Transcript Computational Methods for Management and Economics Carla Gomes Module 7b

Computational Methods for
Management and Economics
Carla Gomes
Module 7b
Duality and Sensitivity Analysis
Economic Interpretation of Duality
(slides adapted from: M. Hillier’s, J. Orlin’s, and H. Sarper’s)
Post-optimality Analysis
• Post-optimality – very important phase of
modeling.
• Duality plays and important role in postoptimality analysis
• Simplex provides several tools to perform
post-optimality analysis
Post-optimality analysis for LP
Task
Purpose
Model Debugging
Model Validation
Final Managerial on
resource allocations
Evaluate estimates of
model parameters
Evaluate parameter
trade-offs
Find errors and weaknesses in the model
Demonstrate validity of final model
Allocation of organizational resources
Technique
Re-optimization
Analysis results
Dual (shadow)
prices
Determine if changes in parameters change Sensitivity
optimal solution
Analysis
Determine best trade-offs between model Parametric
Linear
parameters
Programming
Economic Interpretation of Duality
• LP problems – quite often can be interpreted as
allocating resources to activities.
• Let’s consider the standard form:
xi >= 0 , (i =1,2,…,n)
• Resources – m (plants)
• Activities – n (2 products)
• Wyndor Glass problem optimal product mix --allocation of resources to activities i.e., choose the levels
of the activities that achieve best overall measure of
performance
What if we change our resources – can we
improve our optimal solution?
Sensitivity Analysis
How would changes in the problem’s objective function
coefficients or right-hand side values change the optimal
solution?
Dual Variables (Shadow Prices)
• y1*= 0  dual variable (shadow price) for resource 1
• y2*= 1.5  dual variable (shadow price) for resource 2
• y3*= 1  dual variable (shadow price) for resource 3
How much does Z increase if we increase resource 2 by
1 unit (i.e., b2 = 12  b2=13)?
Graphical Analysis of Dual variables – Variation in RHS
Increasing level of resource 2 (b2)
Production rate for windows
W
10
3 D + 2 W = 18
8
(5/3,13/2)
D= 4
2w=13  Z=3(5/3)+5(13/2)=37.5
6
2 W =12
Z=3(2)+5(6)=36
(2,6)
∆ Z=1.5
= y2*
4
Feasible
2
0
Why is y1*=0?
region
2
4
Production rate for doors
6
8
D
Economic Interpretation of Dual Variables
The dual variable associated with resource i
(also called shadow price), denoted by yi*, measures
the marginal value of this resource, i.e., the rate at
which Z could be increased by (slightly) increasing
the amount of this resource (bi), assuming everything
else stays the same. The dual variable yi* is identified
by the simplex method as the coefficient of the ith slack
variable in row 0 of the final simplex tableau.
Dual Variables: binding and non-binding
constraints
• The shadow prices (dual variables) associated with
non-binding constraints are necessarily 0
(complementary optimal slackness)  there is a
surplus of non-binding resource and therefore
increasing it will not increase the optimal solution.
Economist refer to such resources as free
resources (shadow price =0)
• Binding constraints on the other hand correspond
to scarce resources – there is no surplus. In
general they have a positive shadow price.
Does Z always increase at the same rate if we
keep increasing the amount of resource 2?
Production rate for windows
W
10
(0,9)
b2=18
3 D + 2 W = 18
8
(5/3,13/2)
D= 4
2w=13  Z=3(5/3)+5(13/2)=37.5
6
2 W =12
Z=3(2)+5(6)=36
(2,6)
∆ Z=1.5
= y2*
4
What if b2 > 18 (i.e.,
2W>18)?
Feasible
2
0
region
2
4
Production rate for doors
6
8
D
 the optimal solution will stay at (0,9) for b2>=18
Does Z always decrease at the same rate
if we decrease resource 2?
Production rate for windows
W
10
3 D + 2 W = 18
8
(5/3,13/2)
D= 4
2w=13  Z=3(5/3)+5(13/2)=37.5
6
2 W =12
(2,6)
4
Feasible
2
0
b2=6
region
2
4
Production rate for doors
6
Z=3(2)+5(6)=36
∆ Z=1.5
= y2*
If b2 < 6 the solution will no longer
vary proportionally. The optimal
solution varies proportionally to the
variation in b2 only if 6 <= b2 <=18.
In other words, the current basis remains
optimal for 6 ≤ b2 ≤ 18, but the decision
variable values and z-value will change.
8
D
• A dual variable yi* gives us the rate at which Z
could be increased by increasing the amount of
resource i slightly.
• However this is only true for a small increase in the
amount of the resource. I.e., this definition applies
only if the change in the RHS of constraint i leaves
the current basis optimal. It also assumes
everything else stays the same.
• Another interpretation of yi* is: if a premium price
must be paid for the resource i in the market place,
yi* is the maximum premium (excess over the
regular price) that would be worth paying.
Optimal Basis in the Wyndor Glass
Problem
• How can we characterize (verbally) the
optimal basis of the Wyndor Glass problem?
– Plant 1 – unutilized capacity (non-binding
constraint)
– Plant 2 – fully utilized capacity (binding
constraint)
– Plant 3 - fully utilized capacity (binding
constraint)
How do we interpret the intervals?
• If we change one coefficient in the RHS, say
capacity of plant 2, by D the “basis” remains
optimal, that is, the same equations remain
binding.
• So long as the basis remains optimal, the shadow
prices are unchanged.
• The basic feasible solution varies linearly with D.
If D is big enough or small enough the basis will
change.
The dual price or shadow price for the i th constraint
of an LP is the amount by which the optimal z-value
is improved (increased in a max problem or
decreased in a min problem) if the rhs of the i th
constraint is increased by one. This definition
applies only if the change in the rhs of constraint i
leaves the current basis optimal.
The dual variables or shadow prices are valid in a
given interval.
Sensitivity analysis for c1
How much can we vary c1 without changing
the current basic optimal solution?
Sensitivity analysis for c1
Production rate
W
for windows
8
P = 3600 = 300D + 500W
P = 3000 = 300D + 500W
Our objective function is:
Z= c1 D+5W=k
slope of iso-profit line is:

c1
D
5
Optimal solution
(2, 6)
6
Feasible
4
P = 1500 = 300D + 500W
region
2
isoprofit line
0
2
4
Production rate for doors
6
8
10
D
How much can c1 vary until the slope of the iso-profit line
equals the slope of constraint 2 and constraint 3?
• How much can c1
vary until the slope of
the iso-profit line
equals the slope of
constraint 2 and
constraint 3?
• Slope of constraint
2 0
• Slope of constraint 3
 -3/2
 c1 D  0  c1 0
5
 c1 D   3  c115
5
2
2
0  c1 7.5
Importance of Sensitivity Analysis
Sensitivity analysis is important for several reasons:
• Values of LP parameters might change. If a parameter changes,
sensitivity analysis shows it is unnecessary to solve the problem again.
For example in the Wyndor problem, if the profit contribution of
product 1 changes to 5, sensitivity analysis shows the current solution
remains optimal.
• Uncertainty about LP parameters. In the Wyndor problem for
example, if the capacity of plant 1 decreases to 2, the optimal solution
remains a weekly rate of 2 doors and 6 windows. Thus, even if
availability of capacity of plant 1 uncertain, the company can be fairly
confident that it is still optimal to produce a weekly rate of 2 doors and
6 windows.
Does the shadow price always have an
economic interpretation?
• Not necessarily
• For example,there is no economic interpretation
for dual variables associated with ratio constraints
Glass Example
• x1 = # of cases of 6-oz juice glasses (in 100s)
• x2 = # of cases of 10-oz cocktail glasses (in 100s)
• x3 = # of cases of champagne glasses (in 100s)
max 5 x1
s.t
6 x1
10 x1
x1
x1 
+ 4.5 x2 + 6 x3 ($100s)
+
5 x2 + 8 x3  60 (prod. cap. in hrs)
+ 20 x2 + 10 x3  150 (wareh. cap. in ft2)

8 (6-0z. glass dem.)
0, x2  0, x3  0
(from AMP and slides from James Orlin)
• Z* = 51.4286
Decision Variables
• x1 = 6.4286 (# of cases of 6-oz juice glasses (in 100s))
• x2 = 4.2857 (# of cases of 10-oz cocktail glasses (in 100s))
• x3 = 0 (# of cases of champagne glasses (in 100s))
Slack Variables
• s1* = 0
• s2* = 0
• s3* = 1.5714
Complementary
Dual Variables
optimal slackness
• y1* = 0.7857
conditions
• y2* = 0.0286
• y3* = 0
• Consider constraint 1. 6 x1 +
5 x2 + 8 x3  60
(prod. cap. in hrs)
• Let’s look at the objective function if we change the
production time from 60 and keep all other values the
same.
Production Optimal
obj. value
hours
60
51 3/7
difference
61
52 3/14
11/14
62
53
11/14
63
53 11/14
11/14
The dual
/shadow
Price is
11/14.
More changes in the RHS
Production Optimal
obj. value
hours
64
54 4/7
difference
11/14
65
55 5/14
11/14
66
56 1/11
*
67
56 17/22
15/22
The
shadow
Price is
11/14 until
production
= 65.5
What is the intuition for the shadow price staying
constant, and then changing?
• Recall from the simplex method that the
simplex method produces a “basic feasible
solution.” The basis can often be described
easily in terms of a brief verbal description.
The verbal description for the optimum basis for the
glass problem:
1. Produce Juice Glasses and
cocktail glasses only
2. Fully utilize production
and warehouse capacity
z = 5 x1 + 4.5 x2
6 x1 + 5 x2 = 60
10 x1 + 20 x2 = 150
x1 = 6 3/7 (6.4286)
x2 = 4 2/7 (4.2857)
z = 51 3/7 (51.4286)
The verbal description for the optimum basis
for the glass problem:
1. Produce Juice Glasses and
cocktail glasses only
2. Fully utilize production and
warehouse capacity
z = 5 x1 + 4.5x2
6 x1 + 5 x2 = 60 + D
10 x1 + 20 x2 = 150
For D = 5.5,
x1 = 8, and the
constraint x1  8
becomes binding.
x1 = 6 3/7 + 2D/7
x2 = 4 2/7 – D/7
z = 51 3/7 + 11/14 D
How do we interpret the intervals?
• If we change one coefficient in the RHS, say
production capacity, by D the “basis” remains
optimal, that is, the same equations remain
binding.
• So long as the basis remains optimal, the shadow
prices are unchanged.
• The basic feasible solution varies linearly with D.
If D is big enough or small enough the basis will
change.
Illustration with the glass example:
max 5 x1
s.t 6 x1
10 x1
ft2)
x1
x1 
+ 4.5 x2 + 6 x3 ($100s)
+
5 x2 + 8 x3  60 (prod. cap. in hrs)
+ 20 x2 + 10 x3  150 (wareh. cap. in

0, x2  0, x3  0
8 (6-0z. glass dem.)
The shadow price is the “increase” in the optimal value per
unit increase in the RHS.
If an increase in RHS coefficient leads to an increase in
optimal objective value, then the shadow price is positive.
If an increase in RHS coefficient leads to a decrease in
optimal objective value, then the shadow price is negative.
Illustration with the glass example:
max 5 x1
s.t 6 x1
10 x1
ft2)
x1
x1 
+ 4.5 x2 + 6 x3 ($100s)
+
5 x2 + 8 x3  60 (prod. cap. in hrs)
+ 20 x2 + 10 x3  150 (wareh. cap. in

0, x2  0, x3  0
8 (6-0z. glass dem.)
Claim: the shadow price of the production capacity
constraint cannot be negative.
Reason: any feasible solution for this problem remains
feasible after the production capacity increases. So, the
increase in production capacity cannot cause the optimum
objective value to go down.
Illustration with the glass example:
max 5 x1
s.t
6 x1
10 x1
ft2)
x1
x1 
+ 4.5 x2 + 6 x3 ($100s)
+
5 x2 + 8 x3  60 (prod. cap. in hrs)
+ 20 x2 + 10 x3  150 (wareh. cap. in

0, x2  0, x3  0
8 (6-0z. glass dem.)
Claim: the shadow price of the “x1  0” constraint
cannot be positive.
Reason: Let x* be the solution if we replace the constraint
“x1  0” with the constraint “x1  1”. Then x* is feasible
for the original problem, and thus the original problem has
at least as high an objective value.
Signs of Shadow Prices for
maximization problems
• “  constraint” . The shadow price is non-negative.
• “  constraint” . The shadow price is non-positive.
• “ = constraint”. The shadow price could be zero
or positive or negative.
Signs of Shadow Prices for
minimization problems
• The shadow price for a minimization problem is the
“increase” in the objective function per unit increase
in the RHS.
• “  constraint” . The shadow price is non-positive.
• “  constraint” . The shadow price is non-negative
• “ = constraint”. The shadow price could be zero
or positive or negative.
The shadow price of a non-binding constraint is 0.
“Complementary Slackness.”
max 5 x1
s.t 6 x1
10 x1
ft2)
x1
x1 
+ 4.5 x2 + 6 x3 ($100s)
+
5 x2 + 8 x3  60 (prod. cap. in hrs)
+ 20 x2 + 10 x3  150 (wareh. cap. in

0, x2  0, x3  0
8 (6-0z. glass dem.)
In the optimal solution, x1 = 6 3/7.
Claim: The shadow price for the constraint “x1  8” is zero.
Intuitive Reason: If your optimum solution has x1 < 8, one
does not get a better solution by permitting x1 > 8.
Is the shadow price the change in the
optimal objective value if the RHS
increases by 1 unit.
• That is an excellent rule of thumb! It is true
so long as the shadow price is valid in an
interval that includes an increase of 1 unit.
The shadow price is valid if only one right hand
side changes. What if multiple right hand side
coefficients change?
• The shadow prices are valid if multiple
RHS coefficients change, but the ranges are
no longer valid.
Reduced Costs
Do the non-negativity constraints
also have shadow prices?
• Yes. They are very special and are called
reduced costs?
• Look at the reduced costs for
– Juice glasses
reduced cost = 0
– Cocktail glasses reduced cost = 0
– Champagne glasses red. cost = -4/7
What is the managerial interpretation of
a reduced cost?
• There are two interpretations. Here is one of them.
• We are currently not producing champagne glasses.
How much would the profit of champagne glasses
need to go up for us to produce champagne glasses
in an optimal solution?
• The reduced cost for champagne classes is –4/7. If
we increase the revenue for these glasses by 4/7
(from 6 to 6 4/7), then there will be an alternative
optimum in which champagne glasses are produced.
Why are they called the reduced costs?
Nothing appears to be “reduced”
• The reduced costs can be obtained by
treating the shadow prices are real costs.
This operation is called “pricing out.”
Pricing Out
max 5 x1
s.t
6 x1
10 x1
1 x1
x1 
+ 4.5 x2 + 6 x3 ($100s)
+
5 x2 + 8 x3  60
+ 20 x2 + 10 x3  150

8
0, x2  0, x3  0
Pricing out treats shadow prices as
though they are real prices. The
result is the “reduced costs.”
shadow price
……11/14
……1/35
…….0
Pricing Out of x1
shadow price
max 5 x1
s.t
6 x1
10 x1
1 x1
x1 
+ 4.5 x2 + 6 x3 ($100s)
+
5 x2 + 8 x3  60
+ 20 x2 + 10 x3  150

8
0, x2  0, x3  0
Reduced cost of x1 =
……11/14
……1/35
…….0
5
- 6 x 11/14
- 10 x 1/35
- 1 x0
= 5 – 33/7 – 2/7 = 0
Pricing Out of x2
shadow price
max 5 x1
s.t
6 x1
10 x1
1 x1
x1 
+ 4.5 x2 + 6 x3 ($100s)
+
5 x2 + 8 x3  60 ……11/14
+ 20 x2 + 10 x3  150 ……1/35

8 …….0
0, x2  0, x3  0
Reduced cost of x2 =
4.5
- 5 x 11/14
- 20 x 1/35
- 0 x0
= 4.5 – 55/14 – 4/7 = 0
Pricing Out of x3
shadow price
max 5 x1
s.t
6 x1
10 x1
1 x1
x1 
+ 4.5 x2 + 6 x3 ($100s)
+
5 x2 + 8 x3  60 ……11/14
+ 20 x2 + 10 x3  150 ……1/35

8
…….0
0, x2  0, x3  0
Reduced cost of x3 =
6
- 8 x 11/14
- 10 x 1/35
- 0 x0
= 6 – 44/7 – 2/7 = -4/7
Can we use pricing out to figure out
whether a new type of glass should be
produced? shadow price
max 5 x1
s.t
6 x1
10 x1
1 x1
x1 
+ 4.5 x2 + 7 x4 ($100s)
+
5 x2 + 8 x4  60
+ 20 x2 + 20 x4  150

8
0, x2  0, x4  0
Reduced cost of x4 =
……11/14
……1/35
…….0
7
- 8 x 11/14
- 20 x 1/35
- 0 x0
= 7 – 44/7 – 4/7 = 1/7
Pricing Out of xj
shadow price
max 5 x1 + 4.5 x2 + cj xj ($100s)
s.t
6 x1 +
5 x2 + a1j xj  60
10 x1 + 20 x2 + a2j xj  150
………..
……….
+ amjxj = bm
x1  0, x2  0, x3  0
Reduced cost of xj = ?
……y1
……y2
………
……ym
Brief summary on reduced costs
• The reduced cost of a non-basic variable xj is the
“increase” in the objective value of requiring that
xj >= 1.
• The reduced cost of a basic variable is 0.
• The reduced cost can be computed by treating
shadow prices as real prices. This operation is
known as “pricing out.”
• Pricing out can determine if a new variable would
be of value (and would enter the basis).
Summary
• The shadow price is the unit change in the optimal
objective value per unit change in the RHS.
• The shadow price for a “ 0” constraint is called the
reduced cost.
• Shadow prices usually but not always have economic
interpretations that are managerially useful.
• Non-binding constraints have a shadow price of 0.
• The sign of a shadow price can often be determined
by using the economic interpretation
• Shadow prices are valid in an interval.
• Reduced costs can be determined by pricing out
Reduced Costs
• The reduced cost of a variable x is the shadow
price of the “x  0” constraint. It is also the
negative of cost coefficient for x in the final
tableau.
• Suppose in the previous example that we required
that x3  1? What is the impact on the optimal
objective value? What is the resulting solution?
By the previous slide, the impact is -4/7.
More on reduced costs
• In a pivot, multiples of constraints are
added to the cost row.
• We will use this fact to determine explicitly
how the cost row in the final tableau is
obtained.
Implications of Reduced Costs
• Implication 1: increasing the cost coefficient
of a non-basic variable by D leads to an
increase of its reduced cost by D.
Implications of Reduced Costs
• Implication 2: We can compute the reduced
cost of any variable if we know the original
column and if we know the “prices” for
each constraint.
FACT: We can
compute the
reduced cost of a
new variable. If
the reduced cost is
positive, it should
be entered into
the basis.
• Every tableau has “prices.” These are
usually called simplex multipliers.
• The prices for the optimal tableau are the
shadow prices.
Quick Summary
• Connection between shadow prices and reduced
cost. If xj is the slack variable for a constraint,
then its reduced cost is the negative of the shadow
price for the constraint.
• The reduced cost for a variable is the negative of
its cost coefficient in the final tableau
Sensitivity Analysis
Computer Analysis
The Computer and Sensitivity Analysis
• If an LP has more than two decision variables,
the range of values for a rhs (or objective
function coefficient) for which the basis remains
optimal cannot be determined graphically.
• These ranges can be computed by hand but this is
often tedious, so they are usually determined by a
packaged computer program. MPL and LINDO
will be used and the interpretation of its
sensitivity analysis discussed.
• Note: sometimes Excel provides erroneous
results
MPL – Sensitivity analysis info
Dual or Shadow
prices are the
amount the
optimal z-value
improves if the
rhs of a
constraint is
increased by one
unit (assuming
no change in
basis).
Dual variables
c1 cost
Reduced
is the amount the
objective function
coefficient for
variable i would
have to be
increased for
there to be an
b2
alternative
optimal solution.
More later…
MPL – Sensitivity analysis info
c1
Allowable ranges
(w/o changing basis)
for the x1 coefficient
(c1) is:
0  c1  7.5
b2
What about c2? And b1 and b3?
Allowable range (w/o
changing basis) for
the rhs (b2) of the
second constraint is:
6  b2  18
Lindo Sensitivity Analysis
Allowable ranges – in
terms of increase and
decrease
(w/o changing basis)
for the x1 coefficient
(c1) is:
0  c1  7.5
The Computer and Sensitivity Analysis
• Consider the following maximization problem. Winco sells
four types of products. The resources needed to produce one
unit of each are:
Raw
material
Hours of
labor
Sales price
Product Product Product Produc Availabl
1
2
3
t4
e
2
3
4
7
4600
3
4
5
6
$4
$6
$7
$8
5000
To meet customer demand, exactly 950 total units must be
produced. Customers demand that at least 400 units of product 4 be
produced. Formulate an LP to maximize profit.
Let xi = number of units of product i produced by Winco.
• The Winco LP formulation:
max z = 4x1 + 6x2 +7x3 + 8x4
s.t.
x1 + x2 + x3 + x4 = 950
x4 ≥ 400
2x1 + 3x2 + 4x3 + 7x4 ≤ 4600
3x1 + 4x2 + 5x3 + 6x4 ≤ 5000
x1,x2,x3,x4 ≥ 0
LINDO output and
sensitivity
analysis
example(s).
Reduced cost
is the amount the
objective function
coefficient for
variable i would
have to be
increased for
there to be an
alternative
optimal solution.
MAX 4 X1 + 6 X2 + 7 X3 + 8 X4
SUBJECT TO
2) X1 + X2 + X3 + X4 = 950
3) X4 >= 400
4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <= 4600
5) 3 X1 + 4 X2 + 5 X3 + 6 X4 <= 5000
END
LP OPTIMUM FOUND AT STEP
4
OBJECTIVE FUNCTION VALUE
1) 6650.000
VARIABLE
X1
X2
X3
X4
ROW
2)
3)
4)
5)
NO. ITERATIONS=
VALUE
0.000000
400.000000
150.000000
400.000000
SLACK OR SURPLUS
0.000000
0.000000
0.000000
250.000000
4
REDUCED COST
1.000000
0.000000
0.000000
0.000000
DUAL PRICES
3.000000
-2.000000
1.000000
0.000000
RANGES IN WHICH THE BASIS IS UNCHANGED:
LINDO sensitivity
analysis example(s).
OBJ COEFFICIENT RANGES
VARIABLE
Allowable range (w/o
changing basis) for
the x2 coefficient
(c2) is:
CURRENT
ALLOWABLE
ALLOWABLE
COEF
INCREASE
DECREASE
X1
4.000000
1.000000
INFINITY
X2
6.000000
0.666667
0.500000
X3
7.000000
1.000000
0.500000
X4
8.000000
2.000000
INFINITY
5.50  c2  6.667
RIGHTHAND SIDE RANGES
Allowable range (w/o
changing basis) for
the rhs (b1) of the first
constraint is:
850  b1  1000
ROW
CURRENT
ALLOWABLE
ALLOWABLE
RHS
INCREASE
DECREASE
2
950.000000
50.000000
100.000000
3
400.000000
37.500000
125.000000
4
4600.000000
250.000000
150.000000
5
5000.000000
INFINITY
250.000000
Shadow prices
are shown in the
Dual Prices
section of
LINDO output.
Shadow prices
are the amount
the optimal zvalue improves if
the rhs of a
constraint is
increased by one
unit (assuming
no change in
basis).
MAX 4 X1 + 6 X2 + 7 X3 + 8 X4
SUBJECT TO
2) X1 + X2 + X3 + X4 = 950
3) X4 >= 400
4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <= 4600
5) 3 X1 + 4 X2 + 5 X3 + 6 X4 <= 5000
END
LP OPTIMUM FOUND AT STEP
4
OBJECTIVE FUNCTION VALUE
1) 6650.000
VARIABLE
X1
X2
X3
X4
ROW
2)
3)
4)
5)
NO. ITERATIONS=
VALUE
0.000000
400.000000
150.000000
400.000000
SLACK OR SURPLUS
0.000000
0.000000
0.000000
250.000000
4
REDUCED COST
1.000000
0.000000
0.000000
0.000000
DUAL PRICES
3.000000
-2.000000
1.000000
0.000000
Interpretation of shadow prices for the Winco LP
ROW
SLACK OR SURPLUS
DUAL PRICES
2)
0.000000
3.000000
(overall demand)
3)
0.000000
-2.000000
(product 4 demand)
4)
0.000000
1.000000
(raw material availability)
5)
250.000000
0.000000
(labor availability)
Assuming the allowable range of the rhs is not violated, shadow (Dual) prices
show: $3 for constraint 1 implies that each one-unit increase in total demand
will increase net sales by $3. The -$2 for constraint 2 implies that each unit
increase in the requirement for product 4 will decrease revenue by $2. The $1
shadow price for constraint 3 implies an additional unit of raw material (at no
cost) increases total revenue by $1. Finally, constraint 4 implies any additional
labor (at no cost) will not improve total revenue.
Shadow price signs
1. Constraints with symbols will always have
nonpositive shadow prices.
2. Constraints with  will always have nonnegative
shadow prices.
3. Equality constraints may have a positive, a
negative, or a zero shadow price.
Managerial Use of Shadow Prices
The managerial
significance of shadow
prices is that they can
often be used to
determine the
maximum amount a
manager should be
willing to pay for an
additional unit of a
resource. Reconsider
the Winco to the right.
What is the most
Winco should be
willing to pay for
additional units of raw
material or labor?
MAX 4 X1 + 6 X2 + 7 X3 + 8 X4
SUBJECT TO
2) X1 + X2 + X3 + X4 = 950
3) X4 >= 400
4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <= 4600
5) 3 X1 + 4 X2 + 5 X3 + 6 X4 <= 5000
END
LP OPTIMUM FOUND AT STEP
raw
material
labor
4
OBJECTIVE FUNCTION VALUE
1) 6650.000
VARIABLE
X1
X2
X3
X4
ROW
2)
3)
4)
5)
NO. ITERATIONS=
VALUE
0.000000
400.000000
150.000000
400.000000
SLACK OR SURPLUS
0.000000
0.000000
0.000000
250.000000
4
REDUCED COST
1.000000
0.000000
0.000000
0.000000
DUAL PRICES
3.000000
-2.000000
1.000000
0.000000
Managerial Use of Shadow Prices
The shadow price for raw
material constraint (row 4)
shows an extra unit of raw
material would increase
revenue $1. Winco could
pay up to $1 for an extra
unit of raw material and be
as well off as it is now.
Labor constraint’s (row 5)
shadow price is 0 meaning
that an extra hour of labor
will not increase revenue.
So, Winco should not be
willing to pay anything for
an extra hour of labor.
MAX 4 X1 + 6 X2 + 7 X3 + 8 X4
SUBJECT TO
2) X1 + X2 + X3 + X4 = 950
3) X4 >= 400
4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <= 4600
5) 3 X1 + 4 X2 + 5 X3 + 6 X4 <= 5000
END
LP OPTIMUM FOUND AT STEP
4
OBJECTIVE FUNCTION VALUE
1) 6650.000
VARIABLE
X1
X2
X3
X4
ROW
2)
3)
4)
5)
NO. ITERATIONS=
VALUE
0.000000
400.000000
150.000000
400.000000
SLACK OR SURPLUS
0.000000
0.000000
0.000000
250.000000
4
REDUCED COST
1.000000
0.000000
0.000000
0.000000
DUAL PRICES
3.000000
-2.000000
1.000000
0.000000