$2.($a$)$ The Ayatola Oil Company controls two oil fields. Field $1$ can produce up to $40$ million barrels of oil per day, and field $2$ can produce up to $50$ million barrels of oil per day. Ayatola sells oil to two countries, England and Japan. The company's profit per barrel of shipping oil from each of the fields is shown in the following table: From$\backslash$ To $|$ England Japan Field $1$ E$1\mathrm{.}5$ E $2\mathrm{.}5$ Field $2$ E$2$ Each day, England is willing to buy $40$ million barrels, and Japan is willing to buy $30$ million barrels. Formulate a balanced transportation problem to maximise Ayatola's profits. Donotsolve the problem. $[10$ marks$]($b$)$ Using the Branch$-$and$-$Bounds method, solve the following instance of the Knap$-$ sack Problem: maximise $10<,+6$k$2+7$X$3+12$x$4+9$X$5$ subject to $3$x$,+2$x$2+4$X$3+6$X$4+4$X$5<10$ X$1,$ X$2,$ X$3,$ X$4,$ X$5=0$ or $1$ Use the ratio choice method to obtain solutions to the LP relaxations of the cor$-$ responding problems. Provide all the necessary details and explanation, and draw a Branch$-$and$-$Bounds tree of sub$-$problems for the solution