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Home Work Set 3

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Home Work Set 3
Home Work Set 3
MSIS 685, Linear Programming
Fall 1998
Due : December 21st
Questions with * are optional and are given so that students who wish to improve their grade have an
opportunity to do so.
Use the simplex method to solve the following transportation problem (red numbers
Question 1
are costs, rows are demand nodes and columns are supply nodes.)
Question 2
a
b
3
9
6
11
15
10
4
7
6
22
8
6
17
3
10
25
15
2
19
a) Use the simplex method to solve the min-cost-flow problem for the network
below. The numbers next to nodes are their demand (supply if negative). Those
nodes without numbers are transshipment nodes
b) Ignore the demand and supply and find the all shortest paths (i.e. all minimum
cost paths) from node a to all other nodes, using Dijkstra's algorithm.
8
b
17
-20
f
7
Question 3
1
5
e
10
4
(10, 7)
10,
7)
5
g
5
6
a
12
15
10
c
d
20
h
1
15
For the network shown below find the maximum flow from source nodes s to sink
node t.
1
Question 3
b
f
9
g
15
6
3
9
3
6
3
10
s
t
e
9
3
15
9
9
c
17
d
2
h
Question 4
Do problem 7.1 on page 111 of Vanderbei's text.
Question 5
Do problem 7.2 on page 111 of Vanderbei's text.
Question 6*
There is a general strong connection between the optimization theory and free
competition. Here is an example: Suppose there are n economic activities (for
example building factories, homes, shopping malls, etc.) that are to be individually
located on n distinct parcels of land. If activity i is located on parcel j that activity
yields sij units (e.g. dollars) value.
If the assignment of activities to the land parcels are is made by a central authority, it
might be made in such a way as to maximize the total value generated. In other words,
the assignment would be made so as to maximize ijsijxij where
 1 if activity i is assigned to parcel j
xij  
otherwise.
0
More explicityly this activitiy leads to the optimization problem
i j sijxij
ixij =1 for all nodes
jxij =1 for all nodes j
0xij xij=0, or 1
Max
s.t.
Actually, it can be shown that the final requirement xij=0 or 1 will be satisfied
automatically.
If one considers the problem from the view point of free competition, it is assumed
that, rather than a central authority determining the assignment, the individual
activities bid for the land and thereby establish prices.
2
Question 6*
a) Show that there exists a set of activity proces pi for i=1,...,n and land prices qj,
j=1,...,n such that for all i,j: pi+qj sij with equality holding if in the optimium
activity i is assigned to parcel j.
b) Show that part a) implies that if activity i is optimially assigned to parcel j, and if
j' is any other parcel then sij-qjsij'-qj'. Give an economic interpreation of this
result and explain the relation between free competition and optimiality in this
context.
c) Assuming that each sij is positive, show that the prices can be assumed to be
nonnegative.
Question 7*
a) In a maximum flow problem suppose that we have multiple source and multiple
sink nodes. How would you change the problem to a single source single sink
maximum flow problem? Give an example.
b) In a multiple (or single) source and sink maximum flow problem suppose that the
capacity of source is not unbounded, but has a limit, say ks and/or the sink has a
limited capacity of kt. How would you transform the problem to the ordinary
unlimited source/sink capacity network flow problem? Give an example.
Question 8
a) Consider the upper-bounded linear programming problem:
Min
s.t.
T
c x
Ax=b
0 xu
Find the dual of this problem. If y is the dual variable corresponding to equations
Ax=b then prove that at the optimum the complementary slackness theorem reduces
to:
 If xi=ui then ci<aiTy
 if 0 < xi < ui then ci=aiTy
 if ci>aiTy the xi=0
b) Now take the special case of maximum flow problem in a network, in particular
its minimum cost flow formulation:
Min
s.t.
-x ts
ixik -  ix ki=0 for all nodes
0xijuij
Show that the complementary slackness relations at the optimum in this case can be
written as
 if yj-yi<cij then xij=0
 if 0 < xij < uij then yj-yi=cij
 if yj-yi>cij then xij=uij
 ys-yt=1
3
Question 9*
The minimum cost flow problem (without capacity constraints on the edges) can be
converted to a transportation problem and thus solved by transportation algorithm.
One way to do this conversion is to is to find the minimum cost path from every
supply node to every demand node, allowing for possible shipping through
intermediate transshipment nodes. The values of these minimum costs become the
effective point-to-point costs in the equivalent transportation problem. Once the
transportation problem is solved, yielding amounts to be shipped from origins to
destinations, the result is translated back to flows in arcs and shipping along the
previously determined minimal cost paths.
Consider the transshipment problem with five shipping points defined by the
symmetric cost matrix and the requirement vector
s=(10,30,0,-20,-20)
0

3
C  4

6

4
3 3 6 4

0 5 4 8
5 0 2 5

4 2 0 5

8 5 5 0
(
(The requirement vector gives supply nodes if negative, demand nodes if positive and
transshipment nodes if zero.)
a) Set up the associated transportation problem and solve it.
b) From the associated transportation problem find out the optimal flow in the
original problem
4
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