Tier II: Case Studies

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(No Transcript)
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Tier II Case Studies

• Section 1
• Lingo Optimization Software

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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

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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

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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 www.lindo.com/cgi/frameset.cgi?leftl
  ingo.htmllingof.html

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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 3x2 is y 3x2

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Lingo Operators

• Many of Lingos mathematical operators are
  similar to what Excel uses
• Addition - Multiplication
• Subtraction - - Division /
• For exponents Xn
• Equals
• Greater than or less than gt or lt
• Note Lingo accepts lt as being lt. It does
  not support strictly less than or greater than.

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Lingo Operators, cont

• Absolute value of x _at_abs(x)
• Natural log of x _at_log(x)
• Trigonometric functions _at_sin(x), _at_cos(x),
  _at_tan(x) (x in radians)
• Exponentials _at_exp(x)
• To return integer portion of decimal number
  _at_floor(x)
• _at_sign(x) returns -1 if x lt 0, or else 1

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Lingo Operators, cont

• Find max or min value in a set _at_smax(x1,x2,xn)
  or _at_smin(x1,x2,xn)
• To find maximum or minimum of a function max or
  min
• To allow negative variables _at_free(x)
• Lingo has a number of other operators, but these
  are the mathematical operators that are most
  likely to be used

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Using Lingo

• Other operators, like logic operators, can be
  found in the help files 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

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Lingo Screenshot
Solve to solve current problem set
If additional help is needed
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Basic Equation Solver
This will find the intersection of the lines y
3x 4 and y 5x 1
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Solution
Note Lingo does not distinguish between small
and capital letters
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Equation Solver 2
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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.
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Solution 2
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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

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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

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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

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Chemical Plant Example

• Objective Maximize 1000x1 1500x2
• Constraints
• 4x1 2x2 lt 80
• 2x1 5x2 lt 60
• 4x1 4x2 lt 75
• x1, x2 gt 0

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Chemical Plant Example
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Lingo Solution
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Transportation Scheme Problem
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Problem 2 Solution
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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
• _at_free(x)
• It is a good idea to declare all variables like
  this, unless of course a negative value is
  infeasible

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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

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Attempting to Obtain a Negative Solution
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Attempting to Obtain a Negative Solution
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Attempting to Obtain a Negative Solution
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Attempting to Obtain a Negative Solution
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Greater Than or Less Than

• Another potential problem that will be
  encountered using Lingo is that it treats lt
  the same as lt, and gt the same as gt
• Thus, if a variable must be strictly greater than
  a value, the constraint is best treated as
  follows
• For x gt A, where A is a solution otherwise,
  use x gt A b
• where b is an arbitrary value, like 0.1, that
  covers a portion where the solution will not lie

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Example of lt or gt
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Example of lt or gt
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Example of lt or gt
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.
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Example of lt or gt
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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