Linear programming using the simplex method

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Linear programming using the simplex method

  1. 1. Linear Programming Application Using Matrices
  2. 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. 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. 4. To minimize: f = c1x+c2y+c3z ... We maximize: g = -f = -(c1x+c2y+c3z ...)
  5. 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. 6. Aim Distribute land to Maximize profit.
  7. 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. 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. 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. 10. Algorithm 1) In constraints, select the column with min. negative value at bottom -6 < -3 Constraints
  11. 11. Algorithm 2) Pivot element in the selected row is min (value/respected value) =7
  12. 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. 13. Algorithm 4) Repeat until all elements in the last row of constraints become >=0
  14. 14. Solution The last element of last row is the optimal solution.
  15. 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. 16. Graphical Interpretation http://fooplot.com/plot/ipyhavtwvc
  17. 17. Simplex method mechanically traverses every corner point starting with (0,0)
  18. 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. 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. 20. Thank You

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