1
1. The Two-phase method solves a Linear Programming (LP) problem in two phases. If the LP
I
2. Consider the following LP.
1
is of the maximization type, do we maximize the sum of the artificial variables in Phase l ?
Explain.
(8%)
(16%)
Z=2X1 - X2+ X3
3 x 1+ X2 + X3+Sl
=60
XI -X2+ 2x3
+S2
=lo
X1+X2-X3
+S3 =20
VXi 2 0 VSi 2 0 i=l,2,3.
(a) Please construct the dual problem.
(b) Given that the basic variables of the optimal solution are XI, X2, and S1, solve the dual
problem.
(c) If the right hand side (RHS) of the constraints is changed from (60, 10, 20) to (59, 11, 22),
how much will the Z* be?
(d) If the constraint X3 2 0 is changed to X3 2 1, how much will the Z* be?
Max
S.T.
3. Given six candidate locations to be selected, formulate the following logical constraints:
(a) Exactly three locations are selected.
(b) If location 2 is selected, then so is location I.
(1 2%)
(c) If location 1 is selected, then location 3 is not selected.
(d) Either location 4 is selected or location 5 is selected, but not both.
4.
In a city, the minimum number of buses needed fluctuates with the time of the day. The
required number of buses is shown in the following figure.
(10%)
Required
Buses
I
0:OO
Hint:+-O
4:OO
I
I
8:OO
1200
Time of the day
16:OO
20:OO
24:OO
To carry out the required daily maintenance, each bus can operate only eight successive
hours a day. With the given information, formulate an Integer Programming (IP) to determine
the minimum number of buses that can handle the transportation needs.
(Clearly define all inputs and all decision variables)
5. Consider the following network (All links are two-way links):
Note: The numbers show distances between two nodes
-
(14%)
(a) Find the minimum spanning tree.
(b) Find the minimum (shortest) path tree rooted at node 1.
6 . Consider the following transportation problem:
w
The costs are given by the following matrix.
From
To
(a) Formulate the transportation problem as a linear programming problem.
(b) Construct the dual problem.
(c) Given the optimal set of flows is as the following:
Xaf' 105
X, = 20
Xbe= 150
G e = 50
Xdfif'35
X, = 140
Find the optimal solution for the dual problem
(d) Compare the primal and dual objective functions
(e) Check the complementary slackness conditions
(f) Suppose the supply at node B is increased to 151 units and the supply at node D is
decreased to 84, what is the change in the objective function?
(g) In general, if the supply at node i is increased by a small amount and the supply at node j is
decreased by an equal amount, what is the change in the objective function in terms of the
supplies, the costs, and the dual variables? (Assume that the change is small enough so that
the basic variables in the primal solution remain basic. Clearly define any notation you
use.)
(h) Now, suppose that the supply at node B is increased to 151 units and that none of the other
supplies and demands change, what is the change in the objective function?