A contractor, Susan Meyer, has to haul gravel to three building sites. She can purchase as much as $18$ tons at a gravel pit in the north of the city and $14$ tons at one in the south. She needs $10,5,$ and $10$ tons at sites $1,2,$ and $3,$ respectively. The purchase price per ton at each gravel pit and the hauling cost per ton are given in the table below:

Pit Hauling Cost per ton at site Price per ton

$123$

North $$100$ $$190$ $$160$ $$330$

South $180110140$ $$450$

Susan wishes to determine how much to haul from each pit to each site to minimize the total cost for purchasing and hauling gravel. Susan now needs to hire the trucks $($and their drivers$)$ to do the hauling. Each truck can only be used to haul gravel from a single pit to a single site. In addition to the hauling and gravel costs specified, there now is a fixed cost of $$50$ associated with hiring each truck. A truck can haul $5$ tons, but it is not required to go full. For each combination of pit and site, there are now two decisions to be made: the number of trucks to be used and the amount of gravel to be hauled.

Formulate an MIP model for this problem.

Let xij be the number of trucks hauling from pit i to site j and yij be the number of tons of gravel hauled from pit i to site j$,$ for i $=$ N$,$ S and j $=1,2,3.$

Minimize C $=$

yN$1+$

yN$2+$

yN$3+$

yS$1+$

yS$2+$

yS$3+$

$($xN$1+$ xN$2+$ xN$3+$ xS$1+$ xS$2+$ xS$3)$

subject to

yN$1+$ yN$2+$ yN$3<=18$

yS$1+$ yS$2+$ yS$3<=14$

yij $<=5$ xij

yN$1+$ yS$1>=10$

yN$2+$ yS$2>=5$

yN$3+$ yS$3>=10$

yij $>=0$ xij $>=0$ integers for i $=$ N$.$ S and j $=1,2,3$