Red Brand Canners

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Case Study: RED BRAND CANNERS
Vice President of Operations Mr. Michell Gorden Controller Mr. William Copper Sale Manager Mr. Charles Myers Production Manager Mr. Dan Tucker

Purpose: Decide the amount of tomato products to pack at this season. Tomato Products Whole Tomato Tomato Juice Tomato Paste

Information: 1. Amount of Tomato: 3,000,000 pounds to be delivered. Tomato quality: 20% (grade A) × 3,000,000 = 600,000 pounds 80% (grade B) × 3,000,000 = 2,400,000 pounds (provided by production manager)

2. Demand forecasts & selling prices (provided by sale manager):

Products Demand

Whole canned tomato no limitation

Others Refer Exhibit 1

1

lbs.

correction (800,000/18) = 44444.5 Cases

Selling prices has been set in light of the long-term marketing strategy of the company. Potential sales have been forecasted at these prices. 3. Purchasing price & product profitability (provided by controller) Purchasing price 6cents/pound Net profit Refer Exhibit 2

Grade A 9 points

Grade B 5 points

Product
Minimum requirement

Whole tomato 8 points

Tomato juice 6 points

Tomato paste

5 points (without grade A)

2

3.8 -(0.54+0.26+0.38+0.77) = 1.85 4.0-(1.18+0.24+0.4+0.7) = 1.48

4.5 - (1.32+0.36+0.85+0.65) = 1.32

0.3

5. 80,000 pounds of grade "A" tomatoes are available at 8.5 cents per pound. (provided by the Vice president of operations) 6. Sale manager re-computes the marginal profits (Exhibit 3).

Linear Programming Solutions
(a) How to use the crop of 3,000,000 lbs. of tomatoes? (b) Whether to purchase an additional 80,000 lbs. of A-grade tomatoes?

Part (a)
Formulation: WA = lbs. of A-grade tomatoes in whole. WB = lbs. of B-grade tomatoes in whole. JA = lbs. of A-grade tomatoes in juice. JB = lbs. of B-grade tomatoes in juice.

3

¸Ñ(1) (2) ¤§Áp¥ß¤èµ{¦¡
1 CASE = 0.0518* 25= 1.295

1 CASE = [(0.0932*(3/4)+0.0518*(1/4)]*18

PA = lbs. of A-grade tomatoes in paste. PB = lbs. of B-grade tomatoes in paste. 600,000 lbs. - 3WB ¡Ù 0 WB ¡Ø 600,000/3 = 200,000 600,000 + 200,000 = 800,000 lbs. Demand of whole tomatoes ¡Ø 800,000 lbs. = 44,444.5 ×18 lbs Demand of tomatoes Juice ¡Ø 50,000 cases = 50,000 × 20 lbs = 1,000,000 lbs Demand of tomatoes paste ¡Ø 80,000 cases = 80,000 × 25 lbs = 2,000,000 lbs

Grade "A" ¡Ø 600,000 Grade "B" ¡Ø 2,400,000

( 3,000,000lbs × 20% ) =

600,000 lbs.

( 3,000,000lbs × 80% ) = 2,400,000 lbs.

4

Quality requirement for whole tomato:
Quality requirement for whole tomato: (0.9×WA + 0.5×WB)/2 ¡Ù 0.8× (WA + WB)/2 ⇒ WA - 3WB ¡Ù 0 Quality requirement for tomato juice: (0.9×JA + 0.5×JB)/2 ¡Ù 0.6× (JA + JB)/2 Constraints: WA CWA 1 WB CWB 1 JA CJA 1 JB CJB 1 1 1 1 1 -3 3 -1 1 1 1 1 1 PA CPA PB CPB ¡Ø ¡Ø ¡Ø ¡Ø ¡Ø ¡Ù ¡Ù 14,400,000 1,000,000 2,000,000 600,000 2,400,000 0 0

⇒ 3JA - JB ¡Ù 0
800,000 lbs.

Coefficients of Objective Function:
Both Cooper's and Myers' figures (Exhibits 2 and 3) are wrong. Contribution = selling price - variable cost (excluding tomato cost) Thus, CWA = CWB = 1.48/18 = 0.0822 CJA = CJB = 1.32/20 = 0.066 CPA = CPB = 1.85/25 = 0.074

The contribution = $225,340 - $180,000 = $45,340.

5

Optimal primal solution WA 525,00 WB
175,000

JA 75,000

JB 225,000

PA 0

PB 2×106

Optimal value = 225340

Optimal dual solution Column
Constraint

7 1 0

8 2 0

9 3
0.0161

10 4
0.0903

11 5
0.0579

12 6
8.1×10-3

13 7
8.1×10-3

Value

Shadow price on constraint 4 = 0.0903

Sensitivity on cost values

variable
Lower limit

1 0.0606 0.2336 0.0822

2 0.0606 0.5454 0.0822

3 -0.0884 0.0876 0.066

4
1.45333×10-2

5

6 0.0579

-∞
0.1064 0.074

Upper limit Current value

0.803111 ×10-2

+∞
0.074

0.066

6

Sensitivity on the right-hand sides
constraint

1
700,000

2
300,000

3
1.45333×10-6

4
133,333 1.2×106 600,000

5
2.2×106 2.8×106 2.4×106

6
-600,000

7
-200,000

Lower limit Upper limit Current value

+∞
1.44×107

+∞...
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