Western Family Steakhouse offers a variety of low-cost meals and quick service. Other than management, the steakhouse operates with two full-time employees who work 8 hours per day. The rest of the employees are part-time employees who are scheduled for 4 hours shifts during peak meal times. On Saturdays the steakhouse is open from 11:00A to 10:00P. Management wants to develop a schedule for part-time employees that will minimize labor costs and still provide excellent customer service. The average wage rate for the part-time employees is $7.60 per hour. The total number of full-time and part-time employees needed varies with the time of day as shown:
Time Total Number of
Employees Needed
11:00A-Noon 9
Noon-1:00P 9
1:00P-2:00P 9
2:00P-3:00P 3
3:00P-4:00P 3
4:00P-5:00P 3
5:00P-6:00P 6
6:00P-7:00P 12
7:00P-8:00P 12
8:00P-9:00P 7
9:00P-10:00P 7
One-full time employee comes on duty at 11:00A, works 4 hours, takes an hour off, and returns for another 4 hours. The other full-time employee comes to work at 1:00P and works the same 4 hours-on, 1-hour off, 4-hours-on pattern.
(a) Develop a minimum-cost schedule for part-time employees
(b) What is the 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-hour shifts.
(c) Assume that part-time employees can be assigned either a 3-hour or 4-hour shift. Develop a minimum-cost schedule for the part-time employees. How many part-time shifts are needed, and what is the cost savings compared to the previous schedule?
This is a challenging assignment and will probably take at least an hour to complete. Good luck! :-)
daddafd
To develop the minimum-cost schedule for part-time employees, we can use linear programming. Linear programming is a mathematical technique used to optimize the allocation of resources to achieve a specific objective, subject to certain constraints.
In this case, our objective is to minimize labor costs, and our constraints include the total number of employees needed at different times of the day and the availability of part-time employees for 4-hour shifts.
Let's start by defining some variables:
x1: Number of part-time employees assigned to the shift from 11:00 AM to 3:00 PM
x2: Number of part-time employees assigned to the shift from 3:00 PM to 7:00 PM
x3: Number of part-time employees assigned to the shift from 7:00 PM to 10:00 PM
Now, let's set up the equations based on the constraints and objective:
Objective: Minimize Labor Costs = 7.60 * (x1 + x2 + x3)
Constraints:
1) Total number of employees needed from 11:00 AM to 3:00 PM:
x1 + 2 ≤ 9 (one full-time employee is already working during this time)
2) Total number of employees needed from 3:00 PM to 7:00 PM:
x1 + x2 + 2 ≤ 12 (two full-time employees are already working during this time)
3) Total number of employees needed from 7:00 PM to 10:00 PM:
x2 + x3 ≤ 7 (two full-time employees are already working during this time)
4) Availability constraint - Part-time employees working at least 4-hour shifts:
x1, x2, x3 ≥ 0 (non-negativity constraint)
Now, let's solve these equations to find the minimum-cost schedule for part-time employees. We can use linear programming software or a spreadsheet tool like Excel Solver to solve this.
b) To determine the total payroll for part-time employees and the number of part-time shifts, we need the values of x1, x2, and x3 from the minimum-cost schedule obtained in part (a). Substitute these values into the equation:
Total Payroll = 7.60 * (x1 + x2 + x3).
Additionally, we need to examine the surplus variables to comment on the desirability of scheduling at least some of the part-time employees for 3-hour shifts. Surplus variables represent the unused portions of the constraints.
For example, in constraint 1:
Surplus1 = 9 - (x1 + 2)
By analyzing the surplus values for all constraints, we can determine if there is any underutilization or excess capacity in the labor schedule. If there are significant surplus values, it may suggest that assigning some part-time employees to 3-hour shifts could be desired to better match labor supply with demand while minimizing costs.
c) To develop a minimum-cost schedule when part-time employees can work either 3-hour or 4-hour shifts, we need to revise the constraints accordingly. We can modify constraint 1 as follows:
x1 + 1 ≤ 9 (since part-time employees can now work 3-hour shifts)
By making this adjustment to the constraints, solve the modified linear programming problem once again to obtain the new minimum-cost schedule.
Calculate the cost savings by comparing the total labor costs of the new schedule with the previous schedule.
Please note that the calculations involved in linear programming can be complex but using appropriate software or tools simplifies the process significantly.