Derivation of closed-form expression for optimal base stock level considering partial backorder, deterministic demand, and stochastic supply disruption

Figures & data

Table 1. The comparison between the existing literature and the present research work

Authors	Continuous variable of supply disruption length	Partial backorder	Exact closed-form expressions for optimal variables
Parlar et al. (1995)
Güllü et al. (1999)			●
Li et al. (2004)			●
Schmitt et al. (2010)
Atasoy et al. (2012)
Schmitt and Snyder (2012)			●
Lewis et al. (2013)			●
Warsing et al. (2013)			●
Ciarallo and Niranjan (2014)
Firouzi et al. (2014)
Skouri et al. (2014)			●
Salehi et al. (2016)		●	●
Taleizadeh (2017)		●	●
Taleizadeh and Dehkordi (2017)		●	●
Hsieh and Putera (2018)	●
Konstantaras et al. (2019)			●
Saithong and Luong (2019)	●
De and Mahata (2020)		●	●
This research work	●	●	●

Table 2. List of notations

Notation	Description
$\lambda$	The arrival rate of supply disruption
$\mu$	Parameter of exponential distribution representing the length of a supply disruption
$\beta$	The fraction of amount of demand that is back-ordered
$D$	Demand per day
$C_H$	Inventory holding costs per unit per day
$C_S$	Costs of lost sale per unit
$C_B$	Costs of backorder per unit per day
$X$	The random variable representing the time until an arrival of supply disruption which is assumed to follow an exponential distribution with rate $\lambda$, i.e., the density function of X is $f_X(x)=\lambda e^{-\lambda x}$
$Y$	The random variable representing the length of a supply disruption which is assumed to follow an exponential distribution with rate$\mu$, i.e., the density function of Y is$f_Y(y)=\mu e^{-\mu y}$
$N$	The random variable representing the number of inventory cycles in a renewal cycle
$T$	Length of an inventory review period
${\left.A\right|}_{A>T}$	The random variable representing the length of the last inventory cycle in a renewal cycle
$Z$	The random variable representing the length of a renewal cycle
$W$	The random variable representing the time elapsed from the last order placing before a disruption occurs until the arrival of the disruption
$S$	Base stock level
$i$	Indices of possible scenarios of a renewal cycle: $i=1$: The case when inventory is shortage in non-disrupted cycles $i=2$: The case when inventory is not shortage in non-disrupted cycles
$TC{U}^{\left\{i\right\}}\left(S\right)$	Total costs per day, as a function of $S$, for case $i$
$ND{C}^{\left\{i\right\}}$	Total costs in a non-disrupted cycle for case $i$
$D{C}^{\left\{i\right\}}$	Total costs in the disrupted cycle, i.e., the last inventory cycle, for case $i$
$S^{\ast \left\{i\right\}}$	Optimal base stock level for case $i$
$S^{\ast }$	Optimal base stock level
$TCU\left(S^{\ast }\right)$	The minimum cost per time unit

Figure 1. Possible cases of renewal cycle in the presence of supply disruption.

Figure 2. The last inventory cycle when $i=1$.

Figure 3. The possible scenarios of the last inventory cycle when $i=2$.

Figure 4. Flow chart of solution method.

Table A1. Optimal base stock level and minimum cost per day for $\beta$= 1.00

$\beta$	$\lambda$	$\mu$	$S^{\ast \left\{1\right\}}$	$S^{\ast \left\{2\right\}}$	$E\left[TC{U}^{\left\{1\right\}}\left(S^{\ast \left\{1\right\}}\right)\right]$	$E\left[TC{U}^{\left\{2\right\}}\left(S^{\ast \left\{2\right\}}\right)\right]$	$S^{\ast }$	$E\left[TCU\left(S^{\ast }\right)\right]$
1.00	0.01	0.05	49.60	45.85	100.94	100.86	49.60	100.94
1.00	0.01	0.10	45.63	17.43	43.51	35.92	45.63	43.51
1.00	0.01	0.50	42.46	28.11	22.12	12.46	42.46	22.12
1.00	0.01	1.00	42.06	35.64	21.26	15.35	42.06	21.26
1.00	0.01	5.00	41.75	45.53	20.88	21.48	41.75	20.88
1.00	0.01	10.00	41.71	47.42	20.86	22.89	41.71	20.86
1.00	0.01	20.00	41.69	48.54	20.84	23.77	41.69	20.84
1.00	0.05	0.05	74.46	147.17	214.10	167.12	147.17	167.12
1.00	0.05	0.10	58.06	76.36	85.73	80.18	76.36	80.18
1.00	0.05	0.50	44.95	41.74	25.40	24.18	44.95	25.40
1.00	0.05	1.00	43.31	42.59	22.41	21.45	43.31	22.41
1.00	0.05	5.00	41.99	46.94	21.03	22.74	41.99	21.03
1.00	0.05	10.00	41.83	48.12	20.92	23.53	41.83	20.92
1.00	0.05	20.00	41.75	48.89	20.88	24.09	41.75	20.88
1.00	0.10	0.05	94.34	170.90	238.11	176.11	170.90	176.11
1.00	0.10	0.10	68.01	92.16	100.23	88.11	92.16	88.11
1.00	0.10	0.50	46.93	46.05	27.31	27.12	46.93	27.31
1.00	0.10	1.00	44.30	44.85	23.17	23.06	44.30	23.17
1.00	0.10	5.00	42.19	47.41	21.14	23.08	42.19	21.14
1.00	0.10	10.00	41.93	48.36	20.98	23.70	41.93	20.98
1.00	0.10	20.00	41.80	49.01	20.90	24.18	41.80	20.90
1.00	0.50	0.05	124.44	188.40	229.63	180.26	188.40	180.26
1.00	0.50	0.10	83.05	104.76	104.23	92.39	104.76	92.39
1.00	0.50	0.50	49.94	49.94	29.14	29.14	49.94	29.14
1.00	0.50	1.00	45.81	46.94	24.04	24.23	45.81	24.04
1.00	0.50	5.00	42.49	47.85	21.30	23.35	42.49	21.30
1.00	0.50	10.00	42.08	48.58	21.05	23.83	42.08	21.05
1.00	0.50	20.00	41.87	49.12	20.94	24.25	41.87	20.94
1.00	1.00	0.05	125.00	188.63	229.17	180.30	188.63	180.30
1.00	1.00	0.10	83.33	104.93	104.17	92.43	104.93	92.43
1.00	1.00	0.50	50.00	50.00	29.17	29.17	50.00	29.17
1.00	1.00	1.00	45.83	46.97	24.05	24.24	45.83	24.05
1.00	1.00	5.00	42.50	47.86	21.30	23.35	42.50	21.30
1.00	1.00	10.00	42.08	48.59	21.05	23.84	42.08	21.05
1.00	1.00	20.00	41.87	49.12	20.94	24.25	41.87	20.94
1.00	5.00	0.05	125.00	188.63	229.17	180.30	188.63	180.30
1.00	5.00	0.10	83.33	104.93	104.17	92.43	104.93	92.43
1.00	5.00	0.50	50.00	50.00	29.17	29.17	50.00	29.17
1.00	5.00	1.00	45.83	46.97	24.05	24.24	45.83	24.05
1.00	5.00	5.00	42.50	47.86	21.30	23.35	42.50	21.30
1.00	5.00	10.00	42.08	48.59	21.05	23.84	42.08	21.05
1.00	5.00	20.00	41.88	49.12	20.94	24.25	41.88	20.94
1.00	10.00	0.05	125.00	188.63	229.17	180.30	188.63	180.30
1.00	10.00	0.10	83.33	104.93	104.17	92.43	104.93	92.43
1.00	10.00	0.50	50.00	50.00	29.17	29.17	50.00	29.17
1.00	10.00	1.00	45.83	46.97	24.05	24.24	45.83	24.05
1.00	10.00	5.00	42.50	47.86	21.30	23.35	42.50	21.30
1.00	10.00	10.00	42.08	48.59	21.05	23.84	42.08	21.05
1.00	10.00	20.00	41.88	49.12	20.94	24.25	41.88	20.94

Table A2. Optimal base stock level and minimum cost per day for $\beta$= 0.50

$\beta$	$\lambda$	$\mu$	$S^{\ast \left\{1\right\}}$	$S^{\ast \left\{2\right\}}$	$E\left[TC{U}^{\left\{1\right\}}\left(S^{\ast \left\{1\right\}}\right)\right]$	$E\left[TC{U}^{\left\{2\right\}}\left(S^{\ast \left\{2\right\}}\right)\right]$	$S^{\ast }$	$E\left[TCU\left(S^{\ast }\right)\right]$
0.50	0.01	0.05	56.80	5.30	67.61	60.32	50.00	67.61
0.50	0.01	0.10	53.40	3.05	37.66	21.53	50.00	37.66
0.50	0.01	0.50	50.68	31.60	25.92	15.94	50.00	25.92
0.50	0.01	1.00	50.34	39.69	25.35	19.41	50.00	25.35
0.50	0.01	5.00	50.07	47.71	25.05	23.66	50.00	25.05
0.50	0.01	10.00	50.03	48.84	25.02	24.32	50.00	25.02
0.50	0.01	20.00	50.02	49.42	25.01	24.65	50.00	25.01
0.50	0.05	0.05	78.10	106.62	130.63	126.57	106.62	126.57
0.50	0.05	0.10	64.05	61.98	62.40	65.80	61.98	65.80
0.50	0.05	0.50	52.81	45.22	28.39	27.67	50.00	28.39
0.50	0.05	1.00	51.41	46.64	26.35	25.51	50.00	26.35
0.50	0.05	5.00	50.28	49.13	25.21	24.92	50.00	25.21
0.50	0.05	10.00	50.14	49.55	25.10	24.95	50.00	25.10
0.50	0.05	20.00	50.07	49.77	25.05	24.97	50.00	25.05
0.50	0.10	0.05	95.15	130.35	147.03	135.56	130.35	135.56
0.50	0.10	0.10	72.58	77.78	72.17	73.73	77.78	73.73
0.50	0.10	0.50	54.52	49.53	29.98	30.60	50.00	29.98
0.50	0.10	1.00	52.26	48.90	27.06	27.12	50.00	27.06
0.50	0.10	5.00	50.45	49.60	25.34	25.27	50.00	25.34
0.50	0.10	10.00	50.23	49.78	25.16	25.12	50.00	25.16
0.50	0.10	20.00	50.11	49.89	25.08	25.06	50.00	25.08
0.50	0.50	0.05	120.95	147.86	148.93	139.71	147.86	139.71
0.50	0.50	0.10	85.47	90.38	77.65	78.00	90.38	78.00
0.50	0.50	0.50	57.09	53.43	31.82	32.63	53.43	32.63
0.50	0.50	1.00	53.55	50.99	27.99	28.28	50.99	28.28
0.50	0.50	5.00	50.71	50.04	25.52	25.53	50.04	25.53
0.50	0.50	10.00	50.35	50.01	25.25	25.26	50.01	25.26
0.50	0.50	20.00	50.18	50.00	25.13	25.13	50.00	25.13
0.50	1.00	0.05	121.43	148.08	148.81	139.75	148.08	139.75
0.50	1.00	0.10	85.71	90.55	77.68	78.05	90.55	78.05
0.50	1.00	0.50	57.14	53.48	31.85	32.65	53.48	32.65
0.50	1.00	1.00	53.57	51.02	28.00	28.30	51.02	28.30
0.50	1.00	5.00	50.71	50.05	25.52	25.54	50.05	25.54
0.50	1.00	10.00	50.36	50.01	25.26	25.26	50.01	25.26
0.50	1.00	20.00	50.18	50.00	25.13	25.13	50.00	25.13
0.50	5.00	0.05	121.43	148.08	148.81	139.75	148.08	139.75
0.50	5.00	0.10	85.71	90.55	77.68	78.05	90.55	78.05
0.50	5.00	0.50	57.14	53.48	31.85	32.65	53.48	32.65
0.50	5.00	1.00	53.57	51.02	28.00	28.30	51.02	28.30
0.50	5.00	5.00	50.71	50.05	25.52	25.54	50.05	25.54
0.50	5.00	10.00	50.36	50.01	25.26	25.26	50.01	25.26
0.50	5.00	20.00	50.18	50.00	25.13	25.13	50.00	25.13
0.50	10.00	0.05	121.43	148.08	148.81	139.75	148.08	139.75
0.50	10.00	0.10	85.71	90.55	77.68	78.05	90.55	78.05
0.50	10.00	0.50	57.14	53.48	31.85	32.65	53.48	32.65
0.50	10.00	1.00	53.57	51.02	28.00	28.30	51.02	28.30
0.50	10.00	5.00	50.71	50.05	25.52	25.54	50.05	25.54
0.50	10.00	10.00	50.36	50.01	25.26	25.26	50.01	25.26
0.50	10.00	20.00	50.18	50.00	25.13	25.13	50.00	25.13

Table A3. Optimal base stock level and minimum cost per day for $\beta$= 0.10

$\beta$	$\lambda$	$\mu$	$S^{\ast \left\{1\right\}}$	$S^{\ast \left\{2\right\}}$	$E\left[TC{U}^{\left\{1\right\}}\left(S^{\ast \left\{1\right\}}\right)\right]$	$E\left[TC{U}^{\left\{2\right\}}\left(S^{\ast \left\{2\right\}}\right)\right]$	$S^{\ast }$	$E\left[TCU\left(S^{\ast }\right)\right]$
0.10	0.01	0.05	79.84	−45.78	32.17	9.24	50.00	32.17
0.10	0.01	0.10	78.25	−12.46	21.89	6.03	50.00	21.89
0.10	0.01	0.50	76.98	33.71	15.59	18.06	50.00	15.59
0.10	0.01	1.00	76.83	41.53	14.94	21.25	50.00	14.94
0.10	0.01	5.00	76.70	48.25	14.45	24.20	50.00	14.45
0.10	0.01	10.00	76.68	49.12	14.39	24.60	50.00	14.39
0.10	0.01	20.00	76.67	49.56	14.36	24.80	50.00	14.36
0.10	0.05	0.05	89.78	55.54	62.36	75.49	55.54	75.49
0.10	0.05	0.10	83.22	46.47	38.53	50.29	50.00	38.53
0.10	0.05	0.50	77.98	47.33	19.22	29.78	50.00	19.22
0.10	0.05	1.00	77.32	48.48	16.78	27.35	50.00	16.78
0.10	0.05	5.00	76.80	49.66	14.82	25.46	50.00	14.82
0.10	0.05	10.00	76.73	49.83	14.58	25.23	50.00	14.58
0.10	0.05	20.00	76.70	49.91	14.46	25.11	50.00	14.46
0.10	0.10	0.05	97.74	79.27	74.11	84.48	79.27	84.48
0.10	0.10	0.10	87.20	62.27	47.14	58.22	62.27	58.22
0.10	0.10	0.50	78.77	51.64	21.83	32.71	51.64	32.71
0.10	0.10	1.00	77.72	50.74	18.17	28.96	50.74	28.96
0.10	0.10	5.00	76.88	50.13	15.12	25.81	50.13	25.81
0.10	0.10	10.00	76.77	50.07	14.73	25.41	50.07	25.41
0.10	0.10	20.00	76.72	50.03	14.53	25.20	50.03	25.20
0.10	0.50	0.05	109.78	96.77	83.55	88.63	96.77	88.63
0.10	0.50	0.10	93.22	74.87	55.79	62.50	74.87	62.50
0.10	0.50	0.50	79.98	55.54	25.36	34.74	55.54	34.74
0.10	0.50	1.00	78.32	52.83	20.15	30.12	52.83	30.12
0.10	0.50	5.00	77.00	50.58	15.56	26.07	50.58	26.07
0.10	0.50	10.00	76.83	50.29	14.95	25.54	50.29	25.54
0.10	0.50	20.00	76.75	50.15	14.64	25.27	50.15	25.27
0.10	1.00	0.05	110.00	97.00	83.67	88.67	97.00	88.67
0.10	1.00	0.10	93.33	75.04	55.92	62.54	75.04	62.54
0.10	1.00	0.50	80.00	55.60	25.42	34.76	55.60	34.76
0.10	1.00	1.00	78.33	52.86	20.19	30.14	52.86	30.14
0.10	1.00	5.00	77.00	50.58	15.56	26.07	50.58	26.07
0.10	1.00	10.00	76.83	50.29	14.95	25.54	50.29	25.54
0.10	1.00	20.00	76.75	50.15	14.64	25.27	50.15	25.27
0.10	5.00	0.05	110.00	97.00	83.67	88.67	97.00	88.67
0.10	5.00	0.10	93.33	75.04	55.92	62.54	75.04	62.54
0.10	5.00	0.50	80.00	55.60	25.42	34.76	55.60	34.76
0.10	5.00	1.00	78.33	52.86	20.19	30.14	52.86	30.14
0.10	5.00	5.00	77.00	50.58	15.56	26.07	50.58	26.07
0.10	5.00	10.00	76.83	50.29	14.95	25.54	50.29	25.54
0.10	5.00	20.00	76.75	50.15	14.64	25.27	50.15	25.27
0.10	10.00	0.05	110.00	97.00	83.67	88.67	97.00	88.67
0.10	10.00	0.10	93.33	75.04	55.92	62.54	75.04	62.54
0.10	10.00	0.50	80.00	55.60	25.42	34.76	55.60	34.76
0.10	10.00	1.00	78.33	52.86	20.19	30.14	52.86	30.14
0.10	10.00	5.00	77.00	50.58	15.56	26.07	50.58	26.07
0.10	10.00	10.00	76.83	50.29	14.95	25.54	50.29	25.54
0.10	10.00	20.00	76.75	50.15	14.64	25.27	50.15	25.27

Table A4. Optimal base stock level and minimum cost per day for $\beta$= 0.00

$\beta$	$\lambda$	$\mu$	$S^{\ast \left\{1\right\}}$	$S^{\ast \left\{2\right\}}$	$E\left[TC{U}^{\left\{1\right\}}\left(S^{\ast \left\{1\right\}}\right)\right]$	$E\left[TC{U}^{\left\{2\right\}}\left(S^{\ast \left\{2\right\}}\right)\right]$	$S^{\ast }$	$E\left[TCU\left(S^{\ast }\right)\right]$
0.00	0.01	0.05	100.00	−64.01	15.99	−9.00	50.00	15.99
0.00	0.01	0.10	100.00	−17.22	8.69	1.26	50.00	8.69
0.00	0.01	0.50	100.00	34.17	1.87	18.52	50.00	1.87
0.00	0.01	1.00	100.00	41.90	0.94	21.62	50.00	0.94
0.00	0.01	5.00	100.00	48.35	0.19	24.30	50.00	0.19
0.00	0.01	10.00	100.00	49.17	0.10	24.65	50.00	0.10
0.00	0.01	20.00	100.00	49.59	0.05	24.82	50.00	0.05
0.00	0.05	0.05	100.00	37.30	44.04	57.26	50.00	44.04
0.00	0.05	0.10	100.00	41.70	28.24	45.53	50.00	28.24
0.00	0.05	0.50	100.00	47.80	7.30	30.25	50.00	7.30
0.00	0.05	1.00	100.00	48.85	3.79	27.72	50.00	3.79
0.00	0.05	5.00	100.00	49.76	0.78	25.56	50.00	0.78
0.00	0.05	10.00	100.00	49.88	0.39	25.28	50.00	0.39
0.00	0.05	20.00	100.00	49.94	0.20	25.14	50.00	0.20
0.00	0.10	0.05	100.00	61.04	55.84	66.24	61.04	66.24
0.00	0.10	0.10	100.00	57.50	38.73	53.46	57.50	53.46
0.00	0.10	0.50	100.00	52.11	11.22	33.18	52.11	33.18
0.00	0.10	1.00	100.00	51.11	5.95	29.33	51.11	29.33
0.00	0.10	5.00	100.00	50.23	1.25	25.91	50.23	25.91
0.00	0.10	10.00	100.00	50.12	0.63	25.46	50.12	25.46
0.00	0.10	20.00	100.00	50.06	0.32	25.23	50.06	25.23
0.00	0.50	0.05	100.00	78.54	66.52	70.40	78.54	70.40
0.00	0.50	0.10	100.00	70.10	49.83	57.73	70.10	57.73
0.00	0.50	0.50	100.00	56.00	16.57	35.20	56.00	35.20
0.00	0.50	1.00	100.00	53.20	9.04	30.49	53.20	30.49
0.00	0.50	5.00	100.00	50.68	1.95	26.17	50.68	26.17
0.00	0.50	10.00	100.00	50.34	0.98	25.59	50.34	25.59
0.00	0.50	20.00	100.00	50.17	0.49	25.30	50.17	25.30
0.00	1.00	0.05	100.00	78.77	66.67	70.43	78.77	70.43
0.00	1.00	0.10	100.00	70.27	50.00	57.77	70.27	57.77
0.00	1.00	0.50	100.00	56.06	16.67	35.23	56.06	35.23
0.00	1.00	1.00	100.00	53.23	9.09	30.51	53.23	30.51
0.00	1.00	5.00	100.00	50.68	1.96	26.17	50.68	26.17
0.00	1.00	10.00	100.00	50.34	0.99	25.59	50.34	25.59
0.00	1.00	20.00	100.00	50.17	0.50	25.30	50.17	25.30
0.00	5.00	0.05	100.00	78.77	66.67	70.43	78.77	70.43
0.00	5.00	0.10	100.00	70.27	50.00	57.77	70.27	57.77
0.00	5.00	0.50	100.00	56.06	16.67	35.23	56.06	35.23
0.00	5.00	1.00	100.00	53.23	9.09	30.51	53.23	30.51
0.00	5.00	5.00	100.00	50.68	1.96	26.17	50.68	26.17
0.00	5.00	10.00	100.00	50.34	0.99	25.59	50.34	25.59
0.00	5.00	20.00	100.00	50.17	0.50	25.30	50.17	25.30
0.00	10.00	0.05	100.00	78.77	66.67	70.43	78.77	70.43
0.00	10.00	0.10	100.00	70.27	50.00	57.77	70.27	57.77
0.00	10.00	0.50	100.00	56.06	16.67	35.23	56.06	35.23
0.00	10.00	1.00	100.00	53.23	9.09	30.51	53.23	30.51
0.00	10.00	5.00	100.00	50.68	1.96	26.17	50.68	26.17
0.00	10.00	10.00	100.00	50.34	0.99	25.59	50.34	25.59
0.00	10.00	20.00	100.00	50.17	0.50	25.30	50.17	25.30

Figure B1. Optimal base stock level and minimum cost per time unit for $C_H$= 0.1.

Figure B2. Optimal base stock level and minimum cost per time unit for $C_H$= 1.

Figure B3. Optimal base stock level and minimum cost per time unit for $C_H$= 5.

Figure B4. Optimal base stock level and minimum cost per time unit for $C_H$= 10.

Figure B5. Optimal base stock level and minimum cost per time unit for $C_H$= 25.

Figure B6. Optimal base stock level and minimum cost per time unit for $C_H$= 50.

Parlar, M., Wang, Y., & Gerchak, Y. (1995). A periodic review inventory model with Markovian supply availability. International Journal of Production Economics, 42(2), 131–136. https://doi.org/10.1016/0925-5273(95)00115-8

 Web of Science ®Google Scholar

Güllü, R., Önol, E., & Erkip, N. (1999). Analysis of an inventory system under supply uncertainty. International Journal of Production Economics, 59(1–3), 377–385. https://doi.org/10.1016/s0925-5273(98)00024-3

 Web of Science ®Google Scholar

Li, Z., Xu, S. H., & Hayya, J. (2004). A periodic-review inventory system with supply interruptions. Probability in the Engineering and Informational Sciences, 18(1), 33–53. https://doi.org/10.1017/s0269964804181035

 Web of Science ®Google Scholar

Schmitt, A. J., Snyder, L. V., & Shen, Z. J. M. (2010). Inventory systems with stochastic demand and supply: Properties and approximations. European Journal of Operational Research, 206(2), 313–328. https://doi.org/10.1016/j.cor.2010.08.004

 Web of Science ®Google Scholar

Atasoy, B., Güllü, R., & Tan, T. (2012). Optimal inventory policies with non-stationary supply disruptions and advance supply information. Decision Support Systems, 53(2), 269–27. https://doi.org/10.1016/j.dss.2012.01.005

 Web of Science ®Google Scholar

Schmitt, A. J., & Snyder, L. V. (2012). Infinite-horizon models for inventory control under yield uncertainty and disruptions. Computers & Operations Research, 39(4), 850–862. https://doi.org/10.1016/j.cor.2010.08.004

 Web of Science ®Google Scholar

Lewis, B. M., Erera, A. L., Nowak, M. A., & Chelsea, C. W., III. (2013). Managing inventory in global supply chains facing port-of-entry disruption risks. Transportation Science, 47(2), 162–180. https://doi.org/10.1287/trsc.1120.0406

 Web of Science ®Google Scholar

Warsing, D. P., Jr, Wangwatcharakul, W., & King, R. E. (2013). Computing optimal base-stock levels for an inventory system with imperfect supply. Computers & Operations Research, 40(11), 2786–2800. https://doi.org/10.1016/j.cor.2013.04.001

 Web of Science ®Google Scholar

Ciarallo, F. W., & Niranjan, S. (2014). Properties of optimal order-up-to levels for the newsvendor problem with random capacity. International Journal of Advanced Operations Management, 6(4), 353–376. https://doi.org/10.1504/ijaom.2014.066830

 Google Scholar

Firouzi, F., Baglieri, E., & Jaber, M. Y. (2014). Two-product inventory management with fixed costs and supply uncertainty. Applied Mathematical Modelling, 38(23), 5635–5650. https://doi.org/10.1016/j.apm.2014.03.018

 Web of Science ®Google Scholar

Skouri, K., Konstantaras, I., Lagodimos, A. G., & Papachristos, S. (2014). An EOQ model with backorders and rejection of defective supply batches. International Journal of Production Economics, 155, 148–154. https://doi.org/10.1016/j.ijpe.2013.11.017

 Web of Science ®Google Scholar

Salehi, H., Taleizadeh, A. A., & Tavakkoli-Moghaddam, R. (2016). An EOQ model with random disruption and partial backordering. International Journal of Production Research, 54(9), 2600–2609. https://doi.org/10.1080/00207543.2015.1110634

 Web of Science ®Google Scholar

Taleizadeh, A. A. (2017). Lot‐sizing model with advance payment pricing and disruption in supply under planned partial backordering. International Transactions in Operational Research, 24(4), 783–800. https://doi.org/10.1111/itor.12297

 Web of Science ®Google Scholar

Taleizadeh, A. A., & Dehkordi, N. Z. (2017). Economic order quantity with partial backordering and sampling inspection. Journal of Industrial Engineering International, 13(3), 331–345. https://doi.org/10.1007/s40092-017-0188-8

 Google Scholar

Hsieh, C. C., & Putera, R. R. (2018). Mitigating supply disruption with ordering and supply restoration decisions. Computers & Industrial Engineering, 126, 681–690. https://doi.org/10.1016/j.cie.2018.10.025

 Web of Science ®Google Scholar

Konstantaras, I., Skouri, K., & Lagodimos, A. G. (2019). EOQ with independent endogenous supply disruptions. Omega, 83, 96–106. https://doi.org/10.1016/j.omega.2018.02.006

 Web of Science ®Google Scholar

Saithong, C., & Luong, H. T. (2019). Effect of supply disruption on inventory policy. European Journal of Industrial Engineering, 13(2), 178–212. https://doi.org/10.1504/ejie.2019.10020067

 Web of Science ®Google Scholar

De, S. K., & Mahata, G. C. (2020). A production inventory supply chain model with partial backordering and disruption under triangular linguistic dense fuzzy lock set approach. Soft Computing, 24(7), 5053–5069. https://doi.org/10.1007/s00500-019-04254-2

 Web of Science ®Google Scholar