IIE Transactions (2014) 46, 230–248
C “IIE”
Copyright ISSN: 0740-817X print / 1545-8830 online
DOI: 10.1080/0740817X.2013.813093
Algebraic expression of system configurations and
performance metrics for mixed-model assembly systems
ANDRES G. ABAD1, WEIHONG GUO2 and JIONGHUA (JUDY) JIN2,*
1
Escuela Superior Politecnica del Litoral, Campus Gustavo Galindo Velasco, Km. 30.5 Via Perimetral,
Guayaquil 09-01-5863, Ecuador
2
Industrial and Operations Engineering Department, University of Michigan, Ann Arbor, MI 48109-2117, USA
E-mail: [email protected]
Received December 2010 and accepted April 2013
One of the challenges in the design and operation of a mixed model assembly system (MMAS) is the high complexity of the station
layout configuration due to the various tasks that have to be performed to produce different product variants. It is therefore desirable
to have an effective way of representing complex system configurations and analyzing system performances. By overcoming the
drawbacks of two widely used representation methods (block diagrams and adjacency matrix), this article proposes to use algebraic
expressions to represent the configuration of an MMAS. By further extending the algebraic configuration operators, algebraic
performance operators are defined for the first time to allow systematic evaluation of system performance metrics, such as quality
conforming rates for individual product types at each station and process capability for handling complexity induced by product
variants. Therefore, the benefits of using the proposed algebraic representation are not only their effectiveness in achieving a compact
storage of system configurations but also its ability to systematically implement computational algorithms for automatically evaluating
various system performance metrics. Examples are given in the article to illustrate how the proposed algebraic representation can be
effectively used in assisting the design and performance analysis of an MMAS.
Keywords: Algebraic expression, mixed model assembly, product variety, system configuration model, complexity, modular assembly
systems
1. Introduction
In recent decades, market demand has changed from being
fairly homogenous and relatively stable to highly variable
and rapidly changing. As a response, production systems
have gone from a mass production paradigm to a mass
customization model. Implementing a mass customization
scheme, however, requires overcoming many technological
challenges (Da Silveira et al., 2001). From a production
point of view, the correct implementation of modular production (Sturgeon, 2002) is the key to addressing these
challenges.
In a modular production assembly system, different
product variants are produced in a system that is referred
to as a Mixed Model Assembly System (MMAS). The station configurations of an MMAS become highly complex
because different product types use different production
paths in the same assembly system. For example, the assembly system shown in Fig. 1 is used to produce three different
∗
Corresponding author
C 2014 “IIE”
0740-817X product types. It can be seen that the first two stations, S1
and S2 , are basic operation stations, which means that every
product of any type will be processed at stations S1 and S2 .
In contrast, stations S3 , S4 , S5 , S6 , S7 , and S8 are considered as variant operation stations, which implies that only
certain types of products will be processed at these related
stations. Specifically, the first product type has a production path of S1 → S2 → S3 → S4 → S5 ; the second product type follows the path S1 → S2 → S3 → S6 ; and the third
product type is produced via the path S1 → S2 → S7 → S8 .
It should also be noted that in an MMAS, parallel station configurations are not necessarily used to duplicate the
same tasks as they are used in single-model assembly systems for the purpose of line balancing. As shown in Fig. 1,
the path of station 6 is used to produce a product type that
is different from the type produced in the parallel path of
station 4 and station 5. This may also be true for two single
parallel stations since each of the two parallel stations may
be designed to perform different tasks to produce different
product components or different types of the same component. Consequently, adding product variants in an MMAS
increases the complexity in operating and evaluating
manufacturing systems.
Algebraic expression of MMAS configurations
Fig. 1. Basic operation stations and variant operation stations in
a mixed-model assembly system.
As the demand for MMASs increases, the need for effectively modeling and analyzing the operation performance
of such a system increases. Traditionally, assembly systems
have been represented by the use of block diagrams, as
shown in Fig. 1; this type of diagram shows the advantages of intuitive visual perception of the station layout
configuration. Block diagrams themselves, however, do not
have the computational capability to permit the mathematical manipulation of system configurations or evaluation of
system performances. The use of adjacency matrices representing block diagrams is another common method of
representing assembly systems, as noted in the literature. Adjacency matrices are mathematical structures that
emerge in graph theory as a way to more easily manipulate
and represent graphs. The capability of intuitive visual representation of the system configuration (serial or parallel relationships between stations) cannot, however, be preserved
in the adjacency matrices representation. Furthermore, adjacency matrices tend to result in largely sparse matrices,
especially as the number of stations increases. As a consequence, adjacency matrices are not an effective compact
method for representing an assembly system.
To overcome the drawbacks of block diagrams and adjacency matrices methods, Webbink and Hu (2005) recently
introduced a novel way of representing complex assembly systems by using a string representation. The proposed
method uses characters to represent the stations and parentheses to denote whether stations are connected in a serial or parallel relationship. Although the string representation provides a compact way to represent the system
configuration, it lacks a computational capability to use
computer programs to incorporate the system configuration into mathematical operators to evaluate system performances. Therefore, this article intends to overcome the
shortcoming of the string representation by using an algebraic representation. The advantage of an algebraic representation is that it not only can store and manipulate
231
serial or parallel configuration information so that it can
be easily handled by computer programs but it also can be
transferred to the mathematical operators for evaluating
system performances.
The use of binary operators to deal with graph problems has been proposed in the literature. Path algebras, also
known as (max, +) algebras or dioid algebras, are mathematical structures used to solve a large number of pathfinding and network problems in graph theory (Carre, 1971;
Gondran and Minoux, 1983). The key idea is to use binary
operators to rewrite, in a compact way, the algorithms used
to solve these problems, thus achieving a pseudo-linear system of equations.
The application of path algebras to manufacturing systems was initiated in Cohen et al. (1985). Since then, it
has been used to solve numerous kinds of problems in this
domain; e.g., resource optimization (Gaubert, 1990), production planning and control (Xu and Xu, 1988; Yurdakul
and Odrey, 2004), and modeling of discrete-event systems
(Cohen et al., 1989; Cofer and Garg, 1993). To the best of
our knowledge, however, path algebra has not been directly
used to represent complex assembly system configurations.
Zissos and Duncan (1971) proposed the use of algebraic
operators to represent logic circuits with the advantage that
the symbols stand in a one-to-one correspondence with the
physical elements of the system. Furthermore, a postfix
or reverse Polish notation of this algebraic representation
was proposed in Duncan et al. (1975) with the advantage
that the postfix notation eases implementation in computer
languages by inherently determining the order in which
operations are to be resolved. Recently, Freiheit et al. (2004)
proposed to use Boolean operators as a way to determine
the system states (operative or not operative) in arbitrary
station configurations. This was achieved by representing serial relations among stations by a disjunctive AND
operator and parallel relations among stations by a conjunctive OR operator. As a consequence, all of the states
where production is achieved are determined when a value
of TRUE is achieved by the Boolean expression representing the system. This article further extends previous
work to achieve an intuitive representation of complex assembly system configurations using algebraic expressions.
This algebraic expression will be consequently transferred
into computational operators for evaluating complex systems performances.
For the purpose of evaluating system performance for
MMASs, increasing research has been conducted in recent years. For example, based on information entropy,
Zhu et al. (2008) proposed metrics to quantitatively assess the process complexity induced by product variants.
Subsequently, Abad and Jin (2011) further studied how to
assess the manufacturing system’s capability in handling
such complexity through the linkage with a communication system framework. In their work, a set of new metrics
was proposed, including Process Capability for Complexity (PCC), Normalized Process Capability for Complexity
232
(NPCC), and quality conforming matrix. In this article, these metrics will be directly used to evaluate manufacturing system performances. Going beyond the existing
work, this article further studies how to use the algebraic
expression to mathematically represent the system configurations in order to achieve computational simplicity in
computing those performance metrics. This consequently
permits a scalable computation capability by decomposing
a complex assembly system into hierarchical multi-levels of
subsystems. The detailed merits of the algebraic representation and examples will be shown in later Sections 3 to 5
of this article.
The rest of this article is organized as follows. In Section
2, an algebraic representation of assembly system configurations is introduced. The transferring algorithms that
are used to obtain the algebraic representation from a
traditionally used block diagram or an adjacency matrix
are provided. Section 3 discusses how to extend the algebraic expressions of system configurations by defining the
performance operators for evaluating system performance
metrics. Then, Section 4 introduces the concept of inverse
operators and illustrates how to use the inverse operators
to adjust individual station requirements to improve system performances. A case study is presented in Section 5 to
show some potential applications of the proposed algebraic
modeling method. Finally, conclusions and future work are
provided in Section 6.
2. Algebraic representation of assembly
system configurations
In this section, we describe how to effectively model the
station configurations for a general assembly system with a
hybrid configuration structure. Specifically, we propose the
use of an algebraic expression with two binary operators,
⊗ and ⊕, to represent the serial and parallel relationship
between two stations, respectively. The operands of these
binary operators are two associated stations. For example,
S1 ⊗ S2 is used to represent two stations with a serial configuration layout, and S1 ⊕ S2 is used for two stations with
a parallel configuration layout.
To enable comparison with the existing methods of block
diagram and adjacency matrix, Table 1 shows these three
equivalent ways for representing five simple assembly system configurations consisting of three stations labeled as S1 ,
S2 , and S3 . It can be seen that the proposed algebraic expression keeps the explicit representation of the serial/parallel
configuration, as in block diagrams, thus providing a better
representation than the adjacency matrix method. Furthermore, mathematical computation algorithms can be easily
added into these algebraic operators for evaluating system
performance metrics. For example, ⊗ Q and ⊕ Q will be defined later for evaluating the quality conforming rate of
the tasks performed at the given stations. Therefore, the
Abad et al.
Table 1. Three equivalent representations of system configuration
Block diagram
Adjacency
matrix H
Algebraic
expression
⎡S1
S1 0
H = S2 ⎣ 0
S3 0
S2
1
0
0
S3 ⎤
1
0⎦
0
S1 ⊗ (S2 ⊕ S3 )
⎡S1
S1 0
H = S2 ⎣ 0
S3 0
S2
0
0
0
S3 ⎤
1
1⎦
0
(S1 ⊕ S2 ) ⊗ S3
⎡S1
S1 0
H = S2 ⎣ 0
S3 0
S2
1
0
0
S3 ⎤
0
1⎦
0
S1 ⊗ S2 ⊗ S3
⎡S1
S1 0
H = S2 ⎣ 0
S3 0
S2
1
0
0
S3 ⎤
0
0⎦
0
S1 ⊕ (S2 ⊗ S3 )
⎡S1
S1 0
H = S2 ⎣ 0
S3 0
S2
0
0
0
S3 ⎤
0
0⎦
0
S1 ⊕ S2 ⊕ S3
proposed algebraic expression method also shows a better computational representation than the block diagram
method.
Because the block diagram and adjacency matrix
representations are commonly used to represent system
configurations in practice, it will be practical to develop
transferring algorithms for automatically obtaining an
equivalent algebraic expression from either a block diagram
or an adjacency matrix. The following two subsections will
discuss these transferring algorithms.
2.1. Algebraic representation transferred from system
block diagram
This subsection shows how an algebraic expression can
be directly obtained from a block diagram representation. Let us consider an example of an assembly system
with a system block diagram as shown in Fig. 2. Figure 3 shows the detailed step-by-step transferring procedures. At each step, every pair of stations is grouped
with a serial or parallel configuration by using operator ⊗
or ⊕ accordingly; this generates an equivalent station to
represent the sub-grouped stations. For example, we first
combine station S2 with S3 and generate the equivalent station S2,3 = S2 ⊗ S3 , as shown in Fig. 3(b). Next, we combine group S2,3 with station S4 and generate the equivalent
station S2,3,4 = (S2 ⊗ S3 ) ⊕ S4 , as shown in Fig. 3(c). By
233
Algebraic expression of MMAS configurations
Table 3 illustrates an example of how the algorithm is
used to transfer the adjacency matrix of the assembly system in Fig. 2 into the algebraic expression representation.
2.3. Algebraic representation of multiple tasks
within a station
Fig. 2. A modular assembly system configuration.
performing all of the steps shown in Fig. 3(a) to 3(e) the
algebraic expression of the whole system is
S1,2,3,4,5 = S1 ⊗ [(S2 ⊗ S3 ) ⊕ S4 ] ⊗ S5 .
(1)
The detailed results of the transferred algebraic expressions
at each step are summarized in Table 2. The flowchart
shown in Fig. 4 provides the standard procedures for
transferring a system block diagram into an algebraic
expression.
When multiple tasks are executed within a single station, as
shown in Fig. 6, we assume in this article that those tasks
are executed in a sequential order. Therefore, an equivalent serial configuration can be used for representing those
tasks. For example, if two tasks A and B are executed within
station k for producing different components of products,
the algebraic expression of station k is then represented as
Sk = SkA ⊗ SkB , where Ski represents task i (i = A, B) at station k. As shown in Fig. 2, station 1 is used to produce both
components 1 and 2; thus, two different tasks need to be
executed at station 1. The detailed discussion will be given
through an illustrative example in Section 3.2.
3. Algebraic representation of system performance
2.2. Algebraic representation transferred
from adjacency matrix
In an adjacency matrix H representing the configuration
of the stations in an assembly system, every column and
every row of matrix H have been labeled according to the
station for which they stand. The transferring algorithm is
proposed by iteratively combining two stations (i.e., simultaneously combining two rows and two columns of matrix
H) until all stations are combined into a single station (single entry in matrix H). At each step, we group two stations
(or equivalent stations) by the appropriate algebraic expression based on their configuration relationship (either
serial or parallel). The transferring algorithm is shown by
the pseudo-code in Fig. 5.
Fig. 3. Equivalent station groupings.
It should be noted that the previously defined algebraic operators ⊗ and ⊕ are mainly used to represent the station
layout configurations, which use stations as their operands
and imply no mathematical operations. In this section, we
will discuss how to assign specific mathematical operations
to the operators ⊗ and ⊕ used to evaluate assembly system performances. Specifically, we will show how to use the
proposed algebraic expressions to calculate quality conforming rates, which will then be used to obtain the PCC
following the procedures presented in Abad and Jin (2011).
To obtain the quality conforming rates of the different
product types, we introduce the quality performance operators, represented by adding the subscript Q into the station configuration operators ⊗ and ⊕; i.e., ⊗ Q and ⊕ Q . In
234
Abad et al.
Table 2. Obtaining algebraic expression of system configuration
Step 1
Step 2
Step 3
Step 4
Station(s)
included
Relationship
S2 and S3
S4
S1
S5
Serial
Parallel
Serial
Serial
Operator used
Algebraic expression
⊗
⊕
⊗
⊗
S2,3 = S2 ⊗ S3
S2,3,4 = S2,3 ⊕ S4 = (S2 ⊗ S3 ) ⊕ S4
S1,2,3,4 = S1 ⊗ S2,3,4 = S1 ⊗ [(S2 ⊗ S3 ) ⊕ S4 ]
S1,2,3,4 = S1 ⊗ S2,3,4 = S1 ⊗ [(S2 ⊗ S3 ) ⊕ S4 ] ⊗ S5
contrast to station configuration operators ⊗ and ⊕, quality performance operators ⊗ Q and ⊕ Q use the conforming rates of the tasks assigned to the associated stations
as their operands and convey mathematical operations for
computing quality conforming rates of the corresponding
combined tasks. It should be noted that in a mixed-model
assembly process, the quality conforming rate should be
analyzed for each product type throughout all related stations. The details of those performance metrics will be given
in the following subsections.
3.1. Representation of quality metric for a single station
The input mix ratio at each station is represented by vector
IN,k
πIN,k = [π0IN,k, . . . , π N−1
, πεIN,k]T , where πiIN,k is the input
mix ratio of part type i at station k. The element πεIN,k
corresponds to the portion of nonconforming products
produced at the immediately previous station(s); this is considered as the pseudo-input of station k for consistency of
the model representation of the whole manufacturing sys-
tem. Since π IN,k is an input vector of station k containing
the percentage
of every product type, it is constrained to
satisfy πiIN,k + πεIN,k = 1.
i
Similarly, the output of the produced part mix
ratio at station k can be represented as πOUT,k =
OUT,k
[π0OUT,k, . . . , π N−1
, πεOUT,k]T , with πiOUT,k corresponding
to the output mix ratio of part type i produced at station
k. The element πεOUT,k corresponds to the portion of nonconforming products produced at station k, which is considered as the pseudo-output of station k for consistency
of the model representation of the whole manufacturing
system. The following relationship between the input mix
ratio πIN,k and the output mix ratio πOUT,k holds:
πOUT,k = { k}T × πIN,k.
(2)
The element can be considered as a transfer function
to represent station k’s quality performance as shown in
Fig. 7.
Based on Abad and Jin (2011), if a mixed-model
production process is required to produce N types of
Fig. 4. Flowchart for transferring system block diagram into algebraic expression.
k
235
Algebraic expression of MMAS configurations
Fig. 5. Algorithm for transferring adjacency matrix into algebraic expression.
different parts, the quality transformation matrix at station
k (k = 1, . . . , M, where M is the total number of stations)
can be represented by an (N + 1) × (N + 1) square matrix
k as
⎡
k
ψ00
⎢ 0
⎢
⎢ .
k
=⎢
⎢ ..
⎢
⎣ 0
0
0
k
ψ11
..
.
0
0
···
···
..
.
0
0
..
.
k
· · · ψ N−1,N−1
···
0
⎤
k
ψ0ε
k
⎥
ψ1ε
⎥
.. ⎥
⎥
. ⎥,
⎥
k
⎦
ψ N−1,ε
1
(3)
where ψiik = Prob{Producing a conforming product type i
at station k} and ψikε = Prob{Producing a nonconforming
product type i at station k}; thus, ψiik = 1 − ψikε . Here,
k
≡ 1 stands for the fact that there is no rework or
ψ NN
correction performed on nonconforming parts entering at
station k. Also, for consistency with the matrix formulation
of the model, if station k has no production operation on
part type i, we will set ψiik = 1 and ψikε = 0, which means
no quality loss at station k for part type i. Since k is a
diagonal matrix if the last column and row are ignored, we
call station k a diagonal station.
Fig. 6. Algebraic representation of multiple tasks in a single
station.
3.2. Algebraic operators for quality metric of multiple
stations using equivalent station representation
The concept of the equivalent station is defined for iteratively calculating the quality transfer function when
products are manufactured by assembly (sub)systems
with multiple stations i 1 , i 2 , . . . , i n ; this is denoted by
E(i 1 , i 2 , . . . , i n ). By using such an equivalent station representation, the overall quality transfer function, represented
by the conforming matrix E(i 1 ,i 2 ,...,i n ) , is used to describe
the output of the conforming rate after parts pass through
multiple stations i 1 , i 2 , . . . , i n . Similarly, the input mix ratio
and the output mix ratio for the equivalent station can be
denoted as πIN,E(i 1 ,i 2 ,...,i n ) and πOUT,E(i 1 ,i 2 ,...,i n ) , respectively.
By extending the quality transfer function from a single
station to an equivalent station including multiple stations,
we can obtain the following relationship:
πOUT,E(i 1 ,i 2 ,...,i n ) = { E(i 1 ,i 2 ,...,i n ) }T × πIN,E(i 1 ,i 2 ,...,i n ) ,
E(i 1 ,i 2 ,...,i n )
(4)
is defined in a similar way as matrix k;
where i.e., the matrix of E(i 1 ,i 2 ,...,i n ) is a diagonal matrix if the last
column and row are ignored. The calculation of E(i 1 ,i 2 ,...,i n )
will be conducted step by step by iteratively calculating the
quality transfer function between two sub-grouped equivalent stations with either a serial or parallel configuration. In
the following discussions, the corresponding algebraic operators will be defined for serial and parallel configurations,
respectively, and the propositions are used for the simple
case with two stations. For the general case with more than
two stations, the algebraic operators are defined in terms of
Fig. 7. Quality transfer function.
236
Abad et al.
Table 3. Transfer of an adjacency matrix into an algebraic expression
Adjacency matrix H
Step
S1
S2
S3
S4
S5
S1
S2
S3
S4
S5
S
⎡1
0
⎢0
⎢
⎢0
⎣0
0
S2
1
0
0
0
0
S3 S4
0 1
1 0
0 0
0 0
0 0
S1 (S2 ⊗ S3 )
⎡
S1
0 1 1
(S2 ⊗ S3 ) ⎢ 0 0 0
⎣0 0 0
S4
S5
0 0 0
S4
0
1
1
0
S
⎡1
0
⎢0
⎢
⎢0
⎣0
0
S2
1
0
0
0
0
S5
⎤
0
0⎥
⎥ →
1⎥
1⎦
0
S3 S4
0 1
1 0
0 0
0 0
0 0
S5
⎤
0
0⎥
⎥
1⎥
1⎦
0
⎥
⎦
→
Step 0 (initial)
S1 (S2 ⊗ S3 )
⎡
S1
0 1 1
(S2 ⊗ S3 ) ⎢ 0 0 0
⎣0 0 0
S4
0 0 0
S5
S
⎤5
S1
((S2 ⊗ S3 ) ⊕ S4 )
S5
S4
0
1
1
0
S
⎤5
Step 1
S2,3 = S2 ⊗ S3
⎥
⎦
S4 ) S5
S1 ((S
⎡2 ⊗ S3 ) ⊕ ⎤
0 1 0
⎣0 0 1⎦
0 0 0
S1 ((S
⎡ 2 ⊗ S3 ) ⊕
⎤ S4 ) S5
0 1 0
S1
→
((S2 ⊗ S3 ) ⊕ S4 ) ⎣ 0 0 1 ⎦
0 0 0
S5
(S1 ⊗ ((S2 ⊗ S3 ) ⊕ S4 ))
S5
(S1 ⊗ ((S2 ⊗ S3 ) ⊕ S4 ))
S5
Lines: 10–17
i = S2
j = S3
Type: serial
Operator: ⊗
New row/column
label: S2 ⊗ S3
Step 2
S2,3,4 = S2,3 ⊕ S4
Lines: 2–9
i = S2,3
j = S4
k = S1
Type: parallel
Operator: ⊕
New row/column
label: (S2 ⊗ S3 ) ⊕ S4
Step 3
S1,2,3,4 = S1 ⊗ S2,3,4
Lines: 10–17
i = S1
j = S2,3,4
Type: parallel
Operator: ⊗
New row/column
label:
S1 ⊗ ((S2 ⊗ S3 ) ⊕ S4 )
(S1 ⊗ ((S2⊗ S3 ) ⊕ S4 ))S5
0 1
0 0
(S1 ⊗ ((S2 ⊗ S3 ) ⊕ S4 )) S5
0 1
→
0 0
(S1 ⊗ ((S2 ⊗ S3 ) ⊕ S4 )) ⊗ S5
Algorithm line #
Step 4 (final)
S1,2,3,4,5 = S1,2,3,4 ⊗ S5
(S1 ⊗ ((S2 ⊗ S3 ) ⊕ S4 )) ⊗ S5
[0]
corollaries, which are proved via the mathematical induction method.
3.2.1. Serial Configuration
Serial stations may be used to produce the same type of
product, or different types of products.
Proposition 1. Let i and j be two quality conforming
matrices corresponding to station i and station j, respectively.
The quality conforming matrix, denoted as E(i, j ) , can be
Lines: 10–17
i = S1,2,3,4,5
j = S5
Type: serial
Operator: ⊗
New row/column label:
(S1 ⊗ ((S2 ⊗ S3 ) ⊕ S4 ))
⊗ S5
calculated by
E(i, j )
= ⊗Q = × =
i
j
i
j
s
ψri s ψsvj
, (5)
rv
where {·}rv is the rth row and vth column element of matrix
E(i, j ) .
Justification for Proposition 1. Suppose that two stations,
denoted by i and j, are in a serial configuration and station
237
Algebraic expression of MMAS configurations
i directly precedes station j. Based on Equation (2) we have
πOUT,i = { i }T × πIN,i
(6)
and
conforming matrix, denoted as E(i, j ) , can be calculated by
E(i, j )
i
j
i
j
= ⊕Q =
ωi ψrv + ω j ψrv , (9)
s
π
OUT, j
= { } × π
j T
IN, j
.
(7)
The output mix ratio of station i, denoted by πOUT,i , is
considered as the input mix ratio of station j, denoted by
πIN, j . Combining Equation (6) and Equation (7) yields:
πOUT, j = { j }T × { i }T × πIN,i ,
πOUT, j = { i × j }T × πIN,i .
Hence, Equation (5) in Proposition 1 is justified.
We can further extend Proposition 1 to apply operator
⊗ Q to a serial configuration with more than two stations,
as given by Corollary 1.
Corollary 1. For a serial configuration among stations
1, 2, . . . , n, the quality conforming matrix, denoted as
E(1,2,...,n) , can be calculated by the matrix multiplication:
E(1,2,...,n) = 1 ⊗ Q 2 ⊗ Q . . . ⊗ Q n
= 1 × 2 × · · · × n.
(8)
The detailed justification for Equation (8) is performed via
mathematical induction, which is given in Appendix A.
3.2.2. Parallel Configuration
Parallel stations may be constructed to perform the same
task and act as redundant stations or to perform different
tasks that are used to produce different types of products or
different parts used in the same product type. Depending
on whether the performed tasks are the same, the quality
conforming matrix is calculated according to Proposition
2 and Proposition 3, respectively.
Proposition 2. Let i and j be two quality conforming
matrices corresponding to parallel connected station i and
station j. If both stations i and j are used to perform the same
tasks with production rate of ri and rj , respectively, the quality
Fig. 8. System configuration layout associated with Si ⊕ Sj .
rv
where {·}rv is the rth row and vth column element of matrix E(i, j ) . ωi and ω j are the probability of parts passing
through station i and station j, respectively, which are usually
determined to satisfy the conditions of ωi /ω j = ri /r j and
ωi + ω j = 1.
Proposition 3. Let i and j be two quality conforming
matrices corresponding to parallel connected station i and
station j, respectively. If stations i and j are used to perform
different tasks, the quality conforming matrix, denoted as
E(i, j ) , can be calculated by
E(i, j ) = i ⊕ Q j
j N+1
min ψrvi , ψrv rv , If ν =
= ,
i
j max ψrv , ψrv rv , If ν = N + 1
(10)
where {·}rv is the rth row and vth column element of matrix
E(i, j ) .
Justification for Proposition 3. Two parallel connected stations, denoted by i and j, are used to perform different tasks,
which can be further categorized by whether the outputs of
the parallel stations are the final products with individual
types (e.g., variant stations in Fig. 1) or intermediate parts
that will be used together for the following station (e.g.,
parallel stations before station 5 in Fig. 2). For the case of
producing intermediate parts, it can be further classified by
whether the outputs of the parallel stations are used for the
same type of products. Figure 8 shows these three possible
cases in the parallel configurations, which are defined as
follows.
1. Station i and station j are used to produce type l
and type l of final products, respectively, as shown in
Fig. 8(a).
2. Station i and station j are used to produce type l and
type l of the intermediate parts, respectively, which will
be used together in the next station as shown in Fig.
8(b).
238
Abad et al.
3. Station i and station j are used to produce the same type
of intermediate parts, which will be used together in the
next station as shown in Fig. 8(c).
input generated from the previous stations. Based on Equation (13), we have
The following discussion will first show that Proposition 3
is applicable to all three cases that have two parallel stations.
Afterwards, a general conclusion for n-parallel stations is
given in Corollary 2, which will be proved via mathematical
induction in Appendix B.
= πlIN,i × ψlεi + πl × ψl ε + πεIN,E(i, j ) × 1
IN, j
= πlIN,i × max ψlεi , 0 + πl j
× max 0, ψl ε + πεIN,E(i, j ) × 1
⎡ IN,E(i,j) ⎤
⎡
j ⎤T
πl
max ψlεi , ψlε
⎢ IN,E(i,j) ⎥
i
⎢
j ⎥
⎥ . (16)
= ⎣ max ψl ε , ψl ε ⎦ × ⎢
⎣ πl ⎦
IN,E(i,j)
max(1, 1)
πε
Case 1. Two types of products are produced: suppose that
the outputs of stations i and j correspond to final products
with type l and type l , respectively. The quality conforming
matrix at station i is
⎤ ⎡ i
⎡ i
⎤
ψll
0
ψlεi
ψll 0 ψlεi
⎥ ⎢
⎢
⎥
i = ⎣ 0 ψli l ψli ε ⎦ = ⎣ 0 1 0 ⎦ . (11)
0
0
1
0 0 1
The quality conforming matrix at station j is
⎡
⎤
⎡ j
j ⎤
1
0
0
ψll
0
ψlε
⎥ ⎢
⎥
⎢
j = ⎣ 0 ψ j ψ j ⎦ = ⎣ 0 ψlj l ψlj ε ⎦ .
ll
l ε
0
0
1
0
0
1
(12)
The input mix ratio is given as
T
π IN,E(i, j ) = π IN,i = π IN, j = πlIN,i , πlIN,i
, πεIN,i
IN, j IN, j
T
= πl , πl , πεIN, j .
(13)
Since the proportion of conforming products of type l coming out of equivalent station E(i,j) is exclusively determined
by the conforming matrix and input proportion at station
i, we have
OUT,E(i, j )
πl
= πlOUT,i = πlIN,i × ψlli
j
= πlIN,i × min ψlli , 1 = πlIN,i × min ψlli , ψll
⎤T ⎡ IN,E(i, j ) ⎤
⎡
j
πl
min ψlli , ψll
⎥
⎢ IN,E(i, j ) ⎥
⎢
⎥ × ⎢π ⎥.
(14)
=⎢
⎦
⎣ l
⎦
⎣ min (0, 0)
IN,E(i,
j
)
min(0, 0)
πε
Based on the similar principle, the output of product type
l can be obtained by Equation (15):
OUT,E(i,j)
πl OUT, j
= πl IN, j
= πl j
IN, j
× ψl l = πl j IN, j
j × min 1, ψl l = πl × min ψli l , ψl l ⎡
⎤T ⎡ IN,E(i,j) ⎤
min (0, 0)
πl
⎥
⎢ IN,E(i,j) ⎥
⎢
j
⎢
⎥ . (15)
i
= ⎣ min ψ , ψ ⎦ × ⎣ π ⎦
l
ll
ll
min(0, 0)
IN,E(i,j)
πε
To calculate the proportion of nonconforming products
coming out of equivalent station E(i, j ), it is necessary to
consider the nonconforming parts coming from stations
i and j and the transferred nonconforming parts from the
πεOUT,E(i,j) = πεOUT,i + πεOUT, j + πεIN,E(i, j )
IN, j
j
Combining Equations (14), (15), and (16), we have
πOUT, E(i,j)
⎡ OUT,E(i,j) ⎤
πl
⎢ OUT,E(i,j) ⎥
⎥
=⎢
⎦
⎣ πl OUT,E(i,j)
πε
⎤T
⎡
j
j
max ψlεi , ψlε
min ψlli , ψll min (0, 0)
⎢
⎥
j min (0, 0)
min ψli l , ψl l max ψ i , ψ j ⎥
=⎢
⎣
l ε ⎦
l ε
min (0, 0)
min (0, 0)
max (1, 1)
⎡ IN,E(i,j) ⎤
πl
⎢ IN,E(i,j) ⎥
⎥
×⎢
⎦
⎣ πl IN,E(i,j)
πε
⎡
⎤T
j
j
min ψlli , ψll min (0, 0)
max ψlεi , ψlε
⎢
⎥
j min (0, 0)
=⎢
min ψli l , ψl l max ψ i , ψ j ⎥
⎣
l ε
l ε ⎦
min (0, 0)
min (0, 0)
max (1, 1)
× π IN,E(i,j) = E(i,j) × π IN,E(i,j) .
(17)
Hence, Equation (10) in Proposition 3 is justified under
case 1.
Case 2. Two types of products are produced: suppose station i is used to produce product type l and station j is
used to produce product type l . The quality conforming
matrices at station i and station j are the same as i and
j given by Equation (11) and Equation (12), respectively.
OUT,E(i,j)
Based on similar considerations as in case 1, πl
,
OUT,E(i,j)
OUT,E(i,j)
πl , and πε
can be derived in expressions
similar to Equation (14), Equation (15), and Equation (16),
respectively. Hence, case 2 in Proposition 3 can be justified
in the same manner as for case 1.
Case 3. Both stations i and j are used to produce different
components A and B that are combined in the following
OUT, j
station to produce product type l. Let πlOUT,i and πl
denote the proportion of component A and component
B produced at station i and station j, respectively. Since
both components are needed to produce a product, the
239
Algebraic expression of MMAS configurations
Fig. 9. Example of parallel stations and algebraic operators for
quality metric.
effective output of the conforming rate from the equivalent station (combining stations i and j ) is equal to the
minimum conforming rate of component A and component B produced at station i and station j, respectively; i.e.,
OUT, j
).
min(πlOUT,i , πl
For example, Fig. 9 shows a simple process for producing
a table, which consists of three stations: station 1 produces
the top of the table, station 2 produces a set of four legs, and
station 3 assembles the top of the table and the four legs.
Suppose that the proportion of conforming rates produced
at stations 1 and 2 is ψ 1 = 95% and ψ 2 = 98%, respectively. In this case, we can only obtain a 95% conforming
rate when entering station 3; i.e., the effective output of
combining stations 1 and 2 is equal to the combined conforming rate between the top and the set of four legs, thus
yielding min(ψ 1 , ψ 2 ) = 95%,
Based on Equation (2), we see that Equation (18) holds.
OUT,E(i,j)
OUT, j = min πlOUT,i , πl
πl
IN,E(i,j)
j
.
(18)
= min ψlli , ψll × πl
Based on a similar argument, we can calculate the proportion of nonconforming products of type l produced at
OUT, j
station i and station j, denoted by πεOUT,i and πε
,
respectively. Since every product of type l must be processed by both station i and station j, the resultant nonconforming rate of product type l produced by these two
stations is equal to the maximum of the nonconforming rates of products of type l at these two stations; i.e.,
OUT, j
max(πεOUT,i , πε
). Based on Equation (2), it can be written that:
πεOUT,E(i,j) = max πεOUT,i , πεOUT, j
j
(19)
= max ψlεi , ψlε · πεIN,E(i,j) .
Based on Equations (18) and (19), Equation (10) in Proposition 3 is justified.
It should be noted that Proposition 3 for cases 2 and
3 holds when the parallel configuration is followed immediately by an assembly station that adopts the “check &
assemble” procedure; i.e., at such an assembly station (e.g.,
station 3 in Fig. 9), the quality of intermediate parts to
be assembled is checked before the assembly task is performed. If a defective component is found at station 3, the
station waits for the next arriving good component as its
replacement to complete the assembly operation. Such a
check & assemble procedure removes the conflict or defective components before the assembly operation starts,
which is sometimes essential in order to reduce the high
risk of damaging expensive tools and fixtures and/or to
avoid the destruction of defect-free intermediate parts. In
the contrary case, if the parallel configuration is immediately followed by an assembly station that is not equipped
with this check & assemble procedure, a defect-free intermediate part will be destroyed if it is about to be assembled
with a defective component. The quality conforming matrix calculation of such a structure would be the same as in
Equation (5) presented for the serial configuration. Without losing generality of the algebraic representation, in this
article we consider the parallel configuration described in
cases 2 and 3 is followed immediately by an assembly station which adopts the check & assemble procedure, and
thus Proposition 3 holds.
We can further extend Proposition 3 to apply operator
⊕ Q to a parallel configuration with more than two stations,
as given by Corollary 2.
Corollary 2. For a parallel configuration among stations 1, 2,
. . ., n, the quality conforming matrix, denoted as E(1,2,...,n) ,
can be calculated by
E(1,2,...,n) = 1 ⊕ Q 2 ⊕ Q . . . ⊕ Q n
min ψrv1 , ψrv2 , . . . , ψrvn rv
= max ψrv1 , ψrv2 , . . . , ψrvn rv
If ν =
N+1
,
If ν = N + 1
(20)
where {·}rv is the rth row and vth column element of matrix
E(1,2,...,n) .
The justification of Equation (20) is given in Appendix
B via a mathematical induction proof.
Illustrative example: An assembly system, such as the one
shown in Fig. 2, is used to produce a product consisting of two components, each with two different variants,
for a total of four possible product types (four possible
combinations of two components). The last column in
Table 4 gives the input mix ratio of these four product
IN,E(·)
(i = 1, 2, 3, 4), where E(·) corresponds to
types, πi
an equivalent station containing every station in the system. For production planning, the input mix ratio of all
products having component 1 with variant 1 and variIN,E(·)
IN,E(·)
ant 2 can be calculated by P11 = π1
+ π2
and
IN,E(·)
IN,E(·)
+ π4
, respectively. Similarly, the input
P12 = π3
240
Abad et al.
Table 4. Task assignment, conforming rates, and input mix ratio
7. Conforming rates (%)
Component
Station 1
Station 2
Station 3
Station 4
Station 5
Component 1
processed
Variant 1
Variant 2
Variant 1
Variant 2
1 and 2
1
1
2
1 and 2
92
94
96
—
93
93
94
98
—
94
94
—
—
96
93
90
—
—
92
90
√
√
Product type 1
Product type 2
Product type 3
Product type 4
Input mix ratio
(in terms of component variant)
1,1
1,2
5,1
0.9200
⎢ 0
⎢
=⎢ 0
⎣ 0
0
⎡
0.9400
⎢ 0
⎢
=⎢ 0
⎣ 0
0
⎡
0.9300
⎢ 0
⎢
=⎢ 0
⎣ 0
0
0
0
0.9200
0
0
0.9300
0
0
0
0
0
0
0.9000
0
0
0.9400
0
0
0
0
0
0
0.9300
0
0
0.9400
0
0
0
0
√
√
√
√
√
√
P11 = 70% P12 = 30% P21 = 65% P22 = 35%
mix ratio of all products having component 2 with variants
IN,E(·)
IN,E(·)
+ π3
and
1 and 2 can be calculated by P21 = π1
IN,E(·)
IN,E(·)
P22 = π2
+ π4
, respectively. The results are given
in the last row of Table 4.
Table 4 also describes the required operations at each
station and their corresponding quality conforming rates.
As shown in Fig. 2, stations 1 and 5 are basic operation
stations, whereas stations 2, 3, and 4 are variant operation
stations.
Since station 1 and station 5 have more than one task
(two tasks are need to produce two different components),
based on Fig. 6, we can denote Ski as the individual task for
producing component i at station k. In this way, stations
1 and 5 are represented in terms of their assigned tasks as
S1 = S11 ⊗ S12 and S2 = S51 ⊗ S52 , respectively. Furthermore,
we can define k,i to represent the quality conforming matrix of individual task i at station k. Based on Table 4,
matrices 1,1 , 1,2 , 5,1 , and 5,2 are represented as
⎡
Component 2
0
0
0
0.9300
0
0
0
0
0.9000
0
0
0
0
0.9400
0
⎤
0.0800
0.0800 ⎥
⎥
0.0700 ⎥ ,
0.0700 ⎦
1
⎤
0.0600
0.1000 ⎥
⎥
0.0600 ⎥ ,
0.1000 ⎦
1
⎤
0.0700
0.0700 ⎥
⎥
0.0600 ⎥ ,
0.0600 ⎦
1
⎡
5,2
0.9300
⎢ 0
⎢
=⎢ 0
⎣ 0
0
Input mix ratio
(in terms of product type)
IN,E(·)
π1
= 45%
IN,E(·)
π2
= 25%
IN,E(·)
= 20%
π3
IN,E(·)
π4
= 10%
0
0
0.9000
0
0
0.9300
0
0
0
0
0
0
0
0.9000
0
⎤
0.0700
0.1000 ⎥
⎥
0.0700 ⎥ .
0.1000 ⎦
1
The derivation of 1,1 is given in detailed steps in Appendix C. The conforming matrix of stations 1 and
5 can be obtained by 1 = 1,1 ⊗ Q 1,2 and 5 =
1,1
1
5,1 ⊗ Q 5,2 , respectively. For example, ψ11
= ψ11
×
1,2
ψ11 = 0.92 × 0.94 = 0.8648. Therefore,
⎡
0.8648
0
0.8280
⎢ 0
⎢
0
1 = ⎢ 0
⎣ 0
0
0
0
⎡
0.8649
0
0.8370
⎢ 0
⎢
0
5 = ⎢ 0
⎣ 0
0
0
0
0
0
0.8742
0
0
0
0
0.8742
0
0
0
0
0
0.8370
0
0
0
0
0.8460
0
⎤
0.1352
0.1720 ⎥
⎥
0.1258 ⎥ ,
0.1630 ⎦
1
⎤
0.1351
0.1630 ⎥
⎥
0.1258 ⎥ .
0.1540 ⎦
1
For stations 2, 3, and 4 with a single task, the corresponding conforming matrices 2 , 3 , and 4 are directly
obtained from Table 4 as
⎡
0.9400
0
0.9400
⎢ 0
⎢
0
2 = ⎢ 0
⎣ 0
0
0
0
0
0
0.9400
0
0
0
0
0
0.9400
0
⎤
0.0600
0.0600 ⎥
⎥
0.0600 ⎥ ,
0.0600 ⎦
1
241
Algebraic expression of MMAS configurations
⎡
0.9600
⎢ 0
⎢
3 = ⎢ 0
⎣ 0
0
⎡
0.9600
⎢ 0
⎢
4 = ⎢ 0
⎣ 0
0
0
0.9600
0
0
0
0
0.9200
0
0
0
0
0
0
0
0.9800
0
0
0.9800
0
0
0
0
0
0
0.9600
0
0
0.9200
0
0
⎤
0.0400
0.0400 ⎥
⎥
0.0200 ⎥
0.0200 ⎦
1
⎤
0.0400
0.0800 ⎥
⎥
0.0400 ⎥ .
0.0800 ⎦
1
The system’s quality transfer function, based
on
the
equivalent
station
E(1,2,3,4,5) = 1 ⊗ Q
2
3
4
5
[( ⊗ Q ) ⊕ Q ] ⊗ Q , can be calculated by iteratively applying the operators defined in Equation (5)
and Equation (10) on two subgrouped stations as follows:
i.
ii.
iii.
iv.
E(2,3) = 2 ⊗ Q 3 (Equation (5));
E(2,3,4) = 4 ⊕ Q E(2,3) (Equation (10));
E(1,2,3,4) = 1 ⊗ Q E(2,3,4) (Equation (5));
E(1,2,3,4,5) = E(1,2,3,4) ⊗ Q 5 (Equation (5)).
The final resultant quality conforming matrix is
E(1,2,3,4,5)
⎡
0.6750
⎢ 0
⎢
=⎢ 0
⎣ 0
0
0
0.6254
0
0
0
0
0
0
0
0.7040
0
0
0.6515
0
0
⎤
0.3250
0.3746 ⎥
⎥
0.2960 ⎥ . (21)
0.3485 ⎦
1
E(i ,i ,...,i )
j,l
IN,OUT,E(i 1 ,i 2 ,...,i n )
× log
j <N
(22)
Therefore, the quality conforming rate of the whole manufacturing system can be calculated by considering all stations S1 , S2 ,. . .,SM, as Q E(1,2,...,M) .
The PCC, defined in Abad and Jin (2011), is a performance metric that assesses how well a production process
can handle the demand variety of products in a mixed
model manufacturing process. PCC is calculated based on
the mutual information index (Cover and Thomas, 2006),
which is used to quantify the amount of information that
two random variables share.
Assume that the input and output mix ratios are considered as the marginal probability distribution functions
πIN,E(i 1 ,i 2 ,...,i n ) and πOUT,E(i 1 ,i 2 ,...,i n ) of two categorical random variables, respectively. If the joint probability matrix
is denoted by πIN,OUT,E(i 1 ,i 2 ,...,i n ) , where the element in the
IN,OUT,E(i 1 ,i 2 ,...,i n )
ith row and jth column corresponds to πi j
=
IN,E(i 1 ,i 2 ,...,i n )
πj
OUT,E(i 1 ,i 2 ,...,i n )
πl
NPCC = PCC/H (D) ,
(23)
(24)
where H (D) is the entropy of the input demand random
variable D, given by
IN,E(i ,i ,...,i )
IN,E(i 1 ,i 2 ,...,i n )
1 2
n
πi
log πi
.
(25)
H (D) = −
i
Illustrative example: We now continue the example in Section 3.2 to show how to calculate the system performance
metrics Q E(·) , PCC, and NPCC for the assembly system
shown in Fig. 2. Based on Table 4, the vector of input mix
ratio is represented as
T
(26)
π IN,E(1,2,3,4,5) = 0.45 0.25 0.20 0.10 .
By substituting Equation (21) and Equation (26)
into Equation (22), Q E(1,2,3,4,5) is calculated as
IN,OUT,E(i 1 ,i 2 ,...,i n )
0.6661. Based on the equation πi j
=
E(i 1 ,i 2 ,...,i n )
Based on Equation (4), the total quality conforming rate
of all product types for the equivalent station E(i 1 ,i 2 ,...,i n ) ,
denoted as Q E(i 1 ,i 2 ,...,i n ) , can be calculated by
E(i 1 ,i 2 ,...,i n )
π IN
= 1 − πεOUT,E(i 1 ,i 2 ,...,i n ) .
Q E(i 1 ,i 2 ,...,i n ) =
j ψjj
π j,l
A normalized value of PCC, ranging from zero to one,
called the NPCC, was also proposed in Abad and Jin (2011)
based on the concept of coefficient of constraint (Coombs
et al., 1970); that is,
ψi j
3.3. Quality and PCC for whole assembly system
IN,E(i ,i ,...,i )
1 2
n
ψi j 1 2 n πi
. Thus, PCC can be calculated by
the mutual information index as follows
IN,OUT,E(i ,i ,...,i )
1 2
n
PCC =
π j,l
IN,E(i 1 ,i 2 ,...,i n )
πi
, πIN,OUT,E(1,2,3,4,5) is obtained as
πIN,OUT,E(1,2,3,4,5)
⎡
0.3038
0
0
0
0.1563
0
0
⎢ 0
⎢
0
0.1408
0
=⎢ 0
⎣ 0
0
0
0.0652
0
0
0
0
⎤
0.1462
0.0937 ⎥
⎥
0.0592 ⎥ . (27)
0.0349 ⎦
0
By substituting Equations (21), (26), and (27) into Equation
(23), PCC is obtained as 1.2076. Furthermore, we obtain
the results of H (D) = 1.8150 based on Equation (25) and
NPCC = 0.6653 based on Equation (24). Because NPCC
is far below one, it indicates that this assembly system has a
fair capability of handling the mixed product varieties, but
it needs to be further improved. The following section will
discuss how to adjust the individual station requirements to
improve system performances by defining inverse algebraic
operators.
4. Inverse algebraic operators for improving system
performance
This section is used to show how to define the inverse
algebraic expressions to systematically analyze the effect
of individual stations on the defined performance metric
242
Abad et al.
of an equivalent system station. Such results can be used
to further improve the design of a manufacturing system
to achieve a desired system performance or to identify the
weakest station in an assembly system under a particular
performance criterion, such as quality, throughput, etc.
Based on the previously defined algebraic operators ⊗ Q
−1
and ⊕ Q , the inverse operators ⊗−1
Q and ⊕ Q will be defined
as follows.
−1
4.1. Algebraic operators ⊗−1
Q and ⊕ Q for inverse
computation of quality conforming rate
The following proposition will be used to describe the op−1
erations corresponding to ⊗−1
Q and ⊕ Q .
Proposition 4. Let i and j be two quality conforming
matrices corresponding to station i and station j, respectively,
and let E(i, j ) be the equivalent station formed by stations i
and j. The relationships between each quality operator and
its corresponding inverse operator are defined as follows.
1. For a serial configuration, E(i, j ) = i ⊗ Q j , it yields:
−1
Q j ≡ E(i, j ) ⊗ Q { j }−1
i = E(i, j ) ⊗
(28)
−1
← −1
j = i ⊗ Q E(i, j ) ≡ i
⊗ Q E(i, j ) ,
(29)
and
−1
Q in Equation
where the arrow → on the top of operator ⊗
j
(28) indicates that the operand on the right-hand side
−1
of the operator ⊗
Q should be computed by the inverse op← −1
erator. Similar interpretation is given to ⊗ Q in Equation
(29).
2. For a parallel configuration, E(i, j ) = i ⊕ Q j , it
yields:
j
i = E(i, j ) ⊕−1
Q ⎧ E(i, j ) ⎪
ψrv
⎪
⎪
rv
⎪
⎪
⎪
⎪ ψ i ≥ ψ E(i, j )
⎪
⎪
rv
rv
⎪
rv
⎨
E(i, j )
≡
ψrv
⎪
⎪
rv
⎪
⎪
⎪
i ≥ 1 − ψ E(i, j )
⎪
ψ
⎪
rv
⎪ rv
⎪
rv
⎪
⎩
unfeasible
E(i, j )
< ψrv AND v = N + 1
E(i, j )
= ψrv AND v = N + 1
if ψrv
if ψrv
E(i, j )
if ψrv
j
> ψrv AND v = N + 1 .
j
j
= ψrv AND v = N + 1
otherwise
(30)
Justification for Proposition 4. Since operator ⊗ Q corresponds to the conventional matrix multiplication operation, ⊗−1
Q should naturally correspond to the inverse matrix
operation. Algebraic operators ⊗−1
Q and ⊗ Q are not commutative, which is shown in Equations (28) and (29). For
E(i,j)
j
j
= ψrvi ⊕ Q ψrv = min(ψrvi , ψrv )
operator ⊕−1
Q , since ψrv
E(i,j)
j
NEW
E(1,2,3,4,5),
⎡
0.7000
0
0
0.6254
0
⎢ 0
⎢
0
0
0.7040
=⎢
⎣ 0
0
0
0
0
0
j
for v = N + 1 and ψrv
= ψrvi ⊕ Q ψrv = max(ψrvi , ψrv ) for
v = N + 1; thus Equation (30) is justified.
0
0
0
0.6515
0
⎤
0.3000
0.3746 ⎥
⎥
0.2960 ⎥ .
0.3485 ⎦
1
It is assumed that it is feasible to improve only station
3 to meet the new system requirement of E(1,2,3,4,5),NEW .
The question that remains is what must be the new quality
requirement at station 3 in order to achieve E(1,2,3,4,5),NEW .
The step-by-step inverse operations are illustrated as
follows:
(i)
(ii)
(iii)
(iv)
(v)
SE(1,2,3,4,5) = S1 ⊗ [(S2 ⊗ S3 ) ⊕ S4 ] ⊗ S5 ,
S1 ⊗−1 SE(1,2,3,4,5) = [(S2 ⊗ S3 ) ⊕ S4 ] ⊗ S5 ,
(S1 ⊗−1 SE(1,2,3,4,5) ) ⊗−1 S5 = (S2 ⊗ S3 ) ⊕ S4 ,
[(S1 ⊗−1 SE(1,2,3,4,5) ) ⊗−1 S5 ] ⊕−1 S4 = S2 ⊗ S3 ,
S2 ⊗−1 [[(S1 ⊗−1 SE(1,2,3,4,5) ) ⊗−1 S5 ] ⊕−1 S4 ] = S3 .
(31)
Now, replacing Si for i and using the corresponding algebraic operators yields:
3,NEW
← −1
= 2⊗Q
j
E(i, j )
I f ψrv
Illustrative example: We continue the example of the assembly system shown in Fig. 2, in which the algebraic expression representation of the equivalent station is SE(1,2,3,4,5) =
S1 ⊗ [(S2 ⊗ S3 ) ⊕ S4 ] ⊗ S5 . Suppose that the plant is not
satisfied with the current quality conforming performance
as given in Equation (21) and would like to increase the
quality conforming rate of product type 1, since product
type 1 has the largest demand. For example, the decision is
E(1,2,3,4,5)
made to increase the quality rate from ψ11
= 0.675
E(1,2,3,4,5)
= 0.7. Therefore, the new quality conforming
to ψ11
rate is represented by E(1,2,3,4,5),NEW as
1 ← −1 E(1,2,3,4,5),NEW −1 5 −1 4 Q ⊕Q .
⊗Q ⊗
(32)
−1
After substituting operators ⊗−1
Q and ⊕ Q into Equations
3,NEW
can be obtained as
(28), (29), and (30), 3,NEW
⎡
0.9956
0
0.9600
⎢ 0
⎢
0
=⎢ 0
⎣ 0
0
0
0
0
0
0.9800
0
0
0
0
0
0.9787
0
⎤
0.0044
0.0400 ⎥
⎥
0.0200 ⎥ .
0.0213 ⎦
1
By comparing 3,NEW with 3 , we can see that in order to
E(1,2,3,4,5)
=
achieve the new quality conforming rate of ψ11
0.7, the quality conforming rate of product type 1 at station
3,NEW
3
= 0.96 to ψ11
= 0.9956.
3 should be improved from ψ11
243
Algebraic expression of MMAS configurations
Table 5. Conforming rate for individual stations
Conforming rates (%)
Station
Fig. 10. System block diagram for producing three product types.
5. Case study
In this section, we consider a complex assembly system
consisting of 20 stations for producing three product types
(Ko and Hu, 2008). The block diagram of the system configuration is shown in Fig. 10. In arbitrary order, stations
are numbered from S1 to S20 (M = 20).
In order to efficiently obtain the conforming rate of
each product type and their corresponding production cycle time, we further represent the assembly system by two
sets: the assembly sub-system consisting of the basic operation stations denoted by SB and the assembly sub-system
consisting of the variant operation stations denoted by SV .
In this example, assembly sub-system SB contains stations
S1 , S2 , S3 , S4 , S5 , S6 , S7 , S8 , S9 , S10 , and S11 , while assembly sub-system SV contains stations S12 , S13 , S14 , S15 ,
S16 , S17 , S18 , S19 , and S20 . Figure 10 shows the SB configuration, while different SV configurations for each product
type are shown in Fig. 11.
By means of the proposed algebraic representation of the
system configuration, we can further represent equivalent
sub-systems SB and SV as
SB = S1 ⊗ (S2 ⊗ S3 ) ⊗ [(S4 ⊗ S5 ) ⊕ [(S7 ⊗ (S6 ⊕ S8 ))
(33)
⊕ (S9 ⊗ S10 ⊗ S11 )]],
SV = [(S12 ⊕ S14 ) ⊗ (S13 ⊕ S15 )] ⊕ [S16 ⊗ S17 ]
(34)
⊕ [S18 ⊗ S19 ⊗ S20 ].
Fig. 11. Block diagram for individual product types.
Station 1
Station 2
Station 3
Station 4
Station 5
Station 6
Station 7
Station 8
Station 9
Station 10
Station 11
Station 12
Station 13
Station 14
Station 15
Station 16
Station 17
Station 18
Station 19
Station 20
Product type
processed
Product
type 1
Product
type 2
Product
type 3
1, 2, and 3
1, 2, and 3
1, 2, and 3
1, 2, and 3
1, 2, and 3
1, 2, and 3
1, 2, and 3
1, 2, and 3
1, 2, and 3
1, 2, and 3
—
1 and 2
1
1 and 2
2
1 and 2
1
3
3
3
99
99
97
98
99
98
98
98
99
99
—
97
98
98
—
99
99
—
—
—
98
98
99
98
98
99
99
97
97
99
—
99
—
99
99
99
—
—
—
—
99
97
98
98
99
98
99
96
98
99
—
—
—
—
—
—
—
96
97
98
The equivalent station of the whole assembly system, denoted by SB,V , is given by
SB,V = SB ⊗ SV .
The production throughput is designed to be five units
per minute. Specifically, two units per minute are allocated
for product type 1 and product type 3 and one unit per
minute for product type 2. Thus, the input mix ratio is
πIN,E(·) = π1IN,E(·) = 0.40 π2IN,E(·) = 0.20
T
IN,E(·)
= 0.40 πεIN,E(·) = 0 .
π3
Based on Fig. 11, Table 5 provides the task assignments
to each station for each product type and their corresponding quality conforming rates and production cycle times. It
can be seen in Table 5 that station 11 is regarded as a redundant station and no production task is assigned to it. Also,
244
Abad et al.
Fig. 12. Layer-by-layer system decomposition.
we assume that the station’s cycle times are independent of
the product type being processed.
5.1. Analysis of quality for whole assembly system
We first compute the equivalent matrix E(B) corresponding to the basic operations subsystem SB by replacing ⊗
and ⊕ in Equation (33) with ⊗ Q and ⊕ Q , respectively, thus
yielding
⎡
⎤
0.9130
0
0
0.0870
0.9039
0
0.0961 ⎥
⎢ 0
E(B) = ⎣
.
0
0
0.8944 0.1056 ⎦
0
0
0
1
Similarly, based on Equation (34), E(V) corresponding to
the variant operations subsystem SV is calculated as
⎡
⎤
0.9506
0
0
0.0494
0.9801
0
0.0199 ⎥
⎢ 0
E(V) = ⎣
.
0
0
0.9126 0.0874 ⎦
0
0
0
1
Finally, based on SB,V = SB ⊗ SV , we can calculate
= E(B) ⊗ Q E(V) , corresponding to the whole assembly system, as
⎡
⎤
0.8679
0
0
0.1321
0.8859
0
0.1141 ⎥
⎢ 0
E(B,V) = ⎣
.
0
0
0.8162 0.1838 ⎦
0
0
0
1
E(B,V)
Therefore, based on the proposed methodology, the final
conforming rates of three product types can be obtained
E(B,V)
E(B,V)
E(B,V)
as ψ11
= 86.79%, ψ22
= 88.59%, and ψ33
=
81.62%.
5.2. Identification of weak stations via layer-by-layer tree
decomposition
We now illustrate the use of the expression tree representation of the assembly system SB,V to determine weak sub-
assembly systems with a low process quality Q. Consider
Fig. 12(a), where the system SB,V has been expressed at
its highest level of grouping as SB,V = SB ⊗ SV . Since we
are interested in determining sub-systems that have a low
Q performance, we further decompose, layer by layer, the
equivalent station SB , which is the equivalent station with
the lowest process quality, given by Q = 0.9038. As shown
in Fig. 12(b), the lowest process quality Q = 0.9545 within
SB corresponds to S4−11 . Continuing with a step-by-step
decomposition, we further expand equivalent station S4−11
into S4,5 , S6,7,8 , and S9,10,11 , as shown in Fig. 12(c). Based
on this decomposition, we have that sub-system S6,7,8 has
the lowest performance at this level, given by Q = 0.9564,
and thus further attention should be given to this subsystem to improve the overall process quality of the system.
One should note that we have followed a greedy search approach and hence we cannot claim that sub-system S6,7,8
is a global minimum; nonetheless, the greedy heuristic
search yields a locally optimal solution that may closely
approximate a global optimum under a reasonable short
computation time. If sufficient computational source is
available, one may also follow dynamic programming or
other optimization algorithms to find the guaranteed global
optimum or better approximation solution (Cormen et al.,
2001).
6. Conclusions and future work
This article introduces an efficient way to represent complex mixed-model assembly system configurations using
algebraic expressions. It is applicable to a general hybrid
asymmetric configuration system consisting of mixed serial and parallel stations. Moreover, the article presents
a systematic method to transfer the algebraic expression
into computational algebraic operators, which are assigned
with specific mathematical operations to calculate the system performance metrics, such as quality conforming rate
and process capability in handling the demand variety.
245
Algebraic expression of MMAS configurations
Furthermore, the corresponding inverse algebraic operators are also defined, which provides a systematic way to
guide the adjustment of individual stations for improving
system performances.
The proposed method is restricted to the serial and parallel structure systems. For other complex structure systems,
such as a bridge structure system, one possible approach is
to first decompose or transform the system into serial and
parallel structures by adding pseudo-redundant stations in
the system diagrams, which is considered as future work to
be further studied.
The research presented in this article can be further extended by defining algebraic operators for obtaining other
system performance metrics such as process cycle times,
operational states of a system, reliability, etc. Furthermore,
the algebraic expressions may be mathematically formalized into interesting algebraic fields and thus provide a
way to solve problems such as finding bottleneck stations
and achieving line balancing for a complicated production
system.
ference on Analysis and Optimization of Systems, Springer, Berlin,
pp. 957–966.
Gondran, M. and Minoux, M. (1983) Graphs and Algorithms, Wiley,
New York, NY.
Ko, J. and Hu, J. (2008) Balancing of manufacturing systems with complex configurations for delayed product differentiation. International
Journal of Production Research, 46, 4285–4308.
Sturgeon, T.J. (2002) Modular production networks: a new American
model of industrial organization. Industrial and Corporate Change,
11, 451–496.
Webbink, R.F. and Hu, S.J. (2005) Automated generation of assembly
system-design solutions. IEEE Transactions on Automation Science
and Engineering, 2, 32–39.
Xu, X. and Xu, P. (1988) Working on production planning and control
in flexible manufacturing system with path-algebra, in Proceedings
of the 1988 IEEE International Conference on Systems, Man, and
Cybernetics, IEEE Press, Piscataway, NJ, pp. 1192–1195.
Yurdakul, M. and Odrey, N.G. (2004) Development of a new dioid algebraic model for manufacturing with the scheduling decision making
capability. Robotics and Autonomous Systems, 49, 207–218.
Zhu, X., Hu, S.J., Koren, Y. and Marin, S.P. (2008) Modeling of manufacturing complexity in mixed-model assembly lines. Journal of
Manufacturing Science and Engineering, 130(5), 051013–051013–10.
Zissos, D. and Duncan, F.G. (1971) NOR and NAND operators in
Boolean algebra applied to switching circuit design. The Computer
Journal, 14, 413–417.
Acknowledgement
The authors would like to gratefully acknowledge the financial support of NSF-CMMI: 0825438 and General Motors.
References
Abad, A.G. and Jin, J. (2011) Complexity metrics for mixed model manufacturing systems based on information theory. International Journal
of Information and Decision Sciences, 3(4), 313–334.
Carre, B.A. (1971) An algebra for network routing problems. IMA Journal of Applied Mathematics, 7, 273–294.
Cofer, D.D. and Garg, V.K. (1993) Generalized max-algebra model for
performance analysis of timed and untimed discrete event systems,
in American Control Conference, IEEE, pp. 2288–2292.
Cohen, G., Dubois, D., Quadrat, J.P. and Viot, M. (1985) Linear-systemtheoretic view of discrete-event processes and its use for performance
evaluation in manufacturing. IEEE Transactions on Automatic Control, 30, 210–220.
Cohen, G., Moller, P., Quadrat, J.P. and Viot, M. (1989) Algebraic tools
for the performance evaluation of discrete event systems. IEEE Proceedings, pp. 39–58.
Coombs, C.H., Dawes, R.M. and Tversky, A. (1970) Mathematical
Psychology: An Elementary Introduction, Prentice-Hall, Englewood
Cliffs, NJ.
Cormen, T.H., Leiserson, C.E., Rivest, R.L. and Stein, C. (2001) Introduction to Algorithms, MIT Press, Cambridge, MA.
Cover, T.M. and Thomas, J.A. (2006) Elements of Information Theory,
Wiley, New York, NY.
Da Silveira, G., Borenstein, D. and Fogliatto, F.S. (2001) Mass customization: literature review and research directions. International Journal
of Production Economics, 72, 1–13.
Duncan, F.G., Zissos, D. and Walls, M. (1975) A postfix notation for
logic circuits. The Computer Journal, 18, 63–69.
Freiheit, T., Shpitalni, M. and Hu, S.J. (2004) Productivity of paced
parallel-serial manufacturing lines with and without crossover. Journal of Manufacturing Science and Engineering, 126, 361–367.
Gaubert, S. (1990) An algebraic method for optimizing resources in
timed event graphs, in Proceedings of the Ninth International Con-
Appendices
Appendix A: Justification for Corollary 1
Suppose stations 1, 2, . . . , n are in a serial configuration and that station i directly precedes station i + 1,
i = 1, 2, . . . , n − 1. Based on Equation (2) we have
πOUT,i = { i }T × πIN,i
(A1)
πOUT,i +1 = { j }T × πIN,i +1 .
(A2)
and
The output mix ratio of station i, denoted by πOUT,i , is
considered as the input mix ratio of station i + 1, denoted
by πIN,i +1 . Combining Equation (A1) and Equation (A2),
we have
πOUT,n = { n }T × { n−1 }T × πIN,n−1
= { n }T × { n−1 }T × { n−2 }T × πIN,n−2
= · · · = { n }T × { n−1 }T × · · · × { 1 }T × πIN,1
= { 1 × 2 × · · · × n }T × πIN,1 .
Hence, Equation (8) in Corollary 1 is justified. We
have E(1,2,...,n) = 1 ⊗ Q 2 ⊗ Q . . . ⊗ Q n = 1 × 2 ×
· · · × n.
Appendix B: Justification for Corollary 2
Suppose stations 1, 2, . . . , n are in a parallel configuration.
Similar to the three cases in Proposition 3, there also exist
three parallel configuration cases for multiple stations.
246
Abad et al.
Fig. A1. Three parallel configurations between stations 1, 2, . . . , n.
1. Stations 1, 2, . . . , n are used to produce different types
of final products, respectively, as shown in Fig. A1(a).
2. Stations 1, 2, . . . , n are used to produce different types
of intermediate parts, respectively, which will be used
together in the next station as shown in Fig. A1(b).
3. Stations 1, 2, . . . , n are used to produce the same type
of intermediate parts, which will be used together in the
next station as shown in Fig. A1(c).
In cases 1 and 2, different product types, denoted by
l1 , l2 , . . . , ln , follow different routes of the parallel configuration. In the following discussion we elaborate the
justification for Corollary 2 under cases 1 and 2 using mathematical induction. The justification for case 3 in Corollary
2 is very similar to the justification for case 3 in Proposition
3, and thus will not be discussed further.
We have proved Equation (20) for a two-station scenario
through Equations (11) to (17). Assume that Equation (20)
holds when k stations are in a parallel configuration and different product types follow different routes. Let l1 , l2 , . . . , lk
denote the product types that have been produced through
stations 1, 2, . . . , k. Based on Equation (20) with k stations,
we have
E(1,2,...,k) = 1 ⊕ Q 2 ⊕ Q . . . ⊕ Q k
min(ψrv1 , ψrv2 , . . . , ψrvk ) rv , If ν = N + 1
= max(ψrv1 , ψrv2 , . . . , ψrvk ) rv , If ν = N + 1
⎤
⎡ E(1,2,...,k)
E(1,2,...,k)
···
···
ψl1 ε
ψl1 l1
⎥
⎢
E(1,2,...,k)
E(1,2,...,k)
⎥
⎢
0
ψl2 l2
···
ψl2 ε
⎥
⎢
⎥
⎢
=⎢
⎥.
..
..
..
..
⎥
⎢
.
.
.
.
⎥
⎢
E(1,2,...,k)
E(1,2,...,k)
⎦
⎣
0
···
ψlklk
ψlk ε
0
···
···
···
1
(A3)
Stations 1, 2, . . . , k can be represented by equivalent station
E (1, 2, . . . , k) as Fig. A2 shows.
Now we have station k + 1 performing tasks for product
type lk+1 . Stations 1, 2, . . . , k, k + 1 are in a parallel configuration. Based on Equation (A3) and Fig. A2, this parallel
configuration can be represented as a parallel configuration
between station E (1, 2, . . . , k) and station k + 1. Since now
we have one more product type lk+1 to produce, the quality
conforming matrix E(1,2,...,k) becomes
E(1,2,...,k)
⎡
E(1,2,...,k)
ψl1 l1
0
···
⎢
E(1,2,...,k)
⎢
0
ψl2 l2
⎢
..
⎢
..
.
.
= ⎢
⎢
E(1,2,...,k)
⎢
ψlklk
⎢
⎣
···
0
···
⎡
1
⎢
⎢
k+1 = ⎢ 0
⎣0
0
0
..
.
0
0
0
0
ψlk+1
k+1 l k+1
0
E(1,2,...,k)
0 ψlk ε
1
1
1
0
⎤
⎥
⎥
⎥.
k+1 ⎦
ψlk+1 ε
1
0
⎤
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎦
(A4)
(A5)
The input mix ratio is given as
πIN,E(1,2,...,k,k+1) = πIN,E(1,2,...,k) = πIN,k+1
T
IN,E(1,2,...,k)
IN,E(1,2,...,k)
= πl1
, . . . , πlk+1
, πεIN,E(1,2,...,k)
T
= πlIN,k+1
, . . . , πlIN,k+1
, πεIN,k+1 .
(A6)
1
k+1
Since the proportion of conforming products of
type l1 , l2 , . . . , lk coming out of equivalent station
E(1, 2, . . . , k, k + 1) is exclusively determined by the
conforming matrix and input proportion at station
E(1, 2, . . . , k), for product type li (i = 1, 2, . . . , k), we have
OUT,E(1,2,...,k,k+1)
πli
IN,E(1,2,...,k)
= πli
OUT,E(1,2,...,k)
= πli
E(1,2,...,k)
× ψli li
E(1,2,...,k)
× min ψli li
,1
IN,E(1,2,...,k)
E(1,2,...,k)
× min ψli li
, ψlk+1
= πli
l
i i
IN,E(1,2,...,k)
= πli
Fig. A2. Equivalent station E(1, 2, . . . , k).
0
E(1,2,...,k)
0 ψl1 ε
E(1,2,...,k)
ψl2 ε
..
..
.
.
247
Algebraic expression of MMAS configurations
⎡
⎤T
⎡ IN,E(1,2,...,k,k+1)
πl
⎢
⎥
⎢ 1
..
⎢
⎥
⎢
⎢
⎥
.
⎢
⎢
⎥
min
0)
(0,
⎢ IN,E(1,2,...,k,k+1)
⎢
⎥
π
⎢
E(1,2,...,k)
li
=⎢
, ψlk+1
)⎥
⎢ min(ψli li
⎥ ×⎢
..
i li
⎢
⎢
⎥
.
⎢
min (0, 0)
⎢
⎥
⎢ IN,E(1,2,...,k,k+1)
⎢
⎥
..
⎣
π
⎣
⎦
lk+1
.
IN,E(1,2,...,k,k+1)
π
ε
min (0, 0)
min (0, 0)
..
.
⎤
⎥
⎥
⎥
⎥
⎥
⎥.
⎥
⎥
⎥
⎦
(A7)
The proportion of conforming products of type lk+1 coming
out of equivalent station E(1, 2, . . . , k, k + 1) is exclusively
determined by the conforming matrix and input proportion
at station k + 1. Hence, for product type lk+1 , we have
OUT,E(1,2,...,k,k+1)
ing parts from input. Based on Equation (A.6), we have
OUT,E(1,2,...,k)
πεOUT,E(1,2,...,k,k+1) =
πli ε
=
=
i =1,2,...,k
OUT,k+1
IN,E(1,2,...,k,k+1)
+ πlk+1 ε
+ πε
IN,E(1,2,...,k)
E(1,2,...,k)1
πl
× ψli ε
i =1,2,...,k
×ψlk+1
+ πεIN,E(1,2,...,k,k+1) × 1
k+1 ε
IN,E(1,2,...,k)
πli
+ πlIN,k+1
k+1
E(1,2,...,k)
× max ψli ε
, 0 + πlIN,k+1
k+1
i =1,2,...,k
× max(0, ψlk+1
) + πεIN,E(1,2,...,k,k+1) × 1
k+1 ε
⎡
⎤T
⎡ IN,E(1,2,...,k,k+1) ⎤
E(1,2,...,k)
, ψlk+1
max ψl1 ε
ε
πl
1
⎥
⎢
⎢ 1
⎥
⎢
⎥
.
..
..
⎢
⎥
⎢
⎥
.
⎢
⎢
⎥
⎥
IN,E(1,2,...,k,k+1)
⎢
⎥
⎢
⎥
E(1,2,...,k)
π
, ψlk+1
⎢
⎥
⎢ max ψli ε
⎥
l
i
iε
×
=⎢
⎢
⎥
⎥.
..
⎢
⎥
⎢
⎥
..
.
⎢
⎥
⎢
⎥
.
⎢ IN,E(1,2,...,k,k+1) ⎥
⎢
⎥
⎢
E(1,2,...,k)
⎣ πlk+1
⎦
k+1 ⎥
, ψlk+1 ε ⎦
⎣ max ψlk+1 ε
IN,E(1,2,...,k,k+1)
πε
max(1, 1)
(A9)
= πlOUT,k+1
= πlIN,k+1
× ψlk+1
k+1
k+1 l k+1
k+1
IN,k+1
k+1
= πlk+1
× min 1, ψlk+1 lk+1
E(1,2,...,k)
k+1
= πlIN,k+1
×
min
ψ
,
ψ
lk+1 lk+1
lk+1 lk+1
k+1
⎡ IN,E(1,2,...,k,k+1) ⎤
πl
⎡
⎤T
min (0, 0)
⎢ 1
⎥
..
⎢
⎥
.
..
⎢
⎥
⎢
⎥
.
⎢
⎥
⎢ IN,E(1,2,...,k,k+1) ⎥
⎢
⎥
⎢ πli
⎥ Combining Equation (A7), Equation (A8), and Equation
min (0, 0)
=⎢
⎥ ×⎢
⎥.
.
⎢
⎥
⎢
⎥ (A9), we have
..
k+1
⎢
⎥
⎣ min ψlE(1,2,...,k)
⎦
, ψlk+1 lk+1
k+1 l k+1
⎢ IN,E(1,2,...,k,k+1) ⎥
⎡ OUT,E(1,2,...,k,k+1) ⎤
⎣
⎦
πlk+1
min (0, 0)
πl
IN,E(1,2,...,k,k+1)
⎢ 1
⎥
πε
..
⎢
⎥
.
⎢
⎥
(A8)
⎢ OUT,E(1,2,...,k,k+1) ⎥
π
⎢
⎥
πOUT,E(1,2,...,k,k+1) = ⎢ li
⎥
.
⎢
⎥
The proportion of nonconforming products coming out
..
⎢
⎥
of equivalent station E(1, 2, . . . , k, k + 1) consists of the
⎢ OUT,E(1,2,...,k,k+1) ⎥
⎣ πlk+1
⎦
πlk+1
OUT,E(1,2,...,k,k+1)
πε
⎤T
⎡ IN,E(1,2,...,k,k+1) ⎤
E(1,2,...,k)
..
, ψlk+1
max ψl1 ε
.
ε
·
·
·
min
(0,
0)
πl
1
⎥
⎢
⎥
⎢ 1
⎥
⎢
E(1,2,...,k)
..
..
, ψlk+1
⎥
⎢
⎥
⎢ min ψli li
.
.
i li
⎥
⎢
⎢
..
⎥
⎢ IN,E(1,2,...,k,k+1) ⎥
⎢
E(1,2,...,k)
k+1 ⎥
..
.
π
max
ψ
,
ψ
⎥
⎢
⎥
⎢
.
li ε
li ε
=⎢
⎥.
⎥ × ⎢ li
..
⎥
⎢
⎥
⎢
.
.
.
⎥
⎢
⎥
⎢
.
E(1,2,...,k)
..
k+1
⎢
⎥
⎢
IN,E(1,2,...,k,k+1) ⎥
min ψlk+1 lk+1 , ψlk+1 lk+1
.
⎢
E(1,2,...,k)
⎦
⎣
π
k+1 ⎥
lk+1
max ψlk+1 ε
, ψlk+1 ε ⎦
⎣
IN,E(1,2,...,k,k+1)
πε
min (0, 0) · · ·
min (0, 0)
max(1, 1)
⎡
⎤T
E(1,2,...,k)
..
0
max ψl1 ε
, ψlk+1
.
1ε
⎥
⎢
..
⎥
⎢
E(1,2,...,k)
.
k+1
.
.
, ψli li
⎥
⎢ min ψli li
.
⎥
⎢
..
⎥
⎢
E(1,2,...,k)
k+1
.
..
.
max ψli ε
, ψli ε
⎥
⎢
=⎢
⎥ × πIN,E(1,2,...,k,k+1)
⎥
⎢
..
⎥
⎢
.
E(1,2,...,k)
k+1
⎢
⎥
min ψlk+1 lk+1 , ψlk+1 lk+1
⎢
E(1,2,...,k)
k+1 ⎥
max ψlk+1 ε
, ψlk+1 ε ⎦
⎣
0
1
= E(1,2,...,k,k+1) × πIN,E(1,2,...,k,k+1) .
(A10)
nonconforming parts coming from E(1, 2, . . . , k), nonconHence, Corollary 2 is justified for any n(n ≥ 2).
forming parts coming from station k + 1, and nonconform⎡
248
Abad et al.
Appendix C: Derivation of 1,1
1,1 represents the quality conforming matrix corresponding to four variants of component 1 at station 1. According to the definition of k in Equation (3), we
have
⎤
⎡ 1,1
0
0
0
1 − ψ11,1
ψ1
⎥
⎢
⎢ 0
ψ21,1
0
0
1 − ψ21,1 ⎥
⎥
⎢
⎥
⎢
1,1 = ⎢ 0
0
ψ31,1
0
1 − ψ31,1 ⎥ ,
⎥
⎢
⎢
1,1
1,1 ⎥
⎣ 0
0
0
ψ4
1 − ψ4 ⎦
0
0
0
0
1
where ψ11,1 = Prob{Producing a conforming type 1 product
using component 1 at station 1}, ψ21,1 = Prob{Producing a
conforming type 2 product using component 1 at station 1},
ψ31,1 = Prob{Producing a conforming type 3 product using
component 1 at station 1}, and ψ41,1 = Prob{Producing a
conforming type 4 product using component 1 at station
1}.
Based on Table 4, it can be seen that “product type 1”
and “product type 2” consist of variant 1 of component
1, while “product type 3” and “product type 4” consist of
variant 2 of component 1. Therefore, it further yields that:
ψ11,1 = Prob{Producing a conforming part
of component 1 at station 1} = 92%,
1,1
ψ2 = Prob{Producing a conforming part
of component 1 at station 1} = 92%,
ψ31,1 = Prob{Producing a conforming part
of component 1 at station 1} = 93%,
ψ41,1 = Prob{Producing a conforming part
of component 1 at station 1} = 93%.
with variant 1
with variant 1
with variant 2
with variant 2
Substituting these elements into matrix 1,1 , we obtain
⎡
⎤
0.92
0
0
0
0.08
0.92
0
0
0.08 ⎥
⎢ 0
⎢
⎥
0
0.93
0
0.07 ⎥ .
1,1 = ⎢ 0
⎣ 0
0
0
0.93 0.07 ⎦
0
0
0
0
1
Biographies
Andres G. Abad is an Assistant Professor at Escuela Superior Politecnica del Litoral (ESPOL). He holds a Ph.D. in Industrial and Operations
Engineering from the University of Michigan, Ann Arbor. His research
interests include modeling and analysis of complex manufacturing systems, mathematical modeling of human decision-making behavior, and
advanced statistical analysis of high-dimensional, large datasets. He received the ScholarPOWER academic award in 2008 and 2009. He has
conducted various consulting projects for multinational firms. In 2012 he
was appointed Advisor to the Minister of Production of Ecuador, working on the transformation of the country’s productive structure through
foreign trade data analysis.
Weihong Guo received her B.S. degree in Industrial Engineering from
Tsinghua University, Beijing, China, in 2010 and an M.S. degree in Industrial and Operations Engineering from the University of Michigan,
Ann Arbor, in 2012. She is currently a Ph.D. candidate in Industrial and
Operations Engineering at the University of Michigan. Her research interests are in the areas of quality engineering and performance modeling
for manufacturing and assembly systems using statistical process control
techniques, machine learning methods, and the integration of SPC and
automatic control. She is a member of INFORMS.
Jionghua (Judy) Jin is a Professor in the Department of Industrial and
Operations Engineering at the University of Michigan. She received her
Ph.D. degree from the University of Michigan in 1999. Her recent research focuses on data fusion for effective decision making to improve
system operational quality and efficiency. Her research emphasizes the
integration of applied statistics, signal processing, reliability, system modeling, and decision-making theory. She has received a number of awards
including an NSF CAREER Award in 2002 and a PECASE Award in
2004 and eight Best Paper Awards between 2005 and 2012. She is a Fellow
of ASME and a member of ASQ, IEEE, IIE, INFORMS, ISI, and SME.