Download Linear Programming MSIS 683 Homework 4

Linear Programming
MSIS 683
Homework 4
Instructor: Farid Alizadeh
Due Date: Monday December 11, 2000
before 4pm in my mailbox
(or slide under my door in Ackerson 200m)
last updated on December 3, 2000
1. Consider the transportation problem

10 5 6
C=8 2 7
9 3 4
with the following data:

7
6 a = (25, 25, 50) b = (15, 20, 30, 35)
8
Using the tabular format discussed in class, Find an initial feasible solution using the northwest rule,
solve the transportation problem staring from your initial feasible solution. At each iteration show
the current primal feasible and also dual reduced costs. Indicates which variables enter and which
leave the the basis at each iteration.
2. Using Dijkstra’s algorithm find the shortest path from S to H. Show all labels clearly. At each
iteration indicate the order in which each node was labeled. Mark the final shortest path in the
graph.
E
6
F
4
@ 6
@
3
5
@
@
aa
"
1
D
S ""
8
4
G
12
,
, HH
S
,
7
9
10
,
H1
S6
HH
,
S
H S
3
,
16
C
A
aa
aa
2
"
"
a
B
H
1
MSIS 683, Fall 2000
Homework 4
3. Consider the following network
Due date: 12/18/00
2 1,100 6
1
4
3
J 3,3
0,100
6
3,100
2,2
J
1
0 Q
Q
J
^
0
0,5
5
Q
4,5
1,5
Q
Q
s 2?
Q
- -7
2,100 3
4
Numbers outside nodes are their labels, those inside nodes are their demands (supply if negative).
The first number on each arc is its cost and the second is its capacity.
(a) (AMPL project Formulate the initial phase for finding a feasible solution and solve it using
AMPL with the node/arc formulation. Save your AMPL problem1.mod and problem1.dat files
in your accompanying floppy disk.
(b) Formulate the upper bounded transshipment problem and solve it by hand using the network
simplex method and the initial solution AMPL produced for you. At each iteration, show two
graphs, one showing the basic feasible tree arcs with solid line, and the saturated arcs with
dotted line. In the second graph show reduced costs on the non-tree edges and the dual values
on the nodes. Clearly indicate, the incoming and outgoing edges in iteration.
4. Find the maximum flow and the minimum cut in the following network from source S to source T.
The first number on each edge is the capacity. Use the second number as your initial feasible flow
and improve on it or show it is optimal. Cleanly label every node, and indicate the order in which
the nodes were labeled.
A
D
Z
>
*
6 Z 3,3
Z
2,2
2,2
2,2
Z
1,1
Z
~ Z
S
B
?
1,0
1,0 -
1,1
E
1,0 T
3 Z
6
6
Z4,3
>
Z
1,0
1,1
1,0
Z
3,2
Z
~
Z
2,2
C
F
2