$[\frac{-}{13}$ Points$]$

CAMMIMS$1614.E\mathrm{.}003.$

The $L.$ Young & Sons Manufacturing Company produces two products, which have the following profit and resource

requirement characteristics.

Last month's production schedule used $350$ hours $of$ labor $in$ department A and $1000$ hours $of$ labor $in$ department $B.$

Young's management has been experiencing workforce morale and labor union problems during the past six months

because $of$ monthly departmental workload fluctuations. New hiring, layoffs, and interdepartmental transfers have been

common because the firm has not attempted $to$ stabilize workload requirements.

$M$ Management would like $to$ develop a production schedule for the coming month that will achieve the following goals.

Goal $1$: Use $350$ hours $of$ labor $in$ department $A.$

Goal $2$: Use $1,000$ hours $of$ labor $in$ department $B.$

Goal $3$: Earn a profit $ofat$ least $$1,300.$

$(a)$ Assuming that goals $1$ and $2$ are $P_1$ level goals and goal $3isa{P}_2{x}_{i}be$ the number $of$

units $of$ product i produced for $i=1,2,$ let $d_{pj}be$ the deviation variable which $is$ greater than the value $of$ goal $j,$ and

let $d_{nj}be$ the deviation variable which $is$ less than the value $of$ goal $j,$ for $j=1,2,3.Min{P}_1(d_{p1})+P_1(d_{n1})+P_2(d_{p2})+P_2(d_{n2})+P_3(d_{n3})$

$Min{P}_1(d_{n1})+P_1(d_{n2})+P_2(d_{n3})$

$Min{P}_1(d_{p1})+P_1(d_{p2})+P_2(d_{n1})+P_2(d_{n2})+P_2(d_{n3})$

$Min{P}_1(d_{p1})+P_1(d_{p2})+P_2(d_{n3})$

$Min{P}_1(d_{p1})+P_1(d_{n1})+P_1(d_{p2})+P_1(d_{n2})+P_2(d_{n3})$

Formulate the constraints $of$ the goal programming model for the objective function above.

Goal $1$

Goal $2$

Goal $3$

$x_i,d_{n{j}^{\text{'}}}{d}_{pj}0,$ for $i=1,2$ and $j=1,2,3$

$(b)$ Solve the model formulated $in$ part $(a)$ using the graphical goal programming procedure and determine the profit $(in$

dollars$).$

$

$at(x_1,x_2)=()$

$(c)$ Suppose that the firm ignores the workload fluctuations and considers the $350$ hours $in$ department A and the $1,000$

hours $in$ department $Bas$ the maximum available. Formulate linear programming problem $to$ maximize profit subject

$to$ these constraints.

Max

$s.t.$

Department A Hours

Department $B$ Hours

$x_{i}0,$ for $i=1,2$

Find the optimal solution $(in$ dollars$)to$ this linear program.

$

$at(x_1,x_2)=()$

$(d)$ Reconsider part $(a)$ assuming that the priority level $1$ goal $is$ goal $3$ and the priority level $2$ goals are goals $1$ and $2$;

$as$ before, assume that goals $1$ and $2$ are equally important. Solve this revised problem using the graphical goal

programming procedure and determine the profit $(in$ dollars$).$

$

$at(x_1,x_2)=(,)anagement$ would like $to$ develop a production schedule for the coming month that will achieve the Use $350$ hours $of$ labor $in$ dep $2$: Use $1,000$ hours $of$ laborepartment $B.$

Earn a profit $ofat$ least $$1,300$