IMAP-VISION

We compare the presented models in terms of travel time, allocation of the storage lanes and rearrangement characteristics.

Figure 4.21 shows the surface representing the travel time of the general model depending on P(SSRR) and the filling level. The surface representing the travel time for the model with the modified tango (formula 4.70) shows the same behavior and is not shown. The total difference in travel time is small for varying P(SSRR). In general, we observe the longest travel times for P(SSRR) = 0 and the shortest travel times for P(SSRR) = 1. The difference arises from changed mean load handling times for storage and retrieval as well as changed rearrangement and tango effort. While the difference from load handling times is minor, the differences in rearrangement times, both regular rearrangements and tango, makes up the largest part of the total difference observed. The reason for this behavior is the greater amount of tango rearrangements, occuring with increasing P(SSRR).

Time in seconds

Figure 4.21: Travel time in seconds depending on P(SSRR) and the filling level

An increasing filling level causes an increase of the rearrangement time, while the tango time remains unchanged. This explains the constantly in-creasing difference in travel time between the limit values of P(SSRR)= 0 and P(SSRR)= 1 for a raising filling level. Consequently, the difference is greatest for highest filling levels.

Time in seconds

P(SSRR) P(SSRR) P(SSRR)

Filling level 80% Filling level 90%Filling level 90% Filling level 99%

Figure 4.22: Travel time of the general model and the modified tango model depending on P(SSRR) for different filling levels

Figure 4.22 shows a comparison of the travel time for the general model and the modified tango model over P(SSRR) and for three different cases of filling levels. The travel time of the model with the modified tango

consis-tently lies below the travel time of the general model. With an increasing amount of tango (increasing P(SSRR)), the travel time difference to the gen-eral model increases. However, the maximum difference gets smaller the higher the filling level becomes. While for small and medium filling levels, the chance to perform the modified tango is relatively high, it lessens with increasing filling level, causing the convergence of the modified model and the general model.

On a quantitative basis, the general model performs best with increasing amount of tango rearrangements, i.e., increasing P(SSRR), for the chosen parameter configuration. However, differences in travel time are rather small and will not create significant variances in throughput, which is why we subsequently discuss whether a greater amount of tango is always preferable and should be aimed for.

Evaluation for changed parameters: Role of the Tango

For P(SSRR) = 1, shorter travel times are observed due to smaller rearrange-ment times. Figure 4.23 illustrates the mean rearrangerearrange-ment time, both for the general model with P(SSRR) = 1 and P(SSRR) = 0, as well as the tango time. For all filling levels, the time needed to perform a tango rearrange-ment is smaller than any rearrangerearrange-ment time.

Filling level

Time in seconds

Figure 4.23: Comparison of mean rearrangement time and tango time for basic and SSRR model

With such constellations, an operating strategy with a greater amount of tango leads to shorter travel times. With the modified tango, this situation is further enhanced. However, whether a higher amount of tango rearrange-ments leads to an improved performance is dependent on specific config-uration parameters. Important drivers in this context are

• The access time to the rear position (t_{LH D,r}).

• The rearrangement distance, which itself is depending on the filling level and the stationary allocation of the storage lanes.

To explain the first driver, consider a (rather slow) load handling device, which needs twice the time to access the rear storage position than the front storage position, hence ^t_t^{LH D,r}

LH D, f = 2. In this case, tango is disadvanta-geous, because the re-storing is always performed to the rear position of the storage lane, while regular rearrangements allow re-storing into a free front position as well. Additionally, for high filling levels, the ratio of rear-rangements into front positions increases. This situation can be analyzed depending on the state of a storage lane that is chosen for rearrangement, on the one hand, and the time needed to travel to the respective position on the other hand.

State of the storage lane chosen for rearrangement

Condition for favor of tango

E (^{E (RC )}/2) − td> tLH D,r− tLH D,r

H (^{E (RC )}/2) − td> tLH D,r− tLH D, f Table 4.11: General condition for tango to have an advantage over regular rearrangements for

the different states of the rearrangement storage lane

The first inequation in Table 4.11 is always true. Compared to rearrange-ments into an empty storage lane, a tango is always beneficial, as the time needed to travel to the rearrangement position (^{E (RC )}/2) can not fall below the positioning time the LHD requires during the tango (t_d). For a half-filled storage lane, it depends on the difference between the access times to the front and the rear position of a storage lane as well as the time needed to travel to the chosen rearrangement position. The larger the difference

LH D,r− tLH D,r, the higher is the chance a regular rearrangement is prefer-able over a tango. Figure 4.24 and 4.25 illustrate the comparison in further detail. Remember, for high filling levels, the ratio of rearrangements into half-filled storage lanes increases. Figure 4.24 shows the comparison ap-plied in Figure 4.23 with a changed access time to the rear position. t_{LH D,r} is increased by one second to 6.5 seconds, whereas all other parameters remain unchanged. We see, in this situation the tango rearrangement is only beneficial for extreme filling levels (>95%). In Figure 4.25, this com-parison is generalized by showing the same terms depending on the ratio

tLH D,r

t_{LH D, f} as a variable with the filling level set to 90%. The ratio in the changed situation (tLH D,r = 6.5) is 1.44 ( =^6.5/4.5), while in the former configuration it was 1.22. This shows that the benefit of the tango depends on the load handling times.

Filling level

Time in seconds

Figure 4.24: Mean times for rearrangement and tango with increased tLH D,rof 6.5 seconds

Utilization of the modified tango makes the tango’s benefit more resilient towards an increased ratio of_t^{tLH D,r}

LH D, f. Figure 4.25 shows that the mean time of the tango rearrangement in the modified tango model is more robust with increasing relations of ^t_t^{LH D,r}

LH D, f than the standard tango. The purple line intersects the lines for regular rearrangements later, i.e., for higher ratios, than the black line. Recall the example of the slow LHD with a ratio of

t_{LH D,r}

tLH D, f = 2. In this situation, both tango variants are outperformed by

reg-ular rearrangements, as we can observe both tango lines being above the rearrangement lines in Figure 4.25.

However the quantitative result may not be the only fact considered when addressing the employment of tango. Tango offers the following additional advantages that are independent of the discussed parameters:

• Tango is independent of storage assignment policies. For tango re-arrangements, no additional rearrangement position is needed. This is of increasing interest for very high filling levels or in cases in which an available position for a rearrangement may be further away than the mean analytical distance, e.g. for turnover based storage assign-ment, with SKU identical lanes or any kind of dedicated storage.

• Tango causes horizontal movements only. Regular rearrangements require horizontal and vertical movements and thus are more energy-consuming, as lifting in general needs more energy than driving (Braun 2016, p.302 f.).

• Tango produces more half-filled storage lanes on average. This is beneficial for the overall performance, as half-filled storage lanes serve as potential rearrangement positions and prevent rearranging.

This fact is already included in the quantitative results and therefore also in our analysis. However it may effect other policies that require potential storage lanes, e.g. some of the operating policies discussed in the following chapter.

QC with greater amount of tango appear most efficient, especially when including the modified tango and the additional advantages. We conclude that it is generally favorable to allow for the greatest possible amount of tango.

Transfer of the models to other LHD configurations

To end this section we shortly discuss the relaxation of the LHD’s design condition. In section 4.1.1 we formulate the assumption that the LHDs are horizontally arranged, having the same distance between each other as two storage lanes.

Time in seconds

𝑡𝐿𝐻𝐷,𝑟 𝑡𝐿𝐻𝐷,𝑓

Figure 4.25: Mean times for rearrangement and tango depending on the relation of t_{LH D,r}and t_{LH D, f} for a filling level of 90 %

Vertical Arrangement of the LHDs A possible relaxation of the LHD design is a vertical arrangement of the two load handling devices above each other. All models presented in this chapter are still valid with this as-sembly. Tango is still possible, where the difference is that the LHD moves vertically instead of horizontally. Figure 4.26 shows the lateral view of the tango for this case. If a storage lane either above or below the lane of re-trieval unit has an empty position, the modified Tango is possible, which is why also that model is valid.

a b c

1

d

Storage lane Load

handling device

1 1 1

Mast

Figure 4.26: Tango for vertical arranged LHDs

Varying distance between the LHDs A possible relaxation is that the distance between the two load handling devices is changed. The QC in the general model is not affected by this change and Tango is possible. The difference is that the time needed to perform a Tango changes according to the actual distance between the LHDs. Figure 4.27 shows how a tango rearrangement works in this case.

a b c

1 1 1

d

1 Storage

lane

Load handling

device

Figure 4.27: Tango for changed distance between LHDs

We can see that only one of the LHDs is positioned in front of a storage lane.

As a consequence, the modified Tango is not possible. The same implica-tions hold for the case of vertically arranged LHDs with a changed distance between each other.