MULTIVARIATE CONTROL CHARTS BASED ON DATA DEPTH FOR SUBGROUP LOCATION AND SCALE-CASE STUDY

The purpose of the article is to present a method for determining control charts, which allow to control few interrelated quality characteristics. Often, in production practice, there is a need to simultaneously control several interrelated quality characteristics. The use of univariate control charts, separately for each quality characteristics, may lead to inadequate corrective actions of a production process. In such situations, it is recommended to use multivariate control charts, for example, the T2 control chart. However, the use of this classic approach involves making complicated calculations. Therefore, the author of this paper suggests using multivariate control charts based on data depth proposed by Liu. In this paper, the author presented the idea and principles of the multivariate control chart based on data depth and then, using it to assess the statistical stability of the process in a manufacturing company, engaged in the production of window fittings. UDC Classification: 338.3; DOI: http://dx.doi.org/10.12955/cbup.v6.1292

Introduction chart, using a spatial rank and a VS-MEWMA control chart, to reduce the amount of data from the process control state, due to which the use of a simulation (when in-control process state distribution parameters are unknown), to define control boundaries is not required. Due to the assumptions and limitations of the above-mentioned multivariate control charts, the author decided to use the one Q proposed by Liu (1995) and discussed this in more detail in the next chapter. Mahalanobis data depth in the construction of control charts for subgrouped data A multivariate control chart, based on Mahalanobis data depth, was proposed by Liu (1995). The idea of this control chart is primarily to reduce multivariate qualitative measurements to a univariate index, determined by rank statistics called the depth measure. According to Liu, Parelius and Singh (1999), the depth measure is a way of evaluating the square distance of a given point ∈ ℝ (where ≥ 1the number of KPCs tested) or a sample { 1 , 2 , … , } with respect to the average continuous probability distribution F (in-control process). Liu and Shingh (1993) emphasize that these types of charts simultaneously detect changes in position and variability. What is more, according to Kobylińska (2014), the undoubted advantage of these charts based on the depth measure is the fact that they do not require an assumption regarding the compatibility of the process distribution with the normal distribution or linear relationship between the analyzed KPC, such as in the case of the Hotelling T 2 chart. Liu, Parelius and Singh (1999) describe the following depth measures: Mahalanobis depth, half-space depth, convex hull peeling depth, Oja depth, simplicity depth, majority depth and likelihood depth. For the purposes of calculations in the case study, the author limited herself to discussing Mahalanobis depth, in case when the probability distribution is unknown and only the measurements from this distribution are given (an empirical case). Mahalanobis depth, according to the work of Hamurkaroglu, Mert and Saykan (2004), for ̅ constituting a random vector from the distribution and and for any point in relative to the distribution , is expressed by the following relationship: where: -KPC observation vector with the dimension of × 1, ̅sample vector medium with the dimension of × 1, −matrix that is inverse to the covariance matrix with the dimension of × . Liu, Singh and Teng (2004) argue that using the depth measure provides center-outward ordering of data points within a given sample. A lower value ( , ) for in means greater distance from the average distribution F, and higher value ( , ) for in means a smaller distance from the average distribution of F. Other properties of the depth measure were discussed by Hamurkaroglu, Mert and Saykan (2004) and Zou and Serfling (2000). Let and be distribution functions of two independent -dimensional populations. In turn { 1 , 2 , … , } is an -elemental sample taken from population , which is referred to as a reference sample of the control process. Whereas { 1 , 2 , … , } is the new -element sample from the manufacturing process with a G-distribution. Checking whether the process is controllable or whether the deterioration of the product's quality took place consists in a comparison of the distribution and using observation { 1 , 2 , … , } and assesement of the depth measure relative to the distribution . In a situation where { 1 , 2 , … , } is more distant from the distribution it should be recognized that the process has shifted or there has been a change in the variability of the distribution. In short, to assess whether a process is controllable, the following null hypothesis should be verified against an alternative hypothesis: Relative rank statistics specifying the depth of point in from a set of observations { 1 , 2 , … , } in relation to a given theoretical distribution F, when ~ and ~ and data based on the extent of depth is an expression of: where: #{⋅} -size of the set under consideration, number of observations, ( ; ) -Mahalanobis depth measure for sample{ 1 , 2 , … , } relative to the distribution . ( ; ) -Mahalanobis depth measure for sample { 1 , 2 , … , } relative to the distribution . According to the work of Liu (1995) discussed in this article, multivariate chart is the equivalent of an average univariate chart ̅ , in which the test statistic is the sample average ( , ): which means that if an alternative hypothesis is adopted, on average more than 50% of the population examined from the distribution is deeper (more centrally) than any observation from the distribution . Consequently, the control limits for a small sample ( < 5) can be determined based on: where: number of w elements in the observation, acceptable probability of false alarm (in production practices, ≤ 0.0027 is applied). More detailed information, as well as justification for determining control limits using the dependence (6) for chart can be found in the study of Liu and Singh (1993).

Case of study
The research was carried out in a manufacturing enterprise in the Opolskie voivodship (Poland), dealing in the production of window fittings. The end of an espagnolette was tested using a control chart . This is the fitting element responsible for the functionality, and more specifically the clamping of the window sash to the frame (Fig. 1).
Figure 1: Removal of the window fitting element to be tested Source: Image from a production company The manufacturing process of the tested element is reduced to the following operations ( Figure 2): 1. Riveting of bearing diverticulum, steady bar and centering element. 2. Assembly of a spring.
3. Riveting of a locking bolt, stabilizing element and an element made after operations 1-2 in a zero position. 4. Embedding and riveting of a decorative casing. Riveting of a guiding element and an element made in operations 1-4.  The key features for quality, interrelated and subject to control are: ▪ The height of rivet head (control after operation 3), whose height should not exceed 0.3mm (Fig.   3a) -the control and measurement device used to perform measurements is an altimeter with an accuracy of 0.01mm, a) b) The torque on the bolt (control after operation 3), which should be in the range of 2.0-4.0 Nm (Fig. 3b) -the control and measurement device used to perform the measurements is a torque wrench with an accuracy of 0.01mm. A torque which is too low will result in insufficient clamping of the window sash to the frame, as well as the window's self-adjustment. The height of the rivet's head is influenced by the setting of the riveting machine (including riveting time, stroke, pressure, shape of the riveting tool's mandrel) as well as the quality of the rivet itself. In addition, it is known that the lower the height of the rivet head, the higher the torque. The analysis covered 3-piece samples from two different orders dated 01/09/2017 and 24/10/2017 (Fig.  4), which were collected in accordance with internal procedures (a total of 40 measurements for each of the examined KPCs). Between the orders, the machines were retooled, and other products were manufactured at that time.
In Table. 1, the values of the Mahalanobis depth measure for the measurement data from the order dated 01/09/2017 are presented, determined on the basis of dependence (1) separately for each measurement. On the other hand, in Table. 2, the values of the Mahalanobis depth measure for the measurement data from the order of 24/10/2017 and the relative rank statistics based on the dependence (1) and (2) for each measurement are listed.   206 .147 .125 .130 .531 .206 .320 .256 .125 .192 .105 .286 .131 .202 .330 .345 .330 .206 ( , ) .  101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 ( , ) .206 . 099 .144 .216 .355 .330 .105 .125 .162 .228 .107 .255 .162 .195 .266 .626 .345 .256 .716 .579 ( , ) .000 . 033 .050 .150 .350 .342 .033 .042 .075 .167 .033 .217 .075 .133 .225 .717 .350 .217 .792 .650 Source: Author Then, according to the relationship (4), the test statistic was determined for chart (sample average ( , ) -Tab. 3) and a control chart for = 0.0027 (Fig.5) was made, for which the control limits have been determined based on equation (6), respectively: = 0.5, = (3!⋅0.0027) 1/3 3 = 0.084.  Fig. 5, it can be clearly seen that four points exceed the lower control limit, which may suggest a shift in the mean value. Additionally, the sample averages of the analyzed rank statistics ( , ) show large "jumps" of the value, which may be another proof of changes in the production process in the field of distribution (some measurements are deeply immersed in the reference schedule -points above the central control limit, and others much shallower -points near the lower control limit). Additional analyses carried out in the production company to explain the situation indicate that first of all, unambiguous procedures should be introduced for machine parameter setters that rivet components of the product under analysis -it has been ensured that the values of the tested KPCs are within the tolerance limits, which was caused because at one time the measurements were closer to the lower range and sometimes they were closer to the upper tolerance range, which simultaneously affects the stability of the production process.

Conclusion
The article presents a method for determining control charts based on Mahalanobis data depth. The presented approach makes it possible to transform multivariate measurements of the KPC investigated from the production process into univariate indicators of rank statistics.
( , ). Presented control chart allows simultaneous monitoring of the average value shift and change of distribution in the examined process without preliminary assumptions such as multi-normality, autocorrelation or linear relationship of the examined KPC. The graph from Fig. 5 indicates a change in the distribution of measurements before and after conversion of machines, both in relative changes of position and dispersion, which means that it is necessary to take corrective actions or improve the production process. In addition, the case study presented in this article allows to determine whether a change in the manufacturing process takes place during a changeover of the machine. An undoubted disadvantage of this type of approach is the fact that it was not clearly stated if the reference distribution prior to the changeover is the in-control state of the production process (Phase II of the implementation of control charts). In short, the solution to the reference distribution, which will represent the in-control state of the studied production process remains an issue to be clarified (Phase I of implementation of control charts). The solution to this problem may be (2014) a control chart proposed by Li et al. dedicated to the Phase I implementation of control charts, whose determination of control boundaries is also based on Mahalanobis data depth, which will be the subject of further research by the author.