Duke Test Center administers French language tests to job candidates who are app

Question

Duke Test Center administers French language tests to job candidates who are applying for work in international operations. Employees called raters grade the tests. The center employs five senior raters and nine junior raters.

Each French test consists of three parts: written, computer, and tape-recorded. The time, in minutes, required by each kind of rater to grade each type of test is as follows:

Rater

Test Type

Written

Computer

Recorded

Senior

20

5

15

Junior

30

6

18

Each rater can work up to 40 hours in the coming week. 800 candidates have just taken the test, and the results have to be announced after a week. Work can also be assigned to outside part-time raters, who charge $25 for each written test, $15 for each computer test, and $20 for each recorded test.

For quality control purposes, senior raters must grade at least 10% of all tests rated in-house in each category.

Formulate an algebraic linear programming model to minimize the amount paid to outside raters, while still meeting the deadline for grading all the tests. Clearly define all your decision variables.  

Rater

Test Type

Written

Computer

Recorded

Senior

20

5

15

Junior

30

6

18

Explanation / Answer

LP model:

S1, S2, S3 = Number of each of the three types of tests - Written, Computer and Recorded respectively assigned to Senior raters

J1, J2, J3 = Number of each of the three types of tests - Written, Computer and Recorded respectively assigned to Junior raters

P1, P2, P3 = Number of each of the three types of tests - Written, Computer and Recorded respectively assigned to Part-time raters

Objective: Min (25*'30*P1 + 15*6*P2 + 20*18*P3)/60 (Part-time raters time to grade each test is not provided in the question. so Juniors raters time is considered)

s.t.

20S1+5S2+15S3 <= 5*40*60 (total capacity of Senior raters)

30J1+6J2+18J3 <= 9*40*60 (total capacity of Junior raters)

S1+J1+P1 = 800 (total 800 tests of Written type)

S2+J2+P2 = 800 (total 800 tests of Computer type)

S3+J3+P3 = 800 (total 800 tests of Recorded type)

S1+J1 >= 0.1*(S1+J1+P1) (at least 10% of all Written tests to be rated in-house)

S2+J2 >= 0.1*(S2+J2+P2) (at least 10% of all Computer tests to be rated in-house)

S3+J3 >= 0.1*(S3+J3+P3) (at least 10% of all Recorded tests to be rated in-house)

S1, S2, S3, J1, J2, J3, P1, P2, P3 >= 0

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