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Dual Resource Constrained Scheduling for Quality Control Laboratories
9th IFAC Conference on Manufacturing Modelling, Management and 9th IFAC 9th IFAC Conference Conference on on Manufacturing Manufacturing Modelling, Modelling, Management Management and and Control Control 9th IFAC Conference on Manufacturing Modelling, Management and Control Available online at www.sciencedirect.com Berlin, Germany, August 28-30, 2019 9th IFAC Conference on Manufacturing Modelling, Management and Control Berlin, Germany, August 28-30, Berlin, Germany, August 28-30, 2019 2019 Control Berlin, Germany, August 28-30, 2019 Berlin, Germany, August 28-30, 2019
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Resource Constrained Scheduling Resource Constrained Scheduling Resource Constrained Scheduling Resource Constrained Scheduling Quality Control Laboratories Quality Control Laboratories Resource Constrained Scheduling Quality Control Laboratories Quality Control Laboratories ∗ ∗ Quality Control Laboratories Mariana Mariana M. M. Cunha Cunha ∗∗ Joaquim Joaquim L. L. Viegas Viegas ∗∗
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Mariana M. Cunha L. Viegas ∗ ∗ ∗∗ ∗ Joaquim ∗ ∗ Tiago ∗ Andrea ∗∗ Miguel S. E. Martins Costigliola Mariana M. Cunha Joaquim L. Viegas ∗ Tiago Coito ∗ Andrea ∗∗ Miguel S. E. Martins Coito Costigliola Miguel S. E. Martins Tiago Coito Andrea Costigliola ∗∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗∗ ∗ Jo˜ ao o Figueiredo Figueiredo Jo˜ ao oTiago M. C. C. SousaAndrea Susana M. Vieira Vieira∗∗∗∗∗ Mariana M.Jo˜ Cunha Joaquim Viegas Miguel S. E. Martins Coito Costigliola ∗∗∗ ∗ L. Jo˜ a a M. Sousa Susana M. Jo˜ a o Figueiredo Jo˜ a o M. C. Sousa Susana M. Vieira ∗ ∗ ∗ ∗∗ ∗∗∗ Miguel S. E. Martins Coito Jo˜ ao Figueiredo Jo˜ aoTiago M. C. SousaAndrea SusanaCostigliola M. Vieira ∗∗ ∗∗∗ ∗ ∗ Jo˜ o Figueiredo Jo˜ a o M. C. Sousa Susana M. Vieira ∗aIDMEC, Instituto Superior T´ e cnico, Universidade de Lisboa ∗ IDMEC, Instituto Superior T´ Universidade de InstitutoFarmaCiencia Superior T´eecnico, cnico, Universidade de Lisboa Lisboa ∗∗ ∗ IDMEC, ∗∗ Hovione S.A, Lisbon, Portugal IDMEC, Instituto Superior T´ e cnico, Universidade de Lisboa ∗∗ Hovione FarmaCiencia S.A, Lisbon, Portugal Hovione FarmaCiencia S.A, Lisbon, Portugal ∗ ∗∗∗ ´ Lisbon, ´ IDMEC, Instituto Superior T´ede cnico, Universidade de Lisboa ∗∗∗∗∗ IDMEC, Universidade Evora, Evora, Portugal ´ ´ Hovione FarmaCiencia S.A, Portugal ∗∗∗∗∗ IDMEC, Universidade de Evora, Evora, Portugal ´ ´ Universidade deS.A, Evora, Evora, Portugal Hovione FarmaCiencia Lisbon, Portugal ∗∗∗ IDMEC, ´ ´ IDMEC, Universidade de Evora, Evora, Portugal ∗∗∗ ´ ´ IDMEC,aUniversidade de Evora, Evora, Portugal Abstract: This work presents novel formulation for quality control laboratory scheduling Abstract: This work presents a novel formulation for quality control scheduling Abstract: This work presents a novel formulation for quality control islaboratory laboratory scheduling considering both equipment and analysts as constraints. The problem modelled as aa dualAbstract: This work presents a novel formulation for quality control laboratory scheduling considering both equipment and analysts as constraints. The problem is modelled as considering both equipment and analysts as constraints. The problem is modelled astasks a dualdualresource constrained flexible job shop problem. The formulation considers analyst at Abstract: This work presents a novel formulation for quality control laboratory scheduling considering both equipment and analysts as constraints. The problem is modelled as a dualresource constrained flexible job shop problem. The formulation considers analyst tasks at resource constrained flexible job shop problem. The formulation considers analyst tasks at different times during the processing of samples. The problem is formulated as a mixed integer considering both equipment and analysts as constraints. The problem is modelled as a dualresource constrained flexible job shop problem. The formulation considers analyst tasks at different times during the processing of samples. The problem is formulated as a mixed integer different times during the processing of samples. The problem is formulated as a mixed integer linear programming model (MILP) aiming to minimise makespan. Two sets of instances for the resource constrained flexible job shop problem. Theproblem formulation considers at different times during the processing of samples. The is formulated asanalyst a mixedtasks integer linear model (MILP) aiming to minimise makespan. Two sets instances for linear programming programming model (MILP)The aiming to minimise makespan. Two example sets of of instances for the the scheduling problem are proposed. instance of small illustrates different times during the processing offirst samples. Theconsists problem isaa formulated asinstances athat mixed integer linear programming model (MILP) aiming to minimise makespan. Two sets of for the scheduling problem are proposed. The first instance consists of small example that illustrates scheduling problem are proposed. The first to instance consists of a small example that illustrates the proposed formulation and is solved optimality. The instance mimics the linear programming model (MILP) aiming minimise makespan. Two sets of instances for real the scheduling problem are proposed. The firstto instance consists ofsecond a small example that illustrates the proposed formulation and is solved to optimality. The second instance mimics the the proposed formulation and is solved to optimality. The second instance mimics the real real cc industrial problem and shows the challenges resulting from growing complexity. Copyright scheduling problem are proposed. The first instance consists of a small example that illustrates the proposed formulation andthe is solved to optimality. The growing second complexity. instance mimics the real industrial problem and challenges resulting Copyright c industrial problem and shows shows the challenges resulting from from growing complexity. Copyright 2019 IFAC the proposed formulation and is solved to optimality. The second instance mimics the real c industrial problem and shows the challenges resulting from growing complexity. Copyright 2019 IFAC 2019 IFAC © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. c industrial 2019 IFACproblem and shows the challenges resulting from growing complexity. Copyright 2019 IFAC Scheduling; Keywords: dual resource constrained; quality control; flexible job shop; mixed Keywords: Scheduling; dual resource constrained; quality control; flexible job shop; mixed Keywords: Scheduling; dual resource constrained; quality control; flexible job shop; mixed integer programming. Keywords: Scheduling; dual resource constrained; quality control; flexible job shop; mixed integer linear linear programming. integer linear programming. Keywords: Scheduling; dual resource constrained; quality control; flexible job shop; mixed integer linear programming. integer1.linear programming. INTRODUCTION to find find the the best best possible possible set set of of decisions decisions to to optimise optimise propro1. INTRODUCTION INTRODUCTION to 1. to find theminimising best possible setand/or of decisions to optimise productivity, costs maximise revenues. 1. INTRODUCTION to find the best possible set of decisions to optimise productivity, minimising costs and/or maximise revenues. ductivity, minimising costs and/or maximise revenues. INTRODUCTION find theminimising best possible setand/or of decisions to optimise proRecent trends trends are are1.shifting shifting industrial paradigms paradigms and and enen- to ductivity, costs maximise revenues. Recent industrial Quality control control (QC) (QC) laboratories laboratories are are one one of of the the key key Recent trends aretoshifting industrial paradigms and seren- Quality ductivity, minimising costs and/or maximise revenues. Quality control (QC) laboratories are one of the key abling companies cut on costs and save time on both Recent trends areto industrial paradigms and seren- structures abling companies companies toshifting cut on on costs costs and save save time on on both both serin many chemical process industries. Because Quality control (QC) laboratories are one of the key abling cut and time structures in many chemical process industries. Because structures in many chemical process industries. Because vices and production through better planning and humanRecent trends are shifting industrial paradigms and enabling companies to cut on costs and save time and on both ser- Quality vices and and production through better planning and humanthey are upstream and downstream operations control (QC) laboratories are one of(suppliers the key structures in many chemical process industries. Because vices production through better planning humanthey are upstream and downstream operations (suppliers are upstream and downstream operations machine interfaces, better production control, raw mateabling companies to cut on costs and save time and onraw both ser- they vices and production through better planning machine interfaces, better production control, mateto and close to the a structures in many chemical industries.(suppliers Because are upstream and downstream operations (suppliers machine interfaces, better production control, rawhumanmateto manufacture manufacture and close toprocess the point-of-sale), point-of-sale), a loss loss to manufacture and close to the point-of-sale), loss rial availability availability control, inventory levels, and and energy con- they vices andinterfaces, production through better planning humanmachine better production control, raw material control, inventory levels, and energy conin efficiency efficiency mayand compromise production start(suppliers or a delay they are upstream andclose downstream operations to manufacture to the point-of-sale), adelay loss rial availability control, inventory levels, and energy conin may compromise production start or in efficiency may compromise production start or delay sumption, R¨ u ßmann et al. (2015). With the right decision machine interfaces, better production control, raw material availability control, inventory levels,the andright energy con- to sumption, R¨ ußmann ßmann et al. al. (2015). With With the right decision shipments, Maslaton (2012). QC laboratories manufacture and close toTherefore, the point-of-sale), adelay loss in efficiency may compromise production start or sumption, R¨ u et (2015). decision shipments, Maslaton (2012). Therefore, QC laboratories Maslaton (2012). Therefore, QC laboratories making in R¨ place, the et relationship between customers and shipments, rial availability control, inventory levels,the and energy consumption, u ßmann al. (2015). With right decision making in place, the relationship between customers and have been appointed as an area where scheduling could in efficiency may compromise production start or delay shipments, Maslaton (2012). Therefore, QC laboratories making incan place, the relationship between customers and been appointed as an area where scheduling could have appointed as on an efficiency, area where scheduling could suppliersin be significantly significantly improved and, withdecision theand in- have sumption, R¨ ußmann al. (2015). With and, thecustomers right making place, the et relationship between suppliers can be improved with the inhave aabeen very high impact Sch¨ aafer (2004). shipments, Maslaton (2012). Therefore, QC laboratories have been appointed as an area where scheduling suppliers can be significantly improved and, with the invery high impact on efficiency, Sch¨ fer (2004). have a very high impact on efficiency, Sch¨ a fer (2004). crease in ininprofit profit margins, companies haveand, the flexibility to have been appointed as an area where scheduling could making place, the relationship between customers and suppliers can be significantly improved with the increase margins, companies have the flexibility to could a veryfocuses high impact onlaboratory efficiency,scheduling Sch¨afer (2004). crease in profit margins, companies have the flexibility to have work on QC considerbetter fulfil fulfil their social and environmental responsibilities. suppliers can bemargins, significantly improved with the inThis work on QC considercrease in profit companies haveand, the flexibility to This better their social and environmental responsibilities. have a veryfocuses high impact onlaboratory efficiency,scheduling Sch¨afer (2004). This work focuses on QC laboratory scheduling considerbetter fulfil their social and environmental responsibilities. ing as do crease in profit margins, companies have the flexibility to This workequipment focuses onand QCanalysts laboratory scheduling To considerbetter fulfil socialsectors and environmental ing both both equipment and analysts as constraints. constraints. To do so, so, both as constraints. To do so, Although alltheir industry are affected affected by byresponsibilities. these changes, changes, ing Although all industry sectors are these aThis mathematical formulation of aa dual-resource constrained workequipment focuses onand QCanalysts laboratory scheduling considerbetter fulfil their socialsectors and environmental responsibilities. ing both equipment and analysts as constraints. To do so, Although all industry are affected by these changes, a mathematical formulation of dual-resource constrained a mathematical formulation of a dual-resource constrained one where it is inevitable is the chemical sector. Chemistry Although industry sectors arechemical affectedsector. by these changes, ing one where whereall it is is inevitable is the the chemical sector. Chemistry flexible shop problem is proposed, where assignment of bothjob equipment and analysts as constraints. To do so, mathematical of a dual-resource constrained one it inevitable is Chemistry job shop problem is where of flexible job shopformulation problem is proposed, proposed, where assignment assignment of 4.0, where involving the chemical and pharmaceutical industries Although all industry sectors arechemical affectedsector. by these changes, aflexible one it is inevitable is the Chemistry 4.0, involving the chemical and pharmaceutical industries jobs to equipment and analysts is performed in order to a mathematical formulation of a dual-resource constrained flexible job shop problem is proposed, where assignment of 4.0, involving the chemical and pharmaceutical industries jobs to equipment and analysts is performed in order to jobs to equipment and analysts is performed in order to is experiencing changes with great potential impact, Falter one where it is inevitable is the chemical sector. Chemistry 4.0, involving the chemical and pharmaceutical industries is experiencing experiencing changes with great potential impact, Falter minimise maximum completion time. The peculiarity peculiarity flexible job shop problem is proposed, where assignment of jobs to equipment and analysts is performed in order to is changes with great potential impact, Falter minimise maximum completion time. The of minimise maximum completion time. The peculiarity of et experiencing al.involving (2017). the With the with electric car, aa decline in demand 4.0, chemical and pharmaceutical is changes great potential impact, Falter the et al. (2017). With the electric car, decline inindustries demand model is that it considers analysts’ tasks, not only in jobs to equipment and analysts is performed in order to minimise maximum completion time. The peculiarity of et al. (2017). With the electric car, a decline in demand the model is that it considers analysts’ tasks, not only in the model is that it considers analysts’ tasks, not only in for many chemical products, such as gasoline-resistant is experiencing changes with great potential impact, Falter et (2017). With the electricsuch car, aas in demand minimise for al. many chemical products, such asdecline gasoline-resistant mounting and dismounting, but also at specific times durmaximum completion time. The peculiarity of the model is that it considers analysts’ tasks, not only in for many chemical products, gasoline-resistant mounting and dismounting, but also at specific times durmounting and dismounting, butanalysts’ alsomodel at specific times durplastics, oilchemical and fuelthe additives, is expected. Similarly, in et al. (2017). With electricsuch car, aasdecline in demand for many products, gasoline-resistant plastics, oil and fuel additives, is expected. Similarly, ing processing. The mathematical is implemented the model is that it considers tasks, not only in in and dismounting, but alsomodel at specific times durplastics, oilchemical and fuelsector additives, is expected. Similarly,and in mounting ing processing. The mathematical is Thelinear mathematical is implemented implemented the pharmaceutical pharmaceutical with such the patent expiration for manyoil products, as gasoline-resistant plastics, and fuelsector additives, is expected. Similarly,and in ing the sector with the patent expiration and as aa processing. mixedand integer programming model (MILP). To mounting dismounting, but alsomodel at model specific times during processing. The mathematical model is implemented the pharmaceutical with the patent expiration as mixed integer linear programming (MILP). To as a mixed integer linear programming model (MILP). To pricepharmaceutical controls increasing in Europe and the the US, profit profit plastics, oil andincreasing fuelsector additives, is expected. Similarly, in ing the with the patent expiration and price controls in Europe and US, the best of our knowledge the mathematical formulation processing. The mathematical model is implemented as a mixed integer linear programming model (MILP). To price controls increasing in Europe and the US, profit the best of our knowledge the mathematical formulation the best of our knowledge the mathematical formulation margins are under threat, Maslaton (2012). The prospect the pharmaceutical sector with the patent expiration and price controls increasing Europe(2012). and the profit as margins are under under threat, in Maslaton (2012). TheUS, prospect developed and application to the QC laboratory scheduling a best mixed integer linear programming model (MILP). To the of our knowledge the mathematical formulation margins are threat, Maslaton The prospect developed and application to the QC laboratory scheduling and application tothe themathematical QC laboratory scheduling of these these changes pressure companies to invest and try to developed price controls increasing in Europe(2012). and the US, profit margins are under threat, Maslaton The prospect of changes pressure companies to invest and try to problem has not been presented previously. Additionally, the best of our knowledge formulation andnot application to the QC laboratory scheduling of these changes pressure companies to invest and try to developed problem has been presented previously. Additionally, not been are presented previously. findthese solutions, both in process process improvement and product margins are under threat, Maslaton (2012). The prospect of changes pressure companies to invest and try to problem find solutions, both in improvement and product two sets sets has ofand instances proposed. The first firstAdditionally, isscheduling an illusillusdeveloped application to the QC laboratory problem has not been presented previously. Additionally, find solutions, both in process improvement and product two of instances are proposed. The is an two sets of instances are proposed. The first is ansecond illusportfolio diversification. of these changes pressure companies to investand andproduct try to problem find solutions, both in process improvement portfolio diversification. trative example solved to optimality. The has not that beenis presented previously. Additionally, two sets of instances are proposed. The first is an illusportfolio diversification. trative example that is solved to optimality. The second trative example that is solved to optimality. The second find solutions, both in process improvement and product portfolio diversification. simulates real QC laboratory scheduling two setsexample ofthe instances are proposed. The firstproblem. is ansecond illusIt is within Industry 4.0 scope, and more specifically trative that is solved to optimality. The simulates the real QC laboratory scheduling problem. It is within Industry 4.0 scope, and more specifically simulates the realthat QCislaboratory scheduling problem. portfolio diversification. It is within Industry 4.0 scope, and and rescheduling more specifically trative example solved to optimality. The second Chemistry 4.0, that the scheduling probsimulates the real QC laboratory scheduling problem. It is within scope, and and rescheduling more specifically Chemistry 4.0,Industry that the the 4.0 scheduling probThe remainder of this paper is organised as follows. The Chemistry 4.0, that scheduling and rescheduling probThe remainder of this paper as follows. The simulates the real laboratory scheduling problem. The remainder of QC this paper is isa organised organised asrelated follows.work. The lemisarises. arises. Knowing the4.0 current state ofmore operations, deIt within Industry scope,state and rescheduling specifically Chemistry 4.0, that the scheduling and problem Knowing the current of operations, rest of this section presents review on deThe remainder of this paper is organised as follows. The lem arises. Knowing the current state of operations, derest of this section presents a review on related work. rest of this section presents a review on related work. scribed by data on demand and plant state, it is possible Chemistry 4.0, that the scheduling and rescheduling problem arises. Knowing the current statestate, of operations, de- The scribed by data data on demand Section 22this describes the QC laboratory scheduling problem. remainder of this paper isa organised asrelated follows. The and plant it is possible rest of section presents review on work. scribed by on demand and plant state, it is possible Section describes the QC laboratory scheduling problem. Section 2 describes the QC laboratory scheduling problem. lem arises. Knowing the current statestate, of operations, de- rest scribed by data on demand and plant it is possible Section 3describes gives a the detailed description of the developed of 2this section presents a review on the related work. Section QC laboratory scheduling problem. 3 gives a detailed description of developed Section givesmodel. a the detailed description of the the developed developed scribed by data on demandbyand plant state,IDMEC, it is possible ⋆ mathematical Section 44 presents Section 233describes QC laboratory scheduling problem. This work work was was supported supported by FCT, through through IDMEC, under ⋆ Section givesmodel. a detailed description of the mathematical Section presents developed FCT, under ⋆ This mathematical model. Section 4 presents the developed This work was supported by FCT, through IDMEC, under instances and results obtained. Finally, in Section 5, conLAETA, project UID/EMS/50022/2019. Section 3 gives a detailed description of the developed ⋆ mathematical model. obtained. Section 4Finally, presents the developed instances and in 5, LAETA, project UID/EMS/50022/2019. This work was supported by FCT, through IDMEC, under instances and results results in Section Section 5, conconLAETA, project UID/EMS/50022/2019. ⋆ Corresponding author e-mail: [email protected] [email protected] mathematical model. obtained. Section 4Finally, presents the developed This work was supported by FCT, through IDMEC, under Corresponding author e-mail: instances and results obtained. Finally, in Section 5, conLAETA, project UID/EMS/50022/2019. Corresponding author e-mail: [email protected] instances and results obtained. Finally, in Section 5, conLAETA, projectauthor UID/EMS/50022/2019. Corresponding e-mail: [email protected] Corresponding author e-mail: [email protected] Copyright IFAC 1439 2405-8963 © © 2019 2019, IFAC (International Federation of Automatic Control) Copyright © 2019 IFAC 1439Hosting by Elsevier Ltd. All rights reserved. Copyright © 2019 IFAC 1439 Peer review under responsibility of International Federation of Automatic Control. Copyright © 2019 IFAC 1439 10.1016/j.ifacol.2019.11.398 Copyright © 2019 IFAC 1439
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cluding remarks are made and directions for future work are presented. 1.1 Related work In terms of dual-resource constrained scheduling, ScholzReiter et al. (2009) did a comparison for static dualresource constrained scheduling between MILP and various dispatching rules. Dispatching rules were also compared using long-term simulation. The application use case was a semiconductor manufacturing system. However, workers only had tasks at the beginning of task processing, for loading, and at the end, for unloading. Lei and Guo (2014) presented a variable neighbourhood search (VNS) algorithm for a flexible job shop scheduling with workers. Their algorithm was tested against extensions of classic flexible job shop instances to include workers. Although the presented VNS yielded good results, they assume a worker has to be assigned to the task during the whole machine processing time. Ciro et al. (2015) studied an open shop scheduling with worker flexibility. They implemented a MILP model and compared it with a proposed ant colony algorithm tuned with fuzzy logic controllers. In the implemented scenario, workers were assigned to jobs for the full length of their processing time on the machine. Furthermore, because their application is an open shop problem, jobs have just one operation to perform. In terms of quality control specific applications, RuizTorres et al. (2012) addressed the QC laboratories scheduling of technicians by minimizing flow-time, using a heuristic consisting of a combination of dispatching rules. In Ruiz-Torres et al. (2017) the assignment of work to technicians is dependent on preferences and train related certification. In Costigliola et al. (2017) and Lopes et al. (2018) simulation models of QC laboratories were developed. Costigliola et al. (2017) developed a simulation model for the analytical workflow in the laboratory. Through the analysis of different simulation scenarios, Lopes et al. (2018) studied the impact of different governance strategies in process flow efficiency in the laboratories. In this paper, a dual resource constrained scheduling problem formulation is proposed to model the scheduling problem in QC laboratories as a MILP. The formulation considers cases where analysts/workers are not only required for setup and teardown activities, but also for tasks at specific times during processing operations. 2. PROBLEM DESCRIPTION Quality control receives samples for analyses, depending on the plant type, before, during and after manufacturing to certify product characteristics, Ruiz-Torres et al. (2012). Attaining the specific characteristics of a product requires various analytical procedures. When a sample arrives at the laboratory it may need to be analysed through one or more techniques, involving, or not, precedence between them, Sch¨ afer (2004). A sample analysis requires not only the specific equipment to perform the analysis but various interventions from an analyst. The analyst is usually needed to prepare the sample, setup the equipment, run the test and dismount the equipment, Lopes et al. (2018). These tasks may be
required at specific times during the analysis or may take the whole time an analysis is run. In this study, there is partial flexibility in terms of equipment (i.e. for each operation, there is an equipment subset able to perform the analysis) and, depending on the product, only some of the analysts are able to assist the analysis. Furthermore, the equipments and analysts were considered to be free for all jobs, having no dedicated lines. Quality control laboratories are structures that support production. Hence, the main objective when considering this facilities workflow is to process the samples sent from production as fast as possible, Maslaton (2012). The quality control laboratory scheduling is optimally designed by solving the following problem: • Given: (1) a set of samples that require a set of analytical procedures to be performed in a given order; (2) available equipment and corresponding suitability for each sample analysis procedure; (3) each analytical procedure has a specific, fixed processing time, dependent on which equipment the analysis will be done; (4) available analysts and corresponding suitability for each sample analysis procedure; (5) each analytical procedure requires an analyst at a given time during a specific duration; (6) an analytical procedure may require the analyst more than once during its processing. • Determine: (1) an optimal allocation to an equipment; (2) an optimal allocation to an analyst; (3) an optimal sequence of samples and analytical procedures for an equipment; (4) an optimal sequence of samples and analytical procedures for an analyst. • So as to: (1) minimize maximum completion time. In the following section the mathematical formulation to solve the problem stated above is described. 3. MATHEMATICAL FORMULATION In this section the proposed MILP is presented for the dual resource constraint scheduling problem, and adapted to the case-study presented. Both modelling assumption, constraints and objective function are thoroughly described. 3.1 Time modelling Scheduling formulations are very dependent on the time modelling strategy. Time can be modelled in a discrete or a continuous way, Floudas and Lin (2005). The time discretisation approach involves splitting the time horizon in a number of equal duration time intervals. Events such as the start/end of a task can only take place at those time intervals. It is usually necessary to use a short length for the time units to achieve a good time approximation. However, depending on the relation between time unit duration and time horizon, this may imply very large models.
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In continuous-time approaches, the durations of intervals are treated as variables. An interval starts or ends at an event and its duration is a continuous variable. The time grid can be common to all resources or different time grids for each resource can be used. In this work a continuous-time approach using the two different time grid strategies was used. A common time grid was employed for the worker resources and resource differing time grids were adopted for machines. This way, not only was time modelled in a very accurate way, but a much smaller number of events need to be processed for the machines. The same option was not taken for workers because these are required to be present in various distinct times during the processing of operations and can, in the meantime, engage in other operations.
Equation (4) sets the starting time of an analyst to be at least the time the tests requests the analyst’s presence. Equation (5) enforces that, if a test is assigned to an analyst at a time point n, the analyst needs to be there at exactly that time. Otherwise the analyst’s time is not bounded. Ts (s, a, n) ≥ t(s, o) − M (1 − Xa (s, o, a, θ, n))+ +
(4)
θ
∀s ∈ S, o ∈ Os , a ∈ AOs Ts (s, a, n) ≤ t(s, o) + M (1 − Xa (s, o, a, θ, n))+ +
min
Cmax
(1)
As (s, o, a, θ)Xa (s, o, a, θ, n),
(5)
θ
∀s ∈ S, o ∈ Os , a ∈ AOs Equation (6) defines the ending time of an analyst intervention as the starting time of the intervention, T s(s, a, n), plus its duration. Equation (7) is a bounding equation that limits the ending time of an intervention of an analyst in a test to the end of the test processing time.
3.3 Constraints
Te (s, a, n) =Ts (s, a, n)+
τ (s,o)
Test precedence constraints Precedence constraints enforce that a test can only start after the completion of the previous test, (2). Since a test processing time is considered deterministic and only depends on the equipment processing it, the completion time of the test is given by its starting time plus the processing time on the chosen equipment.
Ad (s, o, a, θ)Xa (s, o, a, θ, n),
(6)
θ
∀s ∈ S, o ∈ Os , a ∈ AOs Te (s, a, n) ≤ t(s, o) + M (1 − Xa (s, o, a, θ, n))+ µ(s, o, k)X(s, o, k, l),
(7)
k,l
µ(s, o, k)X(s, o, k, l) ≤ t(s, o + 1),
∀s ∈ S, o ∈ Os , a ∈ AOs , k ∈ KOs
(2)
k,l
∀s ∈ S, o ∈ Os , o �= |Os | , k ∈ KOs Equipment precedence constraints In an equipment, a test can only start after the previous test was completed. Similarly to (2), the time a test can start on an equipment has to be at least the time a previous test performed by the same equipment was completed, (3). T (k, l) +
As (s, o, a, θ)Xa (s, o, a, θ, n),
τ (s,o)
The objective in this case study was to perform the best scheduling that enables the optimum completion time for the tests in the laboratory. Hence, the objective is to minimise makespan, equation (1). The nomenclature is presented in Annex A.
τ (s,o)
3.2 Objective function
t(s, o) +
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µ(s, o, k)X(s, o, k, l) ≤ T (k, l + 1), (3)
s,o
∀s ∈ S, o ∈ Os , k ∈ KOs , l �= llast Analysts start and end time constraints The analyst allocation variable, Xa(s, o, a, θ, n), is dependent on θ = {1, .., τ (s, o)}, where τ (s, o) is the maximum number of times the analyst’s presence is requested by that test. The indices θ indicates the sequence request within a test, ex. if the first time an analyst is requested for that test is at the time of setup, then the θ for setup will be 1.
Finally, equation (8) sets the precedence constraints for each analyst to ensure no overlap of tasks. Ts (s, a, n + 1) ≥ Te (j, a, n), ∀s, j ∈ S, o ∈ Os , a ∈ AOs , n �= nlast
(8)
Allocation constraints Allocation constraints ensure that the real system capacity is respected and demand is met. Both the equipment allocation variable and the analyst allocation variable are bounded by this constraints. Equations (9) and (10) are equipment related. Equation (9) ensures a sample test is always performed on an equipment suitable to it. Equation (10) limits the number of tests being allocated to an equipment to be at most 1 at a time.
For the definition of the analyst’s starting and ending time for processing of a sample test, five constraints are enforced. 1441
X(s, o, k, l) = 1,∀s ∈ S, o ∈ Os , k ∈ KOs
(9)
k,l
s,o
X(s, o, k, l) ≤ 1,∀s ∈ S, o ∈ Os , k ∈ KOs
(10)
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Table 1. Job types and associated characteristics.
Equations (11) and (12) are equipment related. Equation (11) ensures a sample test is always allocated to an analyst each time the analyst’s presence is requested. Equation (12) states that an analyst can only be allocated to one test at a time.
Job type A
B
Xa (s, o, a, θ, n) = 1, (11)
a,n
C
∀s ∈ S, o ∈ Os , a ∈ AOs , θ ≤ τ (s, o)
(12)
∀s ∈ S, o ∈ Os , a ∈ AOs , θ ≤ τ (s, o) Linking constraints There is the need to link the starting time of a test in an equipment with the equipment starting time, so equations (13) and (14) are defined.
Job type
Oper.
A
1 2 3 1 2 1
B C
T (k, l) ≤ t(s, o) + M (1 − X(s, o, k, l)), ∀s ∈ S, o ∈ Os , k ∈ KOs
(13)
T (k, l) + M (1 − X(s, o, k, l)) ≥ t(s, o), ∀s ∈ S, o ∈ Os , k ∈ KOs
(14)
Makespan definition Makespan is the completion time of the last test to be completed. This constraint cannot be formulated using a max operator since it would be non-linear, and so it has to be enforced using multiple constraints, equation (15). Cmax ≥ t(s, o) +
µ(s, o, k)X(s, o, k, l),
k,l
Processing time 3 2 1 5 4 1
Table 2. Job analyst requirements in terms of start time and duration of the activity.
Xa (s, o, a, θ, n) ≤ 1,
s,o,θ
Operation 1 2 3 1 2 1
(15)
∀s ∈ S, o ∈ Os , k ∈ KOs
Setup Start End 0 0.25 0 0.25 0 0.25 0 0.25 0 0.25 0 0.25
Intermediate Start End 1 2 2 3 1 2 -
Teardown Start End 2.75 3 0.75 1 0.75 1 4 5 3 4 0.75 1
Testing scenarios were characterized by number of jobs A, B and C. Two types of problems were tested, small and medium-large instances. The former was intended as an illustrative example. The latter are instances created to mimic a daily workload in a real QC laboratory, based on the data reported by Costigliola et al. (2017). The total number of jobs considered for the small instance was 3, one of each type. For medium-large instances the number of jobs varied between 30 and 40 jobs. However, since this is a flexible job shop problem and jobs are comprised of more than one operation, it is also important to consider the number of operations in each instance. Therefore, the small instance has 6 operations, while the medium-large instance has between 35 and 52 operations. 4.1 Small instance scenario
4. RESULTS AND DISCUSSION From the presented mathematical formulation, a mixed integer linear programming model was implemented. It was implemented in the General Algebraic Modelling System (GAMS) and solved using an Intel core i7 2.50 GHz processor and 16GB RAM. The commercial solver used was CPLEX. Due to the operational nature of the scheduling problem, the stopping criterion chosen was a maximum computational time of one hour. To develop the scenarios, three types of jobs were created. These were based on the product types in Ruiz-Torres et al. (2012), and are presented in Table 1. Job type A has three associated operations that have precedence between them (e.g. operation 2 can only start after the completion of operation 1), job type B has two operations with precedence constrains as well, and job type C has only one associated operation. Each job type requires an analyst at different times throughout its processing. The starting time and duration of these analyst tasks is presented in Table 2. For job type C and operations 2 and 3 of job type A analysts are only required for setup and teardown. In both operations of job type B and operation 1 of job type A, analysts are needed during a specific part of the processing as well.
The small instance scenario was created as a simple illustrative example. Therefore, three jobs, one of each type, and three machines were used. The association between machines and job types can be seen on Table 3. Furthermore, these instances because of their size had only one analyst, as to be sure the problem was constrained by human resources. Table 3. Resource suitability for the small instance No. jobs
Job type
<3
A B C
Suitable analysts a1 a1 a1
Suitable machines m1 or m2 m1, m2 or m3 m1 or m3
The results for the small instance are shown in Table 4. The model was solved to optimality in 34.52 seconds. In Figure 1 and Figure 2, an optimal solution is shown, having a makespan of 11. It is possible to observe in the figures both the analyst’s presence in intermediate time points during the operations’ processing and the analyst shifting between operations. The model statistics are presented in Table 5. With higher complexity problems, not just in terms of number of operations but also in terms of number of resources,
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Table 4. Results for small instances No. Jobs A B C 1
1
Total no. operations 6
1
4.2 Medium-large instances
Cmax
Gap
CPU time (s)
11
0
34.52
Medium-large instances have between 30 and 40 jobs and were developed to mimic reality. This range of jobs yield a number of operations between 35 and 52, which is in accordance with the values stated by Costigliola et al. (2017). In this study it is considered that 85% of the samples have only one test to perform (similarly to sample type C), around 10% have two tests (resembling sample type B), and around 5% have three tests to perform.
Machine Gantt Job A Job B Job C
Machine
m3
As resources, fifteen machines and five analysts were considered. Because the amount of samples of type C is significantly larger than the amount of samples of the other types, there are more machines able to analyse type C samples. Furthermore, there were more analysts considered able to analyse samples of type C than of type A and B, as presented in Table 6. Compared to the previous instance, the medium-large instances consist of a significantly complexity increase not just because of the higher amount of resources and jobs but also because of the higher flexibility present, namely in job type C.
m2
m1
0
2
4
6 Time
8
10
Table 6. Resource suitability for medium-large instances
Fig. 1. Equipment Gantt chart for the small instance scenario with one job of each type. An optimal solution was obtained (Cmax = 11).
No. jobs
Job type
≥ 30
A B C
Analyst Gantt Job A Job B Job C
Analysts
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Suitable analysts a1 a2 a3 to a5
Suitable machines m15 m12 to m15 m1 to m12
The results are shown in Table 7. The model was tested for a total of 30 and 40 jobs. It is possible to see that both instances were not solved optimally for a maximum computational time of 1h. The gap presented in Table 7 corresponds to the relative gap between the best known integer solution and the lower bound calculated through the linear relaxation. It is enough that the linear relaxation is not very strong for the computed gap to be very different from the actual distance between the best known integer solution and the optimal.
a1
0
2
4
6 Time
8
Table 7. Results for medium-large instances
10
No. Jobs A B C
Fig. 2. Analyst Gantt chart for the small instance scenario. An optimal solution was obtained (Cmax = 11). the model statistics increase significantly. This is due to the fact that there is at least one variable per resource and there are variables associated with the resource time points. The number of time points is a tuning parameter, if it is too large it significantly increases the model size, and if it is too small it may compromise the optimality of the solution.
1 2
26 34
Cmax
Gap
CPU time (h)
27 36
66.67% 75.0%
1 1
In Table 8 the statistics for the medium-large instances are presented. By comparing the two instances it is possible to see the large increase in equations and variables, evidencing the whole problem complexity.
Table 5. Model statistics for the small instance Scenario (NA ,NB ,NC ) Block of equations Single equations Blocks of variables Single variables Non zeros elements Discrete variables Computational time (s)
3 4
Total no. operations 35 52
(1,1,1) 15 1231 8 830 5811 624 34.52
Table 8. Model statistics for the medium-large instances Scenario Block of equations Single equations Blocks of variables Single variables Non zeros elements Discrete variables Computational time (h)
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(1,3,26) 15 118698 8 31747 893219 22410 1
(2,4,34) 15 189586 8 42710 1240706 30360 1
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5. CONCLUSIONS This paper develops a mathematical model for scheduling problem in quality control laboratories. The problem is formulated as a dual resource constrained flexible job shop problem, in which workers are not only required for setup and teardown operations, but also for intermediate points throughout the processing of a task. From the proposed mathematical formulation, a MILP was developed and implemented. The model was then tested on small and medium-large instances. Results show that the model was able to achieve optimality for the small instance. For the two medium-large instances the model was able to find a solution, but optimality was not guaranteed. The problems with finding the optimal solution were related to the instances increase in complexity. As future work, research should focus on exploring additional solution methods for the quality control scheduling problem. Constraint programming and hybrid approaches, the latter combining constraint and integer programming, may be more effective in such a large model. Furthermore, due to the highly combinatorial nature of this problem, we believe heuristics or metaheuristics will also be able to achieve good solutions in a shorter computational time than MILP. Further testing may also be of interest to ascertain the model complexity and how it is affected by the test instance structure. REFERENCES Ciro, G.C., Dugardin, F., Yalaoui, F., and Kelly, R. (2015). A fuzzy ant colony optimization to solve an open shop scheduling problem with multi-skills resource constraints. IFAC-PapersOnLine, 48(3), 715–720. Costigliola, A., Ata´ıde, F.A., Vieira, S.M., and Sousa, J.M. (2017). Simulation model of a quality control laboratory in pharmaceutical industry. IFAC-PapersOnLine, 50(1), 9014–9019. Falter, W., Keller, A., Nickel, J.P., and Meincke, H. (2017). Chemistry 4.0 - growth through innovation in a transforming world. Technical report, Deloitte and German Chemical Industry Association. Floudas, C.A. and Lin, X. (2005). Mixed integer linear programming in process scheduling: Modeling, algorithms, and applications. Annals of Operations Research, 139(1), 131–162. Lei, D. and Guo, X. (2014). Variable neighbourhood search for dual-resource constrained flexible job shop scheduling. International Journal of Production Research, 52(9), 2519–2529. Lopes, M.R., Costigliola, A., Pinto, R.M., Vieira, S.M., and Sousa, J.M. (2018). Novel governance model for planning in pharmaceutical quality control laboratories. IFAC-PapersOnLine, 51(11), 484–489. Maslaton, R. (2012). Resource Scheduling in QC Laboratories. Pharmaceutical Engineering, 32(5). Ruiz-Torres, A.J., Ablanedo-Rosas, J.H., and Otero, L.D. (2012). Scheduling with multiple tasks per job–the case of quality control laboratories in the pharmaceutical industry. International Journal of Production Research, 50(3), 691–705. Ruiz-Torres, A.J., Mahmoodi, F., and Kuula, M. (2017). Quality assurance laboratory planning system to maxi-
mize worker preference subject to certification and preference balance constraints. Computers & Operations Research, 83, 140–149. R¨ ußmann, M., Lorenz, M., Gerbert, P., Waldner, M., Justus, J., Engel, P., and Harnisch, M. (2015). Industry 4.0: The future of productivity and growth in manufacturing industries. Boston Consulting Group, 9. Sch¨afer, R. (2004). Concepts for dynamic scheduling in the laboratory. JALA: Journal of the Association for Laboratory Automation, 9(6), 382–397. Scholz-Reiter, B., Heger, J., and Hildebrandt, T. (2009). Analysis and comparison of dispatching rule-based scheduling in dual-resource constrained shop-floor scenarios. In Proceedings of the World Congress on Engineering and Computer Science, volume 2, 20–22. Appendix A. NOMENCLATURE Sets - S is the set of samples - Os is the set of tests from sample s - KOs is the set of equipment suitable for processing the test Os - AOs is the set of analysts suitable for processing the test Os Indices - s, j = samples - o = test - k = equipment - a = analyst - l = event point for equipment - n = event point for analysts - θ = relative time for tests Parameters - µ(s, o, k) = processing time of the test o of sample s in equipment k - As (s, o, a, θ) = time after the start of the processing of test o of sample s that intervention θ by analyst a is requested - Ad (s, o, a, θ) = duration of intervention θ by analyst a requested by test o of sample s - M = big number - τ (s, o) = maximum number of analyst intervention needed for test o of sample s Variables - X(s, o, k, l) = 1 if test o of sample s starts in equipment k at event point l; 0 otherwise - t(s, o) = starting time of test o from sample s - T (k, l) = starting time of time point l of equipment k - Xa (s, o, a, θ, n) = 1 if intervention θ of test o from sample s is assisted by analyst a at time point n; 0 otherwise - Ts (s, a, n) = starting time of sample s on time point n of analyst a - Te (s, a, n) = end time of sample s on time point n of analyst a - Cmax = makespan
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Dual Dual Dual Dual Dual
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Resource Constrained Scheduling Resource Constrained Scheduling Resource Constrained Scheduling Resource Constrained Scheduling Quality Control Laboratories Quality Control Laboratories Resource Constrained Scheduling Quality Control Laboratories Quality Control Laboratories ∗ ∗ Quality Control Laboratories Mariana Mariana M. M. Cunha Cunha ∗∗ Joaquim Joaquim L. L. Viegas Viegas ∗∗
for for for for for
Mariana M. Cunha L. Viegas ∗ ∗ ∗∗ ∗ Joaquim ∗ ∗ Tiago ∗ Andrea ∗∗ Miguel S. E. Martins Costigliola Mariana M. Cunha Joaquim L. Viegas ∗ Tiago Coito ∗ Andrea ∗∗ Miguel S. E. Martins Coito Costigliola Miguel S. E. Martins Tiago Coito Andrea Costigliola ∗∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗∗ ∗ Jo˜ ao o Figueiredo Figueiredo Jo˜ ao oTiago M. C. C. SousaAndrea Susana M. Vieira Vieira∗∗∗∗∗ Mariana M.Jo˜ Cunha Joaquim Viegas Miguel S. E. Martins Coito Costigliola ∗∗∗ ∗ L. Jo˜ a a M. Sousa Susana M. Jo˜ a o Figueiredo Jo˜ a o M. C. Sousa Susana M. Vieira ∗ ∗ ∗ ∗∗ ∗∗∗ Miguel S. E. Martins Coito Jo˜ ao Figueiredo Jo˜ aoTiago M. C. SousaAndrea SusanaCostigliola M. Vieira ∗∗ ∗∗∗ ∗ ∗ Jo˜ o Figueiredo Jo˜ a o M. C. Sousa Susana M. Vieira ∗aIDMEC, Instituto Superior T´ e cnico, Universidade de Lisboa ∗ IDMEC, Instituto Superior T´ Universidade de InstitutoFarmaCiencia Superior T´eecnico, cnico, Universidade de Lisboa Lisboa ∗∗ ∗ IDMEC, ∗∗ Hovione S.A, Lisbon, Portugal IDMEC, Instituto Superior T´ e cnico, Universidade de Lisboa ∗∗ Hovione FarmaCiencia S.A, Lisbon, Portugal Hovione FarmaCiencia S.A, Lisbon, Portugal ∗ ∗∗∗ ´ Lisbon, ´ IDMEC, Instituto Superior T´ede cnico, Universidade de Lisboa ∗∗∗∗∗ IDMEC, Universidade Evora, Evora, Portugal ´ ´ Hovione FarmaCiencia S.A, Portugal ∗∗∗∗∗ IDMEC, Universidade de Evora, Evora, Portugal ´ ´ Universidade deS.A, Evora, Evora, Portugal Hovione FarmaCiencia Lisbon, Portugal ∗∗∗ IDMEC, ´ ´ IDMEC, Universidade de Evora, Evora, Portugal ∗∗∗ ´ ´ IDMEC,aUniversidade de Evora, Evora, Portugal Abstract: This work presents novel formulation for quality control laboratory scheduling Abstract: This work presents a novel formulation for quality control scheduling Abstract: This work presents a novel formulation for quality control islaboratory laboratory scheduling considering both equipment and analysts as constraints. The problem modelled as aa dualAbstract: This work presents a novel formulation for quality control laboratory scheduling considering both equipment and analysts as constraints. The problem is modelled as considering both equipment and analysts as constraints. The problem is modelled astasks a dualdualresource constrained flexible job shop problem. The formulation considers analyst at Abstract: This work presents a novel formulation for quality control laboratory scheduling considering both equipment and analysts as constraints. The problem is modelled as a dualresource constrained flexible job shop problem. The formulation considers analyst tasks at resource constrained flexible job shop problem. The formulation considers analyst tasks at different times during the processing of samples. The problem is formulated as a mixed integer considering both equipment and analysts as constraints. The problem is modelled as a dualresource constrained flexible job shop problem. The formulation considers analyst tasks at different times during the processing of samples. The problem is formulated as a mixed integer different times during the processing of samples. The problem is formulated as a mixed integer linear programming model (MILP) aiming to minimise makespan. Two sets of instances for the resource constrained flexible job shop problem. Theproblem formulation considers at different times during the processing of samples. The is formulated asanalyst a mixedtasks integer linear model (MILP) aiming to minimise makespan. Two sets instances for linear programming programming model (MILP)The aiming to minimise makespan. Two example sets of of instances for the the scheduling problem are proposed. instance of small illustrates different times during the processing offirst samples. Theconsists problem isaa formulated asinstances athat mixed integer linear programming model (MILP) aiming to minimise makespan. Two sets of for the scheduling problem are proposed. The first instance consists of small example that illustrates scheduling problem are proposed. The first to instance consists of a small example that illustrates the proposed formulation and is solved optimality. The instance mimics the linear programming model (MILP) aiming minimise makespan. Two sets of instances for real the scheduling problem are proposed. The firstto instance consists ofsecond a small example that illustrates the proposed formulation and is solved to optimality. The second instance mimics the the proposed formulation and is solved to optimality. The second instance mimics the real real cc industrial problem and shows the challenges resulting from growing complexity. Copyright scheduling problem are proposed. The first instance consists of a small example that illustrates the proposed formulation andthe is solved to optimality. The growing second complexity. instance mimics the real industrial problem and challenges resulting Copyright c industrial problem and shows shows the challenges resulting from from growing complexity. Copyright 2019 IFAC the proposed formulation and is solved to optimality. The second instance mimics the real c industrial problem and shows the challenges resulting from growing complexity. Copyright 2019 IFAC 2019 IFAC © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. c industrial 2019 IFACproblem and shows the challenges resulting from growing complexity. Copyright 2019 IFAC Scheduling; Keywords: dual resource constrained; quality control; flexible job shop; mixed Keywords: Scheduling; dual resource constrained; quality control; flexible job shop; mixed Keywords: Scheduling; dual resource constrained; quality control; flexible job shop; mixed integer programming. Keywords: Scheduling; dual resource constrained; quality control; flexible job shop; mixed integer linear linear programming. integer linear programming. Keywords: Scheduling; dual resource constrained; quality control; flexible job shop; mixed integer linear programming. integer1.linear programming. INTRODUCTION to find find the the best best possible possible set set of of decisions decisions to to optimise optimise propro1. INTRODUCTION INTRODUCTION to 1. to find theminimising best possible setand/or of decisions to optimise productivity, costs maximise revenues. 1. INTRODUCTION to find the best possible set of decisions to optimise productivity, minimising costs and/or maximise revenues. ductivity, minimising costs and/or maximise revenues. INTRODUCTION find theminimising best possible setand/or of decisions to optimise proRecent trends trends are are1.shifting shifting industrial paradigms paradigms and and enen- to ductivity, costs maximise revenues. Recent industrial Quality control control (QC) (QC) laboratories laboratories are are one one of of the the key key Recent trends aretoshifting industrial paradigms and seren- Quality ductivity, minimising costs and/or maximise revenues. Quality control (QC) laboratories are one of the key abling companies cut on costs and save time on both Recent trends areto industrial paradigms and seren- structures abling companies companies toshifting cut on on costs costs and save save time on on both both serin many chemical process industries. Because Quality control (QC) laboratories are one of the key abling cut and time structures in many chemical process industries. Because structures in many chemical process industries. Because vices and production through better planning and humanRecent trends are shifting industrial paradigms and enabling companies to cut on costs and save time and on both ser- Quality vices and and production through better planning and humanthey are upstream and downstream operations control (QC) laboratories are one of(suppliers the key structures in many chemical process industries. Because vices production through better planning humanthey are upstream and downstream operations (suppliers are upstream and downstream operations machine interfaces, better production control, raw mateabling companies to cut on costs and save time and onraw both ser- they vices and production through better planning machine interfaces, better production control, mateto and close to the a structures in many chemical industries.(suppliers Because are upstream and downstream operations (suppliers machine interfaces, better production control, rawhumanmateto manufacture manufacture and close toprocess the point-of-sale), point-of-sale), a loss loss to manufacture and close to the point-of-sale), loss rial availability availability control, inventory levels, and and energy con- they vices andinterfaces, production through better planning humanmachine better production control, raw material control, inventory levels, and energy conin efficiency efficiency mayand compromise production start(suppliers or a delay they are upstream andclose downstream operations to manufacture to the point-of-sale), adelay loss rial availability control, inventory levels, and energy conin may compromise production start or in efficiency may compromise production start or delay sumption, R¨ u ßmann et al. (2015). With the right decision machine interfaces, better production control, raw material availability control, inventory levels,the andright energy con- to sumption, R¨ ußmann ßmann et al. al. (2015). With With the right decision shipments, Maslaton (2012). QC laboratories manufacture and close toTherefore, the point-of-sale), adelay loss in efficiency may compromise production start or sumption, R¨ u et (2015). decision shipments, Maslaton (2012). Therefore, QC laboratories Maslaton (2012). Therefore, QC laboratories making in R¨ place, the et relationship between customers and shipments, rial availability control, inventory levels,the and energy consumption, u ßmann al. (2015). With right decision making in place, the relationship between customers and have been appointed as an area where scheduling could in efficiency may compromise production start or delay shipments, Maslaton (2012). Therefore, QC laboratories making incan place, the relationship between customers and been appointed as an area where scheduling could have appointed as on an efficiency, area where scheduling could suppliersin be significantly significantly improved and, withdecision theand in- have sumption, R¨ ußmann al. (2015). With and, thecustomers right making place, the et relationship between suppliers can be improved with the inhave aabeen very high impact Sch¨ aafer (2004). shipments, Maslaton (2012). Therefore, QC laboratories have been appointed as an area where scheduling suppliers can be significantly improved and, with the invery high impact on efficiency, Sch¨ fer (2004). have a very high impact on efficiency, Sch¨ a fer (2004). crease in ininprofit profit margins, companies haveand, the flexibility to have been appointed as an area where scheduling could making place, the relationship between customers and suppliers can be significantly improved with the increase margins, companies have the flexibility to could a veryfocuses high impact onlaboratory efficiency,scheduling Sch¨afer (2004). crease in profit margins, companies have the flexibility to have work on QC considerbetter fulfil fulfil their social and environmental responsibilities. suppliers can bemargins, significantly improved with the inThis work on QC considercrease in profit companies haveand, the flexibility to This better their social and environmental responsibilities. have a veryfocuses high impact onlaboratory efficiency,scheduling Sch¨afer (2004). This work focuses on QC laboratory scheduling considerbetter fulfil their social and environmental responsibilities. ing as do crease in profit margins, companies have the flexibility to This workequipment focuses onand QCanalysts laboratory scheduling To considerbetter fulfil socialsectors and environmental ing both both equipment and analysts as constraints. constraints. To do so, so, both as constraints. To do so, Although alltheir industry are affected affected by byresponsibilities. these changes, changes, ing Although all industry sectors are these aThis mathematical formulation of aa dual-resource constrained workequipment focuses onand QCanalysts laboratory scheduling considerbetter fulfil their socialsectors and environmental responsibilities. ing both equipment and analysts as constraints. To do so, Although all industry are affected by these changes, a mathematical formulation of dual-resource constrained a mathematical formulation of a dual-resource constrained one where it is inevitable is the chemical sector. Chemistry Although industry sectors arechemical affectedsector. by these changes, ing one where whereall it is is inevitable is the the chemical sector. Chemistry flexible shop problem is proposed, where assignment of bothjob equipment and analysts as constraints. To do so, mathematical of a dual-resource constrained one it inevitable is Chemistry job shop problem is where of flexible job shopformulation problem is proposed, proposed, where assignment assignment of 4.0, where involving the chemical and pharmaceutical industries Although all industry sectors arechemical affectedsector. by these changes, aflexible one it is inevitable is the Chemistry 4.0, involving the chemical and pharmaceutical industries jobs to equipment and analysts is performed in order to a mathematical formulation of a dual-resource constrained flexible job shop problem is proposed, where assignment of 4.0, involving the chemical and pharmaceutical industries jobs to equipment and analysts is performed in order to jobs to equipment and analysts is performed in order to is experiencing changes with great potential impact, Falter one where it is inevitable is the chemical sector. Chemistry 4.0, involving the chemical and pharmaceutical industries is experiencing experiencing changes with great potential impact, Falter minimise maximum completion time. The peculiarity peculiarity flexible job shop problem is proposed, where assignment of jobs to equipment and analysts is performed in order to is changes with great potential impact, Falter minimise maximum completion time. The of minimise maximum completion time. The peculiarity of et experiencing al.involving (2017). the With the with electric car, aa decline in demand 4.0, chemical and pharmaceutical is changes great potential impact, Falter the et al. (2017). With the electric car, decline inindustries demand model is that it considers analysts’ tasks, not only in jobs to equipment and analysts is performed in order to minimise maximum completion time. The peculiarity of et al. (2017). With the electric car, a decline in demand the model is that it considers analysts’ tasks, not only in the model is that it considers analysts’ tasks, not only in for many chemical products, such as gasoline-resistant is experiencing changes with great potential impact, Falter et (2017). With the electricsuch car, aas in demand minimise for al. many chemical products, such asdecline gasoline-resistant mounting and dismounting, but also at specific times durmaximum completion time. The peculiarity of the model is that it considers analysts’ tasks, not only in for many chemical products, gasoline-resistant mounting and dismounting, but also at specific times durmounting and dismounting, butanalysts’ alsomodel at specific times durplastics, oilchemical and fuelthe additives, is expected. Similarly, in et al. (2017). With electricsuch car, aasdecline in demand for many products, gasoline-resistant plastics, oil and fuel additives, is expected. Similarly, ing processing. The mathematical is implemented the model is that it considers tasks, not only in in and dismounting, but alsomodel at specific times durplastics, oilchemical and fuelsector additives, is expected. Similarly,and in mounting ing processing. The mathematical is Thelinear mathematical is implemented implemented the pharmaceutical pharmaceutical with such the patent expiration for manyoil products, as gasoline-resistant plastics, and fuelsector additives, is expected. Similarly,and in ing the sector with the patent expiration and as aa processing. mixedand integer programming model (MILP). To mounting dismounting, but alsomodel at model specific times during processing. The mathematical model is implemented the pharmaceutical with the patent expiration as mixed integer linear programming (MILP). To as a mixed integer linear programming model (MILP). To pricepharmaceutical controls increasing in Europe and the the US, profit profit plastics, oil andincreasing fuelsector additives, is expected. Similarly, in ing the with the patent expiration and price controls in Europe and US, the best of our knowledge the mathematical formulation processing. The mathematical model is implemented as a mixed integer linear programming model (MILP). To price controls increasing in Europe and the US, profit the best of our knowledge the mathematical formulation the best of our knowledge the mathematical formulation margins are under threat, Maslaton (2012). The prospect the pharmaceutical sector with the patent expiration and price controls increasing Europe(2012). and the profit as margins are under under threat, in Maslaton (2012). TheUS, prospect developed and application to the QC laboratory scheduling a best mixed integer linear programming model (MILP). To the of our knowledge the mathematical formulation margins are threat, Maslaton The prospect developed and application to the QC laboratory scheduling and application tothe themathematical QC laboratory scheduling of these these changes pressure companies to invest and try to developed price controls increasing in Europe(2012). and the US, profit margins are under threat, Maslaton The prospect of changes pressure companies to invest and try to problem has not been presented previously. Additionally, the best of our knowledge formulation andnot application to the QC laboratory scheduling of these changes pressure companies to invest and try to developed problem has been presented previously. Additionally, not been are presented previously. findthese solutions, both in process process improvement and product margins are under threat, Maslaton (2012). The prospect of changes pressure companies to invest and try to problem find solutions, both in improvement and product two sets sets has ofand instances proposed. The first firstAdditionally, isscheduling an illusillusdeveloped application to the QC laboratory problem has not been presented previously. Additionally, find solutions, both in process improvement and product two of instances are proposed. The is an two sets of instances are proposed. The first is ansecond illusportfolio diversification. of these changes pressure companies to investand andproduct try to problem find solutions, both in process improvement portfolio diversification. trative example solved to optimality. The has not that beenis presented previously. Additionally, two sets of instances are proposed. The first is an illusportfolio diversification. trative example that is solved to optimality. The second trative example that is solved to optimality. The second find solutions, both in process improvement and product portfolio diversification. simulates real QC laboratory scheduling two setsexample ofthe instances are proposed. The firstproblem. is ansecond illusIt is within Industry 4.0 scope, and more specifically trative that is solved to optimality. The simulates the real QC laboratory scheduling problem. It is within Industry 4.0 scope, and more specifically simulates the realthat QCislaboratory scheduling problem. portfolio diversification. It is within Industry 4.0 scope, and and rescheduling more specifically trative example solved to optimality. The second Chemistry 4.0, that the scheduling probsimulates the real QC laboratory scheduling problem. It is within scope, and and rescheduling more specifically Chemistry 4.0,Industry that the the 4.0 scheduling probThe remainder of this paper is organised as follows. The Chemistry 4.0, that scheduling and rescheduling probThe remainder of this paper as follows. The simulates the real laboratory scheduling problem. The remainder of QC this paper is isa organised organised asrelated follows.work. The lemisarises. arises. Knowing the4.0 current state ofmore operations, deIt within Industry scope,state and rescheduling specifically Chemistry 4.0, that the scheduling and problem Knowing the current of operations, rest of this section presents review on deThe remainder of this paper is organised as follows. The lem arises. Knowing the current state of operations, derest of this section presents a review on related work. rest of this section presents a review on related work. scribed by data on demand and plant state, it is possible Chemistry 4.0, that the scheduling and rescheduling problem arises. Knowing the current statestate, of operations, de- The scribed by data data on demand Section 22this describes the QC laboratory scheduling problem. remainder of this paper isa organised asrelated follows. The and plant it is possible rest of section presents review on work. scribed by on demand and plant state, it is possible Section describes the QC laboratory scheduling problem. Section 2 describes the QC laboratory scheduling problem. lem arises. Knowing the current statestate, of operations, de- rest scribed by data on demand and plant it is possible Section 3describes gives a the detailed description of the developed of 2this section presents a review on the related work. Section QC laboratory scheduling problem. 3 gives a detailed description of developed Section givesmodel. a the detailed description of the the developed developed scribed by data on demandbyand plant state,IDMEC, it is possible ⋆ mathematical Section 44 presents Section 233describes QC laboratory scheduling problem. This work work was was supported supported by FCT, through through IDMEC, under ⋆ Section givesmodel. a detailed description of the mathematical Section presents developed FCT, under ⋆ This mathematical model. Section 4 presents the developed This work was supported by FCT, through IDMEC, under instances and results obtained. Finally, in Section 5, conLAETA, project UID/EMS/50022/2019. Section 3 gives a detailed description of the developed ⋆ mathematical model. obtained. Section 4Finally, presents the developed instances and in 5, LAETA, project UID/EMS/50022/2019. This work was supported by FCT, through IDMEC, under instances and results results in Section Section 5, conconLAETA, project UID/EMS/50022/2019. ⋆ Corresponding author e-mail: [email protected] [email protected] mathematical model. obtained. Section 4Finally, presents the developed This work was supported by FCT, through IDMEC, under Corresponding author e-mail: instances and results obtained. Finally, in Section 5, conLAETA, project UID/EMS/50022/2019. Corresponding author e-mail: [email protected] instances and results obtained. Finally, in Section 5, conLAETA, projectauthor UID/EMS/50022/2019. Corresponding e-mail: [email protected] Corresponding author e-mail: [email protected] Copyright IFAC 1439 2405-8963 © © 2019 2019, IFAC (International Federation of Automatic Control) Copyright © 2019 IFAC 1439Hosting by Elsevier Ltd. All rights reserved. Copyright © 2019 IFAC 1439 Peer review under responsibility of International Federation of Automatic Control. Copyright © 2019 IFAC 1439 10.1016/j.ifacol.2019.11.398 Copyright © 2019 IFAC 1439
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cluding remarks are made and directions for future work are presented. 1.1 Related work In terms of dual-resource constrained scheduling, ScholzReiter et al. (2009) did a comparison for static dualresource constrained scheduling between MILP and various dispatching rules. Dispatching rules were also compared using long-term simulation. The application use case was a semiconductor manufacturing system. However, workers only had tasks at the beginning of task processing, for loading, and at the end, for unloading. Lei and Guo (2014) presented a variable neighbourhood search (VNS) algorithm for a flexible job shop scheduling with workers. Their algorithm was tested against extensions of classic flexible job shop instances to include workers. Although the presented VNS yielded good results, they assume a worker has to be assigned to the task during the whole machine processing time. Ciro et al. (2015) studied an open shop scheduling with worker flexibility. They implemented a MILP model and compared it with a proposed ant colony algorithm tuned with fuzzy logic controllers. In the implemented scenario, workers were assigned to jobs for the full length of their processing time on the machine. Furthermore, because their application is an open shop problem, jobs have just one operation to perform. In terms of quality control specific applications, RuizTorres et al. (2012) addressed the QC laboratories scheduling of technicians by minimizing flow-time, using a heuristic consisting of a combination of dispatching rules. In Ruiz-Torres et al. (2017) the assignment of work to technicians is dependent on preferences and train related certification. In Costigliola et al. (2017) and Lopes et al. (2018) simulation models of QC laboratories were developed. Costigliola et al. (2017) developed a simulation model for the analytical workflow in the laboratory. Through the analysis of different simulation scenarios, Lopes et al. (2018) studied the impact of different governance strategies in process flow efficiency in the laboratories. In this paper, a dual resource constrained scheduling problem formulation is proposed to model the scheduling problem in QC laboratories as a MILP. The formulation considers cases where analysts/workers are not only required for setup and teardown activities, but also for tasks at specific times during processing operations. 2. PROBLEM DESCRIPTION Quality control receives samples for analyses, depending on the plant type, before, during and after manufacturing to certify product characteristics, Ruiz-Torres et al. (2012). Attaining the specific characteristics of a product requires various analytical procedures. When a sample arrives at the laboratory it may need to be analysed through one or more techniques, involving, or not, precedence between them, Sch¨ afer (2004). A sample analysis requires not only the specific equipment to perform the analysis but various interventions from an analyst. The analyst is usually needed to prepare the sample, setup the equipment, run the test and dismount the equipment, Lopes et al. (2018). These tasks may be
required at specific times during the analysis or may take the whole time an analysis is run. In this study, there is partial flexibility in terms of equipment (i.e. for each operation, there is an equipment subset able to perform the analysis) and, depending on the product, only some of the analysts are able to assist the analysis. Furthermore, the equipments and analysts were considered to be free for all jobs, having no dedicated lines. Quality control laboratories are structures that support production. Hence, the main objective when considering this facilities workflow is to process the samples sent from production as fast as possible, Maslaton (2012). The quality control laboratory scheduling is optimally designed by solving the following problem: • Given: (1) a set of samples that require a set of analytical procedures to be performed in a given order; (2) available equipment and corresponding suitability for each sample analysis procedure; (3) each analytical procedure has a specific, fixed processing time, dependent on which equipment the analysis will be done; (4) available analysts and corresponding suitability for each sample analysis procedure; (5) each analytical procedure requires an analyst at a given time during a specific duration; (6) an analytical procedure may require the analyst more than once during its processing. • Determine: (1) an optimal allocation to an equipment; (2) an optimal allocation to an analyst; (3) an optimal sequence of samples and analytical procedures for an equipment; (4) an optimal sequence of samples and analytical procedures for an analyst. • So as to: (1) minimize maximum completion time. In the following section the mathematical formulation to solve the problem stated above is described. 3. MATHEMATICAL FORMULATION In this section the proposed MILP is presented for the dual resource constraint scheduling problem, and adapted to the case-study presented. Both modelling assumption, constraints and objective function are thoroughly described. 3.1 Time modelling Scheduling formulations are very dependent on the time modelling strategy. Time can be modelled in a discrete or a continuous way, Floudas and Lin (2005). The time discretisation approach involves splitting the time horizon in a number of equal duration time intervals. Events such as the start/end of a task can only take place at those time intervals. It is usually necessary to use a short length for the time units to achieve a good time approximation. However, depending on the relation between time unit duration and time horizon, this may imply very large models.
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In continuous-time approaches, the durations of intervals are treated as variables. An interval starts or ends at an event and its duration is a continuous variable. The time grid can be common to all resources or different time grids for each resource can be used. In this work a continuous-time approach using the two different time grid strategies was used. A common time grid was employed for the worker resources and resource differing time grids were adopted for machines. This way, not only was time modelled in a very accurate way, but a much smaller number of events need to be processed for the machines. The same option was not taken for workers because these are required to be present in various distinct times during the processing of operations and can, in the meantime, engage in other operations.
Equation (4) sets the starting time of an analyst to be at least the time the tests requests the analyst’s presence. Equation (5) enforces that, if a test is assigned to an analyst at a time point n, the analyst needs to be there at exactly that time. Otherwise the analyst’s time is not bounded. Ts (s, a, n) ≥ t(s, o) − M (1 − Xa (s, o, a, θ, n))+ +
(4)
θ
∀s ∈ S, o ∈ Os , a ∈ AOs Ts (s, a, n) ≤ t(s, o) + M (1 − Xa (s, o, a, θ, n))+ +
min
Cmax
(1)
As (s, o, a, θ)Xa (s, o, a, θ, n),
(5)
θ
∀s ∈ S, o ∈ Os , a ∈ AOs Equation (6) defines the ending time of an analyst intervention as the starting time of the intervention, T s(s, a, n), plus its duration. Equation (7) is a bounding equation that limits the ending time of an intervention of an analyst in a test to the end of the test processing time.
3.3 Constraints
Te (s, a, n) =Ts (s, a, n)+
τ (s,o)
Test precedence constraints Precedence constraints enforce that a test can only start after the completion of the previous test, (2). Since a test processing time is considered deterministic and only depends on the equipment processing it, the completion time of the test is given by its starting time plus the processing time on the chosen equipment.
Ad (s, o, a, θ)Xa (s, o, a, θ, n),
(6)
θ
∀s ∈ S, o ∈ Os , a ∈ AOs Te (s, a, n) ≤ t(s, o) + M (1 − Xa (s, o, a, θ, n))+ µ(s, o, k)X(s, o, k, l),
(7)
k,l
µ(s, o, k)X(s, o, k, l) ≤ t(s, o + 1),
∀s ∈ S, o ∈ Os , a ∈ AOs , k ∈ KOs
(2)
k,l
∀s ∈ S, o ∈ Os , o �= |Os | , k ∈ KOs Equipment precedence constraints In an equipment, a test can only start after the previous test was completed. Similarly to (2), the time a test can start on an equipment has to be at least the time a previous test performed by the same equipment was completed, (3). T (k, l) +
As (s, o, a, θ)Xa (s, o, a, θ, n),
τ (s,o)
The objective in this case study was to perform the best scheduling that enables the optimum completion time for the tests in the laboratory. Hence, the objective is to minimise makespan, equation (1). The nomenclature is presented in Annex A.
τ (s,o)
3.2 Objective function
t(s, o) +
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µ(s, o, k)X(s, o, k, l) ≤ T (k, l + 1), (3)
s,o
∀s ∈ S, o ∈ Os , k ∈ KOs , l �= llast Analysts start and end time constraints The analyst allocation variable, Xa(s, o, a, θ, n), is dependent on θ = {1, .., τ (s, o)}, where τ (s, o) is the maximum number of times the analyst’s presence is requested by that test. The indices θ indicates the sequence request within a test, ex. if the first time an analyst is requested for that test is at the time of setup, then the θ for setup will be 1.
Finally, equation (8) sets the precedence constraints for each analyst to ensure no overlap of tasks. Ts (s, a, n + 1) ≥ Te (j, a, n), ∀s, j ∈ S, o ∈ Os , a ∈ AOs , n �= nlast
(8)
Allocation constraints Allocation constraints ensure that the real system capacity is respected and demand is met. Both the equipment allocation variable and the analyst allocation variable are bounded by this constraints. Equations (9) and (10) are equipment related. Equation (9) ensures a sample test is always performed on an equipment suitable to it. Equation (10) limits the number of tests being allocated to an equipment to be at most 1 at a time.
For the definition of the analyst’s starting and ending time for processing of a sample test, five constraints are enforced. 1441
X(s, o, k, l) = 1,∀s ∈ S, o ∈ Os , k ∈ KOs
(9)
k,l
s,o
X(s, o, k, l) ≤ 1,∀s ∈ S, o ∈ Os , k ∈ KOs
(10)
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Table 1. Job types and associated characteristics.
Equations (11) and (12) are equipment related. Equation (11) ensures a sample test is always allocated to an analyst each time the analyst’s presence is requested. Equation (12) states that an analyst can only be allocated to one test at a time.
Job type A
B
Xa (s, o, a, θ, n) = 1, (11)
a,n
C
∀s ∈ S, o ∈ Os , a ∈ AOs , θ ≤ τ (s, o)
(12)
∀s ∈ S, o ∈ Os , a ∈ AOs , θ ≤ τ (s, o) Linking constraints There is the need to link the starting time of a test in an equipment with the equipment starting time, so equations (13) and (14) are defined.
Job type
Oper.
A
1 2 3 1 2 1
B C
T (k, l) ≤ t(s, o) + M (1 − X(s, o, k, l)), ∀s ∈ S, o ∈ Os , k ∈ KOs
(13)
T (k, l) + M (1 − X(s, o, k, l)) ≥ t(s, o), ∀s ∈ S, o ∈ Os , k ∈ KOs
(14)
Makespan definition Makespan is the completion time of the last test to be completed. This constraint cannot be formulated using a max operator since it would be non-linear, and so it has to be enforced using multiple constraints, equation (15). Cmax ≥ t(s, o) +
µ(s, o, k)X(s, o, k, l),
k,l
Processing time 3 2 1 5 4 1
Table 2. Job analyst requirements in terms of start time and duration of the activity.
Xa (s, o, a, θ, n) ≤ 1,
s,o,θ
Operation 1 2 3 1 2 1
(15)
∀s ∈ S, o ∈ Os , k ∈ KOs
Setup Start End 0 0.25 0 0.25 0 0.25 0 0.25 0 0.25 0 0.25
Intermediate Start End 1 2 2 3 1 2 -
Teardown Start End 2.75 3 0.75 1 0.75 1 4 5 3 4 0.75 1
Testing scenarios were characterized by number of jobs A, B and C. Two types of problems were tested, small and medium-large instances. The former was intended as an illustrative example. The latter are instances created to mimic a daily workload in a real QC laboratory, based on the data reported by Costigliola et al. (2017). The total number of jobs considered for the small instance was 3, one of each type. For medium-large instances the number of jobs varied between 30 and 40 jobs. However, since this is a flexible job shop problem and jobs are comprised of more than one operation, it is also important to consider the number of operations in each instance. Therefore, the small instance has 6 operations, while the medium-large instance has between 35 and 52 operations. 4.1 Small instance scenario
4. RESULTS AND DISCUSSION From the presented mathematical formulation, a mixed integer linear programming model was implemented. It was implemented in the General Algebraic Modelling System (GAMS) and solved using an Intel core i7 2.50 GHz processor and 16GB RAM. The commercial solver used was CPLEX. Due to the operational nature of the scheduling problem, the stopping criterion chosen was a maximum computational time of one hour. To develop the scenarios, three types of jobs were created. These were based on the product types in Ruiz-Torres et al. (2012), and are presented in Table 1. Job type A has three associated operations that have precedence between them (e.g. operation 2 can only start after the completion of operation 1), job type B has two operations with precedence constrains as well, and job type C has only one associated operation. Each job type requires an analyst at different times throughout its processing. The starting time and duration of these analyst tasks is presented in Table 2. For job type C and operations 2 and 3 of job type A analysts are only required for setup and teardown. In both operations of job type B and operation 1 of job type A, analysts are needed during a specific part of the processing as well.
The small instance scenario was created as a simple illustrative example. Therefore, three jobs, one of each type, and three machines were used. The association between machines and job types can be seen on Table 3. Furthermore, these instances because of their size had only one analyst, as to be sure the problem was constrained by human resources. Table 3. Resource suitability for the small instance No. jobs
Job type
<3
A B C
Suitable analysts a1 a1 a1
Suitable machines m1 or m2 m1, m2 or m3 m1 or m3
The results for the small instance are shown in Table 4. The model was solved to optimality in 34.52 seconds. In Figure 1 and Figure 2, an optimal solution is shown, having a makespan of 11. It is possible to observe in the figures both the analyst’s presence in intermediate time points during the operations’ processing and the analyst shifting between operations. The model statistics are presented in Table 5. With higher complexity problems, not just in terms of number of operations but also in terms of number of resources,
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Table 4. Results for small instances No. Jobs A B C 1
1
Total no. operations 6
1
4.2 Medium-large instances
Cmax
Gap
CPU time (s)
11
0
34.52
Medium-large instances have between 30 and 40 jobs and were developed to mimic reality. This range of jobs yield a number of operations between 35 and 52, which is in accordance with the values stated by Costigliola et al. (2017). In this study it is considered that 85% of the samples have only one test to perform (similarly to sample type C), around 10% have two tests (resembling sample type B), and around 5% have three tests to perform.
Machine Gantt Job A Job B Job C
Machine
m3
As resources, fifteen machines and five analysts were considered. Because the amount of samples of type C is significantly larger than the amount of samples of the other types, there are more machines able to analyse type C samples. Furthermore, there were more analysts considered able to analyse samples of type C than of type A and B, as presented in Table 6. Compared to the previous instance, the medium-large instances consist of a significantly complexity increase not just because of the higher amount of resources and jobs but also because of the higher flexibility present, namely in job type C.
m2
m1
0
2
4
6 Time
8
10
Table 6. Resource suitability for medium-large instances
Fig. 1. Equipment Gantt chart for the small instance scenario with one job of each type. An optimal solution was obtained (Cmax = 11).
No. jobs
Job type
≥ 30
A B C
Analyst Gantt Job A Job B Job C
Analysts
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Suitable analysts a1 a2 a3 to a5
Suitable machines m15 m12 to m15 m1 to m12
The results are shown in Table 7. The model was tested for a total of 30 and 40 jobs. It is possible to see that both instances were not solved optimally for a maximum computational time of 1h. The gap presented in Table 7 corresponds to the relative gap between the best known integer solution and the lower bound calculated through the linear relaxation. It is enough that the linear relaxation is not very strong for the computed gap to be very different from the actual distance between the best known integer solution and the optimal.
a1
0
2
4
6 Time
8
Table 7. Results for medium-large instances
10
No. Jobs A B C
Fig. 2. Analyst Gantt chart for the small instance scenario. An optimal solution was obtained (Cmax = 11). the model statistics increase significantly. This is due to the fact that there is at least one variable per resource and there are variables associated with the resource time points. The number of time points is a tuning parameter, if it is too large it significantly increases the model size, and if it is too small it may compromise the optimality of the solution.
1 2
26 34
Cmax
Gap
CPU time (h)
27 36
66.67% 75.0%
1 1
In Table 8 the statistics for the medium-large instances are presented. By comparing the two instances it is possible to see the large increase in equations and variables, evidencing the whole problem complexity.
Table 5. Model statistics for the small instance Scenario (NA ,NB ,NC ) Block of equations Single equations Blocks of variables Single variables Non zeros elements Discrete variables Computational time (s)
3 4
Total no. operations 35 52
(1,1,1) 15 1231 8 830 5811 624 34.52
Table 8. Model statistics for the medium-large instances Scenario Block of equations Single equations Blocks of variables Single variables Non zeros elements Discrete variables Computational time (h)
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(1,3,26) 15 118698 8 31747 893219 22410 1
(2,4,34) 15 189586 8 42710 1240706 30360 1
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5. CONCLUSIONS This paper develops a mathematical model for scheduling problem in quality control laboratories. The problem is formulated as a dual resource constrained flexible job shop problem, in which workers are not only required for setup and teardown operations, but also for intermediate points throughout the processing of a task. From the proposed mathematical formulation, a MILP was developed and implemented. The model was then tested on small and medium-large instances. Results show that the model was able to achieve optimality for the small instance. For the two medium-large instances the model was able to find a solution, but optimality was not guaranteed. The problems with finding the optimal solution were related to the instances increase in complexity. As future work, research should focus on exploring additional solution methods for the quality control scheduling problem. Constraint programming and hybrid approaches, the latter combining constraint and integer programming, may be more effective in such a large model. Furthermore, due to the highly combinatorial nature of this problem, we believe heuristics or metaheuristics will also be able to achieve good solutions in a shorter computational time than MILP. Further testing may also be of interest to ascertain the model complexity and how it is affected by the test instance structure. REFERENCES Ciro, G.C., Dugardin, F., Yalaoui, F., and Kelly, R. (2015). A fuzzy ant colony optimization to solve an open shop scheduling problem with multi-skills resource constraints. IFAC-PapersOnLine, 48(3), 715–720. Costigliola, A., Ata´ıde, F.A., Vieira, S.M., and Sousa, J.M. (2017). Simulation model of a quality control laboratory in pharmaceutical industry. IFAC-PapersOnLine, 50(1), 9014–9019. Falter, W., Keller, A., Nickel, J.P., and Meincke, H. (2017). Chemistry 4.0 - growth through innovation in a transforming world. Technical report, Deloitte and German Chemical Industry Association. Floudas, C.A. and Lin, X. (2005). Mixed integer linear programming in process scheduling: Modeling, algorithms, and applications. Annals of Operations Research, 139(1), 131–162. Lei, D. and Guo, X. (2014). Variable neighbourhood search for dual-resource constrained flexible job shop scheduling. International Journal of Production Research, 52(9), 2519–2529. Lopes, M.R., Costigliola, A., Pinto, R.M., Vieira, S.M., and Sousa, J.M. (2018). Novel governance model for planning in pharmaceutical quality control laboratories. IFAC-PapersOnLine, 51(11), 484–489. Maslaton, R. (2012). Resource Scheduling in QC Laboratories. Pharmaceutical Engineering, 32(5). Ruiz-Torres, A.J., Ablanedo-Rosas, J.H., and Otero, L.D. (2012). Scheduling with multiple tasks per job–the case of quality control laboratories in the pharmaceutical industry. International Journal of Production Research, 50(3), 691–705. Ruiz-Torres, A.J., Mahmoodi, F., and Kuula, M. (2017). Quality assurance laboratory planning system to maxi-
mize worker preference subject to certification and preference balance constraints. Computers & Operations Research, 83, 140–149. R¨ ußmann, M., Lorenz, M., Gerbert, P., Waldner, M., Justus, J., Engel, P., and Harnisch, M. (2015). Industry 4.0: The future of productivity and growth in manufacturing industries. Boston Consulting Group, 9. Sch¨afer, R. (2004). Concepts for dynamic scheduling in the laboratory. JALA: Journal of the Association for Laboratory Automation, 9(6), 382–397. Scholz-Reiter, B., Heger, J., and Hildebrandt, T. (2009). Analysis and comparison of dispatching rule-based scheduling in dual-resource constrained shop-floor scenarios. In Proceedings of the World Congress on Engineering and Computer Science, volume 2, 20–22. Appendix A. NOMENCLATURE Sets - S is the set of samples - Os is the set of tests from sample s - KOs is the set of equipment suitable for processing the test Os - AOs is the set of analysts suitable for processing the test Os Indices - s, j = samples - o = test - k = equipment - a = analyst - l = event point for equipment - n = event point for analysts - θ = relative time for tests Parameters - µ(s, o, k) = processing time of the test o of sample s in equipment k - As (s, o, a, θ) = time after the start of the processing of test o of sample s that intervention θ by analyst a is requested - Ad (s, o, a, θ) = duration of intervention θ by analyst a requested by test o of sample s - M = big number - τ (s, o) = maximum number of analyst intervention needed for test o of sample s Variables - X(s, o, k, l) = 1 if test o of sample s starts in equipment k at event point l; 0 otherwise - t(s, o) = starting time of test o from sample s - T (k, l) = starting time of time point l of equipment k - Xa (s, o, a, θ, n) = 1 if intervention θ of test o from sample s is assisted by analyst a at time point n; 0 otherwise - Ts (s, a, n) = starting time of sample s on time point n of analyst a - Te (s, a, n) = end time of sample s on time point n of analyst a - Cmax = makespan
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