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Simplex method

Simplex method. Operational Research By TMJA Cooray. General LP problem is to find the values of X 1 ,X 2 ,…..X n which maximizes (or minimizes) the objective function Max (or Min) Z= C 1 X 1 +C 2 X 2 +………C n X n While satisfying the constraints

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Simplex method

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  1. Simplex method Operational Research By TMJA Cooray

  2. General LP problem is to find the values of X1,X2,…..Xn which maximizes (or minimizes) the objective function • Max (or Min) Z= C1X1+C2X2+………CnXn • While satisfying the constraints • a11X1+…….a1nXn ≤b1 or ≥ b1 • a21X1+…….a2nXn ≤b2 or ≥ b2 • …….. • am1X1+…….amnXn ≤bm or ≥ bm • andX1, ….Xn≥0 MA 4020 Level 4 OPerational research

  3. This problem can be put in the canonical form as follows. • Inequalities can be converted to equalities. • a11X1+…….a1nXn ±S1 =b1 • a21X1+…….a2nXn ±S2 =b2 • …….. • am1X1+…….amnXn ±Sm =bm • ( by adding a slack or a surplus variable) MA 4020 Level 4 OPerational research

  4. Minimization or maximization • Max (min) Z= C1X1+C2X2+………CnXn can be converted to a min (max) Z’=-{ C1X1+C2X2+………CnXn} by multiplying the objective function by -1. MA 4020 Level 4 OPerational research

  5. Variables unrestricted in sign • Xi can take any value, either positive or negative. • In such a case it can be replaced by • Xi = Xi’-Xi’’ MA 4020 Level 4 OPerational research

  6. DEFINITIONS Slack variables: are defined, when there are ≤ inequalities. In the example discussed , the available capacities of the three machines M1,M2 and M3 are 40,40 and 40 respectively.. Unused amounts of the three machines are denoted by X3,X4 and X5. (They are ≥0.) Some books denote them by Sj . MA 4020 Level 4 OPerational research

  7. Surplus variables: • are defined when there are ≥ inequalities. • In the diet planning problem of the dog, The minimum protein requirements are specified on the r.h.s.. If you feed more, the dog gets more than what is required. The excess amount of protein is denoted by Xj or Sj .(they are ≥ 0) The amount overfed is the surplus variable. By subtracting that amount we get the equality. MA 4020 Level 4 OPerational research

  8. Non basic variables: • The variables which have the value zero are called non basic variables. • Basic variables: • Variables which are positive are called basic variables. However some times the basic variables can have zero values and then the solution is said to be degenerate . • . MA 4020 Level 4 OPerational research

  9. Simplex method. • Consider the same problem solved using the graphical method. • This procedure is equivalent to find the cdts of the corner points of the f. region. This method consists of changing set of basic variables one at a time until Z (or f) is maximized. MA 4020 Level 4 OPerational research

  10. The first step is to determine an initial basic feasible solution: an obvious solution is x1=0,x2=0, giving x3=40,x4=40 and x5=40. • This is equivalent to corner point O in the graphical solution. • Step 2: • Solve for the basic variables in terms of the non basics and express f in terms of non basics. MA 4020 Level 4 OPerational research

  11. X3=40-.25x1-.5x2 • X4=40-.4x1-.2x2 • X5 =40 -.8x2 f=2x1+3x2 ……. MA 4020 Level 4 OPerational research

  12. MA 4020 Level 4 OPerational research

  13. PIVOT COLUMN 40/.5 = 80 40/.2 =200 40/.8 =50 PIVOT ROW MA 4020 Level 4 OPerational research

  14. PIVOT COLUMN Row- Pivot row *3 pivot elt 2 0 0 0 -3.75 -150 0 .25 0 1 0 -.625 15 Row- Pivot row *.5 pivot elt 40/.5 = 80 40/.2 =200 40/.8 =50 Row- Pivot row *.2 pivot elt 0 4 0 0 1 -.25 30 0 0 .8/.8 0 0 1/.8 50 PIVOT ROW… PIVOT ELEMENT X2 PIVOT ROW MA 4020 Level 4 OPerational research

  15. PIVOT COLUMN Row- Pivot row *3 pivot elt 2 0 0 0 -3.75 -150 0 .25 0 1 0 -.625 15 Row- Pivot row *.5 pivot elt 40/.5 = 80 40/.2 =200 40/.8 =50 Row- Pivot row *.2 pivot elt 0 4 0 0 1 -.25 30 PIVOT ROW… PIVOT ELEMENT 0 0 .8/.8 0 0 1/.8 50 PIVOT ROW MA 4020 Level 4 OPerational research

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