Introduction to mathematical optimization

optimization techniques

1. Introduction to Mathematical Optimization

2. So… what is mathematical optimization, anyway? “Optimization” comes from the same root as “optimal”, which means best . When you optimize something, you are “making it best”.

3. So… what is mathematical optimization, anyway? “Optimization” comes from the same root as “optimal”, which means best . When you optimize something, you are “making it best”. But “best” can vary. If you’re a football player, you might want to maximize your running yards, and also minimize your fumbles. Both maximizing and minimizing are types of optimization problems.

4. Mathematical Optimization in the “Real World” Mathematical Optimization is a branch of applied mathematics which is useful in many different fields. Here are a few examples:

5. Mathematical Optimization in the “Real World” Mathematical Optimization is a branch of applied mathematics which is useful in many different fields. Here are a few examples: Manufacturing Production Inventory control Transportation Scheduling Networks Finance Engineering Mechanics Economics Control engineering Marketing Policy Modeling

6. Optimization Vocabulary Your basic optimization problem consists of… The objective function, f(x) , which is the output you’re trying to maximize or minimize.

7. Optimization Vocabulary Your basic optimization problem consists of… The objective function, f(x) , which is the output you’re trying to maximize or minimize. Variables, x 1 x 2 x 3 and so on, which are the inputs – things you can control. They are abbreviated x n to refer to individuals or x to refer to them as a group.

8. Optimization Vocabulary Your basic optimization problem consists of… The objective function, f(x) , which is the output you’re trying to maximize or minimize. Variables, x 1 x 2 x 3 and so on, which are the inputs – things you can control. They are abbreviated x n to refer to individuals or x to refer to them as a group. Constraints, which are equations that place limits on how big or small some variables can get. Equality constraints are usually noted h n (x) and inequality constraints are noted g n (x).

9. Optimization Vocabulary A football coach is planning practices for his running backs. His main goal is to maximize running yards – this will become his objective function . He can make his athletes spend practice time in the weight room; running sprints; or practicing ball protection. The amount of time spent on each is a variable . However, there are limits to the total amount of time he has. Also, if he completely sacrifices ball protection he may see running yards go up, but also fumbles, so he may place an upper limit on the amount of fumbles he considers acceptable. These are constraints . Note that the variables influence the objective function and the constraints place limits on the domain of the variables.

10. Imagine a company that manufactures two types of products: chairs and tables. The company has limited resources such as labor hours and materials, and wants to maximize its profit from production. The profit earned per unit for each product is: Chairs: $10 Tables: $15 However, there are constraints on the resources available: Labor Hours : The company can allocate up to 80 hours of labor per day. Wood : The company has 100 units of wood available per day

11. Additionally, there are non-negativity constraints because you cannot produce a negative number of products: x 1 ≥0 x 2 ≥0 The objective is to maximize the total profit: Maximize Z=10x 1 +15x ​2

12. Types of Optimization Problems Some problems have constraints and some do not. unlimited limited

13. Types of Optimization Problems Some problems have constraints and some do not. There can be one variable or many. x 1 x 3 x 2 x 6 x 8 x 5 x 4 x 7

14. Types of Optimization Problems Some problems have constraints and some do not. There can be one variable or many. Variables can be discrete (for example, only have integer values) or continuous.

15. Types of Optimization Problems Some problems have constraints and some do not. There can be one variable or many. Variables can be discrete (for example, only have integer values) or continuous. Some problems are static (do not change over time) while some are dynamic (continual adjustments must be made as changes occur).

16. Types of Optimization Problems Some problems have constraints and some do not. There can be one variable or many. Variables can be discrete (for example, only have integer values) or continuous. Some problems are static (do not change over time) while some are dynamic (continual adjustments must be made as changes occur). Systems can be deterministic (specific causes produce specific effects) or stochastic (involve randomness/ probability).

17. Types of Optimization Problems Some problems have constraints and some do not. There can be one variable or many. Variables can be discrete (for example, only have integer values) or continuous. Some problems are static (do not change over time) while some are dynamic (continual adjustments must be made as changes occur). Systems can be deterministic (specific causes produce specific effects) or stochastic (involve randomness/ probability). Equations can be linear (graph to lines) or nonlinear (graph to curves)

18. Why Mathematical Optimization is Important Mathematical Optimization works better than traditional “guess- and- check” methods M. O. is a lot less expensive than building and testing In the modern world, pennies matter, microseconds matter, microns matter.

19. Solution Methods Least Square Methods Least square method is the process of finding a regression line or best-fitted line for any data set that is described by an equation. This method requires reducing the sum of the squares of the residual parts of the points from the curve or line and the trend of outcomes is found quantitatively. Hey students who spend more time on their assignments are getting better grades

20.  A student wants to estimate his grade for spending 2.3 hours on an assignment . Through the magic of the least-squares method, it is possible to determine the predictive model that will help him estimate the grades far more accurately. This method is much simpler because it requires nothing more than some data and maybe a calculator . The least-squares method is a statistical method used to find the line of best fit of the form of an equation such as y = mx + b to the given data . The curve of the equation is called the regression line. Our main objective in this method is to reduce the  sum  of the squares of errors as much as possible. This is the reason this method is called the least-squares method. 

21. Limitations for Least Square Method Even though the least-squares method is considered the best method to find the line of best fit, it has a few limitations. They are: This method exhibits only the relationship between the two  variables . All other causes and effects are not taken into consideration. This method is unreliable when data is not evenly distributed. This method is very sensitive to outliers.

22. Least Square Method Formula Let us assume that the given points of data are (x 1 , y 1 ), (x 2 , y 2 ), (x 3 , y 3 ), …, ( x n , y n ) in which all x’s are independent variables, while all y’s are dependent ones . This method is used to find a linear line of the form y = mx + b, where y and x are variables, m is the slope, and b is the y-intercept. 

23. Graphical method