LECTURE NOTES Duality Theory, Sensitivity Analysis, and Parametric Programming
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1 CEE 60 Davd Rosenberg p. LECTURE NOTES Dualty Theory, Senstvty Analyss, and Parametrc Programmng Learnng Objectves. Revew the prmal LP model formulaton 2. Formulate the Dual Problem of an LP problem (TUES) 3. Relate the dual program outputs to the prmal problem (TUES) 4. Identfy how LP outputs change wth changes n model nputs (TUES/THURS) o Rght-hand sde constrant values o Objectve functon coeffcents o Addng a new constrant o Systematcally changng a model nput. Revew Optmzaton and LP model formulaton General Formulaton Maxmze (or mnmze) Z = f(x) Subject to: g x b, m m LP Formulaton m LP Formulaton (vector-matrx format) c = by n row vector of objectve functon coeffcents x = n by column vector of decson varable values A = m by n matrx of constrant coeffcents b = m by column vector of constrant rght-hand-sde values
2 CEE 60 Davd Rosenberg p. 2 The PRIMAL formulaton s the formulaton we have been talkng about all along so far: How does the soluton change (the outputs Z and x) f we change varous nputs such as b, c, and A? 2. Formulate the Dual Problem of an LP Every prmal LP problem (.e., every LP problem!) has a correspondng dual formulaton. Prmal Dual n decsons (actvty levels) m constrants (resource avalabltes) Max Zx = cx s.t. Ax b x = vector of actvty levels c = unt product per actvty level ($/level) b = resource levels A = Constrant coeffcents (resource/actvty) Zx = (product/level)*(level) = product (e.g., $) Zy = b T y = Maxmze benefts from actvtes At the optmum,
3 CEE 60 Davd Rosenberg p. 3 Example. What s the dual of the followng prmal problem? Max Z x = 6 x egg + 7 x tom xegg s.t. 4 3 xtom 2000 Dual: Example 2. Solve and compare the solutons to the prmal and dual problems n Example #.
4 CEE 60 Davd Rosenberg p. 4 Example 3. What s the dual of the followng prmal problem? Mn Z x = 0 x + 4 x 2 s.t x x Dual:
5 CEE 60 Davd Rosenberg p Relate the dual program outputs to the prmal problem Interpretaton of dual varable y: Margnal mprovement to objectve functon Your wllngness to Shadow Value Complementary Slackness How are the shadow value and constrant slack varables related? At optmalty, f a prmary constrant s slack (slack varable > 0), then the correspondng shadow value for the constrant = 0. Smlarly, f a prmary constrant s bndng, then the correspondng shadow value > 0. Smlarly for the converse. Ether the slack varable or shadow value have to zero! Interpretaton: Bndng constrants reflect lmted (scare) resources. Scare resources have value. The value of the scarce resource s the shadow value. Relaxng a bndng constrant wll mprove the objectve functon value by the shadow value for the constrant. Show graphcally You can solve ether the Prmal or Dual problem to get all the outputs for both problems (Z, x, y). What s the dual of the dual formulaton?
6 CEE 60 Davd Rosenberg p Identfy how LP outputs change wth changes n model nputs General Method Determne whch decson varables are Determne whch constrants are A. Changes n Rght Hand Sde Constrant Values How? [Hnt look at the Dual Problem outputs] Example 4. How wll the objectve functon value n Example change f: () The rght-hand value n the frst constrant s decreased from 4,000,000 to 3,900,000 gallons? () The rght-hand value of the second constrant s decreased from 2,000 to,900 sq ft? Soluton: In Excel, called Shadow Prce. In GAMS, margnal assocated wth constrants. In Smplex, the objectve functon coeffcents of slack varables n the fnal tableau. Range of bass the change n rght-hand-sde coeffcent values allowed before the bass soluton wll change. In Excel, ths s called the allowable ncrease and allowable decrease. In GAMS, follows nstructons at ex. B. Changes n Objectve Functon Coeffcents Graphcally the rotaton of the objectve functon hyperplane (slope) that wll move the optmal soluton to a new bass. Rotaton so slope s less than (greater than) slope of one of the bndng constrants Mathematcally the Reduced Cost From smplex, the reduced cost s the coeffcent n the fnal row assocated wth the objectve functon
7 CEE 60 Davd Rosenberg p. 7 Can also calculate drectly as ~ c c constrant n the dual formulaton! m j Allowable ncrease before change the bass: a j y j,.e., the slack assocated wth the th c new ~ c c Reduced Cost n Excel; In GAMS, Margnal assocated wth decson varables Interpretaton: C. Addng a New Constrant Two optons:. Optmal soluton stll feasble (satsfes the new constrant) => same soluton 2. Optmal soluton now nfeasble New optmal soluton Objectve functon value wll decrease (for a maxmzaton problem) => smaller feasble regon D. Parametrc Programmng Systematcally change one nput parameter value and see how results (optmal objectve functon and decson varable values) change. Most easly done by solvng the optmzaton model multple tmes (.e., embed n a programmng loop). Brute force approach to senstvty analyss. Cases to use: Change multple model parameters smultaneously Changes for one parameter value are far beyond ranges of values allowed by margnal (shadow value and reduced cost) analyss. Do not use when: Changes n parameter values are very small and wthn the ranges covered by margnal (shadow value and reduced cost) analyss. Example 5. How wll the objectve functon and decson varable values n Example change f the water requrement for tomatoes decreases from 2,000 gal/plant to,500, 000, and 500 gal/plant? At what water requrement level does the soluton bass change? Soluton: See GAMS fle Ex2--parametrc.gms.
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