Question
(in Stat 250 introduction to Operation Research )
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10, n the Ma-and-Pa grocery store, shelf space is limited and must be used effectively t o and Whéatie, compete for a tetal shelf space of ase profit. Two cereal item 60 A box of Grano occupies 2 fti2and a box of Wheatie needs 4 fi The maximum Grano nets1.00 profit and a box of Wheatie SL.35, Ma-and-Pa thinks that because the Grano and Wheaue are 200 andú20 boxes, respectively. A box of nit profit ofwheatie is 5%higher than that of Grano, wheatieshould be allocated more space than Grano, which amounts to allocating about 57% to wheatie and to Grano. What do you think? 11. Jack is an aspiring freshman at Ulern University. He realizes that "all work and no play make Jack a dull boy" Jack wants to apportion his available time of about 10 hours a day between work and play. He estimates that play is twice as much fun as work. He also wants to study at least as much as he plays. However, Jack realizes that if he is going to get all his homework assignments done, he cannot play more than 4 hours a day. How should Jack allocate his time to maximize his pleasure from both work and play? 12 Wild West produces two types of cowboy hats. A Type 1 hat requires twice as much labor time as a Type 2. If all the available labor time is dedicated to Type 2 alone, the company can produce a total of 400 Type 2 hats a day. The respective market limits for Type 1 and Type 2 are 150 and 200 hats per day, respectively. The profit is $8 per Type 1 hat and $5 per Type 2 hat. Determine the number of hats of eaçh type that maximizes profit. 13. Show & Sell can advertise its products on local fadio and television (TV). The advertising budget is limited to $10,000, month. Each minute of radio advertsing costs $15 and each minute of TV commercals S300 Show & Sell likes to advertise on radio at least twice as much as on TV. In the meantime, it is not practical to use more than 400.minutes of radio advertising a month. From past experience, advertising on TV is estimated to b as effective as on radio. Determine the optimum allocation of the budget to radio an advertising.
Explanation / Answer
Solution to question no. 10
let us denote grano and wheatie by x and y respectively.
According to the constraints,
Space constraint : 0.2x + 0.4y = 60
Demand Constraint : x =< 200 , y =< 120.
To maximaize profit Z = 1x+1.35y under the above constraints.
If only X: then Xmax = 60/0.2 = 300 ; If only Y: then Ymax = 150.
Putting the demand constraints, x =< 200 , y =< 120.
So, intersection of these give the extremum points.
{x,y} : {60,120} or {200,50}
Profit at {60,120} = 60*1 + 120*1.35 = 222
profit at {200,50} = 200*1 + 50*1.35 = 267.5
SO maximum profit is found when 200 boxes of Grano and 50 boxes of wheatie are kept.
space allocation in this situation = 200 * 0.2 = 40 sqft. = 66.67% for grano
and 50 * 0.4 = 20 sqft which is 33.33% for wheatie.
Thus, i disagree with the space allocation done.
Note: By Providing 43% and 57% respectively, the profit made would be = 244 approx
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