1) A farmer has a 1,000 acre farm. He can raise and sell any quantity of potatoes, tomatoes, lettuce and onions. The table below gives the technical data.

Crop Price received Labor required Water required Fertilizer

(cents/kg)per acre (days) (gallons/acre) (kg/acre)

Potatoes 5 2 500 80

Tomatoes 14 5 1,500 40

Lettuce 19 6 3,000 60

Onions 8 4 1,000 50

The resources available are as follows:

Labor: 3,000 man-days at $16/day

Water: 2 cents per 100 gallons up to 10⁶gallons

Fertilizer: 15 cents per kilogram, no limit to quantity

Average yield (kg. per acre):potatoes - 3,500

tomatoes - 2,400

lettuce - 1,500

onion - 2,000

a) Formulate the problem as a linear program.

b) Find the optimal solution using simplex method.

c) Identify the basis enterprises (crops to be included), non-basis enterprises (crops not to be included) that can be used by the farmer in his farm business decision.

d) solve for the optimal solution including the area to be planted to the basis enterprises (crops) and the maximum total revenue that can be generated from the basis enterprises.

2) A cattle feeder wishes to purchase four special feeds, F1, F2, F3, and F4, to supply the proper vitamin content to his feed mix. The table below gives the number of units of vitamin content in a pound of each type of feed and the cost per pound of the feed. The feed mix must contain at least 12 units of vitamin A, 14 units of vitamin B, and 8 units of vitamin C. Find the minimum cost of feed mix using the simplex method. The following are the other information related to the technical coefficients of the 4 types of feeds and the cost of each pound of the feeds:

Feed Vitamin Cost per pound of feed

A B C (cents)

F14 1 0 2

F26 1 2 5

F31 7 1 6

F4 2 5 3 8

a) Formulate the problem as a linear program.

b) Find the optimal solution (minimum cost of feed formulation) using simplex method.

c) Identify the best feed mix that can satisfy the proper vitamin contents and minimize the cost of the feed formulation.