5.2.1. Notations

The workers (P1, P2, P3, ... P80) (*i*) in this department were divided into three shifts (*k*); Morning (S1) shift 8 a.m. to 4 p.m., Evening (S2) shift—4 p.m. to 12 a.m., and Night (S3) shift—12 a.m. to 8 a.m. Scheduling horizon is 30 days (*j*). Workers in the power plant have four different seniority level. These are:


There are some hard and weak constraints about our mathematical model. There are 10 sets of constraints, six of them correspond to the hard constraints (Constraint 1–Constraint 6 under 5.2.4 Constraints subsection) and the four corresponds to weak constraints (Goal 1–Goal 4 under 5.2.5 Goal Constraints subsection):

*Constraint 1:* Number of personnel needed for each shift every day. (3 shifts 20 people) Number of personnel assigned to their seniority in each shift.


*Constraint 2*: A staff working any day at night shift should not work in the morning and evening shifts the next day.

*Constraint 3:* A person working on any day of the evening shift should not work the next day in the morning.

*Constraint 4:* This constraint indicates that every staff member should take one day day-off (at least) in a week. In other words, every staff member should not work more than 6 days in a week:

*Constraint 5:* Every staff member should not work on his/her the day off.

*Constraint 6:* In the evening shift, the staff cannot be operated more than 9 days.

Decision variables to be used are *Xijk* and *hij*, here notation of *i*, *j* and *k* are the indices for 80 workers, 30 days and three shifts, respectively. Variable *Xijk* indicates worker *i* is assigned to work on day *j* for shift *k* and *hij*, indicate the assignment of worker *i* to be in day-off, respectively, on day *j*.

The complete formulation of the SSP-GP model is as follows:

5.2.2. Parameters: All Parameters Are Given in the Model Below

$$i \text{: Personnel index}, \qquad i = 1, 2, \dots, l \tag{10}$$

$$j \colon \text{Day index}, \qquad j = 1, 2, \ldots, m \tag{11}$$

$$k \text{ Stifft index}, \qquad k = 1, 2, \dots, n \tag{12}$$

$$\text{g: Goal index}, \qquad \text{g} = 1, 2, \dots, z \tag{13}$$

	- *m:* Number of Day, *m = 30* (15)

$$n \text{: Number of Shifts, } \qquad n = 3 \tag{16}$$


 =

$$h\_{\varnothing} \text{: Decision variable for day-off of } i^{\text{th}} \text{ personnel, } j^{\text{th}} \text{ day} \quad i = 1, 2, \dots, l \quad j = 1 \dots m \tag{19}$$

$$d\_{\mathbf{g}|\mathbf{k}}^{+}: \text{ Positive deviation variable of } \mathbf{g}^{\text{th}} \text{ goal, } \mathbf{j}^{\text{th}} \text{ day, for shift } \mathbf{kg} = 1, 2 \dots z \\ \tag{20}$$

$$j = 1, 2 \dots m \\ \mathbf{k} = 1, 2 \dots n$$

 . . *n*

$$\begin{aligned} d\_{\mathbf{g}|\mathbf{k}}^{-}: \text{ Negative deviation variable of } \mathbf{g}^{\text{th}} \text{ goal} \mathbf{j}^{\text{th}} \text{ day, for shift } \mathbf{kg} = 1, 2 \dots z \\ \mathbf{j} = 1, 2 \dots m \mathbf{k} &= 1, 2 \dots n \end{aligned} \tag{21}$$

 5.2.3. Decision Variables: There Are Two Decision Variables on the Model. Those Are *Xijk* and *hij*

 =

 . . .

$$\mathbf{X}\_{\text{ijk}} = \begin{cases} 1, \text{ } \text{If } \text{personnel } i \text{ is assigned to day } j \text{ on shift } \mathbf{k} \\ 0, \text{ } \text{ otherwise} \end{cases} \\ \text{if } i = 1, 2, \dots, l \text{\textquotedblleft } 1, 2, \dots, m \text{\textquotedblright} \\ = 1, 2, \dots, m \text{\textquotedblleft } 2, 2, \dots, n \text{\textquotedblleft } 2, \dots, m \text{\textquotedblright}$$

$$\mathbf{h}\_{ij} = \begin{cases} 1, & \text{if } f \text{ the personnel i is on day} - off \text{ in day} \text{ } j \\ 0, & \text{otherwise} \end{cases} \quad \mathbf{i} = 1, 2, \dots, l \mathbf{j} = 1, 2, \dots, m \tag{23}$$
