Red Brand Canners
Burhan Sadiq
December 3, 2009
Outline
Statement of Problem
Dilemma
LP Formulation
Results
Optimum Product Mix
Additional Grade A Tomatoes
Advertising Campaign
Tomato Juice Price Increase
Conclusion
Red Brand Canners
Red Brand Canners used to be a medium-size company canning
and distributing a variety of fruit and vegetable products in the
western part of USA in the 1960s. In one year, they signed an
agreement at planting time, to purchase the entire tomato crop in
a large field at an average delivered price of 6 cents per lb.
At harvest time, Produce Inspection estimates that the total crop
will be 3 ∗ 106 lb, of which 20% is expected to be Grade A and the
remaining portion expected to be Grade B.
Data
The company makes three different tomato products, and they set
the selling prices of these products in light of the long-term
marketing strategy of the company.
Data
The company makes three different tomato products, and they set
the selling prices of these products in light of the long-term
marketing strategy of the company.
Moreover, the company uses a numerical scale to record the
quality of both the raw produce and prepared products. The scale
runs from 0 to 10 points, higher numbers representing better
quality. On this scale, Grade A tomatoes were valued at 9
points/lb, and Grade B tomatoes 5 points/lb.
Data Cont.
Product (cases) SP TU VC
DF
MAIQR(per pound)
Whole Tomatoes 4
18 2.52 Unlimited
8
Tomato Juice
4.5 20 3.18
50,000
6
Tomato Paste
3.8 25 1.95
80,000
5
SP:Selling Price($/case), TU:Tomatoes (lb) used/case,
VC:Variable Costs(excluding tomato purchase costs) ($/case) ,DF:
Demand Forecast (cases) at these prices and MAIQR: Minimum
Average Input Quality Requirement.
Problem
The company hires you as a consultant to determine the following:
I i) How to allocate the already purchased tomatoes among the
three tomato products in order to maximize profit.
Problem
The company hires you as a consultant to determine the following:
I i) How to allocate the already purchased tomatoes among the
three tomato products in order to maximize profit.
I ii) Red Brand Canners has been approached by another
distribution company that will supply additional Grade A
tomatoes at a price of .085$ per lb. Should the company
purchase additional Grade A tomatoes and if so, what is the
maximum amount the company can purchase? If the
company wishes to purchase only 80,000 lbs of Grade A
tomatoes, what is the new optimum product mix?
Problem
The company hires you as a consultant to determine the following:
I i) How to allocate the already purchased tomatoes among the
three tomato products in order to maximize profit.
I ii) Red Brand Canners has been approached by another
distribution company that will supply additional Grade A
tomatoes at a price of .085$ per lb. Should the company
purchase additional Grade A tomatoes and if so, what is the
maximum amount the company can purchase? If the
company wishes to purchase only 80,000 lbs of Grade A
tomatoes, what is the new optimum product mix?
I iii) The Marketing Department feels that it can increase
demand for tomato juice by 25,000 cases. How much should
the company spend on such a campaign?
Problem
The company hires you as a consultant to determine the following:
I i) How to allocate the already purchased tomatoes among the
three tomato products in order to maximize profit.
I ii) Red Brand Canners has been approached by another
distribution company that will supply additional Grade A
tomatoes at a price of .085$ per lb. Should the company
purchase additional Grade A tomatoes and if so, what is the
maximum amount the company can purchase? If the
company wishes to purchase only 80,000 lbs of Grade A
tomatoes, what is the new optimum product mix?
I iii) The Marketing Department feels that it can increase
demand for tomato juice by 25,000 cases. How much should
the company spend on such a campaign?
I iv) If the selling price of tomato juice is increased by .30
$/case, how will the optimum product mix change if the
demand remains fixed.
Problem
The company hires you as a consultant to determine the following:
I i) How to allocate the already purchased tomatoes among the
three tomato products in order to maximize profit.
I ii) Red Brand Canners has been approached by another
distribution company that will supply additional Grade A
tomatoes at a price of .085$ per lb. Should the company
purchase additional Grade A tomatoes and if so, what is the
maximum amount the company can purchase? If the
company wishes to purchase only 80,000 lbs of Grade A
tomatoes, what is the new optimum product mix?
I iii) The Marketing Department feels that it can increase
demand for tomato juice by 25,000 cases. How much should
the company spend on such a campaign?
I iv) If the selling price of tomato juice is increased by .30
$/case, how will the optimum product mix change if the
demand remains fixed.
How do you solve this problem?
Problem
The company hires you as a consultant to determine the following:
I i) How to allocate the already purchased tomatoes among the
three tomato products in order to maximize profit.
I ii) Red Brand Canners has been approached by another
distribution company that will supply additional Grade A
tomatoes at a price of .085$ per lb. Should the company
purchase additional Grade A tomatoes and if so, what is the
maximum amount the company can purchase? If the
company wishes to purchase only 80,000 lbs of Grade A
tomatoes, what is the new optimum product mix?
I iii) The Marketing Department feels that it can increase
demand for tomato juice by 25,000 cases. How much should
the company spend on such a campaign?
I iv) If the selling price of tomato juice is increased by .30
$/case, how will the optimum product mix change if the
demand remains fixed.
How do you solve this problem? Linear Programming
LP Formulation
3
X
xai + xbi
max z =
(si − vi )
−c ∗S
li
i=1
subject to:
3
X
xai ≤ Sa
Supply Restriction on Grade A tomatoes
i=1
3
X
xbi ≤ Sb
Supply Restriction on Grade B tomatoes
i=1
qa ∗ xai + qb ∗ xbi ≥ Qi (xai + xbi ) ∀i
Quality Restriction
xai + xbi
≤ di
Demand Restriction
li
xai , xbi ≥ 0
Non-Negativity
si , li , vi : selling price, lbs of tomatoes used, variable costs per case
of ith product. xai , xbi : Amounts of Grade A, Grade B tomatoes
used in ith product.
Results
Optimum Mix
Product
Whole Tomatoes
Tomato Juice
Tomato Paste
Grade A
525,000
75,000
-
Grade B
175,000
225,000
2,000,000
Results
Optimum Mix
Product
Whole Tomatoes
Tomato Juice
Tomato Paste
Grade A
525,000
75,000
-
Grade B
175,000
225,000
2,000,000
Marginal Values
Item
Grade A tomatoes
Grade B tomatoes
Demand of WholeTomatoes
Demand of Tomato Juice
Demand of Tomato Paste
Marginal Values ($/unit)
0.090
0.058
0.000
0.000
0.403
Results
Optimum Mix
Product
Whole Tomatoes
Tomato Juice
Tomato Paste
Grade A
525,000
75,000
-
Grade B
175,000
225,000
2,000,000
Marginal Values
Item
Grade A tomatoes
Grade B tomatoes
Demand of WholeTomatoes
Demand of Tomato Juice
Demand of Tomato Paste
Marginal Values ($/unit)
0.090
0.058
0.000
0.000
0.403
Optimum Objective Value: $45,355.56
Results
Makes sense to purchase additional Grade A tomatoes for 8.5
cents/lb. How much? Turns out: 600,000 lbs of Grade A
Tomatoes. Optimum Objective Value: $48,555.56
Optimum Mix
Product
Whole Tomatoes
Tomato Juice
Tomato Paste
Grade A
1,200,000
-
Grade B
400,000
2,000,000
Results
Makes sense to purchase additional Grade A tomatoes for 8.5
cents/lb. How much? Turns out: 600,000 lbs of Grade A
Tomatoes. Optimum Objective Value: $48,555.56
Optimum Mix
Product
Whole Tomatoes
Tomato Juice
Tomato Paste
Grade A
1,200,000
-
Grade B
400,000
2,000,000
Updated Marginal Values
Item
Grade A tomatoes
Grade B tomatoes
Demand of Whole Tomatoes
Demand of Tomato Juice
Demand of Tomato Paste
Marginal Values ($/unit)
0.085
0.074
0.000
0.000
0.003
Results
If the company wishes to purchase only an additional 80,000 lbs of
Grade A tomatoes, the new optimum mix is given by:
Optimum Product Mix
Product
Whole Tomatoes
Tomato Juice
Tomato Paste
Grade A
615,000
65,000
-
Grade B
205,000
195,000
2,000,000
Results
If the company wishes to purchase only an additional 80,000 lbs of
Grade A tomatoes, the new optimum mix is given by:
Optimum Product Mix
Product
Whole Tomatoes
Tomato Juice
Tomato Paste
Grade A
615,000
65,000
-
Grade B
205,000
195,000
2,000,000
Optimum Objective Value: $45,782.22
Advertising Campaign
Given that the Marginal Values at the optimum product mix are:
Marginal Values
Item
Grade A tomatoes
Grade B tomatoes
Demand of Whole Tomatoes
Demand of Tomato Juice
Demand of Tomato Paste
Marginal Values ($/unit)
0.090
0.058
0.000
0.000
0.403
Company should NOT pay any money for an advertising campaign
to increase the demand for Tomato Juice.
Results
Tomato Juice price is increased by 30 cents per case.
Optimum Product Mix
Product
Whole Tomatoes
Tomato Juice
Tomato Paste
Grade A
350,000
250,000
-
Grade B
116,667
750,000
1,533,330
Results
Tomato Juice price is increased by 30 cents per case.
Optimum Product Mix
Product
Whole Tomatoes
Tomato Juice
Tomato Paste
Grade A
350,000
250,000
-
Grade B
116,667
750,000
1,533,330
Marginal Values
Item
Grade A tomatoes
Grade B tomatoes
Demand of Whole Tomatoes
Demand of Tomato Juice
Demand of Tomato Paste
Marginal Values ($/unit)
0.085
0.074
0.000
0.085
0.000
Optimum Objective Value: $52,837.04
Conclusions
I
i) Optimum Objective Value: $45,355.56
I
ii) Company should purchase an additional 600,000 Grade A
tomatoes. Optimum Objective Value: $48,555.56
I
iii) Company should NOT pay any money for an advertising
campaign to increase the demand for Tomato Juice.
I
iv) If the selling price of tomato juice is increased by .30
$/case and if the demand remains fixed, the new Optimum
Objective Value: $52,837.04