Inteligencia Artificial. Revista Iberoamericana de Inteligencia Artificial ISSN:

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1 Inteligencia Artificial. Revista Iberoamericana de Inteligencia Artificial ISSN: Asociación Española para la Inteligencia Artificial España Zaballos, Lis J.; Henning, Gabriela P. A Constraint Programming Approach to FMS Schedling. Consideration of Storage and Transportation Resorces Inteligencia Artificial. Revista Iberoamericana de Inteligencia Artificial, vol. 9, núm. 26, verano, 2005, pp Asociación Española para la Inteligencia Artificial Valencia, España Available in: How to cite Complete isse More information abot this article Jornal's homepage in redalyc.org Scientific Information System Network of Scientific Jornals from Latin America, the Caribbean, Spain and Portgal Non-profit academic proect, developed nder the open access initiative

2 ARTÍCULO A Constraint Programming Approach to FMS Schedling. Consideration of Storage and Transportation Resorces Lis J. Zeballos and Gabriela P. Henning INTEC (Universidad Nacional del Litoral - CONICET) Güemes 3450 (3000) Santa Fe, Argentina {zeballos, ghenning}@intec.nl.ed.ar Abstract Flexible manfactring systems (FMS) reqire efficient schedling strategies to achieve their principal advantage of being able to combine high flexibility with high prodctivity. This paper presents a novel constraint programming (C formlation addressing the FMS schedling problem for systems that work nder the operational policy of part movement. The formlation addresses some critical featres fond in these problems and proposes a search strategy to speed/p the soltion process. This work also presents an ad-hoc procedre, based on different initialisation points, that aims at making problems tractable when their size is big enogh to make difficlt the attainment of good qality soltions in a reasonable CPU time. Keywords: Constraint satisfaction, Flexible manfactring systems, Schedling, Resorce Limitations. 1. Introdction Flexible manfactring systems (FMS) reqire efficient schedling strategies to achieve their principal advantage of being able to combine high flexibility with high prodctivity [Li & MacCarthy 97]. In most FMSs there are two operational policies [Gamilla & Motavalli 03]. One is the tool movement policy, nder which parts are assigned to machines and the necessary tools are transported to the varios machines to perform the reqired operations. Ths, a tool transportation system is reqired. The other one is the part movement policy, nder which the parts are transferred from one machine to another since a given machining center does not have all the reqired tools to process a part. Therefore, a part s handling device is necessary. De to their inherent complexity, a wide variety of soltion techniqes have been employed to tackle the varios schedling problems that can be fond in several indstrial domains. Ths, researchers and practitioners have resorted to mathematical programming and heristic approaches, network algorithms, dynamic programming, intelligent agent methodologies, genetic algorithms, etc. Constraint Programming (C [Marriot & Stckey 99] is one of the techniqes that has gained increased attention in the last years and that has been sccessflly applied to the schedling domain [Alfonso & Barber 00]. In spite of that, very few contribtions have made se of CP in FMS schedling problems [El Khayat et al. 03]. Moreover, these contribtions are very preliminary. This paper presents a novel CP formlation addressing a part movement class of FMS schedling problem. It involves two components. The CP model itself, consisting of the set of constraints that represents the problem being tackled, and a search procedre. The proposed search strategy takes advantage of some domain featres that improve the performance of the standard domain redction and constraint propagation algorithms inclded in the adopted CP langage. The proposed formlation is based on the

3 40 Inteligencia Artificial V. 9, Nº 26, 2005 OPL langage, spported by ILOG Solver [ILOG03a], and employs some specific schedling constrcts available in ILOG Schedler [ILOG03b]. The CP model addresses some important featres fond in indstrial problems, sch as eqipment ready-times as well as release times and de-dates of obs. Bt, more importantly, it sccessflly takes into accont constraints on resorces sch as storage capacity (machine bffers) and material transport devices, which represent critical aspects of practical problems. It is worth to remark that the proposed formlation does not decople two essential decisions associated to the problem: (i) the assignment of ob operations to machines (i.e. loading) and (ii) the seqencing of the assigned operations. Ths, the FMS schedling problem is tackled in a global way, proving better qality soltions than the widely advocated loading then seqencing method. Regarding the search strategy, the paper proposes distinct approaches for different size problems. For smaller ones, the aim is to start the search process from a good qality soltion that exhibits a balanced machine load. For big size ones, an ad-hoc algorithm, based on different initialisation points, is proposed. This domain specific procedre aims at making problems tractable by speeding-p the soltion process. The contribtion incldes comptational reslts for test problems reported in the open literatre [Li & MacCarthy 97]. Moreover, problems of different sizes, that consider a variety of obective fnctions (sch as minimizing the maximm completion time, mean completion time and tardiness measres), are tackled. The obtained reslts show both, very good qality and a remarkable comptational efficiency. The rest of this paper is organized as follows. Section 2 characterizes the problem nder stdy and Section 3 introdces the CP model. In this section, the adopted nomenclatre is presented, the basic problem constraints and obective fnctions are introdced. Section 4 shows the implemented search strategy and the proposed iterative procedre to find satisfactory soltions for big size problems. To illstrate the application of the proposed CP model, Section 5 presents comptational reslts and discsses experiences associated to varios case stdies. Finally, section 6 reports conclsions and ftre work. 2. Problem Definition This work considers a flexible manfactring cell with a fixed nmber of machines, a single material handling device and a given nmber of obs or parts that reqire several processing operations. In order to bild the CP model, the following assmptions are taken into accont: All parts reqire the same nmber and seqence of operations. The operations precedence is fixed and known in advance. Processing times do not depend on the alternative machines. Pre-processed parts entering the system and finished parts leaving the system are stored in different bffers having infinite capacity. In trn, in-process parts are stored in machines inpt bffers having limited capacity. Processing times, order release-times and dedates, machine ready-times, part transportation times and sizes of local bffers are all fixed and known. The transport device cannot be sed as a temporary storage bffer. Transportation times are eqal for all types of travelling movements. When the transport device goes from the bffer for pre-processed parts to the bffer for finished parts, or vice versa, it mst cross the area where machines are located. Then, it can be assmed that the transportation time taken by the transport device to go from one of these bffers to the other is eqal to the time taken by the transport device to go from the bffer for pre-processed (or finished) parts to the machine s zone, pls the time taken by the transport device to go from the machine s zone to the bffer for finished (or pre-processed) parts. 3. Constraint Programming Model The proposed model will be gradally introdced; starting with the nomenclatre and a basic formlation, that will later incorporate additional featres. Then, constraints for handling both, parts transportation and machine bffers will be presented. Finally, different obective fnctions will be posed Nomenclatre Sbscripts, k Parts or obs to be processed. p, p Operations associated to parts, that need to be schedled Machines to process the reqired operations.

4 Inteligencia Artificial V. 9, Nº 26, Sets U J P s Set of machines and their associated inpt bffers. Set of parts. Set of operations. Set of alternative machines. In addition, the model handles the following extra variables. MK T Maximm completion time or Makespan. Tardiness of order with respect to its promised de-date. Model Variables Task-Timing Related Variables: In most schedling approaches, tasks to be schedled (activities to be done) and resorces are two essential modeling elements. ILOG OPL Stdio and ILOG Schedler have incorporated them as pre-defined variables. In this contribtion, Task, TaskSt and TaskTr are three types of activities to be inclded in the model. The first one represents processing tasks associated to parts (machining operations). Ths, activity Task models the exection of operation p on part. The second one represents storage tasks that take place in bffers and the third one represents part handling tasks that are exected by the transport device. All of the activities are characterized by means of dration, starting time and ending time variables (e.g. Task.dration, Task.start and Task.end), that are related among themselves. The activity type TaskSt has two kinds of tasks associated to it: TaskStIni corresponds to the storage of part previos to the exection of the first reqired operation on part. It occrs in the bffer of the machine in which the first operation is done. TaskStInter p is related to the storage task of part performed between two sccessive operations (p and p ). It takes place in the bffer of the machine to which the second operation is assigned. In the same way, the activity type TaskTr originates three different kinds of tasks: TaskTrIni related to the transportation activity of part from the storage place for pre-processed parts to the machine that will execte the first operation. er p corresponds to the transportation of part between two different processing nits where operations p and p are respectively carried ot. TaskTrFin is related to the transportation activity of part from the machine that has exected the last operation on the part to the storage place for finished parts. Parameters d De date of part t Processing time of operation p on part. rt Release time of part. tp Transportation time between two machines. rd Ready time of machine. sb Capacity of machine s bffer. In trn, the model handles the following special parameters. mach ILOG parameter indicating that each processing nit mach is a nary resorce (cannot be shared by two activities at a given time) that can execte st one operation on a part at any time. Unary resorces having similar capabilities can be groped into resorce sets; this is the case of the machines mach, which are groped into the s set. c ILOG parameter indicating that the handling device c is a nary resorce and that can execte st one transportation task on a part at any time. bffer ILOG parameter indicating that each storage bffer bffer is a discrete resorce (one having a fixed capacity). In this case, it is called sb and represents the maximm amont of parts that can be stored in the resorce at a given time Model Characteristics The CP model has been written in the OPL constraint programming langage, spported by ILOG Solver, employing some specific schedling constrcts available in the ILOG Schedler package. One of them is reqires which, as shown in Eq. (1), enforces the assignment of resorces (on the right hand side) demanded by activities (on the left hand side). Another special constrct is precedes, which has the prpose of imposing a proper seqence of non-overlapping activities. As shown in Eq. (2), the activity located at the right hand side cannot be initiated ntil the activity on the left hand side has been finished. One of the most important special constrcts is ActivityHasSelectedResorce, which

5 42 Inteligencia Artificial V. 9, Nº 26, 2005 easily allows handling alternative resorces. It acts like a predicate that evalates to tre (assmes a vale eqal to one) when an activity has been assigned to a specific resorce belonging to a set of alternative resorces Model Constraints and Performance Measres Assignment, Timing, Seqencing and Precedence constraints Task reqires s J, p P Task precedes Task ' J, p, P, ord ( ) = ord ( + 1 Task start rt + tp J, p P, = first( activityhasselectedresorce Task start rd mach ) J, p P, = first( U (1) (2) (3) (4) Constraint (1) is an assignment relation prescribing that each machining operation p associated to a ob mst be assigned to st one processing nit mach belonging to the set of alternative resorces s. Constraint (2) enforces a proper seqencing of all the consective machining operations (p and p ) that need to be performed on each part. Eqs. (3)-(4) place lower bonds on the start time of the first operation p exected on part. Eq. (3) sets a bond de to the release time of ob (rt ) pls the time reqired to move sch part from the bffer for preprocessed parts to the machine execting the first operation (t. Eq. (4) fixes another bond that depends on the ready time of the machine assigned to the first operation (rd ). In this case ActivityHas SelectedResorce evalates to tre when Task has been assigned to mach, which belongs to the set of alternative nits s. Transportation constraints TaskTrIni. dration = 2* tp J (5) TaskTrFin. dration = 2* tp J (6) Constraints (5)-(6) set the dration of the initial and final transportation tasks, respectively. Eq. (5) prescribes that the dration of part s initial movement is two times the basic transportation time. It incldes the time taken by the nloaded handling device to first go from the processing nit where it is crrently located to the bffer for pre-processed parts where is placed and then, from sch bffer to the one of the machine that will execte the first operation on. In the case of Eq. (6), it incldes the time taken by the transport device to go from the processing nit execting the last operation on to the finished parts bffer and from sch bffer to any machine that wold reqire it. activityhasselectedresorce activityhasselectedresorce ' mach p '. start = Task. end &. end Task '. start & mach ) ) p '. dration = tp J, p, P ) = + 1, U activityhasselectedresorce mach ) = 1& activityhasselectedresorce Task mach ) =1 ( ' p '. start = Task. end &. end Task '. start & p '. dration = 0 J, p, P ) = + 1, U (7) (8) Constraint (7) indicates that when two consective operations (p and p ) associated to a given ob are processed in different nits, an intermediate transportation task takes place and its dration assmes a positive vale. In sch a case, the p activity shold start when the exection of operation p on part has finished and shold end when the next processing operation p, is abot to start. On the other hand, as indicated by constraint (8), when two consective operations belonging to a given ob are exected in the same nit, there is no need for an intermediate transportation activity and its dration is set to nll. TaskTrIni reqires c J (9) TaskTrFin reqires c J p '. dration > 0 p ' reqires c J p, P, ) = + 1,. (10) (11) Eqs. (9) - (11) enforce the assignment of the three kinds of transportation tasks to the handling device c carrying ot the movements of parts. The definitive assignment of p depends on its dration. Indeed, this kind of activity is actally assigned if and only if its dration is greater than zero; otherwise, the activity does not exist and there is no need to assign it to the transport device. TaskTrFin. start = Task. end J, p P, = last( (12)

6 Inteligencia Artificial V. 9, Nº 26, Constraint (12) prescribes that the final movement of a given part shold start st at the time the last operation performed on it has finished. p '. dration> 0 & kqq '. dration > 0 & activityhasselectedresorce mach ) activityhasselectedresorce precedes kq' kqq' kqq '. end + tp. mach ) & start, k J, k, p,, q, q' P, ) = + 1, q' ) = q) + 1, q), U kpp '. dration > 0 & activityha sselectedresorce Task mach activityhasselectedresorce ( q k mach kp precedes TaskTrFin. end + tp TaskTrFin start kpp '. ), k J, k, p,, q P, q) = last( ) = + 1, U (13) ) & (14) Constraints (13) - (15) prescribe the temporal relationships existing between varios transportation activities carried ot by the same handling device on two different parts. Constraint (13) represents the case in which two actal intermediate movements are demanded to transport parts and k. When the intermediate transportation activity corresponding to part k is performed before the intermediate transportation activity corresponding to part, a tp time is needed by the transport device to go from the machine where the part k is nloaded to another machine where the part is to be picked-p in order to be transported. On the other hand, constraint (14) prescribes the temporal relationship between an actal intermediate transportation activity and a final one. When the intermediate transportation activity is performed before the final one, a tp time is needed by the material handling device to go from the machine where part k is to be nloaded (to execte the p operation) to the nit where part is to be picked-p to be transported to the finished parts bffer. This constraint does not prescribe the sitation in which the final transportation activity takes place before the intermediate one. In this case, a tp time is also needed by the transport device to go from the finished parts bffer, where part is nloaded to a machine where part k is to be loaded (to be moved to another nit). However, provided the definition of the final transportation task given in eq. (6), this time is inclded in the dration of the activity and therefore this sitation does not need to be taken into accont in the expression. kpp '. dration > 0 & activityhasselectedresorce Task mach activityhasselectedresorce ( q kp mach TaskTrIni precedes kp TaskTrIni end + tp start. kp. ) ) &, k J, k, p,, q P, q) = first( ) = + 1, U (15) Constraint (15) models the temporal relationship associated to the case in which the initial transportation activity takes place before the intermediate one. Its explanation and nderlying rationale is analogos to the one of constraint (14). Storage constraints TaskStIni reqires ( activityha sselectedresorce( Task mach )) bffer J, p P, = first( U TaskStInt reqires activityha sselectedresorce( Task mach )) bffer J, p, P, ) = + 1, U ( ' (16) (17) Eqs. (16)-(17) enforce the assignment of storage tasks to the proper machine inpt bffers, depending on the assignment of operations to machines. Ths, when a given operation on a certain part is assigned to machine mach, the corresponding storage activity mst also be assigned to this machine s bffer (bffer ). This applies to both types of storage tasks. TaskStIni. start = TaskTrIni. end TaskStIni. end = Task. start J, p P, = First( J (18) (19) Constraints (18)-(19) prescribe the starting and ending times of initial storage activities. Eq. (18) indicates that any initial storage activity mst start when its associated initial transportation task ends, and Eq. (19) sets that it finishes when the first processing operation performed on the part begins. TaskStInt. start =. end J, p, P, ) = + 1, TaskStInt. end = Task'. start J, ord ( ) = + 1, (20) (21) In trn, Eqs. (20)-(21) model the starting and ending times of intermediate storage activities. Eq. (20) states that any intermediate storage activity mst start when its associated intermediate transportation

7 44 Inteligencia Artificial V. 9, Nº 26, 2005 task ends, Eq. (21) asserts that it ends when the corresponding operation on the part starts. Performance measres constraints Makespan Task precedes MK J, p P, = last( Min MK Mean Completion Time Min J Task. end np p P, ord ( = last( np = card( J ) Tardiness T Max 0, Task (. end d ) J, p P, = last( Min J T (22) (23) (24) (25) (26) The CP model can deal with several obective fnctions sch as makespan MK, mean completion time and performance indexes related to ob tardiness T. If makespan is chosen as the problem obective fnction, constraints (22) and (23) shold also be inclded in the model since (22) defines the makespan concept. Constraint (24), which comptes the mean completion time, becomes part of the model if this obective fnction is adopted. Finally, Eq. (25) defines the tardiness concept by assigning a positive vale to variable T if the finishing time of the last operation on ob is greater than the ob s de-date d. This eqation shold be inclded in the model if the total tardiness defined by Eq. (26) becomes the problem performance measre. 4. Search Strategies Constraint programming systems provide defalt search procedres; however, most of them allow to easily program ser defined search strategies. This type of spport is fndamental for hard combinatorial optimisation problems demanding special search procedres [Van Hentenryck 99]. This fnctionality, represents one of the most important characteristics of constraint programming langages, which can lead to improvements in performance by implementing search strategies tailored to particlar domains. Ad-hoc search strategies may explore the entire search space (total strategies) or a portion of it (partial strategies), as in those cases in which the vales that model variables can adopt when instantiated are restricted and only a portion of the search space is actally explored Search Strategy for Small Size Problems A global domain-specific search strategy, that specifies the mechanism by which operations are assigned to machines, has been defined for low dimensionality problems. Conceptally, this strategy starts from an initial balanced assignment of operations to machines, avoiding neven loads. Moreover, provided there is an operations precedence, the search strategy prses the assignment of operations to machines according to sch ordering. This preliminary assignment allows to obtain a good qality soltion in a short time as well as to have a good pper bond that permits to prne poor qality soltions. The initial assignment of operations to machines is done in seqential order, starting from the first operation p 1 exected on each part, following with the second one and contining in the same way with the remaining ones. To achieve this ordered trial of assignments, the procedre C_Constraint_Bilder (see Fig. 1), assembles a list, named C, of length np*n, containing as elements several primitive constraints of the form c i = (Task x). This list acts as a constraint dring the search process. At the very beginning, all assignments belonging to C are free (Task x); bt then, at any point of the global search process, C will keep track of the already fixed variables. Let n card(u) Let n card(j) Let np card( Let C primitive constraints list: c 1..c n Let c 1,c 2,,c n primitive constraints of the form Task x for non-assigned tasks or Task for tasks assigned to machines Let n = np*n, nmber of primitive constraints Let x variable denoting a machine to be assigned Let PartList list of parts Let OperList list of operations Let MachList list of machines C_ Const raint_bilder ( ) Let i = 1 for(p in OperList) do for ( in PartList) do append c i = (Task x) to C i = i + 1 retrn C MachWorkList_Bilder( ) = first element of MachList for (p in OperList) do for ( in PartList) do Let MachWorkList Empty list for (k in 1 n) do append to MachWorkList if ( = last element of MachList) then = first element of MachList

8 Inteligencia Artificial V. 9, Nº 26, else = next element to in MachList if ( = last element of MachList) then = first element of MachList else = next element to retrn MachWorkList in MachList F igre 1. Conceptal definition of axiliary procedres employed dring the search strategy For example, let s consider a problem in which for parts, reqiring three operations each one, are to be schedled in three machines. If at a given point dring the exploration of the search space, only the first operation p1 exected on parts 1 and 2, has been tentatively allocated to machines 1 and 2, and all other assignments have not been attempted yet, the corresponding C constraint, having 12 primitive constraint elements, will adopt the following format: {c 1 =(Task 1p1 1 ), c 2 = (Task 2p1 2 ), c 3 = (Task3p1 x), c 4 = (Task1p2 x), c 5 = (Task 2p2 x),, c 12 = Task 4p3 )} As seen, the order in which tasks appear in C reflects the seqence in which the assignment of tasks to machines will be tried ot dring the search process. Therefore, each task Task represents a node of the search space in which varios machine assignments will be explored. In order to prse a balanced machine load, the order in which machines are attempted to be assigned at each Task node, will be different. Ths, each node has a particlar list of assignable machines, referred as MachWork List. The seqence of elements in sch list represents the order in which machines wold try to be assigned to Task dring the search process. The set of machine lists is created by the Mach WorkList_Bilder procedre shown in Fig. 1. Referring to the previos example, the lists of machines associated to the first three tasks are: Task 1p1 { 1, 2, 3 }, Task 2p1 { 2, 3, 1 }; Task 3p1 { 3, 1, 2 }. Ths, at the start of the search process the first three tasks in C are allocated to three different machines, ths initiating a balanced assignment. The code that formalizes the proposed domain specific search strategy is presented in Figre 2. Let c 1,c 2,,c n primitive constraints of the form Task x for non-assigned tasks or Task for tasks assigned to machines Let C and C1 constraints Let x variable representing a machine to be assigned Search_Strategy(C) If (all tasks have st been assigned) then retrn Partial_Satisfiable( C) else choose c i first not assigned task (Task x) in C Let MachworkList List of machines associated to Task for in MachworkList do let C1 be obtained from C by instantiating x with : c i = (Task ) if Partial_Satisfiable(C1) then if Search_Strategy(C1) then retrn tre else ci = (Task x) retrn false Figre 2. Search strategy for small problems Search_Strategy is a recrsive procedre that resorts to another one that verifies if assignments are feasible. Sch procedre, named Partial_ Satisfiable (see Fig. 3), operates on constraint C testing whether each one of the primitive constraints in which x has already been instantiated is satisfiable. Partial_Satisfiable (C) for i in 1..n do if the task in c i is assigned then if satisfiable(c i ) = false then retrn false retrn tre Figre 3. Constraint feasibility test When the test that is being applied to a given c i primitive constraint does not scceed, the Partial_Satisfiable procedre retrns false and conseqently, the Search_Strategy procedre disengages the last assignment (See Fig. 2) that was done in relation to the Task corresponding task. Ths, it starts a backtracking step, nder which a new machine belonging to MachworkList wold be tried. If all the machines belonging to this list have been tried and have also failed, then, the backtracking step wold be more profond and wold affect the task associated to the previosly treated primitive c i-1

9 46 Inteligencia Artificial V. 9, Nº 26, 2005 constraint. Therefore, if necessary, all the assignments can be revised and the backtracking process can nwrap p to the very first machine allocation that was initially performed Search Strategy for Large Problems When problems are of bigger size, an iterative strategy is proposed to obtain good qality soltions in a reasonable CPU time. This strategy consists in solving the problem several times, as many as the nmber of parts associated to the problem being solved, bt each time applying a different partial search process. The most important change among the processes is the starting point; ths, each one begins from a different balanced assignment of tasks to machines. Each partial search process belonging to the strategy begins from a distinct starting point and stays searching dring a limited time; when the time limit is over the best obtained soltion is taken and added to the model as an pper bond to be applied on the next partial search procedre. This process is repeated ntil the last partial procedre reaches its time limit. The rationale behind this approach is based on the fact that for each partial process, soltions are qickly obtained. However, after a short time no more soltions are fond in a fast fashion and the next soltion may be obtained in an ncertain time (order of several mintes, hors or days, depending on nmerical aspects and problem sizes). Each partial search is conceptally similar to the one implemented for small problems. In fact, the first search procedre is identical to the one applied in the previos section. The remaining ones change in the order in which tasks Task appear in the primitive constraints belonging to C. Provided sch distinct ordering of tasks in C, the search space will be differently explored by Search_Strategy(C). It shold be remarked that the policy of first assigning the first operation to be exected on each part, then the second one, and so on, is to be kept. Ths, in order to generate a different seqencing of tasks in C, a different arrangement of parts in PartList is st necessary. If the ordering of elements in this list changes, then, the otpt of the C_Constraint_Bilder procedre will be different. So, each partial search procedre will be associated to a different ordering of parts, that will be referred as InitWorkList. Fig. 4 shows the procedre that generates the varios part orderings associated to the different partial search processes. Let n card(j) Let PartList list of parts InitW orklist_bilder() = first element of PartList for (l in 1 to n) do Let InitWorkList l empty list for ( in PartList) do append to InitWorkList l if ( = last element of PartList) then ( = first element of PartList) else ( = next element to in PartList) = next element to in PartList retrn InitWorkList Figre 4. Conceptal definition of the initial working list for each partial search 5. Examples and Comptational Reslts In order to examine the effectiveness of the proposed CP model and of the corresponding search strategies, a set of test problem instances of different sizes have been generated in a random fashion. Also, a small problem reported by [Li & MacCarthy 97] has been considered, too. Small problems comprise case-stdies having a range of 2 to 4 machines, 3 to 5 obs, 1 to 4 operations per ob (where any machine can be selected to process any operation), and machine storage bffers having capacities that range from 0 to 2. They correspond to the first five data sets inclded in Table 1. In large problems, in which the strategy introdced in section 4.2 was applied, the nmber of machines was increased. It ranges from 5 to 7, as well as the nmber of obs, that ranges from 6 to 10. However, all the other problem characteristics were kept the same. They correspond to the last three data sets located at the bottom of Table 1.

10 Inteligencia Artificial V. 9, Nº 26, Figre 5. Gantt diagram corresponding to a case stdy having six obs, five operations, five machines and a bffer capacity of one part. Makespan (1) was adopted as the schedling criterion Obective Obective Vale CPU Time Data Set Fnction CP MILP CP MILP J=4, P=2, U=2, sb= J=5, P=2, U=3, sb= * J=5, P=3, U=3, sb= * J=3, P=4, U=3, sb= * J=5, P=3, * U=4, sb= * * 490* J=6, P=5, 2 330* 384* U=5, sb= * J=5, P=7, U=5, sb= * J=8, P=5, * U=5, sb= * * Sboptimal soltion; --- No soltion was fond. Table 3. Comptational reslts corresponding to several examples Table 1 incldes reslts obtained with the proposed approach as well as reslts of the mixed-integer linear programming (MIL model introdced by [Li & MacCarthy 97]. The adopted schedling criteria are makespan (1), mean completion time (2) and tardiness (3). As seen, the comptational effort demanded by the CP approach is not affected by the choice of the obective fnction. The comptation was carried ot on a Pentim VI, 1.8 Ghz, PC with 1Gbyte of RAM memory. The time limit to obtain soltions was set to 2000 seconds of CPU time. Then, to obtain comparative reslts between the CP and MILP approaches, the time limit sed for the large problems search strategy was compted as the previos one divided by the nmber of particlar searches to be

11 48 Inteligencia Artificial V. 9, Nº 26, 2005 performed. From Table 1, it can be observed that in all the cases CP reslts are better than the MILP approach ones. In addition, while the problem sizes are of redced dimensionality, the times of both soltion methodologies are similar, bt when the problem sizes increase, the CP model is the only one that prodces good soltions in small CPU times. Figre 5 depicts the Gantt diagram corresponding to one of the large problems (Data set 6: J=6, P=5, U=5, sb=1). For illstrative prposes, dotted lines help tracing the path associated to part Conclsions A novel CP formlation that addresses a class of FMS schedling problems has been proposed. It can handle many critical featres sally fond in indstrial problems, sch as eqipment ready-times, release times and de-dates of obs, machine bffers of limited capacity, transportation activities carried ot by a material handling device, etc. The approach incldes a CP model as well as domain specific search strategies. Regarding the model, it shold be remarked that it does not decople two essential decisions associated to the problem: (i) the assignment of ob operations to machines and (ii) the seqencing of the assigned operations. Therefore, the problem can be tackled in a global way. Ths, better qality soltions than the widely advocated loading then seqencing heristic method can be obtained. The approach has been tested with a variety of case stdies and has rendered excellent soltions, reqiring very little comptational effort. In the case of small and medim size examples optimal soltions have been reached. For large problems, an ad-hoc partial search strategy has been introdced. It allows reaching sboptimal soltions in reasonable CPU times. References [Alfonso & Barber 00] M.I. Alfonso and F. Barber. Combinación de Procesos de Clasra y CSP para la Resolción de Problemas de Schedling. Revista Iberoamericana de Inteligencia Artificial, 9. pp (2000). [El Khayat et al. 03] G. El Khayat, A. Langevin and D. Riopel. Integrated Prodction and Material Handling Schedling sing Mathematical Programming and Constraint Programming Proceedings of CPAIOR 03. Montreal, Canada, May (2003). [Gamilla & Motavalli 03] M.A. Gamila and S. Motavalli. A Modeling Techniqe Loading and Schedling Problems in FMS. Robotics and Compter Integrated Manfactring, 19. pp (2003). [ILOG 03a] ILOG OPL Stdio 3.7. User s Manal, France. (2003). [ILOG 03b] ILOG Schedler 6.0. User s Manal, France. (2003). [Li & MacCarthy 97] J. Li and B.L. MacCarthy. Theory and Methodology. A Global MILP Model for FMS Schedling. Eropean Jonal of Operations Research, 100. pp (1997). [Marriot & Stckey 99] K. Marriot and P. Stckey. Programming with Constraints. An Introdction. The MIT Press, Cambridge, Massachsetts. (1999). [Van Hentenryck 99] P. Van Hentenryck. The OPL Optimization Programming Langage. The MIT Press, Cambridge, Massachssets. (1999). Ftre work incldes improving the search strategy by making a more intelligent choice of the search procedre s starting points. Moreover, a more clever assignment of operations to machines will be attempted. In addition, the seqencing of the transportation activities will be stdied in detail. Acknowledgments Athors acknowledge financial spport from ANPCyT nder Grant , from CONICET (PEI 6191) and Universidad Nacional del Litoral (CAI+D 2002, 5-30).