## `linopt::Transparent::phaseI_tableau` -- start an ordinary phase one of a 2-phase simplex algorithm

#### Introduction

`linopt::Transparent::phaseI_tableau(`tableau`)` starts an (ordinary) phase one of the 2-phase simplex algorithm on the given simplex tableau `tableau`.

#### Call(s)

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

#### Parameters

`tableau` | - | a simplex tableau of domain type `linopt::Transparent`

#### Returns

a simplex tableau of domain type `linopt::Transparent`.

#### Related Functions

`linopt::Transparent`, `linopt::Transparent::autostep`, `linopt::Transparent::convert`, `linopt::Transparent::clean_basis`, `linopt::Transparent::dual_prices`, `linopt::Transparent::phaseII_tableau`, `linopt::Transparent::result`, `linopt::Transparent::simplex`, `linopt::Transparent::suggest`, `linopt::Transparent::userstep`

#### Details

- `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`).

#### 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)

                      +-                                   -+
                      |  "linopt", "restr", slk[1],  x,  y  |
                      |                                     |
                      |    "obj",     0,       0,    1,  0  |
                      |                                     |
                      |   slk[1],     -2,      1,   -1, -1  |
                      +-                                   -+

                  +-                                           -+
                  |  "linopt", "restr", slk[2], slk[1],  x,  y  |
                  |                                             |
                  |    "obj",     -2,      0,      1,   -1, -1  |
                  |                                             |
                  |   slk[2],     2,       1,     -1,    1,  1  |
                  +-                                           -+

We can see that a new slack variable, slk[2] 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)

                                     slk[2], x

                      "linopt::Transparent::phaseII_tableau"

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)

                       +-                                 -+
                       |  "linopt", "restr", slk[1], x, y  |
                       |                                   |
                       |    "obj",     0,       0,   1, 0  |
                       |                                   |
                       |      y,       2,      -1,   1, 1  |
                       +-                                 -+

                                      OPTIMAL

    >> delete t:

#### Background

- 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.

#### Changes

- new function
- The linopt library was completely rewritten in MuPAD version 2.0.

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