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DETAILS

ASWQUANT$137.$E$\mathrm{.}029.$

Autolgnite produces electronic ignition systems for automobiles at a plant in Cleveland, Ohio. Each ignition system is assembled from' two components produced at AutoIgnite's plants in Buffalo, New York, and Dayton, Ohio. The Buffalo plant can produce $2,000$ units of component $1,1,000$ units of component $2,$ or any combination of the two components each day. For instance, $60\%$ of Buffalo's production time could be used to produce component $1$ and $40\%$ of Buffalo's production time could be used to produce component $2$ ; in this case, the Buffalo plant would be able to produce $0\mathrm{.}6(2,000)=1,200$ units of component $1$ each day and $0\mathrm{.}4(1,000)=400$ units of component $2$ each day. The Dayton plant can produce $300$ units of component $1,1,700$ units of component $2,$ or any combination of the two components each day. At the end of each day, the component production at Buffalo and Dayton is sent to Cleveland for assembly of the ignition systems on the following workday.

$($a$)$ Formulate a finear programming model that can be used to develop a daily production schedule for the Buffalo and Dayton plants that will maximize daily production of ignition systems at Cleveland. $($Assume $B$ is the proportion of Buffalo's time used to produce component $1$ and $D$ is the proportion of Dayton's time used to produce component $1.)$

Max

s$.$t$.$

Buffelo maximum

Dayton maximum

equal number of components $1$ and $2$

$B,D0$

$($b$)$ What is the value of the objective function $($in units per day$)$ at the optimal solution?

units per day $at(B,D)=(,x)$