Buffer Capacity

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.