Transcript Н.Г. Егорова, Ю.Н. Сотсков

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
Outline of the Talk

1. Introduction

2. Problem Setting
3. Stability Box
4. Computational Results
5. Conclusion and Further Work



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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)
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2. Problem setting
J  {J1,..., J n } - set of n jobs
wi  0 - weight for a job Ji  J
pi - processing time of a job J i , pi  [ piL , piU ], 0  piL  piU
T  { p  Rn piL  pi  piU , i  {1,..., n} } - set of vectors p  ( p1 ,..., pn )
of the possible job processing times
S  { 1 ,..., n!},  k  ( J k1 ,..., J kn )
Problem 1 | piL  pi  piU |  wiCi
Find an optimal permutation  t  S such that
 wi Ci ( t , p)
J i J
  tp


 min   wi Ci ( k , p)
 k S J J
i

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Problem 1 | piL  pi  piU |  wiCi is not correct.
Deterministic case:
piL  piU  pi for each job J i  J
1 | piL  pi  piU |  wi Ci
1 ||  wi Ci
The deterministic problem 1 ||  wi Ci 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
 k  ( J k1 ,..., J kn )  S
wk n
wk1
 ... 
p k1
p kn
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3. Stability box
J (k i )  {J k1 ,..., J ki 1 }
J [ki ]  {J k i 1 ,..., J k n }
S ki - set of permutations ( ( J (ki )), J (ki ), ( J [ki ])  S
 (J ) - permutation of the jobs J   J
Nk
- subset of set N  {1,...,n}
S max - set of permutations  k  S with the largest dimension and
volume of the stability box
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Definition 1. (Sotskov and Lai, 2011)
The maximal closed rectangular box
SB( k , T )   k N [l ki , u ki ]  T
i
k
is a stability box of permutation  k  ( J k1 ,..., J kn )  S , if permutation
.
. e  ( J e1 ,..., J en )  S ki being optimal for the instance 1 | p |  wi Ci
with a scenario p  ( p1 ,..., pn )  T remains optimal for the instance
1. | p |  wi Ci with a scenario p {nj 1, j i [ pk j , pk j ]  [lki , uki ]}
for each ki  N k .
If there does not exist a scenario p  T such that permutation  k is
optimal for the instance 1 | p |  wi Ci , then SB( k , T )   .
.
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3.1. Precedence-dominance relation
on the set of jobs J and the
solution concept of a minimal dominant set
Definition 2. (Sotskov and al., 2009)
The set of permutations S (T )  S is a minimal dominant set for a
problem 1 | piL  pi  piU |  wi Ci , if for any fixed scenario p  T , the
set S(T) contains at least one optimal permutation for the instance
1 | p |  wi Ci , provided that any proper subset of set S(T) loses such a
property.
,
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Definition 3. (Sotskov and al., 2009)
Job J u dominates job J v , if there exists a minimal dominant set S(T)
for the problem 1 | piL  pi  piU |  wi Ci such that job J u precedes
job J v in every permutation of the set S(T).
Theorem 1. (Sotskov and al., 2009)
For the problem 1 | piL  pi  piU |  wi Ci , jobJ u dominates job J v if
and only if the following inequality holds:
wu
puU

wv
pvL
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3.2. Calculating the stability box
Lower (upper) bound d k ( d ki), J ki  J , on the maximal range of
i
possible variations of the weight-to-process ratio
wk i
p ki
preserving
the optimality of permutation  k  ( J k1 ,..., J kn )  S :
d ki
w
 wk
 ki
j
 max  U , max  L
 p ki i  j n  p k j
d kn 

 , i  {1,...,n  1}

wkn
pUkn
w
 wk 

k
d ki  min  Li , min  Uj  , i  {2,...,n}
 pki 1 j i  pk j 
wk1

d k1  L
pk1
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Theorem 2. (Sotskov and Lai, 2011)
If there is no job J ki , i  {1,..., n  1} , in permutation  k  ( J k1 ,..., J k n )  S
such that inequality
wki
p kLi

wk j
pUkj
holds for at least one job J k j , j  {i  1,...,n} , then the stability box
SB( k , T ) is calculated as follows:
SB( k , T )   d  d 
ki
ki
 wk wk 
 i , i 
 d ki d ki 
Otherwise, SB( k , T )  
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3.3. Illustrative example
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Stability box for 1
w w  w w 
SB( 1 , T )   2 , 2    4 , 4  
d2 d2  d4 d4 
w w  w w 
  6 , 6    8 , 8  
 d 6 d 6   d8 d8 
 3,6  9,10  12,15  19,20 
Relative volume of a
stability box

 wi wi  U
     : pi  piL
d

d
 i
i 

3 1 3 1
1
   
8 4 9 5 160
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3.4. Properties of a stability box
Property 1.
For any jobs J i  J and J v  J , v  i
 wi wi   wv wv 
 ,    U , L   
 ui li   pv pv 
Case (I)
wv
pvU

wi
piU
,
wv
pvL

wi
piL
Property 2.
max
For case (I), there exists a permutation  t  S , in which job J v
proceeds job J i .
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Case (II)
wv
pvU

wi
piU
,
wv
pvL

wi
piL
(1)
Property 3.
For case (II), there exists a permutation  t  S max , in which jobs J i and J v
are located adjacently: i  t r and v  t r 1.
Remark 1.
Due to Property 3, while looking for a permutation  t  S max , we shall
treat a pair of jobs {Ji , J v} satisfying (1) as one job (either job J i or J v ).
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Case (III)
wv
wi wv wi

, L  L
U
U
pv
pi pv
pi
(2)
Property 4.
(i) For a fixed permutation  k  S , job J i  J may have at most one
maximal segment[li , ui ] of possible variations of the processing time
pi  [ piL , piU ] preserving the optimality of permutation  k .
(ii) For the whole set of permutations S, only in case (III), a job J i  J
may have more than one (namely: J (i)  1  1) maximal segments [li , ui ]
L
U
of possible variations of the time pi  [ pi , pi ] preserving the
optimality of this or that particular permutation from the set S.
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L - the set of all maximal segments [li , ui ] of possible variations of the
processing times pi for all jobs J i  J preserving the optimality of
permutation  t  S max .
Property 5.
| L | n
Property 6.
There exists a permutation  t  S with the set L  L of maximal
segments [li , ui ] of possible variations of the processing time pi , Ji  J ,
preserving the optimality of permutation  t .
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3.5. A job permutation with the largest
volume of a stability box
Algorithm MAX-STABOX
Input: Segments [ piL , piU ] , weights wi , J i  J .
max
Output: Permutation  t  S , stability box SB( t , T ) .
 wu1
wun 

Step 1: Construct the list M (U )  ( Ju1 ,...,Jun ) and the list w(U )  U ,..., U 
 pu
pun 
1

w
in non-decreasing order of
ur
U
ur
p
. Ties are broken via increasing
wur
L
ur
p
 wl
wln
1

Step 2: Construct the list M (L)  ( J l1 ,...,J ln ) and the list w( L)  L ,..., L
 pl
p ln
 1
in non-decreasing order of
wl r
p
L
lr
. Ties are broken via increasing
.




wlr
p
U
lr
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.
Step 3: FOR j = 1 to j = n DO
compare job J u and job J l .
Step 4: IF J u  J l THEN job J u has to be located in position j in
permutation  t  S max. GOTO step 8.
Step 5: ELSE job Ju  Ji satisfies inequalities (2). Construct the set
J (i)  {Jur1 ,...,Jlk1 } of all jobs J v satisfying inequalities (2), where
J i  J u j  J lk
Step 6: Choose the largest range [lu j , uu j ] among those generated for job
J. u j  Ji
Step 7: Partition the set J(i) into subsets J  (i ) and J  (i ) generating the
largest range [lu j , uu j ] . Set j = k+1 GOTO step 4.
Step 8: Set j := j+1 GOTO step 4.
END FOR
max
Step 9: Construct the permutation  t  S
via putting the jobs J in the
positions defined in steps 3 – 8.
Step 10: Construct the stability box SB( t , T ). STOP.
j
j
j
j
j
j
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Remark 2.
Algorithm MAX-STABOX constructs a permutation  t  S such that
the dimension| Nt | of the stability box
SB( t ,T )  t N [lti , uti ]  T
i
t
is the largest one for all permutations S, and the volume of the stability
box SB( t , T ) is the largest one for all permutations  k  S having the
largest dimension | Nk || Nt | of their stability boxes SB( k ,T ) .
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4. Computational results
L
U
C - integer center of a segment [ pi , pi ] was generated using the
uniform distribution in the range [L;U]: L  C  U .
 - maximal possible error of the random processing times pi
piL ( piU ) - lower (upper) bound for the possible job processing time


piL  C  (1 
) , piU  C  (1 
)
100
100
n(n  1) 

 | A |:
  100% - the
2 

average relative number | A | of the arcs in the
dominance digraph

 tp*   p*
 p*
 p*
 tp*
- relative error of the objective function value
- the optimal objective function value of the actual scenario
n
  wiCi ( t , p* )
i 1
Each series contains 100 solved instances
Processor AMD Athlon (tm) 64 3200+, 2.00 GHz; RAM 1.96 GB
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5. Conclusion and further work
 An O(n 2 ) algorithm has been developed for calculating a permutation
. t  S with the largest dimension and volume of a stability box
.SB( t , T ) ;
.
 Properties of a stability box were proved allowing to derive an
O(nlogn) algorithm for calculating a permutation  t  S max ;
 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.
Further research concerning the use of a stability box for other
uncertain scheduling problems appear to be promising.
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6. References
1.
2.
3.
4.
5.
6.
7.
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.
Smith, W. (1956). Various optimizers for single-stage production. Naval Research
Logistics Quarterly, V. 3(1):59–66.
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.
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.
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.
Sotskov, Y., Sotskova, N., Lai, T.-C., and Werner, F. (2010b). Scheduling under
Uncertainty. Theory and Algorithms. Belorusskaya nauka, Minsk, Belarus.
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.
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