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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
21
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.