a$)$ In XYZ distributors, operating in a county with four regions A$,$ B$,$ C$,$ and D$.$ customer demands and be met from three supply points; X$,$ Y and Z with capacities $50,60$ and $40$ respectively. The costs of transporting a unit from each supply point each destination are X$=(3,2,1,2),$ Y$=(1,4,3,1)$ and Z$=(4,2,3,2).$ In a given week the demand for the four regions were $20,50,40,$ and $20$ respectively.

i$)$ Formulate this into a transportation problem. $(4$ marks$)$

ii$)$ Is it possible to allocate this transportation problem for an initial feasible solution without making further assumptions? You must justify your answer. $(2$ marks$)$

iii$)$ Obtain an initial feasible solution for this problem to cost as least as possible.

$(6$ marks$)$

iv$)$ Do you believe the answer to c$)$ above is optimal? Explain. $(1$ mark$)$

v$)$ Apply a method of choice to prove the answer in d$)$ above. $(7$ marks$)$