Decision making insights

Aside from the performance improvement shown in the previous subsection, we also analyzed the results from a managerial perspective. For this, all problem instances created, i.e., 42 have been solved in 16 combinations each and analyzed as described in the following paragraphs.

Costs of line feeding

To investigate the costs of feeding parts to an assembly line, we aimed to create a comparable cost measure by calculating the average costs of feeding a single item to the line. For this, we divide the overall costs by demand over all partsP

i∈Iλi.

Investigating differences in costs, we used a mixed-model two-way ANOVA test evaluating the significance of the impact of BoL length, space borrowing, moving vs. non-moving lines, family vs. part decisions, and all their combinations. A mixed-model ANOVA is used as the dataset consists of repeated measurements of multiple problem instances. Therefore, effects can be tested between different subjects, but also within the same subject. The assumptions for this statistical analysis, namely normal distribution and homogeneity of the variance-covariance matrices were tested and satisfied. A test of sphericity was not conducted as all observations have only two different levels (e.g., two different lengths of the BOL).

The effect of the factors described above is investigated on the results obtained from the best found solution but also on the lower bounds of costs, namely the LP-relaxed costs. The results of this analysis can be found in Table 3.7 and read as follows: If the length of the BoL or the amount of space that can be borrowed is increased, ↑ describes an increase in cost and ↓ a decrease in costs. For the other two categories it means switching from the first to the second option leads to an increase (↑) or decrease (↓) in costs. The effect size shows the interval of increase/decrease for all problem instances when switching from the first to the second option.

As one can observe, costs are relatively robust against changes in the factors tested: However, some things may be observed: (i) a longer border of line slightly decreases costs on average. But it should be noted, that an increase in the length of BoL can also increase costs. The actual effect depends on the dataset, i.e, the part, family, and station characteristics (ii) The allowance of space borrowing lowers costs in most cases. However, this effect is statistically not significant.

An explanation might be, that the amount of space borrowing allowed is not sufficient to lower the cost to a significant point. This also matches previous results of Schmid et al. (2018), where

1We used a 0.005 signifance threshold due to the standards proposed by Benjamin et al. (2018)

Table 3.7Measuring the effect of high level decisions on costs

Setting Observed costs Effect size [%] LP costs Effect size LP [%]

Length BoL ↓^∗ [−8.44; +8.11] ↓^∗ [−8.44; +6.46]

Space borrowing ↓ [−6.82;−0.06] ↓ [−6.05;−0.06]

Moving vs. non-moving line - [−1.81; +2.3] - [−1.81; +0.41]

Family vs. part decision - [−1.42; 0] - [−1.42;−0.002]

∗ ∗ ∗refers to a significance level ofp= 0.005,∗∗to a significance level ofp= 0.01 and∗to a significance level ofp= 0.05¹

significant cost reductions were only achieved by allowing space borrowing of up to 50%. (iii) Both other settings tested do not drastically influence the feeding cost. The two-way ANOVA also did not reveal any significant interaction terms. Therefore, we do not report on them in Table 3.7. Those results are especially relevant for future research as it allows to neglect the decision about a moving or non-moving line (when not considering assembly line balancing).

Moreover, problems can be solved on a family level without risking much higher costs. Lastly, the knowledge that interaction between these decisions are not significant allows for simpler experimental settings.

Figure 3.3Cost shares of feeding processes

Furthermore, the cost percentage of every process can be analyzed. The results can be seen in Figure. 3.3. As one can see, transportation clearly has the greatest share. This can be explained by the long distances in a factory as well as by lower density of transportation when kitting and sequencing are applied. Usage costs are relatively low with an average of 10% and a maximum of 22%. As the wage of assembly operators is assumed to be equal to the wage of logistical operators, it is obvious that in scenarios where assembly operators earn higher wages (for example in the automotive industry) this share would increase. But even with equal costs for assembly and logistics operators taking the usage costs into account is relevant as we are going to show in the next section.

Comparing to a stepwise approach

One may argue that usage costs do not play a significant role in this decision making process, as somewhat indicated in Figure 3.3. Therefore, we also tested decision making in a stepwise approach. For this, we assumed that usage costs are equal over all possible locations along the BoL. Solving this problem (using a simplified model) randomly assigns parts to any location. In a second step, we fixed the assignments of parts to policies resulting from the first stage and just optimized the location to be used for each and every part.

Figure 3.4Comparing a stepwise to the integrated approach

Figure 3.4 shows the relation between costs obtained from the stepwise approach in comparison to the approach proposed within this chapter. The left graph shows results when the minimum

costs over all locations are assumed for every part with respect to usage and replenishment costs, whereas in the right graph the average cost over all locations was assumed. As one can easily see, the integrated approach reduces costs in almost all cases (657/672 cases when using average costs in the first step; 650/672 cases when using minimum costs in the first step). In cases where no costs could be saved, the optimal solution could not be found within the time limit of 3600s and thus costs were marginally higher. Nevertheless, this shows that neglecting usage costs may increase costs by up to 7%. Another interesting observation is that comparing the results from both approaches (minimum versus average costs) were mostly close but never ended up in the exact same solution. Thus, it is also not clear a priori which approach should be chosen when solving the problem in two steps.

Making line feeding decisions

Our results indicate that all line feeding policies are used in most of the examined datasets.

Hence, one can conclude that consider all line feeding policies is key to minimize costs. Figure 3.5 shows the relative usage of line feeding policies over all datasets examined indicating median (dark line), first to third quartile (box) and the minimal and maximal values (whiskers) as well as outliers (circles) by means of a box plot.

Figure 3.5 Line feeding policy mix

One can observe a low usage of line stocking as a feeding policy. As line stocking is often considered to be the basic line feeding policy, this is quite surprising. Taking into account, however, that in this study often multiple models are produced on a single line, many parts are needed and therefore, space is becoming extremely scarce, this is an explainable result.

Furthermore, there are a few datasets where approximately 40% of the parts are line stocked.

This occurs in datasets with only one to two models being produced on a line or the number of parts being low due to a high overlap. The average percentage of parts in stationary kits is

around 10-15%, with a peak of around 25% of the parts. This may also be considered to be a surprising result as a lot of research in literature considers stationary kits to be an important line feeding policy (e.g., Lim`ere et al. (2012)). Thus, it may be stated that stationary kitting is a relevant concept, especially when multiple models are produced on a single line but it does not replace other feeding options such as sequencing and boxed-supply. Sequencing is also a relatively scarcely used concept as it mostly makes sense for extremely large part families. The most prevalent line feeding policies are boxed-supply and traveling kitting. However, one should note the large deviations in traveling kitting.

As described in Section 3.6.1, we used a factorial experiment design. This allows us to investigate the effect of the available space, space borrowing, moving lines and family vs. part decisions on the decisions for line feeding policies similarly to the cost analysis in the previous section. We used a mixed-model two-way ANOVA test and found that space borrowing and space along the line have the most significant effect on decision making. The use of a mixed-model ANOVA is motivated by the same reasons as discussed above and assumptions were tested and satisfied before the application. As interaction terms did not show to be significant, we do not discuss them here. Results can be seen in Table 3.8 and read as follows: If the length of the BoL or the amount of space that can be borrowed is increased,↑describes an increase in the use of a policy and↓a decrease. For the other two categories it means going from the first to the second option leads to an increase (↑) or decrease (↓) in the use of a policy. The differences in decision making can be seen in Figure 3.6 indicating the fraction of parts assigned to a line feeding policy in a violin plot.

Table 3.8Measuring the effect of high level decisions on line feeding policies Line Boxed-supply Sequencing Stationary Traveling

stocking Sequencing kitting kitting

Length BoL ↑^∗∗∗ ↑^∗∗∗ ↓^∗∗∗ ↓^∗∗∗ ↓^∗∗∗

Space borrowing ↑^∗ ↑^∗∗∗ ↓^∗∗ - ↓^∗∗∗

Moving vs. non-moving line - - - - ↑^∗∗∗

Family vs. part decision - - ↑^∗ - ↓^∗∗∗

∗ ∗ ∗refers to a significance level ofp≤0.005,∗∗to a significance level ofp≤0.01 and∗to a significance level ofp≤0.05

Figure3.6Linefeedingpolicymixdependingonfactorialdesign

Results shown in this section so far are rather on a macro level, depending on factory-wide decisions. However, one might also be interested in the way part characteristics lead to different policy assignments. Thus, we show some descriptive statistics of part characteristics on the feeding policy assignment in Table 3.9 by stating mean and standard deviation. These part characteristics may be physical properties such as part volume and weight. However, we also consider some organizational aspects such as the expected demand, and the number of parts that constitute a family. Lastly, we look at station characteristics such as the number of parts and part families that is required at a station.

Table 3.9Part characteristics of parts assigned to a line feeding policy

Line Boxed-supply Sequencing Stationary Traveling

stocking Sequencing kitting kitting

Part volume [m³ 0.032±0.14 0.005±0.007 0.13±0.28 0.008±0.015 0.002±0.006 Part weight [kg] 3.06±5.6 4.03±6.47 3.27±6.13 3.30±4.81 0.7±1.26 Demand [pcs.] 544.28±592.3 161.87±246.29 94.36±261.25 122.63±213.41 391.55±574 Parts in family [pcs.] 2.28±1.85 4.59±4.10 9.75±5.15 6.01±4.77 4.86±4.34 Parts at station [pcs.] 18.52±14.36 39.95±22.29 51.28±24.6 99.07±29.37 54.59±27.47 Families at station [pcs.] 7.79±4.93 13.24±6.42 14.85±6.33 24.88±7.11 16.63±6.76

From Table 3.9, one can see that line stocking and sequencing is mostly applied for relatively large and heavy parts. The other line feeding policies contain generally rather small parts.

Furthermore, traveling kits also contain rather light parts whereas boxed-supplied and stationary kitted parts can also be heavy. Demand seems to vary also drastically for the different line feeding policies. For example, line stocked parts have a low demand and sequenced parts have even lower demand. This makes also sense when looking at the number of parts within a family as demand is linked to this characteristic: the more parts there are in a family, the lower the demand of an individual part on average. The last characteristics are somewhat linked as a large number of families at a station usually results in a large number of families at that station. One can see that both kitting policies and sequencing are applied more often as the number of parts is strongly increasing. This can simply be explained by a lack of space.

Generally note, that the standard deviations are quite high which means that the mean of a characteristic might give an indication on the line feeding policy but can certainly not be consulted as a single criterion for decision making. Furthermore, some general decisions, such as space borrowing, might additional affect this decision (see Table 3.8) and thus, have to be considered.

Multiple models in line feeding

One of the main contributions of this chapter is the incorporation of multiple model assembly on a single assembly line. Obviously, this has an effect on the decision making of line feeding policies. We also investigated this by means of an ANOVA. Here, a regular ANOVA is sufficient as no within-subject effects need to be measured. As one can see in Table 3.10, decision making for all line feeding policies strongly depends on the number of product models and overlap of

families.

Table 3.10Measuring the effect of high level decisions on line feeding policies Line Boxed-supply Sequencing Stationary Traveling

stocking kitting kitting

Number products ↓^∗∗∗ ↓^∗∗∗ - ↑^∗ ↑^∗∗∗

Overlap ↑^∗∗∗ ↑^∗∗∗ ↓^∗∗∗ ↓^∗∗∗ ↓^∗∗∗

Products↔Overlap − ↓↓^∗∗∗ − - ↑↓^∗∗∗

∗ ∗ ∗refers to a significance level ofp≤0.005,∗∗to a significance level ofp≤0.01 and∗to a significance level ofp≤0.05

The results indicated in Table 3.10 can also be observed in Figure 3.7, representing the average usage of line feeding policies in dependency of number of product models and overlap. ↑describes that increasing overlap or the number of products is proportional to the usage of a line feeding policy and↓denotes that changes behave anti-proportional. For the interaction effect of overlap and product number, we use the notation of two arrows. The first one shows, starting from the base case (no overlap, one product) if an increase in both leads to an increase (↑) or a decrease (↓) in the use of a line feeding policy. The second arrow shows if the value is higher (↑) or lower (↓) than the expected effect. The violin plots show the distribution of averages weighted with the number of occurrences as well as their median. E.g., an increasing overlap leads to more space consuming line feeding policies such as line stocking and sequencing. This can be explained by the lower number of parts that are required per station. The requirement for space saving policies is decreasing drastically with increasing overlap. This can be observed especially for boxed-supply and stationary kitting, but to a minor extent also for traveling kitting.

On the contrary, an increase in models leads to an increase in the assignment to kits, both stationary and traveling, but also to boxed-supply. A remarkable result is the slight increase in line stocking when two product models are produced on a single line. This might be explained by datasets with high overlap or a small number of parts at some stations.

Space borrowing

Another novelty of this model is the possibility that stations may borrow space from adjacent stations. As indicated in Table 3.7 the allowance of space borrowing does reduce the observed costs on average. The extent, however is not always very large. This might be explained by the relatively small allowance for space borrowing, i.e., 25% of the preceding and succeeding stations, chosen due to computational tractability. Therefore, increasing the possibilities for space borrowing might lead to a stronger decrease in costs. In Schmid et al. (2018) we could observe that doubling the allowance for space borrowing does increase cost savings to a larger extent.

In Figure 3.8 it is indicated how much space, measured in number of locations, is used by stations on average. One can observe, that all locations are used all the time if no space borrowing is allowed. This is easily explainable as a reduction in space consumption usually requires more

Figure 3.7Line feeding policy mix depending on dataset characteristics

expensive line feeding policies. Thus, the available space is almost always used in entirety.

The allowance of space borrowing on the other hand, indicates that stations have a higher variety in using space when space borrowing is allowed. Every station is still using some of its locations and at no station all parts are fed with a traveling kit. The distribution of locations used looks like it is normally distributed with its mean being equal to the maximum number of locations per station. The amount of space borrowing seems quite high. Lastly, it can be observed that all locations are used as well when allowing space borrowing.

Moving lines vs. non-moving lines

In assembly systems, assembly lines may be designed to move during the assembly process or to be standing still. As shown at the beginning of this section, this decision does neither influence the cost to a significant amount nor the decision for line feeding policies (see above). However, it does influence the use of locations for line feeding policies. As shown in Figure 3.9, this decision does drastically influence where stationary kits are positioned along the BoL. The figure shows for every relative location of a station, i.e., the position of a location in relation to other locations of the same station, how it is used. For every line feeding policy the values of the bars show how

Figure 3.8Space usage in relation to regular station size

many parts are assigned to a specific feeding policy at that relative location. There are a few things to observe here. Starting with the use of locations that do not belong to a station but can be borrowed by that station. Those locations (0,6,7 for short lines; -1,0,9,10 for long lines) are mostly used for line stocking and boxed-supply. Stationary kits are almost never assigned to these locations. Line stocked parts seem to be rather assigned to earlier locations along a station.

This might be explained when considering that line stocked parts are often high demand parts, which makes it reasonable to position them as close as possible to the required point to reduce walking. The left skew is likely a result of the left skewed demand points for parts at moving lines as parts are needed as soon as the product enters the station but no parts are needed at the very end of a station. Stationary kits are mostly placed in the center of the station which facilitates a relative short walking distance on average if the demand points for parts in the kit are spread over the entire station or agglomerate in the center. Surprisingly, this is not true for long non-moving lines. In those cases, the central locations are mostly used for boxed-supply and line stocking. This may be explained by a different constellation of kits as demands for parts in the kits are higher (120 vs. 100). At the same time fewer parts are in the kit as part volumes are almost twice the volume in comparison to a short line (0.0082 vs. 0.0049m³). Sequenced parts

seem to be spread quite equally in a flat U shape over all locations for moving lines, whereas at non-moving lines sequenced parts are rather distributed in a V shape as central locations are used for boxed-supply and line stocking.

Figure3.9Usageofrelativelocationfordifferentlinefeedingpolicies

Another interesting observation w.r.t. the positioning of sequenced parts can be made: In non-moving lines, sequenced parts are positioned mostly in the center and fade slightly out towards the sides. In moving lines in contrast, the center of the station seems to be the least favorable position. This could be explained by the usage of those positions for boxed-supply which should be positioned centrally. Moreover, they may contain parts for multiple operations throughout the station.

Family versus part decisions

Based on Table 3.7 one can conclude that costs are not changing heavily when parts are assigned to line feeding policies on a part level instead of a family level. This might be explained as only relatively few families are supplied using multiple feeding policies and that splitting a family into multiple policies requires additional space unless some parts in the family are box-supplied and

Based on Table 3.7 one can conclude that costs are not changing heavily when parts are assigned to line feeding policies on a part level instead of a family level. This might be explained as only relatively few families are supplied using multiple feeding policies and that splitting a family into multiple policies requires additional space unless some parts in the family are box-supplied and