A consulting firm has hired three consultants: Alan, Beatrice, and Charlie. Each consultant is available to work for $20$ days in the upcoming month. The firm has secured contracts for three projects $-$ P$1,$ P$2,$ and P$3-$ to be completed within the same timeframe. The daily remuneration for each consultant, measured in monetary units, varies depending on the specific project they are assigned to$.$ The corresponding figures are detailed in the table below:

The bottom row of the table also shows the person$-$days required for the completion of each project.

The consulting firm has agreed a fixed fee with its client for all three projects. Consequently, the firm wishes to minimize the total remuneration to be paid to the three consultants $($Alan$,$ Beatrice and Charlie$)$ for their work on these projects.

Question $2\mathrm{.}1$

Formulate the above problem as a Linear Programming model. Provide a complete description of all the decision variables, the objective function, and the constraints.

Question $2\mathrm{.}2$

Use Excel to find the optimal solution to the problem. Provide the practical significance of the optimal values of the decision variables. What is the minimum remuneration cost for the firm? Which of the three consultants is$/$are expected to have available time during the next month, allowing them to be engaged in other projects?

Question $2\mathrm{.}3$

Is there any indication of multiple optimal solutions? What impact would it have on the total remuneration cost if Alan insists on working on project P$2?$

Question $2\mathrm{.}4$

Alan is proposing to extend his work by an extra $5$ days during the next month, incurring an additional cost of $10$ monetary units. Should the consulting firm consider and accept this offer?