This exam consists of three pages
Faculty of Engineering and Natural Sciences
Industrial Engineering
Solution Final exam MS 301
june 8th 2011, 9.0011.00, FENS L045, G035 and G032.
Remark: keep your answers as short as possible but of course complete. Also read very
carefully the questions. The total number of points of this exam is 100. At the beginning
of each part of the exercise the total number of points are listed. Please write your student
number at each page and write clearly. To avoid noise during the whole exam it is only
allowed during the ﬁrst hour of the exam to ask questions. The results of the ﬁnal will be
announced before the makeup on monday june 13th.
Exercise 1. (30 points)
Suppose that there are
m
factories that generate waste and
n
disposal sites. The amount of
waste generated at factory
i,
1
≤
i
≤
m,
is given by
a
i
and the capacity of disposal site
j
is given
by
b
j
.
The transport of waste should ﬁrst go from a factory to a transfer facility (these facilities
need to be constructed) and then from this transfer facility to a disposal site. In order to minimize
cost one now needs to determine at which of the
K
possible locations a transfer facility should be
constructed. If a transfer facility is build on site
k,
1
≤
k
≤
K,
it operates at a ﬁxed cost
f
k
at
that site, has a capacity of
q
k
of taking waste material from factories and the unit processing cost
per ton waste at site
k
is given by
α
k
.
Also let
c
ik
be the unit shipping cost (cost per unit of waste)
from factory
i
to site
k
and
d
kj
the unit shipping cost from site
k
to disposal site
j.
The problem
now is to choose at which sites to build a transfer facility so that the total ﬁxed and operating costs
of the transfer facilities and the total transportation costs of the waste are minimized.
1.
(15 points)
Give a deﬁnition of the used decision variables.
2. (
15 points)
Construct the associated optimization problem and explain in detail how you
obtained the objective function and the restrictions describing the feasible region.
Solution.
1. Introduce for every
1
≤
k
≤
K
the binary decision variables
y
k
=
1
if a transfer facility is build at site
k
0
otherwise
1
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View Full DocumentAlso introduce for every
1
≤
i
≤
m
and
1
≤
k
≤
K
the decision variables
x
ik
=
units of waste transported from factory
i
to transfer site
k
and for every
1
≤
k
≤
K
and
1
≤
j
≤
n
z
kj
=
units of waste transported from transfer site
k
to disposal site
j
2. The restrictions are now given by the following:
•
Amount of waste produced by factory
1
≤
i
≤
m
∑
K
k
=1
x
ik
=
a
i
.
•
Units of waste arriving at transfer facility
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 Spring '12
 H
 Optimization, objective function, dual simplex method, transfer facility

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