2.2.4. Modelling

Based on the above descriptions, the integrated scheduling model of AS/RS and hybrid flowshop can be given as follows:

$$\min(F) = \min(w\_1 \cdot f\_1 + w\_2 \cdot f\_2) \tag{7}$$

which is subject to:

$$\sum\_{j=1}^{X \cdot Y \cdot Z} \alpha\_{nj} = 1 \tag{8}$$

$$ct\_{noka} = st\_{noka} + T\_{pk\varepsilon}; (p = p\_{no})\tag{9}$$

$$st\_{n's} \ge ct\_{ns} - (1 - \beta\_{nn's}) \cdot \Omega \tag{10}$$

$$\text{spt}\_{n'o'kx} \ge \text{ct}\_{n'o'kx} - (1 - \chi\_{nn'kx}) \cdot \Omega \tag{11}$$

$$st\_{no'c} \ge ct\_{ns} + R\_s + \frac{(2S - 1) \cdot \left(L + \frac{bl}{2}\right)}{v\_x} \tag{12}$$

$$\text{sst}\_{n\nu(k+1)\mathfrak{a}'} \ge \mathfrak{c}t\_{\text{mokv}}; (k \ne K) \tag{13}$$

Equation (7) indicates the bi-objective that contains the total operation time in AS/RS and the maximum makespan in production. Equation (8) represents that each task can only correspond to one position in racks. Equation (9) indicates the relationship between the start operation time and the end operation time of the retrieval tasks. Equations (10) and (11) each represent the constraints in the crane and production equipment at the start of the operation time of the next task and at the end of the operation time of the previous task. Equation (12) represents that the start operation time of the production task is larger than the arrival time of the production material. Equation (13) indicates that the production phase in which the start time of the next stage of the task is greater than or equal to the end time of the previous stage of the operation.

#### **3. GA-MBO Design**

Intelligent optimization algorithms are more suitable than accurate algorithms to solve NP-hard problems and are conducted to deliver fast solutions. GA is a classic intelligent optimization algorithm which has been widely used in production scheduling, combinatorial optimization, etc. To fix the poor local search ability of GA, this paper adopts the MBO algorithm which has high local search efficiency and outstanding convergence performance. GA-MBO algorithm is proposed to solve the integrated scheduling optimization problem in AS/RS and workshop.

The GA-MBO algorithm consists of three modules, including coding rules, GA rules, and MBO rules. The flow chart of the GA-MBO is shown in Figure 3. The following parameters are defined as: *NG* represents the number of populations in the GA phase; *Mgen* represents the number of iterations in the GA phase; *Pc* represents the probability of crossover; *Pm* represents the probability of mutation; *NM* is the number of flocks in the MBO phase; *a* represents the number of neighborhood solutions generated by an individual; *b* indicates the number of neighborhood solutions that each individual passes to the next individual; and *G* represents the number of tours.

**Figure 3.** Flow chart of GA-MBO.

#### *3.1. Coding Rules*

The real coding method is used to solve the problems, such as multiple storage locations in the AS/RS, various material types of storage and retrieval tasks, mixed operation of storage and retrieval tasks, and complex operation sequences of production tasks. The coding and the decoding mechanism are illustrated with the warehousing information in the AS/RS with 6 rows, 6 columns, and 5 tiers, which are shown in Table 1. The storage and retrieval tasks are numbered as shown in Figure 4. The storage racks are successively numbered with rows, columns, and tiers, starting from 1. The total number of racks is:

$$J = X \cdot Y \cdot Z \tag{14}$$

**Table 1.** The warehousing information.



**Figure 4.** Mixed storage and retrieval tasks and material numbers.
