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Prediction of variable technological operation times in production jobs scheduling
9th IFAC Conference on Manufacturing Modelling, Management and 9th IFAC Conference on Manufacturing Modelling, Management and Control 9th IFAC Conference on Manufacturing Modelling, Management and Control 9th IFAC Conference on Manufacturing Modelling, Management and Berlin, Germany, August 28-30, 2019 Control Berlin, Germany, August 28-30, 2019 Available online at www.sciencedirect.com 9th IFAC Conference on Manufacturing Modelling, Management and Control Berlin, Germany, August 28-30, 2019 Control Berlin, Germany, August 28-30, 2019 Berlin, Germany, August 28-30, 2019
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IFAC PapersOnLine 52-13 (2019) 1301–1306 Prediction Prediction of of variable variable technological technological operation operation times times Prediction of variable technological operation in production jobs scheduling Prediction in of variable technological operation times times production jobs scheduling Prediction in of variable technological operation times production jobs scheduling inŁukasz production jobs scheduling Sobaszek*, Arkadiusz Gola* inŁukasz production jobs scheduling Sobaszek*, Arkadiusz Gola*
Kozłowski** Łukasz Edward Sobaszek*, Arkadiusz Gola* Kozłowski** Łukasz Edward Sobaszek*, Arkadiusz Gola* Kozłowski** Edward Łukasz Edward Sobaszek*, Arkadiusz Gola* Kozłowski** *Faculty Lublin University *Faculty of of Mechanical Mechanical Engineering, Engineering, Lublin University of of Technology, Technology, Lublin, Lublin, Poland Poland Edward Kozłowski** (Tel: +48-81-538-42-76; e-mail: [email protected], a.gola.pollub.pl) *Faculty of Mechanical Engineering, Lublin University of Technology, Lublin, Poland (Tel: +48-81-538-42-76; e-mail: [email protected], a.gola.pollub.pl) *Faculty of Mechanical Engineering, Lublin University of Technology, Lublin, Poland **Faculty of Lublin University of Lublin, Poland (Tel: +48-81-538-42-76; e-mail: [email protected], a.gola.pollub.pl) **Faculty of Management, Management, Lublin University of Technology, Technology, Lublin, Poland *Faculty of Mechanical Engineering, Lublin University of Technology, Lublin, Poland (Tel: +48-81-538-42-76; e-mail: [email protected], a.gola.pollub.pl) (Tel: +48-81-538-46-19; e-mail: **Faculty of Management, Lublin University of Technology, Lublin, Poland (Tel: +48-81-538-46-19; e-mail: [email protected]) [email protected]) (Tel: +48-81-538-42-76; e-mail: [email protected], a.gola.pollub.pl) **Faculty of Management, Lublin University of Technology, Lublin, Poland +48-81-538-46-19; e-mail: [email protected]) **Faculty (Tel: of Management, Lublin University of Technology, Lublin, Poland (Tel: +48-81-538-46-19; e-mail: [email protected]) (Tel: +48-81-538-46-19; e-mail: [email protected]) Abstract: Abstract: This This paper paper presents presents aa methodology methodology for for determining determining the the variability variability of of technological technological operations operations times, which is suitable for predictive scheduling production tasks. The paper with the Abstract: This paper presents a methodology forof determining the variability of begins technological operations times, which is suitable for predictive scheduling of production tasks. The paper begins with outlining outlining the Abstract: This paper presents a methodology for determining the variability of technological operations problems related to production scheduling in real production systems and the process of robust job times, which is suitable for predictive scheduling ofdetermining production tasks. The and paper begins withofoutlining the problems related to production scheduling in real production systems the process robust job Abstract: This paper presents a methodology for the variability of technological operations times, whichInis the suitable for predictive scheduling of production tasks. The paper beginswith withthe outlining the scheduling. aa special attention was paid to associated variability problems related towork, production scheduling in real production systems and thebegins process ofoutlining robust job scheduling. Inis the work, special attention was paid to the the uncertainties uncertainties associated with variability times, which suitable for predictive scheduling of production tasks. The and paper withthe the problems related to production scheduling in real production systems the process of robust job of the times of technological operations of individual jobs. the proprietary prediction algorithm scheduling. In the towork, a special attention was paid to Subsequently, the uncertainties associated with the variability of the times of technological operations of individual jobs. Subsequently, the proprietary prediction algorithm problems related production scheduling in real production systems and the process of robust job scheduling. In the of work, a special attention was paidwas to the uncertainties associated with the variability for the variability technological operations times discussed, and results its were of the times of technological operations of individual jobs. Subsequently, thethe proprietary prediction algorithm forthe thetimes variability technological operations times was discussed, andthe the results of ofprediction its operation operation were scheduling. In the of work, a special attention was paid to Subsequently, the uncertainties associated with the variability of of technological operations of individual jobs. proprietary algorithm verified using historical production data. The work was completed with the analysis of the obtained for the variability of technological operations times was discussed, and the results of its operation were verified using historical production data. The work was completed with the analysis of the obtained of the times of technological operations of individual jobs. Subsequently, the proprietary prediction algorithm for the variability of technological operations times was discussed, and theproblem results of its operation were results, as well as outlining the of further in area. verified using historical production data. The work work wasdiscussed, completed with the analysis ofoperation the obtained results, asusing well historical as the the outlining the direction direction of times further work in the the analysed analysed area. for the variability of technological operations and theproblem results of itsof were verified production data. The workwas was completed with the analysis the obtained results, as well as the outlining the direction of further work in the analysed problem area. verified using historical production data. The work was completed with the analysis of the obtained results, as well as the outlining the direction of further work in the analysed problem area. Copyright © IFAC © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2019 2019 IFAC results, as well as the outlining the direction of further work in the analysed problem area. Copyright © 2019 IFAC Copyright 2019scheduling, IFAC Keywords:©robust robust scheduling, production planning, planning, operation Keywords: production operation times times uncertainty uncertainty Copyright © 2019 IFAC Keywords: robust scheduling, production planning, operation times uncertainty Keywords: robust scheduling, production planning, operation times uncertainty Keywords: robust scheduling, production planning, operation times uncertainty 1. 2. 1. INTRODUCTION INTRODUCTION 2. RELATED RELATED WORKS WORKS 1. INTRODUCTION 2. RELATED WORKS 1. INTRODUCTION RELATED WORKS Scheduling jobs Scheduling production production jobs is is aa widely-studied widely-studied problem problem 2.1 The general job2. problem and 1. INTRODUCTION 2.scheduling RELATED WORKS in the scientific literature, and the studies typically follow one 2.1 The general job scheduling problem and robust robust sche sche Scheduling production jobs is a widely-studied problem in the scientific literature,jobs and the studies typically follow one Scheduling production is a widely-studied problem 2.1 The general job scheduling problem and robust sche of the two distinct trends. The research in the field revolves in the scientific literature, and the studies typically follow one 2.1 The general job scheduling problem and robust sche of of the scientific two distinct trends.jobs Thethe in the field revolves Scheduling production isresearch a widely-studied problem in the literature, and studies typically follow one The general problem of is allocation around investigating effective task scheduling algorithms 2.1 The general job scheduling problem and robust sche of of the scientific two distinct trends. Thethe research in the field revolves The general problem of job job scheduling scheduling is the the allocation around investigating effective task scheduling algorithms in the literature, and studies typically follow one of theettwo distinct trends. The research in thetarget field revolves elements of the n-element set of jobs JJ between the elements The general problem of job scheduling is the allocation of (Kai al., 2015), on the other hand, it may efficient elements of the n-element set of jobs between the elements around investigating effective task scheduling algorithms (Kai al., 2015), on the other hand, it in may efficient The of theettwo distinct trends. The research thetarget field revolves general problem ofmachines job scheduling isavailable the allocation of around investigating effective task scheduling algorithms of the m-element set of M, i.e. machines elements of the n-element set of jobs J between the elements task under various which (Kai et al., 2015), on the otherconstraints/disruptions, hand, it may target efficient of thegeneral m-element set ofofmachines M, Ji.e. machines The problem job scheduling isavailable the allocation of task scheduling scheduling under various constraints/disruptions, which and elements of the n-element set of jobs between the elements around investigating effective task scheduling algorithms (Kai et al., 2015), on the other hand, it may target efficient on aa machine M job j is of theworkstations. m-element setPerforming of machines M,JJiiJi.e. available machines affect the quality of the constructed schedules (Umang, Erera on machine M is task scheduling under various constraints/disruptions, which and workstations. Performing job elements of the n-element set of jobs between the elements j affectscheduling ofon thethe constructed schedules (Umang, Erera of (Kai etthe al.,quality 2015), otherconstraints/disruptions, hand, it may target efficient the m-element set of machines M, i.e. available machines task under various which referred to as an operation. The fundamental of the onavailable afeature machine Mjjob is workstations. Performing jobM,Jii.e. and Bierlaire, 2014). For practical considerations, this approach referred to as an operation. The fundamental feature of the job affect the quality of the constructed schedules (Umang, Erera and of the m-element set of machines machines and Bierlaire, 2014). For practical considerations, this approach on a machine M task scheduling under various constraints/disruptions, which workstations. Performing joborder Ji of operations (deterj is affect themeans quality of the constructed schedules (Umang, Erera and scheduling process is defining the referred to as an operation. The fundamental feature of the job is by all suitable, as only following an in-depth analysis scheduling process is defining the of aoperations (deterand 2014). For constructed practical considerations, this approach machine Mjjob is workstations. Performing joborder Ji on is byBierlaire, all suitable, as only following an in-depth analysis referred to the as an operation. The fundamental feature of the affect themeans quality of the schedules (Umang, Erera and and Bierlaire, 2014). Foraccounting practical considerations, this approach mined technologist) and processing times. main scheduling process is defining order of operations (deterof an environment and for potential limitations of mined by by technologist) andthe processing times. The The main is by all means2014). suitable, as only following an in-depth analysis referred to the as an operation. The fundamental feature of the job of an environment and accounting for potential limitations of scheduling process is defining the order of operations (deterand Bierlaire, For practical considerations, this approach is byproduction all means suitable, as only following an in-depth analysis purpose of scheduling is the value mined technologist) andtheprocessing times. The(determain the system may facilitate the of aa scheduling purposeby ofthe scheduling is to to obtain obtain the optimal/minimal optimal/minimal of anproduction environment and accounting for potential limitations process is defining order of operations the system may facilitate theandevelopment development of of mined by the technologist) and processing times. The value main is by all means suitable, as only following in-depth analysis of an environment and accounting for potential limitations of of the objective function, which is usually the makespan, purpose of scheduling is to obtain the optimal/minimal value schedule that will exhibit the much-desired stability and of the objective function, which is usually the makespan, the production system may facilitate the development of a mined by the technologist) and processing times. The main schedule that will exhibit the much-desired stability and purpose of scheduling is to obtain the optimal/minimal value of an environment and accounting for potential limitations of the production system may facilitate the development of a of Othman and Rohmah, 2012; Sitek and given C max (Kaban, theas objective function, which is usually the makespan, robustness to occurring disturbances (Gola, 2019; Sitek, (Kaban, Othman and Rohmah, 2012; Sitek and given as C schedule that will exhibit the much-desired stability and purpose of scheduling is to obtain the optimal/minimal value max robustness to will occurring disturbances (Gola, 2019; Sitek, of the objective function, which is usually the makespan, the production system may facilitate the development of a Wikarek, schedule that exhibit the much-desired stability and 2018). (Kaban, Othman and Rohmah, 2012; Sitek and given as C max Wikarek and Nielsen, 2017). Wikarek, 2018). robustness to occurring disturbances (Gola, 2019; Sitek, of the objective function, which is usually the makespan, Wikarek and 2017). schedule that will exhibit the much-desired and given as Cmax (Kaban, Othman and Rohmah, 2012; Sitek and robustness toNielsen, occurring disturbances (Gola, stability 2019; Sitek, Wikarek, 2018). Wikarek andtoNielsen, 2017). (Kaban, Othman and Rohmah, 2012; Sitekbeing and given as C2018). max robustness occurring disturbances (Gola, times 2019;analysis Sitek, Wikarek, Scheduling production Wikarek and Nielsen, 2017). This paper proposed new method of operation Scheduling production tasks tasks may may make make an an impression impression of of being This paper proposed new method of operation times analysis Wikarek, 2018). Wikarek and Nielsen, 2017). aScheduling fairly problem. Nevertheless, scheduling subject production tasks may make an impressionis being and Firstly of all presented the general fairly simple simple problem. Nevertheless, scheduling isof This prediction. paper proposed new method of operation and prediction. Firstly all was was presentedtimes the analysis general ato Scheduling production tasks may limit makeand an impression ofsubject being This paper proposed newof method of operation times analysis numerous constraints, which prevent the use of a fairly simple problem. Nevertheless, scheduling is subject information about production scheduling and robust scheduling numerous constraints, which prevent the use of and prediction. all was presented the general ato Scheduling production tasks may limit makeand an impression being information aboutFirstly production scheduling and robust scheduling fairly simple problem. Nevertheless, scheduling isofsubject This paper proposed newof method of operation times analysis and prediction. Firstly of all was presented the general effective scheduling methods in a real production envito fairly numerous constraints, which in limit prevent the use of process. Secondly was described problem uncertainty of scheduling methods a and real production enviinformation aboutFirstly production scheduling and of robust scheduling aeffective simple problem. Nevertheless, scheduling is subject process. Secondly was described problem of uncertainty of to numerous constraints, which limit and prevent the use of and prediction. of all was presented the general information about production scheduling and robust scheduling ronment (Sobaszek, Gola and Świć, Therefore, from effective scheduling methods in a 2018). real prevent production envioperation times and methods proposed in the literature. In the ronment (Sobaszek, Gola and Świć, 2018). Therefore, from process. Secondly was described problem of uncertainty of to numerous constraints, which limit and the use of operation times and methods proposed in the literature. In the effective scheduling methods in a real production enviinformation about production scheduling and robust scheduling process. Secondly was described problem the of uncertainty of ronment the literature in the field of scheduling emerges an important (Sobaszek, Gola and Świć, 2018). Therefore, from main part of that paper was presented algorithm for the literature in the field of scheduling emerges an important operation times and methods proposed in the literature. In the effective scheduling methods in a real production envimain partSecondly of that paper was presented the algorithm for (Sobaszek, Gola and Świć, 2018). Therefore, from process. described problem ofliterature. uncertainty of ronment operation times andwas methods proposed in the In the trend – This type scheduling into the literature inscheduling. the field scheduling emerges an takes important prediction of times variability. In final part of trend – robust robust scheduling. This type of of scheduling takes into main parttimes of operation that paper was presented thethe algorithm for ronment (Sobaszek, Golaof and Świć, 2018). Therefore, from prediction of operation times variability. In the final part of the literature in the field of scheduling emerges an important operation and methods proposed in the literature. In the main part of that paper was presented the algorithm for account a number of limitations and disruptions in production trend – robust scheduling. This type of scheduling takes into the paper the verification of the proposed algorithm, results account a number of limitations and disruptions in production prediction of operation times variability. In the final part of the literature in the field of scheduling emerges an important the verification of proposedInthe algorithm, results – robust scheduling. This type of scheduling takes into mainpaper part the of operation that paper wasthe presented for prediction of times variability. thealgorithm final part of trend processes (Deepu, 2008). account a number of limitations and disruptions in production and conclusions was outlined. processes (Deepu, 2008). the the verification of the proposedInalgorithm, results – robust scheduling. This and typedisruptions of scheduling takes into and paper conclusions was outlined. account a number of limitations in production prediction of operation times variability. the final part of trend the paper the verification of the proposed algorithm, results processes (Deepu, of 2008). and conclusions was outlined. account a number limitations and disruptions in production processes 2008). the the verification of the proposed algorithm, results 2.2 Robust scheduling and paper conclusions was outlined. 2.2 Robust(Deepu, scheduling processes 2008). and conclusions was outlined. 2.2 Robust(Deepu, scheduling 2.2 Robust scheduling Robust scheduling of Robust scheduling of jobs jobs is is aa process process focused focused on on creating creating Robust scheduling aa2.2 schedule that accounts for the variability of production Robust scheduling of jobs for is athe process focused creating schedule that accounts variability of on production Robust scheduling of jobs is a process focused on creating system parameters (Hong, 1996). The purpose of building asystem schedule that accounts for the variability of on production parameters (Hong, 1996). The purpose of building Robust scheduling of jobs is a process focused creating asuch schedule that accounts for the variability of production schedules is to counteract uncertainty and nervousness. system parameters (Hong, 1996). The purpose of building such schedules is to counteract uncertainty and nervousness. asystem schedule that accounts for the variability of production parameters (Hong, 1996). The purpose of building such schedules is to (Hong, counteract uncertainty and nervousness. system parameters 1996). The purpose of building such schedules is to counteract uncertainty and nervousness. such schedules is to counteract uncertainty and nervousness. Copyright © 2019 IFAC 1318 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Copyright 2019 IFAC 1318Hosting by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic Copyright 2019 responsibility IFAC 1318Control. Copyright © 2019 IFAC 1318 10.1016/j.ifacol.2019.11.378 Copyright © 2019 IFAC 1318
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The subsequent sections of this paper present the problem of variable technological operations times and report on the development of an algorithm for the analysis of operation times, which can be applied in the predictive phase of robust scheduling of production tasks. 2.3 Uncertainty of operation times The processing time of technological operations depends on many factors. It is affected by e.g. poor condition of tools, indisposition of personnel, difficult-to-machine workpiece, etc. Experience teaches that the discrepancy between real processing times of technological operations and nominal times assumed by technologists are quite common (Al-Hinai and ElMekkawy, 2012).Such a change has a negative impact on the executed processes, mainly causing the production schedule to become outdated (Figure 1). Therefore, there emerges a distinct need to study the nature of the variability of technological operations times and to include them in the future production planning processes.
Fig. 1. Consequences of delaying technological operations processing times (* – delayed operations). One of the major trends in the analysis of processing times is the application of the distribution function. However, no specific subgroups can be classified in this field as researchers consider various typical distribution functions – from the uniform distribution to mixed distributions. It is, nevertheless, infrequent to approach job-shop problems with multiplemachines (Deepak et al., 2016), which are relatively common in the industrial conditions, and the majority of analytical works focus on single-machine or two-machine problems (Chung-Cheng, Kuo-Ching and Shih-Wei, 2014). Processing time uncertainty is also analysed by means of fuzzy logic. The body of literature on fuzzy numbers in robust scheduling in job-shop systems is quite extensive. In work (Gonzalez-Rodriguez et al., 2009), authors approach scheduling under uncertainty in a two-machine environment, executed by means of an improved version of a simple local search. The solution is tested for typical test scenarios. Another work (Kai et al., 2016) attempts to verify the efficiency of fuzzy-logic-based solutions with the bee colony approach. The fuzzy number theory finds implementation in other scheduling problems as well, however, the number of published works is significantly lower (Hamed, 2017). The two above-mentioned methods for solving scheduling problems under processing time uncertainty in job-shop systems are currently the most popular in such applications. The literature offers other alternatives to these methods; however, the other solutions are rather limited to individual
case scenarios. In several papers, the processing time uncertainty is considered deterministically, such as in (AlHinai and ElMekkawy, 2012), where the approach is based on the arithmetic mean of registered longest and shortest times. A similar approach is shown in (Shafia, Pourseyed and Jamili, 2011), where different scenarios of processing times are analysed. Another notable solution proposes limiting the considerations to shortening processing time by compression (Karimi-Nasab and Seyedhoseini, 2013). Although the majority of the papers above implement the distribution function, their authors fail to specify why they have selected the analysed functions and the distribution of random variables, i.e. processing times. Another problem is that the proposed solutions do not satisfactorily explore the necessity to identify the distribution and prediction of its future values. This paper proposed new method of operation times uncertainty analysis and prediction. Different methods proposed in the literature allow to analyse processing times uncertainty only when there is a possible of theoretical probability distribution fitting. In that case there is a problem with prediction, especially when historical data have untypical character. That stands a major disadvantage of existing methods. Comparatively with solutions proposed in the literature, approach described in this paper helps to predict processing time of operations even when the nature of the historical data not allow for an unambiguous prediction of the theoretical probability distribution. Moreover, proposed method helps to conduct preliminary identification of the probability distribution, which allow to determine not only one potential theoretical distribution but a group them. That approach allow to select the best solution according to the test of the equality of empirical and theoretical distributions. Mentioned elements are a major advantages of proposed method. 3. PROBLEM DESCRIPTION Let , F, P be a probabilistic space, and a continuous random variable T : 0, represent real times of technological operations. The distribution of a random variable T is derived from the density function f(t) and distribution function F(t), where t 0 . In order to approximate the nature of variability of the technological operation times, the distribution of the random variable T is estimated. After performing the point estimation, the hypotheses regarding matching the distribution to the empirical data are additionally verified. If f(t) is a function of density of a random variable T, then for any ta tb ,
ta , tb 0, the equality tb
f (s)ds P(ta T tb )
ta
is fulfilled (Figure 2).
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4.2 Prediction of operation times variability Proper analysis and implementation of the prediction process developed on data collected in the set of processing times Pij will help determine the predicted time Tij of operation j of job i, and consequently the variability Δtij of the operation time given by the dependence:
t ij T ij tˆ ij Fig. 2. An example of density function. In the process of predicting operation times, we also use a cumulative distribution function. For any value of t 0, we designate as t
F (t ) P(T t ) f ( s)ds
(2)
0
This relationship is shown graphically in Figure 3.
(4)
where tˆ ij is the nominal processing time of operation j of job i assumed by the technologist (time in the nominal schedule). In order to carry out the prediction process, we propose an algorithm for the prediction of the variability of technological operations times of scheduled jobs. It consists of individual stages the execution of which leads to obtaining the values in question. Stage 1 of the algorithm defines the operation number ofa specific job and loads the historical data from the set of processing times Pij. Stage 2 concerns the preparation of data for further analyses. The data are sorted according to the relationship tij(1) < tij(2) <…< tij(n) and are subsequently subjected to preliminary statistical analysis (determination of minimum, maximum, average, range and quartiles values) and data filtering (removal of outliers).
Fig. 3. An example of cumulative distribution function. Both the density function f(t) and the cumulative distribution function F(t) of the random variable T are applicable in the prediction of technological operations times variability. Given the probability distribution of the time T of a given technological operation, one can estimate the probability of future operation time. Thus obtained data will be further implemented to create a robust schedule.
Stage 3 of the proposed algorithm consists in initial identification of the hypothetical probability distribution of the technological operation times. for this purpose, skewness γ1 and kurtosis γ2 are determined. These two commonly used parameters belong to the group of measures describing the shape of distribution. They are determined on the basis of empirical data by determining moments μk of the appropriate order:
1
3 ( 2
4. PROPOSED METHODS
k
3 )2
;
1 n ij (ts ) k ; n s 1
2
4 3 ( 2 ) 2
1 n ij ts n s 1
(5)
4.1 Data source
where:
No investigation of the nature of technological operation times variability could be even approximately valid without referencing an appropriate source of historical data. The implementation of the algorithm proposed in the work should be preceded by collecting operation time data, which will be applied in the scheduling process.
Having established the values of 1 and 2 , one can predetermine the shape of the distribution. It indicates the position of an empirical probability distribution between typical parametric distri-butions (Figure 4).
(6)
The collected data should be saved in an appropriate form so that it could be used at a later stage of predicting variability. Information about historical processing times should, therefore, be recorded individually for each operation j of the job i in the set Pij:
Pij [t1ij , t 2ij ,...,t nij ]
(3)
where tijk, 1 ≤ k ≤ n is the k-th real time of operation j and job i. Fig. 4. Cullen-Frey graph for preliminary identification of the probability distribution (own work based on (Chi, 2018)). 1320
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Stage 4 of the algorithm is the statistical inference. This process consists in the estimation of the distribution parameters (distribution matching) and the procedures for testing the compatibility of distributions.
If the working hypothesis is rejected (inequality of empirical and theoretical distributions), the parameter estimation procedure and compliance testing are performed for a different type of distribution (indicated by the Cullen-Frey graph).
The preliminary selection of the distribution enables only to make assumptions on its possible course offering rough data about it. Therefore, it is necessary to perform point estimation. Point estimators are characteristics obtained from a random sample, around which real population parameters will be located (parameters of the distribution in question).
Stage 5 of the algorithm is executed depending on the results of the statistical inference. If the data exhibits an atypical distribution then the interval-linear nonparametric estimation of the cumulative distribution function is performed. The results from the estimation are then employed to predict the time variability. Let Fn(t), t 0 be an empirical distributor with discontinuities at points k t(1) < t(1) < … < t(k), k n . The interval~ linear distribution function Fk (t ) is determined from (10)
The next step is the implementation of the test of the equality of empirical and theoretical distributions, which in the analysed case is the non-parametric Kolmogorov–Smirnov test. The advantage of the KS test is that it is suitable even for very small samples (StatSoft, 2013). The procedure of the Kolmogorov-Smirnov test includes the verification of the hypothesis concerning the compatibility of the empirical distribution Fn(t) from the n-element sample with the cumulative distribution F(t) for the selected theoretical distribution. At the level of significance 0,1 , we set a null hypothesis:
H 0 : Fn (t ) F (t )
0, Fn t1 t , 2 t 1 ~ Fk (t ) g t , 1 Fn t k 1 Fn t k 1 2 1,
(7)
In order to verify the working hypothesis, we determine the biggest difference between the cumulative distributors, as in Fig. 5:
Dn max Fn s F s s0,
g t Fn t i 1
(8)
where: against the alternative hypothesis:
H1 : Fn (t ) F (t )
t 0,
0 t t 1 , ti t t i 1 ,
1
Fn ti Fn t i 1 2
1 i k 1,
(10)
t t k , t k t 2tk t k 1 , t k 1 t k t 2t k t k 1 ,
Fn ti1 Fn ti1 2
t t i t i 1 t i
The interval-linear cumulative distribution function is shown in Fig. 6 as a dotted broken line marked with a navy blue colour.
(9)
Fig. 6. Interval-linear cumulative distribution function. Fig. 5. Difference between empirical and hypothetical cumulative distribution function.
Based on the designated distribution function, the values of future operation times can be predicted.
Next, based on the tabulated data for n observations and the assumed level of significance α, we determine the critical value d n 1 for statistics Dn .
The execution of the algorithm in question produces the predicted operations times t ij , which are subsequently employed to determine the variability of the analysed operation time t ij .
Given that Dn d n 1 , then at the level of significance α, there are no grounds to reject the hypothesis H0, and therefore the distribution of operation times is assumed to be equal to the theoretical distribution F(t), which is subsequently implemented in the prediction of future times of given technological operations. Additionally, for a given probability, the range of execution values of operation j for job i is determined.
5. EXPERIMENTAL RESULTS 5.1 Verification conditions The application of the algorithm for predicting the variability of technological operation times is presented below along
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with its verification, which was carried out on actual production data. The analysis includes data on the processing of machining operations with a length of 0.17 h (10 minutes) and 0.08 h (5 minutes). In the case of an operation time of 0.17 h, the data set P11 consisted of 26 observations, while in the case of operation times of 0.08 h, the number of elements of the set P12 was 49 observations. The algorithm was implemented in the RStudio environment, which enabled its execution and handling the obtained results in an accessible form. 5.2 Implementation of the proposed algorithm The first analysis considered the variability of the operation times of 0.17 h. For historical data from the set P11 basic statistics were determined, the values of which are given in Table 1. Table 1. Basic statistics of P11 Min.
1’st quantile
Median
Mean
3’rd quantile
Max.
0.17
0.25
0.33
0.3838
0.50
0.83
Fig. 8. Cumulative distribution functions. For the gamma distribution with the parameters k = 5. 67832, λ = 14. 8374 (Fig. 9), the quantiles of the order of 0.1 and 0.9 were determined, which are respectively q0.1 = 0.197 and q0.9 = 0.598, and therefore:
P q0.1 T 11 q0.9 0.8
(11)
Which means that the 0.8 probability of the variability of operation 11 satisfies the inequality:
q0.9 0.17 t ij q0.1 0.17
(12)
In the next step, the empirical density function was determined (Fig. 7), which was followed by the initial identification of the distribution by means of the Cullen-Frey graph.
Fig. 9. Inference from the selected probability distribution. For operations with a length of pt12 = 0.08 h, the variability analysis process was performed in analogically. The nature of the data, however, did not allow for an unambiguous prediction of the theoretical probability distribution, which could enable describing the process of time variability of the analysed operations. Therefore, in the case of the set P12, the prediction process was performed by means of a nonparametric interval-linear CDF estimation (Fig. 10).
Fig. 7. Initial identification of the distribution. Based on the prepared graphs, a gamma distribution was selected as the theoretical probability distribution. The following distribution parameters were determined: shape parameter k = 5.67832 and scale parameter: λ = 14.8374. Next, we verified the equality of this distribution (Fig. 8) with the empirical distribution by means of the K-S test, which confirmed the compatibility of distributions at a significance level of α = 0.97.
Fig. 10. Interval-linear cumulative distribution function – set P12 1322
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As a result of the performed estimation, it was determined that P T 12 0.27 0.8 : which subsequently leads to the conclusion that the variability of the operation times are t12 0.19 h.
Thus determined values should be implemented into the production schedule in order to increase its robustness and obtain a scheduling corresponding to the real conditions of production. 6. CONCLUSIONS The method for predicting the variability of operation times presented in this work corresponds with a current trend of task scheduling under uncertainty. Acquiring real times of technological operations enables developing schedules corresponding with the reality and setting realistic production deadlines. The verification presented in the article showed that the variability of operation times could be examined and determined from historical data. Further research will include constructing schedules based on the results from the algorithm, as well as the verification of robust schedules by means of the simulation tools. Moreover, there is a need to develop solutions which help take into account another uncertainty factors of production process – e.g. machines and robots failures, appearance of new orders and tasks, materials shortage, etc. The project/research was financed in the framework of the project Lublin University of Technology-Regional Excellence Initiative, funded by the Polish Ministry of Science and Higher Education (contract no. 030/RID/2018/19). REFERENCES Kai, Z. G., Ponnuthurai, Na. S., Tay, J. Ch., Chin, S. Ch., Tian, X. C., and Qan, K. P. (2015). A two-stage artificial bee colony algorithm scheduling flexible job-shop scheduling problem with new job insertion. Expert Systems with Applications, 42(21), pp. 7652–7663. Umang, N., Erera, A. L., Bierlaire, M. (2014). The robust single machine scheduling problem with uncertain release and processing times. Cornell University. Gola A. (2019). Reliability analysis of reconfigurable manufacturing system structures using computer simulation methods. Eksploatacja i Niezawodnosc – Maintenance and Reliability, 21(1), 2019, pp. 90–102. doi: http://dx.doi.org/10.17531/ein.2019.1.11. Unpublished. Sitek, P., Wikarek, J., Nielsen, P. (2017). A constraintdriven approach to food supply chain management. Industrial Management & Data Systems, Vol. 117, Issue: 9, pp. 2115–2138. doi: https://doi.org/10.1108/IMDS-102016-0465 Kaban, A. K., Othman, Z., Rohmah, D. S. (2012). Comparison of dispatching rules in job-shop scheduling problem using simulation: a case study. International Journal of Simulation Modelling, 11(3), pp. 129–140. Sitek, P., Wikarek, J. (2018). A multi-level approach to ubiquitous modeling and solving constraints in combinatorial optimization problems in production and distribution.
Applied Intelligence, Vol. 48, Issue 5, pp. 1344–1367, doi: https://doi.org/10.1007/s10489-017-1107-9 Deepu, P. (2008). Robust schedules and disruption management for job shops. Bozeman, Montana. Hong, G. (1996). Bulding robust schedules using temporal protection – an emipirical study of constraint based scheduling under machine failure uncertainty. Toronto, Ontario. Al-Hinai, N., ElMekkawy, T. Y. (2012). Solving the flexible job shop scheduling problem with uniform processing time uncertainty. International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, 6(4), pp. 848–853. Deepak, K., Yi, M., Gang, C., Mengjie, Z. (2016). Dynamic job shop scheduling under uncertainty using genetic programming. Intelligent and Evolutionary Systems, 8, pp. 195–210. Chung-Cheng, L., Kuo-Ching, Y., Shih-Wei, L. (2014). Robust single machine scheduling for minimizing total flow time in the presence of uncertain processing times. Computers & Industrial Engineering, 74, pp. 102–110. Gonzalez-Rodriguez, I., Vela, C. R., Puente, J., HernandezArauzo, A. (2009). Improved local search for job shop scheduling with uncertain durations. Proceedings of the Nineteenth International Conference on Automated Planning and Scheduling, Thessaloniki, Greece, 19–23 September 2009, pp. 154–161. Kai, Z. G., Ponnuthurai, N. S., Quan, K. P., Tay, J. C., Chin, S. C., Tian, X. C. (2016). An improved artificial bee colony algorithm for flexible job-shop scheduling problem with fuzzy processing time. Expert Systems With Applications, 65, pp. 52–67. Hamed, A. (2017). Apply fuzzy learning effect with fuzzy processing times for single machine scheduling problems. Journal of Manufacturing Systems, 42, pp. 244–261. Al-Hinai, N., ElMekkawy, T.Y. (2012). Solving the flexible job shop scheduling problem with uniform processing time uncertainty. International Journal of Mechanical, Aero-space, Industrial, Mechatronic and Manufacturing Engineering, 6(4), pp. 848–853. Shafia, M. A., Pourseyed, A. M., Jamili, A. (2011). A new mathematical model for the job shop scheduling problem with uncertain processing times. International Journal of Industrial Engineering Computations, 2, 295–306. Karimi-Nasab, M., Seyedhoseini, S.M. (2013). Multi-level lot sizing and job shop scheduling with compressible process times: A cutting plane approach. European Journal of Operational Research, 231, 598–616. Chi, Y. (2018). R tutorial – an R introduction to statistics [Online], Available at: http://www.r-tutor.com (Accessed: 15 September 2018). StatSoft, Inc. (2013). Electronic Statistic Textbook [Online], Available at: http://www.statsoft.pl/textbook/stathome.html (Accessed: 4 December 2018). Sobaszek, Ł., Gola, A., Świć, A. (2018). Preditive scheduling as a part of intelligent job scheduling system. Advances in Intelligent Systems and Computing, Vol. 637, pp. 358–367. doi: 10.1007/978-3-319-64465-3_35
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IFAC PapersOnLine 52-13 (2019) 1301–1306 Prediction Prediction of of variable variable technological technological operation operation times times Prediction of variable technological operation in production jobs scheduling Prediction in of variable technological operation times times production jobs scheduling Prediction in of variable technological operation times production jobs scheduling inŁukasz production jobs scheduling Sobaszek*, Arkadiusz Gola* inŁukasz production jobs scheduling Sobaszek*, Arkadiusz Gola*
Kozłowski** Łukasz Edward Sobaszek*, Arkadiusz Gola* Kozłowski** Łukasz Edward Sobaszek*, Arkadiusz Gola* Kozłowski** Edward Łukasz Edward Sobaszek*, Arkadiusz Gola* Kozłowski** *Faculty Lublin University *Faculty of of Mechanical Mechanical Engineering, Engineering, Lublin University of of Technology, Technology, Lublin, Lublin, Poland Poland Edward Kozłowski** (Tel: +48-81-538-42-76; e-mail: [email protected], a.gola.pollub.pl) *Faculty of Mechanical Engineering, Lublin University of Technology, Lublin, Poland (Tel: +48-81-538-42-76; e-mail: [email protected], a.gola.pollub.pl) *Faculty of Mechanical Engineering, Lublin University of Technology, Lublin, Poland **Faculty of Lublin University of Lublin, Poland (Tel: +48-81-538-42-76; e-mail: [email protected], a.gola.pollub.pl) **Faculty of Management, Management, Lublin University of Technology, Technology, Lublin, Poland *Faculty of Mechanical Engineering, Lublin University of Technology, Lublin, Poland (Tel: +48-81-538-42-76; e-mail: [email protected], a.gola.pollub.pl) (Tel: +48-81-538-46-19; e-mail: **Faculty of Management, Lublin University of Technology, Lublin, Poland (Tel: +48-81-538-46-19; e-mail: [email protected]) [email protected]) (Tel: +48-81-538-42-76; e-mail: [email protected], a.gola.pollub.pl) **Faculty of Management, Lublin University of Technology, Lublin, Poland +48-81-538-46-19; e-mail: [email protected]) **Faculty (Tel: of Management, Lublin University of Technology, Lublin, Poland (Tel: +48-81-538-46-19; e-mail: [email protected]) (Tel: +48-81-538-46-19; e-mail: [email protected]) Abstract: Abstract: This This paper paper presents presents aa methodology methodology for for determining determining the the variability variability of of technological technological operations operations times, which is suitable for predictive scheduling production tasks. The paper with the Abstract: This paper presents a methodology forof determining the variability of begins technological operations times, which is suitable for predictive scheduling of production tasks. The paper begins with outlining outlining the Abstract: This paper presents a methodology for determining the variability of technological operations problems related to production scheduling in real production systems and the process of robust job times, which is suitable for predictive scheduling ofdetermining production tasks. The and paper begins withofoutlining the problems related to production scheduling in real production systems the process robust job Abstract: This paper presents a methodology for the variability of technological operations times, whichInis the suitable for predictive scheduling of production tasks. The paper beginswith withthe outlining the scheduling. aa special attention was paid to associated variability problems related towork, production scheduling in real production systems and thebegins process ofoutlining robust job scheduling. Inis the work, special attention was paid to the the uncertainties uncertainties associated with variability times, which suitable for predictive scheduling of production tasks. The and paper withthe the problems related to production scheduling in real production systems the process of robust job of the times of technological operations of individual jobs. the proprietary prediction algorithm scheduling. In the towork, a special attention was paid to Subsequently, the uncertainties associated with the variability of the times of technological operations of individual jobs. Subsequently, the proprietary prediction algorithm problems related production scheduling in real production systems and the process of robust job scheduling. In the of work, a special attention was paidwas to the uncertainties associated with the variability for the variability technological operations times discussed, and results its were of the times of technological operations of individual jobs. Subsequently, thethe proprietary prediction algorithm forthe thetimes variability technological operations times was discussed, andthe the results of ofprediction its operation operation were scheduling. In the of work, a special attention was paid to Subsequently, the uncertainties associated with the variability of of technological operations of individual jobs. proprietary algorithm verified using historical production data. The work was completed with the analysis of the obtained for the variability of technological operations times was discussed, and the results of its operation were verified using historical production data. The work was completed with the analysis of the obtained of the times of technological operations of individual jobs. Subsequently, the proprietary prediction algorithm for the variability of technological operations times was discussed, and theproblem results of its operation were results, as well as outlining the of further in area. verified using historical production data. The work work wasdiscussed, completed with the analysis ofoperation the obtained results, asusing well historical as the the outlining the direction direction of times further work in the the analysed analysed area. for the variability of technological operations and theproblem results of itsof were verified production data. The workwas was completed with the analysis the obtained results, as well as the outlining the direction of further work in the analysed problem area. verified using historical production data. The work was completed with the analysis of the obtained results, as well as the outlining the direction of further work in the analysed problem area. Copyright © IFAC © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2019 2019 IFAC results, as well as the outlining the direction of further work in the analysed problem area. Copyright © 2019 IFAC Copyright 2019scheduling, IFAC Keywords:©robust robust scheduling, production planning, planning, operation Keywords: production operation times times uncertainty uncertainty Copyright © 2019 IFAC Keywords: robust scheduling, production planning, operation times uncertainty Keywords: robust scheduling, production planning, operation times uncertainty Keywords: robust scheduling, production planning, operation times uncertainty 1. 2. 1. INTRODUCTION INTRODUCTION 2. RELATED RELATED WORKS WORKS 1. INTRODUCTION 2. RELATED WORKS 1. INTRODUCTION RELATED WORKS Scheduling jobs Scheduling production production jobs is is aa widely-studied widely-studied problem problem 2.1 The general job2. problem and 1. INTRODUCTION 2.scheduling RELATED WORKS in the scientific literature, and the studies typically follow one 2.1 The general job scheduling problem and robust robust sche sche Scheduling production jobs is a widely-studied problem in the scientific literature,jobs and the studies typically follow one Scheduling production is a widely-studied problem 2.1 The general job scheduling problem and robust sche of the two distinct trends. The research in the field revolves in the scientific literature, and the studies typically follow one 2.1 The general job scheduling problem and robust sche of of the scientific two distinct trends.jobs Thethe in the field revolves Scheduling production isresearch a widely-studied problem in the literature, and studies typically follow one The general problem of is allocation around investigating effective task scheduling algorithms 2.1 The general job scheduling problem and robust sche of of the scientific two distinct trends. Thethe research in the field revolves The general problem of job job scheduling scheduling is the the allocation around investigating effective task scheduling algorithms in the literature, and studies typically follow one of theettwo distinct trends. The research in thetarget field revolves elements of the n-element set of jobs JJ between the elements The general problem of job scheduling is the allocation of (Kai al., 2015), on the other hand, it may efficient elements of the n-element set of jobs between the elements around investigating effective task scheduling algorithms (Kai al., 2015), on the other hand, it in may efficient The of theettwo distinct trends. The research thetarget field revolves general problem ofmachines job scheduling isavailable the allocation of around investigating effective task scheduling algorithms of the m-element set of M, i.e. machines elements of the n-element set of jobs J between the elements task under various which (Kai et al., 2015), on the otherconstraints/disruptions, hand, it may target efficient of thegeneral m-element set ofofmachines M, Ji.e. machines The problem job scheduling isavailable the allocation of task scheduling scheduling under various constraints/disruptions, which and elements of the n-element set of jobs between the elements around investigating effective task scheduling algorithms (Kai et al., 2015), on the other hand, it may target efficient on aa machine M job j is of theworkstations. m-element setPerforming of machines M,JJiiJi.e. available machines affect the quality of the constructed schedules (Umang, Erera on machine M is task scheduling under various constraints/disruptions, which and workstations. Performing job elements of the n-element set of jobs between the elements j affectscheduling ofon thethe constructed schedules (Umang, Erera of (Kai etthe al.,quality 2015), otherconstraints/disruptions, hand, it may target efficient the m-element set of machines M, i.e. available machines task under various which referred to as an operation. The fundamental of the onavailable afeature machine Mjjob is workstations. Performing jobM,Jii.e. and Bierlaire, 2014). For practical considerations, this approach referred to as an operation. The fundamental feature of the job affect the quality of the constructed schedules (Umang, Erera and of the m-element set of machines machines and Bierlaire, 2014). For practical considerations, this approach on a machine M task scheduling under various constraints/disruptions, which workstations. Performing joborder Ji of operations (deterj is affect themeans quality of the constructed schedules (Umang, Erera and scheduling process is defining the referred to as an operation. The fundamental feature of the job is by all suitable, as only following an in-depth analysis scheduling process is defining the of aoperations (deterand 2014). For constructed practical considerations, this approach machine Mjjob is workstations. Performing joborder Ji on is byBierlaire, all suitable, as only following an in-depth analysis referred to the as an operation. The fundamental feature of the affect themeans quality of the schedules (Umang, Erera and and Bierlaire, 2014). Foraccounting practical considerations, this approach mined technologist) and processing times. main scheduling process is defining order of operations (deterof an environment and for potential limitations of mined by by technologist) andthe processing times. The The main is by all means2014). suitable, as only following an in-depth analysis referred to the as an operation. The fundamental feature of the job of an environment and accounting for potential limitations of scheduling process is defining the order of operations (deterand Bierlaire, For practical considerations, this approach is byproduction all means suitable, as only following an in-depth analysis purpose of scheduling is the value mined technologist) andtheprocessing times. The(determain the system may facilitate the of aa scheduling purposeby ofthe scheduling is to to obtain obtain the optimal/minimal optimal/minimal of anproduction environment and accounting for potential limitations process is defining order of operations the system may facilitate theandevelopment development of of mined by the technologist) and processing times. The value main is by all means suitable, as only following in-depth analysis of an environment and accounting for potential limitations of of the objective function, which is usually the makespan, purpose of scheduling is to obtain the optimal/minimal value schedule that will exhibit the much-desired stability and of the objective function, which is usually the makespan, the production system may facilitate the development of a mined by the technologist) and processing times. The main schedule that will exhibit the much-desired stability and purpose of scheduling is to obtain the optimal/minimal value of an environment and accounting for potential limitations of the production system may facilitate the development of a of Othman and Rohmah, 2012; Sitek and given C max (Kaban, theas objective function, which is usually the makespan, robustness to occurring disturbances (Gola, 2019; Sitek, (Kaban, Othman and Rohmah, 2012; Sitek and given as C schedule that will exhibit the much-desired stability and purpose of scheduling is to obtain the optimal/minimal value max robustness to will occurring disturbances (Gola, 2019; Sitek, of the objective function, which is usually the makespan, the production system may facilitate the development of a Wikarek, schedule that exhibit the much-desired stability and 2018). (Kaban, Othman and Rohmah, 2012; Sitek and given as C max Wikarek and Nielsen, 2017). Wikarek, 2018). robustness to occurring disturbances (Gola, 2019; Sitek, of the objective function, which is usually the makespan, Wikarek and 2017). schedule that will exhibit the much-desired and given as Cmax (Kaban, Othman and Rohmah, 2012; Sitek and robustness toNielsen, occurring disturbances (Gola, stability 2019; Sitek, Wikarek, 2018). Wikarek andtoNielsen, 2017). (Kaban, Othman and Rohmah, 2012; Sitekbeing and given as C2018). max robustness occurring disturbances (Gola, times 2019;analysis Sitek, Wikarek, Scheduling production Wikarek and Nielsen, 2017). This paper proposed new method of operation Scheduling production tasks tasks may may make make an an impression impression of of being This paper proposed new method of operation times analysis Wikarek, 2018). Wikarek and Nielsen, 2017). aScheduling fairly problem. Nevertheless, scheduling subject production tasks may make an impressionis being and Firstly of all presented the general fairly simple simple problem. Nevertheless, scheduling isof This prediction. paper proposed new method of operation and prediction. Firstly all was was presentedtimes the analysis general ato Scheduling production tasks may limit makeand an impression ofsubject being This paper proposed newof method of operation times analysis numerous constraints, which prevent the use of a fairly simple problem. Nevertheless, scheduling is subject information about production scheduling and robust scheduling numerous constraints, which prevent the use of and prediction. all was presented the general ato Scheduling production tasks may limit makeand an impression being information aboutFirstly production scheduling and robust scheduling fairly simple problem. Nevertheless, scheduling isofsubject This paper proposed newof method of operation times analysis and prediction. Firstly of all was presented the general effective scheduling methods in a real production envito fairly numerous constraints, which in limit prevent the use of process. Secondly was described problem uncertainty of scheduling methods a and real production enviinformation aboutFirstly production scheduling and of robust scheduling aeffective simple problem. Nevertheless, scheduling is subject process. Secondly was described problem of uncertainty of to numerous constraints, which limit and prevent the use of and prediction. of all was presented the general information about production scheduling and robust scheduling ronment (Sobaszek, Gola and Świć, Therefore, from effective scheduling methods in a 2018). real prevent production envioperation times and methods proposed in the literature. In the ronment (Sobaszek, Gola and Świć, 2018). Therefore, from process. Secondly was described problem of uncertainty of to numerous constraints, which limit and the use of operation times and methods proposed in the literature. In the effective scheduling methods in a real production enviinformation about production scheduling and robust scheduling process. Secondly was described problem the of uncertainty of ronment the literature in the field of scheduling emerges an important (Sobaszek, Gola and Świć, 2018). Therefore, from main part of that paper was presented algorithm for the literature in the field of scheduling emerges an important operation times and methods proposed in the literature. In the effective scheduling methods in a real production envimain partSecondly of that paper was presented the algorithm for (Sobaszek, Gola and Świć, 2018). Therefore, from process. described problem ofliterature. uncertainty of ronment operation times andwas methods proposed in the In the trend – This type scheduling into the literature inscheduling. the field scheduling emerges an takes important prediction of times variability. In final part of trend – robust robust scheduling. This type of of scheduling takes into main parttimes of operation that paper was presented thethe algorithm for ronment (Sobaszek, Golaof and Świć, 2018). Therefore, from prediction of operation times variability. In the final part of the literature in the field of scheduling emerges an important operation and methods proposed in the literature. In the main part of that paper was presented the algorithm for account a number of limitations and disruptions in production trend – robust scheduling. This type of scheduling takes into the paper the verification of the proposed algorithm, results account a number of limitations and disruptions in production prediction of operation times variability. In the final part of the literature in the field of scheduling emerges an important the verification of proposedInthe algorithm, results – robust scheduling. This type of scheduling takes into mainpaper part the of operation that paper wasthe presented for prediction of times variability. thealgorithm final part of trend processes (Deepu, 2008). account a number of limitations and disruptions in production and conclusions was outlined. processes (Deepu, 2008). the the verification of the proposedInalgorithm, results – robust scheduling. This and typedisruptions of scheduling takes into and paper conclusions was outlined. account a number of limitations in production prediction of operation times variability. the final part of trend the paper the verification of the proposed algorithm, results processes (Deepu, of 2008). and conclusions was outlined. account a number limitations and disruptions in production processes 2008). the the verification of the proposed algorithm, results 2.2 Robust scheduling and paper conclusions was outlined. 2.2 Robust(Deepu, scheduling processes 2008). and conclusions was outlined. 2.2 Robust(Deepu, scheduling 2.2 Robust scheduling Robust scheduling of Robust scheduling of jobs jobs is is aa process process focused focused on on creating creating Robust scheduling aa2.2 schedule that accounts for the variability of production Robust scheduling of jobs for is athe process focused creating schedule that accounts variability of on production Robust scheduling of jobs is a process focused on creating system parameters (Hong, 1996). The purpose of building asystem schedule that accounts for the variability of on production parameters (Hong, 1996). The purpose of building Robust scheduling of jobs is a process focused creating asuch schedule that accounts for the variability of production schedules is to counteract uncertainty and nervousness. system parameters (Hong, 1996). The purpose of building such schedules is to counteract uncertainty and nervousness. asystem schedule that accounts for the variability of production parameters (Hong, 1996). The purpose of building such schedules is to (Hong, counteract uncertainty and nervousness. system parameters 1996). The purpose of building such schedules is to counteract uncertainty and nervousness. such schedules is to counteract uncertainty and nervousness. Copyright © 2019 IFAC 1318 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Copyright 2019 IFAC 1318Hosting by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic Copyright 2019 responsibility IFAC 1318Control. Copyright © 2019 IFAC 1318 10.1016/j.ifacol.2019.11.378 Copyright © 2019 IFAC 1318
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The subsequent sections of this paper present the problem of variable technological operations times and report on the development of an algorithm for the analysis of operation times, which can be applied in the predictive phase of robust scheduling of production tasks. 2.3 Uncertainty of operation times The processing time of technological operations depends on many factors. It is affected by e.g. poor condition of tools, indisposition of personnel, difficult-to-machine workpiece, etc. Experience teaches that the discrepancy between real processing times of technological operations and nominal times assumed by technologists are quite common (Al-Hinai and ElMekkawy, 2012).Such a change has a negative impact on the executed processes, mainly causing the production schedule to become outdated (Figure 1). Therefore, there emerges a distinct need to study the nature of the variability of technological operations times and to include them in the future production planning processes.
Fig. 1. Consequences of delaying technological operations processing times (* – delayed operations). One of the major trends in the analysis of processing times is the application of the distribution function. However, no specific subgroups can be classified in this field as researchers consider various typical distribution functions – from the uniform distribution to mixed distributions. It is, nevertheless, infrequent to approach job-shop problems with multiplemachines (Deepak et al., 2016), which are relatively common in the industrial conditions, and the majority of analytical works focus on single-machine or two-machine problems (Chung-Cheng, Kuo-Ching and Shih-Wei, 2014). Processing time uncertainty is also analysed by means of fuzzy logic. The body of literature on fuzzy numbers in robust scheduling in job-shop systems is quite extensive. In work (Gonzalez-Rodriguez et al., 2009), authors approach scheduling under uncertainty in a two-machine environment, executed by means of an improved version of a simple local search. The solution is tested for typical test scenarios. Another work (Kai et al., 2016) attempts to verify the efficiency of fuzzy-logic-based solutions with the bee colony approach. The fuzzy number theory finds implementation in other scheduling problems as well, however, the number of published works is significantly lower (Hamed, 2017). The two above-mentioned methods for solving scheduling problems under processing time uncertainty in job-shop systems are currently the most popular in such applications. The literature offers other alternatives to these methods; however, the other solutions are rather limited to individual
case scenarios. In several papers, the processing time uncertainty is considered deterministically, such as in (AlHinai and ElMekkawy, 2012), where the approach is based on the arithmetic mean of registered longest and shortest times. A similar approach is shown in (Shafia, Pourseyed and Jamili, 2011), where different scenarios of processing times are analysed. Another notable solution proposes limiting the considerations to shortening processing time by compression (Karimi-Nasab and Seyedhoseini, 2013). Although the majority of the papers above implement the distribution function, their authors fail to specify why they have selected the analysed functions and the distribution of random variables, i.e. processing times. Another problem is that the proposed solutions do not satisfactorily explore the necessity to identify the distribution and prediction of its future values. This paper proposed new method of operation times uncertainty analysis and prediction. Different methods proposed in the literature allow to analyse processing times uncertainty only when there is a possible of theoretical probability distribution fitting. In that case there is a problem with prediction, especially when historical data have untypical character. That stands a major disadvantage of existing methods. Comparatively with solutions proposed in the literature, approach described in this paper helps to predict processing time of operations even when the nature of the historical data not allow for an unambiguous prediction of the theoretical probability distribution. Moreover, proposed method helps to conduct preliminary identification of the probability distribution, which allow to determine not only one potential theoretical distribution but a group them. That approach allow to select the best solution according to the test of the equality of empirical and theoretical distributions. Mentioned elements are a major advantages of proposed method. 3. PROBLEM DESCRIPTION Let , F, P be a probabilistic space, and a continuous random variable T : 0, represent real times of technological operations. The distribution of a random variable T is derived from the density function f(t) and distribution function F(t), where t 0 . In order to approximate the nature of variability of the technological operation times, the distribution of the random variable T is estimated. After performing the point estimation, the hypotheses regarding matching the distribution to the empirical data are additionally verified. If f(t) is a function of density of a random variable T, then for any ta tb ,
ta , tb 0, the equality tb
f (s)ds P(ta T tb )
ta
is fulfilled (Figure 2).
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4.2 Prediction of operation times variability Proper analysis and implementation of the prediction process developed on data collected in the set of processing times Pij will help determine the predicted time Tij of operation j of job i, and consequently the variability Δtij of the operation time given by the dependence:
t ij T ij tˆ ij Fig. 2. An example of density function. In the process of predicting operation times, we also use a cumulative distribution function. For any value of t 0, we designate as t
F (t ) P(T t ) f ( s)ds
(2)
0
This relationship is shown graphically in Figure 3.
(4)
where tˆ ij is the nominal processing time of operation j of job i assumed by the technologist (time in the nominal schedule). In order to carry out the prediction process, we propose an algorithm for the prediction of the variability of technological operations times of scheduled jobs. It consists of individual stages the execution of which leads to obtaining the values in question. Stage 1 of the algorithm defines the operation number ofa specific job and loads the historical data from the set of processing times Pij. Stage 2 concerns the preparation of data for further analyses. The data are sorted according to the relationship tij(1) < tij(2) <…< tij(n) and are subsequently subjected to preliminary statistical analysis (determination of minimum, maximum, average, range and quartiles values) and data filtering (removal of outliers).
Fig. 3. An example of cumulative distribution function. Both the density function f(t) and the cumulative distribution function F(t) of the random variable T are applicable in the prediction of technological operations times variability. Given the probability distribution of the time T of a given technological operation, one can estimate the probability of future operation time. Thus obtained data will be further implemented to create a robust schedule.
Stage 3 of the proposed algorithm consists in initial identification of the hypothetical probability distribution of the technological operation times. for this purpose, skewness γ1 and kurtosis γ2 are determined. These two commonly used parameters belong to the group of measures describing the shape of distribution. They are determined on the basis of empirical data by determining moments μk of the appropriate order:
1
3 ( 2
4. PROPOSED METHODS
k
3 )2
;
1 n ij (ts ) k ; n s 1
2
4 3 ( 2 ) 2
1 n ij ts n s 1
(5)
4.1 Data source
where:
No investigation of the nature of technological operation times variability could be even approximately valid without referencing an appropriate source of historical data. The implementation of the algorithm proposed in the work should be preceded by collecting operation time data, which will be applied in the scheduling process.
Having established the values of 1 and 2 , one can predetermine the shape of the distribution. It indicates the position of an empirical probability distribution between typical parametric distri-butions (Figure 4).
(6)
The collected data should be saved in an appropriate form so that it could be used at a later stage of predicting variability. Information about historical processing times should, therefore, be recorded individually for each operation j of the job i in the set Pij:
Pij [t1ij , t 2ij ,...,t nij ]
(3)
where tijk, 1 ≤ k ≤ n is the k-th real time of operation j and job i. Fig. 4. Cullen-Frey graph for preliminary identification of the probability distribution (own work based on (Chi, 2018)). 1320
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Stage 4 of the algorithm is the statistical inference. This process consists in the estimation of the distribution parameters (distribution matching) and the procedures for testing the compatibility of distributions.
If the working hypothesis is rejected (inequality of empirical and theoretical distributions), the parameter estimation procedure and compliance testing are performed for a different type of distribution (indicated by the Cullen-Frey graph).
The preliminary selection of the distribution enables only to make assumptions on its possible course offering rough data about it. Therefore, it is necessary to perform point estimation. Point estimators are characteristics obtained from a random sample, around which real population parameters will be located (parameters of the distribution in question).
Stage 5 of the algorithm is executed depending on the results of the statistical inference. If the data exhibits an atypical distribution then the interval-linear nonparametric estimation of the cumulative distribution function is performed. The results from the estimation are then employed to predict the time variability. Let Fn(t), t 0 be an empirical distributor with discontinuities at points k t(1) < t(1) < … < t(k), k n . The interval~ linear distribution function Fk (t ) is determined from (10)
The next step is the implementation of the test of the equality of empirical and theoretical distributions, which in the analysed case is the non-parametric Kolmogorov–Smirnov test. The advantage of the KS test is that it is suitable even for very small samples (StatSoft, 2013). The procedure of the Kolmogorov-Smirnov test includes the verification of the hypothesis concerning the compatibility of the empirical distribution Fn(t) from the n-element sample with the cumulative distribution F(t) for the selected theoretical distribution. At the level of significance 0,1 , we set a null hypothesis:
H 0 : Fn (t ) F (t )
0, Fn t1 t , 2 t 1 ~ Fk (t ) g t , 1 Fn t k 1 Fn t k 1 2 1,
(7)
In order to verify the working hypothesis, we determine the biggest difference between the cumulative distributors, as in Fig. 5:
Dn max Fn s F s s0,
g t Fn t i 1
(8)
where: against the alternative hypothesis:
H1 : Fn (t ) F (t )
t 0,
0 t t 1 , ti t t i 1 ,
1
Fn ti Fn t i 1 2
1 i k 1,
(10)
t t k , t k t 2tk t k 1 , t k 1 t k t 2t k t k 1 ,
Fn ti1 Fn ti1 2
t t i t i 1 t i
The interval-linear cumulative distribution function is shown in Fig. 6 as a dotted broken line marked with a navy blue colour.
(9)
Fig. 6. Interval-linear cumulative distribution function. Fig. 5. Difference between empirical and hypothetical cumulative distribution function.
Based on the designated distribution function, the values of future operation times can be predicted.
Next, based on the tabulated data for n observations and the assumed level of significance α, we determine the critical value d n 1 for statistics Dn .
The execution of the algorithm in question produces the predicted operations times t ij , which are subsequently employed to determine the variability of the analysed operation time t ij .
Given that Dn d n 1 , then at the level of significance α, there are no grounds to reject the hypothesis H0, and therefore the distribution of operation times is assumed to be equal to the theoretical distribution F(t), which is subsequently implemented in the prediction of future times of given technological operations. Additionally, for a given probability, the range of execution values of operation j for job i is determined.
5. EXPERIMENTAL RESULTS 5.1 Verification conditions The application of the algorithm for predicting the variability of technological operation times is presented below along
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with its verification, which was carried out on actual production data. The analysis includes data on the processing of machining operations with a length of 0.17 h (10 minutes) and 0.08 h (5 minutes). In the case of an operation time of 0.17 h, the data set P11 consisted of 26 observations, while in the case of operation times of 0.08 h, the number of elements of the set P12 was 49 observations. The algorithm was implemented in the RStudio environment, which enabled its execution and handling the obtained results in an accessible form. 5.2 Implementation of the proposed algorithm The first analysis considered the variability of the operation times of 0.17 h. For historical data from the set P11 basic statistics were determined, the values of which are given in Table 1. Table 1. Basic statistics of P11 Min.
1’st quantile
Median
Mean
3’rd quantile
Max.
0.17
0.25
0.33
0.3838
0.50
0.83
Fig. 8. Cumulative distribution functions. For the gamma distribution with the parameters k = 5. 67832, λ = 14. 8374 (Fig. 9), the quantiles of the order of 0.1 and 0.9 were determined, which are respectively q0.1 = 0.197 and q0.9 = 0.598, and therefore:
P q0.1 T 11 q0.9 0.8
(11)
Which means that the 0.8 probability of the variability of operation 11 satisfies the inequality:
q0.9 0.17 t ij q0.1 0.17
(12)
In the next step, the empirical density function was determined (Fig. 7), which was followed by the initial identification of the distribution by means of the Cullen-Frey graph.
Fig. 9. Inference from the selected probability distribution. For operations with a length of pt12 = 0.08 h, the variability analysis process was performed in analogically. The nature of the data, however, did not allow for an unambiguous prediction of the theoretical probability distribution, which could enable describing the process of time variability of the analysed operations. Therefore, in the case of the set P12, the prediction process was performed by means of a nonparametric interval-linear CDF estimation (Fig. 10).
Fig. 7. Initial identification of the distribution. Based on the prepared graphs, a gamma distribution was selected as the theoretical probability distribution. The following distribution parameters were determined: shape parameter k = 5.67832 and scale parameter: λ = 14.8374. Next, we verified the equality of this distribution (Fig. 8) with the empirical distribution by means of the K-S test, which confirmed the compatibility of distributions at a significance level of α = 0.97.
Fig. 10. Interval-linear cumulative distribution function – set P12 1322
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As a result of the performed estimation, it was determined that P T 12 0.27 0.8 : which subsequently leads to the conclusion that the variability of the operation times are t12 0.19 h.
Thus determined values should be implemented into the production schedule in order to increase its robustness and obtain a scheduling corresponding to the real conditions of production. 6. CONCLUSIONS The method for predicting the variability of operation times presented in this work corresponds with a current trend of task scheduling under uncertainty. Acquiring real times of technological operations enables developing schedules corresponding with the reality and setting realistic production deadlines. The verification presented in the article showed that the variability of operation times could be examined and determined from historical data. Further research will include constructing schedules based on the results from the algorithm, as well as the verification of robust schedules by means of the simulation tools. Moreover, there is a need to develop solutions which help take into account another uncertainty factors of production process – e.g. machines and robots failures, appearance of new orders and tasks, materials shortage, etc. The project/research was financed in the framework of the project Lublin University of Technology-Regional Excellence Initiative, funded by the Polish Ministry of Science and Higher Education (contract no. 030/RID/2018/19). REFERENCES Kai, Z. G., Ponnuthurai, Na. S., Tay, J. Ch., Chin, S. Ch., Tian, X. C., and Qan, K. P. (2015). A two-stage artificial bee colony algorithm scheduling flexible job-shop scheduling problem with new job insertion. Expert Systems with Applications, 42(21), pp. 7652–7663. Umang, N., Erera, A. L., Bierlaire, M. (2014). The robust single machine scheduling problem with uncertain release and processing times. Cornell University. Gola A. (2019). Reliability analysis of reconfigurable manufacturing system structures using computer simulation methods. Eksploatacja i Niezawodnosc – Maintenance and Reliability, 21(1), 2019, pp. 90–102. doi: http://dx.doi.org/10.17531/ein.2019.1.11. Unpublished. Sitek, P., Wikarek, J., Nielsen, P. (2017). A constraintdriven approach to food supply chain management. Industrial Management & Data Systems, Vol. 117, Issue: 9, pp. 2115–2138. doi: https://doi.org/10.1108/IMDS-102016-0465 Kaban, A. K., Othman, Z., Rohmah, D. S. (2012). Comparison of dispatching rules in job-shop scheduling problem using simulation: a case study. International Journal of Simulation Modelling, 11(3), pp. 129–140. Sitek, P., Wikarek, J. (2018). A multi-level approach to ubiquitous modeling and solving constraints in combinatorial optimization problems in production and distribution.
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