Oscar and Olaf are teachers with a total of 108 exams to grade. They have three

Question

Oscar and Olaf are teachers with a total of 108 exams to grade. They have three days to complete the task. Oscar has 12 hours available and Olaf has 10 hours available to do the first pass of grading on the exams. They are trying to figure out how much time they need to set aside for regrading, an unfortunate, but sometimes necessary task. It takes Oscar 7.2 minutes on average to grade an exam and it takes Olaf twelve minutes. However, Oscar will catch errors on the way back that will force him to regrade 10% of his exams, while Olaf will only have to regrade 6% of his. How many exams should each of them grade to minimize the number of exams that they have to regrade?

Formulate this problem as a Linear Program in standard form.

Solve for the optimal solution using graphical techniques. Clearly depict the constraints, feasible region, objective function, and optimal solution.

Answer the following the questions.

1. How many TEEs should Oscar and Olaf each grade?

2. How many exams need to be regraded with this assignment strategy?

3. Which constraints are binding?

Explanation / Answer

Let’s say,

Number of exams graded by Oscar = A

Number of exams graded by Olaf = B

Then

A + B = 108          ………………………………………………………………………………….. eqn-1

Also,

B = 108 – A

Now,

Number of exams regraded by Oscar = 10%, i.e. 0.1 A

Number of exams regraded by Olaf = 6% = 0.06B = 0.06 (108 – A)

Since we want to minimize the number of regraded exam, our function would be

Minimum (0.1 A + 0.06 (108 – A))

C = Minimum (0.04 A + 6.48)                      …………………………………………….    eqn-2

Where C is the minimum number of regraded exams. This is the objective function.

It is also given that

Oscar takes 7.2 minute per exam = 7.2/60 hours/exam

Olaf takes 12 minute per exam = 12/60 hours/exam

And,

Total time Oscar has for grading exams = 12 hours

Total time Olaf has for grading exams = 10 hours

Therefore,

Number of exams graded by Oscar times the rate at which he grades exam should be less than 12

i.e.

A x (7.2/60) < 12

Or A < 100           …………………………………………………………………………….          eqn-3

And, similarly

B x (12/60) < 10

Or (108 – A) x (12/60) < 10

Or 108 –A < 50

Or A > 58             ………………………………………………………………………………        eqn-4

Equation 3 and 4 are the two constraints.

Figure below shows the objective function, constraints and feasible region. It also shows the optimal solution.

The minimum value of equation 3 would be at A = 58

C = 0.04 A + 6.48 = 8.8

Since C can only take integers, C = 9

So,

Number of exams graded by Oscar = 58

Number of exams graded by Olaf = 50

Number of exams regraded by Oscar = 6

Number of exams regraded by Olaf = 3

So,

Answer 1: Oscar = 58, Olaf = 50

Answer 2: Total number of regraded exams = 9

Answer 3: Constraint A > 58 is binding, since changing this constraint will also change the optimal solution.

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