SOLUTION: The head of a management information center wants to staff consulting stations with two shifting teams. Team A will comprise of 3 senior programmers and 3 system analysts and Team

Question 1131909: The head of a management information center wants to staff consulting stations with two shifting teams. Team A will comprise of 3 senior programmers and 3 system analysts and Team B will consists of 2 senior programmers and 5 system analysts.
The director wants to use no more than 42 individuals. There will be at least 48 hours to be filled during the week, with Team A shift serving for 4 hours and Team B shift serving 3 hours. The cost of Team A per shift is 3200$ per hour and 2800$ per hour for a Team B per shift.
Determine the number of shifts of each team in order to minimize the cost.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

When I first read this problem a few days ago, the wording seemed confusing, so I didn't spend any time looking at it. Now several days have passed, and nobody has provided a response; so I thought I would take the time to try to decipher the problem.

Unfortunately, that time was wasted, because I could not find any sensible interpretation of the problem.

But since I spent the time, I thought I would give you my feedback so you can see if you have not stated the problem correctly.

Shifts are staffed by either team A or team B.
team A has 3 senior programmers and 3 system analysts; team B has 2 senior programmers and 5 system analysts.
team A shifts are 4 hours; team B shifts are 3 hours.
The objective is to minimize the cost of a minimum of 48 hours of work at a cost of $3200 per hour for team A and $2800 per hour for team B; that is $12,800 per shift for team A shifts and $8400 per shift for team B shifts.

Let A be the number of shifts for team A
Let B be the number of shifts for team B

Then the requirement of filling 48 hours of shifts gives us the following constraint:

(1) 4A + 3B >= 48

That's as far as I can get making any sense out of the problem.

The other given information is that the director "wants to use no more than 42 individuals". I can't see how to use that; I see two possible interpretations, neither of which leads to a solvable problem.

One possibility is that every team A shift is staffed by the same people, and every team B shift is staffed by the same people. In that case, the total number of people used is only 13; so the requirement to use no more than 42 individuals is satisfied. But that leaves us with only one constraint; and the problem can't be solved with only one constraint.

The other possibility is that every team A shift uses a different 6 people and every team B shift uses a different 7 people. That would lead to the following constraint:

(2) 6A + 7B <= 42

But there are no solutions in positive integers for constraints (1) and (2) -- so again the problem can't be solved.

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