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The work of Han et al. (1987) for dual command cycles in traditional AS/RSs is a frequently referenced study that provides the basis for the work of many other researchers. The authors introduce the Nearest Neighbor heuristic that sequences the nearest retrieval position to an open location with the objective to reduce the travel between distance in a dual command cycle. The objective is to select an efficient pair of storage and retrieval po-sitions. In a first step, they derive the pdf for the smallest of k randomly chosen distances, using the results of Bozer and White (1984), who present pdf and cdf for the distance between two randomly selected positions. The smallest of k randomly chosen distance is a random variable, Z_k, with the pdf r (Z_k).

r (Z_k) = k[1 −Q(Zk)]^k−1q(Z_k) for ≤ Zk≤ 1 (3.58) Q(ζ) and q(ζ) are known from equations 3.48 and 3.49. The expected small-est distance is (Han et al. 1987, p. 59):

E (Z_k) = Z1

0 ζk[1 −Q(ζ)]^k−1q(ζ)dζ (3.59)

Based on this result, the authors present an approximation of the travel be-tween distance for m open locations and a block of n requested retrieval positions:

For the analytical formulation of the proposed dual command cycle, E (T B ) is substituted with E (T B_n,m^{N N}) in expression 3.40.

For a block size of 20 retrieval requests they report increased throughput by 18% for the dual command nearest neighbor policy with one open loca-tion. The throughput can be increased further with a greater number of open locations.

Next, the authors formulate the Shortest Leg heuristic in order to find a lower bound of the DC under the nearest neighbor heuristic. The Shortest Leg heuristic selects storage and retrieval position from m open locations and n retrieval positions that create the least total travel distance between the I/O point and the retrieval position. This means, storage positions from the no-cost zone are selected, if possible. Analytical results show that throughput is improved further when applying the shortest leg heuristic.

Using Monte Carlo simulation, the dynamic behavior of the heuristic is studied. They find that the Shortest Leg heuristic changes the distribution of open locations by moving the open locations away from the I/O point.

On the contrary, the nearest neighbor heuristic shows a constant perfor-mance and therefore outperforms Shortest Leg on the long run.

Note that equation 3.59 is also denoted as the mean travel between distance between one randomly selected position and the nearest of m randomly selected positions, or E (T B_m).

Sarker et al. (1991), Sarker et al. (1994) and Keserla and Peters (1994) present similar approaches suggesting a quadruple command cycle that is executed according to the Flip Flop policy. Based on the approach of Bozer and White (1984), they present an analytical formulation of the cor-responding cycle time. Sarker et al. (1991) adjuste the cycle by minimiz-ing the travel between distances based on the the nearest neighbor idea,

i.e., a storage and a retrieval position are chosen near to the position of the Flip Flop operation. As proposed in Han, McGinnis, Shieh and White (1987), they formulate a lower bound of the travel time by selecting a stor-age and a retrieval position from the no-cost zone. Simulation is used to validate the formulated heuristics and compare the performance to single load handling. They report improvements ranging from 50% to 80% and recommend dual load handling systems. Based on these findings, Sarker et al. (1994) analyze the previous system for class based storage. They present a heuristics for a quadruple command cycle on a two class basis and as-sess their analytical model with a simulation. With both methods, they re-port possible throughput improvements of up to 25% in comparison to the results without class based storage. Keserla and Peters (1994) also com-bine the nearest neighbor idea with the Flip Flop heuristic, but adjust this combination in such a way that the minimum perimeter of the triangle, de-fined by the three stops of the cycle, is chosen. Upper and lower bounds of the heuristic based on the ideas of Han, McGinnis, Shieh and White (1987) are presented. They evaluate the heuristic via Monte Carlo simulation and show a throughput improvement of 25% compared to a nearest neighbor dual command cycle. This is a lower potential compared to the results of Sarker et al. (1991), which can be explained by the allowance for load han-dling times.

Meller and Mungwattana (1997) present three sequencing heuristics, both for quadruple and sextuple command cycles. Based on the results of Bozer and White (1984), they use order statistics to derive expected smallest and expected largest one-way travel time from the I/O point to one of m ran-domly selected locations, E (SWm) and E (SLm), respectively. Using equa-tions 3.34 and 3.35, they derive:

E (SW_m) = Z1

0 ζm[1 −G(ζ)]^m−1g (ζ)dζ (3.61) E (SLm) =

Z₁

0 ζm[G(ζ)]^m−1g (ζ)dζ (3.62)

All heuristics are based on the assumption of a fixed number of retrieval po-sition (i.e., two or three) and m open locations. The first nearest neighbor heuristic composes the cycle in the following way: Nearest storage location

— second nearest storage location — third nearest storage location — near-est retrieval position — second nearnear-est retrieval position — third nearnear-est retrieval position — return. The quadruple command cycle is composed accordingly. In a similar variant, the Reverse Nearest Neighbor heuristic (RNN), the retrieval (storage) location closer to the I/O point is approached lastly (first). In their so called modified command cycle, they combine the Nearest Neighbor principle with the Flip Flop heuristic. Stops within a cy-cle that are not involved in Flip Flop are arranged in the same way as before.

Using the newly derived formulas and the Nearest Neighbor travel between distance from Han et al. (1987) (equation 3.60), travel time estimates are presented. Expected travel times are numerically evaluated for different examples of m. They report that the modified quadruple command cycle in combination with the nearest neighbor idea performs about 36% better than the dual command cycle under nearest neighbor policy, for m ranging between 1 and 10.

Eynan and Rosenblatt (1993) analyze the application of the Nearest Neigh-bor idea in a class-based storage environment by selecting storage and re-trieval pairs from the same class. They derive the mean travel between distance for Nearest Neighbor selection in each of i classes and the total expected travel between distance as the average from all classes. They re-port a reduced travel between distance of up to 65% compared to Han et al. (1987) for six classes.

Grafe (1997) presents a qualitative consideration of AS/RS with multiple load handling and addresses possible advantages of a Flip Flop policy and class based storage. No mathematical travel time models are formu-lated. Based on a rule of thumb that approximates the travel distance in x-direction, a method for throughput determination is presented.

Potrˇc, Lerher, Kramberger and Šraml (2004) present an approach to per-form quadruple and sextuple command cycles according to a heuristic called ’Strategy x’. In this heuristic, they propose to randomly choose

stor-age and retrieval positions, which are sequenced in an ascending order in x-direction of the rack. Performance is evaluated by a simulation model in which different system configurations, consisting of rack dimension and velocity profiles, are simulated. In this way, they show improved through-put potential of multiple load handling systems on the one hand and de-pendencies of travel times on rack dimensions and velocities on the other hand.

Kraul (2010) considers performance models for different kinds of AS/RSs.

He presents an adjusted version of the Shortest Leg heuristic for multiple-load handling devices to overcome the reported drawbacks of Han et al.

(1987). He proposes to chose a random storage location first, before loca-tions from the no-cost zone are selected. In this way, the shift of available positions away from the I/O point is prevented.