TT Produces 2 types of Tablets(10.1\" & 8\"), each of which must go through 3 wo

Question

TT Produces 2 types of Tablets(10.1" & 8"), each of which must go through 3 workstations:manufacturing,assembly, & quality control.The amount of time required on each type of Table, in each of the workstation, is given by the following model manufacturing assembly Quality 10.1" 11 minutes 4min 1.5min 8" 15 min 2min 2.5min TT employs 15 workers, who each work 7.5 hours/day. Of these, 2400 man-minutes per day have been allocated to manufacturing,3100 to assembly and 1250 to quality control. Accounting has stated that the cost of each manufacturing minute is $7.50, each Assembly minute $4.25, and each Quality Control minute $15, for a 10.1". For a 8" type, the cost are $7.75.$3,and $18. If TT sells for $299 the 8" type and $349 the 10.1" type, what should be the optimal daily production schedule.

Explanation / Answer

Let's call the tablet T for 10.1" and E for 8"

Tsells for 349 and costs 11*7.5+4*4.25+1.5*15, so the profit is 227

E sells for 299 and costs 15*7.75 + 2*3 + 2.5*18, so the profit is 131.75

Thus, we maximize 227 T + 131.75 E

11T + 15E <= 2400 (manufacturing)

4T + 2E <= 3100 (assembly)

1.5T + 2.5E <= 1250 (QC)

T >= 0 E >= 0

We may write our point as (E, T)

Since our objective function and our constraints all have positive coefficients, and all functions are linear, our solution is at a corner point of the feasible region not equal to (0, 0)

Setting E = 0, we find T <= min(2400/11, 3100/4, 1250/1.5) = min(218.1818.775, 833.333) =

218.181818)   

(0, 218.181818) is a corner point

Setting T = 0, we have

min(2400/15, 3100/2, 1250/2.5) = min(160, 1550, 500) = 160

(160, 0) is a corner point.

Then, we could consider the intersection of the other 3 lines.

However, note that both of these corner points came from the intersection with the same constraint, 11T + 15E <= 2400 (manufacturing), so this is the only constraint we need to consider.

We only need to consider

(0, 218.181818) and (160, 0)

The value of the objective function is

160 * 131.75= 21080 at (160, 0)

218.181818 * 227 = 49527.2727272727 at (0, 218.18181818)

We should sell only 10 " monitors. We should sell 218.18 and make a profit of 49527.27

(The 15 workers at 7.5 hours/day does not impact the problem; I can note, though, that 2400*3100+1250 = 7.5*15*60)

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