# Operations Research

<<< Previous Linear Programming:ARTIFICIAL VARIABLE TECHNIQUE Next >>>

Operations Research (MTH601)
127
ARTIFICIAL VARIABLE TECHNIQUE
If slack variables do not provide an initial basic feasible solution then the question may arise as to how to
start the initial table of simplex method and proceed. This is the case when the slack variables have negative values.
For example, let us consider a constraint
2x + 3y > 15
The method of converting this inequaly (with greater than equal to) into an equation, is to subtract a slack variable
so that we have  2x + 3y - S = 15
Now if x and y are non-basic variables in the problem, then S is taken as the starting basic variable. But the value of
S = -15 which is infeasible. We cannot proceed with the further iteration of the simplex method with infeasible basic
solution.
So to obtain a starting solution, we adopt the 'artificial variable' technique. Two methods are available using
the artificial variables. They are,
(1)
The "Big M technique" or the Charnes method of penalty.
(2)
The two phase technique.
The 'Big M' Technique
If some of the constraints in the linear programming problems are of the type ( > ) or ( = ), we have to use
the M technique for maximization as well as minimization of an objective function. The various steps of the M
technique are given below.
STEP 1 Express the given linear programming problem in the equation form by bringing all the terms in the
objective function to the left hand side and the constraints are also expressed in the equation form by including slack
variables (Add slack variable for constraint of the type < and subtract slack variable for constraint of the type > ).
Now obtain a basic solution for the problem, which will be an infeasible one as the basic variable is
negative in the cases where the constraints are of the type ( > ).
STEP 2 To get a starting basic feasible solution, add n0n-negative variables to the left hand side of each of the
equations corresponding to the constraints of the types ( > ) and ( = ). These variables are called artificial variables.
Thus we change the constraint to get a basic solution. This violates the corresponding constraints. This is only for
the starting purpose. But in the final solution (if it exists) if the artificial variables will become non-basic, (their
values will be zero) then we are coming back to the original constraints. This method or driving the artificial
variables out of the basis is called the Big M technique. This result is achieved by assigning a very large (big) per
unit penalty to these variables in the objective function. Such a penalty will be a -M for maximization and a +M for
minimization problems, on the right hand side, the value of M being strictly positive. By attaching these per unit
penalties to the artificial variables we ensure that they will never become the candidates for entering variables once
they are driven out.
STEP 3 For the starting basic solution; use the artificial variables in the basis. Now the starting table in the simplex
procedure should not contain the terms involving the basic variables, (one of the conditions to be satisfied by the
127
Operations Research (MTH601)
128
simplex method). But we will have the terms like +MA or -MA in a maximization or minimization problem
respectively in the left hand side of the objective row. In other words, the objective function must be expressed in
terms of non-basic variables only. This leads us to have the coefficients of the artificial variables (starting basic
variables) equal to zero in objective row. This result is obtained by adding suitable multiples of the constraint
equations involving artificial variables to the objective row.
STEP 4 Proceed with the regular steps of the simplex method. If the artificial variables leave the basis in the final
solution, then we come back to the original problem. But if any or all of the artificial variables do not leave the basis
in the final solution, then this indicates that the problem does not have a solution.
Example
Consider the problem
Maximize
Z = 2x + 5y
Subject to
< 40,
< 30,
x + y > 60,
x, y > 0
x
y
Solution
Bring the problem to the standard form by including slack variables.
STEP 1
Z - 2x - 5y
=0
+ S1
= 40
x
+ S2
= 30
y
x+y
+ S3 = 60
Solution at this stage is
Z =0
S1 = 40
S2 = 30
Basic variables
S3 = -60
x = 0
Non-basic variables
y = 0
STEP 2 Since S3 = -60 and as such is infeasible, add an artificial variable A to get an initial basic solution. Then we
x + y - S3 + A = 60
have the third constraint equation changed to
STEP 3 Modify the objective function by including a very large per unit penalty M. Thus for maximization problem
we add -MA to RHS of the objective function which will be
Z - 2x - 5 y = -MA
or
128
Operations Research (MTH601)
129
Z - 2x - 5 y + MA = 0
with the constraints
+ S1
= 40
x
+ S2
= 30
y
- S3 + A = 60
x+ y
Now the required condition is that the objective function must be expressed in terms of the non-basic variables only.
Or the coefficients of the basic variables in the objective function must be zero. In the problem, the starting basic
variables are S1, S2 and A. The coefficients of A must be made 0 in the objective function, a result which is obtained
by multiplying the corresponding constraint equation including the artificial variable by -M and adding to the
objective function.
Now Z equation = old Z equation + (-M) A equation
Thus we have the revised objective function as,
(Z - 2x - 5y + MA = 0) + (-M) (x + y - S3 + A = 60)
(i.e.)
Z - 2x - Mx - 5y - My + MS3 = -60 M
(i.e.)
Z + (-2 - M)x + (-5 - M) y + MS3 = -60M
Now the initial table in simplex method is presented below and further iterations are carried out to obtain a
final feasible solution.
Starting Table:
Coefficient of
RHS
Ratio
Eqn.
Basic
No.
Variable
S1  S2
S3
A
Z
x
y
0
-
1
-2-M
-5-M  0  0
M
0
-
60M
1
0
1
0
1
0
0
0
40
S1
2
0
0
1
0
1
0
0
30
30
S2
3
0
1
1
0
0
-1
1
60
60
A
First Iteration: y enters and S2 leaves the basis.
RHS
Ratio
Eqn.
Basic
Coefficient of
No.
Variable
S2
S3
A
Z
x
y  S1
0
-
1
-2-M  0  0  5+M
0
150-30M
M
1
0
1
01
0
0
0
40
40
S1
2
0
0
10
1
0
0
30
y
3
0
1
00
-1
-1
1
30
30
A
129
Operations Research (MTH601)
130
Second Iteration: x enters and A leaves the basis.
RHS
Ratio
Eqn.
Basic
Coefficient of
No.
Variable
S2
S3
A
Z
x
y
S1
0
-
1
0
0
0
7
-2
2+M
210
1
0
0
0
1
1
1
-1
10
10
S1
2
0
0
1
0
1
0
0
30
y
3
0
1
0
0
-1
-1
1
30
-30
x
Third Iteration: S3 enters and S1 leaves the basis.
Eqn.
Basic
Coefficient of
RHS
No.
Variable
S2
S3
A
Z
x
y
S1
0
-
1
0
0
2
9
0
230*
M
1
0
0
0
1
1
1
-1
10
S3
2
0
0
1
0
1
0
0
30
y
3
0
1
0
1
0
0
0
40
x
From the above table, we see that there is no negative coefficient in the objective row. This indicates that
we have reached the optimal solution to the problem.
Another fact which can be noticed that the artificial variable A has left the basis.
Hence we have the original constraint and the original objective function preserved.
The optimal solution
Z* = 230
x = 40
y = 30
S3 = 10
and
S1 = S2 = A = 0
Using surplus and artificial variable, solve the following:
Example
Minimize
Z = 5x1 + 6x2
Subject to
2x1 + 5x2 > 1500
3x1 + x2 > 1200
x1, x2 > 0
Solution
Introducing slack (surplus) variables S1 and S2 and artificial variables A1 and A2 to the two constraints the
problem becomes,
130
Operations Research (MTH601)
131
Minimize
Z = 5x1 + 6x2 + MA1 + MA2
Subject to
2x1 + 5x2 - S1 + A1 = 1500
3x1 + x2 - S2 + A2 = 1200
Since the objective function should not involve coefficient of basic variables A1 and A2, we multiply the
constraint equation with M and add to the objective function. The revised objective function will be
Z - 5x1 + 5Mx1 - 6x2 + 6Mx2 - MS1 - MS2 = 2700M
We prepare the simplex table as follows:
Initial table:
RHS
Ratio
Eqn.
Basic
Coefficient of
No.
variable
x2
S1
S2
A1
A2
Z
x1
0
-
1
5M-5
6M-6  -M  -M
0
0
2700M
1
0
2
5
-1
0
1
0
1500
300
A1
2
0
3
1
0
-1
0
1
1200
1200
A2
Divide the equation 1 by 5 throughout.
RHS
Ratio
Eqn.
Basic
Coefficient of
No.
variable
x2
S1
S2
A1
A2
Z
x1
0
-
1
5M-5
6M-6
-M
-M
0
0
2700M
1
0
2/5
1
-1/5
0
1/5
0
300
300
A1
2
0
3
1
0
-1
0
1
1200
1200
A2
First Iteration: x2 enters and A1 leaves the basis.
RHS
Ratio
Eqn.
Basic
Coefficient of
No.
variable
x2
S1
S2
A1
A2
Z
x1
0
-
1
13M-5
0
M-1  -M
-6M+6
0
900M
5
5
5
+1800
1
0
2/5
1
-1/5
0
1/5
0
300
750
x1
2
0
13/5
-1
1/5
-1
-1/5
1
900
346
A2
Second Iteration: x1 enters and A2 leaves the basis.
131
Operations Research (MTH601)
132
Multiply Eqn. 2 by 5/13 to make the key number 1. Then we have
RHS
Rati
Eqn.
Basic
Coefficient of
o
No.
variable
x2
S1
S2
A1
A2
Z
x1
0
-
1
13M-
0
M-
-M
-
0
900M
13
1
6M+6
+1800
5
5
5
1
0
2/5
1
-
0
1/5
0
300
750
x2
1/5
2
0
1
0
1/1  -5/13
-1/13
346
346
A2
3
5/13
132
Operations Research (MTH601)
133
Eqn.
Basic
Coefficient of
RHS
No.
variable
x2
S1
S2
A1
A2
Z
x1
0
-
1
0
0
0
-1
-M+1
-M+1
2700M
1
0
0
1
-15/65
2/13
3/13
-2/13
2100/1
x2
3
2
0
1
0
1/13
-5/13
-1/13
5/13
4500/1
x1
3
Since the equation 0 does not contain positive coefficient of the variables, the solution found is the optimum.
Z* = 2700,
x1 = 4500 / 13,
x2 = 2100 / 13
Solution:
Example
Minimize
Z = 4x1 + x2
Subject to
3x1 + 4x2 > 20
-x1 - 5x2 < -15
x1, x2 > 0
Solution:
The second constraint can be changed into inequality of the type by multiplying by -1 throughout. Then
introduce slack variable and artificial variable to the two constraints. The equations are transformed into
Z -4x1 - x2 - MA1 - MA2
=0
3x1 + 4x2 - S1 + A1
= 20
x1 + 5x2 - S2 + A2
= 15
The objective function should not involve the coefficients of basic variables A1 and A2. So, multiply the
constraint equation by M and add to the objective equation. Then we get the following equations.
Z - 4x1 + 4Mx1 - x2 + 9Mx2 - MS1 - MS2
= 35M
3x1 + 4x2 - S1 + A1
= 20
x1 + 5x2 - S2 + A2
= 15
The above equations can be conveniently set down in the Simplex table as shown below.
Initial table:
RHS
Ratio
Eqn.  Basic
Coefficient of
No.
variable
x2
S1
S2
A1
A2
Z
x1
0
-
1
4M-4
9M-1  -M
-M
0
0
35M
1
0
3
4
-1
0
1
0
20
5
A1
2
0
1
5
0
-1
0
1
15
3
A2
133
Operations Research (MTH601)
134
Divide equation 2 by 5 to make the key No. 1. Thus we have
134
Operations Research (MTH601)
135
RHS
Ratio
Eqn.
Basic
Coefficient of
No.
variable
x2
S1
S2
A1
A2
Z
x1
0
-
1
4M-
9M-  -M
-M
0
0
35M
4
1
1
0
3
4
-1
0
1
0
20
5
A1
2
0
1/5
1
0
-1/5
0
1/5
3
3
A2
First Iteration: x2 enters and A2 leaves the basis.
RHS
Ratio
Eqn.
Basic
Coefficient of
No.
variable
x2   S1
S2
A1
A2
Z
x1
0
-
1
11M-19
0  -M  4M-
0
-
8M+3
5
1
9M+1
5
5
1
0
11/5
0
-1
4/5
0
-4/5
8
40/11
A1
2
2
0
1/5
1
0
-1/5
0
1/5
3
15
Multiply the equation 1 by 5/11 to make the key No. 1 and the resulting table is given below:
RHS
Ratio
Eqn.
Basic
Coefficient of
No.
variable
x2
S1
S2
A1
A2
Z
x1
0
-
1
11M-
0
-M  -M-
0
-
8M+3
185/11
1
1
9M+1
5
5
5
1
0
1
0
-
4/11
0
-4/11
40/11
40/11
A1
5/11
2
0
1/5
1
0
-1/5
0
1/5
3
25/11
x2
Second Iteration: x1 enters and A1 leaves the basis.
Ratio
Eqn.
Basic
Coefficient of
No.
variable
x2
S1
S2
A1
A2
Z
x1
0
-
1
0
0
-
-
0
-
185/11
19/11  3/11
56M+14
25
1
0
1
0
-5/11
4/11
0
-4/11
40/11
x1
2
0
0
1
1/11
-
0
3/11
25/11
x2
3/11
Optimum solution:
135