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Understanding Duality and Sensitivity Analysis in Linear Programming

This lecture focuses on the concepts of duality and sensitivity analysis within linear programming. It reviews the Simplex Method, exploring how changes in parameters affect the solutions and how the dual of a linear program is constructed. Participants will learn about the importance of sensitivity analysis in determining resource allocation, the implications of changing constraints, and the relationship between primal and dual LPs. Key topics include parametric LP solving, optimality conditions, and the interpretation of dual variables, all vital for effective decision-making in operations research.

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Understanding Duality and Sensitivity Analysis in Linear Programming

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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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