Scheduling a project to maximize its net present value: An integer programming

approach

$($a$)$ write a brief report summarizing the paper; and$/$or$,$

$($b$)$ solve aSolve the following problem:

Consider operating a tutoring small business $($e$.$g$.$ a Mathnasium franchise$).$ You want

to know how many tutors you need to hire and what days to assign them in order to

minimize your labor cost while meeting your staffing needs at the Tutoring Lab. There

are two limiting factors that govern how many people you hire and what days they

work.

First, each of your workers receives $$300$ per week for a regular schedule, with $$25$

extra for Saturday work and $$35$ extra for Sunday work. For regulatory reasons each

worker can work only five days a week and must have two consecutive days off.

Second, you have minimum staffing needs, which must be met. In table form, that

requirement looks like this:

DAY M T W Th F S Su

STAFF NEEDED $20131012161820$

Under the restrictions outlined above, the following schedules are available for a staff

member, with varying weekly pay rates, where the symbol $($X$)$ denotes a day worked

and the symbol $($O$)$ denotes a day off:

Schedule Options

Start Day Mon Tue Wed Thu Fri Sat Sun Weekly Pay$($$$)$

Mon X X X X X O O $300$

Tue O X X X X X O $325$

Wed O O X X X X X $360$

Thu X O O X X X X $360$

Fri X X O O X X X $360$

Sat X X X O O X X $360$

Sun X X X X O O X $335$

Use linear programming to identify an optimal staffing schedule for a week $($number of

staff members working each of the $7$ schedule options and the minimal weekly staffing

cost$)$ that minimizes your labor cost.

$3.$ Identify any specific application of integer programming, describe the application, and

illustrate it with an example.n example to illustrate the results in the paper