Agricultural Studies 3300, Fall 2004
Agricultural Systems Modelling
Assignment # 6
Due: Friday October 22nd
Instructor: Jeff Davidson
Set the following problems up as linear programs, and do the following.
1)
2)
Solve using the Optimizer in Excel. Be sure to print out your excel spreadsheet + Formulas.
Solve the problem by hand using the graphing method by performing the following steps:
1.
Define the decision variables, objective function, constraints and sign restrictions.
2.
Graph the inequality constraints
3.
Re-graph and darken in the feasible region
4.
Compute the slope of the objective function. Set a ruler with this slope and move it to the point where it is tangent to the feasible
region and construct a dashed line.
5.
Read the basic feasible solution for X1 and X2 at the point of tangency and evaluate the objective function at these values
Question #1
David the dairy farmer wants his cows to get a minimum of 54 milligrams of iodine, 126 mg of iron and 24 mg of
zinc each day. One type of supplement (X1) provides 4.5 mg of iodine, 9mg of iron and 1.5 mg of zinc. The other
type of supplement (X2) provides 3 mg of iodine , 9 mg of iron and 6 mg of zinc. The first type of feed costs $30,
the second $22.50. What is the cheapest combination of feeds that will meet the minimum daily requirements?
Question #2
David reads a new study published by Dr Dan LeRoy at the University of Lethbridge about Vitamin supplements.
Dr. Leroy claims that if a cow gets 15 units of vitamin A, 10 units of Vitamin D and 12 units of Vitamin E per day
the cow will develop a longer lactating period. David thinks this is a good idea so he decides to give it a whirl.
There are two vitamin supplements available (X1and X2) the first provides 1 units of vitamin A, 2.5 units of Vitamin
D and 1 unit of vitamin E and costs $40 per kg.. The second costs $80 per kg but provides 3 units of vitamin A 0.5
unit of vitamin D and 1.5 units of vitamin E. What is the least cost combination of supplements that will meet the
vitamin requirements?
Question #3
A toy tractor factory makes two type of toy tractors: John.Deere and Minneapolis Moline. A John Deere takes 6
hours for melting 3 hours for rolling and 1 hour for cutting. The minny molines take 2 hours for melting 5 hours for
rolling and 4 hours of cutting. The plant has 36 hours of melting time available, 30 hours of rolling time and 20
hours of cutting time. If the profit margin for a jd tractor is $100 and the profit margin for a mm tractor is $80, how
many of each tractor should the factory produce, in order to make the most cash ?
Question #4
Five linear programming problems are presented below. One is unbounded, one is infeasible, one is degenerate, and
one yields multiple solutions. Using the graphical solution method, identify which special case(s) applies for each of
the following five problems. Solve using q-pro to check your answers.
1.
Joe can grow Cabbage or cauliflower. He gets $3 for each box of cauliflowers and $4 for each box of cabbage.
Each box of cabbage requires 1 hours of harvesting 0.5 hour of cleaning and 1.5 hours of packaging, a box of
cauliflower requires 1.5 hours of harvesting 0.5 hour of cleaning and 0.5 hour of packaging. If joe only has 5
hours of harvest time available, 2 hours of cleaning time and 4 hours of packaging time, how much of each type
should he produce?
2.
Heather produces beefsteak tomatoes and plum tomatoes in a greenhouse. She has to provide a minimum of 16
pounds per month in any combination. 1 pound of Beefsteak tomatoes take 1 hour to pick and 4 hours to
package while 1 pound of plum tomatoes take 2 hours to pick and 2 hours to package. If she has 8 hours of
picking time and 10 hours of packaging time and she gets paid $4 per pound for Beefsteak, and $2 per pound
for plums how many of each should she produce?
3.
Fred produces 2 types of fertilizer. He gets $9 a bag for type 1 and $6 per bag for type 2. Each bag of type 1
requires 1.5 units of potash, 6 units of Nitrogen and 9 units of phosphate. Each type of bag 3 requires 6 units of
potash, 1.5 unit of nitrogen and 6 units of phosphate. If jack only has 12 units of potash 13.5 units of nitrogen
and 18 units of phosphate how much of each should he produce to maximise his profits.
4.
MAXIMIZE Z = 3 x1 + 6 x2
ST
2 x1 + 4 x2  4
3 x1 + 4 x2  5
2 x1 - 1 x2  3
5.
Dan has 7 acres of land that he wishes to plant to either canola or wheat. The profit margin for canola is $4
per acre and $6 per acre for wheat. Dan wishes to pay jeff to come and tend his crops, but for it to be worth
jeffs time He needs to spend at least 7 hours in the field weeding and 7 hours harvesting. Each acre of
canola requires 3 hours of weeding and 1 hour of harvest while each acre of wheat requires 1 hour of
weeding and 3 hours of harvest time.