*3.3. Mathematical Models*

After understanding the basic concepts of FHFGSP and TFN, mathematical modelling of FHFGSP from the perspective of optimization objectives is needed to facilitate a better understanding of the problem to solve it. The interpretation of the relevant symbols appearing in the FHFGSP is shown in Table 1.

Objective:

Subject to:

$$\dim\{MS, TEC\}$$

$$\sum\_{k \in M\_j} \sum\_{\upsilon \in V\_j} x\_{i,j,k,\upsilon} = 1, \forall i \in I, j \in J \tag{5}$$

(4)

$$x\_{i,j} - b\_{i,j} = \sum\_{k \in M\_j} \sum\_{\upsilon \in V\_j} x\_{i,j,k,\upsilon} \cdot pt\_{i,j,\upsilon} \tag{6}$$

$$b\_{i,j} - e\_{i,j-1} \ge 0, \ j \in \{2, \dots, m\} \tag{7}$$

$$z\_{i,i^\*,j,k} + z\_{i^\*,i,j,k} \le 1, \ \forall i, i^\* \in I, k \in \mathcal{M}\_j \tag{8}$$

$$b\_{i,j} - \sum\_{i^\* \in I} e\_{i^\*,j} \cdot z\_{i^\*,i,j,k} - \sum\_{k \in M\_j} \sum\_{i^\* \in I} z\_{i^\*,j,j,k} \cdot s t\_{i^\*,j,j} \ge 0 \tag{9}$$

$$\mathfrak{e}\_{i,j} - \sum\_{k \in M\_j} \sum\_{v \in V\_j} \mathfrak{x}\_{i,j,k,v} \cdot \mathfrak{p} t\_{i,j,v} - \sum\_{k \in M\_j} \sum\_{i^\* \in I} z\_{i^\*,i,j,k} \cdot \mathfrak{sl}\_{i^\*,j,j} = 0 \tag{10}$$

$$MS = \max \mathfrak{e}\_{i,\mathfrak{m}} \tag{11}$$

$$TEC = PE + SE + IE\tag{12}$$

$$pp\_{i,j,v} = p\_{i,j} / c\_{j,v} \tag{13}$$

$$PE = \sum\_{j \in J} \sum\_{k \in M\_j} \sum\_{i \in I} \sum\_{v \in V\_j} x\_{i,j,k,v} \cdot T\_{i,j,v} \cdot pp\_{i,j,v} \tag{14}$$

$$SE = \sum\_{j \in J} \sum\_{k \in M\_j} \sum\_{i \in I} \sum\_{i^\* \in I} \sum\_{v \in V\_j} x\_{i, j, k, v} \cdot z\_{i^\*, j, j, k} \cdot st\_{i^\*, i, j} \cdot sp \tag{15}$$

$$IE = \sum\_{j \in I} \sum\_{k \in M\_j} \left[ E\_{k, \bar{j}} - B\_{k, \bar{j}} - \sum\_{i \in I} \sum\_{v \in V\_j} x\_{i, \bar{j}, k, v} \cdot (pt\_{i, \bar{j}, v} + \sum\_{i^\* \in I} z\_{i^\*, i, \bar{j}, k} \cdot st\_{i^\*, \bar{j}, \bar{j}}) \right] \cdot ip \tag{16}$$

where (4) gives the objective of the FHFGSP to minimize both MS and TEC (5)–(10) give the associated constraints. (5) guarantees that each job i can be assigned to a specific machine k for processing at speed v at each stage j. (6)–(9) guarantees that no interruptions and preemptions by jobs are allowed during the processing and setup phases. (10) indicates that the machine starts processing as soon as setup is complete. (11) indicates that MS is determined by the end time of the last job to be processed in the final stage. (12) indicates that the TEC consists of three components, PE indicates the energy consumption of the machine while processing the job, SE indicates the energy consumption of the machine during the setup time, IE indicates the energy consumption of the machine during the idle time. (13) denotes the actual power of job i when it is processed at speed v in stage j. (14)–(16) are the specific information of PE, SE, and IE, respectively, all energy consumption is obtained by multiplying power by time.



#### **4. SDABC of FHFGSP**

This section presents the proposed SDABC algorithm for solving FHFGSP. The basic framework of the ABC algorithm is first presented, and then the encoding and decoding scheme and the energy saving procedure are described, followed by the details of SDABC, and finally a summary.

#### *4.1. The Framework of ABC*

In ABC, during the initialization phase, a set of food source locations are randomly selected by bees and their nectar amount is determined, then these bees enter the colony and share nectar information. Each search cycle consists of three steps. In the first phase, after information sharing, each employed bee searches for information in the vicinity of the food source location and abandons the old food source to choose a new food source if a better one is found. In the second phase, the onlooker bee selects a food source to follow based on the nectar distribution information sent by the employed bee, the better the food source the more likely it is to be followed. If the current food source is not updated for a long time, the employed bee will abandon the current food source and become a scout bee. The scout bee randomly selects a new food source to replace the abandoned food source. The overall framework of the basic ABC framework is shown in Algorithm 1.

#### **Algorithm 1** Framework of the basic ABC framework

**Input:** population *P*;

**Output:** results;

1: Initialize population *P*;

 2: **while** requirements are met **do**



In this paper, the linear weighted sum method is used as the decomposition method. For a multi-objective optimization problem with m objectives, a weight vector *λ* = (*λ*1, *λ*2, ... , *λm*) *T* is added, where *i* represents the sum of the weight values of the *i*-th objective. As shown in (17).

$$\begin{aligned} \min \quad F(X) &= \sum\_{i=1}^{m} \lambda\_i f\_i(x\_i) \\ \text{s.t. } x \in \Omega \end{aligned} \tag{17}$$

where *fi*(*xi*) is the objective value for the *i*-th objective. Since this paper is a two-objective problem, *λ* = (*λ*1, *λ*2,) *T* the values of *λ*1 are taken in {0/*H*, 1/*H*, ... , *i*/*H*, ... , *H*/*H*}, where *H* = *N* − 1 and *λ*2 = 1 − *λ*1.
