Measures of problem uncertainty for scheduling with interval processing times

OR Spectrum manuscript No.(will be inserted by the editor)

Measures of problem uncertainty for scheduling withinterval processing times

Yuri N. Sotskov · Tsung-Chyan Lai ·Frank Werner

Received: date / Accepted: date

Abstract The paper deals with scheduling under uncertainty of the job pro-cessing times. The actual value of the processing time of a job becomes knownonly when the schedule is executed and may be equal to any value from thegiven interval. We propose an approach which consists of calculating mea-sures of problem uncertainty to choose an appropriate method for solving anuncertain scheduling problem. These measures are based on the concept ofa minimal dominant set containing at least one optimal schedule for eachrealization of the job processing times. For minimizing the sum of weightedcompletion times of the n jobs to be processed on a single machine, it is shownthat a minimal dominant set may be uniquely determined. We demonstratehow to use an uncertainty measure for selecting a method for finding an ef-fective heuristic solution of the uncertain scheduling problem. The efficiencyof the heuristic O(n log n)-algorithms is demonstrated on a set of randomlygenerated instances with 100 ≤ n ≤ 5000. A similar uncertainty measure maybe applied to many other scheduling problems with interval processing times.

Keywords Scheduling · Interval processing times · Uncertainty measure

Yu. N. SotskovUnited Institute of Informatics Problems, National Academy of Sciences of Belarus,Surganova Street, 6, 220012 Minsk, BelarusE-mail: [email protected]

T.-C. LaiNational Taiwan University, Department of Business Administration,2nd Management Building, 85, Sector 4, Roosevelt Road, 10672 Taipei, TaiwanE-mail: [email protected]

F. WernerOtto-von-Guericke-University, Faculty of Mathematics,Universitatsplatz 2, 39106 Magdeburg, GermanyE-mail: [email protected]

2 Yuri N. Sotskov et al.

1 Introduction

In real-life scheduling, the precise processing time may remain unknown untilthe completion of a job. Due to this reason, it may be impossible to implementa deterministic method [14,24,25] for constructing a really optimal schedule. Inthe OR literature, there have been developed several methods for solving un-certain scheduling problems [10,14,16,20]. In a stochastic method [4,7,14,26],it is assumed that the processing times are random variables with known prob-ability distributions. If the processing times may be defined as fuzzy numbers,a fuzzy logic is used for scheduling under fuzziness [6,16,21]. If the processingtimes can be represented neither as random variables nor as fuzzy numbers,other methods are needed for scheduling under uncertainty [1,10,15,20]. Arobust method [2,8–10,27] assumes that the decision-maker prefers a schedulehedging against the worst-case scenario. A stability method [11,18–20] com-bines a stability analysis with a multi-stage decision framework.

Each of the above methods and other ones have a specific field for anefficient implementation, a method suitable for one class of scheduling prob-lems involving uncertainty may fail for another class. A decision on selecting amethod for solving a concrete scheduling problem influences the quality of theschedule obtained and the time needed for constructing an effective schedule.

In this paper, we propose to use a measure of problem uncertainty forselecting a suitable method and for estimating the quality of a schedule whichmay be constructed. In Sections 2–7, we consider the single-machine problemof minimizing the sum of weighted completion times with a continuum ofpossible scenarios (Section 2). A survey of various methods is presented inSubsections 3.1–3.4. The uncertainty measures, which we suggest, are based onthe cardinality of a minimal dominant set of job permutations (Subsection 3.4).For any feasible scenario, a minimal dominant set contains at least one optimalpermutation. In Section 4, we present a criterion for the largest cardinalityof a minimal dominant set. We show that a minimal dominant set may beuniquely determined (Section 5). Several examples are discussed in Section 6.The efficiency of the O(n log n)-algorithms is tested on randomly generatedinstances with 100 ≤ n ≤ 5000 (Section 7). In Section 8, we describe howto introduce a similar uncertainty measure for other more general schedulingproblems. Section 9 is devoted to an application perspective in predictive-reactive scheduling. Concluding remarks are given in Section 10.

2 Problem description and preliminaries

In real-world scheduling, it is often necessary to manage a bottleneck machine.So, we first consider the following single-machine scheduling problem. A set ofjobs J = {J1, J2, ..., Jn}, n ≥ 2, has to be processed on a single (bottleneck)machine, a weight wi > 0 being given for a job Ji ∈ J to reflect its importance.All the jobs J are ready for processing from the start time t0 = 0 of theplanning horizon. No preemption is allowed while processing a job Ji ∈ J .

OR Spectrum 3

It is assumed that the processing time of a job Ji can take any real valuepi ∈ [pL

i , pUi ], only the bounds pL

i ≥ 0 and pUi ≥ pL

i of the segment [pLi , pU

i ]being given before scheduling. The precise value pi remains unknown untilthe completion of job Ji. Such an assumption is realistic for most real-lifescheduling problems since rough bounds pL

i and pUi may be given for a practical

job Ji ∈ J : For a most uncertain job Ji ∈ J , one can let pUi to be equal to

the length of the planning horizon and pLi = 0. The set T of possible vectors

p = (p1, p2, . . . , pn) of the job processing times is called the set of scenarios:T = {p | p ∈ Rn

+, pLi ≤ pi ≤ pU

i , i ∈ {1, 2, . . . , n}}. Note that there is acontinuum of possible scenarios p ∈ T . Let S = {π1, π2, . . . , πn!} be the set ofpermutations πk = (Jk1 , Jk2 , . . . , Jkn

) of the jobs J .If the scenario p ∈ T is fixed and the permutation πk ∈ S is chosen, then

one can calculate the completion time Ci = Ci(πk, p) of job Ji ∈ J in a semi-active schedule (Ck1 , Ck2 , . . . , Ckn) uniquely determined by the permutationπk. (A schedule (Ck1 , Ck2 , . . . , Ckn

) is called semi-active if no job Ji ∈ J canstart earlier without delaying the completion time of another job or alteringthe order πk for processing the jobs J [14,25].) Since the jobs J are readyfor processing from time t0 = 0 and job preemptions are not allowed, one cancalculate: Ck1 = pk1 , Cki = Cki−1 + pki−1 , i ∈ {2, 3, . . . , n}.

Using the three-field notation α|β|γ [14], the problem considered is denotedas 1|pL

i ≤ pi ≤ pUi |

∑wiCi, where the criterion γ =

∑wiCi denotes the

minimization of the sum of the weighted completion times: γ =∑

Ji∈J wiCi =∑Ji∈J wiCi(πt, p) = minπk∈S

{∑Ji∈J wiCi(πk, p)

}, permutation πt = (Jt1 ,

Jt2 , . . . , Jtn) ∈ S being optimal for the scenario p ∈ T .As the scenario p ∈ T remains unknown until the completion of the jobs

J , the values Ci, Ji ∈ J , cannot be calculated before the schedule realization.Thus, the problem 1|pL

i ≤ pi ≤ pUi |

∑wiCi of finding an optimal permutation

πt is not correct (in general, it is not possible to find an exact solution tothis problem since a scenario is not fixed). If a scenario p ∈ T is fixed, i.e.,pL

i = pUi = pi ∈ [pi, pi], then problem 1|pL

i ≤ pi ≤ pUi |

∑wiCi reduces to the

problem 1||∑ wiCi, which can be solved in O(n log n) time [17].In what follows, the problem α|pL

i ≤ pi ≤ pUi |γ with the objective function

γ = f(C1, C2, . . . , Cn) is called uncertain in contrast to the problem α||γwhich is called deterministic. Note that there is a principal difference betweena deterministic problem and its uncertain counterpart. While for solving anyindividual deterministic problem α||γ, one can apply the same method andan optimal schedule will be obtained (provided that an exact method will beapplied), any two individual uncertain problems may need different methodsto obtain a schedule which will be an ex-post optimal or will provide objectivefunction value close to an ex-post optimal one. (An ex-post optimal schedule isthe best one that could been obtained if exact processing times of jobs J wouldhave been available before scheduling.) First, there may be no informationavailable for implementing a particular method. Second, a method may be notefficient or not effective for a specific instance of the problem α|pL

i ≤ pi ≤ pUi |γ.

Furthermore, while using a multi-stage decision framework, the same instancemay need a different method at different stages of the decision-making.


3 Literature review

While an optimal sequencing rule for the deterministic problem 1||∑wiCi hasbeen known since 1956 [17], its uncertain counterpart 1|pL

i ≤ pi ≤ pUi |

∑wiCi

continues to attract the researchers developing algorithms to solve it exactly(in some sense) or approximately, see [2,8,9,14,18,19,26,27] among others.

3.1 Robust method

For the problem α|pLi ≤ pi ≤ pU

i |γ, there usually does not exist a semi-activeschedule (a job permutation for the problem 1|pL

i ≤ pi ≤ pUi |γ) that remains

optimal for all scenarios from the set T . Due to this reason, an additionalcriterion is often introduced for a problem α|pL

i ≤ pi ≤ pUi |γ. In particular,

a robust schedule minimizing the worst-case deviation from optimality wasintroduced in [2,10]. Let γt

p denote the optimal value of the non-decreasingobjective function γ = f(C1, C2, . . . , Cn) for the deterministic problem α||γwith the scenario p: γt

p = minπk∈S γkp = minπk∈S f(C1(πk, p),. . . ,Cn(πk, p)).

For the permutation πk ∈ S and a scenario p ∈ T , the difference γkp − γt

p =r(πk, p) is called the regret for πk, where permutation πt ∈ S is optimal. Thevalue Z(πk) = max{r(πk, p) | p ∈ T} is the worst-case absolute regret. Theworst-case relative regret is as follows: Z ′(πk) = max

{r(πk,p)

γtp

| p ∈ T}

, γtp 6= 0.

In [2,8,9,27], the problem 1|pLi ≤ pi ≤ pU

i |∑

Ci has been considered (the jobweights are equal). For a specific scenario pj ∈ T , the deterministic problem1||∑ Ci can be solved using the shortest processing time (SPT) rule: Processthe jobs J in non-decreasing order of their processing times pj

i , Ji ∈ J . Whilethe deterministic problem 1||∑Ci is solvable in O(n log n) time, finding apermutation πt ∈ S of minimizing either the worst-case absolute regret Z(πt)or the relative regret Z ′(πk) for the uncertain problem 1|pL

i ≤ pi ≤ pUi |

∑Ci

is NP-hard [2,27]. For the latter problem, a 2-approximation algorithm wasdeveloped to minimize the worst-case regret [9] and several algorithms weretested [2,27]. A few special cases are known to be polynomially solvable forminimizing the worst-case regret for the problems α|pL

i ≤ pi ≤ pUi |γ [8].

3.2 Stochastic method

Part II of monograph [14] is devoted to a stochastic method, allowing a sched-uler to minimize the expectation E(f(C1, C2, . . . , Cn)) of the objective func-tion γ = f(C1, C2, . . . , Cn) for random processing times with known probabil-ity distributions. An optimal schedule for problem 1|pL

i ≤ pi ≤ pUi |E(

∑wiCi)

in the class of non-preemptive static list policies [14] (the jobs have to be or-dered at time t0 = 0 according to a chosen priority rule) may be constructedby the weighted shortest expected processing time (WSEPT) rule: Processthe jobs in non-increasing order of the ratios wi

E(pi), where E(pi) denotes the

expected value of the random processing time pi of the job Ji ∈ J [14,26].

OR Spectrum 5

In [23], stochastic integer programming is used for airline crew schedul-ing (it is necessary to find a set of pairings for a crew group to minimize theplanned cost of operating a schedule of aircraft flights). Taking into accountthe uncertain nature of aircraft delays, the authors of [23] looked for a robustairline crew schedule which is less likely to be disrupted in the case of an air-craft delay. The probability of a flight delay was modelled by logistic regressionand a flight delay time by multi-variable regression.

3.3 Fuzzy method

If the processing times of the jobs J are given as fuzzy numbers, a fuzzymethod is available for solving a scheduling problem under fuzziness. Thebook [16] is devoted to the fuzzy method used in a scheduling environment.

In [6,22], the authors propose a scheduling model, where the due-datesfor the jobs are random variables. The job processing times are script, andone assigns satisfaction levels to the job completion times according to non-increasing membership functions, their positions depending on the expecteddue-dates, which are exponentially distributed.

3.4 Stability method

If no additional criterion is introduced (contrary to a robust method), thereis no information on the probability distributions of the random processingtimes (contrary to a stochastic method) and there are no membership func-tions for non-script processing times (contrary to a fuzzy method), then astability method [20] may be used for a problem α|pL

i ≤ pi ≤ pUi |γ. This

method combines a stability analysis, a multi-stage decision framework andthe solution concept of a minimal dominant set [11,18,20].

Definition 1 A set of permutations S(T ) ⊆ S is a minimal dominant setfor the problem 1|pL

i ≤ pi ≤ pUi |γ, if for any scenario p ∈ T , the set S(T )

contains at least one optimal permutation for the deterministic counterpartα||γ associated with the scenario p, this property being lost for any propersubset of set S(T ).

We recall some known results for the deterministic problem 1||∑ wiCi andfor the uncertain problem 1|pL

i ≤ pi ≤ pUi |

∑wiCi.

Theorem 1 [17] The permutation πk = (Jk1 , Jk2 , . . . , Jkn) ∈ S is optimal forthe problem 1|| ∑

wiCi if and only if the following inequalities hold:

wk1

pk1

≥ wk2

pk2

≥ . . . ≥ wkn

pkn

. (1)


Due to Theorem 1 [17], the deterministic problem 1||∑wiCi can be solvedby the weighted shortest processing time (WSPT) rule: Process the jobs in non-increasing order of the weight-to-process ratios wki

pki. For the uncertain problem

1|pLi ≤ pi ≤ pU

i |∑

wiCi, a minimal dominant set S(T ) may be determineddue to condition (2) for a dominance relation defined as follows.

Definition 2 Job Ju dominates job Jv, if there exists a minimal dominantset S(T ) for the problem 1|pL

i ≤ pi ≤ pUi |

∑wiCi such that job Ju precedes

job Jv in each permutation of the set S(T ).

Theorem 2 [18] For the problem 1|pLi ≤ pi ≤ pU

i |∑

wiCi, job Ju dominatesjob Jv if and only if

wu

pUu

≥ wv

pLv

. (2)

In the simplest case, a minimal dominant set for the uncertain problem1|pL

i ≤ pi ≤ pUi |

∑wiCi is a singleton, {πk} = S(T ), which is an exact solution

to the deterministic counterpart 1||∑ wiCi associated with any scenario p ∈ T .

Theorem 3 [18] There exists a dominant singleton S(T ) = {πk} = {(Jk1 ,Jk2 , . . . , Jkn

)} for the problem 1|pLi ≤ pi ≤ pU

i |∑

wiCi if and only if

wk1

pUk1

≥ wk2

pLk2

;wk2

pUk2

≥ wk3

pLk3

; . . . ;wkn−1

pUkn−1

≥ wkn

pLkn

. (3)

In what follows the notation 1|p|∑wiCi is used for indicating an instance(individual problem) associated with the scenario p ∈ T .

4 A minimal dominant set with cardinality n!

The cardinality |S(T )| of a minimal dominant set may be used to measure theuncertainty of the problem 1|pL

i ≤ pi ≤ pUi |

∑wiCi. E.g., equality |S(T )| = 1

means that there is no influence of uncertainty to the instance 1|pLi ≤ pi ≤

pUi |

∑wiCi at all. Such an instance may be solved exactly via the WSPT rule

similarly as its deterministic counterpart 1|p|∑ wiCi with any scenario p ∈ T .If S(T ) = {πk}, then the permutation πk provides an exact solution to theinstance 1|pL

i ≤ pi ≤ pUi |

∑wiCi since this permutation is optimal for an

instance 1|p|∑wiCi with any scenario p ∈ T . The permutation πk may beconstructed in O(n log n) time using the proof of Theorem 3 given in [18]: Onecan use the WSPT rule for ordering the jobs J with the processing time pi ofa job Ji ∈ J equal either to pL

i or to pUi . Due to (3), in both orderings, the

same permutation πk is obtained.The most uncertain instance of the problem 1|pL

i ≤ pi ≤ pUi |

∑wiCi is

that with |S(T )| = n!. Let a = min{

wi

pUi

| Ji ∈ J}

, b = max{

wi

pLi

| Ji ∈ J}

,and r be a real number from the segment [a, b]. The following subsets Jr

of the set J are used for establishing the largest cardinality of a set S(T ):Jr =

{Ji ∈ J | r = wi

pUi

= wi

pLi

}.

OR Spectrum 7

Theorem 4 Assume that there does not exist a number r ∈ [a, b] with |Jr| ≥2. A minimal dominant set S(T ) for the instance 1|pL

i ≤ pi ≤ pUi |

∑wiCi

attains the minimal cardinality |S(T )| = n! if and only if

max{

wi

pUi

| Ji ∈ J}

< min{

wi

pLi

| Ji ∈ J}

. (4)

Corollary 1 Assume that there does not exist a number r ∈ [a, b] with |Jr| ≥2. For any permutation πk ∈ S, there exists a scenario p ∈ T such that per-mutation πk is the unique optimal one for the instance 1|p|∑wiCi if and onlyif inequality (4) holds.

The proofs of Theorem 4 and Corollary 1 are omitted since they are similarto the proofs of their special cases proven in [18], where the strict inequalitypL

i > pUi has been assumed for each job Ji ∈ J .

5 The uniqueness of a minimal dominant set

Let Z denote the set of all instances 1|pLi ≤ pi ≤ pU

i |∑

wiCi. Each instancezk ∈ Z is totally determined by the set of scenarios Tk = {p | p ∈ Rn

+, pLi ≤

pi ≤ pUi , i ∈ {1, 2, . . . , n}} and the set of weights {w1, w2, . . . , wn}. We shall

consider a mapping ϕ : Z → R+, where

ϕ(zk) = 1− n!− |S(Tk)|n!− 1

, zk ∈ Z. (5)

The numerator n!−|S(Tk)| (denominator n!−1) of the fraction in (5) is equalto the number (to the maximal number) of permutations of the set S which arenot necessary to treat for searching an optimal permutation within a minimaldominant set S(Tk) (within the set S(T ) of minimal cardinality: |S(T )| = 1).

The value ϕ(zk) may be defined as a measure for the instance zk ∈ Z,if the mapping ϕ : Z → R+ is an additive non-negative function. It is clearthat inequality ϕ(zk) ≥ 0 holds for any instance zk ∈ Z. The additivity of themapping ϕ : Z → R+ is determined by setting that the measure of uncertaintyof the sequence X = (z1, z2, . . . , zk, . . .) ⊂ Z is equal to

ϕ(X ) =∞∑

i=1

ϕ(zi) (6)

provided that there are no jobs with identical data in any two instances zg ∈ Xand zh ∈ X . For a finite set X = {z1, z2, . . . , zk}, we define

ϕ(X) =k∑

i=1

ϕ(zi). (7)

For the extreme cases (characterized by Theorem 4 for the hardest case,|S(T )| = n!, and by Theorem 3 for the easiest case, |S(T )| = 1), the mappingϕ : Z → R+ is clearly unique. Next, we show how to provide the uniquenessof the mapping ϕ : Z → R+ in the general case.


5.1 A criterion for the uniqueness of a minimal dominant set

A mapping ϕ : Z → R+ is unique if the minimal dominant set S(T ) for aninstance 1|pL

i ≤ pi ≤ pUi |

∑wiCi is unique.

Theorem 5 For the instance 1|pLi ≤ pi ≤ pU

i |∑

wiCi, a minimal dominantset S(T ) is unique if and only if there is no r ∈ [a, b] with |Jr| ≥ 2.

Thus, the set S(T ) is unique if and only if there are no jobs in the set J withboth fixed processing times and equal weight-to-process ratios. The next claimfollows from Lemma 2 and the proof of Theorem 5 given in the Appendix.

Corollary 2 For any permutation πk ∈ S(T ), there exists a scenario p ∈ Tsuch that πk is the unique optimal permutation for the instance 1|p|∑wiCi ifand only if there is no r ∈ [a, b] with |Jr| ≥ 2.

Due to Theorem 5, if there is no r ∈ [a, b] with |Jr| ≥ 2), then the valueϕ(zk) may be defined as an uncertainty measure of the instance zk ∈ Z.Otherwise, a minimal dominant set is not uniquely determined. Next, we showhow to overcome this drawback.

5.2 How to provide the uniqueness of a minimal dominant set?

We propose a modification of the mapping ϕ : Z → R+ which is a func-tion on the whole set Z and show that the size n of the instance 1|pL

i ≤pi ≤ pU

i |∑

wiCi may be reduced by |Jrq | − 1 for each non-singleton Jrq . Dueto Theorem 1, in any optimal permutation πl ∈ S existing for the instance1|p|∑ wiCi, all the jobs of the set Jrq ⊆ J must be adjacently located oneby one: πl = (. . . , π(Jrq ), . . .), where π(Jrq ) denotes a permutation of thejobs Jrq . Moreover, the order of the jobs {Jq(1), Jq(2), . . . , Jq(|Jrq |)} = Jrq inthe permutation π(Jrq ) does not influence the value of the objective func-tion γ =

∑Ji∈J wiCi calculated for any permutation πk ∈ S of the form

πk = (. . . , π(Jrq ), . . .) (since the processing time of a job Jq(v) ∈ Jrq is fixedand the weight-to-process ratios are the same for all jobs in the set Jrq ). Thus,while looking for an optimal permutation for any instance 1|p|∑wiCi gener-ated by the instance 1|pL

i ≤ pi ≤ pUi |

∑wiCi via fixing a scenario p ∈ T , one

can treat all the jobs {Jq(1), Jq(2), . . . , Jq(|Jrq |)} = Jrq as one job with theweights and processing times equal to those of any job of set Jrq . By choos-ing only one job from each of such sets Jrq , rq ∈ {r1, r2, . . . , rm}, |Jrq | ≥ 2,the original instance 1|pL

i ≤ pi ≤ pUi |

∑wiCi can be transformed into an

equivalent instance (we denote this instance as 1∗|pLi ≤ pi ≤ pU

i |∑

wiCi)with a smaller cardinality of the set of jobs J ∗ to be scheduled: |J ∗| =|J | − ∑m

q=1(|Jrq | − 1) = n + m − ∑mq=1 |Jrq |. Let 1∗|p| ∑

wiCi denote thedeterministic instance generated by the uncertain instance 1∗|pL

i ≤ pi ≤ pUi |∑

wiCi via fixing a scenario p ∈ T . An instance 1∗|pLi ≤ pi ≤ pU

i |∑

wiCi

is equivalent to the original instance 1|pLi ≤ pi ≤ pU

i |∑

wiCi in the sense

OR Spectrum 9

that for any fixed scenario p ∈ T , an optimal permutation πk for the instance1∗|p|∑ wiCi may be obtained from the corresponding optimal permutation πt

for the instance 1|p|∑ wiCi of the original uncertain instance via deleting alljobs of the set J \J ∗ from the permutation πt. Along with a smaller size, theequivalent instance 1∗|pL

i ≤ pi ≤ pUi |

∑wiCi has a unique minimal dominant

set S(T ) (Theorem 5). Consequently, the set S(T ) constructed for the instance1∗|pL

i ≤ pi ≤ pUi |

∑wiCi is a minimal dominant set with respect to both inclu-

sion and cardinality. Thus, in order to measure the uncertainty of the problem1|pL

i ≤ pi ≤ pUi |

∑wiCi, we propose to use the following modification of the

mapping ϕ : Z → R+:

ϕ∗(zk) = 1− n∗!− |S(Tk)|n!− 1

, zk ∈ Z. (8)

5.3 Calculation of the uncertainty measures

Since the cardinality of a minimal dominant set could range from 1 (Theorem3) to n! (Theorem 4), it is impossible to generate all the elements of the setS(T ) in polynomial time of n. On the other hand, due to Theorem 2, one canobtain a compact presentation of a minimal dominant set S(T ) for a problem1|pL

i ≤ pi ≤ pUi |

∑wiCi in the form of a dominance digraph G = (J ,A) with

the vertices J and the arcs A. To this end, one can check condition (2) foreach pair of jobs Ju ∈ J and Jv ∈ J and construct a digraph G = (J ,A) asfollows: The arc (Ju, Jv) belongs to the set A if and only if job Ju dominatesjob Jv. It takes O(n2) time to construct the digraph G = (J ,A).

The digraph G = (J ,A) is uniquely determined for any instance zk ∈ Z(Theorem 2). Therefore, the mapping µ : Z → R+

µ(zk) = 1− 2|A|n(n− 1)

, zk ∈ Z, (9)

is a function. In the fraction |A|(n(n−1)

2

) , the numerator is equal to the cardinality

|A| of the setA and the denominator(

n(n−1)2

)is equal to the maximal possible

number of arcs in the circuit-free digraph of order n. The additivity of thefunction µ(zk) may be settled similarly as that of the mapping ϕ : Z → R+

(see (6) and (7)). Thus, the value µ(zk) may be defined as an uncertaintymeasure for any instance zk ∈ Z.

The instance zk ∈ Z has the largest value µ(zk) = 1 = ϕ(zk) if |S(Tk)| = n!.The set of arcs A is empty for the instance zk.

A zero value ϕ(zl) = 0, zl ∈ Z, corresponds to a zero value µ(zl) = 0. Theequality |S(Tl)| = 1 holds for the instance zl.

6 Illustrative examples

One can check the conditions of Theorems 1–5 in polynomial time of n whichis important while solving a large scheduling problem. If condition (3) holds,


Table 1 Data for Example 1 and Example 2

i pLi pU

i wiwi

pLi

wi

pUi

Examples 1 and 2 1 5 6 300 60 502 4 8 80 20 103 1 3 180 180 604 3 4 120 40 305 4 5 200 50 40

Example 2 6 10 20 80 8 47 20 25 100 5 4

it is possible to construct a permutation for such an instance 1|pLi ≤ pi ≤

pUi |

∑wiCi, which remains optimal for any scenario of T . We demonstrate

this simplest case of the problem 1|pLi ≤ pi ≤ pU

i |∑

wiCi using Example 1.

6.1 Example 1

Let the input data for Example 1 of problem 1|pLi ≤ pi ≤ pU

i |∑

wiCi withn = 5 = n1 be given in Table 1 (columns 2–5, lines 2–6). The weight-to-processratios for the processing times which are equal to the lower (upper) boundsare given in column 6 (column 7). We test condition (2):

w3

pU3

≥ w1

pL1

;w1

pU1

≥ w5

pL5

;w5

pU5

≥ w4

pL4

;w4

pU4

>w2

pL2

. (10)

Since conditions (2) are transitive, we do not present them, if they may beobtained via the transitivity from those already presented in (10).

Since condition (3) of Theorem 3 holds, the minimal dominant set S(T1)is a singleton: S(T1) = {πk}, where the scenario set T1 is defined in Table1 (columns 2–5, lines 2–6) and πk = (J3, J1, J5, J4, J2). The permutation πk

is obtained due to the WSPT rule used for the ordering of the jobs J withthe processing times pi = pL

i (or pi = pUi ) of a job Ji ∈ J (due to (3)

in both cases, the same permutation πk is obtained). The permutation πk

is optimal for any instance 1|p|∑ wiCi generated from Example 1 via fixinga scenario p ∈ T1. We calculate the uncertainty measure using formula (5):ϕ(z1) = 1− n!−|S(T1)|

n!−1 = 1− 5!−15!−1 = 0. Here the instance z1 denotes Example

1. A zero value of the uncertainty measure means that there is no influence ofthe uncertainty while one looks for a solution of such a problem.

6.2 Example 2

Example 2 is obtained from Example 1 via adding the jobs J6 and J7 (Table1). The number of jobs is n = 7 = n2. After checking the validity of condition(2) for all jobs Ju ∈ {J1, J2, . . . , J5} and jobs Jv ∈ {J6, J7}, we obtain theconditions wu

pUu

> w6pL6

and wu

pUu

> w7pL7

in addition to conditions (10). For Example2, condition (3) does not hold since job J6 does not dominate job J7 (Theorem

OR Spectrum 11

2) and vice versa. A minimal dominant set is as follows: S(T2) = {πh, πl} ={(J3, J1, J5, J4, J2, J6, J7), (J3, J1, J5, J4, J2, J7, J6)}. Due to Theorem 5, thisminimal dominant set is unique, and one can calculate the uncertainty measureusing formula (5): ϕ(z2) = 1 − n!−|S(T2)|

n!−1 = 1 − 7!−27!−1 = 1

7!−1 = 15039 , where

z2 denotes Example 2. Since the value ϕ(z2) is close to zero, we propose touse a stability method for solving Example 2. The stability method allows thescheduler to select the permutation πh = (J3, J1, J5, J4, J2, J7, J6) from the setS(T2) since this permutation has the larger volume of a stability box [19] incomparison with permutation πl = (J3, J1, J5, J4, J2, J6, J7) ∈ S(T2).

6.3 Example 3

We illustrate Theorem 4 by Example 3 with the input data given in Table 2(columns 2–5, lines 2–6). The number of jobs is n = 5 = n3. Condition (4)holds: max

{wi

pUi

| Ji ∈ J}

= 50 < 60 = min{

wi

pLi

| Ji ∈ J}

. Due to Theorem4, the minimal dominant set has the maximal cardinality with n = 5: |S(T3)| =5! = 120. Thus, the set S(T3) coincides with the set S of all permutations of thejobs J = {J1, J2, . . . , J5}: S(T3) = S. Due to Corollary 1, for any permutationπk ∈ S, there exists a scenario p ∈ T3 such that πk is the unique optimalpermutation for the instance 1|p|∑ wiCi generated from Example 3 via fixinga scenario p ∈ T3. Since the minimal dominant set S(T3) = S is unique,one can calculate the uncertainty measure for Example 3 using formula (5):ϕ(z3) = 1 − n!−|S(T3)|

n!−1 = 1 − 5!−1205!−1 = 1, where z3 denotes Example 3. Since

the value ϕ(z3) is the largest possible one with n = 5, we propose to use arobust method for solving Example 3.

6.4 Example 4

Example 4 is obtained from Example 3 via adding the job J6 (Table 2). So,n = 6 = n4. We test condition (2) for each pair of jobs from the set J ={J1, J2, . . . , J6} and obtain only one valid condition:

w1

pU1

= 50 ≥ 50 =w6

pL6

. (11)

The cardinality of a minimal dominant set for Example 4 is as follows:|S(T4)| = 6!

2 = 7202 = 360. Due to Theorem 5, the minimal dominant set S(T4)

for Example 4 is unique, and one can calculate its uncertainty measure usingformula (5): ϕ(z4) = 1 − n!−|S(T4)|

n!−1 = 1 − 6!−3606!−1 = 359

719 , where z4 denotesExample 4. Since the value ϕ(z4) is close to 0.5, we propose to use either afuzzy method or a stochastic method (depending on whether fuzzy or randomprocessing times are known for Example 4). In the latter case, one can assumethat the random processing times have uniform distributions (if there is noother information). The expected values E(pi) of the random processing times


Table 2 Data for Examples 3, 4, 5 and Example 5∗ with J ∗ = {J1, J2, . . . , J7}i pL

i pUi wi

wi

pLi

wi

pUi

E(pi)wi

E(pi)

Examples 3, 4, 5 and 5∗ 1 5 6 300 60 50 5.5 54 611

2 4 6 240 60 40 5 483 6 14 420 70 30 10 424 2 7 140 70 20 4.5 31 1

95 10 35 700 70 20 22.5 31 1

9

Examples 4, 5 and 5∗ 6 5 10 250 50 25 7.5 33 13

Examples 5 and 5∗ 7 5 5 200 40 40Example 5 8 10 10 400 40 40

9 1 1 40 40 40

Ji ∈ J = {J1, J2, . . . , J6} are given in column 8 of Table 2, and the valueswi

E(pi)are given in column 9. Using the WSEPT rule, we obtain the following

permutation: πh = (J1, J2, J3, J6, J4, J5).

6.5 Example 5

Example 5 is obtained from Example 4 via adding the jobs J7, J8 and J9 (Table2). So, we have n = 9 = n5 for Example 5. In contrast to Examples 1–4, foreach of which a minimal dominant set is unique, there are several minimaldominant sets for Example 5. This follows from Theorem 5: There exists thenumber r = 40 ∈ [a, b] = [20, 70] such that |J40| = |{J7, J8, J9}| = 3 > 2.

As it has been shown in Subsection 5.2, we have to consider an instance1∗|pL

i ≤ pi ≤ pUi |

∑wiCi (we call it Example 5∗) instead of the Example

5. In Example 5∗, the set of jobs J40 is substituted by one job from theset J40. Let the job J7 be chosen for such a substitution. Thus, the set J ∗ ={J1, J2, . . . , J7} of jobs has to be scheduled in Example 5∗. Along with a smallersize (|J ∗| = n∗ = 7 < 9 = n = |J |), the equivalent Example 5∗ has a uniqueminimal dominant set (Theorem 5). We test condition (2) for the pairs ofjobs from the set J ∗ = {J1, J2, . . . , J7}. Since only condition (11) holds, thecardinality of a minimal dominant set for Example 5∗ is equal to |S(T5)| =7!2 = 5040

2 = 2520. To measure the uncertainty of Example 5, one can use(8) instead of (5): ϕ∗(z5) = 1 − n∗!−|S(T5)|

n!−1 = 1 − 7!−25209!−1 = 360359

362879 , wherethe instance z5 denotes Example 5. Since the value ϕ(z5) is close to one, wepropose to use a robust method for solving Example 5.

6.6 Example 6

Let the input data for Example 6 be given in columns 1–4 of Table 3. Thenumber of jobs is equal to 10 = n6. For each pair of jobs Ju ∈ J and Jv ∈ J ,we check the validity of condition (2) as follows: w1

pU1≥ w2

pL2; w2

pU2≥ w3

pL3; w2

pU2

>w4pL4; w3

pU3≥ w5

pL5; w4

pU4

> w5pL5; w5

pU5≥ w6

pL6; w6

pU6≥ w7

pL7; w6

pU6≥ w8

pL8; w6

pU6

> w9pL9; w7

pU7

>

OR Spectrum 13

Table 3 Data for Example 6

i pLi pU

i wiwi

pLi

wi

pUi

E(pi)wi

E(pi)

1 1 3 180 180 60 – –2 5 6 300 60 50 – –3 4 10 200 50 20 – –4 3 5 150 50 30 – –5 4 8 80 20 10 – –6 10 20 100 10 5 – –

7 20 26 100 5 3 1113

23 4 823

8 10 15 50 5 3 23

12.5 4

9 5 6 20 4 3 13

5.5 36 411

10 10 20 30 3 1.5 15 2

mJ1- mJ2

mJ4

HHHj

mJ3

©©©*

mJ5

HHHj

©©©*mJ6

-

mJ8

HHHHj

mJ7

©©©©*

mJ9

@@

@@@R

mJ10

HHHj

©©©*

¡¡

¡¡¡µ

Fig. 1 Digraph G0 = (J ,A0) constructed for Example 6.

w10pL10

; w8pU8

> w10pL10

; w9pU9

> w10pL10

. Due to Theorem 2, the following dominance re-lations hold. Job J1 dominates job J2. Job J2 dominates jobs J3 and J4. JobJ3 dominates job J5. Job J4 dominates job J5. Job J5 dominates job J6. JobJ6 dominates jobs J7, J8 and J9. Job J7 dominates job J10. Job J8 dominatesjob J10. Job J9 dominates job J10. No job from the set {J3, J4} dominatesanother one from this set. No job from the set {J7, J8, J9} dominates anotherone from this set. Thus, a minimal dominant set S(T6) for Example 6 is definedby the digraph G = (J ,A). In Fig. 1, the reduction digraph G0 = (J ,A0)of the digraph G = (J ,A) is presented. (The digraph G0 is obtained fromdigraph G via deleting the transitive arcs.) Example 6 may be decomposedinto two subproblems. In the first subproblem, the set of jobs {J1, J2, . . . , J6}has to be scheduled (we call this subproblem as instance z7). In the secondsubproblem, the set of jobs {J7, J8, J9, J10} has to be scheduled (we call thissubproblem as instance z8). We have n7 = 6, n8 = 4 and calculate the valuesϕ(z7), ϕ(z8) using formula (5): ϕ(z7) = 1 − 6!−|S(T7)|

6!−1 = 1 − 720−2720−1 = 1

719 ,

ϕ(z8) = 1 − 4!−|S(T8)|4!−1 = 1 − 24−6

24−1 = 523 . Due to the above values of the

uncertainty measure, we propose to use a stability method for solving theinstance z7, and a stochastic method (or a fuzzy method) for solving theinstance z8 (depending on whether fuzzy or random processing times areknown). For the instance z7, the stability method allows the scheduler toselect the permutation (J1, J2, J4, J3, J5, J6) having the larger volume of astability box [19] in comparison with permutation (J1, J2, J3, J4, J5, J6). Let


Table 4 The values of the uncertainty measures µ(zk), ϕ(zk) and ϕ∗(zk)

zk ϕ(zk) ϕ∗(zk) µ(zk)z1 0 – 0z2

15039

– 121

z3 1 – 1z4

359719

– 1415

z5 – 360359362879

3536

z71

719– 1

15z8

523

– 0.5

a stochastic method be used for the instance z8. One can assume that therandom processing times have uniform distributions. The expected valuesE(pi) of the random processing times Ji ∈ {J7, J8, J9, J10} are given in col-umn 7 of Table 3, and the values wi

E(pi)are given in column 8. Using the

WSEPT rule, we obtain permutation: (J9, J7, J8, J10). The job permutation(J1, J2, J4, J3, J5, J6, J9, J7, J8, J10) for the whole instance z6 is defined via con-catenating the permutations (J1, J2, J4, J3, J5, J6) and (J9, J7, J8, J10). Exam-ple 6 demonstrates that an instance 1|pL

i ≤ pi ≤ pUi |

∑wiCi may need several

methods, if it has parts with different values of the uncertainty measure.

6.7 Calculation of the uncertainty measure µ(zk)

Since it is impossible to generate all elements of a set S(T ) in polynomial timeof n, the calculation of the measure ϕ(zk) or ϕ∗(zk) based on the cardinality|S(T )| may be time-consuming for a large n. So, the calculation of µ(zk) ispreferable in comparison with the calculation of ϕ(zk) or ϕ∗(zk). The measureµ(zk) is based on the cardinality of the set of arcs Ak in the dominance digraphGk = (J ,Ak) constructed for the instance zk. Using (9), we calculate thevalues of the measure µ(zk) for the examples zk ∈ Z, k ∈ {1, 2, . . . , 8}, asfollows: µ(z1) = 1 − 2|A1|

n1(n1−1) = 1 − 2·105(5−1) = 0; µ(z2) = 1 − 2|A2|

n2(n2−1) =121 ; µ(z3) = 1 − 2|A3|

n3(n3−1) = 1; µ(z4) = 1 − 2|A4|n4(n4−1) = 14

15 ; µ(z5) = 1 −2|A5|

n5(n5−1) = 3536 ; µ(z7) = 1− 2|A7|

n7(n7−1) = 115 ; µ(z8) = 1− 2|A8|

n8(n8−1) = 0.5.

The values of the uncertainty measures µ(zk), ϕ(zk) and ϕ∗(zk) for theinstances zk, k ∈ {1, 2, 3, 4, 5, 7, 8}, are given in Table 4.

6.8 Calculation of the uncertainty measure for a sequence of instances

If one needs to solve a sequence (or a set) of instances, then formula (6) (or(7)) has to be used for calculating the uncertainty measure of a sequence (aset) of instances instead of formula (5) used to measure the uncertainty ofa single instance. For example, for the set X = {z1, z3, z4, z7}, we obtainϕ(X) = 0 + 1 + 369

719 + 1719 = 1 360

719 . Using a formula analogous to (6) for theuncertainty measure µ(zk), we obtain µ(X) = 0 + 1 + 14

15 + 115 = 2. The

OR Spectrum 15

Table 5 Set W of six O(n log n)-algorithms for the problem 1|pLi ≤ pi ≤ pU

i |∑

wiCi

Notation Heuristic rules underlying the algorithm

L WSPT rule for 1|pL|∑

wiCi with pL = (pL1 , pL

2 , . . . , pLn) ∈ T

U WSPT rule for 1|pU |∑

wiCi with pU = (pU1 , pU

2 , . . . , pUn ) ∈ T

E WSEPT rule with uniform probability distribution of each pi ∈ [pLi , pU

i ]SL Constructing the set of permutations S∗ with the largest stability box;

if |S∗| > 1, then choosing permutation πb ∈ S∗ via Algorithm LSU Constructing the set of permutations S∗ with the largest stability box;

if |S∗| > 1, then choosing permutation πb ∈ S∗ via Algorithm USE Constructing the set of permutations S∗ with the largest stability box;

if |S∗| > 1, then choosing permutation πb ∈ S∗ via Algorithm E

knowledge of the uncertainty measure of a sequence or a set of instances is ofparticular importance for stochastic and fuzzy methods. For a single instancezk ∈ Z, we have the inclusions ϕ(zi) ∈ [0, 1] and µ(zk) ∈ [0, 1]. For a finite setX of instances with |X| ≥ 2, we have ϕ(X) ∈ [0, |X|] and µ(X) ∈ [0, |X|]. Fora sequence X of instances, we have ϕ(X ) ∈ [0,∞) and µ(X ) ∈ [0,∞).

7 Computational results

For the tests, we have chosen the set W of six algorithms each of whichhas complexity O(n log n) (see Table 5). Algorithm L (Algorithm U) is basedon the lower-point (upper-point) heuristic for the uncertain problem 1|pL

i ≤pi ≤ pU

i |∑

wiCi, namely: Algorithm L (Algorithm U) uses the WSPT rule(Subsection 3.4) for generating an optimal permutation πl ∈ S (permuta-tion πu ∈ S) for the deterministic instance 1|pL|∑wiCi (1|pU |∑wiCi) withpL = (pL

1 , pL2 , . . . , pL

n) ∈ T (with pU = (pU1 , pU

2 , . . . , pUn ) ∈ T , respectively). The

stochastic Algorithm E realizes the WSEPT rule (Subsection 3.2) with the as-sumption of a uniform probability distribution for the random processing timepi within the segment [pL

i , pUi ]. The stability method constructs a set S∗ ⊆ S

of permutations with the largest relative volume of a stability box [19]. Sincethere might be several permutations with the largest volume of a stabilitybox, i.e., |S∗| > 1, the stability method was combined with the lower-pointheuristic (this combination of two heuristics is called Algorithm SL), with theupper-point heuristic (this combination is called Algorithm SU) and with astochastic algorithm (this combination is called Algorithm SE).

Algorithms L, U, E, SL, SU and SE were coded in C++ and tested on aPC with AMD Athlon (tm) 64 Processor 3200+, 2.00 GHz, 1.96 GB of RAM.

Table 6 represents the computational results for series of randomly gener-ated instances 1|pL

i ≤ pi ≤ pUi |

∑wiCi with n ∈ {100, 300, 500, 700, 1000, 2000,

3000, 4000, 5000}. Each series contains 10 randomly generated instances withthe same number n of jobs and the same maximal possible error δ% of therandom processing times. The number n is given in parenthesis before thecorresponding part of Table 6. The integer lower and upper bounds pL

i and pUi

on the processing times pi ∈ [pLi , pU

i ] have been generated as follows.


Table 6 Computational results for randomly generated instances 1|pLi ≤ pi ≤ pU

i |∑

wiCi

(n) Average Methods Exact Average Maximal CPUMaximal measure with the best solu- error error time

δ% µ(zi) results tions ∆ ∆ in s(n = 100)

0.1% 0 L,U,E,SL,SU,SE 10 0 0 ≈ 00.2% 0.00153535 SL,SU,SE (E) 3 0.000002 0.000006 ≈ 00.3% 0.00220202 SL,SU,SE (E) 2 0.000002 0.000005 ≈ 00.5% 0.00478788 L (U,E,SU,SE) 1 0.000005 0.00001 0.10.7% 0.00668687 E,SL,SU,SE (L) 0 0.000011 0.000018 ≈ 0

1% 0.01012121 E,SE (SU) 0 0.000019 0.000047 ≈ 05% 0.05042424 SE 0 0.000384 0.000558 0.1

10% 0.1019596 (L,E,SE) 0 0.001895 0.002751 ≈ 020% 0.20391919 (L,E,SE) 0 0.006626 0.009321 ≈ 030% 0.30424242 SU (U) 0 0.015807 0.020013 ≈ 040% 0.39694949 (E,SL) 0 0.02853 0.034695 ≈ 050% 0.49450505 (SL,SU) 0 0.048033 0.063097 ≈ 060% 0.60012121 (SL,SU) 0 0.072783 0.091943 0.170% 0.68371717 (E,SU) 0 0.098158 0.119246 ≈ 080% 0.78563636 SE (SL) 0 0.119808 0.143733 ≈ 090% 0.8910101 SL 0 0.168114 0.21722 ≈ 0

100% 0.99822222 U,E 0 0.223485 0.289013 ≈ 0(n = 300)

0.1% 0.00000669 L,SL,SU,SE (U,E) 9 ≈ 0 ≈ 0 0.30.2% 0.00151394 E (SL,SU,SE) 0 0.000001 0.000002 0.40.3% 0.00247938 E,SL,SU,SE (U) 0 0.000003 0.000004 0.30.5% 0.0046845 L,E,SL (SE) 0 0.000007 0.000009 0.30.7% 0.0066845 E,SL,SU,SE (L,U) 0 0.000012 0.000015 0.4

1% 0.00968785 U (E,SL,SU,SE) 0 0.00002 0.000023 0.35% 0.04991973 SE 0 0.000473 0.000519 0.4

10% 0.10339353 (U,E,SU,SE) 0 0.0019 0.002287 0.420% 0.19759197 (E,SL) 0 0.00706 0.008042 0.330% 0.31923077 U 0 0.017417 0.020331 0.440% 0.40127313 (SU,SE) 0 0.029662 0.036427 0.450% 0.48740691 (U,SL) 0 0.048282 0.057296 0.360% 0.59911037 (SU,SE) 0 0.071866 0.085643 0.470% 0.69057302 (E,SE) 0 0.10173 0.127591 0.480% 0.7919621 (L,SL) 0 0.138258 0.162324 0.490% 0.89793311 (L,SU) 0 0.175432 0.1996 0.4

100% 0.99797324 U,E 0 0.215273 0.270714 0.2(n = 500)

0.1% 0 L,U,E,SL,SU,SE 10 0 0 1.60.2% 0.00163447 E,SE (SL,SU) 0 0.000002 0.000002 1.60.3% 0.00252986 E (SL,SU,SE) 0 0.000003 0.000003 1.60.5% 0.00467174 E,SE (L,U,SU) 0 0.000006 0.000008 1.60.7% 0.00675912 E,SE (U,SU) 0 0.000011 0.000012 1.6

1% 0.01009218 L,SL (SE) 0 0.000021 0.000025 1.55% 0.05115992 (U,E,SE) 0 0.000504 0.000594 1.6

10% 0.10262124 (U,SE) 0 0.001896 0.002235 1.620% 0.20945812 (L,E) 0 0.007964 0.009152 1.730% 0.31091222 E 0 0.017104 0.020882 1.640% 0.41385571 L 0 0.031959 0.034964 1.850% 0.49991022 (U,E) 0 0.049547 0.055293 1.760% 0.60406974 SE 0 0.072576 0.081611 1.870% 0.70436152 SL 0 0.100393 0.109795 1.980% 0.78992946 (U,SU) 0 0.138255 0.147368 1.890% 0.89408257 (SU,SE) 0 0.179828 0.198332 0.7

100% 0.99810501 (U,E,SU) 0 0.239236 0.269457 0.7

OR Spectrum 17

Table 6 (continued)


δ% µ(zi) results tions ∆ ∆ in s(n = 700)

0.1% 0.00000082 L,E,SL,SU,SE (U) 10 0 0 4.20.2% 0.00161496 E,SE (L,U,SL,SU) 0 0.000002 0.000002 4.40.3% 0.00245371 SE (L,E,SL,SU) 0 0.000003 0.000003 4.40.5% 0.00470713 E,SE (SL) 0 0.000006 0.000008 4.40.7% 0.00674596 SE (E,SU) 0 0.00001 0.000012 4.4

1% 0.01007889 E,SE 0 0.000021 0.000024 4.45% 0.05198896 (U,E,SU,SE) 0 0.000485 0.000517 4.5

10% 0.10569916 (U,E,SU,SE) 0 0.001958 0.002068 4.520% 0.20332434 (SL,SE) 0 0.007599 0.008466 4.530% 0.30600041 (E,SE) 0 0.016981 0.019418 4.740% 0.40656203 L 0 0.031444 0.034532 4.750% 0.49323932 L 0 0.047403 0.052001 5.360% 0.59994605 (E,SE) 0 0.072108 0.078825 5.170% 0.70093399 (U,SU) 0 0.10145 0.110735 5.380% 0.79877948 (U,E) 0 0.140561 0.156577 4.990% 0.89469937 (E,SU) 0 0.178536 0.193659 4

100% 0.99735745 U,E 0 0.229305 0.241953 5.5(n = 1000)

0.1% 0.0000016 SL,SU,SE (L,U,E) 10 0 0 12.90.2% 0.00155035 SE (L,U,E,SL,SU) 0 0.000002 0.000002 13.70.3% 0.00248969 SE (U,E,SL,SU) 0 0.000003 0.000003 13.50.5% 0.00472693 E,SE (SL,SU) 0 0.000006 0.000007 13.50.7% 0.00695716 (E,SU,SE) 0 0.000011 0.000013 13.7

1% 0.01011672 E,SE 0 0.000021 0.000023 13.45% 0.05161121 (E,SL,SE) 0 0.000472 0.000528 13.4

10% 0.10269069 (L,E,SE) 0 0.001861 0.001989 13.320% 0.20588408 SE 0 0.007654 0.008093 13.830% 0.3074981 SE 0 0.017078 0.018493 13.940% 0.40773894 (SU,SE) 0 0.030148 0.032429 14.450% 0.50351592 SU 0 0.04974 0.056701 13.960% 0.59407508 (E,SL) 0 0.071938 0.077806 14.870% 0.70174695 (U,SU) 0 0.099364 0.104965 15.180% 0.796204 (U,E) 0 0.132693 0.144993 14.190% 0.89633453 (L,SL) 0 0.176485 0.202118 10.3

100% 0.99826366 (E,SU,SE) 0 0.222582 0.243893 0.7(n=2000)

0.1% 0.0000037 U (L,E,SL,SU,SE) 6 ≈ 0 ≈ 0 111.80.2% 0.00164992 E (SE) 0 0.000004 0.000005 116.80.3% 0.00254732 E (SE) 0 0.000007 0.000008 1180.5% 0.0047036 E (SE) 0 0.000017 0.000019 117.70.7% 0.00687439 E,SE 0 0.00003 0.000033 117.3

1% 0.01007814 E,SE 0 0.000058 0.000066 116.45% 0.05151766 E,SE 0 0.001277 0.001341 117.2

10% 0.10272756 E,SE 0 0.005371 0.005739 116.420% 0.20334027 E,SE 0 0.021531 0.022791 115.730% 0.30503497 (L,E,SE) 0 0.049827 0.054749 11640% 0.40568724 (U,E,SE) 0 0.089643 0.095126 115.950% 0.50140565 SL 0 0.147399 0.160469 112.260% 0.59558989 (L,SL) 0 0.231728 0.248719 114.970% 0.69086248 (L,SL) 0 0.343252 0.38621 109.880% 0.7934938 (SL,SU) 0 0.489548 0.551081 10590% 0.89744117 (SL,SU) 0 0.748658 0.868509 69.4

100% 0.99799405 (U,E,SE) 0 1.176711 1.515202 4.5


Table 6 (continued)


δ% µ(zi) results tions ∆ ∆ in s(n=3000)

0.1% 0.00000247 E (U,SL,SU,SE) 3 ≈ 0 ≈ 0 389.50.2% 0.00160816 E,SE 0 0.000011 0.000013 407.90.3% 0.00255936 E,SE 0 0.000017 0.000022 407.60.5% 0.00468716 E,SE 0 0.000041 0.000056 410.90.7% 0.00681705 E,SE 0 0.000075 0.000088 406.9

1% 0.01004475 E,SE 0 0.000146 0.000179 409.55% 0.05158631 E,SE 0 0.003067 0.003397 407.2

10% 0.10357401 E,SE 0 0.012403 0.013468 407.520% 0.20410181 (U,SU) 0 0.0569 0.066686 404.830% 0.30496243 (E,SU,SE) 0 0.133455 0.170783 399.240% 0.40507643 E,SE 0 0.241445 0.2832 405.550% 0.50361674 (U,SU) 0 0.436796 0.587684 389.460% 0.59769299 (U,E,SE) 0 0.817924 1.836476 398.670% 0.69568187 (U,SU) 0 2.116842 4.18651 388.880% 0.79250197 (L,SL) 0 3.022982 7.487985 35890% 0.89401658 (U,E,SE) 0 2.983179 10.779577 231.3

100% 0.99781916 SL 0 0.695643 0.781601 14.5(n=4000)

0.1% 0.0000016 U,E (L,SL,SU,SE) 2 ≈ 0 ≈ 0 952.80.2% 0.00160898 E (SE) 0 0.000034 0.000057 1000.70.3% 0.00252076 E,SE 0 0.000078 0.000253 1012.40.5% 0.00476307 E,SE 0 0.000225 0.001142 10070.7% 0.00684522 E,SE 0 0.000466 0.00241 1005.2

1% 0.01006218 E,SE 0 0.000633 0.001936 1002.85% 0.05161885 E,SE 0 0.012658 0.024245 989.7

10% 0.10340346 (E,SU,SE) 0 0.055193 0.091543 980.120% 0.20639744 (U,SU) 0 0.190061 0.400541 965.930% 0.30469691 (L,E,SE) 0 0.518807 1.077145 97740% 0.404109 (E,SU,SE) 0 1.1377 5.209055 956.850% 0.50131099 (E,SU,SE) 0 2.964854 10.325552 943.860% 0.59888098 (E,SL,SE) 0 0.482764 0.557009 939.370% 0.69457833 (E,SU,SE) 0 0.79346 0.955864 931.180% 0.7978888 (SL,SU) 0 1.444882 1.901383 854.790% 0.89782297 (L,U) 0 3.551268 6.287494 547.4

100% 0.99811072 (U,SU) 0 1.933561 4.743262 31.7(n=5000)

0.1% 0.00000201 E,SL,SU,SE (L,U) 1 ≈ 0 ≈ 0 1950.90.2% 0.00159402 E,SE 0 0.000021 0.000023 20350.3% 0.00253486 E (SE) 0 0.000044 0.000096 2045.10.5% 0.00475245 E (SE) 0 0.001461 0.013593 2022.20.7% 0.00688566 E (SE) 0 0.000299 0.001262 2018.6

1% 0.01016814 SE (E) 0 0.000695 0.00367 2038.95% 0.05147954 E,SE 0 0.005636 0.006811 2059.7

10% 0.10271562 (U,E,SE) 0 0.024547 0.030565 1968.520% 0.20644317 (E,SU,SE) 0 0.097151 0.130148 1892.130% 0.30734107 (L,U) 0 0.265321 0.386807 1924.340% 0.40436974 (U,SU) 0 0.647339 1.192235 1873.550% 0.50312959 (E,SL,SE) 0 1.294992 2.305626 1899.960% 0.59650079 (E,SU,SE) 0 1.761138 3.729997 1847.870% 0.69465952 E,SE 0 1.639203 6.923092 1827.780% 0.79293199 (E,SU,SE) 0 0.273271 0.362255 1633.390% 0.89629171 (E,SU,SE) 0 1.718265 2.642256 1012

100% 0.99804363 SL 0 3.707979 9.025701 60.1

OR Spectrum 19

An integer center C of a segment [pLi , pU

i ] was generated using the uniformdistribution in the range [L, U ]: L ≤ C ≤ U . The lower bound was defined aspL

i = C · (1− δ100 ) and the upper bound as pU

i = C · (1 + δ100 ). The maximum

possible error of the random processing time in percentages is equal to δ%given in column 1 of Table 6. The same range [L,U ] = [1, 500] for the varyingcenter C was used for all jobs Ji ∈ J in all instances presented in Table 6.For each job Ji ∈ J , the weight wi was uniformly distributed in the range[1, 50]. (In contrast to the processing times pi, the weights wi were assumed tobe known precisely before scheduling.) The computational experiments wereaimed to answer the following two questions. Which algorithms of the six onestested give the best objective function value for a series of instances? Howlarge may the error of the optimal value of the objective function be providedby such an algorithm for the concrete scheduling problem?

The average value of the measure µ(zk) for the 10 instances in a series isgiven in column 2 of Table 6. Columns 3 indicates one or several algorithmsproviding the best value of the objective function γ =

∑Ji∈J wiCi for a series

of instances. The best results obtained by such algorithms are characterizedin columns 4–6. Column 4 presents the number of instances (from 10 ones ina series) solved exactly by the best algorithm for this series. Column 5 (col-umn 6) presents the average (maximal) relative error ∆ obtained by the best

algorithm: ∆ =γt

p∗−γ∗p∗γ∗

p∗, where γt

p∗ denotes the value of the objective function

γ =∑

Ji∈J wiCi for the permutation πt constructed by the best algorithm.The value γ∗p∗ denotes the ex-post optimal value of the objective function, i.e.,that calculated for the instance 1|p∗|∑wiCi, where p∗ = (p∗1, p

∗2, . . . , p

∗n) ∈ T

is the actual scenario (unknown before scheduling).The best algorithm or algorithms (indicated in column 3) provide the best

results for all three characteristics presented in columns 4–6. If for a series,there is no algorithm with the best of all three characteristics, then the algo-rithms providing Pareto optimal characteristics are indicated in column 3 inparenthesis. For example, the line of Table 6 given for n = 100 and δ% = 70%contains (E, SU). This means that Algorithm E provides one minimal error∆ (either the average error or the maximal error) of the objective function,while Algorithm SU provides another minimal error ∆. To be precise, Al-gorithm E (Algorithm SU) gives the following characteristics: 0.09804 and0.11954 (0.098158 and 0.119246). In columns 5 and 6, the characteristics ofAlgorithm SU are presented, since their sum is smaller.

In column 3 in parenthesis, there are indicated the algorithms which givean objective function value worse than that obtained by the best algorithm by1 last digit of the one of the three characteristics given in columns 4–6. E.g.,the line of Table 6 given for n = 100 and δ% = 30% contains SU (U) whichmeans that Algorithm SU provides the minimal average error 0.015807 andthe minimal maximal error 0.020013 of the objective function, while AlgorithmU provides only one minimal error (the same one as Algorithm SU) whileanother error is larger than that provided by Algorithm SU by 1 last digit.To be precise, Algorithm U gives the following characteristics: 0.015808 and


Table 7 Summary of the computational results

Number of jobs n ∈ {100, 300, 500, 700, 1000}Range of the average Methods with Minimal Maximal Maximal

l values of measure µ(zi) the best results aver. error aver. error error ∆1 2 3 4 5

1 [0, 0.00252986] E,SL,SU,SE 0 0.000003 0.0000062 [0.00478788, 0.10569916] SE 0.000005 0.001958 0.0027513 [0.19759197, 0.20945812] (E ∨ SE) 0.006626 0.007964 0.0093214 [0.30424242, 0.31923077] (L ∨ SL ∨ SU) 0.015807 0.017417 0.0208825 [0.39694949, 0.41385571] (U ∨ E ∨ SE) 0.02853 0.031959 0.0364276 [0.48740691, 0.50351592] (L ∨ E ∨ SU) 0.047403 0.04974 0.0630977 [0.59407508, 0.60406974] (SL ∨ SE) 0.071866 0.072783 0.0919438 [0.68371717, 0.70436152] (SL ∨ SU ∨ SE) 0.098158 0.10173 0.1275919 [0.78563636, 0.79877948] (SL ∨ SU) 0.119808 0.140561 0.16232410 [0.8910101, 0.89793311] (U ∨ SU) 0.168114 0.179828 0.2172211 [0.99735745, 0.99826366] E 0.215273 0.239236 0.289013

Number of jobs n ∈ {2000, 3000, 4000, 5000}Range of the average Methods with Minimal Maximal Maximal

l values of measure µ(zi) the best results aver. error aver. error error ∆12 [0.0000016, 0.0000037] U,E,SL,SU,SE ≈ 0 ≈ 0 ≈ 013 [0.00159402, 0.10357401] E,SE 0.000004 0.055193 0.09154314 [0.20334027, 0.10569916] (SU ∨ SE) 0.021531 0.190061 0.40054115 [0.30496243, 0.30734107] (L ∨ SE) 0.049827 0.518807 1.07714516 [0.404109, 0.40568724] (SU ∨ SE) 0.089643 1.1377 5.20905517 [0.50131099, 0.50361674] (SL ∨ SU) 0.147399 2.964854 10.32555218 [0.59558989, 0.59888098] (SL ∨ SE) 0.231728 1.761138 3.72999719 [0.69086248, 0.69568187] (SL ∨ SU ∨ SE) 0.343252 2.116842 6.92309220 [0.79250197, 0.7978888] (SL ∨ SU) 0.273271 3.022982 7.48798521 [0.89401658, 0.89782297] (U ∨ SU) 0.748658 2.983179 10.77957722 [0.99781916, 0.99811072] (U ∨ SL) 0.695643 3.707979 9.025701

0.020013. Column 7 presents the average CPU-time (in seconds) needed forsolving an instance zk ∈ Z. This time is spent for the calculation needed for allsix algorithms of the set W (it takes O(n log n) time) and for calculating theuncertainty measure µ(zk) of the instance of zk (O(n2) time). A summary ofthe computational results is given in Table 7, where column 1 for line l containsthe range of the average values of the uncertainty measures µ(zi) for all seriesof the set D of instances for which the computational results are summarizedin line l. Column 2 indicates the best algorithm (or algorithms) for each seriesof the set D. If there is no best algorithm for all series of the set D, then aset of several algorithms including at least one of the best algorithms for eachseries of the set D is indicated in parenthesis, these algorithms being joinedby the sign ∨. For example, column 2 of line l = 1 contains E, SL, SU, SE,which means that any of the Algorithms E, SL, SU and SE provides the bestcomputational results for the instances zi ∈ Z with µ(zi) ∈ [0, 0.00252986].Column 2 of line l = 3 contains (E ∨ SE), which means that either algorithmE or algorithm SE provides the best computational results for the instanceszi ∈ Z with µ(zi) ∈ [0.19759197, 0.20945812]. Column 3 (column 4) containsthe minimal (maximal) average error ∆ among all series of the set D. Column5 contains the maximal error ∆ for an instance from all series of the set D.

OR Spectrum 21

The experiments have shown that for a number of jobs n not greaterthan 1000, one can select the best algorithm for an instance zi with µ(zi) ∈[0, 0.00252986] (line l = 1) or µ(zi) ∈ [0.00478788, 0.10569916] (line l = 2) orµ(zi) ∈ [0.99735745, 0.99826366] (line l = 11). If µ(zi) ∈ [0.19759197, 0.89793311],one has to choose an algorithm among two alternatives (lines l ∈ {3, 7, 9, 10})or among three alternatives (lines l ∈ {4, 5, 6, 8}). The average and even themaximal errors ∆ of the objective function values provided by the best algo-rithm are sufficiently small for a practical use. Even for the instances zi withthe largest uncertainty measure µ(zi) ∈ [0.99735745, 0.99826366], AlgorithmE provided a schedule with an average value of ∆ belonging to the segment[0.215273, 0.239236] and with a maximal value of ∆ equal to 0.289013.

For n ∈ {2000, 3000, 4000, 5000}, one can select the best algorithm foran instance zi with µ(zi) ∈ [0, 0.00252986] (line l = 12) or with µ(zi) ∈[0.00478788, 0.10569916] (line l = 13). If µ(zi) ∈ [0.20334027, 0.99811072],one has to choose an algorithm among two alternatives (with one exception ofthree alternatives, line l = 19). We are forced to recognize that the measureµ(zi) is not sufficient for choosing one dominating algorithm (among the sixones of the set W) for some instances with µ(zi) ∈ [0.19759197, 0.89793311],if 100 ≤ n ≤ 1000, and with µ(zi) ∈ [0.20334027, 0.99811072], if 2000 ≤ n ≤5000. We can explain the above drawback as follows.

First, the set W of the six algorithms is clearly not complete (our restric-tion to include into the set W only algorithms with complexity O(n log n)was too strong). Second, there are some similarities in the algorithms of theset W. Third, for a large instance, it is useful to use different algorithms atdifferent decision points, since the measure µ(zi) may take different valuesfor different parts of the instance (see Example 6 in Subsection 6.6). Combi-nations of a stability algorithm with Algorithm L, U or E have been tested,however each of these combinations was used for the whole instance in spiteof different parts of the instance having different values of µ(zi). Nevertheless,as we convinced, in all our experiments Algorithm SL outperforms AlgorithmL, Algorithm SU outperforms Algorithm U, Algorithm SE outperforms Algo-rithm E (with only a few exceptions). Fourth, the measure µ(zi) is definedfor an exact solution of the problem 1|pL

i ≤ pi ≤ pUi |

∑wiCi (see Definition

1). Therefore, it allows us to distinguish the quality of the algorithms betterwhen testing algorithms, which allows a scheduler to find an ex-post optimalschedule for the instance 1|pL

i ≤ pi ≤ pUi |

∑wiCi. Unfortunately, in our ex-

periments, it was not possible to find an ex-post optimal schedule for mostinstances with µ(zi) > 0.005 and n ∈ {100, 300, 500}, and for most instanceswith µ(zi) > 0.001 and n ∈ {700, 1000, 2000, 3000, 4000, 5000}. This is impliedby the fact that the objective function

∑wiCi is very sensitive: If at least

one job has a wrong place in a permutation πi ∈ S with respect to a factualscenario, then permutation πi cannot be ex-post optimal (the error ∆ may belarger, if the wrong position of a job is closer to the beginning of permutationπi). It should also be noted that the maximal and even the average errors ∆of the objective function value provided by the best algorithm from the set Wmay be large for instances zi with an uncertainty measure µ(zi) ≥ 0.3. The


maximal errors ∆ and the average errors (among the instances of the seriesD) of the objective function values provided by the best algorithm are equalto 10.7795577 (line l = 21) and to 3.707979 (line l = 22), respectively. Thismeans that for such hard instances zi of the problem 1|pL

i ≤ pi ≤ pUi |

∑wiCi,

one has to test other algorithms, probably with a larger running time thanthe O(n log n)-algorithms of the set W. In our experiments, we did not test arobust algorithm and a fuzzy algorithm since they are time-consuming. How-ever, a time-consuming heuristic may provide a permutation of better qualitythan those provided by the O(n log n)-algorithms included into the set W.

8 Other scheduling problems with interval processing times

We show that measures similar to ϕ(zk) and µ(zk) may be used while solvingother uncertain scheduling problems α|pL

i ≤ pi ≤ pUi |γ.

8.1 Priority rules

Let the objective function γ = f(C1, C2, . . . , Cn) allow a scheduler to find anoptimal permutation πt ∈ S for the deterministic problem 1||γ using a priorityrule. A priority (a rational number) λ(Ji) is assigned to any job Ji ∈ J andthe jobs J are sorted due to their priorities (e.g., in non-increasing order ofλ(Ji)). In particular, the objective function γ =

∑Ji∈J wiCi of the problem

1||∑ wiCi considered in Sections 2–7 is minimized due to the WSPT rule,where the priorities of the jobs Ji ∈ J are defined as λ(Ji) = wi

pi(Theorem 1).

We refer to [24] for other deterministic problems 1||γ with non-decreasingobjective functions γ = f(C1, C2, . . . , Cn) each of which may be minimizedusing an appropriate priority rule. For the uncertain counterpart 1|pL

i ≤ pi ≤pU

i |γ of each of such problems 1||γ, the uncertainty measure ϕ(zk) (measureµ(zk)) may be introduced using formulas (5)–(7) (formula (9), respectively).These uncertainty measures may be used for choosing a suitable method forsolving an instance 1|pL

i ≤ pi ≤ pUi |γ and for estimating the quality of a

schedule which may be obtained using such a method.Furthermore, if there exists a condition which is both necessary and suffi-

cient for the optimality of a permutation πt ∈ S for the deterministic problem1||γ (like condition (1) of Theorem 1 for the problem 1||∑wiCi), then Theo-rems 2–5 and Corollaries 1 and 2 given in Sections 3–5 may be generalized tothe problem 1|pL

i ≤ pi ≤ pUi |γ with the objective function γ = f(C1, C2, . . . ,

Cn) which may be minimized using an appropriate priority rule.

8.2 Priority-generating objective functions

In real-life scheduling, not all permutations of the jobs to be scheduled may befeasible due to technological constraints, assembly restrictions, market require-ments, etc. In such a case, a deterministic problem α|prec|γ or an uncertain

OR Spectrum 23

problem α|prec, pLi ≤ pi ≤ pU

i |γ has to be solved, where prec means thatthe precedence constraints → are given on the set of jobs J a priori (beforescheduling). If the relation Ji → Jl is given, then job Ji ∈ J must be com-pleted before job Jl ∈ J starts. To handle a lot of scheduling problems withthe given precedence constraints, a priority λ(Ji) for individual jobs Ji ∈ Jhas to be generalized to a priority function Λ(π) for subsequences π of thejobs J (see Chapter 3 of the book [24] for a systematic exposition of thisapproach). Let πab = (π′abπ′′) ∈ S and πab = (π′baπ′′) ∈ S denote two per-mutations of the jobs J differing in the order of the subsequences a and b.One can present the objective function γ = f(C1, C2, . . . , Cn) for a problem1||γ as γ = f(C1, C2, . . . , Cn) = F (πk), πk ∈ S.

Definition 3 [24] An objective function γ = F (πk) is called priority-genera-ting if there exists a priority function Λ(π) such that for any two permutationsπab and πba, the inequality Λ(a) > Λ(b) implies F (πab) ≤ F (πba) and theequality Λ(a) = Λ(b) implies F (πab) = F (πba).

In [24], it is shown that a priority-generating function can be minimized inpolynomial time for some types of given precedence constraints →. In partic-ular, a priority-generating function γ = F (πk) can be minimized in O(n log n)time if the priority function Λ(π) can be calculated in O(n) time, the re-duction digraph of the precedence constraints → is series-parallel and itsdecomposition tree is given (see [24]). Thus, the measures ϕ(zk) and µ(zk)may be used to choose a suitable method for solving an uncertain prob-lem 1|prec, pL

i ≤ pi ≤ pUi |γ if the objective function γ = F (πk) is priority-

generating and the reduction digraph of the order → is series-parallel.For the problem 1|prec, pL

i ≤ pi ≤ pUi |γ, the corresponding digraph G =

(J ,A) has to contain the arc (Jv, Jr) only if this arc does not violate theprecedence constraint →, which is a priori given between the jobs Jv and Jr.

8.3 Two-machine minimum-length flow shop problem

The measures ϕ(zk) and µ(zk) may be used to choose a suitable method forsolving an uncertain two-machine minimum-length flow-shop problem F2|pL

i ≤pi ≤ pU

i |Cmax. The processing time pij of a job Ji ∈ J on machine Mj ∈{M1,M2} belongs to the given segment [pL

ij , pUij ]. The objective function is as

follows: γ = minπt∈S Cmax(πt) = minπt∈S{max{Ci(πt) | Ji ∈ J }}.For the deterministic counterpart (problem F2||Cmax), an optimal permu-

tation πt may be constructed in O(n log n) time [5] using the inequalities

min{ptr1, ptm2} ≤ min{ptm1, ptr2}, (12)

1 ≤ r < m ≤ n, which are sufficient for the optimality of a permutationπt = (Jt1 , Jt2 , . . . , Jtn) (such a permutation is called a Johnson one). Since theinequalities (12) are not necessary for the optimality of a permutation πt ∈ S,a J-solution was used in [13] for the problem F2|pL

i ≤ pi ≤ pUi |Cmax instead

of a minimal dominant set S(T ) (Definition 1).


Definition 4 [13,20] A set of permutations SJ(T ) is called a J-solution tothe uncertain problem F2|pL

i ≤ pi ≤ pUi |Cmax, if for any scenario p ∈ T ,

the set SJ(T ) contains at least one permutation that is a Johnson one forthe deterministic counterpart F2|p|Cmax associated with the scenario p, thisproperty being lost for any proper subset of the set SJ(T ).

In [12,13,20], analogues of Theorems 2–5 (Sections 3–5) have been provenfor the problem F2|pL

i ≤ pi ≤ pUi |Cmax using a J-solution SJ(T ) (Defini-

tion 4) instead of a minimal dominant set S(T ). Due to these results, theuncertainty measure ϕ(zk) (measure µ(zk)) may be introduced for the prob-lem F2|pL

i ≤ pi ≤ pUi |Cmax using formulas (5)–(7) (formula (9), respectively).

These measures may be used for choosing an appropriate method for solvingan instance F2|pL

i ≤ pi ≤ pUi |Cmax and estimating its effectiveness.

9 How to use an uncertainty measure in practice?

Because of real-life uncertainty, a schedule is usually not executed exactly as itwas planned (predicted). Therefore, in practice, predictive scheduling (off-linephase) is often supplemented by reactive scheduling (on-line phase), where asuitable rescheduling may be made while a schedule evolves [1,3,15]. Next,we show how one can use an uncertainty measure like the measure µ(zk) inpredictive-reactive scheduling in order to increase the probability to constructa schedule with an objective function value closer to an ex-post optimal one?

As our experiments show for the problem 1|pLi ≤ pi ≤ pU

i |∑

wiCi (Section7), if the value µ(zk) is rather close to 0 (namely, µ(zk) ∈ [0, 0.1975)), then thisuncertainty measure allows a scheduler to select an algorithm from the set Wwhich has a higher probability to generate an ex-post optimal schedule or toprovide an objective function value very close to an ex-post optimal one (lines1, 2, 12 and 13 in Table 7). Thus, if µ(zk) ∈ [0, 0.1975], a superior algorithmmay be chosen from the set W at the off-line phase and a predictive scheduleconstructed by the chosen algorithm will provide an objective function valueclose to an ex-post optimal one.

Otherwise (if µ(zk) ∈ [0.1975, 1]), a scheduler has to choose an appropri-ate algorithm among two or three alternatives within the set W (or anotheralgorithm outside the set W may be used). In such a more uncertain case, thedecision-making process may be continued at the on-line phase. Since the cal-culation of the value µ(zk) takes O(n2) time, this uncertainty measure may bequickly recalculated at one or several decision points during the on-line phase.Let a subset J1 of the job set J be already completed at a time t1 > t0 = 0,while the remaining jobs J \ J1 are waiting for processing. A scheduler cancalculate the value µ(zr) for the corresponding sub-problem zr of the originalproblem zk with the job set J \J1 and (s)he can find an interval in column 1of Table 7 including the calculated value µ(zr). The line of Table 7 with thisinterval indicates one or several algorithms in column 2, one of which may bechosen and used for ordering the jobs of the set J \ J1 which are waiting forprocessing. As Example 6 (Subsection 6.6) shows, the latter algorithm may

OR Spectrum 25

differ from that used at the off-line phase completed before time t0 = 0. Thenumber of decision points for such a decision-making at the on-line phase de-pends on the structure of the dominance digraph G = (J ,A) which may beconstructed in O(n2) time at the off-line phase.

Thus, an uncertainty measure like µ(zk) may be included into the frame-work of predictive-reactive scheduling [1,3]. Due to computational experiments(or a simulation), one can obtain intervals of an uncertainty measure, wherethe algorithms from the set W outperform the other ones tested (such com-putational results are summarized in Table 7). Furthermore, a threshold d ofthe uncertainty measure may be experimentally defined for the division of theset Z into a subset Z ′ of instances with relatively small values of µ(zk) (eachinstance zk ∈ Z ′ may be efficiently solved by a superior algorithm just at theoff-line phase) and into a subset Z\Z ′ of instances with higher values of µ(zk).

To solve an instance zk ∈ Z \Z ′, a scheduler has to realize both an off-linephase for constructing the dominance digraph for the problem 1|pL

i ≤ pi ≤pU

i |∑

wiCi and an on-line phase for selecting appropriate algorithms dynam-ically at suitable time points t1, t2, . . . when the value of the measure µ(zk)changes. Since the time for decision-making is restricted at the on-line phase,an appropriate algorithm has to be chosen just from the best alternatives (col-umn 2 of Table 7) at a decision point ti > t0 for ordering the correspondingsubset of non-dominant jobs in the digraph G = (J ,A). The threshold d ofthe uncertainty measure µ(zk) depends on the set W of algorithms tested (inSection 7, it was experimentally found that d = 0.1975 is appropriate).

In a similar way, an uncertainty measure like µ(zk) may be used at theoff-line phase and at the on-line phase of real-life scheduling if a more generalscheduling problem considered in Section 8 arises. Note that it takes polyno-mial time to calculate the uncertainty measure and to construct the dominancedigraph for each problem α|pL

i ≤ pi ≤ pUi |γ described in Section 8.

At the on-line phase, the dominance digraph G = (J ,A) may be used asa solution to an uncertain problem α|pL

i ≤ pi ≤ pUi |γ since it defines a min-

imal dominant set containing at least one optimal permutation of jobs J foreach scenario from the set T . So, the final schedule actually executed is cre-ated step-by-step via sequencing the jobs at the on-line phase while additionalinformation on the job processing times and their completion times becomesavailable to a scheduler. The final permutation πk = (Jk1 , Jk2 , . . . , Jkn) defin-ing the final schedule is a linear extension of the partial order A, i.e., inclusion(Jku , Jkv ) ∈ A implies inequality u < v.

10 Concluding remarks

In a competitive marketplace, an enterprize needs to use scheduling decisionsclose to an optimal one. However, there is no method which outperforms allother ones in an uncertain environment. Probably, it will be impossible toconstruct such a method in the future since there exists a great variety ofspecific particularities of the different instances even for the same uncertain


scheduling problem. In Section 3, we surveyed methods developed for uncertainscheduling problems. Each of these methods may generate a bad schedule fora factual scenario. So, it is important to find a mechanism for selecting anappropriate algorithm just for a concrete instance of an uncertain problem.

In Sections 5–8, we derived measures of problem uncertainty which arebased on the cardinality of a minimal dominant set of job permutations (mea-sure ϕ(zk)) and on the cardinality of the set of arcs in the dominance digraph(measure µ(zk)). The proposed measures may be calculated automaticallywhich allows a scheduler to incorporate a procedure for selecting a methodwithin a scheduling software.

The measure µ(zi) allows a scheduler to answer the following two questions.Which method may be used for searching a schedule which is close to an ex-post optimal one? How large may the error of the optimal value of the objectivefunction be provided by such a method for the concrete scheduling problem?The numerical tests for the problem 1|pL

i ≤ pi ≤ pUi |

∑wiCi showed that

measure µ(zi) is a good invariant providing a confident answer for the latterquestion. For the former question, the measure µ(zi) gave a clear answer if 0 ≤µ(zi) < 0.1975. For many more uncertain instances zi (if µ(zi) ∈ [0.1976, 1]),a scheduler has to choose the most suitable algorithm among two or even threealternatives from the set of six algorithms tested.

In further research, it will be useful to test the measure µ(zi) (or anotherone) for other objective functions, e.g., for the criterion Cmax, which is not sosensitive to a wrong position of a job in an optimal permutation as the criterion∑

wiCi. It would be also useful to investigate other factors influencing thequality of the algorithms applicable to uncertain scheduling problems.

Acknowledgements This research was supported by the National Science Council of Tai-wan (first and second authors) and by DAAD (first and third authors). The authors aregrateful to N.G. Egorova for coding the algorithms developed and to anonymous referees forvery helpful remarks and suggestions on an earlier draft of the paper.

Appendix

In the proof of Theorem 5, Lemmas 1–3 from [19] are used.

Lemma 1 [19] In any optimal permutation for the instance 1|p|∑ wiCi, jobJu ∈ J precedes job Jv ∈ J if and only if wu

pu> wv

pv.

Lemma 2 [19] For the instance 1|p|∑wiCi, an optimal permutation is uniqueif and only if wu

pu6= wv

pvfor any jobs Ju ∈ J and Jv ∈ J .

Lemma 3 [19] For the instance 1|p|∑ wiCi, there exist both an optimal per-mutation with job Ju ∈ J preceding job Jv ∈ J and one with job Jv precedingjob Ju if and only if wu

pu= wv

pv.

OR Spectrum 27

Proof of Theorem 5

For any pair of jobs Jt ∈ J and Jv ∈ J , we examine all arrangements of thesegments

[wt

pUt

, wt

pLt

]and

[wv

pUv

, wv

pLv

]. W.l.o.g. we assume wv

pUv≤ wt

pUt

. Due to thejob symmetry, it is sufficient to examine the following cases (a)–(i).

Case (a): wt

pUt

< wt

pLt, wv

pUv

< wv

pLv

, wv

pLv

< wt

pUt

. Thus, inequality wv

pv< wt

ptholds

for each scenario p ∈ T . Due to Lemma 1, in all optimal permutations for theinstance 1|p|∑wiCi with a scenario p ∈ T , job Jt precedes job Jv. Thus, ineach permutation of a minimal dominant set S(T ) job Jt precedes job Jv.

Case (b): wt

pUt

< wt

pLt, wv

pUv

< wv

pLv

, wv

pLv≤ wt

pUt

. If for the scenario p ∈ T at leastone of the conditions wv

pv6= wv

pLv

or wt

pt6= wt

pUt

holds, then arguing as in case (a),we obtain that in each permutation of any set S(T ), job Jt precedes job Jv. Forthe remaining scenario p′ = (p′1, p

′2, . . . , p

′n) ∈ T with wv

p′v= wv

pLv

and wt

p′t= wt

pUt

,we obtain wv

p′v= wt

p′t. Due to Lemma 3, for the instance 1|p′|∑ wiCi, there

exist both an optimal permutation of the form πl = (. . . , Jt, . . . , Jv, . . .) ∈ Sand one of the form πm = (. . . , Jv, . . . , Jt, . . .) ∈ S. However, no permutationof the form πm may be contained in a minimal dominant set, since such apermutation will be redundant.

Indeed, a permutation of the form πl will be definitely contained in any setS(T ) because of the scenario p ∈ T with p 6= p′. The permutation πl providesan optimal solution for the instance 1|p′|∑ wiCi. If the set S(T ) contains apermutation of the form πm, then the dominant set S(T ) is not minimal. Thus,in each permutation of any set S(T ), job Jt has to precede job Jv.

Case (c): wt

pUt

< wt

pLt, wv

pUv

< wv

pLv

, wv

pLv

> wt

pUt

. The length of the intersection

of the segments[

wt

pUt

, wt

pLt

]and

[wv

pUv

, wv

pLv

]must be strictly positive. There exist

a scenario p ∈ T with wt

pt> wv

pvand a scenario p′ ∈ T with wt

p′t< wv

p′v. Due

to Lemma 1, in all optimal permutations for the instance 1|p|∑wiCi, job Jt

precedes job Jv, and in all optimal permutations for the instance 1|p′|∑ wiCi,job Jv precedes job Jt. Thus, any minimal dominant set for the instance 1|pL

i ≤pi ≤ pU

i |∑

wiCi contains both a permutation of the form πl = (. . . , Jt, . . . ,Jv, . . .) ∈ S and one of the form πm = (. . . , Jv, . . . , Jt, . . .) ∈ S.

Case (d) with wt

pUt

= wt

pLt, wv

pUv

< wv

pLv

, wv

pLv

< wt

pUt

is examined as case (a).

Case (e) with wt

pUt

= wt

pLt, wv

pUv

< wv

pLv

, wv

pLv

= wt

pUt

is examined as case (b).

Case (f) with wt

pUt

= wt

pLt, wv

pUv

< wv

pLv

, wv

pLv

< wt

pUt

< wv

pUv

is examined as case (c).

Case (g) with wt

pUt

= wt

pLt, wv

pUv

< wv

pLv

, wv

pUv

= wt

pUt

is examined as case (b).

Case (h): wt

pUt

= wt

pLt, wv

pUv

= wv

pLv

, wv

pLv

< wt

pUt

. The processing times of the jobs Jt

and Jv are fixed and wv

pv< wt

pt. Due to Lemma 1, in all optimal permutations for

the instance 1|p|∑ wiCi, job Jv precedes job Jt. Hence, in each permutationof any minimal dominant set, job Jt has to precede job Jv.

Case (i): wt

pUt

= wt

pLt, wv

pUv

= wv

pLv

, wv

pLv

= wt

pUt

. The processing times of the jobs

Jt and Jv are fixed: pLt = pU

t = pt, pLv = pU

v = pt, and wv

pv= wt

pt. Due to Lemma

2 (Lemma 3, respectively) for each instance 1|p|∑ wiCi, p ∈ T , an optimal


permutation is not unique (there exist both an optimal permutation πl ∈ Swith job Jt preceding job Jv and an optimal permutation πm ∈ S with job Jv

preceding job Jt). The jobs Jt and Jv generate two different minimal dominantsets S(T ) and S′(T ). If case (i) occurs, then |Jr| ≥ 2, r = wt

pUt

= wt

pLt

= wv

pUv

= wv

pLv

,

and vice versa. Since the above treatment in cases (a)–(i) is applicable to anypair of jobs of the set J , we complete the proof as follows.

Sufficiency. Assume that there does not exist an r ∈ [a, b] with |Jr| ≥ 2.This means that there is no pair of jobs Jt and Jv in the set J with their

weight-to-process ratios satisfying case (i). In each of the remaining cases (a)–(h), the order of the jobs Jt ∈ J and Jv ∈ J is defined in any permutationfrom a set S(T ) as follows. Job Jt precedes job Jv in the permutation πl ∈S(T ) in the cases (a), (b), (d), (e) and (h). Job Jv precedes job Jt in thepermutation πm ∈ S(T ) in case (g). Both orders of these two jobs are realizedin the permutations πl ∈ S(T ) and πm ∈ S(T ) which have to be contained inany minimal dominant set in each of the cases (c) and (f). Thus, a minimaldominant set S(T ) is unique for an instance 1|pL

i ≤ pi ≤ pUi |

∑wiCi if there

is no r ∈ [a, b] with |Jr| ≥ 2.Necessity. Let the minimal dominant set S(T ) be uniquely determined for

the instance 1|pLi ≤ pi ≤ pU

i |∑

wiCi. By contradiction, we assume that thereexists an r ∈ [a, b] with |Jr| ≥ 2. The inequality |Jr| ≥ 2 holds only if case (i)occurs for the jobs from Jr. There exist at least two minimal dominant setsS(T ) and S′(T ). This contradiction completes the proof.

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Measures of problem uncertainty for scheduling with interval processing times

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