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 $is500$ 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

$xm$

denote the number $of$ pumps produced $in$ month $m$ and

$sm$

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.

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 $d_{p1}+d_{n1}+d_{p2}+d_{n2}+d_{p3}+d_{n3}+d_{p4}+d_{n4}+d_{p5}+d_{n5}+d_{p6}+d_{n6}$

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

$(x1,x2,x3,x4,s1,s2,s3,s4)=$

PLEASE USE EXCEL SOLVER $TO$ FIND THE OPTIMAL SOLUTIONS