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  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 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

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 X jk = quantity of nut type j in blend k Let M jk = margin for nut type j in blend k Let π = profit to company So π =  for k =  for j = (M jk X jk )

The Linear Program On future slides, X jk 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 = X jk ≤ 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 ≤ Ar -.2Pr +.8Cr -.2Wr ≤ Ar +.75Pr -.25Cr -.25Wr ≤ Ad -.35Pd +.65Cd -.35Wd ≤ 0 -.5Ab -.5Pb +.5Cb -.5Wb ≤ 0.4Ar +.4Pr +.4Cr -.6Wr ≤ Ad +.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: $ Buy 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 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)

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 = $ ($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 Reveliotis, Spyros. “An introduction to linear programming and the simplex algorithm.” 20 June Waner, Stefan and Steven R. Costenoble. “Tutorial for the simplex method.” May 2000.

Questions?