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 of $200$ 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

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.

$($a$)$

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

$$

BeginningInventory

$+$

CurrentProduction

$$

EndingInventory

$=$

This Month'sDemand

$$

May

x$1$s$1=200$

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June

s$1+$x$2$s$2=600$

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July

s$2+$x$3$s$3=600$

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August

s$3+$x$4$s$4=600$

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xi$,$ si $0,$ i $=1,2,3,4$

$($b$)$

Write goal equations that represent the fluctuations in the production level from May to June, June to July, and July to August. $($Let

dni

be the deviation variable below the target value of goal i$,$ and

dpi

be the deviation variable above the target value of goal i for

i $=1,2,3.)$

May to June

Your answer$($s$)$ should be in the form of expression$($s$).$June to July

July to August

xi$,$ si$,$ dpi, dni $0,$ i $=1,2,3,4$

$($c$)$

Inventory carrying costs are high and Standard Pump would like to keep their monthly ending inventories low. Develop goal equations with a target of zero for the ending inventory in May, June, and July. $($Let

dni

be the deviation variable below the target value of goal i$,$ and

dpi

be the deviation variable above the target value of goal i for

i $=4,5,6.)$

May

June

July

si$,$ dpj$,$ dnj $0,$ 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

xi $0,$ 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$4)=$