1. LP SENSITIVITY

2. Graphical Sensitivity Analysis A.Objective Function B.Left-hand side of constraint C.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. PLAYER RECEIVER AUDIO 0,60 20,0 ELEC 0,20 40,0 16,12

10. MAXIMUM PROFIT PLAYERS=X1RECEIVERS =X2 PROFIT= 50X1+120X2 0202400=MAX 16122240 2001000

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

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 PLAYERS=X1RECEIVERS =X2 PROFIT= 50X1+80X2 0201600 16121760=NEW MAX 2001000

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

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

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 PLAYERS=X1RECEIVERS =X2 PROFIT= 50X1+120X2 0161920 179.21954=NEW MAX 2001000

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

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

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

33. NEW MAXIMUM X1X2PROFIT= 50X1+120X2 0607200=MAX 2001000

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

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 SLACK VARIABLE ORIGNEW ELECS1=0S1>0 AUDIOS2>0S2=0

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 X1X2PROFIT= 50X1+120X2 080/4=202400=OLD MAX 081/4=20.252430=NEW MAX

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