Chapter 10
The primal linear program solution answers
the tactical question when it tells us how
much to produce. But the dual can have far
greater impact because it addresses
strategical issues regarding the structure of
the business itself.
1
Sensitivity Analysis and
Duality in
Linear Programming
Shadow Prices and
Opportunity Costs
 The Redwood furniture problem involves XT and XC denote the number
of tables and chairs to be made.
Maximize P = 6XT + 8XC
Subject to:
30XT + 20XC < 300 (wood)
5XT + 10XC < 110 (labor)
where
XT, XC > 0
 What is the effect of one additional unit of the respective resources?
 It will allow more product to be made, increasing profit.
 That is the resource’s marginal value or shadow price.
 The marginal value of a resource can be found by changing the
available level and solving the revised problem.
 There is a faster way to find the marginal value.
2
 Knowing levels for marginal values enhances planning capabilities.
 They can be used to “price” out a new product.
 Consider the wood used in making furniture. Raising the available
level from 300 to 301 board feet will change the feasible solution
region, as shown on the following slide. It will increase profit by $.10.
Shadow Prices and
Opportunity Costs
3
The Dual Linear Program
 The original problem is here referred to as the primal linear program.
 The dual linear program expresses the problem with a resource
orientation.
 Let UW and UL denote the marginal values of wood and labor. The
objective is to
Minimize C = 300UW + 110UL
Subject to:
30UW + 5UL > 6
(table)
20UW + 10UL > 8
(chair)
where
UW, UL > 0
 The total value of available resources is minimized.
 The constraint left-hand sides express the opportunity cost of making
one unit of the respective product.
 It is an opportunity cost because making the product may divert resources
from their optimal utilization, and theUs measure their value.
 Each product is required to have an opportunity cost > its unit profit.
4
 The dual has a column (variable) for each primal row (constraint).
Primal right-hand sides are dual objective coefficients and vice versa.
Redwood Furniture Problem:
Graphical Solution of Dual
5
Meaning of Dual Solution
 The values UW = .10 and UL = .60 tell us
that additional wood (acquired at present
costs) would raise profit by $.10. Similarly,
an additional hour of labor raises P by $.60.
 Redwood would benefit by more resources.
They would even pay a small premium to
get it (not more than the respective U.)
 The Us are shadow prices for resource in
pricing out a new product, such a desk.
 Making a desk would divert resources from
tables and chairs, and fewer would be made.
6
Evaluating New Products
Using the Dual
 Redwood evaluates new products:
 Bench having profit of $7, needing 25 board
feet of wood and 7 hours of labor.
 Planter box having profit of $2, needing 10
board feet of wood and 2 hours of labor.
 The opportunity costs for one of each are:
 Bench: $.10(25) + .60(7) = $6.70 (< $7). Make
it, because doing so increases P by $.30/unit.
 Planter box: $.10(10) + .60(2) = $2.40 (> $2).
Do not make. Resources are too valuable.
7
Sensitivity Limits
for Right-Hand Sides
 QuickQuant provides the following report
that lists for each constraint the optimal
level of the applicable dual variable.
wood
labor
8
 Included are the sensitivity limits for the
right-hand sides. The dual values will
be the same for any available quantity of
wood and labor falling within those limits.
Sensitivity Limits
for Objective Coefficients
 QuickQuant also gives sensitivity limits for
objective coefficients (unit profits, costs).
 The present primal optimal quantities will
be exactly the same for any level of profit
falling with those limits.
9
 For example, if the table and chair profits are
both $10, only P changes, to $10(4) + 10(9) =
$130.
Post-Optimality Analysis
 Perhaps the most important role of the dual is
helping the decision maker improve the business
itself.
 That can be done with a post-optimality analysis
in which a series of changes are made to the
problem structure ( and hence business
environment).
 Find economic bottlenecks, get more resources:
 Authorize overtime or train new employees.
 Install faster machines or expand plant.
 Target marketing, expand profitable products.
 Adjust pricing.
 Solve New LPs after each possible change.
 New Us will suggest further possible improvements.
10
Solver’s Sensitivity Report
Solver’s Sensitivity Report yields the values
of the:
 dual variables
 allowable increases and decreases for the
right-hand sides (upper and lower limits).
11
 allowable increases and decreases for the
coefficients of the objective function (upper
and lower limits).
Solver’s Sensitivity Report
To get Solver’s Sensitivity Report, highlight
Sensitivity Report in the Report box of the Solver
Results dialog box before clicking the OK button.
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Sensitivity Report for
Swatville Sluggers (Figure 10-3 )
3. Allowable increases and decreases for right-hand sides.
2. Dual slack
or surplus
variables (in
the Reduced
Cost column).
Adjustable Cells
Final
Reduced
Objective
Allowable
Allowable
Cell
Name
Value
Cost
Coefficient
Increase
Decrease
$D$17 X30
80.46
0.000
12.05
3.68
0.857
$E$17 X32
0
0.000
11.75
0.50
2.537
$F$17 X34
80.46
0.000
13.19
3.57
0.516
$G$17 X36
0
-1.269
12.64
1.27
1E+30
$H$17 X38
0.000
2.01
1.94
4. Allowable increases
and decreases80.46
for coefficients
of the 15.48
objective function.
$I$17 X40
0
-1.202
15.18
1.20
1E+30
Constraints
Final
Shadow
Constraint Allowable
Allowable
1. Shadow
Cell
Name
Value
Price
R.H. Side
Increase
Decrease
prices.
$N$6 Wood
8206.90
0.000
12000
1E+30
3793.10
$N$7 Lathe
2655.17
0.000
6000
1E+30
3344.83
$N$8 Finishing
7000.00
0.468
7000
3235.29
7000.00
$N$9 Boxes
241.38
0.000
1000
1E+30
758.62
$N$10 Stain
563.22
0.000
1500
1E+30
936.78
$N$11 Varnish
1689.66
0.000
3000
1E+30
1310.34
$N$12 Production NB: Multiple
0.00
0.252solutions 0exist to the 0dual.
164.71
optimal
$N$13 <
0.00solution 0.635
0 is different
0
250.00
This dual
found by Excel
$N$14 Restrictions than the
0.00
0.601given in the
0 primal. 0
114.75
dual solution
13
Sensitivity Report for
Redwood Furniture (Figure 10-6)
2. Dual slack or surplus variables
(in the Reduced Cost column).
3. Allowable increases
and decreases for righthand sides. These give
lower and upper limits.
Adjustable Cells
1. Shadow
prices.
Cell
Name
$B$9 XT
$C$9 XC
Final Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase Decrease
4
0
6
6
2
9
0
8
4
4
Constraints
Final Shadow Constraint Allowable Allowable
Cell
Name
Value
Price
R.H. Side Increase Decrease
$G$5 Wood used
300
0.1
300
360
80
$G$6 Labor used
110
0.6
110
40
60
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4. Allowable increases and
decreases for the
coefficients of the objective
function. These give the
lower and upper limits.