PLANNING
UNDER UNCERTAINTY
REGRET THEORY
MINIMAX REGRET ANALYSIS
Motivating Example
Traditional way
Maximize Average…select A
Optimistic decision maker
s1
High
s2
Medium
s3
Low
Average
A
19
14
-3
10
B
16
7
4
9
C
20
8
-4
8
D
10
6
5
7
Max
20(C)
14(A)
5(D)
10(A)
MaxiMax … select C
20
Pessimistic decision maker
C
$ Million
MaxiMin … select D
A
15
10
5
B
D
0
1
-5 s
1
2
s2
3
s3
MINIMAX REGRET ANALYSIS
Regret s
Pr ofit from
 Pr ofit from  

  

 
 Best Alternative s  Chosen Alternative s
 If chosen decision is the best  Zero regret
Nothing is better than the best  No negetive Regret
MINIMAX REGRET ANALYSIS
Motivating Example
Calculate regret:
find maximum regret
A … regret = 8 @ low market
C … regret = 9 @ low market
s1
High
s2
Medium
s3
Low
Maximum
Regret
A
1
0
8
8
B
4
7
1
7
C
0
6
9
9
D
10
8
0
10
D … regret = 10 @ high market
10
8
In general, gives conservative decision
but not pessimistic.
$ Million
B … regret = 7 @ medium market
MINIMAX  B
D
B
6
C
4
2
A
0
s11
s22
s33
MINIMAX REGRET ANALYSIS
Two-Stage Stochastic Programming Using Regret Theory
Two-Stage Model
Optimal Profit
Here & Now (HN)
Uncertainty Free
Optimal Profit
Wait & See (WS)
MINIMAX REGRET ANALYSIS
Two-Stage Stochastic Programming Using Regret Theory
Minimize MR x   MaxWSs - HN s x 
sS|x
where:
WS s  Max qsT ys  c T x
x, ys


subject to:
Ax  b , x  0
ys  0
Ts x  Wy s  hs


HN s ( x)  Max qsT ys  c T x
ys | x
subject to:
Ax  b , x  0
ys  0
Ts x  Wy s  hs
s
s
s1
High
s2
Medium
s3
Low
Average
A
19
14
-3
10
B
16
7
4
9
C
20
8
-4
8
D
10
6
5
7
Max
20(C)
14(A)
5(D)
10(A)
MINIMAX REGRET ANALYSIS
Two-Stage Stochastic Programming Using Regret Theory
Minimize MR x   MaxWSs - HN s x 
sS|x
where:
WS s  Max qsT ys  c T x


s1
High
s2
Medium
s3
Low
Average
A
19
14
-3
10
ys  0
B
16
7
4
9
Ts x  Wy s  hs
C
20
8
-4
8
D
10
6
5
7
Max
20(C)
14(A)
5(D)
10(A)
x, ys
subject to:
Ax  b , x  0

HN s ( x)  Max
ys | x
qsT

ys  c x
subject to:
Ax  b , x  0
ys  0
Ts x  Wy s  hs
T
s
s
MINIMAX REGRET ANALYSIS
Two-Stage Stochastic Programming Using Regret Theory
Minimize MR x   MaxWSs - HN s x 
sS|x
where:
WS s  Max qsT ys  c T x
x, ys


subject to:
Ax  b , x  0
ys  0
Ts x  Wy s  hs


HN s ( x)  Max qsT ys  c T x
ys | x
subject to:
Ax  b , x  0
ys  0
Ts x  Wy s  hs
s
s
s1
High
s2
Medium
s3
Low
Maximum
Regret
A
1
0
8
8
B
4
7
1
7
C
0
6
9
9
D
10
8
0
10
MINIMAX REGRET ANALYSIS
Two-Stage Stochastic Programming Using Regret Theory
s1
d1
19.01
d2
11.15
d3
12.75
d4
5.41
d5
15.09
Max 19.01
s2
10.38
14.47
7.81
9.91
7.40
14.47
s3
10.57
8.87
16.02
12.63
8.81
16.02
NPV
s4
15.48
20.54
22.25
32.02
12.48
32.02
s5
10.66
10.58
9.16
8.08
15.05
15.05
ENPV
13.22
13.12
13.60
13.61
11.77
13.61
Max Min
19.01 10.38
20.54 8.87
22.25 7.81
32.02 5.41
15.09 7.40
32.02 10.38
s1
d1
d2
d3
d4
d5
0.00
7.86
6.26
13.60
3.92
s2
4.09
0.00
6.66
4.56
7.07
Regret
s3
s4
s5
5.45 16.54 4.39
7.15 11.48 4.47
0.00 9.77 5.89
3.39 0.00 6.97
7.21 19.54 0.00
Min
35
25
15
5
s1
s2
d1
s3
d2
d3
s4
d4
s5
d5
Max
16.54
11.48
9.77
13.60
19.54
9.77
MINIMAX REGRET ANALYSIS
Limitations on Regret Theory
It is not necessary that equal differences
in profit would always correspond
to equal amounts of regret:
A small advantage in one scenario
may lead to the loss of larger
advantages in other scenarios.
May select different preferences if one
of the alternatives was excluded
or a new alternative is added.
$1000 - $1050 = 50
$100 - $150 = 50
s1
s2
s3
Max.
Regret
A
100
0
5
100
B
99
95
40
99
C
0
100 200
200
D
150
85
150
0
CONCLUSION
Suggested improvements to minimax-regret criterion:
 Minimizing the average regret instead
of minimizing the maximum.
 Minimizing the upper regret average
instead of the maximum only.
 Measure relative regret instead
of absolute regret:
s1
s2
s3
Max.
Regret
Avrg.
Regret
Upper
Regret
A 100
0
5
100
35
52.5
B
99
95
40
99
78
97
C
0
100 200 200
100
150
85
78.3
117.5
D 150
0
150
150  100
1050  1000
 33 %
 4.8% versus
150
1050
instead of:
1050-1000 = 50
versus
150-100 = 50