Question

Anderson Family Steakhouse offers a variety of low-cost meals and quick service. Other than management, the Steakhouse operates with full time employees work 8 hrs/ day. The rest of employees are part time employees who are scheduled for 4 hrs shifts during peak meal times. On the Saturdays the Steakhouse is open from 11:00AM to10:00PM management want to develop a schedule for part time employees that will minimize labor cost and provide excellent customer service. The average wage rate for the part time employees is $7.60/ hr. The total number of full time and part-time employees needed varies with time of a day as shown.

time	total # of employees needed
11 AM - 12 PM	9
12 PM - 1 PM	9
1 PM - 2 PM	9
2 PM - 3 PM	3
3 PM - 4 PM	3
4 PM - 5 PM	3
5 PM - 6 PM	6
6 PM - 7 PM	12
7 PM - 8 PM	12
8 PM - 9 PM	7
9 PM - 10 PM	7

One full time employee comes on duty at 11:00 AM works 3 hrs.takes one hr. off, and returns for another 5 hrs. The other full-time employee comes to work at 1:00 PM and works the same 3 hours-on, 1-hour-off, 5-hours-on pattern.

a. Write the complete linear programming model (objective function and constraints) and develop a minimum cost schedule for part time employees?

b. What is total payroll for the part time employees? How many part time shifts are needed? (Use the surplus variables to comment on the desirability of scheduling at least some of the part time employees for 3 hrs shifts.)

c. The restaurant wants to open an extra hour from 10 PM-11 PM with six part time employees needed during this hour. How much does payroll cost change?

d. Assume that part-time employees can be assigned either 3hr or 4 hr shifts. Write the complete linear programming model and develop a minimum cost schedule for the part time employees. How many part time shifts are needed and what is the cost saving compared to the previous schedule? (restaurant closing time is 10 PM)

e. The restaurant decides to give part-time employees a raise to $8.50 per hour. How much does the raise cost in part d?

Use the results of ranges of optimality and feasibility for the Model in part d to answer the following questions:

6. Explain the meaning of the dual values. (-15.2, -7.6 and 0)

7. Explain the meaning of nonzero values for reduced cost.

Answer 1

solution:

d.

e.

In the part d the optimal solution dictates 9 part time workers. If we increase their wage from 7.5 to 8.5 then the overall wage increase is 9*3*(8.5-7.5) = 27

The total payroll cost will be 529.5

f.

The dual values (also known as shadow price) refers to the constraints and their impact on the objective function value. Dual values represent the impact on objective function value due to change in 1 unit of the constraint. For example in a minimizing problem if the dual value is -15.2 for a constraint then it means that for every 1 unit change in constraint’s RHS, the impact on the objective function will be -15.2

g.

Reduced costs represent the impact of coefficients of decision variables on the objective function. When a reduced cost is 0 it means that the decision variable has no impact on optimal solution and objective function at current level. When a decision variable value is 0, the reduced cost is usually 0. This means that since the decision variable is not part of the optimal mix, it has been omitted and its coefficient has no impact on objective function.