Use of Promethee method to determine the best alternative for

Int J Adv Manuf Technol (2014) 70:1615–1624
DOI 10.1007/s00170-013-5405-z
ORIGINAL ARTICLE
Use of Promethee method to determine the best alternative
for warehouse storage location assignment
Marcele Elisa Fontana & Cristiano Alexandre Virgínio Cavalcante
Received: 27 June 2011 / Accepted: 4 October 2013 / Published online: 27 October 2013
# Springer-Verlag London 2013
Abstract Storage includes all activities involved in holding
products at a point for temporary custody and subsequent
distribution. The criteria that must be considered in determining the location in a storage facility for each product are often
conflicting. Thus, the main objective of this paper is to determine the best alternative for assigning a product to a warehouse storage location. A class formation and allocation model and a preference ranking organization method for enrichment evaluation (Promethee) multicriteria method are used to
achieve this goal. By using the model proposed, the manager
can learn of all the possible non-dominated allocations of the
products in the warehouse and can then change the allocation
when needed. The Promethee method is efficient at ranking
alternatives which facilitates the decision process.
Keywords Cube-per-order index . Warehouse storage
location assignment . Time required . Promethee method
1 Introduction
Storage is one of the most traditional aspects of logistics. A
good storage system facilitates the identification of physical
material, thereby increasing productivity and reducing the cost
of manpower, because it facilitates the management of inventories. According to Daniels et al. [1], changes in demand, and
the consequent reallocation of warehouse space, often require
movement of stock that can cause severe disruptions to warehouse operations, especially when the warehouse is used
intensively.
M. E. Fontana (*) : C. A. V. Cavalcante (*)
Federal University of Pernambuco, Av. Acadêmico Hélio
Ramos, s/n - Cidade Universitária, UFPE-CTG, Recife,
Pernambuco 50740-530, Brazil
e-mail: [email protected]
e-mail: [email protected]
Warehouse management plays an important role for many
efficient supply chains [2]. With the advent of supply chain
management, the strategic role of a warehouse changed. The
activities involved in processing orders in today’s warehouses
are conducted at a faster pace than in the recent past. In order to
satisfy clients’ demands for shorter cycle times when meeting
orders, warehouse managers are interested in finding the most
economical way of fulfilling orders: the one which minimizes
the costs involved in terms of distance traveled or travel time [3].
An efficient warehouse should take into account, principally,
the reduction in costs, space used, and distance traveled. Therefore, the objective of finding solutions for the stock location
problem is to reduce the requirement for space and to minimize
the total distance traveled or travel time throughout the
warehousing process [4]. For many products, these criteria
can be conflicting. Thus, the main objective, in this paper, is
to investigate an approach that uses a multicriteria method in
order to reach the best alternative for allocating a product in a
warehouse, according to the decision maker’s preferences.
For the decision maker (DM), the criteria are not compensatory, i.e., it is not possible for trade-offs to occur when
evaluating the criteria. Therefore, the preference ranking organization method for enrichment evaluation (Promethee)
multicriteria method was chosen in this paper. The specific
objective of the study is to compare the results provided by the
Promethee method when the DM’s preferences on the criteria
used change.
The paper is organized as follows. Section 2 provides a
review of the literature specific to the problem studied and
focuses on those papers which are fundamental to the model
proposed. Section 3 presents how the problem is formulated
and gives a simulated description of a warehouse, i.e.,
what both the criteria used and the scenarios simulated
are. The results are given and discussed in Section 4. Finally,
Section 5 makes some concluding remarks and suggests further lines of study.
1616
2 Warehouse storage location assignment
The function of a distribution warehouse is to receive and
store products and to fulfill orders from external clients which
typically comprise a large number of order lines (where each
order line specifies a quantity of one particular product). The
number of different products in a distribution warehouse may
be large, while the quantities per order line may be small, and
this often results in a complex and relatively costly order
picking process [5, 6]. One way to decrease handling time is
by changing the operational procedures [7].
Warehouse storage decisions influence almost all the key
performance indicators of a warehouse such as order picking
time and the cost of using storage space, labor, etc. [8]. Typical
planning issues in warehouses are inventory management and
storage location assignment. Intelligent inventory management may well result in reducing warehousing costs. On the
other hand, an effective policy for assigning warehouse storage location may reduce the mean travel times for storage/
retrieval and order picking [9].
Order picking—the retrieval of stock keeping units (SKUs)
from a warehouse to satisfy customer orders—is a vital supply
chain component for many companies, both from the production
system point of view (i.e., the supply of assembly stations with
assembly kits) and from the point of view of physical distribution activities (i.e., fulfilling a customer’s order [10, 11]).
A crucial link between order picking and service level is
that the faster the items on an order can be retrieved, the
sooner they are available for shipping to the client [12].
Successful firms are those that provide the right products to
the right clients, in the right place, at the right time, and for the
right price [13]. The time spent on warehousing activities is an
important factor in the total time spent on the request cycle.
Therefore, it is essential to study viable and sustainable means
to minimize this time.
Different storage strategies can be used. The implementation of each storage strategy is an operational issue. The
selection of an order picking method is a strategic decision,
since it has a wide impact on many other decisions in the
design and operation of a warehouse [14]. Since warehousing
activities are frequent and numerous, even small improvements can achieve significant savings [15]. Order picking is
often considered the most critical operation in warehousing.
The efficiency of an order picking process greatly depends on
the storage policy used, i.e., on where products are located
within the warehouse [16].
According to Brynzér and Johansson [17], components
have many characteristics that can be used to assign items to
locations or zones in warehouses, such as frequency, size,
weight, part number, supplier, etc. Thus, three storage location
assignment policies were put forward by Hausman et al., in
1976, [6, 8], namely: dedicated or fixed storage, randomized
or variable storage, and class-based storage.
Int J Adv Manuf Technol (2014) 70:1615–1624
A dedicated storage policy prescribes a particular location
for the storage of each product [5], such that no other item can
be stored there, even if the space is empty. Under a dedicated
storage policy, each storage area may only be used for a
specific item. The materials are placed in existing open spaces.
A randomized storage policy allows items to be stored anywhere in the storage area. Randomized and dedicated storage
are extreme cases of a class-based storage policy: randomized
storage considers a single class, and dedicated storage considers one class for each item [18]. Under class-based storage
rack assignment, products having similar turnover rates are
stored in the same class, and the rack is divided into several
classes of different sizes [19, 20].
For the formation of classes, Hesket, in 1963, proposed the
cube-per-order index (COI), which is expressed as the ratio of
storage space required (cube) per SKU and the order frequency of the SKU [17], i. e., COIp =f p *[Max I t p ]/D p , where
COIp is the cube-per-order index for product p ; f p is the
density (area required to store one unit load of product p);
I t p is the storage level in unit loads planned for product p
during period t; and D p is the total demand for the product p
in the planning period (in unit loads).
The rule ranks the items in ascending order of the index,
and then assigns them in that order to the locations nearest to
the input/output (I/O) point in order to reduce the cost of order
picking [21]. The location of the items must follow the ascending order of the index. Based on the COI index, there is a
tendency for products with lower space requirements and high
demand to be located at the front of the warehouse, i.e., next to
I/O, and products with greater space requirements and low
demand to be located far from the I/O. Thus, the COI tries to
make storage more efficient, by minimizing the distances with
a single control for order picking.
A class formation and allocation model was put forward by
Muppani and Adil [18; 22]. This model uses the COI index to
rank items. The problem can be defined as: given P products/
items, their average demand D p and planned inventory levels
I p for T periods and the layout of the storage area divided into
a lattice of storage locations, establish classes of products and
allocate them to storage locations so that the total cost of order
picking and storage space is minimized in a single command
warehouse, thus exploiting the reduction in area. Therefore,
we have:
Minimize
"( X
#
)
X
XX
X
ða :d :y Þ
l l l lc
X
ðal :ylc Þ þ 2h:
Dp xpc
Z¼f ða :y Þ
c
c
p
l
l l lc
ð1Þ
Subject to:
COIp xpc ≤ COIp ; xp ;c ;
l ylc ≤ l 0 yl0 c0
∀p≠p0 and c < c0
∀l≠l 0 and c < c0
ð2Þ
ð3Þ
Int J Adv Manuf Technol (2014) 70:1615–1624
"
Max
X
t
#
I tp
f p xpc ≤
p
X
xpc ¼ 1
X
ðal ylc Þ ∀c; t
1617
ð4Þ
p
∀p;
ð5Þ
c
X
ylc ≤ 1 ∀l;
ð6Þ
c
xpc ; ylc ∈f1; 0g
∀p; c; l:
ð7Þ
where:
c
l
p
t
al
dl
f
h
– (c = 1, 2, 3, . . . , C =P) for classes
– (l = 1, 2, 3, . . . , L) for storage locations
– (p = 1, 2, 3, . . . , P ) for products/items
– (t = 1, 2, 3, . . . , T) for time periods.
– Footprint area of location l (in square meters)
– Distance of location l from the input/output point
(in meters)
– Space cost for the planning horizon (in US dollars
per square meter)
– Order picking cost per unit load (in US dollars
per meter)
Decision variables:
xpc ¼ 1; if product p is assigned to class c;
xpc ¼ 0; otherwise
:
ylc ¼ 1; if location l is assigned to class c;
ylc ¼ 0; otherwise
:
The objective function (1) minimizes the sum of storage
space cost and order picking cost over the planning horizon.
Constraints (2) and (3) together ensure that if products that
have a lower COI are assigned to class c and products with a
higher COI are assigned to class c′, then c is located nearer to
the I/O point than c ′. Constraint (4) ensures that there is
adequate storage space to hold the items in class c in each
planning period t. Constraint (5) ensures that each product is
assigned to one and only one class. Constraint (6) ensures that
is assigned to not more than one class. Constraint (7) imposes
binary restrictions on the decision variables.
Based on class formation and an allocation model, the three
criteria needed to determine the best alternative for warehouse
storage location assignment may be reached, namely: the
required space, the order picking distance and the total operation cost (include just the required space cost and the order
picking cost). Other criteria will be mentioned in Section 3.
Moreover, due its features, the Promethee method was
chosen to evaluate the alternatives by each criterion in order
to reach the best alternative as per the DM’s preferences. The
procedures of this method, in the order to be followed, are
given below.
2.1 Promethee method
The Promethee method is a relatively simple ranking method in
conception and application compared with other methods for
multicriteria analysis. It is well adapted for problems where a
finite number of alternatives are to be ranked considering
several, sometimes conflicting, criteria [23]. The method has
two phases: the construction of an outranking relation (aggregating information about the alternatives and about the criteria)
and the exploitation of that relation for decision aid [24].
The additional information requested to run Promethee is
particularly clear and easy for both the analysts and the DMs
to understand. It consists of [25] information between the
criteria and information within each criterion. This approach
does not provide specific guidelines for determining these
weights but assumes that the DM is able to weigh the criteria
appropriately, at least when the number of criteria is not too
large [26]. The weights must be normalized, given j the
number of criteria, where j = {1, 2, 3,…,k} and w j is the
weight of criterion j, since ∑ kj = 1w j =1.
The ranking of alternatives is carried out by pairwise
comparison of the alternatives for each criterion. The
comparison is measured using a predefined preference
function [27]. The preference is expressed by a number in
the interval [0, 1] (0 for no preference or indifference to 1 for
strict preference). The function relating the difference in performance to preference is called the generalized criterion and
is determined by the DM [23].
In order to facilitate the selection of a specific preference
function, Vincke and Brans in 1985 proposed six basic types:
(1) usual criterion, (2) U-shape criterion, (3) V-shape criterion,
(4) level criterion, (5) V-shape with indifference criterion, and
(6) Gaussian criterion. So, for each criterion, the preference
function P j (.) translates the difference between the evaluations obtained by two alternatives [28].
Since g j (a) is the preference function of alternative a on
criterion j, and g j (b) the preference function of alternative b
also on criterion j, the difference between these is d j (a,b)=g j
(a)−g j (b).
The larger the deviation, the larger the is preference. For
small deviations, the DM will allocate a low preference to the
best alternative and even possibly indifference if he/she considers that this deviation is negligible. In some cases, it is
interesting to amplify the notion of indifference for the
Promethee method (see [29]). In each case (Table 1 adapted
from [25]), 0, 1, or 2 parameters have to be defined so that their
significance is clear [30]: “q” is a threshold or indifference; “p”
is a threshold of strict preference; and “s” is an intermediate
value between “q” and “p”.
The indifference threshold is the largest deviation which
the DM considers as negligible, while the preference threshold
is the smallest deviation which is considered as sufficient to
generate a full preference. The identification of a generalized
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Int J Adv Manuf Technol (2014) 70:1615–1624
Table 1 Types of generalized criteria (preference function)
criterion is then limited to the selection of the appropriate
parameters.
Thereafter, a multicriteria preference index is formed for
each pair of alternatives as a weighted average of the corresponding preferences computed in step (1) for each criterion.
The index π(a, b) (in the interval [0, 1]) expresses the preference of alternative a over b considering all criteria. Thus,
the alternatives can be ranked according to [23]:
&
&
The sum of indices π(a, i) indicating the preference of
alternative a over all the others. This is deemed as the
“leaving flow” Φ +(a) and shows how “good” alternative
a is.
The sum of indices π(i, a) indicating the preference of all
other alternatives compared to a. This is deemed as the
“entering flow” Φ −(a) and shows how “inferior” alternative a is.
Promethee I partial ranking provides a ranking of alternatives. In some cases, this ranking may be incomplete. This
means that some alternatives cannot be compared and, therefore, cannot be included in a complete ranking. However,
Promethee II provides a complete ranking of the alternatives
from best to worst [26].
3 Problem formulation
In addition to the required space, distance traveled, and storage cost mentioned in papers arising from the literature review, another important criterion in warehouse operations is
the time required for picking operations. Thus, for each scenario simulated, what should be determined are the following:
(a) the average time taken to go to each product p; (b) the
average time spent meeting a client’s orders; (c) the average
time taken to meet the orders of a group of clients which has
asked for the same product p. Items (b) and (c) are important
since the orders for product p do not arrive at the same time
and, consequently, they are not met at the same time.
Thus, the average time taken to go to each product p
(Time_prod) is the ratio between the sum of the time needed
to pick all products only once and the quantity of different
types of products (P ) there are in the warehouse, as per
Eq. (8). This time is expressed in seconds.
Time prod ¼
P
X
distp
v
p¼1
P
ð8Þ
where:
distp
v
– Distance until product p (in meters)
– Velocity of pick (in meters per second)
To continue formulating the problem, there is a need to
define the time spent serving all clients in the same group
(Time_C p ), i.e., all clients that demand the product p, is the
sum of the time needed to pick all orders for product p, as per
Eq. (9).
Time C p ¼ k p distp
v
ð9Þ
where:
Dp
C
kp ¼
CAPp
CAPp ¼
CAPnominal
fd
kp
– Number of trips needed to pick all orders for
product p
ð10Þ
ð11Þ
Int J Adv Manuf Technol (2014) 70:1615–1624
Dp
C
CAPp
1619
– Total demand for product p
– Total number of clients who ordered product p
– Total capacity of the equipment used to pick the
products. The CAPp is the ratio between the nominal
capacity of the equipment and the density of product
p. The equipment has a specific capacity to pick each
product. The capacity should be rounded towards −∞
and the number of trips (k p ) should be rounded
towards +∞.
Therefore, the average time spent serving each client
(Time_m) is the ratio between the sum of the time taken to
serve each client group and the total number of clients (N), as
in Eq. (12). This time is also expressed in seconds.
p
X
Time m ¼
Time C p
p¼1
N
ð12Þ
Finally, the average time to meet the orders of a group of
clients (Time_G) is the ratio between the sums of the time taken
to serve each group of client by the total number of distinct
products in warehouse, which is expressed by exp. (13).
P
X
Time G ¼
Time C p
p¼1
P
ð13Þ
Figure 1 shows how the model proposed finds the best
alternative, using the Promethee method.
Therefore, based on the characteristics in each warehouse,
the alternatives by each criterion may be evaluated and the
best solution for optimizing the warehouse operations by the
proposal methodology may be reached.
3.1 Description of the warehouse simulated
Initially, this description assumed that the picking operations
are performed under a single command and all items
(products) are stored and transported in identical supports.
Each storage location is uniformly used, and the points
awarded are distributed homogeneously in the space allotted
for the class. This assumption implies that the geometric
center of the class is the same as the load center.
It is assumed that the inventory decision is made independently of the storage decision and all the time required in the
process of storage, except the time for picking, are independent of storage allocation. The warehouse simulated is rectangular. It is divided into cells of 1.0×1.0 m, as per Fig. 2 which
is adapted from [31]. It was divided into four periods, each
period representing a week, totaling a month.
The layout used in the warehouse considers five rows (x
direction in Fig. 2). The use of space in direction y was not
restricted, which does not represent a significant point in the
simulation. It was found that all products are stacked to a
maximum of ten levels.
Storage costs are considered: US$ 1.00/m2 for used space
and US$ 0.0025/m for the distance to pick all products. Other
storage costs were not considered.
The scenario was generated randomly. In these, the quantities requested were in the range of between 100 and 1,200 units
weekly; the demand used in the calculations is the average of
the four periods; and the number of clients is between 1 and 30.
The inventory, i.e., the quantity of items in stock for the
period, is the ratio of the average demand for the product divided
by a random variable, which simulates the variations of units
stored in each period. The values of the quantities demanded, the
numbers of clients, and units in the inventory were rounded
towards +∞, because decimal values make no sense.
The space required is related to the inventory. Thus, the
space used by each type of product is the multiplication between the number of units stored by their density and divided
by ten (total number of levels on which each product can be
stacked). Therefore, it can be seen that there is variation of the
space required in the four periods. In this context, the density of
the products is a value between 0 and 1 m2/un (randomly
generated) and it must always be different from zero.
In the simulated warehouse, there are eight distinct products
and it was established that all products considered in the first
period are the same as those in the subsequent periods. The
values presented in the course of the simulations refer to the
average, i.e., these values represented only one period (a week).
The distances to picking represent the path between the
point of origin (I/O) and the place that the product is stored
(the round trip). Therefore, the picker always starts the process
at I/O. The average speed of picking is considered 1 m/s, so
the data obtained represent both the distance (in meters) and
the time spent (in seconds).
The criteria selected to determine the best alternative for the
formation of class and locations in the warehouse were:
Cr1
Cr2
Cr3
Cr4
– Space: this is the total space required to locate
products, respecting the areas reserved for each class.
The value is given in square meters
– Picking: this is the total distance traveled when the
pick is issued from a single command, or it is caught by
a unit of production time to meet clients’ orders. Its
value is related to fetching the product and returning it
to the origin (I/O); it is given in meters walked
– Total cost of picking a single command: this is the
sum of the costs incurred on the space required and the
distance traveled in a single command. Values are
expressed in US dollars
– Time to products: this is the average time to go to the
product and return to the source (I/O), i.e., the round
trip. Values are in seconds and also represent the
distance (in meters)
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Fig. 1 Proposed model using the
Promethee method
Cr5
Cr6
– The average time it takes to serve a client: the average
time required to meet the orders of each client in
seconds
– The average time it takes to serve a group: the average
time required to meet the orders of a group of clients in
seconds, provided the group is formed by clients who
request the same type of product.
Based on the search of the literature, it was realized that the
choice of warehouse design, i.e., determining the type of class,
formation, and location of the warehouse, depends exclusively
on the scenario in which the warehouse is inserted; the costs
involving its activities; and, especially, on the objectives of its
management.
For example, if the company’s goal is to offer a client
service that takes less time, the importance of this criterion
will be greater and will tend to alternatives that are more
efficient at this. Another important aspect to be considered is
whether the company gives priority to reducing the space
used, or minimizing the distance traveled, or both. This point
is influenced by the costs and the level of service that the
company intends to offer.
Based on these statements, this paper reports on three
simulations about the distinct preferences of the DM. In the
first case, the DM prefers an alternative that offers the shortest
total distance traveled and needs to offer, as far as possible, the
lowest average time to serve each client. This type of storage
generally has a high turnover of products and therefore the
total space required does not have as much impact as the
picking operations do.
In the second case, the manager is interested in an efficient
design in terms of less time; so much attention is paid to each
client and client group in terms of the average time of each
product. This type of company aims to raise the level of
service offered to clients and/or client group as much as
possible.
In the third and last case, the DM wants to lessen the space
used, because he uses a rented area and also expects to
minimize its total cost with storage, since clients are not as
sensitive to service time and the warehouse has a low turnover
Int J Adv Manuf Technol (2014) 70:1615–1624
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Fig. 2 Illustration of the layout of the storage area in the warehouse
of products. The weights assigned to each criterion, for the
three cases, are presented in Table 2.
The weights presented in Table 2 are normalized, i.e., their
sum is equal to 1. The data on the warehouse is the same for all
three cases, since the goal is to compare the results provided
by the Promethee method when the DMs change their
preferences.
4 Results and discussion
In total, 128 alternatives were generated by a class formation
and allocation model. The dominance between alternatives
when comparing the results in all criteria selected for the
problem was evaluated. Only six alternatives are nonTable 2 The weights of the criteria according to the DMs’ preferences
Criteria
Cr1 – Space
Cr2 – Picking
Cr3 – Total cost
Cr4 – Time to products
Cr5 – The average time it takes to serve a client
Cr6 – The average time it takes to serve a group
Cases
dominated. The non-dominated alternatives are a subset of
actions which are better in at least one criterion when compared to another alternative and there is no alternative better
than them in all criteria. Table 3 shows these alternatives. The
simulation made here is just illustrative. The complexity and
number of non-dominated alternatives will increase as the
number of alternatives increases.
For the purposes of Promethee method, the usual criterion
for comparing alternatives was used. Thus, it is not necessary
to change the range of values of the alternatives on different
criteria, since the response will be only 0 or 1 (worse/same or
better). The ranking results from Promethee I and II for the
three simulated cases of weights can be seen in Table 4. In the
first and third case, the most efficient alternative according to
the weights adopted by the DM is a 2. In the second case of
weight, the best alternative is a 4.
In the first scenario, the DM’s objective was to get
the lowest distance in picking under a single command
and the lowest average time taken to serve each client.
In this case, the method enables the DM to obtain the
best alternative, since it offers the third alternative of
shorter distance traveled and lowest average time taken
of both, serve a client and a client group. Note that
alternative a 2 is the best alternative considering the two
criteria simultaneously (Cr2—picking and Cr5—the average time it takes to serve a client). There is incomparability between alternatives a 5 and a 6 in the ranking of
Promethee I. However, these alternatives are ranked last.
In the second case, the DM considers as priorities the
shortest average time to reach each product and lowest average time taken to serve both, the client and the client group.
The result from the Promethee method was a 4 as the best
alternative. This shows the second shortest average time taken
to serve a client (Cr5) and a group of client (Cr6), and the
shortest average time to picking each product (Cr4). The
ranking of the Promethee I is the same as Promethee II,
because there were no incomparability or reversals of order
in this simulated case.
In the third case, the DM prioritizes using less space and
incurring the lowest total cost for storage. Based on these
points, the method gives a 2 as the most efficient alternative.
It has the lowest total cost (Cr3) and the second smallest space
Table 3 Subset of the non-dominated alternatives
1
2
3
Alternatives
Cr1
Cr2
Cr3
Cr4
Cr5
Cr6
0.05
0.35
0.10
0.05
0.35
0.10
0.05
0.10
0.10
0.25
0.25
0.25
0.45
0.05
0.35
0.05
0.05
0.05
a1
a2
a3
a4
a5
a6
189.08
163.63
170.29
170.29
162.57
188.08
125.270
128.430
129.330
132.330
141.510
127.650
502.25
484.70
493.61
501.11
516.35
508.20
30.38
31.45
31.30
29.33
30.03
34.46
181.85
173.95
174.96
174.56
181.06
188.53
2773.26
2652.71
2668.07
2662.09
2761.14
2875.11
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Table 4 Ranking: Promethee I and II
CASE 1
Alternatives
a2
a4
a3
a1
a5
a6
CASE 2
Alternatives
a4
a2
a3
a5
a1
a6
CASE 3
Alternatives
a2
a3
a5
a4
a1
a6
Promethee I
Φ +(a)
4.05
2.80
2.65
Table 5 Product classes
formed by each
alternative
Φ −(a)
0.95
2.15
2.30
Promethee II
Φ(a)
3.10
0.65
0.35
2.45
3.55
3.55
0.10
−2.10
−2.10
Φ −(a)
1.20
1.25
2.25
2.65
3.05
4.55
Promethee II
Φ(a)
2.55
2.50
0.45
−0.30
−1.10
−4.10
2.55
1.45
1.45
Promethee I
Φ +(a)
3.75
3.75
2.70
2.35
1.95
0.45
Promethee I
Φ +(a)
4.25
2.80
3.00
2.65
1.20
0.65
Φ (a)
0.75
1.75
2.00
Promethee II
Φ(a)
3.50
1.05
1.00
1.90
3.80
4.35
0.75
−2.60
−3.70
−
required (Cr1). There is incomparability between alternatives
a 3 and a 5 in the ranking of Promethee I.
To sum up, in the real world, the process of order
picking, in general, is complex, and it has a large
number of different products and customers order. However,
it is important to realize that the benefits of the proposed
model are tangible in large warehouses, although the example
used is a simple one.
The reality of a warehouse is not static. Using the proposed
model, it is possible to discover all the possible nondominated allocations for the products in a warehouse. The
manager can determine, by ranking the alternatives, which
alternative of the allocation is the most efficient and appropriate for the current time. In other words, when the objectives of
the warehouse change, the manager has previous knowledge
of the most efficient allocation for each case. The final configuration of the warehouse for all non-dominated alternatives
is given in Table 5, for the example shown.
The model’s gains may be significant when considering a warehouse which handles a large volume of
materials. The calculations must be redone whenever
Alternative a 1
Class and rank
1
2
3
4
5
Alternative a 2
Class and rank
1
2
3
–
–
Alternative a 3
Class and rank
1
2
3
4
–
Alternative a 4
Class and rank
1
2
3
–
Alternative a 5
Class and rank
1
2
–
–
Alternative a 6
Class and rank
1
2
3
4
Products
P8
P7 P6 P5
P4
P1 P3
P2
Products
P8
P7 P6 P5 P4 P1
P3 P2
–
–
Products
P8
P7 P6 P5 P4 P1
P3
P2
–
Products
P8
P7 P6 P5 P4 P1 P3
P2
Products
P8
P7 P6 P5 P4 P1 P3 P2
Products
P8
P7 P6 P5
P4 P1
P3 P2
a change in the warehouse occurs, such as: the characteristics and quantity of different products, demand,
number of clients, and available space for storage
change. This is unnecessary only when the priorities
of the managing change, i.e., it is not needed only when
the importance of each criterion changes. Moreover,
other multicriteria methods can be used to rank alternatives, but the comparison between them is not relevant
in this paper.
Int J Adv Manuf Technol (2014) 70:1615–1624
5 Conclusion
This study made use of a methodology using a mathematical model and a simulation so that processes in
different scenarios could be evaluated. The model presented is likely to be used in real cases, after adapting it
to the characteristics of a given warehouse and will be
helpful in the search for efficient storage at the lowest
overall cost to the company.
The rankings resulting from Promethee I and II were equal.
This is due to the characteristics of the alternatives evaluated,
where the very good alternatives, i.e., the greatest positive
flow, were the alternatives with the least negative flow for that
particular simulation of weights. Thus, the net flow of
Promethee II resulted in a complete ranking identical to the
partial ranking of Promethee I.
It can be concluded that the weighting given to the criteria
can completely alter the configuration chosen for the warehouse. However that may be, it can be stated that modeling a
warehouse is not static and it must be reviewed when the
objectives of the warehouse change. As seen, this revision is
facilitated by previous knowledge of non-dominated alternatives. This is possible by applying the class formation and
allocation model.
In addition, the Promethee method was effective in
determining the best alternative location of the items
stored. One advantage of this method is the ranking of
alternatives. Thus the DM can make his/her activities
flexible, through the ranking, between one alternative
and another among the best, when it is necessary, i.e.,
they consider which criterion can be left aside so full
benefit may be derived from another one.
Looking to future studies, there are possibilities for further
evaluating how best to elicit weights of criteria, so that the
alternative chosen will meet the DM’s real needs.
Acknowledgments This paper is part of research studies funded by the
Brazilian Research Bureau (CAPES) and the Brazilian Research Council
(CNPq).
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