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LP SENSITIVITY

LP SENSITIVITY. Graphical Sensitivity Analysis. Objective Function Left-hand side of constraint Right-hand side of constraint. SENSITIVITY ANALYSIS. Does optimal solution change if input data changes?. INSENSITIVE. SENSITIVE. SENSITIVITY ANALYSIS. Estimation error Change over time?

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LP SENSITIVITY

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  1. LP SENSITIVITY

  2. Graphical Sensitivity Analysis Objective Function Left-hand side of constraint Right-hand side of constraint

  3. SENSITIVITY ANALYSIS Does optimal solution change if input data changes?

  4. INSENSITIVE

  5. SENSITIVE

  6. SENSITIVITY ANALYSIS • Estimation error • Change over time? • Input might be random variable • Should we remove constraint? • “What if?” questions

  7. GRAPHICAL • NEW EXAMPLE: RENDER AND STAIR • QUANTITATIVE ANALYSIS • X1 = NUMBER OF CD PLAYERS • X2 = NUMBER OF RECEIVERS

  8. ORIGINAL PROBLEM • MAX PROFIT = 50X1 + 120X2 • SUBJECT TO CONSTRAINTS • (1) ELECTRICIAN CONSTRAINT: • 2X1 + 4X2 < 80 • (2) AUDIO TECHNICIAN CONSTR: 3X1+ X2 < 60 • NEXT SLIDE: REVIEW OF LAST WEEK

  9. RECEIVER 0,60 AUDIO 0,20 16,12 ELEC PLAYER 20,0 40,0

  10. MAXIMUM PROFIT

  11. RECEIVER 0,60 MAX AUDIO 0,20 16,12 ELEC PLAYER 20,0 40,0

  12. ORIGINAL PROBLEM • MAKE RECEIVERS ONLY • NOW WE WILL BEGIN TO CONSIDER “WHAT IF” QUESTIONS • IF NEW OPTIMUM IS “MAKE RECEIVERS ONLY”, WE CALL IT “OUTPUT INSENSITIVE” • IF NEW OPTIMUM IS A DIFFERENT CORNER POINT, “OUTPUT SENSITIVE”

  13. SENSITIVITY ANALYSIS I.A. OBJECTIVE FUNCTION

  14. NEW OBJECTIVE FUNCTION • 50X1 + 80X2 • NEW PROFIT PER RECEIVER = $80 • OLD PROFIT “ “ WAS $120 • IS OPTIMUM SENSITIVE TO REDUCED PROFIT, PERHAPS DUE TO LOW PRICE (INCREASED COMPETITION) OR HIGHER COST?

  15. NEW MAXIMUM PROFIT

  16. RECEIVER 0,60 OLD MAX AUDIO 0,20 =NEW MAX 16,12 ELEC PLAYER 20,0 40,0

  17. OUTPUT SENSITIVE • WE NOW MAKE BOTH RECEIVERS AND PLAYERS (MIX) • ORIGINAL: RECEIVERS ONLY • REASON: LOWER PROFIT OF EACH RECEIVER IMPLIES PLAYERS MORE DESIRABLE THAN BEFORE

  18. SENSITIVITY ANALYSIS I.B LEFT –HAND SIDE OF CONSTRAINT

  19. ELECTRICIAN CONSTRAINT • OLD: 2X1 + 4X2 < 80 • NEW: 2X1 + 5X2<80 • REASON: NOW TAKES MORE TIME FORELECTRICIAN TO MAKE ONE RECEIVER(5 NOW VS 4 BEFORE), LOWER PRODUCTIVITY PERHAPS DUE TO WEAR AND TEAR

  20. BACK TO ORIGINAL OBJECTIVE FUNCTION • WE COMPARE WITH ORIGINAL PROBLEM, NOT FIRST SENSITIVITY • BUT FEASIBLE REGION WILL CHANGE

  21. RECEIVER 0,60 AUDIO 0,20 16,12 0,16 OLD NEW ELEC ELEC 17,9.2 PLAYER 20,0 40,0

  22. SMALLER FEASIBLE REGION • CAN MAKE FEWER RECEIVERS THAN BEFORE • NEW INTERCEPT (0,16) REPLACES (0,20) • (0,20) NOW INFEASIBLE • ALSO, NEW MIX CORNER POINT: • (17,9.2)

  23. NEW MAXIMUM PROFIT

  24. OUTPUT SENSITIVE • COMPARED TO ORIGINAL PROBLEM, A NEW OPTIMUM • INCREASED LABOR TO MAKE RECEIVER MAKES PLAYER MORE DESIRABLE

  25. I.C. RIGHT-HAND SIDE OF CONSTRAINT

  26. SLACK VARIABLES • S1 = NUMBER OF HOURS OF ELECTRICIAN TIME NOT USED • S2 =NUMBER OF HOURS OF AUDIO TIME NOT USED • (1) ELEC CONSTR: 2X1+4X2+S1=80 • (2) AUDIO CONSTR: 3X1+X2+S2=60

  27. RECEIVER BACK TO ORIGINAL OPTIMUM OPTIMUM ON ELEC CONSTR 0,60 OPTIMUM NOT ON AUDIO CONSTR MAX AUDIO 0,20 16,12 ELEC PLAYER 20,0 40,0

  28. X1=0,X2=20 • (1)ELEC: 2(0)+4(20)+S1=80 • S1 = 0 • NO IDLE ELECTRICIAN • (2) AUDIO: 3(0)+20+S2=60 • S2 = 60 –20= 40 • AUDIO SLACK

  29. INPUT SENSITIVE • INPUT SENSITIVE IF NEW OPTIMUM CHANGES SLACK • ORIGINAL ELEC SLACK = 0 • IF NEW ELEC SLACK > 0, INPUT SENSITIVE • IF NEW ELEC SLACK STILL ZERO, INPUT INSENSITIVE

  30. CHANGE IN RIGHT SIDE • OLD ELEC CONSTR: 2X1+4X2<80 • NEW ELEC CONSTR 2X1+4X2<300 • NEW INTERCEPTS: (0,75) & (150,0)

  31. RECEIVER 0,75 REDUNDANT CONSTR 0,60 MAX NEW ELEC AUDIO 0,20 16,12 OLD ELEC PLAYER 150,0 20,0 40,0

  32. RECEIVER 0,75 REDUNDANT CONSTR 0,60 OLD MAX AUDIO 0,20 PLAYER 150,0 20,0 40,0

  33. NEW MAXIMUM

  34. RECEIVER INFEASIBLE 0,75 REDUNDANT CONSTR =NEWMAX 0,60 OLD NEW ELEC MAX AUDIO 0,20 PLAYER 150,0 20,0 40,0

  35. NEW OPTIMUM • OUTPUT INSENSITIVE SINCE WE STILL MAKE RECEIVERS ONLY (SAME AS ORIGINAL) • BUT INPUT SENSITIVE • ORIGINAL: OPTIMUM ON ELEC CONSTR • NEW: OPTIMUM ON AUDIO CONSTR

  36. SLACK

  37. INTERPRET • INCREASE IN ELECTRICIAN AVAILABILITY • TOO MANY ELECTRICIANS • THEREFORE ELEC SLACK

  38. SHADOW PRICE • VALUE OF 1 ADDITIONAL UNIT OF RESOURCE • INCREASE IN PROFIT IF WE COULD INCREASE RIGHT-HAND SIDE BY 1 UNIT

  39. THIS EXAMPLE • SHADOW PRICE = MAXIMUM YOU WOULD PAY FOR 1 ADDITIONAL HOUR OF ELECTRICIAN • OLD ELEC CONSTR: 2X1+4X2<80 • NEW ELEC CONSTR: 2X1+4X2<81

  40. OPTIMUM

  41. SHADOW PRICE = 2430-2400=30 Electrician “worth” up to $30/hr “Dual” value

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