The following is a linear programming formulation of a labor

The following is a linear programming formulation of a labor planning problem. There are four overlapping shifts, and management must decide how many employees to schedule to start work on each shift. The objective is to minimize the total number of employees required while the constraints stipulate how many employees are required at each time of day. The variables X₁ - X₄ represent the number of employees starting work on each shift (shift 1 through shift 4).

MIN  X₁ + X₂ + X₃ + X₄

s.t. X₁ + X₄ 12 ........(on duty during shift 1)

X₁ + X₂ 15 ........(on duty during shift 2)

X₂ + X₃ 16 ........(on duty during shift 3)

X₃ + X₄ 14 ........(on duty during shift 4)

X₁, X₂, X₃, and X₄ 0

Given the optimal solution: X₁* = 13, X₂* = 2, X₃* = 14, X₄* = 0, how many workers would be assigned to shift 2?

Select one:

a. 13

b. 12

c. 14

d. 15

e. 2

(The correct answer is: 2, but I would like to know how that is the answer)

Solution

In the optimality criteria it is given that for x2 =2 the objective function is minimized so corresponding to x2 we get the value 2