A linear programming computer package is needed.

EZ$-$Windows, Inc., manufactures replacement windows for the home remodeling business. In January, the company produced $15,000$ windows and ended the month with $9,000$ windows in inventory. EZ$-$Windows' management team would like to develop a production schedule for the next three months. A smooth production schedule is obviously desirable because it maintains the current workforce and provides a similar month$-$to$-$month operation. However, given the sales forecasts, the production capacities, and the storage capabilities as shown, the management team does not think a smooth production schedule with the same production quantity each month possible.

February March April

Sales forecast $15,00016,50020,000$

Production capacity $14,00014,00018,000$

Storage capacity $6,0006,0006,000$

The company's cost accounting department estimates that increasing production by one window from one month to the next will increase total costs by $$1\mathrm{.}00$ for each unit increase in the production level. In addition, decreasing production by one unit from one month to the next will increase total costs by $$0\mathrm{.}65$ for each unit decrease in the production level. Ignoring production and inventory carrying costs, formulate a linear programming model that will minimize the cost $($in dollars$)$ of changing production levels while still satisfying the monthly sales forecasts. $($Let F $=$ number of windows manufactured in February, M $=$ number of windows manufactured in March, A $=$ number of windows manufactured in April, I$1=$ increase in production level necessary during month $1,$ I$2=$ increase in production level necessary during month $2,$ I$3=$ increase in production level necessary during month $3,$ D$1=$ decrease in production level necessary during month $1,$ D$2=$ decrease in production level necessary during month $2,$ D$3=$ decrease in production level necessary during month $3,$ s$1=$ ending inventory in month $1,$ s$2=$ ending inventory in month $2,$ and s$3=$ ending inventory in month $3.)$

Min

s$.$t$.$

February Demand

March Demand

April Demand

Change in February Production

Change in March Production

Change in April Production

February Production Capacity

March Production Capacity

April Production Capacity

February Storage Capacity

March Storage Capacity

April Storage Capacity

Find the optimal solution.

$($F$,$ M$,$ A$,$ I$1,$ I$2,$ I$3,$ D$1,$ D$2,$ D$3,$ s$1,$ s$2,$ s$3)=$

Cost $=$ $