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Quality improvement through product redesign and the learning curve
OMEGA Int. J. of Mgmt Sci., Vol. 20, No. 2, pp. 161-168, 1992 Printed in Great Britain. All rights reserved
0305-0483/92 $5.00 + 0.00 Copyright © 1992 Pergamon Press plc
Quality Improvement Through Product Redesign and the Learning Curve C KOULAMAS Florida International University, USA (Received December 1990; in revisedform June 1991) This paper presents a dynamic programming model for studying the effects of product redesign on the value, coot, and quality control processes in a single product environment. The model provides the optimal redesign policy, that is the optimal depth and the optimal timing for implementing the redesign, so the accumulated net product value can be maximiT~l. The accumulated net product value is used as a measure of product quality, however the model formulation allows for the use of other quality functions as well. The model can be used with different sets of learning rates and cost data. It can be also used with non-uniform learning rates among the different processes, and non-uniform redesign effects on the value, cost, and quality control learning curves. Selective results demonstrate that the early implementation of the optimal redesign level enhances the accumulated net product value.
Key words--learning curve, dynamic programming, quality improvement, product redesign
1. INTRODUCTION
of learning on productivity is also studied in [9], and that effect is then incorporated into THE IMPACT of learning curves in various an aggregate planning model. Shared learning aspects of production has been thoroughly among different activities has been considered in examined in the literature. Earlier contributions [10] for advanced manufacturing technologies are surveyed in [15]. A significant portion of the and in [7] for a group technology setting. literature deals with the determination of optiThe learning phenomenon can also impact mal lot sizes under learning effects. The early the quality level of a production process. In [14] work in [8] was followed by the model in [1] a linkage is established between quality control which assumes equal lot sizes or equal pro- and the production learning process. A stochasduction intervals for deriving an analytical tic dynamic programming approach is develexpression for the optimal lot size. In [6] and oped for computing quality control policies and later in [12] the equal lot size assumption is lifted justifying the intensive use of quality control in by introducing a dynamic programming formu- manufacturing. lation. In [13] an analytical expression is develThere are various measures of quality in a oped for the optimal lot size assuming that there production process. A measure of quality is the is a linear relationship between production time increased value of the product as the production and lot size under learning effects. Another process progresses. Another measure of quality aspect of a production environment which has is the reduction of the number of quality control been thoroughly studied under learning effects is related problems as the production process proassembly line balancing. Procedures for design- gresses. Additional factors for measuring qualing assembly lines under learning effects have ity may be applicable with the overall quality been presented in [4] and [3] among others. The being measured by the compounded effect of impact of learning on production planning has these factors. These factors are usually subject also been studied in [11] and [2]. The effect to learning effects resulting in improved overall 161
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Koulamas--Quality Improvement
quality as the production process progresses, However, in some production processes the product quality is not satisfactory even when most of the learning effects have been incorporated. An alternative for further improving product quality is the early introduction of partial or complete product redesign. The introduction of product redesign will both improve the value of the product and reduce the number of quality control related problems. On the negative side, it will induce some unlearning on the production process itself. This unlearning can be quantified by assuming that the learning curve associated with the product cost is shifted backward when product redesign is introduced, To our knowledge, there are no quantitative models in the literature for examining the effect of product redesign on product quality when learning is applicable, The objective of this paper is to present a quantitative model for studying the effects of product redesign on product quality in a single product environment. Product quality is measured by both the increasing value of the product and the decreasing number of quality control related problems as the production process progresses. The overall objective is the maximization of the net production output value which is measured as the accumulated product value minus the summation of related costs. These costs are the production cost, the cost due to quality control related problems, and the cost of implementing the product redesign (if any). A forward dynamic programming formulation is presented with a two-dimensional stage vector. One dimension represents all feasible intervals at which the production process can be interrupted for implementing product redesign and the other dimension represents all feasible portions of redesign to be implemented. The states are defined as all possible combinations with respect to the depth and the timing of redesign in order to reach a specific stage, The output is the optimal redesign policy implemented during the production of a predetermined quantity. The model can be used for providing management with insights about the effect of product redesign on the overall production process, and for determining the optimal timing and depth of product redesign, The rest of the paper is organized as follows, Section two presents the production process
under study. Section three presents the dynamic programming formulation. Section four presents an example problem for applying the dynamic programming algorithm. Section five summarizes the resulting conclusions.
2. THE PRODUCTION PROCESS Consider a production process for manufacturing a single product operating under the following assumptions: (1) The performance of the process is measured by the accumulated net value of its output. (2) The gross value of the production output is a measure of quality. This value increases as the production process goes on due to learning effects. Usually, the gross value of a product is its selling price. (3) During the production process a small number of quality control related problems are encountered. These problems may be the realignment of a machine part which was accidentally moved during production, the reworking of a unit which turned out defective, etc. Each time a quality control related problem occurs, there is an associated penalty cost due to additional needed work, lost production time, etc. The number ofquality control related problems decreases as time progresses due to learning effects. (4) The production cost is also subject to learning resulting in lower costs as the production process goes on. (5) The possibility of partially (or completely) redesigning the product is available at discrete production intervals. The depth and the frequency of product redesign is subject to technological constraints and these constraints are treated as input to the proposed model. There is a fixed redesign cost incurred whenever product redesign is implemented and a variable redesign cost proportional to the depth of the implemented redesign.
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Omega, Vol. 20, No. 2
(6) The implementation of product redesign results in increased overall product quality. Specifically, product redesign increases the gross value (e.g. the selling price) of the product and also decreases the number of quality control related problems, (7) The implementation of product redesign induces some unlearning in the cost process, that is the learning curve associated with the cost is shifted backward whenever product redesign is implemented, (8) A predetermined production quantity must be produced. (9) The product is produced in small batches with a predetermined batch size. This assumption is justified by the fact that about 75% of dollar volume of the metal-worked products are manufactured in batches of less than 50 parts [5]. Each batch is sufficiently small and continuously produced so it can be assumed that learning becomes effective as the p r o duction process goes on to the next batch but not within a batch due to the relatively small batch size. (10) The objective is the maximization of the accumulated net value of the production output by implementing partial (or complete) product redesign subject to assumptions (1)--(9).
Assumption (4) deals with the production cost. Using learning curve theory, the cost of producing lot (batch) m may be expressed as c,,= C,m -c
(1)
where c = - l o g St~log 2, Cm is the cost (time) for the m th batch, C~ is the cost (time) for the first batch, and St = C~/Cm = 2 -c. The latter term (St) expresses the learning associated with the cost process and it is often called the slope of the learning curve. The relationship between Cm and m can be shown as a straight line when using a log scale for m. Notice that it is possible to assume that only a small proportion of the cost is subject to learning. For example, the part of the cost representing material cost is not subject to learning. In this research we consider only the part of the cost which is subject to learning. The cumulative cost of producing any sequence of Q consecutive batches (L + 1, L + 2. . . . . L + Q) can be approximated as rL+Q JL
C~m -c dm = C~[(L + Q)"-c) _ L o -c)]/(1 _ c).
(2)
Equation (2) approximates an otherwise discrete function by a continuous function inducing an error. The error can be reduced if the integration limits are adjusted resulting in the cumulative cost given as rL+e+0s /
C, m -c dm
dL-O.5
There are other decisions usually associated with production processes like lot sizing decisions. Similarly there are other costs associated with these decisions like setup costs and inventory carrying costs. However these decisions are not directly related with redesign decisions which is the focus of this study. Furthermore, the problem of determining optimal lot sizes under leafing effects has been thoroughly studied in the literature with the main contributions being mentioned in the introduction. Consideration of lot sizing policies will unnecessarily complicate the model and obscure the focus of the paper. Therefore lot sizing decisions are not considered, and it is assumed that the optimal lot (batch) size is predetermined, We will now present quantitative expressions for the above assumptions, OME 20/2--C
=C~[(L+Q+O.5)(~-c)-(L-0.5)~-c)]/(1-c).
(3)
The technique of changing the integration limits will be applied to all remaining similar cases in order to reduce the approximation error. Assumption 3 deals with the occurrence of a number of quality control related problems during the production process. The frequency of occurrence for these problems decreases due to learning effects as the production process goes on according to the following pattern: d,, = d i m -e (4) where d = - l o g Sd/log 2, d,, is the percentage of time the production process is temporarily interrupted due to quality control problems while the m th batch is produced, d~ is the percentage of time the production process is
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Koulamas--Quality Improvement
temporarily interrupted due to quality control problems while the first batch is produced, and Sa = d~/dm = 2 -a. The latter term (Sa) expresses the learning associated with quality control related problems. Notice that Sd does not have to coincide with Sc introduced earlier. In other words the cost process may experience a learning rate different than the one experienced by the quality control process, The penalty cost due to quality control related problems can be computed during any segment of the production process, For example, the penalty cost due to quality control problems incurred during the production of a sequence of Q consecutive batches (L + 1, L + 2 . . . . . L + Q) is P
d~m a dm
= Pd t [(L + Q + 0.5) I~- ~
wL-0.5 -(L-0.5)"-~l/(l-d)
(5)
where P is the penalty cost per quality control problem and the integrated expression is the accumulated percentage of time the pro duction process is temporarily interrupted due to quality control problems while producing Q consecutive batches. Assumption (2) deals with the gross value of the production output. In the cost and quality control processes, the presence of learning results in decreasing the cost and the number of quality control problems respectively, as the production process goes on. However, in the product value process, the presence of learning results in increasing the gross value of the output as the production process goes on. Increased gross value can also be thought as improved quality. From a mathematical point of view, an increasing learning curve function is needed. As a result, using learning curve theory, the gross value of the mth batch produced may be expresses as Vm= V~mt (6) where v = - l o g Sv/log 2, Vm is the gross value of the mth batch, V~is the gross value of the first batch, and S~ = V2m/V,, = 2~. The latter term (Sv) expresses the learning associated with the value (quality). Notice that the value (quality) process may be subject to a learning rate different than the rates applicable to the cost and/or the quality control processes,
The cumulative gross value of any sequence of Q consecutive batches (L + 1, L + 2 . . . . . L + Q) can be computed as ~L+Q+05Vlm~ dm /JL_0.5
= VI[(L + Q +0.5) 0+~0
-(L-0.5)"+°)]/0 + v). (7) The possibility of partially (or completely) redesigning the product and the redesigning effect on all the processes considered thus far will now be discussed. The redesign process is by definition product specific. For some products, partial redesign is not an option at all. For other products the redesign process is not continuous and only specific predetermined portions of redesign are feasible. However, in the general case the only assumption needed is that (some) partial product redesign is a feasible option. The formulation we propose in the next section is general enough to handle different redesign policies from continuous redesign to any discrete redesign policy implemented at any point of the production process as long as some partial redesign is allowed. There is a cost associated with product redesign which can be expressed as Kf+ rKv
(8)
where Kf is a fixed cost incurred whenever an arbitrary level of product redesign is implemented (Kf is independent of the redesign depth) and K~ is a variable cost proportional to the depth r (0 < r ~< 1) of the implemented redesign. Introducing a (0, 1) variable, equation (8) can be rewritten as uK:+rKo
(9)
with u = 0, when r = 0, and u = 1, when 0 < r ~< 1. Let us now study the effect of redesigning the product on all the factors (cost, value, and quality control related problems) associated with the production process. Redesigning the product is supposed to improve the value (quality) of the product and also reduce the number of quality control problems. At the same time the implementation of product redesign will incur some unlearning on the production process. This unlearning will be equivalent to shifting the cost learning curve backward. In this
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Omega, Vol. 20, No. 2
case equation (3) (giving the cost of producing Q consecutive batches) is written as f t.'+Q+O.5C1m-¢dm
(10)
being implemented at various stages of the production process. Consequently, a dynamic programming formulation is appropriate with the optimal value function
'-0.5
where L ' = L ( 1 - z c r ) , and zc is a coefficient
f(x,,,y,,)
(13)
(0 ~
expressing the accumulated net value of the production process after x~ units have been produced with a Yn portion of product redesign already implemented. The two-dimensional stage vector is determined by the combinations of all feasible intervals at which the production process can be interrupted for implementing product redesign and all feasible portions of redesign. For example, let us assume that a production process can be interrupted for redesign purposes only when one or more quarters of the projected output have been completed. Let us also assume that the redesign
process requires that a complete product r+o+O-Sd,m_ddm redesign can be implemented in either one or • o.5 (11) two steps. In that case the two-dimensional stage vector consists of 4 x 2 = 8 points. where L ' = L(1 + Zdr), and Zd is a coefficient The objective is the maximization of the (z a >~0) measuring the redesign effect on reducaccumulated net value f(x,,,y,,). Usually, the ing the number of quality control related prob- accumulated net value is the realized profit. lems. High Zd values correspond to an increased The dynamic programming recursive equation such effect. can now be developed. We can use either a Finally, the redesign process has a positive forward or a backward formulation. Howeffect on the value (quality) of the production ever, for computational purposes a forward output. This positive effect is equivalent to formulation is preferred. Let us first comshifting forward the learning curve associated pute the accumulated net value ( C V ) resultwith the product value process. Equation (7) ing from producing a sequence of batches (giving the value of Q consecutive batches) can (x,,-Q,,- 1,x,-Q,-2 ..... x,,-Q,) assumbe rewritten as ing that a portion y, - r of redesign has been implemented prior to the production of this -0.5 Vtm*dm (12) sequence of batches and that a portion r of product redesign will be implemented during the where L ' = L(1 +zvr), and z,, is a coefficient current production run. The use of equations (z,~>0) measuring the redesign effect on (9)-(12) results in increasing the gross product value. In both equations (11) and (12) maxi~L,+Q~+05 mum improvement is realized when a complete C V(Q,,, r)= rIi l l - 0 , 5 V~mv dm redesign (r = 1) is implemented. In the next f(DL2+Qn+0.5 ) section we present a dynamic programming -P~/ dtm-ddm [ . J L2 -- 0.5 formulation for studying the overall redesign rL3+Q~+0.5 effect on the production process. - | C,m-Cdm-(uK:+rK,,) J L3-0.5 (14)
f:
, + e + o . ~
3. THE DYNAMIC PROGRAMMING FORMULATION The objective of this paper is to study the potential effects of various levels of redesign
where L 1 = (x~- Qn)(1 + zvy~) L2 = (x,--Q~)(I + zdyn) L3 =(x~--Q~)(I-z~y~)
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Koulamas--Quality Improvement
and the rest of the symbols are as defined previously. Using equations (13) and (14) we write the forward dynamic programming recursive equation as
f(x.,y.)=
m a x {cv(a.,r)+f(x.-a.,y.-r)} (15) 0~,.~y. 0~Q..~x. The two dimensional stage vector was defined earlier. The state vector at a particular stage f(x,,y,) is defined by the inequalities (0 ~
The other one-dimensional problem is when the timing of the redesign is predetermined and the objective is to find the optimal depth of the redesign. The presented formulation can solve this problem without any modifications. In this case the state vector is also one-dimensional with its elements being different levels of product redesign. In the next section the use of the dynamic programming algorithm is demonstrated through an example problem. 4. EXAMPLE-APPLICATIONS The forward dynamic programming algorithm was programmed in FORTRAN. The FORTRAN code can accommodate arbitrary redesign policies and arbitrary timings for implementing these redesigns. However, if a continuous redesign policy is in effect with a continuous timing pattern for implementing the redesign, then the well known dimensionality problem of dynamic programming will arise. Let us apply the dynamic programming algorithm to some simple example problems with the following date. The overall production quantity is Q = 200 batches with implementations of redesign being allowed every time 10 consecutive batches are produced. Complete redesign of the product is allowed, with the restriction that when a redesign operation is undertaken, the level of redesign should be either a 20% or a multiple of 20% improvement over the initial design. These conditions result in a (21 × 6) stage vector. The gross value of the first batch produced is Vt = $60.0 and its cost is C~ = $45.0. The learning rate is assumed to be the same for all three processes subject to learning (value, cost and quality control processes).The learning rate is treated as a problem parameter with two different levels. When low learning is in effect, Sv = Sc = Sd = 0.95. When high learning is in effect Sv = Sc = Sd = 0.9. The number and penalty cost of quality control problems are also treated as a problem parameter with two different impacts on the production process. For a low impact, the initial percentage of time the production process is temporarily interrupted due to quality control problems is d~ = 0.05 and the penalty cost for fixing these problems is P = $60.0 per problem. For a high impact these values are d~ = 0.1 and P = $150.0 respectively. The redesign cost was also treated as a problem parameter with
Omega, Vol.
two levels. When the redesign cost is low, Ks= $150.0 and K ~ = $45.0. When the redesign cost is high KI = $750.0 and K~ = $225.0. Also, it is initially assumed (for the first eight problems considered) that the redesign effect is uniform on the learning curves of all three processes (value, cost, and quality control), or equivalently zv = z~ = Zd = 1. The results are tabulated in Table 1. In each case we computed the accumulated net value of not only the optimal redesign policy, but also the accumulated net value of the two extreme redesign policies (complete redesign and no redesign at all) for sensitivity analysis purposes. A general conclusion is that when the optimal policy calls for redesigning the product, most of that redesign should take place during the early stages of the production process. This is attributed to the fact that the exponential learning curves considered in this study experience an early drastic change, and afterwards they relatively stabilize. The cost curve and the curve associated with the number of quality control problems experience early drastic drops while the gross value curve experi-
20, N o . 2
167
ences an early drastic increase. Implementing the redesign process early results in moving further forward on both the value process learning curve and on the learning curve associated with the number of quality control problems while these curves experience their drastic changes. The early redesign is not advantageous with respect to the cost learning curve, since it causes a backward shift on that curve before it experiences its drastic drop. However the realized benefits on the other two curves from early redesign compensate for the backward shift on the cost curve. Another indication supporting this argument is that when higher learning rates are in effect the optimal redesign policy calls for an increased optimal level of redesign. This indicates that the realized benefits from an increased redesign level, that is the increased product value and the reduced number of quality control problems compensate for the resulting additional costs on the cost process. Also, when higher learning is in effect the production process is more sensitive to the various problem parameters and the redesign option becomes more attractive. For example,
Table 1. Results for the example problems Problem S~ = S c = S d = 0.95 K f = 7 5 0 , K,~=225 d 1=0.05, P=60 S~. = Sc = S d = 0.95 K f = 150, 1(,.=45 d~ = 0.05, P = 60 S, = S C= S d = 0.9 K I = 750, K~ = 225 d 1=0.05, P=60
Optimal value Redesign policy
Value for 0% redesign
Value for 100% redesign
10095 r* = 4 0 % after Q = 5 %
10006
9888
10166 after Q = 5 %
10006
10068
19152
19910
19152
20090
r*=40%
20088 r* = 60% after Q = 5%
S~ = S~ = S d = 0.9 K I = 150, K ~ = 4 5 dl = 0.05, P = 60
20198 r* = 6 0 % after Q = 5 % r~ = 20% after Q = 55%
S, = S, = S d = 0.95 K f = 750, K,.= 225 d I = 0 . 1 , P = 150
8296 after Q = 5 %
8167
8095
r* = 4 0 %
S, = S,~ = S d = 0.95 K / = 150, [ ( , = 4 5 dt=0.1, P=150
8368 after Q = 5 %
8167
8275
r* = 4 0 %
S~ = S,. = S d = 0.9 K f = 750, K~ = 225 dI=0.1, P=150
18840 rl* = 60% after Q ~ 5 %
17824
18670
S,. = S~ = S a = 0.9 K I = 150, K , = 4 5 d~ = 0 . 1 , P = 150
18956 r* = 6 0 % after Q = 5 % r~' = 20% after Q ~ 5 5 %
17824
18850
S~ = S¢ = S a = 0.9 K f = 150, K~.=45 d I = 0 . 1 , P = 150 z~= 3, z~= 0.5, z,t= 3
22927 r* = 100% after Q = 5%
17824
Same as optimal
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Koulamas--Quality Improvement
under higher learning (So = S t = Sd = 0.9), when the redesign cost is low the overall optimal redesign level is higher (equal to 60% + 20% = 80%) compared to the optimal redesign level of 60% when the redesign cost is high. In contrast, under a lower learning rate (So = Sc = Sa = 0.95), there is no difference in the optimal redesign level when the redesign cost varies, We wanted to examine a case where the optimal redesign policy is a complete redesign, In order to do so, we experimented with different redesign effects on the value, cost, and quality control processes (the last problem in Table 1). Specifically we allowed a more significant redesign effect on both the gross value and the quality control processes by letting za = zv = 3, and a less significant redesign effect on the cost process by letting zc = 0.5. In this case additional redesign of the product should become more desirable because it further reduces the number of quality control related
value, cost, and quality control processes in a single product environment. The dynamic programming formulation allows management to experiment with different sets of learning rates and cost data. It also allows experimenting with non-uniform learning rates as well as non-uniform redesign effects on the learning curves associated with the value, cost, and quality control processes. In any case the model provides the optimal redesign policy, that is the optimal redesign level and the optimal timing for implementing the redesign. Selective results demonstrate that the early implementation of the optimal redesign level enhances the accumulated net product value. Finally the dynamic programming algorithm is easily modifiable in the case when other quality functions must be used.
problems, it further increases the gross value of the product, and at the same time it has a less negative impact on the cost process. This was
14-20. 2. BehnezhadAR and KhoshnevisB (1988)The effectsof manufacturingprogress function on machine requirements and aggregate planning problems. Int. J. Prodn Res. 26(2), 309-326. 3. Boucher TO (1987) Choice of assembly line design
verified by the results of Table 1. The second to last problem in Table 1 is identical to the problem currently considered except that zc = Zd = Zv = 1. In that problem the optimal policy calls for a 60% + 20% = 80% product redesign. As expected, when Zd = Z~,= 3, and z, = 0.5 (last problem in Table 1), the optimal policy calls for a complete (100%) product redesign. Finally, the production process accumulated
net values resulting from the extreme redesign policies (complete redesign or no redesign at all) can be used for sensitivity analysis purposes so we can study their deviation from the optimal policy's accumulated net value for different learning rates and different parameter values. In general, management has the option to experiment with different data sets since the algorithm treats the learning rates, the initial number of quality control problems, and the initial value and cost data as input. Furthermore, the redesign effects on the value, cost, and quality control processes are also treated as input. This point is further emphasized in the conclusions section. 5. CONCLUSIONS A quantitative model was developed for studying the product redesign effects on the
REFERENCES 1. Adler G and Nanda R (1974) The effects of learning on optimal lot size determination--Single product case.
AIlE Trans. 6(1),
under task learning. Int. J. Prodn Res. 24(4), 513-524.
4. Chakravarty AK (1988) Line balancing with task learning effects, liE Trans. 20(2), 186-193. 5. Cook NH (1975) Computer-managed part manufacture. Sci. Am. 232(2), 22. 6. Fisk JC and Ballou DP (1982) Production lot sizing under a learning effect, l i e Trans. 14(4), 257-264. 7. Globerson S and Millen R (1989) Determining learning curves in group technology settings. Int. J. Prodn Res. 27(10), 1653-1664. 8. Keachie EC and Fontana RJ (1966) Effects of learning on optimal lot size. Mgmt Sci. 13(2), 102-108. 9. Khoshnevis B and Wolfe PM (1983) An aggregate production planning model incorporating dynamic productivity: Part I. Model development, l i e Trans. 15(2),
111-118. 10. Meredith J and Camm J (1989) Modelling synergy and
learning under multiple advanced manufacturing technologies. Decis. Sci. 20, 258-271. 11. Smunt T (1987) The impact ofworker forgetting on production scheduling. Int. J. Prodn Res. 25(5), 689-701. 12. Smunt TL and Morton TE (1985) The effects of learning
on optimal lot sizes: Further developmentson the single product case. lIE Trans. 17(1), 33-37. 13. Sule DR (1978) The effect of alternate periods of learning and forgetting on economic manufacturing quality. AI1E Trans. 10(3), 338-343. 14. Tapiero CS (1987) Production learning and quality control, llE Trans. 19(4), 362-369. 15. Yelle LE (1979) The learning curve: Historical review and comprehensive survey. Decis. Sci. 10, 302-328. FOR CORRESPONDENCE:Professor C Koulamas, Department of Decision Sciences and Information Sciences, FloridalnternationalUniversity, University Park, FL 33199, USA.
ADDRESS
0305-0483/92 $5.00 + 0.00 Copyright © 1992 Pergamon Press plc
Quality Improvement Through Product Redesign and the Learning Curve C KOULAMAS Florida International University, USA (Received December 1990; in revisedform June 1991) This paper presents a dynamic programming model for studying the effects of product redesign on the value, coot, and quality control processes in a single product environment. The model provides the optimal redesign policy, that is the optimal depth and the optimal timing for implementing the redesign, so the accumulated net product value can be maximiT~l. The accumulated net product value is used as a measure of product quality, however the model formulation allows for the use of other quality functions as well. The model can be used with different sets of learning rates and cost data. It can be also used with non-uniform learning rates among the different processes, and non-uniform redesign effects on the value, cost, and quality control learning curves. Selective results demonstrate that the early implementation of the optimal redesign level enhances the accumulated net product value.
Key words--learning curve, dynamic programming, quality improvement, product redesign
1. INTRODUCTION
of learning on productivity is also studied in [9], and that effect is then incorporated into THE IMPACT of learning curves in various an aggregate planning model. Shared learning aspects of production has been thoroughly among different activities has been considered in examined in the literature. Earlier contributions [10] for advanced manufacturing technologies are surveyed in [15]. A significant portion of the and in [7] for a group technology setting. literature deals with the determination of optiThe learning phenomenon can also impact mal lot sizes under learning effects. The early the quality level of a production process. In [14] work in [8] was followed by the model in [1] a linkage is established between quality control which assumes equal lot sizes or equal pro- and the production learning process. A stochasduction intervals for deriving an analytical tic dynamic programming approach is develexpression for the optimal lot size. In [6] and oped for computing quality control policies and later in [12] the equal lot size assumption is lifted justifying the intensive use of quality control in by introducing a dynamic programming formu- manufacturing. lation. In [13] an analytical expression is develThere are various measures of quality in a oped for the optimal lot size assuming that there production process. A measure of quality is the is a linear relationship between production time increased value of the product as the production and lot size under learning effects. Another process progresses. Another measure of quality aspect of a production environment which has is the reduction of the number of quality control been thoroughly studied under learning effects is related problems as the production process proassembly line balancing. Procedures for design- gresses. Additional factors for measuring qualing assembly lines under learning effects have ity may be applicable with the overall quality been presented in [4] and [3] among others. The being measured by the compounded effect of impact of learning on production planning has these factors. These factors are usually subject also been studied in [11] and [2]. The effect to learning effects resulting in improved overall 161
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Koulamas--Quality Improvement
quality as the production process progresses, However, in some production processes the product quality is not satisfactory even when most of the learning effects have been incorporated. An alternative for further improving product quality is the early introduction of partial or complete product redesign. The introduction of product redesign will both improve the value of the product and reduce the number of quality control related problems. On the negative side, it will induce some unlearning on the production process itself. This unlearning can be quantified by assuming that the learning curve associated with the product cost is shifted backward when product redesign is introduced, To our knowledge, there are no quantitative models in the literature for examining the effect of product redesign on product quality when learning is applicable, The objective of this paper is to present a quantitative model for studying the effects of product redesign on product quality in a single product environment. Product quality is measured by both the increasing value of the product and the decreasing number of quality control related problems as the production process progresses. The overall objective is the maximization of the net production output value which is measured as the accumulated product value minus the summation of related costs. These costs are the production cost, the cost due to quality control related problems, and the cost of implementing the product redesign (if any). A forward dynamic programming formulation is presented with a two-dimensional stage vector. One dimension represents all feasible intervals at which the production process can be interrupted for implementing product redesign and the other dimension represents all feasible portions of redesign to be implemented. The states are defined as all possible combinations with respect to the depth and the timing of redesign in order to reach a specific stage, The output is the optimal redesign policy implemented during the production of a predetermined quantity. The model can be used for providing management with insights about the effect of product redesign on the overall production process, and for determining the optimal timing and depth of product redesign, The rest of the paper is organized as follows, Section two presents the production process
under study. Section three presents the dynamic programming formulation. Section four presents an example problem for applying the dynamic programming algorithm. Section five summarizes the resulting conclusions.
2. THE PRODUCTION PROCESS Consider a production process for manufacturing a single product operating under the following assumptions: (1) The performance of the process is measured by the accumulated net value of its output. (2) The gross value of the production output is a measure of quality. This value increases as the production process goes on due to learning effects. Usually, the gross value of a product is its selling price. (3) During the production process a small number of quality control related problems are encountered. These problems may be the realignment of a machine part which was accidentally moved during production, the reworking of a unit which turned out defective, etc. Each time a quality control related problem occurs, there is an associated penalty cost due to additional needed work, lost production time, etc. The number ofquality control related problems decreases as time progresses due to learning effects. (4) The production cost is also subject to learning resulting in lower costs as the production process goes on. (5) The possibility of partially (or completely) redesigning the product is available at discrete production intervals. The depth and the frequency of product redesign is subject to technological constraints and these constraints are treated as input to the proposed model. There is a fixed redesign cost incurred whenever product redesign is implemented and a variable redesign cost proportional to the depth of the implemented redesign.
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(6) The implementation of product redesign results in increased overall product quality. Specifically, product redesign increases the gross value (e.g. the selling price) of the product and also decreases the number of quality control related problems, (7) The implementation of product redesign induces some unlearning in the cost process, that is the learning curve associated with the cost is shifted backward whenever product redesign is implemented, (8) A predetermined production quantity must be produced. (9) The product is produced in small batches with a predetermined batch size. This assumption is justified by the fact that about 75% of dollar volume of the metal-worked products are manufactured in batches of less than 50 parts [5]. Each batch is sufficiently small and continuously produced so it can be assumed that learning becomes effective as the p r o duction process goes on to the next batch but not within a batch due to the relatively small batch size. (10) The objective is the maximization of the accumulated net value of the production output by implementing partial (or complete) product redesign subject to assumptions (1)--(9).
Assumption (4) deals with the production cost. Using learning curve theory, the cost of producing lot (batch) m may be expressed as c,,= C,m -c
(1)
where c = - l o g St~log 2, Cm is the cost (time) for the m th batch, C~ is the cost (time) for the first batch, and St = C~/Cm = 2 -c. The latter term (St) expresses the learning associated with the cost process and it is often called the slope of the learning curve. The relationship between Cm and m can be shown as a straight line when using a log scale for m. Notice that it is possible to assume that only a small proportion of the cost is subject to learning. For example, the part of the cost representing material cost is not subject to learning. In this research we consider only the part of the cost which is subject to learning. The cumulative cost of producing any sequence of Q consecutive batches (L + 1, L + 2. . . . . L + Q) can be approximated as rL+Q JL
C~m -c dm = C~[(L + Q)"-c) _ L o -c)]/(1 _ c).
(2)
Equation (2) approximates an otherwise discrete function by a continuous function inducing an error. The error can be reduced if the integration limits are adjusted resulting in the cumulative cost given as rL+e+0s /
C, m -c dm
dL-O.5
There are other decisions usually associated with production processes like lot sizing decisions. Similarly there are other costs associated with these decisions like setup costs and inventory carrying costs. However these decisions are not directly related with redesign decisions which is the focus of this study. Furthermore, the problem of determining optimal lot sizes under leafing effects has been thoroughly studied in the literature with the main contributions being mentioned in the introduction. Consideration of lot sizing policies will unnecessarily complicate the model and obscure the focus of the paper. Therefore lot sizing decisions are not considered, and it is assumed that the optimal lot (batch) size is predetermined, We will now present quantitative expressions for the above assumptions, OME 20/2--C
=C~[(L+Q+O.5)(~-c)-(L-0.5)~-c)]/(1-c).
(3)
The technique of changing the integration limits will be applied to all remaining similar cases in order to reduce the approximation error. Assumption 3 deals with the occurrence of a number of quality control related problems during the production process. The frequency of occurrence for these problems decreases due to learning effects as the production process goes on according to the following pattern: d,, = d i m -e (4) where d = - l o g Sd/log 2, d,, is the percentage of time the production process is temporarily interrupted due to quality control problems while the m th batch is produced, d~ is the percentage of time the production process is
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temporarily interrupted due to quality control problems while the first batch is produced, and Sa = d~/dm = 2 -a. The latter term (Sa) expresses the learning associated with quality control related problems. Notice that Sd does not have to coincide with Sc introduced earlier. In other words the cost process may experience a learning rate different than the one experienced by the quality control process, The penalty cost due to quality control related problems can be computed during any segment of the production process, For example, the penalty cost due to quality control problems incurred during the production of a sequence of Q consecutive batches (L + 1, L + 2 . . . . . L + Q) is P
d~m a dm
= Pd t [(L + Q + 0.5) I~- ~
wL-0.5 -(L-0.5)"-~l/(l-d)
(5)
where P is the penalty cost per quality control problem and the integrated expression is the accumulated percentage of time the pro duction process is temporarily interrupted due to quality control problems while producing Q consecutive batches. Assumption (2) deals with the gross value of the production output. In the cost and quality control processes, the presence of learning results in decreasing the cost and the number of quality control problems respectively, as the production process goes on. However, in the product value process, the presence of learning results in increasing the gross value of the output as the production process goes on. Increased gross value can also be thought as improved quality. From a mathematical point of view, an increasing learning curve function is needed. As a result, using learning curve theory, the gross value of the mth batch produced may be expresses as Vm= V~mt (6) where v = - l o g Sv/log 2, Vm is the gross value of the mth batch, V~is the gross value of the first batch, and S~ = V2m/V,, = 2~. The latter term (Sv) expresses the learning associated with the value (quality). Notice that the value (quality) process may be subject to a learning rate different than the rates applicable to the cost and/or the quality control processes,
The cumulative gross value of any sequence of Q consecutive batches (L + 1, L + 2 . . . . . L + Q) can be computed as ~L+Q+05Vlm~ dm /JL_0.5
= VI[(L + Q +0.5) 0+~0
-(L-0.5)"+°)]/0 + v). (7) The possibility of partially (or completely) redesigning the product and the redesigning effect on all the processes considered thus far will now be discussed. The redesign process is by definition product specific. For some products, partial redesign is not an option at all. For other products the redesign process is not continuous and only specific predetermined portions of redesign are feasible. However, in the general case the only assumption needed is that (some) partial product redesign is a feasible option. The formulation we propose in the next section is general enough to handle different redesign policies from continuous redesign to any discrete redesign policy implemented at any point of the production process as long as some partial redesign is allowed. There is a cost associated with product redesign which can be expressed as Kf+ rKv
(8)
where Kf is a fixed cost incurred whenever an arbitrary level of product redesign is implemented (Kf is independent of the redesign depth) and K~ is a variable cost proportional to the depth r (0 < r ~< 1) of the implemented redesign. Introducing a (0, 1) variable, equation (8) can be rewritten as uK:+rKo
(9)
with u = 0, when r = 0, and u = 1, when 0 < r ~< 1. Let us now study the effect of redesigning the product on all the factors (cost, value, and quality control related problems) associated with the production process. Redesigning the product is supposed to improve the value (quality) of the product and also reduce the number of quality control problems. At the same time the implementation of product redesign will incur some unlearning on the production process. This unlearning will be equivalent to shifting the cost learning curve backward. In this
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case equation (3) (giving the cost of producing Q consecutive batches) is written as f t.'+Q+O.5C1m-¢dm
(10)
being implemented at various stages of the production process. Consequently, a dynamic programming formulation is appropriate with the optimal value function
'-0.5
where L ' = L ( 1 - z c r ) , and zc is a coefficient
f(x,,,y,,)
(13)
(0 ~
expressing the accumulated net value of the production process after x~ units have been produced with a Yn portion of product redesign already implemented. The two-dimensional stage vector is determined by the combinations of all feasible intervals at which the production process can be interrupted for implementing product redesign and all feasible portions of redesign. For example, let us assume that a production process can be interrupted for redesign purposes only when one or more quarters of the projected output have been completed. Let us also assume that the redesign
process requires that a complete product r+o+O-Sd,m_ddm redesign can be implemented in either one or • o.5 (11) two steps. In that case the two-dimensional stage vector consists of 4 x 2 = 8 points. where L ' = L(1 + Zdr), and Zd is a coefficient The objective is the maximization of the (z a >~0) measuring the redesign effect on reducaccumulated net value f(x,,,y,,). Usually, the ing the number of quality control related prob- accumulated net value is the realized profit. lems. High Zd values correspond to an increased The dynamic programming recursive equation such effect. can now be developed. We can use either a Finally, the redesign process has a positive forward or a backward formulation. Howeffect on the value (quality) of the production ever, for computational purposes a forward output. This positive effect is equivalent to formulation is preferred. Let us first comshifting forward the learning curve associated pute the accumulated net value ( C V ) resultwith the product value process. Equation (7) ing from producing a sequence of batches (giving the value of Q consecutive batches) can (x,,-Q,,- 1,x,-Q,-2 ..... x,,-Q,) assumbe rewritten as ing that a portion y, - r of redesign has been implemented prior to the production of this -0.5 Vtm*dm (12) sequence of batches and that a portion r of product redesign will be implemented during the where L ' = L(1 +zvr), and z,, is a coefficient current production run. The use of equations (z,~>0) measuring the redesign effect on (9)-(12) results in increasing the gross product value. In both equations (11) and (12) maxi~L,+Q~+05 mum improvement is realized when a complete C V(Q,,, r)= rIi l l - 0 , 5 V~mv dm redesign (r = 1) is implemented. In the next f(DL2+Qn+0.5 ) section we present a dynamic programming -P~/ dtm-ddm [ . J L2 -- 0.5 formulation for studying the overall redesign rL3+Q~+0.5 effect on the production process. - | C,m-Cdm-(uK:+rK,,) J L3-0.5 (14)
f:
, + e + o . ~
3. THE DYNAMIC PROGRAMMING FORMULATION The objective of this paper is to study the potential effects of various levels of redesign
where L 1 = (x~- Qn)(1 + zvy~) L2 = (x,--Q~)(I + zdyn) L3 =(x~--Q~)(I-z~y~)
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Koulamas--Quality Improvement
and the rest of the symbols are as defined previously. Using equations (13) and (14) we write the forward dynamic programming recursive equation as
f(x.,y.)=
m a x {cv(a.,r)+f(x.-a.,y.-r)} (15) 0~,.~y. 0~Q..~x. The two dimensional stage vector was defined earlier. The state vector at a particular stage f(x,,y,) is defined by the inequalities (0 ~
The other one-dimensional problem is when the timing of the redesign is predetermined and the objective is to find the optimal depth of the redesign. The presented formulation can solve this problem without any modifications. In this case the state vector is also one-dimensional with its elements being different levels of product redesign. In the next section the use of the dynamic programming algorithm is demonstrated through an example problem. 4. EXAMPLE-APPLICATIONS The forward dynamic programming algorithm was programmed in FORTRAN. The FORTRAN code can accommodate arbitrary redesign policies and arbitrary timings for implementing these redesigns. However, if a continuous redesign policy is in effect with a continuous timing pattern for implementing the redesign, then the well known dimensionality problem of dynamic programming will arise. Let us apply the dynamic programming algorithm to some simple example problems with the following date. The overall production quantity is Q = 200 batches with implementations of redesign being allowed every time 10 consecutive batches are produced. Complete redesign of the product is allowed, with the restriction that when a redesign operation is undertaken, the level of redesign should be either a 20% or a multiple of 20% improvement over the initial design. These conditions result in a (21 × 6) stage vector. The gross value of the first batch produced is Vt = $60.0 and its cost is C~ = $45.0. The learning rate is assumed to be the same for all three processes subject to learning (value, cost and quality control processes).The learning rate is treated as a problem parameter with two different levels. When low learning is in effect, Sv = Sc = Sd = 0.95. When high learning is in effect Sv = Sc = Sd = 0.9. The number and penalty cost of quality control problems are also treated as a problem parameter with two different impacts on the production process. For a low impact, the initial percentage of time the production process is temporarily interrupted due to quality control problems is d~ = 0.05 and the penalty cost for fixing these problems is P = $60.0 per problem. For a high impact these values are d~ = 0.1 and P = $150.0 respectively. The redesign cost was also treated as a problem parameter with
Omega, Vol.
two levels. When the redesign cost is low, Ks= $150.0 and K ~ = $45.0. When the redesign cost is high KI = $750.0 and K~ = $225.0. Also, it is initially assumed (for the first eight problems considered) that the redesign effect is uniform on the learning curves of all three processes (value, cost, and quality control), or equivalently zv = z~ = Zd = 1. The results are tabulated in Table 1. In each case we computed the accumulated net value of not only the optimal redesign policy, but also the accumulated net value of the two extreme redesign policies (complete redesign and no redesign at all) for sensitivity analysis purposes. A general conclusion is that when the optimal policy calls for redesigning the product, most of that redesign should take place during the early stages of the production process. This is attributed to the fact that the exponential learning curves considered in this study experience an early drastic change, and afterwards they relatively stabilize. The cost curve and the curve associated with the number of quality control problems experience early drastic drops while the gross value curve experi-
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167
ences an early drastic increase. Implementing the redesign process early results in moving further forward on both the value process learning curve and on the learning curve associated with the number of quality control problems while these curves experience their drastic changes. The early redesign is not advantageous with respect to the cost learning curve, since it causes a backward shift on that curve before it experiences its drastic drop. However the realized benefits on the other two curves from early redesign compensate for the backward shift on the cost curve. Another indication supporting this argument is that when higher learning rates are in effect the optimal redesign policy calls for an increased optimal level of redesign. This indicates that the realized benefits from an increased redesign level, that is the increased product value and the reduced number of quality control problems compensate for the resulting additional costs on the cost process. Also, when higher learning is in effect the production process is more sensitive to the various problem parameters and the redesign option becomes more attractive. For example,
Table 1. Results for the example problems Problem S~ = S c = S d = 0.95 K f = 7 5 0 , K,~=225 d 1=0.05, P=60 S~. = Sc = S d = 0.95 K f = 150, 1(,.=45 d~ = 0.05, P = 60 S, = S C= S d = 0.9 K I = 750, K~ = 225 d 1=0.05, P=60
Optimal value Redesign policy
Value for 0% redesign
Value for 100% redesign
10095 r* = 4 0 % after Q = 5 %
10006
9888
10166 after Q = 5 %
10006
10068
19152
19910
19152
20090
r*=40%
20088 r* = 60% after Q = 5%
S~ = S~ = S d = 0.9 K I = 150, K ~ = 4 5 dl = 0.05, P = 60
20198 r* = 6 0 % after Q = 5 % r~ = 20% after Q = 55%
S, = S, = S d = 0.95 K f = 750, K,.= 225 d I = 0 . 1 , P = 150
8296 after Q = 5 %
8167
8095
r* = 4 0 %
S, = S,~ = S d = 0.95 K / = 150, [ ( , = 4 5 dt=0.1, P=150
8368 after Q = 5 %
8167
8275
r* = 4 0 %
S~ = S,. = S d = 0.9 K f = 750, K~ = 225 dI=0.1, P=150
18840 rl* = 60% after Q ~ 5 %
17824
18670
S,. = S~ = S a = 0.9 K I = 150, K , = 4 5 d~ = 0 . 1 , P = 150
18956 r* = 6 0 % after Q = 5 % r~' = 20% after Q ~ 5 5 %
17824
18850
S~ = S¢ = S a = 0.9 K f = 150, K~.=45 d I = 0 . 1 , P = 150 z~= 3, z~= 0.5, z,t= 3
22927 r* = 100% after Q = 5%
17824
Same as optimal
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Koulamas--Quality Improvement
under higher learning (So = S t = Sd = 0.9), when the redesign cost is low the overall optimal redesign level is higher (equal to 60% + 20% = 80%) compared to the optimal redesign level of 60% when the redesign cost is high. In contrast, under a lower learning rate (So = Sc = Sa = 0.95), there is no difference in the optimal redesign level when the redesign cost varies, We wanted to examine a case where the optimal redesign policy is a complete redesign, In order to do so, we experimented with different redesign effects on the value, cost, and quality control processes (the last problem in Table 1). Specifically we allowed a more significant redesign effect on both the gross value and the quality control processes by letting za = zv = 3, and a less significant redesign effect on the cost process by letting zc = 0.5. In this case additional redesign of the product should become more desirable because it further reduces the number of quality control related
value, cost, and quality control processes in a single product environment. The dynamic programming formulation allows management to experiment with different sets of learning rates and cost data. It also allows experimenting with non-uniform learning rates as well as non-uniform redesign effects on the learning curves associated with the value, cost, and quality control processes. In any case the model provides the optimal redesign policy, that is the optimal redesign level and the optimal timing for implementing the redesign. Selective results demonstrate that the early implementation of the optimal redesign level enhances the accumulated net product value. Finally the dynamic programming algorithm is easily modifiable in the case when other quality functions must be used.
problems, it further increases the gross value of the product, and at the same time it has a less negative impact on the cost process. This was
14-20. 2. BehnezhadAR and KhoshnevisB (1988)The effectsof manufacturingprogress function on machine requirements and aggregate planning problems. Int. J. Prodn Res. 26(2), 309-326. 3. Boucher TO (1987) Choice of assembly line design
verified by the results of Table 1. The second to last problem in Table 1 is identical to the problem currently considered except that zc = Zd = Zv = 1. In that problem the optimal policy calls for a 60% + 20% = 80% product redesign. As expected, when Zd = Z~,= 3, and z, = 0.5 (last problem in Table 1), the optimal policy calls for a complete (100%) product redesign. Finally, the production process accumulated
net values resulting from the extreme redesign policies (complete redesign or no redesign at all) can be used for sensitivity analysis purposes so we can study their deviation from the optimal policy's accumulated net value for different learning rates and different parameter values. In general, management has the option to experiment with different data sets since the algorithm treats the learning rates, the initial number of quality control problems, and the initial value and cost data as input. Furthermore, the redesign effects on the value, cost, and quality control processes are also treated as input. This point is further emphasized in the conclusions section. 5. CONCLUSIONS A quantitative model was developed for studying the product redesign effects on the
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on optimal lot sizes: Further developmentson the single product case. lIE Trans. 17(1), 33-37. 13. Sule DR (1978) The effect of alternate periods of learning and forgetting on economic manufacturing quality. AI1E Trans. 10(3), 338-343. 14. Tapiero CS (1987) Production learning and quality control, llE Trans. 19(4), 362-369. 15. Yelle LE (1979) The learning curve: Historical review and comprehensive survey. Decis. Sci. 10, 302-328. FOR CORRESPONDENCE:Professor C Koulamas, Department of Decision Sciences and Information Sciences, FloridalnternationalUniversity, University Park, FL 33199, USA.
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