Serial production line performance under random variation: Dealing with the ‘Law of Variability’

TL;DR: Reading this paper provides readers the foundational knowledge needed to develop intuition and insights on the complexities of stochastic simple serial lines, and serves as a guide to better understand and manage the effects of variability and design factors related to improving serial production line performance.

About: This article is published in Journal of Manufacturing Systems.The article was published on 2019-01-01 and is currently open access. It has received 32 citations till now.

Summary

1. Introduction

• T results on the effects of variability in serial production line performance, and provide a reference manual of sorts for practitioners and researchers alike.
• The remainder of this paper is organised as follows: Section 2 presents fundamental definitions and describes the paper’s general scope.

2. Scope of the study

• The fields of Queueing Theory and Production Management have comprehensively investigated the behaviour of serial production lines under variability.
• Most research investigating variability has concentrated on studying the effect of a single variable on the performance of a stochastic production line.
• In addition, intuition can be more easily gained by first studying simpler systems before more realistic but also more complex systems [4].
• Since their main objective is to assist practitioners and researchers in gaining fundamental understanding, it was decided to concentrate on describing the behaviour of serial lines rather than describing specific formulas for different analytical and approximation models of serial lines.
• If that subject is of interest, the reader is referred to Buzacott and Shanthikumar [9].

2.1. Unit of analysis

• Serial, stochastic production lines are systems with n number of stations of single resources.
• They have few constraints and variable production processes, i.e. the processing time of each task is random.
• Fig. 1 represents a serial line where Si is the distribution of processing times in station i, for values of i between 1 and n (n being the total number of stations in the serial line);.
• A major difference between them is the saturation of the line.
• Conversely, an unsaturated line is a system that is limited by a stochastic arrival pattern, due to either material supply constraints or customer arrivals.

2.2. Performance measures

• This subsection defines three complementary measures that have most commonly been used to assess the performance of simple serial lines: throughput, inter-departure times and flow times.
• These three particular performance measures were selected because the effects of variability and design factors on these measures are, for the most part, straightforward.
• Other performance measures such as flexibility, quality, cost, and delivery speed and dependability are not covered in this work because that would require studying more complex systems and interactions and would be outside the scope of this study, i.e. simple serial lines.

2.2.1. Throughput

• The throughput (TH) rate of a simple serial line is a random variable that measures the actual number of finished product units coming out of the production line per time unit.
• TH is important for a company because it is the rate at which a factory produces money per time unit.
• As a random variable, characteristics of the TH distribution can be measured, such as the mean and the variance.
• While TH̄ is commonly considered the main performance measure for saturated lines [13,14], the variance of throughput rate (Var(TH)) has also been deemed important because it measures the predictability of the total output per unit time of a factory and the related firm’s revenue.
• Given this value, results for Var(TH) are also included in this paper.

2.2.2. Inter-departure time

• The inter-departure time (D) describes the process of two consecutive departures from the production line.
• The mean inter-departure time (D̄) is the inverse of TH̄ and is the mean time in which two consecutive departures occur.
• Previous studies have been primarily interested in investigating the variance of inter-departure times (Var(D)), since it describes the regularity of production output in both saturated and unsaturated lines.
• The regularity of production output is also relevant for supply chain performance because an irregular output process upstream will affect operations downstream [15].
• Fig. 1. Graphical representation of a simple serial production line.

2.2.3. Flow time

• The flow time (FT) of an order or customer is the total time the order spent in the production line, either waiting in queue (W) or being processed by a resource.
• Flow time (also known as cycle time (CT) [16–18]) is substantially related to the quality of service so the effect of variability on flow time and waiting time has been a main topic in unsaturated lines research as well as in assembly line balancing problems [10].
• TH̄ and mean FT ( FT̄ ) for stochastic production lines with work conserving properties, i.e. lines that don’t experience system losses or re-entrant flow, are related by Little’s Law [20] through the mean workin-process (WIP¯ ) of the production line, i.e., =TH WIP FT¯ ¯ / ¯ .
• Similarly, FT̄ for a serial line is equal to the sum of the total waiting times plus the sum of the total processing times of all stations.
• For ease of reference, Table 1 shows a summary of all abbreviations used throughout the paper considering the performance measures and different factors included in the study.

3. How different factors affect the performance of production lines under random variation

• The ‘Law of Variability’ posits that the random variability intrinsic to the manufacturing process is detrimental to the productivity of the process.
• To address this concern, this section first pinpoints how different variability factors inherently associated with the process affect line performance.
• Second, the authors relate how some line design factors impact performance.
• Third, the authors analyse the effects of some of the most widely known production control techniques on line performance.

3.1. Effects of variability factors

• This subsection covers the most widely known effects caused by the variance of inter-arrival and processing times, and the impact of resource unreliability.
• It also describes several less known factors, e.g., the skewness and autocorrelation of input distributions.

3.1.1. Inter-arrival and processing time variance

• Similarly, Tan [27] suggested that for unbuffered saturated lines, inter-departure time mean and variance increase as SCVS increases.
• These formulas show that higher Var(A) and/or Var(S) result in higher FT̄ as well as higher Var(D).
• They found that a 100% processing times standard deviation improvement equates to reductions of 83% and 67% of mean processing times for infinite and finite buffer sizes, respectively, with exponential processing times.
• This contrasted to the level of starving, which was lower at the first station and higher at the last one.

3.1.2. Resource unreliability

• Perhaps one of the most widely studied factors regarding variability has been the topic of machine unreliability in saturated serial lines.
• It has been shown that these lines have the property of reversibility, which describes how the TH̄ for a saturated serial line with particularly ordered Bernoulli (or exponential) machines and a given set of buffers is equal to the TH̄ of an equivalent serial line but with inversely ordered Bernoulli (or exponential) machines and buffers.
• Thus, the monotonicity property states that increasing machine reliability (decreasing unreliability) and/or buffer capacity will result in higher throughput.
• Li and Meerkov [30] also suggested that production processes with Bernoulli machines should distribute total production work among several machines arranged in series, instead of one single machine capable of doing all the operations since ‘longer lines smooth out the production and result in a variability lower than that of one-machine systems’.
• Secondly, shorter mean repair times with frequent failures are preferred over longer mean repair times with infrequent failures for the TH̄ of serial lines with the same average downtimes.

3.1.3. Inter-arrival and processing times skewness

• Distributions of inter-arrival and processing times can also have higher moments that affect the mean waiting time [36,37].
• W̄ in unsaturated single queues for different values of the squared coefficient of variation of inter-arrival times (SCVA) and SCVS.
• Atkinson [40] showed that when SCVA<1 considering a Gamma distribution, W̄ increases with increasing processing time (with Erlang distribution) skewness but when SCVA>1, W̄ decreases with increasing processing time skewness.
• Lau and Martin [41] suggested that lower values of Skew(S) result in higherTH̄ in saturated lines.
• They also found that the effect of kurtosis on TH̄ is more difficult to interpret since its effect depends on interactions with other factors.

3.1.4. Auto-correlated processes

• Most studies investigating unsaturated lines assume that the arrival process is a renewal process [43], meaning consecutive customer or part arrivals are identically distributed and independent of each other, i.e. not correlated.
• A serial production line embedded in a downstream supply chain process usually receives arrivals from an upstream process.
• But for some scenarios, negative Corr(A) reduce.
• W̄ in a single-station queue, as indicated by Nielsen [49] in what he called ‘more realistic’ auto-correlation patterns.
• These conclusions are also shared by Takahashi and Nakamura [51], Altiok and Melamed [52] and Pereira et al. [53], who found that both positive and negative Corr(A) and Corr(S) have a negative impact on the performance of serial production lines.

3.1.5. Resource utilisation

• While resource utilisation (ρ) is not a direct source of variability, but instead, is the result of a combination of random variables (i.e. = S A¯/ ¯ , where Ā is the mean inter-arrival time and S̄ is the mean service time), it is worth studying since it is subject to random variation and does affect serial line performance.
• Note that Var(D) tends also to be more dependent on Var(S) in a single-station line when ρ is high.
• This behaviour is shown by the linking equation proposed by Hopp and Spearman [3].
• This effect has been called BRAVO - “Balancing ( =A S¯ ¯) Reduces Asymptotic Variance of Outputs”.
• Lambrecht and Segaert [57] also showed that buffer content and resource utilisation were not equal for all stations in saturated lines modelled with different processing time distributions, even when all the stations of the line were balanced and no buffer limits were present.

3.2.1. Work and variability allocation along the line

• One of the most widely accepted conclusions about serial line design is that balanced stochastic serial lines, if comparing two saturated lines with the same assumed TH̄ , perform worse than unbalanced lines [25,59], which have a protective capacity to deal with processing time uncertainty.
• These studies generated a field of research on the bowl phenomenon, which states that faster stations (non-) of saturated serial lines should be assigned to the middle of the line, while slower stations should be positioned at the beginning and end of the line.
• These and related results can be found in the review papers of Hudson et al. [60] and Mcnamara et al. [14].
• Furthermore, equivalent results can be inferred from the reversibility property in Bernoulli (or exponential) lines with unreliable machines [31], as the most reliable machine should be placed in the middle of the line to improve TH̄ .
• Suresh and Whitt [61] suggested arranging the stations in a decreasing order of processing time variability, i.e. assigning the station with the lowest SCVS at the beginning of the line and the station with the highest SCVS at the end of the line, although the arrangement is not optimal for every configuration.

3.2.2. Buffer size and line length

• The effects of buffer size (B) and line length (n) under variability have commonly been studied concurrently so their effects are considered jointly here.
• They are sometimes considered so by experimental design, e.g., if all the stations have equal B and you increase n, then total buffer capacity for the serial line will increase.
• Hillier and Boling [64] and Conway et al. [26] showed that longer saturated lines with uniform and exponential processing times and smaller buffers reduced TH̄ in balanced saturated lines, because longer lines and smaller buffers resulted in more station blocking and starving.
• Kalir and Sarin [67] present similar conclusions with longer lines with uniform and exponential processing times reducing TH̄ and increasing Var(D), and an increase in B increasing TH̄ and decreasing Var(D).
• When considering the amount of B placed in front of stations in balanced saturated serial lines with limited buffer capacity, Lambrecht and Segaert [57] suggested that if possible, buffers should be placed evenly along the whole serial line.

3.2.3. The complex interactions between variability and design factors

• Real serial lines are subject to these factors in an intertwined manner, as Atkinson [40] showed regarding the effects of Skew(S).
• This means that caution must be exercised since particular factor combinations can produce unexpected results.
• Hillier found that when the cost of inventory increased, the best pattern regarding work allocation was to assign work towards the start of the line and allocate buffer capacity towards the end of the line, versus following the bowl pattern for work allocation and a balanced pattern for buffer allocation to singularly maximise TH̄ .
• Due to the resulting complexity caused by variability and design factor interactions, some authors [56,66,76] have proposed measuring the overall presence of variability on simple serial lines to assess its impact on the performance of serial lines.

3.3. Single measures to assess the impact of variability on the performance

• Measuring the variance of critical production line performance measures can provide a good estimate of variability on a production line, but the magnitude of these measures can vary greatly between different production lines, even in the same industrial sector.
• These measurement differences limit the ability to assess the true impact of variability on performance.
• They proposed an ‘X-factor’ estimate, by dividing the overall mean flow time of a production line by the sum of all tasks’ processing times.
• Wu et al. [56] suggested that the variability of a production line (which they termed α) could be estimated by the ratio between the sum of all the waiting times in all the stations and the hypothetical mean waiting time of the bottleneck station considered as an M/M/1 queue.
• Applied to serial production lines, the measure can assess how variability is dampened or amplified throughout the production line by comparing the variance of the arrival rate against the variance of throughput rate or the variance of inter-arrival times against the variance of inter-departure times.

3.4. Effects of production control techniques

• Multiple techniques have proposed dealing with variability in simple serial production lines without modifying line design.
• These techniques generally focus on the principle of reducing WIP¯ in the production line, because increased WIP¯ creates longer FT̄ for any arriving orders (from Little’s Law).
• While Kanban assigns specific buffer sizes to all stations and then triggers production in each, contingent on the consumption of the next downstream station, CONWIP assigns an overall buffer capacity to minimise starving and blocking and triggers production at the start of the line whenever one part leaves the last line’s station.
• Other authors have studied the differential performance between popular techniques using theoretical settings of simple serial lines.
• Lambrecht and Segaert [57] suggested that a DBR-like strategy for pacing the line (i.e. order release) produces betterTH̄ than a strategy of specifically limiting the buffer on each station.

3.5. Summary of relevant results

• Table 2 provides a summary of serial line factors and performance measures, as well as key authors.
• In Table 2, “↑” entails an increase in the values of the factor or performance measure, while “↓” entails a decrease; “X” marks whether an increase or decrease in a factor results in an increase or decrease of a singular performance measure.
• As reflected in the ‘Law of Variability’, Table 2 shows that an increased variance (either from inter-arrival or processing times) adversely impacts all performance measures relayed in this paper.
• Therefore, to provide with a quick reference for these complex interactions, Table 3 summarises the results from various studies where multiple factors interact to have an effect on a singular performance measure.
• Hence, no general conclusions about the effects of these factors can be extended to all simple serial lines, despite the fact that Table 3 provides concise details of the results of the cited references.

4. Discussion

• Table 2’s summary of the effects of variability and various design factors on different performance measures provides clear and concise guidance to better understand the behaviour of simple serial production lines under the effects of variability.
• On the contrary, to develop a comprehensive understanding of the intricacies of complex manufacturing and service systems, researchers and practitioners should use every tool at their disposal, as it has been shown that combining analytical and simulation models [94,95] can reap better results when trying to deal with the management of an intricate system.
• A manufacturing firm might be simultaneously concerned with increasing throughput (to increase revenue) while decreasing inventory (to decrease cost).
• In spite of these simplifications, the authors feel that this paper provides the OM field a simple and single referent source that explains the overall behaviour of stochastic serial lines.
• It can help readers to gain insight into the effects of variability on the performance of manufacturing firms and further understand the implications of the ‘Law of Variability’.

5. Managerial implications and opportunities for future research

• For managers, the summaries provided in Tables 2 and 3 and discussed in Section 3.5.
• Overall, it can be said that performance of a serial line is dependent on the interactions among variability factors, e.g., Var(S), and design factors, e.g., B, as these interactions create an uneven production flow that results in the starvation or blocking of the stations along the line and, consequently, a decrease in performance.
• Some authors [103,106–109] have suggested that processing/service times are better modelled by other probability distributions, such as, lognormal and Weibull.
• Studies comparing the impact on performance of a reduction in the variance of processing times against the effect of reducing the variance of time to failure or time to repair could provide very interesting insights regarding the leverage of each factor to improve performance as studies concerned with the impact of unreliability have rarely considered stochastic processing times.
• Therefore, a stream of research is needed to investigate the trade-off between the estimation accuracy of different modelling paradigms and the time it takes to produce that estimation , depending on system complexity.

6. Conclusions

• The main objective and contribution of this paper is to present and summarise some of the most relevant conclusions on the performance behaviour of serial production lines under the effects of variability, and extend the implications of the ‘Law of Variability’ to improve factory and service management and gain and retain competitive advantage.
• A brief overview of the most meaningful conclusions on different performance measures is presented and serves as a guide to better understand and manage the effects of variability and design factors on production line performance so managers can exploit the leverage points of factory performance.
• This paper fills a gap in literature because few previous efforts have been made to summarise valuable conclusions in a manner which provides practitioners and researchers easier interpretations of the performance effects of various singular variability and design factors.
• This paper assists readers in developing better understanding and intuition of the behaviour of serial lines so they may design and manage more robust and efficient operations management tasks, before embarking on more complex production systems modelling.

Figures

Table 3 Summary of multiple-factor effects on the performance of the serial lines studied by the cited references.

Table 2 Summary of single-factor effects on serial line performance subject to random variation.

Table 1 Summary of abbreviations used throughout the paper.

Frequently asked questions

Q1. What are the contributions in "Serial production line performance under random variation_ dealing with the ‘law of variability’" ?

This paper fills this gap and provides readers the foundational knowledge needed to develop intuition and insights on the complexities of stochastic simple serial lines, and serves as a guide to better understand and manage the effects of variability and design factors related to improving serial production line performance, i. e. throughput, inter-departure time and flow time, under 

Q2. What are the future works mentioned in the paper "Serial production line performance under random variation_ dealing with the ‘law of variability’" ?

Addressing future research directions and opportunities, since most previous studies on simple serial lines focused on the single effects caused by processing time variance in the resulting mean throughput, mean flow time and variance of inter-departure time measures, more efforts are needed to study the impact of specific and various combined factors on particular performance measures. Therefore, further research considering more realistic probability distributions is needed as it has been previously shown that queueing network models are highly sensitive to the choice of the probability distribution modelling input distributions [ 37,39,110,111 ]. In terms of measure development, another worthy research direction would be to extend proposed work by Delp et al. [ 76 ] and Wu et al. [ 56 ], to directly assess the variability of a production line, or more precisely, the impact that the many variability factors ( e. g., variance, skewness and auto-correlation of inter-arrival and processing times, and unreliability, etc. ) have on the overall performance of the production line. A similar reference compilation on more complex production line results, such as assembly lines with merging materials [ 113,118 ], setup times [ 119,120 ], quality concerns [ 121 ], or multiple-product serial lines [ 122–124 ] could be useful to better understand the effects of variability in different production environments and further extend the reach of the ‘ Law of Variability ’. 

Q3. What is the widely accepted conclusion about serial line design?

One of the most widely accepted conclusions about serial line design is that balanced stochastic serial lines, if comparing two saturated lines with the same assumed TH̄ , perform worse than unbalanced lines [25,59], which have a protective capacity to deal with processing time uncertainty. 

Q4. What fields have comprehensively investigated the behaviour of serial production lines under variability?

The fields of Queueing Theory and Production Management have comprehensively investigated the behaviour of serial production lines under variability. 

Q5. What is the effect of MTTR on Var(TH) in unbuffered balanced?

On the other hand, longer lines with lower MTTR values, by having higher TH̄ values, have lesser constraints on the possible values that the throughput can take, and therefore, lower MTTR values will result in higher Var(TH). 

Q6. What did Hillier and Boling and Conway et al. show?

Hillier and Boling [64] and Conway et al. [26] showed that longer saturated lines with uniform and exponential processing times and smaller buffers reduced TH̄ in balanced saturated lines, because longer lines and smaller buffers resulted in more station blocking and starving. 

Q7. Why is TH important for a company?

In Theory of Constraints terms, TH is important for a company because it is the rate at which a factory produces money per time unit. 

Q8. What is the effect of the buffer size on throughput?

They demonstrated that for saturated lines with n=3, exponential processing times with a mean processing rate equal to 1 and a significantly big buffer size per station equal to 100, TH̄ of the production line was less than 1 (i.e. 0.9866), indicating that, even when the buffer size is not a significant constraint on the system, throughput is affected by the processing times’ variance. 

Q9. How many buffers should be placed along the whole serial line?

When considering the amount of B placed in front of stations in balanced saturated serial lines with limited buffer capacity, Lambrecht and Segaert [57] suggested that if possible, buffers should be placedevenly along the whole serial line. 

Q10. What is the significance of the arrival effects of a single-resource line?

Since it has been shown that the departure process of single-resource lines is not a renewal process [44,45] (unless the distribution of inter-arrival and service times is exponential [46]), this suggests the importance auto-correlated arrival effects. 

Q11. How did Hillier and Boling show that the variance of processing times increased?

They also showed that by reducing the variance of processing times, overall TH̄ increased, without reaching TH̄ equal to 1, the value expected for deterministic serial lines. 

Q12. Why is the effect not present in shorter lines?

This effect is not present in shorter lines because the throughput is not as constrained as in longer lines, due to the smaller interference among the stations. 

Q13. What are the three complementary measures used to assess the performance of simple serial lines?

This subsection defines three complementary measures that have most commonly been used to assess the performance of simple serial lines: throughput, inter-departure times and flow times. 

Q14. What is the widely studied factor regarding variability in saturated serial lines?

Perhaps one of the most widely studied factors regarding variability has been the topic of machine unreliability in saturated serial lines.