LP SENSITIVITY

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LP SENSITIVITY
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Graphical Sensitivity Analysis

• Objective Function
• Left-hand side of constraint
• Right-hand side of constraint

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SENSITIVITY ANALYSIS

• Does optimal solution change if input data
  changes?

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INSENSITIVE
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SENSITIVE
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SENSITIVITY ANALYSIS

• Estimation error
• Change over time?
• Input might be random variable
• Should we remove constraint?
• What if? questions

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GRAPHICAL

• NEW EXAMPLE RENDER AND STAIR
• QUANTITATIVE ANALYSIS
• X1 NUMBER OF CD PLAYERS
• X2 NUMBER OF RECEIVERS

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ORIGINAL PROBLEM

• MAX PROFIT 50X1 120X2
• SUBJECT TO CONSTRAINTS
• (1) ELECTRICIAN CONSTRAINT
• 2X1 4X2 lt 80
• (2) AUDIO TECHNICIAN CONSTR 3X1 X2 lt 60
• NEXT SLIDE REVIEW OF LAST WEEK

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RECEIVER
0,60
AUDIO
0,20
16,12
ELEC
PLAYER
20,0
40,0
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MAXIMUM PROFIT
PLAYERSX1 RECEIVERS X2 PROFIT 50X1120X2
0 20 2400MAX
16 12 2240
20 0 1000
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RECEIVER
0,60
MAX
AUDIO
0,20
16,12
ELEC
PLAYER
20,0
40,0
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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

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SENSITIVITY ANALYSIS

• I.A. OBJECTIVE FUNCTION

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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?

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NEW MAXIMUM PROFIT
PLAYERSX1 RECEIVERS X2 PROFIT 50X180X2
0 20 1600
16 12 1760NEW MAX
20 0 1000
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RECEIVER
0,60
OLD
MAX
AUDIO
0,20
NEW MAX
16,12
ELEC
PLAYER
20,0
40,0
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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

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SENSITIVITY ANALYSIS

• I.B LEFT HAND SIDE OF CONSTRAINT

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ELECTRICIAN CONSTRAINT

• OLD 2X1 4X2 lt 80
• NEW 2X1 5X2lt80
• REASON NOW TAKES MORE TIME FORELECTRICIAN TO
  MAKE ONE RECEIVER(5 NOW VS 4 BEFORE), LOWER
  PRODUCTIVITY PERHAPS DUE TO WEAR AND TEAR

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BACK TO ORIGINAL OBJECTIVE FUNCTION

• WE COMPARE WITH ORIGINAL PROBLEM, NOT FIRST
  SENSITIVITY
• BUT FEASIBLE REGION WILL CHANGE

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

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NEW MAXIMUM PROFIT
PLAYERSX1 RECEIVERS X2 PROFIT 50X1120X2
0 16 1920
17 9.2 1954NEW MAX
20 0 1000
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OUTPUT SENSITIVE

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

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I.C. RIGHT-HAND SIDE OF CONSTRAINT
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SLACK VARIABLES

• S1 NUMBER OF HOURS OF ELECTRICIAN TIME NOT
  USED
• S2 NUMBER OF HOURS OF AUDIO TIME NOT USED
• (1) ELEC CONSTR 2X14X2S180
• (2) AUDIO CONSTR 3X1X2S260

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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
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X10,X220

• (1)ELEC 2(0)4(20)S180
• S1 0
• NO IDLE ELECTRICIAN
• (2) AUDIO 3(0)20S260
• S2 60 20 40
• AUDIO SLACK

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INPUT SENSITIVE

• INPUT SENSITIVE IF NEW OPTIMUM CHANGES SLACK
• ORIGINAL ELEC SLACK 0
• IF NEW ELEC SLACK gt 0, INPUT SENSITIVE
• IF NEW ELEC SLACK STILL ZERO, INPUT INSENSITIVE

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CHANGE IN RIGHT SIDE

• OLD ELEC CONSTR 2X14X2lt80
• NEW ELEC CONSTR 2X14X2lt300
• NEW INTERCEPTS (0,75) (150,0)

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RECEIVER
0,75
REDUNDANT CONSTR
0,60
MAX
NEW ELEC
AUDIO
0,20
16,12
OLD
ELEC
PLAYER
150,0
20,0
40,0
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RECEIVER
0,75
REDUNDANT CONSTR
0,60
OLD
MAX
AUDIO
0,20
PLAYER
150,0
20,0
40,0
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NEW MAXIMUM
X1 X2 PROFIT 50X1120X2
0 60 7200MAX
20 0 1000
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RECEIVER
INFEASIBLE
0,75
REDUNDANT CONSTR
NEWMAX
0,60
OLD
NEW ELEC
MAX
AUDIO
0,20
PLAYER
150,0
20,0
40,0
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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

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SLACK
SLACK VARIABLE ORIG NEW
ELEC S10 S1gt0
AUDIO S2gt0 S20
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INTERPRET

• INCREASE IN ELECTRICIAN AVAILABILITY
• TOO MANY ELECTRICIANS
• THEREFORE ELEC SLACK

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SHADOW PRICE

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

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THIS EXAMPLE

• SHADOW PRICE MAXIMUM YOU WOULD PAY FOR 1
  ADDITIONAL HOUR OF ELECTRICIAN
• OLD ELEC CONSTR 2X14X2lt80
• NEW ELEC CONSTR 2X14X2lt81

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OPTIMUM
X1 X2 PROFIT 50X1120X2
0 80/420 2400OLD MAX
0 81/420.25 2430NEW MAX
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SHADOW PRICE 2430-240030

• Electrician worth up to 30/hr
• Dual value