Product Mix Problem

Description
Product Mix Problem
Your company grows two types of plants, A and B (e.g., roses and begonias).
Both plant types require two types of fertilizer throughout the growing season, Fertilizer 1 and Fertilizer 2.
A single plant A will require 2 pounds of fertilizer 1 and 1 pound of fertilizer 2.
A single plant B will require 1 pound of fertilizer 1 and 2 pounds of fertilizer 2.
Your distributer has 4000 pounds of fertilizer 1 and 5000 pounds of fertilizer 2 available for delivery at the beginning of the season.
You know that every plant A you grow will generate $2.25 profit, and every plant B you grow will generate $2.60 profit.

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How many of each plant should you grow if you want to maximize profit for the season?

What amount of change in profit for plant A and plant B will change your decision about product mix?

How much will you be willing to pay for additional amounts of fertilizer?

If your fertilizer storage capacity is 8000 pounds, what is your new product mix, and how much of each fertilizer should you order?  What would you be willing to pay for additional storage capacity?

Lockbox Problem
A major credit card company (call it “MasterDebt”) receives checks from all different regions in the country on a daily basis.  Once these checks are mailed, the time a check spends in the mail (called “float”) creates loss for MasterDebt, for as soon as they receive the checks they can cash them and collect interest on the funds.  MasterDebt can make 15% annual interest on their cash holdings (since that’s what they charge their customers in credit card interest).
In order to reduce the amount of float loss for these checks, MasterDebt is considering opening “Lockbox” locations across the country where the checks can be received and processed.  The locations and the projected annual cost of operations (labor and overhead) at each location are as follows:

ANNUAL COSTS (X$1000)

Sacramento

Denver

Chicago

Dallas

New York

Atlanta

25

60

35

35

30

35

The average number of days that a check would float between each region and each lockbox location is shown in the following chart.

AVERAGE FLOAT DAYS

REGION

Sacramento

Denver

Chicago

Dallas

New York

Atlanta

Central

4

2

2

2

3

3

Mid-Atlantic

6

4

3

4

2

2

Midwest

3

2

3

2

5

4

Northeast

6

4

2

5

2

3

Northwest

2

3

5

4

6

7

Southeast

7

4

3

2

4

2

Southwest

2

3

6

2

7

6

The average daily payments received from each region are shown in the following chart (in thousands of dollars).

REGION

Payments

Central

45

Mid-Atlantic

65

Midwest

50

Northeast

90

Northwest

70

Southeast

80

Southwest

60

The annual interest lost can be computed for each region-lockbox location by taking the average daily payments times the float time and multiplying by fifteen percent.  For example, if payments from the Central region are sent to New York, then on any given day there is an average of $135,000 of undeposited checks, which costs MasterDebt $20,250 annually in interest.
Where should MasterDebt open lockbox locations in order to save the most money each year?
Which regions should be assigned to those lockbox locations?  (This will be implemented by providing different return addresses on the payment envelopes that are sent to the different regions.)