Pre-Positioning Disaster Relief Shelters) A disaster relief agency plans to estab- lish shelters in preparation for a hurricane that has been forecast for the coming days. The agency wishes to choose shelters from a set J of potential locations in order to cover every population center in the set I. A shelter covers a population center if it is within r miles of it. As in the set covering and maximal covering models, we define the parameter ajj to equal 1 if a shelter at site je J covers population center i = I. If we locate a shelter at site j, we incur a fixed cost of f;, as well as an "assignment cost" of w, for each population center assigned to the shelter at j (regardless of the size of these population centers). For example, if shelter j serves 12 population centers, then we pay an assignment cost of 12w,. Ij = Yij Zjn 1, if we locate a shelter at site j 0, otherwise 1, if a shelter at site j serves population center i 0, otherwise a) Formulate this problem as a linear integer optimization problem. If you introduce any new notation, define it clearly. Briefly explain your objective function and constraints. b) In part (a), the assignment cost is a linear function of the number of population centers assigned to each shelter: It equals win, where n is the number of popula- tion centers assigned to j. Suppose instead that the assignment cost is a nonlinear function g; (n), where n is the number of population centers assigned to j. Define the following decision variables: 1, if exactly n population centers are assigned to a shelter at j -{ otherwise Formulate this problem as a linear integer programming problem. Define any new notation clearly, and explain the objective function and any new constraints. Pre-Positioning Disaster Relief Shelters) A disaster relief agency plans to estab- lish shelters in preparation for a hurricane that has been forecast for the coming days. The agency wishes to choose shelters from a set J of potential locations in order to cover every population center in the set I. A shelter covers a population center if it is within r miles of it. As in the set covering and maximal covering models, we define the parameter ajj to equal 1 if a shelter at site jEJ covers population center i I. If we locate a shelter at site j, we incur a fixed cost of fj, as well as an "assignment cost" of w, for each population center assigned to the shelter at j (regardless of the size of these population centers). For example, if shelter j serves 12 population centers, then we pay an assignment cost of 12w;. Ij = Yij Zjn 1, if we locate a shelter at site j 0, otherwise 1, if a shelter at site j serves population center i 0, otherwise a) Formulate this problem as a linear integer optimization problem. If you introduce any new notation, define it clearly. Briefly explain your objective function and constraints. b) In part (a), the assignment cost is a linear function of the number of population centers assigned to each shelter: It equals win, where n is the number of popula- tion centers assigned to j. Suppose instead that the assignment cost is a nonlinear function g; (n), where n is the number of population centers assigned to j. Define the following decision variables: 1, if exactly n population centers are assigned to a shelter at j -{ otherwise Formulate this problem as a linear integer programming problem. Define any new notation clearly, and explain the objective function and any new constraints.