# Math/Management

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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.

• Math/Management -

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?

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