The analytical model presented in section 4 is able to evaluate the average energy need and its variability for all the possible cycles (single-, dual-, and multi-command cycles) of an AVS/RS consisting of one lift, one shuttle and one satellite, on the basis of the real criteria used to store and retrieve ULs from the rack in the form of probability distributions. In some configurations, the shuttle and the satellite can be provided with the capability of performing simultaneous trips on the corresponding axes: this feature enables the cycle time to be shortened and the system throughput to be increased; however, from the energy point of view, there is no difference between this case and the scenario in which vehicles wait for each other.

In this section, a set of case-studies is introduced to determine the average energy needs and their standard deviation, as provided by the mathematical models. The technical capabilities presented here represent systems that are currently available on the market. In order to validate the models over the widest number of configurations, three factors have been varied in the study: (i) the layout of the rack; (ii) the type of cycle to be performed; (iii) the criteria used to store and retrieve the ULs.

The same experimental setup has been used in (D’Antonio et al., 2017) to validate an analytical model for the evaluation of the AVS/RS cycle time.

TThree rack layouts with different scale-factor values (i.e. the ratio between the length and the width of the rack). The capacity of all the racks is close to 2000 ULs, and each rack is made up of 10 tiers; the properties of the racks are reported in details in Table 1. Each UL has the following size: x = 1:5 m, y = 2:0 m, z = 1:2 m.

Seven cycles that represent different operating conditions: 4 homogeneous cycles and 3 heterogeneous cycles have been considered. The most popular cycles in literature were studied first (single command cycles consisting of 1 storage or 1 retrieval, and double command cycles consisting of both tasks). The analysis was then extended to more complex cycles involving up to 4 ULs. These cycles are more meaningful for AVS/RS designers and sellers, as they are able to better exploit the AVS/RS capabilities: an appropriate performance evaluation of realistic cases is crucial for successfully supply such systems. A detailed summary of the parameters that describe each cycle is provided in Table 2.

Three different storage and retrieval criteria were chosen to test the analytical model under different deployment scenarios:

Random. The position to store or retrieve an UL is chosen randomly, according to a uniform distribution. This is the criterion that is most commonly used in literature.

Closest Floor (CF). The position of the UL is selected according to the following hierarchy: (a) the tier closest to the bay; (b) the channel closest to the lift hosting the same type of item;

Closest Channel (CC). The UL position is selected according to the following hierarchy: (a) the channel closest to the lift hosting the same type of item; (b) the tier closest to the bay. The performances of the vehicles are synthesized in Table 3.

In order to validate the model, the analytical results are compared with the output provided by a set of simulations. In order to be closer to the real deployment scenarios, the rack is assumed to store 4 different classes of ULs. Each channel is dynamically allocated for a single type of UL: any class of item can be stored into an empty channel; conversely, the units stored in the channel have to be of the same type. The simulations have been performed by implementing a Matlab routine. The initialization is made by filling the rack to 100% of its capacity; then, a set of retrieval orders is used to achieve a filling ratio equal to 50%. The rack initialization phase is the same for each simulation made on the same layout. After this step, the observation of the system began: a sequence of 20,000 ULs to be stored and retrieved is generated; each UL is

provided with a class randomly assigned according to a uniform distribution. The same sequence is used for all the case-studies. The ratio between storage and retrieval orders is kept equal to 1 in order to simulate a steady state scenario. To ensure the robustness of the results, 10 repetitions are performed for each case; a different sequence of ULs was

generated for each repetition.

Each combination of the values was simulated for the three varied factors. The distributions a, b, and c are obtained by analyzing the positions used for storing and retrieving the ULs. The average energy needs and the standard deviations provided by the simulations are also stored to perform a comparison with the results provided by the mathematical models. The results comparison is shown in Table 4: the minimum and the maximum values of the average energy need are provided, for both the analytical models and the corresponding simulations, for each layout, positioning criterion and type of cycle. The interval of values for the relative difference between the two approaches is also listed; the complete set of results is shown in Figure 3.

According to the results of cycles 1 and 2 (which are currently the most frequently used in practice), the difference between the analytical estimation and the corresponding simulation is below 4%. For all the homogeneous cycles (cycles 1-4), this difference is usually lower than 6%. For the heterogeneous cycles, the error increases a little: however, the dfference is usually within 10%; the largest relative error (in absolute value) is detected for cycle 6: it reached 20% in some configurations. No huge deviations are found among the different criteria for storing and retrieving ULs, or among the different rack layouts. Conversely, the results in Table 4 show that the energy needs may vary significantly when the storage and retrieval criterion are changed. This highlights the importance of taking into account the real deployment conditions in order to achieve a more accurate evaluation of the AVS/RS performance.

The results provided in Table 5 and Figure 4 show the difference between the standard deviations provided by the analytical model and those of the simulation. The difference is usually below 6% for the homogeneous cycles. Conversely, the deployment of the lift makes this value grow. Among the heterogeneous cycles, the smallest differences are obtained for the CF criterion, which is also the one that minimized the utilization of the lift. Tables 6-8 compare the average values of the energy need to store/retrieve a single UL, the system throughput, and the cycle time obtained through the analytical approach. The values for the latter two parameters are taken from (D’Antonio et al., 2017). These results show how the chosen configuration (1 lift, 1 shuttle, 1 satellite) can behave differently according to the deployment scenario, and they highlight the importance of properly modelling the operating conditions in order to obtain reliable performance estimations.

This paper has been aimed at extending the state of the art concerning the estimation of the performance of AVS/RS. In a former work, an analytical model was introduced for the evaluation of the cycle time. Here, the model has been extended to also take into account the energy features of the system. At the moment of writing this paper, to the best of

the authors’ knowledge, no paper in literature has provided an analytical estimation of the energy need of AVS/RS. The key features introduced into the cycle time model have been maintained: (i) the criteria for UL allocation, in the form of a probability distribution; (ii) the evaluation of both the average values and the standard deviation.

The results presented in Section 6 show that the analytical technique is an effective tool for the evaluation of the performance of AVS/RS systems: the maximum relative error with respect to a set of simulations is usually below 10%. These models have the aim of supporting AVS/RS designers in the evaluation of the performance of the system over

a wide range of scenarios. The probability distributions introduced in Section 4 may be a set of weights with a (normalized) sum equal to 1, moreover, any kind of cycle that is technically feasible can be evaluated.

Further work is necessary to improve the accuracy of the model, in particular concerning the standard deviation; additional efforts must be made to enlarge this kind of analysis to include systems consisting of more than one shuttle, in a tier-to-tier configuration. The models that should be developed will have to consider that the lift must serve several shuttles; hence, the performance evaluation technique must also be able to identify the bottleneck of the system, which determines the takt-time.