Demeulemeester, Erik

Manzini, Massimo

Urgo, Marcello

Massimo Manzini

Erik Demeulemeester

Marcello Urgo

Archivio istituzionale della ricerca - Politecnico di Milano

A proactive-reactive approach to schedule anautomotive assembly lineMassimo Manzini1, Erik Demeulemeester2 and Marcello Urgo11 Politecnico di Milano, Mechanical Engineering Department, Milano, Italymassimo.manzini, marcello.urgo@polimi.it2 KU Leuven, Faculty of Business and Economics, Department of Decision Sciences andInformation Management, Leuven, Belgiumerik.demeulemeester@kuleuven.beKeywords: flow-shop, proactive-reactive scheduling, assembly.1 Introduction and problem statementThe assembly of bodywork parts for the automotive sector is operated in dedicatedassembly lines implementing the sequence of assembling operations through specific join-ing technologies (e.g., spot welding, clinching, hemming, etc.). These assembly lines areorganized as a set of stations executing assembly operations, input/output stations to loadcomponents and unload final parts, and a transportation device moving parts within theline. The latter is usually a 7-axis robot shared among the stations. In this paper we con-sider an assembly line where a batch of parts has to be processed. Assembly operations areexecuted by automatic devices while load/unload operations are executed manually. Theline has a single transportation robot to be shared among the stations and the proposedapproach aims at scheduling its missions. Due to the manual execution of load/unloadoperations, uncertain process times must be considered, thus, the problem under studyis a Stochastic Resource-Constrained Flow-Shop Scheduling Problem to minimize the timeneeded to complete a batch of products, i.e., the makespan. In the need to address uncer-tainty, specific approaches must be adopted. Examples are the ones optimizing the expectedvalue of the makespan (Fernandez 1995, Igelmund and Radermacher 1983). Nevertheless,the minimization of the expected value does not protect against rare but very extreme sce-narios, as discussed in (Alfieri et al. 2012) and (Manzini and Urgo 2015) for Make-to-Orderprocesses. To this aim, we propose a proactive-reactive approach providing a baseline sched-ule and looking for the optimal sequence of the robot considering the actual duration ofoperations during the execution of the assembly process. Differently from other approachesof this class, e.g., (Davari and Demeulemeester 2016), the proposed approach identifiesdisjunctive constraints without explicitly deciding the starting times of operations.2 Solution approachConsider anActivity-on-Node (AoN ) representation of a flow-shop where V = {0, 1, ..., n}is the set of nodes representing operations and E = (i, j), i, j ∈ V the set of arcs modelingprecedence constraints. Operation durations are modeled through general and indepen-dent random distributions p˜ = p˜0, . . . , p˜n, pi being a realization of distribution p˜i andp = p0, . . . , pn a realization of the entire set p˜. Notice that, if an operation is determin-istic, the described formulation still applies with a single value as support. The flow-shopunder study has a limited availability of the transporter and hence we consider a singleresource with unary availability. We address the scheduling of shared transporter’s mis-isons through the decisions over a set of disjunctive constraints named EDC (additionalto the ones in E), resolving resource utilization conflicts.The uncertainty embedded in theproblem is addressed by adopting a proactive-reactive approach made up of two steps. Thefirst step provides the baseline schedule as the optimal sequence of the robot considering agiven duration of the uncertain operations (e.g., a quantile can be used). The second oneis supposed to operate while the baseline schedule is being operated, every time an incon-gruity between the fixed operation duration and the one experienced in the execution ofthe schedule occurs. It checks whether the baseline schedule is supposed to remain optimaland, if needed, reacts by inverting some of the disjunctive constraints previously selected.The two steps are described in detail in the following.2.1 Proactive stepThe proactive step hypothesizes that the duration of operations is fixed. In case of un-certain durations this value can be decided by fixing a quantile q obtaining pq = pq0, . . . , pqn,without considering any anticipation of associated uncertainty. The scheduling problem issolved using the deterministic approach presented in (Demeulemeester and Herroelen 1992).The baseline schedule obtained provides the set of additional constraints EDC . In additionto this, a sensitivity analysis on the solution is also executed. For each precedence con-straint in EDC , the range of variability of operation durations is calculated such that, ifthe durations go outside this range, then the decision taken for the considered disjunctiveconstraint is not optimal anymore, and thus the opposite constraint should be considered.Consider the constraint (i, j) ∈ EDC assuming durations pq, and the eligible times of op-erations i and j, Qpqi and Qpqj , defined as the instants on which each operation can startin terms of all the precedence constraints in E, without considering any of those in EDC .Define ∆pqi,j = Qpqi − Qpqj as the difference between the eligible times of two operationslinked with a disjunctive constraint (i, j). If the decision on this disjunctive constraintis optimal, the associated makespan is shorter than the one considering the opposite di-rection, i.e., S(i,j)n ≤ S(j,i)n , where S(i,j)n is the starting time of operation n, consideringdisjunctive constraint (i, j). Clearly, this depends on the duration of the operations in pq.The makespan takes advantage of an inversion of the disjunctive constraint if and onlyif the lateness of i, compared to Qpqi , is enough to cause a delay of the makespan thatis longer than the delay caused by an inversion without any lateness of i. More formally,the inversion is effective if there is a difference between the eligible times that is greaterthan ∆Ti,j = ∆pqi,j − (S(i,j)n − S(j,i)n ). The threshold ∆Ti,j will be used in the reactive step forevaluating the optimality of the disjunctive constraint (i, j) during the process execution.2.2 Reactive stepThe reactive step considers a vector of realizations p for the durations of the operationand grounds on the definition of a state space Ω modeling the execution of the operations inthe flow-shop. The execution of the operations can be modeled through a sequence of statesover time t, ω(p, t) = (O,F, S, dO) ∈ Ω. Each state is fully described by the set of operationsin execution O, their starting times S and their durations dO(i),∀i ∈ O, as well as the setof completed ones F . Algorithm 1 models the execution of operations starting from t = 0with initial state ω(p, 0) = (0, ∅, 0, 0) and finishes when all the operations are completed,i.e., F = V (steps 1-2). Every time an operation is completed, the set F is updated (step 4)and, if there is an operation i that can start because all its predecessors are completed (step6), it is put into execution and added to the set of ongoing operations O (step 11). On thecontrary, if its execution is constrained by the completion of another operation k througha decision on one disjunctive constraint (k, i) ∈ EDC (step 7), then the algorithm checkswhether (k, i) remains optimal in relation to the realizations in p. This evaluation is donethrough the estimation of the probability that the actual difference between the eligibleReactive-Procedure1 ω(p, 0) = (0, ∅, 0, 0)2 While F ! = V3 t = t+ 14 If dO(i)− S(i) = pi,∀i ∈ O → F = F + i5 Else dO(i) = dO(i) + 16 If i 6∈ O, i 6∈ F and j ∈ F,∀j ∈ (j, i)7 If (k, i) ∈ EDC and P(∆pk,i(t) > ∆Tk,i) > T8 EDC = EDC − (k, i) + (i, k)9 O = O + i, S(i) = t10 Else11 O = O + i , S(i) = tAlgorithm 1: Reactive step algorithm.Operation Mode Min MaxI 6 5 29T1 13 − −A 10 − −T2 9 − −O 5 4 21Table 1: Operation duration inseconds.times exceeds the threshold previously identified: P[∆pk,i(t) > ∆Tk,i]. If this probabilityexceeds a threshold T , the reaction is applied by inverting the constraint (k, i) (steps 8-9).The P[∆pk,i(t) > ∆Tk,i] is estimated considering the duration of the operations in O precedingk and their distributions p˜. The probability that ∆pk,i(t) is greater than ∆Tk,i is equal tothe probability that the difference between the finish time of the last preceding operationof k and the eligible time of i is greater than ∆Tk,i, conditioned on the ongoing durationsin dO. We are looking at the residual duration probability of the operations preceding k:P[∆pk,i(t) > ∆Tk,i] = P[maxl∈prec(k)(dF (l)) − Qi > ∆Tk,i | dO(l)] = P[maxl∈prec(k)(dF (l) −dO(l)) > ∆Tk,i −Qi], where prec(k) indicates an operation preceding k.3 ApplicationThe proposed approach is applied on a single product flow-shop assembling a hoodbodywork. The execution of the process is modeled using the AoN representation in Fig-ure 1. The process consists of five operations, the first and the last ones model the loading(I) and unloading (O) of the parts, executed manually. In the third operation (A), a re-inforcement bar is added through a spot welding process, while the second and fourthoperations are handling tasks (T1 and T2 respectively) operated by the 7-axis robot mov-ing the hood in the line. The two manual operations follow a triangular distribution, whilethe others are deterministic (Table 1). The triangular distributions consider an average ex-ecution duration as the mode, very close to the minimum value, and a worst-case durationas the maximum value, modeling the occurrence of a problem or a delay. The approachaddresses the conflicts between transport operations in the production of a whole batch.These conflicts are depicted with dotted arcs in Figure 1 for a single transport of the firstjob, only (T21), but are repeated for the whole batch. In addition, we set the threshold T to0.5, but let the quantile q, used for fixing the duration in the proactive step, vary between0.1 and 0.9. We evaluate the performances of the approach in terms of the mean squareerror compared to the minimum makespan solution obtained with complete knowledgeof the durations of operations using 10000 runs. In addition, we estimate the approach’sperformances without the reactive step and compare the results. Aggregated performancesfor different lengths of the batch (from 5 to 50 jobs) are included in Table 2. Grounding onthese results, the proactive-reactive approach always performs as good or better than theproactive schedule without reaction (PR and P-only in Table 2). Indeed, if the reactive stepP-only PRQuantile 0.1 0.5 0.9 0.1 0.5 0.9# jobs5 5.473 5.473 0.917 0.917 0.917 0.91710 4.963 4.963 0.980 0.980 0.980 0.98020 7.445 7.445 1.347 1.347 1.347 1.34750 8.456 8.456 1.865 1.865 1.865 1.865Table 2: Aggregated results of the application. Fig. 1: AoN process representation.does not apply any modification, the baseline solution is automatically applied, as depictedfor the 5 jobs and 90th percentile case. The impact of the number of jobs and the percentileis also analyzed: the percentile impacts on results of the only-P approach, with better per-formances for high values. On the other hand, this parameter does not affect the reaction’sperformance due to the uncertainty source being limited to the first and last operations.The performances get worse as the number of jobs increases for both approaches. As aconclusion, the proactive approach provides a good baseline schedule, nevertheless, the re-action step improves the performances when used to manage the occurrence of unexpectedevents, providing a good support in the line’s real-time management.4 ConclusionsIn this article we propose a proactive-reactive approach to schedule a semi-automaticassembly system, with a specific focus on the definition of the reaction policy. The approachhas been tested on a five-operation process with good results, demonstrating that theapplication of the reactive step significantly improves the performances of the baseline one.Future developments will address the investigation of (i) completely manual processes or(ii) tuning the threshold for the reactive step to match user’s aversion to risk and (iii) theapplication of additional disjunctive constraints modeling the schedule of machines besideshandling operations.AcknowledgmentsThis research has been supported by ReCaM EU project, grant agreement No: 680759.ReferencesAlfieri, A., Tolio, T. and Urgo, M., 2012, "A two-stage stochastic programming project schedulingapproach to production planning", Int J Adv Man Technol, Vol. 62, pp. 279-290.Davari, M. and Demeulemeester, E., 2016, "The proactive and reactive resource-constrainedproject scheduling problem", Working paper.Demeulemeester, E. L. and Herroelen, W. S., 1992, "A Branch-and-Bound Procedure for theMultiple RCPSP", Man Sci, Vol. 38, pp. 1803-1818.Fernandez, A. A., 1995, "The Optimal Solution to the Resource-Constrained Project SchedulingProblem with Stochastic Task Durations", Unpublished Doctoral Dissertation.Igelmund, G. and Radermacher, F. J., 1983, "Preselective Strategies for the Optimization ofStochastic Project Networks under Resource Constraints", Networks, Vol. 13, pp. 1-28.Manzini, M. and Urgo, M., 2015, "Makespan estimation of a production process affected by un-certainty: Application on MTO production of NC machine tools", J Man Syst, Vol. 37, No.1, pp. 1-16.

A proactive-reactive approach to schedule an automotive assembly line

https://re.public.polimi.it/bitstream/11311/1053132/1/PMS2018_manzini_rev_v1.pdf

A proactive-reactive approach to schedule an automotive assembly line

Abstract

Similar works

Full text

Available Versions

Archivio istituzionale della ricerca - Politecnico di Milano