International Journal of Computers, Communications & Control
Vol. I (2006), No. 2, pp. 53-60
Importance of Flexibility in Manufacturing Systems
Héctor Kaschel C., Luis Manuel Sánchez y Bernal
Abstract: Flexibility refers to the ability of a manufacturing system to respond cost
effectively and rapidly to changing production needs and requirements. This capabil-
ity is becoming increasingly important for the design and operation of manufacturing
systems, as these systems do operate in highly variable and unpredictable environ-
ments.
It is important to system designers and managers to know of different levels of flex-
ibility and / or determine the amount of flexibility required to achieve a certain level
of performance. This paper shows several representations schemes for product flexi-
bility and discusses the usefulness and limitations of each.
Keywords: Flexibility, Fuzzy Logic, Disjunctive Graph, Constraints Temporal Net-
work, Complexity in Manufacturing, Non Lineal Mathematic Programming
1 Introduction
Present market demands require that Manufacturing Systems develop their activities under a dynamic
and uncertain production environment (Calinescu et al., 2003). To understand the problems affecting
Flexible Manufacturing Systems (FMS), it is necessary to incorporate the concept of flexibility in the
scheduling (Deshmukh et al. 2002). Flexibility constitutes a strategic topic in decision making to give
quick and efficient answers to the demands of the national and international markets (Pelaez, and Ruiz,
2004).
In order to define the flexibility in FMS, the scientific literature reports a wide variety of concepts. For
example, flexibility for Sethi and Sethi (Sethi and Sethi, 1992) represents the capacity of FMS to modify
manufacturing resources to produce different products efficiently maintaining an acceptable quality.
For Gupta (Gupta, 2004), it is necessary to make the difference between flexibility inherent to the
FMS and flexibility required during the production cycle. The first one is defined as the capacity of
resources and human operators to implement decisions of the factory company. While required flexibility
is defined by the changes in the environment that surrounds the FMS and for the implementation of new
manufacturing strategies. Starting from these differences, it is possible to determine the flexibility type
and measure of the FMS quantitatively.
For (Benjaafar and Ramakrishman, 1996), it is important to differentiate the types of flexibility in
the FMS. They define the flexibility of the FMS as product flexibility and process flexibility. The first
one refers to the variety of factory options for a certain product. While process flexibility is defined as
a characteristic of an industrial process to operate under diverse dynamic conditions of operation. Both
general flexibilities can be grouped hierarchically as shown in the figure 1.
The classification proposed by (Benjaafar and Ramakrishman, 1996) shows flexibility as a conse-
quence of the physical and logical characteristics that the manufacturing system presents To quantify the
operative flexibility of job i, a function is defined starting from the time of dependent configuration of
the sequence between operations:
Φ(pi) =
−∑Nij=1 ∑
ki j
l=1
(
1
si jl
∑
ki j
l=1
1
si jl
)
log
(
1
si jl
∑
ki j
l=1
1
si jl
)
Ni
(1)
where:
Copyright c© 2006 by CCC Publications
54 Héctor Kaschel C., Luis Manuel Sánchez y Bernal
Figure 1: Classification of the FMS Flexibility
• si jl represents the time of configuration of job i during the commutation of the operation j to the
operation l.
• Ni represents the total number of operations associated to job i.
In the present paper different models were analyzed as described in several studies dealing with
product flexibility in FMS.
2 Product flexibility
In all FMS, each job or jobs group is associated to a certain scheduling, consisting in an orderly
sequence of operations to be executed in a group of machines. An example of scheduling of the FMS is
illustrated in figure 2(a).
If we incorporate conditions of flexibility to the scheduling (operational flexibility), an operation can
be carried out in other machines. These, in general, can be different or they can have different operational
levels (figure 2 (b)).
The classification of the operations can also be varied (sequence flexibility), where some operations
should be carried out in sequence while others don’t have this restriction. An example of this type of
flexibility is shown in figure 2(c), where operations 2 and 3 can be carried out in any order after executing
operation 1.
Finally, in a highly flexible FMS, a job can have a group of schedulings (processing flexibility). This
means that the group of operations required to manufacture a job can be variable and depending on the
operation conditions of the FMS. For example, in figure 2(d), operations 1 and 2 can be replaced by the
operation 5 while operation 3 can be divided in operations 6 and 7.
3 Analysis models for product flexibility
The product flexibility constitutes an important aspect in the functionality of the FMS, where the
variability in the scheduling and the operability of the machines allow several alternatives to process the
jobs. In what follows some models of analysis of the product flexibility are presented.
Importance of Flexibility in Manufacturing Systems 55
Figure 2: Types of Product Flexibility
3.1 Disjunctive Graph
It is a model of directed graph, whose nodes have a certain weight assigned. Mathematically it is
expressed as 3 - tupla (Benjaafar and Ramakrishman, 1996):
G = (N, A, E) (2)
where:
• N: represents the group of nodes that represent the operations. In this model are defined two
particular nodes, called initial and final. The positive weight assigned to each node represents
the processing time of the corresponding operation. The initial node is connected with the first
operation of each job, and in the same manner the last operation of each job is connected with the
final node.
• A: represents the group of conjunctive directed arcs that indicate the precedence restrictions among
the operations of each job.
• E: represents the group of disjunctive directed arcs that indicate the restrictions of machines oper-
ative capacity.
To illustrate the application of the disjunctive graph, an example of scheduling of 4 different jobs is
presented (J1, J2, J3, J4) in three types of machines (M1, M2, M3). The distribution of processing times
and assignment of machines is shown in table 1, starting from this information we proceed to generate the
disjunctive graph of figure 3, that shows the precedence restrictions between the operations (conjunctive
arcs) and the flexibility in the machines operative sequence (disjunctive arcs).
56 Héctor Kaschel C., Luis Manuel Sánchez y Bernal
Processing time Machines assignment
M1 M2 M3 1 2 3
J1 5 8 2 M1 M2 M3
J2 3 9 7 M3 M1 M2
J3 1 10 7 M1 M3 M2
J4 7 4 11 M2 M3 M1
Table 1: Machines assignment and processing times
Figure 3: Disjunctive Graph generated by table 1
3.2 Temporal Constraints Network
It is a model of directed graph that allows the representation of temporary constraints assigned to the
precedence relationships between operations that are part of a scheduling of a FMS. The nodes of the
graph represent the operations and the arcs have associated temporary intervals to denote the temporary
constraints that exist between two nodes. All the times are calculated starting from of the scheduling
initial state. The development of the network is based mathematically from Disjunctive Metric Temporal
Algebra (Alfonso, 2001).
This methodology is used to solve scheduling problems characterized from a group of temporary
constraints, consequently, the techniques developed in areas like Temporary Reasoning and Constraints
Satisfaction can be applied. In general, two techniques are used: Closure and Constraint Satisfaction
Problem (CSP) (Barber, 2000).
Closure is a deductive process by means of which new constraints are inferred, starting from existing
constraints. It also allows to detect possible inconsistencies (value of the variables that don’t lead to any
solution) that can be eliminated. The main advantage of the closure techniques consists in the reduction
of the search space. Therefore, the closure is used fundamentally as a previous step to a process of search
of solutions.
Techniques of Constraints Satisfaction Problem constitutes a process of search of solutions by means
of the successive assignment of values to the variables that are part of the constraints of the problem. In a
process of Constraints Satisfaction Problem, it is important to use different heuristics to make the search
process more efficient (Beck and Fox, 2000).
To illustrate the use of the Temporary Constraints Network, a distribution of temporary constraints is
presented among four operations of a FMS (see table 2).
Starting from the information in table 2, we proceed to establish the group of operations (O), the
Importance of Flexibility in Manufacturing Systems 57
O1 O2 O3 O4
O0 initial state (10, 20) (10, 20) (20, 50) (60, 70)
O1 (30,40), (60,60) (10,30), (40,40) (40, 60)
O2 (0, 30)
O3 (10, 20) (20,30), (40,50)
Table 2: Precedence Temporary Constraints between operations
group of precedence relationships between operations (P) and the group of temporal relationships be-
tween operations (R). Finally, starting from the definitions of the groups we proceed to generate the
corresponding Graph of Temporal Constraints (see figure 4):
G = (O, P) (3)
O = (O0, O1, O2, O3, O4) (4)
P =
(00, O1), (00, O2), (00, O3), (00, O4)
(01, O2), (01, O3), (01, O4), (02, O4)
(03, O2), (03, O4)
(5)
R =
(O0(10, 20)O1), (O0(10, 20)O2),
O0(20, 50)O3), (O0(60, 70)O4,
(O1(30, 40)(60, 60)O2),
(O1(10, 30)(40, 40)O3), (O1(40, 60)O4),
(O2(0, 30)O4), (O3(10, 20)O2)
(O3(20, 30)(40, 50)O4)
(6)
Figure 4: Temporary Constraints Network generated by Table 2
3.3 Model based on Mathematical Programming
Chandra and Tombak (Chandra and Tombak, 1993) propose a operations sequence flexibility model,
based on the theory of lineal programming. The model incorporates the concepts of machines reliability
and machines processing capacity. Machine reliability is defined as the probability of an operation to be
executed by a machine in a certain interval of time. Machine processing capacity, represents the total
number of jobs that a machine can execute in a interval of time.
58 Héctor Kaschel C., Luis Manuel Sánchez y Bernal
The FMS base for the development of the model consists of a finite group of machines that can
present independent fault events. The model is formulated considering the jobs flow on a fixed scheduling
and it is expressed mathematically as:
RF = max ∑
i
∑
h
cihxih (7)
subject to the following constraints:
∑
i
∑
h|bihk=1
tik
Pk
xih ≤ Tk,∀k (8)
∑
h
xih > di,∀i (9)
xih ≥ 0,∀i, h (10)
where,
• tik represents the processing time of job i in machine k;
• Tk represents the total processing time of machine k;
• Pk represents the probability that machine k is operative in a certain time;
• bihk represents the possibility that job i is processed in machine k along route h;
• Ch represents the factor of cost of job i when processed in route h. This factor depends on the
reliability machines located along route h;
• di represents the minimum demand of job i and,
• xih represents the flow of job i on route h.
The model represents the processing capacity of each machine considering its reliability, where the
factor tikPk represents the time required to process job i in machine k. The model considered n + m con-
straints and m(1 + ∑n−2k=1 k!C
n−2
k ) variables in case of full machines connections.
3.4 Model based on Fuzzy Logic
In (Tsourveloudis and Phillis, 1998) and (Fortemps, 2000), analyze the flexibility starting from a
methodology based on the knowledge that the human expert has on the structural and dynamic behavior
of the FMS. For (Tsourveloudis and Phillis, 1998) the flexibility is represented by means of a group of
diffuse linguistic rules of the type:
IF 〈antecedent fuzzy〉 THEN 〈consequent fuzzy〉 (11)
This proposal is based on the fuzzy behavior that the FMS presents, in cases of concurrence and
synchronization of operations or fault machines events. For (Tsourveloudis and Phillis, 1998), a FMS
presents seven different types of flexibility:
1. Materials Manipulation System Flexibility: It measures the capacity of the system to transfer
several jobs types efficiently between two neighboring points.
2. Product Flexibility: It measures the capacity of the system to produce mixed jobs.
Importance of Flexibility in Manufacturing Systems 59
3. Operative flexibility: It measures the capacity to modify the sequence of operations to produce a
job.
4. Process Flexibility: It measures the capacity of the system to produce different jobs without mod-
ifying the FMS structure.
5. Volume Flexibility: It measures the operative capacity of the FMS for different levels of produc-
tivity.
6. Scalability Flexibility: It measures the cost or time required to enlarge a FMS.
7. Labor Flexibility: It measures the operator’s capacity to develop different operative tasks of FMS.
Each one of these types of flexibility is expressed by means of a fuzzy rule that represents the expert’s
knowledge about this flexibility. The fuzzy rule is expressed mathematically as:
IF F1 is A1 AND . . . AND FN is AN THEN FMF is MF (12)
where: Ai = {Low, About_Low, Average, About_High, High} represents the group of linguistic values for
the flexibility parameter. For example, the following fuzzy rule expresses the Operative Flexibility of the
FMS:
IF C0 is TC0 AND SB is TSB THEN FR is TFR (13)
For this rule, C0 represents the number of operations common to machines group and SB represents
the FMS capacity to route a jobs group under conditions of machines faults.
4 Conclusions
The models described in this paper define a set of basic principles that should satisfy all parameters
associated to the FMS flexibility, such as:
• They should be specified and derived according to the methodology proposed for the analysis of
the flexibility.
• They should incorporate structural and operational aspects of the FMS.
• They should incorporate and accumulate the human expert’s knowledge.
Of the models, we can conclude that the problem of the Flexibility is analyzed starting from one of the
three big current problems affecting the FMS [CALINESCU et al 2003]:
• Making Decisions Problem, analyzed by the methods based on Mathematical Programming and
Fuzzy Logic.
• Structural Problem, analyzed by the method of the Disjunctive Graph and
• Behavior Problem, analyzed by the method of the Temporary Constraints Network.
60 Héctor Kaschel C., Luis Manuel Sánchez y Bernal
References
[1] Alfonso, M., An integration model of Temporary Constraints based on methodology CLOSURE
and CSP: Application to Scheduling Problems, Doctoral Thesis presented at University of Alicante,
Spain, 2001.
[2] Barber, F., "Reasoning on interval and point-based disjunctive metric constraints in temporal con-
texts". Journal of Artificial Intelligence Research, pp:35-86, 2000.
[3] Beck, J. and Fox M., "Constraint directed techniques for scheduling alternative activities". Journal
of Artificial Intelligence, pp: 211-250, 2000.
[4] Benjaafar, S. and Ramakrishman, R., "Modeling, measurement and evaluation of sequencing flexi-
bility in manufacturing systems", International Journal of Production Research, Vol. 34, pp:1195-
1220, 1996
[5] Chandra, P. and Tombak, M., Models for the evaluation of routing and machine flexibility, Technical
Report of Decision Craft Analytics, www.decisioncraft.com, 1993.
[6] Deshmukh, A., Talavage, J. and Barash, M., "Complexity in Manufacturing Systems", IIE Trans-
action on Manufacturing Systems, Vol 30, pp: 645 - 655, 2002.
[7] Fortemps, P., "Introducing Flexibility in Scheduling: The Preference Approach", Advances in
Scheduling and Sequencing under Fuzziness, Ed. Springer Verlag, pp:61-79, 2000.
[8] Gupta, A., "Approach to Characterize Manufacturing Flexibility", Second World Conference on
Production & Operations Management Society, Cancun - México, pp:100-120, 2004.
[9] Calinescu, A., Sivadasan, S., Schirn, J. and Huaccho, L., Complexity in Manufacturing: An Infor-
mation Theoretic Approach, Technical Report of Manufacturing System Research Group, Depart-
ment of Engineering Science, University of Oxford, England, 2003.
[10] Pelaez, J. and Ruiz, J., Measuring Operational Flexibility, Technical Report, University of Murcia,
Faculty of Economy and Enterprise, Spain, 2004.
[11] Sethi, A. and Sethi, S., "Flexibility in Manufacturing: A survey", International Journal of Opera-
tions and Production Management, pp:35-45, 1992.
[12] Tsourveloudis, N. and Phillis, Y., "Fuzzy Assessment of Machine Flexibility", IEEE Transaction
on Engineering Management, Vol. 45, pp:78-87, 1998.
Héctor Kaschel C. and Luis Manuel Sánchez y Bernal
Depto. de Ingeniería Eléctrica and
Depto. de Matemática y Ciencia de la Computación
Universidad de Santiago de Chile
Avda. Ecuador ] 3519 Estación Central. Santiago - Chile
E-mail: hkaschel@lauca.usach.cl o msanchez@lauca.usach.cl