Answers to the questions...

[Emaleth]

<font color="770000">T1: Go to store (10)--T4: Get water (12)--T5: Cook soup (15) = 37 Min.
T2: Collect firewood (8)--Wait (2) for matches to arrive from store…T3: Light fire (6)--T6: Make hamburgers (9)--T7: Cook hamburgers (7)=32 Min.

T1: Go to store (10)--T3: Light fire (6)--T6: Make hamburgers (9)--T7: Cook hamburgers (7)=32 Min.
T2: Collect firewood (8)--T4: Get water (12)--T5: Cook soup (15)=35 Min.

T1: Go to store (10)--T2: Collect firewood (8)--T3: Light fire (6)--T5: Cook soup (15)=39 Min.
T4: Get water (12)--T6: Make hamburgers (9)--T7: Cook hamburgers (7)=28 Min.

With only two people what is the best way to do this problem? Obviously, you can't light the fire until you get matches, so there is a waiting period of 2 minutes inherent in collecting wood to being able to light the fire. Also, you cannot cook the soup until you get the water and you cannot cook the hamburgers until you make the patties. Lastly, you cannot cook the hamburgers and cook the soup until the fire is lit! Therefore, time 16 is critical before the soup and hamburgers can start.

Since T1, T2 & T3 are critical they also require two people. So, depending upon whom gets the matches or wood, work on getting water or making hamburger patties cannot start until at least time 8. If time 8 is added to task T4 to T5 (27) we get 35 minutes as the critical time since the other person would not be available for T6 to T7 (16) until time 16 and this is 32 minutes and less than the critical time of 35.

Conversely, if the person after time 8 chose to do T6 to T7 (16) and got 24 minutes, the other person at time 16 would have to do T4 to T5 (27) or 43 minutes (a longer critical time).

Ahh, isn't it grand to have a Math's Wiz for a Father! Well, I was close.. :-| LOL!</font color>

[Emaleth]

<font color="600099">Cookie/Muffin Batch Problem: Each batch of cookies requires 2 pounds of flour and 1 pound of chocolate chips, and each batch of muffins uses 3 pounds of flour and 2 pounds of chocolate chips. You have 60 pounds of flour and 36 pounds of chocolate chips on hand. Your profit on each batch of cookies is $8, and on muffins your profit is $13 per batch.

60# Flour 36# Chocolate Chips Profit/Batch Total Profit
Cookie 2#/Batch(30) 1#/Batch(36) $8.00 $240.00(limited by 30)
Muffins 3#/Batch(20) 2#/Batch(18) $13.00 $234.00(limited by 18)
As you can see above, you are limited to only 30 batches of cookies as you run out of flour and you are limited to only 18 batches of muffins as you are limited by the chocolate chips.

Mathematically, by solving the two equations: 2x+3y=60 and x+2y=36 for the batch requirements of the flour and chocolate chips respectively, gives a value of 12 batches of cookies and 12 batches of muffins as the best mix for profit.

Using the total number of batches of muffins (the lower of the two possible batches) as the limiting factor in determining a mix, you can figure out the total profit made by various combinations of cookie and muffin batches. As an example; with 1 batch of muffins you use 3 pounds of flour and 2 pounds of chocolate chips. This leaves 57 pounds of flour and 34 pounds of chocolate chips for cookies. With 57 pounds of flour you can only make 57/2 batches and only whole batches are allowed so you can make 28 batches out of the flour. You are limited to 34/1 batches of chocolate chips or 34 batches. Thus, you are limited to only 28 batches of cookies as the poundage of flour is the limiting factor. The total profit would be with this combination:

1 batch of Muffins ($13.00) + 28 batches of Cookies(S8.00) = $13.00 + $224.00=$237.00

Doing the same thing with 2, 3, 4 ..... 17 batches of muffins and getting the number of batches of cookies that are possible with the remaining flour and chocolate chips will allow you to get a listing of total profits available with the various combinations. I have presented a number of graphs and listings on the following page to show these values.

The interesting thing about the combinations is that the total profit fluctuates near the middle values. The reason for this is that you cannot always get even batches our of the flour. You must round down to the lower integer batch and this causes the total profit to go up and down until the best total profit is reached at 12 batches of cookies and 12 batches of muffins. This combination was proven by the mathematical solution to the best combination (the maximum corner point).

The geometry of this situation is crucial to the solution. A fundamental result, the Corner Point principle, states that the optimal solution occurs at a comer of the feasible region. This result suggests a possible solution strategy: Find all corners of the feasible region; determine the profit at each; choose the production policy of any corner producing the maximum profit. Computationally, this procedure suffers from the same problem as the brute force method applied to the TSP; for large problems there are just too many corners to check quickly.

Hmm, I wasn't even close on this one..lol</font color>