Identification of Annual Work Hour Elements in Production Cycle and Experimental Assessment of Flow Coefficient and Optimal Series

In the metalwork production a large part of gross domestic product is being achieved by automating the production in large factories, the finalists, but even a larger part comes from subcontractors and companies that have small-scale and mass production of maintenance elements and other services. Therefore, the production with higher degree of stochastics and organization mostly occurs during the production, and it happens, but to a lesser extent, that everything is planned and set in advance, and according to those facts this survey is going to analyze one of the most important production indicators the production cycle. In this paper we are experimentally demonstrating the original stochastic method of determining 10 different production cycle working hours, and production cycle itself is determined out of factory driving records. The research also enables determining the flow coefficient which represents the function between the size of the series and technological or real time of the production cycle. Within the framework of a broader research we are contributing results of four years long record, from 2011 to 2014 for a large company with high level of organization. The production cycle per years (2011-2014) amounts to 326, 233, 248 and 236 minutes, while the flow coefficient is Kp=(277/x+0.24) the one that enables and experimentally determines optimal series which is in total 9 items for the company in question.


INTRODUCTION
This paper aims to track the time elements using the modified method -work sampling which is described in detail in [1], and which differentiates from the original work sampling method, applied firstly by Tippett (1936) in the textile industry in England in such a way that in the metalwork production we have stochastic dependency according to the normal distribution law where with a large number of separate working elements, and Tippet's method according to binomial distribution law (that is questionable in practice) the machine operates for only three possible elements of working time (+, x, −).
The difference is that with production cycle (PC) the product goes through the plant, and with Tippet the machine produces it's in place.
The PC division with elements of time of work that we apply herein is given in Fig. 1.

Figure 1 Production cycle elements
Screening performance requires the definition not only of technological and mathematical problems, but also of the practical screening process and the establishment of working time elements. Thereafter, the elements of production cycle working time should be defined and, in particular, the difference when compared with the elements of working time related to machinery, i.e. for the purpose of establishing the machine capacity only or within the production cycle, because these two differ. The elements of working time are determined according to authors [2÷8] and may be serial, parallel or combined.
Theoretical and practical studies have shown increasing interest in using different methods and techniques in solving the problem of shortening duration of production cycle [8÷18].
The subjected company in this survey is under German ownership with good organization, operations are parallel but on a lower level operation is of a consecutive type with all its shortcomings given in Fig. 2. The representativeness of the screening sample per number and time of screening was established by mathematical parameters, SD and control limits, where the elements of PC time are observed as the elements of the process function.

APPLICATION OF THE MODEL FOR THE STOCHASTIC DETERMINATION OF PRODUCTION CYCLE TIME ELEMENTS
The screenings were performed from September to November. The production program was not changed during the period of four years since we got the sample size together with representativeness.
The screenings sheet is related to one PC, and the number of individual elements of work i.e. the frequencies are recorded on it. Using the frequencies, we first calculate the percentage of the individual elements against the total PC time, and then based on the analytical screening of the PC time duration, the time duration of individual elements of working time is calculated.

RESULTS AND DISCUSSION
Production cycle time per item and series for the year 2014 for the representative German factory that works in Serbia is given in Tab. 1.
The recorded results for each year are grouped in tables and diagrams according to elements of production cycle time of work in percentage and minutes, and according to number of items in series in minutes. The overall results in four years time are given in Tab. 2.
For easier understanding of this work we will first give an example for the movement of the productive time with controlling limits only for the year 2014, as shown in diagram in Fig. 3 and with performed estimation in Tab. 1 with Fig. 4. Results and methodology from this work were published in [3].
All cycle results, according to the working time elements individually and in total, according to production cycles in percentages (%) and in minutes on yearly basis are given in Tab   The diagram in Fig. 3 shows that there are significant fluctuations in times of some of the time elements of PC depending on cycle, so we have some kind of saw tooth diagram. The average production time is in total 78% PC or 263 minutes per cycle given in Tab  Production cycle time per item and series and experimentally closely specified optimal series is derived from the flow coefficient formula for data in Tab. 1.
Some most important results per cycles, per working time elements, individually and in total, per year production cycles in percentages % and in minutes are given in Tab. 1% which makes the total amount of technological time t t . It can be noticed that for the metal working industry the total percentage of preliminary-final time is high, and with its decline the production cycle could be reduced.  The movement of serial average time per number of items for the year 2014 is given in Fig. 4.
Representation and quality of the research is, besides control limits AC and BC, standard deviation SD, normal distribution of work time elements movement and total amount of production time established (as given in Fig. 5), the different values of Δti of production cycle time elements can be also seen, according to the number of records and number of cycles expressed in percents and minutes with values for four years (n = 4) and average values reduced to the number of cycles per years (N/N i , 46/159, 30/159, 39/159, 43/159). How small the differences are can be noticed in the diagram in Fig. 5, too. This is demonstrated by representativeness because the data movement is the same per cycle per minute and in percentages.
In Fig. 6 there is shown PC time elements movement per years, and it can be noticed that there is averaging of all elements in comparison to those average elements based on the cycles given in Fig. 3, where there are considerable oscillations of PC time elements and total average production time and where the control limits are wider, since the error had to be calculated with ±3SD, while by monitoring per years they were calculated with ±2SD; AC = 93.6%, BC = 64.9%. Generally, for the production where human factor is very important it is better to have 1/3 of narrower limits with 95.45% of probability, than for the small increase of probability of 99.97% (often unnecessary) to increase imprecision of control limits.
Movement of total production time tp and per item can be seen in Tabs. 1, 2, 3 and in Fig. 7. pieces. In addition, in the years 201 and 2016 there was an increase in t m , a decrease in t pt which is due to the better organization of work and changing preparatory finishing times (Fig. 6).
There was also a decrease in production time per piece in Tab. 4 and in percentage in Fig. 8 and Fig. 9 in relation to the increase in the number of pieces in the series, which is practically the second criterion for optimizing the number of pieces.
Both functions per years have the same trend, firstly from 2011 to 2012 there is a considerable decrease, from the year 2012 tpunit in percentage continues to decrease vaguely but t min increases, so that in the year 2013 there is a considerable increase again, while t punit percentage considerably increases, as well. Both functions could be approximated by parabola with existing minimum, which for axis of symmetry has the y axis.
According to the calculation of the flow coefficient for the whole four years using formula , a Y b X = + based on the data from Tab. 1 and using average values for four years, the data are got according to the number of items in series in function, from the production cycle time per items in minutes. Using normalized equations, the parameters a and b are obtained: In the diagram in Fig. 8 it can be noticed that the theoretical function has the real values with 9 points, while in the part of the last 3 points with greater deviations values are approximated, in a way that these are really the shortest periods of time per items in series with 9, 12 and 10 items, 27, 28 and 27.6 minutes. In this way, the real minimum value of the function of production time per item is 9 items, while according to the theoretical function, minimum value is endless number of items in series. This is the reason for the experimental determination of optimal series, because the flow coefficient is affected by a number of factors in the production, and function value after certain minimum increases abruptly. In addition to this one, in our calculations the main criterion is production period, and the other important criteria the costs of frozen funds are not considered, which in higher level criteria analysis would be done, in future works.

CONCLUSION
In this paper, it is proved that it is possible to experimentally determine production cycle by modifying method of work sampling, and based on the analysis of its five work time elements, which make production time t p and certain number of break periods during the nonproductive time t np (in our company is 5, as well).
Monitoring the function of time was performed by using the recorded periods of time per cycles in percentages and minutes. In production planning and production management monitoring per years is very important. Analysis has shown that the parameters of stochastic process of control limits and standard deviation representation are much preferred with monitoring per years.
By monitoring through years and cycles, values of the most important parameter of production time for 2011, 2012, 2013, 2014, 2015 and 2016 are 76.4; 83.6; 79.77, 78, 81.2 and 79.31% or the average for 6 years amounts to 79.7% with standard deviation SD = 6.69%, but control limits are CC2SD = 79.7 ± 2⋅79.7⋅0.669 = 79.7 ± 10.1 or CC 3SD = 79.7 ± 16. This means that for greater sample size we get better precision of all parameters, which also applies to experimental determination of optimal series which is approximately 9 items in series with minimum time per item p min = 27 min/unit and is expressed by flow coefficient function got from average value of the data of Y and X for four years.