A Review and Perspective on Neutrosophic Statistical Process Monitoring Methods

We review the literature on statistical process monitoring methods based on neutrosophic principles. We question some of the underlying assumptions and raise important questions about these and other neutrosophic statistical methods that need to be addressed before the methodology could be taken seriously.

Statistical process monitoring is based on samples of pro-23 cess data collected over time. Historical data are collected 24 in Phase I in order to understand the process variability, 25 to identify opportunities for process improvement, and, if sta- 26 bility is achieved, to fit a statistical model. Important Phase I 27 issues were discussed by Jones-Farmer et al. [13]. Data are 28 collected sequentially in Phase II in order to detect changes 29 from the baseline model fitted in Phase I. Only Phase II 30 methods are considered in our paper. 31 The associate editor coordinating the review of this manuscript and approving it for publication was Derek Abbott .
It is assumed in NSPM that the observed data are interval-32 valued. Typically, the sample sizes and the control chart 33 design constants are also assumed to be interval-valued. 34 It is often assumed that one is sampling from a particu-35 lar neutrosophic probability distribution, i.e., a distribution 36 for which the parameter, or parameters, are interval-valued. 37 Many neutrosophic control charts have been proposed. After 38 our review of these methods, we question some of the 39 underlying assumptions and raise important questions about 40 these and other neutrosophic statistical methods that need 41 to be addressed before the methodology could be taken 42 seriously. 43 Neutrosophic control charts for attribute data have 44 been proposed by Aslam [14], Aslam et al. [15], Albassam 45 and Aslam [3], and Aslam et al. [16]. These methods are 46 designed for monitoring proportions based on sampling from 47 a binomial distribution where the sample sizes, counts of 48 defective items and possibly the probability of a defective 49 item are all imprecisely known and represented by inter-50 vals. Aslam [14] modified the basic neutrosophic approach 51 of Aslam et al. [15] to allow repetitive sampling, i.e., addi-52 tional samples if the neutrosophic control chart statistic 53 at a given time was not outside the neutrosophic con-54 trol limits or sufficiently close to the neutrosophic cen-55 terline. Albassam and Aslam [3] incorporated runs rules 56 into the neutrosophic p-chart while Aslam et al. [16] used 57 an exponentially weighted moving average (EWMA) chart 58 approach.
represent the neutrosophic scale parameter, the neutrosophic 117 density function (PDF N ) and cumulative distribution func-118 tion (CDF N ) of the RD N . Based on the neutrosophic version 119 of the Rayleigh distribution, the mean and variance of the 120 neutrosophic random variable Z were given respectively as The graphical displays of f(Z , θ N ), and F(Z , θ N ) for the 123 neutrosophic Rayleigh random variable Z with imprecise 124 scale parameter θ N = [0.5, 0.75] are shown in Fig. 1, which 125 was reproduced from Khan et al. [2].

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From the general description, it seems that the underly-127 ing distribution could be any Rayleigh distribution with a 128 parameter in the interval [θ l , θ u ]. In constructing their control 129 chart, Khan et al. [2] proposed neutrosophic estimators of the 130 in-control parameter value, an estimator of the parameter at 131 each sampling point, and then derived their neutrosophic dis-132 tributions. Probability-based control limits were based on an 133 underlying neutrosophic chi-distribution. In the neutrosophic 134 approach, there are intervals for each control limit, on the 135 center line of the control chart and for the plotted statistics. 136 Their method was justified using neutrosophic power curves 137 and neutrosophic average run length (ARL) profiles, where 138 the ARL is the expected number of samples until a con-139 trol chart signal is given. An ARL curve reproduced from 140 Khan et al.
[2] is shown in Fig.2 where α is the probability 141 of a false alarm and m represents the sample size.

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The neutrosophic Shewhart control chart resulting from the 143 data in their example is shown in Fig. 3. The control limits and 144 chart statistic are represented by intervals at each sampling 145 time.

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In our view there are some important issues to be addressed 148 with respect to the neutrosophic methodology. Some of our 149 questions apply to neutrosophic statistical methods in gen-150 eral while other questions and concerns are more focused 151 on NSPM issues. We have the following comments and 152 questions: It is imperative to clarify the rules of neutrosophic arithmetic 155 and extensions of the basic neutrosophic distribution theory 156 discussed by Alhabib et al. [40]. It is not clear, for exam-157 ple, how to add an indeterminate number of neutrosophic 158 numbers.

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According to Zhang et al. [41], the neutrosophic rules of 160 arithmetic for interval-valued data are the same as those 161 VOLUME 10, 2022 As an example, consider the two neutrosophic numbers 194 [4, 6] and [2, 4] represented as 4+2I and 4−2I , respectively. 195 Then using the approach of Smarandache [46], the average of 196 these two neutrosophic numbers would be 4 + 0I , or simply 197 the precise value 4. This result does not seem reasonable. 198 Using the approach of Zhang et al. [41] and interval arith-199 metic, however, the interval for the average would be [3,5]. 200 We consider the interval arithmetic approach to lead to the 201 much more useful and realistic results. As a reviewer pointed 202 out, the neutrosophic approach would also yield an average 203 of [3, 5] under the restriction that the coefficient of I is 204 non-negative. 205 is typically assumed that the sample sizes are not known pre-241 cisely even after the sample is collected. With NSPM methods 242 the level of imprecision in the sample sizes is assumed to be 243 maintained over time. We know of no application in quality 244 control, however, where the sample size would not be known 245 precisely after the sample is collected. The examples involv-246 ing imprecise sample sizes given in Smarandache [46] all 247 involve attribute data without carefully expressed operational 248 definitions. It seems impossible to have a sample of variables 249 data without knowing the sample size. , 255 causes their control limits to be 29% and 41% wider at the 256 lower sample size compared to the width at the higher sample 257 size, respectively. The width of control limits tends to be 258 inversely proportional to the square root of the sample size. 259 Note, however, that in all the case study examples involving 260 variables data presented in the neutrosophic literature, the 261 sample sizes are assumed to be known precisely.

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The use of neutrosophic sample sizes is made more 263 explicit in the neutrosophic acceptance sampling literature. 264 Aslam [10] considered an example with n N = [26,34] 265 and simply chose the sample size to be within this interval. 266 In particular, he selected n = 28. In this situation, however, 267 the sample size is known and the neutrosophic approach to the 268 sample size would not be needed. This approach of selecting 269 a sample size to be some value in the interval was also used 270 by Aslam [11]. If the sample size is randomly or arbitrarily 271 selected to be a fixed value within the neutrosophic interval, 272 there is no reason to base the further calculations on the 273 neutrosophic sample size interval or to use the neutrosophic 274 representation of the sample size at all.

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Sample sizes in statistical process monitoring applications 276 often vary, especially when monitoring proportions, but the 277 sample sizes are always assumed to be known. There are stan-278 dard control charting methods for handling variable sample 279 sizes, as described, for example, by Montgomery [50].

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The control chart limit multiplier with the various neutro-282 sophic Shewhart control charts is always considered to be a 283 neutrosophic number. With the neutrosophic EWMA charts 284 of Aslam et al. [16], [25], [28], both the smoothing parame-285 ter and the control limit multiplier are neutrosophic numbers. 286 Aslam et al.
[23] considered the span of a moving average 287 used in a moving average chart to be unknown precisely. 288 Since the EWMA smoothing parameter and the moving aver-289 age span are both selected by the practitioner, these values 290 would be known precisely. Thus it is unclear why they are 291 represented as neutrosophic numbers. We see no advantage 292 in considering the control limit multiplier to be imprecise. 293 Representing control chart limit multiplier and other con-294 trol chart design constants as neutrosophic numbers adds 295 VOLUME 10, 2022 unnecessary imprecision to the control chart limits and the 296 control chart statistics.  Knoth et al. [62] showed that the double and triple EWMA 348 chart approach taken by Shafqat et al. [26] should not be 349 used. These charts give much greater weight to some past data 350 values than to recent data values, resulting in poor detection 351 capability for delayed shifts in the process. Khan et al. [63] 352 recommended using a moving average statistic as input to 353 an EWMA chart. Knoth et al. [62] showed that this added 354 complication of using one control chart statistic as input in to 355 another control chart statistic has weak justification at best. 356 In the repetitive sampling generalizations proposed by 357 Aslam [14], [22], [29], and any other methods that allow 358 multiple samples at each time point, the fixed sample size 359 chart competitor should have a sample size set equal to the 360 expected sample size under the repetitive sampling. This has 361 not been done, so the ARL comparisons have been biased 362 toward favoring the repetitive sampling method since it is 363 based on more data. The rules for implementing repetitive 364 sampling, however, are not explained clearly.

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In our view the justification for NSPM methods, as currently 382 proposed, is very weak at best. This is due in part to the lack 383 of explanations of the mathematical details. Other issues need 384 to be addressed. The sample sizes and control chart design 385 constants will be known precisely in practice. Representing 386 them as neutrosophic numbers in NSPM adds unnecessary 387 imprecision to the control chart statistics and the control 388 limits. If only the data themselves are interval-valued, then 389 we see value in obtaining conservative bounds on the control 390 chart statistics and control limits using interval statistical 391 methods.

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The neutrosophic arithmetic, distribution theory, and 393 NSPM methods need to be fully explained so that results 394 can be reproduced. Once this is done, we think it would be 395 useful to compare the neutrosophic control chart methods for 396 variables data to standard methods based on the midpoints 397 of the data intervals. It would be interesting to investigate 398 how much of the neutrosophic control limit uncertainty is due 399 solely to the uncertainty in the data values.