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The Simplex Method

createsimplexproblem

OperationsResearchModels.Simplex.createsimplexproblem — Function
createsimplexproblem(obj::Vector, amat::Matrix, rhs::Vector, dir::Vector, opttype::OptimizationType)::SimplexProblem

Description

This function creates a SimplexProblem object from the given parameters.

Arguments

  • obj::Vector: The objective function coefficients.
  • amat::Matrix: The LHS of the constraints.
  • rhs::Vector: The RHS of the constraints.
  • dir::Vector: The directions of the constraints. Can be a vector of LE (<=), GE (>=), or EQ (==).
  • opttype::OptimizationType: The type of the optimization. Can be Maximize or Minimize.

Returns

A SimplexProblem object.

Example

Suppose the linear programming problem is as follows:

Maximize: 1.0x1 + 2.0x2 + 3.0x3
Subject to:
1.0x1 + 2.0x2 + 3.0x3 <= 10.0
3.0x1 + 1.0x2 + 5.0x3 <= 15.0
x1, x2 >= 0

The following code creates a SimplexProblem object for the above problem:

using OperationsResearch.Simplex

obj = Float64[1.0, 2.0, 3.0]
amat = Float64[1.0 2.0 3.0; 3.0 1.0 5.0]
rhs = Float64[10.0, 15.0]
dir = [LE, LE]
opttype = Maximize

s = createsimplexproblem(obj, amat, rhs, dir, opttype)
iters = simplexiterations(s)
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solve!

OperationsResearchModels.solve! — Method
solve!(s::SimplexProblem)::SimplexProblem

Description

Solves the given SimplexProblem using the Simplex algorithm. The solve! function modifies the input SimplexProblem in place.

Arguments

  • s::SimplexProblem: The SimplexProblem to solve.

Returns

  • SimplexProblem: The solved SimplexProblem.
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simplexiterations

OperationsResearchModels.Simplex.simplexiterations — Function
simplexiterations(s::SimplexProblem)::Vector{SimplexProblem}

Description

This function performs the Simplex iterations on the given SimplexProblem and returns the history of SimplexProblem states.

Arguments

  • s::SimplexProblem: The SimplexProblem to iterate.

Returns

  • Vector{SimplexProblem}: The history of SimplexProblem states (by iterations).
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Gauss Jordan steps for matrix inversion

OperationsResearchModels.Simplex.gaussjordan — Function
gaussjordan(A::Matrix; verbose::Bool = true)::Matrix

Description

Attaches an Identity matrix to the right of the given matrix A and applies the Gauss-Jordan elimination method to find the inverse of the given matrix.

Arguments

  • A::Matrix: The matrix to find the inverse.
  • verbose::Bool: If true, the intermediate steps are displayed. Default is true.

Returns

The inverse of the given matrix.

Example

julia> A = [1.0 2.0 3.0; 4.0 5.0 6.0; 7.0 8.0 10.0]
julia> invA = gaussjordan(A, verbose = false)
3×3 Matrix{Float64}:
 -0.666667  -1.33333   1.0
 -0.666667   3.66667  -2.0
  1.0       -2.0       1.0
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