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2005, Control Engineering Practice

Abstract-An Automated Storage and Retrieval System (AS/RS) consists of several aisles with storage racks, serviced by cranes. Each aisle is equipped with a storage and a retrieval conveyor and is serviced by rail guided vehicles. This paper focuses on operational control of AS/RSs and presents a management strategy taking decisions on resource allocations in order to avoid conflicts and deadlocks. A Colored Timed Petri Net models the system dynamics describing the system in a resource oriented and modular framework.

2003 •

Abstract The industrial manufacturing environment is nowadays characterized by fierce global competition, rapid market changes and short product life cycles. Such a complex scenario originated a vast demand for sophisticated techniques guaranteeing adequate planning and control of warehouses. A widely used solution is to adopt Automated Storage and Retrieval Systems (AS/RSs). A typical AS/RS comprises a number of parallel aisles with storage racks, serviced by automated stacker cranes and rail guided vehicles.

Considers a modern warehousing system composed of a number of aisles, each one served by a crane and sided by a pair of racks, and a shuttle which moves along a 1D path placed orthogonally with respect to the aisles' axis. The main operations performed by the control system concern the optimization of the cranes and shuttle movements. Regarding the cranes, we propose an algorithm for optimizing the sequencing of dual command (storage and retrieval) cycles, so to maximize the throughput. As for the shuttle, an algorithm is proposed to optimize the scheduling of the shuttle operations; the performance of the algorithm is evaluated and compared with the case in which no optimization is considered. A further contribution of the paper is the derivation of the model of a real warehousing system, by means of a colored Petri net approach. Such a model is then used to show and validate the effectiveness of the proposed algorithms

2005 •

An Automated Storage and Retrieval System (AS/RS) automatically stores incoming material and retrieves stored parts with no direct human handling. This paper proposes a modular and unified modelling framework for heterogeneous automated storage and retrieval systems, comprising rail guided vehicles and narrow aisle cranes.

Computers & Industrial Engineering

Development of timed Colour Petri net simulation models for air cargo terminal operations2006 •

IEEE Transactions on Automation Science and Engineering

A Hybrid Model of Complex Automated Warehouse Systems—Part I: Modeling and Simulation2012 •

INGENIERÍA Y COMPETITIVIDAD

Modeling of an automated integral logistics system using colored petri netsA system is a set of heterogeneous elements that work in direct relation with each other for a specific purpose;through its automation, it is possible to carry out productive processes with machines, without thedirect participation of a human operator. The Automated Integrated Logistics System (SALI by it’s acronymin Spanish) is a didactic environment that allows to emulate the activities and operation of a Logistic DistributionCenter (DC), developed at the New Granada Military University as a teaching and learning toolfor engineering. The current system programming in the software system allows a serial processing of apurchase order, from the moment in which the customer, passing through all the stations, until it is sent to adelivered point, requires the product. The present investigation presents a modeling of the system throughColored Petri Nets (CPN) it’s made, with the purpose of achieve a better understanding of the dynamics ofthe system, the relationships between the station...

2019 Winter Simulation Conference (WSC)

Coordinated Control of Multi-Zone Avs/Rs, Conveyors And Pick-Up Operations In Warehouse System2019 •

Modeling of an AS/RS serviced by rail-guided vehicles with colored Petri nets: a control perspective

2002 •

An automated storage and retrieval system (AS/RS) automatically stores incoming material and retrieves stored parts with no direct human handling. This paper proposes a modular and unified modeling framework for heterogeneous automated storage and retrieval systems, comprising rail guided vehicles and narrow aisle cranes. We employ colored timed Petri nets, representing a concise and computationally efficient tool for modeling the system dynamic behavior, particularly suitable for real time control implementation. Indeed, the model can be utilized in a discrete event simulation to apply control policies in order to solve scheduling problems, as well as to avoid deadlock and collision occurrences.

An analysis of the typical manufacturing/storage system, which represents a group of the discrete event systems, is introduced in the paper. For that system appropriate modelling tools are described where the main attention is devoted to the coloured Petri nets. They are used for the solution of the storage system modelling, which is one of the most important tasks on the DES higher level.

ARTICLE IN PRESS
Control Engineering Practice 13 (2005) 1223–1241
www.elsevier.com/locate/conengprac
An approach to control automated warehouse systems
Francesco Amatoa, Francesco Basileb,, Ciro Carboneb, Pasquale Chiacchiob
a
School of Computer Science and Biomedical Engineering, Università degli Studi Magna Græcia di Catanzaro,
Via Tommaso Campanella 115, 88100 Catanzaro, Italy
b
Dipartimento di Ingegneria dell’Informazione ed Ingegneria Elettrica, Università degli Studi di Salerno, Via Ponte Don Melillo 1, 84084,
Fisciano (SA), Italy
Received 18 March 2003; accepted 28 October 2004
Available online 29 January 2005
Abstract
The goal of this paper is the development of control algorithms for the management of an automated warehouse system. As usual,
the implementation of a control algorithm requires three preliminary steps: development of a reliable model; design of control
procedures according to some optimality criteria; validation of these control procedures. As for modelling, a new level in the control
architecture, namely an optimizer system, is introduced which performs real-time optimization thus simplifying the low-level control
and improving the overall performance. In this context, the role of a detailed model of the whole warehouse is discussed and such a
model is built up by using the colored timed Petri nets framework. As for control, we propose two control algorithms, derived under
the simplifying continuity assumption of the rack locations, to optimize the operations of the cranes (moving within the aisles of the
warehouse) and the operations of the shuttle (moving on a straight line placed between the aisles and the picking/refilling area),
respectively. To evaluate the performance of the proposed control algorithms we define three different cost indices. As for the
validation, extensive simulations are performed on the model in order to prove the effectiveness of the proposed control algorithms.
A further validation of the algorithms has been performed on a test bed in order to take into account communication delays and
computation times. Finally, the proposed architecture and control algorithms have been applied to a real plant.
r 2004 Elsevier Ltd. All rights reserved.
Keywords: Discrete event systems modelling; Colored timed Petri nets; Warehouse automation; Factory automation; Optimization
1. Introduction
In the last fifteen years, a big effort has been made to
find optimal strategies for planning and control of
warehouse systems. These issues have become more and
more challenging with the growth and diffusion of
modern computer technology which allows the implementation of complex operations in a completely
automatic way. Planning involves high level decisions,
like assignment of goods to the storage locations
(random, class-based, correlated product clustering
policies) (VandenBerg, 1999) or the designing of the
Corresponding author. Tel.: +39 089 964400;
fax: +39 06 233227957.
E-mail addresses: fbasile@unisa.it, balraj@macmill.com (F. Basile).
0967-0661/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.conengprac.2004.10.017
warehouse system itself. Control problems involve
optimal sequencing and scheduling of storage and
retrieval requests resulting in the so-called dispatching
control.
We refer to a general warehouse architecture consisting of a number of aisles, each one served by a crane,
shuttles, picking/refilling positions and in/out buffers
(see Fig. 1 for an example of a real layout).
On both sides of each aisle there is a storage rack
composed of nr rows and nc columns; moreover, as said,
each aisle is served by a crane, capable of moving both
vertically and horizontally at the same time, which
performs the following operations: (i) picking of the
Stock Unit (SU) to be stored at the buffer input of the
aisle, referred as the input/output (I/O) point of the
aisle; (ii) storage of the SU into the assigned location S
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F. Amato et al. / Control Engineering Practice 13 (2005) 1223–1241
Fig. 1. Warehouse lay-out.
of the rack; (iii) movement to location R where a
retrieval has been requested; (iv) retrieval of the SU
stored in R; (v) coming back to the I/O point. This set of
operations is called, in the warehousing system context,
a dual command (DC) machine cycle (Graves, Hausman, & Shieh, 1977; Bozer & White, 1984; Han,
McGinnis, Shieh, & White, 1987; Lee, de Souza, &
Ong, 1996).
As to the shuttle, it moves along a mono-dimensional
path placed orthogonally with respect to the aisle axis
and performs picking actions (from the main input
buffer, the aisles output locations and the picking/
refilling output locations) and deposit actions (into the
main output buffer, the aisles input locations, the
picking/refilling input locations).
The picking–refilling areas represent an interface with
the shop floor of the factory. A picking–refilling bay
(denoted as ‘‘BayPR’’ in Fig. 1) consists of a picking
location, a deposit location and a shop floor interface
location connected via conveyors so that an SU (carried
on a pallet) can be partially emptied by a human
operator at the shop floor location and then carried
back on the same pallet to an aisle rack location.
The in/out buffers represent the interface of the
warehouse with the rest of the plant.
In this paper an approach to the control of warehouse
systems is presented.
The first contribution of this paper is the development
of a new hierarchical level in the control architecture
(Basile, Carbone, & Chiacchio, 2003). Indeed, in
previous applications, only long-term optimization was
performed: it usually has a day or week time horizon
and it is based on simplified models of warehouse
systems and on statistical characterization of the system
performance. Conversely, we introduce a further level
constituted by a real time optimizer, whose core is the
dispatcher unit, with the aim of performing the shortterm optimization of handling sequences, that usually
has the objective to minimize the time to complete a
little number of picking or storage missions and is based
on the current state of the system. The role of a detailed
model for the designing and implementation of this level
is also discussed. The main advantages obtained are:
The short-term optimization leads to better performance.
Design, implementation and testing of low-level
handling sequences is greatly simplified since all the
optimization issues are taken in charge by the
dispatcher.
Since the optimizer relies on the current state of the
system (it works in real time) it can also manage faults
and degradation of the system.
Roughly speaking, the dispatcher output is the order
to start handling sequences whose control code is
executed at low-level; therefore, it can be easily
adapted to different plants.
The second contribution of this paper is the development of a detailed formal reliable model of an automated
warehouse, as shown in Fig. 1, in the framework of
discrete event systems; model building is fundamental for
both control design and performance evaluation. The
obtained model is based on the colored timed Petri net
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F. Amato et al. / Control Engineering Practice 13 (2005) 1223–1241
(CTPN) approach (Jensen, 1995; Hsieh, HWang, &
Chou, 1998). In this context the key contributions of the
paper can be summarized as follows:
A detailed formal model of the whole warehouse is
built up, while the previous literature mainly considered the formal modelling of the individual
components (Hsieh, HWang, & Chou, 1998) or only
implemented a simplified model for simulation
purposes (Lee et al., 1996; Chincholkar & Krishnaiah
Chetty, 1996).
The model is highly modular and each module is
parameterized: the reuse of model components to
model different warehouse systems is very easy.
The interface with a higher level scheduler (dispatcher) is embodied in the obtained model, thus allowing
off-line performance evaluation of state-dependent
dispatching control algorithms.
There is a clear correspondence between model
entities and layout, thus allowing the model to be
used for on-line monitoring (see (Feldmann &
Colombo, 1999) for monitoring of flexible manufacturing systems).
The detailed model can be used on-line in complex
control algorithms which are based on look-ahead (or
what-if) techniques (Hsieh & Chen, 1999).
The third contribution is the development of the
control algorithms implemented in the optimizer system
(OS). More precisely we focus on simultaneous optimization of the DC cycles operations of the cranes and the
picking/deposit actions performed by the shuttle.
Previous papers (Bozer & White, 1984) and (Han,
McGinnis, Shieh, & White, 1987) considered the DC
optimization problem under the continuous rack
assumption, i.e. the rack is viewed as an infinite number
of pointwise locations each one addressed by a pair of
real numbers.
In (Han, McGinnis, Shieh, & White, 1987) the nearestneighbor algorithm to optimize the sequence of retrievals
was proposed. This algorithm assumes that, for any
given DC operation, a number of locations is available
for storage and one of the empty locations is paired with
a retrieval location in an optimal way. The throughput
derived by the application of such policy is evaluated
analytically by an approximated formula, and then
validated via computer simulations on both continuous
and discrete racks. In the same context of Bozer and
White (1984) and Han, McGinnis, Shieh, and White
(1987), Lee and Schaefer (1996) proposed a procedure to
optimize the sequencing of retrieval requests based on
the solution of a linear assignment problem.
By following the guidelines of Bozer and White
(1984), Han, McGinnis, Shieh, and White (1987) and
Lee and Schaefer (1996), in this paper we develop an
algorithm to optimally sequence the retrieval orders.
1225
However, we start from the more restrictive assumption
(which is based on real world experience) that the SU to
be stored arrives at the I/O point with the address of the
destination location already assigned. This means that it
is not possible, at this level, to choose the destination
among a number of empty locations, as assumed in
previous papers. Moreover we propose a procedure,
based on parametric identification techniques, to compute precisely the throughput of the crane/aisle subsystem which comes out from the application of our
algorithm. The continuous approximation allows us to
express the throughput improvement as a function of a
small number of meaningful parameters.
The problem of shuttle optimization in warehouse
system of the type in Fig. 1 has not been previously
discussed in the literature. We propose an algorithm to
optimize the sequence of picking and deposit operations
performed by the shuttle, so as to minimize the time
required by each cycle. Assuming again a continuous
structure for the locations placed along the shuttle path,
we evaluate the theoretical improvement of the shuttle
sub-system throughput obtained by the application of
the proposed algorithm, with respect to the case in
which no optimization is considered.
A key point in the development of our optimization is
that, as we shall show, the cranes optimization is
independent (in a sense to be specified) from the shuttle
optimization and therefore they can be performed
separately.
The CTPN model of the warehouse is used for the
performance evaluation of the control algorithms.
2. Automated warehouse control: the role of the optimizer
system
The main task of a warehouse system is to store and
retrieve goods. We refer to the goods in terms of SUs,
which are the entities the system can handle.
In most part of the systems operating today the
automated warehouse management is carried out by a
control system which consists of three different levels:
PS (Planning System—Level 3),
MS (Management System—Level 2),
HS (Handling System—Level 1).
In this context, the MS receives the loading/unloading
planning from the PS and based also on the location map
(i.e. the list of the SUs stored in the warehouse and the
coordinates of their location), provides to the HS a set
of complex handling requests, hereafter named missions,
to be executed. It also communicates to the PS the
inventory items.
The role of PS can be resumed as follows: PS does not
know where an SU is, but it only knows whether an SU
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F. Amato et al. / Control Engineering Practice 13 (2005) 1223–1241
is or is not in the warehouse, thus it can order to pick it
by sending a retrieval mission to MS; PS does not know
where an SU has to be stored but it only knows that
there is at least an empty location in the warehouse, thus
it can send a storage mission to MS. PS works on a
statistical characterization of the system performance.
Its main task is material planning and lot-sizing keeping
in mind the constraints concerning the inventory levels,
buffer capacities and production times.
The MS at level 2 knows only the warehouse location
map, but not the actual state of the whole plant (crane
and shuttle positions, tracking position state). Therefore, level 2 schedules the missions to be performed
exclusively on the basis of the state of the warehouse
location map by using its optimal management algorithm.
The HS is dedicated to the control and coordination
of the plant devices (conveyors, buffers, cranes, shuttle,
etc.). As said, it receives by the MS the set of missions,
which must be executed by controlling and coordinating
different devices. For instance, a mission could request
to move an SU from the main input buffer to a specific
location in one of the aisles. Moreover, the HS has to
manage multiple missions at the same time. Finally it
communicates to the MS the result of the missions.
A number of physical locations where an SU can
stand and be detected are distributed within the plant. A
logical record corresponds to each location where the
data concerning the SUs and their state (present/absent)
are stored; these locations are called tracking positions.
The HS task is to manage the shifting of tracking data
according to the real SU movements in the plant. To this
end the HS knows all plant information (actual location
of cranes and shuttles, the state of SUs, etc.).
On the other hand, performing real-time optimization
at this level presents some disadvantages, as it is
discussed afterwards.
On the basis of the above considerations, we propose
to separate the handling phase from the optimization
phase by introducing another intermediate level, the OS,
between the MS and the HS (Fig. 2). This reflects the
recent trend to bridge the gap between the planning/
management system and the control system of a factory
using on-line information to manage the current
application of manufacturing resources; more in general
this level is referred in technical literature as Manufacturing Execution System.
In our approach, the SU movements in an automated
warehouse are carried out by subdividing the given
mission into several basic handling sequences. In this
context, the HS is used for the sole basic handling
sequences (for example crane_goes_to_position_X, shuttle_picks_SU_from_buffer, etc.)
while the OS takes charge of the real-time optimization
based on the current state of the plant. Therefore, the
OS receives the list of missions from level 2 MS,
Fig. 2. Optimization module in the software architecture.
reschedules and segments them into basic sequences
according to the plant state and the appropriate
optimization policy; such sequences are then communicated to the HS to be executed. To perform the
movement, simple functions which implement those
basic sequences are needed.
The expected advantages are:
The development of the HS code is greatly simplified.
Testing and debugging of the HS is simplified and can
also be performed if other levels have not been
developed yet, thus reducing the start-up time for the
plant.
The basic sequences are similar for different warehouse systems; therefore code reusability is easily
achieved.
The OS can be developed and tested via computer
simulation if a detailed model of the plant, as
controlled by the HS, is available.
Currently the HS is implemented on a programmable
logic controller which does not always allow the
coding of complex algorithms. Since, by our approach, the OS can be physically separated from the
HS, it is possible to implement the OS on a different
machine.
As will be presented later, the interface between the
OS and the HS is very simple, thus making software
portability easier.
3. Model requirements
The modelling of a warehouse system can be
performed at various levels of complexity. The level of
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modelling can range from very aggregate to very
detailed, according to the user’s requirements for system
analysis and control. In this paper, a detailed modelling
approach is presented.
The proposed model can be used at different steps of
the automated warehouse system design.
In the initial phase of warehouse sizing the availability
of a detailed model allows the evaluation of the storage
area sizes. At this level the user needs to determine the
number of shuttles and cranes required to handle the
expected amount of materials. This is often an
important design issue due to costs and lead times for
special vehicles. The user may also wish to model the
interactions of the warehouse with other areas of the
factory in order to see how these interactions affect the
overall system performance.
During the test phase the logic of the warehouse
control system has to be verified. The purpose is to
check if all the movements of the warehouse components are executed according to the specifications and
that no deadlock occurs. Moreover, in the presence of
decisions based on priorities, it is very important to
evaluate the validity of the control algorithm, in order to
observe how mission dispatching affects the system
performance.
In this paper we refer to the automated warehouse
architecture shown in Fig. 1. The main problems
encountered in warehouse system modelling are listed
below.
Resource attributes: SUs are identical in their functionality. However the state attributes (e.g. loaded,
empty), the destination of the load (coordinates of the
rack location), the kind of operation to be performed
(storage operation, retrieval operation) make them
different from each other.
Higher-level scheduler integration: It is important to
remark that the model must represent the plant ‘‘as
controlled by the HS’’. This means that the communication interface between the HS and the OS must be
included in the model.
Operation time: The model must include timing
information. For instance, all loading/unloading activities of cranes take time; moreover, the time required to
pick-up a pallet depends on the distance between the
current position of the crane and the rack location
where it is stored.
Modular design: During the system life-cycle, it may
happen that a component (e.g. a shuttle, a crane, etc.) is
added or removed; moreover, it is important to achieve
the component re-use. Therefore, it is convenient to
divide the warehouse model into sub-models and define
a standard interface for the communication between
them; also, this approach may reduce the simulation
complexity.
We use colored timed Petri nets for the model since
they allow us to fulfill all the above requirements.
1227
4. The model
For the sake of brevity, we present in detail only
the obtained sub-model of the cranes. See (Basile,
Carbone, & Chiacchio, 2001) for more details about
the whole model. Indeed this sub-system is present in
almost all warehouse systems and therefore it is of
general interest. Moreover, most of the complexity
in modelling a warehouse system relies on the above
sub-systems. Indeed the input and output buffers,
their size, number and placement are strictly dependent
on the particular warehouse; in addition, they do
not present any conceptual difficulty in the modelling
phase.
The CTPN model of the generic crane sub-system,
called CRANEi ; and of the interface between a generic
crane and the shuttle sub-systems (with two bays) are
shown in Figs. 3 and 4, respectively. In Tables 1 and 2
the interpretation of all transitions and places of the
crane CTPN model is described.
The sub-system models are all parameterized. For
instance, the CRANEi net model is parameterized with
respect to the size of aisle input/output buffers (index p
in Fig. 3 and Table 2). Other parameters are related to
the operation timing.
After recalling some technical details concerning PN
modelling (DiCesare, Harhalakis, Proth, Silva, &
Vernadat, 1993; Murata, 1989; Silva & Teruel, 1997;
Ramaswamy & Valavanis, 1994), we show how the four
problems described in the previous section are solved for
each sub-system of the warehouse.
4.1. Background
CTPNs extend the framework of PNs by adding
color, time and modular attributes to the net. The color
attribute is developed to deal with systems that have
similar or redundant logical structures. The time
attribute allows various time-based performance measures to be added in the system model.
We assume that the reader already knows CPN
concepts and terminology as introduced in Jensen
(1995). Let us recall here the main characteristics of
CPNs we deal with in this work.
A multi-set m is a set which may contain multiple
occurrences of elements of a non-empty set S. Formally,
a multi-set is a function m : S ! N; where N is the set of
non-negative integers, which we represent as a formal
sum
X
mðsÞs:
s2S
We denote by: S MS the set of all multi-sets over S;
TypeðvÞ the type of a variable v; TypeðexprÞ the type of
an expression expr; VarðexprÞ the set of variables in an
expression expr. In addition we define the binding of a
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F. Amato et al. / Control Engineering Practice 13 (2005) 1223–1241
Fig. 3. CRANEi CTPN module.
set of variables V as the function b associating to each
variable v 2 V an element bðvÞ 2 TypeðvÞ and exprhbi as
the value obtained by evaluating an expression expr in a
binding b, i.e. substituting for each variable v 2
VarðexprÞ the value bðvÞ 2 TypeðvÞ determined by the
binding.
Now we recall the following definition (Jensen, 1995).
Definition 1. A CPN is
ðS; P; T; A; N; C; G; E; IÞ where
a
9-tuple
CPN ¼
S is a set of non-empty types, called color sets;
P is a finite set of places drawn by circles;
T is a finite set of transitions including immediate
transitions drawn by black bars and timed transitions drawn by empty boxes;
A is a finite set of arcs such that P \ T ¼ P \ A ¼
T \ A ¼ ;;
N is a node function defined from A to P T [ T P;
C is a color function defined from P into S;
G is a guard function defined from T into expressions
such
that
8t 2 T : ½TypeðGðtÞÞ
is
boolean
^TypeðVarðGðtÞÞÞ S;
E is an arc expression function defined from A into
expressions such that 8a 2 A : ½TypeðEðaÞÞ ¼
CðpÞMS ^ TypeðVarðEðaÞÞÞ S; where p is a place
of NðaÞ;
I is an initialization function defined from P into closed
expressions such that 8p 2 P : ½TypeðIðpÞÞ ¼ CðpÞMS :
In order to define the binding of a transition t we
introduce the following notation:
ARðtÞ ¼ fa 2 AjNðaÞ 2 P
ftg [ ftg
Pg;
VarðtÞ ¼ fvjv 2 VarðGðtÞÞ _ 9a 2 ARðtÞ
: v 2 VarðEðaÞÞg:
A binding of a transition t is a function b defined on
VarðtÞ such that
v 2 VarðtÞ : bðvÞ 2 TypeðvÞ and GðtÞhbi ¼ 1;
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F. Amato et al. / Control Engineering Practice 13 (2005) 1223–1241
1229
changes according to
M 2 ðpÞ ¼ M 1 ðpÞ
X
Eðp; tÞhbi
ðt;bÞ2Y
þ
X
Eðt; pÞhbi
8p 2 P:
ðt;bÞ2Y
In this case we say that M 2 is reachable by M 1 : This is
written as M 1 ½Y 4M 2 :
The introduction of a timing specification is essential
in order to use Petri net models for performance
evaluation of distributed systems. We consider nets with
deterministic timing and one phase firing rule, i.e. a
timed enabling (called the service time of the transition)
followed by an atomic firing. Service times of transitions
are supposed to be mutually independent and time
independent.
A time function f : BðtÞ ! R is introduced; it is the
time required by the timed transition t associated with a
binding b to complete the firing. After time function has
been introduced, we speak of CTPN models.
In a colored Petri net model two kinds of conflicts
may arise: conflicts among transitions and conflicts
among bindings. In this paper the following solution has
been adopted:
Fig. 4. Interface between shuttle and Cranei CTPN modules.
i.e. the guard function is true. BðtÞ denotes the set of all
bindings for t.
A token is a pair ðp; cÞ where p 2 P and c 2 CðpÞ;
while a binding element is a pair ðt; bÞ where t 2 T and
b 2 BðtÞ: The set of all token elements is denoted
by TE while the set of all binding elements is
denoted by BE. A marking is a multi-set over TE
while a step is a non-empty and finite multi-set
over BE. The sets of all markings and steps are
denoted by M and Y, respectively. We assume
S ¼ S1 S 2 S n and that a function between
each set S i and the integer number set Z is defined;
this leads us to represent a token in a place p by a
n-tuple hti ¼ ðc1 ; c2 ; . . . ; cn Þ; where cj is an integer.
We represent by ð1Þ the black token, i.e. the token
having no color.
A step Y is enabled in a marking M iff the following
property is satisfied:
X
8p 2 P :
Eðp; tÞhbipMðpÞ:
ðt;bÞ2Y
When a step is enabled in a marking M 1 it can occur
and, if the transition t fires in binding b, the net marking
In order to avoid the coupling between resolution of
conflicts1 and duration of activities,2 transitions in
conflict are supposed to be immediate except for
timed transitions modelling a watchdog timer or
immediate transitions modelling a failure detection;
only in these special cases it is allowed that a timed
transition, modelling a watchdog timer, can interrupt
another timed transition or that an immediate
transition, modelling an error detection, can be in
conflict with a timed transition.
In general, in a PN model there is no need of
removing tokens from a place in the same order as
they were added. However, in our case, when the
tokens contained in a place represent SUs, it is
important to preserve the incoming order for their
removing. Therefore, we assume each place as being a
queuing place with a FIFO policy. Thus, no conflict
among bindings may arise.
We denote a net place by Pfx; yg; where x denotes the
sub-system of the model and y denotes the index of the
place inside the sub-model; we assume that, in general, a
number of nt aisles is present and therefore nt þ 1
1
We say that two or more transitions are in conflict if they have
common input places.
2
When a timed activity is associated with transitions, a conflict
resolution policy may be a race between conflicting transitions, that
has no sense in presence of a physical plant.
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1230
denotes the shuttle sub-system net model. Fusion places,
source places and sink places (see Sections 4.3 and 4.5)
are denoted by Pf fxg; Psrfxg; Pskfxg; respectively, and
are all drawn by double circles.
The following expression functions, used in the arc
expressions and guard functions, are defined:
(i) prðp; c1 ; c2 ; . . . ; ck Þ: for each token hti in the place
p the function builds a new token formed only
by the specified components c1 ; c2 ; . . . ; ck of the
token hti;
(ii) concðht1 i; ht2 i; . . . ; htk iÞ: the function builds the new
token ðht1 i; ht2 i; . . . ; htk iÞ formed by the concatenation of the specified components;
(iii) id: this function selects all the components of a
token.
4.2. Resource attributes
The use of a CPN allows us to completely solve the
problem of the resource attribute representation, since
the token color univocally identifies a pallet or the state
of a device.
For example, in the crane model (see Fig. 3), place
Pf f2ig has format ðsh; bay; IDSU ; comÞ: Thus, the
presence of a token in this place means that a shuttle
is aligned to the input buffer of the crane with pallet
IDSU on bay bay; moreover, command com (explained
later) has to be performed on pallet IDSU. Without
using CPN formalism we would need a different place
for each identification number of the pallet and for each
bay of the shuttle.
4.3. Higher level scheduler integration
The modelling of the communication between the OS
and the sub-systems as controlled by the HS can be
performed via sink and source places; source (sink)
places are places having no input (output) arcs; in the
source (sink) places command tokens are delivered by
(got from) the dispatcher (Fig. 5). Further on we focus
on some details about the integration of the subsystems.
For example, concerning the communication between
the OS and CRANEi ; the OS sends a sub-mission order
by putting in place Psrfig (that is a source place) of the
CRANEi model a token whose format is ðop; x; y; z; wÞ
with
op: operation type (0 homing operation, 1 storage
operation, 2 retrieval operation);
x: x-coordinate of rack location;
y: y-coordinate of rack location;
z: rack location depth;
w: rack side.
4.4. Operation time
A CTPN is a CPN where a time delay is associated
with each transition; such delay represents the time
required to complete the firing.
In this paper timed transitions, having a finite time
delay and drawn by empty boxes, are used to model the
vehicles travelling time and loading/unloading time. The
time delay may be either constant, like for loading/
OPTIMIZER
Psr{4}
Psr{1}
Psk{1}
Psr{2}
Psk{2}
Psr{3}
Psk{3}
Psk{4}
Part OUT
Part IN
BufferIN
(i = 5)
CRANE 1
(i = 1)
CRANE 2
(i = 2)
CRANE 3
(i = 3)
Buffer
OUT
(i = 6)
SHUTTLE (i=4)
Pf{1}
Pf{2}
Pf{3}
BayPR 1
(i = 7)
Pf{4}
Pf{5}
BayPR 2
(i = 8)
Pf{6}
Pf{7}
BayPR 3
(i = 9)
Pf{8}
Pf{9}
BayPR 4
(i = 10)
Fig. 5. Interaction between CTPN modules of the whole model and the optimizer sub-system.
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unloading operations, or depending on the distance
between the current position of the crane machine and
the rack location of the pallet destination/location, for
the retrieval/storage time. Immediate transitions, having
a null time delay and drawn by bars, are used to model
the logical activities and failure detection. An example
of logical activity is the choice between storage, homing
or retrieval missions which is modelled by a conflict
between three immediate transitions: the start of each
mission is conditioned by the value of the token color
(the component Op of the token in place Pfi; 2g
determines which of the three transitions Ts1i; Th1i;
Tr1i is enabled, see Fig. 3.)
4.5. Distributed and modular design
The interface among modules is performed via fusion
places. Each fusion place is a set of places that are
considered to be identical, i.e. they all represent a single
conceptual place even though they are drawn as
individual places.
Fig. 5 represents the merging of the sub-models to
obtain the whole model of the warehouse system
depicted in Fig. 1. Note that for each sub-system
present in the warehouse there is a corresponding submodel and the interfaces among the sub-modules are
represented by fusion places. When a token is added to
(removed from) one of these places, an identical token is
added to (removed from) all other places in the fusion
set.
For example, let us consider more in detail the
communication between the shuttle and the CRANEi
during a delivering operation (see Figs. 3 and 4). As
said, the command token format for place Pf f2ig is
ðsh; bay; IDSU; comÞ where:
sh: shuttle identifier;
bay: number of shuttle bay aligned with CRANEi
input buffer;
IDSU: stock unit identifier (pallet number);
com: command to execute on pallet IDSU: (e.g. 0
nothing, 1 picking, 2 delivering) or mission status (e.g.
-3 bay already busy, -2 buffer full, -1 end mission).
In the shuttle model the com component is used to
distinguish a delivering operation from a picking one.
In the crane model the IDSU component is used to
identify the pallet. Once the pallet has reached the
position represented by place Pfi; 15 þ 4p 4g; it is
possible to assign the correct coordinates of the
destination to the crane machine (see the token format
of place Pfi; 5g).
Remark 2. To further clarify this point, we show how a
loading/unloading operation can be executed only when
the pallet colors and the buffer location place colors are
1231
matched. For example in the crane model (see Fig. 3), a
pallet can be loaded on the input buffer if and only if: (a)
the first location of the input buffer is free since a black
token is present in place Pfi; 16g; (b) in place Pf f2ig
token ðsh; bay; IDSU ; comÞ is present, i.e. bay bay of the
shuttle is aligned with the input buffer; (c) guard
½prðPf f2ig; 4Þ ¼ 2 and prðPf f2ig; 3Þa0 associated with
transition TbIi is true, i.e. the component com of the
pallet token must correspond to a delivering operation
ðprðPf f2ig; 4Þ ¼ 2Þ and the pallet is present on the shuttle
bay ðprðPf f2ig; 3Þa0Þ:
Notice that token ðIDSUÞ moves from place Pfi; 15g
to place Pfi; 15 þ 4p 4g having no rack destination
location. When transition TbIpi fires, token ðx; y; z; wÞ is
merged with token ðIDSU Þ producing token
ðx; y; z; w; IDSUÞ: This solution has been chosen in order
to give the IDSU a rack destination location at the last
moment, according to the CRANEi optimization policy.
5. Optimizer system architecture
For the OS we propose the architecture shown in
Fig. 6.
Its inputs are mission queues, while its outputs are
order (sub-mission) queues. All the missions created at
the same time by level 2 MS are placed into the FIFO
queue called Mission. First of all, the OS makes a
separation between storage and retrieval missions. The
storage missions are all saved in the Storage queue,
which is composed of the missions starting from
BufferIN and the missions whose starting point is on
one of the picking/refilling bays. On the other hand, the
retrieval missions have as their starting point a location
in the rack. Therefore, for each aisle there is a different
Retrievalfig queue. In general these queues are not FIFO
since, on the basis of the implemented control policies,
there could be a dynamic rescheduling of the missions
depending on the plant state.
From each mission, the following information are
taken out: SU identifier (IDSU), starting point (SP) and
end point (EP), as shown in Table 3. These information
are transferred to the MissionSeg function which derives
the basic sequences to be performed in order to move
the SU from the SP to the EP, by segmenting the
mission itself in basic handling sequences (Table 3).
For example, as for the mission from BufferIN to
Aislei ; the dispatcher produces the following orders
(segments of mission):
a movement order for the Shuttle sub-system from its
current position to the SP (BufferIN);
a movement order for the Shuttle sub-system from the
SP to the input buffer of Aislei ;
a movement order for the CRANEi from its current
position to its input buffer;
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Fig. 6. Optimizer model.
a movement order for the CRANEi from its input
buffer to the EP (Aislei rack location).
As said, the OS core is the dispatcher block, in which
the control policies are implemented. It decides the
order of the mission segments to be executed, by
dynamically rescheduling the missions inside the order
queues. Therefore, the dispatcher controls the system
progression by adding appropriate tokens to the source
places of the shuttle and of the cranes (Psrfnt þ 1g and
Psrfig; i ¼ 1; 2; 3) (see Fig. 6).
The dispatcher bases its decisions also on the results
of previous sequences assigned to the shuttle and the
cranes (Pskfnt þ 1g and Pskfig; i ¼ 1; 2; 3).
6. Automated warehouse systems control: background
The goal of this section is to propose an optimal
strategy for improving the throughput of the warehouse;
we focus first on the cranes/aisles sub-system and then
on the shuttle sub-system.
6.1. The crane/aisle sub-system
We abstractly assume to deal with a single aisle, onesided rack warehouse under the assumption that a DC
cycle is adopted.
For a single aisle one-sided rack warehouse, the
throughput is defined as the inverse of the mean time
required to perform the cycle; obviously, a simple way to
improve the throughput could be that of reducing the
dimension of the aisle, and therefore of the corresponding rack. However, in a real warehouse (consisting of
two or more two-sided aisles), this approach would lead
to build more aisles if one desires the storage capacity of
the warehouse to remain unchanged. This way of
proceeding requires further replication of expensive
sub-systems and therefore should be avoided; in
particular, in an automatic system, each aisle is
equipped with its own crane, which generally mounts a
rather expensive technology on-board.
The above discussion leads to the following conclusion: it is mandatory to improve the throughput without
reducing the rack dimension.
From now on we shall make the following assumptions:
A1 We deal with an SU system; an SU is regarded as an
indivisible part (at least from the system point of
view) of a certain product. The crane can carry one
SU at the time (one in the storage phase and one in
the retrieval phase). Each location of the rack can
contain exactly one SU of the given product;
A2 The crane travels simultaneously in both rack
directions, with constant speed (acceleration and
deceleration ramps are not considered);
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1233
Table 1
Transition description of CRANEi CTPN model
Table 2
Place description of CRANEi CTPN model
Label
Transition guard
Label
Token color—place interpretation
Transition activity description
Psrfig
Pfi; 2g
ðOp; x; y; z; wÞ—List of crane missions.
ðOp; x; y; z; wÞ—The crane is waiting the choice
of mission to be executed.
ðx; y; z; wÞ—The crane is moving to the input
buffer to pick up a pallet to be stored at
location ðx; y; z; wÞ
ðx; y; z; wÞ—The crane is waiting for a pallet
from the input buffer
ðx; y; z; w; IDSUÞ—The crane is moving to
location ðx; y; z; wÞ with pallet IDSU
ðx; y; z; w; IDSUÞ—The crane is delivering the
pallet at location ðx; y; z; wÞ
ð1Þ—The crane is going back to the home
position
ð1Þ—End mission
ðx; y; z; wÞ—The crane is moving to location
ðx; y; z; wÞ to execute a retrieval operation
ðx; y; z; wÞ—The crane is picking up a pallet
from rack location ðx; y; z; wÞ
ðIDSUÞ—The crane is moving to the output
buffer with pallet IDSU
ðIDSUÞ—The crane is delivering the pallet
IDSU to the output buffer
ð1Þ—Crane ready to execute a new mission
ðsh; bay; IDSU; comÞ—Bay ‘‘bay’’ of shuttle
aligned with input buffer delivery station
ðIDSUÞ—Pallet IDSU is on the
input buffer
ð1Þ—State of input buffer locations
(if marked the associated location is empty)
ðsh; bay; IDSU; comÞ—Bay ‘‘bay’’ of shuttle
aligned with output buffer picking station
ðIDSUÞ—Pallet IDSU is on the
output buffer
ð1Þ—State of output buffer locations
(if marked the associated location is empty)
ðemÞ—Mission result
Topi
Absent
Start crane mission.
Ts1i
½prðPfi; 2g; 1Þ ¼ 1
Start storage mission
Ts2i
Absent
The crane is moving to the input buffer
TbIpi
Absent
The crane is loading the pallet from input buffer
Ts3i
Absent
The crane is moving to the rack location
Ts4i
Absent
The crane is delivering the pallet
Th1i
½prðPfi; 2g; 1Þ ¼ 0
Start homing mission
Th2i
Absent
The crane is going back to
the home position
Tr1i
½prðPfi; 2g; 1Þ ¼ 2
Start retrieval mission
Tr2i
Absent
The crane is moving to the rack location
Tr3i
Absent
The crane is picking up the pallet from the rack location
Tr4i
Absent
The crane is moving to the output buffer
TbO1i
Absent
The crane is delivering the pallet
TbOji
Absent
j ¼ 2::p
A pallet is moving to the next output buffer location
TbOi
½prðPf f2i þ 1g; 3Þ ¼ 0 and prðPf f2i þ 1g; 4Þ ¼ 1
The shuttle bay is free and there is a load command
A pallet is moving onto the shuttle bay from the
output buffer
TnOi
½prðPf f2i þ 1g; 3Þa0 and prðPf f2i þ 1g; 4Þ ¼ 1
The shuttle bay is not free and there is a load command
Error detection
TbIji
Absent
j ¼ 1::p 1 A pallet is moving to the next input buffer location
TbIi
½prðPf f2ig; 4Þ ¼ 2 and prðPf f2ig; 3Þa0
Shuttle bay loaded and there is an unload command
A pallet is moving onto the input buffer from the
shuttle bay
TeOi
Absent
End crane mission
TnIi
½prðPf f2ig; 4Þ ¼ 2 Unload command
Watchdog timer
A3 Storage assignment is performed by the upper level
and the policy is unknown; to evaluate our
algorithms we suppose that a completely random
storage assignment is used. This means that no
specific rule is followed when assigning a storage
location to an incoming SU; the problem of
throughput optimization under more sophisticated
storage strategies (for example class based (Hausman, Schwarz, & Graves, 1976)) will be discussed in
future papers;
Pfi; 3g
Pfi; 4g
Pfi; 5g
Pfi; 6g
Pfi; 7g
Pfi; 8g
Pfi; 9g
Pfi; 10g
Pfi; 11g
Pfi; 12g
Pfi; 13g
Pf f2ig
Pfi; 15 þ ð4j 4Þg
j ¼ 1::p
Pfi; 15 þ ð4j 3Þg
j ¼ 1::p
Pf f2i þ 1g
Pfi; 15 þ ð4j 2Þg
j ¼ 1::p
Pfi; 15 þ ð4j 1Þg
j ¼ 1::p
Pskfig
A4 The rack is continuous; this means that it is
composed of an infinite number of virtual locations.
Each location is addressed by a pair of real numbers.
Remark 3. Assumption A4 is standard in the study of
warehouse systems (see for example, Graves, Hausman,
& Shieh, 1977; Bozer & White, 1984). Indeed it is not
possible to develop a general theory if we try to take into
account the discrete nature of the rack structure.
The following notation will be used:
sh
sv
L
H
th
horizontal speed of the crane [m/min]
vertical speed of the crane [m/min]
rack length [m]
rack height [m]
L=sh time to reach the horizontal end
of the rack [min]
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1234
Table 3
Possible missions in lay-out of Fig. 1
SP
EP
BufferIN
Aislei rack
location
Aislei rack
location
BayPRj
Aislei rack location
BayPRj
E½TB
BufferOUT
tC
Aislei rack location
tW
E½DC C
Table 4
Coefficients of the function d k ðbÞ
k
a0
a1
a2
a3
a4
a5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0.4997
0.3336
0.2503
0.1993
0.1663
0.1422
0.1245
0.1114
0.0999
0.0904
0.0825
0.0764
0.0712
0.0662
0.0622
0.0584
0.0551
0.0519
0.0494
0.0471
0.4864
0.5136
0.4968
0.4615
0.4376
0.3980
0.3737
0.3503
0.3137
0.2839
0.2518
0.2300
0.2109
0.1834
0.1687
0.1469
0.1308
0.1116
0.1006
0.0867
0.4237
1.1575
1.5516
1.7505
1.9227
1.9223
1.9613
1.9492
1.8515
1.7755
1.6601
1.5835
1.5145
1.3838
1.3286
1.2216
1.1479
1.0574
1.0039
0.9283
0.1586
1.0501
1.9574
2.5666
3.1653
3.3451
3.5894
3.6822
3.5835
3.5201
3.3441
3.2365
3.1397
2.8832
2.8057
2.5918
2.4591
2.2877
2.1874
2.0307
0.1525
0.5085
1.2351
1.8095
2.4564
2.6994
3.0003
3.1439
3.1070
3.1010
2.9727
2.9048
2.8425
2.6121
2.5649
2.3682
2.2594
2.1111
2.0245
1.8795
0.0202
0.0811
0.2876
0.4742
0.7127
0.8114
0.9296
0.9917
0.9916
1.0025
0.9670
0.9526
0.9380
0.8611
0.8518
0.7844
0.7514
0.7037
0.6756
0.6263
NS
bS 100%
2
3
4
5
6
7
8
9
10
3.4084
5.6305
7.2506
8.5113
9.5347
10.3904
11.1218
11.7575
12.3176
tv
T
b
E½SC
Without loss of generality we assume that th otv ;
therefore the rack is the rectangular region depicted in
Fig. 7(a). If we locate the I/O point at the left-lower
corner of the rack and fix a pair of coordinate axis with
origin at the I/O point, the rack region can be identified
with the set of points belonging to the interval ½0; b
½0; 1:
According to Fig. 7(b), it is simple to recognize that
E½DC C can be computed according to the following
expression:
E½DC C ¼ E½SC þ E½TB þ E½SC
¼ 2E½SC þ E½TB:
In Bozer and White (1984) it is shown that
1 1 2
þ b T;
E½SC ¼
2 6
1 1 2
1 3
þ b
b T:
E½TB ¼
3 6
30
Table 5
Behavior of bS versus N S
from the I/O point (the retrieve
location) to the storage location (the I/
O point) [min]
expected time for Travel-Between (TB),
i.e. time to travel from the storage
location to the retrieve location [min]
time the crane needs to store or retrieve
an SU [min]
average time the crane has to wait at
the aisle input buffer for an incoming
SU [min]
expected time for the crane DC cycle
[min].
H=sv time to reach the vertical end of
the rack [min]
maxfth ; tv g [min]
minfth =T; tv =Tg rack shape factor
(adimensional)
expected time for the crane Single
Command (SC) cycle, i.e. time to travel
Therefore, from (1) we have
4 1 2
1 3
þ b
b T:
E½DC C ¼
3 2
30
ð1Þ
(2a)
(2b)
(3)
If we consider a set of n DC cycles the crane
throughput gC is defined as
2n
nE½DC C þ 4ntC þ ntW
2
¼
:
E½DC C þ 4tC þ tW
gC ¼
ð4Þ
Now let us assume obtaining a reduction of the travelbetween (TB) time equal to a 100%; the new throughput can be written as
g~ C ¼
2
:
E½DC C aE½TB þ 4tC þ tW
(5)
Obviously, g~ C 4gC ; therefore we let g~ C ¼ ð1 þ bC ÞgC ;
where bC 100 represents the percentage throughput
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F. Amato et al. / Control Engineering Practice 13 (2005) 1223–1241
improvement. We have
g~ C gC
gC
E½DC C þ 4tC þ tW
¼
1:
E½DC C aE½TB þ 4tC þ tW
bC ¼
ð6Þ
Remark 4. Note that bC depends on the design
parameters b, th and tC which, for a given warehouse, are fixed, and on the optimization parameters a
and tW ; a is increased when the TB time is optimized,
while tW is reduced when the shuttle operations are
optimized.
Remark 5. It is important to recall that, while the values
of E½DC C ; E½TB are exactly evaluated in (2b) and (3),
the value tc is known and a will be evaluated in the next
section, the parameter tW can only be evaluated via
simulation and therefore only when a whole model of
the warehouse is available.
In Fig. 8 the behavior of the function bC ðaÞ
parameterized in tW is depicted for b ¼ 0:316; th ¼
1235
0:360 and tC ¼ 0:04 (these are the parameter
values of the real warehouse system we shall consider
later in the paper). Note that, as expected, bC is an
increasing function of a and a decreasing function
of tW :
We note that for tW ¼ 0:5 a TB time reduction of
60% (that is a ¼ 0:6) corresponds to a throughput
improvement of 12%; which, according to Han,
McGinnis, Shieh, and White (1987), leads in a large
warehouse, to the elimination of an aisle without
reducing the overall warehouse performance. Also note
that in correspondence to a smaller tW there is a strong
improvement of the throughput; for example for tW ¼
0:2 we have that bC 40:14:
6.2. The shuttle sub-system
We assume that the shuttle cycle is composed of a
picking operation followed by a deposit operation.
Therefore, also in the shuttle case, we talk of DC cycle.
We make the following further assumptions:
A5 The shuttle travels with constant speed;
A6 Along the mono-dimensional path of the shuttle
there is a continuous distribution of picking/deposit
location; we place the origin of the path abscissa in
correspondence of the main input buffer and the end
in correspondence of the main output buffer.
The following notation will be used:
Fig. 7. (a) Rack shape; (b) Computation of E½DC:
sS
speed of the shuttle [m/min]
0.18
0.16
τw=0.0
βC (α) parameterized in τw
0.14
τ =0.1
w
τ =0.5
w
τ =0.6
w
τ =0.3
τ =0.7
w
w
τ =0.4
τw=0.8
w
τ =0.9
w
τw=1.0
0.12
τw=0.2
0.1
0.08
0.06
0.04
0.02
0
0.2
0.3
0.4
0.5
α
0.6
Fig. 8. Behavior of bC ðaÞ parameterized in tW :
0.7
0.8
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1236
LS
tS
E½P
E½D
tS
E½DC S
length of the shuttle path [m]
LS =sS time to go from the origin of the
path to the end of the path [min]
expected time to go from a given point to
the next picking location [min]
expected time to go from the picking
location to the next deposit location [min]
(E½P ¼ E½D when no optimization
algorithm is used)
time the shuttle needs to pick up or
deposit an SU [min]
expected time to go from a given point to
the deposit location in the shuttle DC
cycle [min] (E½DC S ¼ 2E½P ¼ 2E½D
when no optimization algorithm is used).
At the moment, we assume to have a sequence of n
pair of pick-up and deposit orders LIST ord ¼ fðPi ; Di Þ;
i ¼ 1; . . . ; ng; where Pi and Di denote the location of the
ith pick-up and deposit, respectively, and to follow a
FIFO policy; therefore each pair ðPi ; Di Þ is executed
according to the ordered list LIST ord :
In this case we have
E½DC S ¼ E½P þ E½D ¼ 2E½P:
(7)
E½P ¼ 13 tS ;
@Gðx; zÞ
¼
>
1;
@z
>
>
:
0;
zo0;
0ozox;
xo1 x;
z41 x
(12)
The expected value, EðxÞ; of the distance of an
arbitrary chosen point in the interval ½0; 1 from x 2 ½0; 12
is therefore given by
EðxÞ ¼
Z
1x
0
1
zgðx; zÞ dz ¼ x2 x þ :
2
(13)
By symmetric arguments, it is simple to recognize that
EðxÞ; for x 2 ½12; 1; has the same expression as (13).
Therefore, by integrating over x, we obtain E½P as
follows
E½P ¼ 2tS
Z
1=2
EðxÞ dx ¼
0
1
tS :
3
(14)
6.3. A performance index for the whole warehouse
(8)
2
E½DC C þ 4tC þ tW
gC ¼
therefore, from (7) we have
(15)
and the shuttle/picking refilling sub-system throughput
E½DC S ¼ 23 tS :
(9)
Therefore, when no optimization is performed, the
shuttle throughput is
2
3
¼
:
E½DC S þ 2tS tS þ 3tS
(10)
To prove formula (8), without loss of generality, we
consider a path of unit length; the final result will be
then multiplied by tS : Let us denote by x 2 ½0; 12 the
position of the picking location.
Denote by Zx the stochastic variable whose realization represents the distance of an arbitrary chosen point
of the interval ½0; 1 from the point x. The probability
density function (pdf) Gðx; zÞ of Zx is given by
8
0;
>
>
>
< 2z;
Gðx; zÞ:¼ProbðZ x pzÞ ¼
> z þ x;
>
>
:
1;
gðx; zÞ:¼
8
0;
>
>
>
< 2;
So far we have introduced two partial performance
indices: the crane/aisle sub-system throughput
In the sequel we shall show that
gS ¼
Therefore the cumulative distribution function (cdf)
gðx; zÞ has the following expression:
zo0;
0ozox;
xozo1 x
z41 x:
(11)
gS ¼
2
:
E½DC S þ 2tS
(16)
From the above formulae it is evident that an
improvement of the crane throughput gC can be
obtained, as observed in Section 6.1, via the reduction
of E½DC C ; in the same way an improvement of the
shuttle throughput gS can be obtained by reducing
E½DC S ; note that the improvement of gS also brings a
benefit to the index gC through the reduction of tW :
In any case, from the point of view of the indices
defined above, the crane and shuttle optimizations can
be performed independent of each other.
Now we shall introduce a further index which looks at
the warehouse as a whole. This index is a measure of the
warehouse system throughput, since it considers complete missions, starting from the main input buffer and
ending at the main output buffer or at some of the rack
locations. This index is defined as follows:
number of complete operations
:
(17)
time
From the point of view of index (17) there is
interaction between the crane and shuttle optimizations.
gW ¼
ARTICLE IN PRESS
F. Amato et al. / Control Engineering Practice 13 (2005) 1223–1241
For this reason the separate optimization of the two
sub-systems leads to a sub-optimization of index (17).3
On the other hand, finding an optimization policy which
looks at the system as a whole is a very complex
objective (and may be not achievable by simple
strategies), and is beyond the scope of this paper.
According to the above considerations, in the sequel
we shall develop two strategies: (i) reduction of the TB
time in order to minimize E½DC C ; i. e. to maximize the
parameter a introduced in Section 6.1; (ii) reduction of
tW by optimization of the shuttle operations. The
impact of the optimization policies on the index (17)
will be a posteriori evaluated in Section 9.
7. An optimization algorithm for the crane/aisle subsystem
In this section we shall describe the proposed
optimization policy, under the following further assumption.
A7 Each SU arrives at the I/O point with the storage
location already assigned.
Due to Assumption A7, the only degree of freedom,
in the optimization procedure, is the clever sequencing
of the retrieval orders. The approach followed in this
paper can be roughly described as follows: consider a list
of N C storage and retrieval orders; optimize the
sequencing of the N C retrievals; when the N C DCs
have been performed consider a new list of N C orders
and repeat the same procedure. This approach belongs
to the category of the so-called static approaches.
An alternative way to proceed, which is not considered
here, is known in the literature as the dynamic approach
(Seidmann, 1988; Linn & Xie, 1993). It consists of resequencing the list of orders whenever a new pair storage/
retrieval is available; in this case it is necessary to assign
due dates to the retrievals, to avoid that retrievals from
locations at the end of the rack are never executed.
To describe the algorithm we propose, let us denote
by R the set composed of the retrieval location
addresses, by S the ordered list of storage location
addresses and by N C the cardinality of R and S.
Algorithm 1.
S1 Take the first element s 2 S: Select the element of
R with the minimum TB distance from s measured
by the Chebichev metric. Recall that, given
two points P1 ¼ ðx1 ; y1 Þ and P2 ¼ ðx2 ; y2 Þ;
the Chebichev distance d C ðP1 ; P2 Þ between P1
3
Simulation experiments show that an almost perfect decoupling
between crane and shuttle optimizations is reached for very small
values of tW :
1237
and P2 is defined as
d C ðP1 ; P2 Þ:¼ maxfjx1 x2 j; jy1 y2 jg:
We need to use Chebichev metric because the crane
moves simultaneously in both directions.
S2 Perform the DC cycle with storage in s and retrieval
from r.
S3 Update
R ¼ R frg;
S ¼ S fsg:
S4 If R ¼ ; then stop; otherwise goto Step S1.
The performance of an optimization procedure
similar to Algorithm 1 has been evaluated in Han,
McGinnis, Shieh, and White (1987). However, in that
study it was assumed that the destination address of the
incoming SU could be chosen among a number of
empty locations; moreover, the theoretical computation
of the expected performance of that procedure was
obtained in an approximate way.
The aim of this section is that of computing exactly
the expected performance of the procedure described in
Algorithm 1 under the Assumptions A1–A4, A7. This
result will be then validated in Section 9 on a real
warehouse.
Note that, at step S1 of Algorithm 1, it is chosen,
between the elements of R, the one which guarantees the
minimum distance between the storage address and the
retrieval address. Therefore the first question to be
answered is the following: What is the expected mean
value d k ðbÞ of the distance between a point in the rectangular
region ½0; b ½0; 1 and the nearest (in the Chebichev sense)
of k arbitrary chosen points in the same region?
Experience (see formulae (2a)–(2b)) suggests that
average distances typically are polynomial functions of
the shape factor b. Therefore, we assume the following
structure for the function d k ðbÞ:
d k ðbÞ ¼ a0k þ a1k b þ a2k b2 þ a3k b3 þ a4k b4 þ a5k b5 :
(18)
Moreover, note that a fifth-order polynomial is, in any
case, a very good approximation of any continuous
function around zero (remember that b 2 ð0; 1Þ).
In (Amato & Basile, 2002) a procedure which makes
use of a parametric identification based method to
evaluate the coefficients of the vector ak ¼
ða0k a1k a2k a3k a4k a5k ÞT has been presented. By
applying this procedure we found the coefficients for
d̄ k ðbÞ; k ¼ 1; y, 20, reported in Table 4.
8. An optimization algorithm for the shuttle sub-system
The goal of this sub-section is the development of an
optimal strategy for improving the throughput of the
ARTICLE IN PRESS
F. Amato et al. / Control Engineering Practice 13 (2005) 1223–1241
1238
sub-system composed by the shuttle and the picking/
refilling locations.
The proposed algorithm makes a clever sequencing of
the scheduled picking/deposit operations to improve the
throughput of the shuttle system.
Algorithm 2. (See Section 6.2 for the notation)
distance of an arbitrary chosen picking location from
the starting point x 2 ½0; 1=2 plus the distance of an
arbitrary chosen deposit location from the picking
location.
The cdf G O ðx; zÞ of the stochastic variable Z O
x has the
following expression:
G O ðx; zÞ
S0 (Initialization) Choose N S such that n=N S ¼: nbatch 2
N: Let
START ¼ The initial location of the shuttle
COUNT ¼ 0:
S1 Consider the first N S elements of LIST ord ; ðPi ; Di Þ;
i ¼ 1; . . . ; N S : Let
N left ¼ N S
LIST ¼ fðPi ; Di Þ; i ¼ 1; . . . ; N left g
LIST ord ¼ LIST ord LIST:
S2 Compute:
DIST i ¼ distðSTART; Pi Þ þ distðPi ; Di Þ
i ¼ 1; . . . ; N left :
k ¼ argmin DIST i
i¼1;...;N left
Pcurr ¼ Pk
Dcurr ¼ Dk :
S3 Execute ðPcurr ; Dcurr Þ: Let
START ¼ Dcurr
N left ¼ N left 1
LIST ¼ LIST fðPcurr ; Dcurr Þg
COUNT ¼ COUNT þ 1:
S4 if N left a0 goto Step S2; elseif COUNT onbatch goto
Step S1; else end of the algorithm.
:¼ProbðZO
x pzÞ
8
0;
>
>
>
>
>
>
2z2 ;
>
>
>
>
>
3 2
3
1 2
>
>
4 z þ 2 xz 4 x ;
>
>
>
>
< 12 z2 þ 32 z 12 x2
¼
>
þ 12 x 14 ;
>
>
>
>
>
>
> 14 z2 þ 12 x þ 1 z
>
>
>
>
>
þ 14 x2 þ x ;
>
>
>
>
:
1;
zo0;
0pzox;
xpzo1 x;
1 xpzo1 þ x;
:
ð19Þ
1 þ xpzo2 x;
zX2 x:
The corresponding pdf is
gO ðx; zÞ
d
:¼ G O ðx; zÞ
dz
8
0;
>
>
>
>
>
> 4z;
>
>
>
>
>
< 32 z þ 32 x;
¼
>
z þ 32 ;
>
>
>
>
>
>
> 12 z þ 12 x þ 1 ;
>
>
>
:
0;
zo0;
0pzox;
xpzo1 x;
1 xpzo1 þ x;
ð20Þ
1 þ xpzo2 x;
zX2 x:
Now, by ZO
x;k we denote the stochastic variable whose
realization represents the minimum between k distances
(corresponding to k picking/deposit pairs) starting by
point x. It is simple to recognize that the pdf of Z O
x;k is
given by
k
O
GO
k ðx; zÞ ¼ 1 ð1 G ðx; zÞ Þ:
(21)
Remark 6. Note that Pi ’s represents the coordinates of
the buffer locations where the SUs have to be picked
and Di ’s the coordinates of the buffer locations
where the SUs have to be stored; therefore, in order
to implement the control algorithm described above,
the knowledge of current position of the shuttle is
necessary. Therefore, as in the classical control
context, the control algorithm we finally use is state
dependent.
Therefore the expected travel time, starting from x,
corresponding to the minimum travel time between k
possible picking/deposit pairs is
To evaluate analytically the expected time to execute a
DC under the application of Algorithm 2, let us assume
to deal with a unit length path and denote by ZO
x the
stochastic variable whose realization represents the
EO
k ½DC S ðxÞ
Z 2x
¼ tS
zkð1 G O ðx; zÞÞk1 gO ðx; zÞ dz:
From (21) it is possible to derive the cdf of Z O
x;k
d O
G ðx; zÞ
dz
¼ kð1 G O ðx; zÞÞk1 gO ðx; zÞ:
k1
O
gO
k ðx; zÞ ¼ kð1 G ðx; zÞÞ
0
ARTICLE IN PRESS
F. Amato et al. / Control Engineering Practice 13 (2005) 1223–1241
Finally, by integration over x, we obtain E O
k ½DC S ; the
expected time to perform a shuttle DC cycle by choosing
among k picking/deposit pairs the one with minimum
travel time from an arbitrary starting point in ½0; 1
Z 1=2
EO
½DC
¼
2
EO
S
k
k ½DC S ðxÞ dx:
0
Now if we have a block of N S picking/deposit pairs,
the first time we select the minimum between N S
distances, the second time the minimum between N S
1 distances, etc. Therefore the expected time to perform a
set of N S DC cycles, in presence of optimization, is given
by
PN S O
k¼1 E k ½DC S
:
(22)
EO
½DC
¼
S
N
NS
Now the shuttle throughput in presence of optimization is given by
To test our algorithm we have integrated it with the
CTPN model of the real warehouse as developed in the
first part of the paper.
A simulation experiment has been conducted under
the following assumptions.
therefore, the shuttle throughput improvement bS is
given by
g~ S gS 2
tS þ 3tS
¼
1;
gS
3 EO
½DC
S þ 2tS
N
where E O
N ½DC S is given by (22).
In Table 5 the values of bS for N S ¼ 2; 3; . . . ; 10; tS ¼
0:067 and tS ¼ 0:181 (these are the parameter values of
the real warehouse system we shall consider later in the
paper) are reported.
9. Case study: control of a medium-size warehouse
In this section we refer to the automated warehouse of
a real plant. The main characteristics of the warehouse
model are the following:
sh ¼ sv ¼ 50 ½m= min;
rack location size¼ 1 ½m 1 ½m 1 ½m;
number of x location¼ 18; thus L ¼ 18 ½m;
number of y location¼ 57; thus H ¼ 57 ½m;
speed of shuttle machine ss ¼ 100 ½m= min;
length of the shuttle path LS ¼ 18:10 ½m;
time to move the SU to the next buffer position tB ¼
0:083 ½min;
time to load/unload the pallet on the crane machine
tC ¼ 0:04 ½min;
time the shuttle needs to load/unload a pallet tS ¼
0:067 ½min:
Note that, according to the notation of Section 6, we
have:
th ¼ 0:36 ½min;
tv ¼ 1:14 ½min;
T ¼ tv ¼ 1:14 ½min;
b ¼ 0:316;
tS ¼ 0:181 ½min:
9.1. Simulation results
2
g~ s ¼ O
;
E N ½DC S þ 2tS
bS ¼
1239
A number of 120 complete DC missions is considered.
In this set of missions a number of 40 complete DC
missions is considered for each crane machine.
A complete DC mission consists of a storage mission
starting from the first location of the main input
buffer or from a picking/refilling area output location
and terminating at an aisle rack location and of a
retrieval mission starting from an aisle rack location
and terminating at the last location of the main
output buffer or at a picking/refilling area input
location.
A partial DC mission consists of the part of a
complete mission which starts at the input buffer of
the aisle and terminates at the output buffer of the
same aisle (this is the classical DC cycle considered in
Section 6.1).
The experiment is repeated a number of times by
permuting the list of storage orders S.
For each set of the 40 complete DC missions, two
simulation experiments, according to Algorithm 1 with
N C ¼ 1 and 20, have been executed. In both simulations
we did not perform the shuttle optimization. We denote
by DC N C ¼1 and DC N C ¼20 the average time to perform a
partial DC mission in the two cases. We obtained
DC N C ¼1 ¼ 2:80;
DC N C ¼20 ¼ 2:53;
tW ¼ 0:9:
Now let us compare the theoretical and simulation
results. According to the theory developed in Section
6.1, in the case of one-sided aisle and b ¼ 0:3 we have
that a20 ð0:3Þ ¼ 0:69 (see Fig. 9). From Fig. 8 we have
that the theoretical improvement bC of the throughput
should be about 11% (for tW ¼ 0:9 and a ¼ 0:69).
Concerning the simulation results, taking into account that the computed average time to perform a
simulated partial DC cycle also incorporates the term
4t þ tW ; we have (see formulae (4)–(6))
gC ¼
2
2
¼ 0:714;
¼
DC N C ¼1 2:80
ARTICLE IN PRESS
F. Amato et al. / Control Engineering Practice 13 (2005) 1223–1241
1240
Therefore, concerning the throughput improvement
of the whole warehouse, we have
0.7
b=0.3
0.6
b=0.5
b=0.7
α N (b) parameterised in b
bW 100% ¼
0.5
b=0.9
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
16
18
20
N
Fig. 9. Behavior of aN ðbÞ parameterized in b.
g~ C ¼
2
DC N C ¼20
¼
2
¼ 0:791:
2:53
Therefore, the real improvement is
bC 100% ¼
g~ C gC
100% ¼ 10:8%
gC
which is in accordance with the theoretical result.
For the same sets of DC missions considered above,
we have executed another simulation experiment in
which also the shuttle optimization has been considered;
indeed Algorithm 2 with N S ¼ 5 has been applied. We
denote by DC N C ¼20;N S ¼5 the average time to perform a
partial DC mission in this case.
We obtained
DC N C ¼20;N S ¼5 ¼ 2:42;
tW ¼ 0:5:
Thus we have
g~ 0 C ¼
2
DC N C ¼20;N S ¼5
¼
2
¼ 0:826:
2:42
In this case the real improvement is
0
bC 100% ¼
g~ 0 C gC
100% ¼ 13:6%
gC
g~ W gW
100% ¼ 12:8%:
gW
In order to test the algorithm on the real architecture
to be used on the plant, the test bed represented in Fig.
10 has been built. The plant low-level control is
performed by the Handling Systems which is a
Programmable Logic Controller connected to a process
emulator via Profibus DP. The Management System is
the upper level which sends storage and retrieval
missions, where storage locations are specified. The
optimizer system has been implemented in the C++
language and interfaced with the Management System
and Handling System, according to the architecture
described in the first part of the paper. It has been
executed on a Personal Computer with a Pentium III
processor at 800 MHz and 512 Mb of RAM on board.
The purpose of these further tests is to show that some
approximations used in algorithm synthesis, e.g. neglecting the communication delays and the computation
times, do not affect system performances. Test bed
validation results have confirmed the validity of these
approximations.
Finally, simulation results as well as test bed
validation results have been confirmed by the application of the proposed architecture and control algorithms
to the real automated warehouse in the sense that no
considerable difference has been measured in the real
automated warehouse.
10. Conclusions
The introduction of a new level, the OS, in the control
architecture of warehouse systems has been illustrated.
In this way it is possible to separate handling from
short-term optimization. The development, as well as
the testing and debugging, of handling sequences code is
very simplified; the basic sequences code is very similar
for different warehouse systems, so its reuse is easily
achieved. In addition, the OS can be developed and
which is in accordance with the theoretical result,
obtained by looking at Fig. 8 with a ¼ 0:69 and tW ¼
0:5:
Concerning the throughput improvement index (17)
associated with the global optimization of the warehouse, we compare the case in which no optimization is
performed (N C ¼ 1; N S ¼ 1) with the full optimization
case (N C ¼ 20; N S ¼ 5); we obtain
gW ¼ 0:465;
g~ W ¼ 0:525;
where gW and g~ W are the throughputs before and after
the optimization, respectively.
Fig. 10. Testing environment architecture.
ARTICLE IN PRESS
F. Amato et al. / Control Engineering Practice 13 (2005) 1223–1241
tested using simulations of the plant since a detailed
model of the plant—as controlled by handling sequences
code—is available.
CTPNs have resulted to be an effective tool for
automated warehouse modelling. The main problems
encountered in modelling such systems (resource
attributes, higher level scheduler integration, operation
time, modular design) can be solved by using a unique
formal model. This makes possible to test the whole
control architecture of an automated warehouse and to
work on the same model for the development of
complex control strategies and the on-line plant
monitoring.
The availability of a detailed model opens the door to
the implementation of fault detection functions, after
failures—not considered in this paper—have been
modelled, as well as to the application of modern
optimization techniques to the warehouse low-level
control field, thus obtaining more efficient warehouse
control systems.
In this paper, we have proposed two control
algorithms to manage the cranes and the shuttle
movements; the expected performance of such control
algorithms has been derived under the simplifying
assumption that the distribution of the locations within
the warehouse is continuous; such assumption is
standard in the warehouse system context and is
necessary in order to obtain general results holding for
any rack shape and depending on a few meaningful
parameters.
The evaluation of the control system performance on
a real world warehouse, both validates the theoretical
analysis previously obtained and shows the effectiveness
of the proposed control algorithms.
Open problems which are not discussed in this paper
and will be investigated in future works are:
(i) Use of the cranes optimization algorithm in a
dynamic way, but assigning due dates to the retrieval
orders.
(ii) Development of control algorithms which take
into account the different working conditions of
the warehouse. Indeed it may happen that a set of
only retrieval orders or only storage orders can be
assigned from the high-level scheduler; in this case by
using an ad hoc algorithm better results could
be achieved.
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