Models for warehouse management Classification and - Tài liệu tham khảo | Đại học Hoa Sen

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/223401472
Models for warehouse management: Classification and examples
ArticleinInternational Journal of Production Economics · February 1999
DOI: 10.1016/S0925-5273(98)00114-5 CITATIONS READS 323 45,845 2 authors: Jeroen P. van den Berg W.H.M. Zijm Vrije Universiteit Amsterdam University of Twente
18 PUBLICATIONS1,363 CITATIONS
190 PUBLICATIONS3,589 CITATIONS SEE PROFILE SEE PROFILE
Some of the authors of this publication are also working on these related projects:
Maintenance Logistics: joint optimization of the required spare parts and service engineers, outsourcing, and settling of contract with game- theoretic view View project
Supply chain design for 2020 View project
All content following this page was uploaded by Jeroen P. van den Berg on 17 November 2017.
The user has requested enhancement of the downloaded file.
Int. J. Production Economics 59 (1999) 519 528 —
Models for warehouse management: Classification and examples
J.P. van den Berg, W.H.M. Zijm*
University of Twente, Faculty of Mechanical Engineering, Production and Operations Management Group, P.O. Box 217, 7500 AE Enschede, The Netherlands Abstract
In this paper we discuss warehousing systems and present a classification of warehouse management problems. We
start with a typology and a brief description of several types of warehousing systems. Next, we present a hierarchy of
decision problems encountered in setting up warehousing systems, including justification, design, planning and control
issues. In addition, examples of models supporting decision making at each of these levels are discussed, such as
distribution system design, warehouse design, inventory management under space restrictions, storage allocation, and
assignment and scheduling of warehouse operations. ( 1999 Elsevier Science B.V. All rights reserved.
Keywords: Inventory systems; Warehouse management; Storage allocation and assignment; Forward/reserve problem; Logistics. 1. Introduction
several relatively small distribution centers (DCs) by
a small number of large DCs with an extensive
According to the principles of supply chain man-
distribution network. Often, an entire continent,
agement, modern companies attempt to achieve
like North America or Europe, is serviced by
high-volume production and distribution using
a small number of DCs at strategic positions.
minimal inventories throughout the logistic chain These developments have significantly in-
that are to be delivered within short response times.
fluenced the existing paradigms in inventory
The changes outlined above have had a dramatic
research. Unfortunately, the attention paid by
impact on warehouse management. Low volumes
researchers in inventory theory to the management
have to be delivered more frequently with shorter
of storage systems such as warehouses has been
response times from a significantly wider variety of
relatively limited. Often, it was considered to be
stock keeping units (SKUs). In a further attempt to
a mainly technical issue and therefore belonging to
decrease total inventory, many companies replaced
a different field, i.e., material handling research. The
goal of this paper is to show that, apart from the
close relationship between inventory and ware-
house management problems, the latter often lend
* Corresponding author. Tel: 31 53 489 3621; fax: 31 53 489
themselves to a profound and elegant quantitative
3471; e-mail: w.h.m.zijm@wb.utweule.nl. analysis.
0925-5273/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 8 ) 0 0 1 1 4 - 5 520
J.P. van den Berg, W.H.M. Zijm /Int. J. Production Economics 59 (1999) 519 —528
The new market forces, together with the fast
2. Warehousing systems: A typology and a review
technological developments in material handling,
have affected the operation within warehouses tre-
Material Handling is defined as the movement of
mendously. Shorter product life cycles impose
materials (raw materials, scrap, emballage, semi-
a financial risk on high inventories and, conse-
finished and finished products) to, through, and
quently, on the purchase of capital intensive high-
from productive processes; in warehouses and stor-
performance warehousing systems. Centralized in-
age; and in receiving and shipping areas [1]. Mater-
ventory management, on the other hand, requires ial handling concerns material flow and
an increased productivity and short response times
warehousing. Typical material flow devices are: con-
of the warehousing systems. The aim of this paper is
veyors, fork lifts, automated guided vehicles (AGVs),
to show that sophisticated models and decision
shuttles, overhead cranes and power-and-free con-
support systems for the planning and control of
veyors. Warehousing concerns those material
warehousing systems may significantly contribute
handling activities that take place within the ware-
to the overall research in inventory management.
house, receiving and shipping areas, i.e., receiving of
The developments have been made possible due
goods, storage, order-picking, accumulation and
to recent advances in information technology and sorting and shipping.
the introduction of business information systems.
Basically, we may distinguish three types of
Business information systems support the adminis- warehouses:
trative processes of enterprises. For instance f Distribution warehouses,
enterprise resources planning (ERP) systems are f Production warehouses,
MRP-based business information systems that reg- f Contract warehouses.
istrate all processes concerning finances, human
resources, production planning and inventory
A distribution warehouse is a warehouse in which
management. Other functions that often are sup-
products from different suppliers are collected (and
ported by ERP-systems are, e.g., transportation
sometimes assembled) for delivery to a number of
planning, warehouse management, production
customers. A production warehouse is used for the scheduling and order-entry/order processing.
storage of raw materials, semi-finished products
Besides ERP-systems there are specialized systems
and finished products in a production facility.
that support these functions in complex operations.
A contract warehouse is a facility that performs the
These various systems are linked together using
warehousing operation on behalf of one or more
electronic data interchange (EDI). Examples of customers.
such specialized systems are warehouse management
systems that facilitate the registration, planning and
control of warehouse processes, and inventory man-
2.1. Warehousing activities
agement systems. The models that are presented in
this paper may be implemented in inventory man-
In this section we consider the flow of materials
agement and warehouse management systems and
in a warehouse. Goods are delivered by trucks,
thereby provide significant performance improve-
which are unloaded at the receiving docks. Here
ments in warehouse operations in comparison with
quantities are verified and random quality checks
the methods and models that are currently used.
are performed on the delivered loads. Sub-
This paper is organized as follows. In Section 2,
sequently, the loads are prepared for transportation
we present a typology and a short review of
to the storage area. This means that a label is
warehousing systems. Next, we discuss warehouse
attached to the load, e.g., a bar code or a magnetic
planning problems in Section 3. Section 4 is de-
label. If the storage modules (e.g., pallets, totes or
voted to examples of models for decision support
cartons) for internal use differ from the incoming
for warehouse planning decisions. In Section 5, we
storage modules, then the loads must be reassem-
conclude the paper and discuss opportunities for
bled. After this, the loads are transported to a loca- further research. tion within the storage area.
J.P. van den Berg, W.H.M. Zijm /Int. J. Production Economics 59 (1999) 519 —528 521 systems:
1. Manual warehousing systems (picker-to-prod- uct systems),
2. Automated warehousing systems (product-to- picker systems),
3. Automatic warehousing systems.
We will discuss the three types of warehousing systems in the above sequence.
2.3. A short review of warehousing systems
Fig. 1. Warehousing cost by activity.
A warehouse generally consists of a number of
parallel aisles with products stored alongsides.
A large variety of storage equipment and methods
Subsequently, whenever a product is requested, it
are in use. The most simple storage method is block
must be retrieved from storage. This process is
stacking as is used, e.g., for the stacking of crates of
called order picking. An order lists the products
beer or soft drinks. Bin shelving and modular storage
and quantities requested by a customer or by a
drawers are often used for the storage of small
production/assembly workstation, in the case of
items. For larger items, stored on pallets, pallet
a distribution center or a production warehouse,
racks, gravity flow racks or mobile storage racks are
respectively. When an order contains multiple
often used. For a more elaborate discussion on
SKUs, these must be accumulated and sorted be-
storage methods we refer to [3].
fore being transported to the shipping area or to the
In the preceding section we distinguished be-
production floor. Accumulation and sorting may tween manual, automated and automatic
either be performed during or after the order-pick-
warehousing systems. Below, we describe each of ing process. these in some more detail.
Hence, we may subdivide the activities in a ware-
house into four categories: receiving, storage, or-
2.3.1. Manual warehousing systems
der-picking and shipping. A study in the United
In a manual warehousing system or picker-to-
Kingdom [2] revealed that order-picking is the
product system, the order picker rides a vehicle
most costly among these activities. More than 60%
along pick locations. A wide variety of vehicles is
of all operating costs in a typical warehouse can be
available: we mention pick carts or container carts
attributed to order-picking (Fig. 1).
for manual horizontal item picking and man aboard
storage/retrieval (S/R) machines for both horizontal
and vertical item picking (often, but not necessarily,
2.2. A typology of warehousing systems
restricted to a specific aisle). For storage/retrieval
operations (of complete pallet loads, see Sec-
An item picking operation is an operation in
tion 2.2), fork lift trucks and a variety of reachtrucks
which single items are picked from storage posi- are often used.
tions (less-than-case picking), as opposed to a pal-
Recall that an order may contain a list of quant-
let-picking operation in which pallet loads are
ities of different SKU’s (each SKU in an order
moved in and out. A warehousing system refers to
corresponds to a unique item of supply). Two
the combination of equipment and operating pol-
fundamental approaches may be distinguished in
icies used in an item picking or storage/retrieval
manual order picking: single-order-picking and
environment. With respect to the level of automa-
batch-picking. The former approach indicates that
tion, we may distinguish three types of warehousing
the order-picker is responsible for the picking of 522
J.P. van den Berg, W.H.M. Zijm/Int. J. Production Economics 59 (1999) 519 528 —
a complete order. The latter approach indicates
independently, thus reducing the waiting time of
that multiple orders are picked simultaneously by
the order picker significantly.
one order-picker, who is typically restricted to
The automated storage/retrieval system (AS/RS)
a certain zone in the warehouse (zoning). Batch
is also a product-to-picker system. The AS/RS con-
picking reduces the mean travel time per pick.
sists of one or multiple parallel aisles with two high
However, it requires that orders are to be sorted
bay pallet racks alongside each aisle. Within the
afterwards. The order-picker may either sort the
aisle travels a storage/retrieval (S/R) machine or
orders while traversing the warehouse (sort-while-
automated stacker crane. The S/R machine travels
pick) or the items may be lumped together and
on rails that are mounted to the floor and the
sorted afterwards (pick-and-sort). To apply the
ceiling. In a typical configuration, the S/R machine
sort-while-pick strategy, the order-picking vehicle
may carry at most one pallet at the same time.
must be equipped with separate containers for indi-
Pallets for storage arrive at the input station and
vidual orders. ¼ave picking is a popular strategy if
wait at an accumulator conveyor until the S/R
batching and zoning are both applied. This strategy
machine transports them to a storage location in
implies that all order-pickers start picking in their
the racks. Consequently, storages are performed
respective zones at the same time. Only after all
according to a first come first served (FCFS) rou-
pickers have completed their tour, the next wave
tine. The S/R machine deposits retrieved loads starts.
at the output station, after which a transportation
Instead of a vehicle we may also use a conveyor
system routes them to their destination. The S/R
for the transportation of the picked products. The
machine has three independent drives for horizon-
order-picker directly deposits the picked items on
tal, vertical and shuttle movement. Due to the
a conveyor that is positioned within the aisle. Such
independent horizontal and vertical travel, the
an operation is referred to as pick-to-belt.
travel time of the S/R machine is measured by the
maximum of the isolated horizontal and vertical
2.3.2. Automated warehousing systems
travel times. In many applications the S/R machine
The systems that we discussed so far, were
is confined to one aisle. We may enable movement
picker-to-product systems. A carousel is an example
of the S/R machines between aisles by providing
of a product-to-picker system. A carousel is a com-
curves in the rails that connect the aisles. To main-
puter-controlled warehousing system that is used
tain stability in the giant construction, the cranes
for storage and order picking of small-to medium-
have to assume creep speed in the curves. Another
sized products. A carousel may hold many different
possibility that enables the S/R machine to enter
products stored in bins or drawers that rotate
multiple aisles, is to use a shuttle device that trans-
around a closed loop. The order picker occupies
fers the S/R machine between the aisles.
a fixed position at the front of the carousel. Upon
Due to its unit-load capacity, the operational
request, the carousel automatically rotates the con-
characteristics of the S/R machine are limited to
tainer with the requested product to the position of
single-command cycles and dual-command cycles. In
the order picker. The order picker may effectively
a single-command cycle either a storage or a re-
use the rotation time of the carousel for activities
trieval is performed between two consecutive visits
such as sorting, packaging and labeling of the
of the input and output station. In a dual-command retrieved goods.
cycle the S/R machine consecutively performs
In some situations the order picker serves two to
a storage, travels empty to a retrieval location and
four carousels in parallel. The advantage of this
performs a retrieval. The empty travel between the
configuration is that while the order picker is ex-
storage and retrieval location is referred to as inter-
tracting items from one carousel, the other carou- leaving travel.
sels are rotating. This reduces the waiting time of
A miniload AS/RS is an AS/RS that is designed
the order-picker. The rotary rack is a more expen-
for the storage and order picking of small items.
sive version of the horizontal carousel, with the
The items are stored in modular storage drawers or
extra feature that every storage level can rotate
in bins. These containers may be subdivided into
J.P. van den Berg, W.H.M. Zijm/Int. J. Production Economics 59 (1999) 519—528 523
multiple compartments each containing a specific
scans SKUs that enter the loop. SKUs correspond-
SKU. In a typical miniload AS/RS operation, the
ing to the same order are then automatically
order-picker resides at the end of the aisle at a pick
diverted into one lane. Also carousels and rotary
station. The pick station contains at least two con-
racks are used for the accumulation and sorting of
tainer positions. While the order-picker extracts orders.
items from the container in one pick position, the
S/R machine stores the container from the other
pick position at its location in the rack and 3. Warehouse management
retrieves the next container. Also miniload AS/RS’s
with more than two pick positions per pick station
Typical planning issues in warehouses are inven-
do exist, as well as systems with a conveyor delivery
tory management and storage location assignment.
system to transport containers to remote order
Intelligent inventory management may result in
pickers. A miniload AS/RS is generally referred to
a reduction of the warehousing costs. For example,
as an end-of-aisle order-picking system, as opposed
by applying sophisticated production planning and
to in-the-aisle order-picking systems such as
ordering policies we may reduce the total inven-
the manual order-picking systems discussed in
tory, while guaranteeing a satisfactory service level. Section 2.3.1.
The service level specifies the percentage of the
orders to be supplied directly from stock. Reduced
2.3.3. Automatic warehousing systems
inventory levels not only reduce inventory costs,
Automatic order-picking systems perform high-
but also improve the efficiency of the order-picking
speed picking of small- or medium-sized non-fragile
operation within the warehouse. Clearly, in a small-
items of uniform size and shape, e.g., compact disks
er warehouse, the travel times for order-picking are
or pharmaceuticals. If we replace the order picker smaller.
of a carousel system or rotary rack by a robot, then
Furthermore, an effective storage location as-
we obtain an automatic order-picking system.
signment policy may reduce the mean travel times
An A-frame automatic dispenser machine is
for storage/retrieval and order-picking. Also, by another order-picking device without order-
distributing the activities evenly over the ware-
pickers. The A-frame consists of a conveyor belt
house subsystems, congestion may be reduced and
with magazines arranged in A-frame style on either
activities may be balanced better among subsys-
side of the belt. Each magazine contains a powered
tems, thus increasing the throughput capacity.
mechanism that automatically dispenses items onto
The planning policies define a framework for the
the belt. Each order is assigned a certain section on
control of the warehouse processes. Inventory man-
the conveyor (a cell). When the cell passes a maga-
agement and storage location assignment policies
zine that contains an item requested by the corre-
determine which products arrive and where these
sponding order, the item is automatically dispensed
should be stored. Control problems typically deal
upon the passing cell. At the end of the belt the
with the sequencing of order picking and stor-
items belonging to the same order fall down into
age/retrieval operations, and hence with the rout- a bin or carton.
ing of manual order pickers or S/R machines, the
allocation of products to storage positions in
2.3.4. Order accumulation and sorting systems
a class-based or random location system, the inter-
Order accumulation and sorting systems (OASSs)
nal movement of items to more attractive retrieval
are used to establish order integrity when orders
positions, the dwell point of S/R machines, etc.
are not picked in a single-order fashion. OASSs
exist in various types, ranging from manual staging
using a kitting matrix to high volume automatic 4. Warehousing models
systems. An automatic OASS usually consists of
a closed-loop conveyor with automatic divert
In this section, we discuss examples of models
mechanisms and accumulation lanes. A sensor
that have been presented in the literature or have 524
J.P. van den Berg, W.H.M. Zijm/Int. J. Production Economics 59 (1999) 519 528 —
been developed recently, to illustrate the applica-
occupied by one unit-load of product j, and F the
tion of operations research techniques for the available storage space.
planning of warehousing operations.
If the unconstrained solution exceeds the avail- Inventory management/production planning
able storage space, then a ¸agrangian multiplier
decide which products are to be stored in the ware-
technique is used to find the optimal ordering pol-
house and in what quantities. Storage location
icies. Here, the storage space estimation is based on
assignment decides where the products are to be
the possibility of receiving all deliveries at the same
stored. Here we may distinguish between a forward
epoch. However, by properly staggering the delive-
and a reserve area while also the basic storage
ries in time, the peak demand for warehouse space
policy in S/R systems is determined (e.g., dedicated,
may be moderated. The combined problem of order
class-based or random storage). First, we discuss
sizing and delivery staggering is known as the Eco- inventory management.
nomic ¼arehouse ¸ot Scheduling Problem (EWLSP).
For a survey on the EWLSP we refer to [5].
All models discussed so far assume fixed cost
4.1. Reduction of inventory levels
parameters, a constant demand rate, no delivery
leadtimes and no backlogging. Clearly, the problem
Intelligent inventory management/production
of order sizing and staggering deliveries becomes
planning may reduce the inventory levels and
much more complicated in a stochastic setting.
thereby the operational costs for storage/retrieval
Suppose for example that pallet loads for each
and order picking. Inventory reductions may be
SKU are ordered according to a (continuous
established by having smaller ordering quantities
review) (s, Q)-policy (cf. [6]). Under certain condi-
delivered more frequently. However, the total stor-
tions, the number of pallets per SKU is uniformly
age space needed may still be considerable if all
distributed at an arbitrary point in time. Assuming
deliveries occur at the same time. Hence, we may
stochastic independence of the demands for differ-
further reduce the need for storage space by care-
ent SKUs, the total number of pallets can then
fully scheduling the deliveries. Ultimately, products
be approximated by a normal distribution. Hence,
from incoming trucks are immediately transferred
under a random storage policy, the necessary stor-
to outgoing trucks, a phenomenon known as cross
age space is determined by specifying a probability docking.
on stock overflow (cf. [7]). However, under rigid
Classical inventory management and production
space restrictions, the orders for the different SKUs
planning models determine ordering and produc-
are no longer independent. Besides, many ware-
tion policies for a single product. Hadley and
house managers follow a can-order policy (cf. [6,8])
Whitin [4] consider inventory models for multiple
for groups of products to be delivered by the same
products with a constraint on the total storage
supplier, thereby taking advantage of shared fixed
space. They determine ordering policies for all
costs or combined transport facilities. Hence, in
products which minimize the long-run inventory
such a situation, various orders of different SKUs
holding and ordering costs per unit time by solving arrive at the same time. the following problem: Min + C # # jDj AjD Q j/ j rCjQj/2 (1)
4.2. Storage allocation and assignment s.t. + f Q "
A popular approach to reduce the amount of j j F, (2)
work associated with order picking is to divide the
where Dj is the demand rate in units per year for
warehouse into a forward area and a reserve area.
product j, Aj the fixed ordering costs for product j,
The forward area is used for efficient order picking. C j
j the unit variable purchase costs for product ,
The reserve area holds the bulk storage and is used
r the annual inventory carrying cost rate, Qj the
for replenishing the forward area and for picking
order quantity for product j, fj the amount of space
the products that are not assigned to the forward
J.P. van den Berg, W.H.M. Zijm/Int. J. Production Economics 59 (1999) 519—528 525
area. The forward and reserve area may be distinct
Rij random variable representing the number of
areas within the warehouse or the forward and
concurrent replenishments for product i, if the
reserve area may be located in the same (pallet)
forward area contains j unit-loads of product
rack. In the latter case, the lower levels represent
i at the beginning of the picking period,
the forward area, the higher levels represent the i"1,2, N, j"1,2, mi,
reserve area. In some facilities the reserve area is
ºi random variable representing the number of
once again subdivided into two separate areas: one
unit-loads of product i that is needed to fulfil
for order-picking and one for replenishing.
demand during the picking period.
The forward-reserve problem (FRP) is the prob-
The expected number of picks from the forward
lem of deciding which products should be stored in
area and the reserve area are given by expressions
the forward area and in what quantities. If a prod- (3) and (4), respectively.
uct is not assigned to the forward area, then it is
picked from the reserve area. Hackman and Rosen- + E(P
blatt [9] describe a heuristic for the FRP that i), (3) i|S
attempts to minimize the total costs for picking and + E(P
replenishing. Frazelle et al. [10] incorporate the i). (4) ibS
heuristic into a framework for determining the size Let z
of the forward area together with the allocated
i denote the number of unit-loads of product
i that is stored in the forward area at the beginning
products. The costs in the model for picking in the
of the picking period. Accordingly, the expected
forward area and for replenishing depend on the
number of concurrent replenishments is given by size of the forward area. expression (5).
Van den Berg and Sharp [11] focus on opera-
tions that observe busy and idle periods. In these + E(R ). (5)
operations, it is possible to reduce the number of izi i|S
replenishments in busy periods, by performing
We derive an expression for E(R ).
replenishments in the preceding idle periods. This iz
not only increases the throughput during the busy = E(R + (k!z) ) P(º "k)
periods, it also reduces possible congestion and iz)" i k/z`1
accidents. A typical example is a distribution center
in which trucks are loaded during the afternoon, so = " + P(º k *
that the workforce is available in the morning i ) k/z`1
hours for replenishing the forward area. The z
authors consider a picking period during which the "E(º + P(º *
order-picking operation takes place. Prior to the i)! i k). (6) k/1
picking period, the forward area is replenished in
Subsequently, they formulate the FRP as the
advance. Their objective is to find an allocation of
binary programming problem (B-FRP), using the
product quantities to the forward area, which min- following notation:
imizes the expected labor time during the picking period. m i
i number of unit-loads available of product ,
The authors consider a situation observed in i"1,2, N,
many operations (e.g. pallet storage), where unit p E P i ( i),
loads are replenished one at the time. They use the u E * i (ºi)!P(ºi 1), following notation: u P ij (º * i j), i"1,2, N, j"2,2, m , i
» available storage space in the forward area, S
set of products assigned to the forward area,
¹1& average time for performing one pick from the
Pi random variable representing the number of forward area,
picks for product i during the picking period,
¹13 average time for performing one pick from the i"1,2, N, reserve area (¹13'¹1&), 526
J.P. van den Berg, W.H.M. Zijm/Int. J. Production Economics 59 (1999) 519 528 —
¹#3 average time for performing one concurrent
to storage locations. For automated storage/ replenishment.
retrieval systems, Hausman et al. [12] present three
storage location assignment policies: class-based
They define decision variables xi for i"1,2, N,
storage, randomized storage and dedicated storage.
and yij for i"1,2, N, j"2,2, mi.
The class-based storage policy distributes the
products, based on their demand rates, among
a number of classes and reserves a region within the x " i G1ifproductiisassignedtothe forward area,
storage area for each class. Accordingly, an incom- 0 otherwise,
ing load is stored at an arbitrary open location
within its class. The class-based storage policy and i
the dedicated storage policy attempt to reduce the
yij" G1if the jth unit-load of product is replenished in advance,
mean travel times for storage/retrieval by storing 0 otherwise.
products with high demand at locations that are easily accessible.
Van den Berg [13] presents a polynomial time (B-FRP)
dynamic programming algorithm that partitions N
products and locations into classes such that the
Min + G¹1&pixi#¹13pi(1!xi)
mean single command cycle time is minimized. The i/1
algorithm works under any demand curve, any mi
travel time metric, any warehouse layout and any #¹#3Au ! ixi + u y ij ijBH,
(7)positions of the input station and output station. j/2 We use the following notation: s.t.
Qi independent random variables representing N m
the number of unit-loads present of product i + v + y i at an arbitrary epoch, i/1 i(xi# j/2 ij))», (8) P k K
k set of products in class "1,2, . yi2)xi, i"1,2, N, (9)
Due to the demand and supply processes the inven- y y
tory level fluctuates. We estimate the storage space ij) i(j~1), i"1,2, N, j"3,2, mi, (10)
requirement such that the storage space in every xi3M0, 1N, i"1,2, N, (11)
class suffices for at least a fraction 0(a(1 of the y i
time. In other words, the probability of a stock ij3M0, 1N, "1,2, N, j"2,2, mi. (12)
overflow is less than 1!a. Let Qk be a random
The objective function follows from expres-
variable representing the inventory level of class
sions (3)—(6) after substituting pi, ui and u and ij
k at an arbitrary epoch, i.e., Qk"+ Q
multiplying each term with the corresponding i|P k i. Now, we
want to find the smallest size Sk for the class-region
labor-time average. Constraint (8) stresses that the of class k such that
space occupied by the unit-loads allocated to the
forward area may not exceed the available space. P(Qk)Sk)*a. (13)
The remaining set of constraints (9) and (10) allows
the jth unit-load of product i to be stored in ad- Let t*/
j denote the travel time between the input
vance, only if unit-loads 1,2, ( j!1) of product
station and location j and let t065 j denote the travel
i are assigned to the forward area, for i"1,2, N.
time between the output station and location j.
Every stored unit-load is retrieved some time later,
so that over a long time period half of the single
4.3. Storage location assignment
command cycles are storages and half are re-
trievals. Accordingly, the mean single command The storage location assignment problem
cycle time to location j3¸ equals: 1 # 2(2t*/ j 2t065 j )"
(SLAP) concerns the assignment of incoming stock (t*/# j t065 j ).
J.P. van den Berg, W.H.M. Zijm/Int. J. Production Economics 59 (1999) 519—528 527
The single command cycle time, E(SC), is defined
discussed examples of models in some specific areas as
that in particular highlight the relation between K + E(D (t*/#t065
inventory control decisions and product allocation E(SC)" + i|P k i)) + j j ) , (14)
and assignment problems. Other fields of interest, k/1 + D¸ i|PE(Di) j|Lk kD
not discussed here, include warehouse justification
where ¸k denotes the set of storage locations of
and design problems, as well as operational short- class . k
term routing problems. For instance, Gross et al.
The first factor represents the probability that
[14] outline the relation between multi-echelon
a request concerns class k. The second factor rep-
inventory control policies and the choice of ware-
resents the mean travel time to a location in class . k
house locations on a strategic level. Many authors
In order to minimize the expected single command
concentrate on the development of smart order-
cycle time, we assign the products i that constitute
picking strategies (both for manual orderpickers
the largest demand per reserved space and the
and automatic storage and retrieval machines).
locations j with the smallest (t*/# j t065 j ) to the first
Indeed, also the examples discussed here focus on
class and we assign the products i that constitute
a maximum reduction of retrieval time, e.g., the
the next largest demand per reserved space and the
forward/reserve policy discussed in Section 4.2 has
locations j with the next smallest (t*/# j t065 j ) to the
led to a reduction of the orderpick time of more
second class, and so on. Accordingly, the locations
than 40% in a warehouse with 200 products and
are ranked according to non-decreasing (t*/# j t065 j )
800 storage locations. In one particular case study
and the products are ranked according to non-
carried out at a distribution center of Yamaha
increasing demand per reserved space. We define
Motor Co. at Amsterdam Airport, the class-alloca- g p
k( , l) as the contribution of classes 1,2, k to
tion method discussed in Section 4.3 led to a 10%
Eq. (14), when products 1,2, p and storage loca-
travel time reduction compared with the current
tions 1,2, l are distributed among these classes
four class-based strategy while the algorithm also such that g p k( , l) is minimal. Then g p k( , l) satisfies
compared favorably with other recent procedures g p
(see e.g. [15]). In addition, a sophisticated class- k( , l)"min1xi ,1 p x xj l x Mhj` l 1, i` p 1, #gk~1(i, j)N, (15)
allocation leads to a higher overall service level, where hj`1,l
i`1,p denotes the contribution to Eq. (14) if
since storage space is better used (i.e., for the right the products i#1,2, p and the locations
products). For a more detailed discussion of these
j#1,2,l form one class k. Recalling that the num-
results, as well as for an extensive literature review,
ber of locations required in each class is determined
the reader is referred to [16].
by Eq. (13), the values gk(p, l) are found by iterat-
It will be clear that a higher warehouse service
ively solving the dynamic programming equation
level and shorter response times may lead to addi-
(15). Each gk(p, l) corresponds to an optimal solu-
tional savings downstream the logistic chain as
tion of the subproblem with k classes and the first
well. For instance, in the case of a production
p products and the first l storage locations when
warehouse supplying a two-bin operating assembly ranked as indicated before.
line, shorter response times may significantly
We may use the algorithm to determine the opti-
reduce the total amount of stock placed along the
mal class-partition for 1,2, K classes. Sub-
line. In the food and retail sector, where many
sequently, the number of classes among 1,2, K
stores have moved towards just-in-time delivery,
may be selected that constitutes an acceptable
there is a constant pressure to improve response
mean travel time and space requirement.
times of the warehouses. Wall Mart, a major retail
chain in the U.S., has adopted cross docking (i.e.,
receive, sort and regroup, and ship) as the leading
5. Conclusions, trends and further developments
principle in their supply chain, as opposed to con-
ventional storage in distribution warehouses. As
In this paper, we have presented a review of
a result, the interest in new, sophisticated sorting
warehouse management systems and subsequently
techniques is rapidly growing. ICA, the leading