## `linopt::Transparent::phaseII_tableau` -- start phase two of a 2-phase simplex algorithm

#### Introduction

`linopt::Transparent::phaseII_tableau(`tableau`)` starts the second phase of the simplex algorithm on the given simplex tableau `tableau`.

#### Call(s)

`linopt::Transparent::phaseII_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::phaseI_tableau`, `linopt::Transparent::result`, `linopt::Transparent::simplex`, `linopt::Transparent::suggest`, `linopt::Transparent::userstep`

#### Details

- After the explicitly started first phase of the simplex algorithm (see `linopt::Transparent::phaseI_tanleau`) terminated with an optimal tableau with associated costs 0 and no phase one slack variables in the basis (see `linopt::Transparent::clean_basis`) this procedure can be used to start phase II. The procedure eliminates all artificial variables of phase I and their associated columns and reenters the old objective function modified for the new basis.

#### Example 1

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

    >> t := linopt::Transparent([[x + y >= 2], x, NonNegative]):
       t := linopt::Transparent::simplex(
               linopt::Transparent::phaseI_tableau(t))

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

One sees that the artificial slack variable slk[2] of the first phase is removed by
`linopt::Transparent::phaseII_tableau`. In this example it is not necessary to use
`linopt::Transparent::clean_basis` for cleaning the basis:

    >> linopt::Transparent::phaseII_tableau(t)

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

    >> delete t:

#### Example 2

Again the first simplex tableau is created and the first phase of the simplex algorithm is finished:

    >> t := linopt::Transparent([[x <= 1, y <= 1, x + y >= 2],
                                0, NonNegative]):
       t := linopt::Transparent::phaseI_tableau(t):
       t := linopt::Transparent::simplex(t)

                               array(1..5, 1..10,
                                 (1, 1) = "linopt",
                                 (1, 2) = "restr",
                                 (1, 3) = slk[4],
                                 (1, 4) = slk[5],
                                 (1, 5) = slk[6],
                                 (1, 6) = slk[1],
                                 (1, 7) = slk[2],
                                 (1, 8) = slk[3],
                                 (1, 9) = x,
                                 (1, 10) = y,
                                 (2, 1) = "obj",
                                 (2, 2) = 0,
                                        .
                                        .
                                 (4, 10) = 1,
                                 (5, 1) = slk[6],
                                 (5, 2) = 0,
                                 (5, 3) = -1,
                                 (5, 4) = -1,
                                 (5, 5) = 1,
                                 (5, 6) = -1,
                                 (5, 7) = -1,
                                 (5, 8) = -1,
                                 (5, 9) = 0,
                                 (5, 10) = 0
                               )

In this example the artificial slack variable `slk[6]` is an element of the optimal basis. So we have to use `linopt::Transparent::clean_basis` before continuing with `linopt::Transparent::phaseII_tableau`, otherwise we will get an error message. To get a smarter output in this example one should use at least a `TEXTWIDTH` of 80:

    >> linopt::Transparent::phaseII_tableau(t)

          Error:  Clean the basis from phase I slack variables first! [l\
          inopt::Transparent::phaseII_tableau]

    >> t := linopt::Transparent::clean_basis(t):
       linopt::Transparent::phaseII_tableau(t)

               +-                                                 -+
               |  "linopt", "restr", slk[1], slk[2], slk[3], x, y  |
               |                                                   |
               |    "obj",     0,       0,      0,      0,   0, 0  |
               |                                                   |
               |      x,       1,       0,     -1,     -1,   1, 0  |
               |                                                   |
               |      y,       1,       0,      1,      0,   0, 1  |
               |                                                   |
               |   slk[1],     0,       1,      1,      1,   0, 0  |
               +-                                                 -+

    >> 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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