Scheduling under Uncertainty: Solution Approaches Frank Werner Faculty of Mathematics

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


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 ,...,,,


• 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


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

)(')(


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.


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


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


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)


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


(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


(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 ||


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


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


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


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


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


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



• 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


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


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


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


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~


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


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



• 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



• 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


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


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


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


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


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 ̂


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


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 -


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


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


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


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


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


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


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


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


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


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


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


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


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 ,),(


Example:Data for calculating ),...,(,, 8111 JJTSB


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


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


Computational resultsRandomly generated instances with 50,1,100,1, iwUL


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


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


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


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


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


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


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]

Documents

Scheduling under Uncertainty: Solution Approaches Frank Werner Faculty of Mathematics