- Email: [email protected]
Comparison of the order picking processes duration based on data obtained from the use of pseudorandom number generator
Available online at www.sciencedirect.com Available online at www.sciencedirect.com
ScienceDirect ScienceDirect
Transportation Research Procedia 00 (2019) 000–000 Available online at www.sciencedirect.com Transportation Research Procedia 00 (2019) 000–000
ScienceDirect
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Transportation Research Procedia 40 (2019) 317–324 www.elsevier.com/locate/procedia
13th International Scientific Conference on Sustainable, Modern and Safe Transport 13th International 2019), Scientific Conference on Sustainable, and Safe Transport (TRANSCOM High Tatras, Novy Smokovec –Modern Grand Hotel Bellevue, (TRANSCOM 2019),Slovak High Tatras, Novy Smokovec – Grand Hotel Bellevue, Republic, May 29-31, 2019 Slovak Republic, May 29-31, 2019
Comparison of the order picking processes duration based on data Comparison of the order picking processes duration based on data obtained from the use of pseudorandom number generator obtained from the use of pseudorandom number generator a Mariusz Kostrzewskia* Mariusz Kostrzewski *
*Warsaw University of Technology, Faculty of Transport, Koszykowa 75, Warsaw 00-662, Poland *Warsaw University of Technology, Faculty of Transport, Koszykowa 75, Warsaw 00-662, Poland
a a
Abstract Abstract In this paper a simulation model of a part of logistics facility, i.e. an order picking area in a high-bay warehouse, is considered. In paper a simulation modelwas of apreceded part of logistics i.e. anoforder picking area in aashigh-bay considered. Thethis construction of this model by the facility, development a conceptual model a result warehouse, of the studyis for selected warehouses of thisoftype. The researched simulation is not a of reflection of a logistics functioning in reality. This The construction this model was preceded by themodel development a conceptual model asfacility a result of the study for selected warehouses of this type. warehouse. The researched is not a reflection logistics facility functioning in among reality. other This facility is a hypothetical The simulation model includes generatorsofofapseudorandom numbers, which, facility is a hypothetical The simulation model includes generators pseudorandom numbers, among other things, serve to obtain andwarehouse. introduce randomly generated picking orders (picking of lists) into the model. And atwhich, the same time these generators reflect and provide the real, often unforeseen, size of orders. This allows, other things, thetime capacity things, serve to obtain and introduce randomly generated picking orders (picking lists)among into the model. Andtoatanalyze the same these generators reflect and provide real,presents often unforeseen, sizeofofresults orders.obtained This allows, other things, to analyze the capacity of the considered facility. The the paper a comparison with among use of simulation model for pseudorandom of the considered facility. The paper presents a comparison of results obtained with use of simulation model for pseudorandom number generator with different data at the input to these model. number generator with different data at the input to these model. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published byof Elsevier B.V. committee of the 13th International Scientific Conference on Sustainable, Peer-review under responsibility the scientific Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM Peer-review under responsibility of the scientific Modern and Safe Transport (TRANSCOM2019). 2019).committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM 2019). Keywords: order picking process; picking list; high-bay warehouse; pseudorandom number generator; PRNG Keywords: order picking process; picking list; high-bay warehouse; pseudorandom number generator; PRNG
1. Introduction 1. Introduction Simulation is a term, the sources of which are seen in Latin terms simulatio, i.e. ”pretending” (also: an Simulationfalse is ashow, term,feigning, the sources of which seen in Latin simulatio, i.e. ”pretending” (also: and an assumption, shamming, feint,are insincerity, deceit,terms hypocrisy, according to Perseus 2018a), assumption, false show, feigning, shamming, according feint, insincerity, deceit, hypocrisy, according to Perseus 2018a), also similis ”similar” (also: like, resembling, to Perseus 2018b). Therefore, the common meaning of and the also ”similar” (also: like, resembling, according to Perseus 2018b).orTherefore, meaning ofwith the term similis simulation is understood as an approximate recreation of phenomena behaviorsthe of common some object/system term simulation is understood as an approximate recreation of phenomena or behaviors of some object/system with * Corresponding author. Tel.: +48-22-234-7337; fax: +48-22-234-1597. address:author. [email protected] * E-mail Corresponding Tel.: +48-22-234-7337; fax: +48-22-234-1597. E-mail address: [email protected] 2352-1465 © 2018 The Authors. Published by Elsevier B.V. Peer-review©under responsibility of the scientific committee 2352-1465 2018 The Authors. Published by Elsevier B.V. of the 13th International Scientific Conference on Sustainable, Moder n and Safe Transport (TRANSCOM 2019). Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Moder n and Safe Transport (TRANSCOM 2019). 2352-1465 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM 2019). 10.1016/j.trpro.2019.07.047
318 2
Mariusz Kostrzewski / Transportation Research Procedia 40 (2019) 317–324 Kostrzewski / Transportation Research Procedia 00 (2019) 000–000
the use of the model of this system/object itself. In the literature, a lot of attention is paid to the definitions of simulation, simulation modeling with the use of simulation methods, conceptual model, simulation model – therefore these issues are omitted in this paper. Notwithstanding, it is important that simulation is believed as well proven technology for analyzing stochastic and dynamic objects/systems as given in Wenzel et al. (2010) and in the same time might be treated as support for objects/systems optimization, Rabe and Goldsmann (2019). Discrete event simulation (DES) was mentioned in Rabe and Goldsmann (2019) even as “well proven”, Law (2015). This is also one of several tools and methods which allow managers and decision makers to decide about processes realization in digital factory before making arrangements in real one, Bučková et al. (2017). Simulation modeling is of great impact in different branches of transport as railway – e.g. Dižo et al. (2018), Dižo et al. (2017) – and road – Bęczkowska and Grabarek (2017), Olteanu et al. (2015) – and air etc., so does in internal transport and internal logistics, which is main area of the described research interest, especially in the case of order picking process. As it is noticed in Kostrzewski (2018) and several papers, books mentioned there, over the years, the costs incurred during order picking process still fluctuate around half to three quarters of the total operating costs incurred for all logistics processes cost. This is one of the reasons because of which it is worth to consider analyses of order picking process time, make this time shorter (sub-optimised) and in the same time decrease cost of process operation. In this paper, the problem is discussed on the example of the pseudorandom numbers generator (PRNG), in which a discrete equivalent of logistic differential equation is applied – full model is described in Kostrzewski (2018). Logistic differential equation is known as the simplest model of chaos. In Gutenbaum (2003, p. 96), it was found that relatively simple, strongly nonlinear deterministic differential equations are good models of some complex dynamic processes. In addition, it was noted that this type of mathematical formulae, within a certain range of initial conditions and coefficients, allows to obtain solutions with features of random processes. Processes of this type are called as chaos, in fact deterministic chaos, and their scientific basis derives from hydrodynamic flow studies in meteorology, Lorenz (1963). Deterministic chaos usually refers to nonlinear deterministic differential equations describing dynamic systems, and it is understood as an irregular motion derived from a nonlinear system which dynamics uniquely determines evolution of a system in time, if system’s history is known, and the real cause of irregularities is characteristics of nonlinear systems which is the exponential divergence of initially close trajectories in a limited area of a phase space, Schuster (1993). The solution of logistic differential equation resembles a random sequence that can be used successfully to generate random numbers. The use of this generator ensures the generation of picking lists close to real-world processes, Kostrzewski (2018). What is more, in real-world warehouses, order picking process can be performed in individual orders or in batches of orders, Bučková et al. (2017). In this paper, batch orders are subjects of interest. The paper presents two samples of experiments on the simulation model which is described in Kostrzewski (2018.) They are generated for fixed and unchanged data except for entering two different picking lists (Fig. 1, Fig. 8) and different MTTR (mean time to repair) values with simultaneous constant values of f (the estimated percentage of simulation time during which means of transport may be idle). 2. Simulation model – experiments and results As it was mentioned, the research are based on two samples of simulation experiments. In the case of first sample picking list is given in Fig. 1 and in the case of second one in Fig. 5. Each picking list consists of j = 100 orders. Parameter j is designated to the consecutive number of the experiments in the sample, parameter i is designated to the quantities of the items, and parameter pij is a stochastic value generated according to the PRNG. Several experiments with changes in parameters MTTR and f values were executed. Results of consecutive order picking process times computed for every jth experiment are given in graphs (Figs. 2-4 for first sample and Fig. 6 for second sample). In the case of first sample, every single graph is described below. The graph given in Fig. 2(a) presents total simulation time of order picking process in function of orders given in picking list for MTTR = 1 [min]. At first, the plot apparently shot up slightly from 20.3 minutes and it reached 21 minutes and then it plunged to 19.4 minutes. It decreased a little bit in case of several following orders (without any significant blip) and for the next 80% of orders the values of time fluctuated between 16 and 17 minutes.
Mariusz Kostrzewski / Transportation Research Procedia 40 (2019) 317–324 Kostrzewski / Transportation Research Procedia 00 (2019) 000–000
319 3
The trend equation is the exponential equation (base and exponent are given in Fig. 2(a)) with the coefficient of determination equals to more than 0.91.
Fig. 1. Order picking list – sample 1.
The graph given in Fig. 2(b) presents total simulation time of order picking process in function of orders given in picking list for MTTR = 3 [min]. At first, the plot apparently shot up from 18.5 minutes and it reached 21.7 minutes and then it plunged to 17.5 minutes. It fluctuated a little bit in case of few following orders and then it has fall again, this time from 17.5 minutes to 16.8 minutes. For the next 70% of orders fluctuated between 16.5 and 17.5 minutes. The trend equation is the exponential equation (base and exponent are given in Fig. 2(b)) with the coefficient of determination equals to a little bit less than 0.78.
(a)
(b) Fig. 2. Order picking process time for f = 10% and (a) MTTR = 1 [min] and (b) MTTR = 3 [min].
The graph given in Fig. 3(a) presents total simulation time of order picking process in function of orders given in picking list for MTTR = 9 [min]. In the case of first picking orders the plot apparently shot up and plunged alternatively between 19 and 21 minutes and then it started to decreasing to 17 minutes. Truthfully, the mentioned decreasing was slightly fluctuating between 17 and 18 minutes, however it can be observed that exponential trend line slowly converged to 17 minutes (therefore, it can be said that order picking time was decreasing). The trend equation is the exponential equation (base and exponent are given in Fig. 3(a)) with the coefficient of determination equals almost to 0.76. As it can be observed in Figs. 2-3(a), with the increase in MTTR, the execution times of successive orders are more and more unstable and varied, next orders in relation to the previous ones.
Mariusz Kostrzewski / Transportation Research Procedia 40 (2019) 317–324 Kostrzewski / Transportation Research Procedia 00 (2019) 000–000
320 4
This can be observed, for example, by the coefficient of determination of trend line which gets lower. In the case of experiment presented in Fig. 3(b) it was slightly different. At first it shoot up from 18.5 to more than 20 minutes and then it slumped with the blip c.a. 16.5 minutes. Later on, it took off to 18 minutes and fluctuated around this value for a couple of picking orders. Then, it took off again up to 19.5 minutes and only then it started decreasing as in the previous experiments. This time, the trend equation is the logarithmic equation (given in Fig. 3(b)) with very low value of the coefficient of determination, which is equal to less than 0.07. It was the best coefficient of determination in the case of this experiment.
(a)
(b) Fig. 3. Order picking process time for f = 10% and (a) MTTR = 9 [min] and (b) MTTR = 27 [min].
(a)
(b) Fig. 4. Order picking process time for f = 10% and (a) MTTR = 81 [min] and (b) MTTR = 234 [min].
The graph given in Fig. 4(a) presents total simulation time of order picking process in function of orders given in picking list for MTTR = 81 [min]. At first, the plot apparently shot up slightly from 18.5 minutes and it reached 20.2 minutes and then it plunged to 17 minutes. It decreased a little bit in case of several following orders (without any significant blip) and for the next 70% of orders the values of time fluctuated between 16 and 17 minutes, similarly to experiment with MTTR = 3 [min] (Fig. 2(b)). Situation changed in the case of order j = 82, in the case of which value of time apparently shot up to 17.4 minutes. Later on, it fluctuated around the last value.
Mariusz Kostrzewski / Transportation Research Procedia 40 (2019) 317–324 Kostrzewski / Transportation Research Procedia 00 (2019) 000–000
321 5
Fig. 5. Order picking list – sample 2.
The trend equation is the polynomial equation (given in Fig. 4(a)) with the coefficient of determination equals to a little bit more than 0.80. The graph given in Fig. 4(b) presents total simulation time of order picking process in function of orders given in picking list for MTTR = 243 [min]. At first, the plot apparently shot up slightly from 18.5 minutes and it reached 20.2 minutes and then it plunged to 17 minutes. It decreased a little bit in case of several following orders (without any significant blip) and for the next 80% of orders the values of time fluctuated between 16 and 17 minutes, similarly to experiment with MTTR = 3 [min] (Fig. 2(b)) and with MTTR = 81 [min] (Fig. 4(a)). The trend equation is the exponential equation (base and exponent are given in Fig. 4(b)) with the coefficient of determination equals to a little bit more than 0.87. Table 1. Comparison of order picking process time values for two samples obtained with use of simulation model and order picking process time value calculated in analytical model. k
f = f (k ) [%]
MTTR [min]
t(EX) [min]
t(PS) f (k ) [min]
s t ( PS ) f (k ) [min]
1.1
10
1
19.85
16.84
0.94
1.2
10
3
19.85
17.37
1.07
1.3
10
9
19.85
18.01
1.01
1.4
10
27
19.85
18.09
0.77
1.5
10
81
19.85
16.82
0.95
1.6
10
243
19.85
16.66
0.98
2.1
10
1
19.85
15.36
0.41
2.2
10
3
19.85
15.42
0.36
2.3
10
9
19.85
15.62
0.41
2.4
10
27
19.85
15.80
0.91
2.5
10
81
19.85
14.79
0.46
2.6
10
243
19.85
14.72
0.28
Mariusz Kostrzewski / Transportation Research Procedia 40 (2019) 317–324 Kostrzewski / Transportation Research Procedia 00 (2019) 000–000
322 6
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 6. Order picking process time – second sample – for f = 10% and (a) MTTR = 1 [min], (b) MTTR = 3 [min], (c) MTTR = 9 [min], (d) MTTR = 27 [min], (e) MTTR = 81 [min], (f) MTTR = 243 [min].
The experiment was repeated for different picking list than the one presented in Fig. 1. New picking list is given in Fig. 5. These two samples of experiments are compared in Table 1. Parameter k is understood as.. Parameter t(EX) is order picking process time calculated in analytical model (without any PRNG application). Parameter t(PS) f (k ) is order picking process time computed in simulation model and parameter s t ( PS ) f (k ) is standard deviation for computed time.
Mariusz Kostrzewski / Transportation Research Procedia 40 (2019) 317–324 Kostrzewski / Transportation Research Procedia 00 (2019) 000–000
323 7
3. Discussion The simulation model used for research in this paper was verified in comparison to analytic model with positive conclusions given in Kostrzewski (2018). Herein, time of order picking process obtained from analytical model t(EX) = 19.85 [min] is used in order to compare results obtained from simulation model with PRNG. It is done in order to verify whether range of simulation time (mean value from a sample) – with taking into consideration standard deviation of this time – oscillates around t(EX) value obtained from analytical model. Results of the comparison are given in Table 1 and Fig. 7. Pearson correlation coefficient between times occurred from sample 1 & 2 with value 0.80 is strong and positive and between sample 1 and t(EX) with value -9.23E-16 is weak and negative, meanwhile between sample 2 and t(EX) with value 2.73E-16 is weak and positive.
Fig. 7. Graph of comparison order picking process time for samples 1 & 2.
Large differences between samples 1 & 2 occur due to the fact that the MTTR value does not always correspond to the exact values indicated in column 3 of Table 1. This means that in the simulation model of this kind values are randomly selected and are maximally equal to that given in the descriptions in Figs. 2-4, 6 and Table 1. This is not a negative effect of the simulation model, on the contrary. In real processes in logistics facilities, first of all, uncontrolled interruptions (defects) of a process may occur (and the parameter f is “responsible” for such situations), and secondly, the time of repair of such a defect may be different than one assumes. Thus, the simulation model is even more good reflection of real systems, despite the fact that it is a model referring to a hypothetical warehouse. Moreover, it is worth noting that calculations which use the analytical model are oversized. In the analytical model the duration of the order picking process is oversized – this has already been shown in earlier works, e.g. Kostrzewski (2014). For this reason, its values in Fig. 7 are higher than the values of the duration of order picking processes in the case of several experiments carried out on samples 1 & 2. 4. Conclusion The use of simulations in logistics systems is a key issue, especially in the era of the so-called Industry 4.0 or Logistics 4.0 (which thematically corresponds to the Industry 4.0 concept). The oversized values obtained with the use of simulation models, with PRNG implemented, take into account accumulations and uncertainties. This oversizing gives the impression that the simulation models reflect reality less precisely than the analytical models. However, as it was proved in earlier studies (as referred to in this paper), the results obtained using simulation models are more appropriate to those obtained in real time logistics systems.
324 8
Mariusz Kostrzewski / Transportation Research Procedia 40 (2019) 317–324 Kostrzewski / Transportation Research Procedia 00 (2019) 000–000
Acknowledgements Financed by funds of the statutory work of Educational and Scientific Team of Modeling and Diagnostics of Technical Means of Transport (Faculty of Transport, Warsaw University of Technology). References Bęczkowska, S., Grabarek, I., 2017. Monitorowanie jako element bezpieczeństwa przewozów towarów niebezpiecznych - koncepcja systemu. Zeszyty Naukowe Małopolskiej Wyższej Szkoły Ekonomicznej w Tarnowie 35.3, 67–78. Bučková, M., Krajčovič, M., Edl, M., 2017. Computer simulation and optimization of transport distances of order picking processes. Procedia Engineering 192, 69–74. https://doi.org/10.1016/j.proeng.2017.06.012 Dižo, J., Blatnický, M., Pavlík, A., 2018. Process of modelling the freight wagon multibody system and analysing its dynamic properties by means of simulation computations. MATEC Web of Conferences 235. 00027. https://doi.org/10.1051/matecconf/201823500027 Dižo, J., Blatnický, M., Steišūnas, S., 2017. Assessment of negative effects of a coach running with the wheel-flat on a track by means of simulation computations. Diagnostyka 18.3, 31–37. Grabarek, I., Bęczkowska, S., 2012. Analysis of Factors Determining Ergonomic Conditions of Driver's Workplace and Safety in Transport of Dangerous Goods. Archives of Transport 24.3, 297–306. Gutenbaum, J., 2003. Modelowanie matematyczne systemów. Akademicka Oficyna Wydawnicza EXIT, Warszawa. Kostrzewski, M., 2018. Modelowanie i badanie wybranych elementów i obiektów logistycznych z wykorzystaniem metod symulacyjnych. Oficyna Wydawnicza Politechniki Warszawskiej, Warszawa. Kostrzewski, M., 2014. Mathematical Models of Time Computing in Two-Dimensional Order-Picking Process in High-Bay Warehouses, in “Quantitative Methods in Logistics Management”. In: Sas, J. (Ed.). AGH University of Science and Technology Press, Kraków, pp. 55-69. Law, A.M., 2015. Simulation Modeling and Analysis, 5th ed. McGraw-Hill, New York. Lorenz, E.N., 1963. Deterministic Nonperiodic Flow. Journal of Atmospheric Sciences 20.2, 130-148. https://doi.org/10.1175/15200469(1963)020<0130:DNF>2.0.CO;2 Olteanu, S., Popa, M., Burciu, S., Rusca, F.V., 2015. Mathematical modeling of a private intermodal terminal activity with preferential priorities flows, 7th International Conference of Management and Industrial Engineering (ICMIE 2015) on Management - the Key Driver for Creating Value. International Conference of Management and Industrial Engineering, 302-308. Perseus, 2018a. http://www.perseus.tufts.edu/hopper/resolveform?type=begin&lookup=simulatio&lang=la (accessed on-line: January 15, 2019). Perseus, 2018b. http://www.perseus.tufts.edu/hopper/resolveform?type=begin&lookup=similis&lang=la (accessed on-line: January 15, 2019). Rabe, M., Goldsman, D., 2019. Decision Making Using Simulation Methods in Sustainable Transportation, in “Sustainable Transportation and Smart Logistics, Decision-Making Models and Solutions”. In: Faulin, J., Grasman, S.E., Juan, A.A., Hirsch, P. (Eds.). Elsevier Inc., pp. 305333. https://doi.org/10.1016/B978-0-12-814242-4.00012-0 Schuster, H.G., 1993. Chaos deterministyczny. Wydawnictwo Naukowe PWN, Warszawa. Wenzel, S., Boyaci, O., Jessen, U., 2010. Simulation in production und logistics: Trends, solutions and applications, in “Advanced Manufacturing and Sustainable Logistics, Proceedings of the 8th International Heinz Nixdorf Symposium IHNS 2010”. In: Dangelmaier, W., Blecken, A., Delius, R., Kloepfer, S. (Eds.). Paderborn, pp. 73–84.
ScienceDirect ScienceDirect
Transportation Research Procedia 00 (2019) 000–000 Available online at www.sciencedirect.com Transportation Research Procedia 00 (2019) 000–000
ScienceDirect
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Transportation Research Procedia 40 (2019) 317–324 www.elsevier.com/locate/procedia
13th International Scientific Conference on Sustainable, Modern and Safe Transport 13th International 2019), Scientific Conference on Sustainable, and Safe Transport (TRANSCOM High Tatras, Novy Smokovec –Modern Grand Hotel Bellevue, (TRANSCOM 2019),Slovak High Tatras, Novy Smokovec – Grand Hotel Bellevue, Republic, May 29-31, 2019 Slovak Republic, May 29-31, 2019
Comparison of the order picking processes duration based on data Comparison of the order picking processes duration based on data obtained from the use of pseudorandom number generator obtained from the use of pseudorandom number generator a Mariusz Kostrzewskia* Mariusz Kostrzewski *
*Warsaw University of Technology, Faculty of Transport, Koszykowa 75, Warsaw 00-662, Poland *Warsaw University of Technology, Faculty of Transport, Koszykowa 75, Warsaw 00-662, Poland
a a
Abstract Abstract In this paper a simulation model of a part of logistics facility, i.e. an order picking area in a high-bay warehouse, is considered. In paper a simulation modelwas of apreceded part of logistics i.e. anoforder picking area in aashigh-bay considered. Thethis construction of this model by the facility, development a conceptual model a result warehouse, of the studyis for selected warehouses of thisoftype. The researched simulation is not a of reflection of a logistics functioning in reality. This The construction this model was preceded by themodel development a conceptual model asfacility a result of the study for selected warehouses of this type. warehouse. The researched is not a reflection logistics facility functioning in among reality. other This facility is a hypothetical The simulation model includes generatorsofofapseudorandom numbers, which, facility is a hypothetical The simulation model includes generators pseudorandom numbers, among other things, serve to obtain andwarehouse. introduce randomly generated picking orders (picking of lists) into the model. And atwhich, the same time these generators reflect and provide the real, often unforeseen, size of orders. This allows, other things, thetime capacity things, serve to obtain and introduce randomly generated picking orders (picking lists)among into the model. Andtoatanalyze the same these generators reflect and provide real,presents often unforeseen, sizeofofresults orders.obtained This allows, other things, to analyze the capacity of the considered facility. The the paper a comparison with among use of simulation model for pseudorandom of the considered facility. The paper presents a comparison of results obtained with use of simulation model for pseudorandom number generator with different data at the input to these model. number generator with different data at the input to these model. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published byof Elsevier B.V. committee of the 13th International Scientific Conference on Sustainable, Peer-review under responsibility the scientific Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM Peer-review under responsibility of the scientific Modern and Safe Transport (TRANSCOM2019). 2019).committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM 2019). Keywords: order picking process; picking list; high-bay warehouse; pseudorandom number generator; PRNG Keywords: order picking process; picking list; high-bay warehouse; pseudorandom number generator; PRNG
1. Introduction 1. Introduction Simulation is a term, the sources of which are seen in Latin terms simulatio, i.e. ”pretending” (also: an Simulationfalse is ashow, term,feigning, the sources of which seen in Latin simulatio, i.e. ”pretending” (also: and an assumption, shamming, feint,are insincerity, deceit,terms hypocrisy, according to Perseus 2018a), assumption, false show, feigning, shamming, according feint, insincerity, deceit, hypocrisy, according to Perseus 2018a), also similis ”similar” (also: like, resembling, to Perseus 2018b). Therefore, the common meaning of and the also ”similar” (also: like, resembling, according to Perseus 2018b).orTherefore, meaning ofwith the term similis simulation is understood as an approximate recreation of phenomena behaviorsthe of common some object/system term simulation is understood as an approximate recreation of phenomena or behaviors of some object/system with * Corresponding author. Tel.: +48-22-234-7337; fax: +48-22-234-1597. address:author. [email protected] * E-mail Corresponding Tel.: +48-22-234-7337; fax: +48-22-234-1597. E-mail address: [email protected] 2352-1465 © 2018 The Authors. Published by Elsevier B.V. Peer-review©under responsibility of the scientific committee 2352-1465 2018 The Authors. Published by Elsevier B.V. of the 13th International Scientific Conference on Sustainable, Moder n and Safe Transport (TRANSCOM 2019). Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Moder n and Safe Transport (TRANSCOM 2019). 2352-1465 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM 2019). 10.1016/j.trpro.2019.07.047
318 2
Mariusz Kostrzewski / Transportation Research Procedia 40 (2019) 317–324 Kostrzewski / Transportation Research Procedia 00 (2019) 000–000
the use of the model of this system/object itself. In the literature, a lot of attention is paid to the definitions of simulation, simulation modeling with the use of simulation methods, conceptual model, simulation model – therefore these issues are omitted in this paper. Notwithstanding, it is important that simulation is believed as well proven technology for analyzing stochastic and dynamic objects/systems as given in Wenzel et al. (2010) and in the same time might be treated as support for objects/systems optimization, Rabe and Goldsmann (2019). Discrete event simulation (DES) was mentioned in Rabe and Goldsmann (2019) even as “well proven”, Law (2015). This is also one of several tools and methods which allow managers and decision makers to decide about processes realization in digital factory before making arrangements in real one, Bučková et al. (2017). Simulation modeling is of great impact in different branches of transport as railway – e.g. Dižo et al. (2018), Dižo et al. (2017) – and road – Bęczkowska and Grabarek (2017), Olteanu et al. (2015) – and air etc., so does in internal transport and internal logistics, which is main area of the described research interest, especially in the case of order picking process. As it is noticed in Kostrzewski (2018) and several papers, books mentioned there, over the years, the costs incurred during order picking process still fluctuate around half to three quarters of the total operating costs incurred for all logistics processes cost. This is one of the reasons because of which it is worth to consider analyses of order picking process time, make this time shorter (sub-optimised) and in the same time decrease cost of process operation. In this paper, the problem is discussed on the example of the pseudorandom numbers generator (PRNG), in which a discrete equivalent of logistic differential equation is applied – full model is described in Kostrzewski (2018). Logistic differential equation is known as the simplest model of chaos. In Gutenbaum (2003, p. 96), it was found that relatively simple, strongly nonlinear deterministic differential equations are good models of some complex dynamic processes. In addition, it was noted that this type of mathematical formulae, within a certain range of initial conditions and coefficients, allows to obtain solutions with features of random processes. Processes of this type are called as chaos, in fact deterministic chaos, and their scientific basis derives from hydrodynamic flow studies in meteorology, Lorenz (1963). Deterministic chaos usually refers to nonlinear deterministic differential equations describing dynamic systems, and it is understood as an irregular motion derived from a nonlinear system which dynamics uniquely determines evolution of a system in time, if system’s history is known, and the real cause of irregularities is characteristics of nonlinear systems which is the exponential divergence of initially close trajectories in a limited area of a phase space, Schuster (1993). The solution of logistic differential equation resembles a random sequence that can be used successfully to generate random numbers. The use of this generator ensures the generation of picking lists close to real-world processes, Kostrzewski (2018). What is more, in real-world warehouses, order picking process can be performed in individual orders or in batches of orders, Bučková et al. (2017). In this paper, batch orders are subjects of interest. The paper presents two samples of experiments on the simulation model which is described in Kostrzewski (2018.) They are generated for fixed and unchanged data except for entering two different picking lists (Fig. 1, Fig. 8) and different MTTR (mean time to repair) values with simultaneous constant values of f (the estimated percentage of simulation time during which means of transport may be idle). 2. Simulation model – experiments and results As it was mentioned, the research are based on two samples of simulation experiments. In the case of first sample picking list is given in Fig. 1 and in the case of second one in Fig. 5. Each picking list consists of j = 100 orders. Parameter j is designated to the consecutive number of the experiments in the sample, parameter i is designated to the quantities of the items, and parameter pij is a stochastic value generated according to the PRNG. Several experiments with changes in parameters MTTR and f values were executed. Results of consecutive order picking process times computed for every jth experiment are given in graphs (Figs. 2-4 for first sample and Fig. 6 for second sample). In the case of first sample, every single graph is described below. The graph given in Fig. 2(a) presents total simulation time of order picking process in function of orders given in picking list for MTTR = 1 [min]. At first, the plot apparently shot up slightly from 20.3 minutes and it reached 21 minutes and then it plunged to 19.4 minutes. It decreased a little bit in case of several following orders (without any significant blip) and for the next 80% of orders the values of time fluctuated between 16 and 17 minutes.
Mariusz Kostrzewski / Transportation Research Procedia 40 (2019) 317–324 Kostrzewski / Transportation Research Procedia 00 (2019) 000–000
319 3
The trend equation is the exponential equation (base and exponent are given in Fig. 2(a)) with the coefficient of determination equals to more than 0.91.
Fig. 1. Order picking list – sample 1.
The graph given in Fig. 2(b) presents total simulation time of order picking process in function of orders given in picking list for MTTR = 3 [min]. At first, the plot apparently shot up from 18.5 minutes and it reached 21.7 minutes and then it plunged to 17.5 minutes. It fluctuated a little bit in case of few following orders and then it has fall again, this time from 17.5 minutes to 16.8 minutes. For the next 70% of orders fluctuated between 16.5 and 17.5 minutes. The trend equation is the exponential equation (base and exponent are given in Fig. 2(b)) with the coefficient of determination equals to a little bit less than 0.78.
(a)
(b) Fig. 2. Order picking process time for f = 10% and (a) MTTR = 1 [min] and (b) MTTR = 3 [min].
The graph given in Fig. 3(a) presents total simulation time of order picking process in function of orders given in picking list for MTTR = 9 [min]. In the case of first picking orders the plot apparently shot up and plunged alternatively between 19 and 21 minutes and then it started to decreasing to 17 minutes. Truthfully, the mentioned decreasing was slightly fluctuating between 17 and 18 minutes, however it can be observed that exponential trend line slowly converged to 17 minutes (therefore, it can be said that order picking time was decreasing). The trend equation is the exponential equation (base and exponent are given in Fig. 3(a)) with the coefficient of determination equals almost to 0.76. As it can be observed in Figs. 2-3(a), with the increase in MTTR, the execution times of successive orders are more and more unstable and varied, next orders in relation to the previous ones.
Mariusz Kostrzewski / Transportation Research Procedia 40 (2019) 317–324 Kostrzewski / Transportation Research Procedia 00 (2019) 000–000
320 4
This can be observed, for example, by the coefficient of determination of trend line which gets lower. In the case of experiment presented in Fig. 3(b) it was slightly different. At first it shoot up from 18.5 to more than 20 minutes and then it slumped with the blip c.a. 16.5 minutes. Later on, it took off to 18 minutes and fluctuated around this value for a couple of picking orders. Then, it took off again up to 19.5 minutes and only then it started decreasing as in the previous experiments. This time, the trend equation is the logarithmic equation (given in Fig. 3(b)) with very low value of the coefficient of determination, which is equal to less than 0.07. It was the best coefficient of determination in the case of this experiment.
(a)
(b) Fig. 3. Order picking process time for f = 10% and (a) MTTR = 9 [min] and (b) MTTR = 27 [min].
(a)
(b) Fig. 4. Order picking process time for f = 10% and (a) MTTR = 81 [min] and (b) MTTR = 234 [min].
The graph given in Fig. 4(a) presents total simulation time of order picking process in function of orders given in picking list for MTTR = 81 [min]. At first, the plot apparently shot up slightly from 18.5 minutes and it reached 20.2 minutes and then it plunged to 17 minutes. It decreased a little bit in case of several following orders (without any significant blip) and for the next 70% of orders the values of time fluctuated between 16 and 17 minutes, similarly to experiment with MTTR = 3 [min] (Fig. 2(b)). Situation changed in the case of order j = 82, in the case of which value of time apparently shot up to 17.4 minutes. Later on, it fluctuated around the last value.
Mariusz Kostrzewski / Transportation Research Procedia 40 (2019) 317–324 Kostrzewski / Transportation Research Procedia 00 (2019) 000–000
321 5
Fig. 5. Order picking list – sample 2.
The trend equation is the polynomial equation (given in Fig. 4(a)) with the coefficient of determination equals to a little bit more than 0.80. The graph given in Fig. 4(b) presents total simulation time of order picking process in function of orders given in picking list for MTTR = 243 [min]. At first, the plot apparently shot up slightly from 18.5 minutes and it reached 20.2 minutes and then it plunged to 17 minutes. It decreased a little bit in case of several following orders (without any significant blip) and for the next 80% of orders the values of time fluctuated between 16 and 17 minutes, similarly to experiment with MTTR = 3 [min] (Fig. 2(b)) and with MTTR = 81 [min] (Fig. 4(a)). The trend equation is the exponential equation (base and exponent are given in Fig. 4(b)) with the coefficient of determination equals to a little bit more than 0.87. Table 1. Comparison of order picking process time values for two samples obtained with use of simulation model and order picking process time value calculated in analytical model. k
f = f (k ) [%]
MTTR [min]
t(EX) [min]
t(PS) f (k ) [min]
s t ( PS ) f (k ) [min]
1.1
10
1
19.85
16.84
0.94
1.2
10
3
19.85
17.37
1.07
1.3
10
9
19.85
18.01
1.01
1.4
10
27
19.85
18.09
0.77
1.5
10
81
19.85
16.82
0.95
1.6
10
243
19.85
16.66
0.98
2.1
10
1
19.85
15.36
0.41
2.2
10
3
19.85
15.42
0.36
2.3
10
9
19.85
15.62
0.41
2.4
10
27
19.85
15.80
0.91
2.5
10
81
19.85
14.79
0.46
2.6
10
243
19.85
14.72
0.28
Mariusz Kostrzewski / Transportation Research Procedia 40 (2019) 317–324 Kostrzewski / Transportation Research Procedia 00 (2019) 000–000
322 6
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 6. Order picking process time – second sample – for f = 10% and (a) MTTR = 1 [min], (b) MTTR = 3 [min], (c) MTTR = 9 [min], (d) MTTR = 27 [min], (e) MTTR = 81 [min], (f) MTTR = 243 [min].
The experiment was repeated for different picking list than the one presented in Fig. 1. New picking list is given in Fig. 5. These two samples of experiments are compared in Table 1. Parameter k is understood as
Mariusz Kostrzewski / Transportation Research Procedia 40 (2019) 317–324 Kostrzewski / Transportation Research Procedia 00 (2019) 000–000
323 7
3. Discussion The simulation model used for research in this paper was verified in comparison to analytic model with positive conclusions given in Kostrzewski (2018). Herein, time of order picking process obtained from analytical model t(EX) = 19.85 [min] is used in order to compare results obtained from simulation model with PRNG. It is done in order to verify whether range of simulation time (mean value from a sample) – with taking into consideration standard deviation of this time – oscillates around t(EX) value obtained from analytical model. Results of the comparison are given in Table 1 and Fig. 7. Pearson correlation coefficient between times occurred from sample 1 & 2 with value 0.80 is strong and positive and between sample 1 and t(EX) with value -9.23E-16 is weak and negative, meanwhile between sample 2 and t(EX) with value 2.73E-16 is weak and positive.
Fig. 7. Graph of comparison order picking process time for samples 1 & 2.
Large differences between samples 1 & 2 occur due to the fact that the MTTR value does not always correspond to the exact values indicated in column 3 of Table 1. This means that in the simulation model of this kind values are randomly selected and are maximally equal to that given in the descriptions in Figs. 2-4, 6 and Table 1. This is not a negative effect of the simulation model, on the contrary. In real processes in logistics facilities, first of all, uncontrolled interruptions (defects) of a process may occur (and the parameter f is “responsible” for such situations), and secondly, the time of repair of such a defect may be different than one assumes. Thus, the simulation model is even more good reflection of real systems, despite the fact that it is a model referring to a hypothetical warehouse. Moreover, it is worth noting that calculations which use the analytical model are oversized. In the analytical model the duration of the order picking process is oversized – this has already been shown in earlier works, e.g. Kostrzewski (2014). For this reason, its values in Fig. 7 are higher than the values of the duration of order picking processes in the case of several experiments carried out on samples 1 & 2. 4. Conclusion The use of simulations in logistics systems is a key issue, especially in the era of the so-called Industry 4.0 or Logistics 4.0 (which thematically corresponds to the Industry 4.0 concept). The oversized values obtained with the use of simulation models, with PRNG implemented, take into account accumulations and uncertainties. This oversizing gives the impression that the simulation models reflect reality less precisely than the analytical models. However, as it was proved in earlier studies (as referred to in this paper), the results obtained using simulation models are more appropriate to those obtained in real time logistics systems.
324 8
Mariusz Kostrzewski / Transportation Research Procedia 40 (2019) 317–324 Kostrzewski / Transportation Research Procedia 00 (2019) 000–000
Acknowledgements Financed by funds of the statutory work of Educational and Scientific Team of Modeling and Diagnostics of Technical Means of Transport (Faculty of Transport, Warsaw University of Technology). References Bęczkowska, S., Grabarek, I., 2017. Monitorowanie jako element bezpieczeństwa przewozów towarów niebezpiecznych - koncepcja systemu. Zeszyty Naukowe Małopolskiej Wyższej Szkoły Ekonomicznej w Tarnowie 35.3, 67–78. Bučková, M., Krajčovič, M., Edl, M., 2017. Computer simulation and optimization of transport distances of order picking processes. Procedia Engineering 192, 69–74. https://doi.org/10.1016/j.proeng.2017.06.012 Dižo, J., Blatnický, M., Pavlík, A., 2018. Process of modelling the freight wagon multibody system and analysing its dynamic properties by means of simulation computations. MATEC Web of Conferences 235. 00027. https://doi.org/10.1051/matecconf/201823500027 Dižo, J., Blatnický, M., Steišūnas, S., 2017. Assessment of negative effects of a coach running with the wheel-flat on a track by means of simulation computations. Diagnostyka 18.3, 31–37. Grabarek, I., Bęczkowska, S., 2012. Analysis of Factors Determining Ergonomic Conditions of Driver's Workplace and Safety in Transport of Dangerous Goods. Archives of Transport 24.3, 297–306. Gutenbaum, J., 2003. Modelowanie matematyczne systemów. Akademicka Oficyna Wydawnicza EXIT, Warszawa. Kostrzewski, M., 2018. Modelowanie i badanie wybranych elementów i obiektów logistycznych z wykorzystaniem metod symulacyjnych. Oficyna Wydawnicza Politechniki Warszawskiej, Warszawa. Kostrzewski, M., 2014. Mathematical Models of Time Computing in Two-Dimensional Order-Picking Process in High-Bay Warehouses, in “Quantitative Methods in Logistics Management”. In: Sas, J. (Ed.). AGH University of Science and Technology Press, Kraków, pp. 55-69. Law, A.M., 2015. Simulation Modeling and Analysis, 5th ed. McGraw-Hill, New York. Lorenz, E.N., 1963. Deterministic Nonperiodic Flow. Journal of Atmospheric Sciences 20.2, 130-148. https://doi.org/10.1175/15200469(1963)020<0130:DNF>2.0.CO;2 Olteanu, S., Popa, M., Burciu, S., Rusca, F.V., 2015. Mathematical modeling of a private intermodal terminal activity with preferential priorities flows, 7th International Conference of Management and Industrial Engineering (ICMIE 2015) on Management - the Key Driver for Creating Value. International Conference of Management and Industrial Engineering, 302-308. Perseus, 2018a. http://www.perseus.tufts.edu/hopper/resolveform?type=begin&lookup=simulatio&lang=la (accessed on-line: January 15, 2019). Perseus, 2018b. http://www.perseus.tufts.edu/hopper/resolveform?type=begin&lookup=similis&lang=la (accessed on-line: January 15, 2019). Rabe, M., Goldsman, D., 2019. Decision Making Using Simulation Methods in Sustainable Transportation, in “Sustainable Transportation and Smart Logistics, Decision-Making Models and Solutions”. In: Faulin, J., Grasman, S.E., Juan, A.A., Hirsch, P. (Eds.). Elsevier Inc., pp. 305333. https://doi.org/10.1016/B978-0-12-814242-4.00012-0 Schuster, H.G., 1993. Chaos deterministyczny. Wydawnictwo Naukowe PWN, Warszawa. Wenzel, S., Boyaci, O., Jessen, U., 2010. Simulation in production und logistics: Trends, solutions and applications, in “Advanced Manufacturing and Sustainable Logistics, Proceedings of the 8th International Heinz Nixdorf Symposium IHNS 2010”. In: Dangelmaier, W., Blecken, A., Delius, R., Kloepfer, S. (Eds.). Paderborn, pp. 73–84.