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Chemical Engineering Mathematics

Chemical Engineering Mathematics. Topic : Simplex method By Prof. Sonali Singh Department of Chemical Engineering DYPIEMR, Akurdi. Simplex method. Given the initial BFS, the simplex method (a)First examines whether the present BFS is optimal or not

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Chemical Engineering Mathematics

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  1. Chemical Engineering Mathematics Topic : Simplex method By Prof. Sonali Singh Department of Chemical Engineering DYPIEMR, Akurdi

  2. Simplex method • Given the initial BFS, the simplex method • (a)First examines whether the present BFS is optimal or not • (b)If it is not optimal, the simplex method finds an adjacent basic feasible solution with better (or at least equal) value of objective function (Z) • Definition: An adjacent basic solution differs from the present basic solution in exactly one basic variable. In order to obtain an adjacent basic feasible solution, the simplex method makes one of the basic variables a non-basic variable and converts a non-basic variable to a basic variable in its place. • Question: How to select appropriate basic and non-basic variables that gives maximum improvement to the objective function?

  3. Simplex method • A basic feasible solution has non-negative value. The non-basic variables are always zero. Thus when we convert a non-basic variable to a basic variable we increase its value from zero to some positive quantity. • We have to choose that non-basic variable that causes maximum improvement in the objective function. To determine this, we can increase the value of non-basic variable by one unit (0 to 1) and check the change in the objective function value. Note that values of other non-basic variables will remain zero.

  4. Simplex method • Express problem in standard form. • Start with an initial BFS. • Use Inner Product Rule to find the relative-profit coefficients. • If all the relative profit coefficients are non-positive, then the current BFS is optimal. Otherwise select the non-basic variable with most positive relative-profit coefficient to enter as basic variable. • Apply Minimum Ratio Rule to determine the basic variable that will become non-basic variable. • Checks if the current BFS is optimal by calculating the relative-profit coefficients for all non-basic variables and repeat the cycle until optimality conditions are reached.

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  13. Reference • NPTEL Online course: Optimization in Chemical Engineering

  14. Thank you

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