shadow price

Report
Lecture 11:
Sensitivity Part II: Prices
AGEC 352
Spring 2012-February 27
R. Keeney
Constraint Prices

Shadow price = marginal valuation
 Marginal -> last or next unit
 Shadow?
 an internal price with no actual exchange.
 Decision maker is both the supplier and
demander.
Use of the Shadow Price

Answers the question:
 If the RHS limitation were expanded by
one unit how does the objective
variable change?
 Willingness to pay or accept if entering the
external market
 Recall the objective variable is the only
measure of success/benefit to the
decision maker
Simple Model: RHS +1
max Z  x  y
s.t.
max Z  x  y
s.t.
x  10; y  10
x, y  0
x  11; y  10
x, y  0
Expansion of RHS of x constraint means
we can increase our choice of x , which
increases Z
 The change in Z is our benefit
 The increased benefit is the maximum we
would pay for the added unit on RHS of x
constraint

Simple Model: RHS -1
max Z  x  y
s.t.
max Z  x  y
s.t.
x  10; y  10
x, y  0
x  9; y  10
x, y  0

Reduction of RHS of x constraint means we
must reduce our choice of x , which diminishes Z

The change in Z is our loss and is the minimum
we should charge to sell x rather than use it
Signs and interpretation
( NewZ  OldZ)
Shadowprice 
( NewRHSx  OldRHSx )
If increasing the RHS increases Z, the
shadow price will be positive.
If increasing the RHS decreases Z, the
shadow price will be negative.
Signs and Interpretation
( NewZ  OldZ)
Shadowprice 
( NewRHSx  OldRHSx )

What if Z doesn’t change?
◦ Shadow price = 0
◦ This will be the case for any constraint
that does not bind at the optimum…
◦ Think about the question: What would
we pay for one more unit?
Shadow Price Signs
Objective
Direction
Inequality Direction
<=
=>
Maximization
Positive
Negative
Minimization
Negative
Positive
For a max problem:
Increasing the RHS of a <= constraint expands the feasible
space, increases the value of Z, generates a positive shadow
price.
Increasing the RHS of a => constraint contracts the feasible
space, reduces the value of Z, generates a negative shadow
price.
Shadow Price Signs
Objective
Direction
Inequality Direction
<=
=>
Maximization
Positive
Negative
Minimization
Negative
Positive
For a min problem:
Increasing the RHS of a <= constraint contracts the
feasible space, reduces the value of Z, generates a
negative shadow price.
Increasing the RHS of a => constraint expands the
feasible space, increases the value of Z, generates a
positive shadow price.
Mathematical Rule
Expanding or reducing the feasible space by
adjusting a non-binding constraint has no
impact on Z, shadow price = 0.
Let, slack = (Constraint RHS – Constraint LHS)
Then we can state the following rule:
(slack)*(shadow price) = 0
If shadow price is non-zero, slack must be zero.
If shadow price is zero, slack is either
a) non-zero or
b) zero.
Case b: 0 slack & 0 shadow price
The first constraint is redundant because it
does not add a corner point to the problem.
Payoff:
1.0 x +
1.0 y = 20.0
Plugging x = 10 into 2x + y = 30, gives y=10,
which is already a constraint of the problem.
y
12
11
10
9
8
7
6
5
4
3
2
1
0
:
1.0 x +
:
max Z  x  y
0.0 y = 10.0
0.0 x +
:
0
1
2
3
4
5
s.t.
1.0 y = 10.0
6
7
8
9
2.0 x +
1.0 y = 30.0
10 11 12
x
Optimal Decisions(x,y): (10.0, 10.0)
: 1.0x + 0.0y <= 10.0
: 0.0x + 1.0y <= 10.0
: 2.0x + 1.0y <= 30.0
2 x  y  30
x  10; y  10
x, y  0
Objective Variable Prices

Sensitivity of constraints involves
placing an economic value on the
resources in the problem
◦ Look at Excel’s shadow price report later

Sensitivity of objective coefficients
(prices for short) is completely
different
◦ Under what price range does the optimal
plan remain optimal?
Pizza Maker’s Problem

Two pizza types: max P  2.25R  2.65D
Regular (R) and
subject to :
Deluxe (D)
 Use available
sauce, dough,
sausage, cheese,
and mushrooms
to make pizzas.
 Profit is
 2.25 per R pizza,
 2.65 per D pizza
Sauce:
8 R  8 D  440
Dough:
Sausage:
Cheese :
16R  16D  1000
3R  9 D  275
8 R  12D  500
Mushroom s: 4 D  100
Non  neg. : R  0; D  0
54
51
48
45
42
Feasible Space for Pizza Maker
39
36
33
Payoff:
2.3 R +
Optimum is
R = 40, D = 15
2.6 D = 129.7
30
27
24
Deluxe
Pizzas
21
18
How sensitive is this solution
to a change in the price of
Deluxe Pizzas?
15
12
9
6
3
0
0
5
10
15
Optimal Decisions(R,D): (40.0, 15.0)
Sauce:
8.0R +
8.0D <= 440.0
Dough: 16.0R + 16.0D <= 1000.0
Sausage:
Cheese:
3.0R +
9.0D <= 275.0
8.0R + 12.0D <= 500.0
Mushroom:
0.0R +
4.0D <= 1000.0
20
25
30
35
40
45
Regular Pizzas
50
55
How does changing the price of the
deluxe pizza affect this problem?
Objective Equation: 2.25R + 2.65D = P
Rewrite this as:
D = P/2.65 – R*(2.25/2.65)
The slope of the objective line will flatten if we
increase the price of deluxe pizzas above 2.65.
If the objective line gets flat enough, the optimal
point will switch to the next corner point
immediately leftward.
57
54
51
48
Deluxe Price Increase
45
42
Price increase makes this line flatter. If
it changes enough we will have a new
optimal combination of R and D pizzas.
39
36
33
Payoff:
2.3 R +
2.6 D = 129.7
5
10
30
Deluxe
Pizzas
27
24
21
18
15
12
9
6
3
0
0
15
Optimal Decisions(R,D): (40.0, 15.0)
Sauce:
8.0R +
8.0D <= 440.0
Dough: 16.0R + 16.0D <= 1000.0
Sausage:
Cheese:
3.0R +
9.0D <= 275.0
8.0R + 12.0D <= 500.0
Mushroom:
0.0R +
4.0D <= 1000.0
20
25
30
35
40
45
Regular Pizzas
50
55
Called Allowable increase in Excel’s
Sensitivity Report
The size of the price increase determines
whether the slope of the objective line
gets flat enough to shift to the leftward
corner point.
 This is what the allowable increase on
objective coefficients is measuring.
 The allowable decrease does the same in
the opposite direction.

Sensitivity Report on Pizza Prices:
Prices increase->Profit/pizza goes up
The allowable increase says that if the profit/deluxe pizza goes
up by more than 72.5 cents we should shift to a new
combination of R and D pizzas (more D, less R).
If profit/deluxe pizza goes down by more than 40 cents make
more R and less D.
Important point: Any change in the profits/pizza will change the
objective value, but if in the allowable range, the best choices
do not adjust.
Constraint Sensitivity


Cheese and Sauce are binding
constraints with positive shadow prices
We would pay to have more cheese or
sauce available to make pizzas with
because we could increase profits
57
54
Binding constraints
51
48
45
42
This corner point is
where the cheese and
sauce constraints
cross.
39
36
33
Payoff:
2.3 R +
2.6 D = 129.7
Deluxe30
Pizzas27
24
21
18
Cheese constraint
15
12
9
6
Sauce constraint
3
0
0
5
10
15
Optimal Decisions(R,D): (40.0, 15.0)
Sauce:
8.0R +
8.0D <= 440.0
Dough: 16.0R + 16.0D <= 1000.0
Sausage:
Cheese:
3.0R +
9.0D <= 275.0
8.0R + 12.0D <= 500.0
Mushroom:
0.0R +
4.0D <= 1000.0
20
25
30
35
40
45
50
Regular Pizzas
55
6
Constraint Ranges
Excel’s constraint sensitivity report also
reports allowable increase and decrease
 These values indicate the magnitude of
changes allowed to the RHS quantity
without changing the marginal valuation
(shadow price)

Expanding the sauce constraint
Adding 1 to the RHS of the sauce
constraint expands the feasible space
 Moves the corner point rightward
allowing for a higher objective variable
value
 The shadow price says every time we
expand this constraint by one unit, we
gain about $0.18 of profits
 Allowable increase tells us how long we
can keep making these 1 unit moves in
the constraint

36
33
Expanding sauce capacity
30
27
24
21
18
Cheese: 8.0 R + 12.0 D = 500.0
Deluxe 15
12
9
Sauce: 8.0 R + 8.0 D = 440.0
6
3
0
0
5
10
15
20
Sausage: 3.0 R + 9.0 D = 275.0
Optimal Decisions(R,D): ( 0.0, 0.0)
Sausage: 3.0R + 9.0D <= 275.0
Cheese: 8.0R + 12.0D <= 500.0

30
35
40
45
50
55
60
Regular
If we kept moving the Sauce constraint
to the right what would happen?
Eventually, sauce would not be limiting.
Sauce: 8.0R + 8.0D <= 440.0

25
65
Pay
48
45
Sauce is no longer limiting
42
39
36
33
30
27
24
Sausage: 3.0 R + 9.0 D = 275.0
Sauce: 8.0 R + 8.0 D = 510.0
21
D
18
15
12
Cheese: 8.0 R + 12.0 D = 500.0
9
6
3
0
0
5
10
15
20
Optimal Decisions(R,D): ( 0.0, 0.0)
25
30
35
R
40
45
50
55
60
65
70
75
Payoff: 2.3 R +
Sauce: 8.0R + 8.0D <= 510.0
With
a RHS
value
of 510, the sauce constraint is no longer on
Sausage:
3.0R + 9.0D
<= 275.0
8.0R + 12.0D <= 500.0
theCheese:
boundary
of the feasible space. This is the information
provided by the allowable increase of the constraint.
Typical Questions
Pizza maker wants to sell her excess
dough. What is the minimum amount she
can charge?
 Pizza maker can buy 200 units of sauce
for $15.00. Should she do it?
 Pizza maker has a sale on deluxe pizzas
reducing profit per unit by 15%. Should
she change the production plan for this
week?


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