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MIXED-MODEL ASSEMBLY LINE LEVEL
SCHEDULING AT AUTOMOBILE COMPANIES
Ramón Companys, Joaquín Bautista, Manuel Mateo
Departamento de Organización de Empresas. Universidad Politécnica de Cataluña. España
companys@oe.upc.es, bautista@oe.upc.es, mmateo@oe.upc.es
ABSTRACT
Sequencing units on mixed production or assembly lines in order to reduce the
variations in the rates of resource consumption is an old problem that has received growing
attention in scientific publications these recent times, since it is related to JIT
manufacturing. Nevertheless, academic formulations do not seem to coincide with those
companies face up. In this paper, the results shown in bibliography are analysed and some
modifications are suggested in order to be used in practical applications.
Key words: sequencing, mixed-model assembly lines, just-in-time production
systems.
1. INTRODUCTION
In mixed production or assembly lines not all the flowing units are identical. All of
them have a certain degree of similarity, but some aspects may vary and have influence on
the resource consumption of those units (load in workstations and/or component
consumption). Sequencing units with the objective to attenuate variations in rates of
resource consumption is a problem that has received attention for many years. In 1973 one
of the authors knew about an intuitive procedure for sequencing the daily assembly schedule
in the most important factory of car production in Spain. This problem has achieved more
relief in literature since 1983 due to the relation with JIT concepts.
Kubiak (1993) presented an interesting description about the state of the art, in which
sequencing problems were classified in two categories: PRV (product rate variation) and
ORV (output rate variation).
In the PRV problem the objective is to minimize the variation in the rate of several products
which appear in any segment of the sequence. The problem was introduced by Miltenburg
(1989) and later studied by Miltenburg, Steiner and Yeomans (1990), Sumichrast and
Russell (1990), Kubiak and Sethi (1991), Inman and Bulfin (1991), Bautista, Companys and
Corominas (1992b), Steiner and Yeomans (1993), Ding and Cheng (1993a and b), Bautista,
Companys and Corominas (1993), Kubiak and Sethi (1994), Yeomans (1994), Bautista,
Companys and Corominas (1994, 1995, 1996b and c), Cheng and Ding (1996), and
Bautista, Companys and Corominas (1997a, 1997b and 2000), among others
The problem of regularity in component consumption was formulated by Monden (1983)
and called as ORV by Kubiak (1993). Later, it has been treated by Miltenburg and
Sinnamon (1989), Companys (1989), Miltenburg and Goldstein (1991), Bautista (1993),
Bautista, Companys and Corominas (1995 and 1996a), Duplaga, Hahn and Hur (1996),
Steiner and Yeomans (1996) and García Sabater (1999), among others.
A more detailed classification than Kubiak's one can be found in Bautista, Companys and
Corominas (1996d), as is reproduced in Table 1.
2
RESOURCES
COMPONENTS LOADS
object
criteria
PRODUCTS
1-level multilevel
PROPERTY CP CO CMO CL
FUNCTION PRV ORV MORV LRV
MIX CPRV CORV CMORV CLRV
Table 1. Classification table for regularity problems. The meaning for additional letters
(C=Constrained; M=Multilevel; L=Load).
The double classification inherent to the table requires an explanation:
• the columns are referred to the object whose regularity is looked for: products,
components (at one level or several levels), loads; the mix cases found in literature,
products and components at the same time, are considered in multilevel components
(CMO, MORV or CMORV).
• the rows are referred to how regularity is defined: by one or several properties or
constraints, such that sequences accomplishing them are regular by definition, by a
measure (for regularity or non-regularity) that leads us to indicate if a sequence is more
or less regular than another, and finally by a mixture of both, when only sequences that
accomplish some properties are regular, but a regularity measure permits to index such
regular sequences.
Both aspects, specially this last one, define the kind of algorithms, heuristic or exact, useful
to determine the sequences of products, which anyway become the action variables for the
process.
In the bibliography there are some mentions to problems that correspond to the categories
CO and CP, such that Dincbas, Simonis and Van Hentenryck (1988), BULL (1989), Little
(1993), among others.
Before a wider development about the subject of this work, it may be useful to expose some
considerations of practical behaviour. Two groups of people are interested about the
problem we are dealing with: professionals and academicians. We cannot consider that both
points of view coincide, nor those points of view, shown by ones and others, respond to
reality of the problem.
The majority of academicians took the option of measuring non-regularity by a function of
quadratic discrepancy. We suppose this is due, in part, to the comfortable properties of
quadratic functions and the extended use in some fields of science, but we do not know any
serious discussion about which qualities make this function the most appropriated for the
problem. Some alternative proposals appeared, Miltenburg (1989), Companys (1989),
Inman and Bulfin (1991), Bautista, Companys and Corominas (1996c), but they are quite
shy, limited and with little significance.
On the other hand, professionals show resistance to a procedure to make a complete
sequence before starting the working day. They consider it will be rarely put in practice due
to incidences, which may lead to strike availability of a unit for a variant to be sequenced in
the appropriate moment (the JIT concept has not reached this aspect). They do not advice a
lot of constructive heuristic algorithms are adaptable to be used on-line. Besides, they do
not usually see sequences in a discrete form, but in a continuous form, i.e. a sequence
3
started during a working day has a natural continuation next day. At the beginning of this
new day the line is not empty, but contains units introduced during the last hours of the
previous day in different positions.
Independently which is the real problem, if the proposed solution does not agree with the
points of view of professionals, it counts with few possibilities to be implemented and, thus,
tried. So, in consequence, this paper published in a journal of more or less prestige will be
relegated to a simple article. The focus of professionals takes generally enrichment and
contains sensible choices; only in extreme situations and with sufficient motivation, a
consultant may run the risk of contradict this focus.
The present study analyses, first of all, the results usually found in works published in
scientific references. After the general framework about the subject in Section 1, in Section
2 the PRV problem is described with the specific mention of distance or discrepancy,
including the summary of some computational experiments. In Section 3 we present shortly
the ORV problem and establish the similarities with the previous one. In Section 4, starting
from the car sequencing problem, we introduce a complete variation about the concept of
regularity and present the CP and CV problems. Taking our experimentation with the
software ROSINA, in Section 5 we define a concept of synthesis and the CRPV and CORV
problems. In Section 6 we include a summary about our observation on the real situation,
for which we propose some procedure frameworks in Section 7. Finally, Section 8 presents
some conclusions.
2. PRV PROBLEM: FORMULATION OF DISCREPANCY FUNCTIONS
Let consider the PRV problem, where P products or variants of a product are
scheduled in quantities ui (i = 1, 2, ..., P), being sequenced the T units, as "regular" as
possible:
T =
∑
=
P
i1
ui
A way to formalise regularity consists in using the distance between the real number of units
sequenced between positions 1 and t (t = 1, 2, .., T), xt,i, and the ideal value for that number,
t
.
ri, where:
r
i = ui/T
In other words, the distance between the real point of T.P components xt,i (X) and the ideal
point of components t
.
ri (X0). A sequence will be better than another, on this conception,
when the distance for the former is lower than for the latter. Thus, a measure for the non-
regularity of the sequence is defined.
The formulation of the previous idea in a compact way leads to the following mathematical
program:
[MIN] dist( X , X0 )
4
subject to
Xt ≤ Xt+1 t = 1, 2, ...., T−1
O'.
...Xt = t t = 1, 2, ....,T
0 ≤ Xt ≤ U and integers t = 1, 2, ...., T
where:
Xt is the vector (P,1) of components xt,i, for a given value t,
U is the vector (P,1) of components ui,
O is the vector (P,1) whose components are all equal to 1.
This is the implicit concept in Miltenburg (1989). As a complement, only it is necessary to
establish the distance concept. It is quite frequent to apply procedures in which the global
distance is equal to the sum of elementary distances, each one determined for one value t:
dist( X , X0 ) = ∑
=
T
t1
distt(Xt , t.r)
where:
r is the vector (P,1) of components ri.
Let call distt(Xt , t.r) to the function of discrepancy in t. We showed for this case the
problem of determining an optimum sequence is equivalent to determine the minimum path
in an associated multistage graph (Bautista et al., 2000, 2001). The number of vertices
potentially in the graph may convert the use of classical procedures to search for minimum
paths in graphs in prohibitive, but a lot of heuristic procedures can be suggested considering
them.
As it was indicated below, the most popular form consists in defining the distance by a
quadratic formulation:
dist( X , X0 ) = ∑∑
==
T
t
P
i11
(xt,i − t.ri)2 = ∑∑
==
T
t
P
i11
Xt − t.r =
=
∑
=
T
t1
SDQt = ∑
=
T
t1
Xt’⋅A⋅ Xt = SDQ
where:
A is the matrix (P,P) (I - r.O’)’⋅(I - r.O’).
I is the unit matrix (P,P).
5
In order to measure the relative efficiency for heuristics and procedures, susceptible to lead
to an optimum sequence, we have applied some algorithms to several blocks of instances,
for the problem with quadratic function of discrepancy. Some significant results are
indicated in Table 2, Table 3 and Table 4. The number of instances for each couple of
values, P and T, corresponds to sets with P positive integers whose sum is T. Tables 2 and 3
show the mean value of the relative deviation and the maximum value for that deviation,
respectively, applying each specific procedure to all the instances of a collection, being the
relative deviation of an heuristic solution as:
100×(value_heuristic_solution − optimum_value)/optimum_value
For any instance, an optimum sequence and the associated value were determined by a
procedure based on BDP (Bounded Dynamic Programming). We consider it more efficient
for the dimension of the proposed instances than the adaptation of the assignment algorithm
proposed by Kubiak and Seti (1991, 1994), although the polynomial character. Naturally,
the relative deviation is zero for the optimum solution and has no reflection on the table.
The chosen heuristics are:
• H1, 1-step heuristic, equivalent to the heuristic called goal chasing described by
Monden (1983); also described by Miltenburg (1989).
• H1.5, Ding and Cheng’s heuristic (1993 a, 1993b).
• H2, 2-step heuristic described by Miltenburg (1989).
• H2.5, 3-step heuristic with the third one relaxed in analogous form to that done in
H1.5 with the second one (Bautista et al., 2001).
Taking one cell in the following tables, there are two values: in the upper part, that
corresponding to the direct application for the heuristic algorithm as it is described in
literature; and in the lower part, to the application of the same algorithm filtering previously
the candidates through rules to be satisfied the optimum sequences (Bautista et al., 1997a),
calling the heuristics as H1+, H1.5+, H2+ and H2.5+. In column All, the upper value
corresponds to the best solution found for the four heuristics without filtering, and the lower
one, to the best solution found for any of the eight heuristics.
P T Number of
instances
H1 H1.5 H2 H2.5 All
4 45 672 1.04
0.80
0.37
0.36
0.28
0.28
0.07
0.03
0.03
0.01
5 55 3765 1.70
1.21
0.46
0.40
0.44
0.41
0.14
0.12
0.11
0.07
6 80 49342 1.75
1.34
0.64
0.57
0.74
0.70
0.24
0.22
0.19
0.16
Table 2. Mean relative deviation for the instances.
P T Number of
instances
H1 H1.5 H2 H2.5 All
4 45 672 16.08
16.08
6.52
6.52
14.49
14.49
7.63
3.70
3.70
3.70
5 55 3765 19.00
16.06
11.04
11.04
16.06
16.06
11.04
11.04
11.04
11.04
6 80 49342 22.54
17.10
13.09
10.40
13.90
13.90
13.09
13.09
13.09
8.99
Table 3. Maximum relative deviation for the instances.
6
Table 4 shows the proportion of instances (in percentage) in which the heuristic reached an
optimum solution.
P T Number of
instances
H1 H1.5 H2 H2.5 All
4 45 672 62.80
66.96
72.17
72.92
88.54
88.99
95.98
98.51
97.77
98.96
5 55 3765 36.25
44.30
60.05
62.04
75.59
76.44
91.02
93.07
93.33
94.69
6 80 49342 21.27
26.42
35.08
39.19
49.82
52.93
75.20
81.25
80.49
83.33
Table 4. Proportion of optimum solutions (%).
In order to achieve a complete view, Table 5 shows the mean relative values for the time
required by each algorithm to obtain the solution of an instance, taking as basic reference
the mean time necessary for the algorithm H1. The unitary time depends on the dimension
of the instance; for the three used dimensions the mean unitary times with the algorithm H1
are proportional to (1 : 1,880 : 4,144).
P T H1 H1.5 H2 H2.5
4 45 1.00
5.81
2.88
7.93
6.37
6.67
12.51
14.95
5 55 1.00
2.74
2.94
6.99
5.82
8.37
12.37
13.32
6 80 1.00
2.66
2.18
5.55
5.74
7.01
11.92
11.43
Table 5. Relative time consumed by the algorithms, taking as a reference algorithm H1
(the mean time necessary for applying the exact algorithm oscillates
between 165 and 320 times the reference heuristic consumed).
3. ORV PROBLEM
P products or variants of the same product, each one with quantities ui (i=1,2,...,P),
must be scheduled to have a sequence where T is:
T =
∑
=
P
i1
ui
Product i uses a quantity nj,i of component j (j = 1, 2, ..., C). The expected sequence is such
that the component consumption is as regular as possible. One way to formalise regularity
consists in using the distance between the real consumption of units between positions 1 and
t (t = 1, 2, .., T), yt,j, and the ideal value for that number, t
.
rj, where:
yt,j = ∑
=
P
i1
nj,i⋅xt,i
rj = ∑
=
P
i1
nj,i⋅ui /T
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Those ideas formalised in a compact way lead to a mathematical program, where Y is the
vector of components yt,j and Y0 the vector of components t
⋅
rj:
[MIN] dist( Y , Y0 )
subject to
Yt = N⋅Xt t = 1, 2, ....,T
Xt ≤ Xt+1 t = 1, 2, ...., T−1
O'.
...Xt = t t = 1, 2, ....,T
0 ≤ Xt ≤ U for integers t = 1, 2, ...., T
where:
Yt is a vector (C,1) of components yt,j, given a value t.
N is the matrix (C,P) of components nj,i.
This concept is implicit in Monden's conception (1983) and in Miltenburg's (1989), too.
Only it is necessary the concept of distance to use and, once again, procedures in which the
global distance is equal to the sum of elementary distances, each one determined for every
value t, may be used:
dist( Y , Y0 ) = ∑
=
T
t1
distt(Yt , t.r)
where:
r is a vector (C,1) of components rj.
Moreover, we have demonstrated this formulation is useful since the problem to determine
an optimum sequence is equivalent to determine the minimum path in the associated
multistage graph (Bautista et al., 1996a).
We can define quadratic distance as:
dist( X , X0 ) = ∑∑
==
T
t
P
i11
(xt,i − t.ri)2 = ∑∑
==
T
t
P
i11
Xt − t.r =
=
∑
=
T
t1
SDQt = ∑
=
T
t1
Xt’⋅A⋅ Xt = SDQ
where:
A is a matrix (P,P) (N - r.O’)’⋅(N - r.O’)
8
The identity of formats allow us to use methods as heuristics or based on BDP, similar to
those used for the PRV problem (see a description in Bautista et al., 1996a).
4. CP AND CO PROBLEMS
BULL (1989) published an advertising brochure about their software Charme, based
on a mechanism of constrain propagation, which allowed to obtain feasible solutions for
several combinatorial problems. In this brochure, as example of their software potentiality,
an application appeared, whose characteristics were similar to the ones in the subject we are
dealing with: the application was known as car sequencing on a production line, and its
precedents are found in Dincbas et al. (1988).
The instance of the problem car sequencing consists in sequencing 100 units of 18 different
products. Each product has five modules, for each one of them two different options may be
mixed. The first one of those options is called as normal and has no difficulties when
sequencing, while the second one, called as special, demands some requirements. In the
given instance, four of the products have three special and two normal options (see Figure
1). Nine products have two special options, and the remaining five products, a single special
option. No product without special options in modules must be sequenced, which is a
product with no special sequencing difficulties.
The imposed constraints refer to the maximum number of special options along segments of
a sequence; so:
• module one: just a special option within each two contiguous units of the
sequence;
• module two: just two special options within each three contiguous units of the
sequence;
• module three: just a special option within each three contiguous units of the
sequence;
• module four: just two special option within each five contiguous units of the
sequence;
• module five: just a special option within each five contiguous units of the
sequence.
Figure 1. Instance to sequence 100 units of 18 different products.
9
The objective is to find a sequence, which satisfies the constraints. In the brochure, there is
a solution, found by the software package (see Figure 2). Little (1993) reproduces the same
previous results with some changes.
By analogy with the previous case, and collecting the opinions from several professionals
from the car industry, we can define the following two types of car sequencing problems:
CP (constrained product) problem with the following formulation:
Given P products, each product i (i = 1, 2, ..., P) has three integer positive numbers
( ui , ai , bi ), with ai < bi, whose meanings are:
• Number of units to sequence of product i: ui.
• Maximum number, ai, of units of product i sequenced in any segment of the
sequence within bi consecutive positions (maximum constraint for i).
The objective is to find a sequence of T units (T = Σ ui), such that ui belongs to product i,
that satisfies all the maximum constraints for each product. A sequence of this kind is, by
definition, regular.
Given a number of values (an instance), there may exist none, one or more than one regular
sequences.
CO (constrained output) problem with the following formulation:
Given P products, each product i (i = 1, 2, ..., P) has a consumption of nj,i units of
component j (j = 1, 2, ..., C). Each product has associated a positive integer ui, and for each
component two positive integer numbers aj and bj are defined, whose meanings are:
• Number of units to sequence of product i: ui.
• Maximum number, aj , of units of component j consumed by the products
sequenced in any segment of the sequence within bj consecutive positions
(maximum constraint for j).
The objective is to find a sequence of T units (T = Σ ui), where ui belongs to product i, that
satisfies all the maximum constraints for each component. A sequence in this category is, by
definition, regular.
In this situation:
aj may be greater, lower or equal to bj (aj integrity may be eliminated, if nj,i is not
imposed either).
In many circumstances, like those described in the car sequencing problem, it is
considered nj,i can only have 0 or 1 values.
Also in this problem, given an instance, there may exist none, one or many regular
sequences may exist.
Both problems, CP and CO, given the usual conditions in industry, may define additional
minimum constraints.
10
5. CPRV AND CORV PROBLEMS
Similarities between the car sequencing problem and the ORV problems is obvious, as both
pretend to make regular the appearance of the normal and special options for each module
in a sequence. For example, considering the appearance of each special variant in the
module five, an easy calculus (such as Σ nj,i⋅ui) lead us to know just 14 of the 100 units to
sequence have this option, which is less than the 20% of the respective constraint (1 of each
5). In this case, this suggests, given that 100 is divisible by 5, that the constraint would be a
limit, and no feasible sequence is possible above it. In the same way, the rest of modules
gives a similar relation as shown in Table 6.
module constraint maximum required
1 1/2 50 48
2 2/3 67 57
3 1/3 34 28
4 2/5 40 34
5 1/5 20 14
Table 6. Comparison constraints/appearances of special options.
Therefore, a sequence, in which each one of the five modules were regularly distributed in
Miltenburg's way (with rates 0,48; 0,57; 0,28; 0,34 and 0,14; respectively) should satisfy the
constraints. As the values SDQ for the regular sequences, in this way, are lower than those
of other sequences, the value SDQ or SDQt should work as a sign to go straight in the
search for satisfying the constraints.
In order to be helped during the experiments, we designed, implemented and spread a
software package ROSINA (1992). There, instances of the CO problem can be defined to
solve (P products, each one with m modules and a number ui of units per module, together
must reach a sequence with a total of T units; it is necessary to specify the type of module,
normal or special, for each product, and the constraint associated to the special option, too).
In the search for a solution, three procedures were included:
• A direct procedure, with no backtracking, which builds a sequence progressively,
using as indicator to help the next branch selection to visit in the tree, subjected to
the constraints, the quadratic discrepancy; this procedure may end without
sequencing the T units.
• A backtracking procedure, with or without indicator (in the former, the branch
election is given by the order of data input). If an ending state is reached without
sequencing the T units, the algorithm continues the search rejecting the last
branches taken and following another path until a solution is found or the whole
tree is explored without success (in this case, the instance has no feasible solution).
• CBDP procedure (constrained bounded dynamic programming), an adaptation of
the BDP to keep in parallel several paths (as many as a previous defined
"window").
The use of the indicator ("predictor"), if it is necessary, is similar that used in 1-step
procedures (the classical Monden's goal chasing) based on the theoretical rate of
appearance for special variants rj. Once found a full sequence which satisfies the
constraints, ROSINA calculates the related value SDQ, but shows the unitary IRQ:
11
IRQ = SDQ/T
Surprisingly, the solution in Figure 2 has a very high associated IRQ, IRQ=24.431 (i.e.
SDQ=2443.1). It is also curious that ROSINA finds, without effort, such solution using a
backtracking with no indicator, which has a very strong dependence of the original order of
data. Both the direct method and the backtracking method with indicator obtain a solution
(Figure 3), which satisfies the constraints and whose value IRQ = 0.516 points this solution
is more regular, in Miltenburg's sense, than the previous one. Furthermore, the CBDP
procedure obtains 20 better solutions than that in Figure 3, particularly one with IRQ=
0.449 (Figure 4).
Therefore, it is easy to define two new typologies of problems:
The CPRV (constrained product rate variation) problem with the following formulation:
Given P products, each product i (i = 1, 2, ..., P) has three positive integer numbers
associated (ui , ai , bi ), with ai < bi, whose meanings are:
• Number of units to sequence of product i: ui.
• Maximum number, ai, of units of product i which may appear in any segment
of the sequence within bi consecutive positions (restriction of maximum for i).
A sequence of T units (T = Σ ui) is searched, such that ui of them must be of product
i, satisfying all the constraints of maximum for each product (then a sequence is
feasible) and minimises the distance to an ideal sequence in the form of:
[MIN] dist( X , X0 )
Given a set of values (an instance), there may be none, one or many regular sequences.
The CORV (constrained output rate variation) problem with the following formulation:
Given P products, each product i (i = 1, 2, ..., P) has a consumption of nj,i units of
component j (j = 1, 2, ..., C). Each product has associated a positive integer ui and
each component two positive integer numbers aj and bj whose meanings are:
• Number of units to sequence of product i: ui
• Maximum number, aj, of units of component j consumed by the products
included in any segment of the sequence within bj consecutive positions
(maximum constraint for j).
A sequence of T units (T = Σ ui) searched, such that ui must be of product i, satisfies
all the maximum constraints for components (then a sequence is feasible) and
minimises the distance to an ideal sequence in the form of:
[MIN] dist( Y , Y0 )
12
Figure 2. Solution for the problem in Figure 1 provided by the software package Charme
(IRQ=24.43).
13
Figure 3. Solution for the problem in Figure 1 provided by the backtracking method with
indicator (IRQ=0.516).
14
Figure 4. Solution for the problem in Figure 1 provided by the CBDP procedure
(IRQ=0.449).
15
In these conditions:
aj may be higher, lower or equal to bj (aj integrity may be discarded, if nj,i is not
imposed either).
In many circumstances, like those described in the car sequencing problem, it is
considered nj,i can only have 0 or 1 values.
Also for this problem, given a set of values (an instance), there may exist none, one or many
regular sequences.
Both problems, CPRV and CORV, given the usual conditions in industry, may contain
additional minimum constraints.
6. DISTINCTIVE ASPECTS IN REALITY
After some interviews with several professionals in the car industry interested about the
problem of sequencing units in an assembly line, we have detected some differences
between the practical sight of the problem and literature references. Below, there is a list of
the most remarkable ones:
• Continuous Flow.
Limited sequence segments (a shift, a day, a week,...) may be considered an
artificial segmentation of real live.
A generic criticism to the previously presented procedures deals with the indirect
or "unnatural" way of defining a sequence using the values xt,i. Would not it be
more natural to define the positions of the sequence in which each unit of the
different variants should be ideally placed and measure an index of non-regularity
between real and ideal positions? This conception agrees with the focus presented
by Inman and Bulfin (1991), which would be more primary than the Miltenburg’s
one. Though, no justification is needed. Difficulty comes from defining which is
the ideal position. It seems logical to propose the k-th unit of product i should be
placed in:
(k − 1/2)⋅i
u
T
Even though, other considerations are mostly necessary for finite sequences.
We have yet not escaped from the measurement of positions, given a fixed one. A
possibility, for the PRV problem, is the following: let xt,i(L) = xt,i − x
t−L+1,i the
number of units of variant i in a segment of sequence between positions t - L + 1
and t, a measurement of non-regularity in position t may be, for example:
SDQt,L = ∑
=
P
i1
( xt,i(L) − L.ri )2
which may be also described in a matrix form:
SDQt,L = Xt(L)'.A.Xt(L)
where Xt(L) is the vector (P,1) of components xt,i(L).
16
Once L is defined and given a sequence, determining xt,i(L) for t ≥ L and
subsequently evaluating SDQt,L is easy. Given the vectors Xt(L), it is not so easy to
find the sequence nor even impose conditions on continuity for those vectors. Even
though, a coherent form to define regularity is:
SDQt,L ≤ m for each t ≥ L
• Credibility on scheduling.
Establishing in advance a production schedule, with the detail on long segments of
the sequence, is only useful if the effective accomplishment of the schedule (and
the sequence) is close to 100% in reality. In the other way, expected benefits from
scheduling are not reached,
For example, pre-emptive knowledge for suppliers about component and sub-
element orders must be provided, to help to plan workloads and reduce stocks); on
the contrary, modifications and contradictory orders contribute a great deal to
discredit procedures and force to take expensive measures substituting
inefficiencies.
• Broadcast.
The appropriated advance to fix the sequence depends on the required term for
operations set off the known sequence. Consequently, many times it is not
necessary to know in advance the whole sequence, but longer or shorter segments,
which are moved forward progressively.
• Modularity.
Also, as a consequence, the concretion of which product must be placed at each
position of the sequence may be done by phases, with different advances.
• Constraint perception.
Professionals see units or associated vehicles according to some properties in order
to classify them in different ways. Limitations (or constraints) while building a
sequence are usually expressed as the showed above form: at most, a appearances
of property P in b consecutive vehicles; the level of demand depends on the
property. Besides, in order to guarantee the fulfilment of the production schedule
and, though, manufacturing of difficult units, a new general constraint is added: a
minimum of a' appearances of property P in b' consecutive vehicles.
• Criteria.
For the majority of properties Q (including P), a regular distribution along the
sequence is expected, but in few cases (for some properties), on the contrary,
concentration of some certain units is searched in a segment or determined
segments (for example, in the first shift).
• Impossibility to maintain the sequence.
Incidences caused by production flow can limit usefulness to determine a priori the
sequence. Making the sequence at real time implies the use of algorithms ad hoc
based on local criteria. Helpfulness of a sequence can only be established a
posteriori and the construction algorithm, by statistics or simulation. (The heuristic
constructive algorithms used to establish sequences a priori are easily adaptable).
• Imperfect synchronisation.
In real time scheduling, suitable candidates to sequence in a given moment are
limited to those physically waiting in a finite queue. Defining what to do when
17
there are no compatible candidates in the queue may fit to a set of stepped criteria
(empty position, constraint hierarchy, minimisation of a penalisation function, etc.).
• Fear to void.
If no empty positions can be left in the sequence, the violation of some constraints
must be allowed, if it is necessary. Decisions may be automatic, and thus based, as
indicated below, on some priority levels established among constraints or
associated costs, or manually, taken by an expert sequencer (as sometimes
happens).
The analysis of the previous aspects suggests us the following considerations:
• Independently of the consultant's conception about the real problem and the most
comfortable format to deal with it, he must take the manager's view and language, as
this approach generally is more enriching.
• The constraints described as ‘maximum a elements within each segment of b positions’
are rich enough to support an adequate product structure.
• In order to fulfil all the aspects required by the problem, the maximum constraints must
be complemented, in most cases, with minimum constraints.
• The form of the objective function is difficult to explicit, because it is fuzzy defined.
• It is convenient to analyse if investing on devices to guarantee the accomplishment of
the determined a priori sequence is more profitable than favouring quality of a defined
on-line sequence.
• Any constructive algorithm (even bounding the number of potential candidates) must
consider more than one step to be efficient.
• Any efficient ISHS (interactive system to help sequencing) uses a framework of 4
modules (pre-processor, processor, post-processor and database), allows to personalise
weights associated to criteria and interacts with users in its own language.
7. CRITERIA FORMULATION AND INFLUENCE ON ALGORITHM
STRUCTURE
The above criticism leads to a new framework in the context of regularity, based on the
following considerations:
• Each product is associated to some characteristics, which may have different values.
• Characteristics can be classified in three kinds, for example: alpha, beta and gamma.
• Characteristics of kind alpha are associated to constraints with the form: maximum a
appearances in any segment of b consecutive positions.
• Additionally, characteristics of kind alpha are associated to constraints with the form:
minimum a' appearances in any segment of b' consecutive positions.
• Characteristics of kind beta are associated to the objective of maximum regularity
along the sequence.
• Characteristics of kind gamma are associated to the objective of maximum
concentration along the sequence.
Referred to the algorithms, two approaches can be taken depending on the searched
18
objective:
Algorithms to determine a segment of the sequence in advance:
for example, based on the use of CBDP sometimes combined with local search
algorithms.
Algorithms to build sequences at real time:
for example, formed by a filter for potential candidates through some rigid
constraints, sometimes introduction of a second filter based on rules, classification
of selected candidates (taking on account 2 or 3 steps), and multicriteria selection
of the chosen candidate.
8. CONCLUSIONS
We have shown the differences between academician and professional approaches for the
sequencing problem and have suggested some action lines, according to our research work.
This study has been developed in the framework of the research project TAP98-0494,
funded by CICYT.
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