LP EXAMPLES

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LP EXAMPLES
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Gillians RestaurantCHP.2 Problem 34 35

• Two products, ice cream and yogurt
• The freezer capacity is at most 115 gallons.
• A gallon of ice cream costs 0.93 and a gallon of
  frozen yogurt costs 0.75.
• Restaurant budgets 90 each week for these
  products.
• The demand for ice cream is at least twice of the
  demand for frozen yogurt.
• Profit per gallon of ice cream and yogurt are
  4.15 and 3.60 , respectively.

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Sensitivity Analysis

• REDUCED COST
• A decision variable with a positive value in
  the optimum solution will generally have zero
  reduced cost. Likewise, a decision variable with
  value zero in the optimum solution will have a
  non-zero reduced cost.
• The value of reduced cost gives the amount by
  which the objective function coefficient of this
  variable must change for this variable to have a
  non-zero value in the optimal solution

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SENSITIVITY ANALYSIS

• SHADOW PRICE (DUAL VALUE)
• It is the marginal value of increasing the
  right-hand side of any constraint.
• binding constraints have zero slack or zero
  surplus and vice versa.
• Shadow price for not binding constraints is zero.
• Lower and upper bounds given for each constraint
  provide the range over which the shadow price for
  that constraint is valid.

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SENSITIVITY ANALYSIS

• Dual Value RHS max Z (Profit) min Z
  (Cost)
• Positive increase increase
  decrease
• Positive decrease decrease
  increase
• Negative increase decrease
  increase
• Negative decrease increase
  decrease
• Note As long as increases/decreases in RHSs are
  within the given lower and upper bounds

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EXAMPLE 1

• Tucker Inc. needs to produce 1000 Tucker
  automobiles. The company has four production
  plants. Due to differing workforces,
  technological advances, and so on, the plants
  differ in the cost of producing each car. They
  also use a different amount of labor and raw
  material at each. This is summarized in the
  following table
• Plant Cost ('000) Labor
  Material
• 1 15
  2 3
• 2 10
  3 4
• 3 9
  4 5
• 4 7
  5 6
• The labor contract signed requires at least
  400 cars to be produced at plant 3 there are
  3300 hours of labor and 4000 units of material
  that can be allocated to the four plants.

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Questions for Example 1

• 1. What is the optimum solution? What is the
  current cost of production?
• 2. How much will it cost to produce one more
  vehicle? How much will we save by producing one
  less?
• 3. How would our solution change if it costs only
  8,000 to produce at plant 2?
•  4. For what ranges of costs is our optimal
  solution (except for the objective value) valid
  for plant 2?
• 5. How much are we willing to pay for one more
  labor hour?

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Questions for Example 1

• 6. What would be the effect of reducing the 400
  car limit down to 200 cars? To 0 cars? What
  would be the effect of increasing it by 100 cars?
  by 200 cars?
•  7. How much is our raw material worth (to get
  one more unit)? How many units are we willing to
  buy at that price? What will happen if we want
  more?