center/collection l to disposal center o with the transportation mode v in period t under scenario s |

${y}_{r}^{s}$ | If supplier i is selected to supply raw materials, 1 and otherwise 0. |

${y}_{l}^{d}$ | If distribution/collection center l is constructed for product distribution/collection, 1 and otherwise 0. |

${y}_{k}^{h}$ | If a warehouse (silo) k is constructed for the storage of raw materials, 1 and otherwise 0. |

${y}_{j}^{c}$ | If the production center j is constructed to produce the product, 1 and otherwise 0. |

#### 3.1. Jackson Network

#### 3.2. Robust-Fuzzy-Probabilistic Method

#### 3.3. Psedou Code of Meta Heuristics Algorithm

## 4. Analysis of Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Reference | Year | Objective Function | Uncertain | Pick up Delivery Routing | Multi-Depot | Control Method | Solving Method | Multi-Product | Multi-Level | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Time | CO_{2} | Cost | Profit | |||||||||

Govindan et al. [3]. | 2020 | - | - | √ | - | √ | - | √ | Fuzzy | FANP | √ | - |

Pazhani et al. [22]. | 2021 | - | - | √ | - | - | - | - | - | Relaxation | √ | √ |

Hornstra et al. [27]. | 2020 | - | - | √ | - | - | √ | - | - | ALNS Meta-heuristic | - | - |

Olgun et al. [28]. | 2021 | - | - | √ | - | - | √ | - | - | Meta-heuristic | - | - |

Agra et al. [29]. | 2021 | √ | - | - | - | - | √ | - | - | Branching algorithm | - | - |

Sadati et el. [31]. | 2021 | √ | - | - | - | - | - | √ | - | Tabu search | - | - |

Wang et al. [32]. | 2021 | - | - | √ | - | - | - | √ | - | Heuristic | - | √ |

Zhen et al. [33]. | 2020 | √ | - | - | - | - | - | √ | - | Particle Swarm/genetic | - | - |

Soeanu et al. [34]. | 2020 | √ | √ | √ | √ | - | - | √ | - | Heuristic | - | - |

Li et al. [36]. | 2019 | √ | √ | √ | - | - | - | - | - | Ant colony optimization | - | - |

Wang et al. [38]. | 2021 | - | - | - | - | - | √ | - | Genetic | - | - | |

Araghi et al. [41]. | 2021 | - | - | - | √ | - | √ | Robust | ICA and VNS | - | - | |

Bruni et al. [42]. | 2021 | - | - | - | √ | √ | - | Fuzzy | Heuristic | - | √ | |

Zachariadis et al. [45]. | 2015 | √ | - | - | - | - | √ | - | - | Local search | - | - |

Bektaş et al. [46]. | 2020 | - | - | - | √ | - | √ | - | Heuristic | - | - | |

Bertazzi et al. [47]. | 2019 | - | - | - | - | - | - | Clustering | - | - | ||

Manupati et al. [48]. | 2021 | √ | √ | - | - | - | - | Robust | NSGA II | √ | ||

This research | √ | √ | √ | √ | √ | √ | √ | Stochastic-fuzzy-robust | √ | √ | √ |

Scenario | Parameter | Interval Limits |
---|---|---|

All Scenario | $Fix{I}_{i},Fix{J}_{j},Fix{L}_{l},Fix{K}_{k}$ | $~U\left[\mathrm{10,000},\mathrm{12,000}\right]$ |

$Fix{V}_{v}$ | $~U\left[300,400\right]$ | |

$Cap{V}_{v}$ | $~U\left[100,120\right]$ | |

$Cap{L}_{lp},Cap{I}_{ir},Cap{K}_{kr},Cap{J}_{jp},{\mu}_{l}$ | $~U\left[200,220\right]$ | |

$Di{s}_{n{n}^{\prime}}$ | $~U\left[10,100\right]$ | |

$C{T}_{l}$ | $~U\left[120,150\right]$ | |

${\mu}_{l}$ | $~U\left[140,180\right]$ | |

${\theta}_{l},{p}_{s},{r}_{p}^{c}$ | $0.5$ | |

${T}_{n{n}^{\prime}}$ | $~U\left[15,20\right]$ | |

${S}_{c},{C}_{lp}$ | $~U\left[2,5\right]$ | |

$[A{S}_{c},B{S}_{c}]$ | $~U\left[20,50\right]$ | |

$\alpha $ | $6$ | |

$H$ | $3$ | |

${\omega}_{rp}$ | $~U\left[1,2\right]$ | |

${\vartheta}_{l}$ | $~U\left[3,4\right]$ | |

${B}_{l}$ | $~U\left[250,300\right]$ | |

Scenario 1 | $T{r}_{n{n}^{\prime}vs},T{r}_{ikvs},T{r}_{kjvs},T{r}_{jlvs},T{r}_{ljvs},T{r}_{lovs}$ | $~U\left(\left[10,20\right],\left[20,30\right],\left[30,40\right],\left[40,50\right]\right)$ |

${D}_{cpts}$ | $~U\left(\left[15,20\right],\left[20,25\right],\left[25,30\right],\left[30,35\right]\right)$ | |

${R}_{cpts}$ | $~U\left(\left[5,10\right],\left[10,15\right],\left[15,20\right],\left[20,25\right]\right)$ | |

Scenario 2 | $T{r}_{n{n}^{\prime}vs},T{r}_{ikvs},T{r}_{kjvs},T{r}_{jlvs},T{r}_{ljvs},T{r}_{lovs}$ | $~U\left(\left[12,24\right],\left[24,36\right],\left[36,48\right],\left[48,60\right]\right)$ |

${D}_{cpts}$ | $~U\left(\left[20,25\right],\left[25,30\right],\left[30,35\right],\left[35,40\right]\right)$ | |

${R}_{cpts}$ | $~U\left(\left[10,15\right],\left[15,20\right],\left[20,25\right],\left[25,30\right]\right)$ |

$I-J-K-L-C-N-O-R-P-T-V-S$ |

$2-2-2-2-3-2-2-2-1-2-2-2$ |

OBF | Value | CPU-Time |
---|---|---|

OBFV1 | 347,389.77 | 285.54 |

OBFV2 | 12,822.22 | 360.46 |

OBFV2 | 37.54 | 188.39 |

Efficient Solutions | OBFV1 | OBFV2 | OBFV3 |
---|---|---|---|

1 | 352,373 | 131,062 | 41 |

2 | 351,263 | 131,054 | 54 |

3 | 350,043 | 130,998 | 63 |

4 | 348,895 | 130,983 | 74 |

5 | 348,849 | 130,897 | 89 |

6 | 347,795 | 130,839 | 103 |

7 | 346,218 | 129,746 | 112 |

8 | 345,233 | 128,864 | 120 |

**Table 6.**Locating and routing the vehicle in the small size problem for the first efficient solution.

Centers | Location |
---|---|

Supplier | 2 centers with number 2–3 |

Crude oil silo | 1 center with number 2 |

Production center | 1 center with number 2 |

Distribution/collection center | 2 centers with number 1–2 |

Vehicle routing-scenarion1-period1 | $L1\to N2\to N3\to L1\hspace{1em}L2\to N1\to L2$ |

Vehicle routing-scenarion1-period2 | $L1\to N2\to N3\to L1\hspace{1em}L2\to N1\to L2$ |

Vehicle routing-scenarion2-period1 | $L1\to N1\to N2\to L1\hspace{1em}L2\to N3\to L2$ |

Vehicle routing-scenarion2-period2 | $L1\to N1\to N2\to L1\hspace{1em}L2\to N3\to L2$ |

Changes of $H$(%) | OBFV1 | OBFV2 | OBFV3 |
---|---|---|---|

−30 | 349,666 | 130,672 | 41 |

−20 | 350,471 | 130,727 | 41 |

−10 | 350,993 | 130,783 | 41 |

0 | 352,373 | 131,062 | 41 |

10 | 352,817 | 131,204 | 41 |

20 | 353,614 | 131,467 | 41 |

30 | 353,836 | 131,543 | 41 |

Changes of α (%) | OBFV1 | OBFV2 | OBFV3 |
---|---|---|---|

−30 | 348,961 | 129,263 | 41 |

−20 | 350,009 | 129,938 | 41 |

−10 | 351,060 | 130,395 | 41 |

0 | 352,373 | 131,062 | 41 |

10 | 353,162 | 131,256 | 41 |

20 | 353,489 | 131,449 | 41 |

30 | 354,230 | 132,280 | 41 |

Changes of α_{s} | OBFV1 | OBFV2 | OBFV3 |
---|---|---|---|

0.1 | 347,647 | 128,829 | 38 |

0.3 | 350,470 | 130,183 | 39 |

0.5 | 352,373 | 131,062 | 41 |

0.7 | 353,742 | 133,330 | 43 |

0.9 | 356,462 | 134,240 | 44 |

Algorithm | Parameter | Level 1 | Level 2 | Level 3 | Best Value |
---|---|---|---|---|---|

NSGA II | Max it | 100 | 150 | 200 | 200 |

Npop | 50 | 100 | 200 | 200 | |

Pc | 0.75 | 0.80 | 0.85 | 0.80 | |

Pm | 0.03 | 0.035 | 0.04 | 0.03 | |

MOPSO | Max it | 100 | 150 | 200 | 200 |

NParticle | 50 | 100 | 200 | 200 | |

C1 | 1 | 1.5 | 2 | 1 | |

C2 | 1 | 1.5 | 2 | 2 |

Sample Problem | $I$ | $J$ | $K$ | $O$ | $C$ | $R$ | $S$ | $L$ | $P$ | $V$ | $T$ |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 10 | 10 | 10 | 10 | 12 | 2 | 2 | 10 | 6 | 6 | 4 |

2 | 10 | 10 | 10 | 10 | 14 | 2 | 2 | 12 | 6 | 6 | 4 |

3 | 10 | 10 | 10 | 10 | 16 | 2 | 2 | 14 | 6 | 6 | 4 |

4 | 12 | 12 | 12 | 12 | 18 | 2 | 3 | 16 | 7 | 8 | 5 |

5 | 12 | 12 | 12 | 12 | 20 | 3 | 3 | 18 | 7 | 8 | 5 |

6 | 12 | 12 | 12 | 12 | 22 | 3 | 3 | 20 | 7 | 8 | 5 |

7 | 15 | 15 | 15 | 15 | 24 | 3 | 4 | 22 | 8 | 9 | 6 |

8 | 15 | 15 | 15 | 15 | 26 | 3 | 4 | 24 | 8 | 9 | 6 |

9 | 15 | 15 | 15 | 15 | 28 | 4 | 4 | 26 | 9 | 10 | 6 |

10 | 20 | 20 | 20 | 20 | 30 | 4 | 4 | 28 | 10 | 12 | 7 |

11 | 20 | 20 | 20 | 20 | 32 | 4 | 4 | 30 | 12 | 14 | 7 |

12 | 20 | 20 | 20 | 20 | 34 | 4 | 5 | 35 | 14 | 15 | 7 |

13 | 30 | 30 | 30 | 30 | 36 | 4 | 5 | 40 | 16 | 18 | 8 |

14 | 30 | 30 | 30 | 30 | 38 | 5 | 5 | 45 | 18 | 20 | 8 |

15 | 30 | 30 | 30 | 30 | 40 | 5 | 5 | 50 | 20 | 20 | 8 |

Sample Problem | OBFV1 | OBFV2 | OBFV3 | NPF | MSI | SM | CPU-Time |
---|---|---|---|---|---|---|---|

1 | 5,171,405.41 | 2,185,833.07 | 111.22 | 78 | 13,812.22 | 0.47 | 49.50 |

2 | 5,398,994.99 | 2,762,386.33 | 154.71 | 82 | 15,289.78 | 0.35 | 54.96 |

3 | 6,019,488.06 | 3,177,521.50 | 172.35 | 78 | 13,967.83 | 0.49 | 72.67 |

4 | 6,423,286.41 | 4,097,302.32 | 181.45 | 86 | 25,449.48 | 0.70 | 127.99 |

5 | 6,538,955.89 | 4,479,863.43 | 241.63 | 99 | 21,730.77 | 0.53 | 211.19 |

6 | 6,785,707.08 | 5,184,306.28 | 281.41 | 83 | 25,579.07 | 0.35 | 293.14 |

7 | 7,158,871.09 | 5,171,757.20 | 307.80 | 90 | 11,585.14 | 0.45 | 417.93 |

8 | 7,510,548.53 | 5,866,537.72 | 332.89 | 84 | 19,542.07 | 0.58 | 531.23 |

9 | 8,203,753.18 | 6,163,001.94 | 357.24 | 87 | 13,608.73 | 0.41 | 651.28 |

10 | 8,752,921.66 | 7,404,421.13 | 382.46 | 82 | 24,953.18 | 0.50 | 800.82 |

11 | 9,632,491.97 | 7,679,581.39 | 418.87 | 89 | 15,643.39 | 0.45 | 946.15 |

12 | 9,964,265.20 | 7,753,593.48 | 430.39 | 93 | 20,804.43 | 0.33 | 1090.87 |

13 | 10,331,524.82 | 8,802,996.24 | 459.11 | 100 | 16,704.49 | 0.48 | 1288.36 |

14 | 10,898,039.49 | 9,128,745.36 | 481.50 | 88 | 17,739.98 | 0.32 | 1484.07 |

15 | 11,591,974.13 | 10,023,438.99 | 498.64 | 100 | 12,786.59 | 0.38 | 1741.28 |

Mean | 8,025,481.86 | 5,992,085.76 | 320.78 | 87.93 | 17,946.48 | 0.45 | 650.76 |

Sample Problem | OBFV1 | OBFV2 | OBFV3 | NPF | MSI | SM | CPU-Time |
---|---|---|---|---|---|---|---|

1 | 4,848,705.20 | 2,246,354.07 | 112.48 | 87 | 13,893.62 | 0.11 | 36.12 |

2 | 5,105,061.04 | 2,774,160.60 | 156.37 | 87 | 15,360.07 | 0.84 | 40.26 |

3 | 6,154,917.34 | 3,229,399.79 | 173.93 | 73 | 14,187.67 | 0.06 | 67.47 |

4 | 6,348,347.09 | 4,077,913.22 | 184.79 | 95 | 24,971.92 | 0.00 | 110.46 |

5 | 6,186,380.36 | 4,385,821.23 | 240.44 | 93 | 21,266.31 | 0.93 | 192.41 |

6 | 6,495,843.04 | 5,102,888.86 | 277.56 | 96 | 24,068.86 | 0.64 | 259.46 |

7 | 7,013,609.31 | 5,301,333.88 | 315.55 | 76 | 11,096.09 | 0.84 | 378.08 |

8 | 7,469,621.09 | 6,087,316.25 | 346.42 | 90 | 19,320.87 | 0.20 | 484.95 |

9 | 8,055,298.63 | 6,615,262.62 | 371.42 | 97 | 13,565.07 | 0.99 | 599.87 |

10 | 8,759,362.34 | 7,444,580.76 | 396.65 | 96 | 23,552.83 | 0.81 | 702.82 |

11 | 9,823,948.57 | 8,024,997.44 | 416.51 | 78 | 15,517.71 | 0.81 | 828.01 |

12 | 9,709,571.06 | 8,182,310.00 | 435.42 | 96 | 21,184.23 | 0.80 | 953.14 |

13 | 9,828,373.25 | 8,925,208.72 | 457.80 | 99 | 15,826.70 | 0.47 | 1093.42 |

14 | 10,171,326.13 | 9,752,619.14 | 480.31 | 83 | 17,106.66 | 0.58 | 1298.98 |

15 | 11,678,009.44 | 9,990,665.72 | 502.79 | 95 | 12,995.98 | 0.86 | 1565.99 |

Mean | 7,843,224.93 | 6,142,722.15 | 324.56 | 89.40 | 17,594.31 | 0.60 | 574.10 |

Index | Difference of Mean | 95% CI for Mean Different | T-Value | p-Value |
---|---|---|---|---|

OBFV1 | 182,257 | (45,034 319,480) | 2.85 | 0.013 |

OBFV2 | 150,636 | (30,215 271,057) | 2.68 | 0.018 |

OBFV3 | 3.78 | (0.42 7.15) | 2.41 | 0.030 |

NPF | 1.47 | (−3.43 6.36) | 0.64 | 0.531 |

MSI | 352 | (36 668) | 2.39 | 0.031 |

SM | 0.143 | (−0.084 0.371) | 1.35 | 0.198 |

CPU-Time | 76.7 | (38.7 114.7) | 4.33 | 0.001 |

Algorithm | OBFV1 | OBFV2 | OBFV3 | NPF | MSI | SM | CPU-Time |
---|---|---|---|---|---|---|---|

NSGA II | 8,025,481.86 | 5,992,085.76 | 320.78 | 87.93 | 17,946.48 | 0.45 | 650.76 |

MOPSO | 7,843,224.93 | 6,142,722.15 | 324.56 | 89.40 | 17,594.31 | 0.60 | 574.10 |

weight | 0.2 | 0.2 | 0.2 | 0.1 | 0.1 | 0.1 | 0.1 |

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## Share and Cite

**MDPI and ACS Style**

Bathaee, M.; Nozari, H.; Szmelter-Jarosz, A.
Designing a New Location-Allocation and Routing Model with Simultaneous Pick-Up and Delivery in a Closed-Loop Supply Chain Network under Uncertainty. *Logistics* **2023**, *7*, 3.
https://doi.org/10.3390/logistics7010003

**AMA Style**

Bathaee M, Nozari H, Szmelter-Jarosz A.
Designing a New Location-Allocation and Routing Model with Simultaneous Pick-Up and Delivery in a Closed-Loop Supply Chain Network under Uncertainty. *Logistics*. 2023; 7(1):3.
https://doi.org/10.3390/logistics7010003

**Chicago/Turabian Style**

Bathaee, Mehrnaz, Hamed Nozari, and Agnieszka Szmelter-Jarosz.
2023. "Designing a New Location-Allocation and Routing Model with Simultaneous Pick-Up and Delivery in a Closed-Loop Supply Chain Network under Uncertainty" *Logistics* 7, no. 1: 3.
https://doi.org/10.3390/logistics7010003