![]() ![]() This concludes the constraints for the problem and so the entire linear programming formulation too. For shorthand, we denote the towers sizes small, medium, and large by the integers 1, 2, and 3 respectively.ġ & \textrm \rho_ \qquad \forall i \in $$ To begin, we will need to define a collection of decision variables. If you are only interested in the practical side of this problem then feel free to skip this section but I would argue that any good Excel model will have a well-thought-out mathematical model behind it. We begin by formulating the problem mathematically. Before we get to this, let’s take a look at this problem through a mathematical lens. To do this we will need to build a model to describe the problem and then use the Solver add-in in Excel to find an optimal solution. We will solve this problem using linear programming in particular, the branch and bound algorithm. ![]() Despite this, I believe that this example captures the essence of the problem with the added bonus that it can be formulated in a visual way in a spreadsheet. Using a more powerful method such as a programming language like Python or Matlab would facilitate the expansion of this problem further. The expanse of this example is not limited by the methods we will be using but rather the tool - Microsoft Excel. This feels very close to the sort of problem a real life telecommunications company may be trying to solve albeit with reduced scale and complexity. Lastly it is added that if two towers cover the same customer then only one of them needs to use its coverage to serve that customer. A further constraint is that due to planning permission issues, you can only build one tower in any given town. There are also limitations on the number of some of the types of satellite you can build. The data goes on to detail that there are three sizes of tower available to be built, each with its own associated range, coverage (that is the number of customers it can sustain), and construction costs (we assume that running costs are negligible). Your objective is to determine the cheapest way in which this can be achieved. You are informed that the company wishes to build new communication towers in a selection of these towns so that every customer is covered by at least one tower (we assume that currently no town has any existing coverage). Suppose that you work at a telecommunications company and one day you are sent an Excel spreadsheet with the following data.Īs you can see, the data concerns the location of 6 towns and the number of customers the company has at each one. The Excel spreadsheet used in this post can be found on the GitHub repository for this blog Formulating the Coverage Problem ![]()
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