Model 4: The Nut Companyand the Simplex Method

AJ Epel

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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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