MANAGING BOTTLENECKS IN FAST AND RANDOMLY CHANGING

SAIIE25 Proceedings, 9th – 11th of July 2013, Stellenbosch, South Africa © 2013 SAIIE

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MANAGING BOTTLENECKS IN FAST AND RANDOMLY CHANGING PRODUCTION ENVIRONMENT

T.B. Tengen

Department of Industrial Engineering and Operations Management Faculty of Engineering and Technology

Vaal University of Technology, South Africa [email protected]

ABSTRACT

Bottlenecks create unnecessary requirements for material handling within production settings. Several approaches have been developed and implemented to address bottlenecks such as line balancing, slowest workstation determining the paces at which other workstations operate, introduction of storage buffer, etc. Unfortunately, the existing models dwell on deterministic parameters such as fixed production rate, fixed demands, fixed production cycle time, fixed workstation cycle time, constant pace of the slowest workstation, etc. With the key word being fixed, this paper improves on existing approaches by randomising the fixed parameters that are all functions of time. This is achieved by establishing the time-increments of the production parameters and followed by making them stochastic by the addition of fluctuation terms. The theory of Ito’s Stochastic Differential Rule is applied.

Results reveal that an increase in demands, at constant flow/production output rate, requires an increase in production resources, increase in storage buffer requirement and increase pressure on the production environment when the previous models are employed. The employment of the proposed improved approach shows consideration reduction in the amount of required resources and storage buffers. The effects of machine degradations are also revealed.

Keyword: random demand, random TAKT time, random cycle time, number of resources


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1 INTRODUCTION

In most production environments, products have to go through many workstations before the end products are finally released to customers. This production layout is commonly known as flow line production, more suitable for mass production. Due to the fact that all the workstations along a flow line mass production may not always operate at the same pace, bottlenecks arise which call for (unnecessary) material handling. Several strategies have been devised to handle production bottlenecks [1,2,3,4,5] which include the application of the theory of line balancing, the slowest workstation determining the pace at which the other workstations works and inclusion of storage buffers between workstations. These theories all work best in environments where everything is supposed to go as planned, although with some shortcomings.

The introduction of storage buffers between workstations is aimed at holding extra “raw” materials in front of the receiving workstations if these workstations are operating at rates slower than the preceding ones. This approach works well if production is not continued for very long period of time or production is being interrupted by frequent short-downs of some (or all) of the workstations or when dealing with seasonal raw materials. If production goes on continuously with time without frequent short downs and the raw materials are readily available, then there will come a time when the buffers will get full thus necessitating an increase in the sizes or number of the storage buffers. This increase cannot continue indefinitely (i.e. for a long time) due to space requirements and monetary implications. One way to improve on these current approach and its shortcomings is to use a model where the slowest workstation determines the pace at which other workstations operate.

The approach that slowest workstation determining the pace of other workstations is common in most manufacturing flow line environments and it is, in fact, recommended in some production system textbooks, [1-3]. This approach is more effective where production is uninterrupted, but it cannot handle seasonal raw materials. It also has the limitation of resource idleness and, if not well managed, overstaffing of operators. The overstaffing of operators is common in production environments where there is job specializations i.e. where workers are more specialized to work on their own machines and they may not want themselves to be deplored to another workstations either through the use of job descriptions or unionisms, or simply because they do not have good knowledge of the operations at other workstations, [6]. Thus, the word “overstaffing” is used in this report to indicate the idleness of non-re-deplorable staff. This limitation, which is a clear indication of low productivity, will always be present in such type of production layout. To improve on this overstaffing or resource idleness, the theory of line balancing can be used, [2].

The theory of line balancing deals with the “appropriate” duplication of slower workstations so that there is no idleness of any resource along the production line. So, in this case the manufacturing plant is designed to release products at the rate of the fastest workstation, which is opposed to the design elaborated in the preceding paragraph where jobs are released at the rate of the slowest workstations. This design is an improvement of the previous two designs as no storage buffer is needed between workstation and no resource is idle. It disadvantages are that it is not designed to handle seasonal raw materials, more fund is initially needed to install the workstations’ equipment and it is not (cost) effective with seasonality or varying demands. The management of the varying (specifically increasing) demand will be dealt with in the rest of this paper.

To avoid the “recurring” issue about seasonal raw materials (“recurring issue” because seasonal raw materials seem to be an issue of the three design approaches), some manufacturing plants build warehouses in the plant or for the plant that hold and preserve sufficient raw materials that can be used throughout the production time until the next season where the raw materials are available. This may be dealt with through the inventory management department or inventory policies.


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One can then see that there has been and there is still a constant need for continuous improvement of flows along production lines. This constant need for improvement in production environments, in general and not only limited to flow improvement, has led to the concept of Lean Manufacturing, [3]. Although this paper is not on the general application of Lean Concepts in flow line manufacturing, this paper focuses on the improvements on bottlenecks following randomly changing/increasing demands and machines (or workstations) degradations. With line balancing seeming to be the best of the of the three strategies, the rest of this report will dwell on how this model may be managed following random demand increase and machine breakdowns.

It should be acknowledged that the sole driving factor for production is demand: either current demand or forecasted (future) demand, [1-3]. In line balancing, knowing the demand needed over a period of time (e.g. hourly, daily, weekly or monthly, etc.) one can determine the TAKT time (i.e. the production time needed to be spent per unit to meet the demand, [1,2]). Further knowing the cycle time at each workstation (called standard time) one can determine the number of workstations or operators needed to meet the demand. So, the two major parameters of this report are the random variations in demand and standard time at workstations. If these two parameters vary randomly then the other production parameters are also bound to vary randomly i.e. the demand for the company’s product may not always be constant and the operating machines might degrade with time and as such may operate at a different pace.

2 METHOD

When a machine does not operate as intended by its fabricator or designer then the machine can be considered to be faulty and has to be repaired, [3, 6]. These abnormal operations do not occur frequently: it is only probable that they would occur, [3,6,7,8]. The machine might operates faster or slower, which leads to “asynchronous” situation since the pace of this machine as prescribed by the machine designer traditionally should be or should have been used when deciding on the number of other machine(s) along the line during line balancing, [3]. In such a case, the line would not be balanced again. Examples of a situation where a machine might operate faster are when a wrong component is used in the assembly of the new machine or during maintenance, or when an electrical component is faulty. Although one may want to claim that if a production machine operates faster then it is an indication of improvement on product-release rate, it should be cautioned that machine components are not designed in isolation since the set of production conditions are interrelated, [3]. For example, if the feed rate (determined by the spindle rotational speed, [3]) of machine suddenly increases, then it can lead to, for example, tool breakage or damage of the work piece. The other situation where a machine may operate slower is more common with more common causes being the wear and tear, [6]. Since maintenance is the way forward to address both cases where a machine might operate faster or slower, the practical cycle time (also called standard time) per workstation along flow-line production that involves maintenance operation is given by, [3]

(1)

where Tp is the practical (or real) cycle time, Tc is the ideal cycle time, F is the probability of fault(s) and Td is the downtime. Note that here one deals with “downtime” as opposed to “average downtime” because the exact value is needed during line balancing (i.e. one deals with the “average downtime” when the slowest workstation determines the pace at which other workstations operate). Note that since the objective of line balancing is to “duplicate’ workstations along the productions line so that products are released at the rate of the fastest workstations, the failure of one workstation may not necessarily bring about work total stoppage at the entire plant. Such failure can be managed by introducing temporal storage buffer(s) and overtime(s). It is important to further recall that the downtimes due to

Tp = Tc +FTd


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probable or abrupt machine failure should be treated differently from those planned downtimes meant for, for example, meetings lunch, power outages like load scheduling, panned maintenance, etc.

A product’s demand variation with time (e.g. daily, weekly, monthly, etc.) should lead to the variation of the “expected production rate” or TAKT time. The demand for a product may vary in several different ways: either increasing or decreasing or fluctuating or with periodic jumps, and so on and randomly most time. The exact nature of the variation of the demand should be obtained from the empirical data through, for example, curve fitting of scatter plots and then forecasting. A parabolic model for increase in random demand (i.e. increasing demand that slows down in a long run) is dealt with in this report. When the demand for a company’s product varies in, for example, a parabolic manner, then TAKT time varies in a similar manner given by

(2)

where Tk02 is the square of initial TAKT time or the square of the initial demand per given

time, a is a constant called the parabolic rate constant, and t is time. The rate of variation of the TAKT time (otherwise called the time increment of the TAKT time) is given by

(3)

The stochastic counterpart of last expression is obtained by the addition of fluctuation terms to obtain, [7-11]

(4)

where fc(Tk)dW(t) is the random fluctuation in demand due to continuous changes, fp(Tk)dN(t) is the random fluctuation in demand due to periodic changes, a/Tk is the drift term, fc(Tk) is the diffusion term, fp(Tk) is the jump term, dW(t) is the increment of the Weiner’s process and dN(t) is the number of increment of stochastic counting process or, specifically, the number of demand increments within an infinitesimal time interval. Note that fluctuations in demand may be given as functions of demand or time or other variable(s). In this report, they are given as functions of demands.

During line balancing, dividing the TAKT time by the standard time, [1,2] gives the needed number of workstations Wsi to meet the demand. Thus, the stochastic counterpart of Ws is obtained by the necessary integration of expression (4) followed by dividing the answer by expression (1). Note that expression (1) is already a random expression due to the inclusion of the probabilistic term, F. The result should normally be rounded up to avoid under capacity that creates further bottlenecks. Workers can then be added to workstations to improve on production while taking into consideration the cost implication of such addition. The effect of this possible workers addition to rounded-up workstations is deal with under the “cumulative effect of rounding-up workstations” section of the results.

3 RESULTS AND DISCUSSIONS

An exampled production situation that requires the application of the theory of line balancing is dealt with. For confidentiality reasons and for the sake of not advertising any particular company, the name of the company whose current data has been used will remain anonymous. Without loss of generality, assume that all the workstations operate scrap-free such that the TAKT time per workstation or demand per time per workstation type is the same for all the workstations along the line (normally the knowledge of scrap rates should be used in scrap estimation or backward analysis, [2], to obtain the TAKT time per preceding

Tk2 = Tk0

2 + a.t

dTkdt

=aTK

dTk =aTK

dt + fc (Tk )dW (t)+ fp(Tk )dN(t)


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workstation). Note that this scrap free assumption would not significantly change the trend of the results per workstation: trends are important to the report.

The company operates three 8-hour shifts per day and experiences on average a routine 16 minutes downtime per shift due to breaks and meeting. The efficiency of the company is average 80% and a product goes through six workstations before it is completed. The sequence for processing jobs and the standard processing times at each workstation are as given in table 1. The company’s current demand is 3600 units per day and the demand is projected to experience a parabolic increase per day with the parabolic rate a given as 1.5x10-3.The probability of fault or failure per workstation that necessitates (reactive) maintenance downtime is as given in Table 1. The company feels that the increase in demand might be effected continuously or periodically with randomly fluctuation terms given as fc(D)= fc(Tk)=b(Tk)-1/3 and fp(D)= fp(Tk)=c.Tk

-1/2 respectively, where b=0.5 and c=1.5. The rate of increase in demand per given time caused by periodic fluctuation is given by

. The fluctuation in demand is expected to follow a lognormal distribution.

Thus, expression (4) is simplified by applying the theory of Ito’s Stochastic Differential Rule and Equations for Moments, followed by taking the expectation of the both sides of the resulting expressions using the properties of the lognormal distribution, [12]. The resulting differential equations are solved using Engineering Equation Solver software, EES, [F-Chart software, Madison, WI53744, USA].

Table1: Workstations operating characteristics or parameters

Operation Time standard (minutes)

Downtime (minutes) Probability of failure (F)

Loading 0.251 30 0.001

Mill flat surface 1.260 37 0.02

Mill oval edge 3.251 25 0.01

Drill a hole 0.258 44 0.04

Mill circular edge 2.521 18 0.01

Unloading/packaging 0.895 14 0.001

Note that the fluctuations in demand are chosen such that these fluctuations diminish as the product becomes famous or after a long period of time. Thus, both choices of parabolic increase in demand and the natures of the fluctuation terms are such that demand becomes stable in the long run. This model is, in general, more suitable for a new company or for the introduction of new product to the market, [1,2]. The results for a projected ten-year period are indicated in the plots below.

Observe from Figure 1 that the demand for the product increases in such a way that the TAKT time increases from 3.233 units per minute to 7.968 units per minute after 10 years, which is about 7.968/3.233= 2.465 three times increase. The increases in the number of required workstations in the order of flow through the flow line are as follows (Figure 2a and Figure 2b): (11.5=12) / (28.36=29) = (2.466=2.417); (1.616=2) / (3.984=4) = (2.466=2); (0.9234=1) / (2.276=3) = (2.465=3); (1.602=2) / (3.949=4) = (2.465=2); (1.197=2) / (2.95=3) = (2.464=1.5) and (3.556=4) / (8.766=9) = (2.465=2.25) where the set (actual=rounded) should be interpreted as the actual values against the rounded value. The first sets made up of normal characters are the initial values, the underlined sets are the final values, and the bold-italic-underlined sets are the number of times increase. It can be seen from these sets that if fractions of workstations could be bought then the rate of increase of the TAKT time will be the same as the rate of increase of the number of required workstations along the

λ(TK ) =1/Tk3


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line (proportionate to 2.47 times). But since fractions of workstations cannot be bought, the effect of rounding-up shows a serious fluctuation in the number of workstations required: ranging from as low as 1.5 or 2 to 3 times the number of workstations required. This fluctuation in required number of workstations also indicates the challenge that should be faced during line balancing and possible addition of operators to less capacitated workstations. Thus, the percentage load that indicates the degree of balances line is also expected to fluctuate, Figure 4. Having the knowledge of “Excess resources” required should further help balancing the line or in capacitating the workstations.

Figure1: Demand variations: deterministic (steady) and stochastic (fluctuating) increases.

Observe that due to random fluctuations, the evolution of the demand per day may be more or less than that of the steady demand increase without fluctuations. Observe also that the dispersion of the demand fluctuation is very low since the plots of Figure 1 are closer to each other. It can then be assumed all the curves span the range of a scatter plot of the projected demand and that the two extreme plots can be assumed to be the range of the results within which all other values fall.

Observe by comparing Figure 1, Figure 2a and Figure 2b that as demand increases, the number of required workstation increases too, which agrees with practical expectations. It can be observed from Figure 2c that the current proposed model reveals that as the fluctuation in demand increases the number of required workstations increases and as the fluctuation in demand decreases the required number of workstation is considerably reduced. But the use of a model that does not consider fluctuation reveals that the result does not vary with demand fluctuations. Thus, the use of the current model, and also taking into consideration the fact that the fluctuation in demand may decay in time and that each product has its life cycle, it can be claimed that the current approach shows an improvement in the results that best correlates with empirical data. Figure 2a reveals the parabolic relationship between the required number of workstations and time while the one (Figure 2b) between the required number of workstation and TAKT time is linear.

0 14000 28000 420003.2

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Time (

Units

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Steady IncrSteady Incr


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Figure 2: Variations in number of required workstations

The excess numbers of workstations, results found in Figure 3, are obtained as the differences between the rounded-up values and the actual fractioned values, (see Figure 2c and Figure 3a). This corresponds to the area under the “triangular region” between the curve in Figure 2c and figure 2d. An interesting observation that can be made from the present paper is that the amount of “excess” workstation required for fast changing demand is smaller than that of the slower changing demand in the long run, Figure 2c, Figure 3a, Figure 3b and Figure 3c. The number of excess workstation, obtained as a consequence of rounding-up calculated number of workstations since a fraction of machine cannot be purchased, should be used as an indication of the number of operators that may be added to the workstation to avoid resource wastage. The excess workstation is actually a resource wastage meant to avoid underproduction that should create backlogs. It should be stated that a decision to increase the number of workstation, as proposed by the results, is cost effective if the workstations are cheaper to buy while the model of having extra operators may work best if there is sufficient space within workstations for parallel operations.

Figure 3: (a) Excess number of workstations due to rounding-up, (b) & (c) cumulative

effects of rounding-up workstations (or waste amount of man-hour due to rounding-up); where fig.3c is meant to indicate long run behaviour.

The plots (Figure 3) of the “cumulative excess” number of workstation is an indication of the total amount of resources (such as man-hour) that should be expected to be wasted if extra operators are not added to workstations. Such significantly large value of “cumulative excess” points out the discomfort that managers usually have with idle facilities. As observed that the amount of excess resources (Figure 3b and Figure 3c) and the number of required workstation (Figure 2c) fluctuate tremendously, the ability of a manager to manage source fluctuations accurately with human judgment is very slim. So, instead of managers trying to respond to the fluctuations by adjusting the number of workstations or employing more operators to excess resource, some of them decide to introduce storage buffers between workstations and recommend overtime for operators. With such an over-time approach and also being informed by the trend that demand and number of workstations will take, they temporally use smaller size storage buffers while awaiting the purchase of new facilities. Of course, this might not be optimal, but taking into consideration the fact “each

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change” has its cost implications, those managers would not always be making the worst decisions.

Figure 4: Effect of Average time and degree of balance

The principle of line balancing has been applied as stated in [2]. The TAKT time has been divided by the standard time and the answer rounded. This has been followed by dividing the “standard time” by the “rounded-up value” to obtain “average time”. The average times has been divided by the smallest of them all to obtain the percentage load. Note that this percentage load indicates whether the line is balanced (i.e. fully capacitated) or not i.e. the degree of balance. Observe that if these values were obtained for fractionated workstations, then the average time will be seen to decrease steadily (Figure 4a) and the percentage load (Figure 4b) would be constant for all the workstations. But for practical situation, since fractions of workstations are not realistic, it can be observed that the average time can be seen to experience some jump-decreases followed by constant values for some time and the subsequent jump-decreases (Figure 4c), while the percentage load is seen to show some jump-increases, constant-ranges and jump-decreases (Figure 4d) with increase in TAKT time for some of the workstations. For all the cases, observe that the workstation with the largest average time (i.e. the slowest) is 100% balanced (i.e. 100% loaded) while the other workstations have poor balance. The less balanced workstations indicate that either operator(s) should be added to the workstations or that they should be outsourced. This also indicates the challenge that should be faced in managing the production line.

4 CONCLUSION

It can be concluded that model for predicting and controlling bottleneck in randomly increasing demand environment has been proposed and tested.

Results reveal that based on the constraint of not being able to purchase fractions of workstations, the rate at which the number of workstations should increase does not always follow from the rate of increase in demand. This rate shows random fluctuations that are difficult to manage from mere human judgment. But the use of the proposed model indicates when to react.

Results shows that in fast changing demand, a manager who follows the proposed model will have less excess resource or will waste little amount of resources.

Results reveal that while some of the stations will be fully loaded (up to 100% load) others will be poorly load (as smaller as 19% load).

It can also be concluded that a mixed employment of the idea derived from the proposed model and the use of some amount storage buffer would minimize the risk of underproductions or over capacity.

4.1 Acknowledgement

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This material is based upon work supported financially by the National Research Foundation. Any opinion, findings and conclusions or recommendations expressed in this material are those of the author and therefore the NRF does not accept any liability in regard thereto.

5 REFERENCE

[1] J. Heizer and B. Render, Operations Management, 10th Ed. Pearson Education Limited, Printed in the United States, 2011

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[3] M.P. Groover, Automation, Production Systems, and Computer-Integrated Manufacturing, 3rd Ed., Pearson education Inc, printed in the United States, 2008

[4] J. Lu, M. Shen and X. Lan, Study of the shifting production bottleneck: possible causes and solutions, IEEE Transaction, pp 684-688, 2006.

[5] V.S. Jorapur, V.S. Puranik and A.S. Deshpand, Research issues in detection of bottlenecks in discrete manufacturing system: A review, International Journal of Engineering Research and Technology, Vol. 1, Issue 8, pp 1-6, 2012.

[6] Mattew P. Stephens, Productivity and Reliability-Based Maintenance Management, Perdue University Press, Printed in United States, 2010.

[7] Snyder, D.L. (1975), Random point Processes, J. Wiley & Sons, New York.

[8] Gardiner, C.W. (1985), Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer-Verlag.

[9] Iwankiewicz, R. and Nielsen, S. R. K. (1999), Vibration Theory, Vol. 4:Advanced Methods in Stochastic Dynamic of Non-Linear Systems, Aalborg TekniskeUniversitetsforlag.

[10] Gikhman, I.I. and Skorokhod, A.A. (1972), Stochastic differential equations, Springer-Verlag.

[11] Arnold, L. (1974), Stochastic Differential Equations: theory and applications, J. Wiley & Sons, New York.

[12] Crow, E.L. and Shimizu, K. (1988), Lognormal Distributions, Theory and Applications, M. Dekker New York.


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MANAGING BOTTLENECKS IN FAST AND RANDOMLY CHANGING