Retrieval–traveltime model for freefallflowrack automated storage and retrieval system
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Abstract
Automated storage and retrieval systems (AS/RSs) are material handling systems that are frequently used in manufacturing and distribution centers. The modelling of the retrieval–travel time of an AS/RS (expected product delivery time) is practically important, because it allows us to evaluate and improve the system throughput. The freefallflowrack AS/RS has emerged as a new technology for drug distribution. This system is a new variation of flowrack AS/RS that uses an operator or a single machine for storage operations, and uses a combination between the freefall movement and a transport conveyor for retrieval operations. The main contribution of this paper is to develop an analytical model of the expected retrieval–travel time for the freefall flowrack under a dedicated storage assignment policy. The proposed model, which is based on a continuous approach, is compared for accuracy, via simulation, with discrete model. The obtained results show that the maximum deviation between the continuous model and the simulation is less than 5%, which shows the accuracy of our model to estimate the retrieval time. The analytical model is useful to optimise the dimensions of the rack, assess the system throughput, and evaluate different storage policies.
Keywords
Automated storage/retrieval systems (AS/RS) Freefallflowrack AS/RS (FFflowrack AS/RS) Travel time Throughput Drug distribution ModellingIntroduction
Over the last years, automated storage/retrieval systems (AS/RSs) have become the focus of many companies and distribution centers, through the major advantages, they offer such as reducing delivery time, savings in labour costs and floor space, faster and more reliable operation of the warehouse, the decrease of handling errors, and improved throughput level. AS/RS are mainly introduced to eliminate walking that accounted for 70% of manual retrieval time (Lee 1997). AS/RS are widely used in a variety of industries, as an essential component in a flexible manufacturing system (FMS) (Van Den Berg et al. 2000). Moreover, AS/RS can be classified as mechatronic systems (Sharma and Sharma 2015).
According to the Material Handling Institute of America, an automated storage and retrieval system (AS/RS) consists of a variety of material and control systems that handle store and retrieve loads with precision, accuracy, and great speed (Material Handling Institute 1977). In general, an AS/RS includes the following main components: storage racks, aisles, storage and retrieval (S/R) machines, pickup/dropoff (P/D) stations, and the control system. Racks are composed of bins that can store loads. Aisles are the spaces between the racks. Storage and retrieval machines are fully automated cranes that can pickup and dropoff loads. Pickup/dropoff (P/D) stations are used for incoming and outgoing loads. The control system is the responsible for managing storage and retrieval operations.
AS/RS structures are designed according to their different size, weight, and volume of loads to be handled, and characterized of the warehouse. These different structures include unitload AS/RS, miniload AS/RS, manonboard AS/RS, carousel AS/RS, deeplane AS/RS, multiaisle AS/RS, mobilerack AS/RS, and flowrack AS/RS.
To reduce the installation cost and to increase the performances of the AS/RSs, a new variant of flowrack AS/RS has been developed, which consists of a freefallflowrack AS/RS (FFflowrack AS/RS). This new kind of AS/RS was introduced by Mekapharm (Mekapharm 2016). It can operate without any storage/retrieval machines, which are replaced by the combination of two movements: vertical movement performed by the free fall of items and horizontal movement achieved by a conveyor. As opposed to the traditional flowrack AS/RS, the system used (S/R) machines that travel simultaneously horizontally and vertically.
The main objective of this paper is to develop an analytical expression of expected retrieval–travel time. This research will be useful for the decisionmaker to optimise the rack dimensions, evaluate the system throughput, and compare different storage policies. To evaluate the developed expression (based on a continuous approach) for its accuracy, we compare the results obtained from the expression with those obtained from the computer simulations (based on a discrete model).
The expected travel time of S/R machine is considered as the most important factor when evaluating the performance of an AS/R system. Since 1976, research on the modelling of travel time has been widely investigated. For this reason, several researchers have produced many articles in this area. Sarker and Babu (1995) and Roodbergen and Vis (2009) had presented a survey of AS/RS.
The literature on the modelling of the travel time for a unitload AS/RS shows a variety of approaches. Hausman et al. (1976) were the first who have proposed the traveltime model of the singlecommand cycle. The authors assumed a squareintime continuous rack. They were considered random, full turnover, and classbased storage assignment policies. This study was extended by Graves et al. (1977) by modelling the travel time model of dualcommand cycle for the same assumptions. The most interesting approach to this issue has been proposed by Bozer and White (1984). They developed the expected single and dual travel time models of rectangular rack under a random storage and for different positions input/output point. In addition, various dwellpoint strategies for the storage/retrieval machine were examined.
These researchers do not take into account all operational characteristics of an AS/RS such as acceleration and deceleration of the S/R machine and the rack configuration. Therefore, Hwang and Lee (1990) presented traveltime models that integrated the operating characteristics of the S/R machine, including constant acceleration and deceleration rate, and the maximumvelocity restriction. Chang et al. (1995) continued Bozer and White (1984) study by including acceleration and deceleration instead of assuming constant speed. This work was extended by Wen et al. (2001); they have considered classbased and fullturnoverbased storage assignment policies. In the paper of Chang and Wen (1997), the authors studied the impact of the rack configuration on the expected travel time. Recently, Bortolini et al. (2015) extended the analytical models that are already proposed by the literature to compute the expected travel time of (AS/RS) in threeclassbased storage. The authors determined the analytical closed form of the mean travel time for both the singlecommand (SC) and the dualcommand (DC) cycles varying the warehouse shape factor and the ABC turnover curve.
For the flowrack AS/RS, much research has been done. By the inspiration of Bozer and White (1984) previous work, Sari et al. (2005) presented mathematical models for the expected travel time for a flowrack AS/RS, which used two (S/R) machines. The first model is developed using a continuous approach and compared with a discrete model for accuracy via simulation. The authors conclude that the expressions based on the continuous approach are extremely practical due to the difference in computation time. After that, Sari (2010) performed a comparative study between unitload AS/RS and flowrack AS/RS. The author considered two comparison parameters: space‘s use and travel time. Xu et al. (2015) developed the dual cycle and quadruple cycle travel time models for the traditional singledeep singleAS/RS and doubledeep dualshuttle AS/RS. Through a comparison between the two systems, the authors find that the efficiency of the system has a significant improvement using dualshuttle S/R machine. In a recent paper, Hachemi and Besombes (2013) extended the problem of retrieval sequencing for flowrack AS/RS by integrating the product expiry date. They introduced an optimisation method as decision process, which performs a realtime optimisation into two phases and formulated as an integer program.
To decrease the travel time for the flowrack AS/RS, many researchers have proposed various methods. Sari et al. (2007) studied the impact of P/D stations and restoring conveyor locations on expected retrievaltime and classified their optimal positions. MeghelliGaouar and Sari (2010) used a two classbased storage policy for the flowrack AS/RS, where each item is assigned to the same bin as closely as possible to the P/D point. This work was extended by Bessenouci et al. (2012) by developing two metaheuristic algorithms (tabu search and simulated annealing) applied to the control of the S/R machine. Hachemi and Alla (2008) presented an optimisation method of retrieval sequencing where the AS/RS was depicted by a coloured Petri net model. By considering dual cycle coupled with lower midpoint S/R crane dwellpoint policy, Xu et al. (2017) has developed a continuous traveltime model for DC in 3D compact storage system; then, he has used the analytical models to optimise system dimensions.
Recently, Sari and Bessnouci (2012) proposed a new kind of flowrack AS/RS: using a single machine for storage and retrieval operations instead of two machines. They developed analyticaltraveltime models of the storage and retrieval machine under randomized storage assignment. Two dwellpoint positions were considered. Otherwise, De Koster et al. (2008) used a lifting mechanism on the opposite face of the S/R machine for a flowrack AS/RS. They presented a closedform expression of expected retrieval–travel time for singlecommand cycles and derived an approximate traveltime expression for dualcommand cycles of the system. In addition, Chen et al. (2015) designed a bidirectionalflowrack (BFR) in which bins in adjacent columns slope to opposite directions. They have developed a traveltime model for BFR systems. An extension of Sari and Bessnouci (2012)work was proposed by Hamzaoui and Sari (2015), which determined optimal dimensions of the same AS/RS design minimizing expected travel times of the S/R machine using an enumeration technique.
Other studies have been undertaken to present travel time for many AS/RS types, such as the autonomous vehicle S/R system (AVS/RS), multiaisle AS/RS, splitplatform AS/RS (SPAS/RS), and the multiple intheaisle pick positions (MIAPPAS/RS).
The first study on the performances of an AVS/RS was proposed by Malmborg (2002), who modelled the travel time of an AVS/RS according to the number of aisles, tiers, vehicles, and lifts, by considering a random storage policy, and a tiertotier configuration. Marchet et al. (2012) criticized the previously mentioned work for the fact that they did not take into account the transaction cycle time and waiting times in the calculation of analytical model, which have been integrated to the model proposed by Marchet et al. (2012). Manzini et al. (2016) present analytic models for expected cycle time of a deeplane unitload AVS/RS that performing single and dualcommand cycles.
Ghomri et al. (2009) proposed an analytical expression for the average time of a single cycle of a multiaisle AS/RS, while Lerher et al. (2010) presented analyticaltraveltime models of multiaisle AS/RS for the computation of travel time for both the singlecommand (SC) and the dualcommand (DC) cycles (these models consider the operating characteristics of the storage and retrieval machine such as acceleration and deceleration and the maximum velocity. Ouhoud et al. (2016) developed traveltime models for singlecycle displacements of S/R machine in multiaisle AS/RS by taking into account classbased storage policy. Guezzen et al. (2013) presented two analytical models to estimate travel time for mobilerack AS/RS; these models are validated for accuracy by simulation.
Hu et al. (2005) have presented a new kind of S/R mechanism (SPAS/RS) that enables AS/RS to efficiently handle very heavy loads. They have presented a continuous traveltime model for the new AS/RS under the stay dwellpoint policy. While Liu et al. (2016) presented a continuous travel time model for the dualcommand cycle of the same system under various input and output (I/O) dwellpoint policy. Ramtin and Pazour (2015) developed expected travel time models for MIAPPAS/RS, with different operating policies and different physical configurations. Dukic et al. (2015) presented the throughput model for dualtray Vertical Lift Module (VLM) AS/RS. Accuracy of the model was tested using comparison with results obtained by simulation models. Khojasteh and Son (2016) addressed an order picking problem in a multiaisle unitload AS/RS served by a single S/R machine, the authors formulated the problem as a nonlinear programming model and developed a heuristic called the STD heuristic to find the optimal solution.
Summarizing, in the literature, several traveltime models have been developed and various types of AS/RS have been presented. To our knowledge, no study deals with the modelling of the retrieval time of the FFflowrack AS/RS due to its recent largescale introduction in the drugs distribution in pharmacies. Therefore, an assessment of its performance is necessary, and for this system, it offers advantages in terms of economy (does not need an S/R machine) and the possibility of simultaneous retrieval of multiple products. For this purpose, this paper develops an analytical retrieval–traveltime model for this type of AS/RS, to evaluate the system performances.
The rest of this paper is structured as follows. In the next section, we will describe the structure and operations of the FFflowrack AS/RS. The retrieval–traveltime models for the FFflowrack AS/RS are presented in Sect. 3. In Sect. 4, a validation study of the continuous model by computer simulations is offered. Our conclusions are proposed in the final section (Sect. 5).
System structure and configuration
A new kind of flowrack AS/RS was introduced by Mekapharm company^{®} (Mekapharm 2016). It is mainly used for the automation of the drug distribution in pharmacies and in all the areas of handlings products that support the freefall movement. We shall refer to the new type of AS/RS, the freefallflowrack AS/RS, or FFflowrack AS/RS.
In the flowrack AS/RS proposed by Sari et al. (2005), the authors used a two S/R machines. While De Koster et al. (2008) used a lifting mechanism on the opposite face of the S/R machine, Sari and Bessnouci (2012) used a single S/R machine for handling loads. In this section, we present a new type of flowrack AS/RS, which consists of FFflowrack AS/RS. The main difference between the FFflowrack AS/RS and the other types of flowrack AS/RS is that the FFflowrack AS/RS is used, for the product transport, a combination between the freefall movement and a transport conveyor, while the other systems use S/R machine that moves according to a Tchebychev travel.
The work method (storage and retrieval operation) of the FFflowrack AS/RS is as follows: The storage operation which is performed by a storage machine or an operator, where the products are handled from the pickup station to store it in the adequate bin. The retrieval operation consists of two phases: The first one is the ejection of the product, where it is achieved by the excitement of the electromagnet of the bin containing the desired product. This will cause the free fall of the product on the transport conveyor. The second phase is the transport of this product by the conveyor until reaching the dropoff station (See Fig. 3).
As detailed in Fig. 1, the dimensions of the rack are: length \((L),\) height \((H),\) and depth \((D).\) The rack has \((N_{\text{L}} )\) bins in each tier and \((N_{\text{c}} )\) bins in each column. Each bin has \((M)\) storage segments. Since the use of a person for the storage operation, the maximum height of the rack should be less than 3 m, for practical and ergonomic reasons.
The dedicated storage assignment policy is used, where each bin is dedicated to a particular product because of the storage policy constraint required in pharmacies and distribution centers.
FFflowrack AS/RS advantages
Differences between the freefall flowrack AS/RS and the classic flowrack AS/RS
Freefall flowrack AS/RS  Flowrack AS/RS  

Throughput  High  Low 
Average retrieval–travel time  Low  High 
Delivery capacity  High  Low 
Fault tolerance  Low  High 
Initial investment  Low  High 
Rapidity  High  Low 

According to a comparative study carried out recently by Metahri and Hachemi (2017) between the FFflowrack and the classic one, improved throughput, which can be up to 90%.

The reduction of average retrieval–travel time can reach 38%, when using the FFflowrack (Metahri and Hachemi 2017).

With the FFflowrack, each bin is independent from the others; therefore, many items can be retrieved at the same time.

In the classic flowrack, all bins are inaccessible once the S/R machine is failed. However, in the FFflowrack, if the electromagnet of a bin is out of service, only that bin is affected.

FFslow rack offers a 40% reduction of the initial investment due to the replacement of the S/R machine by the combination between the freefall movement and the transport conveyor (according to Rosenblatt et al. 1993; it represents approximately 40% or more than according to the system’s cost).

With the classic flowrack, the S/R machine travels back and forth to retrieve an item. While, with the FFflowrack, the destocking of a product is made by the combination between the freefall movement and the conveyor, which are too fast. Therefore, the new FFflowrack offers better rapidity of delivery.
Especially, in the pharmaceutical distribution, this system offers a high speed in the preparation and delivery of customer orders (reducing preparation time). In addition, this system presents the advantage to give to the pharmacists more time to advise their customers for the taking of the prescribed drug (Mekapharm 2016).
In the following section, we will present the retrieval–traveltime models of this system.
Retrieval–traveltime models
This part presents the original contribution of this study. It consists of the modelling of the expected retrievaltime of the FFflowrack AS/RS. For this, we start by giving some assumptions and notations that are needed in our approach, and then, we present the retrieval–traveltime models (continuous model and discrete model).
 (1)
The rack is considered to be a continuous rectangular pick face where dropoff station is located at the lower lefthand corner.
 (2)
The rack length, height, and depth are known.
 (3)
Constant velocity is assumed for the transport conveyor.
 (4)
We consider only the retrieval operations, since the responsetime constraints are stronger on the retrieval operations than those on the storage operations, which are usually not timecritical (Roodbergen and Vis 2009).
 (5)
According to pharmacy practice, a dedicated storage policy is used, which means that each product type is assigned to a fixed bin. However, we assumed a random retrieval policy, which means that any product within the pick face has the same probability to be retrieved. Thus, the storage and retrieval are independent operations.
 (6)
The time of the translation by the gravity of load inside a bin is usually deterministic. Thus, the sliding time can be added after the calculation of the expected retrieval time.
 (7)We assume uniform demands distributions; that is, all the products have the same picking frequency. We consider the uniform demands distribution for these reasons:
 \(L,H,D\)

The length, height, and depth of the rack
 \(l,h,d\)

The length, height, and depth of a storage segment
 \(N_{\text{L}}\)

The number of bins per line
 \(N_{c}\)

The number of bins per column
 \(N_{\text{PF}}\)

The total number of bins in pick face
 \(M\)

The number of segments in each bin
 \(N\)

The total number of bins
 \((i,j)\)

The bin coordinates
 \(V_{\text{c}}\)

The speed of the transport conveyor
 \(g\)

The acceleration due to gravity
 \(t_{\text{v}}^{'}\)

The vertical travel time from a bin to the impact point on the transport conveyor
 \(t_{\text{h}}^{'}\)

The horizontal travel time from the impact point to the dropoff station
 \(t_{g}^{'}\)

The total travel time from a bin to the dropoff station
 \(t_{\text{v}}\)

The vertical travel time from the farthest line to the impact point
 \(t_{\text{h}}\)

The horizontal travel time from the farthest impact point to the dropoff station
 \(E(T_{v}^{'} )\)

The mean of the vertical travel time
 \(E(T_{h}^{'} )\)

The mean of the horizontal travel time
 \(E(T_{g}^{'} )\)

The expected retrieval–travel time
 \(A \left( {t_{g}^{'} } \right)\)

The average retrieval–travel time
It should be mentioned that the retrieval–travel time of all items: \(N\) stored in the rack become the retrieval–travel time of items \(N_{\text{PF}}\) stored only in the pick face, since we have assumed that a dedicated storage policy was used, and the items were automatically moved from the storage to the picking side of the rack by gravity.
Continuous model
Our idea consists in dividing the retrieval–travel time of a location \((i,j)\) in a twotravel time (vertical and horizontal) where the locations \((i,j)\) are uniformly distributed. Therefore, the expected retrieval time is the sum of the mean values of each travel time. As stated in the assumption part, the pick face is supposed as a continuous rectangle. In this case, the discrete coordinates \((i,j)\) become continuous coordinates noted \((i,j).\)
To calculate the expected retrieval–travel time, let \((t_{\text{h}}^{'} ,t_{\text{v}}^{'} )\) denote the temporal coordinate of the retrieval point \((x,y),\) where \(\left( x \right)\) and \((y)\) are independently generated and uniformly distributed. The total travel time \(t_{\text{g}}^{'}\) from this point \((x,y)\) to the dropoff station is: \(t_{\text{g}}^{'} = t_{\text{v}}^{'} + t_{\text{h}}^{'} .\) By assuming that: \(T_{\text{v}}^{'}\) presents the continuous random variable associated with the vertical travel time \(t_{v}^{'}\) and \(T_{\text{h}}^{'}\) presents the continuous random variable associated with the horizontal travel time \(t_{\text{h}}^{'} .\) Now, let \(g (t) = g(T_{\text{v}}^{'} = t)\) and \(G(t) = P\left( {T_{\text{v}}^{'} \le t} \right)\) denote the probability density function and cumulative distribution function of the random variable \(T_{\text{v}}^{'} ,\) respectively. In addition, let \(w\left( t \right) = w\left( {T_{\text{h}}^{'} = t} \right)\) and \(W (t) = P (T_{\text{h}}^{'} \le t)\) denote the probability density function and cumulative distribution function of the random variable \(T_{\text{h}}^{'} ,\) respectively.
The calculation of \(E(T_{v}^{'} )\)
Suppose that Y presents the continuous random variable associated with vertical distance \(y,\) and let \(f (y) = f(Y = y)\) and \(F\left( y \right) = P(Y \le y)\) denote the probability density function and cumulative distribution function of the random variable \(Y,\) respectively. Recognizing that \(Y\) is uniformly distributed between \(0\) and \(H,\) while \(T_{v}^{'}\) follows a probability law that can be determined as follow:
Determination of the cumulative distribution function \(G\;(t)\)
We have \(G\left( t \right) = P\left( {T_{v}^{'} \le t} \right)\) and \(T_{v}^{'} = C \sqrt Y .\)
Recall that, \(F\left( y \right) = P\left( {Y \le y} \right);\) and from Eq. (3): \(y = \frac{1}{{C^{2} }}t^{2} .\)
Determination of the probability density function \(g (t)\)
As we assume that \(T_{\text{v}}^{'}\) is a continuous random variable, we have: \(g\left( t \right) = \frac{{{\text{d}}G(t)}}{{{\text{d}}t}} = G'(t).\)
Therefore, Eq. (13) can be reduced to the following equation as follows:
The calculation of \(E(T_{\text{h}}^{'} )\)
We recognize that the random variable \(T_{h}^{'}\) is uniformly distributed between 0 and \(t_{\text{h}} ,\) where: \(t_{\text{h}} = \frac{L}{{V_{\text{c}} }}.\)
The expected retrieval–travel time \(E(T_{\text{g}}^{'} )\) is obtained as follows.
Discrete model
We use the discrete rack face approach (the coordinates of the locations are integers in this case) to evaluate the expected retrieval–travel time of the FFflowrack AS/RS, which presents an exact model that will be used in a computer simulation to validate the continuous model. For this, we calculate the sum of the retrieval time of each bin, then dividing this sum by the total number of segments.
The vertical travel time (\(t'_{v}\)) of a particular item stored in position \((i,j)\) to reach the impact point can be calculated as follows.
Simulation and validation results
The expected retrieval–travel time model presented in Sect. 4.1 provides approximate values, since it was based on a continuous approximation of the probability laws modelling the displacement of a product. To validate the accuracy of this model, we carried out a comparison between the results obtained from the continuous model with those obtained from a computer simulation that based on the discrete model presented in Sect. 4.2.
In the computer simulation, a sequence of 1 million retrieval operations has been performed \((r = 10^{6} ),\) to calculate the average retrieval time. Twenty different configurations of the FFflowrack AS/RS have been simulated. For each configuration, it is assumed that the width \((l)\) and the height \((h)\) of each storage segment are 20 cm and 10 cm, respectively. Moreover, we will vary the speed velocity of the transport conveyor \((V_{\text{c}} ).\)
Comparison between continuous model and simulation results with V_{c} = 1 m/s
Configurations  \(N_{\text{L}}\)  \(N_{\text{C}}\)  \(N_{\text{PF}}\)  Simulation  \(E(T_{\text{g}}^{'} )\)  Deviation (%) 

1  20  24  480  2.5768  2.4663  4.2855 
2  52  27  1404  5.8061  5.6946  1.9208 
3  54  27  1458  6.0073  5.8946  1.8757 
4  32  25  800  3.7886  3.6759  2.9723 
5  36  26  936  4.1968  4.0854  2.6559 
6  50  25  1250  5.5895  5.4759  2.0322 
7  26  24  624  3.1793  3.0663  3.5518 
8  48  24  1152  5.3812  5.2663  2.1348 
9  28  25  700  3.3888  3.2759  3.3288 
10  44  25  1100  4.9892  4.8759  2.2695 
11  42  26  1092  4.7965  4.6854  2.3174 
12  40  27  1080  4.6044  4.4946  2.3842 
13  38  26  988  4.3990  4.2854  2.5822 
14  56  27  1512  6.2059  6.0946  1.7929 
15  34  24  816  3.9769  3.8663  2.7799 
16  30  26  780  3.5993  3.4854  3.1651 
17  22  24  528  2.7792  2.6663  4.0614 
18  24  26  624  2.9993  2.8854  3.7986 
19  46  25  1150  5.1934  5.0759  2.2620 
20  58  27  1566  6.4117  6.2946  1.8268 
Comparison between continuous model and simulation results with V_{c} = 2 m/s
Configurations  \(N_{\text{L}}\)  \(N_{\text{C}}\)  \(N_{\text{PF}}\)  Simulation  \(E(T_{\text{g}}^{'} )\)  Deviation (%) 

1  20  24  480  1.5289  1.4663  4.0931 
2  52  27  1404  3.1555  3.0946  1.9284 
3  54  27  1458  3.2563  3.1946  1.8954 
4  32  25  800  2.1394  2.0759  2.9664 
5  36  26  936  2.3473  2.2854  2.6390 
6  50  25  1250  3.0404  2.9759  2.1205 
7  26  24  624  1.8302  1.7663  3.4911 
8  48  24  1152  2.9283  2.8663  2.1176 
9  28  25  700  1.9393  1.8759  3.2691 
10  44  25  1100  2.7406  2.6759  2.3573 
11  42  26  1092  2.6508  2.5854  2.4672 
12  40  27  1080  2.5567  2.4946  2.4276 
13  38  26  988  2.4476  2.3854  2.5410 
14  56  27  1512  3.3578  3.2946  1.8820 
15  34  24  816  2.2290  2.1663  2.8103 
16  30  26  780  2.0479  1.9854  3.0547 
17  22  24  528  1.6301  1.5663  3.9115 
18  24  26  624  1.7475  1.6854  3.5577 
19  46  25  1150  2.8388  2.7759  2.2147 
20  58  27  1566  3.4550  3.3946  1.7462 
Comparison between continuous model and simulation results with V_{c} = 3 m/s
Configurations  \(N_{L}\)  \(N_{\text{C}}\)  \(N_{PF}\)  Simulation  \(E(T_{g}^{'} )\)  Deviation (%) 

1  20  24  480  1.1791  1.1330  3.9115 
2  52  27  1404  2.2752  2.2280  2.0782 
3  54  27  1458  2.3405  2.2946  1.9615 
4  32  25  800  1.5895  1.5426  2.9478 
5  36  26  936  1.7318  1.6854  2.6831 
6  50  25  1250  2.1882  2.1426  2.0824 
7  26  24  624  1.3803  1.3330  3.4235 
8  48  24  1152  2.1135  2.0663  2.2333 
9  28  25  700  1.4558  1.4093  3.1937 
10  44  25  1100  1.9886  1.9426  2.3135 
11  42  26  1092  1.9312  1.8854  2.3732 
12  40  27  1080  1.8755  1.8280  2.5329 
13  38  26  988  1.7997  1.7520  2.6499 
14  56  27  1512  2.4068  2.3613  1.8915 
15  34  24  816  1.6456  1.5997  2.7939 
16  30  26  780  1.5303  1.4854  2.9336 
17  22  24  528  1.2453  1.1997  3.6683 
18  24  26  624  1.3319  1.2854  3.4901 
19  46  25  1150  2.0556  2.0093  2.2524 
20  58  27  1566  2.4722  2.4280  1.7904 
We observe from Fig. 6 that the obtained results of both continuous model and simulation are near to each other, which prove that our analytical expression captures the real behaviour of the retrieval–travel time for the FFflowrack AS/RS.
Moreover, as it is illustrated in Fig. 7, it can be seen that the maximum deviation is less than 5%, showing that the continuous model gives a very good approximation to simulation results, which are more close to the realrack nature. Because of lack of space, only the simulation results of 20 system configurations have been presented. However, the proposed model (continuous model) can estimate the retrieval time of a large range of system configurations with a very good accuracy.
We notice that the developed analytical expression allows a fast evaluation of the expected retrieval time, also the developed model can be really used to guide the decisionmaking process about design and control of the FFflowrack AS/RS.

evaluate the throughput of the FFflowrack AS/RS for various configurations;

optimise the dimensions of the rack, where the developed expression represents the objective function to be minimized;

compare different storage assignment policies;

conduct a performance comparison between the FFflowrack and other types of AS/RS.
Conclusion
This paper has presented a new kind of flowrack AS/RS, which is called freefallflowrack AS/RS (FFflowrack AS/RS), which presents a novel material handling technology that can be used in several applications, from the distribution centers to the manufacturing industry.
The advantages of this FFflowrack AS/RS include high throughput, flexible configuration and a considerable reduction of the initial investment due to the replacement of the S/R machine by the combination between the freefall movement and the transport conveyor for the retrieval of items. This system can be used in the case of items with low size and weight, such as the pharmaceutical products, food, textile, and electronic components. However, it is not recommended to use this kind of AS/RS in the case of voluminous and\or fragile products, because the freefall of products can damage them.
In this study, we focused on evaluating the performance of the FFflowrack AS/RS by developing a continuous retrieval–traveltime model. This model has been compared for accuracy with a computer simulation based on a discrete model. It can be concluded that the developed model gives very satisfactory results, since the maximum deviation is less than 5%.
To our knowledge, this is the first study to deal with evaluating FFflowrack performance. Likewise, we hope that this research will be usefully employed in the future studies in this area.
Finally, in our future research, we will investigate the developed model, under different demand distributions to study the impact of the turnover rate of items on the retrieval time, which allows establishing new control techniques. In addition, we suggest optimising the dimensions of the FFflowrack AS/RS to minimize the expected retrieval–travel time.
Notes
Compliance with ethical standards
Conflict of interest
No potential conflict of interest was reported by the authors.
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