Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol67.Issue3.Oct2000:

Int. J. Production Economics 67 (2000) 219}233

Integrated batch size and setup reduction decisions in
multi-product, dynamic manufacturing environments
Moustapha Diaby*
Operations and Information Management Department, College of Business, University of Connecticut, Storrs, CT 06269-0241, USA
Received 24 November 1998; accepted 8 December 1999

Abstract
We present a comprehensive model for simultaneously planning for the setup time reductions and the batch sizes of
several products over a "nite planning horizon in a capacitated manufacturing environment. It is assumed that by
investing in the appropriate amounts of various resources (such as R&D time, equipment, "xtures, tooling, re-layout, etc.)
setup times can be reduced. The problem is to determine how much to cut the setup time for each product and how much
of each product to produce in each period of the planning horizon so that total costs are minimized, subject to limits on
the manufacturing and setup reduction resources. A nonlinear, mixed-integer mathematical programming model of this
problem is formulated and a heuristic method is developed for solving it. The proposed model is broad and can be
directly applied in a variety of practical situations including the case where discrete technology choices must be
made. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Production planning; Setup reduction; Continuous improvement; Kaizen; Large-scale optimization

1. Introduction

In order to meet the #exibility and quality requirements of today's competitive global market,
organizations have had to implement advanced
manufacturing technologies such as Group Technology (GT), Flexible Manufacturing Systems
(FMS) and Just-In-Time (JIT) manufacturing. It is
well known however, that the single most important prerequisite to the successful use of all these
technologies is the shortness of the setup/changeover times. Short setup/changeover times allow for
smaller lots and inventories, which in turn can lead

* Tel.: 860-486-5140; fax: 860-486-4839.
E-mail address: mdiaby@sba.uconn.edu (M. Diaby).

to (i) higher quality, (ii) lower waste and rework, (iii)
increased process yield and productivity, (iv) increased awareness of the causes of errors and delays, and (v) greater #exibility and responsiveness
[1]. Hence, it is no surprise that setup reduction
and quality improvement programs have become
so commonplace in industry over the past two
decades. Applications have been described in the
manufacturing contexts of products ranging from
consumer furniture [2,3], to metal and plastic wires
[4], aluminum cans [5], cutting tools [6], industrial lighting [3], and electronic and automotive

components [7,3,8,9], to name just a few.
Mathematical models to help managers in their
e!orts to integrate operations/process improvement investment decisions within an operational
context began to appear in the mid-1980s. These

0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 0 2 1 - 9

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M. Diaby / Int. J. Production Economics 67 (2000) 219}233

models can be broadly classi"ed either as `technology selectiona models or `setup reductiona models. The technology selection models are essentially
concerned with discrete technology choices in the
contexts of Group Technology and Flexible Manufacturing Systems. These models consider setup
costs as decision variables, but mostly from the
perspective of di!ering technologies [10]. Setup
reduction models on the other hand view setups
mostly from the perspectives of continuous improvement (Kaizen) and JIT manufacturing concepts. In general these models are EOQ-based and
assume the setup reduction cost function to be

continuous. The model developed in this research is
presented in the general framework of a setup
reduction model. However, it considers a dynamic,
capacitated manufacturing environment and can
handle discrete as well as continuous setup reduction functions. It can therefore be directly applied
for making discrete technology choices as well.
Reviews of existing setup reduction models can
be found in the works of Diaby [11] and Kim et al.
[12]. The vast majority of the models have focused
on single-product situations. Moreover, almost all
the models have assumed a continuous manufacturing environment and are, therefore, EOQ based.
Papers that have dealt with multi-product situations are those of Banerjee et al. [13], Freeland
et al. [14], Kim et al. [12], Leschke [15], Leschke
and Weiss [16], and Spence and Porteus [17]. The
only papers that have considered a dynamic manufacturing environment are those of Diaby [11],
Freeland et al. [14], Leschke [15], Mekler [18],
and Zangwill [19]. The focus in Zangwill [19] was
to examine the bene"ts of reduced setups in serial
production systems. A particularly insightful "nding of that paper was the fact that setup reduction
has increasing marginal benexts. This result is

shown to hold for the single-level problem as well
in Diaby [11]-although this was not explicitly
stated in that paper. The primary concern in Diaby
[11] and Mekler [18] respectively, was to develop
procedures to determine optimal lot sizes and setup
reduction amounts. Freeland et al. [14] considered
static as well as dynamic manufacturing environments and developed expressions for the setup reduction required to achieve target lot sizes for
single-product situations and a `savings ratioa

method to prioritize products for setup reduction
in multi-product situations. For the dynamic case,
among other things, they did not consider capacity
constraints nor setup reduction costs. The focus in
Leschke [15] was to compare various setup reduction priority rules (along the lines of Freeland et al.
[14]) against each other. To the best of our knowledge, the only existing model that considers limits
on the setup reduction resources is the model of
Banerjee et al. [13]. However, that model is EOQ
based as mentioned above.
A recent case study of setup reduction programs
at manufacturing organizations in "ve di!erent industries revealed that setup reduction programs

have three di!erent stages [3]. The "rst stage,
called `organization stagea, corresponds essentially
to the "rst two stages of Shingo's (see [7,8]) Single
Minute Exchange of Dies (SMED) system. Its focus
is the work center level with the goal of coordinating the setup process of the di!erent products that use the work center or machine by
improving the housekeeping and procedures. For
example, a company may "rst focus on speci"c
work centers and try to achieve uniform ways to
attach dies to machines by standardizing the sizes
and positions of fasteners. This organization stage
typically requires little or no investment. Improvements beyond the organization stage can be
achieved only through some kind of technological
improvement or innovation focused on speci"c
products. Hence, the second stage, called standardization stage, corresponds essentially to the third
and fourth stages of the SMED system (Shingo [8])
and may require substantial investments. The goal
is to achieve a standardization of the setups on
a machine or at a work center by focusing on individual products. For example, after standardizing
fasteners, a company may focus on acquiring/developing new tools, dies, methods, and/or accessory
equipments for individual products as the next step.

The third (and last) stage, termed rationalization
stage is focused on improving the environment of the
setup process (i.e., material quality, product design,
machine reliabilities and capabilities, etc.). More detailed accounts of speci"c company practices for
each of these can be found in Leschke [3].
The focus of this paper is on the standardization
stage of setup reduction programs. The purpose is

M. Diaby / Int. J. Production Economics 67 (2000) 219}233

to propose a "rst mathematical programming
treatment of this problem in a capacitated, dynamic
manufacturing environment. It is assumed that by
investing in the appropriate amounts of various
resources (such as R&D time, equipment, "xtures,
tooling, re-layout, etc.) setup times can be reduced.
The problem is to determine how much to cut the
setup time of each product and how much of each
product to produce in each period of the planning
horizon so that total costs are minimized, subject to

limits on the manufacturing and setup reduction
resources. A nonlinear, mixed-integer mathematical programming model of this problem is formulated and a heuristic method is developed for
solving it. The model can also be applied directly
for making discrete technology choices.
The paper is organized as follows. We discuss the
formulation of the problem in Section 2. The
solution methodology is developed in Section 3.
Computational results are discussed in Section 4.
Finally, conclusions are discussed in Section 5.

2. Model formulation
The problem considered is that of simultaneously planning for the setup time reduction and
the batch sizes of several products over a speci"ed
time horizon. The time horizon is divided into
a number of discrete periods. Several resources are
available in limited amounts respectively, for production in each period. In addition resources are
available, that can be used to reduce the setup time
of any product. We refer to the resources used for
production as `production resourcesa and to the
resources used for setup time reduction as `setup

reduction resourcesa. The production resources
correspond to the various capacity options typically available in an aggregate planning situation.
Examples of production resources are the regular
time and overtime labor hours available in each
period. Examples of setup reduction resources include Research and Development (R&D) labor
time and cost, the capital available for the acquisition of new technologies, or the time and/or capital
available for process improvements, etc. (see
Leschke [15] or Shingo [8,9]). The total setup cost
is assumed to consist of a `direct costa component

221

and a `labor costa component. The `directa component consists only of the administrative costs and
the cost of the material lost in the calibration process. The labor time component corresponds to the
labor time used for setups. For simplicity, we will
henceforth use the term `setup costa to refer to the
direct component of the setup cost only. Hence, the
(direct) setup cost and the setup time are independent of one another and of the sequence in which
the products are processed. However, the setup
time can be reduced by some amount depending on

the amounts of setup reduction resources used for
such endeavor.
Without loss of generality, we assume that:
(i) each production resource is expressed in terms of
labor time; (ii) the production resources are indexed
in increasing order of their unit costs; (iii) the setup
times do not vary with periods; (iv) initial inventories have been netted out of demands; and that
(v) required "nal inventories have been added to
the demands of the last period of the planning
horizon. It is also assumed that all the setup time
reductions occur only once over the planning horizon, before the beginning of the "rst period. This
assumption is justi"ed by the high investments that
are typically required at the standardization stage of
a setup reduction program, which in turn, makes
substantial planning a necessity. It (the assumption)
is also standard in the setup reduction literature
and consistent with manufacturing practice (see the
work of Byrne [4]). We further assume that for
a given product and a given setup reduction resource, the setup time reduction function is a general, nonincreasing, piecewise-linear function of the
amount of setup reduction resource used (see Fig.

1a). Hence, the model proposed in this research can
readily accommodate any arbitrary setup reduction function that may be encountered in practice.
A special case corresponding to the logarithmic and
power forms commonly used in the literature (see
the work of Porteus [20] or that of Kim et al. [12],
among others) is illustrated in Fig. 1b. A systematic
piecewise-linearization approach that can be adjusted in a straightforward manner to approximate
these functional forms to any desired degree of
accuracy is illustrated in Section 4 of this paper.
Denoting by D the amount by which the
i
setup time of product i is reduced and by ¹ the
i

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M. Diaby / Int. J. Production Economics 67 (2000) 219}233

setup reduction rates over the setup reduction
ranges respectively (i.e., a "(t

!t )/u ;
kri
k~1,i
ki kri
∀k, r, i); the B are decision variables representing
kri
the respective amounts of setup reduction resources
used in each range; the Z are binary variables
kri
used to enforce the piecewise-linear nature of the
setup reduction functions; and the t are as shown
kri
in Figs. 1a and b.
The basic notation used in the paper is de"ned in
Fig. 2. We also use the notations Sn, n#tT to
denote the set Mn, n#1, n#2,2, n#tN and 0(z) to
denote the value of Problem (z). The setup time
reduction model is formulated as follows:
Problem STRP
minimize
Z"+ + h H # + + c ¸ #+ + s >
ij ij
rj rj
i ij
i|I j|J
r|R j|J
i|I j|J
#+ + + qv B
r kri
i|I r|Q k|K(i)
subject to:
Fig. 1. (a) Illustration of the setup cost reduction function
} general case. (b) Illustration of the setup cost reduction } function convex case.

corresponding setup time after reduction, then the
piecewise linear setup reduction functions can be
modeled in a mathematical program as follows:
(1)

¹ "t !D
i
0i
i
with

G

H

+ a B ,
kri kri
k|K(i)
u Z )B )u Z
; k3K(i), r3Q
kri kri
kri
kri k~1,r,i
Z 3M0, 1N, k"0,2, i(i), r3Q,
kri

D "min
i
r|Q

(2)
(3)
(4)

where Q is the index set for the setup reduction
resources; K(i) is the index set for the setup reduction ranges; i(i) is the number of setup reduction
ranges; the u 's are the maximum amounts of
kri
setup reduction resources corresponding to the
setup reduction ranges respectively; the a are the
kri

H
!H #X "d , i3I, j3J
i,j~1
ij
ij
ij

(5)

(6)

+ t > #+ p X !+ D > ! + ¸ )0, j3J,
oi ij
i ij
i ij
rj
i|I
i|I
i|I
r|R
(7)
D ! + a B )0,
i
kri kri
k|K(i)

i3I, r3Q,

(8)

+ + + v B )b,
(9)
r kri
i|I r|Q k|K(i)
X !m > )0, i3I, j3J,
(10)
ij
ij ij
B !u Z
)0, i3I, r3Q, k3K(i), (11)
kri
kri k~1,r,I
u Z !B )0, i3I, r3Q, k3K(i),
(12)
kri kri
kri
¸ )w , r3R, j3J,
(13)
rj
rj
> , Z 3M0, 1N, i3I, j3J, r3Q, k3K(i),
(14)
ij kri
D , H , X , B , ¸ *0, i3I, j3J, r3R, k3K(i).
i ij ij kri rj
(15)
In the objective function of Problem STRP the
"rst term is the total inventory holding cost; the
second term is the total cost of the production

M. Diaby / Int. J. Production Economics 67 (2000) 219}233

223

Fig. 2. Basic notation.

resources used; the third term is the total setup cost
incurred; the last term is the amortized capital used
for setup time reductions (see [20]). The nonnegativity constraints (15) in conjunction with constraints (6) ensure that all demands are met without
backorder. Constraints (7) are the (manufacturing)
capacity constraints. They stipulate that the total
amount of production resources used in any given
period must be at least equal to the sum of the times
used for actually processing products in the period
plus the corresponding `before-reductiona setup
times, minus the applicable amounts of setup reduction.
Note that the setup reduction resources must be
used in the appropriate mix. This is enforced by
constraints (8). Constraint (9) enforces the limitation on the total capital available for setup reductions. Constraints (11) and (12) enforce the
piecewise-linear nature of the setup reduction func-

tions. Constraints (13) ensure that no more production resource is used than is available. Constraints
(14) and (15) are the usual binary and nonnegativity
requirements on the variables.
Because of its nonlinearity and combinatorial
nature, even small-sized instances of Problem
STRP cannot be solved optimally using standard
mathematical programming codes. The approach
we describe in the next section is based on dualizing
the constraint set in order to judiciously exploit the
problem structure, and can be used to e$ciently
solve large-scale instances of the problem to nearoptimality.

3. Solution methodology
The basic idea of the overall approach we use to
solve Problem STRP is to exploit the lot-sizing

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M. Diaby / Int. J. Production Economics 67 (2000) 219}233

sub-structure embedded in it. We do this by dualizing constraints (7) and (9) (see the work of Geo!rion
[21] or of Fisher [22]). This results in nonlinear,
mixed integer independent subproblems for the
products. We propose an e$cient dynamic programming procedure for solving these subproblems. Feasible solutions to the overall problem are
heuristically generated by perturbing the solutions
of the dualized problem. These ideas are developed
in the following discussion.

> , Z 3M0, 1N, j3J, r3Q, k3K(i),
ij kri

(22)

D , H , X , B , ¸ *0, j3J, r3R, k3K(i). (23)
i ij ij kri rj
Clearly, Problem PL is equivalent to N"DID
independent problems of the form:
Problem SP(i)
minimize Z (a, b)
i
subject to (17)}(23).

3.1. Lower bounds

subject to

Hence, Problem PL can be solved e$ciently via
Problem SP(i). Note however, that the setup reduction functions have a structure (i.e., piecewise linear) that is di!erent from those considered in the
single product models of Diaby [11] and Mekler
[18], respectively, and that multiple setup reduction resources are considered in Problem SP(i).
Fortunately however, despite these, Problem SP(i)
still has the well-known regeneration point property
of the standard Dynamic Lot-Sizing Problem, as
shown below.

¸ 3[0, w ] and
rj
rj
(H, X, D, >, B, Z) 3C ; i3I, j3J; r3R,
i
i
where

Proposition 1. There exists an optimal solution to
Problem SP(i) with the property H
X "0
i,j~1 ij
∀j3J.

Z (a, b)"+ h H #+ (s #a t )>
i
ij ij
i
j 0i ij
j|J
j|J

Proof. Using (20)}(23), de"ne inverse functions,
< (D), as
ik

Multiplying constraints (7) and (9) by multipliers
a (j3J) and b respectively, and adding them to the
j
objective of Problem STRP yields
Problem PL
minimize

+ Z (a, b)#+ + (c !a )¸ !MbbN
i
rj
j rj
i|I
j|J r|R

< (D)"w D,
i1
1i

!+ a D > #+ (p a )X
j i ij
i j ij
j|J
j|J
# + + (q#b)v B ;
(16)
r kri
r|Q k|K(i)
and C "M(H, X, D, >, B, Z)D(H, X, D, >, B, Z) satisi
i
"es (17)}(23)N with relations (17)}(23) de"ned by
H
!H #X "d , j3J,
i,j~1
ij
ij
ij

(17)

D ! + a B )0, r3Q,
i
kri kri
k|K(i)
X !m > )0, j3J,
ij
ij ij
B !u Z
)0, r3Q, k3K(i),
kri
kri k~1,r,
u Z !B )0, r3Q, k3K(i),
kri kri
kri

(18)
(19)
(20)
(21)

< (D)"<
((t !t
))!w (t !t
)
ik
i,k~1 0i
k~1,i
ki 0i
k~1,i
#w D for k3S2, i(i)T,
ki

(24)

where w ,+ ((q#b)v /a ), i3I; k3K(i).
ki
r|Q
r kri
Then, the last term of (16) (i.e., the amortized cost
component) can be expressed as a function, C (D),
i
of the total setup reduction amount, D, de"ned as
follows:
C (D)"< (D)
i
ki
if (t !t
))D)(t !t ); k3S1, i(i)T. (25)
0i
k~1,i
0i
ki
Hence, for a given value D of the setup time reducM SP(i) can be written as
tion of product i, Problem

M. Diaby / Int. J. Production Economics 67 (2000) 219}233

G

0;
f (0, D)"
i,t
R;

Problem P1(i, D):
minimize
Z1 (a, b, D)"+ h H #+ (s #a (t !D))>
i
ij ij
i
j 0i
ij
1
M
j|J
j|J

t"¹#1,
t3S1, ¹T,

225

(31)

f (n, D)"minn
Mg (t, n)#f n (n!1, D)N;
it
|Wt,T~n`1X i,D
i, `1
t3S1, ¹T, n3S1, ¹!t#1T,
(32)
where

#+ (p a )X #[C (D)]
i j ij
i M
j|J

(26)

g ( j, k),P ( j, k)!a D,
i,D
i
j

(s #a t )#(p a )d ;
k"j
j 0i
i j ij
P ( j, k), i
i
(s #a t )#+n
(+
h )d #(p a )+n
d ; k'j.
i
j 0i
|Wj`1,kX r|Wj,n~1X ir in
i j |Wj,kX in

G

subject to
H
!H #X "d , j3J,
(27)
i,j~1
ij
ij
ij
X !m > )0, j3J,
(28)
ij
ij ij
> 3M0, 1N j3J, k3K(i),
(29)
ij
H , X *0, j3J, k3K(i).
(30)
ij ij
Noting that Problem P1(i, D) is essentially
a Wagner}Whitin [23] problem,M it follows from
standard (extreme #ow) arguments that there exists
an optimal solution to Problem P1(i, D) such
1 such
that H
X "0 ∀j3J. Now, consider one
i,j~1 ij
solution (H, X, >) and let (H, X, >, D) be any
i
1 1 1 by1 ithe opti1 1 1 to
feasible solution
SP(i). Clearly,
mality of (H, X, >) for P1(i, D) we must have
i
1 , X, >) ), where
p (i, D) ((H, 1 X, 1 >) 1 ))p (i, D)((H
i
P1 1
1 (zz)
1 1in Problem
1 i
1 1 cost of P1
1 the
pz (zz) denotes
solution
(z).
Hence, clearly, we must have p ((H, X, >, D) ))
SP(i)
1 1 i
1 arbitrarip ((H, X, >, D) ) which implies (by1 the
SP(i) 1 1 1 1 i
ness of (H, X, >, D) ) that (H, X, >, D) is optimal
1 1 SP(i).
1 1 i
1 1 1 1 i
for Problem
Proposition 1 suggests that Problem SP(i) can be
solved e$ciently using a dynamic programming
approach if the setup reductions are treated parametrically. The reason for this is the discreteness
and relatively small size of the resulting state space.
In fact, let time periods correspond to stages. De"ne the states at a given stage, t, in terms of two
decision variables k (k*t) and n (n*1) representing the last period covered by setup at t and the
total number of setups between t and the last stage,
T, respectively. Finally, associate to the states return
functions, denoted f (n, D)'s, and de"ned by
it

(33)

(34)

Then, it is shown in Diaby [6] that Problem SP(i)
is solved optimally by "rst recursively applying
the relations (31) and (32) in a backward
dynamic programming pass to obtain f (D),
i0
min
Mf (r, D)N and then solving the univariate
r|W1,TX i1
optimization problem obtained by adding the
setup reduction cost function C (D) to f (D).
i
i0
Note that the order of computation of this
dynamic programming approach is roughly the
same as that of the standard DLSP, irrespective
of the number of intervals that comprise the range
of C (D). Hence, Problem SP(i) can be solved
i
e$ciently. A Minimum-Cost Network Flow representation corresponding to the dynamic programming reformulation described above is illustrated
in Fig. 3.
The overall lower bound for Problem STRP is
obtained using a subgradient optimization procedure. Initially, the multipliers for the manufacturing
capacity constraints (constraints (7)) are set equal
to the regular time costs; the initial multiplier for
the setup reduction budget (constraint (9)) is set
equal to zero. Multipliers at other steps of the
subgradient procedure are generated according to
the classic formula:
uk`1"max[0; uk#t (Axk!b) ]
i
k
i
i
where A and b are the matrix of coe$cients, and the
right-hand-side vector for constraints 2.7 and 2.9,
respectively; xk is the solution Problem PL at iteration k; uk, the vector of multipliers at iteration k;
and t , a positive scalar given by
k
t "ak(Z0!Zk)/(Axk!b)2
k

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M. Diaby / Int. J. Production Economics 67 (2000) 219}233

Fig. 3. Illustration of the network representation of problem SP(i) (T"5).

where ak3(0, 2], Z0 is an upper bound on Problem
STRP, and Zk is the value of Problem PL at iteration k. We use an initial step size of a "1.8 and
0
multiply a by 0.8 if the improvement in the lower
k
bound is not more than 0.5% in 5 consecutive
trials. The overall subgradient optimization procedure is terminated if the improvement in the
bound is not more than 0.5% in 25 consecutive
trials or if the gap ratio (i.e., the lower bound to
upper bound ratio) is 98% or more.
Note that if C (D) } and therefore, the setup
i
reduction functions } is a discrete function of D,
then the cost of each of the corresponding discrete
alternatives can be determined by simply evaluating f (D) for the corresponding setup reduction
i0
amounts and adding it to C (D). Hence, our proi
posed lower bounding scheme can be used directly
when discrete technology decisions must be made

as well, as indicated earlier in this paper (see
Section 1).
3.2. Upper bounds
The procedure we use for generating upper
bounds consists of smoothing the solutions to Problem PL at chosen iterations of the subgradient
optimization procedure. The basic idea is to use the
setup decisions of the solutions (to Problem PL) to
formulate and solve transportation problems in
order to obtain feasible setup reduction amounts
and production quantities.
The subgradient procedure described above is
performed twice. During both passes we check for
feasibility at each iteration } as required by a normal subgradient optimization step. We record the
solution at hand as the new incumbent if it is

M. Diaby / Int. J. Production Economics 67 (2000) 219}233

feasible and if its cost is lower than that of the
incumbent at hand. Smoothing is undertaken only
during the second pass, starting at the (0.95n)th
iteration and whenever the lower bound improves,
where n is the iteration at which the highest lower
bound was obtained during the "rst pass. A similar
procedure was used in Diaby et al. [24]. Here
however, the setup reduction amounts must be
perturbed also (in addition to the batch sizes) in
order to achieve feasibility. The approach used for
this is as follows.
Given an optimal solution (HH, XH, DH, >H, BH, ZH)
to Problem PL, in order to obtain feasible setup
reduction amounts for Problem STRP, the products are "rst sorted in a descending order of the
ratios:

C

D

(35)
R " + + a (>H) (DH) ]/[C ((DH) ) ,
j
ij
i
i
i
i
r|R j|J
where C (z) is as given by relation (31) (see Section
i
3.1). Then, the setup reductions are set equal to
their values in the optimal solution to Problem PL,
respectively, starting with the "rst product in the
list and continuing on until the setup reduction
budget is used up. Speci"cally, let the ordered
product indices be i(1), i(2),2, i(N); let k be
such that +
[C ((DH) )](b, and
s|W1,k~1X i(s)
i(s)
+
[C ((DH) )]*b; denote by K (z) the ins|W1,kX i(s)
i(s)
i
verse function of C (z). The perturbed setup reduci
tion amounts are obtained as follows:
(DH) "
i(s)
(D@) ,
s3S1, k!1T;
i(s)
K (b! + (C ((DH) ))), s"k;
i
i(s)
i(s)
s|W1,k~1X
0,
s3Sk#1, NT.

G

(36)
A set of perturbed setup decision vectors is generated next, based on the perturbed setup reduction
amounts. This is done by solving Problem P (i, D)
1
for products i(k) through i(N) } or i(k) through i(M),
where M is such that (DH) "0 with (DH))
i(M)
i(M~1)
'0 } with D set equal to (D@) (s3Sk, NT) (see
i(s)
1
Section 3.1). These perturbed setup decisions (for
products i(k) through i(N)) along with the optimal
setup decisions of Problem PL (for products i(1)
through i(k!1)) are then used to formulate trans-

227

portation problems in order to obtain feasible solutions to the overall problem (Problem STPR).

4. Testing procedure and computational results
Although the problem of determining optimal
batch sizes in multi-product, dynamic-parameter
manufacturing environments } a special case of
Problem STRP } has been extensively researched,
realistically sized test problems are not readily
available from the literature. This is particularly
true for the case where setup times are positive.
The test problems we used in this research were
generated according to the scheme described in
Diaby et al. [24], except for the setup reduction
parameters. We solved a set of 24 problems to
evaluate the performance of the model. We also
undertook some statistical analyses of the results in
an attempt to uncover possible general relationships between the setup time reduction and various
problem parameters. These are discussed in the
following.
4.1. Experimental design and test problems
For the purpose of testing we only considered
one setup reduction resource } namely, capital
} and assumed the setup reduction function of each
product to be logarithmic (see the work of Diaby
[11] and of Porteus [20]). Hence, for a given product i, every time a basic incremental investment
(BII) x dollars is made, the setup time is reduced by
i
a "xed reduction fraction (FRF), y . So, for
i
example, if x "$500, y "0.10 and the setup time
i
i
before reduction is t "10 hours, then it would
0i
cost $500 to reduce the setup time to 9 hours, $1000
to reduce to 8.1 hours, $1500 to reduce to
7.29 hours, etc. We performed a piecewise-linear
approximation of these logarithmic functions by
assuming the setup reduction function to be linear
over the ranges [0, x ], [x , 2x ], [2x , 3x ], etc. (see
i
i
i
i
i
Figs. 1a and b). Hence, the slope of the setup reduction function over the kth range is given by
(37)
!a z "!(y (1!y )k~1)t )/x
i
i
0i i
ki
and the length of each interval is u z "x (see
ki
i
Figs. 1a and b).

228

M. Diaby / Int. J. Production Economics 67 (2000) 219}233

By using the explicit expressions developed in
Diaby [11] for the logarithmic setup reduction cost
function, the number of reduction ranges required
to allow for up to a (100d )% reduction can be
i
easily derived as

G

log(1!d )/log(1!y ),
i
i
if log(1!d )/log(1!y ) is integer;
i
i
(38)
z (d )"
i i
int (log(1!d )/log(1!y ))#1,
i
i
otherwise
where int(z) and log(z) stand for the integer part and
the natural logarithm of (z) respectively, and d is
i
some arbitrary real number between 0 and 1. For
example, if y "0.10 and it is desired to allow for
i
up to a 99% setup time reduction, then the number
of reduction ranges required is 44. Clearly, the
maximum number of feasible reduction ranges
(with respect to the budget constraint) is

G

b/x ,
if b/x is integer;
i
i
(z@) "
(39)
i
int(b/x )#1, otherwise.
i
Hence, the maximum number of reduction ranges
that need to be considered in Problem STRP for
product i is
i(i)"minMz (d ), (z@) N.
(40)
i i
i
We set d "0.95 for all products in all the test
i
problems and generated the x and y from uniform
i
i
distributions with parameters [BIILO, BIIHI] and
[FRFLO, FRFHI], respectively. We "xed FRFLO
and FRFHI at 0.1 and 0.4 respectively and considered three levels for the unit setup reduction cost
by considering three ranges for the BII's. The
low level of setup reduction cost corresponds to
[BIILO, BIIHI]"[$500, $1500]; The medium
level corresponds to [BIILO, BIIHI]"[$1500,
$3000]; and the high level, to [BIILO,
BIIHI]"[$3000, $4500].
The setup reduction budget was generated by
"rst computing the total amount of capital, b ,
.!9
required to reduce the setup time of each product
by 95% and then multiplying this number by
a `budget factora, BF. The budget factor BF is
essentially a measure of what fraction of the products can be reduced by 95%. We considered two

levels for the budget constraint tightness. The low
level corresponds to a BF randomly generated between 0.4 and 0.7. For the high level BF was taken
from a uniform distribution with parameters 0.1
and 0.4.
Only one level was considered for the demand
variability and the unit overtime cost, respectively.
Two levels (low and high) were considered for the
capacity constraint tightness. The number of products was either 100 or 150; the number periods was
15. Finally, the amortization fraction of the setup
reduction capital was generated from a uniform
distribution on [0.1, 0.3] in all the problems.
4.2. Computational performance of the model
The computational results are summarized in
Tables 1}3. Note that the "rst two letters in the
`Problem Namesa shown in these tables corresponds to the capacity constraint tightness and the
budget constraint tightness respectively, with `La
indicating a low level and `Ha indicating a high
level, as described above. Similarly, the third letter
indicates the setup reduction cost level with `Ha,
`Ma, and `La denoting the high, medium and low
levels, respectively. Finally, the last three digits in
the `problem namesa indicate the number of products (either 100 or 150).
The performance of the overall procedure is
shown in Table 1. Computational times are quite
impressive. They range between 4.20 and 34.69 seconds (IBM ES9000/580) with an average of
20.11 seconds for the 100-product problems. The
times for the 150-product problems averaged
30.78 seconds with a range going from 6.58 to
52.13 seconds. All the solutions obtained were
within 2% of lower bounds. The gap ratio (i.e., the
ratio of lower bound to upper bound) was 0.9848
on average for the 100-product problems and
0.9866 for the 150-product problems; the corresponding ranges were 0.9808 to 0.9927, and 0.9802
to 0.9997, respectively. Hence, it appears that solution quality is equally good with respect to three of
the factors of our experimental design (namely,
problem size, setup reduction budget constraint
tightness, and setup reduction cost). On the other
hand, however, computational times appear to be
sensitive to the capacity constraint tightness and

M. Diaby / Int. J. Production Economics 67 (2000) 219}233
Table 1
Summary of computational performance
Problem!
name

Gap"
ratio

CPU seconds
(IBM9000)

Number Percent
improvement
of iterations

LLL100
LLM100
LLH100
LHL100
LHM100
LHH100
HLL100
HLM100
HLH100
HHL100
HHM100
HHH100

0.9898
0.9911
0.9835
0.9820
0.9817
0.9830
0.9817
0.9927
0.9808
0.9813
0.9846
0.9858

17.88
5.75
4.20
20.36
19.81
22.99
22.73
24.83
20.75
34.69
25.27
22.07

13
2
1
12
12
13
16
14
11
23
17
11

5.15
7.56
8.39
18.72
16.28
14.18
9.24
5.37
6.24
26.21
21.80
17.06

Average

0.9848

20.11

12

13.02

LLL150
LLM150
LLH150
LHL150
LHM150
LHH150
HLL150
HLM150
HLH150
HHL150
HHM150
HHH150

0.9822
0.9997
0.9880
0.9843
0.9819
0.9869
0.9837
0.9924
0.9855
0.9855
0.9894
0.9802

25.97
7.87
30.21
41.35
28.18
19.08
27.20
6.58
52.13
39.36
42.84
48.61

12
2
12
18
11
6
12
1
21
17
15
16

6.79
6.47
6.09
19.55
16.50
14.65
9.06
8.38
8.18
23.53
18.12
14.80

Average

0.9866

30.78

12

12.68

!See Section 4 for explanation.
"(lower bound)/(upper bound).

the combination of the setup reduction budget
tightness and the setup reduction cost level.
Greater times are incurred when the budget constraint tightness is high with the setup reduction
cost at the low or medium level. A justi"cation for
this may be that all the products may tend to have
high levels of setup reduction in the solution to the
relaxed problem (Problem PL) } because of the low
costs } whereas this may not be feasible } because
of the tight budget constraint.
Resource utilizations are shown in Table 2. The
average regular time utilization is 84.32% for the
100-product problems and 84.57% for the 150product problems; the corresponding ranges are

229

75.70% to 91.93% for the 100-product problems
and 74.22% to 93.27% for the 150-product problems. The average overtime utilizations (in the
periods where overtime was used) are 9.44% and
7.22% for the 100-product and 150-product problems, respectively. In general, it appears that overtime tends to be used only when the capacity
constraint tightness is high with either a tight setup
reduction budget constraint or a high setup reduction cost. An explanation for this may be that both
overtime and setup reduction contribute to increased e!ective capacity as pointed out in Kim et
al. [12] and Spence and Porteus [17]. The average
percents setup time reduction are 32.78% and
31.32% for the 100-product and 150-product problems, respectively. They appear to be sensitive to
the setup reduction budget constraint tightness
only.
Table 3 shows the resource allocation in the "nal
solution. Regular time production contributes
72.43% and 73.25% on average to total costs with
ranges going from 62.07% to 80.05% and from
63.56% to 79.62% for the 100-product and 150product problems, respectively. In general, it
appears that the percent contributions of inventory
and setups to total costs vary in the same direction
as the budget constraint tightness is changed. Speci"cally, it appears they both increase when the
setup reduction budget constraint tightness is increased. The amortized setup reduction cost as
a percentage of total costs is 0.53% on average for
the 100-product problems and ranges between
0.11% and 1.38%; the corresponding average for
the 150-product problems is 0.67% with a range
from 0.14% to 1.89%. We also computed the percent contribution of the setup reduction program
to the operating cash requirement over the horizon.
This is shown in the last column of the table
(Table 3). The average values for these are 2.69%
and 2.79% for the 100-product and 150-product
problems respectively, with ranges going from
0.56% to 5.70% for the 100-product problems
and from 0.55% to 6.65% for the 150-product
problems. Hence, the amount of money allocated
to setup reductions is relatively small compared to
the overall manufacturing budget, which is
consistent with basic principles of Continuous Improvement.

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M. Diaby / Int. J. Production Economics 67 (2000) 219}233

Table 2
Summary of resource utilization
Problem!
name

Regular
time (%)

Overtime (%)

Periods with
overtime

LLL100
LLM100
LLH100
LHL100
LHM100
LHH100
HLL100
HLM100
HLH100
HHL100
HHM100
HHH100

75.70
75.84
78.39
81.19
83.63
82.77
87.92
85.55
86.28
91.57
91.93
91.05

0.00
0.00
0.00
0.00
8.16
0.50
17.13
17.89
9.26
21.17
24.14
14.98

0
0
0
0
2
1
5
1
4
10
9
5

99.48
98.80
87.27
98.11
96.79
98.56
99.86
94.55
84.79
99.12
99.48
96.92

42.11
45.95
43.57
20.78
21.21
22.16
43.82
47.89
42.53
22.61
19.80
20.90

Average

84.32

9.44

3

96.14

32.78

LLL150
LLM150
LLH150
LHL150
LHM150
LHH150
HLL150
HLM150
HLH150
HHL150
HHM150
HHH150

74.22
76.46
77.58
81.39
81.78
81.38
87.53
87.42
88.29
93.27
92.91
92.61

0.00
0.00
0.60
9.09
10.50
0.00
17.88
20.68
0.00
12.91
5.74
9.27

0
0
1
1
1
0
5
3
0
9
4
5

94.97
99.77
89.83
97.42
97.11
97.43
98.76
92.27
89.12
99.52
100.00
82.51

48.13
42.60
40.70
21.11
17.60
20.95
42.33
41.11
41.46
20.38
20.61
18.80

Average

84.57

7.22

94.89

31.32

2.4

Setup reduction
budget (%)

Average % setup
reduction

!See Section 4 for explanation.

4.3. Statistical analyses
In an attempt to gain some more general insight
into the problem, we performed some statistical
analyses of the computational results. First, we
made trend plots of the fraction setup reductions
e!ected in our "nal solutions against the ratio of
the setup time-to-carrying cost (ST/CC) and the
ratio of the Basic Incremental Investment to the
Fixed Reduction Fraction (BII/(100*FRF)).
A sample of these plots is shown in Fig. 4. In
general, these plots seem to indicate that on average the amount of setup reduction e!ected increases as the ST/CC ratio increases and decreases
as the BII/FRF ratio increases.

We hypothesized the following relation between
the setup reduction amount, R , and the various
i
problem parameters:
tc1 yc4 dc5 pc6
R "c 0i i i i
i
0 hc2 xc3
i i
where i is the product index; c through c , are
0
6
scalar coe$cients; d , the average period-demand;
i
p , the standard deviation of the demand per perii
od; x , the Basic Incremental Investment (BII); y ,
i
i
the Fixed Reduction Fraction (FRF); h , the unit
i
holding cost per period, and t , the setup time
0i
before reduction.
We then went about estimating the coe$cients
by performing 24 (`log}loga) regressions (i.e., one

M. Diaby / Int. J. Production Economics 67 (2000) 219}233

231

Table 3
Summary of resource allocation
Setup#
reduction
cash (%)

Setup" reduction
cost (%)

4.01
4.35
5.10
6.55
8.84
6.16
4.99
3.76
4.02
9.44
7.73
6.87

1.77
4.14
5.70
0.63
1.29
2.24
1.84
4.59
5.67
0.56
1.33
2.47

0.44
0.83
1.38
0.16
0.17
0.55
0.48
0.73
0.95
0.11
0.21
0.39

19.89

5.99

2.69

0.53

0.00
0.00
0.02
0.24
0.28
0.00
2.21
1.59
0.00
2.46
0.52
1.08

15.52
17.38
18.95
22.60
23.03
21.85
17.57
16.87
16.95
23.42
22.15
21.12

4.33
4.29
4.34
6.94
6.72
5.76
4.92
2.89
3.91
10.56
7.39
6.35

1.95
3.97
6.23
0.69
1.30
2.53
2.04
3.92
6.65
0.55
1.54
2.05

0.52
0.81
1.32
0.16
0.14
0.54
0.60
1.01
1.89
0.09
0.43
0.55

0.70

19.78

5.70

2.79

0.67

Problem!
name

Regular"
time cost
(%)

Over-time"
cost(%)

Setup2
cost (%)

LLL100
LLM100
LLH100
LHL100
LHM100
LHH100
HLL100
HLM100
HLH100
HHL100
HHM100
HHH100

78.75
78.56
74.57
71.39
65.30
70.65
75.17
80.05
78.40
62.07
64.82
69.42

0.00
0.00
0.00
0.00
0.40
0.01
2.13
0.47
0.95
4.48
4.80
1.72

17.24
16.26
18.95
21.90
25.30
22.63
17.71
15.00
15.68
23.90
22.45
21.61

Average

72.43

1.25

LLL150
LLM150
LLH150
LHL150
LHM150
LHH150
HLL150
HLM150
HLH150
HHL150
HHM150
HHH150

79.62
78.33
75.37
70.06
69.83
71.84
74.70
77.64
77.25
63.56
69.94
70.91

Average

73.25

Holding"
cost (%)

!See Section 4 for explanation.
"Based on total cost including the amortized capital used for setup reduction.
#Based on total cost including the cash expended for setup reduction.

for each problem of the experimental design). Except for the unit holding cost and the standard
deviation of the period-demand, all the coe$cients
had the sign hypothesized for them. The average
values we found for c through c are 0.3850,
0
6
0.8601, !0.2007, 0.4132, 0.4138, 0.6942, !0.5672.
The average R-square for the regressions was
0.5969 with a range going from 0.2590 to 0.8555.
The most signi"cant factor (based on the p-values)
a!ecting the setup reduction appears to be the
length of the setup time, followed in order by the
average demand, the standard deviation of demand, the FRF, the BII, and the unit holding cost.

Hence, the practice of reducing the longest setups
"rst which is often used by practitioners [8,9] appears to be somewhat justi"ed, although as indicated by the plots of Fig. 4 and the regression
results above, this can yield solutions that may be
far from optimal.

5. Conclusions
We have presented a comprehensive model for
simultaneously planning for the setup time reductions and the batch sizes of several products in

232

M. Diaby / Int. J. Production Economics 67 (2000) 219}233

Fig. 4. Sample trend line plots for the setup reduction amounts.

capacitated, dynamic manufacturing environments
and a heuristic method for solving it. The model
incorporates setup costs, setup times, multiple
manufacturing resources, and multiple setup reduction resources. It considers an arbitrary setup reduction cost structure including the case where
discrete alternative technologies must be selected.
Because of its broad generality and computational
e$ciency we believe the model can be used directly
in many practical situations. In particular, it can be
used by managers to make `what}ifa analyses in
order to gain context-speci"c insights into their
particular setup reduction problems. The model
can also serve as a useful tool to researchers in
evaluating the performance of existing `rules of
thumba and more complex heuristics for solving
the problem, or in guiding the development of new
rules-of-thumb and heuristics. Finally, the proposed model and solution methodology can be
extended with relative ease to more general decision-making contexts such as hierarchical production planning contexts (see [25]) or multistage
manufacturing contexts (see [26,27]).
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