IEOR E4007
G. Iyengar
September 26, 2018
Homework #3
1. Continuation of the toy LP from Homework 2
Consider the following linear program
max 2x1 + x2 ,
subject to 12x1 + 3x2 ≤ 60,
3x1 − x2 ≥ −7,
x2 ≤ 10,
x1 ≥ 0, x2 free.
The standard form LP is given by
max 2y1 + (y2 − y3 ),
subject to 12y1 + 3(y2 − y3 ) + y4 = 60,
3y1 − (y2 − y3 ) − y5 = −7,
(y2 − y3 ) + y6 = 10,
y ≥ 0.
(a) Show that the constraints imply that
y2 = 7 + 3y1 + y3 − y5
y4 = 39 − 21y1
+ 3y5
y6 = 3 − 3y1
+ y5
From this set of equations it follows that B = {y2 , y4 , y6 } is a basis. Using only
the information in the above set of equalities, and no matrix inversions compute
the reduced cost with respect to each of the non-basic directions N = {y1 , y3 , y5 }.
Recall that the reduced cost is the impact on the objective when any one of the
non-basic variables is increased.
(b) Choose the non-basic direction that corresponds to the largest reduced cost, and
compute the next corner, i.e. the new basis. You are only allowed use the information in the set of equations in part (a).
(c) Suppose the objective function is given by max 3y1 − y2 + y3 . Start from the basis
B = {y2 , y4 , y6 } to conclude the problem is unbounded. You are only allowed to
use the information in part (a).
(d) Show that there does not exist any vector c such that the basis B = {y2 , y4 , y6 }
is optimal for the linear program max c1 y1 + c2 (y2 − y3 ).
1
(e) The basis B ∗ = {y1 , y2 , y5 } is optimal for the objective max 2y1 +y2 −y3 . Suppose
the RHS
 
 
60
1



−7
bθ =
+ θ 2
10
1
What is the range of θ for which this basis remains optimal.
−1
Note that since the reduced costs c̄> = c> − c>
B AB A does not change as θ is
changed, the current basis will remain optimal provided xB = A−1
B bθ ≥ 0.
2. Exercise 3.14 in in Optimization Methods in Finance
Sensitivity table
Time
-----1
2
3
4
5
6
7
8
9
Shadow price
--------------1.0829
-1.0775
-1.0657
-1.0397
-1.0345
-1.0294
-1.0243
-1.0050
-1.0000
RHS Low
---------299.8099
198.6910
0.0000
-912.2537
-793.0497
-94.5149
304.0125
-inf
-inf
2
RHS
--------100.0000
500.0000
100.0000
-600.0000
-500.0000
200.0000
600.0000
-900.0000
0.0000
RHS Up
--------535.4807
937.6581
300.8861
-394.0917
-293.0622
407.9725
809.0124
-430.7829
471.5631