model 4: the nut company and the simplex method aj epel thursday, oct. 1
TRANSCRIPT
Model 4: The Nut Companyand the Simplex Method
AJ Epel
Thursday, Oct. 1QuickTime™ and a
decompressorare needed to see this picture.
Contents
The ProblemAssumptions and ConstraintsThe Linear ProgramStep-by-step Review: Simplex MethodSolution by ComputerConclusion
The Problem
Three different blends for sale Regular - sells for $0.59/lb Deluxe - sells for $0.69/lb Blue Ribbon - sells for $0.85/lb
Four kinds of nuts can be mixed in each Almonds - costs $0.25/lb Pecans - costs $0.35/lb Cashews - costs $0.50/lb Walnuts - costs $0.30/lb
The Problem
How should the company maximize weekly profit?
What amounts of each nut type should go into each blend?
Use a linear model!
Assumptions and Constraints
Non-negative quantities of nuts and blendsContinuous model: fractions okayCosts, quantities supplied constant from
week to weekCan sell all blends made at their listed
selling pricesNot every nut needs to be in each blend
Assumptions and Constraints
Max. quantities of supplied nuts Almonds: 2000 lbs. altogether Pecans: 4000 lbs. altogether Cashews: 5000 lbs. altogether Walnuts: 3000 lbs. altogether
Assumptions and Constraints
Proportions of one nut to the whole blend Regular
No more than 20% cashews No more than 25% pecans No less than 40% walnuts
Deluxe No more than 35% cashews No less than 25% almonds
Blue Ribbon No more than 50% cashews No less than 30% cashews No less than 30% almonds
The Linear Program
Let Xjk = quantity of nut type j in blend k
Let Mjk = margin for nut type j in blend k
Let π = profit to companySo π = for k = 1...3for j = 1...4 (MjkXjk)
The Linear Program
On future slides, Xjk may be written as Jk J is the nut type: A(lmond), P(ecan), C(ashew),
W(alnut) k is the blend: r(egular), d(eluxe), b(lue ribbon)
The Linear Program
Quantity constraints for j = 1...4Xjk ≤ Max. quantity. for j Example: Ar + Ad + Ab ≤ 2000
Proportion constraints Example: Cr ≤ 0.2(Ar + Pr + Cr + Wr) 0.8Cr - 0.2Ar - 0.2Pr - 0.2Wr ≤ 0
“No less than” constraints Multiply everything by -1
The Linear Program
Max π = .34Ar + .44Ad + .6Ab + .24Pr + .34Pd + .5Pb + .09Cr + .19Cd + .35Cb +.29Wr +.39Wd + .55Wb subject to
Ar + Ad + Ab ≤ 2000 Pr + Pd + Pb ≤ 4000 Cr + Cd + Cb ≤ 5000 Wr + Wd + Wb ≤ 3000 -.2Ar - .2Pr + .8Cr - .2Wr ≤ 0 -.25Ar + .75Pr - .25Cr - .25Wr ≤ 0 -.35Ad - .35Pd + .65Cd - .35Wd ≤ 0 -.5Ab - .5Pb + .5Cb - .5Wb ≤ 0 .4Ar + .4Pr + .4Cr - .6Wr ≤ 0 -.75Ad + .25Pd + .25Cd + .25Wd ≤ 0 .3Ab + .3Pb - .7Cb + .3Wb ≤ 0 -.7Ab + .3Pb + .3Cb + .3Wb ≤ 0
The Tableau: Setup
Step 1 and Step 2
Step 3 and Step 4
Solution by Computer
Conclusion
Maximum weekly profit: $4524.24Buy these:
Almonds: 2000 lbs. Pecans: 4000 lbs. Cashews: 3121 lbs. Walnuts: 3000 lbs.
Conclusion
Blend 5455 lbs. of Regular this way: 1364 lbs. pecan (25% of blend) 1091 lbs. cashew (20% of blend) 3000 lbs. walnut (55% of blend)
Eliminate Deluxe blendBlend 6667 lbs. of Blue Ribbon this way:
2000 lbs. almond (30% of blend) 2636 lbs. pecan (39.55% of blend) 2030 lbs. cashew (30.45% of blend)
Conclusion: What if Deluxe can’t be eliminated?New constraints:
Ar + Pr + Cr + Wr ≥ 1 lb. Ad + Pd + Cd + Wd ≥ 1 lb. Ab + Pb + Cb + Wb ≥ 1 lb.
Solved again Profit = $4524.14 ($0.10/week less) Only 1 lb. of Deluxe manufactured!
75% pecan, 25% almond 1 less lb. of Blue Ribbon
Sources used on the Simplex method
Shepperd, Mike. "Mathematics C: linear programming: simplex method.” July 2003. <http://www.teachers.ash.org.au/miKemath/mathsc/linearprogramming/simplex.PDF>
Reveliotis, Spyros. “An introduction to linear programming and the simplex algorithm.” 20 June 1997. <http://www2.isye.gatech.edu/~spyros/LP/LP.html>
Waner, Stefan and Steven R. Costenoble. “Tutorial for the simplex method.” May 2000. <http://people.hofstra.edu/Stefan_Waner/RealWorld/tutorialsf4/frames4_3.html>
Questions?
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