Designing of multiple dependent state sampling plan for Type-II generalized half logistic distribution

Figures & data

Table 1. Best parameters of the proposed MDS plan for TGHLD with $\mathit{\theta }$ = 1.5

$\mathit{\beta }$	$t_q/t_q^0$	$\mathit{a}=0.5$					$\mathit{a}=0.7$					$\mathit{a}=1.0$
		n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$
0.25	2	31	6	11	2	0.9633	21	5	9	1	0.9510	14	5	13	2	0.9543
	4	10	1	9	2	0.9538	7	1	6	2	0.9584	5	1	2	1	0.9583
	6	9	1	3	4	0.9835	7	1	6	2	0.9890	5	1	2	1	0.9875
	8	6	0	2	1	0.9599	4	0	1	1	0.9533	3	0	2	1	0.9595
	10	5	0	4	2	0.9651	4	0	1	1	0.9689	3	0	2	1	0.9732
0.10	2	50	9	19	2	0.9517	36	9	19	2	0.9578	25	8	15	1	0.9500
	4	18	2	6	2	0.9658	13	2	3	1	0.9524	9	2	8	2	0.9714
	6	13	1	5	2	0.9697	9	1	2	2	0.9590	7	1	3	1	0.9789
	8	13	1	5	2	0.9882	9	1	2	2	0.9824	7	1	3	1	0.9920
	10	13	1	5	2	0.9945	6	0	2	1	0.9508	7	1	3	1	0.9964
0.05	2	-	-	-	-	-	-	-	-	-	-	31	10	14	1	0.9501
	4	26	3	8	2	0.9759	16	2	5	1	0.9529	11	2	5	1	0.9599
	6	17	1	4	1	0.9607	12	1	4	1	0.9631	8	1	3	1	0.9665
	8	16	1	2	1	0.9708	11	1	10	2	0.9796	8	1	3	1	0.9870
	10	16	1	2	1	0.9846	11	1	10	2	0.9903	8	1	3	1	0.9940
0.01	2	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-
	4	39	4	7	1	0.9738	28	4	8	1	0.9787	17	3	8	1	0.9546
	6	28	2	5	1	0.9758	20	2	5	1	0.9770	14	2	6	1	0.9795
	8	22	1	3	1	0.9585	16	1	5	1	0.9629	11	1	5	1	0.9658
	10	22	1	3	1	0.9795	15	1	2	1	0.9668	11	1	5	1	0.9832

Table 2. Best parameters of the proposed MDS plan for TGHLD with $\mathit{\theta }$ = 2

$\mathit{\beta }$	$t_q/t_q^0$	$\mathit{a}=0.5$					$\mathit{a}=0.7$					$\mathit{a}=1.0$
		n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$
0.25	2	30	6	16	3	0.9533	22	6	8	2	0.9529	16	6	8	2	0.9588
	4	11	1	4	1	0.9621	7	1	6	2	0.9549	5	1	2	1	0.9554
	6	9	1	4	3	0.9862	7	1	6	2	0.9879	5	1	2	1	0.9864
	8	6	0	2	1	0.9575	4	0	1	1	0.9506	3	0	2	1	0.9573
	10	6	0	2	1	0.9718	3	0	2	4	0.9518	3	0	2	1	0.9716
0.10	2	49	9	16	2	0.9505	36	9	19	2	0.9500	26	9	19	2	0.9547
	4	18	2	12	2	0.9616	13	2	12	2	0.9625	9	2	8	2	0.9683
	6	13	1	11	2	0.96	9	1	3	2	0.9693	7	1	3	1	0.9769
	8	13	1	11	2	0.9868	9	1	3	2	0.9881	7	1	3	1	0.9912
	10	13	1	11	2	0.9939	9	1	3	2	0.9945	7	1	3	1	0.9960
0.05	2	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-
	4	26	3	4	1	0.9505	16	2	6	1	0.9500	11	2	5	1	0.9558
	6	17	1	5	1	0.9577	12	1	4	1	0.9596	8	1	3	1	0.9635
	8	16	1	2	1	0.9681	11	1	2	1	0.9729	8	1	3	1	0.9856
	10	16	1	2	1	0.9830	11	1	2	1	0.9858	8	1	3	1	0.9933
0.01	2	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-
	4	38	4	14	2	0.9601	27	4	9	2	0.9646	19	4	14	2	0.9687
	6	27	2	12	2	0.9593	20	2	6	1	0.9752	14	2	6	1	0.9769
	8	22	1	4	1	0.9600	16	1	6	1	0.9594	11	1	5	1	0.9626
	10	21	1	5	2	0.9686	15	1	2	1	0.9638	11	1	5	1	0.9815

Table 3. Best parameters of the proposed MDS plan for TGHLD with $\mathit{\theta }$ = 2.5

$\mathit{\beta }$	$t_q/t_q^0$	$\mathit{a}=0.5$					$\mathit{a}=0.7$					$\mathit{a}=1.0$
		n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$
0.25	2	30	6	9	2	0.9564	22	6	9	2	0.9596	16	6	8	2	0.9559
	4	11	1	4	1	0.9597	7	1	6	2	0.9523	5	1	2	1	0.9533
	6	9	1	8	3	0.9852	7	1	2	1	0.9842	5	1	2	1	0.9856
	8	6	0	2	1	0.9559	7	1	2	1	0.9933	3	0	2	1	0.9557
	10	5	0	4	2	0.9616	4	0	1	1	0.9657	3	0	2	1	0.9706
0.10	2	51	9	14	1	0.9517	37	9	13	1	0.9523	26	9	19	2	0.9507
	4	18	2	12	2	0.9584	13	2	12	2	0.9596	9	2	8	2	0.9661
	6	13	1	2	1	0.9590	9	1	4	2	0.9690	7	1	3	1	0.9755
	8	13	1	2	1	0.9817	9	1	4	2	0.9879	7	1	3	1	0.9906
	10	13	1	2	1	0.9905	9	1	4	2	0.9944	7	1	3	1	0.9957
0.05	2	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-
	4	26	3	5	1	0.9736	19	3	5	1	0.9739	11	2	5	1	0.9529
	6	16	1	3	1	0.9568	12	1	5	1	0.9577	8	1	3	1	0.9613
	8	16	1	3	1	0.9826	11	1	2	1	0.9713	8	1	3	1	0.9846
	10	15	1	5	3	0.9842	11	1	2	1	0.9849	8	1	3	1	0.9928
0.01	2	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-
	4	38	4	7	1	0.9703	27	4	14	2	0.9609	19	4	14	2	0.9656
	6	27	2	4	1	0.9671	19	2	12	2	0.9604	14	2	6	1	0.9751
	8	22	1	5	1	0.9582	15	1	3	1	0.9591	11	1	5	1	0.9603
	10	21	1	2	1	0.9602	15	1	3	1	0.9798	11	1	5	1	0.9802

Table

0.014	0.034	0.059	0.061	0.069	0.080	0.123	0.142	0.165	0.210
0.381	0.464	0.479	0.556	0.574	0.839	0.917	0.969	0.991	1.064
1.088	1.091	1.174	1.270	1.355	1.397	1.477	1.578	1.649	1.702
1.893	1.932	2.001	2.161	2.292	2.326	2.337	2.628	2.785	2.811
2.993	3.122	3.248	3.715	3.790	3.857	3.912	4.100	4.106	4.116
4.315	4.510	4.580	5.267	5.299	5.583	6.065	0.9701

Figure 1. Histogram superimposed empirical density plots and Q-Q plot of the fitted TGHLD for cycles to failure for electrical appliances data.

Table 4. Best parameters of the proposed MDS plan for TGHLD with $\mathit{\theta }$ = 0.5749

$\mathit{\beta }$	$t_q/t_q^0$	$\mathit{a}=0.5$					$\mathit{a}=0.7$					$\mathit{a}=1.0$
		n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$
0.25	2	28	5	8	2	0.9565	20	5	15	2	0.9630	14	5	13	2	0.9676
	4	10	1	3	2	0.9666	7	1	6	2	0.9708	5	1	2	1	0.9684
	6	10	1	3	2	0.9916	7	1	6	2	0.9929	5	1	2	1	0.9911
	8	5	0	1	2	0.9526	4	0	1	1	0.9632	3	0	2	1	0.9677
	10	5	0	1	2	0.9684	4	0	1	1	0.9758	3	0	2	1	0.9790
0.10	2	48	8	18	2	0.9531	32	7	12	1	0.9517	21	7	11	2	0.9529
	4	19	2	6	2	0.9743	13	2	3	2	0.9601	9	2	8	2	0.9813
	6	14	1	11	2	0.9747	10	1	2	1	0.9705	7	1	3	1	0.9855
	8	14	1	11	2	0.9904	10	1	2	1	0.9874	7	1	3	1	0.9948
	10	9	0	2	1	0.9570	7	0	6	1	0.9504	4	0	1	1	0.9529
0.05	2	63	10	14	1	0.9557	41	9	13	1	0.9511	29	9	15	1	0.9581
	4	22	2	5	2	0.9505	16	2	4	1	0.9627	11	2	5	1	0.9732
	6	17	1	11	2	0.9531	12	1	3	1	0.9719	8	1	3	1	0.9767
	8	17	1	2	1	0.9764	12	1	3	1	0.9894	8	1	3	1	0.9914
	10	17	1	2	1	0.9877	12	1	3	1	0.9952	8	1	3	1	0.9962
0.01	2	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-
	4	36	3	7	1	0.9610	24	3	13	2	0.9510	17	3	8	1	0.9717
	6	29	2	3	1	0.9567	20	2	4	1	0.9826	14	2	6	1	0.9876
	8	23	1	3	1	0.9689	16	1	4	1	0.9751	11	1	5	1	0.9768
	10	23	1	3	1	0.9850	16	1	4	1	0.9881	11	1	5	1	0.9891

Table 5. ASN values of the proposed MDS plan, repetitive sampling plan (RSP) and single sampling plan (SSP) for the TGHLD when $\mathit{a}=0.5$

$\mathit{\beta }$	$t_q/t_q^0$	$\mathit{a}=1.5$			$\mathit{a}=2.0$			$\mathit{a}=2.5$
		MDS	RSP	SSP	MDS	RSP	SSP	MDS	RSP	SSP
0.25	2	31	43.53	50	30	39.92	54	30	42.96	49
	4	10	14.32	18	11	17.02	18	11	13.43	18
	6	9	8.98	14	9	8.93	14	9	8.89	14
	8	6	8.98	9	6	8.93	9	6	8.89	9
	10	5	8.98	9	6	8.93	9	5	8.89	9
0.10	2	50	58.32	79	49	59.94	82	51	57.74	81
	4	18	21.58	28	18	21.93	27	18	16.86	27
	6	13	16.42	23	13	16.32	23	13	16.26	22
	8	13	16.42	18	13	10.37	18	13	10.33	18
	10	13	11.13	18	13	10.37	18	13	10.33	18
0.05	2	-	66.39	–	-	65.86	–	-	16.85	–
	4	26	22.46	36	26	22.21	36	26	23.73	35
	6	17	17.06	26	17	16.83	26	16	16.68	26
	8	16	17.06	21	16	16.83	21	16	16.68	21
	10	16	11.82	21	16	16.83	21	15	11.71	21
0.01	2	-	79.17	–	-	78.53	–	-	81.91	–
	4	39	28.69	54	38	28.25	53	38	27.99	53
	6	28	23.32	39	27	22.99	38	27	22.78	38
	8	22	18.08	34	22	17.92	33	22	17.82	33
	10	22	18.08	28	21	17.92	27	21	17.82	27

Table 6. ASN values of the proposed MDS plan, repetitive sampling plan (RSP) and single sampling plan (SSP) for the TGHLD when $\mathit{a}=1.0$

$\mathit{\beta }$	$t_q/t_q^0$	$\mathit{a}=1.5$			$\mathit{a}=2.0$			$\mathit{a}=2.5$
		MDS	RSP	SSP	MDS	RSP	SSP	MDS	RSP	SSP
0.25	2	14	23.05	27	16	23.05	27	16	23.05	27
	4	5	7.84	10	5	10.24	10	5	10.00	10
	6	5	4.80	7	5	4.80	7	5	4.80	7
	8	3	4.80	5	3	4.80	5	3	4.80	5
	10	3	4.80	5	3	4.80	5	3	4.80	5
0.10	2	25	29.81	39	26	30.97	44	26	30.97	44
	4	9	9.41	14	9	11.76	14	9	11.76	14
	6	7	8.37	9	7	8.37	12	7	8.37	12
	8	7	5.33	9	7	5.33	9	7	5.23	9
	10	7	5.33	9	7	5.33	9	7	5.23	9
0.05	2	31	32.41	51	-	32.41	51	-	34.25	53
	4	11	11.96	18	11	11.75	18	11	11.76	18
	6	8	8.93	13	8	8.93	13	8	8.93	13
	8	8	8.93	11	8	8.93	11	8	8.93	11
	10	8	5.93	11	8	5.93	11	8	5.93	11
0.01	2	-	38.86	–	-	43.23	–	-	43.23	–
	4	17	14.83	27	19	14.83	27	19	14.83	27
	6	14	12.03	19	14	12.03	19	14	12.03	19
	8	11	9.31	17	11	9.31	17	11	9.31	17
	10	11	9.31	14	11	9.31	14	11	9.31	14

Figure 2. OC curves for comparison between MDS and SSP.

Figure 3. OC curves for comparison between MDS, RSP and SSP.