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 re-quirements of today's competitive global market, organizations have had to implement advanced manufacturing technologies such as Group Tech-nology (GT), Flexible Manufacturing Systems (FMS) and Just-In-Time (JIT) manufacturing. It is well known however, that the single most impor-tant prerequisite to the successful use of all these technologies is the shortness of the setup/change-over times. Short setup/changesetup/change-over times allow for smaller lots and inventories, which in turn can lead
*Tel.: 860-486-5140; fax: 860-486-4839.
E-mail address:[email protected] (M. Diaby).
to (i) higher quality, (ii) lower waste and rework, (iii) increased process yield and productivity, (iv) in-creased awareness of the causes of errors and de-lays, 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], indus-trial 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 improve-ment investimprove-ment decisions within an operational context began to appear in the mid-1980s. These
models can be broadly classi"ed either as` techno-logy selectionamodels or`setup reductiona mod-els. The technology selection models are essentially concerned with discrete technology choices in the contexts of Group Technology and Flexible Manu-facturing 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 im-provement (Kaizen) and JIT manufacturing con-cepts. 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 reduc-tion funcreduc-tions. 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 manufac-turing environment and are, therefore, EOQ based. Papers that have dealt with multi-product situ-ations 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 adynamic manu-facturing 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 "nd-ing 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 environ-ments and developed expressions for the setup re-duction 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 reduc-tion priority rules (along the lines of Freeland et al. [14]) against each other. To the best of our know-ledge, 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 in-dustries 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 co-ordinating the setup process of the di!erent prod-ucts 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.
Improve-ments 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
standard-ization 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 indi-vidual products. For example, after standardizing fasteners, a company may focus on acquiring/devel-oping new tools, dies, methods, and/or accessory equipments for individual products as the next step. The third (and last) stage, termed rationalization
stageis focused on improving the environment of the
setup process (i.e., material quality, product design, machine reliabilities and capabilities, etc.). More de-tailed accounts of speci"c company practices for each of these can be found in Leschke [3].
The focus of this paper is on thestandardization
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 mathemat-ical programming model of this problem is for-mulated 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 simulta-neously 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 pro-duction 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 typi-cally 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 in-clude Research and Development (R&D) labor time and cost, the capital available for the acquisi-tion 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 costacomponent
and a`labor costacomponent. The`directa com-ponent consists only of the administrative costs and the cost of the material lost in the calibration pro-cess. The labor time component corresponds to the labor time used for setups. For simplicity, we will henceforth use the term`setup costato refer to the direct component of the setup cost only. Hence, the (direct) setup cost and the setup time are indepen-dent 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 invento-ries 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 hor-izon, before the beginning of the"rst period. This assumption is justi"ed by the high investments that are typically required at thestandardization stageof 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 re-source, the setup time reduction function is a gen-eral, 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 reduc-tion funcreduc-tion that may be encountered in practice. A special case corresponding to thelogarithmicand
powerforms 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 ad-justed 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
i the amount by which the
Fig. 1. (a) Illustration of the setup cost reduction function
}general case. (b) Illustration of the setup cost reduction} func-tion convex case.
corresponding setup time after reduction, then the piecewise linear setup reduction functions can be modeled in a mathematical program as follows:
¹ where Q is the index set for the setup reduction resources;K(i) is the index set for the setup reduc-tion ranges; i(i) is the number of setup reduction ranges; the u
kri's are the maximum amounts of
setup reduction resources corresponding to the setup reduction ranges respectively; thea
kri are the
setup reduction rates over the setup reduction ranges respectively (i.e., a
kri"(tk~1,i!tki)/ukri; ∀k,r,i); theB
kri are decision variables representing the respective amounts of setup reduction resources used in each range; the Z
kri are binary variables used to enforce the piecewise-linear nature of the setup reduction functions; and thet
kri are as shown 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 setMn,n#1,n#2,2,n#tNand0(z) to
denote the value of Problem (z). The setup time
reduction model is formulated as follows:
Problem STRP
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 non-negativity constraints (15) in conjunction with con-straints (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 re-duction.
Note that the setup reduction resources must be used in the appropriate mix. This is enforced by constraints (8). Constraint (9) enforces the limita-tion on the total capital available for setup reduc-tions. Constraints (11) and (12) enforce the piecewise-linear nature of the setup reduction
func-tions. Constraints (13) ensure that no more produc-tion 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$ciently solve large-scale instances of the problem to near-optimality.
3. Solution methodology
sub-structure embedded in it. We do this by dualiz-ing constraints (7) and (9) (see the work of Geo!rion [21] or of Fisher [22]). This results in nonlinear, mixed integer independent subproblems for the products. We propose an e$cient dynamic pro-gramming procedure for solving these subprob-lems. 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.
3.1. Lower bounds
Multiplying constraints (7) and (9) by multipliers a
j(j3J) andbrespectively, and adding them to the objective of Problem STRP yields
Problem PL
"es (17)}(23)Nwith relations (17)}(23) de"ned by
H Clearly, Problem PL is equivalent to N"DID independent problems of the form:
Problem SP(i)
minimizeZ i(a,b) subject to (17)}(23).
Hence, Problem PL can be solved e$ciently via Problem SP(i). Note however, that the setup reduc-tion funcreduc-tions have a structure (i.e., piecewise lin-ear) that is di!erent from those considered in the single product models of Diaby [11] and Mekler [18], respectively, and that multiple setup reduc-tion resources are considered in Problem SP(i). Fortunately however, despite these, Problem SP(i) still has the well-knownregeneration point property
of the standard Dynamic Lot-Sizing Problem, as shown below.
Proposition 1. There exists an optimal solution to
Problem SP(i) with the property H
i,j~1Xij"0 ∀j3J.
Proof. Using (20)}(23), de"ne inverse functions,
< Then, the last term of (16) (i.e., the amortized cost component) can be expressed as a function,C
i(D), of the total setup reduction amount,D, de"ned as follows:
C
i(D)"<ki(D) if (t
0i!tk~1,i))D)(t0i!tki);k3S1,i(i)T. (25) Hence, for a given valueD
M of the setup time
P
Noting that Problem P1(i,D
M) is essentially
a Wagner}Whitin [23] problem, it follows from standard (extreme#ow) arguments that there exists
an optimal solution to Problem P1(i,D
1) such feasible solution to SP(i). Clearly, by the opti-mality of (H Hence, clearly, we must havepSP(
i)((H11 ,X11 ,>11 ,D1)i))
Proposition 1 suggests that Problem SP(i) can be solved e$ciently using a dynamic programming approach if the setup reductions are treated para-metrically. 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 variablesk(k*t) andn(n*1) represent-ing the last period covered by setup at t and the total number of setups betweentand the last stage,
T, respectively. Finally, associate to the statesreturn
functions, denotedf
it(n,D)'s, and de"ned by is solved optimally by "rst recursively applying the relations (31) and (32) in a backward
dynamic programming pass to obtain f
i0(D),
min
r|W1,TXMfi1(r,D)Nand then solving the univariate
optimization problem obtained by adding the setup reduction cost function C
i(D) to fi0(D).
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
i(D). Hence, Problem SP(i) can be solved e$ciently. A Minimum-Cost Network Flow repres-entation corresponding to the dynamic program-ming reformulation described above is illustrated in Fig. 3.
The overall lower bound for Problem STRP is obtained using a subgradient optimization proced-ure. Initially, the multipliers for the manufacturing capacity constraints (constraints (7)) are set equal to the regular time costs; the initial multiplier for the setup reduction budget (constraint (9)) is set equal to zero. Multipliers at other steps of the subgradient procedure are generated according to the classic formula:
uk`1
i "max[0;uki#tk(Axk!b)i]
whereAandbare the matrix of coe$cients, and the right-hand-side vector for constraints 2.7 and 2.9, respectively;xkis the solution Problem PL at iter-ationk;uk, the vector of multipliers at iterationk; andt
k, a positive scalar given by t
Fig. 3. Illustration of the network representation of problem SP(i) (T"5).
whereak3(0, 2],Z0is an upper bound on Problem STRP, and Zkis the value of Problem PL at iter-ationk. We use an initial step size ofa
0"1.8 and
multiplya
kby 0.8 if the improvement in the lower bound is not more than 0.5% in 5 consecutive trials. The overall subgradient optimization pro-cedure 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
i(D) } and therefore, the setup reduction functions } is a discrete function of D, then the cost of each of the corresponding discrete alternatives can be determined by simply evaluat-ing f
i0(D) for the corresponding setup reduction
amounts and adding it to C
i(D). Hence, our pro-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 ofsmoothingthe solutions to Prob-lem 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.
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, wherenis 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 prod-ucts are "rst sorted in a descending order of the ratios: 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 +s|W1,k~1X[C
i(s)((DH)i(s))](b, and
+s|W1,kX[C
i(s)((DH)i(s))]*b; denote by Ki(z) the in-verse function ofC
i(z). The perturbed setup reduc-tion amounts are obtained as follows:
(DH)
A set of perturbed setup decision vectors is gener-ated next, based on the perturbed setup reduction amounts. This is done by solving Problem P
1(i,D)
for productsi(k) throughi(N)}ori(k) throughi(M), whereMis such that (DH)
i(M)"0 with (DH))i(M~1)
'0 } with D
1 set equal to (D@)i(s)(s3Sk,NT) (see Section 3.1). These perturbed setup decisions (for productsi(k) throughi(N)) along with the optimal setup decisions of Problem PL (for products i(1) throughi(k!1)) are then used to formulate
trans-portation problems in order to obtain feasible solu-tions 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 relation-ships 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 prod-uct i, every time a basic incremental investment (BII)x
idollars is made, the setup time is reduced by
a "xed reduction fraction (FRF), y
i. So, for example, ifx
i"$500,yi"0.10 and the setup time before reduction is t
0i"10 hours, then it would 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
i], [xi, 2xi], [2xi, 3xi], etc. (see Figs. 1a and b). Hence, the slope of the setup reduc-tion funcreduc-tion over thekth range is given by
!a
kzi"!(yi(1!yi)k~1)t0i)/xi (37) and the length of each interval is u
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
i)% reduction can be easily derived as
the natural logarithm of (z) respectively, anddi is
some arbitrary real number between 0 and 1. For example, if y
i"0.10 and it is desired to allow for 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
(z@)
Hence, the maximum number of reduction ranges that need to be considered in Problem STRP for productiis
i(i)"minMz
i(di), (z@)iN. (40)
We set d
i"0.95 for all products in all the test problems and generated thex
iandyifrom uniform distributions with parameters [BIILO, BIIHI] and [FRFLO, FRFHI], respectively. We"xed FRFLO and FRFHI at 0.1 and 0.4 respectively and con-sidered 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 prod-ucts can be reduced by 95%. We considered two
levels for the budget constraint tightness. The low level corresponds to a BF randomly generated be-tween 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 prod-ucts 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 corres-ponds 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`Ladenoting the high, medium and low levels, respectively. Finally, the last three digits in the`problem namesaindicate the number of prod-ucts (either 100 or 150).
Table 1
Summary of computational performance Problem!
LLL100 0.9898 17.88 13 5.15
LLM100 0.9911 5.75 2 7.56
LLH100 0.9835 4.20 1 8.39
LHL100 0.9820 20.36 12 18.72 LHM100 0.9817 19.81 12 16.28 LHH100 0.9830 22.99 13 14.18
HLL100 0.9817 22.73 16 9.24
HLM100 0.9927 24.83 14 5.37
HLH100 0.9808 20.75 11 6.24
HHL100 0.9813 34.69 23 26.21 HHM100 0.9846 25.27 17 21.80 HHH100 0.9858 22.07 11 17.06 Average 0.9848 20.11 12 13.02
LLL150 0.9822 25.97 12 6.79
LLM150 0.9997 7.87 2 6.47
LLH150 0.9880 30.21 12 6.09
LHL150 0.9843 41.35 18 19.55 LHM150 0.9819 28.18 11 16.50
LHH150 0.9869 19.08 6 14.65
HLL150 0.9837 27.20 12 9.06
HLM150 0.9924 6.58 1 8.38
HLH150 0.9855 52.13 21 8.18
HHL150 0.9855 39.36 17 23.53 HHM150 0.9894 42.84 15 18.12 HHH150 0.9802 48.61 16 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 con-straint 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 150-product problems; the corresponding ranges are
75.70% to 91.93% for the 100-product problems and 74.22% to 93.27% for the 150-product prob-lems. The average overtime utilizations (in the periods where overtime was used) are 9.44% and 7.22% for the 100-product and 150-product prob-lems, respectively. In general, it appears that over-time tends to be used only when the capacity constraint tightness is high with either a tight setup reduction budget constraint or a high setup reduc-tion cost. An explanareduc-tion for this may be that both overtime and setup reduction contribute to in-creased 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 prob-lems, respectively. They appear to be sensitive to the setup reduction budget constraint tightness only.
Table 2
Summary of resource utilization Problem!
name
Regular time (%)
Overtime (%) Periods with overtime
Setup reduction budget (%)
Average % setup reduction
LLL100 75.70 0.00 0 99.48 42.11
LLM100 75.84 0.00 0 98.80 45.95
LLH100 78.39 0.00 0 87.27 43.57
LHL100 81.19 0.00 0 98.11 20.78
LHM100 83.63 8.16 2 96.79 21.21
LHH100 82.77 0.50 1 98.56 22.16
HLL100 87.92 17.13 5 99.86 43.82
HLM100 85.55 17.89 1 94.55 47.89
HLH100 86.28 9.26 4 84.79 42.53
HHL100 91.57 21.17 10 99.12 22.61
HHM100 91.93 24.14 9 99.48 19.80
HHH100 91.05 14.98 5 96.92 20.90
Average 84.32 9.44 3 96.14 32.78
LLL150 74.22 0.00 0 94.97 48.13
LLM150 76.46 0.00 0 99.77 42.60
LLH150 77.58 0.60 1 89.83 40.70
LHL150 81.39 9.09 1 97.42 21.11
LHM150 81.78 10.50 1 97.11 17.60
LHH150 81.38 0.00 0 97.43 20.95
HLL150 87.53 17.88 5 98.76 42.33
HLM150 87.42 20.68 3 92.27 41.11
HLH150 88.29 0.00 0 89.12 41.46
HHL150 93.27 12.91 9 99.52 20.38
HHM150 92.91 5.74 4 100.00 20.61
HHH150 92.61 9.27 5 82.51 18.80
Average 84.57 7.22 2.4 94.89 31.32
!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 aver-age the amount of setup reduction e!ected in-creases as the ST/CC ratio inin-creases and dein-creases as the BII/FRF ratio increases.
We hypothesized the following relation between the setup reduction amount, R
i, and the various problem parameters:
R i"c0
tc1
0iyci4dci5pci6 hc2
i xci3
where i is the product index; c0 through c6, are scalar coe$cients;d
i, the average period-demand; p
i, the standard deviation of the demand per peri-od;x
i, the Basic Incremental Investment (BII);yi, the Fixed Reduction Fraction (FRF); h
i, the unit holding cost per period, and t
0i, the setup time before reduction.
Table 3
Summary of resource allocation Problem!
name
Regular"
time cost (%)
Over-time"
cost(%)
Setup2 cost (%)
Holding"
cost (%)
Setup#
reduction cash (%)
Setup"reduction cost (%)
LLL100 78.75 0.00 17.24 4.01 1.77 0.44
LLM100 78.56 0.00 16.26 4.35 4.14 0.83
LLH100 74.57 0.00 18.95 5.10 5.70 1.38
LHL100 71.39 0.00 21.90 6.55 0.63 0.16
LHM100 65.30 0.40 25.30 8.84 1.29 0.17
LHH100 70.65 0.01 22.63 6.16 2.24 0.55
HLL100 75.17 2.13 17.71 4.99 1.84 0.48
HLM100 80.05 0.47 15.00 3.76 4.59 0.73
HLH100 78.40 0.95 15.68 4.02 5.67 0.95
HHL100 62.07 4.48 23.90 9.44 0.56 0.11
HHM100 64.82 4.80 22.45 7.73 1.33 0.21
HHH100 69.42 1.72 21.61 6.87 2.47 0.39
Average 72.43 1.25 19.89 5.99 2.69 0.53
LLL150 79.62 0.00 15.52 4.33 1.95 0.52
LLM150 78.33 0.00 17.38 4.29 3.97 0.81
LLH150 75.37 0.02 18.95 4.34 6.23 1.32
LHL150 70.06 0.24 22.60 6.94 0.69 0.16
LHM150 69.83 0.28 23.03 6.72 1.30 0.14
LHH150 71.84 0.00 21.85 5.76 2.53 0.54
HLL150 74.70 2.21 17.57 4.92 2.04 0.60
HLM150 77.64 1.59 16.87 2.89 3.92 1.01
HLH150 77.25 0.00 16.95 3.91 6.65 1.89
HHL150 63.56 2.46 23.42 10.56 0.55 0.09
HHM150 69.94 0.52 22.15 7.39 1.54 0.43
HHH150 70.91 1.08 21.12 6.35 2.05 0.55
Average 73.25 0.70 19.78 5.70 2.79 0.67
!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). Ex-cept for the unit holding cost and the standard deviation of the period-demand, all the coe$cients had the sign hypothesized for them. The average values we found for c0 through c6 are 0.3850, 0.8601,!0.2007, 0.4132, 0.4138, 0.6942,!0.5672. The average R-square for the regressions was 0.5969 with a range going from 0.2590 to 0.8555. The most signi"cant factor (based on thep-values) a!ecting the setup reduction appears to be the length of the setup time, followed in order by the average demand, the standard deviation of de-mand, the FRF, the BII, and the unit holding cost.
Hence, the practice of reducing the longest setups "rst which is often used by practitioners [8,9] ap-pears to be somewhat justi"ed, although as in-dicated by the plots of Fig. 4 and the regression results above, this can yield solutions that may be far from optimal.
5. Conclusions
Fig. 4. Sample trend line plots for the setup reduction amounts.
capacitated, dynamic manufacturing environments and a heuristic method for solving it. The model incorporates setup costs, setup times, multiple manufacturing resources, and multiple setup reduc-tion resources. It considers an arbitrary setup re-duction cost structure including the case where discrete alternative technologies must be selected. Because of its broad generality and computational e$ciency we believe the model can be used directly in many practical situations. In particular, it can be used by managers to make `what}ifaanalyses 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 pro-posed model and solution methodology can be extended with relative ease to more general deci-sion-making contexts such as hierarchical produc-tion planning contexts (see [25]) or multistage manufacturing contexts (see [26,27]).
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