3.2. The Outbound Time Model
The operation mode of a double-deep multi-tier shuttle system is divided into two types: single operation and compound operation. Single operation mode refers to that the system only carries out outbound or inbound tasks in the single time window. Compound operation refers to the system’s outbound operations and inbound operations proceeded simultaneously in the same time window. The double-deep multi-tier shuttle system gives priority to picking process and takes storage tasks as the auxiliary part. Therefore, outbound picking is its most important function. In the actual operation of the system, the priority of outbound operation is higher than inbound operation. Therefore, the research is based on the single operation mode.
Assuming that there are
retrieval tasks in the system and these tasks are distributed in different aisles and different tiers. The number of aisles is
(the number of lifts is also
), and the number of retrieval tasks in each aisle is
, where
. The batch of outbound tasks are all dealt in the same time window. The arrival time of the 1st retrieval task is 0.
N retrieval tasks’ total time
can be expressed as the maximum time to complete
N retrieval tasks, as shown in Equation (3).
In Equation (3), represents the completion time of the i-th retrieval task in N retrieval tasks.
In addition to Equation (3), can also be expressed by the maximum completion time of the A lifts finish all retrieval tasks of each aisle, shown as Equation (4).
In Equation (4),
represents the completion time when the
j-th lift completes all the retrieval tasks in its aisle.
can also express the total outbound time
for the
j-th lift to complete
retrieval tasks of its aisle, and the total outbound time
for the
n tasks can be expressed as the Equation (5):
N retrieval tasks are distributed in different aisles and different tiers, which are finished by shuttles and lifts in serial or parallel operation. But for the
retrieval tasks in
j-th aisle, they are finished with shuttles’ parallel retrieval operation and lift’s serial operation; that is, the
outbound tasks of the
j-th aisle are completed by the lift serially. Therefore, the total outbound time
of the
j-th aisle can be obtained from the lift’s serial operation, which is shown as Equation (6).
In Equation (6), represents the outbound operation time of the k-th retrieval task in j-th aisle.
From the lift’s point of view, the time on the lift of the
k-th outbound task can be divided into the waiting time
before the lift responds to the shuttle’s dispatching, and the working time
after the lift responds to the shuttle’s dispatching, that is Equation (7):
Compositing Equations (6) and (7),
can be expressed as Equation (8):
For the
j-th aisle, the serial operation time sequence of lift is shown in
Figure 5.
The following part analyzes detail of the lift’s waiting time and lift’s operation time .
1. Lift’s waiting time .
When the shuttle is applied for dispatching lift, lift may be in idle or busy state. When the lift is idle, the lift can immediately respond to the shuttle’s dispatching request. So, the lift’s waiting time is calculated from the completion of the last outbound task to the response of the shuttle’s dispatching application. When the lift is busy, after completing the last outbound task, the lift immediately responds to the shuttle’s dispatching and executes the outbound task. Thus, the lift is always in a busy state and there is no lift’s waiting time. For more detailed analysis of lift waiting time
, according to the outbound operation flow described above, we will analyze and defined time nodes of the
k-th outbound task in the
j-th aisle. The timing of the outbound task are shown in
Table 1.
Combining with the analysis of the lift’s waiting time and the time nodes of the outbound tasks described in the
Table 1, it can be concluded that the lift’s waiting time is divided into the following two cases:
- (1)
when , the lift is idle, and there is a waiting time .
- (2)
when , the lift is busy and the waiting time is .
Set decision variable
denotes whether the lift needs to wait at the time of the
k-th outbound task of the
j-th aisle in the system needing retrieval, such as shown in Equation (9).
The time that lift waits for the shuttle,
, can be expressed as Equation (10):
The
in Equation (10) can be expressed by the total time of the
1st k−1 outbound tasks in the aisle, namely Equation (11):
The in Equation (10) can be expressed as Equation (12) according to the time nodes table of outbound task.
in Equation (12) indicates the operation time after the shuttle responds to dispatching.
The shuttle’s operation after responding to dispatching can be divided into two following situations. When the outbound task needs rearrangement, includes two parts, i.e., taking the turnover box from the target position and rearranging the blocked turnover box. When the outbound task does not require rearrangement, the only includes the operation time of fetching the turnover box from the target position without rearrangement time. Therefore, a decision variable is set to indicate whether the k-th task needs rearrangement or not, such as Equation (13). When the outbound task one and the task two are in the same batch of outbound tasks, the turnover box locations required for tasks one and two are determined, and the locations are located at different depths of the same container. The depth of the target turnover box of the task one is 1, and the depth of the target turnover box of the task two is 2. When the sequence of outgoing tasks is task one to task two, the rearrangement operation is need; when the sequence of outgoing tasks is task two to task one, there is no need to rearrangement.
Based on the above analysis, the operating time of shuttle
can be expressed as Equation (14):
In Equation (14), denotes when the k-th outbound task is operating, the time the shuttle runs from the dwell point (i.e., the I/O station) to the target unit; denotes the k-th outbound task’s rearrangement time; and denotes the time the shuttle takes (puts) the turnover box, which is a constant.
According to Equations (10)–(12) and (14), we can get the waiting time of the lift
as Equation (15):
2. Lift’s operation time .
According to the outbound operation process, the lift’s operation time is mainly composed of four parts: (1) : The time lift runs from the dwell point (I/O point in the first tier) to the corresponding tier’s outbound platform. (2) : The time to complete the transfer of the turnover box between the lift and the shuttle, which is a constant. (3) : The time lift delivers the target turnover box to the I/O point at first tier, which is the same as the value of . (4) : The time it takes for the lift to put down the turnover box is same as the time for the shuttle to take or put a SKU and the time for the lift and shuttle exchange the turnover box. It is only related to the hardware characteristics of the equipment and is constant. Therefore, lift’s operation time can be expressed as Equation (16).
Through the analysis of lift’s waiting time and lift’s operation time, the k-th outbound task’s time in the j-th aisle, as shown in Equation (17), can be obtained via Equations (7), (15) and (16).
By synthesizing Equations (16) and (17), the total outbound time
or the
-th outbound tasks in the
j-th aisle is obtained as Equation (18):
Through Equations (4), (5), and (18), the total outbound time
of
N outbound tasks in the system is Equation (19):
It can be seen from Equation (19) that the system efficiency can be improved by reducing the rearrangement distance when the equipment operation characteristics, task scheduling sequence and warehouse layout are fixed. In other words, transporting the block turnover box to the nearest spare location can reduce the distance of rearrangement.