1. 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

2. 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

3. 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?

4. Example 1 – continued (0,0) 10 20 30 40 20 40 Corner Points, Profit 50 60 80 100 60

5. 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?

6. Example 2 – continued 4 8 12 16 2 4 Corner Points, C 20 68 10 Intersection?