It is well-known that different scheduling problems may be represented as extremal prob-lems on mixed (disjunctive) graphs.
As it was mentioned in [CCLe95, pp. 277-293], the disjunctive graph approach is the most suitable one for traditionally difficult scheduling problems.
A mixed (or disjunctive) graph is often introduced to model a deterministic scheduling problem (see [BDP96, Pin95b, RS64, TSS94]).
Problem G//Φ is unary NP-hard for any given regular criterion Φ considered in scheduling theory, [LLRKS93, TSS94], the running time of calculating an optimal sched-ule s = (c1(s), c2(s), . . . , cq(s)) may be restricted by an O(q2)-algorithm (see [[TSS94], p.
285]).
Due to the particular importance of set Es, it is called a signature of a schedule s [BSW96, Sot91b, SLG95, STW98, Sus72].
The results of [BSW96, KSW95, Sot91b, SSW97] were devoted to the stability of an optimal digraph Gs(p) which represents an optimal solution to problem G//Φ.
Next, we introduce these notions in a formal way (see [Sot91b]).
In [Sot91b], the calculation of %s(p) has been reduced to a non-linear programming problem. Next, we give the presentation from [Sot91b, STW98] for the case of the general shop problem G//Φ when the set of all operations Q is partitioned into n technological routes Q(i) of a job Ji, i ∈ {1, 2, . . . , n}.
Formulas for calculating the stability radius for the makespan criterion and the char-acterization of the extreme values of %bs(p) have been proven in [Sot91b, 84]. The same questions for the mean flow time criterion have been considered in [BSW96].
a formal definition of the stability radius, which is the maximal value of the radius of such a stability ball (see [Sot91b]).
In order to calculate %s(p), it is sufficient to know the optimal value of the objective function f (x1, x2, . . . , xq) of the following non-linear programming problem (see [Sot91b]):
In [Sot91b, 84], the stability radius for criterion Cmax has been considered and here we survey these results.
Next, we present necessary and sufficient conditions for equality %bs(p) = 0 proven in [Sot91b].
In [Sot91b], the following characterization of an infinitely large stability radius was proven.
In [KSW95], it has been shown that for problem J //Cmax, there are necessary and sufficient conditions for%bs(p) = ∞ which can be verified in O(q2) time.
Moreover, In [KSW95], the analogies to Theorems 1.2 and 1.3 for the job shop problem J //Lmax with minimizing maximum lateness (see [LLRKS93]) have been proven and it has been shown that there does not exist an optimal schedule s with %s(p) = ∞ for all other regular criteria (see [LLRKS93]), which are considered in classical scheduling theory.
Formulas for calculating %bs(p) have been derived in [Sot91b].
Formulas for calculating the stability radius for criterion Cmax have been derived in [Sot91b, 84].
In this section, we survey results from [BSW96, STW98], where the stability radius
%s(p) for criterion PCi has been studied.
Continuing similarly as in the above proof of Theorem 1.4 the following claim may be proven in [BSW96].
The extreme values of %s(p) were considered in [BSW96, STW98]. The following necessary and sufficient conditions for equality %s(p) = 0 have been derived in [BSW96].
Section 1.5 was based on the results proven in [SD01, SDP05].
Assembly line balancing problem with variable operation times was considered, e.g., in [GTL96, LJ91, SEDE99, TGK95]. In [GTL96, TGK95], fuzzy set theory was used to represent uncertainty of the operation times. Genetic algorithms were used either to mini-mize the total operation time for each station [GTL96] or to minimini-mize the efficiency of the fuzzy line balance [TGK95]. In [LJ91], the entire decision process has been decomposed into two parts: Deterministic problem and stochastic problem. For the former problem, integer programming is used to minimize number of stations. For the latter problem, which takes into account variations of the operation times over different products, queu-ing network analysis is used to determine the necessary capacity of the material-handlqueu-ing system. In [SEDE99], dynamic programming and branch-and-bound methods have been used to minimize the total labor cost and the expected incompletion cost arising from operations not completed within cycle time c.
(e.g., via branch-and-bound algorithm [BT00, PDK83, SEDE99, SK98], or via integer programming [DGHL02, LJ91, PDK83], etc.).
Some preliminary results for such a stability analysis have been obtained in [SD01]. In [SD01], slightly different definitions of stability radius and stability ball of an optimal line balance have been used. Namely, it was assumed that . Therefore in that paper generally smaller stability radii were obtained (in particular, it cannot be greater than ). The above Definition 1 from Section 2 seems to be more appropriate for practical assembly lines.
The scheduling theory has received a lot of attention among OR practitioners, man-agement scientists, production and operations research workers and mathematicians since the early 1950s. However, the utilization of classical scheduling theory in most produc-tion environments is minimal (see [ML93, PL94, Pin95a]). MacCarthy and Liu [ML93]
aim the gap between scheduling theory and scheduling practice. They also discuss some research issues which attempt to make scheduling theory more useful in practice. Next, we describe some recent trends in scheduling research which try to make it more relevant and applicable.
For an uncertain scheduling environment stochastic models are introduced, where the processing times (and some other parameters) are assumed to be random variables with known probability distributions. For example, such stochastic models for a single machine with the minimization of mean flow time are considered by Chand et al. [CTU96], by Li and Cao [LC95], and with the minimization of earliness-tardiness penalties by Cai and Tu [CT96] as well as by Robb and Rohleder [RR96]. Since it is possible for a company to estimate the times at which jobs are expected to arrive, Chand et al. [CTU96] develop a decomposition approach such that a large problem can be solved by combining optimal solutions of several smaller problems. The model of Robb and Rohleder [RR96] consists of a probabilistic dynamic scheduling problem with non-regular performance measures. Using simulation, they explore the robustness of the heuristics with respect to uncertainty in the durations of the operations.
Schmidt [Sch00] reviews some results related to deterministic scheduling problems where the machines are not continuously available for processing. The complexity of single and multi-machine problems is analyzed considering criteria depending on the completion times and the due dates. Chu and Gordon [CG00] consider a single machine problem including both due date assignment and the scheduling decision. It is assumed that the due dates are proportional to the job processing times. The objective is to minimize the weighted earliness-tardiness and the penalty related to the size of the dates with respect to the processing times. Jain and Meeran [JM99] present a concise overview of job shop scheduling techniques and the best computational results obtained.
The scheduling problem with an availability constraint is very important, as it happens often in the industry. For example, a machine may not be available during the scheduling horizon due to a breakdown (stochastic) or preventive maintenance (deterministic). In an on-line setting, machine availabilities are not known in advance. Unexpected machine breakdowns are a typical example of events that arise on-line. Sometimes schedulers have partial knowledge of the availabilities, i.e. they have some ‘look-ahead’ information.
They might know the next time interval where a machine requires maintenance or they might know when a broken machine will be available again [San95]. In an off-line setting, one assumes complete information, i.e. all machine availabilities are known prior to the schedule generation [Sch00].
Several on-line models have been proposed, and the main difference between these models are the assumptions on the information that becomes available to the scheduler.
For a description of these on-line models, we refer to the survey by Sgall [Sga98]. According to [CV97], on-line means that jobs arrive over time, and all job characteristics become known at their arrival time [CV97]. Jobs do not have to be scheduled immediately upon arrival. At each time a machine is idle and a job is available, the algorithm decides which one of the available jobs is scheduled, if any. An on-line algorithm for the problem
of scheduling jobs on identical parallel machines with the objective of minimizing the makespan is proposed and analyzed by Chen and Vestjens [CV97]. This problem is NP-hard when the off-line version is considered, although it can be solved in polynomial time by an on-line algorithm if preemption is allowed [CV97].
Seiden [Sei98] studies on-line scheduling of jobs with fixed start and completion times.
Jobs must be scheduled on a single machine which runs at most one job at a given time.
The problem is on-line since jobs are unknown until their start times. Each job must be started or rejected immediately when it becomes known. The goal is to maximize the sum of the value the payoff (the sum of the values of those jobs which run to completion).
Scheduling problems with controllable processing times have received an increasing attention during the last decade. It is often assumed that the actual possible processing time of a job can be continuously controlled, i.e. it can be any number in a given interval.
Recent results are presented in [DHM96, KDV00, Str95, Tri94].
Traditional scheduling procedures consider static and deterministic future conditions even though this may not be the case in actual scheduling problems. After a descrip-tion, the preplanned schedule can become inapplicable to the new conditions. As Graves [Gra81] stated, there is no scheduling problem but rather a rescheduling problem. Re-sponding to such dynamic factors immediately as they occur is called real-time schedul-ing. An on-line simulation methodology is proposed by Davis and Jones [DJ88] to ana-lyze several scheduling rules in a stochastic job shop. The job shop rescheduling problem is considered as a particularly hard combinatorial optimization problem (Parunak and van Dyke [PD91]). The production rescheduling problem deals with uncertainty caused by the exterior business environment and interior production conditions. Since it has practical applications, the rescheduling problem is studied by many authors (see e.g.
[LLLH00, PD91, SK94]).
A reactive scheduling approach is developed by Smith et al. [SOM+90], which uses different knowledge sources and aims to make decisions faster with less emphasis on opti-mality. For the knowledge-based systems, the most difficult operation is to decide which knowledge source has to be activated. A discussion of knowledge-based reactive schedul-ing systems can be found in Blazewicz et al. [BESW93] as well as Szelke and Kerr [SK94].
Bean et al. [BBMN91] propose a ‘match-up’ heuristic method for scheduling problems with disruptions. They show that assuming enough idle time is present in the original schedule and disruptions are sufficiently spaced over time, the optimal rescheduling strat-egy is to match-up with the preschedule at some time in the future. The objective in [AG99] is to create a new schedule that is consistent with the order production planning decisions like material flow, tooling and purchasing. When a machine breakdown forces a modified flow shop out of the prescribed state, the proposed strategy reschedules a part of the initial schedule to match-up with the preschedule at some point.
Fuzzy scheduling techniques proposed in the literature either fuzzify directly the ex-isting scheduling rules, or solve mathematical programming problems to determine the optimal schedules. The optimality of a fuzzy logic alternative to the usual treatment of uncertainties in a scheduling system using probability theory was examined by Ozelkan and Duckstein [OD99]. The purpose of the latter paper was to investigate necessary optimality conditions of fuzzy counterparts of ‘classical’ dispatching rules, such as the shortest processing time (SPT) and the earliest due date (EDD). Essentially, any element of a scheduling problem may be uncertain.
Dumitru and Lubau [DL82] propose fuzzy mathematical models to solve the job shop problem. Grabot and Geneste [GG94] use a fuzzy rule-based approach to find a com-promise between different job shop dispatching rules. Kuroda and Wang [KW96] also analyze fuzzy job shop problems using a branch-and-bound algorithm to obtain results for lateness related criteria. A mathematical programming approach to a single machine scheduling problem with fuzzy precedence relation is given in [IT95]. Job shop scheduling with both fuzzy processing times and fuzzy due dates are proposed in [SK00]. Sakawa and Kubota [SK00] formulate a multiobjective fuzzy job shop problem as three-objective ones which not only maximizes the minimum agreement index but also maximizes the average agreement index and minimizes the maximum fuzzy completion time. Generally, the topic of fuzzy scheduling has received much attention during the last decade. Slowinski and Hapke [SH99] collect the main works.
In most of the classical shop scheduling models, it is assumed that an individual processing time incorporates all other time parameters (lags) attached to a job or to an operation. In practice, however, such parameters often have to be viewed separately from the actual processing times. For example, if for an operation some pre-processing and/or post-processing is required, then we obtain a scheduling model with set-up and/or removal times separated. Strusevich [Str99] considers a two-machine open shop problem with involved interstage transportation times. He assumes that there is a known time lag (transportation time) between the completion of an operation and the beginning of the next operation of the same job.
The majority of scheduling research assumes set-up as negligible or as a part of the processing time. While this assumption simplifies the analysis, it adversely affects the solution quality for many applications which require an explicit treatment of set-up times.
Such applications, coupled with the emergence of production concepts like time-based competition and group technology, have motivated an increasing interest to include set-up considerations in scheduling problems. The paper [AGA98] provides a comprehensive review of the literature on scheduling problems involving set-up times (set-up costs).
In [All97], Allahverdi considers a two-machine flow shop problem with the objective to minimize the expected makespan where machines suffer breakdowns and the job set-up and removal times are separated from the processing times. The same author [All95]
proposes a dominance relation where no assumption about the breakdown processes is made. In general, such a dominance relation does not yield optimal schedules. However, if certain assumptions about the breakdowns distributions and counting processes hold, it is possible to obtain an optimal schedule.
Decision-makers often consider multiple objectives when making scheduling decisions.
However, very little research has been done in multiple machine environments with mul-tiple objectives. Allahverdi and Mittenthal [AM98] consider a two-machine flow shop scheduling problem, where machines suffer random breakdowns and processing times are constant, with respect to both the makespan and the maximum lateness objective func-tions. Kyparisis and Koulamas [KK00] study the two-machine open shop problem with a hierarchical objective: Minimize the total completion time subject to minimum makespan O2//PCi|Cmax.
Cheng and Shakhlevich [CS98a] consider a special class of flow shop problems, known as the proportionate flow shop. In such a shop, each job flows through the machines in the same order and has equal processing times on the machines. It is assumed that
all operations of a job may be compressed by the same amount which will incur an additional cost. The objective is to minimize the makespan of the schedule together with a compression cost function which is nondecreasing with respect to the amount of compression. A bicriterion approach to solve the single machine scheduling problem in which the job release dates can be compressed while incurring additional costs, is considered in [CS98b].
Stein and Wein [SJ97] give a proof that, for any instance of a rather general class of scheduling problems, there exists a schedule with a makespan at most twice that of the optimal value and of a total weighted completion time at most twice that of the optimal value.
Brucker and Kr¨amer [BK96b] derive complexity results for resource-constrained scheduling problems with a fixed number of operation types in which either the processing times are bounded or the number of processors is fixed. They consider shop problems with multiprocessor operations, in which either the number of jobs or the number of stages is fixed. They present polynomial time algorithms for these problems with makespan, mean flow time, weighted number of tardy operations, and sum of tardiness as objective func-tions.
The papers above address problems of practical importance in planning, scheduling, and control. It is therefore important to produce schedules that are both stable (robust) and adaptable to system disturbances. More importantly, it offers unique properties that lead to a more effective planning and control methods for systems under uncertainty.
An extensive survey of the obtained results within such an a posteriori analysis is given in [SLG95]. Greenberg [Gre97] categorizes types of postoptimal sensitivity analyses and gives a survey of the literature started in the late 1970’s. A primary concern of sensitivity analysis is how the optimal solution values change when the data changes. The subject of post-solution analysis includes debugging a scenario, such as when it is anomalous, unbounded or infeasible.
In spite of obvious practical importance, the literature on stability analysis in schedul-ing is rather small. Outside the considered approach, one can mention [KRKvHW94, Mel78, PQ78]: In [KRKvHW94], the sensitivity of a heuristic algorithm with respect to the variation of the processing time of one job is investigated, in [Mel78] the stability of an optimal schedule for the flow shop problem F //Cmax is considered, and in [PQ78] the results for the traveling salesman problem are used for a one machine scheduling problem with minimizing tardiness (see [LLRKS93]).
In general, studying a scheduling problem with uncertain processing times and its sensitivity analysis is of importance. The reasons can be illustrated by giving references to practical applications. In many cases the data used are imprecise due to uncertainty with respect to the exact parameter values or due to errors in the measurement. In industrial applications of mathematical programming models, there are almost always uncertain elements that are assumed away or suppressed in the formal description of the model (see [Wag95]).
We have to emphasize that the random processing times pi, i ∈ Q, in problem G/ai≤ pi≤ bi/Φ are due to external forces in contrast to scheduling problems with controllable processing times, see e.g. [DHM96, IMN87, IN86, Jan88, Str95, Tri94], where the objective is to choose both the optimal processing times (which are under the control of a decision-maker) and an optimal schedule for the chosen processing times. Both of the above parts of
a solution are supposed to be arguments in the objective function which is non-decreasing in the job completion times and non-increasing in the operation processing times.
To model scheduling in an uncertain environment, a two-person non-zero sum game is introduced by Chrysslouris et al. [CDL94], where the decision-maker was considered as player 1 and the ‘nature’ as player 2.
Next, we observe known results for makespan minimization under ‘strict uncertainty’
of the numerical input data. Lai and Sotskov [LS99] use a weighted mixed graph G for representing the input data of a job shop problem which implies a one-to-one correspon-dence between the set of semiactive schedules S and circuit-free digraphs Λ(G). Since the optimality of a schedule s ∈ S for the makespan criterion depends on the critical path in the corresponding digraph Gs, the analysis in [LS99] is focused on the set of paths in Gs∈ Λ(G) which may be critical.
In [LS99], the critical path method [Dij59] is modified for constructing a minimal digraph containing only possible candidates of critical paths. A minimal set of makespan optimal schedules for uncertain numerical input data is characterized in [LS99], where an exact and a heuristic algorithm are developed for problem J /ai≤ pi≤ bi/Cmax. Note that the approach developed in [LS99] is based on the stability property of a makespan optimal schedule, which is theoretically investigated in [KSW95, Sot91b, SWW98] and in some other papers (see [SLG95, STW98] for surveys of stability analysis for scheduling problems).
Briefly, the main issue of the research presented in [LS99] is to simplify the digraph Gs due to the existence of two types of dominance relations between its paths (see Section ??
and 3.2). In this dissertation, we perform a further step in this direction by focusing on two types of dominance relations between feasible digraphs (schedules) (see Section 3.4 below). This step is useful for shop scheduling problems under ‘strict uncertainty’ with both CmaxandPCi criteria since it allows to reduce significantly the number of schedules
and 3.2). In this dissertation, we perform a further step in this direction by focusing on two types of dominance relations between feasible digraphs (schedules) (see Section 3.4 below). This step is useful for shop scheduling problems under ‘strict uncertainty’ with both CmaxandPCi criteria since it allows to reduce significantly the number of schedules