An application of Kernel estimation method to the demand in two-stage inventory model

Figures & data

Table 1. Descriptive statistics of monthly DPUTs of illustrative product

mean	SD	IQR	0%	25%	50%	75%	100%	n
2,699.083	322.7705	438.25	2,115	2,477.25	2,647	2,915.5	3,331	500

Table 2. Modeling of DPUT

Distribution	PDF	CDF	Parameters
NO	$\mathit{\varphi }\left(\frac{y_t-{\mathit{\mu }}_t}{{\mathit{\sigma }}_{\text{NO}}}\right)\frac{1}{{\mathit{\sigma }}_{\text{NO}}}$	$\mathrm{\Phi }\left(\frac{y_t-{\mathit{\mu }}_t}{{\mathit{\sigma }}_{\text{NO}}}\right)$	${\mathit{\mu }}_t\in \mathbb{R},{\mathit{\sigma }}_{\text{NO}}>0$
GA	$\frac{y_t^{\frac{1}{{\mathit{\sigma }}_{\text{GA}}^2}-1}}{\mathrm{\Gamma }\left(\frac{1}{{\mathit{\sigma }}_{\text{GA}}^2}\right){\mathit{\sigma }}_{\text{GA}}^2{\mathit{\mu }}_t^{\frac{1}{{\mathit{\sigma }}_{\text{GA}}^2}}}$$exp\left(-\frac{y_t}{{\mathit{\sigma }}_{\text{GA}}^2{\mathit{\mu }}_t}\right)$	$\frac{1}{\mathrm{\Gamma }\left(\frac{1}{{\mathit{\sigma }}_{\text{GA}}^2}\right)}\mathit{\gamma }\left(\frac{1}{{\mathit{\sigma }}_{\text{GA}}^2},y_t/{\mathit{\sigma }}_{\text{GA}}^2{\mathit{\mu }}_t\right)$	${\mathit{\mu }}_t>0,{\mathit{\sigma }}_{\text{GA}}>0$

Table 3. Modeling of LTD

Distribution	PDF	CDF	Parameters
NOsum	$(\mathit{\varphi }\left(\frac{s-{\mathit{\mu }}_t{\mathit{\mu }}_k}{{\mathit{\sigma }}_{\text{NO}}\sqrt{(}{\mathit{\mu }}_k)}\right)\frac{1}{{\mathit{\sigma }}_{\text{NO}}\sqrt{(}{\mathit{\mu }}_k)}$	$\mathrm{\Phi }\left(\frac{s-{\mathit{\mu }}_t{\mathit{\mu }}_k}{{\mathit{\sigma }}_{\text{NO}}\sqrt{(}{\mathit{\mu }}_k)}\right)$	${\mathit{\mu }}_t{\mathit{\mu }}_k\in \mathbb{R},{\mathit{\sigma }}_{NO}\sqrt{(}{\mathit{\mu }}_k)>0$, ${\mathit{\mu }}_k\geqq 2$
GAsum	$\frac{s^{\frac{1}{{\mathit{\sigma }}_{\text{GA}}^2{\mathit{\mu }}_k}-1}}{\mathrm{\Gamma }\left(\frac{1}{{\mathit{\sigma }}_{\text{GA}}^2{\mathit{\mu }}_k}\right){\mathit{\sigma }}_{\text{GA}}^2{\mathit{\mu }}_k{({\mathit{\mu }}_k{\mathit{\mu }}_t)}^{\frac{1}{{\mathit{\sigma }}_{\text{GA}}^2{\mathit{\mu }}_k}}}$$exp\left(-\frac{s}{{\mathit{\sigma }}_{GA}^2{\mathit{\mu }}_t}\right)$	$\frac{1}{\mathrm{\Gamma }\left(\frac{1}{{\mathit{\sigma }}_{GA}^2}{\mathit{\mu }}_k\right)}\mathit{\gamma }\left(\frac{1}{{\mathit{\sigma }}_{\text{GA}}^2}{\mathit{\mu }}_k,s/{\mathit{\sigma }}_{\text{GA}}^2{\mathit{\mu }}_k{\mathit{\mu }}_t{\mathit{\mu }}_k\right)$	${\mathit{\mu }}_t{\mathit{\mu }}_k>0,{\mathit{\sigma }}_{\text{GA}}{\mathit{\mu }}_k>0$, ${\mathit{\mu }}_k\geqq 2$

Figure 1. Comparison of DPUT density modeling.

Figure 2. Comparison of LTD density modeling.

Table 4. Effect of modeling of DPUTs with PDF Kernel estimation, NO IID and GA IID on SP of continuous review with shortage in two stages

NO scenario	${\mathit{\omega }}^1$	${\mathit{\omega }}^2$	${\mathit{\omega }}^3$	${\mathit{\omega }}^4$	${\mathit{\omega }}^5$	${\mathit{\omega }}^6$	${\mathit{\omega }}^7$	${\mathit{\omega }}^8$	${\mathit{\omega }}^9$	${\mathit{\omega }}^{10}$	${\mathit{\omega }}^{11}$	${\mathit{\omega }}^{12}$
$Y_{t_{95}}$ (unit/month)	3,045	3,426	2,838	3,321	3,636	4,133	3,527	4,518	3,194	3,856	3,981	3,743
Weight (%)	8.1	12.5	4.9	12.3	10.9	3.3	11.4	0.3	11.9	7.0	6.1	11.3
$Q_{95}$ (unit)	1,721
$r_{95}$ (without unit)	1.75	1.54	1.88	1.59	1.44	1.24	1.49	1.12	1.66	1.35	1.30	1.39
$S(r_{95})$(unit shortage/cycle)	733	651	786	671	613	539	632	493	698	578	560	596
$T{C}_{95}$ (USD/year)	415.73
$Y_{t_5}$ (unit/month)	1,469	1,850	1,262	1,745	2,060	2,557	1,950	2,942	1,618	2,280	2,405	2,166
Weight (%)	8.1	12.5	4.9	12.3	10.9	3.3	11.4	0.3	11.9	7.0	6.1	11.3
$Q_5$ (unit)	507
$r_5$ (without unit)	1.16	0.86	1.39	0.93	0.74	0.50	0.80	0.36	1.03	0.62	0.57	0.68
$S(r_5)$ (unit shortage/cycle)	509	404	593	428	362	291	383	253	462	327	310	344
$T{C}_5$ (USD/year)	230.81
GA scenario	${\mathit{\omega }}^1$	${\mathit{\omega }}^2$	${\mathit{\omega }}^3$	${\mathit{\omega }}^4$	${\mathit{\omega }}^5$	${\mathit{\omega }}^6$	${\mathit{\omega }}^7$	${\mathit{\omega }}^8$	${\mathit{\omega }}^9$	${\mathit{\omega }}^{10}$	${\mathit{\omega }}^{11}$	${\mathit{\omega }}^{12}$
$Y_{t_{95}}$ (unit/month)	3,458	3,562	2,444	3,923	4,325	4,890	3,127	5,180	2,994	4,021	4,567	3,256
Weight (%)	8.1	12.5	4.9	12.3	10.9	3.3	11.4	0.3	11.9	7.0	6.1	11.3
$Q_{95}$ (unit)	1,932
$r_{95}$ (without unit)	2.57	2.84	1.12	1.34	2.01	0.98	2.01	0.91	1.86	1.52	1.67	1.87
$S(r_{95})$ (unit shortage/cycle)	753	676	805	694	637	544	636	497	721	602	582	615
$T{C}_{95}$ (USD/year)	444.92
$Y_{t_5}$ (unit/month)	1,507	1,872	1,285	1,766	2,083	2,570	1,973	2,965	1,641	2,303	2,428	2,189
Weight (%)	8.1	12.5	4.9	12.3	10.9	3.3	11.4	0.3	11.9	7.0	6.1	11.3
$Q_5$ (unit)	531
$r_5$ (without unit)	1.11	1.06	1.09	0.74	0.52	0.37	0.68	0.22	0.93	0.42	0.44	0.55
$S(r_5)$ (unit shortage/cycle)	520	421	614	443	373	303	400	275	489	351	331	376
$T{C}_5$ (USD/year)	233.21
PDF Kernel estimation scenario	${\mathit{\omega }}^1$	${\mathit{\omega }}^2$	${\mathit{\omega }}^3$	${\mathit{\omega }}^4$	${\mathit{\omega }}^5$	${\mathit{\omega }}^6$	${\mathit{\omega }}^7$	${\mathit{\omega }}^8$	${\mathit{\omega }}^9$	${\mathit{\omega }}^{10}$	${\mathit{\omega }}^{11}$	${\mathit{\omega }}^{12}$
$Y_{t_{95}}$ (unit/month)	2,990	2,970	3,047	2,907	2,885	2,858	2,856	2,862	2,848	2,900	3,103	2,924
Weight (%)	8.3	8.3	8.3	8.3	8.3	8.3	8.3	8.3	8.3	8.3	8.3	8.3
$Q_{95}$ (unit)	1,443
$r_{95}$ (without unit)	2.51	2.53	2.47	2.59	2.61	2.63	2.63	2.63	2.64	2.59	2.42	2.57
$S(r_{95})$ (unit shortage/cycle)	525	528	515	540	544	549	549	548	551	541	505	536
$T{C}_{95}$ (USD/year)	347.44
$Y_{t_5}$ (unit/month)	2,198	2,178	2,256	2,115	2,094	2,066	2,064	2,071	2,056	2,108	2,312	2,132
Weight (%)	8.3	8.3	8.3	8.3	8.3	8.3	8.3	8.3	8.3	8.3	8.3	8.3
$Q_5$ (unit)	1,077
$r_5$ (without unit)	1.89	1.91	1.84	1.97	1.99	2.02	2.02	2.01	2.03	1.98	1.80	1.95
$S(k_5)$(unit shortage/cycle)	397	400	387	412	417	422	423	421	424	414	377	409
$T{C}_5$ (USD/year)	258.3