ESI 4313 Operations Research 2

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ESI 4313Operations Research 2

• Nonlinear Programming Models
• (Lecture 2)

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Linear vs. nonlinear models

• In OR1 we focused mainly on formulating and
  solving linear programming problems.
• That is, optimization problems with the property
  that
• The objective function is a linear function of
  the decision variables
• The constraints are linear functions of the
  decision variables

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Linear vs. nonlinear models

• Due to the fact that LP models are very
  efficiently solvable, it is usually preferable to
  formulate a problem as an LP if at all possible.
• Sometimes this can be achieved at the expense of
  introducing more decision variables.
• Even in these cases, the advantages of the LP
  model usually outweigh the increase in size of
  the model.
• However, in many cases it is impossible to
  represent the decision problem using linear
  functions only.

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Example 1Transport of hazardous waste

• A company is responsible for transporting
  hazardous waste produced at their 5 production
  plants to 3 waste processing plants.
• Problem data
• Cost per unit transported from production plant i
  to processing plant j cij (i1,,5 j1,2,3)
• Number of loads of waste produced by production
  plant i Li (i1,,5)

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Example 1 (contd.)Transport of hazardous waste

• Suppose first that we are interested in finding
  the solution that minimizes the total costs of
  all 5 plants
• Decision variables
• xij number of loads transported from production
  plant i to processing plant j cij (i1,,5
  j1,2,3)

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Example 1 (contd.)Transport of hazardous waste

• Objective
• Constraints
• Ensure that all production plans transport all
  waste
• Nonnegativity

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Example 1 (contd.)Transport of hazardous waste

• Full LP model

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Example 1 (contd.)Transport of hazardous waste

• This problem is actually easy to solve by
  inspection due to the absence of capacity
  constraints at the processing plants
• How?
• Now assume in addition that
• At most U loads may be transported along each
  route (i ?j ) (i1,,5 j1,2,3)
• The (absolute) difference between the number of
  loads to be processed by any pair of processing
  plants may not be larger than ?

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Example 1 (contd.)Transport of hazardous waste

• Additional constraints
• Ensure that no more than U loads are transported
  from i to j
• Ensure that the difference in loads to process
  between plant j and plant j is no more than ?
• This latter constraint is nonlinear!

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Example 1 (contd.)Transport of hazardous waste

• How to convert to linear constraints?
• Realize that they are equivalent to
• or even
• Why is the latter true?

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Example 1 (contd.)Transport of hazardous waste

• Full LP model (2)

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Example 1 (contd.)Transport of hazardous waste

• Now also assume that each of the 5 production
  plants is responsible for the transportation
  costs of their own waste
• However, the optimal solution to the model
  developed so far may yield very different costs
  per unit of waste transported for the different
  plants
• This means some plants will not be happy with the
  overall minimum cost solution

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Example 1 (contd.)Transport of hazardous waste

• As a mechanism for finding a solution that makes
  the costs per unit of waste similar between the
  different plants, suppose that we change our
  objective to
• minimize the maximum cost per unit waste paid by
  the plants
• How would you model this?

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Example 1 (contd.)Transport of hazardous waste

• Additional decision variables
• Ki cost per unit waste transported faced by
  production plant i (i 1,,5)
• Additional constraints
• Objective

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Example 1 (contd.)Transport of hazardous waste

• The objective function
• is nonlinear!
• Why?
• Is it continuous?
• Is it differentiable?

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Example 1 (contd.)Transport of hazardous waste

• Full NLP model (3)

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Example 1 (contd.)Transport of hazardous waste

• Can we formulate this model in a linear way?
• Trick define yet another decision variable
• K maximum cost paid by any of the 5 plants
• Then the objective becomes
• which is linear!

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Example 1 (contd.)Transport of hazardous waste

• Did this really buy us something?
• Note that we need to introduce a constraint
  linking K to Ki (i 1,,5)
• Additional constraint
• So now we have a nonlinear constraint

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Example 1 (contd.)Transport of hazardous waste

• Another trick
• Suppose that we allow choosing K larger than
  necessary, i.e., we replace the additional
  constraint by
• Since K does not appear in any other constraint,
  and it is the only term in the objective, we know
  that this constraint will be binding in the
  optimal solution!

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Example 1 (contd.)Transport of hazardous waste

• Finally, note that the single constraint
• is equivalent to the 5 constraints
• In fact, we can now get rid of the variables Ki
  by substituting their definition

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Example 1 (contd.)Transport of hazardous waste

• Full LP model (4)