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### linear programming GIPALS Start

As mentioned in Linear Program almost every linear program has to be converted to Standard Form. There are several situations could be encountered:

Maximization Linear Program

Maximization linear program lies to find such values of variables that maximize the objective function. Such type of linear program can be converted to minimization  linear program simply by multiplying the objective function by a negative sign. For example,

`Maximize f(X) = 12x1 + 4x2 - 6x3. `
`Converted linear program is the following: `
`Minimize f(x) = -12x1 - 4x2 + 6x3`

Both the linear programs are the same.

Negative Values on the Right-Side of Constraints

All constraints in Standard Form of linear program are required to have positive right-sides. If any constraint has negative quantity at the right-side then this constraint has to be multiplied by -1 and the direction of inequality has to be swapped (that <= becomes >=, and vice versa). For example:

`4x1 + 5x2 - 6x3 >= 4.2 is the same as -4x1 - 5x2 + 6x3 <= 4.2`

Less than Type Constraints

Linear program constraint of this type has to be converted by adding one extra positive variable (called a slack variable) to convert to Less or Equal (<=) type. Slack variable has 0 as costs coefficient in appropriate position in the linear program objective function. For example:

`Minimize f(x) = 2x1 + 3x2 - 4x3 `
`Subject to 	x1 + 2x2 < 4`
`		x1 - x3 <= 5`
`		x1,x2,x3 >= 0`

The slack variable x4 is added to the first constraint and the linear program is converted as the following:

`Minimize f(x) = 2x1 + 3x2 - 4x3 + 0x4`
`Subject to 	x1 + 2x2 + x4 <= 4`
`		x2 - x3 <= 5`
`		x1,x2,x3,x4 >= 0`

Greater (Greater or Equal) than Type Constraints

Linear program constraint of this type has to be converted by subtracting one extra positive variable (called a surplus variable) to convert to Equal (=) type. Surplus variable has 0 as costs coefficient in appropriate position in the linear program objective function. For example:

`Minimize f(x) = 2x1 + 3x2 - 4x3`
`Subject to 	x1 + 2x2 >= 4`
`		x2 - x3 <= 5`
`		x1,x2,x3 >= 0`

The surplus variable x4 is subtracted from the first constraint and the linear program is converted as the following:

`Minimize f(x) = 2x1 + 3x2 - 4x3 + 0x4`
`Subject to 	x1 + 2x2 - x4 <= 4`
`		x2 - x3 <= 5`
`		x1,x2,x3,x4 >= 0`

Unrestricted Variables

The Standard Form of linear program requires all variables to be positive. If some of variables are not restricted to be positive only then those variables have to be substituted by the difference of two positive variables. For example, if variable x2 is unrestricted in sign, it is replaced by two new positive variables x21 and x22 with x2 = x21- x22.

Example:

`Minimize f(x) = 2x1 + 3x2 - 4x3`
`Subject to 	x1 + 2x3 >= -40`
`		x1 - x2 <= 5`
`		x1,x2 >= 0 `

So, x3 is unrestricted in sign. Substitute x3 by x31 - x32. The linear program is the following now:

`Minimize f(x) = 2x1 + 3x2 - 4x3`
`Subject to 	x1 + 2x31 - 2x32 >= -40`
`		x1 - x2 <= 5`
`		x1,x2,x31,x32 >= 0 `

If a variable has a negative lower bound then another method is applied. Let assume that variable x2 has lower limit L = -20. Then all linear program constraints contained x2 have to be modified as the following:

`		x1 + 2x2 >= -40`
`		x1 >= 0, x2 >= L`

Introduce variable y2 = x2 - L (hence, x2 = y2 + L) and rearrange the constraint as:

`		x1 + 2y2 + 2L >= -40 `

Finally:

`		x1 + 2y2 >= -40 - 2L`
`		x1, y2 >= 0`

linear programming GIPALS Start

Linear programming examples

There are several examples of linear programming intended to make the users of GIPALS familiar with it. These examples are included in GIPALS installation and can be found in ..\GIPALS\Examples folder.

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