Question

please simplify answers as factions if possible

Use the simplex method to solve the problem. Maximize subject to P = 12X1 + 16x2 2x1 + x2 = 16 X1 +5x2 5 16 X1, X220 The maximum value is P = when X1 and X2 = (Simplify your answers.)

Answer 1

In a simplex method problem, the first step is to develop standard form of new Linear Programming Problem.

We transform the old(unequal) constraints equation into new(equal) constraints equation by adding slack variables:

So,

2.11 + 12 < 16

LT1+522 < 16

becomes

2.11 + x2 + S1 = 16

11 + 5.02 + S2 = 16

(Note: S₁ and S₂ are slack variables)

We also change the old objective function(OOF) i.e.

Maximize

P= 12.01 + 16.0

into new objective function(NOF) by adding slack variables with zero coefficients, i.e.

PMaxima 12x1 + 16х2 + 0s1 + 0s-

Now, if we compare the new objective function and the new constraints equation, we find that the constraints equation are not balanced. To do so, we insert slack variables with zero coefficients.

So, the standard form of new LPP is:

PMaxima 12x1 + 16х2 + 0s1 + 0s-

Subject to:

2.61 +62 + S1 +0.S2 = 16

11 + 5.02 +0S1 + S2 = 16

Now, we assume that the 'Initial basic solution' of the Old Objective Function is:

x₁, x₂ = 0. Hence, P = 12 x 0 + 16 x 0 = 0

Next, we prepare 'First Stage Simplex Table' where:

CV = Constraint Value

NV = New variables

Ci = Coefficient of NV in the New Objective Function(NOF)

Cj = Coefficient of variables in the NOF

The two rows correspond to the two constraint equations.

We find the Zj values for different variables as:

Zj = (Ci of first variable) x (Value of the variable in first equation) + (Ci of second variable) x (Value of the variable in second equation)

Table 1: First Stage Simplex Table

NOF & NE Ratio d 2 1 52 $ NY 12 16 0 0 16 1 1 0 N S1 $2 16 16/5 16 1 5 1 0 4-g -12 -16 0 0

We show the calculations in the above table below:

Zj values:

For x1: (0x2) + (0x1) = 0 + 0 = 0

For x2: (0x1) + (0x5) = 0 + 0 = 0

For S1: (0x1) + (0x0) = 0 + 0 = 0

For S2: (0x0) + (0x1) = 0 + 0 = 0

Zj - Cj values:

For x1: 0 -12 = -12

For x2: 0 - 16 = -16

For S1: 0 - 0 = 0

For S2: 0 - 0 = 0

The optimization condition is that the calculated value of (Zj - Cj) for all variables should be ≥ 0.

Since all values are not greater than or equal to 0, we see that the optimality condition is not yet reached.

Now, the lowest integer of Zj - Cj value is -16, which corresponds to x2 variable(column). So, x2 is the incoming variable and this is the pivot column.

Next, we find the ratio of each new variable as:

(Constraint Value) / ( Corresponding value in the pivot column).

So, ratio

For S1 = 16/1 = 16

For S2 = 16/5

The least positive ratio(LPR) is 16/5, which corresponds to S2.Hence, S2 is the outgoing variable and the row corresponding to this is the pivot row.

The key element is given by the intersection of the pivot row and pivot column i.e. 5.

Now, we create the second stage simplex table for which:

New elements for incoming variable x2 is calculated by dividing all elements in the pivot row by the key element.

For CV = 16/5

Ci for x2 = 16

For x1 = 1/5

For x2 = 5/5 = 1

For S1 = 0/5 = 0

For S2 = 1/5

New elements for the surviving variable, S1, can be calculated by the given formula:

(Corresponding Old value in the previous table) - (Element of surviving variable in pivot column in the prev table) x (Newly calculated elements of incoming variable in the new table)

So,

For CV = 16 - 1 x(16/5) = 16 - 16/5 = 64/5

Ci for S1 = 0

For x1 = 2 - 1x(1/5) = 2 - 1/5 = 9/5

For x2 = 1 - 1x1 = 1- 1 = 0

For S1 = 1 - 1x0 = 1 - 0 = 1

For S2 = 0 - 1x(1/5) = -1/5

Table 2: Second Stage Simplex Table

NOF & NE Ratio d 2 1 52 NY 12 16 0 0 S1 0 1 64/5 16/5 64/9 16 x2 16 1 9/5 1/5 16/5 -44/5 ee -1/5 15 16/5 16/5 16 4-g 0

Zj values:

For x1: (0x9/5) + (16x1/5) = 0 + 16/5 = 16/5

For x2: (0 x0) + (16 x1) = 0 + 16 = 16

For S1: (0 x1) + (16 x0) = 0 + 0 = 0

For S2: (0 x -1/5) + (16 x1/5) = 0 + 16/5 = 16/5

Zj - Cj values:

For x1: 16/5 -12 = -44/5

For x2: 16 - 16 = 0

For S1: 0 - 0 = 0

For S2: 16/5 - 0 = 16/5

The optimization condition is that the calculated value of (Zj - Cj) for all variables should be ≥ 0.

Since all values are not greater than or equal to 0, we see that the optimality condition is not yet reached.

Now, the lowest integer of Zj - Cj value is -44/5, which corresponds to x1 variable(column). So, x1 is the incoming variable and this is the pivot column.

Next, we find the ratio of each new variable as:

(Constraint Value) / ( Corresponding value in the pivot column).

So, ratio

For S1 = (64/5)/(9/5) = 64/9

For x2 = (16/5)/(1/5) = 16

The least positive ratio(LPR) is 64/9, which corresponds to S1.Hence, S1 is the outgoing variable and the row corresponding to this is the pivot row.

The key element is given by the intersection of the pivot row and pivot column i.e. 9/5.

Now, we create the third stage simplex table for which:

New elements for incoming variable x1 is calculated by dividing all elements in the pivot row by the key element.

For CV = (64/5)/(9/5) = 64/9

Ci for x1 = 12

For x1 = (9/5) / (9/5) = 1

For x2 = 0/(9/5) = 0

For S1 = 1/(9/5) = 5/9

For S2 = (-1/5)/(9/5) = -1/9

New elements for the surviving variable, x2, can be calculated by the given formula:

(Old value in the previous table) - (Element of surviving variable in pivot column in the prev table) x(Newly calculated elements of incoming variable in the new table)

So,

For CV = 16/5 - (1/5) x(64/9) = 80/45 = 16/9

Ci for x2 = 16

For x1 = 1/5 - (1/5) x1 = 1/5 - 1/5 = 0

For x2 = 1 -(1/5) x 0 = 1 - 0 = 1

For S1 = 0 - (1/5) x(5/9) = -1/9

For S2 = 1/5 - (1/5) x(-1/9) = 1/5 + 1/45 = 2/9

Table 3: Third Stage Simplex Table

NOF & NE Ratio d 2 51 52 NY 12 16 0 0 x1 12 1 0 64/9 16/9 x2 16 1 5/9 -1/9 4449 44/9 -1//9 29 20/9 12 16 4-g 0 20/9

Zj values:

For x1: (12 x 1) + (16 x 0) = 12

For x2: (12 x 0) + (16 x 1) = 16

For S1: (12 x 5/9) + (16 x -1/9) = 60/9 - 16/9 = 44/9

For S2: (12 x -1/9) + (16 x 2/9) = -12/9 + 32/9 = 20/9

Zj - Cj values:

For x1: 12 - 12 = 0

For x2: 16 - 16 = 0

For S1: 44/9 - 0 = 44/9

For S2: 20/9 - 0 = 20/9

Since (Zj - Cj) for all variables are ≥ 0, we conclude that the optimality condition has been reached.

We see that in the 'optimality table', the CV values for x1 and x2 are 64/9 and 16/9 respectively.

So, our old objective function was:

Maximize

P= 12.01 + 16.0

Substituting the values for x_1 and x_2, we have:

P = 12 x (64/9) + 16 x(16/9)

   = 1024/9

The maximum value is P = 1024/9 when x_1 = 64/9 and x_2 = 16/9.