$[20]$ Consider a company that needs to assign $6$ tasks $($T$1,...,$ T$6)$ to $4$ workers $($W$1,...,$ W$4).$ Each task must be assigned to exactly one worker, and each worker has a limited capacity in terms of the total time they can dedicate to tasks. The time it takes for a worker to complete a task varies depending on the task$-$worker pair due to different skill levels and efficiencies. The cost of assigning a task to a worker also varies depending on the task$-$worker pair. The objective is to minimize the total assignment cost while ensuring that all tasks are assigned and no worker exceeds their capacity in terms of total time. Use the following data:

Task W$1$ W$2$ W$3$ W$4$ T$15465$ T$27657$ T$34534$ T$46756$ T$58798$ T$63423$ Table $1$: Time $($number of hours$)$ it takes for each worker to finish each task. Task W$1$ W$2$ W$3$ W$4$ T$1$ $$10$ $$15$ $$20$ $$18$ T$2$ $$20$ $$10$ $$25$ $$22$ T$3$ $$30$ $$25$ $$15$ $$28$ T$4$ $$25$ $$30$ $$15$ $$28$ T$5$ $$18$ $$22$ $$28$ $$15$ T$6$ $$22$ $$18$ $$24$ $$20$ Table $2$: Costs for each worker to do each task. W$1$ W$2$ W$3$ W$4$ Capacity $($in hours$)78109$ Table $3$: Capacity $($total time in hours$)$ of each worker

$($a$)$ Formulate this problem as an integer program, defining the decision variables, objective function, and constraints.

$($b$)($Excel$)$ Set up and solve the integer program in part $($a$)$ in the Excel workbook $($Sheet Taskb$).$ What is the optimal assignment of the tasks? What is the optimal total cost? Write down the answers here and save the Answer report in Excel.

Suppose that due to the nature of the tasks, there is an additional penalty of $$15$ if we assign T$3$ and T$4$ to the same worker. How will this change your model formulation? $($Hint: the objective function may become nonlinear$)$

Set up and solve the nonlinear program in part $($c$)$ in the Excel workbook $($Sheet Task$-$d$).$ What is the optimal assignment of the tasks? What is the optimal total cost? Write down the answers here and save the Answer report in Excel.