# Fuzzy Bi-Objective Closed-Loop Supply Chain Network Design Problem with Multiple Recovery Options

^{*}

## Abstract

**:**

## 1. Introduction

Paper | Recovery Options | Goals | Uncertainty | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

REP | REM | RCY | SM | DSP | MC | MP | MS | CSR | STO | FZY | |

Tuzkaya et al. [2] | √ | √ | √ | √ | |||||||

Ozkir and Basligil [5] | √ | √ | √ | √ | |||||||

Pishvaee and Torabi [6] | √ | √ | √ | √ | √ | ||||||

Soleimani et al. [8] | √ | √ | √ | √ | √ | √ | √ | ||||

Pourjavad and Mayorga [9] | √ | √ | √ | √ | √ | √ | |||||

Khajavi et al. [7] | √ | √ | √ | √ | √ | ||||||

Tang et al. [10] | √ | √ | |||||||||

Uster and Hwang [11] | √ | √ | √ | √ | |||||||

Ahmadi and Amin [12] | √ | √ | √ | √ | √ | ||||||

Jindal and Sangwan [15] | √ | √ | √ | √ | √ | ||||||

ÖZceylan and Paksoy [16] | √ | √ | √ | √ | |||||||

Pourjavad and Mayorga [17] | √ | √ | √ | √ | √ | √ | |||||

Jerbia et al. [19] | √ | √ | √ | √ | √ | √ | |||||

Darbari et al. [20] | √ | √ | √ | √ | √ | √ | |||||

Javid et al. [21] | √ | √ | √ | √ | √ | ||||||

Zhen et al. [22] | √ | √ | √ | √ | √ | ||||||

Our work | √ | √ | √ | √ | √ | √ | √ | √ |

## 2. CLSC Network Design Under Fuzziness

#### 2.1. Problem Description

#### 2.2. Mathematical Formulation

## 3. Processing Fuzzy Parameters in the CLSC Model

#### 3.1. Fuzzy Objective Functions

**Definition**

**1.**

**Definition**

**2**

**.**The expected interval (EI) and expected value (EV) of a TFN $\tilde{\xi}=({\xi}^{p},{\xi}^{m},{\xi}^{o})$ are, respectively,

**Theorem**

**1**

**.**Let $\tilde{\xi}$ and $\tilde{\zeta}$ be two TFNs. For any real numbers a and b, we have

#### 3.2. Fuzzy Constraint Functions

**Definition**

**3**

**.**For TFNs $\tilde{\xi}$ and $\tilde{\eta}$, the degree of $\tilde{\xi}>\tilde{\eta}$ is defined as

**Definition**

**4**

**.**For any two TFNs $\tilde{\xi}$ and $\tilde{\eta}$, we have $\tilde{\xi}{\approx}_{\alpha}\tilde{\eta}$, if $\tilde{\xi}{\le}_{\frac{\alpha}{2}}\tilde{\eta}$ and $\tilde{\eta}{\le}_{\frac{\alpha}{2}}\tilde{\xi}$, i.e.,

#### 3.3. Transformation

## 4. Solution Framework for the Fuzzy CLSC Model

#### 4.1. Interactive Approach to Multi-Objective Programming

#### 4.2. Two-Stage Fuzzy Interaction Solution Procedure

**Step****1:****Step****2:****Step****3:**- Calculate the PIS ${Z}_{1}^{\alpha -PIS}$ and ${Z}_{2}^{\alpha -PIS}$ for each objective at the feasibility level $\alpha $ and the corresponding decision schemes ${{\mathit{x}}_{1}}^{\alpha -PIS}$ and ${{\mathit{x}}_{2}}^{\alpha -PIS}$, and then figure out ${Z}_{1}^{\alpha -NIS}={Z}_{1}\left({{\mathit{x}}_{2}}^{\alpha -PIS}\right)$, ${Z}_{2}^{\alpha -NIS}={Z}_{2}\left({{\mathit{x}}_{1}}^{\alpha -PIS}\right)$.
**Step****4:**- Based on the PIS and NIS, compute the realization levels of the objectives in model (26), ${\mu}_{g}\left(x\right)$, which are definitely constructed as follows,$${\mu}_{1}\left(x\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}x<{Z}_{1}^{\alpha -NIS}\hfill \\ {\displaystyle \frac{x-{Z}_{1}^{\alpha -NIS}}{{Z}_{1}^{\alpha -PIS}-{Z}_{1}^{\alpha -NIS}},}\hfill & \mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{Z}_{1}^{\alpha -NIS}\le x\le {Z}_{1}^{\alpha -PIS}\hfill \\ 1,\hfill & \mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}x>{Z}_{1}^{\alpha -PIS},\hfill \end{array}\right.$$$${\mu}_{2}\left(x\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}x>{Z}_{2}^{\alpha -NIS}\hfill \\ {\displaystyle \frac{{Z}_{2}^{\alpha -NIS}-x}{{Z}_{2}^{\alpha -NIS}-{Z}_{2}^{\alpha -PIS}},}\hfill & \mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{Z}_{2}^{\alpha -PIS}\le x\le {Z}_{2}^{\alpha -NIS}\hfill \\ 1,\hfill & \mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}x<{Z}_{2}^{\alpha -PIS}.\hfill \end{array}\right.$$
**Step****5:**- Establish the synthesis evaluation function $\lambda \left(\mathbf{x}\right)$ via the multi-objective interactive TH method to convert the CB-MILP model (26) into a deterministic single-objective MILP model (30), and then determine the compensation factor $\gamma $ and the weight coefficients ${w}_{g}$ of the evaluation function to solve it.
**Step****6:**- Adjust the parameters $\alpha $, $\gamma $ (if necessary, the weight coefficients ${w}_{g}$ can be changed), and repeat Steps 2–6 until the solution is satisfactory from the perspective of the decision-maker.

## 5. Numerical Analysis

#### 5.1. Experimental Data

#### 5.2. Two-Stage Fuzzy Interaction Analysis

#### 5.3. Comparisons with the Deterministic Problem

#### 5.4. Discussions

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CLSC | Closed-loop supply chain |

MILP | Mixed integer linear programming |

TFN | Triangular fuzzy number |

EI | Expected interval |

EV | Expected value |

CB-MILP | Crisp bi-objective mixed 0-1 integer linear programming |

PIS | Positive ideal solution |

NIS | Negative ideal solution |

## Appendix A. Notations Used in the Fuzzy CLSC Model

Sets | |

I | Set of potential locations of plants, indexed by i; |

J | Set of potential locations of distribution centers, indexed by j; |

K | Set of fixed locations of primary customers, indexed by k; |

L | Set of potential locations of disassembly centers, indexed by l; |

M | Set of potential locations of redistribution centers, indexed by m; |

N | Set of fixed locations of secondary customers, indexed by n; |

P | Set of potential locations of disposal centers, indexed by p; |

Parameters | |

${\tilde{f}}_{i}$ | Fixed start-up cost of plant i; |

${\tilde{f}}_{j}$ | Fixed start-up cost of distribution center j; |

${\tilde{f}}_{l}$ | Fixed start-up cost of disassembly center l; |

${\tilde{f}}_{m}$ | Fixed start-up cost of redistribution center m; |

${\tilde{f}}_{p}$ | Fixed start-up cost of disposal center p; |

${\tilde{p}}_{i}$ | Capacity of plant i; |

${\tilde{p}}_{j}$ | Capacity of distribution center j; |

${\tilde{p}}_{l}$ | Capacity of disassembly center l; |

${\tilde{p}}_{m}$ | Capacity of redistribution center m; |

${\tilde{p}}_{p}$ | Capacity of disposal center p; |

${\tilde{tc}}_{ij}$ | Unit transportation cost from plant i to distribution center j; |

${\tilde{tc}}_{jk}$ | Unit transportation cost from distribution center j to customer k; |

${\tilde{tc}}_{li}$ | Unit transportation cost from disassembly center l to plant i; |

${\tilde{tc}}_{lm}$ | Unit transportation cost from disassembly center l to redistribution center m; |

${\tilde{tc}}_{lp}$ | Unit transportation cost from disassembly center l to disposal center p; |

${\tilde{tc}}_{im}$ | Unit transportation cost from plant i to redistribution center m; |

${\tilde{tc}}_{mn}$ | Unit transportation cost from redistribution center m to second customer n; |

${\tilde{vc}}_{i}$ | Unit remanufacturing cost at plant i; |

${\tilde{rc}}_{l}$ | Unit repairing cost at disassembly center l; |

${\tilde{dc}}_{p}$ | Unit disposing cost at disposal center p; |

${\tilde{hc}}_{j}$ | Unit handling cost at distribution center j; |

${\tilde{hc}}_{l}$ | Unit handling cost at disassembly center l; |

${\tilde{hc}}_{m}$ | Unit handling cost at redistribution center m; |

${\tilde{mc}}_{i}$ | Unit manufacturing cost at plant i; |

${\tilde{cc}}_{kl}$ | Unit collecting and transporting costs from primary customer k to disassembly center l; |

${\tilde{d}}_{k}$ | Demand of primary customer k; |

${\tilde{d}}_{n}$ | Demand of secondary customer n; |

$\tilde{\omega}$ | Maximum fractions of collecting waste products; |

${\tilde{\theta}}_{1}$ | Fraction of disposing waste products; |

${\tilde{\theta}}_{2}$ | Fraction of repairing products; |

${\tilde{et}}_{k}$ | Expected delivery time of customer k; |

${\tilde{s}}_{1}$ | Unit revenue of new products; |

${\tilde{s}}_{2}$ | Unit revenue of remanufactured products; |

${\tilde{s}}_{3}$ | Unit revenue of recycled raw materials; |

${\tilde{dt}}_{jk}$ | Delivery time from distribution center j to customer k; |

Decision variables | |

${X}_{ij}$ | Quantity of products shipped from plant i to distribution center j; |

${X}_{jk}$ | Quantity of products shipped from distribution center j to customer k; |

${X}_{kl}$ | Quantity of scrapped products collected from customer k to disassembly center l; |

${X}_{li}$ | Quantity of scrapped products shipped from disassembly center l to plant i; |

${X}_{lm}$ | Quantity of repaired products shipped from disassembly center l to redistribution center m; |

${X}_{lp}$ | Quantity of useless products shipped from disassembly center l to disposal center p; |

${X}_{im}$ | Quantity of products shipped from plant i to redistribution center m; |

${X}_{mn}$ | Quantity of products shipped from redistribution center m to second customer n; |

${X}_{l}$ | Quantity of raw material recycled by disassembly center l; |

${Y}_{i}$ | Binary variable equals 1 if plant is established at location i and 0 otherwise; |

${Y}_{j}$ | Binary variable equals 1 if distribution center is established at location j and 0 otherwise; |

${Y}_{l}$ | Binary variable equals 1 if disassembly center is established at location l and 0 otherwise; |

${Y}_{m}$ | Binary variable equals 1 if redistribution center is established at location m and 0 otherwise; |

${Y}_{p}$ | Binary variable equals 1 if disposal center is established at location p and 0 otherwise. |

## Appendix B. Formulas of the Fuzzy CLSC Model

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**Figure 6.**The realization levels of the two objectives with different weight coefficients and compensation factors.

**Figure 7.**Comparisons of the objective values under fuzzy and deterministic environments with the TH method at different feasibility levels.

**Figure 8.**Comparative results regarding the influence of total cost on the profit and the delayed delivery time under deterministic and fuzzy environments

Parameter | $({\mathit{\tau}}_{1},{\mathit{\tau}}_{2})$ | Parameter | $({\mathit{\tau}}_{1},{\mathit{\tau}}_{2})$ | Parameter | $({\mathit{\tau}}_{1},{\mathit{\tau}}_{2})$ |
---|---|---|---|---|---|

${\tilde{f}}_{i}$ | (0.31, 0.41) | ${\tilde{tc}}_{li}$ | (0.43, 0.41) | ${\tilde{cc}}_{kl}$ | (0.39, 0.49) |

${\tilde{f}}_{j}$ | (0.31, 0.48) | ${\tilde{tc}}_{lm}$ | (0.37, 0.25) | ${\tilde{d}}_{k}$ | (0.28, 0.34) |

${\tilde{f}}_{l}$ | (0.22, 0.41) | ${\tilde{tc}}_{lp}$ | (0.29, 0.24) | ${\tilde{d}}_{n}$ | (0.43, 0.37) |

${\tilde{f}}_{m}$ | (0.41, 0.39) | ${\tilde{tc}}_{im}$ | (0.44, 0.29) | $\tilde{\omega}$ | (0.29, 0.43) |

${\tilde{f}}_{p}$ | (0.33, 0.30) | ${\tilde{tc}}_{mn}$ | (0.29, 0.25) | ${\tilde{\theta}}_{1}$ | (0.23, 0.42) |

${\tilde{p}}_{i}$ | (0.28, 0.40) | ${\tilde{mc}}_{i}$ | (0.21, 0.22) | ${\tilde{\theta}}_{2}$ | (0.37, 0.29) |

${\tilde{p}}_{j}$ | (0.43, 0.28) | ${\tilde{vc}}_{i}$ | (0.43, 0.21) | ${\tilde{dt}}_{jk}$ | (0.49, 0.40) |

${\tilde{p}}_{l}$ | (0.25, 0.32) | ${\tilde{rc}}_{l}$ | (0.29, 0.26) | ${\tilde{et}}_{k}$ | (0.46, 0.33) |

${\tilde{p}}_{m}$ | (0.24, 0.43) | ${\tilde{dc}}_{p}$ | (0.33, 0.21) | ${\tilde{s}}_{1}$ | (0.20, 0.40) |

${\tilde{p}}_{p}$ | (0.48, 0.27) | ${\tilde{hc}}_{j}$ | (0.23, 0.20) | ${\tilde{s}}_{2}$ | (0.31, 0.33) |

${\tilde{tc}}_{ij}$ | (0.26, 0.48) | ${\tilde{hc}}_{l}$ | (0.38, 0.33) | ${\tilde{s}}_{3}$ | (0.29, 0.34) |

${\tilde{tc}}_{jk}$ | (0.37, 0.39) | ${\tilde{hc}}_{m}$ | (0.34, 0.37) |

**Table 3.**Realization levels of the two objectives with different feasibility levels ($\gamma =0.5$).

$\mathit{\alpha}$ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | Average |
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mu}_{1}(\%)$ | 80.29 | 81.64 | 82.98 | 84.30 | 85.06 | 86.81 | 88.46 | 89.93 | 92.20 | 95.18 | 95.32 | 87.47 |

${\mu}_{2}(\%)$ | 80.29 | 81.64 | 82.98 | 84.30 | 85.06 | 86.81 | 88.46 | 89.93 | 92.20 | 95.18 | 98.33 | 87.74 |

$\mathit{\alpha}$ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | Average |
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mu}_{1}(\%)$ | 98.26 | 98.22 | 98.16 | 98.06 | 97.94 | 97.76 | 97.57 | 97.30 | 96.95 | 96.49 | 94.40 | 97.28 |

${\mu}_{2}(\%)$ | 67.59 | 71.30 | 72.48 | 75.01 | 77.36 | 79.93 | 82.75 | 85.90 | 89.67 | 94.54 | 99.98 | 81.50 |

Gap (%) | 30.67 | 26.92 | 25.68 | 23.05 | 20.58 | 17.83 | 14.82 | 11.40 | 7.28 | 1.95 | −5.58 | 15.78 |

$\mathit{\gamma}$ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 |
---|---|---|---|---|---|---|---|---|---|---|---|

${\mu}_{1}(\%)$ | 96.49 | 96.49 | 96.48 | 96.48 | 95.18 | 95.18 | 95.18 | 95.18 | 95.18 | 95.18 | 95.18 |

${\mu}_{2}(\%)$ | 94.54 | 94.54 | 94.54 | 94.54 | 95.18 | 95.18 | 95.18 | 95.18 | 95.18 | 95.18 | 95.18 |

Gap (%) | 1.95 | 1.95 | 1.94 | 1.94 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

**Table 6.**Optimal solutions of the fuzzy and the deterministic model at different feasibility levels.

$\mathit{\alpha}$ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Fuzzy | Profit | 4,518,735 | 4,477,079 | 4,444,595 | 4,410,003 | 4,363,774 | 4,316,022 | 4,274,709 | 4,226,976 | 4,182,241 | 4,136,117 | 4,071,544 |

Time | 14,336.6 | 14,291.9 | 14,285.6 | 14,270.3 | 14,301.8 | 14,233.5 | 14,194.8 | 14,189.8 | 14,144.0 | 14,103.0 | 14,092.1 | |

Determ inistic | Profit | 3,878,633 | 3,878,633 | 3,878,633 | 3,878,633 | 3,878,633 | 3,878,633 | 3,878,633 | 3,878,633 | 3,878,633 | 3,878,633 | 3,878,633 |

Time | 4701.0 | 4701.0 | 4701.0 | 4701.0 | 4701.0 | 4701.0 | 4701.0 | 4701.0 | 4701.0 | 4701.0 | 4701.0 |

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## Share and Cite

**MDPI and ACS Style**

Zhou, J.; Xia, W.; Wang, K.; Li, H.; Zhang, Q.
Fuzzy Bi-Objective Closed-Loop Supply Chain Network Design Problem with Multiple Recovery Options. *Sustainability* **2020**, *12*, 6770.
https://doi.org/10.3390/su12176770

**AMA Style**

Zhou J, Xia W, Wang K, Li H, Zhang Q.
Fuzzy Bi-Objective Closed-Loop Supply Chain Network Design Problem with Multiple Recovery Options. *Sustainability*. 2020; 12(17):6770.
https://doi.org/10.3390/su12176770

**Chicago/Turabian Style**

Zhou, Jian, Wenying Xia, Ke Wang, Hui Li, and Qianyu Zhang.
2020. "Fuzzy Bi-Objective Closed-Loop Supply Chain Network Design Problem with Multiple Recovery Options" *Sustainability* 12, no. 17: 6770.
https://doi.org/10.3390/su12176770