Linear programming solution and sensitivity analysis review

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Linear programming solution and sensitivity
analysis (review)
Nakorn Indra-Payoong
Maritime College, Burapha University
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• Improvement begins with I

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Sensitivity analysis

• Some of the data in a LP problem may change over
  time because of the dynamic nature of the
  business
• For example
• What happens to the optimal solution if market
  price drops
• If labour or raw-material costs rise
• If more employees are hired in a  particular
  production line
• After formulating and solving a LP problem, a
  manager should ask a number of important
  questions
• "What happens if ...?".

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Binding vs non-binding constraint

• A constraint is considered to be binding if
  changing it also changes the optimal solution
• Less severe constraints that do not affect the
  optimal solution are non-binding

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Reduced costs

• The reduced cost for a decision variable is the
  amount by which the objective function value
  would decrease or (increase) if one more unit of
  the decision variables were forced into or (out
  of) the solution

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Example I
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Shadow prices

• The shadow price for a particular resource is the
  value of an additional unit of that resource
• The shadow price for resource 1 (or resource 2)
  is equal to the change in the objective value if
  R1MAX (or R2MAX) increases by one unit

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Calculation of shadow price

• Solve the original LP
• Resolve the LP, after increasing the RHS value by
  one unit
• i.e. R1x X R1y Y lt (R1MAX 1)
• Subtract the original the objective function
  value from the new objective function value to
  find the shadow price

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Surplus value (or slack)

• The surplus value for a resource depends on
  whether the constraint that addresses that
  resource is binding or non-binding
• If binding, the surplus value is always zero
  since all of the resource that is available is
  being consumed
• If not binding, the surplus value is the number
  of unconsumed units of the resource

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Change in objective function coefficient

• Indicate the amount by which the objective
  function coefficients can change before the
  optimal solution changes
• By how much can the objective function
  coefficient on X (or Y) increase before the
  optimal solution changes? (allowable change)

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Change in RHS

• RHS range amount is the amount by which a given
  constraint limit can change before binding or
  non-binding status of the constraint changes

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LP sensitivity

• maximise 3X1 7X2 4X3 9X4
• subject to
• X1 4X2 5X3 8X4 lt 9 (1)
• X1 2X2 6X3 4X4 lt 7 (2)

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LP solution
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Questions?

• Which constraints are tight
• What would you estimate the objective function
  would change to if
• we change the right-hand side of constraint (1)
  to 10
• we change the right-hand side of constraint (2)
  to 6.5
• we add (force in) to the linear program the value
  of X3 0.7

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Answers

• Both constraints are tight because they have no
  slack or surplus
• Objective function change (10 9) 0.5 0.5
• the new value (10) of RHS of constraint (1) is
  within the limits specified so the new value of
  the objective function will be
• 22.0 0.5 22.5

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Answers

• Objective function change (7-6.5) 2.5 1.25
• The new value of RHS of constraint (2) is within
  the limits .. so the new value of the objective
  function will be 22.0 1.25 20.75

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Answers

• Objective function change 0.7 13.5 9.45,
  the objective function would decrease to 22.0
  9.45 12.45
• This is just estimation, the actual change may be
  different from the estimate (but will always be
  gt this estimate)

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Example II
Decision variables x1 kg of corn, x2 kg of
tankage, and x3 kg of alfalfa
Constraints Nutritional ingredients are
carbohydrates, proteins, and vitamins are
expressed in constraint (1) (3)
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LP solution (by Lindo)
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LP solution (by Lindo)
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Results and analysis

• Software introduces slack (or surplus) variables
  to convert inequality constraints to equalities
• This problem, 3 slack (surplus) variables (s1,
  s2, s3) are introduced
• The objective function value 241.714
• The optimal value of variables (x1, x2, x3, s1,
  s2, s3) (1.142, 0, 2.428, 0, 0, 7.142)

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Reduced costs

• x2 (non basic variable) has a reduced cost
  17.714
• Non-basic variable (x2) has a optimal value 0
• It is not optimal to include any amount of x2 in
  the mix
• Minimisation problem, reduced cost is positive,
  maximisation, reduced cost is negative

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Reduced cost

• x1 and x3 (basic variables) have reduced costs
  0, because the basic variables are already
  participating in the current solution
• Min problem, the reduced cost for a non basic
  variable is the amount by which the value of z
  will increase if we decrease the value of that
  non basic variable by 1 (whilst holding all other
  non basic variables at 0)

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Reduced cost

• Max problem, the reduced cost for a non basic
  variable is the amount by which the value of z
  will decrease if we increase the value of that
  non basic variable by 1 (whilst holding all other
  non basic variables at 0)

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Dual price (shadow price)

• Shadow price associates with the RHS constant of
  the original constraint
• The shadow price is an amount by which z will
  improve if we increase the value of constant by 1
• This problem, shadow price is negative because
  increasing a constant is tightening the constraint

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Shadow price (example)

• One unit of increase in Carbohydrate, the cost of
  optimal mix will increase by 0.771
• The shadow price for vitamins (constraint 3) 0
  because the optimal mix is already exceed the
  vitamins requirement by a margin of 7.142 (i.e.,
  the surplus (slack) variable 7.142 in the
  optimal solution

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Change in objective function coefficient

• If the cost of corn (coefficient of x1) is
  revised by a from 84 to 84 a
• New coefficient has to ensure that the solution
  remains optimal
• a should stay within the interval -37.199,
  51.000
• The range for coefficient of x2, cost of x2 is
  revised by a to 72 a

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Change in objective function coefficient

• As x2 is non basic, it follows that a has to be
  less than -17.714 to remain the solution optimal
• Thus a will stay within the interval
  -17.714, infinity
• The max possible decrease of 17.714 the reduced
  cost of x2

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Range for RHS constants

• Constraint 1 (RHS constant 200) if it is
  revised to 200 a
• Solution remains optimal for all a inside the
  interval -80.000, 24.999

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Conclusions

• From a practical viewpoint, the ranges for both
  objective function coefficients and RHS constants
  are rather widethe solution is not sensitive
• The formulation and solution of LP is not
  sensitive good or bad ???