11-1
CHAPTER 11: DYNAMIC PROGRAMMING
11.2-1.
(a) The nodes of the network can be divided into "layers" such that the nodes in the
th
8
layer are accessible from the origin only through the nodes in the
st layer. These
Ð8 "Ñ
layers define the stages of the problem, which can be labeled as
. The nodes
8 œ "ß #ß $ß %
constitute the states.
Let
denote the set of the nodes in the
th layer of the network, i.e.,
,
W
8
W
œ ÖS× W
œ
8
"
#
ÖEß Fß G×
W
œ ÖHß I×
W
œ ÖX×
B
,
and
. The decision variable
is the immediate
$
%
8
destination at stage
. Then the problem can be formulated as follows:
8
0 Ð=Ñ œ
Ò-
0
ÐB ÑÓ ´
0 Ð=ß B Ñ
= − W
8 œ "ß #ß $
‡
‡
8
8"
B −W
B −W
=B
8
8
8
8
for
and
min
min
8
8"
8
8"
8
0 ÐXÑ œ !
‡
%
(b) The shortest path is
.
S F H X
(c) Number of stages: 3
6
=
0 Ð=Ñ
B
H
X
I
(
X
$
‡
‡
$
$
=
0 Ð=ß HÑ
0 Ð=ß IÑ
0 Ð=Ñ
B
E
""
""
H
F
"$
"&
"$
H
G
"$
"$
I
#
#
#
‡
‡
#
#
=
0 Ð=ß EÑ
0 Ð=ß FÑ
0 Ð=ß GÑ
0 Ð=Ñ
B
S
#!
"*
#!
"*
F
"
"
"
"
‡
‡
"
"
Optimal Solution:
,
and
.
B
œ F B
œ H
B
œ H
‡
‡
‡
"
#
$

11-2
(d) Shortest-Path Algorithm:
Solved nodes
Closest
th
Distance to
directly connected
connected
total
nearest
th nearest
Last
to unsolved nodes
8
8
8
unsolved node
distance
node
node
connection
1
S
F
'
F
'
SF
#
S
G
(
G
(
SG
F
H
' ( œ "$
$
S
E
*
E
*
SE
F
H
' ( œ "$
G
I
( ' œ "$
%
E
H
* & œ "%
H
"$
FH
F
H
' ( œ "$
G
I
( ' œ "$
I
GI
&
H
X
"$ ' œ "*
X
"*
HX
I
X
"$ ( œ #!
The shortest-path algorithm required
additions and
comparisons whereas dynamic
)
'
programming required
additions and
comparisons. Hence, the latter seems to be more
(
$
efficient for shortest-path problems with "layered" network graphs.
11.2-2.
(a)
The optimal routes are
and
, the associated sales
S E J X
S G L X
income is
40. The route
corresponds to assigning
,
, and
"
S E J X
"
#
$
salespeople to regions , , and
respectively. The route
corresponds
" #
$
S G L X
to assigning , , and
salespeople to regions , , and
respectively.
$ #
"
" #
$