Consider the following decision scenario for a work environment with characteris

Question

Consider the following decision scenario for a work environment with characteristics remaining the same from week to week:

A post office facility requires a different available labor resource on each day of the week as shown, and can only use integer numbers of full and part time employees. All employees must have planned schedules of 5 consecutive days working followed by 2 days off. Full time employees work 8 hours a day; part time employees only 4 hours a day. Union requirements limit daily part-time employees working their planned schedule are paid #140 per day; part time employees are paid $72 per day. Managment wants to meet weekly labor requirements at a minimum cost. One "flexibility" management has is the option of forcing all or any portion of full time employees to work 1 (further) day of overtime each week. For example, an employee scheduled for Monday through Friday can be forced to work on Saturday. The pay for a day of overtime is $210 and 8 hours are worked.

Provide:

1. a liner algebraic formulation for this scenario, minimizing cost

2. Step by step of using Excel Solver to find an optimal solution

3. A written statement of the optimal solution.

Explanation / Answer

Let x1 = Number of employees who start work on Sunday and work 5 days, x2 = Number of employees who start work on Monday and work 5 days, ... x7 = Number of employees who start work on Saturday and work 5 days.

Also let y1 = Number of employees who start work on Sunday and work 6 days, ... , y7 = Number of employees who start work on Saturday and work 6 days. Then the appropriate LP is

min z=5*50*(x1+x2+...x7)+(5*50+62)*(y1+y2+...y7)=250(x1+x2+...x7) + 312(y1+y2+...y7)

s.t. x1+x4+x5+x6+x7+y1+y3+y4+y5+y6+y7>=11 (Sunday)

x1+x2+x5+x6+x7+y1+y2+y4+y5+y6+y7>=27 (Monday)

x1+x2+x3+x6+x7+y1+y2+y3+y5+y6+y7 >=23 (Tuesday)

x1+x2+x3+x4+x7+y1+y2+y3+y4+y6+y7>=35 (Wednesday)

x1+x2+x3+x4+x5+y1+y2+y3+y4+y5+y7>=19 (Thursday)

x2+x3+x4+x5+x6+y1+y2+y3+y4+y5+y6>=34 (Friday)

x3+x4+x5+x6+x7+y2+y3+y4+y5+y6+y7>=16 (Saturday)

All variables nonnegative

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