Assumptions

In our analytical model, we consider only one side of the aisle and make the following assumptions:

Rack shape

The rack is rectangular and has a fixed number of storage lanes; each lane consists of two storage positions. This means the number of storage posi-tions (l^∗) at one side of the rack is:

l^∗= lhor i zont al· lver t i c al· 2 (4.1) lhor i zont al and lver t i c al are the number of lanes in horizontal and vertical direction, respectively. The number of storage lanes is l =^l∗/2. We assume the rack to be square-in-time, i.e., with a shape factor of 1.

I/O position

The I/O point is located at the bottom left corner of the rack.

Dwell times

We include a term for dwell times that is added to the path-depending travel times. Unless stated otherwise, we assume the following:

• Two t_{d ead}per access cycle of the load handling device.

Storage policy

A random storage assignment policy is applied. For retrieval, any occupied position has an equal probability to be selected. Therefore, in a fully oc-cupied storage lane, both units have the same selection probability. For storage, any storage lane that is not fully occupied has an equal selection probability. Units are always stored in the rearmost position, which is why it is equally likely to choose an empty storage lane or a half-filled storage lane. Since the lanes are selected randomly, fully occupied lanes can also exist below filling levels of 50%.

Storage lane allocation

Storage lanes that are only occupied in the front position are not possible.

This remains valid during rearrangements. The state half-filled front in Fig-ure 4.1 is excluded in our model. For practical applications, this state would only make sense for low filling levels to save handling time. Otherwise stor-age space in the rear positions would be blocked.

Empty (E) Half filled rear (H) Half filled front Filled (F)

Excluded in the model

Access side

Figure 4.1: All potential states of a storage lane

The empty state of a storage lane is denoted as E, the half-filled state as H and the filled state as F.

• One tmastper movement of the S/R machine, thus per travel between different positions.

Load handling

The two load handling devices can handle goods independently and are both able to access the front and the rear position of a storage lane. We distinguish between the following access times:

• t_{LH D, f}: The access time to the front position, meaning the time the load handling device needs to extend to the front position of a lane and move back to the initial position. We assume the handling time at the I/O point equivalent to the access time to the front position.

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

The devices are horizontally arranged. There is a simultaneous process at the beginning and the end of every command cycle for pick-up and deposit of units at the I/O point. The distance between the two load handling de-vices is equivalent to the distance between two storage lanes. Hence, the positioning of one load handling device in front of one storage lane means the other load handling device is necessarily positioned in front of a neigh-boring lane. We discuss the impacts for the relaxation of this assumption in section 4.6.1.

The utilization order of the load handling devices is randomly chosen.

Rearrangements

For each retrieval of units stored at the rear position of a storage lane, there is a positive probability that a rearrangement is required. Rearrangements occur, if a unit at the rear position is retrieved, while the front position is occupied. Storing in the rearmost position also applies during rearrange-ments.

We distinguish between regular rearrangements and tango rearrange-ments:

• Regular rearrangements describe the process of rearranging the blocking unit into an available storage position. Subsequently the retrieval unit is handled. Regular rearrangements are conducted in

case that one handling device is in use. For the selection of the rear-rangement position, the Nearest Neighbor policy applies.

• The tango rearrangement is a combined operation of retrieval and rearrangement. Every time both load handling devices are free and a rearrangement is required, it is possible to perform a tango which works as follows: The blocking unit in the front is picked by the first load handling device (Figure 4.2 a). In the second step, the S/R ma-chine moves sideways to bring the other load handling device in front of the same storage lane. Then, the (formerly blocked) unit in the rear is picked by the second load handling device (b). At last, the S/R ma-chine moves back and the blocking unit is restored to the rear posi-tion of the same storage lane (c). In case a tango is performed, no additional storage position for rearrangement is required. The tango rearrangement is performed whenever possible. The order in which the load handling devices are used is randomly selected.

a b c

1 1 1

d

1 Storage

lane

Load handling

device

Figure 4.2: Rearrangement performed by the load handling devices (tango)

System load

We consider the AS/RS to be operated at full capacity, meaning there are al-ways storage and retrieval requests waiting. Consequently, the AS/RS per-forms only quadruple command cycles.

Filling Level

The filling level, z, of the whole rack depends directly on the condition of the storage lanes. Thus, the filling level can be calculated given the number of half-filled and filled storage lanes in the following way:

z =¹/2· Number of half-filled lanes + Number of filled lanes Total number of lanes

=¹/2· P (H) + P (F )

(4.2)

We refer to the filling level as z.