A framework for integrating shop floor modeling and scheduling applications: an illustrative case

Transcription

1 A framework for integrating shop floor modeling and scheduling applications: an illustrative case Gonzalo Mejía Universidad de los Andes Eliécer Gutiérrez Universidad de los Andes Andrés L. Medaglia Universidad de los Andes Abstract: Scheduling in real-world manufacturing systems is difficult due in part to the lack of modeling and optimization tools which can be efficiently adapted to various shop floor environments. We propose a framework for modeling manufacturing systems which uses a flexible modeling engine and a declarative language. The language allows a high level specification of the system and the engine provides services for creating, storing, and querying the system s model. This paper focuses on the interface between the modeling engine and a flexible Petri Net-based scheduler. Our scheduler pulls data directly from the modeling engine and automatically transforms it into a Petri Net representation. Near-optimal schedules are generated by the Petri Net algorithms and loaded back into the engine for further analysis and processing. Keywords: Petri Nets; scheduling; manufacturing systems modeling; declarative languages; optimization 1 Introduction Industrial scheduling is difficult due to the complexities and particularities at each company. A major problem is the lack of accurate representations of the shop floor for which scheduling methods can be applied. For this reason, generic scheduling software has been difficult to develop.

2 There exist many differences between academic and industrial scheduling applications. Pinedo (2002) highlights some of the key limitations of scheduling frameworks present in the academic literature: 1) the number of jobs do not change over time; 2) lack of emphasis on the rescheduling problem; 3) machine environments are often more complicated than job and flow shops; 4) mathematical models do not take into account human preferences (e.g., a customer requirement may pre-determine an specific machine on which an operation must be performed); 5) most models do not consider machine availability caused by shifts, rest periods, shutdowns or other kind of interruptions; 6) theoretical models concentrate in a single objective, but in real-life scheduling there are present several conflicting objectives; and 7) unlike typical job shop frameworks, in industrial applications jobs can follow alternative routings and each operation can be performed in more than one machine. Moreover, in practice optimal solutions are of little or no interest. What the plant personnel requires is a good schedule which can be followed and trusted. To close the gap between industrial and academic scheduling, some researchers have proposed descriptive languages for manufacturing and production systems. Beck, Currie, & Tate (1993) present a domain description language for scheduling applications which is independent of the implementation details and particular scheduling strategies. Bock & Gruninger (2005) describe a Process Specification Language (PSL), which is a neutral representation for manufacturing processes. This project is being sponsored by the National Institute of Standards and Technology (NIST). Commercial vendors such as ILOG have developed tools that integrate modeling and scheduling. ILOG s Optimization Decision Management System (ODMS) includes ILOG Scheduler, a tool that offers a C++ object model library for representing finite capacity scheduling and resource allocation problems (ILOG, 2003). Other modeling approaches for manufacturing systems are based on state space representations. These methods provide both a graphical representation and a mathematical formalism which can be used for analysis, validation and verification. One of the most powerful state space based tools are Petri Nets which have been used for modeling a wide variety of manufacturing systems due to their capabilities to represent features such as alternative routing, parallel machines, and assembly operations, among others. More recently Petri Nets have been also used for optimization purposes by using Heuristic Search methods in their state space.

3 In a companion paper, we described an architecture for modeling real world manufacturing systems using a modeling engine and a supporting language. The language allows a high level specification of a manufacturing system and the engine provides services for creating, storing, and querying the manufacturing system s model. In this paper we illustrate the integration of the modeling framework with scheduling applications. Such a framework pulls instantiated information of a manufacturing system and translates it into a Petri Net model which supports scheduling algorithms. Two Petri Net based scheduling algorithms, described below, perform the scheduling function. The resulting schedules can be loaded back into the engine for further processing. For instance, the final schedule can be displayed and analyzed with the ILOG Scheduler Viewer. This paper is organized as follows. In Section 2 we describe the architecture of the integration framework. Section 3 describes the Petri Net modeling technique and the scheduling algorithms which lie at the core of the system. Section 4 describes a particular implementation of the integration framework with the Petri Net scheduler. Last, Section 5 describes briefly some lines of research to be explored in the future. 2 Integration Framework Architecture Figure 1 shows an overview of the proposed modeling-scheduling integration framework for manufacturing systems which is fully described in Gutiérrez et al. (2006). A real-world manufacturing system is modeled using either a high-level modeling language or a graphical modeling tool (GMT). The model consists of entities (called plant elements) which capture the attributes and behavior of the elements of a manufacturing system. Such entities include products, machines, manufacturing operations and work orders, among others. When the entities of the real system are described via the language (or the GMT), an interpreter translates them into their corresponding modeling engine objects. This representation of the real system is denoted in this document as the manufacturing system s model.

4 Engine layer Real-world manufacturing system Modeling Language Interpreter Modeling engine Manufacturing System s Model Modeling Λ Scheduling Optimization Simulation Application layer Figure 1: Integration framework The upper part of Figure 1 corresponds to the Engine layer described above; the lower part shows the Application layer which is separated from the Engine layer. The Application layer comprises third-party applications linked to the manufacturing system s model via the application programming interface (API) of the modeling engine. In this paper we focus on developing a prototype implementation for the scheduling module of the Application layer. 3 Petri Net Modeling Framework A Petri Net model of a manufacturing system serves for validation and analysis purposes and is the basis for the scheduling algorithms to be described below. A Petri Net is a particular kind of directed graph, together with an initial state called the initial marking M 0. The underlying structure of a Petri Net is a directed, weighted, bipartite graph consisting of two kinds of nodes, called places and transitions where arcs are either from a place to a transition or from a transition to a place. In a graphical representation, places are drawn as circles and transitions as bars or boxes. Places represent actions or conditions; transitions represent events. Tokens reside in places and represent the truth of the condition associated with such places; tokens move by effect of transition firings. A transition fires if all its input places have tokens. When a transition fires it removes a token from all its input places and puts one token in all its output places 1. Definition. Timed Petri Net (TPN): A TPN is a 6-tuple (P, T, I, O, M 0,, τ) where P = set of n places, T = set of m transitions, P T =, I = a set of input arcs (P T), O = a set of 1 This is the definition of Ordinary Petri Nets in which all arcs weights are 1.

5 output arcs (T P), M 0 = Initial marking, and τ = set of time delays associated with places. The modeling approach at the TPN layer follows the concepts of S 4 R nets (Systems of Sequential Systems with Shared Resources) (Ezpeleta et al., 1995; Abdallah et al., 2002). In S 4 R nets, a number of concurrent serial processes are modeled with an equal number of strongly connected state machines 2. Places belonging to such serial processes are denoted as Operational Places and can be timed, if representing the execution of an operation, or untimed if representing a condition such as part in buffer, ready, or in queue. Resource capacity constraints are incorporated to the model via Resource Places. All resource places are untimed. A full TPN model of a manufacturing system consists of a set of serial processes (job routings) connected by resource places. Conflicts arise by the effect of resource selection and routing flexibility. Resource selection occurs when a resource has the choice of selecting multiple jobs or processes, and routing flexibility refers to the possibility of a job operation to be processed by several resources. Figure 2 illustrates a TPN model of a manufacturing system having two workstations and processing two jobs. Workstation 1 has two machines m 11 and m 12 ; workstation 2 has only one machine m 21. Both jobs follow the same route: first any job is processed by workstation 1 and then by workstation 2. Due to customer constraints job 2 must only be processed by machine m 11, which is a newer machine. When a job finishes its processing on a machine, it moves onto the queue of next machine and the machine is released. In the figure highlighted places represent timed operations. 2 A state machine is a Petri Nets in which all transitions have exactly one input and one output place.

6 p 1 t 1 p 2 t 2 p 3 t 3 p 4 t 4 p 5 p 11 p 13 p 6 t 5 p 7 t 6 p 8 t 7 p 9 t 8 p 10 t 5 p 7 t 6 p 12 Figure 2: TPN model of a job shop with two jobs and two workstations Table 1. Place Description for Petri Net of figure 1 Place Description for Petri Net of figure 1 p1 Job 1 ready p7 p2 Job 1 being processed by machine m11 p3 Operation 1 of Job 1 finished p9 p4 Job 1 being processed by machine m21 p8 p10 Job 2 being processed by machine m12 Operation 1 of Job 2 finished Job 2 being processed by machine m21 Operation 2 of Job 2 finished p5 Operation 2 of Job 1 finished p11 Machine m 11 available p6 Job 2 ready p12 Machine m12 available p7 Job 2 being processed by machine m11 p13 Machine m21 available The TPN can be reconfigured using the engine layer services: For example, suppose that a new machine is added to a bank of parallel machines process. An Engine service re-builds the shop floor specification and the Petri Net is reconfigured automatically. 3.1 Scheduling Algorithms Graph Search Based Methods The states of a Petri Net can be deployed into a (reachability) graph. Nodes in this reachability graph are the states or markings of the net and arcs are transition firings which represent changes (events) in the state of the net. The initial marking for this graph is the node with the state i.e., all jobs are ready for processing. A valid schedule corresponds to a path in the reachability graph which represents a sequence of

7 transition firings) which leads from the initial marking to a goal or final marking representing the state all jobs are finished. As such, an optimization approach may search the net reachability graph aiming to find the minimum cost path from the initial to the final marking. However, as the reachability graph can be huge, even for small nets, performing a full search would not be a practical alternative. In order to reduce the effort required to search the reachability graph, past approaches have used adaptations of the common A* search algorithm. The A* search algorithm for Petri Nets is a branch-and-bound-like algorithm which expands only the most promising nodes of the reachability graph. Seminal papers date back to 1994 (Lee & DiCesare, 1994). Since then, other authors have presented improved versions of the algorithm. For example, Sun et al. (1994) limited the expansion of the net reachability graph by selectively pruning non-promising branches; Xiong & Zhou (1998) implemented the A* search with limited back-tracking capability; Jeng & Chen (1998) used the Petri Net state equations to calculate lower bounds; Reyes-Moro et al. (2002) presented a staged search approach combined with a pruning strategy to reduce the search space; and Mejía & Odrey (2003) combined classical scheduling theory with graph search techniques to reduce the search effort and achieve near-optimal solutions. In this paper we use an implementation of the BAS (Beam A* Search) scheduling algorithm introduced by Mejía & Odrey (2003). Henceforth we call this implementation Petri Net BAS. An advantage of this algorithm is the versatility to adapt to the user requirements: By adjusting the beam or the number of nodes expanded at each level of the reachability graph, the algorithm can run at different speeds. Faster speeds (low beam numbers) produce reasonable schedules in seconds even for large problems, while lower speeds aim to improve the schedule quality. The latest versions of the BAS (Mejía, 2004) algorithm handle several objective functions unlike previous approaches aimed only for the makespan criterion Petri Net based Genetic Algorithms Another approach used for scheduling with Petri Nets is a Genetic Algorithm (GA). Caballero & Mejía (2004) developed a GA algorithm which used the structure of a Petri Net to design a chromosome representation. Such a chromosome is a string of integers

8 which define the transition firing sequence. The length of the chromosome corresponds to the total number of transitions to be fired. The decoding is performed by selecting a transition to fire from the set of enabled transition at each marking state. The decoding strategy always generates valid schedules with any crossover or mutation operators. Henceforth we call this implementation Petri Net GA. 4 Testing the Performance of the Petri-Net Schedulers The performance of both Petri-Net schedulers was tested using a set of classical benchmark Job Shop Scheduling Problems (JSSP) and examples of Flexible Job Shop Problems (FJSSP). A job shop consists of a number of jobs which are processed by a set of machines in some predefined order. In general, jobs follow different routes and no recirculation is allowed. In this project we used the Makespan criterion. Examples of JSSP are the (10 jobs and 10 machines) ABZ5 and ABZ6 problems proposed by Adams et al., (1988) and LA16-LA20 were presented by Lawrence (1984). For these problems the Petri-Net models consisted of 220 places and 200 transitions. A summary of the results is shown in Table 2. Notice that both algorithms show competitive results in terms of solution quality and CPU times. These tests were executed in a Pentium 4 running at 2.4 GHz with 512MB of RAM. Table 2. Job Shop Scheduling Results with the BAS and the Petri Net algorithms. Optimal Petri Net BAS Petri Net GA Problem Cmax Cmax Dev CPU Cmax Dev CPU (sec) (sec) ABZ % ABZ % LA % % 160 LA % % 162 LA % % 163 LA % % 161 LA % % 162 Cmax: Makespan Dev: Deviation from the optimal solution CPU: Computer time in seconds. A Flexible Job Shop is a variation of the job shop problem in which machines are replaced by workstations. In each workstation there are a varying number of identical parallel machines. A problem of this type is ffs3 proposed by Pinedo (2002) in which

9 Total Tardiness is the optimization criterion. This problem consists of 15 jobs and 3 workstations and each workstation has up to 5 machines. The resulting Petri Net model has 106 places and 120 transitions. The objective function of the solution found by the specialized Shifting Bottleneck algorithm (Pinedo & Singer, 1999) was 163 and took 17 seconds. BAS obtained a solution with the same objective value in less than 7 seconds in the same computer. 5 An Implementation of the Integration Framework We developed an implementation that integrates the Petri-Net scheduler with the shop floor modeling engine as described in Section 2. Figure 3 shows the components and their interactions in the application. Real-world manufacturing system Modeling Language Interpreter Modeling engine API Manufacturing System s Model Engine layer Lekin I/O files Engine-Lekin interface module Lekin scheduling system Schedule Viewer module ILOG s Scheduler Viewer Petri Net BAS Scheduler Petri Net GA Scheduler Figure 3: Implementation of the integration framework Application layer The modeling engine provides the language and services required to construct a manufacturing system model. This model includes the specification of the resources and the work orders to be scheduled. As described above, this specification is translated into a Petri Net model. Such a Petri Net model is the framework in which the Petri Net schedulers operate. These schedulers can be linked as plug-in components to third party software packages or to user-interfaces. In this project we used the Lekin scheduling software developed at NYU (Pinedo, 2005; Pinedo & Chao, 1999). The Lekin system allows user-implemented algorithms and supports basic framework configurations ranging from single machine to flexible job shops. Lekin provides a user interface and offers an open specification for input/output file formats used for scheduling problems. More importantly, it allows the integration of user-implemented

10 algorithms. For instance, in our application we integrated into Lekin the Petri Net BAS and Petri Net GA schedulers described in the previous section. Our implementation incorporates an Engine-Lekin interface module which provides services for writing and reading the Lekin files. The writing service generates the Lekin input files for the work orders and resources stored in the modeling engine. The reading service loads into the engine the schedules generated by the algorithms invoked from the Lekin system. The externally produced schedule can be analyzed and visualized by other modules integrated to the engine. For example, a Schedule-Viewer module retrieves a schedule from the engine and displays it graphically using a third-party component, namely, the ILOG Scheduler Viewer (ILOG, 2006). This component, included in the ILOG s Optimization Decision Management System (ODMS), displays a Gantt chart for resources and activities, and calculates the resource loads. The ILOG component reads the schedule from an XML file generated by the Schedule Viewer module. To illustrate the integration framework we use a classical benchmark problem, namely the (10 jobs and 10 machines) ABZ5 problem proposed by Adams et al. (1988). The problem was modeled using the language described in the modeling engine framework (Gutiérrez et al., 2006). Figure 4 shows a fragment of the source file for the problem ABZ5, including definitions for resources, manufacturing operations and process routings. This file was compiled with our own interpreter and the model was loaded into the engine. Next, from the model representation, the Engine-Lekin interface module generated the Lekin input files (see Figure 5).

11 Figure 4: Model description using the modeling language for problem ABZ5 (a) Job Routing (b) WorkStations (c) Schedule Figure 5: Lekin files for problem ABZ5 The Petri-Net schedulers, plugged into Lekin, were used to solve the problems. Figure 6 shows the final schedule for problem ABZ5 in the Lekin scheduling system. This schedule was stored using the Lekin output format shown in Figure 5 (c).

12 Figure 6: Lekin s display of the Petri-Net BAS schedule for problem ABZ5 After a solution is found, it is loaded back into the engine using the Engine-Lekin interface again. We use the Schedule-Viewer module to export the schedule into the XML format shown in Figure 7 required by ILOG s Scheduler Viewer. Finally, we retrieve the schedule from the engine in an XML format to generate the Gantt and load charts shown in Figure 8.

13 Figure 7: ILOG Scheduler Viewer XML file Figure 8: Gantt and load charts in ILOG Schedule Viewer for problem ABZ5

14 6 Future Research and Concluding Remarks In this paper we presented an integrated modeling and optimization framework intended for real-world scheduling problems. This document presents the integration of the modeling engine, Petri Net models and scheduling algorithms. At this stage, we illustrated the application with a classical job shop problem. Future research will be focused on modeling and scheduling systems which include features such as operations with multiple resources, alternative routings, setup times, production batches, machine setups, customer due dates and order releases. Acknowledgments This research project has been funded by: (1) Colciencias (the Colombian Institute for the Development of Science and Technology) under grant ; (2) Sistemas Corporativos S.A.; and (3) Universidad de los Andes. We also thank José T. Hernández, the former Dean of the Engineering School at Universidad de los Andes, and Oscar Ramos who envisioned the project. Last, but not least, we would like to thank Rafael Botero, Juliana Hernández, Andrés Marentes, Juan Carlos Martínez, Andrés Molano, Cristina Sarmiento and Silvia Takahashi, for their valuable contributions to the project. References Abdallah, I. B.; Elmaraghy, H. A.; & Elmekkawy, T. Y. Deadlock-free Scheduling in Flexible Manufacturing Systems using Petri-Nets, International. Journal of Production Research. Vol. 40, No. 12, Adams, J.; Balas, E.; & Zawack, D. The shifting bottleneck procedure for job shop scheduling. Management Science. Vol. 34, Beck H.; Currie, K.; & Tate, A. A Domain Description Language for Job-Shop Scheduling. Artificial Intelligence Applications Institute, University of Edinburgh. Techinical Report AIAI-TR Bock, C.; Gruninger, M. (2005). PSL: A semantic domain for flow models. Software and System Modeling. Vol. 4,

15 Caballero, J. P.; & Mejía, G. A Hybrid Approach that combines Petri Net models and Genetic Algorithms for scheduling of Flexible Manufacturing Systems. Proceedings of the III Congreso Colombiano - I Conferencia Andina de Investigación de Operaciones; Colombia, Ezpeleta, J.; Colom, M.; & Martínez, J. A Petri Net Based Deadlock Prevention Policy for Flexible Manufacturing Systems. IEEE Transactions on Robotics and Automation. Vol. 11, No. 2, Gutiérrez, E.; Botero, R.; Hernández, J.; Takahashi, S.; & Medaglia, A. L. An Object Oriented Modeling Engine of Manufacturing Systems. Centro para la Optimización y la Probabilidad Aplicada, Universidad de los Andes. Working Paper. Available at ILOG. ILOG Scheduler 6.0. User Manual. ILOG Inc, ILOG. ILOG Views. ILOG Inc. From Last Access on February 22, Jeng, M. D.; & Chen, S. C. A Heuristic Search Approach Using Approximate Solutions to Petri Net State Equations for Scheduling Flexible Manufacturing Systems. The International Journal of Flexible Manufacturing Systems. Vol. 10, Lawrence, S. Resource constrained project scheduling: an experimental investigation of heuristic scheduling techniques (Supplement), Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, Pennsylvania Lee, D. Y.; & DiCesare, F. Scheduling Flexible Manufacturing Systems using Petri Nets and Heuristic Search. IEEE Transactions on Robotics and Automation. Vol. 10, No. 2, Mejía, G. A Petri Net Based Algorithm for Minimizing Total Tardiness in Flexible Manufacturing Systems. In Proceedings of the III Congreso Colombiano - I Conferencia Andina de Investigación de Operaciones; Colombia, 2004.

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