A periodic review integrated inventory model for buyer’s unidentified protection interval demand distribution

Figures & data

Figure 1. Inventory level for the vendor

Figure 2. Inventory level for the buyer

Table 1. Lead time data

Lead time component z	Normal duration b z (days)	Minimum duration a z (days)	Unit crashing cost c z ($/day)
1	20	6	0.4
2	20	6	1.2
3	16	9	5.0

Table 2. The results of solution procedure (lead time in year)

E (i)	M	(T, A, k, L)	$\mathit{\rho }\left(T\right),\mathit{\alpha }(L)$	EOCWIb, EHCWIb, ESCWIb, EPCWIb	ESUCWIv, EHCWIv, EPCWIv	JEACWI
0.00	1	0.44, 200.00, 0.52, 0.1,250	3.39, concave	509.28, 2,535.75, 245.91, 36,000	2,477.07, 672.83, 26,666.67	69,107.54
	2	0.27, 138.25, 0.90, 0.1,667	1.89, concave	766.77, 1,655.78, 260.13, 36,000	2,025.27, 1,237.39, 26,666.67	68,609.04
	3	0.19, 100.34, 1.17, 0.1,667	1.35, concave	946.25, 1,282.09, 271.99, 36,000	1,860.32, 1,493.16, 26,666.67	68,520.50*
	4	0.15, 79.11, 1.38, 0.1,667	1.08, concave	1,079.32, 1,079.00, 284.97, 36,000	1,769.47, 1,648.31, 26,666.67	68,527.76
0.04	1	0.49, 200.00, 0.44, 0.0,833	3.40, 2.40	507.00, 2,812.92, 242.97, 36,520.65	2,279.71, 754.00, 26,963.44	70,080.73
	2	0.30, 151.66, 0.81, 0.0,833	1.76, 4.55	797.61, 1,831.23, 236.50, 36,642.16	1,836.13, 1,401.29, 26,906.91	69,651.87
	3	0.22, 110.50, 1.07, 0.0,833	1.25, 6.62	1,005.74, 1,402.37, 239.46, 36,695.65	1,675.89, 1,704.11, 26,879.22	69,602.47*
	4	0.17, 85.93, 1.29, 0.1,250	1.03, 11.90	1,069.50, 1,165.71, 265.89, 36,787.64	1,614.05, 1,857.03, 26,867.08	69,626.93
0.08	1	0.54, 200.00, 0.35, 0.0,833	3.56, 1.06	460.59, 3,158.28, 251.28, 36,970.57	2,071.01, 853.93, 27,207.83	70,973.53
	2	0.34, 168.56, 0.72, 0.0,833	1.83, 1.99	732.45, 2,064.85, 241.09, 37,236.14	1,647.04, 1,602.80, 27,075.77	70,600.16
	3	0.24, 122.00, 0.97, 0.0,833	1.28, 2.96	939.59, 1,571.29, 242.39, 37,356.54	1,503.31, 1,946.74, 27,014.53	70,574.41*
	4	0.19, 96.53, 1.17, 0.0,833	1.01, 3.84	1,097.87, 1,298.99, 246.54, 37,423.61	1,420.30, 2,160.28, 26,973.67	70,621.28
0.12	1	0.62, 200.00, 0.23, 0.0,625	3.88, 0.43	465.81, 3,663.86, 258.91, 37,205.42	1,809.71, 1,001.58, 27,359.57	71,764.89
	2	0.39, 191.11, 0.61, 0.0,833	1.92, 1.11	651.09, 2,383.39, 246.73, 37,754.31	1,435.12, 1,878.77, 27,132.31	71,481.75*
	3	0.28, 138.77, 0.86, 0.0,833	1.33, 1.69	856.93, 1,802.16, 245.64, 37,962.51	1,309.00, 2,279.84, 27,027.80	71,483.91
0.16	1	0.74, 200.00, 0.07, 0.0,625	4.37, 0.27	394.87, 4,355.36, 277.81, 37,312.97	1,534.09, 1,204.38, 27,372.57	72,452.09
	2	0.47, 200.00, 0.46, 0.0,625	2.14, 0.48	630.80, 2,869.68, 249.46, 38,010.92	1,185.34, 2,304.50, 26,979.15	72,229.89*
	3	0.35, 169.84, 0.68, 0.0,625	1.48, 0.73	840.76, 2,210.33, 243.27, 38,318.97	1,045.69, 2,878.21, 26,753.26	72,290.53

* Optimal value for each set of E(i).

Table 3. Summary of optimal results (lead time in year)

E (i)	T*	A*	L*	k*	m*	JEACWI*
0.00	0.19	100.34	0.1,667	1.17	3	68,520.50
0.04	0.22	110.50	0.0,833	1.07	3	69,602.47
0.08	0.24	122.00	0.0,833	0.97	3	70,574.41
0.12	0.39	191.11	0.0,833	0.61	2	71,481.75
0.16	0.47	200.00	0.0,625	0.46	2	72,229.89

Table 4. Optimal values in terms of different lead time (lead time in year)

E (i)	L z	C (L z )	(T z , A z , k z , M z )	$\mathit{\rho }\left(T_z\right),\mathit{\alpha }(L_z)$	JEACWI z
0.00	0.1,667	0	(0.19, 100.34, 1.17, 3)	1.35, concave	68,520.50*
	0.1,250	5.6	(0.19, 100.47, 1.17, 3)	1.28, concave	68,521.97
	0.0,833	22.4	(0.20, 101.91, 1.15, 3)	1.22, concave	68,583.27
	0.0,625	57.4	(0.20, 105.75, 1.12, 3)	1.25, concave	68,755.65
0.04	0.1,667	0	(0.21, 108.85, 1.09, 3)	1.37, 11.92	69,666.37
	0.1,250	5.6	(0.21, 108.97, 1.08, 3)	1.30, 9.44	69,606.76
	0.0,833	22.4	(0.22, 110.50, 1.07, 3)	1.25, 6.62	69,602.42*
	0.0,625	57.4	(0.23, 114.63, 1.04, 3)	1.28, 4.93	69,731.15
0.08	0.1,667	0	(0.24, 120.26, 0.99, 3)	1.40, 5.46	70,765.24
	0.1,250	5.6	(0.24, 120.36, 0.99, 3)	1.33, 4.27	70,644.76
	0.0,833	22.4	(0.25, 122.01, 0.97, 3)	1.28, 2.96	70,574.41*
	0.0,625	57.4	(0.25, 126.51, 0.94, 3)	1.32, 2.18	70,657.83
0.12	0.1,667	0	(0.39, 189.46, 0.62, 2)	2.05, 2.27	71,812.73
	0.1,250	5.6	(0.39, 189.46, 0.62, 2)	1.98, 1.69	71,631.08
	0.0,833	22.4	(0.39, 191.11, 0.61, 2)	1.92, 1.11	71,481.75*
	0.0,625	57.4	(0.40, 195.97, 0.59, 2)	1.97, 0.81	71,483.52
0.16	0.1,667	0	(0.46, 200.00, 0.49, 2)	2.18, 1.45	72,727.89
	0.1,250	5.6	(0.46, 200.00, 0.49, 2)	2.11, 1.05	72,485.85
	0.0,833	22.4	(0.46, 200.00, 0.49, 2)	2.08, 0.67	72,271.52
	0.0,625	57.4	(0.47, 200.00, 0.49, 2)	2.14, 0.48	72,229.89*

Table 5. Effect of changes in various parameters (lead time in year)

Parameter change (%)	−20%	−15%	−10%	−5%	+5%	+10%	+15%	+20%
$\frac{1}{\mathit{\delta }}$
T*	0.24	0.24	0.24	0.24	0.25	0.25	0.25	0.25
A*	94.83	101.49	108.23	115.08	129.02	136.14	143.35	150.68
k*	1.00	0.99	0.99	0.98	0.97	0.96	0.96	0.95
L*	4	4	4	4	4	4	4	4
M*	3	3	3	3	3	3	3	3
$\mathit{\rho }\left(T^{\ast }\right)$	1.27	1.27	1.27	1.28	1.28	1.28	1.29	1.29
$\mathit{\alpha }\left(L^{\ast }\right)$	3.06	3.03	3.01	2.98	2.93	2.91	2.88	2.86
JEACWI*	70,505.41	70,525.34	70,543.43	70,559.75	70,587.46	70,598.97	70,609.01	70,617.63
h b
T*	0.37	0.36	0.35	0.25	0.24	0.24	0.23	0.18
A*	185.02	180.27	175.82	124.10	120.01	118.11	116.29	97.72
k*	0.82	0.79	0.77	1.00	0.95	0.92	0.90	1.06
L*	4	4	4	4	4	4	4	4
M*	2	2	2	3	3	3	3	4
$\mathit{\rho }\left(T^{\ast }\right)$	1.88	1.86	1.85	1.29	1.27	1.27	1.26	0.99
$\mathit{\alpha }\left(L^{\ast }\right)$	2.01	2.01	2.00	2.96	2.95	2.95	2.94	3.82
JEACWI*	70,166.49	70,279.10	70,388.79	70,495.10	70,652.26	70,728.73	70,803.88	70,873.27
π
T*	0.24	0.24	0.24	0.24	0.24	0.24	0.24	0.24
A*	121.94	121.95	121.95	121.97	122.04	122.04	122.06	122.08
k*	0.80	0.84	0.89	0.93	1.02	1.05	1.09	1.13
L*	4	4	4	4	4	4	4	4
M*	3	3	3	3	3	3	3	3
$\mathit{\rho }\left(T^{\ast }\right)$	1.36	1.34	1.32	1.30	1.26	1.25	1.23	1.22
$\mathit{\alpha }\left(L^{\ast }\right)$	2.65	2.74	2.82	2.89	3.01	3.06	3.11	3.16
JEACWI*	75,522.68	70,536.28	70,549.41	70,562.10	70,586.35	70,597.97	70609.29	70,620.32
h v
T*	0.22	0.27	0.26	0.25	0.24	0.33	0.32	0.31
A*	109.45	132.15	128.51	125.14	119.11	163.07	158.83	156.13
k*	1.07	0.91	0.93	0.95	0.99	0.75	0.76	0.78
L*	4	4	4	4	4	4	4	4
M*	4	3	3	3	3	2	2	2
$\mathit{\rho }\left(T^{\ast }\right)$	1.04	1.31	1.30	1.29	1.27	1.81	1.79	1.79
$\mathit{\alpha }\left(L^{\ast }\right)$	3.35	2.69	2.78	2.87	3.04	2.08	2.14	2.19
JEACWI*	70,162.22	70,270.82	70,374.72	70,475.85	70,670.58	70,757.40	70,833.89	70,909.07