The production-line employees at GE work four days a week, 10 hours a day. Each day of the
week, (at least) the following numbers of line employees are needed: Monday–Friday, 7 employees; Saturday and Sunday, 3 employees. GE has 11 production-line employees. Formulate an LP that can be used to maximize the number of consecutive days off received by the employees.
For example, a worker who gets Sunday, Monday, and Wednesday off receives two consecutive days off.
To maximize the number of consecutive days off received by the employees, we can create a binary variable for each day indicating whether an employee is assigned to work on that day or not. Let's denote this binary variable as X_i, where i represents the day of the week.
The objective is to maximize the total number of consecutive days off. Therefore, we need to maximize the sum of consecutive days off across the week.
We also need to consider the constraints:
1. For Monday to Friday, at least 7 employees should work each day.
2. For Saturday and Sunday, at least 3 employees should work each day.
3. The total number of employees working on a particular day cannot exceed the total number of employees available.
With these considerations, the linear programming (LP) formulation can be given as follows:
Maximize: ∑ X_i
Subject to:
X_1 + X_2 + X_3 + X_4 + X_5 ≥ 7
X_6 + X_7 ≥ 3
X_i = 0 or 1, for all i = 1 to 7
∑ X_i ≤ 11, for all i = 1 to 7
In this formulation, X_i represents whether employee i is assigned to work on day i (0 for no, 1 for yes). The objective function maximizes the total number of consecutive days off. The constraints ensure that the minimum required number of employees work on each day and the total number of employees working on any day does not exceed the available number of employees.
To formulate a linear programming (LP) problem to maximize the number of consecutive days off received by the employees, we can represent the problem with decision variables and constraints.
Let's define the decision variables as binary variables, where:
- Xi represents whether employee i starts their consecutive days off on a particular day.
- Yi represents whether employee i ends their consecutive days off on a particular day.
We can set up the decision variables as follows:
- Xi, for i = 1 to 11, and Yi, for i = 1 to 11, assuming there are 11 employees in total.
Now, let's define the objective function and constraints.
Objective function: Maximize the total number of consecutive days off.
To maximize the number of consecutive days off, we want to optimize the sum of Yi - Xi + 1 for all employees i.
Maximize: Σ (Yi - Xi + 1) for i = 1 to 11.
Subject to the following constraints:
1. Each day has the required number of employees:
- For Monday to Friday: X1 + X2 + ... + X11 + Y1 + Y2 + ... + Y11 >= 7.
- For Saturday and Sunday: X1 + X2 + ... + X11 + Y1 + Y2 + ... + Y11 >= 3.
2. Each employee must have exactly one starting day and one ending day for their consecutive days off:
- X1 + X2 + ... + X11 = 11.
- Y1 + Y2 + ... + Y11 = 11.
3. Constraint to ensure consecutiveness of days off:
- Y1 - X1 <= 6.
- Y2 - X2 <= 6.
- ...
- Y11 - X11 <= 6.
4. Non-negativity constraints:
- X1, X2, ..., X11, Y1, Y2, ..., Y11 >= 0.
Once the LP problem is set up, it can be solved using an LP solver to obtain the optimal solution that maximizes the number of consecutive days off received by the employees at GE.