# Incorporating Vehicle-Routing Problems into a Closed-Loop Supply Chain Network Using a Mixed-Integer Linear-Programming Model

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. Model Description and Formulation

- To select the optimum CLSC-network configuration.
- Finding the routes for the limited number of vehicles to serve a group of customers with their demand.

- Facility locations are known beforehand.
- The following of products between HFs is not allowed.
- The set-up cost of facilities is considered as fixed and predefined.

#### 3.1. Objective Function

#### 3.2. Constraints

$\sum _{h\in H}^{}{\displaystyle \sum _{vi\in VI}^{}{X}_{1}{}_{{}_{mhvi}}}}\le C{M}_{m$ | $\forall m\in M$ | (10) |

${R}_{2}{}_{{}_{h}}\le C{D}_{h}{U}_{1}{}_{{}_{h}}$ | $\forall h\in H$ | (11) |

$\sum _{h\in H}^{}{\displaystyle \sum _{vj\in VJ}^{}{X}_{3}{}_{{}_{hcvj}}}}\le C{C}_{c$ | $\forall c\in C$ | (12) |

$\sum _{m\in M}^{}{X}_{4}{}_{{}_{cim}}}\le C{C}_{c}{U}_{i$ | $\forall c\in C;\forall i\in I$ | (13) |

$\sum _{m\in M}^{}{\displaystyle \sum _{h\in H}^{}{X}_{1}{}_{{}_{mhvi}}}}\le {C}_{1}{}_{{}_{vi}$ | $\forall vi\in VI$ | (14) |

${L}_{2}{}_{{}_{k}}\le {C}_{2}{}_{{}_{k}}$ | $\forall k\in K$ | (15) |

$L{t}_{j{j}^{\prime}k}\le {C}_{2}{}_{{}_{k}}$ | $\forall j,{j}^{\prime}\in b;\forall k\in K$ | (16) |

$L{{t}^{\prime}}_{j{j}^{\prime}k}\le {C}_{2}{}_{{}_{k}}$ | $\forall j,{j}^{\prime}\in b;\forall k\in K$ | (17) |

$\sum _{h\in H}^{}{\displaystyle \sum _{c\in C}^{}{X}_{3}{}_{{}_{hcvj}}}}\le {C}_{3}{}_{{}_{vj}$ | $\forall vj\in VJ$ | (18) |

$\sum _{h\in H}^{}{X}_{4}{}_{{}_{cim}}}+{\displaystyle \sum _{s\in S}^{}{X}_{0}{}_{{}_{sim}}}={\displaystyle \sum _{h\in H}^{}{\displaystyle \sum _{vi\in VI}^{}{U}_{i}{X}_{1}{}_{{}_{mhvi}}}$ | $\forall m\in M;\forall i\in I$ | (19) |

$\sum _{h\in H}^{}{\displaystyle \sum _{vi\in VI}^{}{U}_{i}{X}_{3}{}_{{}_{hcvj}}}}={\displaystyle \sum _{m\in M}^{}{X}_{4}{}_{{}_{cim}}$ | $\forall c\in C;\forall i\in I$ | (20) |

$\sum _{h\in H}^{}{\displaystyle \sum _{vj\in VJ}^{}{X}_{3}{}_{{}_{hcvj}}}}={{R}^{\prime}}_{{{}_{2}}_{{}_{h}}$ | $\forall h\in h$ | (21) |

$\sum _{k\in K}^{}{\displaystyle \sum _{j\in b}^{}{X}_{2}{}_{{}_{j{j}^{\prime}k}}=1}$ | $\forall {j}^{\prime}\in R$ | (22) |

$\sum _{{j}^{\prime}\in b}^{}{X}_{2}{}_{{}_{{j}^{\prime}jk}}={\displaystyle \sum _{{j}^{\prime}\in b}^{}{X}_{2}{}_{{}_{j{j}^{\prime}k}}}$ | $\forall j\in R;\forall k\in K$ | (23) |

$\sum _{{j}^{\prime}\in R}^{}{X}_{2}{}_{{}_{j{j}^{\prime}k}}={\displaystyle \sum _{{j}^{\prime}\in R}^{}{X}_{2}{}_{{}_{{j}^{\prime}jk}}}$ | $\forall k\in K;\forall j\in H$ | (24) |

$\sum _{j\in H}^{}{\displaystyle \sum _{{j}^{\prime}\in R}^{}{X}_{2}{}_{{j}_{{j}^{\prime}k}}\le 1}$ | $\forall k\in K$ | (25) |

$\sum _{r\in R}^{}{Y}_{2}{}_{{}_{hr}}^{}}\ge {U}_{1}{}_{{}_{h}$ | $\forall h\in H$ | (26) |

${M}_{2}{}_{{}_{rk}}-{M}_{2}{}_{{}_{{r}^{\prime}k}}+\left({B}_{2}{X}_{2}{}_{{}_{j{j}^{\prime}k}}\right)\le {B}_{2}-1$ | $\forall r,{r}^{\prime}\in R;\forall k\in K$ | (27) |

${X}_{2}{}_{{}_{j{j}^{\prime}k}}=0$ | $\forall k\in K;\forall j,{j}^{\prime}\in H$ | (28) |

$\sum _{j\in b}^{}{\displaystyle \sum _{{j}^{\prime}\in b}^{}{d}_{2}{}_{{}_{j{j}^{\prime}}}{X}_{2}{}_{{}_{{j}_{{j}^{\prime}k}}}}\le M{D}_{2}{}_{{}_{k}}$ | $\forall k\in K$ | (29) |

$\sum _{{j}^{\prime}\in b}^{}{X}_{2}{}_{{}_{j{j}^{\prime}k}}}+{\displaystyle \sum _{{j}^{\prime}\in b}^{}{X}_{2}{}_{{}_{i{j}^{\prime}k}}-{Y}_{2}{}_{{}_{hr}}\le 1$ | $\forall j\in R;\forall i\in H;\forall k\in K$ | (30) |

$\sum _{h\in H}^{}{Y}_{2}{}_{{}_{hr}}}=1$ | $\forall r\in R$ | (31) |

$\sum _{m\in M}^{}{\displaystyle \sum _{vi\in VI}^{}{X}_{1}{}_{{}_{mhvi}}}}\ge {R}_{2}{}_{{}_{h}}^{$ | $\forall h\in H$ | (32) |

${L}_{2}{}_{{}_{k}}\ge {\displaystyle \sum _{j\in b}^{}{\displaystyle \sum _{{j}^{\prime}\in R}^{}{\tilde{D}}_{r}{X}_{2}{}_{{}_{j{j}^{\prime}k}}}}$ | $\forall k\in K$ | (33) |

${R}_{2}{}_{{}_{h}}\ge {\displaystyle \sum _{r\in R}^{}{\tilde{D}}_{r}{y}_{2}{}_{{}_{hr}}}$ | $\forall h\in H$ | (34) |

$L{t}_{j{j}^{\prime}k}\le {X}_{2}{}_{{}_{j{j}^{\prime}k}}{\displaystyle \sum _{r\in R}{D}_{r}}$ | $\forall j,{j}^{\prime}\in b;\forall k\in K$ | (35) |

$\sum _{k\in K}{\displaystyle \sum _{b\in B}L{t}_{brk}}}\le {\displaystyle \sum _{k\in K}{\displaystyle \sum _{b\in B}L{t}_{rbk}+}}{D}_{r$ | $\forall r\in R$ | (36) |

${L}_{{2}^{\prime}}{}_{{}_{k}}\ge {\displaystyle \sum _{j\in b}^{}{\displaystyle \sum _{{j}^{\prime}\in R}^{}{\tilde{P}}_{r}{X}_{2}{}_{{}_{j{j}^{\prime}k}}}}$ | $\forall k\in K$ | (37) |

${R}_{{2}^{\prime}}{}_{{}_{h}}\ge {\displaystyle \sum _{r\in R}^{}{\tilde{P}}_{r}{Y}_{2}{}_{{}_{hr}^{}}}$ | $\forall j\in H$ | (38) |

$L{{t}^{\prime}}_{j{j}^{\prime}k}\le {X}_{2}{}_{{}_{j{j}^{\prime}k}}{\displaystyle \sum _{r\in R}{P}_{r}}$ | $\forall j,{j}^{\prime}\in b;\forall k\in K$ | (39) |

$\sum _{k\in K}{\displaystyle \sum _{b\in B}L{{t}^{\prime}}_{brk}}}\le {\displaystyle \sum _{k\in K}{\displaystyle \sum _{b\in B}L{{t}^{\prime}}_{rbk}-}}{P}_{r$ | $\forall r\in R$ | (40) |

#### 3.3. Proposed Solution

- Parameters need to be determined before the value of a random variable.
- Parameters need to be determined after the random event has happened.

#### 3.4. Numerical Study

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

s | set of suppliers $s\in S$ |

i | set of raw materials $i\in I$ |

m | set of plants $m\in M$ |

h | set of hybrid facilities $(H{F}_{s})$ $h\in H$ |

r&j | set of customers |

b | aggregate set of hybrid facilities and customers ($h\cup r$) |

c | set of recycling centers $c\in C$ |

vi | set of vehicles $vi\in VI$ at node M |

k | set of vehicles $k\in K$ |

vj | set of vehicles $vj\in VJ$ at node RC |

Parameters: | |

${d}_{1}{}_{mh}$ | distance between plants $m\in M$ and hybrid facility $h\in H$ |

${d}_{2}{}_{j{j}^{\prime}}$ | distance between customer in node $j\in J$ and ${j}^{\prime}\in J$ |

${d}_{{3}_{hc}}$ | distance between hybrid facility $h\in H$ and recycling center $c\in C$ |

${\tilde{D}}_{r}$ | demand of customer $r\in R$ |

${\tilde{P}}_{r}$ | pickup customer $r\in R$ |

$M{D}_{2}{}_{{}_{k}}$ | maximum distance which vehicle $k\in K$ covers in a tour |

${C}_{1}{}_{{}_{vi}}$ | capacity of vehicle $vi\in VI$ |

${C}_{2}{}_{{}_{k}}$ | capacity of vehicle $k\in K$ |

${C}_{3}{}_{{}_{vj}}$ | capacity of vehicle $vj\in VJ$ |

$C{S}_{si}$ | capacity of supplier $s\in S$ to supply raw material $i\in I$ |

$C{M}_{m}$ | production capacity of plant $m\in M$ |

$C{D}_{h}$ | distribution capacity of hybrid facility $h\in H$ |

$C{C}_{c}$ | recycling capacity of recycling center $c\in C$ |

$P{S}_{si}$ | purchasing cost per unit of material $i\in I$ from supplier $s\in S$ |

$P{M}_{m}$ | production cost per unit of product at plant $m\in M$ |

$P{R}_{c}$ | recycling cost per unit of product at recycling center $c\in C$ |

$P{C}_{ci}$ | purchasing cost of unit of material $i\in I$ from recycling center $c\in C$ |

${B}_{2}$ | total number of customers |

${F}_{1}{}_{{}_{h}}$ | fixed opening cost of hybrid facility $h\in H$ |

$T{C}_{1}{}_{{}_{mhvi}}$ | transportation cost per unit of product of vehicle $vi\in VI$ between plant $m\in M$ and hybrid facility $h\in H$ |

$T{C}_{2}{}_{{}_{{j}_{{j}^{\prime}k}}}$ | transportation cost per unit of product of vehicle $k\in K$ between customer $j\in J$ and customer ${j}^{\prime}\in J$ |

$T{C}_{3}^{}{}_{{}_{hcvj}}$ | transportation cost per unit of product of vehicle $vj\in VJ$ between hybrid facility $h\in H$ and recycling center $c\in C$ |

${u}_{i}$ | utilization rate of material $i\in I$ |

$SP$ | selling price of end-of-life product |

Decision variables: | |

${X}_{2}{}_{{}_{j{j}^{\prime}k}}$ | binary variable indicating whether vehicle $k\in K$ travels directly from node $j\in J$ to node ${j}^{\prime}\in J$ |

${Y}_{2}{}_{{}_{hr}}$ | binary variable if hybrid facility $h\in H$ is assigned to customer $r\in R$ |

${U}_{1}{}_{{}_{h}}$ | $\mathrm{binary}\mathrm{variable}\mathrm{if}\mathrm{hybrid}\mathrm{facility}h\in H$ is open |

${L}_{2}{}_{{}_{k}}$ | load of vehicle $k\in K$ when leaving hybrid facility |

${L}_{{2}^{\prime}}{}_{{}_{k}}$ | load of vehicle $k\in K$ after having serviced all assigned customers |

${M}_{2}{}_{{}_{rk}}$ | sub-tour elimination variable for customer $r\in R$ $\mathrm{in}k\in K$ |

${R}_{2}{}_{{}_{h}}$ | distribution quantity of hybrid facility $h\in H$ |

${R}_{{2}^{\prime}}{}_{{}_{h}}$ | pick-up quantity for return product of hybrid facility $h\in H$ |

${X}_{0}{}_{{}_{sim}}$ | shipment quantity of raw material $i\in I$ between supplier $s\in S$ and plant $m\in M$ |

${X}_{1}{}_{{}_{mhvi}}$ | shipment quantity between plant $m\in M$ and hybrid facility $h\in H$ with vehicle $vi\in VI$ |

${X}_{3}{}_{{}_{hcvj}}$ | shipment quantity between hybrid facility $h\in H$ and recycling center $c\in C$ with vehicle $vj\in VJ$ |

${X}_{4}{}_{{}_{cim}}$ | shipment quantity of raw material $i\in I$ between recycling center $c\in C$ and plant $m\in M$ |

${L}_{t}{}_{{{}_{j}}_{{j}^{\prime}k}}$ | unload of demand for vehicle $k\in K$ from node $j\in J$ to node ${j}^{\prime}\in J$ |

${L}_{{t}^{\prime}}{}_{{}_{j{j}^{\prime}k}}$ | load of vehicle $k\in K$ from node $j\in J$ to node ${j}^{\prime}\in J$ |

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**Figure 4.**Network configuration when α- cut level is 0.4 for the lower bound for problem number two.

Model Characteristics | Özceylan, Demirel [72] | Farrokh, Azar [73] | Jabbarzadeh, Haughton [26] | Jerbia, Kchaou Boujelben [74] | Almaraj and Trafalis [75] | Zhou, Xia [76] | Diabat and Jebali [77] | Chouhan, Khan [78] | Chiu, Cheng [79] | This Paper |
---|---|---|---|---|---|---|---|---|---|---|

Product | ||||||||||

Single | X | X | X | X | X | |||||

Multiple | X | X | X | X | X | |||||

Period | ||||||||||

Single | X | X | X | |||||||

Multiple | X | X | X | X | X | X | X | |||

Modeling Approach | ||||||||||

Deterministic | X | X | X | |||||||

Stochastic-robust optimization | ||||||||||

Fuzzy programming | X | X | ||||||||

Scenario-based robust optimization | X | |||||||||

Mixed-integer linear programming | X | X | X | X | X | X | X | X | ||

Two-stage stochastic program | X | |||||||||

Robust optimization | X | |||||||||

Fuzzy-stochastic programming | X | X | ||||||||

Solution approach | ||||||||||

Optimization software package | X | X | X | X | X | X | X | |||

Lagrangian relaxation | X | |||||||||

Metaheuristics algorithms | X | |||||||||

Decomposition method | X | |||||||||

Uncertain parameters | ||||||||||

Demand | X | X | X | X | X | |||||

Capacity | X | X | X | X | ||||||

Cost | X | X | X | X | X | |||||

Return | X | X | X | |||||||

Recovery rate | X | |||||||||

Revenue | X | X | ||||||||

Error type | X | |||||||||

Delivery time | X | |||||||||

Disposal and repair | X | |||||||||

Objective-function components | ||||||||||

Transportation cost (min) | X | X | X | X | X | X | X | X | X | |

Inventory cost (min) | X | X | X | X | X | X | ||||

Facility fixed-cost opening (min) | X | X | X | X | X | X | ||||

Penalty cost (min) | X | X | ||||||||

Disposal (min) | X | X | X | |||||||

Purchasing (min) | X | X | X | X | X | |||||

Collection (min) | X | X | X | X | X | |||||

Manufacturing (min) | X | X | X | X | X | X | X | |||

Disassembling (min) | X | |||||||||

Recycling cost (min) | X | X | X | |||||||

Lost cost (min) | X | |||||||||

Profit (max) | X | X | ||||||||

Remanufacturing cost (min) | X | X | ||||||||

Repair cost (min) | X | |||||||||

Distribution cost (min) | X | |||||||||

Procurement cost (min) | X | |||||||||

Labor cost (min) | X | |||||||||

Allocation cost (min) | X | |||||||||

Processing cost (min) | X | X | X |

Problem No. | No. of Potential Suppliers | No. of Potential Manufacturers | No. of Hybrid Facilities | No. of Existing Customers | No. of Potential Recycling Centers |
---|---|---|---|---|---|

1 | 4 | 2 | 6 | 8 | 3 |

2 | 6 | 4 | 11 | 17 | 4 |

3 | 5 | 3 | 10 | 15 | 6 |

4 | 5 | 5 | 15 | 20 | 5 |

Parameter | Value | |||
---|---|---|---|---|

${\tilde{D}}_{r}$ | ~Unif (20,000, 30,000) | ~Unif (30,000, 40,000) | ~Unif (40,000, 50,000) | ~Unif (50,000, 60,000) |

${\tilde{P}}_{r}$ | ~Unif (5000, 10,000) | ~Unif (10,000,15,000) | ~Unif (15,000,20,000) | ~Unif (20,000, 25,000) |

${\tilde{C}}_{1}{}_{{}_{vi}}$ | ~Unif (3,000,000, 4,000,000) | ~Unif (4,000,000, 5,000,000) | ~Unif (5,000,000, 6,000,000) | ~Unif (6,000,000, 7,000,000) |

${\tilde{C}}_{2}{}_{{}_{{}_{k}}}$ | ~Unif (3,000,000, 4,000,000) | ~Unif (4,000,000, 5,000,000) | ~Unif (5,000,000, 6,000,000) | ~Unif (6,000,000, 7,000,000) |

${\tilde{C}}_{3}{}_{vj}$ | ~Unif (3,000,000, 4,000,000) | ~Unif (4,000,000, 5,000,000) | ~Unif (5,000,000, 6,000,000) | ~Unif (6,000,000, 7,000,000) |

$\tilde{C}{S}_{si}$ | ~Unif (2,000,000, 3,000,000) | ~Unif (3,000,000, 4,000,000) | ~Unif (4,000,000, 5,000,000) | ~Unif (5,000,000, 6,000,000) |

$\tilde{C}{M}_{m}$ | ~Unif (2,000,000, 2,500,000) | ~Unif (2,500,000, 3,500,000) | ~Unif (3,500,000, 4,500,000) | ~Unif (4,500,000, 5,500,000) |

$\tilde{C}{D}_{h}$ | ~Unif (900,000, 1,000,000) | ~Unif (1,000,000, 1,500,000) | ~Unif (1,500,000, 2,000,000) | ~Unif (2,000,000, 2,500,000) |

$\tilde{C}{C}_{c}$ | ~Unif (900,000, 1,000,000) | ~Unif (1,000,000, 1,500,000) | ~Unif (1,500,000, 2,000,000) | ~Unif (2,000,000, 2,500,000) |

${\tilde{F}}_{1}{}_{{}_{h}}$ | ~Unif (200,000, 250,000) | ~Unif (250,000, 300,000) | ~Unif (300,000, 350,000) | ~Unif (350,000, 400,000) |

$\tilde{P}{S}_{si}$ | ~Unif (1, 3) | ~Unif (3, 6) | ~Unif (6, 9) | ~Unif (9, 12) |

$\tilde{P}{M}_{m}$ | ~Unif (2, 4) | ~Unif (4, 6) | ~Unif (6, 8) | ~Unif (8, 10) |

$\tilde{P}{R}_{c}$ | ~Unif (2, 4) | ~Unif (4, 6) | ~Unif (6, 8) | ~Unif (8, 10) |

$\tilde{P}{C}_{ci}$ | ~Unif (2, 4) | ~Unif (4, 6) | ~Unif (6, 8) | ~Unif (8, 10) |

$\tilde{T}{C}_{1}{}_{{}_{mhvi}}$ | ~Unif (10, 12) | ~Unif (12, 14) | ~Unif (14, 16) | ~Unif (14, 18) |

$\tilde{T}{C}_{2}{}_{{}_{j{j}^{\prime}k}}$ | ~Unif (10, 12) | ~Unif (12, 14) | ~Unif (14, 16) | ~Unif (14, 18) |

$\tilde{T}{C}_{3}{}_{{}_{hcvj}}$ | ~Unif (10, 12) | ~Unif (12, 14) | ~Unif (14, 16) | ~Unif (14, 18) |

$SP$ | 20 | 30 | 40 | 50 |

Problem No. | ||||
---|---|---|---|---|

$\alpha -cutlevel$ | 1 | 2 | 3 | 4 |

$\alpha =0$ | [190,528,300, 773,505,400] | [474,328,100, 1,719,479,000] | [590,739,600, 2,150,690,000] | [306,343,100, 1,191,767,000] |

$\alpha =0.2$ | [217,807,900, 726,333,400] | [527,852,300, 1,663,605,000] | [662,951,500, 2,150,690,000] | [342,264,400, 1,138,206,000] |

$\alpha =0.4$ | [246,618,800, 680,223,700] | [584,420,000, 1,494,726,000] | [727,251,800, 2,032,501,000] | [421,803,700, 1,076,690,000] |

$\alpha =0.6$ | [276,812,200, 6,302,216,00] | [631,762,400, 1,387,619,000] | [775,098,100, 1,748,506,000] | [464,338,500, 995,462,700] |

$\alpha =0.8$ | [308,375,900, 577,293,400] | [679,946,900, 128,397,3000] | [845,901,700, 1,599,929,000] | [500,146,300, 912,892,800] |

$\alpha =1$ | [341,418,500, 526,630,300] | [748,914,300, 1,183,985,000] | [986,029,000, 1,457,822,000] | [554,242,300, 833,767,800] |

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## Share and Cite

**MDPI and ACS Style**

Pedram, A.; Sorooshian, S.; Mulubrhan, F.; Abbaspour, A.
Incorporating Vehicle-Routing Problems into a Closed-Loop Supply Chain Network Using a Mixed-Integer Linear-Programming Model. *Sustainability* **2023**, *15*, 2967.
https://doi.org/10.3390/su15042967

**AMA Style**

Pedram A, Sorooshian S, Mulubrhan F, Abbaspour A.
Incorporating Vehicle-Routing Problems into a Closed-Loop Supply Chain Network Using a Mixed-Integer Linear-Programming Model. *Sustainability*. 2023; 15(4):2967.
https://doi.org/10.3390/su15042967

**Chicago/Turabian Style**

Pedram, Ali, Shahryar Sorooshian, Freselam Mulubrhan, and Afshin Abbaspour.
2023. "Incorporating Vehicle-Routing Problems into a Closed-Loop Supply Chain Network Using a Mixed-Integer Linear-Programming Model" *Sustainability* 15, no. 4: 2967.
https://doi.org/10.3390/su15042967