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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

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

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)

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

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

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 functi...