Standard Pump recently won $a$ $$14$ million contract with the $U.S.$ Navy $to$ supply $2,000$ custom$-$designed submersible pumps over the next four months. The contract calls for the

delivery $of200$ pumps $at$ the end $of$ May, $600$ pumps $at$ the end $of$ June, $600$ pumps $at$ the end $of$ July, and $600$ pumps $at$ the end $of$ August. Standard's production capacity $is$

$500$ pumps $in$ May, $400$ pumps $in$ June, $800$ pumps $in$ July, and $500$ pumps $in$ August. Management would like $to$ develop a production schedule that will keep monthly ending

inventories low while $at$ the same time minimizing the fluctuations $in$ inventory levels from month $to$ month. $In$ attempting $to$ develop a goal programming model $of$ the problem,

the company's production scheduler let $x_m$ denote the number $of$ pumps produced $in$ month $m$ and $s_m$ denote the number $of$ pumps $in$ inventory $at$ the end $of$ month $m.$ Here,

$m=1$ refers $to$ May, $m=2$ refers $to$ June, $m=3$ refers $to$ July, and $m=4$ refers $to$ August. Management asks you $to$ assist the production scheduler $in$ model development.

$(a)$ Using these variables, develop a constraint for each month that will satisfy the following demand requirement.

$(\frac{\text{B}\text{e}\text{g}\text{i}\text{n}\text{n}\text{i}\text{n}\text{g}}{\text{I}\text{n}\text{v}\text{e}\text{n}\text{t}\text{o}\text{r}\text{y}})+(\frac{\text{C}\text{u}\text{r}\text{r}\text{e}\text{n}\text{t}}{\text{P}\text{r}\text{o}\text{d}\text{u}\text{c}\text{t}\text{i}\text{o}\text{n}})-(\frac{\text{E}\text{n}\text{d}\text{i}\text{n}\text{g}}{\text{I}\text{n}\text{v}\text{e}\text{n}\text{t}\text{o}\text{r}\text{y}})=(\frac{\text{T}\text{h}\text{i}\text{s}\text{M}\text{o}\text{n}\text{t}\text{h}\text{'}\text{s}}{\text{D}\text{e}\text{m}\text{a}\text{n}\text{d}})$

May

June

July

August

$x_i,s_{i}0,i=1,2,3,4d_{ni}be$ the deviation variable below the

target value $of$ goal $i,$ and $d_{}be$ the deviation variable above the target value $of$ goal i for $i=1,2,3.x_i,s_i,d_{{}^{\text{'}}}{d}_{ni}0,i=1,2,3,4d_{ni}be$ the deviation variable below the target value $of$ goal $i,$ and $d_{}be$ the deviation variable above the target value $of$ goal i for

$i=4,5,6.x_i,s_i,d_{{}^{\text{'}}}{d}_{ni}0,i=1,2,3,4d_{ni}be$ the deviation variable below the target value $of$ goal $i,$ and $d_{}be$ the deviation variable above the target value $of$ goal i for

$i=4,5,6s_{i^{\text{'}}}{d}_{p{j}^{\text{'}}}{d}_{nj}0,i=1,2,3$ and $j=4,5,6$

$(d)In$ addition $to$ the goal equations developed $in$ parts $(b)$ and $(c),$ develop constraints for the production capacities for each month.

May

June

July

August

$x_{i}0,i=1,2,3,4$

$(e)$ Assuming the production fluctuation and inventory goals are $of$ equal importance, write $an$ objective function for a goal programming model which, when used with the

constraints constructed $in$ parts $(a)-(d),$ can $be$ used $to$ determine the best production schedule. Develop and solve a goal programming model $to$ determine the best

production schedule.

Min

What $is$ the optimal solution $to$ the goal programming model?

$(x_1,x_2,x_3,x_4,s_1,s_2,s_3,s_4)=(,)$