Buffer Capacity

Due to their large state spaces, exact solutions of merging line systems can only be obtained by analysing the underlying Markov chain using numerical methods, which are not computationally feasible for lines longer than three stations and for non-exponential distributions. To address these constraints, computer simulation is applied in many cases to study such systems. Discrete-event simulation was deemed the most appropriate tool for this study because of the severe limitations of mathematical approaches in dealing with more realistic and complex merging lines. The Simio 10.165 simulation software was used to study the behavior of the unreliable, unbalanced merging lines at the heart of this paper.

Model description

If $B_{i A}$ is full and preceding station $S_{i A}$ has completed a task, then $S_{i A}$ will be blocked until $B_{i A}$ has space. If $S_{(i+1) A}$ has completed a task but preceding buffer $B_{i A}$ is empty, then $S_{(i+1) A}$ will be starved. The first station of a parallel line $\left(S_{1 A}\right.$ and $\left.S_{1 B}\right)$ is never starved and the Final Assembly station is never blocked. Both parallel lines $A$ and $B$ have identical behaviour.

In addition, all stations have the same unreliability profile, depending on the experimental setting. MTTFs are modelled based on machine operation time, as opposed to production running time.

An example of two merging lines with $N=5$ and $B C=2$ is shown in Fig. 1, where the Final Assembly station is starved (in grey) because it has not received a component part from parallel line $\mathrm{B}$ ( $\mathrm{F}_{2}$ is empty). $S_{3 B}$ (shown in red) has failed and is being repaired, causing $S_{4 B}$ and $S_{5 B}$ to be starved and $S_{1 B}$ and $S_{2 B}$ (shown in yellow) to be blocked, as $\mathrm{B}_{1 \mathrm{~B}}$ and $\mathrm{B}_{2 \mathrm{~B}}$ are full.

Fig. 1 Screenshot of a Simio model for two merging lines with $N = 5$ and $BC = 2$

The above assumptions are in agreement with those stated in previous simulation studies as well as empirical findings.

Research design

This investigation utilises a full factorial experimental design, which permits the consideration of all desired levels of a given factor, together with all levels of every other factor, to measure the impact of independent variables on dependent variables.

Experimental factors

In this paper, the independent variables (factors) studied are:

• Number of stations (line length), $N$.

• Mean capacity of each buffer, $BC$, or equivalently, total buffer capacity of the line divided by the number of buffers.

• Buffer allocation patterns, $P_A$ and $P_B$, for parallel lines $A$ and $B$, respectively.

• Degree of machine unreliability, which is made up of two components:

  • Machine efficiency or (un)reliability

    $\varepsilon = \frac{MTTF}{MTTF + MTTR}$

  • Duration of $MTTF$ and $MTTR$ ($alpha$)

The use of fixed patterns of uneven mean buffer size allocation is a well-established method of investigation in previous literature to evaluate their effect on production line behaviour. Furthermore, all independent variables were chosen because of their demonstrated influence on TR and ABL.

Five different uneven buffer capacity allocation policies for lines $A$ and $B$ were considered: balanced, ascending, descending, bowl and inverted bowl. The patterns used in this study correspond to those used in some previous publications. The experimental values used in the simulation analysis can be found in the "Appendix" (Table 3).

$MTTF$ and $MTTR$ were modelled with an exponential distribution, based on the empirical results of Inman. Also based on Inman, a minimum realistic $\varepsilon$ of $70 \%$ was selected, while $90.9 \%$ was regarded as a typical value for $\varepsilon$, i.e. $[MTTF] 1000/([MTTF] 1000 + [MTTR] 100)]$, in accordance with previous work.

Three levels of $\alpha$ ($1$,$2$ and $3$) were estimated for $MTTF$ and $MTTR$. $MTTF_{\alpha \varepsilon}$ and $MTTR_{\alpha \varepsilon}$ model the $MTTF$ and $MTTR$ values used for experiments with machine efficiency $\varepsilon$ and degree (length) of duration $\alpha$. An $MTTF = 1000$ and $MTTR = 100$ were considered as a medium level $\alpha(\alpha=2)$ for $\varepsilon=90.9 \%$. Short $MTTFs$ for a specific value of $\varepsilon\left(\mathrm{MTTF}_{1 \varepsilon}\right)$ were then calculated as $\mathrm{MTTF}_{1 \varepsilon}=1 / 2 \mathrm{MTTF}_{2 \varepsilon}$, while longer $MTTFs$ were calculated as $\mathrm{MTTF}_{3 \varepsilon}=2 \mathrm{MTTF}_{2 \varepsilon}$. For example, $\mathrm{MTTF}_{1,90.9 \%}=500$ and $\mathrm{MTTF}_{3,90.9 \%}=2000$. The calculations were equivalent for $\operatorname{MTTR}_{1 \varepsilon}$ and $\operatorname{MTTR}_{3 \varepsilon}$.

Finally, based on the value of $MTTF_{2,90.9 \%}$, $\mathrm{MTTF}_{2,70 \%}$ was calculated by assuming that a lower efficiency will be the result of a proportionally shorter mean time to failure, whereas a higher efficiency will be the result of a proportionally higher mean time to failure. For instance,

$M T P F_{2,70 \backslash \%}=\frac{0.7}{0.909} M T P F_{2,90.9 \backslash \%}=770$

For parallel lines A and B, the levels (experimental values) are summarised in Table 1 below.

Table 1 Experimental factors and their levels

Thus, taking into account all levels for the 6 factors (considering $P_{A}$ and $P_{B}$ as two different factors), a total of $3^{*} 2^{*} 5^{*} 5^{*} 3^{*} 2+3^{*} 2^{*} 5^{*} 5=1050\left(\mathrm{~N}^{*} \mathrm{BC}^{*} \mathrm{PA}^{*} \mathrm{~PB}^{*}(\varepsilon < 1)^{*} \alpha\right.$ [for unreliable lines]$+\mathrm{N}^{*} \mathrm{BC}^{*} \mathrm{PA}^{*} \mathrm{~PB}$ [for reliable lines]) experimental points were studied.

Performance measures

Similar to Hillier's approach, a profit function ($Z$) was used to evaluate the performance in terms of both TR and ABL, whereby a unit produced by the system generates revenue ($r$), while an inventory unit stored in a period of time incurs a holding cost ($c_1$). Since additional expenses are often incurred to maintain certain levels of buffer capacity, investment and/or maintenance costs per average unit of buffer capacity per time unit ($c_2$) were also considered.

However, to simplify the analysis, both $c_{1}$ and $c_{2}$ were considered as relative values of $r$, leading to simplified versions of holding $\left(h=c_{1} / r\right)$ and investment/maintenance $\left(i=c_{2} / r\right)$ costs, resulting in the following profit function:

$Z=T R-h A B L-i B C$

Dunnett's $t$ test and Tukey's HSD test were carried out to statistically assess the differences among the experimental results. ANOVA tests were carried out to determine the statistical significance of each factor for the resulting TR and ABL. The 'R' package (The R Foundation 2016) agricolae was utilised to statistically analyse TR and ABL data.

Simulation run parameters

To generate representative simulation data, a suitable warm-up/transient period is needed to ensure that observations are very close to normal operating conditions. Law suggested running a preliminary system simulation, selecting one output variable for observation. A trial procedure for this system found that after an initial simulation run of 20,000 min, acceptable steady-state behaviour for TR was established. In this regard, all data gathered during the first 20,000 min were discarded, and 300 independent runs of 120,000 min each were carried out, excluding the first 20,000 min of non-steady state data. Thus, TR and ABL estimations presented in this paper are in fact the average values of TR and ABL over the 300 replications.

Moreover, to reduce experimental variance, specific random number streams were assigned to each random variable (factor) at each station, i.e. processing times, time-to-failure and time-to-repair probability distributions; and common random numbers were used for each stream throughout the 300 replications.