**3. FHFGSP**

This section first details the problem definition of FHFGSP, then the rules of TFN operations are explained, and finally the symbolic representation of FHFGSP is given and the mathematical model of FHFGSP is developed.

#### *3.1. Description of the Problem*

FHFGSP combines the features of fuzzy scheduling and HFSP. In FHFGSP, *n* jobs will be processed in *m* (*m* ≥ 2) stages in the same order. Each stage *j* has at least one machine *Mj,k* (*k* ≥ 1) and at least one stage has multiple machines [1,35,36]. The processing time *Ti,j,v* of job*i* on machine *Mj,k* is uncertain and is given by the triple [37] (*t o i*,*j*,*v*, *t m i*,*j*,*v*, *t p <sup>i</sup>*,*j*,*<sup>v</sup>*) where *t o i*,*j*,*<sup>v</sup>* ≤ *t m i*,*j*,*<sup>v</sup>* ≤ *t p i*,*j*,*v*. *t o i*,*j*,*<sup>v</sup>* denotes the optimal processing time, *t m i*,*j*,*<sup>v</sup>* denotes the most probable processing time, and *t p i*,*j*,*<sup>v</sup>*denotes the worst processing time.

The constraints for FHFGSP are formulated as follows:



The objective to be optimized by FHFGSP is to minimize MS and TEC. In this paper, the TEC is divided into three parts: when the machine is idle, when the machine is in the setup phase, and when the machine is processing jobs. There are three ways to reduce MS: (1) reduce machine idle time, which is influenced by the job sequence. (2) Reduce machine setup time, which also reduces TEC, which is also influenced by the job sequence. (3) Reducing the time of the job being processed, which means increasing the processing speed of the job. However, the energy consumption of the machine when processing a job is proportional to the processing speed of the job [38], and reducing the job processing time increases the TEC. Since the two objectives to be optimized are in conflict with each other, this paper solves the FHFGSP by adjusting the job sequence and the speed of the machine when processing the job.

#### *3.2. TFN Concepts and Operations*

The concept of fuzzy sets was introduced by Zadeh [39] and the basic idea is to fuzzily the absolute affiliation in classical sets. It can be used to solve real-life uncertainty problems [40]. This subsection gives the rules for the operation of the TFN to facilitate the solution of the GFHSP.

For any two TFNs *A* = (*<sup>a</sup>*1, *a*2, *a*3) and *B* = (*b*1, *b*2, *b*3) the rules for each operation are as follows:

1. Additive operations

$$A + B = \begin{pmatrix} a\_1 + b\_1 \ \ a\_2 + b\_2 \ \ a\_3 + b\_3 \end{pmatrix} \tag{1}$$

2. Multiplication operations

$$A \times B = \left(a\_1 \times b\_1, a\_2 \times b\_2, a\_3 \times b\_3\right) \tag{2}$$

3. Comparative operations

−

−

−

−

$$\bar{A} = (\frac{a\_1 + 2a\_2 + a\_3}{4})\tag{3}$$

The TFN comparison operation is divided into three steps and has three judgement criteria. Step 1: Get − *A* and − *B* by (3). If − *A* > (<) − *B*, then *A* > (<) *B*.

Step 2: If *A* = *B*, then compare *a*2 and *b*2. If *a*2 > (<) *b*2, then *A* > (<) *B*.

Step 3: If *A* = *B* and *a*2 = *b*2, then compare the difference between *a*3 and *a*1. If *a*3−*a*1 > (<) *b*3−*b*1, then *A* < (>) *B*.
