Question

LA WU25 l. 37 Wildl حالة إكمال الأسئلة 3 Jl لانرجات Farmer Jones must determine how many acres of corn and wheat to plant. You make a profit of SAR 10 and SR 14 respectively from selling the plants. What should your production levels be for both plants? You have 6 farmers (3 labors and 2 engineers), each working 40 hours/week. A corn requires 3 hours of labors and 1 hour of engineer A Wheat requires 2 hours of labors and 2 hours of engineer Let X be the number of corns planted, and LAY be the number of wheats planted. Using decision variables, formulate an LP whose solution will tell farmer Jones how to maximize the total revenue Profit 800 SR Corn 30 Plants: Wheat 30 plants.a Profit 500 SR. Corn 19 plants: Wheat 23 planta Profit 620 SR Corn 20 plants: Wheat 30 Plants a Profit 600 St Corn 21 plants. Wheat 31 plants a

Answer 1

Let 'X' be the no of cons planted and 'y' be the no. of wheats planted. Let the objective function be z (Total Revenue). Profit seen coon = SAR 10 Profit from wheat = SAR 14. ours profit per unite gorening hours 14 Ous objective is to maximize the total Revenue. so the objective function is :- 0 Maximize z = 10x + 14Y - Equation ④ Pubjected to the following constraints :- Product - tabour houes Labour Engineering hones con 10 Wheat Total hours available - 3x40 = 120 2x 40 - 50 (1.8 labours work (Since 2 enginees pe 40 hes each) coock pe to hours each) " to the constraints are :- 3x + 2y = 120 a Eqn 24 < 80 - > San. Ⓡ. y zo Ean Solving the Linear Programming on for the equation 6 as the objective function and S ., & and ② as the constraints. Polving graphically :-

Take there the equation 3x +2y = 120. . the co-oedinates are 140,0), and (0,60) * 3x + 2y = 120. * yo 24 - 120 => zu = 120 y = 60. (0,60) x=40 (0.40) x + 2y = 80, the co-ordinates similaily for are (800) the equation, and (0,40). tof Go (0,60) B (20,30) . 3x +24 x+2y = 80 (80,0 0 10 20 30 40 50 60 to 80 90 7 So the region is OABC. The co-ordinates of poist B can be obtained by solving the equations : 3x + 2y = 120 9 + 24 = 40 2y = 80-20 = 30

are The boundary points 0 (0,6) A (0,40) B (20,30) e (40,0) z at various poroke z = lox + 14y. The value of ase - 0 :- 7 = Ox10 + 0x14 = 0. A ! ZA = 10x0 +14 %40 = 560 B : ZB = 10x20 + 14 x 30 = 620 C - Zc = 10x40 + 14 x0 - 400 at the . The maximum value of z is 620 which is point B , where co-ordinates are (20,30). ie, inoider to maximize the total Revenue, farmee Jones need to plant 20 cons and so wheats so that he will get a profit of SR 6 20. Ans: Option 3'. Prohit - 620 SR Coon - 20 plante , Wheat - 30 plants.