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Mathl. Comput. Modelling Vol. 26, No. 3, pp. 67-86, 1997 Copyright@1997 Elsevier Science Ltd Printed ln Great Britain. All rights rsserved PII: SO8957177(97)00132-S 98957177/97 817.99 + 0.06 Optimal Makespan Scheduling with Given Bounds of Processing Times TSUNG-CHYAN LAI+ Department of Industrial and Business Management National Taiwan University, Taipei 106, Taiwan, R.O.C. Y. N. SOTSKOV+$ Institute of Engineering Cybernetics, Belarusiau Academy of Sciences Surgauov St. 6, 220012 Minsk, Belarus N. Yu. SOTSKOVA Faculty of Applied Mathematics and Computer Science Belarusiau State University 220080 Minsk, Belarus F. WERNER~ Fakultiit fiir Mathematik, Otto-von-Guericke-Universitiit PSF 4120, 39016 Magdeburg, Germany (Received and accepted June 1997) Abstract-This paper deals with the general shop scheduling problem with the objective of mini- mizing the makespan under uncertain scheduling environments. The proceasing time of au operation is usually assumed to take a known probability distribution function when dealing with uncertain scheduling environments. The scheduling environments that we consider in this paper are so uncer- tain that all information available about the processing time of an operation is an upper and lower bound. We present an approach to deal with such a situation based on an improved stability analysis of an optimal makespan schedule and demonstrate this approach on an illustrative example of the job shop scheduling problem. Keywords--General shop scheduling, Makespan, Mixed graph, Uncertain processing times. 1. INTRODUCTION Deterministic sequencing and scheduling models are introduced for scheduling environments in which the processing time (duration) of each operation processed by a machine is assumed to be a constant. Difficulties arise when the processing time of an operation may vary due to a change in a dynamic scheduling environment. In such an uncertain environment, stochastic models are often introduced, where the duration of an operation is assumed to be a random variable with a known probability distribution function. Difficulties may still arise in some scenarios. First, we may not have enough prior information to characterize the probability distribution of a random processing time. Second, even if the probability distribution function of a random processing time is known a priori, the distribution function is useful only for a rather large number of realizations iSupported by National Science Council of Taiwan under NSC 862416H992-002. tSupported by Deutsche Porschungsgemeinschaft (Project ScheMA) and by INTAS (Project 93257). 67

Optimal makespan scheduling with given bounds of processing times

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Page 1: Optimal makespan scheduling with given bounds of processing times

Mathl. Comput. Modelling Vol. 26, No. 3, pp. 67-86, 1997 Copyright@1997 Elsevier Science Ltd

Printed ln Great Britain. All rights rsserved

PII: SO8957177(97)00132-S 98957177/97 817.99 + 0.06

Optimal Makespan Scheduling with Given Bounds of Processing Times

TSUNG-CHYAN LAI+ Department of Industrial and Business Management

National Taiwan University, Taipei 106, Taiwan, R.O.C.

Y. N. SOTSKOV+$ Institute of Engineering Cybernetics, Belarusiau Academy of Sciences

Surgauov St. 6, 220012 Minsk, Belarus

N. Yu. SOTSKOVA Faculty of Applied Mathematics and Computer Science

Belarusiau State University 220080 Minsk, Belarus

F. WERNER~ Fakultiit fiir Mathematik, Otto-von-Guericke-Universitiit

PSF 4120, 39016 Magdeburg, Germany

(Received and accepted June 1997)

Abstract-This paper deals with the general shop scheduling problem with the objective of mini- mizing the makespan under uncertain scheduling environments. The proceasing time of au operation is usually assumed to take a known probability distribution function when dealing with uncertain scheduling environments. The scheduling environments that we consider in this paper are so uncer- tain that all information available about the processing time of an operation is an upper and lower bound. We present an approach to deal with such a situation based on an improved stability analysis of an optimal makespan schedule and demonstrate this approach on an illustrative example of the job shop scheduling problem.

Keywords--General shop scheduling, Makespan, Mixed graph, Uncertain processing times.

1. INTRODUCTION

Deterministic sequencing and scheduling models are introduced for scheduling environments in

which the processing time (duration) of each operation processed by a machine is assumed to be a constant. Difficulties arise when the processing time of an operation may vary due to a change

in a dynamic scheduling environment. In such an uncertain environment, stochastic models are often introduced, where the duration of an operation is assumed to be a random variable with a known probability distribution function. Difficulties may still arise in some scenarios. First, we may not have enough prior information to characterize the probability distribution of a random processing time. Second, even if the probability distribution function of a random processing time is known a priori, the distribution function is useful only for a rather large number of realizations

iSupported by National Science Council of Taiwan under NSC 862416H992-002. tSupported by Deutsche Porschungsgemeinschaft (Project ScheMA) and by INTAS (Project 93257).

67

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68 T.-C. LAI et al.

of similar scheduling environments, but is of little practical sense for a unique realization or for a small number of similar realizations.

This paper deals with a model of one of the more realistic scheduling scenarios in which in a practical realization, the processing time of an operation may take any value between a known lower and upper Lund. A de~rministic model is a special case of this considered model in which lower and upper bounds are equal. This considered model can also be interpreted as a stochastic model under ‘strict uncertainty’, where there is no sufficient a priori information about the probability distribution function of a random processing time, or more precisely, the random processing time will fall between a given lower and upper bound with probability one.

Let us consider a general shop scheduling problem in which there is a set of partially ordered

operations & = (1,2, . . . ,q) to be processed without operation interruptions by a given set of

machines M = {Ml,Mz,... , M,,,}. We assume that each operation is assigned exactly to a machine and each machine at any time can process at most one operation. Let pj denote the processing time (duration) of operation j E & and cj denote its completion time. We assume that pj 2 0 for any j E Q, where j E Q is a dummgt operation if pj = 0.

For each machine Mk E M, our goal is to determine a sequence of operations Qk to be processed

by Mk, where Q = Ur!rQk and Qk n QZ = 0, if k # 1. Such a set of m sequences satisfying both

given precedence and capacity constraints is a fecrsible schedule. The objective is to determine an optimal (makespan) schedule, i.e., a feasible schedule with a minimum value of the makespan max(ci : i E Q} among all feasible schedules.

Precedence constr~nts are defined as follows: given two operations i, j E Q, we assume that i ---t j denotes that operation i is a predecessor of operation j, i.e., if i + j, then the inequality

ci + Pj 5 Cj (1)

holds for any feasible schedule. Given that {Qk : k = 1,2,. . . , m} is a partition of Q, we have the following capacity constraints:

Ci +pj 5 Cj or Cj +pi I Ci, (2)

wherei,jEQandk=1,2 ,..., m. A mixed (disjunctive) graph is often introduced to model a difficult determini~ic scheduli~

problem (see [l-3]). We follow this approach and represent the input data of a general shop scheduling problem by a mixed graph G = (Q, A, D), where the set Q of operations is the set of vertices, precedence constraints (1) are represented by the set A of (directed) arcs:

A={(& j) : i ---t j; i, j E Q; there is no k E Q such that i -_* k and k + j simult~~~ly hold),

and capacity constraints (2) are represented by the set D of (undirected) edges:

D = {[i, j] : i, j E Qk; k = 1,2,. . . , m; neither i -+ j nor j 3 i holds}.

If the processing times of all operations are known, we associate each nonnegative weight pi (operation duration) with each vertex i f Q in G = (Q, A, I)) to obtain the weighted mixed graph denoted by G(p) = (Q(p), A, D).

To accomodate dummy operations in the framework of the mixed graph, we assume that each dummy operation ‘has to be processed’ by a special dummy machine with a zero processing time, where the number of dummy machines is equal to the number of dummy operations. Therefore, each dummy operation is an isolated vertex in the graph (Q, $9, 13).

While solving the scheduling problem, each edge [i, jl E D has to be oriented, where the choice of arc (i, j) (or arc (j, i)) specifies that operation i precedes operation j on their common machine it& E M and the first inequality from (2) holds (or, respectively, j precedes i and the

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Optimal Makerpan Scheduling 69

second inequality from (2) holds). Letting G, = (Q, A U Dd, 0) denote the digraph generated from the mixed graph G by orienting all edges of the set D, digraph G, is called feasible if and only if G, contains no circuits.

It is easy to see that each feasible digmph G, uniquely defines a feasible schedule s, and vice

versa.

Let A(G) = {Gr,Gz, . . . , GA} be the set of all feasible digraphs. Given a vector p = (pi ,pz, . . . ,pq) of processing times, a feasible digraph G, E A(G) corresponding to Gd(p) = (Q(p), A U D,, 0) uniquely defines the earliest completion time G(S) of each operation i E Q along with the makespan value max{ci(s) : i E Q} of schedule s. Since pi is nonnegative for each i E Q, the running time of calculating cl(s), cz(s), . . . , c*(s) may be restricted by O(q2) (see [3, p. 2851). The maximal weight of a path in the digraph G,(p) (called critical weight) defines the makespan max{s(s) : i E Q} of schedule s. The path in G,(p) with a critical weight is called a critical path.

Given a fixed vector p = (p~,pz, . . . , pq) of the processing times, in order to construct an

optimal makespan schedule for the general shop scheduling problem denoted by G//Cm=, one may first enumerate (explicitly or implicitly) all feasible digraphs Gr(p), Gz(p), . . . , GA(~) and then select an optimal digraph with a minimal value of the critical weight among all X feasible

digraphs. It is worthwhile to mention that the feasibility of a schedule s (and the feasibility of a di-

graph G,(p)) is independent of the vector p = (~1, ~2,. . . ,pp) of processing times, while the optimality of a schedule (and the optimal@ of a digraph) is dependent on p. In other words, the

setS={1,2,..., X} of feasible schedules is completely defined by the mixed graph G = (Q, A, D)

(without weights p), while the information on the vector p of processing times is needed (on de- termining) whether a schedule k E S is optimal or not, i.e., the optimality of a schedule is defined by the weighted mixed graph G(p) = (Q(p), A, D).

If vector p of the processing times is not known exactly before scheduling (e.g., the processing times may vary in a practical realization), different realizations may result in different critical

paths in the digraph G,. Let I? and I!&, respectively, denote the sets of all paths in the di- graph (Q, A, 0) and in the digraph G, E A(G). We denote by [p] the set of vertices which form a path p in a digraph and by P’(p) the weight of this path: F’(p) = &,l pi.

Given a digraph G,(p), ci(s) is the maximum weight among all the paths in &a ending in vertex i E Q. While calculating G(S), i E Q, we need only to consider a subset of fi8 due to the following binary relation. Path u E I& ds dominated by path ~1 E fi# if and only if the set [v] is a proper subset of the set [p] (i.e., [v] c [P]), where [v] c [p] means that [v] 5 [cl] and [v] # [P]. The above dominance relation is a strict onier relation, where transitivity holds since [v] C [J.J]

ad b-4 c 171 imply 14 c [ 7 , and also antirefEtivity holds since [v] c [v] does not hold for any 1 path V.

Let Ha denote the set of all paths u E I& such that there is no path Jo E i% dominating path Y:

H, = {V E fi8 : inclusion [u] c [p] does not hold for any path p E I&} .

The set H s fi is defined similarly. Since digraph G, contains no circuits, this dominance relation defines a lattice on the set of paths fi8, and the set of paths H, is uniquely defined: it is a maximal set in the lattice fir,.

Typically, the cardinality X of the set A(G) is very large since the trivial upper bound X 5 21D1 could be tight. However, we need only to consider some subset B of the set A(G) : B G A(G). Since pi 2 0 for all i E Q, we obtain the equality

and digraph G,(p) has the minimal critical weight within the set B E A(G) if and only if

(3)

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70 T.-C. LAI et al.

For the case B = A(G), equality (3) p rovides a criterion of optima&y of a schedule s (if vector p is I&d).

The inadequacy of a deterministic scheduling problem in modelling real-world situations was emphasized in several recent publications (e.g., [4-6]). In this paper, we allow the duration pi of an operation i E Q to assume any value in the fixed closed interval [ai, bi], where 0 5 ai < bi, i.e.,

ai I pi I bi, i E Q. (4)

Such a general shop scheduling problem with uncertain processing times will be denoted by

G/oi < pi 5 W&x. As w&s already mentioned, problem G//Cmax is a special case of problem

G/oi < pi 5 bi/Cmax with ai = bi for each i E Q. Also, one can interpret pi in problem G/ai 5 pi 5 hi/Cm, as a random variable zi with the following cumulative distribution function F*(t):

fi(t) = P(zi < r) = 0, ift<aj, I ,

if t b, = t.

In the framework of stochastic scheduling [6, pp. 167-2521, each random variable zi associated with the processing time of operation i E Q (and perhaps similar random variables associated with release dates and/or due dates) is assumed to have a known probability distribution function. For example, a stochastic variant of problem G//Cmax with exponential distribution functions

with rates cri, i E Q, is denoted by G/pi N exp(ai)/EC,,, where the objective is to minimize

the expected makespan EC,, of a schedule using an appropriate scheduling policy.

Note also that the processing times pi, i E Q, in the problem G/ai 5 pi I hi/Cm, are

determined by ‘external factors’ in contrast to the problem with controllable machine speeds

considered in [7-111, where the processing times of the operations are under the control of a decision maker and the objective is to choose suitable processing times in order to minimize a given function trading off between the profits due to the reduction of the makespan (7-111 (or mean flow time [ll]) and the costs for increasing the machine speeds.

The approach we present in this paper for solving problem G/ai < pi 5 hi/C,,,, is based on an improved stability analysis of an optimal schedule. Stability analysis has been studied in [12-151 for the makespan criterion, in [14,16] for the mean flow time criterion, and surveyed in [17,18]. The paper is organized as follows. In Section 2, we demonstrate some preliminary ideas of our approach using a small example of a job shop scheduling problem. Section 3 deals with the required mathematical background for later presentations. In Section 4, we present the main formula and an algorithm for solving problem G/ai 5 pi 5 hi/Cm,. A summary and some concluding remarks are provided in Section 5.

2. PRELIMINARY ANALYSIS

As follows from Section 1, an optimal digraph G, E A(G) provides a solution to problem

GllCmax. In other words, an optimal digraph defines a set of m sequences of operations Qk pro- cessedbymachineM~,k=1,2,... , m, with a minimal value of the makespan among all feasible schedules given a known vector p = (p~,pz, . . . ,pq) of the processing times of the operations. Hereafter, we often use an optimal digraph G, instead of an optimal schedule s since digraph G, E A(G) represents a set of m optimal sequences in a rather compact form. Note however, that we can manage also without the term ‘digraph G,’ by using only the term ‘schedule s’.

Now, we define a solution to problem G/oi I pi 5 hi/C,,. Let R”+ be the space of q-dimensional nonnegative real vectors

RQ+=(z=(21,52,...,2q):5i50,i~Q},

and T denote a (feasible) polytope in the space Rt defined by inequalities (4)

T = {z = (q, x2,. . . , zq) : ai 2 xi 5 bi, i E Q} .

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Optimal Makespan Scheduling 71

A set A*(G) C A(G) of feasible digraphs is a solution of problem G/ai < pi 5 hi/Cm, if this set contains at least one optimal digraph for each vector p E T of processing times. Note again that instead of considering a set of digraphs A*(G), we can consider directly a set of schedules S.ES={1,2,..., X) which can be induced by A*(G):

s’ = {k : Gk E A*(G)}.

Obviously, the whole set A(G) may be considered as a solution of problem G/ai 5 pi 5

WC,,,, with any given feasible polytope T E R”+. However, such a solution may be redundant: polytope T may contain no point p, where some digraph from the set A(G) is optimal. Moreover, a decision maker may have difficulties dealing with such a vast set of possible candidates for practical realizations. It is, therefore, useful to look for a ‘minimal solution’ A*(G) C A(G) of problem G/ai < pi 5 bi/Cmax+ In other words, A*(G) is a minimal optimal set if and only if any proper subset of A*(G) is not a solution. Note that A*(G) may not be unique since there may exist two or more optimal digraphs for some vector p E T of processing times.

Table 1, which follows, summarizes the mixed graph approach to several formulations of the general shop scheduling problem in accordance with the availability of the information on the vector p of the processing times. Note that row 1 of Table 1 is on the mass general shop scheduling problem, where the information requirement on p is that p E R$.

Table 1. Optimal makespan scheduling with different information about proceaeing times.

Individual problem

Any digmph G, E A(G) may become optimal in some realization of the process. This is true since for each critical path p E H,, we can set pi equal to a sufficiently small real e > 0 for each i E Q” = [cl]\ Uk#s L&H~(~J[v]. Hereafter, &(p) denotes the set of all critical paths in the set Hk for digraph Gk E A(G). For such a setting of the processing times, equality (3) is satisfied with B = A(G). In particular, if Q” = Q, we can get an (artificial) trivial individual problem

G//&X&X with pi = e = 0, i E Q, where any feasible digraph G, is optimal. In this paper, we consider an individual problem G/a+ 5 pi 5 hi/Cm,, which is also very

general (see row 2, in Table 1). In one extreme case when ai = 0 and bi = oo for each i E Q, problem G/ai 2 pi 5 bi/Cmm coincides with the whole mass problem presented in row 1. In the other extreme case when ai = bi for each i E Q, problem G/ai I pi 5 hi/Cm, reduces to problem G//C&= (see row 4), a basic model studied in deterministic scheduling theory. Clearly, the more information about the processing times is available before scheduling, the ‘better solution’ may be obtained. For example, a minimal solution set reduces to a single optimal digraph G, E A(G) in the case of problem G//C,,-,= (see row 4).

Row 3 is on the individual problem G/pi - Fi(t)/ECmax, a basic model studied in stochastic scheduling theory, where each operation i E Q is assumed to be a random variable with a prob- ability distribution function Fi(t) known before scheduling. For problem G/pi N Fi(t))/ECma 9

the optimal solution may be a single digraph G, when one adopts a ‘static’ scheduling policy [6, p, 1781 or a subset of feasible digraphs A*(G) when one adopts a ‘dynamic’ scheduling policy

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72 T.-C. LAI et al.

[6, p. 1791. When a static scheduling policy is adopted, a decision maker seeks and implements an optimal schedule s which minimizes the expected makespan EC,,,, and schedule s remains unchanged during the entire process. In the case of a ‘dynamic’ scheduling policy, an initial sched- ule s is constantly revised during the process based on the updated information available [6]. We note that the minimal solution set A*(G) for problem G/ai 5 pi I hi/C,,,, may be calculated ex- actly before the realization of the process, while for problem G/pi N Fi(t))/EC,,,,, the minimal solution set A*(G) may vary and even assume up to the whole set A(G) for a lot of probability

distribution functions Fi(t). As was noted in [l], the disjunctive graph model “has mostly replaced the solution representcl-

tion by Gantt charts as described in [19]“. Some additional comments are given here to elaborate this kind of preference. While a Gantt chart is useful for the graphical presentation of a particular solution, the mixed (disjunctive) graph model is suitable for the whole scheduling process from the mixed graph G(p) (representing the input data) till a digraph G, (representing a solution s). Second, a Gantt chart is practically a representation of one particular situation when there are no changes either in a ptiori known processing times or in the calculated start times. But such a situation is ‘ideal’ (at least, it occurs very seldom). So, a Gantt chart seems more appropriate ‘after realization’ of the process (when all processing and start times are known) while ‘before re- alization’, a mixed graph and a digraph seem more useful (since they are more stable with respect to possible changes of the ‘times’). Third, while a Gantt chart is a picture in the plane, a digraph is a mathematical (abstract) object, and thus, can assume different graphical presentations. In particular, one can view a Gantt chart as a diagram of the digraph G,(p) in the plane.

It is worth noting that for all four formulations presented in Table 1, the set of feasible solutions remains the same, and therefore, the properties on feasible digraphs A(G) are of particular importance. Our approach for solving problem G/ai 5 pi 5 bi/Cmax is based on a stable property which guarantees a feasible schedule and a digraph to be optimal after some ‘possible’ variations of the processing times.

To facilitate the presentation of our main ideas, let us consider the following job shop sched- uling problem with two jobs, say Ji and Jz, and five machines M = {Ml, &, . . . , AC&}, where job J1 (job Js) consists of the set of completely ordered operations (2,3,4) (operations (5,6,7), respectively). The assignment of operations Q = {1,2,. . . ,8} to the set of machines M is as follows: Qi = (2,6}, QZ = {3,5,7}, Qa = {4}, Q4 = {l}, and Qs = (8). Operations 1 and 8 and machines M4 and Ms are dummy, where operation 1 (operation 8) denotes the start (the finish) of a schedule and so it precedes the others (all other operations precede operation 8). Such a job shop scheduling problem is denoted by J//C ,,,= provided that the processing times are fixed. One version of this problem with given bounds of processing times will be denoted by

Jlai I pi I WG,.

The input data of this instance are represented by the mixed graph G(p) = (Q(p), A, D) (see Figure l), where each processing time pi is given near vertex i E Q and vector p of the processing times is as follows: p = (0,75,50,40,60,55,30,0). For a small example such as the one considered, we can explicitly enumerate all feasible digraphs A(G) = {Gr , Gs, . . . , Gb}, calculate their makespans

max{ci(i) : i E Q} = 165, max(ci(2) : i E Q} = 250, max(ci(3) : i E Q} = 27%

max(s(4) : i E Q} = 280, max (~(5) : i E Q} = 280,

and then select an optimal digraph Gl(p) = (Q(p), AU Di, 8) with Di = {(2,6), (5,3), (3,7)} and a minimal critical weight 165 (see Figure 2). Using the formulas from [13,15], one can calculate the stability radius of the optimal digraph Gr(p). Before going further, let us define the stability radius of an optimal schedule.

Let P be the space of q-dimensional real vectors p with the maximum metric, where the distance r(p, p’) between two vectors p and p’ = (pi ,#!, . . . , pk) E P is equal to the maximal

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Optimal M&span Scheduling

p2 = 75 p3 = 50 p4 = 40

p5 = 60 ps = 55 p7 = 30 Figure 1. Weighted mixed graph G(p) = (Q(p), A, D).

c2 = 75 c3 = 125 c4 = 165

73

Pl = 0 P8 = 0

Cl = 0

c5 = 60 cs = 130 cr = 160

Figure 2. Optimal digraph G1 = (Q, AU&, 0) with completion timea q( 1) presented near vertex i E Q.

I28 = 165

absolute value of the q differences of the corresponding components, i.e.,

Let schedule s be optimal for problem G//Cmax with vector p E RQ, c RQ of processing times. A closed ball O,(p) with the radius Q and the centre p in the space P is a stability ball of schedule s, if for any vector p’ E O,(p) n R”+ of processing times, schedule s remains optimal. Due to the maximum metric used in the space R9, the set O,(p) n RQ, is a polytope for any given Q E R:. The maximum value Q&) of the radius Q of a stability ball O,(p) of schedule s is called the stability radius of schedule s, where

es(p) = max (8 E R$ : schedule s is optimal,

for any vector of processing times in the polytope O,(p) n R$} .

In what follows, we will often use (whenever appropriate) the notion ‘stability radius of an optimal digraph G, E A(G) ’ instead of ‘stability radius of an optimal schedule s E S’.

In the above example, we obtain ~1 (p) = 30, where digraph Gr remains optimal if independent variations of the processing times are no more than 30. Thus, in solving problem J/ai 5 pi 5

bi/Gsx, digraph Gi = (4, AU&, 0) remains optimal if for all possible variations of the processing

times x = (21, x2,. . . ,x9) E O,,(,)(P) = 030(p), the following inequality holds:

$% {xi - ai, bi - xi} 5 30.

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74 T.-C. LAI et al.

In such a case, the given polytope T defined by inequalities (4) in the space RI is completely contained in the stability ball &s(p) of the optimal digraph Gr : T C 030(p). In other words, digraph Gr provides a solution of problem J/ui h pi 5 hi/C,, as long as inequality (5) is

satisfied: A*(G) = {G1}.

In this case, a decision maker needs to use only schedule 1 from the set S = {1,2,3,4,5} in any possible realization and so the solution of problem Jfai 5 pi I b$&,ax turns out to be the same as for problem J//Cmax with the fixed vector of processing times p E T. The minimal optimal set consists of one schedule: S* = { 1).

Otherwise (if inequality (5) does not hold), the optimality of digraph Gr is not guaranteed within the whole given polytope T. Say there exists another feasible digraph Gk, k # 1, (we shall call it a corn@tive digraph of Gr) with a critical weight being smaller than that of digraph 01 in some realization of the process. If such a ‘superiority’ of the competitve digraph Gk occurs

when the processing times are equal to p* = (pi, p& . . . ,P, *) E T (i.e., digraph Gk instead of Gi is optimal with vector p*), we have to calculate the stability radius Q&P) of digraph Gk with the new vector p* of processing times. In the case when &v) is strictly positive, we can consider the union Os&) U O,,(,.)(p*) of the two balls instead of one ball @s(p). If the inclusion T C 03,&f U O,,(,*)(p*) holds, the problem J,/a+ < pi < bi/Gmax is solved. In such a case, a decision maker needs to use either schedule 1 or schedule k for any possible realization:

A*(G) = {Gr, Gk}.

Otherwise, we have to calculate the stability radius of a competitive digraph of digraph Gk with a new vector of processing times. Continuing in this manner, we may cover the given polytope 2’

by the union of the stability balls of some feasible digraphs. And ss a result, for any vector of the processing times in T (i.e., whenever inequalities (4) hold), we shall have at least one optimal

(makespan) schedule.

c2 = 75 c3 = 125 ~4 = 165

C8 = 270

c5 = 185 c6 = 240 c7 = 270

Figure 3. Competitive d&mph G3 = (Q, A U D3,0) of d&mph Gl = (Q, A U DI,%)

which is optimal with p = (0,75,50,40,60,55,30,0).

For the above exsmple with the original vector p = (0,75,50,40,60,55,30,0) and the corre- sponding optimal digraph Gl, two competitive digraphs of digraph G1 axe G3 = (Q, A U D3,fl)

and G4 = (Q, A U Dd, 0), where Ds = {(2,6), (3,5), (3,7)) and Lt4 = ((6,2), (5,3), (3,7)). We note that the arcs except arc (3,5) are the same for digraphs GI and Gs (ses Figure 3). As the collation of the stability radius shows, at the boundary of the ball Oso~) (namely, in the point

P* = @;,P&..., p;) = (0,45,50,70,60,25,0,0) E RF), both digraphs Gr and Gs are optimal. Note that vector p* is defined during the calculation of the stability radius on the basis of the for- mulas from [13,15]. Specifically, vector p* is obtained from vector p by decreasing the processing

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Optimal Makeapan Scheduling 75

times of operations 2,6, and 7 by the value ei (p) of the stability radius, where pz = 75 - 30 = 45, pt; = 55 - 30 = 25, and p; = 30 - 30 = 0, and by increasing the processing time of operation 4 by the value ,el (p), where pz = 40 + 30 = 70. Due to such changes in the processing times, the critical weight of digraph Gi is increased from 165 to 180 and that of digraph Gs is decreased from 270 to 180. We find that both stability radii are equal to zero for both digraphs Gi and Gs given the new vector p* of processing times.

Ram [13], it follows that the existence of two or more optimal digraphs is a necessary condition (but not a sufficient one> for the stability radius to be equal to zero. Nevertheless, the ‘u~abi~~’ of an optimal digraph is possible at the boundary of a stability @on (the stability region of schedule s is the whole set of vectors p E R$ with schedule s being optimal), where there exist at least two optimal digraphs. Such a situation occurs for the above example, namely, @iv) = es(pf) = 0. Note also that the only competitive digraph for digraph Gs is digraph Gr (and vice versa), where the stability radius of Gi at the original point p E R”+ has been already calculated.

Considering the competitive digraph G4 instead of the competitive digraph Gs also gives zero stability radii for both digraphs Gi and G4 with the corresponding vector p’ = (0,75,20,10,30,

55,60,0) of processing times. From the above discussion, it follows that another notion of the stability radius is required for

solving problem G/at 2 pi 5 hi/C,,,. While es(p) denotes the largest radius of a ball O,(p)

within which digraph G, is ‘the best’ for the whole set A(G), we need to determine the largest ball within which digraph G, is ‘the best’ for some subset B of the set of feasible digraphs A(G). Indeed, for the above example, we need to calculate the largest radius of the ball within which digraph Gs has the minimal critical weight among the feasible digraphs A(G) except diiraph Gi, which is optimal within the ball 0 p1 cP) (p) and which is already contained in the set of candidates for a practical realization: Gi E A*(G). Thus, in this case, we need to consider the set B =

A(G)\{G& In Section 3, we propose a new definition of a stability radius. Note also that taking into

account the given bounds ai and bi of possible variations of the processing time xi, i E Q may

enlarge the stability ball of an optimal digraph G,. Thii is true for the above example since inequality (5) become8 only a sufficient condition for the optima&y of digraph Gr (but not a necessary one). In Section 3, we provide both necessary and sufficient conditions for a zero (and for an infinitely large) stability radius. In Section 4, the formulas from [13,15] are generalized from the case of calculating the stability radius with 0 I pi < 00, i E &, to the case when ~iations of the processing times are given by inequ~ities (4) and some feasible digraphs have

to be excluded from the comparisons with ‘the best’ one.

3. MATHEMATICAL BACKGROUND

In [13-151, the stability radius &(p) of an optimal digraph has been investigated which denotes the largest quantity of independent variations within the interval (0, co) of each processing time pi of operation i E & such that digraph G, remains ‘the best’ (i.e., the weighted digraph G,(p) has the minimal critical weight) among all feasible digraphs A(G). For solving problem G/oi I

pi < bi/Gm,,, we need a more general notion of a stability radius since the processing time of operation i E Q falls within the given closed interval [ai, bi], 0 5 ai < ba, and competitive digraphs only belong to some subset B of the set A(G). The following generalization of the stability radius (we call it relet&e stability radius) is defined by considering the closed interval [ai, bi] instead of [O,oo) and the set B C A(G) instead of A(G).

DEFINITION 1. Let for each vector p’ E O,(p) 17 T, digraph G, E B E A(G) with p’ have the

minimal critical weight among ail digraphs of the set B. The maximal w&e of the radius e of such a ball O,(p) is denoted by ~f(p f T) and is caBed tire ~~a~~ve stub~~t~ mdOw of dj~8p~ G, (or the relative stability radius of schedule s) with respect to polytope T.

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76 T.-C. LAI et al.

Let 13 be the critical weight of digraph G, E A(G) with the vector p of processing times, I$! = max,,eH, 1*(p) = 1*(~*), where p* E H,(p). As follows from Definition 1, the relative stability radius is equal to the mazimal ewur of the given processing times pi (oi < pi 5 bi, i E Q) within which the ‘superiority’ of digraph G, is still preserved over the given set B of feasible digraphs. The following two extreme csses of such an error are of particular importance for problem G/oi 5 pi 5 hi/C,,,,. On one hand, if for any real c > 0 which may be as small as

desired, there exist a vector p’ E O,(p) f~ T and a digraph Gk E B such that lf < 1?, we have a zero relative stability radius

e,B(p E T) = 0.

On the other hand, if If’ 5 1:’ for any vector p’ E T and for any digraph Gk E B, we have an infinitely large relative stability radius

$(p E T) = 00.

Even if the maximal error of pi, in the case of bi < 00, for each i E Q is restricted by

emax=m&X{{pi-ai,bi-pi}:iEQ},

it is still possible for & @ E T) to be infinitely large as implied by Definition 1. Problem G//Cmax is one trivial such example with an infinitely large relative stability radius, where ai = pi = bi for each i E & and so polytope T degenerates to a single point, T = {p}.

To characterize the extreme values of #(p E T), we define the following binary relation which generalizes the dominance relation introduced in Section 1. Path /J dominates path v in polytope T if and only if there is no compatibility (no solution) in the following system of linear inequalities:

W/J) < l”(y),

ai 5 Xi < bi, i E Q. (6)

The relation introduced here is an extension of the dominance relation introduced in Section 1 in the sense that path p dominates path u in any polytope T C Rt if path p dominates path V. Indeed, if [v] c b], then the inequality 1’(p) < Z”(V) does not hold for any vector z E R”+. Note also that both relations coincide at least when ai = 0 and bi = 00 for each i E Q (it is easy to see that inclusion [v] c [/I] holds if and only if system (6) with ai = 0 and bi = 00, i E Q is not compatible).

Thus, we conclude that the dominance relation introdzlced in Section 1, is a special case of the

dominance relation defined by the incompatibility of system (6) when T is equal to the space R$ : ai = 0 and bi = oo for each i E Q. Hence, the phrase “path ~1 dominates path Y” is identical to the phrase “path p dominates path u in R$“.

The following claim gives a simple criterion for the compatibility of system (6).

LEMMA 1. System (6) is compatible if and only if inequality

c ai < C bj ieM\M jeM\[ccl

(7)

holds.

PROOF. By subtracting all common variables from the left- and right-hand sides of the first inequality in (6) and taking into account that a{ 5 bi for each i E Q, we obtain that system (6) is equivalent to the following system:

c xi < c Xj9 ai I xi 5 bit i E [p] U [u]. ieM\I4 jei4\l~l

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Optimal Makeqan Scheduling 77

Furthermore, system (8) is compatible if and only if it has the following solution:

2.=2? = 1 ai, if i E bl\[4 I i

bi, if i E [V]\[fi]. (9)

It is easy to see that set (9) is a solution of system (8) if and only if inequality (7) holds. i

Obviously, if Hk = Hk(P), we have Hk(P’) C Hk = Hk@) for any Vector p' E R”+ of processing times. The following claim shows that, in general, the set of the critical paths does not expand for a sticiently small variation of the processing times.

LEMMA 2. If.& # Hk(~), the ~nC~~j5~ Hk(p') c fylc(p) hoids for 8ny Vi?&Orp' E Own R”+

with Ck > E > 0 defined as fo#ows:

ek = mmc {ip(v) : JI E Hk\Hk(P))) . (10)

PROOF. Since Hk\Hk(P) # 63, we can consider any path V* E H,+ with

lp(u*) = msx{iP(V) : v E &\&(p)} .

From (lo), it follows that 1, ’ - IP( Y*) = Q-Q, and therefore, to make the difference 2; - IP(Y*) equal to zero one needs a vector P’ with a distance from vector P greater than or equal ek : ?-($.I, p’) 2 ek.

But due to the conditions of Lemma 2, we have r(p,P’) 5 e < ck. Consequently, V* # Hk(p’).

Since for any path v E Hk\Hk(P) with UP < Z~(V*), the difference 1: - Zr(v) is still greater than the product Q - ek, such a path v cannot belong to set Hk(P’). I

On the basis of the above path domination, we introduce a domination of sets of paths: the set of paths Hk do~~~u~e~ the set of paths H8 in ~~~to~ T if and only if for any path p E Ha, there exists a path Y E Hk, which dominates path ,U in polytope T.

THEOREM 1. For digraph G,, which has the minimal critical weight I{, p E T, within the

set B E A(G) of feasible digraphs, the equality ,g,“(p E T) = 0 holds if and only if there exists a dj~aph Gk f B such that Ef = li, k # s, md the set of paths Hk~) does not d5~te the set

of paths H,(p) in polytope T.

PROOF. SUFFICIENCY (IF). Let the conditions of the theorem be satisfied if there exists a digraph Gk E B such that 1:: = Ii, k # s, and i!&(P) does not dominate Ha(p) in T. We show that &(P E T) < E for any given E > 0, which may be as small as desired.

Since set Hk(P) does not dominate set H,(p) in polytope I’, there exists a path p* E H*(p)

such that no path v f &(p) dominates path p* in polytope T, i.e., system

(11)

has a solution for any path v E Hk(p). First, we make the following remark: from the compat- ibility of (ll), it follows that for the considered problem G/ai < pi 5 b+/Cmax, the trivial case with ai = bi for each i E Q does not hold, since in this case, the first inequality in (11) transforms into ZP(v) < ~P(/.J*) which is wrong: ~J’(Y) = 11 = 1: = ~P(P*>.

We construct a vector P’ = tp’,,pb, . . . , pt) with the following components:

Pi + E’, if i E [cl*] , Pi # h,

pi = pi - et, if i E {UV~~k(p)[4} \ b*l V Pi # oij

Pi, otherwise,

(12)

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78 T.-C. LAI et al.

where E’ is chosen 8s a strictly positive real number less than both value e and value

emjn = max (0, min {min {p$ --ai:~~>>ai,i~Q),min(bi-pi:&i>pi,i~Q)}).

We can also choose E’ less than eh > 0 defined in (10). More precisely, if & # &(p), then ck > 0, 8nd we can choose r? such that 0 < E.’ < min{e, ek, emin }. Otherwise, if .& = i&(p), we choose E’

such that 0 < e’ < min(6,emin ). Such choices are possible since in both above cases, Emin > 0

holds, due to the remark given after (11). The following laments Ltre the same for both c&se8 of the choice of c’ except the ‘lest step’ since ~k\~~~) = 8 in the latter case.

Since system (11) has a solution for each path v E & the first inequality in (11)

P(Y) < IX (cl*)

has a solution with 5 E R& which implies that incl~ion [p*] C [v] does not hold for 8ny path v f &(p). Therefore, from the equalities lp(y) = 12 = Zz = P(p*) and (lZ), we conclude that vector p’ is a solution of system (11) for each path v E &(p). In other words, vector p’ is a solution of the following joined system:

Thus, P’(V) < d(p*) for each v E i?&(p), and therefore,

max { ip’(V) : Y E &&J)} < 2” (p*) . (13)

The ‘lest step’ is es follows. Since p’ E O,,(p) n R”+ with 0 < E’ < ek, due to Lemma 2, we have &(lp’) 2 &(p) and, as 8 result,

l”(T) < 13f’ = m8X {l”(u) : v E &(p)} , (14

for each path T E &\@&). l?rom inequalities (13) and (Id), it follows that $ C @. Taking into account that r(p’,p) = d < (5, we conclude that @,“(p E 2’) < e.

NECJSSARY (ONLY 1~). We prove necessity by contradiction. Let us suppose that $(p E T) = 0, but the conditions of the theorem do not hold. The following Cases i 8nd ii, violating these condition, may t8ke place.

CASE i. There does not exist 8 digraph Gk: E B such that 1: = It, k # s. In the trivial cese when B = {Gd}, we have #(p E 2’) = 00 due to Definition 1. If B\{G,} # 0, we can calculate the following real number:

which is strictly positive since 9 < 2; for each Gt E B, t # s. Arguing in a similar way ae in the proof of Lemma 2, we can show that the difference I, P - 1: cannot become negative when vector p is replaced by 8r-1 arbitrary vector p” E 0,. (p) n @+. So we conclude that digraph G, remains ‘the best’ (perhaps one of the ‘best’) within the set B for any vector p” of the processing times. Due to Defmition 1, we have @F(p E 2’) 2 e* > 0 which contradicts our essumption of $@ f 2’) = 0.

CASE ii. There exists a digraph Gk E B such that 1: = $, k # s, and for any such digraph Gk, the set of paths H&(p) dominates the set of paths J&(p) in polytope T.

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

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Optimal Make~pau scheduling 79

Due to the inequality that e > e,, we get from Lemma 2 the equalities

(15)

for any vector p” E G,(p) n R$ . The claim that for any digraph Gk E B, k # s with &’ = l:, the set of paths &(p) d ominates the set of paths H,(p) in polytope T means that for any path ~1 E &(p), there exists a path V* E &(p) such that system

lx @*I < l”(P), ai<~i<bi, iEQ

has no solution. Therefore,

1” (v*) 2 ix(P), (16)

for any vector x E T. From (16) and taking into account that e < ek and c < cd, we obtain, due to Lemma 2, the following inequality:

Thus, due to (15) and (17), we have

for any digraph Gk E B, 1: = ii, k # s. Since

E < i min {lf - 1: : 1: > lr, Gt E B} ,

inequality 1: > 1: implies 1:’ > lf . Taking into account (18), we conclude that 1:’ 5 1:’ for any digraph Gk E B, and any vector p” E T with r(p,p’) 5 c. Consequently, $(p E T) 2 E > 0, which contradicts the assumption of $(p E T) = 0. I

From Theorem 1, we obtain the following lower bound of the relative stability radius.

COROLLARY 1. If G, E B and 1: = min{$ : Gk E B}, then e,“(p E T) 2 E*.

PROOF. If there exists a digraph Gk E B such that 1: = Ii, k # s, the equality ef(p E T) 1

e* = 0 holds due to Definition 1. Otherwise, bound ~f(p E T) 2 E* follows from the above proof of necessity (see Case i). I

Theorem 1 identifies a digraph G, E A(G) w h ose ‘superiority’ within the set B is unstable:

even a very small change in the processing times can make another digraph to be ‘better’ than G,. The following theorem identifies a digraph G, whose ‘superiority’ within the set B in polytope T

is ‘absolute’: any changes of the processing times within the polytope T cannot make another digraph from the set B to be ‘better’ than G,.

THEOREM 2. For digraph G, E B, we have $(p E T) = co, if and only if for any digraph Gt E B, t # s, the set of paths Ht dominates the set of paths H,\H in polytope T.

PROOF. SUFFICIENCX (IF). If e is a positive number ss large as desired, we take any vector p E O,(p) n T C RQ, and consider a path Jo E Hd such that 1: = P(p).

CASE j. If /.J E H, then inequality 1: = P(p) < 1: holds for any digraph Gt E A(G).

CASE jj. If /.J E H,\H, then due to the condition of Theorem 2, it follows that for any digraph Gt E B, t # s, there exists a path u* E Ht such that system

1” (v’) < w-4 ai<xiIbi, iEQ

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SO T.-C. LAI et al.

has no solution. In other words, the inequality P(v’) 2 1”(p) holds for any vector z E T (and for the vector p, too). Therefore, we have I$’ = P(p) < P(v*) 5 1:.

Thus, in both Cases j and jj, we have 1: = min{lr : Gt E I?}.

NECESSITY (ONLY IF). We prove necessity by contradiction. Let us suppose that ~,“(p E T) = co, but there exists a digraph Gt E B, t # s, such that the set of paths Ht does not dominate the set of paths H,\H in polytope T. Thus, there exists a path p” E H,\H such that for any path

u E Ht, the system V4 < 1” (PO) ’

ai 5 xi I bi, iEQ (19)

has a solution. Therefore, due to Lemma 1, the inequality

c ai < c bj w4\l~“l jG”l\14

holds. We consider the following vector p* = (pf ,pz, . . . ,pG) E T with

(

ai’ if i E {U[Y~~~t[4} \ [PO] , pi’ = bi, if i E [/JO] ,

pi, otherwise.

(20)

Adding to the left- and to the right-hand sides of (20) the value &lVlnlPOl bj, we obtain that inequality _ _

c ai + C bj < C bj G4\b”l jWn[~“l jEb”l

holds. Thus, we can conclude that vector p’ is a solution of the system of linear inequalities obtained by joining systems (19) for all paths v E Ht, i.e., we have

F(Y) < P’* (/JO) ’ UE Ht,

ai 5 xi 5 bi, i E Q.

Therefore, 1:. < P*(/.L~) 5 If*, and hence, we get a contradiction to the above assumption:

&(JJ E T) < r(p*‘P) I emax < 00. I

From Theorem 2, we obtain the following upper bound of the relative stability radius.

COROLLARY 2. If e,“(p E T) < co, then &(p E T) 5 emax.

PROOF. The desired bound immediately follows from the proof of necessity in Theorem 2. a

In the following section, we shall use Theorem 2 as a stopping rule in our developed algorithm for solving problem G/oi 5 pi 5 hi/Cm,, since the optimality of digraph G, E B with Q? (p E T) = oo does not depend on the vector p E T of the processing times.

4. ALGORITHM FOR PROBLEM G/ai 5 pi 5 bJCmax

From Sections 2 and 3, it follows that problem G/ai 5 pi 5 hi/Cm, may be solved on the basis of a repeated calculation of the relative stability radii #(p E T). The formulas for calculating

@l(P) = @twP E q.) were given in [13,15]. Theorem 3, which follows, generalizes these formulas for any given set B C A(G) and any given polytope T C R’+. To present the new formula, we need the following notations.

Let ~1 and v be paths in the digraphs from the set A(G). We denote the ‘symmetric differ- ence’ [/I] U [v]\b] f~ [v] of sets [p] and [v] by [p] + [Y] and calculate the following values:

A&r 4 = bi -pi, if i E bl\[~l, Pi -ai, ifiE [V]\[p].

(21)

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Optimal Makeapau Scheduling 81

Let P~,(P, ~1 be equal to zero and (pi, (cl, 4,pi2 (cl, v) , . . . ,p~~~+[~~, (p, v)) denote the sequence of the processing times pi of all operations i from the set [p] + [v] and let, for this sequence, the following inequalities hold:

For any two feasible digraphs G, and Gk, we let

&k(T) = {j‘ E Hs : there is no path v E Hk which dominates path p in polytope I!‘}.

THEOREM 3. Given digraph G, with minimal critical weight If, p E T, within the set B c h(G)

of feasible digraphs, we have

where

PROOF. From Definition 1, it follows that

~f(p E T) = inf {~(p, z) : z E T, 1,” > min{l; : Gk E B}} .

Therefore, to find the stability radius ~r(p E T), it is sufficient to construct a vector z E T

which satisfies the following three conditions.

CONDITION 1. There exists a digraph Gk E B, k # s, such that 1: = lz, i.e.,

(22)

CONDITION 2. For any given real E > 0, which may be as small as desired, there exists a vector p’ E T such that T(Z, p’) = e and 1:’ > 1:‘) i.e., inequality

(23)

is satisfied for at least one digraph Gk E B.

CONDITION 3. The distance r(p,z) achieves the minimal value among the distances between vector p and the other vectors in the polytope T which satisfy both Conditions 1 and 2 above.

After having constructed such a vector z E T, one can define the relative stability radius of digraph G,:

e,B(p E T) = ~(P,s),

since the critical path of digraph G, becomes larger than that of digraph Gk for any vector p’ E T

with positive real c, which may be as small as possible (see Condition 2), and so digraph G, no longer has the minimal critical weight among all other feasible digraphs, while in the ball O,(,,)(p E T), digraph G, has the minimal critical weight (see Condition 3). Digraph Gk satisfying Conditions l-3 is a competitive digraph for the optimal digraph G,.

To satisfy Conditions l-3 (except the inclusion z E T), we first search for a vector z = p(r) =

(P1(r),Pz(r), . * * 7 pp(r)) E RQ with the components pi(r) E {pi,pi + r,pi - r} on the basis of a direct comparison of the paths from the set H, and the paths from the sets Hk, where Gk E B.

Let the value UP be greater than the weight of a critical path in an optimal digraph G,. To satisfy equality (22), the weight of a path u E Hk has to be no greater than that of at least one

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82 T.-C. LAI et al.

path /,L E Ha, and there must exist a path v E Hk with a weight equal to the weight of a critical path of G,. Thus, if we have calculated

(24)

we obtain the equality msx IP@)(fi) = lp(‘)(v), /.bEH.

for the vector P(r) = p(ry), with the components

(25)

Pi + T”, if i E [p),

Pi(r) = pi(r”) = Pi - rV, if i E [v]\M, (26)

Pi, if i $! [II] U [v].

REMARK. Due to (24), the vector p(r) calculated in (26) is the closest one to the given vector p

among all vectors z for which equality (25) with p(r) = z holds. Indeed, to make the difference

Y(V) - mex,ea, V(p) equal to zero, one needs a q-dimensional vector z with a distance from vector p greater than or equal ry : r(p, z) 3 ry.

To reach equality (22) for the whole digraph Gk, we have to repeat calculation (24) for each path v E Hk with UP > 1,. P Thus, instead of vector p(ry), we have to consider the vector p(r) = P(r&) calculated according to formula (26), where

(27)

We next consider inequality (23). Since the processing times have to belong to polytope T 2 R:, this inequality may not be valid for a vector p’ E T if path u dominates path ~1 in polytope T. Thus, we can restrict our consideration to the subset H&(T) of the set Ha of all paths, which are not dominated by paths from the set Hk in polytope T. Hence, we can replace Ha in equality (27) by H&f).

To obtain the desired vector CC E Rq, we have to use equality (27) for each digraph Gk E A(G), Ic # s. Let T denote the minimum of such a value rGL;:

T=Tr&. =min{T& :Gk,k#s},

and let V’ E Hk. and p* E H& be paths at which value rGL. has been reached:

TQ. = TV8 = /P(V’) - IQ&u*)

lb1 + MI *

Due to the remark given after formula (26), we have obtained a lower bound of the stability radius

(28)

The bound (28) is tight: if ,#(p E T) 5 min{A&*,v*) : i E [cl*] U [v’]}, then Q,“<P E T) = T.

For example, we have Q,“(P E T) = T in (28) if e,“<p E T) I Emin* To obtain the exact value of Q~(P E T) in the general case, we can use vector z = p*(r) =

(Pi(r)9P4(r)9***9 p:(r)) with the components

{

Pi +min{r,bi -Pi}, if i E [CL],

Pi(r) = PC - min{r,pi - at}, if i E [v] \ [j.~],

Pi? if i $ [CL] u [4,

(29)

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Optimal Makespan Scheduling 83

instead of vector p(r) defined in (26). As it follows from the remark given after formula (26), such a vector p*(r) E T is the closest one to vector p among all vectors z E T which satisfy both Conditions 1 and 2.

For calculating the maximal value T for vector p* (r), we can consider each operation i from the set [p] u [ v , one by one in nondecreasing order of the values Ai(p, V) defined in (21). As a ]

result, formula (28) will be transformed into the formulas given in Theorem 3. I

Note that the formulas in Theorem 3 turn into #(p E T) = oo if &k(T) = 8 for each Gk E B. Returning to the example in Section 2, let us consider the problem J/ai 5 pi 5 bi/Cmax, whose

input data are given by the weighted mixed graph G(p) in Figure 1, together with the vectors a = (ar,as,..., as) and b = (61, bp, . . . , b8) of lower and upper bounds of possible variations of processing times p, where a = (0,35,40,20,50,45,20,0) and b = (0,100,90,110,80,80,40,0).

Since the mixed graph G is the same for the problem J//Cmax considered in Section 2, and for the new problem J/ai 5 pi 5 bi/CmBx, we have the same set A(G) of feasible digraphs. Moreover, if we start with the same initial vector p = (0,75,50,40,60,55,30,0) of processing times, we obtain the same optimal digraph Gr, presented in Figure 2. Using Theorem 3, we

can calculate the relative stability radius of this digraph: ,ottG’(p E T) = 60, where polytope T is defined by the above vectors a and b. Note that due to these bounds ai and bi of possible variations of the processing times pi, i E & = {1,2, . . . , 81, the stability radius of the digraph Gr

increased from 30 to 60 (in Section 2, we calculated pi = ~f’~‘(p E R8,) = 30). One of the two competitive digraphs, namely, digraph Gs (see Figure 3) remains also a competitive digraph of Gr for problem J/ai 5 pi < hi/C,,,,. However, the new vector of processing times p* is ss follows: p* = (0,35,90,100,80,45,20,0). This vector p* is calculated due to (29) with

r=~;(~)(p~T)=60,~=(3,4,5)~H,,andv=(2,3,5,6,7)~H~.

Next we can calculate ~~(~)‘{~lI(p* E T) = co on the basis of Theorem 2 or Theorem 3. Thus, the set of digraphs (Gi, Gs} gives a solution of problem J/a3 5 pi 5 hi/C,,:

s = {1,3}.

Therefore, a decision maker can use either schedule 1 or 3 for all possible realizations of the processing times.

In general, problem G/ai 5 pi I hi/C,, may be solved as follows. Let B denote the set of feasible digraphs which contains the minimal optimal set h*(G) for problem G/ai 5 pi 5 hi/C,,,,. On the basis of our developed algorithm, which follows, we can expand the set A’ c A*(G) starting with A’ = 0 and finishing with A’ = A*(G).

ALGORITHM.

1. Find the set B E A(G) of possible candidates for the set A*(G); 2. A’ := 0; 3. Fix one vector p of processing times, p E T; 4. Find an optimal digraph G, E B for problem G//C,,,=

with the vector p of processing times; 5. Calculate ~f(p E T); 6. If #(p E T) < 00 Then

Begin

7. Select a competitive digraph Gk E B for digraph G,; 8. Find a vectqr p+ of processing times closest to p such that

1;’ = 1: and for any small c > 0, there exists a vector p’ with 1;’ > 1; and TV&) < e;

9. A’ =: A’ u {G,}; 10. B := B \ {G,); 11. s := k;

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84 T.-C. LAI et al.

12. p := p*; 13. Goto 5

End Else

14. A*(G) := A’ u {Gb}.

Now we concretize some steps of the algorithm. In Step 1, the determination of the set B = A(G) of all feasible digraphs by explicit enumeration is possible only for a small number of edges in the mixed graph G. In the computational experiments discussed in [14], a direct enumeration has been used for ID] I 30. These experiments have shown that a competitve digraph has a critical weight that is usually very close to that of an optimal digraph. Moreover, using the simple bound from (141, we can considerably restrict the number of feasible digraphs, with which a comparison has to be done when calculating es(p). Note that a similar bound is valid for the relative stability radius $(p E T). So, for a larger cardinality of the set D, we can use a branch and bound algorithm for constructing the k-best digraphs (see [14]). As was shown for the traveling salesman problem [20,21] and for linear binary programming [22], the running time of such a branch and bound algorithm grows slowly with Ic.

In Step 3, we can 6x the processing times ss any vector from T. For example, we can use a historical vector p of processing times which helps simplify the Steps 3, 4, or 5 (as we have followed in our example). If the input data of the problem are new, we can set pi = (1/2)(bi-ai),

i E Q. Step 4 may be realized by an explicit or implicit (by a branch and bound method) enumeration

of the feasible digraphs B. In Step 4, we can apply Theorem 1 to make sure that the selected optimal digraph G, is stable. If #(p E T) = 0, we can take another optimal digraph (the latter there exists due to Theorem 1) which is stable, or we can change the initial vector p of processing times.

Steps 5,7, and 8 may be done on the basis of Theorem 2 and/or Theorem 3. If $(p E T) = 00,

Theorem 2 can be used as a ‘stopping rule’. Otherwise, we are forced to use Theorem 3 which is more time consuming. A competitive digraph and a new vector p* of the processing times are calculated in parallel with the calculation of ~,“(p E T).

Of course, the above algorithm presents only one possible scheme of solving problem G/oi 5

pi I &/&ax* The order of the above steps and their substance may be modified considerably in accordance with the available information and software.

5. CONCLUSION

In [6], it w&s noted that one “source of uncertainty is processing times, which, typically, are not known in advance. Thus, a good model of a scheduling problem would need to address these forms of uncertainty”. In this paper, we proposed problem G/ai 5 pi 5 bJC,, for dealing with the uncertain scheduling environments in which only lower and upper bounds of processing times are known before scheduling. Such a problem may arise in many practical situations, since even if no specific bounds on an uncertain processing time pi are known, we can set at least oi = 0 and at most bi equal to a horizon of planning.

As far as we know, such a kind of scheduling problem was not considered in OR literature. In Section 2, we defined a solution of problem G/ad 5 pi 5 hi/Cm, as a minimal (with respect to inclusion) set of schedules such that at least one of them is optimal for any fixed processing time pi in the interval [oi, bi], i E Q. We used a mixed graph model for representing the input data, the scheduling process, and the final solution. Our ‘strategy’ was to separate the ‘structural’ input data from the ‘numerical’ input data as much as possible. The precedence and capacity constraints (i.e., structural input data) are given by the mixed graph G, which completely defines the set of feasibile schedules. The set of optimal schedules is defined by the weighted mixed graph G(p) which gives both the structural and numerical input data.

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Optimal Makeepan Scheduling 85

Since the optimality of a schedule s depends on the critical path in the digraph G,, we focused

on the set of paths in digraph G, which may be critical (see Lemma 2 and Theorem 1). To restrict the set of paths which may be critical, we have introduced in Section 1 a dominance relation on the set of paths. Although this relation is based only on the structural input data, its use may considerably reduce the set of paths which may be critical. To deal with problem

Gloi I pi I b&Lax in Section 3, we generalized the dominance relation due to numerical input data. On the basis of this more general dominance relation, we presented a characterization of a zero relative stability radius (Theorem 1) and an infinite relative stability radius (Theorem 2). In Section 4, we have given a formula for calculating the exact value of the relative stability radius (Theorem 3). These results may be considered as a mathematical background for developing algorithms for solving problem G/ai 5 pi 5 bi/Cmax.

In conclusion, we note that the presented approach seems to be particularly useful when the structural input data are fixed before scheduling but the numerical input data are uncertain, especially when a lot of scheduling problems with the same (or close) structural input data have

to be solved. Since problem G/ai 5 pi 5 hi/Cm,, is very general (see Table l), additional research seems necessary based on the ideas described in this paper or on some others.

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