Question

The Bradley family owns 410 acres of farmland in North Carolina on which they grow corn and tobacco. Each acre of corn costs $105 to plant, cultivate, and harvest; each acre of tobacco costs $210. The Bradleys’ have a budget of $52,500 for next year. The government limits the number of acres of tobacco that can be planted to 100. The profit from each acre of corn is $300; the profit from each acre of tobacco is $520. The Bradleys’ want to know how many acres of each crop to plant in order to maximize their profit.

a. Formulate the linear programming model for the problem and solve.

b. How many acres of farmland will not be cultivated at the optimal solution? Do the Bradleys use the entire 100-acre tobacco allotment?

c. The Bradleys’ have an opportunity to lease some extra land from a neighbor. The neighbor is offering the land to them for $110 per acre. Should the Bradleys’ lease the land at that price? What is the maximum price the Bradleys’ should pay their neighbor for the land, and how much land should they lease at that price?

Bradley Family (a) Let X1 = # of acres of corn and X2 = # of acres of tobacco TEST Corn Tobacco Resources Available 300 5 20 Land Budget Govt. Restriction 105 210 01 | 410 52500 100 X1 X2 Decision Variables Objective function

Answer 1

(a) Linear poto grammy Linear programming model is obta shown below: Given, * There is 410 acres of farmland An Acre of corn costs $105 to plarst & An acre of tobacco cost $210. * Budget is $52,600 max timit of acres of tobacco is 100. 1 & profet for con in $300 profit for tobacco is $520 & Objective is to maximize the profit * Obiective to the formellation of linear model: To Let x be the acre of con at ang on luddin 'y be the acre of tobacco Maximizing = 3002 +5904 then, xty < 410 (Arailable form) 105x+210 y $ 52,500 (mar Budget) x, y = 0 (non -negative) hamda 2100 (max tobacco unito nito mit nito bassade 16.00 Linear equation of model : 300x + 5204 - maximize subjected to above conditions bood

E F B - C objective function Maximum" Profit Max prota 320 90 Acres $142,800.00 SUMPRODUCT (B3C3, B45C7) solven Profet $300.00 $520,00 lek todo 6 constante sign Available ocesowice 7 trailable Parmland rah solve dato to turn 410 [-suM PRODUCT (($B$3:$C$3 BF: 40 & max Budget 105 210 Go below under objective а tobacco lint 0 1 1 62500 SUMPRODUCT (09B$3:$$3,38:0 90 - (-SUMPRODUCT ($B$3: 4043, B: Cabo 52,500 Is 100

from solven the output in : corn(x) =320 acres , tobaccoly) = 90 acre Total profit & $142,800. (6) the number of acres of farmland that will not be cultivated at optimal solution is obtained as below from the above mentioned table a They have used entire 410 acres of available farmland for cultivation as shown in the mentioned table under used resources celumn. I Hence no. of acres of land will not be I cultivated at optimal solution in 410 acres the no. of acres of farmland will not be cultivated at optimal solution is calculated by locating no. of available farmlands' & lused resources the interiected I value in the soluer output forom the Bradley mentioned has used table only as acres of land.

re whether in accepted land offered by the neighor cons not is given below! should nitive can be sensitive generated from MS EXCEL scher ensitince seccable report : Cells Name Final valere 320 redesced cost o objected Allow allow Coefficient Increage decre -ase 200 220 40 cel & B$3 Actees + $($3 acres y. Constraint cells 90 final value o shadow price 520 80 220 constraist Allowable Allow RH side Increas able decrease 40 90 10 410 80 $0$7 Available farmland used olesowice 1050 9450 52500 2.0952381 52500 $D$ 8 Mar Budget used resources 10. o 1E+30 90 100 max $0$9 tobacco used und rusowered accept and offered From above accept they They should not should not by neighbour because shadow price are in above in above suport et $80 whereas bowl is price offered by neighbour $110 (price is very high) that price they must pay neighboue is $80 and they must at lease 410 acres at that pouce , lease a P9