Profile-Based Algorithms to Solve Multiple Capacitated Metric Scheduling Problems

Profile-Based Algorithms to Solve Multiple Capacitated MetricScheduling Problems

Amedeo CestaIP-CNK

National Research CouncilViale Marx 15

1-00137 Rome, Italyaaedeo4~scs2, irakant, am. cnr. it

Angelo OddiIP-CNR

National Research CouncilViale Marx 15

1-00137 Rome, Italyoddiepscs2, irakant, rm. cnr. it

Stephen F. SmithThe Robotics Institute

Carnegie Mellon University5000 Forbes Avenue

Pittsburgh, PA 15213, USAsfsQisll .ri.aau. edu

Abstract

Though CSP scheduling models have tended to as-sume fairly general representations of temporal con-straints, most work has restricted attention to prob-lems that require allocation of simple, unit-capacityr~=,ources. This paper considers an extended class ofscheduling problems where resources have capacity tosimultaneously support more than one activity, andresource availability at any point in time is conse-quently a function of whether sufficient unallocatedcapacity remains. We present a progression of algo-rithms for solving such multiple-capacitated schedul-ing problems, and evaluate the performance of eachwith respect to problem solving ability and quality ofsolutions generated. A previously reported algorithm,named the Conflict Free Solution Algorithm (CFSA),is first evaluated against a set of problems of increasingdimension and is shown to be of limited effectiveness.Two variations of this algorithm are then introducedwhich incorporate measures of temporal flexibility asan alternative heuristic basis for directing the search,and the variant making broadest use of these searchheuristics is shown to yield significant performanceimprovement. Observations about the tendency ofthe CFSA solution approach to produce unnecessar-ily overconstrained solutions then lead to developmentof a second heuristic algorithm, named Earliest StartTime Algorithm (ESTA). ESTA is shown to be themost effective of the set, both in terms of its abilityto efficiently solve problems of increasing scale andits ability to produce schedules that minimize overallcompletion time while retaining solution robustness.

Introduction

Over the past few years, constraint satisfaction prob-lem solving (CSP) techniques have been productivelyapplied to several classes of scheduling problem (e.g.,(Cheng & Smith 1994; 1996; Nuijten & Aarts 1994;Oddi & Cesta 1997; Sadeh 1991; Oddi & Smith 1997)).One advantage of CSP approaches to scheduling is thegenerality of their underlying representational assump-tions. Techniques have been proposed, for example,which operate relative to very general temporal con-straint models and support resource allocation under

Copyright (c) 1998 American Association for ArtificialIntelligence (www.aaai.org).

complex quantitative time constraints. Thus, in con-trast to classical scheduling approaches which tend to-ward specialized solutions to idealized problem formu-lations, the representational assumptions underlyingCSP scheduling techniques are well matched to themodeling requirements of practical scheduling prob-lems.

However, current CSP scheduling models are notwithout modeling limitations. Most research has fo-cused on problems that require allocation of simple,"unit-capacity" resources; i.e., resources that must bededicated exclusively to performing any given activ-ity and, at any point in time, are either "available"or "in-use". In many practical domains, resources infact have the capacity to support some number of ac-tivities at any point in time. Indeed, the inability toaccommodate capacity resources and to solve multiple-capacitated scheduling problems is perhaps the princi-pal obstacle to widespread application of most CSPscheduling models and heuristics. In this paper, wetake some steps toward removing this obstacle.

Let us introduce a very intuitive example of the typeof multiple-capacitated scheduling problem we will fo-cus on: suppose that a vacation house has a limitedamount of power supply (a generator with capacityk - 2) and a family (two parents and a child) rents for a week-end. They are unaware on the power limita-tion upon arrival but soon after realize that this will bea serious constraint on their week-end activities. Forexample Tim, the son, is unable to switch on the hair-drier after taking a shower because the water-heater ison and his mother is simultaneously cooking an applepie in the electric oven (we assume that any electric de-vice consumes one unit of power supply). It is easy tosee that it is not possible to solve these resource avail-ability conflicts by simply postponing one or more ofthese conflicting activities, because these activities areconnected to others by temporal relations: Tim cannotpostpone his hair drying for more than 5 minutes afterhaving a shower because it is winter time and he riskscatching a cold; mother cannot indefinitely delay thebaking of the pie once it has been prepared, becausethis may harm its taste. In general it is necessary topostpone or interleave blocks of temporally connectedactivities, and different decisions can have significantimpact on how efficiently resources are used and howefficiently activities can be completed.

214 Scheduling

From: AIPS 1998 Proceedings. Copyright © 1998, AAAI (www.aaai.org). All rights reserved.

The type of scheduling problem illustrated above isdistinguished by the need to re .~o.n abo..ut capacity, x~-sources in the presence of quantitatlve tlme constraintson the starts, ends and durations of individual activi-ties as well aa on the temporal separations between re-lated activities. We call this particular class of schedul-ing problem the Multiple Capacitated Metric Schedul-ing Problem (MCM-SP). Variants of the MCM-SP arecommonplace in practical domains. Aircraft trans-portation scheduling applications, for example, gener-ally require management of capacity resources at air-ports (e.g., onload/offload capacity, aircraft parkingspace, cargo storage area) while simultaneously enforc-ing complex temporal constraints on airlift activities(e.g., minimum time on ground, takeoff/landing sepa-ration times). Similarly, in most manufacturing envi-ronments, production activities must be synchronizedto account for the finite processing capacity of variousmachining centers and operator pools while respect-ing the ordering, timing and separation constraints onvarious production steps.

In this paper, we define and evaluate a set of ~profile-based~ methods for solving MCM-SP. Profile.baaedsolution methods rely centrally on a temporal func-tion called a demand profile Dj (t) which describes thequantity of resource rj required by various activitiesat time t. Generally speaking, profile baaed solutionmethods proceed by detecting time instants in whichthe profile Dj (t) is greater than the available capac-ity of resource rj and then "leveling" these utilizationpeaks by posting additional ordering constraints be-tween those activities competing for rj’s capacity.

Our starting point is a solution procedure baseddirectly on the techniques described in (E1-Kholy Richards 1996). This approach, which emphasizescontention-baaed criteria for prioritizing the levelingdecisions that must be made, is found to be ineffectiveaa problem size is inc~eaaed. We then propose variantsof this procedure which alternatively use the heuristiccriteria proposed in (Cheng & Smith 1994) (baaed the idea of maintaining sequencing flexibility) to pri-oritize and make leveling decisions. These variants areshown to yield significant performance improvements.Finally, we propose a new algorithm, also incorporat~ing the heuristic ideas of (Cheng & Smith 1994) but this case designed specifically to emphasize construc-tion of compact solutions. We show that this approachoutperforms all other procedures, both in terms of thenumber of solved problems and overall solution quality.

The remainder of the paper is organized aa follows:first, the MCM-SP is formally defined, and connectionsand distinctions with related literature are discussed;second, the problem of generating reproducible randominstances of MCM-SPs is addressed and performanceevaluation metrics are defined; third, a progression ofprofile.baaed solution methods is presented and evalu-ated on the controlled set of problems. Some conclud-ing remarks close the paper.

Definition of MCM-SPThe Multiple-Capacitated Metric Scheduling Problem(MCM-SP) considered in this work involves synchro-

nizing the use of a set of resources R = {rl... rm} toperform a set of jobs J = {jl...j,} over time. Theprocessing of a job j~ requires the execution of a se-quence of ni activities (ail...ainu}, a resource rj canprocess at most cj activities at the same time (withcj >_ 1) and execution of each activity a~j is subject tothe following constraints:

¯ resource availability- each aij requires the use of asingle resource r,,j for its entire duration.

¯ processing time constraints- each aij has a mini-mum and maximum processing time, proc~in andproc~ija’, such that proc~i/n < e,j -- so <_ proe~or,

where the variables s~j and eij represent the startand end times respectively of a~j.

¯ separation constraints- for each pair of successiveactivities a0 and aio+l), j = 1... (n~ - 1), in job there is a minimum and maximum separation time,sep~’’~ and sep~ka’, such that {se~’n <_ s,<,+,~ -e,, <_ sept*Z: k-- l...(n,- 1)}.

¯ job release and due dates - Every job j~ has a releasedate rdi, which specifies the earliest time that anyaij can be started, and a due date dd~, which desig-nates the time by which all alj must be completed.

A solution to a MCM-SP is any temporally consistentassignment of start and end times which does not vio-late resource capacity constraints.

It is worth noting the main features of the problem:(a) the resources are discrete (and not consumable)with capacity e$ greater than or equal to one; (b) alltemporal constraints are quantitative and expressed asa bound between a maximal and minimal duration ofa distance; (c) an important role is played by the pres-ence of separation constraints between activities; theseconstraints in fact have a crucial role in increasing theintrinsic difficulty of the problem. They constrain the’~fluidity" of a single activity in the solution to be de-pendent on a set of connected activities. This factintroduces a further level of interdependency betweenactivity start and end time assignments.

Time and Resource Constraints. The style of so-lution investigated here is based on the explicit man-agement of sets of constraints that the solution shouldsatisfy. To sharpen the basis for comparison of alterna-tive algorithms, we will assume a common constraintrepresentation and management infra-structure.

Time Ontology. To support the search for a consis-tent solution, a directed graph Gd(V, E), named dis-tance graph, is defined, wherein the set of nodes Vrepresents time.points tpi and the set of edges E rep-resents temporal distance constraints. The origin, to-gether with the start time s,~ and end time e,~ ofeach activity a~, comprise the set of represented timepoints in Gd for any given MCM-SP. Activity process-ing time constraints, aa well aa separation constraintsand precedence constraints between pairs of activities,are encoded (naturally) aa distance constraints. Ev-ery constraint in Gd is expressed as a bound on thedifferences between time.points a < tpj - tp~ <_ b andis represented in Gd(V, E) with two weighted edges:

Cesta 215


the first one directed from tpi to tpj with weight b,the second one from tpj to tp~ with weight -a. Thegraph Gd(V, E) corresponds directly to a Simple Tem-poral Problem (STP) (Dechter, Meiri, & Pearl 1991)and its consistency can be efficiently determined viashortest path computations (Ausiello et al. 1991;Oddi 1997). Thus, the search for a solution to MCM-SP can proceed by repeatedly adding new prece-dence constraints into Gd to resolve detected con-flicts and recomputing shortest path lengths to con-firm that Gd remains consistent (i.e., no negativeweight cycles). We let d(tpi, tpj) designate the short-est path length in graph Gd(V~ E) from node tpl tonode tpj. Note that, given a Gd, the distance be-tween any pair of time points satisfies the constrainttpj - tpi = [-d( t pj , tpi ) , d( tpi , tpj ) (Dechter, Meiri, &Pearl 1991).

Resource Ontology. Each problem relies on a set ofresources R. We denote the required capacity (alsocalled resource demand) of a resource rj E R by anactivity ak as rcah,j , and the set of activities ak thatdemand resource rj as Aj (i.e., Aj = {at,[ rc, hj ~ 0}).Then for each resource rj a Demand Profile Dj(t)can be defined, representing the total capacity require-ments of ak E Aj at any instant of time. For rj then,the resource capacity constraint requires Dj(t) < cjfor all t. For purposes of this paper, we assume thatrca~ j = 1 for all at,.

Related Literature. An earlier work that formal-izes the scheduling problem as a CSP (Constraint Sat-isfaction Problem) is (Sadeh 1991). This work focuseson the JSSP (Job-Shop Scheduling Problem) where:(a) temporal constraints represent constant durationsof activities; (b) separation between activities are sim-ple qualitative ordering constraints; (c) capacity con-straints are binary (a resource is either busy or free).

An extension of JSSP to multi-capacitated problemsis developed in (Nuijten & Aarts 1994). The paperintroduces and presents a solution procedure for theMulti-Capacitated JSSP (MCJSSP), which retains thetemporal constraint model of the classical JSSP butallows resource capacities to be greater than one. In(Smith & Westfold 1996), a similar problem involvinga single, multi-capacitated resource is addressed.

An extension of JSSP to include the more generaltemporal constraint assumptions of an STP (while re-taining the assumption of binary capacity constraints)is introduced and solved in (Cheng ~k Smith 1994), andalso subsequently considered in (Oddi & Smith 1997).A variant of this problem where temporal constraintsmay be relaxed is discussed in (Oddi g~ Cesta 1997).

Representations and solution procedures that simul-taneously account for STP-style temporal constraintsand multi-capacitated resource constraints have re-cently been proposed within the planning literature,principally to provide resource reasoning subcompo-nents within larger planning architectures (Laborie GhMlab 1995; EI-Kholy & Richards 1996; Cesta &Stella 1997). These efforts will be considered later onin the paper.

216 Scheduling

A Controlled Experimental SettingThe scheduling problem introduced here has not beencarefully studied from previous research. It is thus im-portant at the outset to consider issues related to es-tablishing a common basis and experimental design forcomparison of alternative solution procedures. Below,we consider in turn two critical issues in this regard:(a) generation of benchmark problems, and (b) devel-opment of measures that characterize the quality of analgorithm and the solution it produces.

Generation of Problem Instances. The first steptoward establishing an experimental design is the im-plementation of a controlled random number gener-ator. We adopt the generator proposed in (Taillard1993) (pag.179) and obtain the uniform distributionfunction U[a,b] which generates a random numbern, where n~ a and b are positive numbers such thata < n < b z. To generate different problem instanceswe use the time seeds reported in Figure I of (Taillard1993) (in particular the first 50 seeds in this paper).

Next we define the dimensions along which probleminstances will be varied. Using the format for formulat-ing job shop scheduling problems, we call Njobs x Nresthe problem with Njoba each of them composed of asequence of Nrea activities that must be executed onone of the Nre~ different resources. For purposes ofthis paper, we create problem sets of 50 instances foreach of the following sizes: 5 x 5, 10 × 5, 15 × 5, 20 × 5,and 25 x 5. To generate random instances of MCM-SP of the above sizes we assign the remaining data asfollows:

¯ Each resource rj has capacity cj generated randomlyas U[2, 3], with full availability over the horizon.

¯ The minimum processing time of activities is drawnfrom a uniform distribution U[10, 50], and the max-imum processing time is generated by multiplyingthe minimum processing time by the value (1 + p),where p -- U[0, 0.4].

¯ The separation constraints [a, b] between every twoconsecutive activities in a job are generated with a =U[0, 10] and b = U[40, 50].

¯ Release and due dates for jobs are not considered ex-plicitly in the current experiments so they are fixedto 0 and H respectively for all the jobs.

Finally, the horizon H is computed by the followingformula adapted from (Cheng ~k Smith 1994): H N 2vr.o .( jobs--1)Pbk+~_1 Pi, where Pbk lS the average min-imum processing t]~ne of the activity on the bottleneckresource, and pi is the average minimum processingtime of the activity on resource rf. The bottleneck re-source is the resource with maximum value of the sumof the minimum processing time of the activities whichrequest the resource.

Evaluation Criteria. The identification of criteriafor assessing the relative effectiveness of alternative so-lution methods and solutions is always somewhat con-

21fa and b are integers then n is obtained as an integer,if one of them is real n is real.


troversial. For purposes of this paper, we consider thequality of a given solution procedure to be given as* number of solved problems from a fixed set (Na);¯ average CPU time spent to solve instances of the

problem ( C PU).

¯ the number of leveling constraints posted in the so-lution S (N~e). This number gives a further struc-tural information of the kind of solution created. Itindicates the number of sequencing constraints in-troduced by the algorithm to resolve competing re-source capacity requests.

¯ the average quality of the solutions generated.We consider two factors as contributing to judgementsof solution quality. From an optimization viewpoint,we will measure the compactness of the solution andfrom an executability viewpoint, we will characterizeits robustness. These notions are formalized below.

One commonly used measure of schedule quality isits overall makespan. Consequently we consider themakespan Mk, (or minimum completion time) of a so-lution to be one indicator of quality.

Robustness is a trickier notion to formalize. It iscommonly believed that robustness is related to thepossibilities for obtaining minor variations to a solution(at execution time for example) without incurring a major disruption to the overall solution. In general,when working with strongly temporalized domains (asis the case in this paper) this capability seems con-nected to the degree of "fluidity" of the solutions. Thekind of measure we are trying to define takes into ac-count the fact that a temporal variation concerning anactivity is absorbed by the temporal flexibility of thesolution instead of generating a deleterious "dominoeffect" in the solution itself.

Building on this intuition, we define the robustnessRB of a solution S to be the average width, relative tothe temporal horizon H, of the distance, for all pairs ofactivities, between the start time of one activity andend time of the other. More precisely:

Id(e,o, s,j) - d(s,~, e,,) I ....RB(S)= ~ ....... x,uu

,,es,~jes,o,#o, H x (N, x (No - 1))

where N6 is the number of activities in the solution, ahis a generic scheduled activity, d(sa,, ea,) is the lengthof the shortest path between the start-time of aj andthe end-time ofal, 100 is a scaling factor. With RB(S),we at least obtain a coarse idea of the reciprocal "shift-ing" potential between pairs of activities in a solution.In fact, we focus on the current average bounds on dis-tances between the end time and start time of any pairof activities in the schedule.

Profile-Based AlgorithmsWe can now define and evaluate alternative proceduresfor solving MCM-SPs. From previous work that hastaken into account multiple capacitated resources, wecan distinguish at least two general approaches to theidentification of resource conflicts s:

SAny classification of existing works might be partialand a bit arbitrary. For example the current classification

¯ profile.based approaches (EI-Kholy & Richards 1996;Cesta ~ Stella 1997): these approaches are exten-sions of a technique quite common in unit-capacityscheduling (e.g. (Sadeh 1991)). Most generally, consist of characterizing resource utilization demandas a function of time, identifying periods of resourceoverallocation in this utilization, and incrementallyperforming "leveling actions" to (hopefully) ensurethat resource usage peaks fall below the total capac-ity of the resource;

¯ clique-based approaches (Laborie & Ghallab 1995):given a current schedule, this approach builds upa "conflicts graph" whose nodes are activities andwhose edges represent possible resource conflicts be-tween the connected activities. Fully connected sub-graphs (cliques) are identified and if the number nodes in the clique is greater than resource capacitya conflict is detected.

In this paper we restrict attention to the profile-basedsolution methods and consider several different algo-rithms based on this idea. All proposed strategies havein common the basic concept of demand profile Dj (t).The goal of a profile-based scheduling algorithm is tocreate a temporally consistent schedule where in everytime instant the demand profile of any resource satis-fies the resource constraints Dj (t) <_ cj.

In (Cesta & Stella 1997) the complementary aspectof Dj(~) , resource availability Qj(t)= cj- Dj(Q, considered, and a set of constraint synthesis techniquesis introduced that perform filtering of the search spaceduring resolution. In (EI-Kholy & Richards 1996) more limited form of resource constraint propagationis used. However, a leveling heuristic is also given sothat a procedure can be reproduced which forms a com-plete scheduling algorithm. For this reason, we havechosen this second procedure as the starting point ofour investigation and analysis.

Conflict Free Solution Algorithm (CFSA)

The Conflict Free Solution Algorithm (CFSA) whichwe first consider is based directly on the proceduredefined in (El-Kholy & Richards 1996). The princi-pal point of departure in our formulation has been toeliminate the backtracking search framework that isemployed in (EI-Kholy & Pdchards 1996) in favor a greedy, backtrack-free search model. This more re-strictive search model obviously places much strongerimportance on the heuristics used to direct the search,and one of our interests in this paper is to improveupon these heuristics. But also, from a pragmaticstandpoint, the scale and complexity of the MCM-SPproblem that is considered here dwarfs the types ofproblems considered in (EI-Kholy & Richards 1996)and a complete, back-tracking search model is not vi-able computationally.

Before presenting the initial CFSA algorithm itself,some preliminary definitions are needed.

is enough for purposes of the paper but does not representwell proposals like the "energetic approach" in (Erschler,Loper, & Thuriot 1990).

Cesta 217


Demand Profile Bounds. Due to the temporal flex-ibility of an intermediate solution, an exact computa-tion of the D/(t) is not possible but upper and lowerbounds for Dj (t) may be defined. Without loss of gen-erality, such bounds are computed taking into accountonly activity start times. This is because it is in suchpoints that a positive variation of the demand profilehappens. Broadly speaking, for each start time aa,,the value of the lower bound profile (upper bound pro-file) is the sum of the capacity requirements of the ac-tivities whose execution interval necessarily (possibly)intersects with the time point sa,.

A more formal definition is based on the temporaldistances on the graph Gd. The lower-bound profile ofa resource rj is defined as:

LBD~(Sa,) = E Pik × rcahjakEAj

where Pi~ = I if d(sa,, sah) _< 0 A d(ea~, s.,) < 0 Pik = 0 otherwise. Similarly the upper-bound profile ofa resource rj is defined as:

uBo,(,..) = ×a~EAi

where Pi’k = 1 if d(sah, s~,) > 0 d(sa,, e~) > 0 anp; = 0 otherwise.

~Resource Conflicts. The concept of peak formalizesthe idea of conflict (or contention) in the use of a re-source. A peak is the couple <aa,, {aj}>, where aa,is the start time of an activity ai and {aj } is a set ofactivities which produces a violation of the resourceconstraints. Three types of peaks are introduced:

¯ lb-peak: a lower-bound peak where LBD, (s~,) = and U BDj(s,,) > cj;

¯ unre.s-peak: an unresolvable peak whereLBD~(Sa,) > cj;

¯ ub-pea& an upper bound peak where LBDi (s,,) cj and UBo, (s,,) > cj.

Conflict Resolution. A solution is always obtainablewhen UBoj (aa,) < ej for each resource rj. In such acase any consistent assignment of values to the solu-tion’s time points gives a solution where resource uti-lizations are consistent. A conflict free solution is builtby iteratively posting leveling constraints on the lb-peaks and ub-peaks until every peak is removed. Givena set of peaks, the leveling action can be formalized intwo steps:

1. conflict selection: a conflict in this case is a couple<aa,, aj> (notice a time-point and an activity) which the condition Pij = 0 A Pi’j = 1 holds;

2. leveling: one of the two leveling constraintss,,{before}sai or e,j{before}sa, is selected andposted in the partial solution.

Note that it may be the case. that one of the two pos-sible leveling constraints cannot be posted, because ofits inconsistency in the graph Gal. In this case, the

218 Scheduling

choice is uniquely determined. When both constraintsare consistent alternatives, a heuristic choice is needed.The strategy followed in (EI-Kholy ~ Richards 1996),according to a least commitment principle, first makeschoices in the area of the solution where the most con-straints have already been posted. The algorithm firstdetects the Ib-peaks, where the lower bound of the de-mand has reached the maximal resource availability,and tries to satisfy UBDi (s,,) <_ cj by adding levelingconstraints. Note that all the lb-peaks must be neces-sarily leveled to ensure a conflict free solution, so theheuristic choice concerns only the order in which theconflicts within a lb-peak are considered and which lev-eling constraint is applied when both possibilities areconsistent. After having leveled the Ib-peaks, the algo-rithm starts to consider the ub-peaks. In this case theheuristic choice concerns not only the conflict choiceand the leveling constraints but also the order in whichthe ub-peaks are considered for leveling.

The CFSA Algorithm. The basic CFSA algo-rithm is sketched in Figure 1. On any given cy-cle, it first looks for unresolvable peaks (functionExists-Unresolvable-Peak). If any are found thealgorithm stops. Otherwise the presence of lb-peaks is detected (gxists-Lb-Peak), the heuristicSelect-Lb-Peak-Lev-Constr is applied to select aleveling constraint, and it is posted in the underlyingtemporal constraint graph Gd.

C onfli et- Free-S olver ( mcmsp 1. loop2. if Exlsts-Unresolvable-Peak(mcmsp)3. then return(Failure)4. else begin5. if Ex~sts-Lb-Peak(mcmsp)6. then begin7. LC := Select-Lb-Peak-Lev-Constr( memsp)8. Post-Lev-Constr(LC)9. end10. else if Exists-Ub-Peak(mcmsp)l 1. then begin12. LC := Select-Ub-Peak-Lev-Constr( mcmsp)13. Post-Lev-Constr(LC)14. end15. else return(Solution)16. end-loop

Figure 1: Conflict Free Solution Algorithm

The choice is implemented as follows: in the set ofall lb-peaks {<aa,, {aj}>} the set of conflicts whichhave a unique solution {only one of Sa,{before}saj,e~{before}na, is consistent) is computed. The con-flicts in the set are randomly chosen and leveled byposting the uniquely determined ordering constraint.In none of these conflicts exist, the set of conflictswhere both leveling constraints are possible is com-puted and iteratively one is randomly chosen and lev-eled by randomly posting one of the two orderingconstraints. If, in the current solution, the Ib-peakshave been leveled, the existence of ub-peaks is checked(Exists-Ub-Peak). If any peaks are found, a differ-


ent leveling heuristic ($ele¢~-Ub-Peak-Lev-Constr)is applied: in the subset of the ub-peaks where the valueLBD~(Sal) is maxima], one of the ub-peaks is selectedwhere the value UBDj(s..) is minimal. Within thisub-peak the same strategy used for lb-peaks is appliedto level conflicts. When all ub-pmks have been leveleda solution is found.

Experimental Evaluation. The CSFA algorithmwas run on generated problem sets of increasing sizeand the results are given in Table 1 4. We call this ver-sion of the algorithm CFSA+PK to stress the composi-tion of a generic conflict free strategy with the particu-lar mechanism that focuses on peaks. Each problem setconsists of 50 randomly generated problem instances.

It is interesting to notice that the performance ofCSFA, although effective on small problems, degradesrather markedly as problem size increases. At the twolargest problem sizes, the algorithm is able to solvevery few problems.

Table 1: Experiments with CFSA+PK

] Probl II N. I RB I Mk. I Jv o [ c,’eu I5x5 50 5.9 [ 241.7 [ 29.3 0.510x5 [ 48 2.4 [424.0 [ 114.5 7.015×5 II 32 1.21606.71212.8 25.820x5 11 5 0.9[785.21365.8 67.925×5 ]] 1 0.5] 884.0 ] 497.0 133.2

Retaining Temporal Flexibility

The heuristics embedded in the basic CFSA procedurejust described attempt to first level a peak where thelower-bound profile is maximal and at the same timethe upper bound profile is minimal. In this way, the lev-eling action is concentrated in the area of the currentsolution where it is more probable that an unres.peakwill appear. However, we observe that the selection ofthe leveling constraints is randomly done without anyevaluation of its temporal commitment on the currentsolution. On the contrary, as observed in (ChengSmith 1994), such analysis can provide crucial guid-ance for fmding solutions to scheduling problems withtemporal separation between activities.

Following this observation, we propose aa extensionof the basic CFSA procedure that utilizes the notion oftemporal flexibility as a heuristic basis for prioritizingleveling decisions. At each (previously random) deci-sion point, we propose instead to non randomly makethe choice which leaves the maximal degree of tempo-ral flexibility. Intuitively, the notion of temporal flex-ibility is related to the idea of how many "temporalpositions" the time points in a solution may assumewith respect to each other (i.e., how many consistentassignments remain). Given two partial solutions Sxand $2 where the degree of temporal flexibility of $1 ismuch greater than $2 the probability to get a complete

4The CPU times given here and in subsequent tablesare in seconds. All procedures described in this paper wereimplemented in Commonlisp and all experiments were runon a SUN Sporcstation I0.

solution from $1 is greater because there is more ’Treetemporal space~ in it.

We can define the notion of temporal flexibilitybetween any pair of time points in a way similarto (Cheng & Smith 1994). Assume tp~ and tpjare two time points in a solution’s temporal net-work where the following relation holds: tpj - tp~ E[-d(tpj, tp~), d(tpi, tpj)]. We define temporal flexibility(flez(tp~, tpj)) associated to the pair (tpi, tpj) the am-plitude of the interval of temporal distances betweentp~, tpj with the condition tpi <_ tpj. For example, inthe case that tpl - tp2 E [-5, 10] the temporal flexibil-ities are flez(tp2, tpl) = 10 and flez(tpl, tp2) = 5. In-stead, in the case tpl -tp~ 6 [12, 16], flez(tp~,tpl) = and flez(tpl,tp2) =

When a temporal constraint is posted between tpiand tpj a fixed amount of temporal flexibility is re-moved from the temporal network. In particular, whensolving MCM-SPs, we use only constraints of the typetp~{before}tpj which removes an amount of tempo-ral flexibility equal to flez(tpj, tpi). This simple ideais used in the algorithms proposed in the next para-graphs where the goal of the heuristic is to improvethe probability to find a solution by maintaining thevalue ~-’~tp,,tp~ flez(tpi, tpj) as high as possible.

Heuristics Using Temporal Flexibility

To incorporate a bias toward retention of temporalflexibility into the basic CFSA procedure, we need tospecify two heuristic estimators. The first estimator isused to select a conflict <aa,, aj> from a set of peaksand it is computed as follows:

¯ Conflict Selection (CS) Given a set of peaks, theselection of a single conflict <aa., aj> is done on thebasis of a measurement of the temporal flexibilityassociated to <sa~,aj>. We use the two temporaldistances on the Gd graph d(sa,, sai) and d(eaj, sa,).In the ease that for each <sa,, aj> both the orderingchoices s.~ {before)s,~ and e,~ {before)s,, are tem-porally consistent (d(sa,, saj) > 0 and d(eoi, s,,) 0) then, the conflict <sah,am> is selected with theminimum value ~0re°(s, h, am) 5 where:

car.s(s..,am) min{d(S~sa’)d(ea.,sak)

, ,/-g }

with S = ~n~(’*~"")’~("’"’~)) Otherwise, raaz{d(,.~,,.,.).d(e.=.~.~))"the case there is at least one conflict which canbe leveled in a unique way (one of the distancesd(sa.,sa#) or d(ea#,s,,) is 0), th e conflict is se-lected with the minimum value w~o(s,~,a~) min{d(sa~, aa,), d(ea,, s~)}--the conflict closer tothe resource consistent status.

SAs suggested in (Cheng & Smith 1994) a balancingfactor ~is used. It is possible to see that S ~ [0,1]: S = 1when d(s.~,s,,,~) = d(e..,s°~) and it is dose to 0 whend(s.,, s,. ~ >> d(e,., aa, ) d(e,., s,, ) > >d(s,,, s,..The aim of this balancing factor is to select first conflictswith small and similar values of flexibility.

Cesta 219


The second estimator is used, given an <sa~, aj>, to se-lect one of the two leveling constraints aa, {before}sa~or eai {before}s.,:

¯ Levehng Constraint Selection (LCS) The choiceis made according to the function pc, such that,pc(s.,,aj) : se,{be/ore)s., when d(s..,sej) d(.eaj, aa,) and pc(se,, aj) -- eej (before}aa. other-wISe.

These two estimators provide a basis for defining twonew algorithms for MCM-SP:

¯ The first variant maintains the CFSA strategy of"peak focusing" and simply utilizes a deterministicchoice procedure based on temporal flexibility in-stead of the previous random choice procedure (wecall this version CFSA+PK+TF). The two heuristicfunctions of CFSA are modified as follows:

Select-Lb-Peak-Lev-Constr - Given the set of alllb-peaks: first, apply the US heuristic to select aconflict; next, apply the LCS heuristic to select aleveling constraint.

Select-Ub-Peak-Lev-Constr- Given the set ofthe ub-peaks, the subset of peaks where the valueof the lower bound profile is maximal and thevalue of the upper-bound profile is minimal is cal-culated. Within this subset of peaks: first, applythe CS heuristic to select a conflict; next, applyLCS to select a leveling constraint.

¯ The second variant is designed to more widely takeadvantage of the temporal flexibility estimators (wecall this algorithm CFSA+TF). In more detail:

Select-Lb-Peak-Lev-Constr - same as above.Seleet-Ub-Peak-Lev-Constr- Given the set of

all ub-peaks: first, apply the CS to select a con-flict;, next, apply the LCS to select a leveling con-straint.

Experimental Evaluation. The two procedures,CFSA+PK÷TF and CFSA+TF, were both run on thesame sets of scheduling problems used to evaluate theoriginal CFSA÷PK algorithm. Results are shown inTable 2 where the column "Alg" distinguishes between"pktf" for CFSA+PK+TF and "tf" for CFSA+TF.

The results provide an interesting insight. They in-dicate quite clearly that the use of demand upper andlower bounds as a focusing mechanism is not an ef-fective strategy for this type of problems. Problemsolving performance does not improve when the basicprocedure is augmented to additionally utilize heuris-tic estimates of temporal flexibility. Yet when estima-tions of temporal flexibility are used more broadly asa basis for prioritizing and selecting leveling decisions(variant CFSA+TF), problem solving effectiveness larger problem sizes increases dramatically. Thus, thepower of look-ahead heuristics which emphasize reten-tion of temporal flexibility in the evolving solution isseen to extend directly to MCM-SP. On the contrary,decision prioritization based on the lower and upperbounds on demand profiles appears to be of limitedeffectiveness.

220 Scheduling

One additional observation about the performanceof CFSA+TF is warranted. It is clear that the timeneeded to produce a solution is very high at the largestproblem size. This fact seems due to the huge num-ber of leveling constraints that the algorithm insertsto produce a solution. This observation motivates thealternative approach described in the next subsection.

Table 2: Experimentswith the CFSA variants

I Probl II Alg IN, IRB I Mk, N~c [ CPU[5 x 5 pktf 50 7.0 236.1 39.5 0.6

tf 50 7.5 228.4 54.4 1.010 x 5 pktf 50 2.8 422.8 144.4 7.7

tf 50 3.6 402.8 354.9 25.815 x 5 pktf 30 1.5 596.6 278.0 28.5

tf 50 2.0 580.9 884.4 137.020 x 5 pktf 6 1.1 776.2 409.0 69.2

tf 49 1.3 750.4 1630.1 430.425x5 pktf 2 0.6 1012.0 556.5 150.3

tf 50 0.9 911.5 2601.6 1058.8

Earliest Start Time Algorithm (ESTA)The rationale behind the CFSA strategy is the idea ofmaintaining a conflict free solution, forcing the condi-tion UBoj(s~,) <_ c~ to be valid for each resource rjand start time sa,.

CFSA Leveling Activity. Observing the solutionsproduced by all the CFSA algorithms, we have ob-served that such strategies post large numbers of level-ing constraints in the process of determining a feasiblesolution. As it turns out, the solutions produced arein fact overconstrained solutions. We can demonstratethis fact with a simple example. Consider a simplescheduling problem consisting of a single resource withcapacity c = 2 and three independent activities al,aa and a3. All the activities have a processing timePti,ne : 1, they demand one unit of r and must bescheduled in the time window [0, 10]. It is easy to ver-ify that to build a conflict free solution it is sufficientto post only one precedence constraint between two ofthe three activities considered. For example, we canpost the constraint al{before}aa and we are ensuredto have no resource conflicts. However, if after postingthis constraint we were to compute the upper-boundprofile in sa~, we obtain UBo(sas) = 3. In fact both aland aa can overlap sa~ even if separately because of theconstraint al{before}a~. The upper-bound demand isbuilt on the basis of pairwise tests, so following theCFSA strategy a new leveling constraint is posted toget UBz7 (sas) --- 2. The problem is due to the fact thata pairwise accounting of possible overlapping gives onlypartial information ~. Trying to overcome this kind oflimitation of the CFSA approach, we propose anotherprofile-based approach which, instead of focusing onthe development of a conflict free solution tries insteadto find a so-called eadiest start time solution.

eAn approach based on cliques detection (Laborie Gba|lab 1995) can avoid this problem because it is global intaking into account overlapping activities. Unfortunatelyit is more time consuming than profile-based approaches.


Earliest Start Time Solution. As is well known from(Dechter, Meiri, & Pearl 1991), the STP temporalmodel associates with any time-point tps (the variablesof a temporal CSP) an interval of possible time values[ib¢p,, ubt~,]. It is known that the extremes of the in-terval, either all the lower bounds ibtp~ or all the upperbounds ubtp,, if chosen as the value for all time vari-ables tpi, identify a consistent solution for the STP. If,in correspondence of these two sets of temporal val-ues, we also have a resource consistent solution (i.e.,all the resource requirements respect the capacity con-straints), then we have obtained a solution for the cor-respondent MCM-SP. We call these two particular so-lutions the Earliest Start Time Solution and the LatestStart Time Solution. Since we have previously identi-fied minimum completion time as an objective criteriait is straightforward to observe that we are more inter-sated in producing Earliest Start Time Solutions. It isinteresting to notice that an earliest start time solutionsolves the previous simple example by posting only oneleveling constraint.

It should also be noted that if, starting from an ear-liest start time solution, we consider an arbitrary timevalue that is consistent for a given time point we cangenerate a conflicting solution. In other words, whilethe CFSA strategy guarantees a conflict free solutionfor any choice of time values for individual time-points,an earliest start time solution guarantees this propertyonly if the values lbtp~ are chosen. Later on we intro-duce an algorithm to obtain a conflict free solution alsoin this second case.

Resource Demand at the Earliest Start Time. Thedevelopment of a general algorithm to create earlieststart time solutions is based on the following observa-tion: having chosen a single value for the time-points(the lbtp,), we obtain a single value for the resourcedemand (instead of an approximate bound) and canperform stronger deductions using this information.

A demand profile for a resource rj is a temporalfunction ESTDj(Sa,) that takes the start time aa, ofactivity as computes the required resource in the in-stant lb(sa~). Given a resource rj and the set of activi-ties Aj which request rj, the earliest start time demandprofile is defined as follows:

ESTDj (so,) Pih ×akEAj

where rca~,j is the required capacity of resource rj fromthe activity ak E Aj, P~j = 1 when lb(saj) <_ lb(sa,) Ib(ea~) and Psi = 0 otherwise.

Resource Conflict. Given an instance of MCM-SP,a peak is a tuple <rj, aa,, ca~>, where rj is a resource,sa~ is the start time point of the activity as and caiis a set of activities ca~ = {ak : Psk = 1}, such thatESTDj Ca.,) > c~.

A peak <rj,so~,cas> can be leveled by postingprecedence constraints between a pair of activitiesas, aj G ca. It is worth noting that in this algorithma conflict <ai,aj> is composed of two activities. Wewill refer generally to <ai, aj> as a pairwise conflict.

On the basis of the shortest paths information con-tained in the graph Gd, as explained in (Cheng

Smith 1994; Oddi & Smith 1997), it is possible to definea set of conditions which identify unconditional deci-sions and promote early pruning of alternatives. Forany pair of activities as and aj that are competing forthe same resource, four poss~le cases of conflict aredefined:

1. d(ea,,saj) < 0 A d(ea~,sa,) 2. d(eo,,soj) < 0 d(ea,,s,,)_>0 A d(sa,,ea~)>03. d(eaj,s,,) < 0 dCe,.,sai) > 0 A d(so~,ee,) > 4. d(e.,, s.j) _> 0 s.,)

Condition 1 represents a pairwise unresolvable con-flict. There is no way to sort ai and aj without in-ducing a negative cycle in graph Gd(V, E). Conditions2, and 3, alternatively, distinguish pairwise uniquelyresolvable conflicts. Here, there is only one feasibleordering of as and aj and the decision of which con-straint to post is thus unconditional. In the case ofCondition 2, only aj{before}ai leaves Gd(V, E) con-sistent and similarly, only as{before}aj is feasible inthe case of Condition 3. Condition 4 designates a fi-nal class of pairwise resolvable conflicts. In this case,both orderings of as and aj remain feasible and it isnecessary to make a choice. Now we can give a suffi-cient condition for detecting an unresolvable peak. Apeak <rj, aa,, cas> is unresolvable if for each pairwiseconflict <as,aj> with a~,aj ~ ca and a~ # aj holdsd(e~,,sa~) < 0 d(ea~,s~,) <

The ESTA Algorithm. We now propose a new algo-rithm for MCM-SP that: (a) finds a solution by level-ing the earliest start time demand in accordance withthe capacity of the resource; (b) extends the heuristicmethod defined in (Cheng & Smith 1994) to this case.The algorithm, named Earliest Start Time Algorithm(ESTA), is shown in Figure 2. ESTA iteratively selectsa pair~ise conflict by a heuristic method until the de-mand on all the resources is less than or equal to theirrespective capacities or until an unresolvable conflictis detected. In this last case the procedure stops withfailure.

Earliest-Start-Time-Solver(mcmsp)1. loop2. if E~dsts-an-Unresolvable-Confiict(mcmsp)3. then return(Failure)4. else begin5. ResolvableCon flict:=Sel-Res-Conf(mcmsp)6. if (ResolvableConflict - NIL)7. then return(Solution)8. else begin9. Pc:= Sel-Prec-Constr( ResolvableCon flict)10. Insert-Prec-Constr (Pc)11. end12. end13. end-loop

Figure 2: Earliest Start Time Algorithm

The Leveling Heuristics. The heuristic methods forselecting the next conflict to resolve and for determin-ing how to resolve it (i.e., which precedence constraintsto post between activities <as, a~>) are derived from

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(Cheng & Smith 1994). Both the selection of pair-wise conflicts and precedence constraints are based onthe principle of leaving maximum amount of tempo-rat flexibility possible in the solution at each step. Weuse the values of distances d(ea,, saj) (or d(eai, sa°)) a quantification (or measurement) of how many timepositions the pair of activities <ai,aj> may assumewith respect to each other while respecting the timeconstraints.

The proposed heuristic estimators operate as follows:

¯ Selection of Pairwise Conflict- We have to distin-guish two cases. When all pairwise conflicts satisfyCondition 4, the conflict <ai, aj > with the minimumvalue tarea (Ui, a j) is selected, where:

Wres(ai, aj ) rain{d(ea’’ saJ) d(ea,,sa,)-

min{d(e*"s’~)’d(e*~’s*i)} In the case wherewith S = max{d(e.,,saj),d(eoi,s.,)}.

a subset of pairwise conflicts satisfying Condition2 or Condition 3 exists, the heuristic selects theconflict with the minimum (and negative) valuew,~,(ai,aj) = min{d{e~,,sai),d{ea,,s,,)}, that isthe pair of activities which are closest to having theirordering decision be forced.

¯ Selection of Precedence Constraints - The choiceprocedure is defined by the function pc, suchthat, pc(ai,aj) = ai{before}aj when d(e,,,s~j) d(ea,, sa,) and pc(at, aj) = aj{before}ai otherwise.

In the case of conflicts <ai, aj > with a negative valueof w~,(ai, aj), we can observe that since the capacityof a resource is in general greater than one, some ac-tivities in a solution can overlap and others have tobe sequenced. From the point of view of the heuristicmethod it is better to resolve pairwise conflicts in theway that minimizes loss of temporal positions.

The time complexity associated with detecting aconflict in the use of a resource rj which satisfies theheuristic method previously defined, is quadratic. Infact, if N~j is the number of activities which request theresource rj, then we have to make O(N~j) comparisonsfor every non sequenced pair of activities <ai, aj>.

Chaining Algorithm. Given an earliest start timesolution, one way to generate a conflict free solution isto create a set of ej chains of activities on the resourcerj. That is, we can partition the set of activities whichrequire rj, into a set of cj linear sequences. This oper-ation earl be accomplished by deleting all the levelingconstraints in the solution and using the establishedlower bound solution to post a new set of constraintsaccording to the division into linear sequences. In thissituation, which we refer to as chain-form, if N~ is thenumber of activities which request rj, then the numberof precedence constraints posted is at most N,i - ej. Incontrast, we would generally expect that the processof determining an earliest start time solution insertsa greater number of leveling constraints. A solutionin chain-form is a different way to represent a solu-tion that. presents two advantages: Ca) the solution is

222 Scheduling

conflict free for on line modifications of start or endtime of activities; (b) there are always O(No) levelingconstraints {where Na is the total number of activi-ties). So, every temporal algorithm whose complexitydepends on the number of distance constraints can gainadvantages from this new form of the solution.

Experimental Evaluation. The ESTA algorithmwas also run on our set of scheduling problems andresults are shown in Table 3. It should be noted thatin this case the parameter Me refers only to the solu-tion produced by the ESTA without the chaining pro-cedure. On the contrary, the other parameters refer tothe production of the conflict free solution (ESTA Chaining).

The results are particularly encouraging. ESTAsolves all the problems presented but more interest-ingly it is very quick in solving them and the quality ofthe solution is much better than the CFSA+TF algo-rithm. The speed in producing a solution is confirmedby the low number of leveling constraints posted. It isalso worth noting that with respect to makespan androbustness, ESTA outperforms all CFSA style solutionstrategy. The result concerning Mk, can be justified bythe search of the earliest start time solution and by thefact that the CFSA has the pathology of overconstrain-ing the solution by inserting unnecessary leveling con-straints (more precisely, these algorithms "over-chain"the solution). The advantage with respect to robust-ness RB confirms the relevance of techniques whichretain solution flexibility in solving MCM-SPs.

Table 3: Experiments with ESTA Algorithm

I Probl II No I RB I Mk, I N,o I CPUI5x5 50 9.1 210.5 1.8 1.010x5 50 7.5 300.0 21.1 5.815x5 50 5.4 430.1 71.9 18.220x5 50 4.9 545.2 143.4 41.025×5 50 4.6 664.1 233.2 81.3

Concluding RemarksIn this paper we have introduced and analyzed a pro-gression of algorithms for solving the multi-capacitatedmetric scheduling problem (MCM-SP): a class that in-volves both metric time constraints and sharable re-sources with the capacity to simultaneously supportmultiple activities. This type of problem is quite com-monplace in real applications, but it has received rel-atively little attention in the planning and schedulingliterature.

To support a comparative analysis of solution meth-ods, we began by defining a procedure for generating acontrolled set of random benchmark problems. A set ofcriteria for judging the effectiveness of alternative so-lution procedures were also identified, including mea-sures relating to both problem solving ability (num-ber of problems solved, average computation time) andquality of the solutions generated. With respect to thelatter point, we considered Makespan minimization (acommon objective) as one target criterion. We also in-troduced a definition of robustness. Although initial,


we believe it provides a useful characterization of the~lluidity" of a solution in these %trongly temporal~

domains.Having established a framework and design for

experimental evaluation, a number of profile-basedscheduling algorithms were then defined and evalu-ated. Each algorithm that was considered conformedto a general solution schema that entailed (1) moni-toring resource demand over time for periods of over-utilization and (2) introducing additional sequencingconstraints between pairs of competing activities tolevel out periods of high demand. The different algo-rithms considered varied in the heuristic criteria usedto prioritize and make sequencing decisions, as well asin assumptions made about the number of sequencingconstraints required for a feasible solution.

A first result of the experimentation showed that agreedy version of a previously reported algorithm (EI-Kholy & Pdchards 1996) (called CFSA in this paper)performed rather poorly on larger sized problems. Thereason for the limited effectiveness of the algorithmstemmed from the weak look-ahead guidance providedby the computation of upper and lower bound on re-source demand profiles.

A second result obtained concerns the extensionof ideas of temporal flexibility introduced in (Cheng& Smith 1994) to the case of multiple-capacitatedsched-llng problems. By substituting the use of tern-pored flexibility as a basis for prioritizing and mak-ing decisions with the basic CFSA procedure, substan-tial performance improvements were obtained at thelarger problem sizes tested. Once again these tech-niques have been shown to be very effective and rele-vant for scheduling problems with quantitative separa-tion constraints.

A final result, triggered by the observation thatCFSA style procedures tend to insert many more level-ing constraints than are necessary to ensure feasibility,concerned the development of an alternative MCM-SP solution approach. In particular, a very quick andeffective solution procedure was obtained by focusingon construction of solution for a precise time instanton the time line as opposed to more general construc-tion of a conflict free solution. The ESTA procedure,which embodies this approach and computes an "ear-liest start time solution", was shown to outperformall variants of CFSA on all dimensions of performanceconsidered. Furthermore, a post-processing algorithmwas also s-mmexised for transforming this early-timesolution into a conflict-free solution; thus achieving thesame level of solution generality as CFSA but withoutthe unnecessary leveling constraints inserted by CFSAstyle approaches. This fact reproduces, in a differentcontext, the well known effect in scheduling that whenreasoning with exact start times it is possible to makestronger deductions than in the case in which fluctu-ating time-points are reasoned about.

There are a number of directions in which the workreported in this paper might be extended. One path ofinterest is to attempt to productively expand the levelof search performed by the greedy, one-pass solutionprocedures developed here. Recent work in iterative

sampling and re-starting search techniques (e.g., (Oddi& Smith 1997)) would seem to provide a natural ba-sis. Another area of future research concerns compari-son of the profile-based solution procedures developedhere with other (non profile-based) approaches. We areparticularly interested in exploring is the clique-basedapproach proposed in (Laborie & Ghallab 1995). Dueto its global nature, this approach offers an alterna-tive basis for eliminating the posting of extra (over-constraining) sequencing constraints. But the implica-tions remain unclear with respect to how much increasein computation time is required.

Acknowledgments.Amedeo Cesta and Angelo Oddi’s work is supportedby Italian Space Agency, and by CNR Committee 12on Information Technology (Project SCI*SIA). AngeloOddi is currently supported by a scholarship from CNKCommittee 12 on Information Technology. StephenF. Smith’s work has been sponsored in part by theNational Aeronautics and Space Administration undercontract NCC 2-976, by the US Department of DefenseAdvanced Research Projects Agency under contractF30602-97-20227, and by the CMU Robotics Institute.

ReferencesAumello, G.; Italiano, G. F.; Marchetti Spaccamela, A.; and Nanni,U. 1991. Incremental Algorithms for Minimal Length Paths. Jour-nal of Algorithms 12:615-638.Cesta, A., and Steiln, C. 1997. A Time end Resource Problem forPlanning Architectures. In Proceedings of the Fonrth EuropeanConference on Planning (ECP 97).Cheng, C., and Smithj S. F. 1994. Generating Feasible Schedulesunder Complex Metric Constraints. In Proceedings J~th NationalConference on AI (AAAI-94).Cheng, C., and Smith, S. F. 1996. A Constraint Satisfaction Ap-proach to Makespan Scheduling. In Proceedings of the 4th Inter-national Conference on AI Planning Systems (AIPS-g6).Dechter, R.; Meiri, I.; and Pearl, J. 1991. Temporal ConstraintNetworks. Artificial Intelligence 49:61-95.El-Kholy, A., and Pdchards, B. 1995. Temporal and ResourceReasoning in Planning: the parcPLAN Approach. In Proc. of thel£th European Conference on Artificial Intelligence (ECAI.96).Ersclder, J.; Lopez, P.; and Thuriot, C. 1990. Temporal reason-ing under resource constrednts: Application to task scheduling. InLasker, G, and Hughes, R., ode., Advances in Support SystemResearch. International Institute for Advanced Studies in SystemsResearch and Cybernetics.Lsborie, P., and Ghallab, M. 1998. Planning with Sharable Re-source Constraints. In Proceedings of the International Joint Con-ference on Artificial Intelligence (IJCAI-95).Nuijten, W. P. M., and Aarte, 1~, H. L. 1994. Constraint Satisfac-tion for Multiple Cnpacitated Job-Shop Scheduling. In Prec. o] thel lth European Con]crones on Artificial lnteiligence(ECAI-9~).Oddi, A., and Cesta, A. 1997. A Tabu Search Strategy to SolveScheduling Problems with Deadlines and Complex Metric Con-attaints. In Prec. ~th European Cony. on Planning (ECP 97).Oddi, A., and Smith, S. F. 1997. Stochastic Procedures for Gener-ating Feasible Schedules. In Proceedings l~th National Con]erencson AI (AAAI-97).Oddi, A. 1997. Sequencing Methods and Temporal Algorithmswith Application in the Management oy Medical Resources. Ph.D.Dissertation, Department of Computer and System Science, Uni-verslty of Rome "La Sapienza’.Sadeh, N. 1991. Look-ahead Techniques for Micro-opportunisticJob.shop Scheduling. Ph.D. Dissertation, School of Computer Sci-ence, Carnegie Mellon University, Pittsburgh, PA.Smith, D. R., and Westfold, S. J. 1996. Scheduling an Aeyn-chronouely Shared Resource. Working Paper, Kestrel Institute.Taillard, E. 1993. Benchmarks for Basic Scheduling problems.European Journal o] Operational Research 64:278-285.

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Profile-Based Algorithms to Solve Multiple Capacitated Metric Scheduling Problems

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