Q$\mathrm{.}6.$ WetLimeStone has two cement production plants $(P1$ and $P2)$ located $500kms$ of each other. Cement from the two production plants are shipped by rail wagons to two regional distribution centres $($D$1$ and D$2),$ and then to three supply houses $($S$1,$ S$2$ and S$3)$ by trucks and vans.

The production capacities of plants $P1$ and $P2$ are $8000$ and $7000$ tons$/$week respectively, and the demand at supply houses S$1,$ S$2$ and S$3$ are $4500,3500$ and $3000$ tons$/$week respectively. The unit cost of transporting cement $($Rs$/$ton$)$ between the production plants and distribution centres, and between the distribution centres and supply houses is given in the table below.

$\backslash$table$[[$From$/$To$,$D$1,$D$2,$S$1,$S$2,$S$3],[$P$1,4,7,,,],[$P$2,5,7,,,],[$D$1,,,6,4,5],[$D$2,,,2,3,6]]$

Due to limited availability of transport and prior commitments, no more than $1500$ tons week can be shipped from D$2$ to S$1,$ and a minimum of $400$ tons week must be shipped from D$1$ to S$2.$

Write a set of linear programming equations to minimize total cost of transportation, and draw a neat network diagram.