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# 68. simplex

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## 68.1 Introduction to simplex

`simplex` is a package for linear optimization using the simplex algorithm.

Example:

```(%i1) load("simplex")\$
(%i2) minimize_lp(x+y, [3*x+2*y>2, x+4*y>3]);
9        7       1
(%o2)            [--, [y = --, x = -]]
10       10      5
```

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## 68.2 Functions and Variables for simplex

Option variable: epsilon_lp

Default value: `10^-8`

Epsilon used for numerical computations in `linear_program`.

See also: `linear_program`.

Function: linear_program (A, b, c)

`linear_program` is an implementation of the simplex algorithm. `linear_program(A, b, c)` computes a vector x for which `c.x` is minimum possible among vectors for which `A.x = b` and `x >= 0`. Argument A is a matrix and arguments b and c are lists.

`linear_program` returns a list which contains the minimizing vector x and the minimum value `c.x`. If the problem is not bounded, it returns "Problem not bounded!" and if the problem is not feasible, it returns "Problem not feasible!".

To use this function first load the `simplex` package with `load(simplex);`.

Example:

```(%i2) A: matrix([1,1,-1,0], [2,-3,0,-1], [4,-5,0,0])\$
(%i3) b: [1,1,6]\$
(%i4) c: [1,-2,0,0]\$
(%i5) linear_program(A, b, c);
13     19        3
(%o5)            [[--, 4, --, 0], - -]
2      2         2
```

See also: `minimize_lp`, `scale_lp`, and `epsilon_lp`.

Function: maximize_lp (obj, cond, [pos])

Maximizes linear objective function obj subject to some linear constraints cond. See `minimize_lp` for detailed description of arguments and return value.

See also: `minimize_lp`.

Function: minimize_lp (obj, cond, [pos])

Minimizes a linear objective function obj subject to some linear constraints cond. cond a list of linear equations or inequalities. In strict inequalities `>` is replaced by `>=` and `<` by `<=`. The optional argument pos is a list of decision variables which are assumed to be positive.

If the minimum exists, `minimize_lp` returns a list which contains the minimum value of the objective function and a list of decision variable values for which the minimum is attained. If the problem is not bounded, `minimize_lp` returns "Problem not bounded!" and if the problem is not feasible, it returns "Ploblem not feasible!".

The decision variables are not assumed to be nonegative by default. If all decision variables are nonegative, set `nonegative_lp` to `true`. If only some of decision variables are positive, list them in the optional argument pos (note that this is more efficient than adding constraints).

`minimize_lp` uses the simplex algorithm which is implemented in maxima `linear_program` function.

To use this function first load the `simplex` package with `load(simplex);`.

Examples:

```(%i1) minimize_lp(x+y, [3*x+y=0, x+2*y>2]);
4       6        2
(%o1)                [-, [y = -, x = - -]]
5       5        5
(%i2) minimize_lp(x+y, [3*x+y>0, x+2*y>2]), nonegative_lp=true;
(%o2)                [1, [y = 1, x = 0]]
(%i3) minimize_lp(x+y, [3*x+y=0, x+2*y>2]), nonegative_lp=true;
(%o3)                Problem not feasible!
(%i4) minimize_lp(x+y, [3*x+y>0]);
(%o4)                Problem not bounded!
```

See also: `maximize_lp`, `nonegative_lp`, `epsilon_lp`.

Option variable: nonegative_lp

Default value: `false`

If `nonegative_lp` is true all decision variables to `minimize_lp` and `maximize_lp` are assumed to be positive.

See also: `minimize_lp`.

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