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Linear Programming
Operations Research – Engineering and Math
Management Sciences – Business
Goals for this section
Modeling situations in a linear environment Linear inequalities (constraints), restrictions Linear objective function, goal to be optimized Minimum cost, Maximum revenue, Maximum profit
1. Write the linear programming problem
2. Solve the problem graphically
Linear Program (LP) Characteristics
LP: optimize objective
subject to
constraints
Need to find the solution(s) in the feasible region that is best.
feasible region is closed and bounded: max & min values exist feasible region is not closed and bounded: max only, min only,
or no solution If LP has a solution, then optimal value can be found at a
corner point. If two corner points are optimal, then any point on the line
connecting them is optimal. (infinitely many optimal solutions)
generates feasible region,
collection of all possible solutions
Example 1
Formulate an LP for this problem.
Apple Pie: 3/4 cup of sugar, 1 egg, $2.5 in profit
Peach Cobbler: 1 ½ cups of sugar, 1 egg, $3 in profit
With only 60 eggs and 80 cups of sugar available,
how many of each pie should you make in order to
maximize your profits?
Example 1 – continued
(0,0)
10
20
30
40
20 40
Corner Points, Profit
50
60 80 100
60
0,
6008 5.175.
subject to
35.2 maximize
yx
yxyx
yxP
Example 2
Formulate an LP for this problem.
Based on the table, that gives mg per
serving for three nutrients, how many
servings of each food is required to
meet the minimal needs and keep the
amount of nutrient C to a minimum? 6354Needs
11232 Food
36101 Food
CBA
Nutrients
Example 2 – continued
4
8
12
16
2 4
Corner Points, C
20
6 8 10
Intersection?