Question $1$: Cell Tower Optimization Challenge $-$ Covering Problem

Imagine you are part of a strategic planning team for a telecom company that aims to enhance mobile signal coverage in a bustling urban environment. The company has identified several potential locations for new cell towers, each with the capacity to provide coverage to surrounding areas. However, due to budget constraints, the company can only afford to build a limited number of towers. To tackle this challenge effectively, the city has been divided into several distinct regions, each with its own population density.

The following table illustrates which regions $($columns$)$ are covered by each cell tower site $($rows$).$

The population at each region is stated in the following table.

The cost to build a cell tower at each location site is stated in the following table.

The allocated budget is $\backslash (\backslash$$ $20,000,000\backslash ).$

Objective:

$-$ The primary goal is to maximize the total population that receives reliable signal coverage from the towers.

Requirements:

$-$ Decision Variables: Clearly define the decision variables that will represent the locations of the cell towers.

$-$ Objective Function: Formulate an objective function that captures the essence of maximizing population coverage.

$-$ Constraints: Identify and explain any constraints that must be considered, including budget limitations and coverage range of the towers.

Your task is:

a$.$ Devise an optimization model $($mathematical formulation$)$ that will help the company determine the most effective locations for the cell towers.

b$.$ Code this model using OPL $+$ Cplex and solve the problem.

c$.$ Present the optimal solution and the optimal value and discuss the percentages of population covered and the budget utilised.