# Documentation

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# `linopt`::`Transparent::phaseI_tableau`

Start an ordinary phase one of a 2-phase simplex algorithm

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## Syntax

```linopt::Transparent::phaseI_tableau(`tableau`)
```

## Description

`linopt::Transparent::phaseI_tableau` explicitly starts an (ordinary) phase one of the simplex algorithm , i.e. rows associated with infeasible basic variables are multiplied with -1 and another identity matrix with new slack variables is added to the given tableau. As soon as an optimal tableau with relative costs 0 is found the calculation can be continued with `linopt::Transparent::clean_basis` and the second phase of the simplex algorithm (`linopt::Transparent::phaseII_tableau`).

## Examples

### Example 1

The first simplex tableau is created and the first phase of the simplex algorithm is started:

```t := linopt::Transparent([{x + y >= 2}, x, NonNegative]); t := linopt::Transparent::phaseI_tableau(t)```

We can see that a new slack variable, slk2, was added to the tableau. And if we now execute `linopt::Transparent::simplex` we can see that we have just finished the first phase of the simplex algorithm:

```linopt::Transparent::suggest(t); t := linopt::Transparent::simplex(t): linopt::Transparent::suggest(t)```

We continue the simplex algorithm by executing `linopt::Transparent::clean_basis`, `linopt::Transparent::phaseII_tableau` and `linopt::Transparent::simplex`. Observe in this special case `linopt::Transparent::clean_basis` is not necessary:

```t := linopt::Transparent::clean_basis(t): t := linopt::Transparent::phaseII_tableau(t): t := linopt::Transparent::simplex(t); linopt::Transparent::suggest(t)```

`delete t:`

## Parameters

 `tableau` A simplex tableau of domain type `linopt::Transparent`

## Return Values

Simplex tableau of domain type `linopt::Transparent`.

## References

Papadimitriou, Christos H; Steiglitz, Kenneth: Combinatorial Optimization; Algorithms and Complexity. Prentice-Hall, 1982.

Nemhauser, George L; Wolsey, Laurence A: Integer and Combinatorial Optimization. New York, Wiley, 1988.

Salkin, Harvey M; Mathur, Kamlesh: Foundations of Integer Programming. North-Holland, 1989.

Neumann, Klaus; Morlock, Martin: Operations-Research. Munich, Hanser, 1993.

Duerr, Walter; Kleibohm, Klaus: Operations Research; Lineare Modelle und ihre Anwendungen. Munich, Hanser, 1992.

Suhl, Uwe H: MOPS - Mathematical OPtimization System. European Journal of Operational Research 72(1994)312-322. North-Holland, 1994.

Suhl, Uwe H; Szymanski, Ralf: Supernode Processing of Mixed Integer Models. Boston, Kluwer Academic Publishers, 1994.