Chapter 6-5 Linear Optimization
Obj: Use linear programming to solve applications and to recognize situations where there are no
solutions, or more than 1 solution of a linear programming application.
Linear Programming – Many applications in business and economics involve optimization (finding a max or min
value for a specific quality). When the quantity to be optimized is represented by a linear function, the process is
called linear programming.
Objective Function – the linear function to be optimized, in the form𝑓(𝑥, 𝑦) = 𝑎𝑥 + 𝑏𝑦 + 𝑐
Constraints – system of linear equations that graphically is their intersection (will OFTEN form a polygonal region)
Feasible Solutions – points in form (𝑥, 𝑦) are in the solution set of inequalities (graphically…it’s the intersecting or
“shaded” region)
Ex. 1: Solve this system of inequalities by GRAPHING. Then, name the vertices of the *polygonal convex set.
(*Often the solution set of a system of equations is a polygonal convex set. Such a set consists of all points on or
inside a convex polygon.)
1
𝑦 ≥ 𝑥−1
2
𝑦≤3
𝑥≥0
𝑦≥0
Vertices: _______________________________________
𝑓(𝑥, 𝑦) = 3𝑥 + 5𝑦
Suppose we were asked to find the maximum value of a function 𝑓(𝑥, 𝑦) = 3𝑥 + 5𝑦 subject to the constraints
given by the system above. Because the shaded region contains infinitely many points, it would be impossible to
evaluate 𝑓(𝑥, 𝑦) for all of them to find a maximum (or min) value.
Fortunately, the Vertex Theorem provides a strategy for finding a solution if it exists.
Vertex Theorem – The maximum or minimum value of 𝑓(𝑥, 𝑦) = 𝑎𝑥 + 𝑏𝑦 + 𝑐 on a polygon convex set occurs at
the vertex of the polygonal boundary.
LINEAR PROGRAMMING
STEPS 1) Graph the region corresponding to the solution of the system of inequalities (constraints).
2) Find the coordinates of the vertices of the region formed.
3) Evaluate the objective function at each vertex to find max and min values (if any)
Ex 2: Find the maximum and minimum values of the objective function 𝑓(𝑥, 𝑦) = 5𝑥 − 3𝑦 and for what
values of x and y they occur subject to the following constraints.
𝑦≥0
𝑦 ≤𝑥+2
0≤𝑥≤5
𝑦 ≤ −𝑥 + 6
𝑓(𝑥, 𝑦) = 5𝑥 − 3𝑦
Vertices:
Now find the value of 𝑓(𝑥, 𝑦) = 5𝑥 − 3𝑦 at each vertex
Max of 𝑓(𝑥, 𝑦) =____________________ @ vertex _____________
Min of 𝑓(𝑥, 𝑦) = ____________________ @ vertex _____________
Ex. 5 Optimization at Multiple Points
Find the maximum value of the objective function 𝑓(𝑥, 𝑦) = 4𝑥 + 2𝑦 and for what values of 𝑥 𝑎𝑛𝑑 𝑦 it
occurs, subject to the following constraints.
𝑦 + 2𝑥 ≤ 18
𝑦≤6
𝑥≤8
𝑥≥0
𝑦≥0
Ex. 3
Vertices:
Ex 4
HW Ch 6-5 – p.410/ #1-4, 10, 12 , 13