A note on the can-order policy for the two-item stochastic joint-replenishment problem

Figures & data

Fig. 1 A realization of can-order policy for two-items: plot of inventory position against time.

Fig. 2 Order of events.

Fig. 3 Enumerative algorithm (ENUM) to find can-order policy variables.

Table 1 Experiment results (μ = 1, h _j = 1, L _j = 2, π _j = 10, i = 1,2)

Exp.	p 1	p 2	K	κ	λ	μ	FGT cost	FGT par.	ENUM cost	ENUM par.	% diff.
1	1	30	3	1	1	1	10.122	(2, 4, 6)	10.040	(2, 4, 6)	0.82
								(3, 5, 7)		(3, 5, 7)
2	1	30	3	1	3	1	13.971	(8, 11, 14)	13.860	(8, 11, 14)	0.81
								(3, 5, 7)		(3, 5, 7)
3	1	30	3	1	10	1	21.976	(26, 31, 36)	21.876	(26, 31, 36)	0.46
								(3, 5, 7)		(3, 4, 7)
4	1	30	10	1	1	1	12.583	(1, 5, 7)	12.225	(1, 5, 7)	2.92
								(3, 6, 8)		(3, 6, 8)
5	1	30	10	1	3	1	17.468	(7, 13, 16)	17.049	(7, 13, 16)	2.46
								(3, 6, 8)		(3, 6, 7)
6	1	30	10	1	10	1	27.805	(24, 36, 41)	27.217	(24, 36, 41)	2.16
								(3, 6, 8)		(2, 5, 7)
7	1	30	100	1	1	1	26.878	(0, 12, 14)	24.549	(0, 11, 12)	9.49
								(2, 14, 16)		(2, 12, 14)
8	1	30	100	1	3	1	37.621	(3, 27, 30)	34.394	(3, 26, 28)	9.38
								(2, 13, 15)		(2, 11, 12)
9	1	30	100	1	10	1	59.602	(20, 63, 68)	56.229	(20, 63, 67)	6.00
								(1, 11, 13)		(1, 8, 10)
10	10	30	3	1	1	1	10.463	(3, 4, 6)	10.395	(3, 4, 6)	0.65
								(3, 5, 7)		(3, 5, 7)
11	10	30	3	1	3	1	14.354	(8, 11, 14)	14.253	(9, 11, 14)	0.71
								(3, 5, 7)		(3, 5, 7)
12	10	30	3	1	10	1	22.290	(26, 31, 36)	22.185	(26, 31, 36)	0.47
								(3, 5, 7)		(3, 4, 7)
13	10	30	10	1	1	1	13.019	(2, 6, 8)	12.681	(2, 6, 7)	2.66
								(3, 6, 8)		(3, 6, 8)
14	10	30	10	1	3	1	17.925	(8, 14, 17)	17.504	(8, 14, 16)	2.40
								(3, 6, 8)		(3, 6, 7)
15	10	30	10	1	10	1	28.230	(25, 37, 42)	27.555	(25, 37, 41)	2.45
								(2, 6, 8)		(2, 5, 7)
16	10	30	100	1	1	1	27.693	(1, 13, 15)	25.321	(1, 12, 13)	9.37
								(2, 14, 16)		(2, 12, 14)
17	10	30	100	1	3	1	38.411	(5, 28, 31)	35.196	(5, 27, 29)	9.13
								(2, 13, 15)		(2, 11, 12)
18	10	30	100	1	10	1	60.182	(21, 63, 68)	56.834	(21, 63, 68)	5.89
								(1, 11, 13)		(1, 8, 10)
19	100	30	3	1	1	1	11.613	(4, 6, 8)	11.531	(4, 5, 7)	0.71
								(3, 5, 7)		(3, 5, 7)
20	100	30	3	1	3	1	15.681	(10, 13, 16)	15.571	(10, 13, 16)	0.71
								(3, 5, 7)		(3, 5, 7)
21	100	30	3	1	10	1	23.891	(28, 33, 38)	23.788	(28, 33, 38)	0.43
								(3, 5, 7)		(3, 4, 7)
22	100	30	10	1	1	1	14.155	(4, 7, 9)	13.862	(4, 7, 9)	2.11
								(3, 6, 8)		(3, 6, 8)
23	100	30	10	1	3	1	19.410	(10, 15, 18)	19.005	(10, 15, 18)	2.13
								(3, 6, 8)		(3, 6, 7)
24	100	30	10	1	10	1	29.976	(27, 38, 43)	29.335	(27, 38, 43)	2.19
								(2, 6, 8)		(2, 5, 7)
25	100	30	100	1	1	1	29.292	(3, 15, 17)	26.850	(3, 13, 15)	9.10
								(2, 14, 16)		(2, 13, 14)
26	100	30	100	1	3	1	40.445	(8, 30, 33)	37.101	(8, 29, 31)	9.01
								(1, 13, 15)		(1, 10, 12)
27	100	30	100	1	10	1	62.587	(25, 66, 71)	59.234	(25, 66, 70)	5.66
								(1, 11, 13)		(1, 8, 10)

Table 2 Experiment results (μ = 1, h _j = 1, L _j = 2, π _j = 10, i = 1,2)

Exp.	p 1	p 2	K	κ	λ	μ	FGT cost	FGT par.	ENUM cost	ENUM par.	% diff.
28	1	30	100	10	1	1	27.459	(0, 9, 14)	25.406	(0, 10, 13)	8.08
								(2, 11, 16)		(2, 11, 15)
29	1	30	100	10	3	1	38.331	(3, 22, 30)	35.547	(3, 22, 29)	7.83
								(2, 10, 15)		(2, 9, 12)
30	1	30	100	10	10	1	60.702	(20, 53, 68)	58.040	(20, 55, 68)	4.59
								(1, 9, 14)		(1, 7, 11)
31	10	30	100	10	1	1	28.268	(1, 10, 15)	26.199	(1, 10, 14)	7.89
								(2, 11, 16)		(2, 11, 15)
32	10	30	100	10	3	1	39.124	(5, 23, 31)	36.362	(5, 23, 30)	7.60
								(2, 10, 15)		(1, 9, 12)
33	10	30	100	10	10	1	61.322	(21, 54, 69)	58.654	(21, 55, 69)	4.55
								(1, 9, 14)		(1, 7, 11)
34	100	30	100	10	1	1	29.823	(3, 12, 17)	27.736	(3, 11, 15)	7.52
								(2, 11, 16)		(2, 11, 14)
35	100	30	100	10	3	1	41.145	(8, 25, 33)	38.283	(8, 25, 32)	7.48
								(1, 10, 15)		(1, 9, 12)
36	100	30	100	10	10	1	63.698	(25, 56, 71)	61.075	(25, 58, 71)	4.29
								(1, 9, 14)		(1, 7, 11)
FGT cost = optimal cost of FGT.
FGT par. = can-order policy parameters for FGT (upper one is for item 1 and the lower one is for item 2).
ENUM cost = optimal cost of ENUM.
ENUM par = can-order policy parameters for ENUM (upper one is for item 1 and the lower one is for item 2).
% diff. = percentage difference between FGT cost and ENUM cost.