Case Study $\backslash$# $2$: Optimizing Air Freight Routes for Widget Co$.($Total marks: $20)$

Widget Co$.$ manufactures a product in two cities, which are Montreal and Ottawa. The daily production capacities at Montreal and Ottawa are $180$ and $220,$ respectively. Products are shipped by air to customers in Vancouver and Toronto. The customers in each city require $150$ units of the product per day. Because of the deregulation of air fares, the company believes that it may be cheaper to first fly some products to Winnipeg or Halifax and then fly the products to their final destinations. The costs of flying one unit of the product between these cities are shown in the table below where $"-"$ denotes no direct flight.

$($a$)$ Draw the transportation network, clearly labelling all nodes. Use arrows to show routes shown in table $(5$ marks$).$

$($b$)$ Formulate a linear programming model for this problem. $(15$ marks$)$