Linear Programming

1
Linear Programming

• Sensitivity of the Right Hand Side Coefficients

2
Sensitivity of RHS Coefficients

• RHS coefficients usually give some maximum limit
  for a resource or some minimum requirement that
  must be met.
• Changes to the RHS can happen when extra units of
  the resource become available or when some of the
  original resource becomes unavailable.
• Or the minimum requirement is loosened (made
  less) or strengthened (made greater).
• Extra units may be available for a price.
• The question becomes how much would an extra unit
  add to the value of the objective function, that
  is, what is the most we would be willing to pay
  for extra units of the resource?

3
Finding the Optimal Point - Review
X1, X2 0
4
Optimal Point WithOne Extra Unit of Plastic
Shadow Price(for Plastic) 4363.40 4360 (new
profit) - (old profit) 3.40
Max 8X1 5X2 s.t.
X1, X2 0
Still determined byPlastic and Time constraints
5
Shadow Prices

• The shadow price for a constraint is the amount
  the objective function value will change given
• 1 additional unit on the RHS of the constraint
• No other changes
• This shadow price is valid as long as the same
  constraints (including the one whose RHS is
  changing) determine the optimal point.
• In this case plastic and production time
• It can be shown that if the RHS for plastic were
  1002 the profit would increase another 3.40 to
  4366.80.
• It can also be shown that if the RHS for plastic
  were 999 the profit would decrease by 3.40 to
  4356.60.

6
Allowable Increase andAllowable Decreaseof a
RHS Value

• The shadow prices remain valid as long as the
  same constraints (called the binding constraints)
  determine the optimal point.
• When the RHS of the constraint is increased or
  decreased to the point that another constraint
  replaces one of the binding constraints to
  determine the optimal point a new shadow price
  becomes valid for the constraint.
• The amount the RHS can increase or decrease
  before another constraint becomes one of the
  binding constraints is what Excel calls the
  Allowable Increase and the Allowable Decrease
  respectively.

7
Increasing the Right Handside for Plastic
Max 8X1 5X2 s.t.
X1, X2 0
8
Further Increasingthe Right Hand Side for Plastic
Max 8X1 5X2 s.t.
X1, X2 0
The shadow priceswill now CHANGE
9
Decreasing the RHS for Plastic
Max 8X1 5X2 s.t.
2X1 1X2 (Plastic)
X1, X2 0
Optimal solution determined by Plastic and Time
Constraints
and by X2 axis!
10
Further Decreasingthe RHS for Plastic
Max 8X1 5X2 s.t.
2X1 1X2 (Plastic)
X1, X2 0
The shadow priceswill now CHANGE
11
Comparison With Excel

• Here is the printout out of the sensitivity
  analysis dealing with the objective RHS
  coefficients for the original Galaxy Industries
  problem.

Range of Feasibility is the range of values that
an RHS coefficient can assume without changing
the shadow prices as long as no other changes are
made.
12
Exact Meaning of Shadow Prices

• A shadow price always means the amount the
  objective function will change given a one unit
  increase in the RHS value of a constraint.
• But does this mean that this is the value (the
  most you would be willing to pay) for an extra
  unit? The answer depends on how the objective
  function coefficients were calculated.
• If the objective function coefficients did not
  take the value of the resource into
  consideration, these are sunk costs.
• Shadow price the value of an extra unit of the
  resource.
• If the objective function coefficients did take
  the value of the resource into consideration,
  these are included costs.
• Shadow price a premium above the current price
  of the item that one would be willing to pay for
  an extra unit.

13
EXAMPLE

• Suppose the 8 objective function coefficient for
  dozens of Space Rays and the 5 objective
  function coefficient for dozens of Zappers were
  calculated as follows

DOZ. DOZ. SPACE RAYS
ZAPPERS Selling Price 24
26 Costs Plastic (3/lb) 6 (2
lbs.) 3 (1 lb.) Other Variable
Costs 10 18

Total Profit Per Dozen 8
5
Production time is a sunk cost
The 3.40 shadow price for plastic means we would
be willing to pay up to 3.40 more than the
current price of 3 per pound (that is up
to 6.40/ lb.) for extra plastic.
It is not included in the objective function
coefficient calculation. The 0.40 shadow price
is the value of an extra minute of production
time.
14
Complementary Slackness

• Complementary slackness also holds for RHS
  values. This property for RHS values states
• Again, it can happen, that both are 0.

Complementary Slackness For RHS Coefficients For
each constraint, either the slack (difference
between RHS LHS) is 0 or its shadow price will
be 0.
Plastic Shadow Price ? 0 Slack 1000-1000
0
Time Shadow Price ? 0 Slack 2400-2400 0
Prod. Limit Slack 700-680 ? 0 Shadow Price
0
Prod. Mix Slack 350-(-40) ? 0 Shadow Price
0
15
Review

• Shadow price
• Found by subtracting the original objective
  function value from the objective function value
  with one more unit of the resource on the RHS
• Meaning
• Included Cost
• Sunk Cost
• Range of Feasibility
• Range of RHS value in which shadow price does not
  change
• The same constraints determine the optimal
  solution in the range of feasibility
• Complementary Slackness
• Either the slack is 0 or the shadow price is 0