- The document discusses optimizing production of three tomato products - whole canned tomatoes, tomato juice, and tomato paste - given constraints around available quantities and qualities of tomatoes. - It presents a linear programming model to maximize total contribution (profit) by determining the optimal allocation of grade A and B tomatoes to the three products, while adhering to demand, quality, and quantity constraints. - Running the model for the contracted 3 million pounds of tomatoes results in a gross profit of $45,340, whereas procuring an additional 80,000 pounds could increase profit by $424 to $45,764.

1. Faculty: Prof J Jha

4. Three types of tomato products.
 Whole canned tomato
 Tomato Juice
 Tomato Paste
Total Quantity
(already
contracted/purchased)
Cost per lb* in US ¢
(1 lb = 453.592 gms)
Total Cost in US $
( 1 $ = 100¢)
*Libra Pondo (lb) (Latin)
3M Lbs
06
1,80,000

5. Types of
tomato
Quantity (in M
lbs)
Quality (in points)
Quality points acceptable for
products
Grade A
0.6
9
Whole Canned - 8
Grade B
2.4
5
Juice - 6

6. Variables
Particulars
AW
A grade/whole canned
BW
B grade/whole canned
AJ
SP per case
(in $)/ lbs per
case
Contribution per lb
(in $)
4.0/18
(1.48/18) = 0.0822
4.5/20
(1.32/20) = 0.066
3.80/25
(1.85/25) = 0.074
A grade/juice
BJ
B grade/juice
AP
A grade/paste
BP
B grade/paste
Since the fruit has already been contracted,
The cost of fruit per pack is taken as a Sunk Cost

7. Max: Z = 0.0822*(AW+BW) + 0.066*(AJ+BJ)
+ 0.074*(AP+BP)

8. Demand constraints (lbs)
AW + BW <= 8,00,000 (Pre-determined because of quality of fruit available)
AJ + BJ <= 10,00,000 (50000 cases * 20 lbs/case)
AP + BP <= 20,00,000 (80000 cases * 25 lbs/case)
Quality constraints
For whole canned tomato
9AW + 5BW >= 8(AW + BW)
Solving, 1AW – 3BW >= 0
For tomato juice
9AW + 5BW >= 6(AW + BW)
Solving, 3AW – 1BW >= 0
For tomato paste . No constraints since paste is acceptable with B grade fruits.

9. Quantity constraints
For A Grade
AW + AJ + AP <= 6,00,000
(since only 0.6M lbs available)
For B Grade
BW + BA + BP <= 24,00,000
(Balance out of 3.0M lbs)

10. AW
BW
AJ
BA
AP
BP
Contribution
profit
225340
45340
525000
175000
75000
225000
0
2000000
0.0822
0.0822
0.066
0.066
0.074
0.074
AW
BW
AJ
BA
AP
BP
USED
AVAILABLE
CANNED
TOMATOES
1
1
0
0
0
0
700000
800000
JUICE
0
0
1
1
0
0
300000
1000000
PASTE
0
0
0
0
1
1
2000000
2000000
CANNED
TOMATOES
1
-3
0
0
0
0
0
0
JUICE
0
0
3
-1
0
0
0
0
A GRADE
1
0
1
0
1
0
600000
600000
B GRADE
0
1
0
1
0
1
2400000
2400000
TOTAL QUANTITY
DEMAND
QUALITY
QUANTITY

12. AW
BW
AJ
BA
AP
BP
contribution
profit
45764
615000
205000
65000
195000
0
2000000
0.0822
0.0822
0.066
0.066
0.074
0.074
Total Quantity
232564
USED
AVAILABLE
CANNED
TOMATOES
1
1
0
0
0
0
820000
880000
JUICE
0
0
1
1
0
0
260000
1000000
PASTE
0
0
0
0
1
1
2000000
2000000
CANNED
TOMATOES
1
-3
0
0
0
0
0
0
JUICE
0
0
3
-1
0
0
0
0
A GRADE
1
0
1
0
1
0
680000
680000
B GRADE
0
1
0
1
0
1
2400000
2400000

13. • Gross Profit with 3.0M lbs= $ 45340.
• Gross Profit with 3.08M lbs= $ 45764.
• That translates into a situation
wherein the company can earn more
profit if it procures 80000 lbs of A
grade tomato @ 8.5¢/lb, thereby,
earning an extra profit of US$ 424.