# A Stochastic Programming Approach with Improved Multi-Criteria Scenario-Based Solution Method for Sustainable Reverse Logistics Design of Waste Electrical and Electronic Equipment (WEEE)

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement and Modeling

- The number and locations of local collection centers, product markets, material markets, and disposal facilities are known.
- The potential locations of regional collection centers and recycling plants are known.
- The fixed costs, unit transportation costs, and unit processing costs are known.
- The capacities of new facilities are predetermined.
- The WEEE can be converted at a fixed rate to new products, recycled parts, disposal fractions, and hazardous materials.
- The carbon emissions rate is mainly determined by the size of facility and technology adopted in the treatment and transportation of WEEE.

Sets: | |

C | Set of local collection sites of WEEE, indexed by c |

R | Set of potential locations of regional collection centers of WEEE, indexed by r |

P | Set of potential locations of recycling plants of WEEE, indexed by p |

H | Set of hazardous waste management systems, indexed by h |

F | Set of product market, indexed by f |

M | Set of material market, indexed by m |

D | Set of disposal sites, indexed by d |

L | Set of products, indexed by l |

S | Set of scenarios, indexed by s |

Parameters: | |

$F{x}_{r}$ | Fixed cost of opening regional collection center at potential location r $\in $ R |

$F{x}_{p}$ | Fixed cost of opening recycling plant at potential location p $\in $ P |

$C{p}_{rl}$ | Unit cost at regional collection center r $\in $ R for processing product l $\in $ L |

$C{p}_{pl}$ | Unit cost at recycling plant p $\in $ P for processing product l $\in $ L |

$C{p}_{d}$ | Unit cost at disposal site d $\in $ D |

$C{p}_{h}$ | Unit cost at hazardous waste management system h $\in $ H |

$PC{p}_{fl}^{s}$ | Unit price of recycled product l $\in $ L at product market f $\in $ F in scenario s $\in $ S |

$PC{m}_{ml}^{s}$ | Unit price of recycled product l $\in $ L at material market m $\in $ M in scenario s $\in $ S |

$Ct{r}_{lcr}$ | Transportation cost per unit product l $\in $ L from local collection site c $\in $ C to regional collection center r $\in $ R |

$Ct{r}_{lrp}$ | Transportation cost per unit recyclable fraction of product l $\in $ L from regional collection site r $\in $ R to recycling plant p $\in $ P |

$Ct{r}_{lrd}$ | Transportation cost per unit disposed fraction of product l $\in $ L from regional collection center r $\in $ R to disposal site d $\in $ D |

$Ct{r}_{lrh}$ | Transportation cost per unit hazardous fraction of product l $\in $ L from regional collection center r $\in $ R to hazardous waste management system h $\in $ H |

$Ct{r}_{lpf}$ | Transportation cost per unit recycled fraction of product l $\in $ L from recycling plant p $\in $ P to product market f $\in $ F |

$Ct{r}_{lpm}$ | Transportation cost per unit recycled fraction of product l $\in $ L from recycling plant p $\in $ P to material market m $\in $ M |

$Ct{r}_{lpd}$ | Transportation cost per unit disposed fraction of product l $\in $ L from recycling plant p $\in $ P to disposal site d $\in $ D |

$\theta $ | Unit cost of carbon credit |

$C{O}_{2}^{cap}$ | Carbon emissions cap for reverse logistics system for WEEE |

$Co{l}_{lc}^{s}$ | Amount of product l $\in $ L collected at local collection site c $\in $ C in scenario s $\in $ S |

$Capacit{y}_{rl}$ | Capacity of regional collection center r $\in $ R for product l $\in $ L |

$Capacit{y}_{pl}$ | Capacity of recycling products center p $\in $ P for product l $\in $ L |

${\phi}_{lp}$ | Recycling fraction of product l $\in $ L |

${\phi}_{ld}$ | Disposed fraction of product l $\in $ L |

${\phi}_{lh}$ | Hazardous fraction of product l $\in $ L |

$L{m}^{\prime}$ | An infinitely large positive number |

${\vartheta}_{lf}$ | Conversion rate of production l $\in $ L to product market f $\in $ F |

${\vartheta}_{lm}$ | Conversion rate of production l $\in $ L to material market m $\in $ M |

${\vartheta}_{ld}$ | Conversion rate of production l $\in $ L to disposal site d $\in $ D |

$Em{s}_{r}$ | Carbon emissions per unit capacity for opening a regional collection site r $\in $ R |

$Em{s}_{p}$ | Carbon emissions per unit capacity for opening a recycling plant p $\in $ P |

$Em{s}_{lcr}$ | Carbon emissions for transporting one unit product l $\in $ L from local collection site c $\in $ C to regional collection center r $\in $ R |

$Em{s}_{lrp}$ | Transportation cost one unit recyclable fraction of product l $\in $ L from regional collection site r $\in $ R to recycling plant p $\in $ P |

$Em{s}_{lrd}$ | Transportation cost one unit disposed fraction of product l $\in $ L from regional collection center r $\in $ R to disposal site d $\in $ D |

$Em{s}_{lrh}$ | Transportation cost one unit hazardous fraction of product l $\in $ L from regional collection center r $\in $ R to hazardous waste management system h $\in $ H |

$Em{s}_{lpf}$ | Transportation cost one unit recycled fraction of product l $\in $ L from recycling plant p $\in $ P to product market f $\in $ F |

$Em{s}_{lpm}$ | Transportation cost one unit recycled fraction of product l $\in $ L from recycling plant p $\in $ P to material market m $\in $ M |

$Em{s}_{lpd}$ | Transportation cost one unit disposed fraction of product l $\in $ L from recycling plant p $\in $ P to disposal site d $\in $ D |

Decision variables (First-level): | |

${Y}_{r}^{s}=\{\begin{array}{c}1\\ 0\end{array}$ | Potential location of regional collection center r $\in $ R is selected in scenario s $\in $ S Otherwise |

${Y}_{p}^{s}=\{\begin{array}{c}1\\ 0\end{array}$ | Potential location of recycling plant p $\in $ P is selected in scenario s $\in $ S Otherwise |

Decision variables (Second-level): | |

$Q{g}_{rl}^{s}$ | Amount of product l $\in $ L processed at regional collection center r $\in $ R in scenario s $\in $ S |

$Q{g}_{pl}^{s}$ | Amount of recycled fraction of product l $\in $ L processed at recycling plant p $\in $ P in scenario s $\in $ S |

$Q{g}_{d}^{s}$ | Amount of disposed fraction of product l $\in $ L processed at disposal site d $\in $ D in scenario s $\in $ S |

$Q{g}_{h}^{s}$ | Amount of hazardous fraction of product l $\in $ L processed at hazardous waste management system h $\in $ H in scenario s $\in $ S |

$Q{g}_{fl}^{s}$ | Amount of recycled fraction of product l $\in $ L sold at product market f $\in $ F in scenario s $\in $ S |

$Q{g}_{ml}^{s}$ | Amount of recycled fraction of product l $\in $ L sold at material market m $\in $ M in scenario s $\in $ S |

$Qt{r}_{lcr}^{s}$ | Amount of product l $\in $ L transported from local collection site c $\in $ C to regional collection center r $\in $ R in scenario s $\in $ S |

$Qt{r}_{lrp}^{s}$ | Amount recycled fraction of product l $\in $ L transported from regional collection site r $\in $ R to recycling plant p $\in $ P in scenario s $\in $ S |

$Qt{r}_{lrd}^{s}$ | Amount of disposed fraction of product l $\in $ L transported from regional collection center r $\in $ R to disposal site d $\in $ D in scenario s $\in $ S |

$Qt{r}_{lrh}^{s}$ | Amount of hazardous fraction of product l $\in $ L transported from regional collection center r $\in $ R to hazardous waste management system h $\in $ H in scenario s $\in $ S |

$Qt{r}_{lpf}^{s}$ | Amount of recycled fraction of product l $\in $ L transported from recycling plant p $\in $ P to product market f $\in $ F in scenario s $\in $ S |

$Qt{r}_{lpm}^{s}$ | Amount of recycled fraction of product l $\in $ L transported from recycling plant p $\in $ P to material market m $\in $ M in scenario s $\in $ S |

$Qt{r}_{lpd}^{s}$ | Amount of disposed fraction product l $\in $ L transported from recycling plant p $\in $ P to disposal site d $\in $ D in scenario s $\in $ S |

$C{O}_{2}^{S}$ | Total amount of carbon emissions from the reverse logistics system for WEEE |

## 3. Multi-Criteria Scenario-Based Solution Method

- (1)
- From a mathematical perspective, Soleimani et al.’s method is only capable of solving the max-min and min-max problems [37]. For example, it is able to resolve the problem considering the maximum profit and the minimum risk or maximum reliability of the supply chain. However, for the problem formulated in this paper, which is a min-min problem aiming at determining the minimum costs and minimum data dispersion, this solution method is ineffective.
- (2)
- From a managerial perspective, the managerial meaning of coefficient of variation and its reciprocal are associated with the relative data dispersion compared with the absolute data dispersion determined by standard deviation, but it is not a dedicated tool for determining the optimal solution of a stochastic optimization problem. The theoretical justification of Soleimani et al.’s method [37] is not strong enough to enable comprehensive managerial interpretation. Furthermore, the data dispersion evaluated by standard deviation is significantly affected by the mean value, and this may lead to misinterpretation of the real shape of data dispersion.

## 4. Numerical Experiment

- (1)
- As shown in Figure 4a, the best solution of individual optimal costs is achieved in scenario s5 with the lowest system costs of 35,306,520 USD. The optimal cost of the deterministic scenario s0 is the median of the problem, and the optimal costs of scenarios s1, s2, s3, s4, and s10 are higher than the median value, while scenarios s5, s6, s7, s8, and s9 have a better performance in their individual optimal costs.
- (2)
- As shown in Figure 4b, when the candidate solutions are evaluated through all the test scenarios, the best solution of the mean costs is 36,997,582 USD, achieved in scenario s4, and the worst solution is found for scenario s1, with 37,107,575 USD. It is noteworthy that, in terms of the mean costs, the performance of scenarios s2, s3, s4, s6, s8, and s9 is close to the best performance. Furthermore, it is also observed that the change of mean costs is not correlated to the change of optimal individual costs, which means the better optimal individual costs may not lead to a better overall economic performance in most cases.
- (3)
- As shown in Figure 4c, when the candidate solutions are evaluated by standard deviation, the best solution is obtained via scenario s1, with the lowest standard deviation at 4,837,063.5 USD, which is far better than the other candidate solutions. This illustrates that the result of candidate solution s1 tested with all possible scenarios has a more centered distribution around the mean value.
- (4)
- As shown in Figure 4d, in terms of the performance of the coefficient of variation, the best solution is obtained in scenario s2, with the lowest value of coefficient of variation at 0.1303524, which is far better than the other candidate solutions. The second best solution in terms of the coefficient of variation is achieved in scenario s7.
- (5)
- Comparing Figure 4c with Figure 4d, it is observed that the change of the performance of candidate solutions with respect to standard deviation and coefficient of variation is quite similar, and the influence of the mean costs seems insignificant in this example. This result can be explained by the significant difference in the ranges of mean values and standard deviation. The range of the mean value is only 0.3%, which means the difference between the best and the worst solution is not significant. However, the range between the best solution and the worst solution in standard deviation is 7.2%, which is 24 times higher than that of the mean value, so standard deviation has a much more significant influence on coefficient of variation in this example.
- (6)
- In this example, $Mea{n}_{min}$ is 36,997,582 USD, obtained from candidate solution s4, and $Coefficient\text{}of\text{}variatio{n}_{min}$ is 0.1303524, obtained from candidate solution s1. ${W}_{M}$ and ${W}_{C}$ denote the weight of the mean and coefficient of variation in the evaluation of the overall performance of the reverse logistics system for WEEE, which reflects the relative importance of the expected objective value and reliability in decision-making. In this example, we test the same weights of the mean (0.5) and coefficient of variation (0.5), and the optimal result is 1.001486, achieved at candidate solution s1; the second and third best solutions equal 1.032073 and 1.035218, obtained through candidate solutions s7, s0 and s10 (s0 and s10 have the same value), respectively. When the optimal overall performance is obtained at candidate solution s1, potential locations r1 and r5 are selected for opening WEEE regional collection centers and potential locations p4 and p5 are selected for opening WEEE recycling plants.

## 5. Sensitivity Analysis

## 6. Conclusions

- (1)
- The paper provides a novel stochastic optimization model for the design of a generic reverse logistics system for WEEE. Reverse logistics is characterized as having a high level of uncertainty, so the modelling and formulation of some uncertain parameters are of significant importance.
- (2)
- Compared with previous mathematical models, the model proposed in this paper considers the environmental impacts of the reverse logistics system for WEEE, and the minimization of carbon emissions is also a very important consideration of the model.
- (3)
- The model is resolved with the multi-criteria scenario-based solution method in order to find the most economically efficient and reliable solution to the stochastic optimization problem. The expected objective value and reliability are evaluated by the mean and coefficient of variation, and normalized weighted sum formulation is applied to combine the two evaluation criteria. The solution method enables interactions between the subjective evaluation from the decision makers and the objective system values, so the result achieved is more reliable and robust. In addition, our improved solution method also resolves the deficiencies of the original solution method, and is capable of solving min-min and max-max optimization problems. In addition, the managerial meaning of the solution method is explicitly explained in this paper.
- (4)
- The numerical experiment and sensitivity analyses provide valuable managerial insights into the design of a reverse logistics system for WEEE. For example, capacity expansion at existing facilities may be a more economically efficient way for dealing with an increased amount of WEEE, and both economic and environmental performance may be improved simultaneously with location optimization and transportation aggregation. In addition, the managerial insight from the system design and planning of a reverse logistics network of WEEE may also provide valuable information for the government in determining a subsidy scheme for companies performing WEEE treatment.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Computational Results of Numerical Example

Sol. s0 | Sol. s1 | Sol. s2 | Sol. s3 | Sol. s4 | Sol. s5 | Sol. s6 | Sol. s7 | Sol. s8 | Sol. s9 | Sol. s10 | |
---|---|---|---|---|---|---|---|---|---|---|---|

s0 | 35,801,630 | 35,952,130 | 35,801,630 | 35,801,630 | 35,801,630 | 35,801,630 | 35,801,630 | 35,875,160 | 35,801,630 | 35,801,630 | 35,801,630 |

s1 | 36,718,370 | 36,548,660 | 36,718,370 | 36,718,370 | 36,718,370 | 36,718,370 | 36,718,370 | 36,746,110 | 36,718,370 | 36,718,370 | 36,718,370 |

s2 | 36,893,640 | 37,110,980 | 36,726,620 | 36,893,640 | 36,893,640 | 36,893,640 | 36,893,640 | 36,921,500 | 36,893,640 | 36,893,640 | 36,893,640 |

s3 | 36,727,900 | 36,783,460 | 36,727,900 | 36,558,310 | 36,727,900 | 36,727,900 | 36,727,900 | 36,755,780 | 36,727,900 | 36,727,900 | 36,727,900 |

s4 | 36,908,510 | 37,181,410 | 36,908,510 | 36,908,510 | 36,736,270 | 36,908,510 | 36,908,510 | 36,936,390 | 36,908,510 | 36,908,510 | 36,908,510 |

s5 | 35,472,470 | 35,472,470 | 35,472,470 | 35,472,470 | 35,472,470 | 35,306,520 | 35,472,470 | 35,472,470 | 35,472,470 | 35,472,470 | 35,472,470 |

s6 | 35,633,550 | 35,825,670 | 35,633,550 | 35,633,550 | 35,633,550 | 35,633,550 | 35,470,410 | 35,633,550 | 35,633,550 | 35,633,550 | 35,633,550 |

s7 | 35,494,550 | 35,519,400 | 35,494,550 | 35,494,550 | 35,494,550 | 35,494,550 | 35,494,550 | 35,328,710 | 35,494,550 | 35,494,550 | 35,494,550 |

s8 | 35,655,000 | 35,871,970 | 35,655,000 | 35,655,000 | 35,655,000 | 35,655,000 | 35,655,000 | 35,655,640 | 35,492,600 | 35,655,000 | 35,655,000 |

s9 | 30,292,120 | 31,104,970 | 30,292,120 | 30,292,120 | 30,292,120 | 30,292,120 | 30,292,120 | 30,607,740 | 30,292,120 | 30,136,120 | 30,292,120 |

s10 | 51,547,900 | 50,812,200 | 51,547,900 | 51,547,900 | 51,547,900 | 51,547,900 | 51,547,900 | 51,573,440 | 51,547,900 | 51,547,900 | 51,547,900 |

MV ^{1} | 37,013,240 | 37,107,575 | 36,998,056 | 36,997,823 | 36,997,582 | 36,998,154 | 36,998,409 | 37,046,045 | 36,998,476 | 36,999,058 | 37,013,240 |

SD ^{2} | 5,162,559.3 | 4,837,063.5 | 5,163,191.8 | 5,163,749.7 | 5,163,169.9 | 5,167,752 | 5,167,151.5 | 5,132,479.4 | 5,167,062.2 | 5,183,042.5 | 5,162,559.3 |

CV ^{3} | 0.1394787 | 0.1303524 | 0.139553 | 0.139569 | 0.1395542 | 0.1396759 | 0.1396587 | 0.1385432 | 0.139656 | 0.1400858 | 0.1394787 |

^{1}MV: Mean value.

^{2}SD: Standard deviation.

^{3}CV: Coefficient of variance (CV = SD/MV).

## Appendix B. Computational Result of Sensitivity Analysis

**Table B1.**Scenario-based overall system costs of the reverse logistics system for WEEE in sensitivity analysis (A).

Sol. s0 | Sol. s1 | Sol. s2 | Sol. s3 | Sol. s4 | Sol. s5 | Sol. s6 | Sol. s7 | Sol. s8 | Sol. s9 | Sol. s10 | |
---|---|---|---|---|---|---|---|---|---|---|---|

s0 | 30,433,820 | 30,710,800 | 30,451,520 | 30,593,350 | 30,334,070 | 30,710,800 | 30,451,520 | 30,593,350 | 30,334,070 | 29,651,950 | 31,215,700 |

s1 | 31,251,300 | 30,937,490 | 31,134,170 | 30,789,760 | 31,176,440 | 30,747,490 | 31,134,170 | 30,789,760 | 31,176,440 | 30,416,190 | 32,086,410 |

s2 | 31,408,420 | 31,530,740 | 31,091,320 | 31,573,010 | 31,294,760 | 31,530,740 | 31,252,500 | 31,573,010 | 31,294,760 | 30,573,310 | 32,243,530 |

s3 | 31,586,510 | 31,183,690 | 31,570,370 | 30,959,830 | 31,511,650 | 31,183,690 | 31,570,370 | 31,124,970 | 31,511,650 | 30,751,400 | 32,421,620 |

s4 | 31,394,940 | 31,618,250 | 31,340,010 | 31,559,530 | 31,113,650 | 31,618,250 | 31,340,010 | 31,559,530 | 31,281,290 | 30,559,830 | 32,230,050 |

s5 | 30,670,490 | 30,210,790 | 30,537,930 | 30,247,360 | 30,574,500 | 30,049,090 | 30,537,930 | 30,247,360 | 30,574,500 | 29,898,040 | 31,442,940 |

s6 | 30,478,020 | 30,585,080 | 30,329,100 | 30,621,640 | 30,365,660 | 30,585,080 | 30,171,130 | 30,621,640 | 30,365,660 | 29,705,560 | 31,250,470 |

s7 | 30,649,910 | 30,293,480 | 30,620,610 | 30,226,780 | 30,553,910 | 30,293,480 | 30,620,610 | 30,065,270 | 30,553,910 | 29,877,450 | 31,422,360 |

s8 | 30,416,720 | 30,644,330 | 30,376,820 | 30,577,630 | 30,310,120 | 30,644,330 | 30,376,820 | 30,577,630 | 30,187,320 | 29,626,270 | 31,207,180 |

s9 | 25,749,000 | 25,939,050 | 25,768,800 | 25,849,950 | 25,679,700 | 25,939,050 | 25,768,800 | 25,849,950 | 25,679,700 | 25,212,750 | 26,285,250 |

s10 | 45,929,180 | 46,175,180 | 45,874,080 | 46,115,780 | 45,814,680 | 46,175,180 | 45,874,080 | 46,115,780 | 45,814,680 | 44,901,680 | 48,997,110 |

MV ^{1} | 31,815,301 | 31,802,625 | 31,735,885 | 31,737,693 | 31,702,649 | 31,770,653 | 31,736,176 | 31,738,023 | 31,706,725 | 31,015,857 | 32,800,238 |

SD ^{2} | 4,950,109.6 | 5,010,178.7 | 4,945,815.7 | 5,013,603.6 | 4,944,127.9 | 5,019,098.1 | 4,948,722.4 | 5,016,440 | 4,946,024 | 4,851,587.7 | 5,632,609.1 |

CV ^{3} | 0.155589 | 0.1575398 | 0.155843 | 0.15797 | 0.1559531 | 0.1579791 | 0.1559332 | 0.1580577 | 0.1559928 | 0.1564228 | 0.1717246 |

^{1}MV: Mean value.

^{2}SD: Standard deviation.

^{3}CV: Coefficient of variance (CV = SD/MV).

**Table B2.**Scenario-based overall system costs of the reverse logistics system for WEEE in sensitivity analysis (B).

Sol. s0 | Sol. s1 | Sol. s2 | Sol. s3 | Sol. s4 | Sol. s5 | Sol. s6 | Sol. s7 | Sol. s8 | Sol. s9 | Sol. s10 | |
---|---|---|---|---|---|---|---|---|---|---|---|

s0 | 35,217,930 | 35,511,820 | 35,252,540 | 35,394,370 | 35,135,090 | 35,511,820 | 35,252,540 | 35,394,370 | 35,135,090 | 34,452,970 | 36,016,720 |

s1 | 36,188,520 | 36,055,660 | 36,071,390 | 35,726,980 | 36,113,660 | 35,684,710 | 36,071,390 | 35,726,980 | 36,113,660 | 35,353,410 | 37,023,630 |

s2 | 36,284,990 | 36,407,310 | 36,111,890 | 36,449,570 | 36,171,330 | 36,407,310 | 36,129,070 | 36,449,570 | 36,171,330 | 35,449,880 | 37,387,200 |

s3 | 36,552,090 | 36,149,270 | 36,535,950 | 36,073,380 | 36,477,230 | 36,149,270 | 36,535,950 | 36,090,550 | 36,477,230 | 35,716,980 | 37,387,200 |

s4 | 36,267,670 | 36,490,970 | 36,212,730 | 36,432,250 | 36,129,610 | 36,490,970 | 36,212,730 | 36,432,250 | 36,154,010 | 35,432,560 | 37,102,780 |

s5 | 35,558,890 | 35,099,190 | 35,426,330 | 35,135,750 | 35,462,890 | 35,081,860 | 35,426,330 | 35,135,750 | 35,462,890 | 34,786,430 | 36,331,340 |

s6 | 35,282,160 | 35,389,220 | 35,133,240 | 35,425,790 | 35,169,800 | 35,389,220 | 35,115,900 | 35,425,790 | 35,169,800 | 34,509,710 | 36,054,620 |

s7 | 35,533,790 | 35,177,360 | 35,504,490 | 35,110,650 | 35,437,790 | 35,177,360 | 35,504,490 | 35,093,320 | 35,437,790 | 34,761,330 | 36,306,240 |

s8 | 35,219,260 | 35,446,870 | 35,179,360 | 35,380,170 | 35,112,660 | 35,446,870 | 35,179,360 | 30,325,590 | 35,127,370 | 34,428,810 | 36,009,720 |

s9 | 30,224,640 | 30,414,690 | 30,244,440 | 30,325,590 | 30,155,340 | 30,414,690 | 30,244,440 | 25,849,950 | 30,155,340 | 29,673,020 | 30,760,890 |

s10 | 54,892,650 | 55,138,650 | 54,837,550 | 55,079,250 | 54,778,150 | 55,138,650 | 54,837,550 | 55,079,250 | 54,778,150 | 53,865,150 | 57,978,850 |

MV ^{1} | 37,020,235 | 37,025,546 | 36,955,446 | 36,957,614 | 36,922,141 | 36,990,248 | 36,955,432 | 36,091,215 | 36,925,696 | 36,220,932 | 38,032,654 |

SD ^{2} | 6,176,398.2 | 6,235,000.1 | 6,171,274.3 | 6,238,725 | 6,168,802.5 | 6,242,297.6 | 6,171,556.3 | 7,084,390.6 | 6,168,063 | 6,081,694.7 | 6,865,778.5 |

CV ^{3} | 0.1668384 | 0.1683972 | 0.1669923 | 0.1688076 | 0.167076 | 0.1687552 | 0.1669999 | 0.1962913 | 0.1670399 | 0.1679055 | 0.1805232 |

^{1}MV: Mean value.

^{2}SD: Standard deviation.

^{3}CV: Coefficient of variance (CV = SD/MV).

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**Figure 4.**Comparison of the evaluation criteria in candidate solutions: (

**a**) Optimal costs; (

**b**) mean costs; (

**c**) standard deviation; (

**d**) coefficient of variation.

**Figure 5.**Comparison of the reliability of different cost components in candidate solutions: (

**a**) Coefficient of variation of the facility costs; (

**b**) coefficient of variation of the transportation costs; (

**c**) coefficient of variation of the profits; (

**d**) coefficient of variation of the carbon cost.

**Figure 7.**Comparison of the evaluation criteria in candidate solutions of sensitivity analysis S-A: (

**a**) Optimal costs; (

**b**) mean costs; (

**c**) standard deviation; (

**d**) coefficient of variation.

**Figure 9.**Comparison of the evaluation criteria in candidate solutions of sensitivity analysis S-B: (

**a**) Optimal costs; (

**b**) mean costs; (

**c**) standard deviation; (

**d**) coefficient of variation.

Parameters | Interval |
---|---|

$F{x}_{r}$ | Unif. (4, 5); million USD |

$F{x}_{p}$ | Unif. (5, 6); million USD |

$Em{s}_{r}$ | Unif. (200, 300); g/ton-capacity |

$Em{s}_{p}$ | Unif. (200, 300); g/ton-capacity |

$C{p}_{d}$ | Unif. (100, 150); USD |

$C{p}_{h}$ | Unif. (500, 600); USD |

$d{s}_{cr}$ | Unif. (20, 30); km |

$d{s}_{rp}$ | Unif. (20, 50); km |

$d{s}_{rd}$ | Unif. (20, 50); km |

$d{s}_{rh}$ | Unif. (20, 50); km |

$d{s}_{pf}$ | Unif. (30, 60); km |

$d{s}_{pm}$ | Unif. (30, 60); km |

$d{s}_{pd}$ | Unif. (20, 50); km |

$C{p}_{rl}$ | Unif. (100, 200); USD |

$C{p}_{pl}$ | Unif. (100, 200); USD |

$Ct{r}_{lcr}$/$d{s}_{cr}$ | Unif. (1, 2); USD/ton/km |

$Ct{r}_{lrp}$/$d{s}_{rp}$ | Unif. (1, 2); USD/ton/km |

$Ct{r}_{lrd}$/$d{s}_{rd}$ | Unif. (4, 5); USD/ton/km |

$Ct{r}_{lrh}$/$d{s}_{rh}$ | Unif. (2, 3); USD/ton/km |

$Ct{r}_{lpf}$/$d{s}_{pf}$ | Unif. (1, 2); USD/ton/km |

$Ct{r}_{lpm}$/$d{s}_{pm}$ | Unif. (1, 2); USD/ton/km |

$Ct{r}_{lpd}$/$d{s}_{pd}$ | Unif. (2, 3); USD/ton/km |

$Em{s}_{lcr}$/$d{s}_{cr}$ | Unif. (100, 200); g/ton/km |

$Em{s}_{lrp}$/$d{s}_{rp}$ | Unif. (100, 200); g/ton/km |

$Em{s}_{lrd}$/$d{s}_{rd}$ | Unif. (100, 200); g/ton/km |

$Em{s}_{lrh}$/$d{s}_{rh}$ | Unif. (100, 200); g/ton/km |

$Em{s}_{lpf}$/$d{s}_{pf}$ | Unif. (100, 200); g/ton/km |

$Em{s}_{lpm}$/$d{s}_{pm}$ | Unif. (100, 200); g/ton/km |

$Em{s}_{lpd}$/$d{s}_{pd}$ | Unif. (100, 200); g/ton/km |

Parameters | Interval | ||
---|---|---|---|

Type A | Type B | Type C | |

$Co{l}_{lc}^{s}$ | Unif. (500, 600); ton | Unif. (1000, 2000); ton | Unif. (1000, 2000); ton |

$PC{p}_{fl}^{s}$ | Unif. (200, 300); USD/ton | Unif. (150, 250); USD/ton | Unif. (150, 250); USD/ton |

$PC{m}_{ml}^{s}$ | Unif. (100, 150); USD/ton | Unif. (100, 200); USD/ton | Unif. (100, 200); USD/ton |

$Capacit{y}_{rl}$ | 4000; ton | 8000; ton | 10,000; ton |

$Capacit{y}_{pl}$ | 3000; ton | 5000; ton | 8000; ton |

${\phi}_{lp}$ | 50% | 60% | 60% |

${\phi}_{ld}$ | 20% | 20% | 30% |

${\phi}_{lh}$ | 30% | 20% | 10% |

${\vartheta}_{lf}$ | 30% | 30% | 40% |

${\vartheta}_{lm}$ | 30% | 40% | 50% |

${\vartheta}_{ld}$ | 40% | 30% | 10% |

Weight Combination | Sol. s0 | Sol. s1 | Sol. s2 | Sol. s3 | Sol. s4 | Sol. s5 | Sol. s6 | Sol. s7 | Sol. s8 | Sol. s9 | Sol. s10 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{W}}_{\mathit{M}}$ | ${\mathit{W}}_{\mathit{C}}$ | |||||||||||

0 | 1 | 1.070012 | 1.000000 | 1.070582 | 1.070705 | 1.070592 | 1.071525 | 1.071393 | 1.062836 | 1.071373 | 1.074669 | 1.070012 |

0.1 | 0.9 | 1.063053 | 1.000297 | 1.063526 | 1.063635 | 1.063532 | 1.064374 | 1.064256 | 1.056683 | 1.064238 | 1.067206 | 1.063053 |

0.2 | 0.8 | 1.056094 | 1.000595 | 1.056469 | 1.056565 | 1.056473 | 1.057223 | 1.057119 | 1.050531 | 1.057103 | 1.059743 | 1.056094 |

0.3 | 0.7 | 1.049136 | 1.000892 | 1.049412 | 1.049495 | 1.049414 | 1.050072 | 1.049982 | 1.044378 | 1.049968 | 1.052281 | 1.049136 |

0.4 | 0.6 | 1.042177 | 1.001189 | 1.042355 | 1.042426 | 1.042355 | 1.042921 | 1.042845 | 1.038225 | 1.042833 | 1.044818 | 1.042177 |

0.5 | 0.5 | 1.035218 | 1.001486 | 1.035298 | 1.035356 | 1.035296 | 1.035770 | 1.035708 | 1.032073 | 1.035699 | 1.037355 | 1.035218 |

0.6 | 0.4 | 1.028259 | 1.001784 | 1.028241 | 1.028286 | 1.028237 | 1.028619 | 1.028571 | 1.025920 | 1.028564 | 1.029892 | 1.028259 |

0.7 | 0.3 | 1.021300 | 1.002081 | 1.021184 | 1.021216 | 1.021177 | 1.021468 | 1.021434 | 1.019768 | 1.021429 | 1.022429 | 1.021300 |

0.8 | 0.2 | 1.014341 | 1.002378 | 1.014127 | 1.014146 | 1.014118 | 1.014317 | 1.014297 | 1.013615 | 1.014294 | 1.014966 | 1.014341 |

0.9 | 0.1 | 1.007382 | 1.002676 | 1.007070 | 1.007076 | 1.007059 | 1.007166 | 1.007159 | 1.007462 | 1.007159 | 1.007503 | 1.007382 |

1 | 0 | 1.000423 | 1.002973 | 1.000013 | 1.000007 | 1.000000 | 1.000015 | 1.000022 | 1.001310 | 1.000024 | 1.000040 | 1.000423 |

**Table 4.**Sensitivity analysis of the results with respect to the change of weight combination (S-A).

Weight Combination | Sol. s0 | Sol. s1 | Sol. s2 | Sol. s3 | Sol. s4 | Sol. s5 | Sol. s6 | Sol. s7 | Sol. s8 | Sol. s9 | Sol. s10 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{W}}_{\mathit{M}}$ | ${\mathit{W}}_{\mathit{C}}$ | |||||||||||

0 | 1 | 1.000000 | 1.012538 | 1.001633 | 1.015303 | 1.002341 | 1.015362 | 1.002212 | 1.015867 | 1.002596 | 1.005359 | 1.103707 |

0.1 | 0.9 | 1.002578 | 1.013821 | 1.003791 | 1.016100 | 1.004321 | 1.016259 | 1.004313 | 1.016609 | 1.004564 | 1.004823 | 1.099090 |

0.2 | 0.8 | 1.005155 | 1.015104 | 1.005949 | 1.016897 | 1.006301 | 1.017156 | 1.006415 | 1.017351 | 1.006532 | 1.004287 | 1.094472 |

0.3 | 0.7 | 1.007733 | 1.016387 | 1.008107 | 1.017694 | 1.008281 | 1.018054 | 1.008516 | 1.018092 | 1.008500 | 1.003752 | 1.089854 |

0.4 | 0.6 | 1.010310 | 1.017670 | 1.010266 | 1.018491 | 1.010262 | 1.018951 | 1.010617 | 1.018834 | 1.010468 | 1.003216 | 1.085237 |

0.5 | 0.5 | 1.012888 | 1.018952 | 1.012424 | 1.019288 | 1.012242 | 1.019849 | 1.012718 | 1.019576 | 1.012435 | 1.002680 | 1.080619 |

0.6 | 0.4 | 1.015465 | 1.020235 | 1.014582 | 1.020085 | 1.014222 | 1.020746 | 1.014819 | 1.020317 | 1.014403 | 1.002144 | 1.076002 |

0.7 | 0.3 | 1.018043 | 1.021518 | 1.016740 | 1.020882 | 1.016202 | 1.021644 | 1.016921 | 1.021059 | 1.016371 | 1.001608 | 1.071384 |

0.8 | 0.2 | 1.020620 | 1.022801 | 1.018898 | 1.021679 | 1.018183 | 1.022541 | 1.019022 | 1.021800 | 1.018339 | 1.001072 | 1.066766 |

0.9 | 0.1 | 1.023198 | 1.024084 | 1.021057 | 1.022476 | 1.020163 | 1.023438 | 1.021123 | 1.022542 | 1.020307 | 1.000536 | 1.062149 |

1 | 0 | 1.025775 | 1.025367 | 1.023215 | 1.023273 | 1.022143 | 1.024336 | 1.023224 | 1.023284 | 1.022275 | 1.000000 | 1.057531 |

Weight Combination | Sol. s0 | Sol. s1 | Sol. s2 | Sol. s3 | Sol. s4 | Sol. s5 | Sol. s6 | Sol. s7 | Sol. s8 | Sol. s9 | Sol. s10 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathbf{W}}_{\mathbf{M}}$ | ${\mathbf{W}}_{\mathbf{C}}$ | |||||||||||

0 | 1 | 1.000000 | 1.009343 | 1.000922 | 1.011803 | 1.001424 | 1.011489 | 1.000968 | 1.176535 | 1.001207 | 1.006396 | 1.082024 |

0.1 | 0.9 | 1.002574 | 1.010998 | 1.003224 | 1.013023 | 1.003584 | 1.012831 | 1.003266 | 1.158882 | 1.003399 | 1.006116 | 1.079201 |

0.2 | 0.8 | 1.005148 | 1.012652 | 1.005527 | 1.014243 | 1.005744 | 1.014173 | 1.005564 | 1.141228 | 1.005590 | 1.005836 | 1.076378 |

0.3 | 0.7 | 1.007722 | 1.014307 | 1.007829 | 1.015464 | 1.007903 | 1.015515 | 1.007861 | 1.123575 | 1.007782 | 1.005555 | 1.073555 |

0.4 | 0.6 | 1.010296 | 1.015961 | 1.010132 | 1.016684 | 1.010063 | 1.016857 | 1.010159 | 1.105921 | 1.009973 | 1.005275 | 1.070732 |

0.5 | 0.5 | 1.012870 | 1.017616 | 1.012434 | 1.017904 | 1.012223 | 1.018199 | 1.012457 | 1.088268 | 1.012164 | 1.004995 | 1.067908 |

0.6 | 0.4 | 1.015445 | 1.019270 | 1.014736 | 1.019125 | 1.014383 | 1.019542 | 1.014755 | 1.070614 | 1.014356 | 1.004715 | 1.065085 |

0.7 | 0.3 | 1.018019 | 1.020925 | 1.017039 | 1.020345 | 1.016543 | 1.020884 | 1.017052 | 1.052961 | 1.016547 | 1.004435 | 1.062262 |

0.8 | 0.2 | 1.020593 | 1.022579 | 1.019341 | 1.021565 | 1.018703 | 1.022226 | 1.019350 | 1.035307 | 1.018739 | 1.004154 | 1.059439 |

0.9 | 0.1 | 1.023167 | 1.024234 | 1.021643 | 1.022785 | 1.020863 | 1.023568 | 1.021648 | 1.017654 | 1.020930 | 1.003874 | 1.056616 |

1 | 0 | 1.025741 | 1.025888 | 1.023946 | 1.024006 | 1.023023 | 1.024910 | 1.023945 | 1.000000 | 1.023121 | 1.003594 | 1.053793 |

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**MDPI and ACS Style**

Yu, H.; Solvang, W.D.
A Stochastic Programming Approach with Improved Multi-Criteria Scenario-Based Solution Method for Sustainable Reverse Logistics Design of Waste Electrical and Electronic Equipment (WEEE). *Sustainability* **2016**, *8*, 1331.
https://doi.org/10.3390/su8121331

**AMA Style**

Yu H, Solvang WD.
A Stochastic Programming Approach with Improved Multi-Criteria Scenario-Based Solution Method for Sustainable Reverse Logistics Design of Waste Electrical and Electronic Equipment (WEEE). *Sustainability*. 2016; 8(12):1331.
https://doi.org/10.3390/su8121331

**Chicago/Turabian Style**

Yu, Hao, and Wei Deng Solvang.
2016. "A Stochastic Programming Approach with Improved Multi-Criteria Scenario-Based Solution Method for Sustainable Reverse Logistics Design of Waste Electrical and Electronic Equipment (WEEE)" *Sustainability* 8, no. 12: 1331.
https://doi.org/10.3390/su8121331