Stability Radius of an Optimal Schedule: a Survey and Recent Developments,

Stability radius of an optimal schedule: asurvey and recent developments 1

Yuri N. Sotskov, Vyacheslav S. TanaevInstitute of Engineering Cybernetics, Surganov St. 6,

220012 Minsk, Belarus

Frank WernerOtto-von-Guericke-Universitat, Fakultat fur Mathematik, PSF 4120,

39016 Magdeburg, Germany

Abstract

The usual assumption that the processing times of the operations are known in advance isthe strictest one in deterministic scheduling theory and it essentially restricts its practicalaspects. Indeed, this assumption is not valid for the most real-world processes. This paperis devoted to a stability analysis of an optimal schedule which may help to extend the signif-icance of scheduling theory for some production scheduling problems. The terms ’stability’,’sensitivity’ or ’postoptimal analysis’ are generally used for the phase of an algorithm atwhich a solution (or solutions) of an optimization problem has already been found, and ad-ditional calculations are performed in order to investigate how this solution depends on theproblem data. We survey some recent results in the calculation of the stability radius of anoptimal schedule for a general shop scheduling problem which denotes the largest quantityof independent variations of the processing times of the operations such that this scheduleremains optimal. We present formulas for the calculation of the stability radius, when theobjective is to minimize mean or maximum flow time. The extreme values of the stabilityradius are of particular importance, and these cases are considered more in detail. Moreover,computational results on the calculation of the stability radius for randomly generated jobshop scheduling problems are briefly discussed. The developed software allows us to showthat the most optimal schedules of the well-known test problem with 6 jobs and 6 machinesare unstable.

Key words - job shop scheduling, disjunctive graph, stability

1Supported by Deutsche Forschungsgemeinschaft (Project ScheMA) and by INTAS (Project 93-257)

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1 Introduction

In deterministic scheduling theory [11, 29] the processing times of the operations, whichhave to be processed, are supposed to be given in advance, i.e. before applying a schedulingprocedure. Unfortunately, such problems do not often arise in practice. Even if the pro-cessing times are known beforehand, OR workers are forced to take into account possiblechanges and errors within the practical realization of a schedule, the precision of equipmentfor calculating the processing times, round-off errors in the calculation of a schedule on thecomputer, machine breakdowns, additionally arriving jobs with high priorities and so on.

More general scheduling settings have been considered in stochastic scheduling [2, pp. 33-59],[20], where the processing times are random variables with a known distribution functionof probabilities. However, in practice such functions also may be unknown. The resultspresented in this survey may be considered as an attempt to initialize investigations ofscheduling problems under conditions of uncertainty. We study the influence of round-offerrors of the processing times on the property of a schedule to be optimal.

Different scheduling problems may be represented as extremal problems on disjunctive graphs(see e.g. [11, 28, 29]). The only requirement for such a representation is the prohibition ofpreemptions of operations. As it follows from recent results [2, pp. 277-293] this approachis the most suitable one for traditionally difficult scheduling problems. In this paper, weuse the disjunctive graph model to represent the input data of the following general shopscheduling problem denoted by G//F .

There are a setQ = 1, 2, . . . , q of operations and a setM = M1,M2, . . . ,Mm of machines.Let Qk be the set of operations that have to be processed on machine Mk:

Q = ∪mk=1Qk, Qk 6= ∅, Qk ∩Ql = ∅, k = 1, 2, . . . ,m, l = 1, 2, . . . ,m, k 6= l.

At any time each machine can process at most one operation. If i ∈ Qk, then the non-negativereal value pi denotes the processing time of operation i on machine Mk ∈ M . Preemptionsof operations are not allowed and a schedule of the operations Q on the machines M maybe defined by the completion times ci or by the starting times ci− pi of all operations i ∈ Q.

The set of operations Q is supposed to be partially ordered by the given precedence con-straints →. We write i → j if operation i ∈ Q is the immediate predecessor of operationj ∈ Q. So, i→ j implies

ci ≤ cj − pj. (1)

In particular, for the job shop problem denoted by J//F the set of operations Q is partitionedinto n chains

Q = ∪nk=1Q(k),

where each chain includes the set Q(k) of all operations of a job Jk, 1 ≤ k ≤ n, and itrepresents the technological route of this job.

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Since at any time a machine can process at most one operation, the conditions i ∈ Qk andj ∈ Qk imply one of the following inequalities:

ci ≤ cj − pj or cj ≤ ci − pi. (2)

The general shop scheduling problem G//F is to find a feasible schedule (c1, c2, . . . , cq) inorder to minimize the value of a given non-decreasing objective function F (C1, C2, . . . , Cn),where Ci is the completion time of job Ji. Such a criterion is called regular. We concentratemainly on two frequently used regular criteria: the maximum flow time (makespan) Cmax =maxni=1Ci and the mean flow time

∑Ci =

∑ni=1 Ci.

The input data of such a problem can be represented by means of a disjunctive graphG = (Q,A,D), where– the set Q of operations is the set of vertices, a non-negative weight pi being assigned toeach vertex i ∈ Q,– A is the set of directed (conjunctive) arcs, representing conditions (1):

A = (i, j) : i→ j, i ∈ Q, j ∈ Q,

– D is the set of pairs of directed (disjunctive) arcs, representing conditions (2):

D = (i, j), (j, i) : i ∈ Qk; j ∈ Qk; i 6→ j; j 6→ i; k = 1, 2, . . . ,m.

While solving problem G//F , each pair of disjunctive arcs (i, j), (j, i) must be settled. Itmeans that one of these arcs must be added to a subset Ds ⊂ D of chosen arcs and the otherone must be rejected. The choice of arc (i, j) ((j, i), respectively) defines a precedence ofoperation i (operation j) over operation j (operation i) on their common machine Mk ∈M ,and a feasible schedule s is defined by a subset Ds ⊂ D such that

(*) (i, j) ∈ Ds if and only if (j, i) ∈ D\Ds, and

(**) digraph Gs = (Q,A ∪Ds, ∅) has no circuits.

Since the objective function is non-decreasing, we may consider only semiactive schedules:a schedule is called semiactive if no operation can start earlier without delaying the process-ing of some other operation or/and without violating the sequence of operations on somemachine. Let P (G) = G1, G2, . . . , Gλ be the set of all digraphs Gs that satisfy both con-ditions (*) and (**). On the one hand, each digraph Gs ∈ P (G) defines a unique semiactiveschedule

s = (c1(s), c2(s), . . . , cq(s)),

where ci(s) is the earliest completion time of operation i ∈ Q with respect to the digraphGs. On the other hand, each semiactive schedule defines a unique digraph Gs ∈ P (G). Inthe following we call the digraph Gs ∈ P (G) optimal if and only if s is an optimal schedule.

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Although problem G//F is NP-hard in the strong sense for any regular criterion consideredin scheduling theory [11, 29], we can find an optimal schedule s = (c1(s), c2(s), . . . , cq(s))in O(q2) time after having constructed an optimal digraph Gs. In other words, the ’maindifficulty’ of problem G//F is to construct an optimal digraph Gs = (Q,A ∪Ds, ∅), i.e. tofind the best set Ds of chosen arcs. Due to the particular importance of the set Ds, it iscalled the signature of a schedule s [1, 23, 26, 27, 28].

This paper is devoted to the stability ball of an optimal digraph Gs, i.e. a ball in the spaceof the numerical input data such that within this ball schedule s remains optimal. Section 2contains a formal definition of the stability radius, i.e. the maximal radius of such a stabilityball.

As already mentioned, the main reason for performing a stability analysis is that in mostpractical cases the processing times of the operations, which have to be processed in the con-sidered processing system, are inexact or uncertain before applying a scheduling procedure.In such cases a stability analysis is necessary to investigate the credibility of an optimalschedule at hand. If possible errors of the processing times are larger than the stabilityradius of an optimal schedule, this schedule may not be the best in a practical realizationand, consequently, there is not much sense in large efforts to construct an ’optimal sched-ule’. In such a case it may be more advisable to restrict the scheduling procedure to theconstruction of an approximate or heuristic solution. On the other hand, this is not the casewhen all real changes of the processing times are less than or equal to the stability radiusof an optimal schedule: an a priori constructed optimal schedule will remain optimal (thebest) in the practical realization.

Another reason for investigating the stability radius may be connected with the frequentneed to solve a set of similar scheduling problems. Since in reality the characteristics of theprocessing systems (the number of machines, the type of technological routes, the range ofvariations of the processing times and so on) do not change quickly, it is sometimes possibleto use previous experience for solving a new similar scheduling problem. It is particularlyimportant for NP-hard problems to have information on the stability because for theseproblems it is undesirable to resolve the problem for each new data set.

Moreover, since the majority of scheduling problems is NP-hard, enumeration schemes suchas branch and bound are often used for finding an optimal schedule. To this end, it isnecessary to construct a large branching tree and most of the information, contained in thistree, is lost after having solved the problem. In such situations the stability radius of theconstructed schedule gives the possibility to use at least a part of this information for furthersimilar scheduling problems.

In Section 3 the calculation of the stability radius of an optimal schedule for problem G//Fis reduced to a non-linear mathematical programming problem if the criterion is regular.Formulas for calculating the stability radius along with characterizations of its extremevalues for problems G//Cmax and G//

∑Ci are presented in Sections 4 and 5, respectively.

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A small example is given in Section 6 to illustrate the notations and some presented results.In Section 7 we briefly discuss computational results on the calculation of the stability radiior at least of their upper bounds for randomly generated job shop scheduling problems withthe mean flow time and the makespan criteria. Section 8 surveys some related approachesto stability analysis in combinatorial optimization.

2 The Definition of the Stability Radius

The main question under consideration is: How can one vary the processing times pi ≥ 0, i ∈Q, in the problem G//F such that an optimal schedule of the problem remains optimal andhow can one calculate the largest quantity of such variations of the processing times?

Note that any small variation pi ± ε, ε > 0, of a processing time pi changes at least thecompletion time ci(s) of operation i in an optimal schedule s = (c1(s), . . . , ci(s), . . . , cq(s))and, as a result, we obtain another semiactive schedule: (. . . , ci(s) + ε, . . .) or (. . . , ci(s) −ε, . . .). However, an optimal digraph Gs = (Q,A ∪Ds, ∅) for the new problem obtained dueto such a variation of pi usually remains the same if ε is sufficiently small. Thus, a signatureDs of an optimal schedule s is essentially more stable. Moreover, in practice it is often moreimportant to keep in mind not the calendar times when the operations have to be startedor have to be completed, but only m optimal sequences in which the operations Qk have tobe processed on machine Mk, k = 1, 2, . . . ,m.

Therefore, the research in [1, 10, 22, 23, 26] was mainly devoted to the stability of anoptimal digraph Gs, which represents an optimal solution of problem G//F in a rathercompact form. Due to these arguments, we can concretize the above question: Under whichlargest independent changes in the components of the vector of the processing times p =(p1, p2, . . . , pq), digraph Gs remains optimal? Next, we introduce these notions in a formalway.

Let Rq be the set of all non-negative q-dimensional real vectors p with the maximum metric,i.e. the distance r(p, p′) between the vectors p ∈ Rq and p′ = (p′1, p

′2, . . . , p

′q) ∈ Rq is defined

as follows:r(p, p′) = max

i∈Q|pi − p′i|,

where |pi − p′i| denotes the absolute value of the difference pi − p′i.

Definition 1 The closed ball O%(p) with the radius % and the centre p in the space of theq-dimensional real vectors is called a stability ball of Gs, if for any vector p′ ∈ O%(p) ∩ Rq,schedule s remains optimal. The maximum value of the radius % of a stability ball O%(p) ofGs is called the stability radius of Gs and it is denoted by %maxs (p) for the makespan, by %Σ

s (p)for the mean flow time and by %Fs (p) for an arbitrarily given regular criterion.

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This definition implies a general approach for calculating %Fs (p), which is discussed in Section3 and concretized for F = Cmax and F =

∑Ci in Sections 4 and 5, respectively.

3 Regular Criterion

Let operation ji ∈ Q(i) ⊂ Q be the last operation of job Ji, 1 ≤ i ≤ n. We denote by µthe set of vertices which form a path µ in the digraph Gs and by lp(µ) the weight of thispath:

lp(µ) =∑i∈µ

pi.

Let H is be the set of all paths in the digraph Gs = (Q,A ∪Ds, ∅) ending in vertex ji ∈ Q(i).

Obviously, the value of cji(s) for a schedule s is equal to the largest weight of a path in H is.

The path µ ∈ H is is called dominant if there is no other path ν ∈ H i

s such that µ ⊂ ν.Otherwise, we shall say that path µ is dominated by path ν. Let H i

s denote the set of alldominant paths in H i

s. Since pi ≥ 0 for all i ∈ Q, we obtain

ci(s) = maxµ∈Hi

s

lp(µ) .

Thus, the schedule s = (c1(s), c2(s), . . . , cq(s)) is optimal if and only if

F (maxµ∈H1

s

lp(µ),maxµ∈H2

s

lp(µ), . . . , maxµ∈Hn

s

lp(µ)) = mink=1,...,λ

F (maxν∈H1

k

lp(ν),maxν∈H2

k

lp(ν), . . . ,maxν∈Hn

k

lp(ν)).

(3)

If φF (p) denotes the set of all optimal schedules with respect to criterion F , then by Definition1 we have

%Fs (p) = infr(p, x) : x ∈ Rq, s /∈ φF (x). (4)

So, it follows from (3) and (4) that, in order to calculate %Fs (p), it is sufficient to know theoptimal value of the objective function f(x1, x2, . . . , xq) of the following non-linear mathe-matical programming problem:

minimizef(x1, x2, . . . , xq) = max

i=1,...,q|xi − pi| (5)

subject to

F (maxµ∈H1

s

lx(µ),maxµ∈H2

s

lx(µ), . . . , maxµ∈Hn

s

lx(µ)) > mink=1,...,λ;k 6=s

F (maxν∈H1

k

lx(ν),maxν∈H2

k

lx(ν), . . . ,maxν∈Hn

k

lx(ν))

(6)xi ≥ 0, i = 1, 2, . . . , q. (7)

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If condition (6) is not satisfied for any x ∈ Rq, then digraph Gs is optimal for any vectorx ∈ Rq of processing times: s ∈ φF (x) for any x ∈ Rq. In this case we shall say that thestability radius is infinitely large and we shall write

%Fs (p) =∞.

In all other cases there exists an optimal value f ∗ of the objective function of problem (5)-(7):

f ∗ = inf maxi=1,...,q

|xi − pi|,

where the infimum is taken over all vectors x satisfying conditions (6) and (7). To find thevalue f ∗, it is sufficient to know a solution x0 = (x0

1, x02, . . . , x

0q) of problem (5)-(7), where

the sign > in inequality (6) is replaced by the sign ≥. It is clear that

f ∗ = maxi=1,...,q

|x0i − pi| = r(x0, p) = %Fs (p)

and for any small ε > 0, there exists a vector xε = (xε1, xε2, . . . , x

εq) such that r(xε, p) =

%Fs (p) + ε and s 6∈ φF (xε). It may occur that solution x0 is equal to p. In the latter case wehave

%Fs (p) = r(p, p) = 0

and the optimal digraph Gs is unstable. If x0 is not equal to p, we have %Fs (p) > 0 and theoptimal digraph Gs is stable.

4 Makespan Criterion

The best studied case of problem G//F is the one with F = Cmax. Let H and Hs denotethe sets of all dominant paths in digraph (Q,A, ∅) and in digraph Gs ∈ P (G), respectively.The value of maxni=1Ci of a schedule s is given by the weight of a maximum-weight (alsocalled critical) path in the digraph Gs ∈ P (G). Obviously, at least one critical path in Gs isdominant and for any µ ∈ H, there exists a path ν ∈ Hs that dominates path µ or µ ∈ Hs.Thus, equality (3) for problem G//Cmax is converted to

maxµ∈Hs

lp(µ) = mink=1,...,λ

maxν∈Hk

lp(ν) (8)

and an optimal makespan schedule s is defined by the digraph Gs ∈ P (G) satisfying equality(8).

Let Hk(p) denote the set of all critical dominant paths in the digraph Gk ∈ P (G) (withrespect to vector p). Obviously, we have Hk(p) ⊆ Hk. To present necessary and sufficientconditions for %maxs (p) = 0 proven in [22, 23], we need the following auxiliary claim from [17].

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Lemma 1 There exists a real ε such that the set Hk\Hk(p) contains no critical path ofdigraph Gk ∈ P (G) for any vector of processing times pε = (pε1, p

ε2, . . . , p

εq) ∈ Oε(p) ∩Rq, i.e.

Hk(pε) ⊆ Hk(p).

Proof. We calculate the positive real number

εk = minν∈Hk\Hk(p)

lpk − lp(ν)

2q. (9)

Hereafter lpk = maxµ∈Hk lp(µ) is the value of the objective function Cmax for schedule k with

the vector p of processing times. For any real ε, which satisfies the inequalities 0 < ε < εk,the difference in the right side of equality (9) remains positive when vector p is replaced byany vector pε ∈ Oε(p)∩Rq. Indeed, the number of vertices in any path ν in digraph Gk is atmost equal to q and, therefore, the difference lpk − lp(ν) may not be ’overcome’ by a vectorpε if r(p, pε) < εk.

Theorem 1 For an optimal schedule s ∈ φmax(p), equality %maxs (p) = 0 holds if and only ifthere exists another optimal schedule k ∈ φmax(p), k 6= s, and there exists a path µ∗ ∈ Hs(p)such that there does not exist any path ν∗ ∈ Hk(p) with µ∗ ⊆ ν∗.

Proof. We prove sufficiency by contradiction. Assume that %maxs (p) = ε0 > 0 holds, but theconditions of the theorem are satisfied: there exists a schedule k ∈ φmax(p), k 6= s, and thereexists a path µ∗ ∈ Hs(p) such that for any path ν∗ ∈ Hk(p) condition µ∗ ⊆ ν∗ does nothold.

We construct a vector p∗ = (p∗1, p∗2, . . . , p

∗q) with the components

p∗i =

pi + ε∗, if i ∈ µ∗pi otherwise,

where ε∗ = minεk, εs, ε0 with εs and εk defined as in (9). If maxν∈Hk lp∗(ν) = lp

∗(ν0), then

due to Lemma 1 and the inequalities p∗i ≥ pi, i ∈ Q, we have

lp∗(ν0) = max

ν∈Hk(p)lp∗(ν) = lp(ν∗) + ε∗|µ∗ ∩ ν∗| = lp(µ∗) + ε∗|µ∗ ∩ ν∗|. (10)

Since µ∗ ⊆ ν∗ does not hold for any ν∗ ∈ Hk(p), the inequality

|µ∗ ∩ ν∗| < |µ∗|

holds (hereafter |a| denotes the number of elements in the set a) and we can continue (10)in the following way:

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lp(µ∗) + ε∗|µ∗ ∩ ν∗| < lp(µ∗) + ε∗|µ∗| = lp∗(µ∗) = max

µ∈Hslp∗(µ).

Therefore, we have lp∗

k < lp∗s and s /∈ φmax(p∗), which contradicts the assumption %s(p) =

ε0 ≥ ε∗ because of r(p, p∗) = ε∗.

Next, we prove necessity by contradiction, too. Suppose that %maxs (p) = 0 but the conditionsof the theorem are not satisfied. We consider two cases i) and ii) of violating these conditions.

i) Assume that there does not exist another optimal makespan schedule: φmax(p) = s.Then we calculate the real number

ε∗ =1

2qminlpt − lps : t = 1, 2, . . . , λ; t 6= s.

Since s is the only optimal makespan schedule, we have ε∗ > 0. For any positive real ε < ε∗,the difference lpt − lps remains (strictly) positive when vector p is replaced by an arbitraryvector p0 ∈ Oε(p) ∩ Rq. So we can conclude that digraph Gs remains optimal for any suchvector p0 of the processing times. Therefore, we have %maxs (p) ≥ ε > 0 which contradicts theassumption %maxs (p) = 0.

ii) Assume that |φmax(p)| > 1 and for any schedule k ∈ φmax(p), k 6= s, and for any pathµ∗ ∈ Hs(p), there exists a path ν∗k ∈ Hk(p) such that µ∗ ⊆ ν∗k. In this case we can takeany ε that satisfies the inequalities

0 < ε < min

mink∈φmax(p)

εk,1

2qminlpt − lps : t = 1, 2, . . . , λ, t 6∈ φmax(p)

.

From Lemma 1, due to inequality ε < εs, we get equality

lp0

s = maxµ∈Hs(p0)

lp0

(µ) (11)

for any vector p0 ∈ Oε(p) ∩ Rq. Since ε < εs and ε < εk and there exists a path ν∗k ∈Hk(p), k ∈ φmax(p), k 6= s, for any path µ∗ ∈ Hs(p) such that µ∗ ⊆ ν∗k, we obtain theinequality

maxµ∈Hs(p)

lp0

(µ) ≤ maxν∈Hk(p)

lp0

(ν). (12)

Thus, due to (11) and (12), we have

lp0

s ≤ maxν∈Hk(p)

lp0

(ν) (13)

for any optimal schedule k ∈ φmax(p), k 6= s. Since

ε <1

2qminlpt − lps : t = 1, 2, . . . , λ, t 6∈ φmax(p),

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condition t 6∈ φmax(p) implies t 6∈ φmax(p0). Taking into account (13) and the latter im-plication, we can conclude that s ∈ φmax(p0) for any vector p0 ∈ Rq with r(p, p0) ≤ ε.Consequently, we have %maxs (p) ≥ ε > 0, which contradicts the assumption %s(p) = 0.

Obviously, the conditions of Theorem 1 are violated if Hs(p) ⊆ H and so we get

Corollary 1 If s ∈ φmax(p) and Hs(p) ⊆ H, then %maxs (p) > 0.

For the following corollary it is not necessary to know the set Hs(p).

Corollary 2 If %maxs (p) = s, then %maxs (p) > 0.

In [22, 23] the following characterization of an infinitely large stability radius was given.

Theorem 2 For an optimal schedule s ∈ φmax(p), we have %maxs (p) = ∞ if and only iffor any path µ ∈ Hs\H and any digraph Gt ∈ P (G), there exists a path ν ∈ Ht such thatµ ⊆ ν.

Proof: Necessity. Following the contradiction method, we suppose that %maxs (p) = ∞but there exist a path µ ∈ Hs\H and a digraph Gt ∈ P (G) such that for any ν ∈ Ht

the relation µ ⊆ ν does not hold. We set ε′ = maxqi=1 pi and consider the vectorp′ = (p′1, p

′2, . . . p

′q) ∈ Rq, where

p′i =

ε′, if i ∈ µ0 otherwise.

For any path ν ∈ Ht, we havelp′(ν) = ε′|µ ∩ ν|.

Since relation µ ⊆ ν does not hold, we have lp′(ν) < lp

′(µ). Therefore, we obtain

lp′

t < lp′(µ) = lp

′s and hence s 6∈ φmax(p′). We get a contradiction:

%maxs (p) < r(p′, p) =q

maxi=1

pi <∞.

Sufficiency. Let ε be a positive real as large as desired. We take any vector p0 ∈ Oε(p) ∩Rq

and suppose thatlp

0

s = lp0

(µ).

If µ ∈ H, then lp0(µ) ≤ lp

0

t holds for every t = 1, 2, . . . , λ. If µ ∈ Hs\H, then, due to theconditions of the theorem, for any feasible schedule t, there exists a path ν ∈ Ht such that

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µ ⊆ ν. Therefore, we get lp0(µ) ≤ lp

0(ν) ≤ lp

0

t . Thus in both cases we have s ∈ φmax(p0).

Directly from the above proof of the necessity we obtain a simple upper bound of the stabilityradius.

Corollary 3 If %maxs (p) <∞, then we have %maxs (p) < maxqi=1 pi.

Due to Theorem 2 one can identify a problem whose optimal makespan schedule is impliedonly by the given precedence constraints and the distribution of the operations to the ma-chines, but independent from the processing times of the operations. However, because ofthe generality of problem G//Cmax, it is difficult to check the above conditions. In [10] ithas been shown that for problem J//Cmax, there are necessary and sufficient conditions for%maxs (p) =∞ which can be verified in O(q2) time. To present the latter conditions, we needthe following notations.

Let Bk (Ck, respectively) be the set of all operations i such that i → j (j → i) andj ∈ Qk, i 6∈ Qk. For a set B of operations, let n(B) be the number of jobs having at leastone operation in B.

Theorem 3 For problem J//Cmax, there exists an optimal digraph Gs with an infinitelylarge stability radius if and only if we have max|Bk|, |Ck| ≤ 1 for any machine Mk withn(Qk) > 1 and, if there exist two operations g ∈ Bk and f ∈ Ck of job Jl, then there existsa path from f to g in the digraph (Q,A, ∅) (possibly f = g).

From Theorem 3 it follows that there are problems J//Cmax with an optimal schedule withan infinitely large stability radius for any given number of jobs n and number of machinesm. On the other hand, for flow shop and open shop problems such a schedule can exist onlyfor n and m not greater than 2. Moreover, in [10] the analogies to Theorems 2 and 3 for thejob shop problem J//Lmax with minimizing maximum lateness have been proven and it hasbeen shown that there does not exist an optimal schedule s with %Fs (p) = ∞ for all otherregular criteria (see [11]), which are considered in scheduling theory.

Formulas for calculating %s(p) have been derived in [22, 23, 26]. Next we prove a formulagiven in [26].

Assume that %maxs (p) < ∞ holds for the given optimal schedule s ∈ φmax(p) of problemG//Cmax. Using equality (8), we can conclude that equality (4) for F = Cmax is convertedto the following:

%maxs (p) = infr(p, x) : x ∈ Rq, max

µ∈Hslx(µ) > min

k=1,...,λ; k 6=smaxν∈Hk

lx(ν).

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Therefore, to find the stability radius %maxs (p) it is sufficient to construct a vector x ∈ Rq

that satisfies the following three conditions:1) there exists a digraph Gk ∈ P (G), k 6= s, such that lxs = lxk , i.e.

maxµ∈Hs

lx(µ) = maxν∈Hk

lx(ν); (14)

2) for any given real ε > 0, which may be as small as desired, there exists a vector pε ∈ Rq

such that r(x, pε) = ε and inequality

maxµ∈Hs

lpε

(µ) > maxν∈Hk

lpε

(ν) (15)

is satisfied for at least one digraph Gk ∈ P (G);3) the distance r(p, x) achieves its minimal value among the distances between vector p andthe other vectors in the space Rq which satisfy both above conditions 1 and 2.

After having constructed such a vector x ∈ Rq one can define the stability radius of digraphGs:

%maxs (p) = r(p, x),

since the critical path in digraph Gs becomes larger than that of digraph Gk for any pε ∈ Rq

with positive real ε, which may be as small as possible (see condition 2), and so digraph Gs

is no longer optimal, while in the ball Or(p,x)(p) digraph Gs remains optimal (see condition3). Digraph Gk, which satisfies conditions 1, 2 and 3 will be called a competitive digraph forthe optimal digraph Gs.

Thus, the calculation of the stability radius may be reduced to a rather sophisticated extremalproblem on the given set of weighted digraphs P (G) = G1, G2, . . . , Gλ with a variablevector of weights assigned to the vertices of each digraph Gk ∈ P (G). As it follows from (14)and (15), the main objects for such a calculation are the sets of dominant paths Hk, k =1, 2, . . . , λ.

Similarly to [23], we look next for a vector p′ = p(r) = (p1(r), p2(r), . . . , pq(r)) ∈ Rq withthe components pi(r) ∈ pi, pi + r, pi − r on the basis of a direct comparison of the pathsfrom the set Hs and the paths from the sets Hk, where k = 1, 2, . . . , λ and k 6= s.

Let the value lp(ν) be greater than the length of a critical path in an optimal digraph Gs.To satisfy equality (14), the length of a path ν ∈ Hk has to be not greater than that of atleast one path µ ∈ Hs and there must be a path ν ∈ Hk with a length equal to the lengthof a critical path in Gs. Thus, if we have calculated

rν = minµ∈Hs

lp(ν)− lp(µ)

|µ ∪ ν| − |µ ∩ ν|, (16)

we obtain equalitymaxµ∈Hs

lp(r)(µ) = lp(r)(ν)

12

for the vector p(r) = p(rν) with the components

pi(r) =

pi + rν , if i ∈ µ,pi − rν , if i ∈ ν\µ,pi, if i 6∈ µ ∪ ν.

(17)

On the other hand, to reach equality (14) for the whole digraph Gk, we have to repeatcalculation (16) for each path ν ∈ Hk with lp(ν) > lps . Thus, instead of vector p(rν) we haveto consider the vector p(r) = p(rGk) calculated according to formula (17), where

rGk = minµ∈Hs

maxν∈Hk; lp(ν)>lps

lp(ν)− lp(µ)

|µ ∪ ν| − |µ ∩ ν|. (18)

Let us now consider inequality (15). Since the processing times are non-negative, this in-equality may not be valid for a vector pε ∈ Rq if path µ is dominated by path ν. Thus wecan restrict our consideration to the subset Hsk of the set Hs of all paths, which are notdominated by paths from Hk:

Hsk =µ ∈ Hs : there is no path ν ∈ Hk such that µ ⊆ ν

.

Hence, we can replace Hs in equality (18) by Hsk.

To obtain the desired vector x ∈ Rq, we have to use equality (18) for each digraph Gk ∈P (G), k 6= s. Let r denote the minimum of such a value rGk :

r = rG∗k

= minrGk : Gk ∈ P (G), k 6= s

and let ν∗ ∈ Hk∗ and µ∗ ∈ Hsk∗ be paths at which value rG∗ has been reached:

rGk∗ = rν∗ =lp(ν∗)− lp(µ∗)

|µ∗ ∪ ν∗| − |µ∗ ∩ ν∗|.

Taking into account (17), we note that, if rν∗ ≤ pi for each i ∈ ν∗\µ∗, vector p(r) =p(rν∗) does not contain negative components, i.e. p(r) ∈ Rq. For the general case we haveobtained a lower bound of the stability radius:

%maxs (p) ≥ r = mink=1,...,λ; k 6=s

minµ∈Hsk

maxν∈Hk; lp(ν)>lps

lp(ν)− lp(µ)

|µ ∪ ν| − |µ ∩ ν|, (19)

which is tight if %maxs (p) ≤ pi for each i ∈ ν∗\µ∗ due to the above remark. For example,we have %maxs (p) = r in (19) if %maxs (p) ≤ minpi : i ∈ Q.To obtain the exact value of %s(p) in the general case, we follow [22, 23]. Let p0

νµ be equalto zero and let (p1

νµ, p2νµ, . . . , p

wνµνµ ) denote a non-decreasing sequence of processing times of

operations from the set ν\µ, where wνµ = |ν\µ|. We obtain the following assertion.

13

Theorem 4 If s ∈ φmax(p) holds, then

%maxs (p) = mink=1,2,...,λ;k 6=s

rks, (20)

where

rks = minµ∈Hsk

maxν∈Hk;lp(ν)>lps

maxβ=0,...,wνµ

lp(ν)− lp(µ)−∑βα=0 p

ανµ

|µ ∪ ν| − |µ ∩ ν| − β.

Note that formula (20) turns into %maxs (p) =∞ if Hsk = ∅ for any k = 1, 2, . . . , λ, k 6= s (seeTheorem 2). Moreover, if only a subset of the processing times (say, P ⊆ p1, p2, . . . , pq)can be changed but the other ones cannot be changed, formulas (19) and (20) remain validprovided that the difference |µ ∪ ν| − |µ ∩ ν| is replaced by |µ ∪ ν ∩ P | −|µ ∩ ν ∩ P |.

5 Mean Flow Time Criterion

If F =∑Ci, conditions (3) and (4) are converted to the following conditions (21) and (22),

respectively.

n∑i=1

maxµ∈Hi

s

lp(µ) = mink=1,...,λ

n∑i=1

maxν∈Hi

k

lp(ν), (21)

%Σs (p) = inf

r(p, x) : x ∈ Rq,

n∑i=1

maxµ∈Hi

s

lx(µ) > mink=1,...,λ;k 6=s

n∑i=1

maxν∈Hi

k

lx(ν). (22)

Therefore, to find the stability radius %Σs (p) it is sufficient to construct a vector x ∈ Rq that

satisfies the following three conditions:1’) there exists a digraph Gk ∈ P (G), k 6= s, such that

n∑i=1

maxµ∈Hi

s

lx(µ) =n∑i=1

maxν∈Hi

k

lx(ν); (23)

2’) for any given real ε > 0, which may be as small as desired, there exists a vector pε suchthat r(x, pε) = ε and inequality

n∑i=1

maxµ∈Hi

s

lpε

(µ) >n∑i=1

maxν∈Hi

k

lpε

(ν) (24)

is satisfied for at least one digraph Gk ∈ P (G);3’) the distance r(p, x) achieves its minimal value among the distances between vector p andthe other vectors in the space Rq which satisfy both above conditions 1’ and 2’.

14

Similarly as in the previous section, after having constructed such a vector x ∈ Rq one candefine the stability radius of digraph Gs: %

Σs (p) = r(p, x). Thus, due to (23) and (24) the

calculation of the stability radius may again be reduced to an extremal problem on the setof weighted digraphs P (G). However, in this case we are forced to consider numerous sets ofrepresentatives of the family of sets H i

k, 1 ≤ i ≤ n, which may be denoted as follows. Let Ωuk

be a set of representatives of the family of sets (H ik)1≤i≤n. More precisely, the set Ωu

k includesexactly one path from each set H i

k, 1 ≤ i ≤ n. Since H ik∩H

jk = ∅ for each pair of different jobs

Ji and Jj, we have |Ωuk| = n and there exist ωk =

∏ni=1 |H i

k| different sets of representativesfor each digraph Gk, namely: Ω1

k,Ω2k, . . . ,Ω

ωkk . For each set Ωu

k we can calculate the integervector n(Ωu

k) = (n1(Ωuk), n2(Ωu

k), . . . , nq(Ωuk)), where nj(Ω

uk), j ∈ 1, 2, ..., q, is equal to the

number of paths in Ωuk which includes vertex j. Since a path ν ∈ H i

k includes vertex j ∈ Qat most once, the value nj(Ω

uk) is equal to the number how often vertex j is contained in the

multiset ν : ν ∈ Ωuk.

In [1], vector x satisfying conditions 1’, 2’ and 3’ was found in the form x(r) = (x1(r), x2(r), . . .,xq(r)) with the components xi(r) ∈ pi, pi + r, pi− r on the basis of a direct comparison ofthe set Ωu

s of representatives of the family of sets (H is)1≤i≤n, and the set Ωu

k of representativesof the family of sets (H i

k)1≤i≤n, where k = 1, 2, . . . , λ and k 6= s. The following lower boundof the stability radius has been obtained:

%Σs (p) ≥ r = min

k=1,...,λ; k 6=smin

Ωvs∈Ωs,kmax

u∈1,...,ωkrΩu

k,Ωvs , (25)

where

rΩuk,Ωvs =

∑ν∈Ωu

klp(ν)−∑

µ∈Ωvslp(µ)∑n

i=1 |ni(Ωuk)− ni(Ωv

s)|.

This bound is tight: we have %Σs (p) = r in (25) if r ≤ minpi : i ∈ Q.

To obtain the exact value of %Σs (p), the vector x∗(r) = (x∗1(r), x∗2(r), . . . , x∗q(r)) with the

components

x∗i (r) =

pi + r, if ni(Ω

uk) < ni(Ω

vs),

max0, pi − r, if ni(Ωuk) > ni(Ω

vs),

pi, if ni(Ωuk) = ni(Ω

vs),

was used in [1]. In contrast to vector x(r), vector x∗(r) may not have negative components.

Let the set of operations Q be ordered in the following way:

i1, i2, . . . , im, im+1, . . . , iq (26)

where niα(Ωuk) ≤ niα(Ωv

s) for each α = 1, 2, . . . ,m and niα(Ωuk) > niα(Ωv

s) for each α =m+ 1,m+ 2, . . . , q. Moreover, for the sequence (26) the inequalities

pim+1 ≤ pim+2 ≤ . . . ≤ piq

have to be satisfied. Following [1], one can derive a formula for calculating %Σs (p).

15

Theorem 5 If s ∈ φΣ(p) holds, then

%Σs (p) = min

k=1,2,...,λ;k 6=sRks, (27)

where

Rks = minΩvs∈Ωs,k

maxu=1,...,ωk

maxβ=0,...,q−m

∑m+βα=1 piα|niα(Ωu

k)− niα(Ωvs)|∑m+β

α=1 |niα(Ωuk)− niα(Ωv

s)|.

If only a subset of the processing times can be changed but the other ones cannot be changed,a formula similar to (27) was derived in [1], too.

Next, we consider the extreme values of %Σs (p). Similarly to the notion of a critical path,

which is important for problem G//Cmax (see Section 4), we introduce the notion of a criticalset Ωu∗

k of digraph Gk ∈ P (G) for problem G//∑Ci. The set Ωu∗

k , u∗ ∈ 1, 2, . . . , ωk, is

critical if the value of the objective function

Lpk = maxu∈1,...,ωk

∑ν∈Ωu

k

lp(ν)

for digraph Gk is reached on this set:∑ν∈Ωu

∗k

lp(ν) = maxu∈1,...,Ωk

∑ν∈Ωu

k

lp(ν) = Lpk.

Obviously, a critical set Ωu∗k may include a path ν ∈ H i

k, i = 1, 2, . . . , n, if and only if

lp(ν) = maxµ∈Hi

k

lp(µ)

and so for different vectors p ∈ Rq of processing times, different sets Ωuk , u ∈ 1, 2, . . . , ωk,

may be critical. Let Ωk(p) denote the set of all critical sets Ωu∗k of digraph Gk ∈ P (G)

with the vector of processing times p = (p1, p2, . . . , pq) ∈ Rq and let Ωk denote the setΩu

k : u = 1, 2, . . . , ωk.To present necessary and sufficient conditions for %Σ

s (p) = 0, we need the following auxiliaryclaim.

Lemma 2 There exists a real ε > 0 such that the set Ωk\Ωk(p) contains no critical set ofdigraph Gk ∈ P (G) for any vector p′ ∈ Oε(p) ∩Rq of processing times, i.e.

Ωk(p′) ⊆ Ωk(p).

Proof: After having calculated the value

εk =1

2qnmin

Lpk −

∑ν∈Ωu

k

lp(ν) : Ωuk ∈ Ωk\Ωk(p)

, (28)

16

one can convince that for any real ε, which satisfies the inequalities 0 < ε < εk, the differencein the right side of equality (28) remains positive when vector p is replaced by any vectorp′ ∈ Oε(p)∩Rq. Indeed, for any u ∈ 1, 2, . . . , ωk, the cardinality of set Ωu

k may be at mostequal to qn. Thus, the difference Lpk −

∑ν∈Ωu

klp(ν) may not be ’overcome’ by a vector p′ if

r(p, p′) < εk.

The following necessary and sufficient conditions for %Σs (p) = 0 have been derived in [1].

Theorem 6 Let s be an optimal schedule of problem G//∑Ci with positive processing times

pi > 0 of all operations i ∈ Q. The equality %Σs (p) = 0 holds if and only if there exists another

optimal schedule k ∈ φΣ(p), k 6= s, and there exists a set Ωv∗s ∈ Ωs(p) such that for any set

Ωu∗k ∈ Ωk(p) there exists a job Ji, 1 ≤ i ≤ n, such that condition

ni(Ωv∗

s ) ≥ ni(Ωuk), Ωu

k ∈ Ωk(p), (29)

holds (or conditionni(Ω

v∗

s ) ≤ ni(Ωuk), Ωu

k ∈ Ωk(p) (30)

holds) and inequality (29) (or inequality (30), respectively) is satisfied as a strict one for theset Ωu∗

k .

Proof: We prove necessity by contradiction. Assume that %Σs (p) = 0 but the conditions

of the theorem are not satisfied. We consider three cases j), jj) and jjj) of violating theseconditions.

j) Assume that there does not exist another optimal schedule, i.e. we have φΣ(p) = s.Then we consider a real ε such that

0 < ε <1

2qnmint6=s

(Lpt − Lps)

holds. Similarly to the proof of Lemma 2, we can show that digraph Gs remains optimalfor any vector p0 = (p0

1, p02, . . . , p

0q) ∈ Rq of the processing times provided that r(p, p0) ≤ ε.

Therefore, we have %Σs (p) ≥ ε > 0 which contradicts the assumption %Σ

s (p) = 0.

jj) Assume that |φΣ(p)| > 1 and for any optimal schedule k ∈ φΣ(p) with k 6= s, and forany set Ωv

s ∈ Ωs(p), there exists a set Ωu∗k ∈ Ωk(p) such that ni(Ω

vs) = ni(Ω

u∗k ) for any job

Ji, i = 1, 2, . . . , n.

In this case, we can take any ε that satisfies the inequalities

0 < ε < minεs, εk,

1

2qnmint6∈φΣ(p)

(Lpt − Lps). (31)

17

From Lemma 2, due to inequality ε < εs, we get that equality

Lp0

s = maxΩvs∈Ωs(p)

∑µ∈Ωvs

lp0

(µ) (32)

holds for any vector p0 ∈ Oε(p) ∩ Rq. Since there exists a set Ωu∗k ∈ Ωk(p) for any set

Ωvs ∈ Ωs(p) and any k ∈ φΣ(p), k 6= s, such that ni(Ω

vs) = ni(Ω

u∗k ), i = 1, 2, . . . , n, we obtain

the inequalitymax

Ωvs∈Ωs(p)

∑µ∈Ωvs

lp0

(µ) ≤ maxΩuk∈Ωk(p)

∑ν∈Ωu

∗k

lp0

(ν),

because of ε < εs and ε < εk. Therefore, due to (32) we have

Lp0

s ≤ maxΩuk∈Ωk(p)

∑ν∈Ωu

∗k

lp0

(ν) (33)

for any optimal schedule k ∈ φΣ(p), k 6= s. Since ε < 12qn

mint6∈φΣ(p)Lpt − Lps, condition

t 6∈ φΣ(p) implies t 6∈ φΣ(p0). So taking into account (31) and the latter implication, weconclude that s ∈ φΣ(p0) for any vector p0 ∈ Rq provided that r(p, p0) ≤ ε. Consequently,we have %Σ

s (p) ≥ ε > 0, which contradicts the assumption %Σs (p) = 0.

jjj) Assume that |φΣ(p)| > 1 and for any optimal schedule k ∈ φΣ(p), k 6= s, and for any setΩvs ∈ Ωs(p), there exists a set Ωu∗

k ∈ Ωk(p) such that for any job Ji with ni(Ωvs) > ni(Ω

u∗k )

there exists a set Ωu0

k ∈ Ωk(p) such that ni(Ωvs) < ni(Ω

u∗k ). Argueing in the same way as

in case jj), we can show that %Σs (p) ≥ ε > 0, where ε is as in (31), since for any vector

p0 ∈ Oε(p)∩Rq, the value∑µ∈Ωv∗s

lp0(µ) is less than or equal to value

∑ν∈Ωu

∗klp

0(ν) or value∑

ν∈Ωu0klp

0(ν).

Next, we prove sufficiency by contradiction, too. Assume that %Σs (p) = ε0 > 0 holds, but the

conditions of the theorem are satisfied.

We construct a vector p∗ = (p∗1, p∗2, . . . , p

∗q) ∈ Rq with components p∗i ∈ pi, pi + ε∗, pi − ε∗,

where ε∗ = minεk, ε0,mini∈Q pi, using the following rule: for each Ωu∗k ∈ Ωk(p), mentioned

in Theorem 6, we set p∗i = pi+ε∗, if inequalities (29) hold, or we set p∗i = pi−ε∗, if inequalities

(30) hold. Note that ε∗ > 0 since pi > 0, i ∈ Q.

After changing |Ωk(p)| components of vector p according to this rule, we obtain a vector p∗

of processing times for which inequality∑µ∈Ωv∗s

lp∗(µ) >

∑ν∈Ωu

∗k

lp∗(ν)

holds for each set Ωu∗k ∈ Ωk(p). Due to ε∗ ≤ mini∈Q pi, we have p∗ ∈ Rq. Since ε∗ ≤ εk, we

have

Lp∗

k = maxu∈1,...,ωk

∑ν∈Ωu

k

lp∗(ν) = max

Ωuk∈Ωk(p)

∑ν∈Ωu

k

lp∗(ν) =

∑ν∈Ωu

∗k

lp∗(ν) <

∑µ∈Ωv∗s

lp∗(µ) ≤ Lp

∗

s .

18

Thus, we conclude that s /∈ φΣ(p∗) with r(p, p∗) = ε∗ which contradicts the assumption%Σs (p) = ε0 ≥ ε∗ > 0.

Theorem 6 directly implies the following assertion.

Corollary 4 If φΣ(p) = s, then %Σs (p) > 0.

The following simple upper bound of the stability radius for problem G//∑Ci was presented

in [1].

Theorem 7 If s ∈ φΣ(p) holds for problem G//∑Ci with λ > 0 and pi > 0 for at least one

operation i ∈ Q, then we have

%Σs (p) < maxpi : i ∈ Q.

Proof: Let us consider vector p0 ∈ Rq with zero components: p0i = 0 for each i ∈ Q. For

this vector of processing times, each feasible digraph Gt ∈ P (G) is optimal and each set ofrepresentatives Ωu

t is critical. We can take a schedule k ∈ φΣ(p0) which has only one arc(j, i) ∈ Dk different from the arcs in Ds, i.e. (i, j) ∈ Ds and Ds\(i, j) = Dk\(j, i). It iseasy to see that there exist sets Ωv

s ∈ Ωs(p) and Ωuk ∈ Ωk(p) such that

ni(Ωvs) > ni(Ω

uk).

Setting pεi = ε > 0 and pεl = 0 for each l ∈ Q\i, we obtain s /∈ φΣ(pε) and r(pε, p) <maxpi : i ∈ Q.

As it follows from Theorem 7, problem J//∑Ci with λ > 0 cannot have an optimal schedule

with an infinitely large stability radius in contrast to problem J//Cmax and problem J//Lmax(see Section 4).

To illustrate the above notations and some results, we consider in Section 6 an example ofa job shop scheduling problem with two jobs and two machines.

6 Example

The job shop problem is specified by the disjunctive graph G = (Q,A,D) given in Fig. 1.The first job consists of operations 1 and 2, and the second job consists of operations 3 and 4.

19

So we have the precedence constraints 1→ 2 and 3→ 4. The assignment of the operationsto the machines is as follows: Q1 = 1, 4, Q2 = 2, 3. The vector p = (10, 30, 20, 40)defines the processing times of the operations Q = 1, 2, 3, 4.For this problem we get P (G) = G1, G2, G3 with the following signatures of semiactiveschedules: D1 = (1, 4), (3, 2), D2 = (1, 4), (2, 3) and D3 = (4, 1), (3, 2). The corre-sponding sets of dominant paths are as follows: H1

1 = (1, 2), (3, 2), H21 = (1, 4), (3, 4),

H12 = (1, 2), H2

2 = (1, 2, 3, 4), H13 = (3, 4, 1, 2) and H2

3 = (3, 4). The optimaldigraph G1 = (Q,A∪D1, ∅) is represented in Fig. 2 and it defines the unique optimal semi-active schedule (10, 50, 20, 60) for both criteria Cmax and

∑Ci. Therefore, due to Corollary

2 and Corollary 4 we have %max1 (p) > 0 and %Σ1 (p) > 0.

-

-

1

)@@@@R

@@I

1 2

3 4

p1 = 10 p2 = 30

p3 = 20 p4 = 40

Figure 1: The disjunctive graph G = (Q,A,D)

We can calculate the exact value of %Σ1 (p) on the basis of Theorem 5. First we compare

the digraphs G1 and G2. We have four sets of representatives for digraph G1, namely:Ω1

1 = (1, 2), (1, 4), Ω21 = (1, 2), (3, 4), Ω3

1 = (3, 2), (1, 4) and Ω41 = (3, 2), (3, 4).

We can calculate vectors n(Ω11) = (2, 1, 0, 1), n(Ω2

1) = (1, 1, 1, 1), n(Ω31) = (1, 1, 1, 1) and

n(Ω41) = (0, 1, 2, 1). Digraph G2 has only one set of representatives: Ω1

2 = (1, 2), (1, 2, 3, 4)with n(Ω1

2) = (2, 2, 1, 1). Obviously, we have n(Ωi1) ≤ n(Ω1

2) for all i ∈ 1, 2, 3. Thus wehave to calculate only

rΩ12,Ω

41

=(40 + 100)− (50 + 60)

| 2− 0 | + | 2− 1 | + | 1− 2 | + | 1− 1 |=

30

4= 7.5

(cf. (25)).

Next we compare the digraphs G1 and G3. Digraph G3 has only one set of representatives:Ω1

3 = (3, 4, 1, 2), (3, 4) with n(Ω13) = (1, 1, 2, 2). Obviously, we have n(Ωi

1) ≤ n(Ω13) for

i ∈ 2, 3, 4. Thus we have to calculate only

rΩ13,Ω

11

=(100 + 60)− (40 + 50)

| 1− 2 | + | 1− 1 | + | 2− 0 | + | 2− 1 |=

70

4= 17.5

Due to (25) we obtain %Σs (p) = min7.5; 17.5 = 7.5 and G3 is a competitive digraph for

digraph G1, which is optimal for p = (10, 30, 20, 40).

20

-

-

1 2

3 4

c1(1) = 10 c2(1) = 50

c3(1) = 20 c4(1) = 60

1

@@@@@R

Figure 2: The optimal digraph G1 = (Q,A ∪D1, ∅)

Next we consider the stability radius of digraph G1 with respect to criterion Cmax. We havethe following sets of dominant paths: H = (1, 2), (3, 4), H1 = (1, 2), (1, 4), (3, 2), (3, 4), H2 =(1, 2, 3, 4) and H3 = (3, 4, 1, 2). It is clear that any path µ ∈ H1\H is dominated bythe path (1, 2, 3, 4) ∈ H2 and by the path (3, 4, 1, 2) ∈ H3. Thus, due to Theorem 2 we have%max1 (p) =∞.

7 Computational Results

The above formulas for calculating the stability radii were coded in the language FORTRAN77. At the worst, calculating %maxs (p) and %Σ

s (p) implies not only to have an optimal digraphGs but to construct all feasible digraphs G1, G2, . . . , Gλ. So, for small scheduling problemsthe program starts with generating all feasible digraphs and for each of them, which has tobe compared with the optimal digraph, it finds all dominant paths. After that formulas (20)and (27) are used for calculating %maxs (p) and %Σ

s (p), respectively.

In order to restrict the number of feasible digraphs with which a comparison of the optimaldigraph Gs has to be done during the calculation of the stability radius %maxs (p), we use thefollowing simple bound from [25].

If there exists a digraph Gk such that

rks ≤lpt − lpsq

, (34)

then digraph Gt need not to be considered during the calculation of %maxs (p). Indeed, it iseasy to show that, due to (34), rts cannot be smaller than rks during the calculation of (20).

Similarly to (34) for the makespan, we use the following bound (35) for the mean flow timecriterion, and as a result, the number of comparisons of digraphs, which need to be performedfor calculating %Σ

s (p), were considerably reduced.

21

If there exists a digraph Gk such that

Rks ≤Lpt − Lpsnq − n

, (35)

then digraph Gt need not be considered during the calculation of %Σs (p).

So, for each considered criterion we compare the optimal digraphGs = Gi1 consecutively withthe digraphs Gi2 , Gi3 , . . . , Giλ from P (G) in non-decreasing order of the objective functionvalues, i.e. F (Gi1) ≤ F (Gi2) ≤ . . . ≤ F (Giλ), where F = Cmax (or F =

∑Ci). If for

the currently compared digraph Gk = Gir inequality (34) holds for Cmax (or inequality (35)for

∑Ci), we have the possibility to exclude the digraphs Gir , Gir+1 , . . . , Giλ from further

considerations.

Note that the developed software is rather general. In principle, it allows to calculate theexact or approximate values of %maxs (p) and %Σ

s (p) for most scheduling problems (since thereexists a possibility to represent them as extremal problems on a disjunctive graph, see Intro-duction). The only ’theoretical’ requirement for the considered problems is the prohibitionof preemptions of operations. However, in the experiments we are forced to take into accountalso ’practical’ requirements: the running time and the memory of the used computers. Itshould be noted that the most critical parameter of the problem under consideration is thenumber |D| of pairs of disjunctive arcs in G because the whole number of feasible (withouta circuit) and infeasible (with circuits) digraphs generated by G is equal to 2|D|. Moreover,for each feasible digraph Gk, we have to find all dominant paths for Cmax and (what isessentially larger) all subsets of the set of dominant paths for

∑Ci. Therefore, we restricted

our experiments to scheduling problems with q = 12, q = 20 and q = 36 operations.

First, we present some computational results for the stability radii of small randomly gener-ated job shop problems. When generating the test problems, we distributed the operationsevenly to the machines and then the operations assigned to the same machine have beenevenly distributed to the jobs. The processing times of the operations are uniformly dis-tributed real numbers from the interval [10,100]. If an instance has more than one optimalschedule, we calculate the stability radius for each of them.

In Table 1 we present the computational results for job shop problems with q = 12 and q = 20operations. For the problems with 12 operations, we generated 50 instances for each of theconsidered 5 format types n x m. The stability radii %maxs (p) and %Σ

s (p) were calculated on aPC 386 usually within some seconds using only internal memory. For the problems with 20operations we generated only 10 instances for each of the considered 5 format types n x m.Since the number of generated feasible and infeasible digraphs was equal to 220 = 1048576,and we had to use external memory on a hard disk, the running time for some of the instanceswith 20 operations achieved two hours on a PC 486. Table 1 gives the minimum (MIN),average (AVE) and maximum (MAX) values of the stability radius (columns 2, 3 and 4) andthe minimum, average and maximum values of the stability radius divided by the average

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processing time of the instance (columns 5, 6 and 7). Moreover, column 8 contains theaverage values of the percentage of digraphs, which may be a competitive digraph (COMP)for the optimal one. In columns 9 and 10 the average and maximum numbers of optimalsolutions (NOS) and in columns 11 and 12 the average and maximum numbers of feasiblesolutions (NFS) of the instances of each considered type are given.

It can be observed that for the small problems with 12 operations an optimal makespanschedule is more stable than an optimal mean flow time schedule. Concerning the averagevalues of the stability radii for the problems with 20 operations, the differences betweenboth criteria are not so large. From column 8 it can be seen that the average percentagevalues of competitive digraphs for the small instances with the mean flow time criterion werebounded by 7.31 % and for the larger instances with 20 operations even by 2.42 %. Whenminimizing the makespan, these values are larger (even 51.93 % for instances with 6 jobs, 6machines and 20 operations), but the latter results are mostly due to the very large numbersof optimal makespan schedules. Moreover, it can be seen that an optimal mean flow timeschedule is usually uniquely determined in contrast to the makespan criterion. Therefore,for the makespan problems it has sense to look for an optimal schedule that has the largeststability radius.

More detailed results for further classes of randomly generated job shop problems havebeen given in [25], where the stability radii have been calculated for about 10000 randomlygenerated job shop problems with no more than 25 pairs of disjunctive arcs. Additionallywe mention that in the experiments we obtained only very seldom a stability radius equalto zero for the makespan criterion (for less than 10 optimal makespan schedules) and neverfor the mean flow time criterion. On the other hand, for the makespan criterion an infinitelylarge stability radius was not seldom obtained, at least essentially more often than a zerostability radius.

From the above results it follows that we can calculate the exact value of the stability radiuson the basis of a direct enumeration of the digraphs G1, G2, . . . , Gλ only for a very smallnumber of pairs of disjunctive arcs (about 30). On the other hand, in most consideredexamples the competitive digraphs had objective function values that are rather close to theoptimal one. So, for larger examples we decided to look for an upper bound of the stabilityradius, using a branch and bound algorithm for constructing k ’best’ schedules (an optimalschedule or schedules and other feasible schedules with objective function values that arerather close to the optimal one).

Next, we considered the well-known job shop test problem J//Cmax from [3] with 6 jobs and6 machines. For this test problem each job has to be processed on each machine exactly onceand so we have q = 6 x 6 = 36 and |D| = 6 x 6

2= 90. By the branch and bound algorithm we

constructed k = 50 best schedules: 22 of them are optimal with Cmax = 55 and the other 28schedules have a makespan value equal to 56. We calculated an upper bound of %maxs (p) foreach optimal makespan schedule. It turned out that 18 of them have a zero stability radius

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Table 1: Computational results for job shop problems with 12 and 20 operations

n x m RADIUS RADIUS/pAV E COMP NOS NFSMIN AVE MAX MIN AVE MAX AVE MAX AVE MAX

1 2 3 4 5 6 7 8 9 10 11 12problems with q = 12 operations; maximum flow time (makespan)

3 x 3 0.01 4.33 15.71 0.02 7.85 31.51 0.96 1.98 12 443.68 8764 x 4 0.01 3.13 20.51 0.02 5.75 33.84 1.03 2.52 11 531.86 12965 x 5 0.01 5.72 21.67 0.02 12.64 53.64 2.49 4.20 40 183.84 2886 x 6 0.00 5.62 34.90 0.00 9.96 63.13 8.41 2.94 12 42.62 647 x 7 0.05 5.57 40.99 0.12 10.64 78.47 11.15 3.84 12 32.00 32

problems with q = 20 operations; maximum flow time (makespan)6 x 6 0.01 0.62 4.26 0.02 1.09 7.40 51.93 21.50 78 100.00 1007 x 7 0.07 1.76 11.16 0.12 3.45 23.59 10.83 15.60 43 100.00 1008 x 8 0.07 3.43 12.66 0.13 6.00 17.80 10.20 17.00 70 3287.30 70209 x 9 0.00 3.97 11.52 0.00 6.91 22.14 4.43 28.90 144 2932.80 3456

10 x 10 0.18 3.33 21.90 0.32 5.97 41.38 1.54 12.40 48 435.90 768problems with q = 12 operations; mean flow time

3 x 3 0.09 2.95 14.57 0.14 5.27 22.98 0.68 1.12 3 443.68 8764 x 4 0.00 2.17 9.35 0.00 3.94 15.47 0.59 1.08 2 531.86 12965 x 5 0.08 3.59 12.33 0.16 6.43 24.88 1.55 1.02 2 183.84 2886 x 6 0.05 5.86 21.72 0.07 10.75 36.34 5.43 1.00 1 42.62 647 x 7 0.03 5.08 20.88 0.07 9.75 43.53 7.31 1.00 1 32.00 32

problems with q = 20 operations; mean flow time6 x 6 0.33 1.28 5.10 0.59 2.25 8.67 2.27 1.10 2 100.00 1007 x 7 0.23 1.33 6.57 0.40 2.32 11.19 2.42 1.20 2 100.00 1008 x 8 0.26 1.86 6.54 0.51 3.28 11.85 0.03 1.20 2 3287.30 70209 x 9 0.60 2.20 4.41 1.10 3.84 8.22 0.10 1.10 2 2932.80 3456

10 x 10 0.46 3.83 8.05 0.75 6.79 13.69 0.57 1.00 1 435.90 768

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Table 2: Computational results for job shop problems with 6 jobs, 6 machines and 36operations

RADIUS (UB) RADIUS (UB)/pAV E COMP NOSMIN AVE MAX MIN AVE MAX MIN AVE MAX

0.00100 0.12939 0.87455 0.01777 2.40814 15.61688 12.30 1 ≥ 15.24 ≥ 50.00

and the other four optimal schedules have an upper bound of %maxs (p) equal to 0.08333. Theexistence of unstable optimal schedules for this test problem is implied mainly by the factthat its processing times are integers from 1 to 10.

We also randomly generated 50 instances with 6 jobs, 6 machines and 36 operations. Againeach job has to be processed on each machine exactly once as in the test problem from[3], but in contrast to latter problem, the processing times were uniformly distributed realnumbers between 1 and 10. For each generated problem with 36 operations we costructed50 best schedules (for makespan criterion) on the basis of the branch and bound algorithmand calculated upper bounds of %maxs (p) for each optimal makespan schedule which wasconstructed. Some obtained results are given in Table 2. Note that 45 instances from Table2 have more than one optimal (makespan) schedule, and among them 7 instances have50 or even more optimal schedules. We calculated also the differences between the upperbounds of %maxs (p) for different optimal schedules s of the same instance from Table 2 ifthis instance has two or more optimal schedules. The maximum such a difference was equalto 0.84636, the average difference was 0.11709 and some optimal schedules had the samestability radius. Among 50 instances presented in Table 2, there were no optimal schedulewith a zero stability radius. However, in other series of instances of the same format type 6x 6 with 36 operations we obtained two instances with zero stability radii: one with two andsecond with four unstable optimal schedules. But again unstable optimal schedules arisedessentially more seldom than for the test example from [3] with integer processing times.

Unfortunately, the developed software did not allow us to find %Σs (p) for most of the above

instances with 36 operations since the calculation of the stability radius for the mean flowtime criterion is essentially more time consuming than for the makespan.

8 Related Approaches

In spite of obvious practical importance, the literature on stability analysis in schedulingis rather small. Outside the considered approach we can mention [9, 17, 19]: in [9] thesensitivity of a heuristic algorithm with respect to the variation of the processing time of onejob has been investigated, in [17] the stability of an optimal permutation schedule for the flow

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shop problem F//Cmax has been considered, and in [19] the results for the traveling salesmanproblem were used for a one machine scheduling problem with minimizing tardiness.

On the other hand, there exist a lot of papers presenting different approaches to stabilityanalysis of optimization problems, and in this section we try to provide only a sketch of someapproaches to stability analysis, which are close to the subject of this paper.

A related approach to stability analysis for so-called linear trajectory problems (such as thetraveling salesman problem, the assignment problem, the shortest path problem and someother discrete optimization problems has been initiated in [6, 12, 13, 14, 21, 30] and developedin some other papers (see [27] for an extensive survey). Most results have been obtained forthe stability radius of the whole set of solutions (optimal trajectories), i.e. for the largestradius %(p) of an open ball in the space of the numerical input data p such that a newoptimal trajectory does not arise. A formula for calculating the stability radius %(p) of theset of all solutions of the traveling salesman problem has been obtained in [12, 13] and theextreme values of %(p) have also been determined. Analogous results for a similar problemwith a bottleneck objective function have been derived in [6]. In [8] a specific transformationof a branch and bound algorithm for a traveling salesman problem for calculating %(p) wassuggested. In [4] a polynomial algorithm has been proposed for calculating the stabilityradius of the whole set of solutions of some extremal problems on matroids and on theintersection of two matroids.

It should be noted that the investigation of the stability radius of one optimal trajectoryof such a problem has the following drawback: the stability radius of an optimal trajectoryis equal to zero if at least one further optimal trajectory exists. As it has been shown inSections 4 and 5, this is not the case for problems G//Cmax and G//

∑Ci, for which the

existence of two or more optimal semiactive schedules is only a necessary condition to have azero stability radius (see Theorems 1 and 6). Such a property follows from essentially morecomplicated graph-theoretical representation of a scheduling problem. This issue is confirmedby our computational experiments: although the instances of problem J//Cmax had usuallya lot of optimal semiactive schedules, only a few of them had zero stability radii. For meanflow time criterion we even did not obtain optimal schedule with zero stability radius in ourextensive experiments (although due to Theorem 6 it is not difficult to construct unstableoptimal mean flow time schedule at least for small number of operations).

The complexity of calculating %(p) has been studied in [5, 7, 18, 24]. In [18] it has beenshown that the problem of determining the arc tolerance for a combinatorial optimizationproblem is as hard as the problem itself (the arc tolerance is the maximum change, i.e.increase or decrease, of a single weight, which does not destroy the optimality of a solution).This means that in the case of the traveling salesman problem the arc tolerance problem isNP-hard even if an optimal tour is given. Moreover, in [5] the NP-hardness of the problemof calculating %(p) for the polynomially solvable shortest path problem in a digraph withoutnegative circuits has been proven. On the other hand, in [24] it has been shown that the

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stability radius of an approximate solution may be calculated in polynomial time if thenumber of unstable components grows rather slowly, namely as O(logN), where N is thenumber of cities in the traveling salesman problem. In [15, 16] it has been argued that it israther convenient from computational point of view to use the set of k shortest tours whenapplying stability analysis to the traveling salesman problem.

9 Conclusions

The calculation of the stability radius based on a direct comparison of the paths for Cmaxand subsets of paths for

∑Ci is very complicated and time consuming. Nevertheless, such an

’unpractical’ calculation for sample problems allows to derive some properties of schedulingproblems, which may be used in practically efficient methods for determining lower andupper bounds of the stability radius of an optimal schedule.

For example, our computational experiments show that the most optimal schedules arestable, i.e. rather small errors in determination of the processing times do not influenceon the property of a schedule to be optimal, or more precisely: stability radius of such aschedule is strictly positive and so there exists a ball with the center p of processing times inthe space of input data, within which the schedule remains optimal. Moreover, on the basisof the above computational results (see [25] for detail), one can make the conclusion that anoptimal schedule for the makespan criterion is usually more stable than that for the meanflow time criterion.

On the other hand, stability radius may be equal to a very small positive real number andthus with high probability a schedule, which is (a priori) optimal, may not be the ’best’ onein reality. In the latter case, a better scheduling strategy consists in restricting computationsto the construction of an approximate or heuristic schedule (or schedules), which is usuallynot very time consuming in comparison with the construction of an optimal schedule.

Moreover, the surveyed approach gives not only the exact value or bounds of the stabilityradius but also ’competitive’ schedules (’competitive’ digraphs), which along with an optimalschedule have to be considered as possible candidates for the practical realization, when thestability radius or its upper bound is less than the possible error of the processing timesknown in advance. Theoretical results presented in Sections 3, 4 and 5 and developedsoftware may be the basis for solving scheduling problems under conditions of uncertaintywhen only lower and/or upper bounds of the processing times of the given operations areknown in advance.

The above computational results also show that an optimal mean flow time schedule isusually uniquely determined, while two or more optimal makespan schedules are very usual(at least for considered job shop scheduling problems with q = 12, q = 20 and q = 36). So,in the latter case it makes sense to look for an optimal makespan schedule with the largest

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value of the stability radius, since the difference of the stability radii for different optimalschedules of the same job shop problem may be very large for the makespan criterion. Anoptimal schedule with larger stability radius has a better chance to be optimal in its practicalrealization. On the other hand, our computational experiments show that this is not validfor the mean flow time criterion, for which one can be satisfied by the first constructedoptimal schedule, because even if there are two or more optimal mean flow time schedules,they usually have the same value of the stability radius.

Moreover, there exist scheduling problems for which one can look for an optimal makespanschedule with an infinitely large stability radius. In particular, if we can influence theproperties of the processing system (i.e. the technological routes of the jobs, the numberof used machines, the distribution of the operations to the machines, etc.), we can design aprocessing system with an optimal makespan schedule that has an infinitely large stabilityradius. In this case the variations of the processing times have no influence of such a scheduleto be optimal. For some real-world scheduling problems, such a property may be desirable.

Since stability radius of an optimal schedule may be very small, it is important to make theerrors in the determination of the processing times as small as possible in order to guaranteethe real optimality of a schedule at hand. Note that often only integer processing timesare considered by scheduling theorists and since most benchmarks for scheduling problemsalso have only integer processing times, many scheduling algorithms use such property ofprocessing times essentially. On the basis of the above results, consideration of real processingtimes is higher appreciated. We can illustrate this issue by the test 6 x 6 problem from[3], which has 18 unstable optimal semiactive schedules and only four stable ones. (Frompractical point of view only stable optimal schedule may be considered as ’really optimal’since there exists a positive error in calculation of processing times.) The existence of largenumber of unstable optimal semiactive schedules for the test 6 x 6 problem was impliedmainly by its processing times, which are integers between 1 and 10 (such an input datamay be considered as very raugh estimation of real processing times). If the processing timesare real numbers between 1 and 10, randomly generated job shop problems of the type 6 x6 have usually only stable optimal makespan schedules (see Table 2).

In conclusion we present some topics for future research. It would be useful to improve thebounds presented in Section 6 in order to restrict further the number of digraphs Gs, withwhich an optimal digraph has to be compared, while calculating its stability radius. A morecomplex question is to find formulas for calculating the stability radius or at least its boundswithout considering the paths of digraphs Gk ∈ P (G).

For practical aims it is useful to refine different branch and bound algorithms for constructingthe k best schedules (instead of one, which is usually constructed) and to combine such acalculation with a stability analysis on the basis of the discussed ideas or some others (notethat as it has been shown for the traveling salesman problem [16] and for binary linearprogramming [31], the running time of such a variant of a branch and bound algorithm

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grows rather slowly with k). After that one can calculate upper or/and lower bounds of thestability radius of the optimal schedule.

Moreover, even if we have not the possibility to find an optimal schedule and only approxi-mate or heuristic schedules have been constructed, we can investigate the ’stability radius’of the best of them in comparison with the other schedules that have been constructed. Aninteresting topic is also to restrict the calculations only to a part of the paths of the digraphsfor the makespan criterion and only to a part of the subsets of paths for the mean flow timecriterion.

Acknowledgement: The authors are grateful to G.V. Andreev for his qualified contribu-tions to the developed software and to the simulation study.

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