Simplex Method
The simplex method was developed by George Dantzig in 1949. This method overcomes the limitation of the graphical method and can be applied to problems where the number of decision variables is more than two. The algorithm for the simplex method is iterative in nature and solves a problem in a systematic manner to determine the optimum solution for a problem.
However, the following points should be kept in mind before solving any problem:

The value of the constraint in the righthand side of each of the constraints should be nonnegative. If not, it should be converted into a nonnegative value.

Each decision variable of the problem should be nonnegative.

Slack variables are introduced in each constraint equation as an idle source to convert inequalities to equalities.
Several operations managers prefer the simplex method because of its advantages:

The algorithm of the simplex method moves from one extreme point to a better (or at least equally good) extreme point, by skipping many suboptimal extreme points. Thus, the problem checks only a limited number of points to determine the optimal solution. This saves the time and computing effort.

The simplex method finds the point where the problem has to be stopped and thereby reduces the effort of verifying for optimality at each and every step.

The algorithm detects whether the problem is infeasible, unbounded or has multiple solutions.

The simplex algorithm can be easily computerized as the method involves simple mathematical calculations.
The simplex method is explained below in Problem.
Problem
Solve the linear programming problem formulated in Table 4.3 by using the Simplex method.
Solution:
The objective of the problem is to maximize the function, Z =700x_{1}+400x_{2 }
The constraints are:
2x_{1 }+ x_{2 }≤ 3000,
x_{1 }+ 2x_{2 }≤ 4000,
x_{1 }+ x_{2 }≤ 2500,
x_{1}, x_{2} ≥0.
Adding slack variables to the problem,
Maximize, Z = 700x_{1} + 400x_{2 }+0 S_{1 }+ 0 S_{2 }+ 0 S_{3}
Subject to
2x_{1} + x_{2} + S_{1} = 3000,
Table: Initial Simplex Table
x_{1}+ 2x_{2} + S_{2} = 4000,
x_{1} + x_{2} + S_{3} = 2500,
x_{1}, x_{2}, x_{3}, S_{1,} S_{2}, S_{3},_{ }≥0.
Where, S_{1}, S_{2}, and S_{3} are slack variables.
The initial simplex table is developed as shown in Table 4.5.
The highest element in the Index or (C_{j } Z_{j}) row is 700. Therefore, the x_{1 }column becomes the key column and x_{1 }is called the entering variable. The ratios are obtained by dividing the solution variables with the corresponding elements of the key column.
The row with minimum ratio is called the key row and the intersection element of the key row and the key column becomes the key element. Here, 'S_{1}' row is the key row and '2' is the key element. The variable S_{1} is called the departing variable.
Now, the new simplex table is developed by using the following procedure.
All the values in the key row are divided by the key element to obtain the new values and the departing variable S_{1 }is replaced by the entering variable x_{1}.
Thus the values of the revised key row are:
The new values for each remaining row (other than the key row) are computed by using the formula:
New row value = Old row value  (Corresponding value in the key row × Corresponding value in the revised key row).
Thus the new values of 'S_{2}' row are obtained as shown below.
'S_{2}' row:
4000

1

2

0

1

0

 (Corresponding element, i.e. 1) × (value in the revised row):

1500

1

1/2

1/2

0

0

Table: Simplex Table
The new 'S_{2}' row is:

2500

0

3/2

1/2

1

0

Similarly, the new 'S_{3}' row is calculated as shown below.
'S_{3}' row:
2500

1

1

0

0

1

 (corresponding element, i.e. 1) × (value in the revised row):

1500

1

½

½

0

0

The new 'S_{3}' row is:

1000

0

1/2

1/2

0

1

These new values are shown in the Simplex Table I.
From the Simplex Table 1, the largest positive value in the (C_{j } Z_{j}) row is 50 and it lies in 'x_{2}' column. So, 'x_{2}' becomes the entering variable and the 'x_{2}' column becomes the key column.
The ratios obtained by dividing the solution variables with the values in the key column are 3000, 5000/3 and 2000. Here, 5000/3 is the minimum ratio. Therefore, the S_{2} row becomes the key row and the variable 'S_{2}' becomes the departing variable. The key element is '3/2.'
Now, the departing variable, S_{2 }is replaced by the entering variable x_{2 }and the_{ }revised key row is obtained by dividing all the values in the key row with the key element.
The revised key row is,
The new 'x_{1}' row can be developed as shown below.
Here, the corresponding element is 1/2.
'x_{1}' row:
1500

1

1/2

1/2

0

0

(corresponding element, i.e. 1/2 × corresponding value in the revised row)

2500/3

0

½

1/6

1/3

0

The new 'x_{3}' row:

2000/3

1

0

2/3

1/3

0

The new 'S_{3}' row is calculated as shown below.
'S_{3}' row:
1000 
0 
1/2 
1/2 
0 
1

(corresponding element, i.e. ½ × corresponding value in the revised row) 
2500/3 
0 
1/2 
1/6 
1/3 
0

The new 'S_{3}' row is: 
500/3 
0 
0 
1/3 
1/3 
1

These values are shown in Simplex Table.
Since there is no positive value in the (C_{j}  Z_{j}) row of the Simplex Table 2, further simplex table cannot be developed. Therefore, the optimum solution is
x_{1 }= 2000/3, x_{2} = 5000/3, and
The maximum value of the profit at this optimum solution is
Z_{max} = 700x_{1 }+ 400x_{2 }
= 700(2000/3) + 400 (5000/3)
= Rs. 11,33,333.
Table: Simplex Table
C_{j }

700

400

0

0

0

C_{B }

Basic variables

Solution variables

x_{1 }

x_{2 }

S_{1 }

S_{2 }

S_{3 }

700

x_{1}

2000/3

1

0

2/3

1/3

0

400

x_{2}

5000/3

0

1

1/3

2/3

0

0

S_{3}

500/3

0

0

1/3

1/3

1



Z_{j }

700

400

1000/3

100/3

0



(C_{j}Z_{j})

0

0

1000/3

100/3

0

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