Foundations 20
6.4 Optimization Problems I: Creating the Model
To ‘optimize’ refers to the selection of a best element from some set of available alternatives. In
other words, it means to make the most out of what’s available.
Optimization Problem:
Constraint:
Objective Function:
Feasible Region:
In this section, we are setting the stage for section 6.5. Here, we are creating models, which
means we are writing and graphing inequalities to represent a real-world situation. To
create a model, we have to:





Ask ourselves, “What is being optimized?”
Define the variables, including the restrictions on the variables
o Whole numbers, integers or real numbers?
Write inequalities that describe the constraints of the situation
Graph the system
Write an objective function that shows how the variables are related to the quantity to be
optimized
Example 1
Create a model to represent this situation.
Step 1: What is being optimized?
Step 2: Define the variables and state the restrictions.
Step 3: Write a system of inequalities to describe all the constraints of the problem.

No more than 40 racing cars

No more than 60 SUV’s

70 or more vehicles can be made each day
(The costs of each item are considered when we get to Step 5!)
Step 4: Graph the system.
Step 5: Write an objective function to represent the relationship between the variables and the
quantity to be optimized.
Step 6: (To be completed with 6.5) What is the maximum and minimum cost for making
vehicles? How many vehicles are made are maximum and minimum cost?
Assignment: p. 330 # 1 – 4, 6a (no graphing)
Foundations 20
6.4 OptimizationProblems I: Creating the Model
To ‘optimize’ refers to the selection of a best element from some set of available alternatives. In
other words, it means to make the most out of what’s available.
Optimization Problem: where a quantity must be maximized or minimized following a set of
conditions.
Constraint: a limiting condition represented by a linear inequality.
Objective Function: the equation that represents the relationship between the 2 variables in the
system of linear inequalities and the quantity to be optimized.
Feasible Region: the solution region for a system of linear inequalities.
In this section, we are setting the stage for section 6.5. Here, we are creating models, which
means we are writing and graphing inequalities to represent a real-world situation. To
create a model, we have to:





Ask ourselves, “What is being optimized?”
Define the variables, including the restrictions on the variables
o Whole numbers, integers or real numbers?
Write inequalities that describe the constraints of the situation
Graph the system
Write an objective function that shows how the variables are related to the quantity to be
optimized
Example 1
Create a model to represent this situation.
Step 1: What is being optimized?
Step 2: Define the variables and state the restrictions.