$1$ Farmer Leary grows wheat and corn on his $45-$acre farm. He can sell at most $140$ bushels of wheat and $120$ bushels of corn. Each acre planted with wheat yields $5$ bushels, and each acre planted with corn yields $4$ bushels. Wheat sells for $$30$ per bushel, and corn sells for $$50$ per bushel. To harvest an acre of wheat requires $6$ hours of labor; $10$ hours are needed to harvest an acre of corn. Up to $350$ hours of labor can be purchased at $$10$ per hour. Let $A1=$ acres planted with wheat; $A2=$ acres planted with

FIGURE $6$ LINDO Output for Wheat and Corn

$\backslash$table$[[$MAX $150,$A$1+,200,$A$2,-,10,$L$],[$SUBJECT TO$],[2),$A$1+,$A A$1,+10,$A$2,-$ LL A A$2,,120,,,]]$

END

LP OPTIMUM FOUND AT STEP

corn; and $L=$ hours of labor that are purchased. To maximize profits, Leary should solve the following LP:

maxz$=150A1+200A2-10L$

$s.t.A1+A245$

$6A1+10A2-L0$

$350$

$5A1140$

$4A2120$

$A1,A2,L0$

Use the LINDO output in Figure $6$ to answer the following questions:

a If only $40$ acres of land were available, what would Leary's profit be$?$

b If the price of wheat dropped to $$26,$ what would be the new optimal solution to Leary's problem?

c Use the SLACK portion of the output to determine the allowable increase and allowable decrease for the amount of wheat that can be sold. If only $130$ bushels of wheat could be sold, then would the answer to the problem change?

OBJECTIVE FUNCTION VALUE

$4250\mathrm{.}00000$

VARIABLE VALUE REDUCED COST

$\backslash$table$[[$A$1,25\mathrm{.}000000,\mathrm{.}000000],[$A$2,20\mathrm{.}000000,\mathrm{.}000000],[$L$,350\mathrm{.}000000,\mathrm{.}000000]]$

$\backslash$table$[[$ROW$,$SLACK OR SURPLUS,DUAL PRICES$],[2),\mathrm{.}000000,75\mathrm{.}000000],[3),\mathrm{.}000000,12\mathrm{.}500000],[4),\mathrm{.}000000,2\mathrm{.}500000],[5),15\mathrm{.}000000,\mathrm{.}000000],[6),40\mathrm{.}000000,\mathrm{.}000000]]$

NO$.$ ITERATIONS$=$

$4$

RANGES IN WHICH THE BASIS IS UNCHANGED:

$\backslash$table$[[,$OBJ,COEFFICIENT$],[$VARANGES$,,,],[$VARIABLE$,$CURRENT,ALLOWABLE,ALLOWABLE$],[,$COEF,INCREASE,DECREASE$],[$A$1,150\mathrm{.}000000,10\mathrm{.}000000,30\mathrm{.}000000],[$A$2,200\mathrm{.}000000,50\mathrm{.}000000,10\mathrm{.}000000],[$L$,-10\mathrm{.}000000,$INFINITY,$2\mathrm{.}500000]]$

$\backslash$table$[[,,$RIGHTHAND$],[$ROW$,$CURRENT,ALLOWABLE RANGES,ALLOWABLE$],[,$RHS$,$INCREASE,DECREASE$],[2,45\mathrm{.}000000,1\mathrm{.}200000,6\mathrm{.}666667],[3,\mathrm{.}000000,40\mathrm{.}000000,12\mathrm{.}000000],[4,350\mathrm{.}000000,40\mathrm{.}000000,12\mathrm{.}000000],[5,140\mathrm{.}000000,$INFINITY,$15\mathrm{.}000000],[6,120\mathrm{.}000000,$INFINITY,$40\mathrm{.}000000]]$