Scheduling under Uncertainty:
Solution Approaches
Frank Werner
Faculty of Mathematics
Outline of the talk
1.
2.
3.
4.
5.
6.
Introduction
Stochastic approach
Fuzzy approach
Robust approach
Stability approach
Selection of a suitable approach
St. Etienne / France | November 23, 2012
2
1. Introduction
Notations
J  J1 ,...,J n  - setof jobs
M  M1 ,...,M m  - setof machines
Q  Oij | Ji  J , j  1,...,ni   1,...,q - setof operations
pij - processingtimeof Oij
wi , ri , di ,..., - furtherdatafor J i  J
St. Etienne / France | November 23, 2012
3
• Deterministic models:
all data are deterministically given in advance
• Stochastic models:
data include random variables
In real-life scheduling: many types of uncertainty
(e.g. processing times not exactly known, machine
breakdowns, additionally ariving jobs with high priorities,
rounding errors, etc.)
Uncertain (interval) processing times:
pijL  pij  pijU
St. Etienne / France | November 23, 2012
for allOij  Q
4


T  p  Rq | pijL  pij  pijU , Oij Q - setof scenarios
 problem
 |  , pijL  pij  pijU | 
pijL  pijU for all Oij  Q
 deterministic problem
 |  |
Relationship between stochastic and uncertain problems:
L
Distribution function

0 if t  p
Density function
Fij (t )  P( pij  t )  

1
0

f ij (t )  F 'ij (t )  ?
0

St. Etienne / France | November 23, 2012
ij
U
ij
if t  p
if t  pijL
if pijL  pij  pijU
if t  pijU
5
Approaches for problems with inaccurate data:
• Stochastic approach: use of random variables with
known probability distributions
• Fuzzy approach: fuzzy numbers as data
• Robust approach: determination of a schedule
hedging against the worst-case scenario
• Stability approach: combination of a stability analysis, a
multi-stage decision framework and the concept of a
minimal dominant set of semi-active schedules
→ There is no unique method for all types of uncertainties.
St. Etienne / France | November 23, 2012
6
Two-phase decision-making procedure
1) Off-line (proactive) phase
construction of a set S * of potentially optimal solutions
before the realization of the activities
(static scheduling environment, schedule planning phase)
2) On-line (reactive) phase
selection of a solution from S * when more information
is available and/or a part of the schedule has already
been realized
→ use of fast algorithms
(dynamic scheduling environment, schedule execution
phase)
St. Etienne / France | November 23, 2012
7
General literature (surveys)
•
Pinedo: Scheduling, Theory, Algorithms and Systems, Prentice Hall,
1995, 2002, 2008, 2012
• Slowinski and Hapke: Scheduling under Fuzziness, Physica, 1999
• Kasperski: Discrete Optimization with Interval Data, Springer, 2008
• Sotskov, Sotskova, Lai and Werner: Scheduling under Uncertainty;
Theory and Algorithms, Belarusian Science, 2010
For the RCPSP under uncertainty, see e.g.
• Herroelen and Leus, Int. J. Prod. Res.. 2004
• Herroelen and Leus, EJOR, 2005
• Demeulemeester and Herroelen, Special Issue, J. Scheduling, 2007
St. Etienne / France | November 23, 2012
8
2. Stochastic approach
•
Distribution of random variables
(e.g. processing times, release dates, due dates)
known in advance
• Often: minimization of expectation values
(of makespan, total completion time, etc.)
Classes of policies (see Pinedo 1995)
• Non-preemptive static list policy (NSL)
• Preemptive static list policy (PSL)
• Non-preemptive dynamic policy (ND)
• Preemptive dynamic policy (PD)
St. Etienne / France | November 23, 2012
9
Some results for single-stage problems (see Pinedo 1995)
Single machine problems
(a) Problem 1 || E wiCi 
pi ~ arbitrarily distributed
WSEPT rule: order the jobs according to non-increasing
ratios wi
E  pi 
Theorem 1: The WSEPT rule determines an optimal
solution in the class of NSL as well as ND policies.
(b) Problem 1 || ELmax 
pi ~ arbitrarily distributed, di fixed
Theorem 2: The EDD rule determines an optimal
solution in the class of NSL, ND and PD policies.
St. Etienne / France | November 23, 2012
10
(c) Problem1 | di  d | E wiUi 
pi ~ exponentiallydistributed, d fixed
Theorem 3: The WSEPT rule determines an optimal
solution in the class of NSL, ND and PD policies.
Remark: The same result holds for geometrically
distributed pi .
Parallel machine problems
pi ~ exponentiallydistributed
(a) Problem P2 || ECmax 
Theorem 4: The LEPT rule determines an optimal
solution in the class of NSL policies.
St. Etienne / France | November 23, 2012
11
(b) Problem P | pmtn| ECmax 
Theorem 5: The non-preemptive LEPT policy
determines an optimal solution in the class of PD
policies.
(c) Problem P | pmtn| ECi 
Theorem 6: The non-preemptive SEPT policy
determines an optimal solution in the class of PD
policies.
St. Etienne / France | November 23, 2012
12
Selected references (1)
•
•
•
•
•
•
•
•
•
•
•
•
•
Pinedo and Weiss, Nav. Res. Log. Quart., 1979
Glazebrook, J. Appl. Prob., 1979
Weiss and Pinedo, J. Appl. Prob., 1980
Weber, J. Appl. Prob., 1982
Pinedo, Oper. Res., 1982; 1983
Pinedo, EJOR, 1984
Pinedo and Weiss, Oper. Res., 1984
Möhring, Radermacher and Weiss, ZOR, 1984; 1985
Pinedo, Management Sci., 1985
Wie and Pinedo, Math. Oper. Res., 1986
Weber, Varaiya and Walrand, J. Appl. Prob., 1986
Righter, System and Control Letters, 1988
Weiss, Ann. Oper. Res., 1990
St. Etienne / France | November 23, 2012
13
Selected references (2)
•
•
•
•
•
•
•
•
•
•
•
•
•
Weiss, Math. Oper. Res., 1992
Righter, Stochastic Orders, 1994
Cai and Tu, Nav. Res. Log., 1996
Cai and Zhou, Oper. Res., 1999
Möhring, Schulz and Uetz, J. ACM, 1999
Nino-Mora, Encyclop. Optimiz., 2001
Cai, Sun and Zhou, Prob. Eng. Inform. Sci., 2003
Ebben, Hans and Olde Weghuis, OR Spectrum, 2005
Ivanescu, Fransoo and Bertrand, OR Spectrum, 2005
Cai, Wu and Zhou, IEEE Transactions Autom. Sci. Eng., 2007
Cai, Wu and Zhou, J. Scheduling, 2007; 2011
Cai, Wu and Zhou, Oper. Res., 2009
Tam, Ehrgott, Ryan and Zakeri, OR Spectrum, 2011
St. Etienne / France | November 23, 2012
14
3. Fuzzy approach
•
Fuzzy scheduling techniques either fuzzify existing
scheduling rules or solve mathematical programming
problems
~
~
• Often: fuzzy processing times pi , fuzzy due dates di
• Examples
triangular fuzzy processing times
 ~p
" pi is aroundpiM "
i
1 .0
trapezoidal fuzzy processing times
 ~p
i
~
pi
1 .0
~
pi
0
piL
piM
piU
St. Etienne / France | November 23, 2012
0 pL
i
p
p
piU
15
Often: possibilistic approach (Dubois and Prade 1988)
PosV  x   ~p ( x), x  R
PosV  a, b  sup  ~p ( x)
xa ,b 
Nec V  a, b  inf 1   ~p ( x) 
xa ,b 
Chanas and Kasperski (2001)
~
~
Problem 1 | prec, pi , di | f max
~
fi Ci ( ) min!
Objective: max
i
~
Assumption: fi F - monotonicw.r.t. Ci
→ adaption of Lawler‘s algorithm for problem 1 | prec | f max
St. Etienne / France | November 23, 2012
16
Special cases:
a)
b)
c)
d)
 

 

~
~
minPosC ( )  d  max!
~
maxE L ( )  min!
~
~
max Pos Ci ( )  d i  min!
i
~
~
max Nec Ci ( )  d i  min!
i
i
i
i
i
i
Alternative goal approach
~
~
~
 ~

  max!
Pos
max
w
L
(

)

G
G - fuzzy goal, Objective:
 i
i i


Chanas and Kasperski (2003)
~
~
~


E Ti ( )  min!
Problem 1| pi , di | maxETi 
Objective: max
i
→ adaption of Lawler‘s algorithm for problem 1 | prec | f max
St. Etienne / France | November 23, 2012
17
Selected references (1)
•
•
•
•
•
•
•
•
•
•
Dumitru and Luban, Fuzzy Sets and Systems, 1982
Tada, Ishii and Nishida, APORS, 1990
Ishii, Tada and Masuda, Fuzzy Sets and Systems, 1992
Grabot and Geneste, Int. J. Prod. Res., 1994
Han, Ishii and Fuji, EJOR, 1994
Ishii and Tada, EJOR, 1995
Stanfield, King and Joines, EJOR, 1996
Kuroda and Wang, Int. J. Prod. Econ., 1996
Özelkan and Duckstein, EJOR, 1999
Sakawa and Kubota, EJOR, 2000
St. Etienne / France | November 23, 2012
18
Selected references (2)
•
•
•
•
•
•
•
Chanas and Kasperski, Eng. Appl. Artif. Intell., 2001
Chanas and Kasperski, EJOR, 2003
Chanas and Kasperski, Fuzzy Sets and Systems, 2004
Itoh and Ishii, Fuzzy Optim. and Dec. Mak., 2005
Kasperski, Fuzzy Sets and Systems, 2005
Inuiguchi, LNCS, 2007
Petrovic, Fayad, Petrovic, Burke and Kendall, Ann. Oper. Res., 2008
St. Etienne / France | November 23, 2012
19
4. Robust approach
Objective: Find a solution, which minimizes the „worst-case“
performance over all scenarios.
Notations (single machine problems)


Fp ( ) - functionvalueof sequence  J k1 ,...,J kn for p T
Fp* - optimalfunctionvaluefor p T
S - setof feasiblejobsequences
maximal regret of   S

Z ( )  max Fp ( )  Fp*
pT

Minmax regret problem (MRP): Find a sequence * such that
 
Z  *  minZ ( )
 S
St. Etienne / France | November 23, 2012
20
Some polynomially solvable MRP
1 | prec, piL  pi  piU , diL  di  diU | Lmax
(Kasperski 2005)
1 | prec, piL  pi  piU , diL  di  diU , wiL  wi  wiU | maxwiTi
(Volgenant and Duin 2010)
(Averbakh 2006)
Fm | n  2, p L  p  pU | C
ij
ij
ij
max
1| pi  1, wiL  wi  wiU | Ui
(Kasperski 2008)
Some NP-hard MRP
1| piL  pi  piU | Ci is NP - hard (Lebedev and Averbakh 2006)
(for a 2-approximation algorithm, see Kasperski and Zielinski 2008)
F 2 | pijL  pij  pijU | Cmax is stronglyNP - hard
(Kasperski, Kurpisz and Zielinski 2012)
St. Etienne / France | November 23, 2012
21
Kasperski and Zielinski (2011)
Consideration of MRP‘s using fuzzy intervals

Z ' ( )  min Fp ( )  Fp*

Deviation interval I  Z ' ( ), Z ( )
Known: deviation z ( )  I
Application of possibility theory (Dubois and Prade 1988)
 possibly optimal if Z ' ( )  0
 necessarily optimal if Z ( )  0
St. Etienne / France | November 23, 2012
22
Fuzzy problem




~
Nec z( ) G  max!
or equivalently
~C
Pos z( )  G  min!
~
~C
~
where G is a fuzzy interval and G is the complement of G
with membership function 1  G~ ( x).
The fuzzy problem can be efficiently solved if a polynomial
algorithm for the corresponding MRP exists.
St. Etienne / France | November 23, 2012
23
Solution approaches
a) Binary search method
- repeated exact solution of the MRP
- applications:
~
1 | prec, ~
pi , di | Lmax : O(n4 log 1 ) algorithm
~ | maxw T : O(n3 log 1 ) algorithm
1 | prec, w
i
i i
~
~
1| prec, wi , di | maxwiTi : O(n4 log 1 ) algorithm
~ | wU : O(n mind , n  dlog 1 ) algorithm
1 | pi  1, di  d , w
i  i i
F2 | ~
pij | Cmax: binary search subroutine in B&B algorithm
St. Etienne / France | November 23, 2012
24
b) Mixed integer programming formulation
- use of a MIP solver
- application:
1| ~
p| C
i

i
c) Parametric approach
- solution of a parametric version of a MRP
(often time-consuming)
- application:
~
1 | prec, di | Lmax
St. Etienne / France | November 23, 2012
25
Selected references (1)
•
•
•
•
•
•
•
•
•
•
Daniels and Kouvelis, Management Sci., 1995
Kouvelis and Yu, Kluwer, 1997
Kouvelis, Daniels and Vairaktarakis, IEEE Transactions, 2000
Averbakh, OR Letters, 2001
Yang and Yu, J. Comb. Optimiz., 2002
Kasperski, OR Letters, 2005
Kasperski and Zielinski, Inf. Proc. Letters, 2006
Lebedev and Averbakh, DAM, 2006
Averbakh, EJOR, 2006
Montemanni, JMMA, 2007
St. Etienne / France | November 23, 2012
26
Selected references (2)
•
•
•
•
•
•
Kasperski and Zielinski, OR Letters, 2008
Sabuncuoglu and Goren, Int. J. Comp. Integr. Manufact., 2009
Aissi, Bazgan and Vanderpooten, EJOR, 2009
Volgenant and Duin, COR, 2010
Kasperski and Zielinski, FUZZ-IEEE, 2011
Kasperski, Kurpisz and Zielinski, EJOR, 2012
St. Etienne / France | November 23, 2012
27
5. Stability approach
5.1. Foundations
5.2. General shop problem
5.3. Two-machine flow and job
shop problems
5.4. Problem
1 | p  pi  p |  wiCi
L
i
U
i
28
5.1. Foundations
Mixed Graph G  (V , A, E)
Example:
p  75
p
11
12
11
 50
12
p13  40
13
00
**
p00  0
p**  0
21
22
p21  60
p22  55
23
p23  30
(G)  Gs | Gs  (V , A  Es , )  G1, G2 ,...,G  - set of digraphs
Gs  (G)  semiactiveschedules  c1 (s), c2 (s),...,cq (s)
St. Etienne / France | November 23, 2012
29
Example (continued)
c11  75
G1
c12  125
11
12
c13  165
13
**
00
21
22
c21  60
c22  130
Cmax G1   165
23
c23  160
C G   325
i
1
G  G1, G2 ,...,G5 
St. Etienne / France | November 23, 2012
30
Stability analysis of an optimal digraph
Definition 1
The closed ball O ( p) with   R1 and p  Rq is called a
stability ball of Gs  (G) if for any p' O ( p)  R q , Gs ( p' )
remains optimal.
The maximal value
s ( p)  max  R1 | Gs optimalfor any p'O ( p)  Rq 
is called the stability radius of digraph Gs .
Known:
• Characterization of the extreme values of s ( p)
• Formulas for calculating s ( p) for   Cmax , Ci
• Computational results for job shop problems with


n  10 and m  8 (see Sotskov, Sotskova and Werner, Omega, 1997)
St. Etienne / France | November 23, 2012
31
L
U
G
|
p

p

p
5.2. General shop problem ij ij ij | 
Definition 2
* (G)  (G) is called a G-solution for problem
G | pijL  pij  pijU |  if for any fixed p T , * (G) contains an
optimal digraph.
*
*
If any (G)   (G) is not a G-solution,  (G) is called a
minimal G-solution denoted as T (G).
Introduction of the relative stability radius:
0  pij    pijL  pij  pijU  polytope T
(G)  B  (G)
St. Etienne / France | November 23, 2012
32
  Cmax
lsp - criticalweightinGs for p T
Definition 3
Let Gs  B  (G) be such that for any p' O ( p) T


lsp'  min lkp' | Gk  B .
The maximal value of  of such a stability ball O ( p) is
B
called the relative stability radius ˆ s  p T .
Known:
• Dominance relations among paths and sets of paths
• Characterization of the extreme values of ˆ sB  p T 
St. Etienne / France | November 23, 2012
33
L
U
Characterization of a G-solution for problem J | pij  pij  pij | Cmax
Definition 4
Gs (strongly) dominates Gk in D  R q
if lsp  lkp lsp  lkp  for any p  D.
→ dominance relation Gs DGk Gs  D Gk 
Theorem 7:
  (G) is a G-solution. There exists a finite covering
q
T
D

R
of polytope by closed convex sets j
 with
d
T   j 1 D j , d   , such that for any Gk  (G) and any
D j , j  1,...,d , there exists a Gs   for which Gs D Gk .
Corollary: T (G)  Gs   Gs T Gk for any Gk  (G).
j
St. Etienne / France | November 23, 2012
34
Theorem 8:
*
*

Let  (G) be a G-solution with (G )  2.
Then:
*
* (G) is a minimal G-solution.
For any Gs   (G)
there exists a vector p( s ) T such that
Gs  p ( s ) Gk for any Gk  * (G ) \ Gs .

L
U
J
|
p

p

p
Algorithms for problem
ij
ij
ij | 
 - regular criterion,e.g.  Cmax , Ci ,...
St. Etienne / France | November 23, 2012
35
Several 3-phase schemes:
• B&B: implicit (or explicit) enumeration scheme for
generating a G-solution B
B  '
•
•
SOL: reduction of B by generating a sequence
ˆ1  ˆ 2  ...  ˆ I of Oˆi ( p) with the same p  T and
different B
(G)  *
MINSOL: generation of a minimal G-solution T (G) by
a repeated application of algorithm SOL
T (G )  T
St. Etienne / France | November 23, 2012
36
Some computational results:
  Cmax
(n, m)
Degree of
uncertainty
(4,4)
1, 3, 5, 7
Exact solution
*
T
'
41.8
6.4
2.4
2, 6, 8, 10
79.0
14.7
5, 10, 15, 20
434.9
43.5
'
Heuristic solution
*
T
19.9
3.8
2.4
9.5
27.3
6.9
4.4
34.8
112.8
25.7
20.0
  Ci
(n, m)
Degree of
uncertainty
(4,4)
1, 3, 5, 7
'
Exact solution
*
T
'
34.2
7.5
6.3
2, 6, 8, 10
88.3
16.1
5, 10, 15, 20
477.7
30.8
Heuristic solution
*
T
24.1
6.5
5.5
14.5
52.9
13.5
12.0
30.1
132.0
24.8
24.0
Exact sol.: n  m  24 , Heuristic sol.: n  m  50 (n  10, m  8)
St. Etienne / France | November 23, 2012
37
5.3. Two-machine problems with interval
processing times
a) Problem F 2 | pijL  pij  pijU | Cmax
pijL  pijU for all (i, j)  Q  O(n logn) algorithmby Johnson(1954)
Johnson permutation:   Ji , Ji ,...,Ji



1

2
n

with min pik ,1, pil ,2  min pil ,1, pik ,2 for 1  k  l  n is optimal
*
Partition of the job set J  J 0  J1  J 2  J with


J  J  J \ J | p  p 
J  J  J \ J | p  p 
J  J  J | p  p , p  p 
J 0  J i  J | piL1  piU1  piL2  piU2
1
i
2
i
*
i
0
U
i1
L
i2
0
U
i2
L
i1
U
i1
L
i2
U
i2
St. Etienne / France | November 23, 2012
L
i1
38
J - solutionS (T ) : minimalset containinga Johnsonpermutation
for any p  T
Theorem 9:
S (T ) 1 
(1) for any Ji , J j  J1 ( J 2 , respectively)
U
L
U
L
p

p
or
p

p
either i1
j1
j1
i1
U
L
U
L
p

p
or
p

p
(either i 2
j2
j2
i 2) and
*
*
*
J

1
J


,
J

J
(2)
and if
satisfies
i*
L
U
p

max
p
*
– i ,1
i1 | J i  J1

 maxp

J 
U
– piL , 2
j2 | J j
2
L
L

 pk for any J k  J 0
max
p
,
p
–
i ,1
i ,2
*
*
*
St. Etienne / France | November 23, 2012
39
Theorem 10:
If maxpijL | Ji  J ,1  j  2 minpijU | Ji  J ,1  j  2
then S (T )  n!
Percentage of instances with S (T )  1 , where pijU  pijL  L
n
L
5
10
15
20
25
30
1
99.2
95.2
91.2
86.1
79.2
72.8
2
97.2
89.8
77.6
63.5
51.0
39.6
3
95.0
80.9
66.4
47.6
32.8
20.6
4
91.8
78.6
56.0
39.2
20.3
10.7
5
91.0
69.4
44.9
28.9
14.6
6.0
St. Etienne / France | November 23, 2012
40
General case of problem F 2 | pij  pij  pij | Cmax
Theorem 11:
There exists an S (T ) with J v  J w inany   S (T ) 
L
p
U
v1
 
U

 pwL1 and pvU1  pvL2 or pUw2  pwL1 and pUw2  pvL2 .
 J v , J w  A , J v  J w
 constructthe dominancegraph G  J, A  in O(n²) time
Theorem 12:
If J 0  , then A transitive.
St. Etienne / France | November 23, 2012
41
Example: n  6
Ji
J1
J2
J3
J4
J5
J6
piL1
1
9
10
5
5
10
piU1
piL2
2
12
11
8
6
11
8
14
13
6
4
4
piU2
9
15
17
7
4
4
A : J1  J 2 , J1  J 3 , J1  J 4 , J1  J 5 , J1  J 6 ,
J 2  J5 , J 2  J 6 , J3  J5 , J3  J6 , J5  J6 , J6  J5
J,A 

without transitive arcs:
J2
J1
J3
J5
J6
J4
St. Etienne / France | November 23, 2012
42
Properties of J, A  in the case of J 0   :
see Matsveichuk, Sotskov and Werner, Optimization, 2011
Schedule execution phase:
see Sotskov, Sotskova, Lai and Werner, Scheduling under
uncertainty, 2010 (Section 3.5)
Computational results for n  100 if J 0  
and for n  1000 if J 0  
b) Problem J 2 | ni  2, pijL  pij  pijU | Cmax
→ Reduction to two F 2 | pijL  pij  pijU | Cmax problems:
see Sotskov, Sotskova, Lai and Werner, Scheduling under
uncertainty, 2010 (Section 3.6)
St. Etienne / France | November 23, 2012
43
5.4. Problem 1| p  pi  p |  wiCi
L
i
U
i
Notations:
J  J1,...,J n  - setof n jobs
wi - weightfor J i  J


T  p  R | p
pi  piL , piU - processingtimeof J i  J , 0  piL  piU
n

L
i
p   p1 ,..., pn 
 k  J k ,...,J k
1
n

 pi  piU , i  1,...,n - setof scenarios
 - jobsequence
S  1,..., n! - setof jobsequences
St. Etienne / France | November 23, 2012
44
1||  wiCi : On logn algorithmby Smith (1956)
 k  J k ,...,J k  S optimal 
1
n
wk1
pk1
 ... 
wkn
pk n
Definition of the stability box:

J (ki )  J k1 ,...,J ki1


J ki   J ki1 ,...,J kn


Ski - setof permutations  J ki , J ki ,  J ki  S

 J ' - permutation of the jobs J '  J
Nk  N  1,...,n
St. Etienne / France | November 23, 2012
45
Definition 5
The maximal closed rectangular box
SB k , T   ki Nk [lki , uki ]  T
is a stability box of permutation  k  J k ,...,J k  S , if permutation  e  J e ,...,J e  Sk being optimal for instance 1| p |  wiCi
with a scenario p   p1,..., pn T remains optimal for the
i 1
n

instance 1| p'|  wiCi with a scenario p'   pk , pk  lk , uk   pk , pk 
j i 1
 j 1

for each ki  Nk .
If there does not exist a scenario p T such that permutation  k
is optimal for instance 1| p |  wiCi , then SB k , T   .
1
1
n
n
i
j
j
i
i
j
j
Remark: The stability box is a subset of the stability region. However, the
stability box is used since it can easily be computed.
St. Etienne / France | November 23, 2012
46
Theorem 13:
For the problem 1 | piL  pi  piU |  wiCi , job J u dominates J v
if and only if the following inequality holds:
wu wv
 L
U
pu
pv
wk i
Lower (upper) bound on the range of
preserving the
pk
optimality of  k  S :
i


 wki
 wk j 


d  max U , max L , i  1,...,n  1
pki i j n 

 pk j 




ki


 wki
 wk j 


d  min L , min U , i  2,...,n
pki 1 j i 
pk j 






ki
St. Etienne / France | November 23, 2012

kn
d 

k1
d 
wk n
pkUn
wk1
pkL1
47
Theorem 14:
If there is no job J ki , i 1,...,n 1 , in permutation
 k  ( J k ,...,J k )  S such that inequality
wki wk j
 U
L
pki
pk j
1
n
holds for at least one job J k j , j i 1,...,n, then the
stability box SB( k , T ) is calculated as follows:
 wki wki 
SB( k , T )  d  d    ,  
ki
ki
 d ki d ki 
otherwise SB( k , T )  .
St. Etienne / France | November 23, 2012
48
Example:
Data for calculating SB1, T , 1  ( J1,...,J8 )
St. Etienne / France | November 23, 2012
49
Stability box for
SB1 , T  
 w2 w2   w4 w4 
  ,    ,  
 d2 d2   d4 d4 
 w6 w6   w8 w8 
  ,    ,   
 d 6 d 6   d8 d8 
3,6 9,10 12,15 19,20
Relative volume of
a stability box
Maximal ranges li , ui  of possible variations
of the processing times pi , i 2,4,6,8 ,
within the stability box SB1, T  are dashed.
St. Etienne / France | November 23, 2012
 wi wi  U
     : pi  piL
 di di 


3 1 3 1
1
   
8 4 9 5 160
50
Sotskov, Egorova, Lai and Werner (2011)
Derivation of properties of a stability box that allow to
derive an O(n log n) algorithm MAX-STABOX for finding a
permutation  t with
• the largest dimension | Nt | and
• the largest volume
of a stability box SB t ,T .
St. Etienne / France | November 23, 2012
51
Computational results
Randomly generated instances with L,U   1,100, wi 1, 50
St. Etienne / France | November 23, 2012
52
Selected references
•
•
•
•
•
•
•
•
Lai, Sotskov, Sotskova and Werner, Math. Comp. Model., vol. 26,
1997
Sotskov, Wagelmans and Werner, Ann. Oper. Res., vol. 38, 1998
Lai, Sotskov, Sotskova and Werner, Eur. J. Oper. Res., vol. 159, 2004
Sotskov, Egorova and Lai, Math. Comp. Model., vol. 50, 2009
Sotskov, Egorova and Werner, Aut. Rem. Control, vol. 71, 2010
Sotskov, Egorova, Lai and Werner, Proceedings SIMULTECH, 2011
Sotskov and Lai, Comp. Oper. Res., vol. 39, 2012
Sotskov, Lai and Werner, Manuscript, 2012
St. Etienne / France | November 23, 2012
53
6. Selection of a suitable approach
Problem 1 | pij  pij  pij |  wiCi
S (T ) - minimaldominantset
L
U
Cardinality of S (T )
Theorem 15:
S (T )   k  J k1 , J k2 ,...,J kn 
 wi

a  min U J i  J 
 pi

wkn1 wkn
wk2 wk2 wk3
 L , U  L ,..., U  L
U
pk1 pk2 pk2 pk3
pkn1 pkn
wk1
 wi

b  max L J i  J 
 pi


wi
wi 
J r   J i  J r  U  L , r  a, b, r  R
pi
pi 

St. Etienne / France | November 23, 2012
54
Theorem 16:
Assume that there is no r a, b with J r  2.
Then: S (T )  n! 
 wi

 wi

max U J i  J   min L J i  J .
 pi

 pi

Theorem 17:
S (T ) uniquelydetermined there is no r a, b with J r  2.
S (T ) not uniquely determined
 Construct an equivalent
instance with less jobs for which S (T ) is uniquely
determined
Assumption: S (T ) uniquely determined.
z - instance with the set T of scenarios
St. Etienne / France | November 23, 2012
55
Uncertainty measures
 ( z)  1 
n! S (T )
n!1
1  S (T )  n!  0  ( z)  1
Dominance graph G J, A 
 ( z)  1 
2A
n(n  1)
0 A 
n(n  1)
2

0   ( z)  1
Recommendations:
 ( z ),  ( z) small 
 ( z ),  ( z ) large 
 ( z),  ( z) around0.5 
use a stability approach
use a robust approach
use a fuzzy or stochastic
approach
St. Etienne / France | November 23, 2012
56
Example: n  6
L
i
U
i
p
wi
wi
piL
wi
piU
i
p
1
5
6
300
60
50
2
4
6
240
60
40
3
6
14
420
70
30
4
2
7
140
70
20
5
10
35
700
70
20
6
5
10
250
50
25
Dominance conditions:
w6
w1
 50  50  L
p1U
p6
  ( z)  1 

n! S (T )
n!1
 S (T ) 
 1
6!
 360
2
6!320 359

 0.5
6!1
719
apply a stochastic or a fuzzy approach
St. Etienne / France | November 23, 2012
57
Example (continued):
L
i
U
i
wi
wi
piL
wi
piU
E pi 
wi
E  pi 
i
p
p
1
5
6
300
60
50
5.5
54 6/11
2
4
6
240
60
40
5
48
3
6
14
420
70
30
10
42
4
2
7
140
70
20
4.5
31 1/9
5
10
35
700
70
20
22.5
31 1/9
6
5
10
250
50
25
7.5
33 1/9


pi ~ U piL , piU for all J i  J  apply WSEPT rule
   J1 , J 2 , J 3 , J 6 , J 5 , J 4 
  ( z)  1 
2A
n(n  1)
 1
2
14

1
6  5 15
n(n  1)
A  1,
 15
2
(apply a robust approach)
Remark:  (z ) easier computable than  (z )
St. Etienne / France | November 23, 2012
58
Announcement of a book
Sequencing and Scheduling with Inaccurate Data
Editors:
To appear at:
Completion:
Yuri N. Sotskov and Frank Werner
Nova Science Publishers
Summer 2013
4 parts: Each part contains a survey and 2-4 further chapters.
Part 1:
Part 2:
Part 3:
Part 4:
Stochastic approach
Fuzzy approach
Robust approach
Stability approach
Contact address:
survey: Cai et al.
survey: Sakawa et al.
survey: Kasperski and Zielinski
survey: Sotskov and Werner
frank.werner@ovgu.de
St. Etienne / France | November 23, 2012
59