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ISM 206 Lecture 4

ISM 206 Lecture 4 . Duality and Sensitivity Analysis. Announcements. Outline. 1. Review Simplex Method 2. Sensitivity analysis: How does the solution change as the parameters change? How much would we ‘pay’ for more resources What is the effect of changing parameters A, b, c

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ISM 206 Lecture 4

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  1. ISM 206Lecture 4 Duality and Sensitivity Analysis

  2. Announcements

  3. Outline • 1. Review Simplex Method • 2. Sensitivity analysis: • How does the solution change as the parameters change? • How much would we ‘pay’ for more resources • What is the effect of changing parameters A, b, c • Sensitivity through parametric LP solving • 2. Duality • What is duality and why does it matter? • The dual of a linear program • Sensitivity through duality

  4. Simplex Method • Objective: Max Z (first line of table) • Rules: • Update basic set of variables • Keep looking at feasible basic solutions • Keep positive x_j values • Zeros in top line correspond to basic variables • Identity matrix rows in table correspond to basic • Aim: Top line is non-negative

  5. The elements of the simplex tableu • After any iteration, the coefficients of the slack variables in each equation immediately reveal how that equation has been obtained from the initial equations. • The text talks about the ‘fundamental insight’: • After any iteration, the coefficients of the slack variables in each equation immediately reveal how that equation has been obtained by the initial equations

  6. Sensitivity Analysis • Changes in b • Changes in c • Changes in A • Introduction of a new variable • Introduction of a new constraint • Parametric Linear Programming • All demonstrated in OR Tutor

  7. Changes in b • Handle by checking optimality conditions under previous basis • How much could b change and still be optimal? • ranging

  8. Changes in c • Change will not affect feasibility! • Different procedure when parameter being changed depends on basic or nonbasic variable • Called ranging again

  9. Introducing a New variable • Same as changing the coefficients to a nonbasic variable

  10. Introducing a New Constraint • Check feasibility of original optimal • Add row to tableu and proceed as

  11. Parametric LP • Are there values of the parameter for which the problem has a solution? • How do the objective and optimal x depend on the parameter?

  12. Questions and Break

  13. The dual Linear Program Dual Primal

  14. Dual Linear Programs • The dual of a LP is another LP • Coefficients of primal objective = rhs of dual constraints • Rhs of primal constraints = coeffs of dual objective • Variable coefficients are the same (transposed)

  15. Translating between primal and dual • The dual of the dual is the primal • Weak duality theorem • Strong duality theorem • Complementary Slackness • Optimality Conditions • Interpretation of dual variables • Dual Simplex algorithm

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