Question

Primal and Dual Problems

Your company grows two types of plants, A and B (e.g., roses and begonias). Both plant types require two types of fertilizer throughout the growing season, Fertilizer 1 and Fertilizer 2.A single plant A will require 2 pounds of fertilizer 1 and 1 pound of fertilizer 2.A single plant B will require 2 pounds of fertilizer 1 and 2 pounds of fertilizer 2.Your distributer has 4000 pounds of fertilizer 1 and 5000 pounds of fertilizer 2 available for delivery at the beginning of the season.You know that every plant A you grow will generate $2.25 profit, and every plant B you grow will generate $2.60 profit.

Solve both the Primal and the Dual problems for this scenario (as if the variables were continuous).

Generate the sensitivity reports for both problems and relate the shadow prices of each to the solutions of the others, to include interpreting the limits.

Interpret and explain the solution values and the variable meanings for both problems, to include the decision variables and the shadow prices.

Please show how you setup the model in excel and include your solvers setup so I may reference

Answer 1

Son x, B 2 pounds of fertilizers and first formulate the problem in equations let be the fertilizers de type A and My be the fertilizer a od type if a will required ne pound of fertilizer a i.e. 24, +42 and distributer has 4000 pounds of fertilizers 2017H2 € 460o Simidarly an, tan2 < 5ooo and total profit of A and B = 2.257,+2.6037 Maximize Z= 2.25 xit 2.6022 Subject to Constraints, one i.e. 27, +72 4050 Soo 22, +222 above problem is primal and its solutien is 4coo govo K B region founded 2010 erot = "htthet C C 1000 A (90) 3000 1000 2000 20750 2*, + x2 = 4032 => if m=0 => x2 = 4000 and if r2 = 0 =) 1.² so points (2000, 2000

and 200+ 202 = 5000 if 4,20 = 72 = 2500 and if n2 =0) = 2500 pointos (2500, 2500) equations 21,7 24, = 4000 221 +242 = get the point Solving equunkiem ( and (2) we ofe intersection is (1500, 1000) region foundled between three pointly A (2000,0) B (0,2500) and C (1500, 1000) Maximize the problem of 2.250, +2.6072 at Ziegrov,«) 2.25 X 2000 +2.60XO 2000,0) $ sooo at B = 710,2500) 2:25 X0 + 2.60 X 2500 = $6500 at C Z isoo, I 2.25X1500+ 2.60x1000 Cisco, 1000) 3375 + 2600 - $5975 rience we get Maximize B is $ 6500 the problem at point

and solution is by graphical method and dual of the primal problem is Minz May (-2) 40004,- 5000m2 and subject to Censtscinta, 2917 342 > 2.25 a + 272 > 2.60 2.25 27.7 272 2.60 2 Ik + 2012 1.125 = ?k to='k fi' So G=0kb t'ke sell='n kostet = be Coolk fog.b = 746 tlk 1.30 if ^2=0 =) M1 = 9.60 and their points of intersection is (-0.35, 1.475) 3 y unboundled region 28, +242 = 2.25 M+424 2.60 nitame A 2

we get unbounded region at three points A (2.60,0), B(0,1.30) and C (-0.35,1.475) Min 7 at A, B and cis de -40000 - 5000 ty 218.60,0) = - 4000 x 260-soou to = - 10400 (in dollar) 2 (0,1.30) - 4000 Xo-sroux 1.30 =-6500 (in dollar) 26.0.3671.975) = -4000 x 60.25-sooox 1.475 - 1400-7375 hence we .8775 (in dollar) eget Min I ofe dual problem at point A is $-10400 Ang