**Unformatted text preview: **Chapter 4
Duality Theory and Sensitivity Analysis
Also see Chapter 6 of the text book. One of the most important discoveries in the early development of linear programming
was the concept of duality and its many important ramifications. This discovery revealed
that every linear programming problem has associated with it another linear programming
problem called the dual. The relationships between the dual problem and the original
problem (called the primal) prove to be extremely useful in a variety of ways. For example,
you soon will see that the shadow prices described in Chapter 3 actually are provided by the
optimal solution for the dual problem. We shall describe many other valuable applications
of duality theory in this chapter as well.
One of the key uses of duality theory lies in the interpretation and implementation of
sensitivity analysis. As we already mentioned many times, sensitivity analysis is a very
important part of almost every linear programming study. Because most of the parameter
values used in the original model are just estimates of future conditions, the effect on the optimal solution if other conditions prevail instead needs to be investigated. Furthermore,
certain parameter values (such as resource amounts) may represent managerial decisions,
in which case the choice of the parameter values may be the main issue to be studied, which
can be done through sensitivity analysis.
For greater clarity, the first three sections discuss duality theory under the assumption
that the primal linear programming problem is in our standard form (but with no restriction
that the bi values need to be positive). Other forms are then discussed in Sec. 4. We
begin the chapter by introducing the essence of duality theory and its applications. We
then describe the economic interpretation of the dual problem and delve deeper into the
relationships between the primal and dual problems (Sec. 3). Section 6.5 focuses on the role
of duality theory in sensitivity analysis. The basic procedure for sensitivity analysis (which
is based on the fundamental insight of Sec. 3 in last chapter) is summarized in Sec. 6 and
illustrated in Sec. 7. 1 CHAPTER
6 6DUALITY
ANALYSIS
CHAPTER
DUALITYTHEORY
THEORYAND
AND SENSITIVITY
SENSITIVITY ANALYSIS 1 THE ESSENCE OF DUALITY THEORY ESSENCE
DUALITYTHEORY
THEORY
ETHE
ESSENCE
OFOF
DUALITY Given our standard form for the primal problem at the left (perhaps after conversion from
Given
our
standard
form
thehas
primal
problem
at
conversion
from
another
form),
its dual
problem
the form
shown
to left
the
right.
Given
our
standard
form
forforthe
primal
problem
at the
the
left(perhaps
(perhapsafter
after
conversion
from
another form), its dual problem has the form shown to the right. another form), its dual problem has the form shown to the right.
Primal Problem Dual Problem Primal Problem Dual Problem
m n Maximize Maximize Z n c j x j, Z j1c j x j,
j1 subject to aij x j bi, j1 aij x j
j1 and bi, and xj 0, i1
W
bi yi, Minimize i1 subject to subject
to
m subject
to
n
n W mbi yi, Minimize for i 1, 2, . . . , m for i 1, 2, . . . , m m aij yi cj, aij yi cj, i1 for j 1, 2, . . . , n for j 1, 2, . . . , n i1
and xj 0, for j 1, 2, . . . , n. and yi 0, for i 1, 2, . . . , m. yi 0, for j 1, 2, . . . , n. for i 1, 2, . . . , m. Thus, with the primal problem in maximization form, the dual problem is in minimization
Thus,
with the
primal problem
in maximization
form, the
is in as
minimization
form instead.
Furthermore,
the dual
problem uses exactly
thedual
sameproblem
parameters
the priThus,
with
the
primal
problem
in
maximization
form,
the
dual
problem
is
in
minimization
formmal
instead.
Furthermore,
the
dual
problem
uses
exactly
the
same
parameters
as
the primal
problem, but in different locations, as summarized below.
form
instead.
Furthermore,
the
dual
problem
uses
exactly
the
same
parameters
as
the priproblem, but in different locations, as summarized below.
1.
The
coefficients
in
the
objective
function
of
the
primal
problem
are
the
right-hand
sides
mal problem, but in different locations, as summarized below.
• The
coefficients
the objective
function
of the primal problem are the right-hand sides
of the
functionalinconstraints
in the
dual problem.
1. The
coefficients
in
the
objective
function
of
the primal
problem
are the right-hand
sides
of
the
functional
constraints
in
the
dual
problem.
2. The right-hand sides of the functional constraints
in the
primal problem
are the coefof the
functional
constraints
in theofdual
problem.
ficients
in the objective
function
the dual
problem.
•
The
right-hand
sides
of
the
functional
constraints
theprimal
primal
problem
arethe
the
coef2. The
right-hand
sides
the functional
constraints
ininthe
are
coef3. The
coefficients
of of
a variable
in the functional
constraints
of
the problem
primal
problem
are
ficients
in the objective
function of the dual
problem.
the coefficients
in a functional
of the
dual problem.
ficients
in the objective
functionconstraint
of the dual
problem.
3. The
coefficients
ofofa avariable
the
functional
ofofthe
primal
problem
•ToThe
coefficients
variable
theat
functional
constraints
the
primal
problem
are
highlight
the comparison,
nowinin
look
these sameconstraints
two problems
in
matrix
notation
(as are
the
coefficients
in
a
functional
constraint
of
the
dual
problem.
the
coefficients
in
a
functional
constraint
of
the
dual
problem.
introduced at the beginning of Sec. 5.2), where c and y [y1, y2, . . . , ym] are row vectorshighlight
but b the
andthe
x are
column now
vectors.
To highlight
comparison,
looklook
at these
samesame
two two
problems
in matrix
notation
(as
To
comparison,
now
at these
problems
in matrix
notation
(as introduced
at the
beginning
of Sec.
2 inwhere
Chapter
3), ywhere
, · · row
· , ymvec] are
introduced
at the
beginning
of Sec.
5.2),
c and
[y1c, and
y2, .y. =
. , [yy1m,]y2are
Primal
Problem
Dual
Problem
row vectors
butx bare
and
x are column
tors
but b and
column
vectors.vectors.
Z cx, Maximize PrimaltoProblem
subject
Ax b Maximize Z cx, Dual
subject
to Problem
yA c Minimize x 0. W yb, subject to y 0. yA c Ax b
and W yb, and and subject to Minimize and To illustrate, the primal and dual problems for the Wyndor Glass Co. example of Sec. 3.1
y 0.
x 0.
are shown in Table 6.1 in both algebraic and matrix form.
The primal-dual table for linear programming (Table 6.2) also helps to highlight
the correspondence between the two problems. It shows all the linear programming paTo illustrate, the primal and dual problems for the Wyndor Glass Co. example of Sec. 3.1
rameters (the aij, bi, and cj) and how they are2 used to construct the two problems. All the
are headings
shown infor
Table
6.1 in problem
both algebraic
and matrix
form.
the primal
are horizontal,
whereas
the headings for the dual probTheare
primal-dual
tablethefor
linear
programming
(Tableproblem,
6.2) alsoeach
helps
to highlight
lem
read by turning
book
sideways.
For the primal
column
(exthe cept
correspondence
between
the
two
problems.
It
shows
all
the
linear
programming
the Right Side column) gives the coefficients of a single variable in the respective pa- rameters (the a , b , and c ) and how they are used to construct the two problems. All the 76299_ch06_195-275.qxd 11/19/08 09:50 AM Rev.Confirming Pages Page 197 6.1 primal
THE ESSENCE
OF DUALITY
THEORY
To illustrate, the
and dual
problems
for the Wyndor Glass Co. example are197
shown
in Fig. 4.1 in both algebraic and matrix form.
■ TABLE 6.1 Primal and dual problems for the Wyndor Glass Co. example
Primal Problem
in Algebraic Form Dual Problem
in Algebraic Form Z 3x1 5x2, Maximize
subject to W 4y1 12y2 18y3, Minimize
subject to 3x1 2x2 4 y12y2 3y3 3 3x1 2x2 12 2y2 2y3 5 3x1 2x2 18
and x1 0, and x2 0. y1 0, Primal Problem
in Matrix Form
x1 x ,
2 Minimize subject to 1
0 3 0
x1
2 x2
2 y3 0. Dual Problem
in Matrix Form Z [3, 5] Maximize y2 0, 4
W [y1, y2, y3] 12 18 subject to 4 12 18 [y1, y2, y3] and 1
0 3 0
2 [3, 5] 2 and x1
0
.
x2
0 [y1, y2, y3] [0, 0, 0]. ■ TABLE 6.2 Primal-dual table for linear programming, illustrated by the Wyndor Figure 4.1: Primal and
dual
problems for the Wyndor Glass Co. example
Glass
Co. example
(a) General Case y1
y2
y3 Coefficients
for
Objective
Function
(Minimize) Right
Side Dual Problem Coefficient
of: The primal-dual table for linear programming (Fig. 4.2) also helps to highlight the correPrimal Problem
spondence between the two problems. It shows all the linear programming parameters (the
Coefficient
of: problems. All the headings for
aij , bi , and cj ) and how they are used to construct
the two
Right
… for the
x1 the x
xn dual
Side
the primal problem are horizontal, whereas
headings
problem are read by
2
turning the note sideways. For they1primal
problem,
each
column
(except
the Right Side
…
a11
a12
a1n
b1
…
y2
a21
a22in the respective
a2n
b2
column) gives the coefficients of a single
variable
constraints
and then in
the objective function, whereas eachy row (except
the bottom
one)
gives
the parameters for a
…
am1
am2
amn
bm
m
single constraint. For the dual problem, each row (except the Right Side row) gives the co…
VI
VI
VI
efficients of a single variable in the respective
in the objective function,
… and then
c1 constraints
c2
cn
whereas each column (except the rightmost one)Coefficients
gives theforparameters for a single constraint.
In addition, the Right Side column gives the Objective
righthand
sides for the primal problem and
Function
(Maximize)
the objective function coefficients for the dual problem, whereas the bottom row gives the
Wyndor Glass for
Co. Example
objective function(b)coefficients
the primal problem and the right-hand sides for the dual
problem.
x1
x2
1
0
3 0
2
2 VI
3 VI
5 4
12
18 3 3 2 AND SENSITIVITY ANALYSIS
CHAPTER 6 DUALITY THEORY and
x1 and 0 x 0. [y , y , y ] [0, 0, 0]. 2
3
the parameters for a 1single
constraint. In addition, the Right Side column gives the rig
hand sides for the primal problem and the objective function coefficients for the d
■ TABLE 6.2 Primal-dual
for linear
illustrated
by the Wyndor
problem,table
whereas
the programming,
bottom row gives
the objective
function coefficients for the prim
Glass
Co. example
problem
and the right-hand sides for the dual problem.
(a) General Case
Consequently, we now have the following general relationships between the prim
and dual problems. Primal Problem Coefficient
of:
1. The parameters
for a (functional)
constraint in either problem are the coefficients o
Right
variablex1in thexother
problem.
…
xn
Side
2
2. The coefficients in the…objective function of either problem are the right-hand sides
y1
a11
a12
a1n
b1
the
…
y2 other
a21problem.
a22
a2n
b2 Coefficients
for
Objective
Function
(Minimize) Coefficient
of: 2 …
Thus,
is a direct
between
these entities in the two problems, as su
ym theream1
am2 correspondence
amn
bm
marized in VITable 6.3.
These
are a key to some of the applications of
… correspondences
VI
VI
…
ality theory,c1 including
sensitivity
c2
cn analysis.
The WorkedCoefficients
Examples
for section of the book’s website provides another example
Objectivetable
Function
using the primal-dual
to construct the dual problem for a linear programming mod
Right
Side Dual Problem 198 (Maximize) (b) Wyndor Glass Co.
Example of
Origin
x1 x2 1
0
3 0
2
2 the Dual Problem Duality theory is based directly on the fundamental insight (particularly with regard
presented
4
row 0)
in Sec. 5.3. To see why, we continue to use the notation introduced
12
Table5.9
for
row
0 of the final tableau, except for replacing Z* by W* and dropping
18
asterisks from z* and y* when referring to any tableau. Thus, at any given iteration of
VI
VI
3
5 simplex method for the primal problem, the current numbers in row 0 are denoted
shown in the (partial) tableau given in Table 6.4. For the coefficients of x1, x2, . . . ,
recall that z (z1, z2, . . . , zn) denotes the vector that the simplex method added to
Figure 4.2: Primal-dual table
programming,
by the
Co. tableau. (Do
vectorfor
of linear
initial coefficients,
in the process
of Wyndor
reaching Glass
the current
c,illustrated
example
confuse z with the value of the objective function Z.) Similarly, since the initial coe
cients of xn1, xn2, . . . , xnm in row 0 all are 0, y (y1, y2, . . . , ym) denotes the v
Consequently, we now
the simplex
following
general
the primal
and [see Eq. (1) in
torhave
that the
method
hasrelationships
added to thesebetween
coefficients.
Also recall
dual problems.
statement of the fundamental insight in Sec. 5.3] that the fundamental insight led to
following relationships between these quantities and the parameters of the original mod
• The parameters for a (functional) constraint in either problem are the coefficients of a
m
variable in the other problem.
W yb bi yi ,
i1
• The coefficients in the objective function
of either problem are the right-hand sides for
m
the other problem.
for j 1, 2, . . . , n.
z yA,
so
zj aij yi ,
i1
Thus, there is a direct correspondence between these entities in the two problems, as
summarized in Fig. 4.3.■ These
are a between
key to some of the applications of
TABLEcorrespondences
6.3 Correspondence
entities in primal and
duality theory, including sensitivity analysis.
y1
y2
y3 dual problems
One Problem Other Problem Constraint i ←→ Variable i
Objective function ←→ Right-hand sides ■ TABLE 6.4between
Notation
for entries
in row
of a simplex
Figure 4.3: Correspondence
entities
in primal
and0 dual
problemstableau
Coefficient of:
Iteration Basic
Variable Eq. Z x1 x2 … xn xn1 xn2 … xnm Any Z (0) 1 z1 c1 z2 c2 … zn cn y1 y2 … ym 4 Rig
Si W variable in the other problem.
Thus, there
is aobjective
direct correspondence
between
these
in the twosides
problems,
2. The coefficients
in the
function of either
problem
areentities
the right-hand
for as summarized
in
Table
6.3.
These
correspondences
are
a
key
to
some
of
the
applications
of duthe other problem.
ality theory, including sensitivity analysis.
Thus, there is a direct
correspondence
the twoprovides
problems,another
as sum-example of
The Worked
Examplesbetween
section these
of theentities
book’sinwebsite
marized in Table
6.3.
These
correspondences
are
a
key
to
some
of
the
applications
of duusing the primal-dual table to construct the dual problem for a linear programming
model.
ality theory, including sensitivity analysis.
Origin of the Dual Problem
The Worked
Examples
of the book’s website provides another example of
Origin
of the section
Dual Problem
using thetheory
primal-dual
tabledirectly
to construct
thefundamental
dual probleminsight
for a linear
programming
Duality
is based
on the
(particularly
withmodel.
regard to
Duality theory is based directly on the fundamental insight (particularly with regard to
row 0) presented in Sec. 3 of Chapter 3. To see why, we continue to use the notation introrow
presented
in Sec. 5.3. To see why, we continue to ∗use the ∗notation introduced in
Origin
the0)
Dual
Problem
duced
in Fig.of3.11
for
row
0
of
the
except
for replacing
Z by
dropping
Table 5.9
for row
0 offinal
the tableau,
final tableau,
except
for replacing
Z*Wby and
W* and
dropping the
∗
∗
the Duality
asterisks
from
z
and
y
when
referring
to
any
tableau.
Thus,
at
any
given
iteration
theory
is
based
directly
on
the
fundamental
insight
(particularly
with
regard
to
asterisks from z* and y* when referring to any tableau. Thus, at any given iteration of the
of the
method
for the
primal
problem,
the current
numbers
in row
0 are0 denoted
rowsimplex
0) presented
in method
Sec.
5.3.for
Tothe
seeprimal
why,
we
continue
use
thenumbers
notation
introduced
in denoted as
simplex
problem,
thetocurrent
in
row
are
as shown
in
the
(partial)
tableau
given
in
Fig.
4.4.
For
the
coefficients
of
x
,
x
,
·
·
· , xn ,
Table 5.9 for
row 0inofthe
the(partial)
final tableau,
replacing
and dropping
1
shown
tableauexcept
given for
in Table
6.4. Z*
Forby
theW*
coefficients
of2 xthe
1, x2, . . . , xn,
recall
that zfrom
=
(z
z2 , ·y*
z(z
vector
thatThus,
the simplex
method
added
to the
asterisks
z*1 ,and
referringthe
to any
tableau.
at any given
iteration
of the
n ) , denotes
recall
that
z· ·,when
1 z2, . . . , zn) denotes the vector that the simplex method added to the
vector
of initial
coefficients,
−c,
in
the process
of reaching
tableau.
(Do
simplex
method
for
primal
problem,
numbers
incurrent
row 0the
are
denoted
as not(Do not
vector
of the
initial
coefficients,
c, incurrent
the
process
ofthe
reaching
current
tableau.
the
confuse
z
with
the
value
of
the
objective
function
Z.)
Similarly,
since
the
initial
coefficients
shown in the
(partial)
tableau
6.4. For function
the coefficients
of x1, xsince
. , xinitial
2, . . the
n,
confuse
z with
the given
value in
of Table
the objective
Z.) Similarly,
coeffiof xrecall
·
· ,(z
xn+m
in
row
0
all
are
0,
y
=
(y
,
y
,
·
·
·
,
y
)
denotes
the
vector
that
thethe vec,
z
,
.
.
.
,
z
)
denotes
the
vector
that
the
simplex
method
added
to
the
n+1 , xthat
n+2 , ·zcients
1
2
m
1of 2x
n, . . . , x
,
x
in
row
0
all
are
0,
y
(y
,
y
,
.
.
.
,
y
)
denotes
n1
n2
nm
1
2
m
simplex
added
to these
recall
[the
seecurrent
Eq. Also
(1)
in
the(Do
statement
vectormethod
of initial
coefficients,
c, incoefficients.
thehas
process
oftoreaching
tableau.
not
method
torhas
that
the simplex
addedAlso
these
coefficients.
recall
[see
Eq. (1) in the
of the
fundamental
insight
in
Sec.
3
of
Chapter
3]
that
the
fundamental
insight
led
to the
confuse
z with
the
value
of
the
objective
function
Z.)
Similarly,
since
the
initial
coeffistatement of the fundamental insight in Sec. 5.3] that the fundamental insight
led to the
following
between
these
quantities
and
the
parameters
of
the
original
model:
cients relationships
of xn1
,
x
,
.
.
.
,
x
in
row
0
all
are
0,
y
(y
,
y
,
.
.
.
,
y
)
denotes
the
vecn2
nm
1
2
m
following relationships between these quantities and the parameters of the original model:
tor that the simplex method has added to these coefficients. Also recall [see Eq. (1) in the
m
statement of theWfundamental
yb insight
bi yi , in Sec. 5.3] that the fundamental insight led to the
following relationships between
i1 these quantities and the parameters of the original model:
m m z byA,
W yb
i yi , zj aij yi , so for j 1, 2, . . . , n. i1 i1 m
■ TABLE 6.3 Correspondence
between awith
yi ,primal
j 1, example,
2, . . . , n.the first equation gives W =
z yA, these
so relationships
zj entities
ijin
and
To illustrate
the for
Wyndor
i1
4y1 +12y2 +18y3 , which is justdual
the problems
objective function for the dual problem shown in the upper
■
TABLE
6.3
Correspondence
between
right-hand boxOne
of Fig.
4.1.
The
second
set of equations give z1 = y1 + 3y3 and z2 = 2y2 + 2y3 ,
Problem
entities
in primal andOther Problem
which are the left-hand sides of the functional constraints for this dual problem. Thus, by
dual
problems
Constraint
i ←→
Variable
i
subtracting the right-hand
sides
of these
≥ constraints
(c1 = 3 and c2 = 5), (z1 − c1 ) and
Objective function ←→ Right-hand sides
One
Problem
(z2 −
c2 )Problem
can be interpreted Other
as being
the surplus variables for these functional constraints.
The remaining key is to express what the simplex method tries to accomplish (according
Constraint i ←→ Variable i
to the
optimality
test)
in6.4
terms
of thesefor
symbols.
Specifically, it seeks a set of basic variables,
Objective
function
←→
Right-hand
■ TABLE
Notation sides
entries in row 0 of a simplex tableau
and the corresponding BF solution, such that all coefficients in row 0 are nonnegative.
Coefficient
of:
It then stops with this optimal solution. Using the notation
in Fig. 4.4:
Basic
Right
■ TABLE 6.4
Notation
for entries
tableau
…
…
Iteration
Variable
Eq. in
Z rowx10 of a xsimplex
x
x
x
x
Side
2
n
n1
n2
nm
Z (0) 1 of: z c
z1 c1Coefficient
z2 c2 …...

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