2.2.2. Notation

In order to describe the problem and build a model by a better way, the notations are as follows:



#### 2.2.3. Objective Function

The common operational efficiency in the AS/RS scheduling problem is to measure the operation time, and in the hybrid flowshop scheduling problem, to measure the makespan. The integrated scheduling optimization of these two scheduling problems presents a conflict that needs to be evaluated in model objects.

The crane operates retrieval tasks from the dwell point to the location of the retrieval task and then moves to the output point in the aisle, which becomes a new dwell point. The retrieval operation time of the crane at SC is:

$$\text{sct}\_{ns} = \text{st}\_{ns} + 2\text{C} + \max\left(\frac{(y\_o - y\_s) \cdot W}{v\_y}, \frac{(z\_o - z\_s) \cdot H}{v\_z}\right) + \max\left(\frac{y\_o \cdot W}{v\_y}, \frac{(z\_o - 1) \cdot H}{v\_z}\right) \tag{1}$$

The crane operates storage tasks from the dwell point to the aisle output point to pick up goods and then moves to the storage task location, which becomes a new dwell point. The storage operation time of the crane at SC is:

$$\text{sct}\_{\text{ns}} = \text{sct}\_{\text{ns}} + 2\text{C} + \max\left(\frac{y\_s \cdot \mathcal{W}}{v\_y}, \frac{(z\_s - 1) \cdot H}{v\_z}\right) + \max\left(\frac{y\_o \cdot \mathcal{W}}{v\_y}, \frac{(z\_o - 1) \cdot H}{v\_z}\right) \tag{2}$$

In the DC operation of the crane, the dwell point of input is the output point, which reduces the operation time. The objective function of operation time in AS/RS is:

$$cf\_1 = \sum\_{s=1}^{S} \max(ct\_{\text{ns}}) \tag{3}$$

The end operation time is the task completion time at the last production stage in hybrid flowshop. The objective function of the makespan is:

$$f\_2 = \max(ct\_{nokx})\tag{4}$$

To eliminate the influence of different objective dimensions, the above two evaluation objective functions need to be normalized as:

$$f(\mathbf{x}) = \frac{f(\mathbf{x}) - \min f(\mathbf{x}) + 0.001}{\max f(\mathbf{x}) - \min f(\mathbf{x}) + 0.001} \tag{5}$$

Normalization needs to determine the extreme value of the objective function. The researched problem is NP-hard and it is difficult to obtain the exact solution for which optimization can be obtained by the single-objective function. The weight coefficient method converts the multi-objective optimization into a single-objective description as:

$$F = w\_1 \cdot f\_1 + w\_2 \cdot f\_2 \tag{6}$$

The total operating time of the warehouse and the makespan in production are the two study targets of the study. The primary goal in enterprise managemen<sup>t</sup> is to increase production efficiency, which is also called the makespan. Increasing the operational efficiency in AS/RS cannot directly improve production efficiency but can help to reduce operating

costs. Therefore, the weights of the two evaluated objectives are taken as *w*1 = 0.3 and *w*2 = 0.7.
