ENM 509 – HW 1
Due:03.03.2016, Thursday
1. You want to place transmitters over a region to cover all the facilities in that region.
Assume that there are n facilities that needs to be covered and there are m different possible
locations that you can place the transmitters. To define the coverages of the possible
locations, the parameter aij is defined to be equal to 1 if location i covers facility j and aij is
equal to 0 otherwise.
Note that, if a facility is covered by more than 1 transmitter, then the transmitters interact with
each other and the quality of the transmissions decrease. Thus, your objective is to minimize
the number of facilities that are covered by more than 1 transmitter while all the facilities are
covered at least once. Define the decision variables and write the mathematical model to solve
this problem.
Note: If 2 or 3 or more transmitters cover the same facility, then that facility has low quality
transmission and your objective is to minimize the number of facilities with low quality
transmission. The number of transmitters being 2 or 3 or more, that cause the low quality,
doesn’t matter.
2. In the Gulf of Mexico, Tell Oil has 4 rigs. The location of each is given on the table below.
Food for all the rigs is brought out once a month by ships to a central warehouse and then
distributed according to the data in the table.
Rig
Coordinates
Trips/Month
1
60,70
7
2
80,60
10
3
100,30
8
4
40,20
12
Determine where the warehouse should be located
a) using euclidean distances
b) using rectilinear distances
3. A new community hospital is to be located which will serve 5 different communities at
locations (8, 10), (30, 15), (2, 4), (25, 18), (14, 20). We want to minimize the maximum time
any patient from these 5 communities will take to reach the hospital. Determine the optimal
location for the hospital if
a) the travel is measured in rectilinear distance
b) the travel is measured in euclidean distance
4. A fishing group found eight probable areas in which fish groups may be present. They can
throw nets that will cover several different areas at one time. For example if they throw a net
in area 1, it will cover areas 1,3 and 6. The coverage of the areas are shown below. Determine
the minimum number of nets required to cover all areas.
S1=1,3,6
S2=1,5,6,8
S3=2,5
S4=3,6,7
S5=4,6,7,8,9
S6=3,4,6,8
S7=1,3,5,7,8
S8=2,5,6,8
5. Dry Ice Inc. İs a mnaufacturer of air conditioners that has seen its demand grow
significantly. The company anticipates nationwide demand for the year 2016 to be 180,000
units in the South, 120,000 units in the Midwest, 110,000 units in the East and 100,000 units
in the West. Managers at DryIce are designing the manufacturing network and have selected
four potential sites – New York, Atlanta, Chicago and San Diego. Plants could have a
capacity of either 200,000 or 400,000 units. The annual fixed costs at the four locations are
shown in the Table below, along with the cost of production and shipping an air conditioner to
each of the four markets. Write a mathematical model to denote where DryIce shoul build its
factories, how large they should be, how should the demand be satisfied?
New York
Atlanta
Chicago
San Diego
Annual Fixed
$6 million
$5.5 million
$5.6 million
$6.1 million
Cost of 200,000
plant
Annual Fixed
$10 million
$9.2 million
$9.3 million
$10.2 million
Cost of 400,000
plant
East
$211
$232
$238
$299
South
$232
$212
$230
$280
Midwest
$240
$230
$215
$270
West
$300
$280
$270
$225