# American Institute of Mathematical Sciences

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
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##### References:
Shuttle-based compact storage system: (a) isometric view [39] and (b) top view
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
A possible way to assign storage locations
Queueing network model for a single-tier system
Queueing network model for a multitier system
Solution approach of the integrated queueing network
Distribution of absolute errors for performance measures
Performance of class-based storage policy under different skewness parameters
Comparison of different storage policies
Comparison of random and class-base storage policies in system with different number of tiers
Comparison of random and class-based storage policies in system with different depth/width ratios
Comparison of system performance under different size of storage position
Comparison of system performance under different velocity of system resources
Side view sketch of the storage area
Flowchart to estimate the departure process from tiers and the lift
Flowchart of simulation model
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
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 $t_{sh}$
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 $P_r \cdot P_{sio} \cdot P_{cin}$ $t_{c1}$ $4t_c+2t_{c1}+2t_{sh1}+t_{sh}$
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 $2t_l+P_{lin} (t_{l2}+t_{l3} )+2P_{lio} t_{l2}$
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
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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