[aahuja1]

Help, I think I have either solved it or come to a roadbloc, any suggestions?

Production Costs: Cauchy Canners produces canned
whole tomatoes and tomato sauce. This season, the company
has available 3,000,000 kg of tomatoes for these two
products. To meet the demands of regular customers, it
must produce at least 80,000 kg of sauce and 800,000 kg of
whole tomatoes. The cost per kilogram is $4 to produce
canned whole tomatoes and $3.25 to produce tomato sauce.
Labor agreements require that at least 110,000 personhours
be used. Each kilogram can of sauce requires 3 minutes
for one worker, and each kilogram can of whole
tomatoes requires 6 minutes for one worker. How many
kilograms of tomatoes should Cauchy use for each product
to minimize cost? (For simplicity, assume production of
kg of canned whole tomatoes and kg of tomato sauce
requires kg of tomatoes.)

We also have to solve this problem, so this is what I have so far. Let me know if this is wrong or were I could go from here, and am I done at this point?



A. Select appropriate variables

Assume production of x kg of canned whole tomatoes and y kg of tomato sauce

B. Write objective functions

Since the cost per kg is $4 to produce canned whole tomatoes and $3.25 to produce tomato sauce,
the total cost is: p(x,y)=4x3.25y

C. Write constraints of inequalities

Constraint 1:
The company has available 3,000,000 kg of tomatoes for these two products

Constraint 2 &3 :
To meet the demands of customers, it must produce at least 80,000 kg of sauce and 800,000 kg of whole tomatoes.

Constraint 4:
Labor agreements require that at least 110,000 person-hours be used.
Each kilogram can of sauce requires 3 minutes for one worker and each kilogram can of whole tomatoes requires
6 minutes for one worker


Note: 110,000 person-hours equal to 6600000 minutes.

D. Solve

We can formulate the following LP:
mminimize St. Solving it by simplex method, we get
Optimal Solution: p = 4500000; x = 1060000, y = 80000
So, Cauchy should use 1060000 kg for canned whole tomatoes and 80000 kg for tomato sauces to minimize cost.

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Tableau #1
x y s1 s2 s3 s4 -p
1 1 1 0 0 0 0 3000000
1 0 0 -1 0 0 0 800000
0 1 0 0 -1 0 0 80000
6 3 0 0 0 -1 0 6600000
4 3.25 0 0 0 0 1 0

Tableau #2
x y s1 s2 s3 s4 -p
0 1 1 1 0 0 0 2200000
1 0 0 -1 0 0 0 800000
0 1 0 0 -1 0 0 80000
0 3 0 6 0 -1 0 1800000
0 3.25 0 4 0 0 1 -3200000

Tableau #3
x y s1 s2 s3 s4 -p
0 0 1 1 1 0 0 2120000
1 0 0 -1 0 0 0 800000
0 1 0 0 -1 0 0 80000
0 0 0 6 3 -1 0 1560000
0 0 0 4 3.25 0 1 -3460000

Tableau #4
x y s1 s2 s3 s4 -p
0 0 1 0 0.5 0.166667 0 1860000
1 0 0 0 0.5 -0.166667 0 1060000
0 1 0 0 -1 0 0 80000
0 0 0 1 0.5 -0.166667 0 260000
0 0 0 0 1.25 0.666667 1 -4500000