A pharmaceutical company manages a complex supply chain involving the production and distribution of various drugs, each with different lead times, production costs, and perishability considerations. The company aims to minimize transportation costs $($in $$)$ and lead times $($in days, assuming that each day is estimated to add $$2)$ while ensuring drug availability.

a$.$ How can the pharmaceutical company construct an advanced LP model $($using the Vogel Method$)$ that considers lead times, multiple objectives, and complex supply chain dynamics, and how can they interpret the results effectively?

b$.$ How does incorporating delivery time constraints affect the formulation of the objective function?

c$.$ Explain the concept of multi$-$objective optimization and why it is important for LP models to balance cost and lead time considerations.

Factory$1$ Factory$2$ Factory$3$ Factory$4$ Lead time Supply

Drug$181012142600$

Drug$211913153750$

Drug$391211162700$

Demand $7008009001000$