1 / 20

Model 4: The Nut Company and the Simplex Method

Model 4: The Nut Company and the Simplex Method. AJ Epel Thursday, Oct. 1. Contents. The Problem Assumptions and Constraints The Linear Program Step-by-step Review: Simplex Method Solution by Computer Conclusion. The Problem. Three different blends for sale Regular - sells for $0.59/lb

amelia
Download Presentation

Model 4: The Nut Company and the Simplex Method

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Model 4: The Nut Companyand the Simplex Method AJ Epel Thursday, Oct. 1

  2. Contents • The Problem • Assumptions and Constraints • The Linear Program • Step-by-step Review: Simplex Method • Solution by Computer • Conclusion

  3. 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

  4. The Problem • How should the company maximize weekly profit? • What amounts of each nut type should go into each blend? • Use a linear model!

  5. Assumptions and Constraints • Non-negative quantities of nuts and blends • Continuous model: fractions okay • Costs, quantities supplied constant from week to week • Can sell all blends made at their listed selling prices • Not every nut needs to be in each blend

  6. Assumptions and Constraints • Max. quantities of supplied nuts • Almonds: 2000 lbs. altogether • Pecans: 4000 lbs. altogether • Cashews: 5000 lbs. altogether • Walnuts: 3000 lbs. altogether

  7. 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

  8. 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 company • So π = for k = 1...3for j = 1...4 (MjkXjk)

  9. 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)

  10. 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

  11. 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

  12. The Tableau: Setup

  13. Step 1 and Step 2

  14. Step 3 and Step 4

  15. Solution by Computer

  16. Conclusion • Maximum weekly profit: $4524.24 • Buy these: • Almonds: 2000 lbs. • Pecans: 4000 lbs. • Cashews: 3121 lbs. • Walnuts: 3000 lbs.

  17. 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 blend • Blend 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)

  18. 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

  19. 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>

  20. Questions?

More Related