Linear Programming Operations Research – Engineering and Math Management Sciences – Business...

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?