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Simplex Method

07-12-18 Arjun Y 2 comments

Simplex Method

When they are one or more then variables in the given LPP, the most widely used Simplex method is adopted.

Slack Variables:
If the inequality having “<=” sign to make this into equality we had positive is also called “Slack Variable”.

Eg: S1=Slack Variable

Surplus Variable:
If the equality having “>=” sign to make this into equality we had negative is also called “Surplus Variable”.

Eg: S2=Surplus Variable.

  • To solve the problem by giving the following steps:
  • Entering Vector or key column: Min(Δ1,Δ2——-)
  • Leaving Vector or key row: Min ratio of ((xB/xj)/xj>0)
  • Key Element: Intersection of key row and key column is key Element.

Eg1: Solve the following LPP by using Simplex Method
Max z=
S to
where  x1, x2>=0.

Sol:                  Given that Max z=
S.to,
where  x1, x2>=0
: This is in standard from equate the inequality by introducing the “Slack Variable”.
Max z=
S.to
x1, x2>=0.

Constract Simplex table.

   Bv     CB             xB  x1            x2              s1            s2 M/R
S1

S2

0                                    6

0                12

 1               2                1              0

4               3                0              1

6/1=6

12/4=3

                       Δ= -7↑          -5               0              0
S1

X1

0                                    3

7                3

 0               5/4            1           -1/4

1               3/4            0            1/4

S1-x1                        Δ=  0               1/4            0            7/4

:-   Here all >=0

:-   The Solution is optimal.

[Δ=> cBx1-cj

=0(1)-7

=-7

=>cBx2-cj

=0(2)-5

=-5   ]

:-  Δ value consider in the least –ve values will be consider.

Must all Δ value are +ve.

[  M/R=always least value will be considered.

=6 ,3

=> M/R=>3 is the least value will be consider
=> key element always 1,divisible,are multiple——–

Anything key value must be ‘1’.

=> Slack variable value must be always zero.
=> cj values consider in the given problem x1,x2 values
=> s2 is the leaving vector and entering vector is x1, but key value is key column in x1.
=> key value top/down value must be zero, any division, multiplication——-]

:  Max z=

X1=3,x2=0

[:- xB column value in the x1 value.]

=7(3)+5(0)
=21
: Max z=21

 

Ex 2:- Solve the following LPP by using simplex Method

Max z =
S.to
where   x1, x2, x3>=0

Sol: Given that
Max z=
S.to
x1, x2, x3>=0

: This is in standard from Equate the inequality by introducing a Slack Variables.
Max z=x1+x2+3x3
S.to 3x1+2x2+x3<=3

2x1+x2+2x3<=2              x1,x2,x3>=0

Bv   CB                       xB X1    x2      x3      s1     s2 M/R
S1

S2

   0                          3

0                          2

3       2        1         1       0

2       1        2         0       1

3/1=3

2/2=1

                             Δ= -1     -1      -3         0       0
S1

X3

   0                          2

3                          1

 2      3/2    0         1  -1/2

1      1/2    1         0   ½

S1-x3                              Δ=  0      1/2    0         0   3/2

:- Here all Δ’s>=0.

:- The Solution is optimal.

:- Max z=x1+x2+3x3

:- x1=0,x2=0,x3=1

=0+0+3(1)

=3

:-Max z=3



Discussion

  • Mandadi Ramanarayana Reddy

    This method is easy to understand..I learned so much from this method with a neat explanation.

    07-12-18 Reply
  • Surendra

    Simple and easy to study

    07-12-18 Reply

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