IEOR E4007
G. Iyengar
August 18th, 2009.
IEOR E4007 Final Exam
Instructions
:
200pts
1. You have 3 hours to do the exam.
2. The exam is closed book. You are allowed one standard sheet of paper as a cheat sheet.
3. Calculators are allowed but no computers.
4.
Important
: Please answer every question in the space provided.
You may use the
reverse sides of the pages as well.
5. If I don’t see you again, the very best of wishes for your success here at Columbia and
later !
Please sign the following declaration.
I declare that I did not receive nor did I give any assistance to anyone else
during this exam.
———————————————————————–
1
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1.
A simple problem on unconstrained optimization
50pts
A retailer is planning her yearly inventory strategy for a commodity that sells at a
steady demand rate
D
= 20.
Her objective is to minimize her total costs which
includes storage costs and ordering costs.
The storage cost is
S
= 10 per unit, the
fixed cost per order is
C
= 100, the variable cost is
P
= 5 per unit and there is no lead
time before orders are filled.
Let
x
denote the order size. Then, the retailers optimization problem is given by (see
figure below)
min
{
x
:
x>
0
}
braceleftbigg
D
x
(
C
+
Px
) +
S
2
x
bracerightbigg
x
D/x
(a) Solve this optimization. Prove that the solution that you obtain is, in fact, the
global minimum.
10pts
(b) Suppose the retailer is concerned that her supplier may not be able to meet her
needs and decides to diversify between two suppliers
(i) Supplier 1: Fixed cost
C
and variable cost
P/
2
(ii) Supplier 2: Fixed cost
C
and variable cost 3
P/
2
Let (
x
1
,x
2
) denote the order sizes for the two retailers. Formulate the new opti
mization problem faced by the retailer.
10pts
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 Summer '09
 OptimizationModelsandMethods
 Optimization, retail location, optimal replacement policy, G. Iyengar

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