Scheduling under Uncertainty:Solution Approaches
Frank WernerFaculty of Mathematics
2St. Etienne / France | November 23, 2012
Outline of the talk
1. Introduction2. Stochastic approach3. Fuzzy approach4. Robust approach5. Stability approach6. Selection of a suitable approach
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1. Introduction
Notations
jobsof set - nJJ ,...,
1J
machinesof set - m
MM ,...,1
M
operationsof set - , qnjJO iiij ,...,1,...,1| JQ
ijij Op of time processing -
J for data further - iiii Jdrw ,...,,,
4St. Etienne / France | November 23, 2012
• 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:
Q all for ijUijij
Lij Oppp
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scenariosof set - Q ijUijij
Lij
q OppppT ,|R
|,| Uijij
Lij ppp problem
Q all for ijUij
Lij Opp
|| problem ticdeterminis
Relationship between stochastic and uncertain problems:Distribution function
Density function
Uij
Lij
ijij pt
pttpPtF
if if
1
0)()(
Uij
Uijij
Lij
Lij
ijij
pt
ppp
pt
tFtf
if if
if
0
?
0
)(')(
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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.
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Two-phase decision-making procedure1) Off-line (proactive) phase
construction of a set of potentially optimal solutions before the realization of the activities(static scheduling environment, schedule planning phase)
2) On-line (reactive) phaseselection of a solution from 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)
*S
*S
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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
9St. Etienne / France | November 23, 2012
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)
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Some results for single-stage problems (see Pinedo 1995)Single machine problems(a) Problem
WSEPT rule: order the jobs according to non-increasing ratios
Theorem 1: The WSEPT rule determines an optimal solution in the class of NSL as well as ND policies.
(b) Problem
Theorem 2: The EDD rule determines an optimal solution in the class of NSL, ND and PD policies.
iiCwE||1ddistribute yarbitraril ~ip
max||1 LEfixed d,distribute yarbitraril ii dp ~
ii
pE
w
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(c) Problem
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
Parallel machine problems
(d) ProblemTheorem 4: The LEPT rule determines an optimal solution in the class of NSL policies.
fixed d,distribute llyexponentia dpi ~ iii UwEdd ||1
.ip
ddistribute llyexponentia ~ip
max||2 CEP
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(b) Problem
Theorem 5: The non-preemptive LEPT policy determines an optimal solution in the class of PD policies.
(c) Problem
Theorem 6: The non-preemptive SEPT policy determines an optimal solution in the class of PD policies.
max|| CEpmtnP
iCEpmtnP ||
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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
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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
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3. Fuzzy approach
• Fuzzy scheduling techniques either fuzzify existing scheduling rules or solve mathematical programming problems
• Often: fuzzy processing times , fuzzy due dates• Examples
triangular fuzzy processing times trapezoidal fuzzy processing times
ip~
id~
Lip
Uip
Mip
0
0.1ip
~
ip~
Lip
Uipp0
0.1ip
~ip
~
p
"" Mii pp around is
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Often: possibilistic approach (Dubois and Prade 1988)
Chanas and Kasperski (2001)ProblemObjective:Assumption:
→ adaption of Lawler‘s algorithm for problem
R xxxVPos p ),(~
)(sup, ~,
xbaVPos pbax
)(1inf, ~,
xbaVNec pbax
max|~,~,|1 fdpprec ii
min!)(~
max iii
Cf
ii Cf~
w.r.t. monotonic-F
max||1 fprec
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Special cases:
a) b) c) d)
Alternative goal approach - fuzzy goal, Objective:Chanas and Kasperski (2003)Problem Objective:
→ adaption of Lawler‘s algorithm for problemmax||1 fprec
min!~
)(~
max iii
dCPos
min!~
)(~
max iii
dCNec
max!~
)(~
min iii
dCPos
min!)(~
max ii
LE
G~ max!
~)(
~max
~
GLwPos ii
i
iii TEdp max|~,~|1 min!)(
~max ii
TE
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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
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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
20St. Etienne / France | November 23, 2012
4. Robust approach
Objective: Find a solution, which minimizes the „worst-case“ performance over all scenarios.
Notations (single machine problems)
maximal regret of
Minmax regret problem (MRP): Find a sequence such that
TpJJFnkkp for sequenceof value function - ,...,)(
1
TpFp for value function optimal - *
sequences job feasibleof set - S
S *)(max)( pp
TpFFZ
* )(min*
ZZ
S
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Some polynomially solvable MRP(Kasperski 2005)
(Volgenant and Duin 2010)(Averbakh 2006)
(Kasperski 2008)
Some NP-hard MRP(Lebedev and Averbakh 2006)
(for a 2-approximation algorithm, see Kasperski and Zielinski 2008)
(Kasperski, Kurpisz and Zielinski 2012)
max|,,|1 Ldddpppprec Uii
Li
Uii
Li
iiUii
Li
Uii
Li
Uii
Li Twwwwdddpppprec max|,,,|1
max|,2| CpppnFm Uijij
Lij
iUii
Lii Uwwwp |,1|1
hard-NP is iUii
Li Cppp ||1
hard-NP strongly is max||2 CpppF Uijij
Lij
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Kasperski and Zielinski (2011)Consideration of MRP‘s using fuzzy intervals
Deviation interval
Known: deviation
Application of possibility theory (Dubois and Prade 1988)
possibly optimal if necessarily optimal if
*)(min)(' pp FFZ
)(),(' ZZI
Iz )(
0)(' Z0)( Z
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Fuzzy problem
or equivalently
where is a fuzzy interval and is the complement of with membership function
The fuzzy problem can be efficiently solved if a polynomial algorithm for the corresponding MRP exists.
max!~
)( GzNec
min!~
)( CGzPos CG
~G~
).(1 ~ xG
G~
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Solution approachesa) Binary search method
- repeated exact solution of the MRP
- applications:
: binary search subroutine in B&B algorithm
algorithm )log(:|~,~,|1 14
maxnOLdpprec ii
algorithm )log(:max|~,|1 13 nOTwwprec iii
algorithm )log(:max|~,~,|1 14 nOTwdwprec iiii
algorithm )log,min(:|~,,1|1 1 dndnOUwwddp iiiii
max|~|2 CpF ij
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b) Mixed integer programming formulation- use of a MIP solver
- application:
c) Parametric approach - solution of a parametric version of a MRP(often time-consuming)
- application:
ii Cp |~|1
max|~,|1 Ldprec i
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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
27St. Etienne / France | November 23, 2012
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
28
5. Stability approach
5.1. Foundations5.2. General shop problem5.3. Two-machine flow and job shop problems5.4. Problem
iiUii
Li Cwppp ||1
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5.1. Foundations
Mixed GraphExample:
),,( EAVG
00
23
13
22
12
21
11
**
000 p
6021 p 5522 p 3023 p
4013 p5012 p7511 p
0** p
digraphsof set - GGGEAVGGG sss ,...,,),,(|)( 21
)(),...,(),()( 21 scscscsGG qs schedule semiactive
30St. Etienne / France | November 23, 2012
Example (continued)
00
23
13
22
12
21
11
**
6021 c 13022 c 16023 c
16513 c12512 c7511 c
1651max GC
1G
3251 GCi
521 ,...,, GGGG
31St. Etienne / France | November 23, 2012
Stability analysis of an optimal digraphDefinition 1The closed ball is called a stability ball of if for anyremains optimal.The maximal value
is called the stability radius of digraphKnown:•Characterization of the extreme values of•Formulas for calculating•Computational results for job shop problems with (see Sotskov, Sotskova and Werner, Omega, 1997)
qppO RR and with 1)( )(GGs )'(,)(' pGpOp s
q R
.sG qss pOpGp RR )('|max)( 1
any for optimal
)( ps is CCp ,)( max for
810 mn and
32St. Etienne / France | November 23, 2012
5.2. General shop problem
Definition 2 is called a G-solution for problem if for any fixed contains an optimal digraph.If any is not a G-solution, is called a minimal G-solution denoted as
Introduction of the relative stability radius:
|| Uijij
Lij pppG
)()(* GG || U
ijijLij pppG )(, * GTp
)()( * GG )(* G
Tpppp Uijij
Lijij polytope 0
)()( GBG
).(GT
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Definition 3Let be such that for any
The maximal value of of such a stability ball is called the relative stability radius
Known: •Dominance relations among paths and sets of paths•Characterization of the extreme values of
maxC
TpGl sps for in weight critical -
)(GBGs TpOp )('
.|min '' BGll kpk
ps
)( pO .ˆ TpB
s
TpBs ̂
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Characterization of a G-solution for problemDefinition 4 (strongly) dominates in
→ dominance relation
Theorem 7: is a G-solution. There exists a finite covering of polytope by closed convex sets with such that for any and any there exists a for whichCorollary:
sG kG
.Dpllll pk
ps
pk
ps any for if
max|| CpppJ Uijij
Lij
kDskDs GGGG
)(G T q
jD R,
1d
j jDT
,d )(GGk ,,...,1, djD j sG .kDs GG
j
).()( GGGGGG kkTssT any for
qD R
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Theorem 8:Let be a G-solution withThen: is a minimal G-solution. For anythere exists a vector such that
Algorithms for problem
)(* G .2)(* G
)(* G )(* GGs Tp s )(
. any for skkps GGGGG s \)(*)(
|| Uijij
Lij pppJ
,...,max iCC e.g. criterion, regular -
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Several 3-phase schemes:•B&B: implicit (or explicit) enumeration scheme for generating a G-solution
• SOL: reduction of by generating a sequence with the same and
different
• MINSOL: generation of a minimal G-solution by a repeated application of algorithm SOL
)(ˆ...ˆˆ ˆ21 pOiI of Tp
*)( G
)(GT
TT G )(
'BB
B
B
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Some computational results:
Exact sol.: , Heuristic sol.:
Degree of uncertainty
Exact solution Heuristic solution
(4,4) 1, 3, 5, 7 34.2 7.5 6.3 24.1 6.5 5.5
2, 6, 8, 10 88.3 16.1 14.5 52.9 13.5 12.0
5, 10, 15, 20 477.7 30.8 30.1 132.0 24.8 24.0
iC
),( mnT T' '* *
24mn )8,10(50 mnmn
Degree of uncertainty
Exact solution Heuristic solution
(4,4) 1, 3, 5, 7 41.8 6.4 2.4 19.9 3.8 2.4
2, 6, 8, 10 79.0 14.7 9.5 27.3 6.9 4.4
5, 10, 15, 20 434.9 43.5 34.8 112.8 25.7 20.0
),( mn' * T ' * T
maxC
38St. Etienne / France | November 23, 2012
5.3. Two-machine problems with interval processing times
a) Problem
Johnson permutation:
Partition of the job set
max||2 CpppF Uijij
Lij
(1954) Johnsonby algorithm all for )log(),( nnOQjipp Uij
Lij
optimal is for with nlkppppkllk iiii 1,min,min 2,1,2,1,
with *210 JJJJ J
UiLi
Ui
Lii ppppJ 22110 | JJ
LiUii ppJ 2101 | J\JJ
LiUii ppJ 1202 | J\JJ
LiUi
Li
Uii ppppJ 1221
* ,| JJ
niii JJJ ,...,,
21
39St. Etienne / France | November 23, 2012
Theorem 9:
(1) for any either (either ) and
(2) and if satisfies– – –
npermutatio Johnsona containing set minimal : solution-J )(TS
1)(TSly)respective ,( 21, JJji JJ
Li
Uj
Lj
Ui pppp 1111 or
Li
Uj
Lj
Ui pppp 2222 or
1* J **, JJ * iJ
111,|max* J i
Ui
L
iJpp
222,|max* J j
Uj
L
iJpp
02,1, ** ,max J kkL
i
L
iJppp any for
Tp any for
40St. Etienne / France | November 23, 2012
Theorem 10:If then
Percentage of instances with , where
21,|min21,|max jJpjJp iUiji
Lij JJ
!)( nTS
1)( TS Lpp Lij
Uij
5 10 15 20 25 301 99.2 95.2 91.2 86.1 79.2 72.82 97.2 89.8 77.6 63.5 51.0 39.63 95.0 80.9 66.4 47.6 32.8 20.64 91.8 78.6 56.0 39.2 20.3 10.75 91.0 69.4 44.9 28.9 14.6 6.0
nL
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General case of problemTheorem 11:There exists an
Theorem 12:
max||2 CpppF Uijij
Lij
)()( TSJJTS wv any in with
.22122111 and or and Lv
Uw
Lw
Uw
Lv
Uv
Lw
Uv pppppppp
wvwv JJJJ A ,,
time in graph dominance the construct ²)(nOG AJ,
.transitive then If AJ ,0
42St. Etienne / France | November 23, 2012
Example:
without transitive arcs:
6n
1 9 10 5 5 102 12 11 8 6 118 14 13 6 4 49 15 17 7 4 4
iJ 1J 2J 3J 4J 5J 6JLip 1
Uip 1Lip 2
Uip 2
,,,,,: 6151413121 JJJJJJJJJJ A566563536252 ,,,,, JJJJJJJJJJJJ
AJ,
J1
J4
J3
J2
J5 J6
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Properties of in the case ofsee 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 and for
b) Problem
→ Reduction to two problems:see Sotskov, Sotskova, Lai and Werner, Scheduling under uncertainty, 2010 (Section 3.6)
AJ, :0 J
0100 Jif n 01000 Jif n
max|,2|2 CpppnJ Uijij
Liji
max||2 CpppF Uijij
Lij
44St. Etienne / France | November 23, 2012
5.4. Problem
Notations:
iiUii
Li Cwppp ||1
jobs of set - nJJ n,...,1J
J for weight - ii Jw
Ui
Lii
Ui
Lii ppJppp 0, , of time processing - J
scenariosof set - nippppT Uii
Li
n ,...,1,| R
nppp ,...,1
sequence job - nkkk JJ ,...,
1
sequences jobof set - !1,..., nS
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Definition of the stability box:
11
,...,)(
ikki JJkJ
ni kki JJkJ ,...,1
SkJJkJS ikik ii nspermutatioof set - ,,
JJJ '' jobs theof npermutatio - nNNk ,...,1
(1956) Smith by algorithm nnOCw ii log:||1
n
n
n
k
k
k
kkkk p
w
p
wSJJ ...,...,
1
1
1 optimal
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Definition 5
The maximal closed rectangular box
is a stability box of permutation , if permu-tation being optimal for instance with a scenario remains optimal for the instance with a scenariofor each If there does not exist a scenario such that permutation is optimal for instance , then
Remark: The stability box is a subset of the stability region. However, the stability box is used since it can easily be computed.
TulTSBiiki kkNkk ],[,
SJJnkkk ,...,
1
in keee SJJ ,...,
1 iiCwp ||1
Tppp n ,...,1
iiCwp |'|1
jjiijj kk
n
ijkkkk
i
jppulppp ,,,'
1
1
1
.ki Nk
Tp k
iiCwp ||1 ., TSB k
47St. Etienne / France | November 23, 2012
Theorem 13: For the problem , job dominates if and only if the following inequality holds:
Lower (upper) bound on the range of preserving the optimality of :
iiUii
Li Cwppp ||1
uJ vJ
Lv
vUu
u
p
w
p
w
i
i
k
k
p
w
Sk
1,...,1,max,max
nip
w
p
wd
Lk
k
njiUk
kk
j
j
i
i
i
nip
w
p
wd
Uk
k
ijLk
kk
j
j
i
i
i,...,2,min,min
1
Uk
kk
n
n
n p
wd
Lk
kk p
wd
1
1
1
48St. Etienne / France | November 23, 2012
Theorem 14:If there is no job , in permutation such that inequality
holds for at least one job , then the stability box is calculated as follows:
otherwise
1,...,1, niJik
SJJ
nkkk ),...,(1
nijJjk
,...,1, ),( TSB k
.),( TSB k
Uk
k
Lk
k
j
j
i
i
p
w
p
w
i
i
i
i
ikikk
k
k
k
ddk d
w
d
wTSB ,),(
49St. Etienne / France | November 23, 2012
Example:Data for calculating ),...,(,, 8111 JJTSB
50St. Etienne / France | November 23, 2012
Stability box for
Relative volume of a stability box
Maximal ranges of possible variations of the processing times , within the stability box are dashed. TSB ,1
ii ul ,
8,6,4,2, ipi
4
4
4
4
2
2
2
2 ,,d
w
d
w
d
w
d
w
8
8
8
8
6
6
6
6 ,,d
w
d
w
d
w
d
w
TSB ,1
20,1915,1210,96,3
LiUi
i
i
i
i ppd
w
d
w
:
160
1
5
1
9
3
4
1
8
3
51St. Etienne / France | November 23, 2012
Sotskov, Egorova, Lai and Werner (2011)Derivation of properties of a stability box that allow to derive an algorithm MAX-STABOX for finding a permutation with•the largest dimension and•the largest volumeof a stability box
)log( nnOt
|| tN
.,TSB t
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Computational resultsRandomly generated instances with 50,1,100,1, iwUL
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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
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6. Selection of a suitable approach
Problem
Cardinality ofTheorem 15:
iiUijij
Lij Cwppp ||1
set dominant minimal - )(TS
)(TS
nkkkk JJJTS ,...,,)(
21 L
k
k
Uk
k
Lk
k
Uk
k
Lk
k
Uk
k
n
n
n
n
p
w
p
w
p
w
p
w
p
w
p
w
1
1
3
3
2
2
2
2
1
1 ,...,,
JiUi
i Jp
wa min
JiLi
i Jp
wb max
R
rbarp
w
p
wrJ
Li
iUi
iir ,,,JJ
55St. Etienne / France | November 23, 2012
Theorem 16:Assume that there is no
Then:
Theorem 17:
not uniquely determined Construct an equivalent instance with less jobs for which is uniquely determined Assumption: uniquely determined. - instance with the set of scenarios
.2, rbar J with
.minmax!)(
JJ iLi
iiU
i
i Jp
wJ
p
wnTS
.2,)( rbarTS J with no is there determined uniquely
)(TS )(TS
)(TS
z T
56St. Etienne / France | November 23, 2012
Uncertainty measures
Dominance graph
Recommendations:use a stability approachuse a robust approachuse a fuzzy or stochastic approach
1!
)(!1)(
n
TSnz 1)(0!)(1 znTS
AJ,G
)1(
21)(
nnz
A 1)(0
2
)1(0
z
nn A
small )(),( zz large )(),( zz
around 5.0)(),( zz
57St. Etienne / France | November 23, 2012
Example:
Dominance conditions:
apply a stochastic or a fuzzy approach
6n
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
iLip
Uip iw L
i
i
p
wUi
i
p
w
LU p
w
p
w
6
6
1
1 5050 3602
!6)( TS
5.0719
359
1!6
320!61
1!
)(!1)(
n
TSnz
58St. Etienne / France | November 23, 2012
Example (continued):
(apply a robust approach)Remark: easier computable than
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
iLip
Uip iw L
i
i
p
wUi
i
p
w ipE ii
pE
w
rule WSEPT apply all for JiUi
Lii JppUp ,~
456321 ,,,,, JJJJJJ 152
)1(,1
nn A
115
14
56
21
)1(
21)(
nnz
A
)(z )(z
59St. Etienne / France | November 23, 2012
Announcement of a book
Sequencing and Scheduling with Inaccurate DataEditors: Yuri N. Sotskov and Frank WernerTo appear at: Nova Science PublishersCompletion: Summer 20134 parts: Each part contains a survey and 2-4 further chapters.
Part 1: Stochastic approach survey: Cai et al.Part 2: Fuzzy approach survey: Sakawa et al.Part 3: Robust approach survey: Kasperski and ZielinskiPart 4: Stability approach survey: Sotskov and WernerContact address: [email protected]