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Linear programming using the simplex method
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Linear programming using the simplex method

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Transcript

  • 1. Linear Programming Application Using Matrices
  • 2. LP History LP first developed by Leonid Kontorovich in 1939 to plan expenditures and returns during WW 2. It was kept secret until 1947. Revealed after publication of Dantzig's Simplex Algorithm.
  • 3. Application To maximize: f = c1x+c2y+c3z ... Subjected to constraints : 0<= ax + by + cz + ... <= P1 0<= dx + ey + fz + ... <= P2 ... STANDARD FORM (x >= 0 y >= 0 ...)
  • 4. To minimize: f = c1x+c2y+c3z ... We maximize: g = -f = -(c1x+c2y+c3z ...)
  • 5. Crop Plantation Problem 1. L acres of land 2. Two crops to be planted : potato and ladyfinger 3. Budget : a. F for fertilisers b. P for pesticides 4. Crops has the following requirements/ returns per acre per season: Crop Water Manure Pesticide Profit Potato W1 M1 P1 R1 Ladyfinger W2 M2 P2 R2
  • 6. Aim Distribute land to Maximize profit.
  • 7. Simplex Algorithm x = Potato area y = Ladyfinger area Constraints : 1. 2. 3. 4. x , y >= 0 x + y <= L 0<= xP1 + yP2 <= P 0<= xM1 + yM2 <= M (non negative) (land) (Pesticide) (Manure) Aim : To Maximize Profit (f) f = xR1 + yR2
  • 8. Simplex Method Introduce slack variables & remove inequalities Constraints 1. x + y <= L 2. xP1 + yP2 <= P 3. xM1 + yM2 <= M x+y xP1 + yP2 xM1 + yM2 -xR1 - yR2 + u + + + v w f =L =P =M =0
  • 9. For solution purpose, let : P1 = 10, P2 = 12, P = 18 |L=6 M1 = 5, M2 = 7, M = 10 | R1 = 3 ; R2 = 6 Constraints Slacks Values
  • 10. Algorithm 1) In constraints, select the column with min. negative value at bottom -6 < -3 Constraints
  • 11. Algorithm 2) Pivot element in the selected row is min (value/respected value) =7
  • 12. Algorithm 3) Apply row operations to make pivot element = 1 and all other elements in that column = 0 1. R3 = R3 + R4 2. R1 = R1 - R3 3. R2 = R2 - 2R4
  • 13. Algorithm 4) Repeat until all elements in the last row of constraints become >=0
  • 14. Solution The last element of last row is the optimal solution.
  • 15. Determining x,y From final matrix we get the following equations : 1. 2. 3. 4. 0.28x + 1u -0.14w = 4.57 10x + 1v = 18 0.7x + 1y + 0.14w = 1.42 1.28x + 0.85w + 1f = 8.57 Therefore f is 8.57 (max) when x = 0, w = 0 y = 1.42 (using x,w,(3))
  • 16. Graphical Interpretation http://fooplot.com/plot/ipyhavtwvc
  • 17. Simplex method mechanically traverses every corner point starting with (0,0)
  • 18. Reference 1. Wikipedia 2. Logic of how simplex method works by Mathnik http://explain-that.blogspot.in/2011/06/logicof-how-simplex-method-works.html 3. Youtube : http://www.youtube.com/watch? v=qxls3cYg8to
  • 19. Credits 1. Matrix images : Roger's Online Equation Editor http://rogercortesi.com/eqn/ 2. Title font : Amatic Sc by Vernon Adams https://plus. google.com/107807505287232434305/posts
  • 20. Thank You