Optimize Cell Tower Locations

Optimize the placement and ranges of cellular towers to provide coverage for a set of known customer locations

We wish to find the positions

and ranges

of various cell towers needed to serve customers located at positions

that minimize the power consumed.

Define the positions of the customers and the number of towers used to cover them:

In[1]:=

customerPos={{0,0},{1,0},{3/2,1},{-1/2,1},{1/2,3/2},{1/2,1/2},{0,3/2},{1,1},{0,1},{3/2,3/2}};nTowers=3;nCustomers=Length[customerPos];

Each cell tower consumes power that depends on its range according to

∝Erf

1.5



. The objective is to minimize the power consumption:

In[2]:=

objective=Sum[Erf[r[i]^1.5],{i,nTowers}];

Let

i,j

be a decision variable indicating that

i,j

if client

is covered by cell tower

In[3]:=

decisionConstraint=0z1;

Each cell tower must be located such that its range covers some of the clients:

In[4]:=

m=20;distConstraint=Table[Norm[{p[j,1],p[j,2]}-customerPos〚i〛]-m*(1-Indexed[z,{i,j}])≤r[j],{i,nCustomers},{j,nTowers}];

Each cell tower can cover multiple clients:

In[5]:=

coverageConstraint=Total[z,{2}]1;

Each cell tower has a minimum and maximum coverage:

In[6]:=

maxCoverageConstraint=Table[0≤r[j]≤1,{j,nTowers}];

Collect all the variables:

In[7]:=

vars=Flatten[{Table[{p[j,1],p[j,2],r[j]},{j,nTowers}],z∈Matrices[{nCustomers,nTowers},Integers]}];

Find the cell tower positions and their ranges:

In[8]:=

Quiet[res=NMinimize[{objective,decisionConstraint,distConstraint,coverageConstraint,maxCoverageConstraint},vars]]

Out[8]=

{1.91992,{p[1,1]0.416665,p[1,2]1.08333,r[1]0.589257,p[2,1]0.999978,p[2,2]1.00002,r[2]0.707107,p[3,1]0.5,p[3,2]0.872896,r[3]1.,z{{0,0,1},{0,0,1},{0,1,1},{0,0,1},{1,1,1},{1,1,1},{1,0,0},{1,0,1},{1,0,1},{0,1,0}}}}

Extract the values:

In[9]:=

towerPos=Table[{p[i,1],p[i,2]},{i,nTowers}]/.res〚2〛;towerRange=Table[r[i],{i,nTowers}]/.res〚2〛;

Visualize the positions and ranges of the towers with respect to client locations:

In[10]:=

Graphics[{{Orange,PointSize[0.04],Point[towerPos]},{Black,PointSize[0.02],Point[customerPos]},{Red,EdgeForm[Thick],MapThread[Circle[#1,Abs[#2]+0.04`]&,{towerPos,towerRange}]}},FrameTrue]

Out[10]=

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Optimize Cell Tower Locations

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Optimize Cell Tower Locations

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