Given a study area with $4$ traffic analysis zones $($TAZs$),$ the following table gives the total numbers of trip

production and attraction for each TAZ.

The second table $($as shown below$)$ gives the travel time $($in minute$)$ matrix from one TAZ to another.

Assume the friction function:

$F_{ij}(t_{ij})=$exp$(+t_{ij})$

Ignore socioeconomic factor $($i$.$e$.,K_{ij}=1).$ And assume $=-0\mathrm{.}15,$ and $=-0\mathrm{.}8$

$(1)$ Determine the matrix of friction factors $($from one TAZ to another$).(30$ pt$)$

$(2)$ Given the convergence error as $0\mathrm{.}1\%,$ using the doubly constraints on the numbers of both trip

productions and attractions, determine the final matrix of trip distributions between TAZs. $(70$ pt$)$

$($Hint: Use Excel to solve these two problems, and copy the matrix of each iteration to the Word

document with necessary explanations. A single Excel document is NOT accepted$)$