Mr. John is an avid gardener, and each spring he plans what to plant in his vege

Question

Mr. John is an avid gardener, and each spring he plans what to plant in his vegetable

garden. He grows tomatoes, eggplants, zucchini and green peppers. He would like his garden

to provide a rich harvest of these vegetables so that he has fresh vegetables throughout the

late summer and fall, as well as frozen vegetables for the winter.

It is now the end of May, and he is going to the local nursery to buy the vegetables to plant.

The table below provides the information needed for the vegetables.

Mr. John is considering three types of tomatoes, along with eggplant, zucchini and green

pepper. For each vegetable, he needs to decide the number of plants to grow. Each plant will

occupy a certain amount of area in the garden, as noted in the second column. His garden

is 2000 square feet in total.

Also, each plant requires a certain amount of labor each week (third column), to weed,

fertilize, water and harvest. For instance, each Beefsteak tomato plant requires 0.1 hour per

week. The labor requirement for each plant doesn't end until the end of the harvest, 20 weeks

later (i.e., the labor requirements continue even after the plant stops yielding produce). Mr.

John can work only 25 hours per week, because this is a part-time job.

Mr. John will plant the entire garden on June 1. The fourth column indicates how many

weeks until he gets the first fresh vegetables from each plant type. Once the plant starts

yielding vegetables, a fixed number of pounds can be harvested each week, as given in the

fifth column.

This harvest continues for a number of weeks, as noted in the sixth column. For instance,

a Rutgers plant produces 1.5 pounds of tomatoes each week for a ten-week period, namely

from week 7 through to week 16.

The growing season is from week 7 to week 20. During the growing season Mr. John will

consume 10 pounds of his fresh tomatoes each week. In addition to eating tomatoes, each

week Mr. John has the option to freeze fresh tomatoes for use in the winter. By week 20,

he needs to have 250 pounds of frozen tomatoes. The tomatoes that he freezes must come

from his garden. Any leftover tomatoes can be sold at the market at the market price. The

market price for tomatoes will be $4 per pound for week 7, and then will drop in price by

$0.20 per pound every week thereafter. Thus, he expects in week 20 the price per pound

will be $1.40.

Mr. John uses the other vegetables to make vegetable soup that he stores for consumption

in the winter. His recipe calls for 1 pound of zucchini and 2 pounds of green peppers for

every pound of eggplant; this recipe makes 4 pounds of soup. To make the soup, all of the

vegetables must be fresh; that is, the vegetables were picked in the same week that the soup

is made. By the end of the season, he needs 200 pounds of soup prepared for the winter.

Any leftover vegetables can be used to make more soup to be sold at the market. Mr. John

will be able to sell up to 2000 pounds of soup for $5 per pound. Excess eggplants, or zucchini

or green peppers cannot be sold and are discarded.

Help Mr. John determine what to plant in order to meet his requirements and maximize his

Profit from his garden. DO NOT WORRY about the integrality of the decision variables -

the garden is large and it's okay to plant fractional bushes.

Formulate this problem as a linear program. Then, solve the problem (presumably with

Excel).

Your project should:

1. Clearly define all of the decision variables.

2. Provide the mathematical formulation in an easy to read form.

3. Provide a description in words of the objective function and each constraint.

4. Explicitly state what Mr. John should plant.

5. Provide, on a separate sheet, the actual computer output with the important answers

Explanation / Answer

I can help you a little bit

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