doi: 10.3934/jimo.2022080
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Modelling and performance analysis of shuttle-based compact storage systems under different storage policies

1. 

School of Information, Beijing Wuzi University, China

2. 

Beijing Municipal Tax Service, State Taxation Administration, China

3. 

College of Economics and Management, China Agricultural University, China

*Corresponding author: Lei Deng

Received  June 2021 Revised  March 2022 Early access May 2022

Fund Project: The first author is supported by the National Social Science Foundation of China grant 21CGL027

Short response time for order processing is important for modern warehouses, which can be potentially achieved by adopting appropriate storage policy. This paper models and analyses shuttle-based compact storage systems under random and class-based storage policies using a probability and queueing based approach. The ABC curve and the basic Economic Order Quantity model are used to determine the assignment of storage locations under class-based storage policy. The performance measures are obtained by using an iterative approach based on parametric-decomposition. The analytical model is validated against simulations and the results show our model can accurately estimate the system performance. Numerical experiments based on a real case are carried out. The results show that the best performance is likely to be provided by a class-based storage policy with a steep ABC curve and a skewed demand rate distribution. Multiple system configurations in terms of number of tiers, depth/width ratio, size of storage positions and velocity of system resources are also analyzed. The results suggest benefits of class-based storage policy regardless of the number of tiers. Moreover, the class-based storage policy outperforms the random storage policy in the system with a small depth/with ratio. However, the random storage policy is recommended for the system with deep storage lanes.

Citation: Lei Deng, Jingjie Zhao, Ruimei Wang. Modelling and performance analysis of shuttle-based compact storage systems under different storage policies. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022080
References:
[1]

K. AzadehR. De Koster and D. Roy, Robotized and automated warehouse systems: Review and recent developments, Transportation Science, 53 (2019), 917-945. 

[2]

M. BorovinšekB. Y. EkrenA. Burinskienė and T. Lerher, Multi-objective optimisation model of shuttle-based storage and retrieval system, Transport, 32 (2017), 120-137. 

[3]

M. BortoliniM. FaccioE. FerrariM. Gamberi and F. Pilati, Design of diagonal cross-aisle warehouses with class-based storage assignment strategy, The International J. Advanced Manufacturing Technology, 100 (2019), 2521-2536. 

[4]

N. BoysenD. Boywitz and F. Weidinger, Deep-lane storage of time-critical items: One-sided versus two-sided access, OR Spectrum, 40 (2018), 1141-1170.  doi: 10.1007/s00291-017-0488-9.

[5]

H. J. Carlo and I. F. A. Vis, Sequencing dynamic storage systems with multiple lifts and shuttles, International J. Production Economics, 140 (2012), 844-853. 

[6]

G. D'AntonioM. D. MaddisJ. S. BedollaP. Chiabert and F. Lombardi, Analytical models for the evaluation of deep-lane autonomous vehicle storage and retrieval system performance, The International J. Advanced Manufacturing Technology, 94 (2018), 1811-1824.  doi: 10.1007/s00170-017-0313-2.

[7]

L. Deng, L. Chen, J. Zhao and R. Wang, Modeling and performance analysis of shuttle-based compact storage systems under parallel processing policy, PLOS ONE, 16 (2021), e259773.

[8]

W. DongM. JinY. Wang and P. Kelle, Retrieval scheduling in crane-based 3D automated retrieval and storage systems with shuttles, Ann. Oper. Res., 302 (2021), 111-135.  doi: 10.1007/s10479-021-03967-8.

[9]

M. Eder, Analytical model to estimate the performance of shuttle-based storage and retrieval systems with class-based storage policy, The International J. Advanced Manufacturing Technology, 107 (2020), 2091-2106. 

[10]

B. Y. Ekren, Graph-based solution for performance evaluation of shuttle-based storage and retrieval system, International J. Production Research, 55 (2017), 6516-6526. 

[11]

B. Y. EkrenZ. Sari and T. Lerher, RWarehouse design under class-based storage policy of shuttle-based storage and retrieval system, IFAC-PapersOnLine, 48 (2015), 1152-1154. 

[12]

B. Y. Ekren and A. Akpunar, An open queuing network based tool for performance estimations in a shuttle-based storage and retrieval system, Appl. Math. Model., 89 (2021), 1678-1695.  doi: 10.1016/j.apm.2020.07.055.

[13]

B. Y. EkrenA. AkpunarZ. Sari and T. Lerher, A tool for time, variance and energy related performance estimations in a shuttle-based storage and retrieval system, Appl. Math. Model., 63 (2018), 109-127.  doi: 10.1016/j.apm.2018.06.037.

[14]

M. Fukunari and C. J. Malmborg, An efficient cycle time Model for autonomous vehicle storage and retrieval systems, International J. Production Research, 46 (2008), 3167-3184. 

[15]

Y. Ha and J. Chae, Free balancing for a shuttle-based storage and retrieval system, Simulation Modelling Practice and Theory, 82 (2018), 12-31. 

[16]

W. H. HausmanL. B. Schwarz and S. C. Graves, Optimal storage assignment in automatic warehousing system, Management Sci., 22 (1976), 629-638. 

[17]

S. S. HeraguX. CaiA. Krishnamurthy and C. J. Malmborg, Analytical models for analysis of automated warehouse material handling systems, International J. Production Research, 49 (2011), 6833-6861. 

[18]

B. JermanB. Y. EkrenM. Küçükyaşar and T. Lerher, Simulation-based performance analysis for a novel AVS/RS technology with movable lifts, Appl. Sci., 11 (2021), 2283. 

[19]

J. Jia and S. S. Heragu, Solving semi-open queuing networks, Oper. Res., 57 (2009), 391-401.  doi: 10.1287/opre.1080.0627.

[20]

T. KriehnF. SchlozK. Wehking and M. Fittinghoff, Impact of class-based storage, sequencing of retrieval requests and warehouse reorganisation on throughput of shuttle-based storage and retrieval systems, FME Transaction, 46 (2018), 320-329. 

[21]

M. KüçükyaşarB. Y. Ekren and T. Lerher, Cost and performance comparison for tier-captive and tier-to-tier SBS/RS warehouse configurations, International Transactions in Operational Research, 28 (2021), 1847-1863. 

[22]

G. L. KumawatD. RoyR. De Koster and I. Adan, Stochastic modeling of parallel process flows in intra-logistics systems: Applications in container terminals and compact storage systems, European J. Ope. Res., 290 (2021), 159-176.  doi: 10.1016/j.ejor.2020.08.006.

[23]

P. KuoA. Krishnamurthy and C. J. Malmborg, Design models for unit load storage and retrieval systems using autonomous vehicle technology and resource conserving storage and dwell point policies, Appl. Math. Modelling, 31 (2007), 2332-2346. 

[24]

B. Lei, F. Hu, Z. Jiang and H. Mu, Optimization of storage location assignment in tier-to-tier shuttle-based storage and retrieval systems based on mixed storage, Math. Problems in Engineering, (2020), 1–17.

[25]

T. Lerher, Aisle changing shuttle carriers in autonomous vehicle storage and retrieval systems, International J. Production Economics, 56 (2018), 3859-3879. 

[26]

T. LerherM. Borovinsek and M. Ficko, Parametric study of throughput performance in SBS/RS based on simulation, International J. Simulation Modelling, 16 (2017), 96-107. 

[27]

T. LerherM. Ficko and I. Palčič, Throughput performance analysis of automated vehicle storage and retrieval systems with multiple-tier shuttle vehicles, Appl. Math. Model., 91 (2021), 1004-1022.  doi: 10.1016/j.apm.2020.10.032.

[28]

Z. LiuY. WangM. JinH. Wu and W. Dong, Energy consumption model for shuttle-based storage and retrieval systems, J. Cleaner Production, 282 (2021), 124480. 

[29]

L. LuoN. Zhao and G. Lodewijks, Scheduling storage process of shuttle-based storage and retrieval systems based on reinforcement learning, Complex System Modeling and Simulation, 1 (2021), 131-144. 

[30]

C. J. Malmborg, Conceptualizing tools for autonomous vehicle storage and retrieval systems, International J. Production Research, 40 (2002), 1807-1822. 

[31]

R. ManziniR. AccorsiG. BaruffaldiT. Cennerazzo and M. Gamberi, Travel time models for deep-lane unit-load autonomous vehicle storage and retrieval system (AVS/RS), International J. Production Research, 54 (2016), 4286-4304. 

[32]

G. MarchetM. MelaciniS. Perotti and E. Tappia, Analytical model to estimate performances of autonomous vehicle storage and retrieval systems for product totes, International J. Production Research, 50 (2021), 7134-7148. 

[33]

M. MirzaeiN. Zaerpour and R. De Koster, The impact of integrated cluster-based storage allocation on parts-to-picker warehouse performance, Transportation Research Part E: Logistics and Transportation Review, 146 (2021), 102207. 

[34]

M. J. Rosenblatt and A. Eynan, Deriving the optimal boundaries for class-based automatic storage/retrieval systems, Management Sci., 35 (1989), 1519-1524. 

[35]

K. RoshanA. Shojaie and M. Javadi, Advanced allocation policy in class-based storage to improve AS/RS efficiency toward green manufacturing, International J. Environmental Science and Technology, 16 (2019), 5695-5706. 

[36]

D. RoyA. KrishnamurthyS. S. Heragu and C. J. Malmborg, Performance analysis and design trade-offs in warehouses with autonomous vehicle technology, IIE Transactions, 44 (2012), 1045-1060. 

[37]

D. RoyA. KrishnamurthyS. S. Heragu and C. J. Malmborg, A multi-tier linking approach to analyze performance of autonomous vehicle-based storage and retrieval systems, Comput. Oper. Res., 83 (2017), 173-188.  doi: 10.1016/j.cor.2017.02.012.

[38]

M. SchenoneG. ManganoS. Grimaldi and A. C. Cagliano, An approach for computing AS/R systems travel times in a class-based storage configuration, Production & Manufacturing Research, 8 (2020), 273-290. 

[39]

E. TappiaD. RoyR. De Koster and M. Melacini, Modeling, analysis, and design insights for shuttle-based compact storage systems, Transportation Science, 51 (2017), 269-295. 

[40]

W. Whitt, The queueing network analyzer, The Bell System Technical Journal, 62 (1983), 2779-2815. 

[41]

Y. WuC. ZhouW. Ma and X. T. R. Kong, Modelling and design for a shuttle-based storage and retrieval system, International J. Production Research, 58 (2020), 4808-4828. 

[42]

Y. Yu and R. De Koster, Optimal zone boundaries for two-class-based compact three-dimensional automated storage and retrieval systems, IIE Transactions, 41 (2009), 194-208. 

[43]

Y. YuR. De Koster and X. Guo, Class-based storage with a finite number of items: Using more classes is not always better, Production and Operations Management, 24 (2015), 1235-1247. 

[44]

N. ZaerpourY. Yu and R. De Koster, Optimal two-class-based storage in a live-cube compact storage system, IISE Transactions, 49 (2017), 653-668. 

[45]

N. ZaerpourY. Yu and R. De Koster, Response time analysis of a live-cube compact storage system with two storage classes, IISE Transactions, 29 (2017), 461-480. 

[46]

X. ZhaoR. ZhangN. ZhangY. WangM. Jin and S. Mou, Analysis of the shuttle-based storage and retrieval system, IEEE Access, 8 (2020), 146154-146165. 

[47]

B. ZouX. XuY. Y. Gong and R. De Koster, Modeling parallel movement of lifts and vehicles in tier-captive vehicle-based warehousing systems, European J. Oper. Res., 254 (2016), 51-67.  doi: 10.1016/j.ejor.2016.03.039.

show all references

References:
[1]

K. AzadehR. De Koster and D. Roy, Robotized and automated warehouse systems: Review and recent developments, Transportation Science, 53 (2019), 917-945. 

[2]

M. BorovinšekB. Y. EkrenA. Burinskienė and T. Lerher, Multi-objective optimisation model of shuttle-based storage and retrieval system, Transport, 32 (2017), 120-137. 

[3]

M. BortoliniM. FaccioE. FerrariM. Gamberi and F. Pilati, Design of diagonal cross-aisle warehouses with class-based storage assignment strategy, The International J. Advanced Manufacturing Technology, 100 (2019), 2521-2536. 

[4]

N. BoysenD. Boywitz and F. Weidinger, Deep-lane storage of time-critical items: One-sided versus two-sided access, OR Spectrum, 40 (2018), 1141-1170.  doi: 10.1007/s00291-017-0488-9.

[5]

H. J. Carlo and I. F. A. Vis, Sequencing dynamic storage systems with multiple lifts and shuttles, International J. Production Economics, 140 (2012), 844-853. 

[6]

G. D'AntonioM. D. MaddisJ. S. BedollaP. Chiabert and F. Lombardi, Analytical models for the evaluation of deep-lane autonomous vehicle storage and retrieval system performance, The International J. Advanced Manufacturing Technology, 94 (2018), 1811-1824.  doi: 10.1007/s00170-017-0313-2.

[7]

L. Deng, L. Chen, J. Zhao and R. Wang, Modeling and performance analysis of shuttle-based compact storage systems under parallel processing policy, PLOS ONE, 16 (2021), e259773.

[8]

W. DongM. JinY. Wang and P. Kelle, Retrieval scheduling in crane-based 3D automated retrieval and storage systems with shuttles, Ann. Oper. Res., 302 (2021), 111-135.  doi: 10.1007/s10479-021-03967-8.

[9]

M. Eder, Analytical model to estimate the performance of shuttle-based storage and retrieval systems with class-based storage policy, The International J. Advanced Manufacturing Technology, 107 (2020), 2091-2106. 

[10]

B. Y. Ekren, Graph-based solution for performance evaluation of shuttle-based storage and retrieval system, International J. Production Research, 55 (2017), 6516-6526. 

[11]

B. Y. EkrenZ. Sari and T. Lerher, RWarehouse design under class-based storage policy of shuttle-based storage and retrieval system, IFAC-PapersOnLine, 48 (2015), 1152-1154. 

[12]

B. Y. Ekren and A. Akpunar, An open queuing network based tool for performance estimations in a shuttle-based storage and retrieval system, Appl. Math. Model., 89 (2021), 1678-1695.  doi: 10.1016/j.apm.2020.07.055.

[13]

B. Y. EkrenA. AkpunarZ. Sari and T. Lerher, A tool for time, variance and energy related performance estimations in a shuttle-based storage and retrieval system, Appl. Math. Model., 63 (2018), 109-127.  doi: 10.1016/j.apm.2018.06.037.

[14]

M. Fukunari and C. J. Malmborg, An efficient cycle time Model for autonomous vehicle storage and retrieval systems, International J. Production Research, 46 (2008), 3167-3184. 

[15]

Y. Ha and J. Chae, Free balancing for a shuttle-based storage and retrieval system, Simulation Modelling Practice and Theory, 82 (2018), 12-31. 

[16]

W. H. HausmanL. B. Schwarz and S. C. Graves, Optimal storage assignment in automatic warehousing system, Management Sci., 22 (1976), 629-638. 

[17]

S. S. HeraguX. CaiA. Krishnamurthy and C. J. Malmborg, Analytical models for analysis of automated warehouse material handling systems, International J. Production Research, 49 (2011), 6833-6861. 

[18]

B. JermanB. Y. EkrenM. Küçükyaşar and T. Lerher, Simulation-based performance analysis for a novel AVS/RS technology with movable lifts, Appl. Sci., 11 (2021), 2283. 

[19]

J. Jia and S. S. Heragu, Solving semi-open queuing networks, Oper. Res., 57 (2009), 391-401.  doi: 10.1287/opre.1080.0627.

[20]

T. KriehnF. SchlozK. Wehking and M. Fittinghoff, Impact of class-based storage, sequencing of retrieval requests and warehouse reorganisation on throughput of shuttle-based storage and retrieval systems, FME Transaction, 46 (2018), 320-329. 

[21]

M. KüçükyaşarB. Y. Ekren and T. Lerher, Cost and performance comparison for tier-captive and tier-to-tier SBS/RS warehouse configurations, International Transactions in Operational Research, 28 (2021), 1847-1863. 

[22]

G. L. KumawatD. RoyR. De Koster and I. Adan, Stochastic modeling of parallel process flows in intra-logistics systems: Applications in container terminals and compact storage systems, European J. Ope. Res., 290 (2021), 159-176.  doi: 10.1016/j.ejor.2020.08.006.

[23]

P. KuoA. Krishnamurthy and C. J. Malmborg, Design models for unit load storage and retrieval systems using autonomous vehicle technology and resource conserving storage and dwell point policies, Appl. Math. Modelling, 31 (2007), 2332-2346. 

[24]

B. Lei, F. Hu, Z. Jiang and H. Mu, Optimization of storage location assignment in tier-to-tier shuttle-based storage and retrieval systems based on mixed storage, Math. Problems in Engineering, (2020), 1–17.

[25]

T. Lerher, Aisle changing shuttle carriers in autonomous vehicle storage and retrieval systems, International J. Production Economics, 56 (2018), 3859-3879. 

[26]

T. LerherM. Borovinsek and M. Ficko, Parametric study of throughput performance in SBS/RS based on simulation, International J. Simulation Modelling, 16 (2017), 96-107. 

[27]

T. LerherM. Ficko and I. Palčič, Throughput performance analysis of automated vehicle storage and retrieval systems with multiple-tier shuttle vehicles, Appl. Math. Model., 91 (2021), 1004-1022.  doi: 10.1016/j.apm.2020.10.032.

[28]

Z. LiuY. WangM. JinH. Wu and W. Dong, Energy consumption model for shuttle-based storage and retrieval systems, J. Cleaner Production, 282 (2021), 124480. 

[29]

L. LuoN. Zhao and G. Lodewijks, Scheduling storage process of shuttle-based storage and retrieval systems based on reinforcement learning, Complex System Modeling and Simulation, 1 (2021), 131-144. 

[30]

C. J. Malmborg, Conceptualizing tools for autonomous vehicle storage and retrieval systems, International J. Production Research, 40 (2002), 1807-1822. 

[31]

R. ManziniR. AccorsiG. BaruffaldiT. Cennerazzo and M. Gamberi, Travel time models for deep-lane unit-load autonomous vehicle storage and retrieval system (AVS/RS), International J. Production Research, 54 (2016), 4286-4304. 

[32]

G. MarchetM. MelaciniS. Perotti and E. Tappia, Analytical model to estimate performances of autonomous vehicle storage and retrieval systems for product totes, International J. Production Research, 50 (2021), 7134-7148. 

[33]

M. MirzaeiN. Zaerpour and R. De Koster, The impact of integrated cluster-based storage allocation on parts-to-picker warehouse performance, Transportation Research Part E: Logistics and Transportation Review, 146 (2021), 102207. 

[34]

M. J. Rosenblatt and A. Eynan, Deriving the optimal boundaries for class-based automatic storage/retrieval systems, Management Sci., 35 (1989), 1519-1524. 

[35]

K. RoshanA. Shojaie and M. Javadi, Advanced allocation policy in class-based storage to improve AS/RS efficiency toward green manufacturing, International J. Environmental Science and Technology, 16 (2019), 5695-5706. 

[36]

D. RoyA. KrishnamurthyS. S. Heragu and C. J. Malmborg, Performance analysis and design trade-offs in warehouses with autonomous vehicle technology, IIE Transactions, 44 (2012), 1045-1060. 

[37]

D. RoyA. KrishnamurthyS. S. Heragu and C. J. Malmborg, A multi-tier linking approach to analyze performance of autonomous vehicle-based storage and retrieval systems, Comput. Oper. Res., 83 (2017), 173-188.  doi: 10.1016/j.cor.2017.02.012.

[38]

M. SchenoneG. ManganoS. Grimaldi and A. C. Cagliano, An approach for computing AS/R systems travel times in a class-based storage configuration, Production & Manufacturing Research, 8 (2020), 273-290. 

[39]

E. TappiaD. RoyR. De Koster and M. Melacini, Modeling, analysis, and design insights for shuttle-based compact storage systems, Transportation Science, 51 (2017), 269-295. 

[40]

W. Whitt, The queueing network analyzer, The Bell System Technical Journal, 62 (1983), 2779-2815. 

[41]

Y. WuC. ZhouW. Ma and X. T. R. Kong, Modelling and design for a shuttle-based storage and retrieval system, International J. Production Research, 58 (2020), 4808-4828. 

[42]

Y. Yu and R. De Koster, Optimal zone boundaries for two-class-based compact three-dimensional automated storage and retrieval systems, IIE Transactions, 41 (2009), 194-208. 

[43]

Y. YuR. De Koster and X. Guo, Class-based storage with a finite number of items: Using more classes is not always better, Production and Operations Management, 24 (2015), 1235-1247. 

[44]

N. ZaerpourY. Yu and R. De Koster, Optimal two-class-based storage in a live-cube compact storage system, IISE Transactions, 49 (2017), 653-668. 

[45]

N. ZaerpourY. Yu and R. De Koster, Response time analysis of a live-cube compact storage system with two storage classes, IISE Transactions, 29 (2017), 461-480. 

[46]

X. ZhaoR. ZhangN. ZhangY. WangM. Jin and S. Mou, Analysis of the shuttle-based storage and retrieval system, IEEE Access, 8 (2020), 146154-146165. 

[47]

B. ZouX. XuY. Y. Gong and R. De Koster, Modeling parallel movement of lifts and vehicles in tier-captive vehicle-based warehousing systems, European J. Oper. Res., 254 (2016), 51-67.  doi: 10.1016/j.ejor.2016.03.039.

Figure 1.  Shuttle-based compact storage system: (a) isometric view [39] and (b) top view
Figure 2.  Operational process of shuttle-based compact storage system under parallel processing policy in the case that (a) shuttle dwells at interior point, (b) and (d) shuttle dwells at l/u point, (c) shuttle dwells at an interior point which is not located at the lane of the retrieval load and (e) shuttle dwells at an interior point which is located at the lane of the retrieval load
Figure 3.  A possible way to assign storage locations
Figure 4.  Queueing network model for a single-tier system
Figure 5.  Queueing network model for a multitier system
Figure 6.  Solution approach of the integrated queueing network
Figure 7.  Distribution of absolute errors for performance measures
Figure 8.  Performance of class-based storage policy under different skewness parameters
Figure 9.  Comparison of different storage policies
Figure 10.  Comparison of random and class-base storage policies in system with different number of tiers
Figure 11.  Comparison of random and class-based storage policies in system with different depth/width ratios
Figure 12.  Comparison of system performance under different size of storage position
Figure 13.  Comparison of system performance under different velocity of system resources
Figure 14.  Side view sketch of the storage area
Figure 15.  Flowchart to estimate the departure process from tiers and the lift
Figure 16.  Flowchart of simulation model
Table 1.  Main notations
Notation Description
$ \lambda_r $, $ \lambda_s $ Arrival rate of retrieval and storage transactions to the system
$ \lambda_{r_t} $, $ \lambda_{s_t} $, $ c_{l, s_t}^a $, $ c_{l, r_t}^a $ Arrival rate of retrieval and storage transactions to tth tier
$ N_c $, $ N_l $, $ N_S $, $ N_T $ Number of storage columns and lanes at each side of cross-aisle, shuttles dedicated to a single tier and tiers
$ u_w $, $ u_d $, $ u_h $ Unit width, depth and height per storage position
$ t_c $, $ t_{sh} $, $ t_l $ Constant time required for transfer car, shuttle and lift to load/unload the shuttle or unit load
$ v_c $, $ v_{sh} $, $ v_l $ Constant velocity of transfer car, shuttle and lift
M The number of product classes
$ P_i $ Probability that a class i product is to be stored or retrieved (e.g., the normalized demand rate of product class i)
$ D(i) $ Ratio of cumulative demand for all product classes up to class i to total demand in the system
$ I(i) $ Percentage of inventoried products
s Skewness parameter
$ t_w^m $, $ t_w $ One-way travel time for the transfer car from l/u point to the most distant storage lane and any storage lane
$ t_h^m $, $ t_h $ One-way travel time for the transfer car from l/u point to the most distant storage lane and any storage lane
$ D^0 $, $ D^1 $ Cumulative demand related to the first and the second boundaries
$ R^i $ Boundary parameter that determines the maximum travel time from locations of storage zone for product class i to the I/O point
$ Q_1 $, $ Q_2 $ the external queue of transactions and idle shuttles in the single-tier model
$ L_{it} $ Number of storage lanes at each side of cross-aisle assigned to product class i at tth tier
$ P_{it} $ Probability that the destination tier of a class i product is tth tier
$ P_t $ Probability of tth tier being visited
$ P_{i|t} $ Conditional probability of a class i product being stored or retrieved in tth tier given that tth tier is visited
$ t_{sh1} $, $ t_{sh2} $ Expected travel time related to shuttles
$ t_{c1} $, $ t_{c2} $ Expected travel time related to the transfer car
$ t_{l1} $, $ t_{l2} $, $ t_{l3} $ Expected travel time related to the lift
$ P_s $, $ P_r $ Probability of storage and retrieval transaction
$ P_{sin} $, $ P_{sio} $ Probability that a transaction is assigned to shuttle dwelling at interior or l/u point
$ P_{cin} $, $ P_{cio} $ Probability that the transfer car dwells at interior or l/u point
$ P_{lin} $, $ P_{lio} $ Probability of the lift dwelling at the first tier or any other tier
$ P_{ss} $, $ P_{sd} $ Probability that the assigned shuttle is or is not present in the lane where the retrieval load is present
$ \mu_{l, s_t}^{-1} $, $ \mu_{l, r_t}^{-1} $, $ c_{l, s_t} $, $ c_{l, r_t} $ Expected service time and their corresponding SCV of the lift for each transaction class
$ c_l^d $, $ c_t^d $ SCV of inter-departure times from the lift and from tier t
$ c_{l, r_t}^a $, $ c_{l, s_t}^a $ SCV of inter-arrival times for retrieval transactions to the lift and inter-arrival times for storage transactions to tier t
$ \mu_t(k_t)^{-1} $ Load-dependent service time of closed single-tier model at tier t
$ U_{sh_t} $, $ U_{c_t} $, $ U_l $ Average utilizations of the shuttle and transfer car at tier t, and of the lift
$ Q_{j_t} $, $ Q_{sh_t} $ Mean queue length of transactions and free shuttles at tier t
$ Q_j $, $ Q_{sh} $ Average number of transactions and shuttles waiting across all tiers
$ U_{sh} $, $ U_c $ Average utilizations of the shuttle and transfer car across all tiers
$ Q_l $ Mean queue length of transactions at the lift
$ W_l $ Expected waiting time at the lift
$ E[T] $ Expected throughput time of the system
$ E[T_s] $, $ E[T_r] $ Expected throughput time of the system for storage and retrieval transactions
$ E[T_{s_t}] $, $ E[T_{r_t}] $ Expected throughput time at tier t for storage and retrieval transactions
$ E[T_{s_l}] $, $ E[T_{r_l}] $ Expected throughput time at the lift for storage and retrieval transactions
$ \varepsilon $ Absolute relative error
A, S Result of analytical and simulation model
$ I(P) $ Improvement percentage of the class-based storage policy over the random storage policy
$ E[T_C] $, $ E[T_R] $ Expected system throughput time under class-based and random storage policy
Notation Description
$ \lambda_r $, $ \lambda_s $ Arrival rate of retrieval and storage transactions to the system
$ \lambda_{r_t} $, $ \lambda_{s_t} $, $ c_{l, s_t}^a $, $ c_{l, r_t}^a $ Arrival rate of retrieval and storage transactions to tth tier
$ N_c $, $ N_l $, $ N_S $, $ N_T $ Number of storage columns and lanes at each side of cross-aisle, shuttles dedicated to a single tier and tiers
$ u_w $, $ u_d $, $ u_h $ Unit width, depth and height per storage position
$ t_c $, $ t_{sh} $, $ t_l $ Constant time required for transfer car, shuttle and lift to load/unload the shuttle or unit load
$ v_c $, $ v_{sh} $, $ v_l $ Constant velocity of transfer car, shuttle and lift
M The number of product classes
$ P_i $ Probability that a class i product is to be stored or retrieved (e.g., the normalized demand rate of product class i)
$ D(i) $ Ratio of cumulative demand for all product classes up to class i to total demand in the system
$ I(i) $ Percentage of inventoried products
s Skewness parameter
$ t_w^m $, $ t_w $ One-way travel time for the transfer car from l/u point to the most distant storage lane and any storage lane
$ t_h^m $, $ t_h $ One-way travel time for the transfer car from l/u point to the most distant storage lane and any storage lane
$ D^0 $, $ D^1 $ Cumulative demand related to the first and the second boundaries
$ R^i $ Boundary parameter that determines the maximum travel time from locations of storage zone for product class i to the I/O point
$ Q_1 $, $ Q_2 $ the external queue of transactions and idle shuttles in the single-tier model
$ L_{it} $ Number of storage lanes at each side of cross-aisle assigned to product class i at tth tier
$ P_{it} $ Probability that the destination tier of a class i product is tth tier
$ P_t $ Probability of tth tier being visited
$ P_{i|t} $ Conditional probability of a class i product being stored or retrieved in tth tier given that tth tier is visited
$ t_{sh1} $, $ t_{sh2} $ Expected travel time related to shuttles
$ t_{c1} $, $ t_{c2} $ Expected travel time related to the transfer car
$ t_{l1} $, $ t_{l2} $, $ t_{l3} $ Expected travel time related to the lift
$ P_s $, $ P_r $ Probability of storage and retrieval transaction
$ P_{sin} $, $ P_{sio} $ Probability that a transaction is assigned to shuttle dwelling at interior or l/u point
$ P_{cin} $, $ P_{cio} $ Probability that the transfer car dwells at interior or l/u point
$ P_{lin} $, $ P_{lio} $ Probability of the lift dwelling at the first tier or any other tier
$ P_{ss} $, $ P_{sd} $ Probability that the assigned shuttle is or is not present in the lane where the retrieval load is present
$ \mu_{l, s_t}^{-1} $, $ \mu_{l, r_t}^{-1} $, $ c_{l, s_t} $, $ c_{l, r_t} $ Expected service time and their corresponding SCV of the lift for each transaction class
$ c_l^d $, $ c_t^d $ SCV of inter-departure times from the lift and from tier t
$ c_{l, r_t}^a $, $ c_{l, s_t}^a $ SCV of inter-arrival times for retrieval transactions to the lift and inter-arrival times for storage transactions to tier t
$ \mu_t(k_t)^{-1} $ Load-dependent service time of closed single-tier model at tier t
$ U_{sh_t} $, $ U_{c_t} $, $ U_l $ Average utilizations of the shuttle and transfer car at tier t, and of the lift
$ Q_{j_t} $, $ Q_{sh_t} $ Mean queue length of transactions and free shuttles at tier t
$ Q_j $, $ Q_{sh} $ Average number of transactions and shuttles waiting across all tiers
$ U_{sh} $, $ U_c $ Average utilizations of the shuttle and transfer car across all tiers
$ Q_l $ Mean queue length of transactions at the lift
$ W_l $ Expected waiting time at the lift
$ E[T] $ Expected throughput time of the system
$ E[T_s] $, $ E[T_r] $ Expected throughput time of the system for storage and retrieval transactions
$ E[T_{s_t}] $, $ E[T_{r_t}] $ Expected throughput time at tier t for storage and retrieval transactions
$ E[T_{s_l}] $, $ E[T_{r_l}] $ Expected throughput time at the lift for storage and retrieval transactions
$ \varepsilon $ Absolute relative error
A, S Result of analytical and simulation model
$ I(P) $ Improvement percentage of the class-based storage policy over the random storage policy
$ E[T_C] $, $ E[T_R] $ Expected system throughput time under class-based and random storage policy
Table 2.  Service time expressions for nodes 1, 2, 3, 5 and 6
Node Mean service time
1 $ t_{shi1} $
2 $ P_{sd} t_{sh1}+P_{ss} (t_{sh1}+t_{sh2}+t_{sh}) $
3 $ t_{sh} $
5 $ t_{sh1}+t_{sh} $
6 $ t_{sh} $
Node Mean service time
1 $ t_{shi1} $
2 $ P_{sd} t_{sh1}+P_{ss} (t_{sh1}+t_{sh2}+t_{sh}) $
3 $ t_{sh} $
5 $ t_{sh1}+t_{sh} $
6 $ t_{sh} $
Table 3.  Service time expressions for node 4
Scenario Transaction type Dwell point Same lane Probability Mean service time
Shuttle Transfer car Setup phase Processing phase
1 Storage Interior l/u $ P_s \cdot P_{sin} \cdot P_{cio} $ $ t_{c1} $ $ 4t_c+2t_{c1}+t_{sh} $
2 Storage Interior Interior $ P_s \cdot P_{sin} \cdot P_{cin} $ $ t_{c2} $ $ 4t_c+2t_{c1}+t_{sh} $
3 Storage l/u l/u $ P_s \cdot P_{sio} \cdot P_{cio} $ 0 $ 2t_c+t_{c1} $
4 Storage l/u Interior $ P_s \cdot P_{sio} \cdot P_{cin} $ $ t_{c1} $ $ 2t_c+t_{c1} $
5 Retrieval Interior l/u yes $ P_r \cdot P_{sin} \cdot P_{cio} \cdot P_{ss} $ $ t_{c1} $ $ 2t_c+t_{c1} $
6 Retrieval Interior Interior yes $ P_r \cdot P_{sin} \cdot P_{cin} \cdot P_{ss} $ $ t_{c2} $ $ 2t_c+t_{c1} $
7 Retrieval Interior l/u no $ P_r \cdot P_{sin} \cdot P_{cio} \cdot P_{sd} $ $ t_{c1} $ $ 4t_c+t_{c2}+2t_{sh1}+t_{sh}+t_{c1} $
8 Retrieval Interior Interior no $ P_r \cdot P_{sin} \cdot P_{cin} \cdot P_{sd} $ $ t_{c2} $ $ 4t_c+t_{c2}+2t_{sh1}+t_{sh}+t_{c1} $
9 Retrieval l/u l/u $ P_r \cdot P_{sio} \cdot P_{cio} $ 0 $ 4t_c+2t_{c1}+2t_{sh1}+t_{sh} $
10 Retrieval l/u Interior $ P_r \cdot P_{sio} \cdot P_{cin} $ $ t_{c1} $ $ 4t_c+2t_{c1}+2t_{sh1}+t_{sh} $
Scenario Transaction type Dwell point Same lane Probability Mean service time
Shuttle Transfer car Setup phase Processing phase
1 Storage Interior l/u $ P_s \cdot P_{sin} \cdot P_{cio} $ $ t_{c1} $ $ 4t_c+2t_{c1}+t_{sh} $
2 Storage Interior Interior $ P_s \cdot P_{sin} \cdot P_{cin} $ $ t_{c2} $ $ 4t_c+2t_{c1}+t_{sh} $
3 Storage l/u l/u $ P_s \cdot P_{sio} \cdot P_{cio} $ 0 $ 2t_c+t_{c1} $
4 Storage l/u Interior $ P_s \cdot P_{sio} \cdot P_{cin} $ $ t_{c1} $ $ 2t_c+t_{c1} $
5 Retrieval Interior l/u yes $ P_r \cdot P_{sin} \cdot P_{cio} \cdot P_{ss} $ $ t_{c1} $ $ 2t_c+t_{c1} $
6 Retrieval Interior Interior yes $ P_r \cdot P_{sin} \cdot P_{cin} \cdot P_{ss} $ $ t_{c2} $ $ 2t_c+t_{c1} $
7 Retrieval Interior l/u no $ P_r \cdot P_{sin} \cdot P_{cio} \cdot P_{sd} $ $ t_{c1} $ $ 4t_c+t_{c2}+2t_{sh1}+t_{sh}+t_{c1} $
8 Retrieval Interior Interior no $ P_r \cdot P_{sin} \cdot P_{cin} \cdot P_{sd} $ $ t_{c2} $ $ 4t_c+t_{c2}+2t_{sh1}+t_{sh}+t_{c1} $
9 Retrieval l/u l/u $ P_r \cdot P_{sio} \cdot P_{cio} $ 0 $ 4t_c+2t_{c1}+2t_{sh1}+t_{sh} $
10 Retrieval l/u Interior $ P_r \cdot P_{sio} \cdot P_{cin} $ $ t_{c1} $ $ 4t_c+2t_{c1}+2t_{sh1}+t_{sh} $
Table 4.  Service time expressions for nodes 7
Transaction type Mean service time
Storage $ 2t_l+P_{lin} (t_{l1}+t_{l2} )+P_{lio} t_{l2} $
Retrieval $ 2t_l+P_{lin} (t_{l2}+t_{l3} )+2P_{lio} t_{l2} $
Transaction type Mean service time
Storage $ 2t_l+P_{lin} (t_{l1}+t_{l2} )+P_{lio} t_{l2} $
Retrieval $ 2t_l+P_{lin} (t_{l2}+t_{l3} )+2P_{lio} t_{l2} $
Table 5.  System parameters related to the real case
Variable Description Value
$ u_w $ Unit width per storage position 1.47m
$ u_d $ Unit depth per storage position 0.9m
$ u_h $ Unit height per storage position 2m
$ t_c $, $ t_{sh} $, $ t_l $ The transfer car, shuttle and lift loading/unloading time 3.5s; 6s; 8s
$ v_c $, $ v_{sh} $ Constant velocity of the transfer car and shuttle 1m/s
$ v_l $ Constant velocity of the lift 0.9m/s
Variable Description Value
$ u_w $ Unit width per storage position 1.47m
$ u_d $ Unit depth per storage position 0.9m
$ u_h $ Unit height per storage position 2m
$ t_c $, $ t_{sh} $, $ t_l $ The transfer car, shuttle and lift loading/unloading time 3.5s; 6s; 8s
$ v_c $, $ v_{sh} $ Constant velocity of the transfer car and shuttle 1m/s
$ v_l $ Constant velocity of the lift 0.9m/s
Table 6.  Abbreviations
Abbreviations Description
AVS/RS Autonomous vehicle–based storage and retrieval systems
AS/RS Automatic storage and retrieval systems
SBS/RS Shuttle-based storage/retrieval systems
SOQN Semi-open queuing network
ABC Pareto-demand
EOQ Economic order quantity
POSC Point-of-service-completion
FCFS First-come-first-served
l/u Load/unload
I/O Input/output
IS Infinite-servers
SCV Squared coefficient variations
NADA Network aggregation dis-aggregation approach
dw Depth/width ratio
Abbreviations Description
AVS/RS Autonomous vehicle–based storage and retrieval systems
AS/RS Automatic storage and retrieval systems
SBS/RS Shuttle-based storage/retrieval systems
SOQN Semi-open queuing network
ABC Pareto-demand
EOQ Economic order quantity
POSC Point-of-service-completion
FCFS First-come-first-served
l/u Load/unload
I/O Input/output
IS Infinite-servers
SCV Squared coefficient variations
NADA Network aggregation dis-aggregation approach
dw Depth/width ratio
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