Chapter 7
Duality and Sensitivity
in Linear Programming
7.1 Generic Activities versus
Resources Perspective
Objective Functions as Costs and Benefits
• Optimization models objective functions usually can be
interpreted as minimizing some measure of cost or
maximizing some measure of benefit. [7.1]
Choosing a Direction for Inequality Constraints
• The most natural expression of a constraint is usually
the one making the right-hand-side constant nonnegative. [7.2]
0.3 x1 + 0.4 x2  2.0 (gasoline)
-0.3 x1 - 0.4 x2  -2.0
7.1 Generic Activities versus
Resources Perspective
Inequalities as Resource Supplies and Demands
• Optimization model constraints of the  form usually
can be interpreted as restricting the supply of some
commodity or resource. [7.3]
x1  9 (Saudi)
• Optimization model constraints of the  form usually
can be interpreted as requiring satisfaction of a demand
for some commodity or resource. [7.4]
0.4 x1 + 0.2 x2  1.5 (jet fuel)
7.1 Generic Activities versus
Resources Perspective
Equality Constraints as Both Supplies and Demands
• Optimization model equality constraints usually can be
interpreted as imposing both a supply restriction and a
demand requirement on some commodity or resource.
[7.5]
Variable-Type Constraints
• Non-negativity and other sign restriction constraints are
usually best interpreted as declarations of variable type
rather than supply or demand limits on resources. [7.6]
7.1 Generic Activities versus
Resources Perspective
Variables as Activities
• Decision variables in optimization models can usually
be interpreted as choosing the level of some activity.
[7.7]
LHS Coefficients as Activity Inputs and Outputs
• Non-zero objective function and constraint coefficients
on LP decision variables display the impacts per unit of
the variable’s activity on resources or commodities
associated with the objective and constraints. [7.8]
Inputs and Outputs for Activities
Availability
1000 barrels
Cost $20000
Two Crude
Example
1000 barrels of
Saudi
petroleum
processed (x1)
.3 unit
gasoline
.4 unit
jet fuel
.2 unit
lubricants
7.2 Qualitative Sensitivity to
Changes in Model Coefficients
Relaxing versus Tightening Constraints
• Relaxing the constraints of an optimization model either
leaves the optimal value unchanged or makes it better
(higher for a maximize, lower for a minimize).
Tightening the constraints either leaves the optimal
value unchanged or makes it worse. [7.9]
Relaxing Constraints
x2
x2
8
7
6
8
7
6
x2  6
5
4
5
4
x1  9
3
3
2
2
1
1
1
2
3
4
5
6
x2  6
7
8x1
x1  9
1
2
3
4
5
6
7
8x1
Tightening Constraints
x2
x2
8
7
6
8
7
6
x2  6
5
4
5
4
x1  9
3
3
2
2
1
1
1
2
3
4
5
6
x2  6
7
8x1
x1  9
1
2
3
4
5
6
7
8x1
Swedish Steel Blending
Example
min 16 x1+10 x2 +8 x3+9 x4 +48 x5+60 x6 +53 x7
s.t.
x1+ x2 + x3+ x4 + x5+ x6 + x7 = 1000
0.0080 x1 + 0.0070 x2 + 0.0085 x3 + 0.0040 x4
0.0080 x1 + 0.0070 x2 + 0.0085 x3 + 0.0040 x4
0.180 x1 + 0.032 x2 + 1.0 x5
0.180 x1 + 0.032 x2 + 1.0 x5
0.120 x1 + 0.011 x2 + 1.0 x6
0.120 x1 + 0.011 x2 + 1.0 x6
0.001 x2 + 1.0 x7
0.001 x2 + 1.0 x7
x1  75
x2  250
x1…x7  0
(7.1)
 6.5
 7.5
 30.0
 30.5
 10.0
 12.0
 11.0
 13.0
Effect of Changes in RHS
Optimal
Value
9900
Slope -
x1  75
9800
Slope -4.98
9700
9600
9500
Slope -3.38
Current 9526.9
Slope 0.00
9400
60.4
75
83.3
RHS
Effect of Changes in RHS
Optimal
Value
9900
0.120 x1 + 0.011 x2 + 1.0 x6  10.0
9800
Slope +
9700
Slope 50.11
9600
Current 9526.9
9500
Slope 0.00
Slope 36.73
Slope 8.57
9400
9.0
10
11.7
12.0
RHS
Effect of Changes in RHS and LHS
• Changes in LP model RHS coefficients affect the
feasible space as follows: [7.10]
Constraint Type
RHS Increase
RHS Decrease
Supply ()
Relax
Tighten
Demand ()
Tighten
Relax
• Changes in LP model LHS constraint coefficients on
non-negative decision variables affect the feasible
space as follows: [7.11]
Constraint Type
Coefficient
Increase
Coefficient
Decrease
Supply ()
Tighten
Relax
Demand ()
Relax
Tighten
Effect of Adding or Dropping
Constraints
• Adding constraints to an optimization model tightens its
feasible set, and dropping constraints relaxes its
feasible set. [7.12]
• Explicitly including previous un-modeled constraints in
an optimization model must leave the optimal value
either unchanged or worsened. [7.13]
Effect of Changing Rates of
Constraint Coefficient Impact
• Coefficient changes that help the optimal value in LP by
relaxing constraints help less and less as the change
becomes large. Changes that hurt the optimal value by
tightening constraints hurt more and more. [7.14]
Effect of Changing Rates of
Constraint Coefficient Impact
Optimal
Value
Maximize objective
Supply
()
Optimal
Value
Demand
()
RHS
RHS
Effect of Changing Rates of
Constraint Coefficient Impact
Optimal
Value
Minimize objective
Supply
()
Optimal
Value
Demand
()
RHS
RHS
Effects of Objective Function
Coefficient Changes
• Changing the objective function coefficient of a nonnegative variable in an optimization model affects the
optimal value as follows: [7.15]
Model Form
(Primal)
Coefficient
Increase
Coefficient
Decrease
Maximize objective
Same or better
Same or worse
Minimize objective
Same or worse
Same or better
Changing Rates of Objective
Function Coefficient Impact
• Objective function coefficient changes that help the optimal
value in LP help more and more as the change becomes
large. Changes that hurt the optimal value less and less.
[7.16]
Maximize objective
Optimal Minimize objective
Optimal
Value
Value
coef
coef
Effect of Adding or Dropping
Variables
• Adding optimization model activities (variables) must
leave the optimal value unchanged or improved.
Dropping activities will leave the value unchanged or
degraded. [7.17]
7.3 Quantitative Sensitivity to
Changes in LP Model Coefficients
Primals and Duals Defined
• The primal is the given optimization model, the one
formulating the application of primary interest. [7.18]
• The dual is a subsidiary optimization model, defined
over the same input parameters as the primal but
characterizing the sensitivity of primal results to
changes in inputs. [7.19]
Dual Variables
• There is one dual variable for each main primal
constraint. Each reflects the rate of change in primal
value per unit increase from the given RHS value of the
corresponding constraint. [7.20]
• The LP dual variable on constraint i has type as follows:
[7.21]
Primal
i is 
i is 
i is =
Minimize objective
i  0
i  0
Unrestricted (URS)
Maximize objective
i  0
i  0
Unrestricted (URS)
Two Crude Example
min 20 x1 + 15 x2
s.t.
0.3 x1 + 0.4 x2
0.4 x1 + 0.2 x2
0.2 x1 + 0.3 x2
x1
x2
x1, x2  0
(7.4)
 2.0
 1.5
 0.5
9
6
: 1
: 2
: 3
: 4
: 5
1  0, 2  0, 3  0, 4  0, 5  0
(gasoline)
(jet fuel)
(lubricants)
(Saudi)
(Venezuelan)
(7.5)
Dual Variables as
Implicit Marginal Prices
• Dual variables provide implicit prices for the marginal
unit of the resource modeled by each constraint as its
RHS limit is encountered. [7.22]
• Variable, 1 , $1000s/1000 barrels, is the implicit price of
gasoline at the margin when demand for gasoline is at 2000
barrels.
• Variable, 4 , reflects the marginal impact of the Saudi
availability constraint at its current level of 9000 barrels.
Implicit Activity Pricing in Terms of
Resources Produced and Consumed
• The implicit marginal value (minimize problems) or price
(maximize problems) of a unit of LP activity (primal
variable) j implied by dual variable values i is 𝒊 𝒂𝒊,𝒋 𝝂𝒊
where ai,j denotes the coefficient of activity j in the LHS
of constraint i. [7.23]
• For j=2 (Venezuelan), its implicit worth is
5
𝑎𝑖,2 𝜈𝑖 = 0.4𝜈1 + 0.2𝜈2 + 0.3𝜈3 + 1𝜈5
𝑖=1
Optimal Value Equality between
Primal and Dual
• For each non-negative variable activity xj in a minimize
LP, there is a corresponding main dual constraint
𝒊 𝒂𝒊,𝒋 𝝂𝒊 ≤ 𝒄𝒋 requiring the net marginal value of the
activity not to exceed its given cost. In a maximize
problem, main dual constraints for xj0 are 𝒊 𝒂𝒊,𝒋 𝝂𝒊 ≥ 𝒄𝒋
which keeps the net marginal cost of the activity at least
equal to its given benefit. [7.24]
• For the Two Crude model,
0.3𝜈1 + 0.4𝜈2 + 0.2𝜈3 + 1𝜈4 ≤ 20
0.4𝜈1 + 0.2𝜈2 + 0.3𝜈3 + 1𝜈5 ≤ 15
(7.6)
Main Dual Constraints to Enforce
Activity Pricing
• If a primal LP has an optimal solution, its optimal value
∗
equals the corresponding optimal dual implicit
𝒋 𝒄𝒋 𝒙𝒋
total value 𝒊 𝒃𝒊 𝝂𝒊 ∗ of all constraint resources. [7.25]
• For the Two Crude model,
20𝑥1 ∗ + 15𝑥2 ∗ = 2𝜈1 ∗ + 1.5𝜈2 ∗ + 0.5𝜈3 ∗ + 9𝜈4 ∗ + 6𝜈5 ∗
Primal Complementary Slackness between
Primal Constraints and Dual Variable Values
• Either the primal optimal solution makes main inequality
constraint i active or the corresponding dual variable
𝝂𝒊 =0. [7.26]
• For the primal optimum (2, 3.5)
0.3 (2) + 0.4 (3.5)
= 2.0
0.4 (2) + 0.2 (3.5)
= 1.5
0.2 (2) + 0.3 (3.5) = 1.45
> 0.5
(2)
=2
<9
(3.5) = 3.5
<6
3 = 0, 4 = 0, 5 = 0
(active)
(active)
(inactive)
(inactive)
(inactive)
Dual Complementary Slackness between
Dual Constraints and Primal Variable Values
• Either a non-negative primal variable has optimal value
xj=0 or the corresponding dual price 𝝂𝒊 must make the
jth dual constraint 𝒊 𝒂𝒊,𝒋 𝝂𝒊 ≤ 𝒄𝒋 (minimize) or
𝒊 𝒂𝒊,𝒋 𝝂𝒊 ≥ 𝒄𝒋 (maximize) active. [7.27]
• For the primal optimum (2, 3.5)
0.3𝜈1 + 0.4𝜈2 + 0.2𝜈3 + 1𝜈4 ≤ 20
0.4𝜈1 + 0.2𝜈2 + 0.3𝜈3 + 1𝜈5 ≤ 15
are both active.
7.4 Formulating LP Duals
Form of the Dual for Non-negative
Primal Variables
• The dual of a minimize primal over xj0 is [7.28]
𝑀𝑎𝑥 𝑖 𝑏𝑖 𝜈𝑖
s.t.
for all primal activities j
𝑖 𝑎𝑖,𝑗 𝜈𝑖 ≤ 𝑐𝑗
𝜈𝑖 ≥ 0
𝜈𝑖 ≤ 0
𝜈𝑖 URS
for all primal ’s i
for all primal ’s i
for all primal =’s i
7.4 Formulating LP Duals
Form of the Dual for Non-negative
Primal Variables
• The dual of a maximize primal over xj0 is [7.29]
𝑀𝑖𝑛 𝑖 𝑏𝑖 𝜈𝑖
s.t.
for all primal activities j
𝑖 𝑎𝑖,𝑗 𝜈𝑖 ≥ 𝑐𝑗
𝜈𝑖 ≥ 0
𝜈𝑖 ≤ 0
𝜈𝑖 URS
for all primal ’s i
for all primal ’s i
for all primal =’s i
Two Crude Example
Primal:
min 20 x1 + 15 x2
s.t.
0.3 x1 + 0.4 x2
0.4 x1 + 0.2 x2
0.2 x1 + 0.3 x2
x1
x2
x1, x2  0
Dual:
Max 2𝜈1 + 1.5𝜈2 + 0.5𝜈3 + 9𝜈4 + 9𝜈4
s.t.
 2.0
0.3𝜈1 + 0.4𝜈2 + 0.2𝜈3 + 1𝜈4 ≤ 20
 1.5
0.4𝜈1 + 0.2𝜈2 + 0.3𝜈3 + 1𝜈5 ≤ 15
 0.5
𝜈1 , 𝜈2 , 𝜈3 ≥ 0
9
𝜈4 , 𝜈5 ≤ 0
6
(7.7)
Duals of LP Models with Non-positive
and Unrestricted Variables
Max
Form
Primal Element
Obj.
𝑀𝑎𝑥
Constraint
𝑗 𝑐𝑗 𝑥𝑗
Corresponding
Dual Element
𝑀𝑖𝑛 𝑖 𝑏𝑖 𝜈𝑖
𝑗 𝑎𝑖,𝑗 𝑥𝑗
≥ 𝑏𝑖
𝜈𝑖 ≤ 0
𝑗 𝑎𝑖,𝑗 𝑥𝑗
= 𝑏𝑖
𝜈𝑖 URS
𝑗 𝑎𝑖,𝑗 𝑥𝑗
≤ 𝑏𝑖
𝜈𝑖 ≥ 0
𝑥𝑗 ≥ 0
𝑖 𝑎𝑖,𝑗 𝜈𝑖
≥ 𝑐𝑗
𝑥𝑗 URS
𝑖 𝑎𝑖,𝑗 𝜈𝑖
= 𝑐𝑗
𝑥𝑗 ≤ 0
𝑖 𝑎𝑖,𝑗 𝜈𝑖
≤ 𝑐𝑗
Duals of LP Models with Non-positive
and Unrestricted Variables
Min
Form
Primal Element
Obj.
𝑀𝑖𝑛
Constraint
𝑗 𝑐𝑗 𝑥𝑗
Corresponding
Dual Element
𝑀𝑎𝑥 𝑖 𝑏𝑖 𝜈𝑖
𝑗 𝑎𝑖,𝑗 𝑥𝑗
≥ 𝑏𝑖
𝜈𝑖 ≥ 0
𝑗 𝑎𝑖,𝑗 𝑥𝑗
= 𝑏𝑖
𝜈𝑖 URS
𝑗 𝑎𝑖,𝑗 𝑥𝑗
≤ 𝑏𝑖
𝜈𝑖 ≤ 0
𝑥𝑗 ≥ 0
𝑖 𝑎𝑖,𝑗 𝜈𝑖
≤ 𝑐𝑗
𝑥𝑗 URS
𝑖 𝑎𝑖,𝑗 𝜈𝑖
= 𝑐𝑗
𝑥𝑗 ≤ 0
𝑖 𝑎𝑖,𝑗 𝜈𝑖
≥ 𝑐𝑗
Dual of the Dual Is the Primal
• The dual of the dual of any linear program is the LP
itself. [7.30]
7.5 Primal-to-Dual Relationships
Weak Duality between Objective Values
𝑐𝑗 𝑥𝑗 −
𝑗
=
𝑏𝑖 𝜈𝑖
𝑖
𝑗 (𝑐𝑗 −
𝑖 𝑣𝑖
𝑎𝑖,𝑗 )𝑥𝑗 +
𝑖( 𝑗 𝑎𝑖,𝑗 𝑥𝑗
− 𝑏𝑖 )𝜈𝑖
(7.8)
Primal objective function – Dual objective function
= (slack in dual constraints)(primal variables) +
(slack in primal constraints)(dual variables)
= (non-positive) + (non-positive) for max OR
(non-negative) + (non-negative) for min
Weak Duality between Objective Values
• The primal objective function evaluated at any feasible
solution to a minimize primal is greater than or equal to
() the objective function value of the corresponding
dual evaluated at any dual feasible solution. For a
maximize primal it is (). [7.31]
Strong Duality
between Objective Values
• If either a primal LP or its dual has an optimal solution,
both do, and their optimal objective function values are
equal. [7.32]
Dual Optimum as a By-product
• 𝕧𝔹 ≡ (𝑐1𝑠𝑡 , 𝑐2𝑛𝑑 , ⋯ , 𝑐𝑚𝑡ℎ )
(7.10)
– 𝕧 is a pricing vector
– 𝔹 is the current basis column matrix of a primal LP
– 𝑐1𝑠𝑡 , 𝑐2𝑛𝑑 , ⋯ , 𝑐𝑚𝑡ℎ are the objective function
coefficients of the 1st, 2nd,…, mth basic variables
• Optimality has been reached (in min) if
(7.11)
𝑐𝑗 = (𝑐𝑗 − 𝑖 𝑣𝑖 𝑎𝑖,𝑗 ) ≥ 0 for all variables j
• If the revised simplex algorithm stops with a
primal optimal solution, the final pricing vector
𝕧 yields an optimal solution in the corresponding
dual. [7.33]
Unbounded and Infeasible Cases
• If either a primal LP model or its dual is unbounded, the
other is infeasible. [7.34]
• The following shows which outcome pars are possible
for a primal LP and its dual: [7.35]
Primal
Dual
Optimal
Infeasible
Unbounded
Possible
Never
Never
Infeasible
Never
Possible
Possible
Unbounded
Never
Possible
Never
Optimal
7.6 Computer Outputs and What If
Changes of Single Parameters
• RHS ranges in LP sensitivity outputs show the interval
within which the corresponding dual variable value
provides the exact rate of change in optimal value per
unit change in RHS (all other data held constant) [7.36]
7.6 Computer Outputs and What If
Changes of Single Parameters
• Dropping a constraint can change the optimal solution
only if the constraint is active at optimality. [7.39]
• Adding a constraint can change the optimal solution
only if that optimum violates the constraint. [7.40]
• An LP variable can be dropped without changing the optimal
solution only if its optimal value is zero. [7.41]
• A new LP variable can change the current primal optimal
solution only if its dual constraint is violated by the current
dual optimum. [7.42]
7.7 Bigger Model Changes, Reoptimization, and Parametric Programing
Ambiguity at Limits of the RHS and Objective Coefficient
Ranges
• At the limits of the RHS and objective function
sensitivity ranges, rates of optimal value change are
ambiguous, with one value applying below the limit and
another above. Computer outputs may show either
value. [7.43]
Two Crude Example
x2
8
B1=2.0
7
6
Venezuelan
Saudi
5
4
Optimal Lower
Dual
Range
Upper
Range
20.000
2.625
1.125
3
2
1
1
2
3
4
5
6
7
8
9
10
x1
Two Crude Example
x2
8
B1=2.625
7
Optimal Lower
Dual
Range
Upper
Range
6
20.000
2.625
1.125
Venezuelan
Saudi
5
4
3
Optimal Lower
Dual
Range
Upper
Range
66.667
5.100
2.625
2
1
1
2
3
4
5
6
7
8
9
10
x1
Two Crude Example
x2
8
B1=3.25
7
6
Venezuelan
Saudi
5
4
Optimal Lower
Dual
Range
Upper
Range
66.667
5.100
2.625
3
2
1
1
2
3
4
5
6
7
8
9
10
x1
Connection between
Rate Changes and Degeneracy
• Rates of variation in optimal value with model constants
change when the collection of active primal or dual
constraints changes. [7.44]
• Degeneracy, which is extremely common in large-scale
LP models, limits the usefulness of sensitivity byproducts from primal optimization because it leads to
narrow RHS and objective coefficient ranges and
ambiguity at the range limits. [7.45]
Re-Optimization to Make Sensitivity
Exact
• If the number of “what-if” variations does not grow too
big, re-optimization using different values of model input
parameters often provides the most practical avenue to
good sensitivity analysis. [7.46]
Parametric Variation of
One Coefficient
• Parametric studies track the optimal value as a function
of model inputs.
• If the number of “what-if” variations does not grow too
big, re-optimization using different values of model input
parameters often provides the most practical avenue to
good sensitivity analysis. [7.46]
Parametric Variation of
One Coefficient
• Parametric studies track the optimal value as a function
of model inputs.
• Parametric studies of optimal value as a function of a
single-model RHS or objective function coefficient can
be constructed by repeated optimization using new
coefficient values just outside the previous applicable
sensitivity range. [7.47]
Parametric Variation of One Coefficient:
Two Crude Example
Optimal
Value
Slope +
(infeasible)
250
Case
150
Slope 66.67
Slope 20.00
92.5
50
Dual
Lower
Rang
Upper
Range
2.000
20.000
1.125
2.625
Variant 1
2.625+
66.667
2.626
5.100
Variant 2
5.100+
+
5.100
+
Variant 3
1.125-
0.000
-
1.125
Base
Model
RHS
Slope 0.00
1.1
2.0
2.6
5.1
RHS
Assessing Effects of Multiple
Parameter Changes
• Elementary LP sensitivity rates of change and ranges
hold only for a single coefficient change, with all other
data held constant. [7.48]
• If demand increase in jet fuel (b2) is twice as the
increase for gasoline (b1).
b1new = (1+) b1base
(7.14)
b2new = (1+2) b2base
binew = bibase + bi
b1=2.0
b2=3.0
Assessing Effects of Multiple
Parameter Changes
The effect of a multiple
min 20 x1 + 15 x2
change in RHS with step  s.t.
0.3 x1 + 0.4 x2 - 2  2.0
can be analyzed
0.4 x1 + 0.2 x2 - 3 
 1.5
parametrically by treating  as
0.2 x1 + 0.3 x2
 0.5
a new decision variable with
x1
9
constraint coefficient -bi that
x2
6
detail the rates of change in

= b6
RHS’s bi and a value fixed by
x1, x2  0,  URS
a new equality constrain.
[7.49]
Parametric Variation of One Coefficient:
Two Crude Example
Optimal
Value
Slope +
(infeasible)
250
Slope 225.00
150
Slope 145.00
92.5
50
0.0
0.9
1.1

Parametric Change of Multiple
Objective Function Coefficients
• The effect of multiple change in objective function with
step  can be analyzed parametrically by treating
objective rates of change -ci as coefficients in a
new equality constraint having RHS zero and a new
unrestricted variable with objective coefficient .
[7.50]