QUESTION ONE $($COMPULSORY$)$

A small construction firm based in Kitwe specializes in building and selling single$-$family homes. The firm offers two basic types of houses, model A and model B$.$ Model A houses require $4,000$ labour hours, $2$ tons of stone, and $2,000$ board feet of lumber. Model B houses require $10,000$ labor hours, $3$ tons of stone, and $2,000$ board feet of lumber. Due to long lead times for ordering supplies and the scarcity of skilled and semiskilled workers in the area, the firm will be forced to rely on its present resources for the upcoming building season. It has $400,000$ hours of labor, $150$ tons of stone and $200,000$ board feet of lumber.

i Using the simplex method, what mix of model A and B houses should the firm construct if model A$$s yield a profit of $$1,000$ per unit and model B$$s yield $$2,000$ per unit? Assume that the firm will be able to sell all the units it builds. $[15$ Marks$]$

ii Which resources are not completely used by this firm? How much of each resource is unused? $[3$ Marks$]$

iii Determine the ranges of optimality for the objective function coefficients $[4$ Marks$]$

iv Write the dual solutions to the problem