From a single bhutta of maize, a farmer generates 200 grains of boon 740 long and 40 small plants, the genotype of these offspring will be?
Answers
A linear programming problem consists of
a linear objective function
a collection of constraints, each in the form of a linear equality or
linear equality.
The goal of a linear programming problem is to maximize or
minimize the objective function, while satisfying all of the
A very simple linear programming problem
A farmer has 100 acres of land.
The farmer can use the land to grow corn or wheat.
For each acre of corn, the farmer earns $651.
For each acre of wheat, the farmer earns $523.
In order to maximize his revenue, how many acres should be used
for corn, and how many acres for wheat.
The solution
We can find the answer to this problem without using any fancy
techniques.
The farmer earns more from corn than from wheat, so farmer
should devote all available land to corn.
Thus, 100 acres, all devoted to corn, $651 per acre, so maximum
revenue is $65, 100
Not all LPs are this straightforward, so lets look at a more robust
method.
Setting up the simple linear programming problem
Let C denote number of acres of corn, W the number of acres of
wheat.
The objective function is R = 651 · C + 523 · W .
There are three constraints:
C ≥ 0
W ≥ 0
C + W ≤ 100
Solving the linear programming problem
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C
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A less simple linear programming problem
A farmer has 100 acres of land to grow corn or wheat.
Farmer earns $651 for each acre of corn and $523 for each acre of
wheat.
Harvesting the corn requires 20 hours of labor per acre.
Harvesting the wheat requires 12 hours of labor per acre.
The farmer has enough workers for 1500 hours of labor.
In order to maximize his revenue, how many acres should be used
for corn, and how many acres for wheat.
Solution?
The “use all 100 acres for corn” is no longer a valid solution, as
this would require 2000 hours of labor, but the farmer only has
1500 hours available.
Setting up the linear programming problem
Let C denote number of acres of corn, W the number of acres of
wheat.
The objective function is R = 651 · C + 523 · W .
There are four constraints:
C ≥ 0
W ≥ 0
C + W ≤ 100
20 · C + 12 · W ≤ 1500
Solving the linear programming problem
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C
W
(0, 0)
(0, 100)
(75, 0)
(37.5, 62.5)
The Method of Corners
Graph the feasible set.
Find the coordinates of all of the corner points of the feasible set.
Evaluate the objective function at each corner.
Theorems 1 and 2 from the text guarantee that the objective
function reaches a maximum at one of these corner points, and a
minimum at another corner point, provided the feasible set is
bounded.
A Nutrition Example
A Food-and-Nutrition-Science student was asked to design a diet
for someone with iron and vitamin B deficiencies
The student said the person should get at least 2400mg of iron,
2100mg of vitamin B1, and 1500mg of vitamin B2 (over 90 days)
The student recommended two brands of vitamins:
Brand A Brand B Min. Req
Iron 40mg 10mg 2400mg
B1 10mg 15mg 2100mg
B2 5mg 15mg 1500mg
Cost: $0.06 $0.08
The client asked the student to recommend the cheapest solution
How many pills of each brand should the person get in order to
meet the nutritional requirements at the minimal cost?
Shipping costs example
You hit the big time, Mr. or Ms. Big Shot.
You’ve got two manufacturing plants and two assembly plants
Your assembly plants A1 and A2 need 80 and 70 engines
Your production plants can produce up to 100 and 110 engines
The shipping costs are:
To assembly plant
From A1 A2
P1 100 60
P2 120 70
How many engines should each production plant ship to each
assembly plant to meet the production goals at the minimum