Multi-objective optimization of recycling and remanufacturing supply chain logistics network with scalable facility under uncertainty

Figures & data

Figure 1. Recycling and remanufacturing logistics network.

Figure 2. The nodes and costs of the recycling and remanufacturing logistics network based on scalable facility.

Table 1. Notation description.

Indices
$r$	Index of recycling center ($r=1,2,\dots ,R$)
$p$	Index of preprocessing center ($p=1,2,\dots ,P$)
$t$	Index of operating cycle ($t=1,2,\dots ,T$)
Parameters
$R$	The number of recycling centers
$P$	The number of preprocessing centers
$T$	Planned operating cycle
$a_r$	The capacity of the recycling center $r$
$s_r(t)$	In the operation cycle $t$, the quantity of used products recycled by the recycling center $r$
$b_p$	The capacity of the preprocessing center $p$
$d_M(t)$	The demand of the manufacturer in operating cycle $t$
${\eta }_M$	The maximum processing capacity of the manufacturer
$u_M(t)$	The maximum stock out quantity allowed by the manufacturer during the operating cycle $t$
$c_r$	Unit expansion cost of the recycling center $r$
$c_p$	Unit expansion cost of the preprocessing center $p$
$e_p$	Expansion level of the preprocessing center $p$
$c_{rp}$	Unit transportation cost from the recycling center $r$ to the preprocessing center $p$
$c_{pM}$	Unit transportation cost from the preprocessing center $p$ to the manufacturer
$c_{pD}$	Unit transportation cost from the preprocessing center $p$ to the processing center
$c_{SM}$	Unit transportation cost from the supplier to the manufacturer
$c_{pG}$	Unit transportation cost from the preprocessing center $p$ to the remanufacturing center
$c_p^o$	Fixed cost of opening the preprocessing center $p$
$c_p^H$	The unit inventory holding cost of the preprocessing center $p$ in each operating cycle
$c_H^M$	The unit inventory holding cost of the manufacturer in each operating cycle
$\rho$	The level of remanufacturing center newly built by manufacturer
$c_G^o$	Fixed cost of the remanufacturing center unit level opening
${\lambda }_p$	Remanufacturing waste rate of the preprocessing center $p$
Decision variables
$x_{rp}(t)$	Shipment volume from the recycling center $r$ to the preprocessing center $p$ in the operation cycle $t$
$x_{pM}(t)$	Shipments from the preprocessing center $p$ to the manufacturer in the operating cycle $t$
$x_{pD}(t)$	Shipment volume from the preprocessing center $p$ to the processing center in operating cycle $t$
$x_{pG}(t)$	Shipment volume from the preprocessing center $p$ to the remanufacturing center in operating cycle $t$
$x_M^B(t)$	The quantity of manufacturer’s out-of-stock in the operating cycle $t$
$y_P^H(t)$	The inventory holdings of the preprocessing center $p$ in the operating cycle $t$
$z_M^H(t)$	The inventory holdings of the manufacturer in the operating cycle $t$
$h_r(t)=\left\{\begin{array}{c}1,S_r(t)>a_r,\mathrm{\forall }t\\ 0,S_r(t)\le a_r,\mathrm{\forall }t\end{array}\right.$
$k_p(t)=\left\{\begin{array}{c}1,\sum _{r=1}^R{S}_r(t)>\sum _{p=1}^P{b}_p,\mathrm{\forall }t\\ 0,\mathrm{O}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\end{array}\right.$
$w_M(t)=\left\{\begin{array}{c}1,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{u}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{r}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{y}\mathrm{d}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}t\\ 0,\mathrm{O}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\end{array}\right.$
$z_P(t)=\left\{\begin{array}{c}1,\mathrm{I}\mathrm{n}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}t,\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}p\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{d}\\ 0,\mathrm{O}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\end{array}\right.$
${\pi }_G(t)=\left\{\begin{array}{c}1,\mathrm{M}\mathrm{a}\mathrm{n}\mathrm{u}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}t\\ 0,\mathrm{O}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\end{array}\right.$

Figure 3. The framework of the improved nonlinear grey Bernoulli-Markov prediction model.

Table 2. The recycled quantity of used products in the first four operating cycles of each recycling center.

Operating cycle t	Recycling center $r$
	1	2	3	4	5	6	7	8
1	86	91.3	115.6	73.6	132.5	94.8	83.4	123.2
2	93	103.5	126.4	96.8	162.7	105.3	97.2	152.7
3	101.2	117.8	143.2	118.1	153.6	162.7	116.3	143.6
4	123.6	143.6	168.3	136.4	192.8	152.6	142.4	191.3

Table 3. The predicted recycled quantity of a certain type of scrap automobile products.

Operating cycle t	Recycling center $r$
	1	2	3	4	5	6	7	8
5	115.8	168.3	195.1	160.2	204.6	187.7	169.3	209.5
6	123.0	199.2	222.3	189.3	221.8	224.6	208.7	245.8

Table 4. Parameter settings of the recycling center $r$.

Recycling center $r$		1	2	3	4	5	6	7	8
Capacity (t)		150	190	230	140	220	180	150	210
Unit expansion cost (RMB/t)		50	60	55	45	70	45	50	60
Quantity of recycling	$t=1$	115.8	168.3	195.1	160.2	204.6	187.7	169.3	209.5
	$t=2$	123.0	199.2	222.3	189.3	221.8	224.6	208.7	245.8

Table 5. Parameter settings of preprocessing center $p$

Preprocessing center $p$	1	2	3	4	5
Capacity (t)	340	210	320	280	360
Fixed cost of opening (RMB)	60000	90000	75000	60000	75000
Unit expansion cost (RMB/t)	60	70	65	70	60
Expansion level (t)	100	50	100	100	150
Unit inventory holding cost (RMB/t)	15	5	15	10	15
Unit transportation cost to manufacturer (RMB/t)	55	110	60	85	75
Unit transportation cost to remanufacturing center (RMB/t)	45	90	55	75	65
Unit transportation cost to the processing center (RMB/t)	25	50	50	75	25
Remanufacturing waste rate of preprocessing center	1.8%	2%	2%	1.8%	1.8%

Table 6. The unit transportation cost (RMB/t) from the recycling center $r$ to the preprocessing center $p$

Preprocessing center $p$	Recycling center $r$
	1	2	3	4	5	6	7	8
1	30	55	25	40	25	50	55	55
2	65	80	40	45	20	35	20	20
3	50	55	15	25	5	25	45	25
4	65	75	45	30	35	10	50	10
5	45	35	15	10	30	25	60	40

Table 7. The relevant parameter settings of the manufacturer.

Operating cycle $t$	Demand (t)	Maximum allowable shortage (t)	Maximum processing capacity (t)	Unit inventory holding cost (RMB/t)
1	1000	80	3000	5
2	1200	100

Table 8. The values of binary variables $h_r$, $z_p$ and $k_p$.

	$h_r$								$z_p$					$k_p$
Operating cycle $t$	$h_1$	$h_2$	$h_3$	$h_4$	$h_5$	$h_6$	$h_7$	$h_8$	$z_1$	$z_2$	$z_3$	$z_4$	$z_5$	$k_1$	$k_2$	$k_3$	$k_4$	$k_5$
1	0	0	0	1	0	1	1	0	1	0	1	1	1	0	0	0	0	1
2	0	1	0	1	1	1	1	1	1	0	1	1	1	1	0	1	0	1

Table 9. Shipment volume from the recycling center $r$ to the preprocessing center $p$.

Recycling center $r$	Preprocessing center $p$
	1	2	3	4	5
1	(15.0236, 19.2365)	(0,0)	(28.1542, 21.3476)	(20.7238, 32.2289)	(51.8982, 50.1870)
2	(58.2345, 50.7256)	(0,0)	(6.7482, 39.4263)	(16.3442, 20.1342)	(86.9731, 88.9139)
3	(13.9352, 33.1928)	(0,0)	(66.3245, 34.3841)	(86.0437, 99.4255)	(28.7966, 55.2976)
4	(32.7382, 34.9312)	(0,0)	(54.1456, 56.3538)	(13.1893, 20.0472)	(60.1269, 77.9678)
5	(63.6571, 60.0968)	(0,0)	(44.8913, 46.4365)	(26.3511, 41.1798)	(69.7005, 74.0869)
6	(16.6577, 82.3864)	(0,0)	(42.3521, 54.8541)	(97.5725, 34.1164)	(31.1177, 53.2431)
7	(21.3142, 48.0712)	(0,0)	(46.6522, 98.2754)	(15.8934, 28.3488)	(85.4402, 34.0046)
8	(113.3889, 97.4438)	(0,0)	(30.7319, 68.9222)	(3.8820, 3.5743)	(61.4972, 75.8597)

Table 10. Shipment volume from the preprocessing center $p$ to the manufacturer.

$p$	1	2	3	4	5
Manufacturer	(296.8676,418.5504)	(0, 0)	(313.7255,411.7647)	(275.0491,274.1209)	(467.1420,500.5507)

Figure 4. The calculated effective solution set.