$1.$ A company makes $2$ products $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ from $2$ resources, labor and material. The products have the following resource requirements and produce the accompanying profits. The available quantity of resources is also shown in the table.

Management has developed the following set of goals:

Goal $1$: $\backslash (\backslash$quad $\backslash )$ Produce approximately $40$ units of product $\backslash ($ A $\backslash ).$

Goal $2$: $\backslash (\backslash$quad $\backslash )$ Produce approximately $70$ units of product $\backslash ($ B $\backslash ).$

Goal $3$: Achieve a profit over $\backslash (\backslash$$ $400\backslash ).$

Goal $4$: Consume less than $150$ hours of labor.

Goal $5$: $\backslash (\backslash$quad $\backslash )$ Consume less than $200$ ounces of material.

a$.$ Formulate a goal programming model of this problem so as to minimize the sum of undesirable deviations of the five goals established by the management. $($Let $\backslash ($ d$\_\{$i$\}^\{-\}\backslash )$and $\backslash ($ d$\_\{$i$\}^\{+\}\backslash )$be the deviational variables of goal i$.)(12$ marks$)$

b$.$ If the manager now wants to implement the MINIMAX ojective, what would be the new objective function and the additional constraints for this nodel? $(6$ marks$)$