THE STABILITY BOX IN INTERVAL DATA FOR MINIMIZING THE SUM OF WEIGHTED COMPLETION TIMES

Yuri N. Sotskov Natalja G. Egorova

United Institute of Informatics Problems, National Academy of Sciences of Belarus

Tsung-Chyan Lai Department of Business Administration, National Taiwan University

Frank Werner Faculty of Mathematics, Otto-von-Guericke-University

2

Outline of the Talk

1. Introduction 2. Problem Setting 3. Stability Box 4. Computational Results 5. Conclusion and Further Work

3

1. Introduction

Approaches dealing with scheduling problems under uncertainty

1. Stochastic method (Pinedo, 2002)

2. Fuzzy method (Slowinski and Hapke, 1999)

3. Robust method (Daniels and Kouvelis, 1995; Kasperski,

2005; Kasperski and Zelinski, 2008)

4. Stability method (Lai and Sotskov, 1999; Lai et al., 1997;

Sotskov et al., 2009; Sotskov et al., 2010a; Sotskov et al.,

2010b)

4

2. Problem setting

- set of n jobs

- weight for a job

- processing time of a job ,

- set of vectors

of the possible job processing times

Find an optimal permutation such that

},...,{ 1 nJJJ

0iw JJi

ip iJ Ui

Li

Ui

Lii ppppp 0],,[

} },...,1{, { nipppRpT Uii

Li

n ),...,( 1 nppp

},...,{ !1 nS ),...,(1 nkkk JJ

iiUii

Li Cwppp ||1

St

Problem

,

JJ

kiiS

tp

JJtii

iki

pCwpCw ),(min),(

Problem is not correct.

The deterministic problem is correct and can be solved exactly in O(n log n) time (Smith, 1956).

Necessary and sufficient condition for the optimality of a permutation

Deterministic case:for each job

iiUii

Li Cwppp ||1

iUi

Li ppp JJ i

iiUii

Li Cwppp ||1 iiCw||1

iiCw||1

SJJnkkk ),...,(

1

n

n

k

k

k

k

p

w

p

w...

1

1

5

6

3. Stability box

- set of permutations

- permutation of the jobs

- subset of set

},...,{)(11

ikki JJkJ },...,{][

1 ni kki JJkJ

ikS SkJkJkJ iii ])[(),()),(((

)(J JJ

kN },...,1{ nN

- set of permutations with the largest dimension and volume of the stability box

Sk maxS

7

Definition 1. (Sotskov and Lai, 2011)

The maximal closed rectangular box

.is a stability box of permutation , if permutation

. being optimal for the instance

with a scenario remains optimal for the

instance . with a scenario

.for each .

If there does not exist a scenario such that permutation is

optimal for the instance , then .

TulTSBiiki kkNkk ],[),(

SJJnkkk ),...,(

1

in keee SJJ ),...,(1

iiCwp ||1

Tppp n ),...,( 1

iiCwp ||1

Tp k

iiCwp ||1 ),( TSB k

]},[],[{ ,1 iijj kkkkn

ijj ulppp

ki Nk

8

3.1. Precedence-dominance relation on the set of jobs J and the

solution concept of a minimal dominant set

The set of permutations is a minimal dominant set for a

problem , if for any fixed scenario , the

set S(T) contains at least one optimal permutation for the instance

, , provided that any proper subset of set S(T) loses such a

property.

Definition 2. (Sotskov and al., 2009)

STS )(

iiUii

Li Cwppp ||1 Tp

iiCwp ||1

9

Theorem 1. (Sotskov and al., 2009)

For the problem , job dominates job if

and only if the following inequality holds:

iiUii

Li Cwppp ||1 uJ vJ

Lv

vUu

u

p

w

p

w

Definition 3. (Sotskov and al., 2009)

Job dominates job , if there exists a minimal dominant set S(T)

for the problem such that job precedes

job in every permutation of the set S(T).

uJ vJ

iiUii

Li Cwppp ||1 uJ

vJ

10

Lower (upper) bound ( ), , on the maximal range of

possible variations of the weight-to-process ratio preserving

the optimality of permutation :

ikd

3.2. Calculating the stability box

ikd JJ

ik

i

i

k

k

p

w

SJJnkkk ),...,(

1

,

Uk

k

ijLk

kk

j

j

i

i

i p

w

p

wd

1min,min ,

Lk

k

njiUk

kk

j

j

i

i

i p

w

p

wd max,max

Uk

kk

n

n

n p

wd

Lk

kk

p

wd

1

1

1

}1,...,1{ ni

},...,2{ ni

11

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,

Theorem 2. (Sotskov and Lai, 2011)

ikJ }1,...,1{ ni SJJnkkk ),...,(

1

Uk

k

Lk

k

j

j

i

i

p

w

p

w

jkJ },...,1{ nij ),( TSB k

i

i

i

i

ikikk

k

k

k

ddkd

w

d

wTSB ,),(

),( TSB k

12

3.3. Illustrative example

13

Stability box for 1

Relative volume of a stability box

Li

Ui

i

i

i

i ppd

w

d

w

:

160

1

5

1

9

3

4

1

8

3

19,2012,159,103,6

,,

,,),(

8

8

8

8

6

6

6

6

4

4

4

4

2

2

2

21

d

w

d

w

d

w

d

w

d

w

d

w

d

w

d

wTSB

14

3.4. Properties of a stability box

Property 1. For any jobs and ,

Case (I)

For case (I), there exists a permutation , in which job

proceeds job .

JJ i JJ v iv

Li

iLv

vUi

iUv

v

p

w

p

w

p

w

p

w ,

vJ

iJ

maxSt

Lv

vUv

v

i

i

i

i

p

w

p

w

l

w

u

w,,

Property 2.

15

Property 3.

For case (II), there exists a permutation , in which jobs and

are located adjacently: and .

Remark 1.

Due to Property 3, while looking for a permutation , we shall

treat a pair of jobs satisfying (1) as one job (either job or ).},{ vi JJ iJ vJ

maxSt iJ vJ

rti 1 rtv

maxSt

Case (II) Li

iLv

vUi

iUv

v

p

w

p

w

p

w

p

w , (1)

Case (III) Li

iLv

vUi

iUv

v

p

w

p

w

p

w

p

w ,

Property 4.

(i) For a fixed permutation , job may have at most one

maximal segment of possible variations of the processing time

preserving the optimality of permutation .

(ii) For the whole set of permutations S, only in case (III), a job

may have more than one (namely: ) maximal segments

of possible variations of the time preserving the

optimality of this or that particular permutation from the set S.

kJJ i

11)( iJ ],[ ii ul

(2)

16

Sk JJ i ],[ ii ul

],[ Ui

Lii ppp

],[ Ui

Lii ppp

17

Property 5.

nL ||

L - the set of all maximal segments of possible variations of the

processing times for all jobs preserving the optimality of

permutation .

],[ ii ul

JJ i

Property 6.

There exists a permutation with the set of maximal

segments of possible variations of the processing time ,

preserving the optimality of permutation .

],[ ii ul JJp ii ,

t

St

maxSt ip

LL

18

3.5. A job permutation with the largestvolume of a stability box

Algorithm MAX-STABOX

Input: Segments , weights , .Output: Permutation , stability box .

Step 1: Construct the list and the list

in non-decreasing order of . Ties are broken via increasing .

Step 2: Construct the list and the list

in non-decreasing order of . Ties are broken via increasing .

JJ i ),( TSB t

),...,()(1 nuu JJUM

Uu

u

Uu

u

n

n

p

w

p

wUw ,...,)(

1

1

Uu

u

r

r

p

wLu

u

r

r

p

w

),...,()(1 nll JJLM

Ll

l

r

r

p

wUl

l

r

r

p

w

maxSt ],[ U

iLi pp iw

Ll

l

Ll

l

n

n

p

w

p

wLw ,...,)(

1

1

19

Step 3: FOR j = 1 to j = n DO compare job and job .Step 4: IF THEN job has to be located in position j in permutation . GOTO step 8.Step 5: ELSE job satisfies inequalities (2). Construct the set of all jobs satisfying inequalities (2), where Step 6: Choose the largest range among those generated for job .Step 7: Partition the set J(i) into subsets and generating the largest range . Set j = k+1 GOTO step 4.Step 8: Set j := j+1 GOTO step 4. END FORStep 9: Construct the permutation via putting the jobs J in the positions defined in steps 3 – 8.Step 10: Construct the stability box . STOP.

juJjlJ

jj lu JJ juJ

iu JJj

},...,{)(11

kr lu JJiJ vJ

],[jj uu ul

iu JJj

),( TSB t

],[jj uu ul

)(iJ )(iJ

maxSt

kj lui JJJ

maxSt

20

Remark 2.

Algorithm MAX-STABOX constructs a permutation such that

the dimension of the stability box

is the largest one for all permutations S, and the volume of the stability

box is the largest one for all permutations having the

largest dimension of their stability boxes .

St || tN

TulTSBiiti ttNtt ],[),(

),( TSB t Sk |||| tk NN ),( TSB k

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4. Computational results

C - integer center of a segment was generated using the uniform distribution in the range [L;U]: . - maximal possible error of the random processing times - lower (upper) bound for the possible job processing time

,

%1002

)1(|:|

nn

A - the average relative number of the arcs in the dominance digraph

- relative error of the objective function value

- the optimal objective function value of the actual scenario

|| A

*

**

p

ptp

Each series contains 100 solved instancesProcessor AMD Athlon (tm) 64 3200+, 2.00 GHz; RAM 1.96 GB

*p

),( *

1* pCw ti

n

ii

tp

UCL

)( Ui

Li pp

)100

1(

CpLi )

1001(

CpU

i

ip

],[ Ui

Li pp

22

23

5. Conclusion and further work

Further research concerning the use of a stability box for other uncertain scheduling problems appear to be promising.

An algorithm has been developed for calculating a permutation

. with the largest dimension and volume of a stability box .

. ;

Properties of a stability box were proved allowing to derive an

O(nlogn) algorithm for calculating a permutation ;

The dimension and volume of a stability box are efficient invariants

of uncertain data T, as it is shown in simulation experiments on a PC.

)( 2nO

St ),( TSB t

maxSt

24

6. References1. Lai, T.-C., Sotskov, Y., Sotskova, N., and Werner, F. (1997). Optimal makespan

scheduling with given bounds of processing times. Mathematical and Computer Modelling, V. 26(3):67–86.

2. Smith, W. (1956). Various optimizers for single-stage production. Naval Research Logistics Quarterly, V. 3(1):59–66.

3. Sotskov, Y., Egorova, N., and Lai, T.-C. (2009). Minimizing total weighted flow time of a set of jobs with interval processing times. Mathematical and Computer Modelling, V. 50:556–573.

4. Sotskov, Y., Egorova, N., and Werner, F. (2010a). Minimizing total weighted completion time with uncertain data: A stability approach. Automation and Remote Control, V. 71(10):2038–2057.

5. Sotskov, Y. and Lai, T.-C. (2011). Minimizing total weighted flow time under uncertainty using dominance and a stability box. Computers & Operations Research. doi:10.1016/j.cor.2011.02.001.

6. Sotskov, Y., Sotskova, N., Lai, T.-C., and Werner, F. (2010b). Scheduling under Uncertainty. Theory and Algorithms. Belorusskaya nauka, Minsk, Belarus.

7. Sotskov, Y., Wagelmans, A., and Werner, F. (1998). On the calculation of the stability radius of an optimal or an approximate schedule. Annals of Operations Research, V. 83:213–252.