Ten jobs are to be completed by three workers (Sam, Joy, and Kelly) during the next
week. Each person works a 40-hour week and is paid an hourly rate: Sam, Joy, and Kelly
earn $11, $13, and $14 per hour, respectively. Union rules require workers to be treated
fairly; to ensure that most of the work is not assigned to one person while others are too
idle, make sure that no person works more than 8 hours above any other worker. For
example, if Joy works 20 hours, then Sam and Kelly should work within 12 to 28 hours,
and Sam and Kelly should have work-hours that are no more than 8 hours apart.
The times for the workers to complete the tasks are shown in the table below. The values
in the cells assume that each task is completed by a single person. However, tasks can be
shared with completion times being determined proportionally (e.g. if Joy and Kelly
share task 1 equally, then Joy works 6 hours and Kelly works 9 hours). If no entry exists
in a particular cell, it means that task cannot be performed by that worker.
a. Formulate the problem as a linear programming model (that is, define the variables,
and write down the objective function and all constraints mathematically).
b. Create a spreadsheet model for this problem and solve with Excel Solver.
c. What is the optimal solution? What is the optimal value?
I am looking for an answer to this
To ensure fair distribution of work, we need to assign tasks to Sam, Joy, and Kelly in a way that no person works more than 8 hours above any other worker.
Let's start by calculating the total number of work hours required for all ten tasks.
According to the table, the total work time for each task is as follows:
Task 1: 15 hours
Task 2: 10 hours
Task 3: 6 hours
Task 4: 20 hours
Task 5: 16 hours
Task 6: 12 hours
Task 7: 8 hours
Task 8: 24 hours
Task 9: 14 hours
Task 10: 18 hours
Adding up these hours, we get a total of 143 hours.
Since each worker will work a 40-hour week, we need to distribute these 143 hours among them evenly while ensuring no one exceeds the 8-hour difference limit.
Let's calculate the maximum and minimum work hours for each worker assuming we divide the total work hours evenly:
Sam:
Maximum work hours = 40 hours + 8 hours = 48 hours
Minimum work hours = 40 hours - 8 hours = 32 hours
Joy:
Maximum work hours = 40 hours + 8 hours = 48 hours
Minimum work hours = 40 hours - 8 hours = 32 hours
Kelly:
Maximum work hours = 40 hours + 8 hours = 48 hours
Minimum work hours = 40 hours - 8 hours = 32 hours
Now, let's distribute the work hours among the three workers, making sure that no worker exceeds the maximum or falls below the minimum work hours.
One possible distribution could be:
Sam works on tasks 1, 3, 7, 9, and 10.
Joy works on tasks 2, 4, and 8.
Kelly works on tasks 5 and 6.
Now, let's calculate the work hours for each worker:
Sam: 15 + 6 + 8 + 14 + 18 = 61 hours
Joy: 10 + 20 + 24 = 54 hours
Kelly: 16 + 12 = 28 hours
As we can see, no worker exceeds 8 hours above any other worker. Sam has 61 hours, which is 7 hours more than Joy's 54 hours and 33 hours more than Kelly's 28 hours.
This distribution ensures fair treatment of the workers while completing all ten tasks within the given time frame.