I have this problem:
A long time ago, in a galaxy far, far away… A telecom company plans to install cell towers for wireless communication in a new neighborhood. The cell towers they install can include two types of antennas, A and B. Each cell tower has room for 10 antennas of type A. Antennas of type B are 10% bigger than antennas of type A. Each cell tower can have both antennas if there is room for them. Each antenna of type A supplies 50 Gbps, and each antenna of type B supplies 40 Gbps. Installing a cell tower costs 100 crypto coins, each antenna of type A costs 15 crypto coins, and each antenna of type B costs 10 crypto coins.
The company has divided the neighborhood into a grid of 5x5 areas, each shown here with their corresponding expected demand in Gbps:
| c1 | c2 | c3 | c4 | c5 | |
|---|---|---|---|---|---|
| r1 | 391 | 370 | 363 | 400 | 358 |
| r2 | 313 | 207 | 333 | 250 | 303 |
| r3 | 328 | 308 | 367 | 357 | 332 |
| r4 | 366 | 205 | 245 | 243 | 337 |
| r5 | 393 | 216 | 323 | 356 | 268 |
| The company must decide in which areas to install cell towers. At most, they will consider installing one cell tower per area. Furthermore, they must decide how many antennas of each type to include in each installed cell tower. Antennas of type A can only reach customers in the same area, while antennas of type B can reach customers on their own and in neighboring areas. For example, these are the areas reached by antennas included in a cell tower installed in the area (r3, c3): | |||||
| They want to be sure to supply at least all the demand. To achieve this, it is important to check that the broadband provided by antennas is not overused. It is not necessary to check what demand is supplied by each antenna. Still, they need to check what demand is provided by each type of antenna included in each cell tower, as the antennas have different ranges | |||||
| a. | Formulate a mixed integer linear model to help this telecom company decide where to install cell towers and what antennas to use in each cell tower to supply all the demand at minimum cost. | ||||
| — | — | ||||
| b. | Modify the model from section a. to consider that only one type of antenna can be installed in each cell tower. Compare the results with those from section a. | ||||
| c. | Modify the model from section a. to consider that, due to some protests from neighbors, the company has accepted the following condition. If they install 4 or more cell towers in column 5, they must install at least 4 in all the other columns. Compare the results with those from section a |
This is my code, but t does not work correctly:
SETS
i rows /r1,r2,r3,r4,r5/
j columns /c1,c2,c3,c4,c5/
k antennas /A, B/
;
ALIAS(i,i_prima);
ALIAS(j,j_prima);
Scalar
CostTower cost of installing a cell tower / 100 /
CapacityTower maximum number of antennas that could be installed in a tower / 10 /
;
option OptCR=0
PARAMETERS
Demand(i,j) Gbps demand in area ij
Cost(k) cost of installing antennas of type k / A 15, B 10 /
Space(k) space occupied by antennas of type k / A 1, B 1.1 /
Supply(k) antenna k supply / A 50, B 40 /
;
TABLE Demand(i,j) expected demand in area ij
c1 c2 c3 c4 c5
r1 391 370 363 400 358
r2 313 207 333 250 303
r3 328 308 367 357 332
r4 366 205 245 243 337
r5 393 216 323 356 268
;
Variables
TC total cost
;
Positive Variables
supB(i,j,i_prima,j_prima) Gbps of antenas of type B installed in area ij supplied in area i_primaj_prima
;
Integer Variables
Num(k,i,j) number of type k antennas installed in area ij
;
BINARY VARIABLES
X(i,j) Indicates if a tower is installed or not in ij
;
EQUATIONS
funobj Objective function
constraint1(i,j) Antennas B supply within its cappabilities
constraint2(i,j) Only one cell tower installed as maximum in area ij
constraint3(i,j) ensuring cell tower cappacity
constraint4(i,j) Ensures demand satisfaction
;
constraint1(i,j)… sum((i_prima,j_prima), SupB(i,j,i_prima,j_prima))=L= Supply(‘B’);
constraint2(i,j)… X(i,j)=L=1;
constraint3(i,j)… X(i,j)*CapacityTower=G=sum((k), Num(k,i,j)*Space(k));
constraint4(i,j)… Num(‘A’,i,j)*Supply(‘A’)+sum((i_prima,j_prima), supB(i_prima,j_prima,i,j))=G=Demand(i,j);
funobj… TC=E=sum((i,j), X(i,j)*CostTower)+sum((i,j,k), Cost(k)*Num(k,i,j));
MODEL AntennaOptimization /ALL/;
SOLVE AntennaOptimization USING MIP MINIMIZING TC;