Repetitive sampling inspection plan for cancer patients using exponentiated half-logistic distribution under indeterminacy

This piece of work deals with a time truncated sampling scheme for cancer patients using exponentiated half-logistic distribution (EHLD) based on indeterminacy. We have studied time truncated schemes like repetitive acceptance sampling plan (RASP) under indeterminacy. We have estimated the projected scheme parameters such as sample size and acceptance and rejection sample numbers for known indeterminacy parameters. In addition to the projected sampling scheme quantities, the corresponding tables are generated for various values of indeterminacy parameters. The results of a sampling scheme show that the average sample number (ASN) decreases as indeterminacy values increase. It leads that the indeterminacy parameter is played a crucial portrayal in ASN. A comparative study is carried out with existing sampling schemes based on indeterminacy and classical sampling schemes. The evaluated sampling schemes are exemplified with the help of cancer data. From tables and exemplification, we wind up that the projected RSP scheme under indeterminacy desired a smaller sample size than the existing schemes.

Exponentiated half-logistic distribution under indeterminacy.We will provide a brief summary of the EHLD.The EHLD was acquainted and contemplated quite comprehensively by 35 , further 36 studied for various acceptance sampling plans for this distribution.Suppose that f (t N ) = f (t L ) + f (t U )I N ; I N ǫ[I L , I U ] be a neutrosophic probability density function (npdf) with determinate part f (t L ) , indeterminate part f (t U )I N and indeterminacy period I N ǫ[I L , I U ] for more details refer 33 .Remember that t N ǫ[t L , t U ] be a neutrosophic random variable (NRV) follows the npdf.The npdf is the oversimplification of pdf under conventional statistics.The anticipated neutrosophic form of f (t N )ǫ f (t L ), f (t U ) turns to pdf under classical statistics when I L =0.Using this background, the npdf of the EHLD is outlined as under where σ and θ are scale and shape parameters, respectively.It is significant to note that the developed npdf of the EHLD is the oversimplification of pdf of the EHLD based on conventional statistics.The neutrosophic form of the npdf of the EHLD reduces to the EHLD when I L =0.The neutrosophic cumulative distribution function (ncdf) of the EHLD is given by The average lifetime of the NEHLD is given by Repetitive sampling plan under indeterminacy.The traditional RASP based on the truncated life test sampling scheme is initiated by 37 .The step-by-step procedure to adopt the repetitive acceptance sampling plan under indeterminacy is stated below: Step 1: From a lot choose a sample of size n.Conduct a life testing for these sample for a pre-specified time say t 0 .Indicate the average µ 0N and indeterminacy parameter I N ǫ[I L , I U ]. (1) www.nature.com/scientificreports/ Step 2: Accept H 0 : µ N = µ 0N if specified average quantity µ 0N is less than or equal to c 1 (i.e., µ 0N ≤ c 1 ).If specified average quantity µ 0N is more than c 2 (i.e., µ 0N > c 2 ) then we reject H 0 : µ N = µ 0N and conclude the test, where c 1 ≤ c 2 .
Step 3: If c 1 < µ 0N ≤ c 2 then go to Step 1 and do again the entire procedure.
The developed RASP based on above indeterminacy methodology is consists of n, c 1 , c 2 and I N , where I N ǫ[I L , I U ] is known as uncertainty level and it is predetermined.RASP is a generalization of SSP under uncer- tainty studied in "Comparative studies".The proposed RASP is reduced to a SSP under uncertainty when c 1 = c 2 .It is a convention to assume that t 0 = dµ 0 where d is the termination factor.The operating characteristic (OC) function would be obtained based on lot acceptance probability for more details refer 10 and it is defined as: where P a p N is the chance of accepting under H 0 : µ N = µ 0N whereas P r p N is the chance of rejecting at H 0 : µ N = µ 0N , these are obtained in the following expressions: where p N is the chance of rejecting H 0 : µ N = µ 0N and it is obtained from Eq. ( 2) and Eq. ( 3) and it is defined by Where ϑ = ln 1+2 −1/θ 1−2 −1/θ .Using Eqs. ( 5) and ( 6) the Eq. ( 4) becomes The researcher is paying attention to concern the developed scheme to test H 0 : µ N = µ 0N such that the chance of accepting H 0 : µ N = µ 0N while it is true ought to be more than 1 − α ( α is type-I ) for µ/µ 0 and the chance of accepting H 0 : µ N = µ 0N while it is wrong ought to be smaller than β (type-II error) for µ N /µ 0N = 1 .In producer opinion, the chance of approval should be greater than or equal to 1 − α at acceptable quality level (AQL), p 1N .In the same way, in consumer opinion the lot rejection chance ought to be less than or equal to β at limiting quality level (LQL),p 2N .The intended quantities would be obtained by solving the following two inequalities simultaneously.
where p 1N and p 2N are respectively given by and (4) Vol:.(1234567890) www.nature.com/scientificreports/ The estimated intended quantities of the developed scheme should be minimizing the average sample number (ASN) at AQL.The ASN of the developed sampling scheme in terms of fraction defective ( p N ) is given below: The intended quantities for the created method would therefore be determined by resolving the nonlinear programming problem for optimization shown below.

Comparative studies
This section's goal is to examine the projected RASP's effectiveness in relation to ASN.The average hypothesis may be examined more affordably the lower the ASN.If no uncertainty or indeterminacy is established while remembering the average value, note that the sampling plan developed is an oversimplification of the plan based on conventional statistics.When I N =0, the developed RSP becomes the on-hand sampling plan.In Tables 1, 2, 3, 4, 5 and 6 the first spell of column i.e. at I N = 0 is the plan parameter of the traditional or existing RASP.From the results from the tables, we would conclude that the ASN is large in traditional RASP as compared with the proposed RASP.For example, when α = 0.10, β = 0.25, µ N /µ 0N =1.3, θ=1.5 and d=0.5 from Table 1, it can be (13) ASN = n P a p N + P r p N . ( Table 1.The RASP parameter of EHLD when α = 0.10; θ = 1.5 and d = 0.50.seen that ASN=107.11 from the plan under classical statistics and ASN=95.82 for the projected RASP when I N = 0.05.Furthermore, when θ =1 the EHLD becomes a half-logistic distribution (HLD), we have constructed Tables 5 and 6 for half-logistic distribution for comparison purpose.Table 5 depicts that EHLD shows less ASN as compared with HLD.For example when α = 0.10, β = 0.10, µ N /µ 0N = 1.5, d = 0.5 and I N = 0.04 the Table 5 shows that the ASN is 100.99where as proposed plan values are ASN = 67.83for θ = 1.5 and ASN = 51.76 for θ = 2.0.From this study, it is concluded that the projected plan under indeterminacy is efficient over the existing RASP under traditional statistics with respect to sample size.We have also compared our proposed RASP under indeterminacy with SSP under indeterminacy developed by 38 .The results show that RASP is superior to the SSP for same specific parameters.For example when α = 0.10, β = 0.10, µ N /µ 0N = 1.4,d = 0.5, I N = 0.04 and θ = 1.5 the ASN in SSP is 105 whereas in RASP the ASN is 67.83.Operating characteristic (OC) curve of plan of the EHLD when α = 0.10, β = 0.10, θ = 2 , µ N /µ 0N = 1.3 and d = 0.5 is depicted in Fig. 1.From Fig. 1, we I U = 0.00 I U = 0.02 I U = 0.04  I U = 0.00 I U = 0.02 I U = 0.04 conclude that indeterminacy parameter shows significant effect on reduce the ASN.Therefore, the application of the proposed plan for testing the null hypothesis H 0 : µ N = µ 0N demands a lesser ASN as compared to the on hand plan.Moreover, the OC curve comparison between SSP and RASP is also displayed in Fig. 2. The OC curve in Fig. 2 also shows that RASP is superior to the SSP for the same specific parameters.The researchers advised as proposed RASP under uncertainty is more economical to apply in a medical study specifically for remission time of the patients due to melanoma cancer.

Applications of proposed plan for remission times of melanoma patients
The present section deals with the postulation of the developed sampling scheme for the EHLD under the indeterminacy obtained by means of a real paradigm.This data set is picked out from 39 and it constitutes the remission times, in months for 30 melanoma cancer patients at stages 2 to 4. For ready reference, the data is given below.I U = 0.00 I U = 0.02 I U = 0.04 Melanoma is a very dangerous kind of skin cancer which develops in the cells (melanocytes) that develop melanin and it creates the color change in the skin.
It is establish that the remission times of melanoma patients data comes from the EHLD with shape parameter θ = 1.4097 and scale parameter σ = 7.3811 and the maximum distance between the real time data and the fitted of EHLD is found from the Kolmogorov-Smirnov test as 0.1324 and also the p-value is 0.6687.The demonstration of the goodness of fit for the given model is shown in Fig. 3, the empirical and theoretical cdfs and Q-Q plots for the EHLD for the remission times of melanoma patients' data.In Tables 7 and 8 presented the plan quantities for the fitted shape parameter.It is assumed there is indeterminacy in measuring remission time and let it is 0.05.The measurements of remission time for cancer patients with respect to interval measures and fuzzy-type Table 6.The RASP parameter of EHLD when α = 0.10; θ = 1.0 and d = 1.00.data sets were studied by various authors, for instance, refer [40][41][42] .For the proposed plan, the shape parameter is θ N = (1 + 0.05) × 1.4097 ≈ 1.4802 when I U = 0.05.Assume that a medical researcher would like to employ the developed RSP for EHLD under indeterminacy to guarantee the remission time of melanoma cancer patients is at least 6 months using the truncated life test for 3 months (thus d = 0.5).Suppose that medical researchers are paying attention to test H 0 : µ N = 4.7893 with the support of the developed RASP when I U = 0.05,α = 0.10 , µ N /µ 0N = 1.5, d = 0.5 and β = 0.10.From Table 7, it can be noted that n = 40, c 1 = 5, c 2 = 9 and ASN = 68.35.Thus, the RASP for EHLD under indeterminacy could be enforced in the following way: picking out a random sample of 40 melanoma cancer patients from the indoor group of patients, and conducting the truncated life test of remission time for 3 months.The developed RASP scheme could be adopted in the following way: hypothesis H 0 : µ N = 4.7893 will be accepted if the average remission time of melanoma cancer patients in 6 months is  less than five patients, but a lot of patients should be rejected as soon as the remission time of melanoma cancer patients exceeds nine patients.Contrary, the experimentation could be repeated.From remission time data shows that seven patients before the average remission time of melanoma cancer patients of 4.7893.Therefore, the medical practitioners would have to repeat the entire procedure until accept/reject the hypothesis.Accordingly, it is competent that the developed sampling will be taken into consideration to check the typical length of remission for melanoma cancer patients based on the real application.

Conclusions
In order to design an exponentiated half-logistic distribution based on indeterminacy for a time-truncated repetitive sampling strategy, a thorough investigation of melanoma cancer patients was conducted.The sample scheme parameters are determined for the identified values of the indeterminacy parameters.For simple reference, we    www.nature.com/scientificreports/have given lengthy tables including the values of the known indeterminacy constants.The developed sampling strategy is compared to the available conventional statistical strategies.The results show that the designed sampling plan is more cost-effective than the on-hand SSP under indeterminacy and conventional sampling plans.Furthermore, the proposed RASP under indeterminacy is more cost effective than the single sample strategy.It is also noticed that indeterminacy values play a vital role in ASN, when the indeterminacy quantities increase at that time the ASN quantity is decreased.Hence, the proposed sample strategies are convenient for researchers, particularly in medical experimentation, because medical experimentation requires more costly and qualified specialists.As a result, the created sampling strategy under indeterminacy is required to be valid for testing the average number of melanoma cancer patients.The real examples based on the melanoma cancer patients for developed sampling scheme under indeterminacy show a piece of evidence.The suggested sampling strategy for big data analytics could be applied to various scientific and technical disciplines.The next step in the research would be to develop multiple dependent state sampling plans and multiple dependent state repeating sampling plans for different lifetime distributions.

Figure 1 .
Figure 1.OC curve plan at different indeterminacy values.

Figure 2 .
Figure 2. OC curves comparison between SSP and RASP under indeterminacy.

Figure 3 .
Figure 3.The empirical and theoretical pdf and Q-Q plots for the EHLD for the remission times of melanoma patients.