Section 7.5 Linear Programming Math in Our World

Learning Objectives  Use linear programming to solve real-world problems.

Linear Programming Linear programming was developed to handle military logistical problems such as efficiently transporting equipment and personnel during wartime. Today the techniques of linear programming are used by business and industry to make decisions and find cost-effective solutions to many problems.

Linear Programming The idea is to write a formula describing some quantity of interest, like profit or cost, which is called the objective function. The objective function must be linear, meaning that all variables appear only to the first power. The goal is to find values of the variables that either maximize (make as large as possible) or minimize (make as small as possible) the objective function.

Linear Programming In most real-world problems, there are limitations on values of the variable, which we call constraints. The constraints of the problem will form a system of linear inequalities, and in most cases the graph of that system will be a polygonal region. Each corner point of the polygonal region is called a vertex (plural “vertices”). Every vertex has a pair of coordinates that represent a potential solution to the problem at hand.

Linear Programming To solve a linear programming problem find the vertices of the region and evaluate the objective function for each, picking out the largest or smallest value.

EXAMPLE 1 Solving a Linear Programming Problem The owner of a small business manufactures playground sets and playhouses. The process involves two steps. First, the lumber must be cut and drilled, and second, the product must be assembled.

EXAMPLE 1 Solving a Linear Programming Problem For a playground set, it takes one worker 2 hours to cut and drill the lumber and another worker 1 hour to assemble the set. For the playhouse, it takes one worker 1 hour to cut and drill the lumber and another worker 1.5 hours to assemble the playhouse. Workers work at most 8 hours per day. The owner makes $100 profit on every playground set that is sold and $75 profit on every playhouse that is sold. Using linear programming techniques, how many of each product should be manufactured in 1 day in order to maximize profit?

EXAMPLE 1 Solving a Linear Programming Problem SOLUTION Step 1 Write the objective function. Since the owner makes a profit of $100 on each playground set sold and a profit of $75 on each playhouse sold, the objective function can be written as P = 100x + 75y where x = the number of playground sets sold, y = the number of playhouses sold, and P = the profit made.

EXAMPLE 1 Solving a Linear Programming Problem SOLUTION Step 2 Write the constraints. Listing the information in a table makes it somewhat easier to find the constraints.

EXAMPLE 1 Solving a Linear Programming Problem SOLUTION Step 2 Write the constraints. Since it takes 2 hours to cut lumber for one playground set, 2 x is the number of hours needed to cut lumber for x playground sets. It only takes 1 hour to cut lumber for a playhouse, so 1 y is the number of hours needed to cut lumber for y playhouses. The cutter works at most 8 hours, so we have our first constraint: 2x + y ≤ 8.

EXAMPLE 1 Solving a Linear Programming Problem SOLUTION Step 2 Write the constraints. Since it takes 1 hour to assemble a playground set and 1.5 hours to assemble a playhouse, the second constraint is x + 1.5y ≤ 8 We can’t make a negative number of playground sets or playhouses, so, there are two additional constraints: x ≥ 0 and y ≥ 0

EXAMPLE 1 Solving a Linear Programming Problem SOLUTION Step 3 Graph the linear system made up of the constraints. Step 4 Find the vertices of the polygonal region, which are (0, 0), (0, 5.33), (2, 4), (4, 0).

EXAMPLE 1 Solving a Linear Programming Problem SOLUTION Step 5 Substitute the vertices into the objective function and find the maximum value. The vertex (2, 4) produces the maximum value 500. The solution is to make two playground sets and four playhouses for a maximum profit of $500 per day.

Linear Programming Procedure for Using Linear Programming Step 1 Write the objective function. Step 2 Write the constraints. Step 3 Graph the constraints. Step 4 Find the vertices of the polygonal region. Step 5 Substitute the coordinates of the vertices into the objective function and find the maximum or minimum value. (Note: The solutions will not always be integers.)

EXAMPLE 2 Solving a Linear Programming Problem An automobile dealer has room for no more than 100 cars on his lot. The dealer sells two models, convertibles and sedans, and he sells at least 3 times as many sedans as convertibles. If he makes a profit of $1,000 on a convertible and a profit of $1,500 on a sedan, how many of each automobile should he have on his lot in order to maximize his profit?

EXAMPLE 2 Solving a Linear Programming Problem SOLUTION Step 1 Write the objective function. Since the dealer makes a profit of $1,000 on each convertible sold and $1,500 on each sedan sold, the objective function is P = 1000x y where x = the number of convertibles sold, y = the number of sedans sold, and P = the profit made.

EXAMPLE 2 Solving a Linear Programming Problem SOLUTION Step 2 Write the constraints. In addition, x and y can’t be negative: x ≥ 0 and y ≥ 0

EXAMPLE 2 Solving a Linear Programming Problem SOLUTION Step 3 Graph the system. Step 4 Find the vertices of the polygonal region, which are (0, 0), (0, 100), and (25, 75).

EXAMPLE 2 Solving a Linear Programming Problem SOLUTION Step 5 Substitute the vertices into the objective function and find the maximum value. The maximum profit occurs at the point (0, 100), so the dealer should stock no convertibles and 100 sedans to maximize his profit.