(a) The Director of the Southeastern Anatolia Program (SAP) has been informed that the Program will be provided with relatively large annual budgets for the next T years. The budgets are also stated by the Government as part of a regional development plan. He is asked to generate a set of possible projects extending over several years in consultation with the stakeholders of the SAP and then choose among these projects a subset to implement over the time horizon T. The projects selected do not need to start simultaneously but they are required to finish within the next $T$ years. Each project $i$ has an execution period $D_i$, in general differing over the projects. Some of the projects have various versions out of which at most one is to be selected, if at all. For modeling purposes these versions are treated as individual projects. The annual expenditure for project $i$ can vary depending on which year of its execution the project is in, i.e., if project $i$ has been initiated in year $\mathrm{j}$, then its annual expenditure in year $k \geq j$ is given as $\mathrm{C}_{\mathrm{i}, \mathrm{k}+1-\mathrm{j}}$. Let $a_i$ be the score of project $i$ determined to represent the contribution of project $i$ to the Program; hence, the higher the better. The Director wants to maximize the sum of the scores of all projects implemented. Write down a mathematical programming model for the problem environment stated above.
(b) Let us assume that we have solved the above mathematical programming formulation and obtained an optimal solution; in other words, an optimal objective function value $Z^*$, an optimal set of projects to be implemented together with their initiation period. Suggest a methodology for finding an alternative optimal solution, if there exists one.