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Simplex method

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Operational Research. By. TMJA Cooray. General LP problem is to find the values of ... is denoted by Xj or Sj .(they are 0) The amount overfed is the surplus variable. ... – PowerPoint PPT presentation

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Title: Simplex method


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 C1X1C2X2CnXn
  • While satisfying the constraints
  • a11X1.a1nXn b1 or b1
  • a21X1.a2nXn b2 or b2
  • ..
  • am1X1.amnXn bm or bm
  • and X1, .Xn 0

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)

4
Minimization or maximization
  • Max (min) Z C1X1C2X2CnXn
  • can be converted to a
  • min (max) Z- C1X1C2X2CnXn
  • by multiplying the objective function by -1.

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

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 .
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.

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 .
  • .

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.

10
  • The first step is to determine an initial basic
    feasible solution an obvious solution is
    x10,x20, giving x340,x440 and x540.
  • 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.

11
  • X340-.25x1-.5x2
  • X440-.4x1-.2x2
  • X5 40 -.8x2
  • f2x13x2
  • .

12
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13
PIVOT COLUMN
40/.5 80 40/.2 200 40/.8 50
PIVOT ROW
14
PIVOT COLUMN
Row- Pivot row 3 pivot elt
2 0 0 0 -3.75 -150
0 .25 0 1 0 -.625 15
40/.5 80 40/.2 200 40/.8 50
Row- Pivot row .5 pivot elt
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
15
PIVOT COLUMN
Row- Pivot row 3 pivot elt
2 0 0 0 -3.75 -150
0 .25 0 1 0 -.625 15
40/.5 80 40/.2 200 40/.8 50
Row- Pivot row .5 pivot elt
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
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