Integrated decision-making in reverse logistics: an optimisation of interacting acquisition, grading and disposition processes

Figures & data

Table 1. Summary of the model notation.

Parameter/Variable	Description
q	Remanufactured quantity
e	Acquisition effort
$\gamma \left(e\right)$	Effort-dependent acquisition function (strictly increasing)
p	Sales price of remanufactured item
c	Max. cost for remanufacturing
N	Max. amount of available used items for acquisition
D	Max. demand for remanufactured items
$f\left(x\right)$, $F\left(x\right)$	Known quality distribution of acquired used items

Table 2. Optimal strategy defined by 8 scenarios.

Scenario	Acquisition	Remanufacturing	Shortages	Conditions
1	Full	Full	Yes	N<D	$p>2m_{acq}+{\displaystyle \frac{c}{2}}$
	$e=m_{acq}$	q=N	D−N	${\lambda }_1>0$, ${\lambda }_2=0$, ${\lambda }_3>0$	p>c
2	Full	Full	No	D=N	$p>2m_{acq}+{\displaystyle \frac{c}{2}}$
	$e=m_{acq}$	q=D=N	–	${\lambda }_1>0$, ${\lambda }_2>0$, ${\lambda }_3>0$	p>c
3	Full	Selective	Yes	p<c	$p<{\displaystyle \frac{cD}{N}}$
	$e=m_{acq}$	$q={\displaystyle \frac{pN}{c}}$	D−q	${\lambda }_1>0$, ${\lambda }_2=0$, ${\lambda }_3=0$	$p^2>4c{m}_{acq}$
4	Selective	Full	Yes	$2p-c<{\displaystyle \frac{4D{m}_{acq}}{N}}$	${\displaystyle \frac{2p-c}{4}}<m_{acq}$
	$e={\displaystyle \frac{2p-c}{4}}<m_{acq}$	$q={\displaystyle \frac{N(2p-c)}{4m_{acq}}}={\displaystyle \frac{eN}{m_{acq}}}$	D−q	${\lambda }_1=0$, ${\lambda }_2=0$, ${\lambda }_3>0$	p>c
5	Selective	Full	No	$p-{\displaystyle \frac{c}{2}}>{\displaystyle \frac{2D{m}_{acq}}{N}}$	$c<{\displaystyle \frac{4D{m}_{acq}}{N}}$
	$e={\displaystyle \frac{D{m}_{acq}}{N}}$	q=D	–	${\lambda }_1=0$, ${\lambda }_2>0$, ${\lambda }_3>0$	p>c, N>D
6	Selective	Selective	Yes	p<c	${\displaystyle \frac{p^3}{c^2}}<{\displaystyle \frac{4D{m}_{acq}}{N}}$
	$e={\displaystyle \frac{p^2}{4c}}<m_{acq}$	$q={\displaystyle \frac{p^{3}N}{4c^2{m}_{acq}}}={\displaystyle \frac{eNp}{c{m}_{acq}}}$	D−q	${\lambda }_1=0$, ${\lambda }_2=0$, ${\lambda }_3=0$	${\displaystyle \frac{p^2}{c}}<4m_{acq}$
7	Selective	Selective	No	${\displaystyle \frac{p}{m_{acq}}}\sqrt[3]{{\displaystyle \frac{D^{2}c{m}_{acq}^2}{4N^2}}}>{\displaystyle \frac{Dc}{N}}$	$m_{acq}>{\displaystyle \frac{D^{2}c}{4N^2}}$
	$e=\sqrt[3]{{\displaystyle \frac{D^{2}c{m}_{acq}^2}{4N^2}}}$	q=D	–	${\lambda }_1=0$, ${\lambda }_2>0$, ${\lambda }_3=0$	N>D
	–	–	–	$\sqrt[3]{{\displaystyle \frac{D^{2}c{m}_{acq}^2}{4N^2}}}>{\displaystyle \frac{Dm}{N}}$	$pD>{\displaystyle \frac{4N{e}^2}{m_{acq}}}$
8	Full	Selective	No	N>D	$D^{2}c>4N^2{m}_{acq}$
	$e=m_{acq}$	q=D	–	${\lambda }_1>0$, ${\lambda }_2>0$, ${\lambda }_3=0$	pN>cD

Figure 1. Strategy maps containing different scenarios.

Table 3. Optimal strategy of the stochastic model defined by 4 scenarios.

Scenario	Acquisition	Remanufacturing	Conditions
1	Selective	Selective	$e<m_{acq}$, $q<{\displaystyle \frac{eN}{m_{acq}}}$
	$e={\displaystyle \frac{qc{m}_{acq}}{Np(1-G_D(q\left)\right)}}$	$q=\sqrt{{\displaystyle \frac{4e^3{N}^2}{c{m}_{acq}^2}}}$	${\lambda }_3=0$, ${\lambda }_4=0$
2	Selective	Full	$e\le m_{acq}$, $q={\displaystyle \frac{eN}{m_{acq}}}$
	$e={\displaystyle \frac{q{m}_{acq}}{N}}$	$G_D\left(q\right)+{\displaystyle \frac{2q{m}_{acq}}{Np}}=1-{\displaystyle \frac{c}{2p}}$	${\lambda }_3>0$, ${\lambda }_4=0$
3	Full	Selective	$q<{\displaystyle \frac{eN}{m_{acq}}}$
	$e=m_{acq}$	$G_D\left(q\right)+{\displaystyle \frac{qc}{pN}}=1$	${\lambda }_3=0$, ${\lambda }_4>0$
4	Full	Full	${\lambda }_3=p(1-G_D(N\left)\right)-c$
	$e=m_{acq}$	q=N	${\lambda }_4={\displaystyle \frac{Nc}{2}}-2m_{acq}N+{\lambda }_{3}N$

Table 4. Setup of parameters for numerical analysis.

Parameter	Value	Base case value
Sales price of remanufactured item	p=10	p=10
Max. remanufacturing cost	$c=1.00,1.05,\dots ,19.95,20.00$	c=5
Max. available used items	$N=5.00,5.05,\dots ,99.95,100.00$	N=30
Lower limit of uniformly distributed demand	a=0,5	a=0
Upper limit of uniformly distributed demand	b=a+20	b=20
Slope parameter of acquisition function	$m_{acq}=1,2,\dots ,9,10$	$m_{acq}=5$

Table 5. Key figures concerning the sensitivity of the stochastic base case over N.

N	Scenario	Acq. effort	Acq. qty.	Offered qty.	Acq. rate	Reman. rate	Revenue	Total acq. cost	Total reman. cost	Profit
5	2	3.00000	3.000	3.000	60.000%	100.000%	27.750	9.000	7.500	11.250
10	2	2.50000	5.000	5.000	50.000%	100.000%	43.750	12.500	12.500	18.750
15	2	2.14285	6.429	6.429	42.860%	100.000%	53.954	13.775	16.071	24.107
20	2	1.87500	7.500	7.500	37.500%	100.000%	60.938	14.063	18.750	28.125
25	2	1.66667	8.333	8.333	33.332%	100.000%	65.972	13.889	20.833	31.250
30	2	1.50000	9.000	9.000	30.000%	100.000%	69.750	13.500	22.500	33.750
35	2	1.36364	9.545	9.545	27.271%	100.000%	72.676	13.017	23.864	35.795
40	2	1.25000	10.000	10.000	25.000%	100.000%	75.000	12.500	25.000	37.500
45	1	1.17801	10.602	10.292	23.560%	97.076%	76.440	12.489	24.979	38.972
50	1	1.11634	11.163	10.550	22.326%	94.509%	77.673	12.462	24.924	40.286

Figure 2. Revenue, profit, and total costs with varying c.

Table 6. Key figures concerning the sensitivity of the stochastic base case over c.

c	Scenario	Acq. effort	Acq. qty.	Offered qty.	Acq. rate	Reman. rate
2	2	1.80000	10.800	10.800	36.000%	100.000%
4	2	1.60000	9.600	9.600	32.000%	100.000%
6	1	1.42270	8.536	8.313	28.453%	97.388%
8	1	1.36628	8.198	6.776	27.327%	82.654%
10	1	1.29611	7.777	5.599	25.923%	71.994%
12	1	1.22225	7.334	4.681	24.447%	63.826%
14	1	1.14958	6.898	3.953	22.993%	57.306%
16	1	1.08042	6.483	3.369	21.610%	51.967%
18	1	1.01581	6.095	2.896	20.317%	47.514%
20	1	0.95610	5.737	2.509	19.123%	43.734%

Table 7. Occurrence of further scenarios of the optimal strategy.

N	Scenario	Acq. effort	Acq. qty.	Offered qty.	Acq. rate	Reman. rate
10	4	1.00000	10.000	10.000	100.000%	100.000%
15	3	1.00000	15.000	12.931	100.000%	86.207%
20	1	0.97482	19.496	14.551	97.482%	74.635%