Model 4: The Nut Company and the Simplex Method

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 p profit to company
• So p ?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 p .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 cant 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.
  lthttp//www.teachers.ash.org.au/miKemath/mathsc/li
  nearprogramming/simplex.PDFgt
• Reveliotis, Spyros. An introduction to linear
  programming and the simplex algorithm. 20 June
  1997. lthttp//www2.isye.gatech.edu/spyros/LP/LP.h
  tmlgt
• Waner, Stefan and Steven R. Costenoble. Tutorial
  for the simplex method. May 2000.
  lthttp//people.hofstra.edu/Stefan_Waner/RealWorld/
  tutorialsf4/frames4_3.htmlgt

20
Questions?