2. Lingo Optimization Software
Tier II: Case Studies
Section 1:
Lingo Optimization Software

3. Optimization Software
Many of the optimization methods previously outlined can be tedious and require a lot of work to solve, especially as models get more complex and move beyond two or three variables, which will often be the case
Software can be used to solve these problems more efficiently

4. Optimization Software
Software that is available usually uses the same methods previously outlined, but can of course perform the calculations quicker, allowing the effect of variations in the model to be studied more easily

5. Optimization Software
Some optimization examples have already been shown using Excel
Another program, Lingo, will now be demonstrated
A trial version of this software can be downloaded at

6. Lingo
Lingo is a program designed specifically for solving optimization problems
It uses a syntax that is similar to what would be written by hand, or what would be used in Excel, not requiring variables to be declared
For example, y = 3*x^2 is y = 3x2

7. Lingo Operators
Many of Lingo’s mathematical operators are similar to what Excel uses:
Addition: + - Multiplication: *
Subtraction: - - Division: /
For exponents: X^n
Equals: =
Greater than or less than: > or <
Note: Lingo accepts ‘<’ as being ‘<=’. It does not support strictly less than or greater than.

8. Lingo Operators, con’t Absolute value of x: @abs(x)
Natural log of
@tan(x) (x in radians)
To return integer portion of decimal
@sign(x): returns -1 if x < 0, or else 1

9. Lingo Operators, con’t
Find max or min value in a
To find maximum or minimum of a function: max or min
To allow negative
Lingo has a number of other operators, but these are the mathematical operators that are most likely to be used

10. Using Lingo
Other operators, like logic operators, can be found in the help file’s complete list of operators
Now that we have the mathematical operators that are likely to be used, we can demonstrate how Lingo works with some examples
Lingo can be used strictly as an equation solver or as an optimizer

11. Solve – to solve current problem set
Lingo Screenshot
If additional help is needed
Solve – to solve current problem set

12. Basic Equation Solver
This will find the intersection of the lines “y = 3x + 4” and “y = 5x + 1”

13. Note: Lingo does not distinguish between small and capital letters
Solution
Note: Lingo does not distinguish between small and capital letters

14. Equation Solver #2

15. Solution #2
Only one solution was found! There should be two solutions to this problem. The solver automatically stops when it finds the first solution.

16. Solution #2

17. Non-linear Difficulties
Lingo is not designed to deal with non-linear equations
It cannot find multiple solutions
There is a problem with solving non-linear problems, especially if the solution is in the negative domain

18. Maximum and Minimum
The maximum and minimum functions are the most important functions needed for optimization problems
These functions are used as follows:
max = objective function;
min = objective function;

19. Solving Optimization Problems
Several optimization examples that were worked through in previous sections will now be solved using Lingo
The first example is from the introduction section

20. Chemical Plant Example
Objective: Maximize 1000x x2
Constraints:
4x1 + 2x2 <= 80
2x1 + 5x2 <= 60
4x1 + 4x2 <= 75
x1, x2 >= 0

21. Chemical Plant Example

22. Lingo Solution
Solution, including value of objective function at optimum and optimum point

23. Transportation Scheme Problem

24. Problem #2 Solution

25. Negative Values
Lingo cannot automatically solve for a negative variable value
If it is suspected that a solution will be negative, then that variable will need to be specifically declared as free:
@free(x);
It is a good idea to declare all variables like this, unless of course a negative value is infeasible

26. Attempting to Obtain a Negative Solution
The following example will demonstrate what happens if a negative value is required to get an optimum solution
Lingo will automatically solve for the optimum solution obtained from only positive variables, even if this is not the true optimum

27. Attempting to Obtain a Negative Solution

28. Attempting to Obtain a Negative Solution
This solution is viable if the variable values must be positive, but this is not the true optimum

29. Attempting to Obtain a Negative Solution
These statements allow negative values to be used for these variables

30. Attempting to Obtain a Negative Solution
Now the true optimum is obtained, with negative variables

31. Greater Than or Less Than
Another potential problem that will be encountered using Lingo is that it treats < the same as <=, and > the same as >=
Thus, if a variable must be strictly greater than a value, the constraint is best treated as follows:
For x > A, where A is a solution otherwise, use x > A + b;
where b is an arbitrary value, like 0.1, that covers a portion where the solution will not lie

32. Example of < or >

33. Example of < or >
Clearly this is not correct as X1 was constrained to be greater than 0!

34. Example of < or >
This will now force X1 and X2 to be greater than 0. We can do this because we know X1 and X2 are also greater than 0.1.

35. Example of < or >
Variables now obey desired constraints. Objective just happens to be the same in this case.

36. Conclusions
Lingo is effective and efficient for solving optimization problems if they are linear
It is not designed to deal with non-linear problems
It is not very good at dealing with non-linear problems, so these must be approached with caution
It does not handle multiple maximum or minimum points very well in non-linear cases