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ESI 6314:
Midterm Take Home Exam
Answers
Question 1 (15 points):
True/False and why/why not
The following five questions are true/false.
Credit is given
only
if a brief rationale is
provided with your answers.
(A sentence or two will suffice.)
Part a (3 points):
All
optimal solutions to a linear program exist at extreme points.
False.
Some linear programs have alternative optimal extreme points.
Every point on the
line between alternative optimal extreme points are also optimal.
In this case, there’s an
infinite number of optimal solutions, and some are not extreme points.
Part b (3 points):
In the simplex method for a minimization problem, the objective
function strictly decreases at each iteration until an optimal solution is found.
(An
iteration consists of entering a nonbasic variable and exiting a basic variable.)
False.
If the problem is degenerate, we could enter a variable with a negative reduced
cost, but zero might win the minimum ratio test, and we would not change our solution
(or objective) at the next iteration.
Part c (3 points):
The minimum ratio test is designed to ensure that no variable becomes
negative in our solution (except maybe for the
z
value itself).
True.
We wish to make the entering variable as large as possible, because increasing the
entering variable improve our objective function.
However, we have to stop increasing
the entering variable before any of our basic variables become negative.
This limit is
determined by the minimum ratio test.
Part d (3 points):
If both a primal problem and its dual have feasible solutions, neither
problem can be unbounded.
True.
A feasible solution to a min problem puts an upper bound on the optimal objective
function value, and a feasible solution to a max problem puts a lower bound on the
optimal objective function value.
Part e (3 points):
Suppose that the shadow price for a constraint is equal to 5, and that its
range allows an increase of 10 and a decrease of 3 before the basis changes.
Then if the
righthandside for that constraint is increased by 5, the optimal objective function value
will increase by 25.
True.
The shadow price of a constraint tells you the rate of change in the objective
function when you change its righthandside. If the shadow price is 5, and you increase
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View Full Documentthe righthandside by 5, the objective function change is going to be 25 in the objective
function
as long as our righthandside change is within the allowable range
.
In this
case, the shadow price is valid as long as we do not decrease the righthandside by more
than 3, or increase it by more than 10.
Question 2 (20 points):
Model the following problem as a linear program.
Consider a cancer patient who is being
treated with radiation therapy.
The doctor has determined that she should receive
treatment in a twodimensional crosssection, represented by a 6 x 6 grid (or matrix).
The following matrix gives the number of units of radiation that should be received:
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 Fall '09
 VLADIMIRLBOGINSKI

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