Linear Programming: Assumptions and Implications of the LP Model

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SMU EMIS 8374
Network Flows
Linear Programming Assumptions and Implications
of the LP Model updated 18 January 2006
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Assumptions of the LP Model

• Proportionality
• The contribution of any decision variable to the
  objective function is proportional to its value.
• For example, in the diet problem, the
  contribution to the cost of the diet from one
  pound of apples is 0.75, 1.50 from two pounds
  of apples, 3.00 for four pounds, and 300.00 for
  four hundred pounds.
• In many cases, a volume discount is available
  whereby the price per pound goes down as more
  apples are purchased.
• Such discounts are often nonlinear which means
  that a linear programming model is either
  inappropriate or is really just an approximation
  of the real world.

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Assumptions of the LP Model

• Additivity
• The contribution to the objective function for
  any variable is independent of the other decision
  variables.
• For example in the NSC production problem, the
  production of P2 tons of steel in Month 2 will
  always contribute 4000 P2 regardless of how much
  steel is produced in Month 1.

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Assumptions of the LP Model

• Proportionality and Additivity are also implied
  by the linear constraints.
• In the diet problem, you can obtain 40 milligrams
  of protein for each gallon of milk you drink. It
  is unlikely, however, that you would actually
  obtain 4,000 milligrams of protein by drinking
  100 gallons of milk.
• Also, it may be the case due to a chemical
  reaction, you might obtain less than 70
  milligrams of Vitamin A by combining a pound of
  cheese with a pound of apples.

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Assumptions of the LP Model

• Divisibility
• The LP model assumes that the decision variables
  can take on fractional variables.
• Thus, it allows for a solution to the GT Railroad
  problem that sends 0.7 locomotives from
  Centerville to Fine Place.
• In many situations, the LP is being used on a
  large enough scale that one can round the optimal
  decision variables up or down to the nearest
  integer and get an answer that is feasible and
  reasonably close to the optimal integer solution.
  
• Divisibility also implies that the decision
  variables can take on the full range of real
  values.
• For example in the diet problem the LP may tell
  you to buy 1.739130 apples.
• For large problems, rounding or truncating of
  the optimal LP decision variables will not
  greatly affect the solution.

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Assumptions of the LP Model

• Certainty
• The LP model assumes that all the constant terms,
  objective function and constraint coefficients as
  well as the right hand sides, are know with
  absolute certainty and will not change.
• If the values of these quantities are not known
  with certainty, for example if the demand data
  given in the NSC are forecasts that might not be
  100 accurate, then this assumption is violated.