publication lamghabbar

int. j. prod. res., 1 november 2004,
vol. 42, no. 21, 4495–4512
Concurrent optimization of the design and manufacturing stages
of product development
A. LAMGHABBAR, S. YACOUT* and M. S. OUALI
The problem of concurrent optimization of the design and the process planning
stages when a new product is developed is addressed. The paper advocates for
a simultaneous approach rather than the traditional sequential one. A mathema-
tical representation of this approach is given for these two stages. A mathematical
programming technique is used to find the optimal values of the design and the
process characteristics. The objective function is a quality loss function. The
constraints are the customer requirements, the product’s specification limits, the
parts’ dimensional limits and the process capability. The traditional sequential
approach of concurrent engineering is compared with the proposed simultaneous
approach. A parametric analysis of the objective function is performed by apply-
ing an interactive multi-objective goal programming technique. A numerical
example of a low-pass electrical circuit is given. It is shown that the proposed
approach leads to better efficient solutions than the sequential approach.
The decision-maker interacts with the optimization process and can choose the
efficient solution that best satisfies the company’s needs.
1. Introduction
Since the late 1980s, the product development process has witnessed a major
change. The separate sequential stages of the product development process have
been replaced by the process of planning these stages simultaneously (Asiedu and
Gu 1998). Concurrent engineering or life cycle engineering has emerged as an effec-
tive approach to improve the design of products and to reduce the series of trial-and-
error dry runs, manufacturing costs and time to market. This approach encourages
the developers to consider interactively all elements of the product’s development
process from the design through to the disposal, including customer requirements,
product quality, manufacturing costs and production time. Despite the wide accep-
tance of the approach an implementation rate of around 50% is reported (Brookes
and Backhouse 1998). The main barrier to implementation is the lack of tools and
techniques available to assist in implementing the approach.
Quality Function Deployment (QFD) is a technique used to implement
Concurrent Engineering (Tsuda 1997). The use of QFD for a product development
process becomes visible with the construction of four Houses of Quality: Product
Planning, Part Planning, Process Planning and Production Planning
(Silvaloganathan and Eubuomwan 1997). Practitioners believe that QFD is a helpful
tool, but much development remains to be undertaken. Based on the literature
review, it is apparent that although mathematical formulations exist for each of
Revision received April 2004.
Mathematics and Industrial Engineering, Department Ecole Polyrechnique de Montre´ al,
PO Box 6079, Montreal, Quebec, Canada H3C3A7.
*To whom correspondence should be addressed. E-mail:soumaya.yacout@polymtl.ca.
International Journal of Production Research ISSN 0020–7543 print/ISSN 1366–588X online # 2004 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080/00207540410001720403

the four stages particularly the first two stages, none of these formulations was
intended to optimize the four stages simultaneously (Vonderembse and
Roghunathan 1997).
This paper proposes a mathematical formulation that links two stages of product
development, the product design and the process planning stages. The simultaneous
optimization of these two stages is presented. Although the paper deals only with
two stages, the general approach is equally applicable for four stages. The objective
is to identify and quantify optimal manufacturing cost features and parts’ tolerance
in the early design stage. Mathematical programming models have been developed
to find the optimal values of parts and process characteristics. These models are
presented in section 2. A numerical example is given in section 3. In section 4,
parametric analysis is given using an interactive goal programming technique.
Conclusions are discussed in section 5.
2. Mathematical formulation of product and process planning
According to the standard QFD houses of quality, a product’s development
process includes four stages. In stage 1, the customer needs are identified (the
What of stage 1) and the technical requirements (the How of stage 1) are established.
In stage 2, the technical requirements (the What of stage 2) are mapped onto the
design requirements (the How of stage 2). The piece parts are thus established and
their characteristics are enumerated. In stage 3, the final process plan is specified.
This plan explains how the product should be manufactured, including the sequence
of machine tools (the How of stages 3). The process plan is determined such that
the components’ assembly guidelines and tolerance specifications are met. These are
dependent on the design requirements (the What of stage 3). In stage 4, a criterion
for quality control is established (the How of stage 4). Inspection plans are defined
such that the final yield and total production costs are optimized.
Currently, the QFD stages are planned sequentially. The output of one stage (the
How) becomes the input of the following stage (the What). Although the QFD,
as described by the four stages, is a powerful way of ensuring the delivery of the
ultimate product characteristics through the design of the subsystems and parts and
their manufacturing, the sequential way of constructing and planning these four
stages may lead to a suboptimal product development process. The earlier stages
become more important than the later, since their outputs are the fixed inputs to the
following stages. One way of overcoming this drawback is by constructing feedback
loops so that all the stages are planned simultaneously. A technique for the simulta-
neous planning and determination of the elements of the four stages is thus needed.
This paper shows that the simultaneous optimization of the four stages may be
achieved by using an appropriate mathematical representation of each stage. In this
section, we consider stages 2 and 3 only. The methodology used can then be extended
to all four stages. In Piadras et al. (2001), the first two stages were also mathema-
tically represented and optimized simultaneously. Based on the literature review,
we first present a mathematical formulation of stages 2 and 3 when planned and
optimized sequentially. Then, a mathematical formulation that takes into consid-
eration the simultaneous optimization of both stages is introduced.
2.1. Mathematical formulation of a product design
The product design planning is the stage that follows the determination of the
customer requirements and their transfer into technical requirements. Having set
4496 A. Lamghabbar et al.

the target values for these requirements, the objective of stage 2 is to incorporate
these values into the final product parameters. One way of doing this is by finding
transfer functions that map the target values into the design elements. Many authors
used the Taguchi’s loss function to establish the design elements that lead to robust
design (Pignatiello 1993, Ames et al. 1997, Tsui 1999). In this paper, this function is
introduced as follows:
Let Y ¼ (y1, y2, . . . , yI) denote the technical requirements and Z ¼ (z1, z2, . . . , zJ)
represent the parts characteristics. At the product design phase, the technical require-
ments are the responses obtained when the product’s design characteristics are varied
at different levels. Different combinations of design variables will lead to different
values of responses. By using some statistical techniques such as the design of experi-
ments (DOE) and the response surface methodology (RSM), an estimate of an input-
response functional form can be obtained, even if this functional relationship is
either not known or very complex. Transfer functions of the following form can
thus be obtained:
yi
¼ Fðz1, z2, . . . , zJÞ ð1Þ
and
yi
¼ Gðz1, z2, . . . , zJÞ, i ¼ 1, 2, . . . , I, ð2Þ
where yi
, i ¼ 1, 2, . . . , I are the mean responses and yi
¼ 1, 2, . . . , I are the standard
deviations. For a comprehensive presentation of the DOE and the RSM, see Box
et al. (1978); for Taguchi’s technique, see Ross (1988).
By assuming that the target values, Tyi
and the upper and lower limits ð yþ
i , yÀ
i Þ
are determined in stage 1, the optimization problem of stage 2 is thus to find z,
z2, . . . , zJ, j ¼ 1, . . . , J, that minimize the square of the deviation from the target
values, and the variation around these targets. This problem is represented as
follows:
P1 : minz1, z2,..., zI
XI
i¼1
ai yi
À Tyi
Â Ã2
þ biðyi
Þ2
ð3Þ
subject to
yi
¼ Fðz1, z2, . . . , zJ Þ, i ¼ 1, 2, . . . , I ð4Þ
yi
¼ Gðz1, z2, . . . , zJÞ, i ¼ 1, 2, . . . , I, ð5Þ
yÀ
i yi
yþ
i , i ¼ 1, 2, . . . , I, ð6Þ
where ai and bi, i ¼ 1, 2, . . . , I are constants.
2.2. Mathematical formulation of process planning
Manufacturing process planning is the process of determining the sequence of
operations required for converting raw materials into parts, then for assembling the
parts into products.
Any given product may be manufactured in more than one way. The choice
of the manufacturing process depends on the specifications of the design charac-
teristics set at the design planning stage. Tighter specifications usually result in
higher manufacturing costs (Wei 1997). In this section, Jeang’s (1997) inverse
4497Concurrent optimization of the design and manufacturing stages

function is used to represent the mathematical relation between parts’ tolerances, tk,
k ¼ 1, 2, . . . , K and manufacturing costs, CM(tk), where CMðtkÞ ¼ =:t

are
constants.
The optimal values of the parts characteristics zÃ
j , i ¼ 1, . . . , J, and the speciﬁca-
tion limits zþ
j , zÀ
j , obtained from stage 2, are used in stage 3 to determine the process
characteristics, wk, and the process tolerance, tk, k ¼ 1, 2, . . . , K. We assume that each
wk should be within the interval [wÀ
k , wþ
k ], and tk within [tÀ
k , tþ
k ].
Taguchi’s loss function is used again to represent the quality of the manufactured
product. The objective is to minimize the deviation from the speciﬁed design target
values, Tzj
¼ zÃ
j , j ¼ 1, . . . , J, the variation around these targets, and the manufac-
turing costs. The mathematical formulation is given as follows:
P2 : min
XJ
j¼1
cjðzj
À Tzj
Þ2
þ djðzj
Þ2
þ
XK
k¼1
CMðtkÞ ð7Þ
subject to
zj
¼ Hðw1, w2, . . . , tk, . . . , tkÞ, j ¼ 1, 2, . . . , J ð8Þ
zj
¼ Eðw1, w2, . . . , wK , t1, . . . , tK Þ, j ¼ 1, 2, . . . , J ð9Þ
zÀ
j zj zþ
j , j ¼ 1, 2, . . . , J ð10Þ
wÀ
k wk wþ
k , k ¼ 1, 2, . . . , K ð11Þ
tÀ
k tk tþ
k , k ¼ 1, 2, . . . , K ð12Þ
CMðtkÞ ¼
.
t

k, ð13Þ
where zj
and zj
are the means and the standard deviations of the design character-
istic zj, j ¼ 1,2, . . . , J after the manufacturing stage. H(.) and G(.) can be obtained
by the least-squares technique. cj and dj, j ¼ 1,2, . . . , J are constants.
To keep the process variability under control, the standard deviation
zj
, j ¼ 1, 2, . . . , J is bounded as follows:
zj
Supzj
: ð14Þ
Optimal values of the process characteristics WÃ
¼ ðwÃ
1, wÃ
2, . . . , wÃ
K Þ and the
process tolerances TolÃ
¼ ðtÃ
1, tÃ
2, . . . , tÃ
K Þ are obtained.
2.3. Mathematical formulation for simultaneous optimization
In the mathematical formulations presented in sections 2.1 and 2.2, priority is
given to the design stage. Reaching the design specifications is important, even
if it means that the manufacturing costs will be very high. At worst, the avail-
able manufacturing process will not be capable of producing the planned
design specifications. This is due to the fact that the design requirements, zj,
j ¼ 1, . . . , J are specified first and then introduced as target inputs, Tzj
, j ¼ 1, . . . , J
to the manufacturing stage. Figure 1 shows the sequential nature of this optimization
approach.

To overcome this drawback, the design and the process characteristics are set
as variables. What were known parameters, Tzj
, j ¼ 1, . . . , J in section 2.2 are now
variables zj, j ¼ 1, . . . , J in the following formulation:
P3 : min
XI
i¼1
ðaiðyi
À Tyi
Þ2
þ biðyi
Þ2
Þ þ
XJ
j¼1
ðcjðzj
À zjÞ2
þdjðzj
Þ2
Þ þ
XK
k¼1

.
t

k

ð15Þ
subject to
yi
¼ Fðz1, z2, . . . , zJ Þ, i ¼ 1, 2, . . . , I ð16Þ
yi
¼ Gðz1, z2, . . . , zjÞ, i ¼ 1, 2, . . . , I ð17Þ
yÀ
i yi
yþ
j , i ¼ 1, 2, . . . , I ð18Þ
zj
¼ Hðw1, w2, . . . , t1, t2, . . . , tK Þ, j ¼ 1, 2, . . . , J ð19Þ
zj
¼ Eðw1, w2, . . . , t1, t2, . . . , tK Þ, j ¼ 1, 2, . . . , J ð20Þ
zÀ
j zj zþ
j , j ¼ 1, . . . , J ð21Þ
wÀ
k wk wþ
k , k ¼ 1, . . . , K ð22Þ
tÀ
k tk tþ
k , k ¼ 1, . . . , K: ð23Þ
Figure 2 shows the simultaneous nature of this formulation. Solving the above
mathematical programming problem gives optimal solutions to both the product
design and the process planning stages simultaneously.
3. Numerical example
We consider a low-pass electrical circuit composed of two parts: a resistance,
R, measured in kOhms (k
), and a capacitance, C, measured in nFarads (F).
The electrical circuit is presented in ﬁgure 3.
Determine
),...,,( **
2
*
1
*
Jzzzz =
Solve 1P
Let
jzT jz j
1,2,...,J,*
==
Solve 2P
Determine
),...,,( **
2
*
1
*
Kwwww = and
),...,,( **
2
*
1
*
Ktttt =
Figure 1. Optimization steps of the sequential approach.

Without loss of generality and to simplify the calculations, we consider only
one customer requirement: the cut-off frequency of the circuit, F (MHz). Floyd
(1999) gives the following transfer functions for the low-pass electrical circuits:
F ¼
1
2pRC
: ð24Þ
R is a function of three parts’ characteristics: material resistivity, r, length Lr,
and cross-section, Sr (figure 4). C is obtained by controlling three product charac-
teristics: material dielectric permittivity, Cc, distance, Uc, between the two surfaces,
and surface area, Ac (figure 5).
The transfer function for R and C are as follows:
R ¼
rLr
Sr
ð25Þ
C ¼
AcCc
Uc
: ð26Þ
Solve P3
Determine
( )**
2
*
1
*
,...,, JzzzZ =
( )**
2
*
1
*
,...,, KwwwW = and ( (**
2
*
1
*
,...,, Ktttt =
,
Figure 2. Optimization step of the simultaneous approach.
Figure 3. Electrical circuit.
W1
W2
W3
Figure 4. Components of resistance.

3.1. Optimization of the design stage
We assume that the target value of the cut-off frequency, F ¼ y, is known and
is equal to 20 MHz, with upper and lower bounds USLy ¼ 30 MHz and
LSLy ¼ 15 MHz. These values are supposed to be obtained from the previous
stage. We conduct experiments at the resistance (R z1) values of zþ
1 ¼ 12 and
zÀ
1 ¼ 8 and the capacitances ðC z2Þ, values of zþ
1 ¼ 1:2 and zÀ
2 ¼ 0:4. The mean
and standard deviation equations are obtained and used in the optimization problem
as follows:
P1 : minz1, z2
ðy À 20Þ2
þ 2
y ð27Þ
subject to
y ¼
103
2pz1z2
ð28Þ
y ¼
103
2pz1z2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
z1
z1
2
þ
z2
z2
2
s
ð29Þ
15 y 30 ð30Þ
8 z1 12 ð31Þ
0:4 z2 1:2: ð32Þ
The optimal solution of this problem is [zÃ
1 ¼ 8, zÃ
2 ¼ 0:995], and the optimal value
function is 0.00031.
3.2. Optimization of the manufacturing stage
In this problem, the optimal solution of P1 is used as the target value
[TZ1
¼ 8, TZ2
¼ 0:995] . We denote k W1, Lr W2, Sr W3, Ac W4, Cc W5
and Uc W6 and their tolerances tk where k ¼ 1, 2, . . . , 6.
The means z1
and z2
are determined by the following equations:
z1
¼
W1W2
W3
ð33Þ
z2
¼
W4W5
5W6
: ð34Þ
W6
W4
W5
Figure 5. Components of capacitance.

The standard deviations, z1
and z2
, are given by
z1
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
1ðt1Þ þ 2
2ðt2Þ þ 2
3ðt3Þ
q
ð35Þ
z2
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
4ðt4Þ þ 2
5ðt5Þ þ 2
6ðt6Þ
q
: ð36Þ
In terms of tolerances, the above equations become
z1
¼
1
3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2
1 þ t2
2 þ t2
3
q
ð37Þ
z2
¼
1
3
t2
4 þ t2
5 þ t2
6
q
ð38Þ
where kðtkÞ ¼ ðtk=3CpÞ, k ¼ 1, 2, . . . , 6, and assuming that the manufacturing pro-
cess capability, Cp, follows a normal distribution with Cp ¼ 1. The following con-
straints can be derived knowing the maximal and minimal tolerance values, tþ
zj
and
tÀ
zj
, of the design characteristics zj, j ¼ 1, 2:
max ðzj
Þ À zj
tþ
zj
, j ¼ 1, 2 ð39Þ
zj
À min ðzj
Þ ! tÀ
zj
, j ¼ 1, 2 ð40Þ
where
max ðzj
Þ ¼
ðW1 þ t1ÞðW2 þ t2Þ
W3 À t3
ð41Þ
minðz1
Þ ¼
ðW1 À t1ÞðW2 À t2Þ
W3 þ t3
ð42Þ
maxðz2
Þ ¼
5ðW6 À t6Þ
ð43Þ
minðz3
Þ ¼
5ðW6 þ t6Þ
ð44Þ
Table 1 presents the data used in solving the optimization problem of the
manufacturing stage.
yi zj wk tk
Index yÀ
i yþ
i zÀ
i zþ
i wÀ
i wþ
i tÀ
i tþ
i
1 15 30 8 12 24.95 25.05 0.01 0.05
2 0.4 1.2 4.5 5.5 0.1 0.5
3 11.5 12.5 0.1 0.5
4 3.5 4.5 0.1 0.5
5 0.88 1.8 0.01 0.05
6 0.55 0.75 0.01 0.05
Table 1. Parameters of problem P2.

Problem P2 is given as follows:
P2 : min w1,..., w6, t1,..., t6
ðz1
À 8Þ2
þ ðz2
À 0:995Þ2
þ 2
z1
þ 2
z2
þ
X6
k¼1
0:01
tk
ð45Þ
subject to
W3 À t3
À z1
0:4 ð46Þ
z1
À
W3 þ t3
! 0:39 ð47Þ
5ðW6 À t6Þ
À z2
0:15 ð48Þ
z2
À
5ðW6 þ t6Þ
! 0:05 ð49Þ
zk
0:08, k ¼ 1, 2 ð50Þ
z1
¼
W1W2
W3
ð51Þ
z2
¼
W4W5
5W6
ð52Þ
z1
¼
1
3
t2
1 þ t2
2 þ t2
3
q
ð53Þ
z2
¼
1
3
t2
4 þ t2
5 þ t2
6
q
ð54Þ
24:95 W1 25:05 ð55Þ
4:5 W2 5:5 ð56Þ
11:5 W3 12:5 ð57Þ
3:5 W4 4:5 ð58Þ
0:88 W5 1:8 ð59Þ
0:55 W6 0:75 ð60Þ
0:01 t1 0:05 ð61Þ
0:1 t2 0:5 ð62Þ
0:1 t3 0:5 ð63Þ
0:1 t4 0:5 ð64Þ

0:01 t5 0:05 ð65Þ
0:01 t6 0:05 ð66Þ
Table 2 gives the optimal solution and optimal value function of the above
problem.
3.3. Simultaneous optimization of the product design and the process
planning stages
Instead of ﬁxing the target values of the design characteristics Tz1
and Tz2
to
the optimal value zÃ
1and zÃ
2, respectively, Tz1
and Tz2
appear as decision variables
in the objective function. P1 and P2 are combined to form P3 as follows:
P3 : min z1, z2, W1,..., W6, t1,..., t6
ðy À 20Þ2
þ 2
y þ ðz1
À z1Þ2
þ ðz2
À z2Þ2
þ 2
z1
þ 2
z2
þ
X6
k¼1
0:01
tk
: ð67Þ
subject to constraints (29–33) and (47–67).
Table 3 shows the results of problem P3.
3.4. Analysis of the results
The solution of problem P3 is a trade-oﬀ between the optimal values of the design
characteristics and the optimal values of the process characteristics. The solution
of the problem P3 gives a higher deviation from the target value Ty and a higher
standard deviation of y. This means a higher objective function value of 0.0008
Independent variables Dependent variables Objective function of
Index wÃ
k tÃ
k zÃ
j zj
zj
yi
yi
P1 P2 P1 þ P2
1 24.95 0.05 9.476 9.474 0.067 20.001 0.022 0.0008 0.8171 0.8179
2 4.75 0.13 0.840 0.821 0.057
3 12.5 0.15
4 3.5 0.16
5 0.88 0.05
6 0.75 0.05
Table 3. Solution to problem P3.
Independent variables Dependent variables
Index wÃ
k tÃ
k zj
zj
Objective function of P2
1 24.95 0.05 8.982 0.068 1.8044
2 4.5 0.14 0.884 0.050
3 12.5 0.14
4 3.5 0.13
5 0.95 0.05
6 0.75 0.05
Table 2. Solution to problem P2.

compared with 0.0003 in problem P1. On the other hand, the manufacturing costs
are lowered from 1.8044 to 0.8171. The sum of these two objective functions is thus
lowered to 0.8179 in comparison with 0.00031 þ 1.8044 ¼ 1.80471. Obviously, the
simultaneous optimization gives an overall better solution to the product design
and the process planning stages.
4. Parametric analysis
In the example of section 3, we assumed that the constants a, b, c1, c2, d1 and d2
were all equal to 1. This is an important assumption because it means that the
deviations from the target values and the variances in the product design and the
process planning stages all have equal weights, that is, they have equal importance
in the optimization problem. It also means that all scaling constants are equal to 1.
Since this assumption is very restrictive, we present in this section a solution
algorithm that allows a relaxation of the above assumption. This algorithm, intro-
duced by Abdel Haleem (1991, 1993, 2004), is based on the theory of multiobjective
goal programming. It generates efficient solutions to a general multicriteria optimi-
zation problem. An efficient solution is defined by Chankong and Haimes (1983) as
one which is not dominated by any other feasible solutions. We say x1
x2
if
and only if v( f (x1
)) ! v( f (x2
)), where v is the value function. These efficient solutions
can be seen as optimal solutions at different values of the parameters a, b, c1, c2, d1
and d2.
The algorithm interacts with the decision-makers and stops when he/she is
satisfied with the solution.
4.1. Interactive multiobjective goal-programming algorithm
The algorithm is described as follows:
Let the goal-programming problem, GP, be:
min
f1ðXÞ
f2ðXÞ
Á
Á
Á
fmðXÞ
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
ð68Þ
subject to
M ¼ fX 2 Rn
=grðXÞ 0, r ¼ 1, . . . , pg: ð69Þ
We suppose that there is an arbitrary real number pi ðgoalsÞ, i ¼ 1, . . . , m such that:
f1ðXÞ p1
f2ðXÞ p2
Á
Á
Á
fmðXÞ pm
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
ð70Þ

We denote F ¼ [ f1, f2, . . . , fm] the vector of objective functions, which should be
minimized. F(X) ¼ [ f1(X), f2(X), . . . , fm(X)], where X ¼ [x1, x2, . . . , xn] is the vector
of independent variables. GP(p) is the problem of minimizing F(X) subject to
X 2 M(p) where M(p) ¼ {X 2 Rmþn
/gr (X) pr, r ¼ 1, 2, . . . , p þ m}.
To solve the GP(p) the following sets are defined: the set of feasible parameters:
A ¼ fp 2 Rmþp
=MðpÞ 6¼ g where Rmþp
is the m þ p dimensional vector space of
parameters.
The solvability set D ¼ fp 2 Rmþp
=UoptðpÞ 6¼ g where Uopt(p) is the set
of all optimal points of the problem GP(p), UoptðpÞ ¼ fXÃ
2 Rnþm
=FðXÃ
Þ ¼
minx 2 MðpÞ FðXÞg; and the stability set LðXÃ
Þ ¼ fp 2 Rnþm
=FðXÃ
Þ ¼ minx 2 MðpÞ FðXÞg.
The algorithm consists of the following steps:
Step 1. Fori ¼ 1,2, . . . , m,solveminx 2 MðpÞ f1ðXÞanddetermineXÃ
i ¼ðxÃ
i1, xÃ
i2, . . . , xÃ
inÞ,
i ¼ 1, . . . , m.
Step 2. Calculate FiðXÃ
i Þ ¼ ½ f1ðXÃ
i Þ, f2ðXÃ
i Þ, . . . , fmðXÃ
i ÞŠi ¼ 1, . . . , m; i ¼ 1. For i ¼ 1,
2, . . . , m, we obtain pimin, pimax, thus we determine the reduced solvability
set D0
¼ {{D}/pimin pi pimax, i ¼ 1, 2, . . . , m. Let pj min ¼ minj fiðxÃ
j Þ,
where i ¼ 1, 2, . . . , m, j ¼ 1, 2, . . . , m. pj max ¼ maxj fiðxÃ
j Þ, where i ¼ 1,
2, . . . , m, j ¼ 1, 2, . . . , m.
Step 3. Set j ¼ 1 and have the decision-maker (DM) select p j
i, i ¼ 1, 2, . . . , m.
Step 4. Solve the problem GPðpj
i Þ and obtain optimal solution XXj.
Step 5. Construct the stability set L ( XXj) and present XXj, L ( XXj) to the DM. If he or
she is satisfied with XXj, stop the algorithm. If the DM is not satisfied with XXj
go to Step 6.
Step 6. Construct the set D0
À
P
Lð XXpÞ, p ¼ 1, 2, . . . , j
È É
, if D0
À
P
Lð XXpÞ,
È
p ¼ 1, 2, . . . , jg and go to Step 8. Otherwise, go to Step 7.
Step 7. Set j ¼ j þ 1 and ask the DM to select another value for the goal vector
p j
i 2 D0
À
P
Lð XXpÞ, p ¼ 1, 2, . . . , ð j À 1Þ
È É
, and go to Step 4.
Step 8. Have the DM select his or her preferred solution.
Step 9. Stop.
4.2. Numerical example
In the numerical example, we had five constants a1, a2, c1, c2 and d1. We consider
that these constants are unknown and create instead five objective functions f1,
i ¼ 1, 2, . . . , 5 and five goals pi, i ¼ 1, 2, . . . , 5 as follows:
f1 ¼ ðy À 20Þ2
p1 ð71Þ
f2 ¼ ðyÞ2
p2 ð72Þ
f3 ¼
X2
j¼1
ðzj
À zjÞ2
p3 ð73Þ
f4 ¼
X2
j¼1
ðzj
Þ2
p4 ð74Þ
f5 ¼
X6
k¼1
0:01
tk
p5 ð75Þ

The multi-objective optimization problem is thus to minimize the ﬁve objective
functions, interactively, subject to constraints (28–32) and (46–66).
The interactive multi-objective goal programming is applied as follows:
Step 1. Min fi, i ¼ 1, 2, . . . , 5 subject to constraints (28–32) and (46–66).
The solutions to these problems are given in table 4.
Step 2. Table 5 shows the values fiðXÃ
i Þ, i ¼ 1, 2, . . . , 5. pi min and pi max, i ¼ 1, 2, . . . , 5
are shown in bold characters and D0
is determined.
Independent variable XÃ
1 Objective functions F1ðXÃ
1 Þ
Index k wÃ
k tÃ
k ZÃ
k f1 f2 f3 f4 f5
1 24.95 0.01 12 0 0.0008 8.690 0.009 3.164
2 4.54 0.15 0.663
3 12.5 0.11
4 3.5 0.22
5 1.07 0.01
6 0.75 0.01
XÃ
2 F2ðXÃ
2 Þ
1 24.95 0.01 8.846 25 0.0001 0.151 0.0088 3.169
2 4.55 0.15 1.2
3 12.5 0.11
4 3.5 0.21
5 1.61 0.01
6 0.75 0.01
XÃ
3 F3ðXÃ
3 Þ
1 25.05 0.01 10.578 2.8 0.0004 0 0.007 3.188
2 5.27 0.14 0.821
3 12.48 0.12
4 3.5 0.17
5 0.88 0.01
6 0.75 0.01
XÃ
4 F4ðXÃ
4 Þ
1 25.05 0.03 12 1.9 0.0006 0.06 0.004 2.646
2 5.5 0.13 0.71
3 11.5 0.1
4 3.5 0.1
5 0.88 0.01
6 0.64 0.01
XÃ
5 F5ðXÃ
5 Þ
1 24.95 0.05 12 1.95 0.0006 9.12 0.0078 0.808
2 4.5 0.14 0.713
3 12.5 0.14
4 3.5 0.16
5 0.88 0.05
6 0.75 0.05
Table 4. Solution to problem GP.

Step 3. Let p1
¼½p1
1 ¼ 0, p1
2 ¼0:0004, p1
3 ¼0:015, p1
4 ¼0:0055, p1
5 ¼0:9Š be the choice
of the decision-makers.
Step 4. We solve GP(p1
):
min
f1ðxÞ p1
1
f2ðxÞ p1
2
Á
Á
Á
f5ðxÞ p1
5
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
subject to constraints (28–32) and (46–66).
Step 5. Table 6 shows the solution XX1 to problem GP(p1
). If the decision-maker is
satisﬁed with the solution, we stop, if not we go to Step 6.
Step 6. with LðXpÞ¼½p1 ¼ 0,p2 ¼ 0:0004,p3 ¼ 0:015,p4 ¼ 0:0055,p5 ¼ 0:9Š. The set
fD0
À LðXpÞg is determined. Table 7 shows this set.
Step 7. Set j ¼ 2. Assume that the DM has chosen. p2
¼ ½p2
1 ¼ 0:0001, p2
2 ¼
0:00035, p2
3 ¼ 0:35, p2
4 ¼ 0:0055, p2
5 ¼ 1Š.
Step 8. Steps 4–7 are repeated until the DM is satisﬁed.
4.3. Analysis of the results
Rows 3–5 of table 8 show three solutions of the simultaneous optimization of the
design and the manufacturing stages, when using the interactive goal-programming
algorithm. Row 2 shows the solution obtained when solving the two stages simul-
taneously and a1, a2, c1, c2 and d1 are all equal to 1. Row 1 of table 8 shows the
Independent variables Objective functions
Index wÃ
1 tÃ
1 ZÃ
k f1 f2 f3 f4 f5
1 24.95 0.04 8.860 0 0.0004 0.015 0.0055 0.95
2 4.5 0.15 0.898
3 12.5 0.1
4 3.5 0.1
5 0.98 0.04
6 0.75 0.04
Table 6. Solution to problem GP(p1
).
FiðXÃ
j Þ f1ðXÃ
j Þ f2ðXÃ
j Þ f3ðXÃ
j Þ f4ðXÃ
j Þ f5ðXÃ
j Þ
XÃ
1 0 0.0008 8.690 0.009 3.164
XÃ
2 25 0.0001 0.151 0.0088 3.169
XÃ
3 2.8 0.0004 0 0.007 3.188
XÃ
4 1.9 0.0006 0.060 0.004 2.646
XÃ
5 1.95 0.0006 9.12 0.0078 0.808
Table 5. pj min and pj max, j ¼ 1, 2, . . . , 5 in bold characters.

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