Linear Programming for Transportation Problems $(5$ points$)$

Let us consider the problem faced by Foster Generators, Inc. Currently, Foster

has three plants: one in Cleveland, Ohio, one in Bedford, Indiana, and one in

York, Pennsylvania. Generators produced at the plants $($origins$)$ are shipped to

distribution centers $($destinations$)$ in Boston, Chicago, St$.$ Louis, and Lexington,

Kentucky.

Using a typical, one$-$month planning period, the production capacities at the

three plants are shown in Exhibit $6\mathrm{.}10.$ Forecasts of monthly demand at the four

distribution centers are shown in Exhibit $6\mathrm{.}11.$ Note that the total production

capacity must be at least as large as the total demand in order to meet the

forecasted demand. The transportation cost per unit for each route is shown in

Exhibit $6\mathrm{.}12.$

According to above information, please help the manager to decide the best

way to distribute the product from the plants to the distribution centers. $\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|\}$

$\backslash$hline $\backslash$multicolumn$\{3\}\{|$l$|\}\{$EXHIIBIT $6\mathrm{.}11$ Foster Generators Monthly Forecast$\}\backslash \backslash$

$\backslash$hline Destination & Distribution Center & Demand Forecast $($units$)\backslash \backslash$

$\backslash$hline $1$ & Boston & $6,000\backslash \backslash$

$\backslash$hline $2$ & Chicago & $4,000\backslash \backslash$

$\backslash$hline $3$ & St$.$ Louis & $2,000\backslash \backslash$

$\backslash$hline $4$ & Lexington & $1,500\backslash \backslash$

$\backslash$hline & & Total $13,500\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

$\backslash$begin$\{$tabular$\}\{|$l$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline EXHIBIT $6\mathrm{.}12$ & $\backslash$multicolumn$\{4\}\{$c$|\}\{$ Foster Generators Transportation Cost per Unit $\}\backslash \backslash$

Destination $\backslash \backslash$

$\backslash$hline Origin & Boston & Chicago & St$.$ Louis & Lexington $\backslash \backslash$

$\backslash$hline Cleveland & $\backslash (\backslash$$ $3\backslash )$ & $\backslash (\backslash$$ $2\backslash )$ & $\backslash (\backslash$$ $7\backslash )$ & $\backslash (\backslash$$ $6\backslash )\backslash \backslash$

Bedford & $7$ & $5$ & $2$ & $3\backslash \backslash$

$\backslash$hline York & $2$ & $5$ & $4$ & $5\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$