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Procedia CIRP 28 ( 2015 ) 131 – 136

Available online at www.sciencedirect.com

2212-8271 © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Selection and peer-review under responsibility of the International Scientific Committee of the “3rd CIRP Global Web Conference” in the person of the Conference Chair Dr. Alessandra Caggiano.

doi: 10.1016/j.procir.2015.04.022

ScienceDirect

3rd CIRP Global Web Conference

Integrating optimisation and simulation to solve manufacturing

scheduling problems

A. Caggiano

a,b,

*, G. Bruno

c

, R. Teti

a,b

aDepartment of Chemical, Materials and Industrial Production Engineering, University of Naples Federico II, P.le Tecchio 80, 80125 Naples, Italy bFraunhofer Joint Laboratory of Excellence on Advanced Production Technology, Fh-J_LEAPT, P.le Tecchio 80, 80125 Naples, Italy

cDepartment of Industrial Engineering, University of Naples Federico II, P.le Tecchio 80, 80125 Naples, Italy

* Corresponding author. Tel.: +39-081-7682371; fax: +39-081-7682362. E-mail address: alessandra.caggiano@unina.it.

Abstract

In this research paper, a novel approach to solve manufacturing scheduling problems based on the integration of optimisation and simulation tools is proposed. The advantages of this approach are illustrated with reference to a manufacturing scheduling case study. The initial optimisation model is formulated as an adaptation of the Capacitated Lot-Sizing Problem (CLSP). Then, a Discrete Event Simulation (DES) software is employed to verify the optimised solution and analyse its robustness taking into account uncertainties in the processing times and arrival rates, and a heuristic method is applied to improve the initial solution. © 2014 The Authors. Published by Elsevier B.V.

Selection and peer-review under responsibility of the International Scientific Committee of the 3rd CIRP Global Web Conference in the person of the Conference Chair Dr. Alessandra Caggiano

Keywords: Scheduling; Simulation; Optimisation; Mathematical programming; Discrete event simulation; Capacitated lot-sizing problem;

1. Introduction

Nowadays, manufacturing systems entail complex decisions concerning short term issues (e.g. daily production) as well as long term investment strategies (e.g. the introduction of new machines, new products, etc.), that can be effectively supported by tools of different nature, such as simulation models and mathematical programming techniques.

As a matter of fact, several simulation software tools have been developed in recent years to support complex decision-making situations. These tools allow to setup a digital model of a real manufacturing system providing detailed information on its behaviour within several different scenarios, without intervening on the physical level. Simulation tools are able to take into account uncertainties that may characterise real systems, such as faults of machines, delays, scraps, etc. In this way, it is possible to evaluate the most relevant performance indicators for each problem and obtain valuable data that suggest potential reconfigurations or improvements to

enhance the system. However, although simulation models are a useful tool to verify given scenarios, if used alone they are not able to help in the research of the optimal (or near-optimal) solutions to a given problem.

On the other hand, mathematical programming models serve the aim of determining optimal (or near-optimal) solutions for decision variables by formally describing problems in terms of objective function and constraints to be satisfied. Nevertheless, the drawbacks of these models are related to their complexity and the difficulty to involve uncertainties in their formulation.

Therefore, it could be useful to combine the different features of the two approaches by defining an integrated conceptual framework able to exploit the feedbacks provided by the joint use of these diverse tools in order to enhance their performance.

In the recent literature, there is a growing number of studies in which optimisation and simulation tools have been employed together to solve complex problems, especially in different industrial environments: in most cases, one of the tools is used as first and the obtained results are then fed to the second tool [1-5]. In few cases,

© 2014 The Authors. Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Selection and peer-review under responsibility of the International Scientifi c Committee of the “3rd CIRP Global Web Conference” in the person of the Conference Chair Dr. Alessandra Caggiano.

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a third software tool is introduced to connect optimisation and simulation tools and realise their integration to deal with industrial case studies [6].

In this research paper, a novel approach integrating optimisation models and simulation techniques and tools through the automatic implementation of a closed loop in which information and data are mutually exchanged in an effective way is proposed. In practice, a valid integration of the optimisation and simulation models can help the search of a good solution characterized by a major robustness in terms of sensitivity to uncertainties.

2. Simulation-Optimisation integrated approach

The aim of this research paper is to define a novel approach where optimisation and simulation models are integrated and combined through the implementation of a feedback loop involving the mutual exchange of information, according to the scheme of Fig. 1.

Starting from an optimisation model, able to determine an optimal or suboptimal solution to a given problem, the approach proceeds with the employment of simulation tools, able to produce information concerning the system behaviour and reaction to different inputs.

In fact, the solution generated by (exact or heuristic) optimisation techniques is used as input of a simulation model with the aim to verify the actual feasibility and robustness of the found solution through the generation of different scenarios able to take into account several kinds of uncertainties which are typical of real systems.

The results of the simulation experiments, allowing an evaluation of the system performance, can support the detection of the current solution weaknesses and the identification of suitable modifications concerning the ingredients of the initial problem formulation.

The feedback loop is then realised going back to the optimisation phase with the new information generated by the simulation model. This information is used to improve the initial optimal solution, e.g. through heuristic techniques or changes in the problem formulation. This approach allows to connect the evaluation of the relevant system performance indicators to the decision making process on the optimal solution.

3. Modelling of Manufacturing Scheduling Problems

The novel approach presented in this research work is implemented with particular reference to the solution of manufacturing scheduling problems, that can be briefly described as follows.

Lots of units belonging to different part numbers, produced by various upstream processes, are delivered according to a specified arrival rate to a manufacturing cell. Here, parallel machines able to process all the part numbers according to different cycle times are available.

Fig. 1. Simulation-Optimisation integrated approach

A typical objective to be optimized in manufacturing scheduling problems is represented by the makespan, i.e. the maximum completion time considering the items to be processed. When a time window to perform the operations is given, a proxy of the makespan can be represented by the number of items in queue at the end of the time window.

In the following, a possible formulation of the introduced problem in terms of mathematical programming is described.

3.1. Formulation of the problem

The time window T is divided in N periods of equal length indexed by t=1...N. Then, the following formulation is adopted: j jt t t j t

z

¦ ¦

h I



s B

Min! (1) subject to: ( 1) jt j t jt jt

I

I





d



q

j

1... ;

J

t

1... ;

N

(2) j jt t j

p q

d

lB

¦

j

1... ;

J

t

1... ;

N

(3)

0

jN

I

j

1... ;

J

(4)

, ,

0

jt jt t

q I B t

j

1... ;

J

t

1... ;

N

(5) t

B



N

j

1... ;

J

t

1... ;

N

(6) The following notation has been adopted:

x J: set of part numbers to be processed;

x pj: average processing time for part number j;

x djt: arrival rate of part number j in period t;

x hj: queue holding cost for a unit of j in period t;

x qjt: units of part number j to be processed in period t;

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x st: cost associated to activating a machine in period t;

x A: JxN matrix; the single coefficient ajt is equal to 1

when units of a part number cannot be processed; 0 in all other intervals;

x Bt: number of machines to be assigned in period t;

x l: average process time per machine;

In the formulation, the assumed objective function (1) represents the total cost to be minimized, made up of two components: the costs associated to the maintenance of the queue on the buffer and the costs associated to the activation of the used machines.

Constraints (2) represent the relationships linking queues of units of a given part number from two consecutive periods. Constraints (3) express capacity constraints. Equations (4) express the fact that all the units of a part number j must be processed within the time window. The other inequalities derive from the physical meaning of the variables.

The model represents a modified version of the well-known CLSP (Capacitated Lot Sizing Problem). The latter model in fact can be effectively used to solve various logistic problems, as recently shown in [7-10].

The model presents J*N decisional variables (qjt), J*N

decisional variables (Ijt), N capacity constraints, J*N

balance constraints, for a total of 2*J*N decisional variables and N*(J+1) constraints.

The solution of the model provides the number of machines to be used at each period to minimize the total number of periods of activity of the available machines. It should be highlighted that constraints (4) assure that the makespan is equal to the time window duration.

4. Case study application

The CLSP model described in subsection 3.1 has been applied to a manufacturing scheduling case study (Fig. 2) where lots of units belonging to J=16 part numbers, produced by diverse upstream processes, are delivered to a manufacturing cell according to a specific arrival rate. The units are then processed in the manufacturing cell where 9 parallel machines able to process all the part numbers with different cycle times are available. The entire time window, T=10 hrs, has been divided into N=40 periods of 15 minutes each.

The resulting model is a linear problem with 16*40=

640 decisional variables (qjt), 16*40=640 decisional

variables (Ijt), 40 capacity constraints, 16*40=640

balance constraints, for a total of 2*16*40=1280 decisional variables and 40*(16+1)=680 constraints.

The model has been solved through the Excel solver. The solution provides the number of machines to be used in each period (see Fig. 3) in order to minimize the total number of periods of activity of the available machines and, at the same time, minimize the waiting time of the various part number units on the buffer.

Fig. 2. Scheme of the manufacturing scheduling problem

Fig. 3. CLSP optimal solution: no. of machines to use in each period

5. Discrete Event Simulation model

According to the proposed integrated approach, the output of the optimisation model can be fed into a simulation model to verify the actual feasibility and robustness of the found solution through the generation of different scenarios able to consider several kinds of uncertainties which are likely to occur in real systems.

To this aim, a Discrete Event Simulation (DES) software was employed to carry out the described performance analysis. DES allows to create and modify a digital model of a real manufacturing system: by properly setting the model parameters (e.g. processing times, arrival rates, failures, etc.) it is possible to perform stochastic simulations taking into account uncertainties that occur in real systems, such as machine faults, delays, etc. Therefore, a manufacturing system can be investigated in terms of the most relevant performance indicators (production flow, bottlenecks, productivity, work in progress, etc.) through the simulation of different “what if” scenarios, and the generation of analytical data and charts [12-17].

The model setup for the scheduling problem under study involved the creation of a Part Class and a corresponding Source for each part number (Fig. 4).

Every Source was connected to a unique common Buffer in charge of routing the units to all the active Machines according to a FIFO logic. Each machine was configured so as to be able process all the part numbers with different process cycle times. The number of active machines for each period t was set according to the solution generated by the optimisation model.

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The described DES model was implemented at first to verify the feasibility of the optimal solution found, and to evaluate its performance in the hypothesis of a deterministic scenario consistent with the assumptions made for the CLSP model. Therefore, arrival rates and processing times were modelled as constant parameters.

This model allowed to verify whether the optimal solution was feasible and capable to process the total number of part number units within the time window. During the simulation, the size of the buffer queue was monitored and plotted vs. time as shown on the chart of Fig. 5. It can be seen that, at the end of the simulation, there was no queue on the buffer, i.e. there were no parts waiting to be completed: the optimal solution was successfully verified in the deterministic scenario.

6. Solution robustness analysis: uncertain scenarios

The verification of the optimised solution in a deterministic model cannot be considered sufficient to assess its robustness. The next step, according to the integrated approach presented in this paper, is represented by the introduction of various sources of uncertainty in the model: the robustness of the optimised solution is then verified in different circumstances. In particular, three scenarios were considered, assuming the following variables as stochastic parameters:

1. Machine processing times; 2. Part number arrival rates;

3. Item arrival rates at manufacturing cell.

Some preliminary experimental results obtained for each considered scenario are described in the followings.

Fig. 4. DES model of the CLSP problem case study

Fig. 5. Size of the buffer queue during simulation time

6.1. Uncertainty of the processing times

In the initial model definition, the machine processing times were assumed to be constant and known. Actually, variations in the processing times, due to systematic (e.g. part geometry complexity, tricky fixturing, etc.) and/or unpredictable factors, could occur in a real system.

In order to take into account this aspect, processing times were modelled according to a Gamma distribution defined as follows: 1 ( ) ( ) x x e f x D D E

E

D

   * (7)

with a shape parameter Į=k and an inverse scale (or rate)

parameter ȕ=1

ș. To test different conditions, three

distinct values of ȕ, respectively equal to 0.25, 0.50 and 0.75, were used.

A total number of 10 simulations for each ȕ value were performed: in 100% of cases, the results showed that the solution was not affected by the process times variability, as the final queue size was always equal to zero and the makespan was not concerned.

6.2. Uncertainty of the arrival rates

In the second scenario, the hypothesis that arrival rates are known and constant for each item and in each period was removed. This condition assumed, in practice, that the upstream processes regularly provide input for the downstream process according to a deterministic schedule. However, faults in the upstream processes may occur and cause uncertainty in the dynamic arrival of units at the manufacturing cell.

To simulate this unpredictable condition, a “delayed arrival” event for randomly selected part numbers was introduced in the DES model. This event was simulated by supposing that, at a certain period, the expected lot of units of the selected part number was not delivered. By assuming an upstream process failure duration equal to a period, all the subsequent lot arrivals were shifted forward in time by one period (Fig. 6). To evaluate the performance of the CLSP optimal solution in the new model in this scenario, 10 distinct simulations were performed. In each of these simulations, 5 different items were randomly selected and delayed.

Fig. 6. Example of delayed arrival of selected part numbers: columns correspond to the time intervals, rows refer to the part numbers

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This perturbation affected the performance of the system in terms of makespan. As above mentioned, the number of items not yet processed at the end of the time window can be assumed as a proxy of the consequent increase of the makespan. In particular, in 8 cases the final queue at the end of the given time window, though very limited, was not zero, ranging from 1 to 17 units (Table 1). These results proved that the optimal solution provided by the CLSP model is not able to assure the target in terms of makespan. However, on the basis of information provided by the simulations, a simple heuristic to boost the robustness of the current solution could be defined. In practice, when the results of the simulations show that the final queue is higher than zero, a minimum number of extra machines to be activated in order to eliminate the final queue was determined. This procedure was performed by repeating the simulations with the same delayed part numbers, and adding one by one a new active machine in the last period. When the last period is saturated, the next required machine is added in the penultimate period, and so on.

Table 1 shows, for each test problem, the number of machines to be added such that the final queue vanishes. In practice, the number of machines to be added can be viewed as an extra cost to be paid in order to make the solution provided by the mathematical model more robust in terms of makespan.

6.3. Uncertainty of the amount of arriving lots

In the third scenario, it was supposed that the arrival rates of the units at the manufacturing cell could be variable. In particular, in order to simulate this cause of uncertainty, an additional lot arrival event was added for some randomly selected part numbers, thus increasing the number of total units for those part numbers.

For example, Fig. 7 shows how the arrival pattern of a given part number was modified according to this option: the red rectangle represents the added lot arrival.

Table 1. Final queue size and minimum number of additional machines required to eliminate the final queue in the different simulations.

Test problem Final Queue No. of Machines

1 1 1 2 9 3 3 0 - 4 1 1 5 6 2 6 4 1 7 4 1 8 6 1 9 0 - 10 17 4

As in the previous case, 10 test problems were carried out by randomly selecting different part numbers: each of these problems showed a diverse behaviour of the system and a different queue development during time.

A comparison between the various results was performed by evaluating the size of the final queue in each case. In none of the 10 cases the final queue size was equal to 0, and much higher values, ranging from 13 to 41 units, were measured.

According to the numerical results of the simulation, it came clear that the perturbation generated by the uncertain scenario involving the variability of the amount of lots arriving to the manufacturing cell had a higher impact on the CLSP solution robustness.

As in the previous case, the DES results were used to improve the solution found through the optimization model with the aim to enhance its robustness. The same heuristic solution improvement method applied in the previous case was adopted to boost the robustness of the current solution.

For each of the 10 simulations, the minimum number of extra machines to be activated so as to eliminate the final queue was determined. This procedure was performed by repeating all the 10 simulations and adding one by one a new active machine starting from the last period and going backward. The number of machines to be added for each simulation is reported in the second column of Table 2. The table shows that a higher number of machines is required to improve the robustness of the optimised solution in this scenario.

Fig. 7. Example of added lot arrival of part numbers (in red): columns correspond to the time intervals, rows refer to the part numbers Table 2. Final queue size and minimum number of additional machines required to eliminate the final queue in the different simulations.

Test problem Final Queue No. of Machines

1 35 7 2 16 4 3 14 4 4 13 3 5 30 7 6 21 5 7 33 7 8 41 12 9 18 4 10 24 6

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However, in this case an averagely higher ratio between effective performance improvement (expressed as number of additional units processed) and cost required to enhance the current optimal solution (expressed as number of additional machines to be activated) was measured.

7. Conclusions

In this research paper, a novel approach to solve manufacturing scheduling problems based on the integration of optimisation and simulation tools was presented. The aim is to implement a closed loop involving feedback from both optimisation and simulation models and combine the strengths of the two different tools so as to allow for a more robust decision making on the optimal problem solution.

In particular, the manufacturing scheduling problem optimisation model was formulated as an adaptation of the Capacitated Lot-Sizing Problem (CLSP). Then, a Discrete Event Simulation (DES) software was employed to verify the optimal solution and analyse its behaviour taking into account different kinds of uncertainties related to the processing times, the arrival rates and the amount of lots arriving to the manufacturing cell.

Some preliminary results provided through the application of the integrated approach to test problems appropriately generated show that DES can produce useful indications on the robustness of the optimal solution found through the CLSP model, and suggest potential improvement strategies.

Further developments on the current research may focus on performing a vast experimentation in order to achieve more general conclusions on the efficiency of the integrated approach.

Acknowledgements

This research work was carried out within the project FORGIARE (FORmazione Giovani Alla RicErca), co-funded by Compagnia di San Paolo, entitled “Digital Modelling, Simulation and Validation of Manufacturing Products, Processes and Systems” at the Department of Industrial Engineering, University of Naples Federico II, (2012-2016).

The Fraunhofer Joint Laboratory of Excellence on Advanced Production Technology (Fh-J_LEAPT) at the Department of Chemical, Materials and Industrial Production Engineering, University of Naples Federico II, is gratefully acknowledged for its support to this research activity.

References

[1] Bachelet, B., Yon, L., 2007. Model enhancement: Improving theoretical optimisation with simulation, Simulation Modelling Practice and Theory 15, p. 703-715

[2] Iassinovski, S., Artiba, A., Bachelet, V., Riane, F., 2003. Integration of simulation and optimisation for solving complex decision making problems, International Journal of Production Economics 85, p. 3-10

[3] Mebarki, N., Castagna, P., 2000. An approach based on Hotelling’s test for multicriteria stochastic simulation-optimisation, Simulation Practice and Theory 8, p. 341-355.

[4] Pierreval, H., Paris, J.L., 2003. From simulation optimisation to simulation configuration of systems, Simulation Modelling Practice and Theory 11, p. 5-19

[5] Bi, Z.M., Wang, L., 2012. Optimisation of machining processes from the perspective of energy consumption: A case study, Journal of Manufacturing Systems 31/4, p.420-428

[6] Syberfeldt, A., Karlsson, I., Amos, N., Svantesson, J., Almgren, T., 2013. A web-based platform for the simulation–optimisation of industrial problems, Computers & Industrial Engineering 64/4, p. 987-998

[7] Bruno, G., Genovese, A., Piccolo, C., 2014. The capacitated Lot Sizing model: A powerful tool for logistics decision making, International Journal of Production Economics, DOI: 10.1016/j.ijpe.2014.03.008.

[8] Bruno, G., Genovese, A., Sgalambro, A., 2012. An extension of the schedule optimisation problem at a public transit terminal to the multiple destinations case, Public Transport 3/3,p. 189-198. [9] Bruno, G., Genovese, A., 2010. A mathematical model for the

optimisation of the airport check-in service problem, Electronic Notes in Discrete Mathematics 36, p. 703-710.

[10]Bruno, G., Improta, G., Sgalambro, A., 2009. Models for the schedule optimisation problem at a public transit terminal, OR Spectrum 31/3, p. 465-481.

[11]Caggiano, A., Teti, R., 2013. Modelling, analysis and improvement of mass and small batch production through advanced simulation tools, 8th CIRP Conference on Intelligent Computation in Manufacturing Engineering - CIRP ICME’12, Ischia, Italy, 18-20 July, Procedia CIRP 12, p. 426-431.

[12]Caggiano, A., Teti, R., 2012. Improving the performance of a real manufacturing cell through advanced digital simulation, 14th Int. Conf. on Modern Information Technology in the Innovation Processes of Industrial Enterprises, MITIP 2012, Budapest, Hungary, 24-26 October, p. 267-279.

[13]Caggiano, A., Teti, R., 2012. Digital Manufacturing Cell Design for Performance Increase, 1st CIRP Global Web Conference on Interdisciplinary Research in Production Engineering - CIRPE 2012, 12-13 June, Procedia CIRP 2, p. 64-69.

[14]Caggiano, A., Teti, R., 2011. Digital Factory Simulation Tools for the Analysis of a Robotic Manufacturing Cell, 7th Int. Conf. on Digital Enterprise Technology - DET 2011, Athens, Greece, 28-30 September, ISBN 978-960-88104-2-6, p. 478-485.

[15]Caggiano, A., Teti, R., 2010. Simulation of a Robotic Manufacturing Cell for Digital Factory Concept Implementation, 7th CIRP Int. Conf. on Intelligent Computation in Manufacturing Engineering – CIRP ICME ‘10, Capri, Italy, 23-25 June, p. 76-79. [16]Caggiano, A., Keshari, A., Teti, R., 2009. Analysis and

Reconfiguration of a Manufacturing Cell through Discrete Event Simulation, 2nd IPROMS International Researchers Symposium, Ischia, 22-24 July, ISBN 978-88-95028-38-5, p. 175-180.

[17]Marzano A., K. Agyapong-Kodua, S. Ratchev. Virtual Ergonomics and Time Optimization of a Railway Coach Assembly Line. In Procedia CIRP 45th Conference on Manufacturing Systems, Volume 3, 2012, Pages 555-560.

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