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.

To show the global effect of each pattern on TR and simplify the analysis, average TR results for experiments with different $P_{A}$ were calculated. For instance, results for a balanced $P_{A}(-)$ were calculated as the average TR of experiments with patterns $(-,-),(-, /),(-,),(-, \Lambda$, and $(-, \mathrm{V})$. Furthermore, as TR results have very different magnitudes for different $\varepsilon$ and $\alpha$ values, Fig. 2 shows "normalised" TR results (nTR), which were calculated by dividing the average TR results of each $P_{A}$ by the average overall TR value of all experiments with the same $N, \varepsilon, \alpha$ and $\mathrm{BC}$ values. It is worth noting that results regarding $\mathrm{P}_{\mathrm{B}}$ are not shown as they were equivalent. Full TR results with their corresponding Dunnett's and Tukey's tests results can be found in the "A ppendix", Tables 4 and 5 , respectively.

Fig. 2 nTR results for $P_A$ with $a N = 5$, $b N = 8$ and $c N = 11$

Results shown in Fig. 2 suggest that the performance of buffer allocation patterns is highly dependent on the values of $N, \varepsilon$ and $\alpha$. For example, the ascending $P_{A}$(/) performed very well with $\varepsilon=70 \%$, which was only outperformed by the balanced pattern when $B C = 6$ and $\alpha < 3$ for lines with $N \geq 8$. Figure 2 also suggests that the balanced pattern performs better with increasing values of $\mathrm{N}, \mathrm{BC}$ and $\varepsilon$, while the ascending pattern performs better with decreasing values of $N, B C$ and $\varepsilon$, and increasing values of $\alpha$. These results are also confirmed by Tables 4 and 5, as experiments with the pattern (/ , /) were found to have statistically significant differences with the control (balanced pattern) only when $\varepsilon = 70 \%$. Thus, increasing values of $\varepsilon$ resulted in lesser relative differences among the patterns, suggesting that the effect of buffer allocation patterns on TR is highly dependent on the reliability of the machines.

On the other hand, the descending $P_{A}$ ( \ ) was the worst pattern in terms of TR for all scenarios, a result confrmed by Tables 4 and 5 as experiments with the pattern (\ , \) had the lowest TR in all experimental conditions. Interestingly, the bowl ($V$) and inverted bowl patterns ($\Lambda$) changed their relative performance with increasing $N$ values, since the ($\Lambda$) pattern was almost on par with the good performance of ($-$) when $N = 8$ for some scenarios; whereas the ($V$) pattern seemed to have an overall good performance for scenarios with $\mathrm{N}=11, \mathrm{BC}=6$ and $\alpha \geq 2$.

Similar to Fig. 2, Fig 3 shows the normalised ABL results ($nABL$) showing the relative performance in each experimental scenario (different values of $N, \varepsilon, \alpha$ and $BC$) per $P_A$.

Fig. 3 $nABL$ results for $P_A$ with $a N = 5, b N = 8$ and $c N = 11$

Figure 3 suggests that ABL results are more consistent than TR results in terms of $P_A$ performance, since the ascending pattern is always the best-performing (with lower ABL) and the descending pattern is always the worst. Tables 6 and 7 in the "Appendix" confirm this conclusion by showing that the best pattern in terms of ABL for all scenarios is (/,/), while the worst pattern for almost all scenarios is (\,\). Contrary to the interaction effect between $P_A$ and $\varepsilon$ on TR, higher values of $\varepsilon$ resulted in a higher influence of $P_A$ on ABL, especially with lower values of $N$.

The Analysis of Variance test (Table 2) shows that both reliability-related factors ($\varepsilon$ and $\alpha$) have the highest influence on $TR$, followed by $BC$ and the interaction between $\varepsilon$ and $\alpha$. As seen in Figs. 2 and 3, the interactions between $BP$ and $BC$, $N$, $\varepsilon$ and $\alpha$ are also significant, albeit they have a lower effect on $TR$ when compared to single factors. For ABL, $BC$ is the most important factor, followed by $BP$ and the interaction between $BP$ and $BC$. Therefore, the performance of ABL is more dependent on selecting a good (bad) pattern.

Table 2 Analysis of variance test for TR and ABL

While TR and ABL results are relevant in isolation for some firms with a concern for either maximising revenue or minimising inventory costs, most firms are more interested in finding a balance between revenue and costs via a profit function. For this reason, studying the effect of buffer allocation patterns on the combined performance of TR and ABL provides a deeper insight into the implications of unbalanced buffer allocation.

Consequently, the profit function ($Z$), defined in Eq. (3), was used to study the combined performance of TR and ABL, as it takes into consideration inventory holding costs and buffer capacity investment/maintenance costs while generating revenue via the production rate. Therefore, the best pattern for different values of $N, \varepsilon$ and $\alpha$ will be the pattern which sufficiently increases TR, outweighing the costs of ABL and BC.

$Z_{BPmaxTR}=TR_{BPmaxTR}−hABL_{BPmaxTR}−iBC=Z_{BPminABL}=TR_{BPminABL}−hABL_{BPminABL}−iBC$

where $Z_{B P_{\max T R}}$ is the profit resulting from the buffer allocation pattern attaining the maximum TR, $T R_{B P_{\max T R}}$ is the maximum TR per scenario, $A B L_{B P_{\max T R}}$ is the ABL resulting from the BP that reached maximum TR, BC is the average buffer capacity for a particular experimental scenario; $Z_{B P_{\min A B L}}$ is the profit obtained from the BP with minimum ABL, $A B L_{B P_{\min A B L}}$ is the minimum ABL per scenario, and $T R_{B P_{\min A B L}}$ is the TR generated from using the BP which resulted in the minimum ABL.

Since the term $iBC$ is equal for both sides of the equation when $BC$ is equal among experiments; then,

$h_{0}=\frac{T R_{B P_{\max } T R}-T R_{B P_{\min A B L}}}{A B L_{B P_{\max } T R}-A B L_{B P_{\min A B L}}}$

This means that when $h > h_{\circ}$ for a particular manufacturing environment, $BP_{minABL}$ has a higher $Z$ than $\mathrm{BP}_{\operatorname{maxTR}}$ because the system is better off minimising holding costs, as they are too high to overcome with higher TR; whereas when $h < h_{\circ}, \mathrm{BP}_{\max \mathrm{TR}}$ will result in higher $Z$ than $\mathrm{BP}_{\min A B L}$ because inventory holding costs are not as penalising. For example, an $h_{\circ}= 0.019$ shown in Fig. 4a for $\mathrm{N}=11, \mathrm{BC}=6, \varepsilon=70 \%$ and $\alpha = 1$, means that if $h=0.02, \mathrm{BP}_{\min A B L}$ $(/ , /)$ has a higher $Z$ than $\mathrm{BP}_{\max \mathrm{TR}}(-,-)$ since

$Z_{(/ /)}=0.2812-0.02(2.7279)=0.2266 > Z_{(-,-)}=0.3010-0.02(3.7827)=0.2253$; whereas if $h=0.018, \mathrm{BP}_{\min \mathrm{ABL}}(/ /)$ has a lower $Z$ than $\mathrm{BP}_{\operatorname{maxTR}}(-,-)$ since

$Z_{(/ /)}=0.2812-0.018(2.7279)=0.2321 < Z_{(-,-)}=0.3010-0.018(3.7827)=0.2329$

Fig. 4 $a h_{\circ}$ and $b i_{\circ}$ values for all experimental scenarios

$TR_{{BPminABL,{\text{BC}}_{2} }} - i_{0} {\text{BC}}_{2} = TR_{{BPminABL,{\text{BC}}_{6} }} - i_{0} {\text{BC}}_{6}$

where $TR_{BPminABL,BC2}$ is the $TR$ reached for a $BC = 2$ while applying $BP_{minABL}$ and $TR_{BPminABL,BC6}$ is the $TR$ reached for a $BC = 6$ using $BP_{minABL}$ for a specific scenario.

Therefore,

It is worth noting that all variables in Eqs. (6) and (7) consider experimental scenarios with equal values of $N, \varepsilon$ and $\alpha$. Moreover, $BP_{minABL}$ was selected as the pattern considered for these equations because the BP with minimum experimental ABL produces the highest $Z$ when $h = 0$.

Thus, lower $i_{\circ}$ values for a given set of $N, \varepsilon$ and $\alpha$ values represent higher penalties for investing in higher buffer capacity, while higher $i_{\circ}$ values depict lower profit penalties with high $i$ values. Similarly to $h_{\circ}$, a higher value of $i$ than $i_{\circ}$ for a given line configuration suggests that it is more profitable not to invest in higher buffer capacity, i.e. stay at $\mathrm{BC}=2$; whereas a lower value of $i$ than $i_{\circ}$ means that it is profitable to invest in the additional 4 buffer spaces, i.e. a $\mathrm{BC}=6$. For instance, an $i_{\circ}=0.010$ shown in Fig. 4b for $\mathrm{N}=5, \varepsilon=70 \%$ and $\alpha=3$, means that a buffer investment/maintenance cost of $i=0.011$ results in a decision of staying with $\mathrm{BC}=2$ as $Z_{B C=2}=0.2055-0.011(2)=0.1835 > Z_{B C=6}=0.2438-0.011(6)=0.1777$; whereas if $i=0.009$, then $\mathrm{BC}=6$ is more profitable than $\mathrm{BC}=2$ since

$Z_{B C=2}=0.2055-0.009(2)=0.1875 < Z_{B C=6}=0.2438-0.009(6)=0.1898$.

Results from Fig. 4a show that higher values of $N$ resulted in higher values of $h_{\circ}$, which suggests that patterns that increase TR are more relevant for longer lines than for shorter lines; whereas patterns that reduce ABL produce better overall results for shorter lines in terms of $Z$. Furthermore, higher values of $\varepsilon$ and lower values of $\alpha$ (shorter MTTF and MTTR) result in higher values for $h_{\circ}$, suggesting that higher machine reliability results in a reduced impact of inventory holding costs.

Further analysis of Fig. 4 a shows that scenarios with smaller buffer capacity $(\mathrm{BC}=2)$ have lower values of $h_{\circ}$, suggesting that profit in these scenarios is highly penalised by ABL and that a pattern that reduces ABL produces higher $Z$ for most inventory holding costs values. The opposite is true for scenarios with $\mathrm{BC}=6$, since $Z$ is less penalised by $\mathrm{ABL}$ as $\mathrm{BP}_{\text {maxTR }}$ (the pattern producing the maximum TR) results in a higher $Z$ even for higher values of $h$. This suggests that the extra TR produced by the extra ABL (resulting from higher BC capacity) allows for $\mathrm{BP}_{\max T R}$ to be more relevant when $\mathrm{BC}=6$. An $h_{\circ}=0$ indicates that the corresponding scenario, e.g. $N=5, \varepsilon=70.0 \%, \alpha=2$ and $B C=2$, produces the highest profit by selecting the buffer allocation pattern that reduces $\mathrm{ABL}$, irrespective of the value of $h$.

The only exception to this general $h_{\circ}$ behaviour occurs in scenarios with reliable merging lines $(\varepsilon=100 \%)$, or with $N=11, \varepsilon=90.9 \%$ and $\alpha=1$, since $h_{\circ}$ is higher for scenarios with $B C=2$ than for experiments with $\mathrm{BC}=6$. This might be due to the fact that the added $\mathrm{ABL}$ produced by higher BC capacity does not result in a sufficiently additional TR to overcome the inventory holding costs. Therefore, for reliable lines with $\mathrm{BC}=2, \mathrm{BP}_{\max T R}$ is more relevant than $\mathrm{BP}_{\min A B L}$ with respect to $Z$.

Results regarding $i_{\circ}$ (see Fig. 4b ) suggest that higher values of $\alpha$ (longer MTTF and MTTR) produce a higher investment/maintenance penalty (lower $i_{\circ}$ values) for systems with higher buffer capacities, which means that the additional throughput produced by the added buffer capacity is more cost-effective for shorter MTTF and MTTR than for longer ones. Similarly, systems with $\varepsilon=90.9 \%$ were less penalised in terms of profit by higher buffer capacities than systems with $\varepsilon=70 \%$, suggesting that the extra throughput produced by the increased buffer capacity is more cost-effective when reliability is $90.9 \%$ than when reliability equals $70 \%$.

An exception to this observation occurred for the reliable merging lines results, as $i_{\circ}$ values for reliable lines were lower than for experiments with $\varepsilon=90.9 \%$ and $\varepsilon=70.0 \%$ with $\alpha=1$, suggesting that even with low buffer capacity investment/maintenance costs, small buffer capacities will be better in terms of profit performance for reliable lines than larger buffer capacities. This result might be caused by the fact that the relative difference between the throughput generated by lines with small and big buffers is smaller for reliable lines than for unreliable ones. For instance, considering $N=5$ and a balanced $B P(-,-)$, the increase in $T R$ between a line with $\mathrm{BC}=2$ and a line with $\mathrm{BC}=6$, considering $\varepsilon=70 \%$ and $\alpha=1$, is $28 \%$; whereas for a reliable line with $\varepsilon=100 \%$, the relative increase in TR between a line with $BC = 2$ and a line with $\mathrm{BC}=6$ is only $4 \%$ (see Table 4 in the "Appendix").

Finally, to investigate the relationship between $h$ and $i$ values in terms of the profit function for different buffer capacity investments levels, Fig. 5 shows a comparison of the suface plots of $Z$ between merging lines with $BC = 2$ (in blue) and merging lines with $BC = 6$ (in orange) for various values of $h$ and $i$, taking $N = 5$ as an example.

Fig. 5 $Z$ function surface plots for $\mathrm{BC}=2$ (blue) and $\mathrm{BC}=6$ (orange) for different values of $h$ and $i$ considering experiments with $\mathrm{N}=5$ and $\mathbf{a} \varepsilon=70.0 \%, \alpha=1, \mathbf{b} \varepsilon = 70.0 \%, \alpha=2, c \varepsilon=70.0 \%, \alpha=3, d \varepsilon=90.9 \%, \alpha=1, e \varepsilon=90.9 \%, \alpha=2, f \varepsilon=90.9 \%, \alpha=3$, and $g \varepsilon=100 \%, \alpha=0$ (colour figure online)

Figure 5 shows similar results than those for Fig. 4b by suggesting that highly unreliable scenarios with longer MTTF and MTTR (e.g. Fig. 5c) scenario are more sensitive to higher costs than moderately unreliable scenarios with shorter MTTR and MTTR (see, e.g. Fig. 5d). Thus, a bigger buffer capacity ($BC = 6$) is only more profitable than a smaller buffer capacity ($BC = 2$) when very low costs (both $h$ and $i$) are present and when lower unreliability exists. Again, a reliable system is the exception, as a smaller buffer capacity is almost always more profitable in reliable scenarios (see Fig. 5g: the blue surface ($BC = 2$) is "above" the orange surface ($BC = 6$) for most of the values of $h$ and $i$).

Note that results pertaining to merging lines with $N = 8$ and $N = 11$ are not shown as they are very similar to the ones presented in Fig. 5 and follow the same general pattern as in Fig. 4, i.e. the profit in longer lines is less penalised by inventory-related costs than for shorter lines. Furthermore, in order to reach the highest possible $Z$ values for Fig. 5, the best pattern for the corresponding $h$ values was considered. That is, for values lower than $h_{\circ}$, $BP_{maxTR}$ was used in the calculation of $Z$, whereas for values higher or equal to $h_{\circ}$, $BP_{minABL}$ was considered.

An overall analysis of Sect. 5's results shows that the performance of buffer allocation patterns is highly dependent on the configuration of the system. The ascending unbalanced buffer allocation pattern (/) led to better TR performance with shorter, more unreliable lines; whereas the balanced pattern attained better TR performance when longer, more reliable lines were considered. These results suggest that different degrees of unreliability do influence which particular pattern is the best-performing, as question (1) of the study postulated.

Concerning ABL, the system's configuration did not influence the best-performing pattern. The ascending pattern was found to be the best pattern for all scenarios and the descending pattern was found to be the worst pattern for almost all scenarios.

In contrast, the effect of buffer allocation patterns on ABL was found to be much higher on reliable lines than on unreliable lines, showing a higher effect in shorter lines than in longer lines (see the comparison between Fig. 3a, c). Results from the Analysis of Variance test (Table 2) show that, overall, buffer capacity and buffer allocation pattern are the most important factors in the resulting ABL. Interaction effects such as buffer allocation pattern with reliability and length ($BP:N:\varepsilon$) were also found to be statistically significant.

Thus, addressing question (2) of the study, results suggest that the relative impact of buffer allocation patterns for TR is low when compared with the impact of reliability, but is moderately high for ABL, since a good or bad pattern will significantly affect the average buffer content of a merging line. It was also found that all of the experimental factors considered in this study are statistically significant, confirming the results of previous studies regarding the effect of $N, BC, ε, α$ and $BP$ on TR.

Section 6 presents very interesting results on the impact of inventory holding costs and buffer capacity investment/maintenance costs on the profit of different merging line configurations. In this regard, shorter merging lines with higher levels of machine unreliability and longer MTTF and MTTR were found to be particularly sensitive to inventory-related costs since the values of both $h_{\circ}$ and $i_{\circ}$ were commonly close to zero. These results answer question (3) and confirm that merging line characteristics influence the performance of merging lines when considering inventory-related costs. Shorter, highly unreliable lines with long MTTF and MTTR tended to achieve a higher profit performance with a smaller buffer capacity (due to the trade-off between additional buffer capacity costs and the additional TR generated by that capacity), and with buffer allocation patterns that reduced ABL (due to their comparatively high levels of average inventory content). Therefore, it can be reasonably concluded that increasing average buffer capacity per station from 2 to 6 in a merging line is more cost-effective for lines with a machine efficiency of 90.9%, shorter MTTF and MTTR, and longer lines than it is for lines with 70% machine efficiency, longer MTTF and MTTR, and shorter lines. Despite these results, reliable merging lines were found to benefit the least by increasing buffer capacity, for the majority of inventory-related values, as only for near-to-zero inventory-related costs did a buffer capacity of 6 produce higher profit than a buffer capacity of 2 (see Fig. 5g).

Regarding the impact of costs on buffer allocation patterns, the results of the current study agree with those of Hillier, who showed that buffer capacity should be significantly reduced and allocated towards the end of the line when inventory holding costs were high (e.g. $h = 0.1$). Thus, investing in additional buffer capacity to increase throughput seems to be subject to the "Law of diminishing returns", according to the more general "Theory of performance frontiers", since investment in extra buffer capacity is seemingly only effective when both inventory holding costs and buffer capacity investment/maintenance costs are quite low (see, e.g. Fig. 5).

Many of the results from the current study confirm and extend previous results which found that inventory holding costs had a higher impact on the profit function of unreliable lines than on the profit of reliable lines, but some results contrast with previous research by finding that reliable lines were more affected by buffer capacity investment/maintenance costs than unreliable lines. These results also contribute new understanding by showing that the profit function of lines possessing machines with longer MTTF and MTTR was more highly impacted by both inventory holding costs and buffer capacity investment/maintenance costs than was the profit of lines having machines with shorter MTTF and MTTR.

These results suggest that managers, working in industries which produce goods through merging lines (e.g. automotive, electronics, window and door factories with unreliable machines, stochastic processing times, and short line lengths, should consider unbalancing their buffers towards an ascending pattern. Furthermore, firms with high inventory-related costs, particularly those working in reliable machine industries, should be extremely careful when investing in additional buffer capacity, because the revenue gained through additional buffer capacity seldom covers the costs of obtaining that extra capacity. In practical terms, this means that industrial sectors with scarce inventory space (hence a high cost of expansion) and/or very high product value (hence higher inventory holding costs should conduct serious cost–benefit analyses when considering buffer capacity expansion and line design (i.e. buffer allocation), particularly in reliable environments.

Limitations of the study and future research

All methodologies have limitations, so conclusions from this study are only applicable to the simulated experimental points and care should be taken when generalising the results. Despite this, these results provide value in supporting a better understanding of the impact of unreliability and inventory-related costs on the performance of uneven buffer allocation patterns. More experimental points need to be explored in future research extensions, particularly regarding the average buffer capacity per station, to better understand the benefits of buffer capacity and average inventory levels on the performance of merging lines under different costs profiles.

The study's considerations on different machine efficiency values and various lengths of mean time to failure and to repair assumed a balanced allocation of these factors along the line, i.e. balanced unreliability. However, since it has been suggested that unreliability patterns have a significant effect on the performance of single serial lines, further research is needed to understand the influence of unreliability allocation on the performance of merging lines.