A Petri Net Technique for Batch Delivery Time Estimation

A Petri Net Technique for Batch Delivery l i m e Estimation f? Xirouchakis’, D. Kiritsisl, C. Gunther’, J.-G. Perssonz(2) Laboratory for Computer Aided Design and Production (LICP), Dept. of Mechanical Engineering, Swiss Federal Institute of Technology - Lausanne (EPFL), Lausanne, Switzerland ’Engineering Design, Dept. of Machine Design, Royal Institute of Technology (KTH), Stockholm, Sweden Received on January 10,1999

Abstract The problem under consideration is delivery time and cost estimation in batch manufacturing of mechanical parts. We consider non-linear process planning with explicit resource modeling. We present a two-level Petri net class which are high level nets extended with time that allow the interleavingof transitions. They consist of a system net modeling the job shop layout (including the machines) and token nets modeling the jobs and setups. We calculate the optimum delivery time and associated cost of a batch of one type of mechanical workpiece together with the associated resources. Keywords: CAPP, cost, Petri net

INTRODUCTION The goal of this paper is the development and application of computer support methods for the delivery time and cost estimation concernina the manufacturina of mechanical parts. In an earlie; ClRP paper [l] a PeGi net technique was presented covering the process planning cost estimation i.e. the focus was the cost estimation of one mechanical part. In this paper we go beyond the cost estimation of single parts and we consider the estimation of the delivery time and cost of a batch of mechanical parts. We are interested in the approximate and simultaneous planning and scheduling of the manufacturing of a batch of mechanical parts. Quite a few papers are available in the published literature [2 to 71 etc., concerning the use of Petri nets for the modeling of the actual planning (and sometimes scheduling) of the fabrication of mechanical parts. However, very few papers address the approximate and simultaneous planning and scheduling during the bid preparation process. The origin of the approximation is in the imprecise and incomplete knowledge of the actual plans and schedules and associated resources at the stage of the bid preparation. We consider some aspects of this uncertainty by the presentation of a non-deterministic Petri net model that simultaneously considers processing alternatives and variations in the start and duration (enabling and duration time intervals) of resource or resource changing operations [a). This approach allows the estimation of upper and lower bounds of batch delivery times and costs. The simultaneous consideration of planning and scheduling non-determinism provides more opportunities for optimization, so that better schedules are obtained. An improvement over our previous ClRP paper is that now we allow the interleaving of transitions,which is necessary in order to obtain optimal schedules and realistic delivery time estimations. Another improvement is the explicif resource modeling (machine and setup changes), which allows the consideration of the restricted availability of resources. We present a two-level Petri net model that supports the modeling of the job-shop layout (including the machines) at the upper level and the modeling of the 1

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jobs and setups at the lower level [8]. This two level modeling Structure SUPPOrtS a modular object-based user friendly interfacing with Our Software prototype. TWO-LEVEL TIMED PETRI NETS The basic constituents of a Petri net are the p/aces, transitions and tokens. The dichotomy between places and transitions serves the dynamic aspects of the Petri net model, where the number of tokens on the places is subject to change by firing transitions. Here we consider the extension of this simple structure by forming a two level hierarchy of nets, where at the top of the hierarchy there is a system net, whose places can hold one or more token nets [8].The token nets are Petri nets again, but of the most basic kind. This approach has its origins in the work of Valk [S]. We give below an informal introduction of two-level timed Petri nets. The basic advantage of this approach over the use of purely colored Petri net models is that now the job shop layout, jobs and setups can be considered as separate objects with their unique identity and behavior. In particular the modeling of autonomous transitions and the introduction of time at the token net level is greatly facilitated as discussed later. A state of a two-level Petri net system is given by the distribution of the token nets (i.e. the marking) in the system net and the markings of all token nets. State changes occur by the firing of synchronized or autonomous transitions. Synchronized transitions require the firing of a synchronized sfep which consists of one system net transition and the associated token net transitions. An example of the use of the concept of synchronized transitions is the modeling of the execution of one operation from a job token net, which requires a setup from the corresponding setup token net and synchronized at the system level net, which identifies the selected machine from the job shop layout. Autonomous transitions can only occur at the token net level and affect the state of the corresponding token net. An example of the use of the concept of an autonomous transition is the modeling of the change of a setup in the corresponding 2

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setup token net, which does not require synchronization with other token nets. We continue with an introductory example consisting of one system net and two token nets, see figure 1, to describe the notions of state and state changes of a twolevel Petri net system. The system net has two transitions, T1 and T2, and three places P1, P2 and P3 (the gray patches). However, places P1 and P2 hold a token net each. As it is seen, the state of the two level Petri net system is given by (i) the arrangement of the token nets on the places of the system net and (ii) the arrangement of the black tokens on the places of the token nets.

P1

T1

Figure 1. System Net with two Token Nets. Consider the step (Tl, U, 01). It says that T1 should occur at the same instant of time as U and 01. Furthermore, only those transitions can occur at the system net level that are synchronized with some token net transitions. Step occurrence changes the state of the system net and also that of the token nets involved. The new state of the system after the Occurrence of the step (Tl, U, 01) is depicted in figure 2. The state of the system net has changed so that both token nets are moved to the place P2. At the token net level, the occurrence of 0 1 and U is reflected through the change in the respective markings. Note that although the two token nets are at the same system net place, they remain two distinct token nets, so that the occurrence of the synchronized step (T2, S, F) moves them back to the places they originally occupied.

introduction of a global clock, associated with the system net state and local clocks, associated with the corresponding token net states. The clocks permit to arrange on a time scale, the transition occurrences, as well as to decide whether a transition is enabled (disabled). There are many ways to introduce time in a Petri net. Our goal is to allow the modeling of variations in the start (interval-enabling time model) and duration (interval-durationtime model) of resource and resource changing operations. In the interval-enabling time model we assign an earliest and latest firing time to each autonomous transition and synchronized step, so that, after becoming enabled at time t, the autonomous transition or synchronized step must occur not sooner than t + earliest firing time and not later than t + latest firing time. In our example there are only two synchronized steps. We assign time information to the transitions 0 1 and F which, in turn, will be inherited by the steps in which they are involved. In the interval-duration time model we assign the minimum and maximum firing time to each autonomous transition and synchronized step.

3 GUIDE COMPONENT CASE STUDY The following case study is used to show the application of the modeling techniques presented above to the delivery time estimation for the manufacturing of a batch of a guide component [lo]. As noted in figure 3 there are four features FJ (J=1,4) to be manufactured: F1, F2, F3 and F4 corresponding to the two steps, the first slope, the second slope and the two tapped holes respectively. We assume that the features F1 and F4 have been already manufactured and we study the completion of the part by the features F2 and F3. We consider a batch of n parts (associated with the jobs J1 to Jn respectively) to be manufactured in a job-shop consisting of two machines (M1 and M2) with two setups (S1 and S2) each.

Figure 3. Guide Component Geometry and Features. Each part is to be manufactured by a process plan description as given by the job token net in figure 4.

Svnchronised Steps = { (Tl, U,Ol), (T2,S,F) }

n

n

Figure 2. The state of the system after the occurrence of step (Tl, U, 011. Time Concepts of the Model 2.1 Now we come to the introduction of time into the model. The addition of time has as a consequence the

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Figure 4. Guide component Job Token Net.

Each part will be manufactured with a sequence of two operations: either operation A or D followed by either operation B or C. Each operation is fully specified: operation A or C are associated with setup 1 and operations B or D are associated with setup 2. The modeling of the setup changes and job-shop layout is done by the setup token net and the job-shop system net shown in figures 5 and 6 respectively.

M&m MlSllb

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lines while the added arcs with thicker lines. The analysis is performed on the unfolded net using the software tool INA [ l l ) .

M1S1IS

WlSlUl

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Figure 7. Unfolding the System Net. M18iu

Figure 5. Guide component Setup Token Net.

M2 table

M1

M1 start

1 ’ M2 start

Figure 6. Guide component Job Shop Net. Cost and Time Data 3.1 On machine M l the setup change duration is’6 min while its use is 100 CHF/hour, while on machine M2 the setup duration is 7 min, while its use is 75 CHFIhour. Operations A, 8,C and D last 2, 4, 5 and 3 minutes respectively. We neglect the time and costs associated with the changing from one machine to the other, the tool changing and the loading and unloading of parts in the setups. The time (in minutes) interval window for the start of each operation is c0,12l i.e. each operation can start no later than 12 minutes after being enabled. This feature allows more freedom for optimization i.e. by permitting the waiting of operations a better schedule could result (see Guide Component Results section below and figure 8). The maximum allowable waiting time was calculated as the sum of the maximum setup and machining time. The operation time durations (also the setup changing operations) were considered fixed i.e. the time window (2.23 in minutes was used for operation A and similarly for the remaining operations.

Unfolding the Two-Level Net 3.2 The principle on which the unfolding algorithm is based is illustrated in figure 7 for the synchronized step (JiA-b, M1S1ub, M1ub). The deleted arcs are shown with dotted

3.3 Guide Component Results Minimum batch delivery times and costs are shown in figures 8 and 9 respectively in the form of Gantt charts. The minimum delivery time is 18 min (figure 8)while the minimum cost is 37.50 CHF (30 min 75/60 CHF/min, figure 9). As can be seen in figure 8 transition firings are allowed to occur concurrently, as they should, for an optimum delivery time estimation. Our interval-enabling time model allows to find better schedules by considering idle machining times as seen in figure 8. Figures 10 and 11 show the dependence of the normalized delivery time and associated cost (i.e. time and cost are divided by the number of jobs) with batch size and time duration variation respectively. M1

S1

11

1

M2 18 Figure 8.Minimum Delivery Time Gantt charts for Batch of two jobs. M1 0 M2 30

S2 7

2D 1D

S1

1C

2C

3 3

7

5

5

Figure 9. Minimum Cost Gantt charts for batch of two jobs. The minimum (maximum) time and associated cost curves reported in figures 10 and 11 were obtained with fixed duration time data by reducing (increasing) all the time data of the previous section by lminute. We used fixed durations to circumvent the difficulties in the application of optimization algorithms on interval duration time Petri net models. This is a subject we intend to investigate in the future.

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Time results c

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1 Pb

2 lob!, Batch sire

3 robs

Figure 10.Dependence of Normalized Delivery Time with Batch Size and Time Duration Variation. cost results

Alting L., Zhang, H., 1989,Computer Aided Process Planning: a State-of-the-Art Survey, Int. J. Prod. Res., 27553-585. Tonshoff, U., Beckendorff, U., Anders, N., 1989, FLEXPLAN-A Concept for Intelligent Process Planning and Scheduling, ClRP International Workshop on Computer Aided Process Planning, Hannover Univ., 87-106. Srihari. K.. Emerson, C.R., 1990, Petri Nets in Dynamic Process Planning, Computers Industrial Engineering, 9:447-451. Kruth, J.P., Detand, J., 1992,A CAPP System for Nonlinear Process Plans, CIRP Annals, 41/1:489-

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Figure 1 1. Dependence of Normalized Cost with Batch Size and Time Duration Variation. Figures 10 and 1 1 show the normalized delivery time and associated cost obtained due to the non-deterministic modeling (routing alternatives and intetval-enabling interval-duration Petri Nets) and simultaneous optimization of planning and scheduling. They show the capability of this modeling approach to allow the estimation of worst case and best case delivery times and associated costs. The two-level Petri Net model allows, in a simple way, the modeling of uncertainties in the start and duration of operations (resource or resource changing operations) by adopting interval-enabling and intervalduration time models.

FUTUREWORK In the future we intend to develop algorithms to group operations together in order to achieve a certain tolerance level. Therefore this will introduce the tolerance parameter (in addition to the cost and de/ivety time). Furthermore it will allow the development of more efficient optimization algorithms that exploit the group formations. We also intend to develop symmetty reduction algorithms with transition time interval duration semantics. As a result we expect to severely reduce the size of the unfolded net for a batch of identical parts. Finally we intend to develop mappings of the two-level Petri net representationto other representations that allow the application of complete schedule improvement algorithms.

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5 ACKNOWLEDGEMENTS We express our thanks to Tamas Kis, EPFL, for his helpful discussions on the elaboration of the test case. REFERENCES Xirouchakis, P., Kiritsis, D. and Persson, J.G., 1998, A Petri Net Technique for Process Planning Cost Estimation, CIRP Annals, 47/1:427-430. Ham, I., Lu, S. C.-U., 1988, Computer-Aided Process Planning: The ,Present and the Future, ClRP Annals, 37/2:591-601.

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Cecil, J.A.. Srihari, K., Emerson, C.R., 1992, A review of Petri Net Applications in Process Planning, The Int. J. of Advanced Manufacturing Technology, 7:1 68-1 77. Neuendorf K.-P., Kiritsis D., Kis T. & Xirouchakis P. 1997,Two-level Petri Net Modelling for Integrated Process and Job Shop Production Planning, XVlll Int. Conf. on Applications and Theory of Petri Nets, ICAPTN.97, Toulouse, 135-150. Valk R. (1995) Petri nets as dynamical objects. 1 st Workshop on Object-Oriented Programming and Models of Concurrency, 27 June 1995,Turin, Italy. Gunther, C., 1998,Batch Delivery Time Calculations Using INA, EPFL report. Starke, P., 1992, Integrated Net Analyzer INA, htto://www.informatik.hu-berlin.de/-starke/ina.html