Cleveland Area Rapid Delivery $($CARD$)$ operates a delivery service in the Cleveland metropolitan area. Most of CARD's business involves rapid delivery of documents and parcels between offices during the business day. CARD promotes its ability to make fast and on$-$time deliveries anywhere in the metropolitan area. When a customer calls with a delivery request, CARD quotes a guaranteed delivery time. The following network shows the street routes available. The numbers above each arc indicate the travel time in minutes between the two locations.

A network diagram between six parties, distributed across four columns, is shown. The left$-$most column contains node $1.$ The second column contains nodes $2$ and $3.$ The third column contains nodes $4$ and $5.$ The right$-$most column contains node $6.$ Lines are shown between various locations. The following list contains the numbers placed on the graph.

$12$: $35$

$13$: $30$

$21$: $35$

$23$: $12$

$24$: $18$

$26$: $39$

$31$: $30$

$32$: $12$

$35$: $15$

$42$: $18$

$45$: $12$

$46$: $16$

$53$: $15$

$54$: $12$

$56$: $30$

$62$: $39$

$64$: $16$

$65$: $30$

$($a$)$

Develop a linear programming model that can be used to find the minimum time required to make a delivery from location $1$ to location $6.($Express your answers in the form xij, where each xij represents the arc from node i to node j as either $1$ or $0.)$

Min

s$.$t$.$

Node $1$ Flows

Node $2$ Flows

Node $3$ Flows

Node $4$ Flows

Node $5$ Flows

Node $6$ Flows

For all xij $=0,1.$

$($b$)$

How long $($in minutes$)$ does it take to make a delivery from location $1$ to location $6?($Round your answer to the nearest whole number.$)$

min

$($c$)$

Assume that it is now $1$:$00$ p$.$m$.$ and that CARD just received a request for a pickup at location $1.$ The closest CARD courier is $8$ minutes away from location $1.$ If CARD provides a $20\%$ safety margin in guaranteeing a delivery time, what is the minimum guaranteed delivery time if the package picked up at location $1$ is to be delivered to location $6?($Enter your answer in standard time. Round your answer to the nearest minute.$)$

:

p$.$m$.$