Simplex Method - Maximization Case

Consider the general linear programming problem

Maximize z = c₁x₁ + c₂x₂ + c₃x₃ + .........+ c_nx_n
subject to
a₁₁x₁ + a₁₂x₂ + a₁₃x₃ + .........+ a_1nx_n ≤ b₁
a₂₁x₁ + a₂₂x₂ + a₂₃x₃ + .........+ a_2nx_n ≤ b₂
.........................................................................
a_m1x₁ + a_m2x₂ + a_m3x₃ + .........+ a_mnx_n ≤ b_m
x₁, x₂,....., x_n≥ 0
Where:
c_j (j = 1, 2, ...., n) in the objective function are called the cost or profit coefficients.
b_i (i = 1, 2, ...., m) are called resources.
a_ij (i = 1, 2, ...., m; j = 1, 2, ...., n) are called technological coefficients or input-output coefficients.

Converting inequalities to equalities

Introducing slack variables to convert inequalities to equalities
a₁₁x₁ + a₁₂x₂ + a₁₃x₃ + .........+ a_1nx_n + s₁ = b₁
a₂₁x₁ + a₂₂x₂ + a₂₃x₃ + .........+ a_2nx_n + s₂ = b₂
..............................................................................
a_m1x₁ + a_m2x₂ + a_m3x₃ + .........+ a_mnx_n + s_m = b_m
x₁, x₂,....., x_n≥ 0
s₁, s₂,....., s_m ≥ 0
An initial basic feasible solution is obtained by setting x₁ = x₂ =........ = x_n = 0
s₁ = b₁
s₂ = b₂
..............
s_m = b_m

The initial simplex table is formed by writing out the coefficients and constraints of a LPP in a systematic tabular form. The following table shows the structure of a simplex table.

Structure of a simplex table

	cj	c1	c2	c3	---	cn	Right Hand Side
cB	Basic variables B	x1	x2	x3	---	Sn	Solution values b (=XB)
cB1	x1	a11	a12	a13	---	a1n	b1
cB2	x2	a21	a22	a23	---	a2n	b2
cB3	x3	a31	a32	a33	---	a3n	b3
---	-----	----	----	-----	---	----	-----
cBm	xn	am1	am2	am3	---	amn	bm
Z		z1	Z2	Z3	---	Zn
cj- zj		c1-z1	C2-z2	C3-z3	---	Cn-zn

Where:
_Cj = coefficients of the variables in the objective function.
c_B = coefficients of the current basic variables in the objective function.
B = basic variables in the basis.
X_B = solution values of the basic variables(Right Hand Side.
c_j- z_j = index row.

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