Transcript Model 4: The Nut Company and the Simplex Method

Model 4: The Nut Company
and the Simplex Method
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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 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)
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: $4524.24
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 = $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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