ENM 509– HW 2
Due:09.03.2017, 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 transmittersbeing 2 or 3 or more, that cause the low quality, doesn’t matter.
2. In the Gulf of Mexico, Tell Oilhas 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.
RigCoordinatesTrips/Month
160,707
280,6010
3100,308
440,2012
Determine where the warehouse should be located in order to minimize the total cost
a) using euclidean distances
b) using rectilinear distances
3. 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=1,3,6S2=1,5,6,8S3=2,5S4=3,6,7
S5=4,6,7,8,9S6=3,4,6,8S7=1,3,5,7,8S8=2,5,6,8
4. Dry Ice Inc. is a manufacturer 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 DiegoAnnual Fixed Cost of 200,000 plant / $6 million / $5.5 million / $5.6 million / $6.1 million
Annual Fixed Cost of 400,000 plant / $10 million / $9.2 million / $9.3 million / $10.2 million
East / $211 / $232 / $238 / $299
South / $232 / $212 / $230 / $280
Midwest / $240 / $230 / $215 / $270
West / $300 / $280 / $270 / $225