# binary optimization

*label*Management

*timer*Asked: Mar 6th, 2016

**Question description**

1. Ten jobs are to be completed by three workers (Sam, Joy, and Kelly) during the next week. Each person works up to 40 hours per 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** optimization
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?