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Handling "="-constraints and unconstrained variables (with simple Mathematical tricks)

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• Handling "="-constrains
  • Sample "="-constraint:

    3×x1 - 4×x2 + 5×x3 + 8×x3 = 56

  • Problem with an equality constraint:

    It is not straighforward to find a basic feasible solution I.e.: When we use row operations to transform some columns into canonical unit column vectors, we cannot guarantee that the resulting Simplex Tableau contains a feasible solution.

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  • Handling "="-constraints:

    An equality constraint can be converted into 2 inequality constraints as follows: 3×x1 - 4×x2 + 5×x3 + 8×x3 ≤ 56 3×x1 - 4×x2 + 5×x3 + 8×x3 ≥ 56 This will result in the following linear equations: 3×x1 - 4×x2 + 5×x3 + 8×x3 + s1 = 56 3×x1 - 4×x2 + 5×x3 + 8×x3 - s2 = 56 And we will need to add an artificial variable for −s2: 3×x1 - 4×x2 + 5×x3 + 8×x3 + s1 = 56 3×x1 - 4×x2 + 5×x3 + 8×x3 - s2 + a1 = 56

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  • We have to do these steps to obtain a basis

    The variables s₁ and a₁ will be part of the basis of the initial Simplex Tableau (for Phase 1)

    Since we added artificial variables, we need to use the 2-phase Simplex method to drive the artificial variables to ZERO (0) to obtain a feasible basic solution before starting with the actual Simplex method (phase 2)

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• Handling unconstrained variables
  • The Simplex method assumes that every variable x_i has the following constraint:

    xi   ≥   0

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  • Unconstrained variables:

    An Unconstrained variables xi can assume any value (including negative values)

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  • Handling unconstrained variables in Simplex:

    Replace every unconstrained variable xi by 2 constrained ones as follows: xi is unconstrained Replace with: xi = xi1 - xi2 where xi1, xi2 ≥ 0

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• Handling "x_i ≥ constant" constraints
  • The Simplex method assumes that every variable x_i satisfies the following constraint:

    xi   ≥   0

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  • We can encounter constraints with non-zero constant of this form:

    xi ≥ ci ( ci ≠ 0 )

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  • This is handle easily by a substitution:

    xi ≥ ci ( ci ≠ 0 ) Substitute: xi' = xi - ci ==> xi ≥ ci <==> xi - ci ≥ 0 <==> xi' ≥ 0

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  • So we simply replace x_i in every equation in the constrained optimalization problem by x_i'   first, before we apply the Simplex Method