Information： Amount of incoming tomatoes: 3,000,000 pounds Grade A：20%×3,000,000 = 600,000 pounds Grade B：80%×3,000,000 = 2,400,000 pounds Product Selling Price Per Case Demand Forecast (Cases)(1) Pounds Per Case(2) Total weight in pounds(1)*(2)
Minimum Quality requirement

Whole tomatoes $4.00 800,000 18 14,400,000
8 $1.48

Tomato juice $4.50 50,000 20 1,000,000
6 $1.32

Tomato paste $3.80 80,000 25 2,000,000
5 $1.85

Contribution per case (3) Contribution per pound (excl Tomato cost) (3)/(2)

$0.082

$0.066

$0.074

WA = Pounds of A-grade tomatoes in canned whole tomatoes.. WB = Pounds of B-grade tomatoes in canned whole tomatoes. JA = Pounds of A-grade tomatoes in tomato juice. JB = Pounds of B-grade tomatoes in tomato juice.

PA = Pounds of A-grade tomatoes in tomato paste. PB = Pounds of B-grade tomatoes in tomato paste.

Z: Total net profit cost per pound  Copper’s objective function(see Table 3-18): Max Z= (WA+WB) *0.12/18-(JA+JB)*0.09/20+(PA+PB)*0.12/25 ==> Max Z=0.0067(WA+WB)-0.0045(JA+JB)+0.0048(PA+PB)


Myer’s objective function(see Table 3-19):

Max Z= (WA+WB) *(-0.01)/18-(JA+JB)*0.08/20+(PA+PB)*0.55/25 ==> Max Z=-0.00056(WA+WB)+0.004 (JA+JB)+0.022(PA+PB)



Correct objective function(see Table 3-19):

Max Z= (WA+WB) *(1.48)/18-(JA+JB)*1.32/20+(PA+PB)*1.85/25-0.06*3,000,000 ==> Max Z=0.0822(WA+WB)+0.066(JA+JB)+0.074(PA+PB)- 180,000

Q1：Verify, why, as per Tucker, the Whole Tomato production is limited to 800,000 pounds (i.e., where does the number 800,000 come from? "A" tomatoes averaged 9 points per pound, "B" tomatoes averaged 5 points per pound, and the minimum average input quality for canned whole tomatoes was 8, so the quality constraints for canned whole tomatoes is WA≥3WB (
9WA +5WB WA +WB

≥8)..

Since only 600,000 pounds of incoming tomatoes was Grade "A" quality, if all go into canned whole