Expert: Paul Klarreich - 11/24/2008

Question
A company has 20 checkouts which can be staffed by either trainee or
experienced operators. Experienced operators serve on average 15 customers
per hour and trainees 10 customers. The company intends to serve a
minimum of 210 customers per hour. At present there have to be more
experienced operators than trainees and the ratio of trainees to experienced
operators must be at least 1 to 3.
Experienced operators are paid £7.50 per hour and trainees are paid £3 per
hour. What is the cheapest arrangement?

What happens if the trainees wage is raised to £4 per hour


So far:.....is this ok

let's have x experienced operators and y trainees.

x + y = 20 (?)
15x + 10y ≥ 210
x - y > 0
x - 3y ≤ 0
x ≥ 0 and y ≥ 0
minimizing 7.5x + 3y

(x,y) = (11,9) is the optimal point.

answer
= 7.5*11 + 3*9
= 82.5 + 27
= £109.50  

Answer
Questioner:   rg
Category:  Advanced Math
Private:  No
 
Subject:  word problem /algebra
Question:  A company has 20 checkouts which can be staffed by either trainee or experienced operators. Experienced operators serve on average 15 customers per hour and trainees 10 customers. The company intends to serve a minimum of 210 customers per hour.

At present:
-- there have to be more experienced operators than trainees and
-- the ratio of trainees to experienced operators must be at least 1 to 3.

Experienced operators are paid £7.50 per hour and trainees are paid £3 per hour. What is the cheapest arrangement?

What happens if the trainees wage is raised to £4 per hour
................................
Hi, rg,

I am not sure how you got your (11,9) as the optimal point.



So far:.....is this ok

let's have x experienced operators and y trainees.

x + y = 20 (?)  (call this A)

>> Make that  x + y <= 20; you might not need them all.  Now this becomes one of your constraints.

15x + 10y >= 210 (B)

OR 3x + 2y >= 42  (B)

 x - y > 0   (C)
OR  x > y     (C)

x - 3y <= 0  (D)

x >= 0 and y >= 0
minimizing 7.5x + 3y

(x,y) = (11,9) is the optimal point.

answer
= 7.5*11 + 3*9
= 82.5 + 27
= £109.50


The usual process is to turn the above into EQUATIONS, find points of intersection, then make sure they satisfy the inequalities.  It is the BOUNDARY points that produce the min or max.


Here are the inequalities to solve, and we look for solutions to the associated equations:

x + y <= 20     (A)
3x + 2y >= 42   (B)
x > y       (C)
x <= 3y      (D)

A+B:

2x + 2y = 40     (A)
3x + 2y = 42   (B)
x = 2, y = 18; (2,18)  No good for C

AC:
x = 10, y = 10; (10,10); OK for all. A boundary point.

AD:
4y = 20
y = 5, x = 15; (15,5); OK;

BC:
5x = 42; x = 8.4; y = 8.4; OK.

BD: 11y = 42; y = 3+9/11; x = 11+5/11; OK

CD: (0,0);  NG.

So we have our four points:

(10,10)
(15,5)
(8.4,8.4)
(3.82, 11.45)

and you are supposed to check these.