red brand canners - university of michiganmillerpd/docs/501_fall09/burhansadiq.pdfred brand canners...

Red Brand Canners

Burhan Sadiq

December 3, 2009

Outline

Statement of ProblemDilemma

LP Formulation

ResultsOptimum Product MixAdditional Grade A TomatoesAdvertising CampaignTomato Juice Price Increase

Conclusion

Red Brand Canners

Red Brand Canners used to be a medium-size company canningand distributing a variety of fruit and vegetable products in thewestern part of USA in the 1960s. In one year, they signed anagreement at planting time, to purchase the entire tomato crop ina large field at an average delivered price of 6 cents per lb.At harvest time, Produce Inspection estimates that the total cropwill be 3 ∗ 106 lb, of which 20% is expected to be Grade A and theremaining portion expected to be Grade B.

Data

The company makes three different tomato products, and they setthe selling prices of these products in light of the long-termmarketing strategy of the company.

Moreover, the company uses a numerical scale to record thequality of both the raw produce and prepared products. The scaleruns from 0 to 10 points, higher numbers representing betterquality. On this scale, Grade A tomatoes were valued at 9points/lb, and Grade B tomatoes 5 points/lb.

Data

The company makes three different tomato products, and they setthe selling prices of these products in light of the long-termmarketing strategy of the company.Moreover, the company uses a numerical scale to record thequality of both the raw produce and prepared products. The scaleruns from 0 to 10 points, higher numbers representing betterquality. On this scale, Grade A tomatoes were valued at 9points/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 5SP: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: MinimumAverage 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.

I ii) Red Brand Canners has been approached by anotherdistribution company that will supply additional Grade Atomatoes at a price of .085$ per lb. Should the companypurchase additional Grade A tomatoes and if so, what is themaximum amount the company can purchase? If thecompany wishes to purchase only 80,000 lbs of Grade Atomatoes, what is the new optimum product mix?

I iii) The Marketing Department feels that it can increasedemand for tomato juice by 25,000 cases. How much shouldthe 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 thedemand remains fixed.

How do you solve this problem? Linear Programming

Problem


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 Atomatoes at a price of .085$ per lb. Should the companypurchase additional Grade A tomatoes and if so, what is themaximum amount the company can purchase? If thecompany wishes to purchase only 80,000 lbs of Grade Atomatoes, what is the new optimum product mix?




Problem






How do you solve this problem?

Linear Programming

Problem







LP Formulation

max z =3∑

i=1

(si − vi )

(xai + xbi

li

)− c ∗ S

subject to:3∑

i=1

xai ≤ Sa Supply Restriction on Grade A tomatoes

3∑i=1

xbi ≤ Sb Supply Restriction on Grade B tomatoes

qa ∗ xai + qb ∗ xbi ≥ Qi (xai + xbi ) ∀i Quality Restriction(xai + xbi

li

)≤ di Demand Restriction

xai , xbi ≥ 0 Non-Negativitysi , li , vi : selling price, lbs of tomatoes used, variable costs per caseof ith product. xai , xbi : Amounts of Grade A, Grade B tomatoesused in ith product.

Results

Optimum MixProduct Grade A Grade B

Whole Tomatoes 525,000 175,000

Tomato Juice 75,000 225,000

Tomato Paste - 2,000,000

Marginal ValuesItem Marginal Values ($/unit)

Grade A tomatoes 0.090

Grade B tomatoes 0.058

Demand of WholeTomatoes 0.000

Demand of Tomato Juice 0.000

Demand of Tomato Paste 0.403

Optimum Objective Value: $45,355.56

Results

Makes sense to purchase additional Grade A tomatoes for 8.5cents/lb. How much? Turns out: 600,000 lbs of Grade ATomatoes. Optimum Objective Value: $48,555.56

Optimum MixProduct Grade A Grade B

Whole Tomatoes 1,200,000 400,000

Tomato Juice - -


Updated Marginal ValuesItem Marginal Values ($/unit)



Demand of Whole Tomatoes 0.000



Results

If the company wishes to purchase only an additional 80,000 lbs ofGrade A tomatoes, the new optimum mix is given by:

Optimum Product MixProduct Grade A Grade B


Tomato Juice 65,000 195,000



Advertising Campaign

Given that the Marginal Values at the optimum product mix are:







Company should NOT pay any money for an advertising campaignto increase the demand for Tomato Juice.

Results

Tomato Juice price is increased by 30 cents per case.

Optimum Product MixProduct Grade A Grade B


Tomato Juice 250,000 750,000









Conclusions

I i) Optimum Objective Value: $45,355.56

I ii) Company should purchase an additional 600,000 Grade Atomatoes. Optimum Objective Value: $48,555.56

I iii) Company should NOT pay any money for an advertisingcampaign 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 OptimumObjective Value: $52,837.04

red brand canners - university of michiganmillerpd/docs/501_fall09/burhansadiq.pdfred brand canners...

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