# American Institute of Mathematical Sciences

• Previous Article
Sparse markowitz portfolio selection by using stochastic linear complementarity approach
• JIMO Home
• This Issue
• Next Article
An improved 2.11-competitive algorithm for online scheduling on parallel machines to minimize total weighted completion time
April  2018, 14(2): 511-539. doi: 10.3934/jimo.2017058

## Integrated recycling-integrated production - distribution planning for decentralized closed-loop supply chain

 1 College of Management, Chongqing University of Technology, NO.69, Hongguang Road, Banan District, Chongqing 400054, China 2 School of Economics and Management, Nanchang Hangkong University, NO.696, Fenghenan Road, Honggutan District, Nanchang 330063, China

* Corresponding authorr: Yi Jing

Received  May 2015 Published  April 2018 Early access  June 2017

Integrated integrated production - distribution planning in traditional forward supply chain has attracted a lot of attention in recent years and its economic advantages are particularly noticeable. However, for closed-loop supply chain, recycling and remanufacturing processes should be taken further into account to the integrated planning. In this paper, we address a planning problem of a multi - echelon decentralized closed-loop supply chain system, which consists of a joint recycling center, multiple manufacturing/remanufacturing factories and multiple distributors decentralized to different regions. For this problem, an integrated recycling-integrated production - distribution multi - level planning model is developed, which considers material flows and decision interactions among members at different echelons in the system, as well as their own operation objectives. And the local interests of members at every echelon would be balanced in order to coordinate the operation of the whole system. According to the characteristics of the planning model, the solution approach is designed by hierarchical iteration strategy based on Self-Adaptive Genetic Algorithm (SAGA). Hierarchical iteration processes, in which SAGA is used to solve every single level model, are corresponding to repeated negotiation behaviors among members at different echelons in closed-loop supply chain. Finally, a numerical example is suggested to demonstrate the applicability and effectiveness of the proposed model and solution approach.

Citation: Yi Jing, Wenchuan Li. Integrated recycling-integrated production - distribution planning for decentralized closed-loop supply chain. Journal of Industrial & Management Optimization, 2018, 14 (2) : 511-539. doi: 10.3934/jimo.2017058
##### References:

show all references

##### References:
Diagrammatic sketch for the two - point crossover
Diagrammatic sketch for the inverted sequence mutation
The changes of computational results with the values of $M$ and $N$
The generation intervals of decision variables in encoding
 Variables Generation interval Variables Generation interval $x_{pit}$ $[0, MA_{pi}/ab_{pi}]$ $fdn_{pijt}$ $\displaystyle \left[0, \left(MT_{i}^{f} -\sum\limits_{N_{p} =1}^{p-1}{\sum\limits_{N_{j} =1}^J {fdn_{pijt} \cdot tpb_{p}}} \right. \right.$ $\displaystyle \left. \left. -\sum\limits_{N_{j} =1}^{j-1}{fdn_{pijt} \cdot tpb_{p}} \right)\Bigg/tpb_{p} \right]$ $y_{pit}$ $[0, MRA_{pi}/rab_{pi}]$ $fdr_{pijt}$ $\displaystyle \left[0, \left(MT_{i}^{f} -\sum\limits_{N_{p} =1}^P {\sum\limits_{N_{j} =1}^J {fdn_{pijt} \cdot tpb_{p}}} \right. \right.$ $\displaystyle -\sum\limits_{N_{p} =1}^{p-1}{\sum\limits_{N_{j} =1}^J {fdr_{pijt} \cdot tpb_{p}}}$ $\displaystyle \left. \left.-\sum\limits_{N_{j} =1}^{j-1} {fdr_{pijt} \cdot tpb_{p}} \right)\Bigg/tpb_{p} \right]$ $v_{cit}$ $[0, MP_{ci}/pb_{ci}]$ $subc_{cit}$ $\displaystyle \left[0, {v_{cit} \cdot pb_{c}} -\sum\limits_{p=1}^P {BOC_{pc} \cdot x_{pit} \cdot ab_{p} +\overline \zeta_{ci}^{f}}\right]$ $z_{cit}$ $[0, MRP_{ci}/rpb_{ci}]$
 Variables Generation interval Variables Generation interval $x_{pit}$ $[0, MA_{pi}/ab_{pi}]$ $fdn_{pijt}$ $\displaystyle \left[0, \left(MT_{i}^{f} -\sum\limits_{N_{p} =1}^{p-1}{\sum\limits_{N_{j} =1}^J {fdn_{pijt} \cdot tpb_{p}}} \right. \right.$ $\displaystyle \left. \left. -\sum\limits_{N_{j} =1}^{j-1}{fdn_{pijt} \cdot tpb_{p}} \right)\Bigg/tpb_{p} \right]$ $y_{pit}$ $[0, MRA_{pi}/rab_{pi}]$ $fdr_{pijt}$ $\displaystyle \left[0, \left(MT_{i}^{f} -\sum\limits_{N_{p} =1}^P {\sum\limits_{N_{j} =1}^J {fdn_{pijt} \cdot tpb_{p}}} \right. \right.$ $\displaystyle -\sum\limits_{N_{p} =1}^{p-1}{\sum\limits_{N_{j} =1}^J {fdr_{pijt} \cdot tpb_{p}}}$ $\displaystyle \left. \left.-\sum\limits_{N_{j} =1}^{j-1} {fdr_{pijt} \cdot tpb_{p}} \right)\Bigg/tpb_{p} \right]$ $v_{cit}$ $[0, MP_{ci}/pb_{ci}]$ $subc_{cit}$ $\displaystyle \left[0, {v_{cit} \cdot pb_{c}} -\sum\limits_{p=1}^P {BOC_{pc} \cdot x_{pit} \cdot ab_{p} +\overline \zeta_{ci}^{f}}\right]$ $z_{cit}$ $[0, MRP_{ci}/rpb_{ci}]$
The bill of each kind of core component to each type of product
 A B-1 B-2 C-1 C-2 D Ⅰ 1 1 0 4 0 1 Ⅱ 1 0 1 0 4 1 Ⅲ 1 0 1 0 4 1
 A B-1 B-2 C-1 C-2 D Ⅰ 1 1 0 4 0 1 Ⅱ 1 0 1 0 4 1 Ⅲ 1 0 1 0 4 1
The demand data for new products in market of each distributor
 $j=$ 1 2 3 4 5 $p=$ 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 $t=1$ 378 384 400 394 395 396 365 367 389 370 391 405 398 408 423 $t=2$ 387 407 402 380 387 394 376 395 399 368 378 393 409 416 416 $t=3$ 380 397 393 376 391 409 384 389 384 374 397 393 365 367 389 $t=4$ 393 395 390 381 388 391 370 395 383 393 395 390 376 395 399 $t=5$ 398 408 423 370 391 405 381 387 394 394 395 396 374 389 384 $t=6$ 409 416 416 368 378 393 385 386 397 380 387 394 370 395 383
 $j=$ 1 2 3 4 5 $p=$ 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 $t=1$ 378 384 400 394 395 396 365 367 389 370 391 405 398 408 423 $t=2$ 387 407 402 380 387 394 376 395 399 368 378 393 409 416 416 $t=3$ 380 397 393 376 391 409 384 389 384 374 397 393 365 367 389 $t=4$ 393 395 390 381 388 391 370 395 383 393 395 390 376 395 399 $t=5$ 398 408 423 370 391 405 381 387 394 394 395 396 374 389 384 $t=6$ 409 416 416 368 378 393 385 386 397 380 387 394 370 395 383
The demand data for remanufactured products in market of each distributor
 $j=$ 1 2 3 4 5 $p=$ 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 $t=1$ 157 163 165 158 166 170 168 172 172 158 160 162 158 167 171 $t=2$ 160 162 169 160 162 173 162 163 165 163 169 171 167 170 172 $t=3$ 155 156 160 162 168 175 158 167 171 162 168 175 152 159 166 $t=4$ 162 163 165 156 159 168 167 170 172 156 159 168 158 163 170 $t=5$ 158 160 162 152 159 166 167 174 174 157 163 165 155 156 160 $t=6$ 163 169 171 158 163 170 151 163 166 160 162 169 162 163 165
 $j=$ 1 2 3 4 5 $p=$ 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 $t=1$ 157 163 165 158 166 170 168 172 172 158 160 162 158 167 171 $t=2$ 160 162 169 160 162 173 162 163 165 163 169 171 167 170 172 $t=3$ 155 156 160 162 168 175 158 167 171 162 168 175 152 159 166 $t=4$ 162 163 165 156 159 168 167 170 172 156 159 168 158 163 170 $t=5$ 158 160 162 152 159 166 167 174 174 157 163 165 155 156 160 $t=6$ 163 169 171 158 163 170 151 163 166 160 162 169 162 163 165
The quantity of EOL products available in market of each distributor
 $j=$ 1 2 3 4 5 $p=$ 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 $t=1$ 186 188 189 186 191 197 191 192 187 184 184 190 187 198 202 $t=2$ 190 198 199 184 195 198 198 193 195 192 195 198 199 207 205 $t=3$ 184 184 190 181 195 195 187 198 202 181 195 195 196 198 205 $t=4$ 192 195 198 197 190 199 199 204 204 197 190 199 182 190 191 $t=5$ 174 198 204 187 179 195 189 184 198 186 191 197 198 193 195 $t=6$ 182 190 191 188 189 191 180 193 194 184 195 198 187 179 195
 $j=$ 1 2 3 4 5 $p=$ 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 $t=1$ 186 188 189 186 191 197 191 192 187 184 184 190 187 198 202 $t=2$ 190 198 199 184 195 198 198 193 195 192 195 198 199 207 205 $t=3$ 184 184 190 181 195 195 187 198 202 181 195 195 196 198 205 $t=4$ 192 195 198 197 190 199 199 204 204 197 190 199 182 190 191 $t=5$ 174 198 204 187 179 195 189 184 198 186 191 197 198 193 195 $t=6$ 182 190 191 188 189 191 180 193 194 184 195 198 187 179 195
The parameters data about joint recycling center
 $c$ $p$ 1 2 3 4 5 6 1 2 3 $UDC_{ct}$ 35 30 30 25 25 30 $SDT_{pt}$ 1000 1200 1200 $ICQC_{ct}^{a}$ 10 10 10 5 5 5 $UDTC_{pt}$ 100 100 100 $\theta_{ct}$ 0.95 0.92 0.92 0.90 0.90 0.85 $ICRP_{pt}^{a}$ 10 10 10
 $c$ $p$ 1 2 3 4 5 6 1 2 3 $UDC_{ct}$ 35 30 30 25 25 30 $SDT_{pt}$ 1000 1200 1200 $ICQC_{ct}^{a}$ 10 10 10 5 5 5 $UDTC_{pt}$ 100 100 100 $\theta_{ct}$ 0.95 0.92 0.92 0.90 0.90 0.85 $ICRP_{pt}^{a}$ 10 10 10
The parameters data about products in manufacturing/remanufacturing factories
 $i=$ 1 2 3 $p=$ 1 2 3 1 2 3 1 2 3 $SA_{pit}$ 25000 30000 35000 25000 30000 35000 25000 30000 35000 $SRA_{pit}$ 25000 30000 35000 25000 30000 35000 25000 30000 35000 $UAC_{pit}$ 5900 7400 7900 6000 7500 8000 6050 7550 8050 $URAC_{pit}$ 5900 7400 7900 6000 7500 8000 6050 7550 8050 $ICNP_{pit}^{f}$ 20 20 20 20 20 20 20 20 20 $ICRMP_{pit}^{f}$ 20 20 20 20 20 20 20 20 20
 $i=$ 1 2 3 $p=$ 1 2 3 1 2 3 1 2 3 $SA_{pit}$ 25000 30000 35000 25000 30000 35000 25000 30000 35000 $SRA_{pit}$ 25000 30000 35000 25000 30000 35000 25000 30000 35000 $UAC_{pit}$ 5900 7400 7900 6000 7500 8000 6050 7550 8050 $URAC_{pit}$ 5900 7400 7900 6000 7500 8000 6050 7550 8050 $ICNP_{pit}^{f}$ 20 20 20 20 20 20 20 20 20 $ICRMP_{pit}^{f}$ 20 20 20 20 20 20 20 20 20
The parameters data about components in manufacturing/remanufacturing factories
 i= 1 2 3 c= 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 SPcit 25000 20000 24000 15000 18000 18000 25000 20000 24000 15000 18000 18000 25000 20000 24000 15000 18000 18000 SRPcit 10000 6000 7000 5000 6000 5000 10000 6000 7000 5000 6000 5000 10000 6000 7000 5000 6000 5000 UPCcit 3900 2400 2700 70 95 950 4000 2500 2800 75 100 1000 4100 2600 2900 80 105 1050 URPCcit 950 550 650 22 28 325 1000 600 700 25 30 350 1050 650 750 28 32 375 ICQCcitf 10 10 10 5 5 5 10 10 10 5 5 5 10 10 10 5 5 5 ICNCcitf 15 15 15 10 10 10 15 15 15 10 10 10 15 15 15 10 10 10 ICRCcitf 15 15 15 10 10 10 15 15 15 10 10 10 15 15 15 10 10 10 UTCacitf 40 30 30 15 15 25 45 35 35 15 15 30 40 30 30 15 15 25 PPCcit 820 520 540 135 135 355 820 520 540 135 135 355 800 500 520 120 120 345 MPci 2400 800 1600 3200 6400 2400 2400 800 1600 3200 6400 2400 2400 800 1600 3200 6400 2400 MRPci 1050 350 700 1400 2800 1050 1050 350 700 1400 2800 1050 1050 350 700 1400 2800 1050
 i= 1 2 3 c= 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 SPcit 25000 20000 24000 15000 18000 18000 25000 20000 24000 15000 18000 18000 25000 20000 24000 15000 18000 18000 SRPcit 10000 6000 7000 5000 6000 5000 10000 6000 7000 5000 6000 5000 10000 6000 7000 5000 6000 5000 UPCcit 3900 2400 2700 70 95 950 4000 2500 2800 75 100 1000 4100 2600 2900 80 105 1050 URPCcit 950 550 650 22 28 325 1000 600 700 25 30 350 1050 650 750 28 32 375 ICQCcitf 10 10 10 5 5 5 10 10 10 5 5 5 10 10 10 5 5 5 ICNCcitf 15 15 15 10 10 10 15 15 15 10 10 10 15 15 15 10 10 10 ICRCcitf 15 15 15 10 10 10 15 15 15 10 10 10 15 15 15 10 10 10 UTCacitf 40 30 30 15 15 25 45 35 35 15 15 30 40 30 30 15 15 25 PPCcit 820 520 540 135 135 355 820 520 540 135 135 355 800 500 520 120 120 345 MPci 2400 800 1600 3200 6400 2400 2400 800 1600 3200 6400 2400 2400 800 1600 3200 6400 2400 MRPci 1050 350 700 1400 2800 1050 1050 350 700 1400 2800 1050 1050 350 700 1400 2800 1050
 j= 1 2 3 4 5 p= 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 URCCpjt 1000 1150 1150 1000 1150 1150 1050 1200 1200 1050 1200 1200 1100 1250 1250 URCDpjt 900 1050 1050 900 1050 1050 950 1100 1100 950 1100 1100 1000 1150 1150 SPNpjt 18160 20670 21340 18160 20665 21340 18150 20680 21345 18150 20680 21345 18170 20690 21365 SPRpjt 12570 14740 15400 12570 14740 15400 12580 14740 15410 12580 14740 15410 12590 14760 15430 USNPpjt 4120 4690 4840 4120 4690 4840 4110 4690 4840 4110 4690 4840 4120 4690 4850 USRPpjt 2850 3340 3490 2850 3340 3490 2850 3340 3500 2850 3340 3500 2860 3350 3500 UTCpjtda 30 30 30 35 35 35 30 30 30 35 35 35 30 30 30 ICNPpjtd 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 ICRMPpjtd 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 ICRPpjtd 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
 j= 1 2 3 4 5 p= 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 URCCpjt 1000 1150 1150 1000 1150 1150 1050 1200 1200 1050 1200 1200 1100 1250 1250 URCDpjt 900 1050 1050 900 1050 1050 950 1100 1100 950 1100 1100 1000 1150 1150 SPNpjt 18160 20670 21340 18160 20665 21340 18150 20680 21345 18150 20680 21345 18170 20690 21365 SPRpjt 12570 14740 15400 12570 14740 15400 12580 14740 15410 12580 14740 15410 12590 14760 15430 USNPpjt 4120 4690 4840 4120 4690 4840 4110 4690 4840 4110 4690 4840 4120 4690 4850 USRPpjt 2850 3340 3490 2850 3340 3490 2850 3340 3500 2850 3340 3500 2860 3350 3500 UTCpjtda 30 30 30 35 35 35 30 30 30 35 35 35 30 30 30 ICNPpjtd 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 ICRMPpjtd 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 ICRPpjtd 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
The unit transportation cost of products from factories to distributors
 i 1 2 3 j 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 MPN1ijt 15440 15440 15460 15460 15500 15840 15870 15780 15780 15840 16100 16070 16100 16100 16070 MPN2ijt 17640 17640 17670 17670 17720 17990 18000 17980 17980 17990 18290 18270 18290 18290 18270 MPN3ijt 18220 18220 18260 18260 18300 18580 18600 18560 18560 18580 18860 18850 18860 18860 18850 MPR1ijt 10730 10730 10770 10770 10810 10970 10990 10950 10950 10870 11090 11070 11090 11090 11070 MPR2ijt 12630 12630 12650 12650 12690 12860 12880 12840 12840 12860 12970 12950 12970 12970 12950 MPR3ijt 13200 13200 13240 13240 13290 13430 13450 13410 13410 13430 13550 13530 13550 13550 13530 UTCpijtfd 40 40 50 50 60 60 50 40 50 60 60 50 50 40 40
 i 1 2 3 j 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 MPN1ijt 15440 15440 15460 15460 15500 15840 15870 15780 15780 15840 16100 16070 16100 16100 16070 MPN2ijt 17640 17640 17670 17670 17720 17990 18000 17980 17980 17990 18290 18270 18290 18290 18270 MPN3ijt 18220 18220 18260 18260 18300 18580 18600 18560 18560 18580 18860 18850 18860 18860 18850 MPR1ijt 10730 10730 10770 10770 10810 10970 10990 10950 10950 10870 11090 11070 11090 11090 11070 MPR2ijt 12630 12630 12650 12650 12690 12860 12880 12840 12840 12860 12970 12950 12970 12970 12950 MPR3ijt 13200 13200 13240 13240 13290 13430 13450 13410 13410 13430 13550 13530 13550 13550 13530 UTCpijtfd 40 40 50 50 60 60 50 40 50 60 60 50 50 40 40
The comparison results among SGA, AGA and SAGA
 Algorithm Running result Convergence generation Best Mean Worst Proportion of Best Result Standard Deviation Best Mean Worst SGA(Pcr = 0:6, Pmu = 0:005) 103014002 102488365 102017694 36% 453083 712 736 754 SGA(Pcr = 0:6, Pmu = 0:02) 104891190 103904980 103309728 20% 672762 616 658 682 SGA(Pcr = 0:8, Pmu = 0:005) 103499360 103141280 102441552 46% 432193 658 688 712 SGA(Pcr = 0:8, Pmu = 0:02) 104514894 103861409 103309728 42% 566313 724 742 766 AGA 106570884 106360212 105911706 54% 262057 458 489 511 SAGA 107302078 107217562 106984452 62% 124847 489 527 557
 Algorithm Running result Convergence generation Best Mean Worst Proportion of Best Result Standard Deviation Best Mean Worst SGA(Pcr = 0:6, Pmu = 0:005) 103014002 102488365 102017694 36% 453083 712 736 754 SGA(Pcr = 0:6, Pmu = 0:02) 104891190 103904980 103309728 20% 672762 616 658 682 SGA(Pcr = 0:8, Pmu = 0:005) 103499360 103141280 102441552 46% 432193 658 688 712 SGA(Pcr = 0:8, Pmu = 0:02) 104514894 103861409 103309728 42% 566313 724 742 766 AGA 106570884 106360212 105911706 54% 262057 458 489 511 SAGA 107302078 107217562 106984452 62% 124847 489 527 557
Definition of the subscripts
 Notation Description $t$ the index set of periods, $\{1, 2, \cdots, T\}$ $p$ the index set of product type, $\{1, 2, \cdots, P\}$ $c$ the index set of component kind, $\{1, 2, \cdots, C\}$ $i$ the index set of manufacturing/remanufacturing factory, $\{1,$ $2,$ $\cdots,$ $I\}$ $j$ the index set of distributor, $\{1, 2, \cdots, J\}$ $lt_{1},lt_{2},lt_{3}$ the accumulative leading time of the joint recycling center, manufacturing/remanufacturing factories and distribution centers, respectively
 Notation Description $t$ the index set of periods, $\{1, 2, \cdots, T\}$ $p$ the index set of product type, $\{1, 2, \cdots, P\}$ $c$ the index set of component kind, $\{1, 2, \cdots, C\}$ $i$ the index set of manufacturing/remanufacturing factory, $\{1,$ $2,$ $\cdots,$ $I\}$ $j$ the index set of distributor, $\{1, 2, \cdots, J\}$ $lt_{1},lt_{2},lt_{3}$ the accumulative leading time of the joint recycling center, manufacturing/remanufacturing factories and distribution centers, respectively
Definition of the variables occurred in the first level model
 Notation Description $af_{cit}$ the quantity of batches of qualified component $c$ to be transported from the recycling center to factory $i$ in period $t$ $da_{pjt}$ the quantity of batches of return product $p$ to be transported from distributor $j$ to the recycling center in period $t$ $\sigma_{pt}$ the binary variable indicating whether return product $p$ is disassembled & tested in batches in period $t$ $dt_{pt}$ the quantity of batches of return product $p$ to be disassembled & tested in period $t$ $d_{ct}$ the quantity of component $c$ to be disposed in period $t$ $\alpha_{pt}^{a},\beta_{ct}^{a}$ the inventory of return product $p$ and qualified component $c$ at the recycling center at the end of period $t$, respectively
 Notation Description $af_{cit}$ the quantity of batches of qualified component $c$ to be transported from the recycling center to factory $i$ in period $t$ $da_{pjt}$ the quantity of batches of return product $p$ to be transported from distributor $j$ to the recycling center in period $t$ $\sigma_{pt}$ the binary variable indicating whether return product $p$ is disassembled & tested in batches in period $t$ $dt_{pt}$ the quantity of batches of return product $p$ to be disassembled & tested in period $t$ $d_{ct}$ the quantity of component $c$ to be disposed in period $t$ $\alpha_{pt}^{a},\beta_{ct}^{a}$ the inventory of return product $p$ and qualified component $c$ at the recycling center at the end of period $t$, respectively
Definition of the variables occurred in the second level model
 Notation Description $fdn_{pijt},fdr_{pijt}$ the quantity of batches of new and remanufacturing product $p$ to be transported from factory $i$ to distributor $j$ in period $t$, respectively $\eta_{pit},\delta_{pit}$ the binary variable indicating whether new and remanufacturing product $p$ is assembled by factory $i$ in batches in period $t$, respectively $x_{pit},y_{pit}$ the quantity of batches of new and remanufactured product $p$ to be assembled at factory $i$ in period $t$, respectively $\pi_{cit},\tau_{cit}$ the binary variable indicating whether component $c$ is newly processed and reprocessed by factory $i$ in batches in period $t$, respectively $v_{cit},z_{cit}$ the quantity of batches of component $c$ to be processed and reprocessed at factory $i$ in period $t$, respectively $\lambda_{pit}^{f},\chi_{pit}^{f}$ the inventory of new and remanufactured product $p$ at factory $i$ at the end of period $t$, respectively $\beta_{cit}^{f},\zeta_{cit}^{f},\xi_{cit}^{f}$ the inventory of qualified, new and remanufactured component $c$ at factory $i$ at the end of period $t$, respectively $subc_{cit}$ the quantity of one-way substitution for component $c$ at factory $i$ in period $t$ $af_{cit}$ this notation has occurred in the first level model
 Notation Description $fdn_{pijt},fdr_{pijt}$ the quantity of batches of new and remanufacturing product $p$ to be transported from factory $i$ to distributor $j$ in period $t$, respectively $\eta_{pit},\delta_{pit}$ the binary variable indicating whether new and remanufacturing product $p$ is assembled by factory $i$ in batches in period $t$, respectively $x_{pit},y_{pit}$ the quantity of batches of new and remanufactured product $p$ to be assembled at factory $i$ in period $t$, respectively $\pi_{cit},\tau_{cit}$ the binary variable indicating whether component $c$ is newly processed and reprocessed by factory $i$ in batches in period $t$, respectively $v_{cit},z_{cit}$ the quantity of batches of component $c$ to be processed and reprocessed at factory $i$ in period $t$, respectively $\lambda_{pit}^{f},\chi_{pit}^{f}$ the inventory of new and remanufactured product $p$ at factory $i$ at the end of period $t$, respectively $\beta_{cit}^{f},\zeta_{cit}^{f},\xi_{cit}^{f}$ the inventory of qualified, new and remanufactured component $c$ at factory $i$ at the end of period $t$, respectively $subc_{cit}$ the quantity of one-way substitution for component $c$ at factory $i$ in period $t$ $af_{cit}$ this notation has occurred in the first level model
Definition of the variables occurred in the third level model
 Notation Description $nss_{pjt},rss_{pjt}$ the quantity of new and remanufactured product $p$ in short supply at distributor $j$ in period $t$, respectively $\gamma_{pjt}$ the quantity of EOL product $p$ to be recycled by distributor $j$ from downstream markets in period $t$ $\lambda_{pjt}^{d},\chi_{pjt}^{d},\alpha_{pjt}^{d}$ the inventory of new, remanufactured and return product $p$ at distributor $j$ at the end of period $t$, respectively $da_{pjt}$ this natation has occurred in the first level model $fdn_{pijt},fdr_{pijt}$ these notations have occurred in second level model
 Notation Description $nss_{pjt},rss_{pjt}$ the quantity of new and remanufactured product $p$ in short supply at distributor $j$ in period $t$, respectively $\gamma_{pjt}$ the quantity of EOL product $p$ to be recycled by distributor $j$ from downstream markets in period $t$ $\lambda_{pjt}^{d},\chi_{pjt}^{d},\alpha_{pjt}^{d}$ the inventory of new, remanufactured and return product $p$ at distributor $j$ at the end of period $t$, respectively $da_{pjt}$ this natation has occurred in the first level model $fdn_{pijt},fdr_{pijt}$ these notations have occurred in second level model
Definition of the parameters occurred in the first level model
 Notation Description $tcb_{c}$ the quantity of qualified component $c$ transported per batch from the recycling center to factories $trb_{p}$ the quantity of return product $p$ transported per batch from distributors to the recycling center $dtb_{p}$ the quantity of return product $p$ disassembled & tested per batch $PPC_{cit}$ the unit purchase cost of qualified component $c$ paid to the recycling center by factory $i$ in period $t$ $URCC_{pjt}$ the unit recycling cost of return product $p$ paid to distributor $j$ by the recycling center in period $t$ $SDT_{pt}$ the set-up cost incurred if return product $p$ is disassembled & tested in batches in period $t$ $UDTC_{pt}$ the unit disassembly & tested cost of return product $p$ in period $t$ $UDC_{ct}$ the unit disposing cost of component $c$ in period $t$ $ICQC_{ct}^{a},ICRP_{pt}^{a}$ the unit inventory cost of qualified component $c$ and return product $p$ at the recycling center in period $t$, respectively $UTC_{cit}^{af}$ the unit transportation cost of qualified component $c$ from the recycling center to factory $i$ in period $t$ $BOC_{pc}$ the bill of component $c$ to product $p$ $\theta_{ct}$ the remanufacturable rate of component $c$ in period $t$ $\overline \alpha_{p}^{a},\overline \beta_{c}^{a}$ the maximum inventory level of return products and qualified components at the recycling center, respectively $MDT_{p}$ the maximum quantity of return product $p$ can be disassembled & tested in every periods $MT^{a}$ the maximum quantity of qualified components can be transported from the recycling center to factories in every periods
 Notation Description $tcb_{c}$ the quantity of qualified component $c$ transported per batch from the recycling center to factories $trb_{p}$ the quantity of return product $p$ transported per batch from distributors to the recycling center $dtb_{p}$ the quantity of return product $p$ disassembled & tested per batch $PPC_{cit}$ the unit purchase cost of qualified component $c$ paid to the recycling center by factory $i$ in period $t$ $URCC_{pjt}$ the unit recycling cost of return product $p$ paid to distributor $j$ by the recycling center in period $t$ $SDT_{pt}$ the set-up cost incurred if return product $p$ is disassembled & tested in batches in period $t$ $UDTC_{pt}$ the unit disassembly & tested cost of return product $p$ in period $t$ $UDC_{ct}$ the unit disposing cost of component $c$ in period $t$ $ICQC_{ct}^{a},ICRP_{pt}^{a}$ the unit inventory cost of qualified component $c$ and return product $p$ at the recycling center in period $t$, respectively $UTC_{cit}^{af}$ the unit transportation cost of qualified component $c$ from the recycling center to factory $i$ in period $t$ $BOC_{pc}$ the bill of component $c$ to product $p$ $\theta_{ct}$ the remanufacturable rate of component $c$ in period $t$ $\overline \alpha_{p}^{a},\overline \beta_{c}^{a}$ the maximum inventory level of return products and qualified components at the recycling center, respectively $MDT_{p}$ the maximum quantity of return product $p$ can be disassembled & tested in every periods $MT^{a}$ the maximum quantity of qualified components can be transported from the recycling center to factories in every periods
Definition of the parameters occurred in the second level model
 Notation Description $MPN_{pijt},MPR_{pijt}$ the middle price of new and remanufactured product $p$ paid to factory $i$ by distributor $j$ in period $t$, respectively $tpb_{p}$ the quantity of new or remanufactured product $p$ transported per batch from factories todistributors $ab_{pi},rab_{pi}$ the quantity of new and remanufactured product $p$ assembled per batch at factory $i$, respectively $pb_{ci},rpb_{ci}$ the quantity of component $c$ processed and reprocessed per batch at factory $i$, respectively $SA_{pit},SRA_{pit}$ the set-up cost incurred if new and remanufactured product $p$ is assembled in batches at factory $i$ in period $t$, respectively $UAC_{pit},URAC_{pit}$ the unit assembly cost of new and remanufactured product $p$ at factory $i$ in period $t$, respectively $SP_{cit},SRP_{cit}$ the set-up cost incurred if component $c$ is newly processed and reprocessed in batches at factory $i$ in period $t$, respectively $UPC_{cit},URPC_{cit}$ the unit processing and reprocessing cost of component $c$ at factory $i$ in period $t$, respectively $ICNP_{pit}^{f},ICRMP_{pit}^{f}$ the unit inventory cost of new and remanufactured product $p$ at factory $i$ in period $t$, respectively $ICQC_{cit}^{f},ICNC_{cit}^{f},ICRC_{cit}^{f}$ the unit inventory cost of qualified, new and remanufactured component $c$ at factory $i$ in period $t$, respectively $UTC_{pijt}^{fd}$ the unit transportation cost of product $p$ from factory $i$ to distributor $j$ in period $t$ $\overline \beta_{ci}^{f},\overline \zeta_{ci}^{f},\overline \xi_{ci}^{f},\overline \lambda_{pi}^{f},\overline \chi_{pi}^{f}$ the maximum inventory level of qualified components, new components, remanufactured components, new products andremanufactured products at factory $i$, respectively $MA_{pi},MRA_{pi}$ the maximum quantity of new and remanufactured product $p$ can be assembled in factory $i$ in every periods, respectively $MP_{ci},MRP_{ci}$ the maximum quantity of component $c$ can be processed and reprocessed in factory $i$ in every periods, respectively $MT_{i}^{f}$ the maximum quantity of products can be transported from factory $i$ to distributors in every periods $PPC_{cit},tcb_{c},BOC_{pc}$ these notations have occurred in the first level model
 Notation Description $MPN_{pijt},MPR_{pijt}$ the middle price of new and remanufactured product $p$ paid to factory $i$ by distributor $j$ in period $t$, respectively $tpb_{p}$ the quantity of new or remanufactured product $p$ transported per batch from factories todistributors $ab_{pi},rab_{pi}$ the quantity of new and remanufactured product $p$ assembled per batch at factory $i$, respectively $pb_{ci},rpb_{ci}$ the quantity of component $c$ processed and reprocessed per batch at factory $i$, respectively $SA_{pit},SRA_{pit}$ the set-up cost incurred if new and remanufactured product $p$ is assembled in batches at factory $i$ in period $t$, respectively $UAC_{pit},URAC_{pit}$ the unit assembly cost of new and remanufactured product $p$ at factory $i$ in period $t$, respectively $SP_{cit},SRP_{cit}$ the set-up cost incurred if component $c$ is newly processed and reprocessed in batches at factory $i$ in period $t$, respectively $UPC_{cit},URPC_{cit}$ the unit processing and reprocessing cost of component $c$ at factory $i$ in period $t$, respectively $ICNP_{pit}^{f},ICRMP_{pit}^{f}$ the unit inventory cost of new and remanufactured product $p$ at factory $i$ in period $t$, respectively $ICQC_{cit}^{f},ICNC_{cit}^{f},ICRC_{cit}^{f}$ the unit inventory cost of qualified, new and remanufactured component $c$ at factory $i$ in period $t$, respectively $UTC_{pijt}^{fd}$ the unit transportation cost of product $p$ from factory $i$ to distributor $j$ in period $t$ $\overline \beta_{ci}^{f},\overline \zeta_{ci}^{f},\overline \xi_{ci}^{f},\overline \lambda_{pi}^{f},\overline \chi_{pi}^{f}$ the maximum inventory level of qualified components, new components, remanufactured components, new products andremanufactured products at factory $i$, respectively $MA_{pi},MRA_{pi}$ the maximum quantity of new and remanufactured product $p$ can be assembled in factory $i$ in every periods, respectively $MP_{ci},MRP_{ci}$ the maximum quantity of component $c$ can be processed and reprocessed in factory $i$ in every periods, respectively $MT_{i}^{f}$ the maximum quantity of products can be transported from factory $i$ to distributors in every periods $PPC_{cit},tcb_{c},BOC_{pc}$ these notations have occurred in the first level model
Definition of the parameters occurred in the third level model
 Notation Description $SPN_{pjt},SPR_{pjt}$ the selling price of new and remanufactured product $p$ at distributor $j$ in period $t$, respectively $DNM_{pjt},DRM_{pjt}$ the demands of new and remanufactured product $p$ in the market of distributor $j$ in period $t$, respectively $URCD_{pjt}$ the unit recycling cost of EOL product $p$ paid to retailers or customers by distributor $j$ in period $t$ $USNP_{pjt},USRP_{pjt}$ the unit shortage cost of new and remanufactured product $p$ paid by distributor $j$ in period $t$, respectively $ICNP_{pjt}^{d},ICRMP_{pjt}^{d},ICRP_{pjt}^{d}$ the unit inventory cost of new, remanufactured and return product $p$ at distributor $j$ in period $t$, respectively $UTC_{pjt}^{da}$ the unit transportation cost of return product $p$ from distributor $j$ to the recycling center in period $t$ $EPA_{pjt}$ the quantity of EOL product $p$ available in the market of distributor $j$ in period $t$ $\overline \lambda_{pj}^{d},\overline \chi_{pj}^{d},\overline \alpha_{pj}^{d}$ the maximum inventory level of new, remanufactured and return products at distributor $j$, respectively $MT_{j}^{d}$ the maximum quantity of return products can be transported from distributor $j$ to the recycling center in every periods $URCC_{pjt},trb_{p}$ these notations have occurred in the first level model $MPN_{pijt},tpb_{p},MPR_{pijt}$ these notations have occurred in the second level model
 Notation Description $SPN_{pjt},SPR_{pjt}$ the selling price of new and remanufactured product $p$ at distributor $j$ in period $t$, respectively $DNM_{pjt},DRM_{pjt}$ the demands of new and remanufactured product $p$ in the market of distributor $j$ in period $t$, respectively $URCD_{pjt}$ the unit recycling cost of EOL product $p$ paid to retailers or customers by distributor $j$ in period $t$ $USNP_{pjt},USRP_{pjt}$ the unit shortage cost of new and remanufactured product $p$ paid by distributor $j$ in period $t$, respectively $ICNP_{pjt}^{d},ICRMP_{pjt}^{d},ICRP_{pjt}^{d}$ the unit inventory cost of new, remanufactured and return product $p$ at distributor $j$ in period $t$, respectively $UTC_{pjt}^{da}$ the unit transportation cost of return product $p$ from distributor $j$ to the recycling center in period $t$ $EPA_{pjt}$ the quantity of EOL product $p$ available in the market of distributor $j$ in period $t$ $\overline \lambda_{pj}^{d},\overline \chi_{pj}^{d},\overline \alpha_{pj}^{d}$ the maximum inventory level of new, remanufactured and return products at distributor $j$, respectively $MT_{j}^{d}$ the maximum quantity of return products can be transported from distributor $j$ to the recycling center in every periods $URCC_{pjt},trb_{p}$ these notations have occurred in the first level model $MPN_{pijt},tpb_{p},MPR_{pijt}$ these notations have occurred in the second level model

2020 Impact Factor: 1.801