decision making under interval uncertainty – selected ... · decision making under interval...

Decision making under interval uncertainty –

selected models and algorithms

Jerzy Józefczyk

Division of Intelligent Decision Support Systems Wrocław University of Technology, Wrocław, POLAND

e-mail: [email protected]

City of Wrocław, POLAND

Wrocław 2013

Wrocław University of Technology

Budget: budget grant for

teaching and research activities own income

from teaching and research activity

Figures

Fields of study 39

Students 35 252

Academic staff 2 000

Professors 395

Associate professors 240

PhD holders 1 223

Campus area (ha) 87.6

Buildings 251

Faculties

Faculty of Architecture Faculty of Civil Engineering Faculty of Chemistry Faculty of Electronics Faculty of Electrical Engineering Faculty of Geoengineering, Mining and Geology Faculty of Environmental Engineering

Faculty of Computer Science and Management Faculty of Mechanical and Power Engineering Faculty of Mechanical Engineering Faculty of Fundamental Problems of Technology Faculty of Microsystems Electronics and Photonics Faculty of Mathematics Regional Branches in Jelenia Góra, Legnica, Wałbrzych

Division of Intelligent Decision Support Systems Problems

Modelling of decision making (management) systems Complex decision making problems Decision making for uncertain systems Big data processing and machine learning

Methods Classic methods of modelling and optimization Methods of computational intelligence Methods of multi-criteria decision making. Methods of uncertain (non-deterministic) knowledge representation, e.g. interval

uncertainty. Applications

Computer systems and networks, e.g. routing, rate control, distribution of data. Complex manufacturing and multi-robot systems, e.g. logistic systems and supply

networks, wireless sensor networks. Biomedical systems.

Content

1. Decision making for input-output plants.

2. Parameter uncertainty and its selected representations.

3. Determinization (substantiation) of evaluation function.

4. Selected decision making problems under interval uncertainty.

a. Time-optimal resource allocation.

b. Task scheduling minimizing the sum of completion times.

c. Task scheduling minimizing the makespan.

d. Permutational flow-shop

5. Final remarks

Decision making for input-output plants

decision u evaluation of decision y

System(Process)

)(u,aFy =

u – decision

– evaluation of the decision u

Special cases of decision making problems

Case A Given: , Find: to fulfil

)(or:)( Yyyyy ∈′=ϕ)(, yF ϕ

)(yϕu′

Case B Given: F, Q, Find to minimise (maximise) Q, i.e. [ ] – optimisation problem

Δ ()(()( u,a)Qu,aFy ==ϕϕ

u′

)(minarg u,aQuUu∈

=′ )(maxarg u,aQuUu∈

=′

Parameter uncertainty and its selected representations (1/2)

It is assumed that set A is the only information on uncertain parameter a.

amaaaaa mk parameterunknownldimensiona–],,...,,,[ T)()()2()1(=

asAa m parameterofvaluesfeasibleofet–R⊆∈

],[ )()(1

kkmk aaA =×=,],,[,particularIn )()()()()( kkkkk aaaaa ≤∈

In discrete case, },...,,,{ )()(2

)(1

)( kn

kkkk

aaaa ∈ }...,,,{ )()(2

)(11

kn

kkmk k

aaaA =×=

A specific realization of parameter a is often called a scenario.

decision evaluation of decision

System(Process)

uu,aF decisiontheofevaluation–)(

//decision– aparameterondependentu

//decisionsfeasibleofset–)( aparameterondependentaU

Parameter uncertainty and its selected representations (2/2)

The most known representations of the uncertainty: • Probabilistic • Fuzzy • Grey • Uncertain variables /credibility • Possibilistic

Bubnicki Z. (2004) Analysis and decision making in uncertain systems. Springer Verlag, Berlin, London, New York. Liu B. (2010) Uncertain Theory. Springer-Verlag Berlin. Li X., Ralescu D. (2008) Credibility Measure of Fuzzy Set and Applications. Proc of IPMU.

Calculation of the evaluation function /in the presence of interval uncertainty/

),(max auFAa∈

),(min auFAa∈

Aggregation of values )(u,aF

),,(min)1(),(max auFauFAaAa ∈∈

−+ ββ ]1,0[∈β

∫A

daauF ),(

The basis for such calculation may be directly evaluation function or any term built on it. Additionally, different operators for aggregation can be used.

)(u,aF

Determinization of the evaluation function (1/8)

Aggregation of terms containing ),( auF

AaaFauF

∈′

,)(),( – relative regret

uAaaFauF givenafor),(),( ∈′− – regret (opportunity loss)

Regret (opportunity loss) is a difference between the cost of a specific decision and the

corresponding cost of the optimal decision – for any given realization of unknown

parameter.

decisionsfeasibleofset–),,(min)(where UauFaFUu∈

=′


Savage L.-J. (1951) The theory of statistical decision. Journal of American Statist. Assoc., 46, 55–67. Kouvelis P. and Yu G., (1997) Robust discrete optimization and its applications. Kluwer Academic Publishers, Dordrecht/Boston/London.

We are interested in decisions acceptable irrespective of a specific scenario a. The notion ’robustness’ is used.

Decision is robust if it is ’good’ or at least ’not so bad’ for all scenarios.

Decision is robust if its realization prevents undesirable or deteriorating behavior of decision object.


Due to uncertain a we do not seek for optimal decisions.

decisionoptimal––)()(),( paUupaFauF ∈→<′−

decisionsoptimal–all–})(),(:)({)( ppaFauFaUuaU p ofset<′−∈=

)(and,)(),( aUupaFauF pAa

Aa∈

∈ ∈<′−∀

If we require the inequality for all scenarios, we have p-robust decision /used mainly for discrete A/.

In general, the set of p-robust decisions can be obtained:

})(),(:{ paFauFUuU Aap ≤′−∀∈= ∈

The solution is close to the optimal one for all scenarios.


p-robustness


Instead of considering all scenarios, it is enough to take into account only the worst scenario, i.e.

UuaFauFuzAa

∈′−=∈

)],(),([max)(

Minmax regret approach (worst-case regret approach)

Instead of postulating seeking for the minimal p, i.e. puz <)(

)](),([maxmin)(min aFauFuzAaUuUu

′−=∈∈∈

)](),([max)( aFauFuzAa

′−=∈

– measure of robustness

)(min)(, ** uzuzuUu∈

= – results of solving minmax regret problem

Other measures of robustness are encountered in literature.

(b,w) – robustness, parametric robustness, used for discrete A /proposed by B. Roy/,

b, w – non-negative parameters )( wb <


Robust decision maximizes the number of scenarios for which the value of evaluation (e.g. the value of regret) does not exceed b as well as this value for all scenarios does not exceed w.

scenariosallofnumber–)],(),(([max 1 NaFauFb iiNiUu

′−−∑ =∈1

NiwauF i ...,,2,1,),(.t.s =≤

Lexicographic – robustness, parametric robustness, used for discrete A α

Non-increasing sequence of the evaluation function values is determined for all scenarios, i.e.:

scenariosallofnumber–,1...,,2,1),,(),( 1 NNiauFauF ii −=≥ +

})(),(:{)(~ αα ≤′−∀∈= ≤ jjnj aFauFUuU

Set of – robust lexicographic decisions is derived. α

Lexicographic – robustness implies p-robustness but not vice versa. α



Other minmax (non-parametric) approaches Robust optimal decision making

),(max auFAa∈

– robust (worst-case) criterion ),(maxmin auFAaUu ∈∈

→

)(),(max

aFauF

Aa ′∈– robust (worst-case) relative regret

criterion )(),(maxmin

aFauF

AaUu ′→

∈∈

Challenge: Three nested optimization problems.

– robust (worst-case) regret criterion

)](),([max aFauFAa

′−∈

)](),([maxmin aFauFAaUu

′−→∈∈

Kasperski A., Zieliński P. (2013), Bottleneck combinatorial optimization problems with uncertain costs and OWA criterion, Operations Research Letters, 41, 639-643. Yager Y.Y. (1988) On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans. on SMC, 18, 183-190.

Special case of continuous functions of the form iini

uaauF,...,2,1

max),(=

=

u F(u,a) where...,,2,1,1 nicp

Tii

i ==

0>ip – parameters / /

C – global amount of resources ,0:]...,,,[{ T21c ≥== ∆

in cccccD Ccni ni i ≤= ∑ =1,...,,2,1

where1max),(,...,2,1 iini cp

cpQ=

= T21 ]...,,,[ Ipppp =

ii

ii c

up

a 1,1==

Deterministic time optimal resource allocation problem )(min)(),(c

Δ' p,cQpQ'cpQDc∈

==

Time-optimal resource allocation (1/5)

,...,,2,1,1

1

1

' Ii

p

pCc I

m m

ii ==

∑=

∑=

=′I

m mpCpQ

1

11)(Analytical solution:

,]...,,...,,,[],,[ T21 niiii pppppppp =∈ ××=∈ ],[],[ 2211 ppppp P ],[... nn pp×

−= ∑

==∈

I

m mii,...,niPp pCcpcz

12,1

111maxmax)(

.11max1111max111max)( 111

−

−=

−−=

−= ∑∑∑

≠=∈≠

=∈=∈

n

smm

mss

sPp

n

smm

msssPp

nm

mssPp pCCcpcC

pCCpcppCcpcz

Let us find such an index s that . Then is,cpup iiss

≠≥11

It is easy to see that is the worst scenario and T1 ]...,,...,,[ ns

c pppp =

,1111)(1)( 1w

+−=′−= ∑

≠=

n

smm

msssss ppCcppQ

cpcz


Averbakh I. (2000) Minimax regret solutions for minimax optimization problems with uncertainty. Operations Research Letters, 27, 57–65. Józefczyk J. (2008) Worst-case allocation algorithms in a complex of operations with interval parameters. Kybernetes, 37, 652-676.

Theorem. The solution of the uncertain time-optimal resource allocation problem is optimal if and only if the conditions

Ccni i =∑ =1

*

isnscpcp ssii

≠== ,...,,2,1,11**

are fulfilled, and s is the longest lasting operation.

*c


1. Calculate and find index s and such that

2. Determine .

The optimal resource allocation for operations is as follows: *c si...,n,,,i ≠= 21

∑≠=+

=n

smm

ms

ii

pp

pCc

1

*11

1

*sc

.ˆmax*i

is cc =

(A)

sin,...,,,i, ≠= 21

The optimal value of the absolute regret . ∑≠=

−= n

smm

mms

msms

ppppppp

Ccz 1

* 1)(

ni

pp

pCcn

smm

mi

ii ...,,2,1,11

1

ˆ1

=+

=∑

≠=


Optimal polynomial solution algorithm

],4,2[2 ∈p

Step 1. ,4

21

31

31

1011

1

ˆ

21

11 =

+=

+=

pp

pCc

Step 2. 6*2 =c

Moreover,

,2

11

41

41

1011

1

ˆ

12

22 =

+=

+=

pp

pCc

Therefore, s = 1, and


Numerical example: ],3,1[,2 1∈= pn )10(1021 ==+ Ccc

.4*1 =c

.81)( * =cz

Task scheduling minimizing the sum of completion times (1/5)

7/23

}...,,,,,{= 21 nj JJJJ J – set of n tasks

}...,,,,,{= 21 mi MMMM M – set of m machines (executors)

njmijipp

,...,2,1,...,2,1, ][

=== – matrix of task execution times

nkjmiikjxx

,1,,1,, ][∈

∈=

=otherwise,0

machinethbyendthefromththeasperformedistaskth,1,,

ikjx ikj

onceexecutedbemusttaskeach–,1,11 1 ,, njxmi

nk ikj ==∑ ∑= =

momenttimeeveryat

taskonemostatperformecanmachineeach–,1,,1,11 ,,∑ ===≤n

j ikj njmix

∑ ∑ ∑= = −= nj

mi

nk ikjji xpkxpF 1 1 1 ,,,1 ),(

∑≤≤ jjijiji CpppR || ,,, – notation of the problem

Problem formulation (deterministic version): Given: J, M, p Find: the optimal schedule , i.e. x′

∑ ∑ ∑= = −==′=′ nj

mi

nk ikjji

xxxpkxpFxpFpF 1 1 1 ,,,111 min),(min),()( Δ


Problem formulation (uncertain version): Given: J, M, p Find: the optimal schedule , i.e. *x

xppFxpFpFxpFxz xxxp

forscenarioworstthe–where),(),()](),([max)( 11111 ′−=′−=∈P

)(min)( 1*

1*

1 xzxzzx

==∆

– evaluation function (robust regret criterion)

jijijijiji ppppp ,,,,, ],,[ ≤∈ – interval execution times

],[...],[ ,,,1,1 nmnmji ppppp ××=∈P

)(),( 11 pFxpF ′− – regret

)](),([max)( 111 pFxpFxzp

′−=∈P


Important results:

• Reducing the calculation of the worst scenario to the weighted bipartite matching problem solved polynomially by the Hungarian algorithm.

• Proofing of the 2-approximate property for the mid-point scenario (MIH).

• Elaboration and experimental evaluation of heuristic Scatter Search (SS) algorithm.

9/23

2,,MIH

,jiji

jipp

p+

= – the mid-point scenario ).(2)( *1

MIH1 xzxz ≤→


2,10 == mn

C )( *1 xz )( SS

1 xz )( MIH1 xz

10 57 57 64 30 95 97 111 50 220 220 248 70 197 202 232

100 322 330 372 150 497 507 565

%100)(

)()(MIH

1

MIH1

SS1

1 xzxzxz −

=δ

n 10 20 50 100

1δ 9.90 8.13 7.66 5.60


Siepak M., Józefczyk J. (2014) Solution algorithms for unrelated machines minmax regret scheduling problem with interval processing times and the total flow time criterion, Annals of Operations Research, 222, 517-533.



njjpp ,...,2,1][ == – vector of execution times

=otherwise,0

machinethbyperformedistaskth,1,

ijc ij

njmiijcc

,...,2,1,...,2,1, ][

===

onceexecutedbemusttaskeach–,1,11 , njcmi ij ==∑ =

∑ == nj ijj

icpcpC 1 ,max max),(

Problem formulation (deterministic version): Given: J, M, p Find: the optimal schedule , i.e. c′

∑ ===′=′ nj ijj

icccpcpCcpCpC 1 ,maxmaxmax maxmin),(min),()( Δ

Task scheduling minimizing the makespan (1/4)

max,,, || CpppR jijiji ≤≤ – notation of the problem


jjjjj ppppp ≤∈ ],,[ – interval execution times

],[...],[ 11 nn ppppp ××=∈P

)(),( maxmax pCcpC ′− – regret

)](),([max)( maxmax2 pCcpCczp

′−=∈P

Problem formulation (uncertain version): Given: J, M, p Find: the optimal schedule , i.e. *c

cp

pCxpCpCcpCcz

c

ccp

forscenarioworstthe–where

)(),()](),([max)( maxmaxmaxmax2 ′−=′−=∈P

)(min)( 2*

2*

2 czczzc

==∆


15/23

Important results:

• Determination of the algorithm for calculating of lower and upper bounds of for a given c.

• Development of heuristic algorithms (SS-based and MIH) as well as their experimental evaluation /lack of any approximate algorithm/.

)(2 cz


%100)(

)()(MIH

2

MIH2

SS2

2 czczcz −

=δ

2,10 == mn

C )( *2 cz )( SS

2 xz )( MIH2 xz

10 4 4 6 30 21 21 31 50 38 40 58 70 56 56 75

100 114 119 155 150 135 136 195

n 10 20 50 100

2δ 44.60 27.48 19.19 9.77


Józefczyk J., Siepak M., (2011) Minmax regret algorithms for uncertain P| | Cmax problem with interval processing times. In: H. Selvaraj and D. Zydek, eds. Proc. of 21st Int. Conf. On Systems Engineering, Las Vegas USA, IEEE Computer Society, Los Alamitos, 115-119.

Permutational flow-shop problem (1/8)

General scheme of the flow-shop problem (tasks move along stationary executors)

M1set of tasks M2 Mm



),,,( ,,2,1 jmjjj OOOJ = – operations for jth task

njmijipp

,...,2,1,...,2,1, ][

=== – matrix of operation execution times

Π∈= )...,,...,,,( 21 nj πππππ – permutation of tasks (decision) where

},...,,21{ n,j ∈π }},...,,2,1{,,:{ kjnkjkj ≠∈≠= πππΠ

Evaluation function (criterion)

where),,(),( ,max ππ π pCpCnm=

njppC jk kj

...,,2,1for,=),( 1 ,1,1 =∑ = ππ π

mippC ik ki ...,,2,1for,=),( 1 ,, 11

=∑ = ππ π

njmipCpCppCjjjj iiii ...,,2,1,...,,2,1for],),(,),(max[=),(

1,,1,, ==+−− πππ ππππ

Problem formulation: Given: J, M, p Find: the optimal permutation , i.e. π ′

),(min),()( maxmaxmax πππ

pCpCpCΠ∈

=′=′ ∆



jijijijiji ppppp ,,,,, ],,[ ≤∈ – interval execution times

],[...],[ ,,,1,1 nmnmji ppppp ××=∈P

)(),( maxmax pCpC ′−π – regret

)](),([max)( maxmax3 pCpCzp

′−=∈

ππP

Problem formulation: Given: J, M, Find: the optimal permutation , i.e.

njmipp jiji ...,,2,1,...,,2,1,, ,, ==∗π

)])(),([max(min)( maxmax*

3 pCpCzp

′−=∈∈

πππ PΠ

This problem is at least NP-hard. Moreover, three nested optimizations.


Case of two machines m = 2

Uncertain flow-shop is NP-hard.

Theorem. For mid-point scenario , 2

,,MIH,

jijiji

ppp

+=

).(2)( *3

MIH3 ππ zz ≤


))((min)]),(min),([max(min 3maxmax πσππσπ

zpCpCp ΠΠΠ ∈∈∈∈

=−P

Worst-case scenario analysis (maximization)

It is easy to justify that /extreme scenarios only/ jijijiji ppppji ,,,,:, =∨=∀

mn2 – number of all scenarios )!1()!1(

)!2(−−

−+nm

nm– number of scenarios to check

Inner minimization

– lower bound of )(πz)),(),((max)( NEHmaxmax,3 σππ pCpCz

pLB −=

∈P

))(),((max)( max,max,3 pCpCz LBp

UB −=∈

ππP

– upper bound of where

++= ∑∑∑

+=≠==

−

===

m

kili

jlnl

n

jjk

k

iji

njmkLB ppppC

1,

,11,

1

1,

,1,1max, minminmax)(

)(πz


Representation of a feasible solution (chromosome): Sequence of numbers from 1 to n.

Fitness function: )(,3 πUBz

Initial population: 10 random permutations + solution determined by constructive MIH algorithm and its 9 random versions.

Crossover: Order crossover operator (Goldberg) giving feasible offsprings.

Mutation: Two random genes are swapped.

Selection: The current population is sorted non-increasingly. First two chromosomes are removed from the list and the result of their crossover is added to the new population. The process is repeated until the list is empty.


],0(, Kp ji ∈ ],[ ,,, Cppp jijiji +∈

Middle interval heuristic (MIH) Application od NEH constructive algorithm for

2,,MIH

,jiji

jipp

p+

=

n . n .

30

50

70

90

110

130

150

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Average upper and lower bounds of z returned by MIH heuristic (triangles) and EVO algorithm (dots) for different values of n.


0

50

100

150

200

250

300

350

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Average upper and lower bounds of z returned by MIH heuristic (triangles) and EVO algorithm (dots) for different values of C.


Józefczyk J., Siepak M., (2013) Scatter Searcg based algorithms for min-max regret task scheduling problems with interval uncertainty, Control and Cybernetics, 42, 667-698. Józefczyk J., Siepak M., (2013) Worst-case regret algorithms for selected optimization problems with interval uncertainty, Kybernetes, 43, 371-382.

Final remarks (1/2)

Current work: • Considering of multi-criteria problems Ehrgott M., Ide J., Schoebel A. (2014) Minmax robustness for multi-objective optimization problems, European Journal of Operational Research, 239, 17-31.

• Investigation of computational complexity and seeking for approximate algorithms for basic problems

Kasperski A., Kurpisz A., Zieliński P. (2014) Approximability of the robust representatives selection problem Operations Research Letters, DOI: 10.1016/j.orl.2014.10.007 • Developing of heuristic algorithms for problem instances more close to

real-world applications Ćwik M., Józefczyk J. (2015) Evolutionary algorithm for minmax regret flow-shop problem, Management and Production Engineering Review 6(3), 3–9. Averbakh I., Pereira J. (2011) Exact and heuristic algorithms for the interval data robust assignment problem. Computers & Operations Research, 38,1153-1163. Kasperski A., Makuchowski M., Zieliński P. (2012) A tabu search algorithm for the minmax regret minimum spanning tree problem with interval data, Journal of Heuristics, 18(4), 593-625.

Applications Non-repetitive and unique processes (systems) as well as non-available and(or) non-existing) historical data.

Final remarks (2/2)

Example Producing of software by small/medium computer company: Tasks: specific programming activities with changing execution times. Aim: Minimization of the total execution time.

decision making under interval uncertainty – selected ... · decision making under interval...

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