REAL-TIME SCHEDULING AND CONTROL OF A FLOW-SHOP USING DIOID ALGEBRA

Transcription

1 REAL-TIME SCHEDULING AND CONTROL OF A FLOW-SHOP USING DIOID ALGEBRA Mustafa YURDAKUL Assistant Professor, Departent of Mechanical Engineering, Faculty of Engineering and Architecture, Gazi University, 657 Maltepe Anara, Turey. Nicholas G. ODREY Professor, Departent of Industrial & Manufacturing Systes Engineering, Lehigh University, Bethlehe, PA 85, USA. ABSTRACT A real-tie scheduling and control ethodology in dioid algebra that allows repeated production of parts over the planning horizon of a flow-shop in the presence of disturbances is proposed in this paper. The behavior of a anufacturing syste as an discrete-event dynaic syste can be copletely characterized by the nowledge of starting and ending ties of its activities. Dioid algebra can be used to represent discrete event dynaic systes where relationships between the starting ties of the activities require aiu and addition operators and are linear in the sense of dioid algebra. To use dioid algebra in a real-tie scheduling and control ethodology, conflict resolution and optiization are introduced into dioid algebra with this study. This paper deals with the cell controller and its activities. A two-level hierarchical control structure is proposed. In this structure, the cell controller deterines a feasible sequence of events, which are identified as start ties of operations that optiize a given scheduling obective. In addition, the cell controller deterines a sequence of events that iniizes the effect of disturbances on eeting the production goal given a disturbance has occurred. The cell controller controls the syste by adusting the start and end of the production periods and sequences at achines given inforation on disturbances and cell status. Machine breadown, priority ob order, and peranent changes in ob and achine types can be handled by the cell controller as disturbances..introduction In real-world applications, various uncertainty aspects of the syste will nullify the schedule as soon as this schedule is released to the shop (Wu & Wys, 99). Uncertainty aspects of a syste can be grouped into five types (Paruna, 99): ()rush order and order change, (2)ob delay or advance, (3)facility breadown, (4)operations error, and (5)unacceptable quality. When any of these abnoral situations coe up, huan intervention is necessary to propt operational decisions and the scheduler has to adust, update, or even reproduce the operations schedule. However, autoated systes require inial huan intervention unlie traditional systes. In this case, it becoes etreely iportant to consider the uncertain aspects of the syste eplicitly (Wu & Wys, 99). The isatch between the atheatical abstractions that schedulers anipulate and the state of the real world and the operational solutions to handle this isatch are subects of real-tie scheduling. Real-tie scheduling processes sense the state of the world and adust to deviations fro previous assuptions (Paruna, 99). Traditionally, real-tie refers to the iediate response to soe event in a syste, such as process copletion, part arrivals, achine breadowns. Responses would include selecting parts for a achine, starting a achining process, rerouting a part, etc. (Haronosy & Robohn, 99). Dispatching

2 rules have been developed and widely used in the dynaic anufacturing environent for real-tie decision aing (Ki & Ki, 994, Chryssolouris, 994, Moser & Engell, 992, Engell, 99). On the other hand, Yaaoto & Nof, 985 proposed a scheduling/rescheduling approach which revises schedules at given points in tie due to certain significant changes in operation requireents. One line of research focuses on revising the pre-coputed optial schedule by a partial or coplete updating in the event of disruptions (Leon, et.al., 994). For instance, when a scheduled syste is disrupted, atch-up scheduling ethods have been offered to handle disturbances (Bean & Birge, 985; Odrey & Wilson, 99; Saleh, et.al.,99). Bean & Birge, 985 proposed the atch-up scheduling approach to ae a generated schedule adaptable to the changing environent. The adaptation process taes place by rescheduling to return to the pre-planned schedule in a finite tie. Bean & Birge utilized the "econoic turnpie theory" as a foundation for atch-up scheduling approach. According to this theory, if any deviations fro the optial traectory are initiated by disturbances, the best course of action is to get bac to this optial traectory (i.e., turnpie) at an optial atch-up point. In an application of atch-up scheduling to periodic loading systes by Odrey & Wilson, 99, Saleh, 988, an original steady-state schedule is pre-coputed by optiizing syste perforance. In the event of an occurrence of a disturbance, the steady-state schedule is no longer valid and the syste enters a transient state. A transient schedule is generated and this transient schedule will eventually atch-up with the steady-state schedule in a finite aount of tie. The atch-up scheduling procedure can handle arrival of parts, which require iediate processing, and achine breadowns as disturbances. A periodic loading echanis facilitates decoupling of production requireents of a anufacturing syste into a series of input sequences to be loaded into the syste at equal tie intervals tered "cycle ties". If the nuber of part types and quantities in an input sequence and the corresponding utilized achines for processing of the input sequence are identical fro one cycle to the net cycle a unique schedule can be prescribed for all input sequences (Hitz, 979, 98). This schedule is called the steady state schedule in this research. Dioid algebra is used in constructing the odel of the anufacturing syste and developent of control actions to use in the real-tie scheduling and control of the anufacturing syste. Both a-plus and in-plus algebra are needed in this developent. A different notation for a and in is necessary: is reserved for a, and Λ will denote in. The notation Θ refers to the ultiplication of two atrices in which Λ operation is used instead of (Cohen et. al, 992). The epressions a b and a Θ b are identical if at least either or is a scalar. + is the zero eleent of Λ and is the unity eleent of Θ. ε denotes - ; - ε will refer to +. e will be the unity eleent of both Θ and. 2. REAL-TIME APPLICATION OF DIOID ALGEBRAIC MODEL AND SOLUTION ALGORITHMS A cell controller for a periodic loading production syste is developed with the capabilities of generating steady-state schedules and taing corrective control actions in the face of disturbances. A flow-chart of this real-tie scheduling and control schee is shown in Figure. Soeties syste disturbances occur, such as achine breadowns, tool breaages, arrivals of urgent obs, late delivery of obs to the cell,

3 changes in the due-dates, or the addition or reoval of obs fro the cell worload. In the event of an occurrence of a disturbance, the steady-state schedule ay no longer be valid and control actions have to be eployed in order to copensate for the effect of the disturbance. These control actions ay tae different fors depending on the type of the disturbances. In this research, the real-tie scheduling and control schee developed by Saleh, 988 and presented in Saleh 988, Saleh, et. al.,99, and Odrey & Wilson, 99 will be iproved upon by adding two new disturbances, peranent and inor disturbances. Minor disturbances will be handled by worstation controllers and will not be discussed in this paper. Generate Steady-State Schedule Eecute Steady-State Schedule Generate & Eecute Matchup Schedule Yes Disturbance Yes M/C Breadown Yes Generate & Eecute Machine Withdrawal Phase Machine Operational No Priority Order No Deterine Initial Conditions Generate A New Steady- State Schedule Figure. Flow chart of the Real-Tie Scheduling and Control Schee 2. Steady-State Schedule Generation The cell controller generates a sequence of operations and idle ties between operations by deterining the values of the control variables. The affects of initial conditions and ready ties of obs and achines are insignificant copared to periodic steady-state schedule over a long tie. An algorith, tered the FIRST ALGORITHM, is developed in dioid algebra which iniizes the aespan given non-unifor ready ties for obs and operational ties. It tries to construct the schedule by adding new obs to the partial schedules at each step by using the obective function. To prevent generating cycles in the schedule, A atri is used in the algoriths. A i shows the availability of ob given ob i is selected at the last stage. For eaple, if a partial schedule includes obs, 4, 5, and 7, where ob 7 is selected at the last stage, A 7, A 74,

4 and A 75 are set to + to prevent this particular partial schedule getting obs, 4, and 5 again. The following steps detail the FIRST ALGORITHM for a flow-shop: I. INITIAL CONDITIONS Step: A n n Step 2: For each ob,,,n use the following forulas to copute the copletion ties of ob at each achine.,) r (, (,2) r,,2 (, ) r,, Step 3 : II. ITERATION Step : For each ob i, calculate For each,, n where i ( ) ( ) ( i,, ) (,) rφ (,) ri i, ( i,2, ) (,2) ( i,, ) φ (,2) ( i,, ) ( ) i, ( i,, ) (, ) ( i,, ) φ (, ) ( i,, ) Step 2: Calculate the obective function value for each ob i using the following forula. v i n i ( i,, ) A (, i) () i,2 Keep trac of the ob, in, which gives iniu obective function value. Step 3:The following calculations are done using in ( i,) ( i,, in ) ( i,2) ( i,2, ) in ( i, ) ( i,, ) in

5 Step 4: Modify the A atri for the (+) th stage by using the following substitutions A ( in, i) + A ( l, i) +, Where, l is any ob that is part of the partial schedule which has in as the last ob. III. STOPPING CRITERION Step : Is n -? Step 2: If yes, stop. Step 3: If no, continue. IV. UPDATE Step : Increase + Step 2: Return to II. With this algorith values of control variables are deterined and these values can be subsituted into the following syste state equation (Yurdaul,997). X ( + ) A Y( ) B S( + ) C X ( + ) D U( + ) X ( + ) (2) where, X (+) the vector of starting ties of the operations in the new period. Y () the vector of the predicted ties when the operations of th period are finished which eans the achines are available again for the new period's operations. U (+) the vector of control (sequencing) variables which deterine when each operation should be started on its designated achine in the th period. S (+) the eternal starting condition vector in the th period A, B, C, D and E atrices are constant atrices that describe the relationships aong different syste variables. 2.2 Transient Analysis A steady-state periodic loading schee is followed in the cell until a disturbance occurs. In that case, the steady-state schedule prescribed by the cell controller is no longer valid and an interi schedule ust be developed. This non steady-state phase refers to a period (or periods) where the overall goal is to handle the disturbance and eventually return to the steady-state schedule. When there is a priority order or peranent changes in obs and achine types, the syste enters a atch-up phase to return to a steady-state ode. On the other hand, in the achine failure case, a resource is withdrawn fro the syste and none of the operations requiring this resource can be carried out. This phase is called the achine withdrawal phase. Only when the syste returns to full capacity, the cell controller generates a atch-up schedule siilar to priority order disturbance to return bac to the steady-state ode. To conclude there are two distinct phases in the transient phase. These are the achine withdrawal phase associated with achine breadown and the atch-up phase which ay be activated by either achine recovery, priority order or peranent changes. The transient phase ends with the reoccurance of the steady-state periodic scheduling.

6 2.2.. Machine-Withdrawal Phase A achine breadown disturbance starts when a achine breas down. When a achine goes down, the syste wors in a reduced capacity. During the achine breadown, the operations that have to be processed on that achine stays idle. The phase that achine reains down is called a achine withdrawal phase. This phase starts with the achine breadown and ends when the down achine is operational again. In this phase, the syste continues to receive the parts at their regular ties and operational achines continue on processing the parts. However, the down achine divides the flow-line into two subsystes. The flow line up to the down achine is not bloced by the down achine and continues processing the parts by using their steadystate schedule. The parts accuulate in the buffer of the down achine. The down strea achines will starve after the parts which are currently being processed are ehausted. A practical strategy that can be eployed by the cell controller is to continue to use the steady-state schedule for the operational achines. This policy will aintain the original sequence and is ideal for situations in which sequence changes are costly. The operational strategy during the achine withdrawal phase is as follows: ) Find out which operation is directly affected by checing start and copletion ties of operations with the tie when breadown occurs. The tie when the breadown happens ust be between start tie and copletion tie of the directly affected operation. 2) By checing dependencies between operations, find out the unaffected operations and continue processing the at their regular state ties. The affected operations can not be started because of their dependency on directly affected operation until the down achine is operational again Match-up Phase In the atch-up phase all achines are operational. The finish ties of achines after eerging fro the atch-up phase should atch the steady-state start ties of achines. The steady-state start ties are treated as a set of due-dates of the achines. To atch-up to the steady-state pattern all operations belonging to the atch-up schedule ust be copleted before or at the given achine due-dates. By using this set of due-dates of achines, d i, the due-dates of obs that will be processed in the atchup phase can be calculated. The following forula is used to calculate each ob's due date which is needed for atch-up scheduling: where, i,, n. ( d ) ( d ) d i d i, i,2 i,3 i, Any violation of due-dates by any achine will result in a forward tie shift. The obective of the atch-up scheduling is to iniize the aiu delay at the achines. The following algorith, tered SECOND ALGORITHM, is developed for the atch-up phase. This algorith iniizes the tardiness. Lower copletion tie is used as a tie-breaer in case of having paths that have the sae tardiness value. Siilar to the FIRST ALGORITHM, A atri is used and odified at each stage to prevent returning bac to the obs that are already scheduled. The following steps detail the SECOND ALGORITHM for a flow-shop: (3)

7 I. INITIAL CONDITIONS Step: A n n Step 2: For each ob,,,n use the following forulas to copute the copletion ties of ob at each achine.,) r (, (,2) r,,2 (, ) r,, Step 3: t Step 4 : II. ITERATION Step : For each ob i, calculate Step 2: Calculate the obective function value for each ob i using the following forula. Tie-braer Rule: ( (, ) φ d ) φ (, ) For each,, n where i t i ( φ d ) ) ( ) ( ) ( i,, ) (,) rφ (,) ri i, ( i,2, ) (,2) ( i,, ) φ (,2) ( i,, ) ( i,, ) (, ) ( i,, ) φ (, ) ( i,, ) i,2 ( ) i, [ t A (, i) φ[ ( ( i,, ) φd i ) φ ( ( i,. ) φ d ) ) ] n i n i ( i,, ) t i Keep trac of the ob, in, which gives iniu.

8 Step 3:The following calculations are done using in ( i,) ( i,, in ) ( i,2) ( i,2, ) in ( i, ) ( i,, ) in Step 4: Modify the A atri for the (+) th stage by using the following substitutions A ( in, i) + A ( l, i) +, where, l is any ob that is part of the partial schedule which has in as the last ob. III. STOPPING CRITERION Step : Is n -? Step 2: If yes, stop. Step 3: If no, continue. IV. UPDATE Step : Increase + Step 2: Return to II. If the production of the transient period taes longer than the cycle length, then the start of the net period should be shifted using the following shifting algorith: If i n p T (cycle length) then cycle length of the transient period is equal to T t i n p and also the start of the net period, where the steady-state production ode begins again, is delayed by the aount of T t φ T. 3. A REAL TIME SCHEDULING EXAMPLE The real-tie scheduling and control ethodology described earlier will be illustrated in this eaple. The FIRST ALGORITHM is applied to obtain the steady-state schedule. Transient schedules will be generated to handle a achine breadown and a priority order disturbances. To generate the atch-up schedules in the transient periods the SECOND ALGORITHM is used. This eaple also shows the application of the dioid algebraic odel in representing the syste and in obtaining the steady-state operational plan. The processing ties of the operations are given as p, p 5, p 3, and p 2 6.., 2 2,2, 3,4 In developing the steady-state schedule, a MPS (, ) is given and no specific cycle length is specified by the shop controller. The FIRST ALGORITHM is used to generate the steady-state schedule. The values of the control variables will be deterined with the application of this algorith. The calculations of the last stage are given to illustrate the application of FIRST ALGORITHM.

9 Calculations for stage : For i v (,,2 ) ( 2,), (,2,2 ) ( 2,2) (,,2 ) φ ( 2,2) (,,2 ) 4 For i 2 v 2 ( 2,, ) (, ) 2, ( 2,2,) (,2) ( 2,, ) φ (,2) ( 2,, ) 3 ( ) 9 5φ ( 9 5) ( ) 7 5φ ( 7 5) Algorith stops at n. Since, v sequence of obs at achines is taen 2 < v as 2 and the corresponding cycle tie is equal to. This scheduling inforation corresponds to following values of control variables in the odel: u and. By subsituting the values of the control variables 3 e, u3 ε, u 42 e, u 24 ε into the dioid algebraic odel, the steady-state operational plan can be obtained. The steady-state operational plan, shown in table 2, contains each operation's start and copletion tie on achine and is used to eep trac of the syste in the steady state operation ode. The steady-state ode of operation is shown in Figure 2. Table. Steady State operational plan 6 3 Operations Start tie Copletion Tie M/C No ,2 2,2 5 4 Machine JOB JOB 2 JOB JOB 2 Machine 2 JOB JOB 2 JOB JOB tie Figure 2. Gantt Chart for the steady-state schedule

10 Operations perfored in the syste are dependent on each other. For eaple, if operation can not be perfored because of a achine breadown, none of the operations can be perfored as long as this disturbance stays in effect. Furtherore, the net cycle's operations can't be perfored. An eaple for the achine withdrawal phase is a achine breadown at achine 2 happening at t 4. By checing table 2, it can be seen that, operation 2 was being perfored at t 4 at the second achine. Operation 2 can't be perfored any further because of the breadown, so that the directly affected operation is the second operation. Operations and 3 are not affected by the breadown and they can be perfored at their regularly scheduled start ties. On the other hand, in addition to operation 2, operation 4 is affected by the disturbance, and both operation 2 and operation 4 should wait for processing until the second achine is up again. When the achine is repaired at t 8, the syste has full capacity and a atch-up schedule is generated to return to the steady-state operational ode. Figure 3 shows the atch-up phase. In this phase, the atch-up point is selected as the copletion of the net cycle ( t 22 ). Because of this selection, ob and ob 2 of the net cycle are included in the ob load of the atch-up phase. The SECOND ALGORITHM which iniizes the aiu tardiness will be applied to return the syste to the steady-state operational ode as soon as possible and the results of the calculations at the last stage, which is stage 3, are suarized in table for and given as follows: I Path 3 t i 3 (i,) 3 (i,2) Match-up points A Machine JOB JOB 2 Machine 2 JOB B Machine 2 breas down Machine 2 is repaired Figure 3. Graphical description of the atch-up phase 24 tie

11 At the last stage, t 2 3 t 4 3 6, which eans either of the schedules can be taen as the outcoe of the second algorith. Also, the atch-up period's length is 22, which is bigger than the steady-state cycle length. This requires a right shift of the start of the steady-state schedule so that the new steady-state schedule starts at achine at t 28 instead of t 22. Figure 4 suarizes and shows these findings. MATCH-UP PERIOD Machine JOB 3 JOB 4 shift of the steady-state period steady-state period Machine 2 JOB JOB 3 JOB 4 JOB tie Figure 4. Gantt Chart for the atch-up phase after achine repair 4. CONCLUSIONS This research was directed towards etending dioid algebraic applications for real-tie scheduling, optiization, and control of a flow-shop. This research introduced decision aing and heuristic optiization techniques into the dioid algebra. Maespan and a tardiness based obective function were eplored in the heuristic algoriths. In addition, a new dioid algebraic odel capable of representing sequencing decisions is developed. In this dioid algebraic odel, operation start tie at a achine is taen as the state variable of the syste. Control variables deterine the sequencing order and the output variables of the odel are achine release ties in a specific period. Both the odel and heuristic algoriths are used in a real-tie scheduling and control schee. In this schee, not only teporary changes such as achine failures or hot obs, but also peranent changes in wor load can be handled. This study shows that the dioid algebraic odel and the solution techniques can be used as an alternative analytical ethod for a wide range of applications in scheduling and control of a flow-shop. REFERENCES.Bean, J. C. and Birge, J. R. (985), "Match-up Real-Tie Scheduling", Departent of Industrial Engineering, The University of Michigan, Ann Arbor, MI. 2.Chryssolouris, G., Dice, K., Lee, M.(994) "An Approach to Real Tie Fleible Scheduling", International Journal of Fleible Manufacturing Systes, Vol. 6, No. 3, p

12 3.Cohen, G., Quadrat, J. P., Olsder, G. J., and Baccelli, F. (992), Synchronization and Linearity, An Algebra for Discrete Event Systes, John Wiley & Sons, New Yor. 4.Engell, S., Kuhn, T., Moser, M. (99), "On Decentralized on-line Scheduling of FMS", Proceedings of the 29th Conference on Decision and Control, Honolulu, Hawaii, Deceber, Haronosy, C. M. and Robohn, S. F. (99), "Real-Tie Scheduling in Coputer Integrated Manufacturing: A Review of Recent Research", International Journal of Coputer Integrated Manufacturing, Vol. 4, No. 6, pp Hitz, K. L.(979), Scheduling of Fleible Flow Shops, LIDS-R-879, Laboratory for Inforation and Decision Systes, M.I.T., Cabridge, Mass. 7.Hitz, K. L. (98), Scheduling of Fleible Flow Shops II, LIDS-R-49, Laboratory for Inforation and Decision Systes, M.I.T., Cabridge, Mass. 8.Ki, M, H., Ki, Y. D. (994), "Siulation-Based Real-Tie Scheduling in a Fleible Manufacturing Syste", Journal of Manufacturing Systes, Vol. 3, No. 2, pp Leon, V. J., Wu, S. D., and Storer, R. H. (994), "Robustness Measures and Robust Scheduling for Job Shops", IEEE Transactions, Vol. 26, No. 5, pp Moser, M., Engell, S. (992), "Avoiding Scheduling Errors by Partial Siulation of the Future", Proceedings of the 3st Conference on Decision and Control, Tucson, Arizona, Deceber, 992, pp Odrey, N. G. and Wilson, G. R., "Hierarchical Planning and Control for a Fleible Manufacturing Shop, Conference Proceedings: NSF Design and Manufacturing Grantees Conference, Arizona State University, Tepe, AR, January, 99, pp Paruna, H. V. D. (99), "Characterizing the Manufacturing Scheduling Proble", Journal of Manufacturing Systes, Vol., No. 3, pp Saleh, A. (988), "Real-Tie Control of a Fleible Manufacturing Cell", Ph.D. Dissertation, Departent of Industrial Engineering, Lehigh University. 4.Saleh, A., Odrey, N. G. and Wilson, G. R. (99), "Design and Algorithic Ipleentation of a Real-Tie Controller for a Manufacturing Cell," PED-Vol. 53, Design, Analysis and Control of Manufacturing Cells ASME, pp Wu, S. D. (99), "Scheduling, Control and Rescheduling Methodologies for Uncertain Manufacturing Environents", Proceedings of the 99 NSF Design and Manufacturing Systes Conference, Austin, Teas. 6.Wu, S. D. and Wys, R. A. (99), "Scheduling, Optiization and Control in Autoated Systes", Control and Dynaic Systes, Vol. 47, pp

13 7.Yaaoto, M. and Nof, S. Y. (985), "Scheduling/Rescheduling in the Manufacturing Operating Syste Environent", IJPR, Vol. 23, No. 4, pp Yurdaul, Mustafa (997), "Real-Tie Scheduling and Control of a Flow Shop Using Dioid Algebra, Ph. D. Dissertation, Departent of Industrial and Manufacturing Systes Engineering, 56 pp., Lehigh University.