Performance Evaluation of Warehousing Units

Some general remarks• In general, a difficult problem due the

– large number of operational issues that must be introduced in the model– stochastic nature of the system operations– unique aspects of the various environments– etc.

• Therefore, simulation is the most extensively used tool• Analytical models exist mainly for some automated modules, because

– they present some better defined structure and behavior (therefore, easier to justify the modeling assumptions)

– the need for good performance estimates for these modules is more critical, due to their high investment cost and inflexibility to modifications

– modeling and analyzing automated (production) systems is a prevailing trend in the scientific community

An example application:“Travel-Time Models for AS/RS”

(Y. Bozer and J. White, IIE Trans., pgs 329-338, 1984)

Modeling assumptions:

• The S/R machine operates either on a single or a dual command basis.

• The S/R machine travels simultaneously in the horizontal and vertical directions with constant velocities.

• Pick-up and deposit times associated with load handling can be ignored. In general, this is a deterministic component of the overall cycle time which can be added to it at the end, if it is deemed significant.

• Randomized storage is used; thus, any location in the pick face is equally likely to be selected for storage or retrieval.

Quantities to be evaluated:

• Expected cycle time and throughput, under SC and DC operation

A “brute force” calculation

Assuming that:

• the total number of storage locations is N

• one-way travel time from I/O point to location i is t_oi = t_io

• one-way travel time between locations i and j is t_ij = t_ji

we have:

• E(SC) = (2/N) * _{i=1}^N t_oi

• E(DC) = (2 / (N * (N-1))) *

* _{i=1}^{N-1} _{j=i+1}^N [t_oi+ t_ij + t_jo]

Bozer & White’s approximating formulae

L

H

sh

sv

th = L / sh

tv = H / sv

T = max{th, tv}b = min{th, tv}/T

E(SC) = (1/3)b^2+1E(DC) = (4/3) + (1/2)b^2-(1/30)b^3

Some interesting follow-up works

• Y. Bozer and J. White, “Design and Performance Models for End-of-Aisle order picking systems” Management Science, Vol. 36, No. 7, pgs 852-866, 1990

• Y. Bozer and J. White, “A generalized design and performance analysis model for end-of-aisle order-picking systems”, IIE Trans., Vol. 28, pgs 271-280, 1996

• R. Foley and E. Frazelle, “Analytical results for miniload throughput and the distribution of dual command travel time”, IIE Trans., Vol. 23, No. 3, pgs 273-281, 1991

• R. Foley, S. Hackman and B. C. Park, “Back-of-the envelope miniload throughput bounds and approximations”, working paper, ISyE, Georgia Tech, 2001