High quality and low costs? An application.

ABSTRACT

The relationship between quality and quality costs is an intriguing one. The usefulness of quality cost data as the primary accounting information source to help managers make quality-improvement decisions is also debatable. In order to tackle these problems, we make use of the historical quality cost data to devise a quality improvement strategy and investigate whether the application of such an optimal strategy will lead to declining quality costs over time while maintaining or even improving quality. In this paper, the prevention and the appraisal costs are assumed to be quadratic in their respective level of efforts exerted. The main advantages of a quadratic cost structure are that interior solutions are ensured and the interactions between control variables (i.e., prevention and appraisal) are simpler. However, no closed form solution is available as the problem involves a system of four differential equations in four variables. Therefore, we rely upon the numerical examples to demonstrate the behaviors of the control and state variables over time. We construct two similar models in this paper. The only difference between them lies in whether the decay factor, c, for the prevention effort is present. With the decay factor absent, Model 2 is expected to perform better than Model 1 in terms of the resulting quality improvement and the amount of quality costs reduced because prevention will be used more extensively as a superior quality improvement alternative than appraisal We also find that both models support a complementary relationship between quality and quality costs.

Keywords: Quality, Quality Costs, Prevention, Appraisal, Internal Failure, External Failure, Quality Improvement, Optimal Control

1. INTRODUCTION

"The success of management accounting depends on whether mangers' decisions are improved by the accounting information provided to them." Nowadays, the issue of quality commands a lot of managers' attention and the need for information to address it is increasing. Quality improvement, though critical, cannot operate on blind faith alone. An information network is needed to handle the process and to gauge the financial impact of its implementation. Quality cost, a key financial measure associated with the product (or service) quality of a firm, is at the core of such a network. However, one of the critical problems concerning quality cost research is that the relationship between quality and quality cost is unclear. Yet it is this fundamental relationship which will influence managers' decisions toward quality improvement efforts and the use of quality cost information in a profound way.

There are two main schools of thought on this issue. The original, "cost first" thinking believes that quality and quality cost involve tradeoffs: higher quality can only be accomplished by spending more on quality costs, and that the goal of the quality program and the corresponding resource allocation decision is to determine the acceptable level of quality that minimizes quality costs. The second, "quality first" thinking believes that the pursuit of providing high quality products and services to the customers is not necessarily accomplished by high quality costs. On the contrary, the increased demand for high quality products and services will generate enough revenue to cover the quality costs. Better yet, the increased efficiency in terms of reduced bottlenecks, less schedule disruptions and smaller inventory buffer will actually drive down the quality costs, leading to a complementary relationship between quality and quality costs.

We adopt a simple quality cost framework to derive quality and quality cost structures for this research. In this paper, two models with quadratic cost structure are formed. The solutions involve a system of four differential equations and we rely upon a numerical example to demonstrate the interactions of control and state variables over time. A complementary relationship between quality and quality costs is evident.

In practice, management accountants help identify and eliminate quality problems by quantifying and ranking failure costs in order to remove the "vital few" defect causes with the largest financial impacts using so called Pareto Analysis. Since accounting research on quality costs seldom treats quality as a separate variable, we can only assume that changes in quality costs, especially failure costs (i.e., the sum of internal and external failure costs), may be viewed as indicators of changes in quality. In particular, failure cost reductions are usually assumed to produce the quality improvement desired. Our numerical results help rationalize this approach as we observe that reductions in failure costs are consistent with improvements in the outgoing quality over time.

This paper is organized as follows. Section 2 reviews the literature on quality and quality costs. Section 3 discusses the model building process and introduces a quadratic cost model with the decay factor for prevention and the model without the decay factor. A comparison of the models with their numerical results is also discussed. Section 4 concludes the paper.

2. LITERATURE REVIEW

Kaplan (1983) mentions quality as one of the missing measurements in accounting, or "perhaps the most important manufacturing performance area." The concept of quality, more often than not, means different things to different people. However, a distinction can be made between conformance quality and design quality. Conformance quality, "[t]he most pervasive understanding of quality in manufacturing organizations," indicates a product's degree of conformance to design specifications, a set of engineering standards. It is usually represented in the form of a target value of interest and tolerance limits. Depending on the extent to which the requirements of conformance quality are met, we can differentiate three levels of conformance quality. Acceptable quality level (AQL, or economic conformance level of quality, ECL) is determined on financial grounds as a result of comparing the marginal cost and marginal benefit of quality improvement efforts. This exercise of cost-benefit analysis usually ends with a level of conformance quality that is far from perfect. Zero defect, on the other hand, means that all output units meet conformance quality requirements (i.e., their configurations are all within the tolerance limits). Crosby (1979) promotes the pursuit of zero defect because "there is absolutely no reason for having errors or defects in any product or service." Finally, according to Taguchi and Clausing (1990), a product is of robust quality if all of its actual outputs meet exactly the target value of interest, not just stay within the tolerance limits.

For the purpose of quality cost research, conformance quality, with its emphasis on the manufacturing side of operations, is consistent with the definition of quality cost and is the quality concept chosen by most researchers. Performance quality as used in Balachandran and Srinidhi (1996) contrasts conceptually with conformance quality and is used far less by researchers. Design quality involves with the marketing function and is seldom used in quality cost research. So from now on, when we mention "quality" costs, the quality part refers to conformance quality.

The general concept of quality costs was first proposed by Juran in his 1951 Quality Control Handbook. Masser (1957) and Feigenbaum (1957) subdivide quality costs into prevention, appraisal, internal failure and external failure costs, the popular categories of quality costs still used today.

Juran describes quality cost as "the sum of all costs that would disappear if there were no quality problems." Similarly, Campanella treats it as "the difference between the actual cost of a product or service, and what the reduced cost would be if there was no possibility of substandard service, failure of products, or defects in their manufacture." The following definitions associated with the concept of quality costs and its components are from Morse et al. (1987).

"Quality costs are costs incurred either because poor quality may exist or because poor quality does exist. In defining and measuring quality costs, a quality product is one that conforms to design specifications. Prevention and appraisal costs are incurred because poor quality of conformance may exist. Prevention costs are incurred to prevent nonconforming units from being produced. Appraisal costs are incurred to identify nonconforming units before they are shipped to customers. Failure costs are incurred because poor quality of conformance does exist. Internal failure costs are incurred when materials, components, or products are identified as nonconforming before they are shipped to customers. External failure costs are incurred when nonconforming products are shipped to customers."

According to Shank's (1989) view on strategic cost management, "... one of the important roles of internal accounting information within a business is to facilitate the development and implementation of business strategies." At the strategic planning level, accountants should be encouraged to help managers devise a proactive, top down approach to quality improvement by applying prevention and appraisal optimally based on the data available to them. So strategic use of quality cost data, in addition to its measurement and reporting, represents an area where accountants are expected to make significant contributions.

As Morse and Roth (1987) put it, "[t]he objective of a quality cost system should be to help improve quality rather than simply measuring and reporting quality cost data." In addition, "[t]he goal of any quality cost system, therefore, is to facilitate quality improvement efforts that will lead to operating cost reduction opportunities." Obviously, measurement and reporting of quality costs are means for other ends.

In a different type of application of quality cost data, Radhakrishnan and Srinidhi (1994) adopt a quality cost framework to study a resource allocation problem regarding whether quality should be designed in or inspected in. They find such quality management decisions to be context specific in the sense that different cost structures for prevention and appraisal (e.g., prevention and appraisal costs are either quadratic or linear in the quality level achieved) will lead to different sampling policies (e.g., for quadratic cost, partial acceptance sampling; for linear cost, zero percent acceptance sampling), which underscores the importance of systematic measurement and reporting of quality costs for management decisions.

Radhakrishnan and Srinidhi (1994) also consider the quadratic cost structure in their static setting. However, they assume that both costs are quadratic in the quality level achieved. Although quality is sometimes used as a surrogate for the amount of quality improvement efforts expended, their formulation leads to the implication that quality "causes" the prevention and the appraisal costs. In our model, we treat the changing quality level achieved as a result of both of the efforts exerted in a quality evolution process. At the same time, prevention and appraisal efforts are assumed to drive the associated prevention and appraisal costs. This alternative formulation leads to a more intuitive relationship between quality costs and their drivers and provides a richer context to work on.

3. THE MODEL

3.1 Model Building Process

In this section, we will assemble the building blocks of our models. We begin with a quality cost framework whose origin can be traced back to Feigenbaum (1957), as shown in Figure 1.

[FIGURE 1 OMITTED]

We consider the quality part of the framework first. For a batch of S units going through the production process, an average of fS units are considered good (or of high quality) and the rest, (1-f)S units, are not acceptable (or of low quality). The percentage f will be called "default quality" (or, in manufacturing term, first-yield rate) for the obvious reason that it represents the innate capability of the manufacturing process to produce defect-free units. We attribute this quality component to the prevention effort, u, exerted. When the appraisal effort is applied to the batch, the associated appraisal activities will not mistake the good units for the bad, but they may incorrectly classify some bad units to be of acceptable quality. Then out of the bad units, some of them (i.e., (1-f)gS units) will be picked up by the appraisal process and reworked, while the rest (i.e., (1-f)(1-g)S units) will slip through the process and become external failures. Here the percentage g will be called "appraisal effectiveness" as a result of the appraisal effort, v, applied. By assuming that all reworked units will meet the internal quality standard, we will be able to ship the whole batch to the customers. Since (1-f)(1-g)S units out of the whole batch of S units are bad, we can then define the "outgoing quality," q, as

q = 1 - (1 - f)(1 - g)S/S = f + g - fg,

representing the percentage of the output which meets the quality specification. As a note on terminology, we use "quality components" as a collective term to stand for both default quality and appraisal effectiveness, in contrast to outgoing quality.

We next consider the quality cost part of the framework. First of all, we note that the prevention and appraisal efforts are costly exercises subject to budget constraints in the form of u [member of] [0,u] and v [member of] [0,v], where u and v represent the level of prevention and appraisal applied, respectively. For a quadratic cost structure, the prevention effort is assumed to cost [k.sub.1][u.sup.2] while the appraisal effort will cost [k.sub.2][v.sup.2]. [k.sub.1] and [k.sub.2] are unit cost of prevention and appraisal, respectively. We all recognize the distinction between "one ounce of prevention" and "one pound of cure," and readily appreciate what prevention can do to stem any possible problem before it goes awry. Pasewark (1991) observes that "[t]he emphasis on prevention reduced the need for appraisal activities to detect defective products, because the probability of producing defects decreased."

In the framework (Figure 1), it is obvious that improvements in the default quality (via prevention effort), as a superior alternative, will bring down both the internal and the external failure while improvements in the appraisal effectiveness (via appraisal effort) will drive up the internal failure. But in reality, prevention is not usually applied as early and as often as it should have been to have any positive impact. On the cost side, we simply assume that the prevention effort is more expensive than the appraisal effort, or [k.sub.1] > [k.sub.2] > 0, to "prevent" it from being exerted more often or at higher level.

For failure costs, we assume that the internal failure (in our case, rework) will cost r per unit, for a total of r(1-f)gS while the external failure (consider, for example, warranty claims) will cost w per unit, for a total of w(1-f)(1-g)S. The relationship between r and w is easier to determine. By assuming w > r > 0, we indicate that the internal failure is cheaper than the external failure. Otherwise, one can shut off the whole appraisal operation, save both the appraisal and the internal failure costs and let the customers do the inspections.

We will now establish the link between the quality components (i.e., f and g) and their associated efforts (i.e., u and v). As mentioned earlier, the default quality is driven by the prevention effort and the appraisal effectiveness is driven by the appraisal effort. Bear in mind that the four variables are indexed by time in a multi-period, dynamic setting. We model the quality evolution process as follows:

f' = au(1 - f) - cf and g' = dv(1 - g) - bg.

This formulation has its root in optimal control literature. Since the natural effect of any quality improvement effort tends to persist beyond the current period although with a diminishing return, we follow both the Vidale-Wolfe (1957) sales advertising response model and the Nerlove-Arrow (1962) advertising capital model to obtain a direct relation between improved quality and its associated effort in the form of a differential equation. According to the quality evolution process, any change in each of the quality components is governed by two factors, a positive effect due to the associated effort exerted (via an efficacy parameter, a or d, where 0 < a < 1, 0 < d < 1) on the yet-to-improve portion and a negative effect due to decay (via a decay parameter, c or b, where 0 [less than of equal to] c < 1, 0 < b < 1) in the form of, say, depreciation of the capital equipment involved, lapse in training over time, transfers between divisions, employee turnover or normal attrition of the work force. Consider, for example, the application of one unit of the prevention effort to the quality improvement process. Only a fraction of that unit, a, will contribute positively to the improvement of default quality, in the amount of a(1-f). As is obvious, the higher the default quality f, the smaller the portion (1-f) and therefore the improvement in f (i.e., f') become. On the other hand, a fraction of default quality already achieved, in the amount of cf, will contribute negatively to the changes in the default quality. The difference between the positive and the negative effects will then determine whether and in what amount the default quality will improve over time, which underlines the importance of long-term application of the prevention effort to ensure the continuous improvement of default quality. The quality evolution process for appraisal effectiveness follows a similar pattern.

Here we continue our search for ways to differentiate the prevention and appraisal efforts. In general, the prevention effort, like R&D, involves more risk and uncertainty than does the appraisal effort. So we assume 0 < a < d = 1, meaning that, relative to appraisal, for each unit of effort exerted in prevention, we expect less than one unit will eventually contribute to the improvement of default quality. Alternatively, according to Ishikawa (1985), broad-based effort like prevention is more difficult to apply than appraisal especially in the U.S. where emphasis on specialization in certain fields undermines the idea that quality is everyone's responsibility. Also, union rules restrict the activities of workers. However, we also expect the impact of prevention to last longer than that of appraisal, so we assume 0 [less than or equal to] c < b < 1, indicating that the negative decay will be less severe on prevention. In the extreme case, the impact of prevention is assumed to last forever so that we may have c = 0. However, this case is somewhat moderated by the finite planning horizon adopted in our model. By focusing the prevention effort on just one aspect, training, we may also interpret the absence of the c factor as the provision of life-time employment made famous by the Japanese system versus downsizing or high turnover regularly seen in U.S. firms. Then the knowledge base of the whole work force will remain in place over time.

This formulation emphasizes the importance of continuous improvement in any quality management program. The basic theme is, if the organization stops exerting efforts, or even does not exert enough, it will fall behind because, after all, its competitors are charging ahead. At a more fundamental level, the presence of negative effects (i.e., c and b) indicates that "backsliding will occur unless [management] maintains a steadfast commitment to quality improvement." It is also consistent with Wasserman and Lindland's (1993) contention that maintaining quality alone will not satisfy customers' heightened expectations over time. In our model, the levels of effort (u and/or v) have to be large enough to ensure improvement in f and/or g, leading to improved outgoing quality. Otherwise, the improvement may be stalled (i.e., f' = 0 and/or g' = 0) or even deteriorated (i.e., f' < 0 and/or g' < 0).

In our model, the roles played by the prevention and appraisal efforts are twofold. As the quality drivers, they help direct changes in default quality and appraisal effectiveness, respectively, which together determine the outgoing quality. As the cost drivers, they directly influence the cost of conformance (i.e., the prevention and appraisal costs) and indirectly contribute to the changes in the cost of nonconformance (i.e., the internal and external failure costs).

The interaction of the production process with the outside market is modeled through the sales volume, S(q), as a function of the outgoing quality, q. Presumably, the higher the outgoing quality, the more units will be demanded and sold, but at a decreasing rate (i.e. S' > 0, S" [less than or equal to] 0). However, we assume S = 1 in our model to normalize the production activities to one unit at a time so that the trends of quality costs over time can be formulated and compared across a variety of models considered. In addition, it is consistent with the use of conformance quality, an internally driven quality concept, in our model where the market force is ignored. Finally, it renders the derivation of the optimal solution more tractable. As we expect the outgoing quality to improve over time as part of the solution, fixing the sales volume underestimates the revenues generated in our model and produces a conservative result.

By fixing the planning horizon at T, we anticipate a certain degree of shirking in terms of reduced level of efforts applied toward the end of the fixed horizon. This is due to the fact that any unit of effort exerted near the end of the horizon costs as much as the units applied earlier but produces less and less benefits in terms of reductions in quality costs. In order to mitigate such possible side effect as a result of the model being a finite one, we attach a reward function, Rq(T), in direct proportion to the end-of-period outgoing quality achieved. This in effect helps the firm maintain its reputation as a player in the market who is willing to stay on doing business even after the current planning horizon ends.

We apply the optimal control theory to solve the problem. We adopt the traditional view of quality costs as the point of departure by choosing the levels of prevention and appraisal effort (the control variables in optimal control theory) to minimize the total quality costs over time, subject to the quality evolution processes for default quality and appraisal effectiveness (the state variables), the constraints on the control variables (if any) and the initial conditions on quality components.

3.2 Model 1 (with decay for prevention)

We will solve the following optimal control problem with quadratic costs in the respective level of efforts exerted.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

(1) f'(t) = au(t)(1 - f(t)) - cf(t), f(0) = [f.sub.0] > 0, f(T) is free,

(2) g'(t) = v(t)(1 - g(t)) - bg(t), g(0) = [g.sub.0] > 0, g(T) is free, u(t), v(t) [greater than or equal to] 0, t [member of] [0, T], T is fixed.

This is an optimal control problem that maximizes the profit streams over a fixed planning horizon. (1) and (2) are the quality evolution processes that establish the connection between the control (i.e., prevention and appraisal, respectively) and the state (i.e., default quality and appraisal effectiveness, respectively) variables. Note that the decay factor, c, is present in (1) in this model. No budget constraint on the applicable level of efforts is imposed.

To solve the problem, we first form the Hamiltonian function

H = p - [k.sub.1][u.sup.2] - [k.sub.2][v.sup.2] - r(1 - f(t))g(t) - w(1 - f(t))(1 - g(t)) + [[lambda].sub.f (t)[au(t)(1 - f(t)) - cf(t)] + [[lambda].sub.g] (t)[v(t)(1 - g(t)) - bg(t)],

where [[lambda].sub.f] and [[lambda].sub.g] are the two continuously differentiable adjoint variables associated with default quality and appraisal effectiveness, respectively, and they obey

(3) [[lambda]'.sub.f = [partial derivative]H/[partial derivative]f = -[rg + w (l - g)] + [[lambda].sub.f] (au + c), [[lambda].sub.f] (T) = R (1 - g (T)) and

(4) [[lambda]'.sub.g] = - [partial derivative]H/[partial derivative]g = (r - w)(l - f) + [[lambda].sub.g] (v + b), [[lambda].sub.g] (T) = R(1 - f(T)).

The optimality conditions are

(5) [partial derivative]H/[partial derivative]u = 2[k.sub.1]u + [[lambda].sub.f]a(1 - f) = 0 and

(6) [partial derivative]H/[partial derivative]v = -2[k.sub.2] v + [[lambda].sub.g] (1 - g) = 0.

We first observe that

Lemma

The Hamiltonian function is strictly concave in the control variables u and v.

Proof. See Appendix A.

Because of the property of concavity, an interior solution is assured for this problem.

From (5) and (6), we can express the control variables as

(7) u = [[lambda].sub.f] a (1 - f)/2[k.sub.1]

and

(8) v = [[lambda].sub.g] (1 - g)/2[k.sub.2].

Substituting (7) and (8) into (1) - (4) and simplifying, we get

(9) f' = [[lambda].sub.f][a.sup.2] [(1 - f).sup.2]/2[k.su.1] - cf,

(10) g' = [[lambda].sub.g] [(1 - g).sup.2]/2[k.sub.2] - bg,

(11) [[lambda]'.sub.f] = -[rg + w(1 - g)] + [[lambda].sub.f.sup.2][a.sup.2] (1 - f)/2[k.sub.1] + c[[lambda].sub.f] and

(12) [[lambda]'.sub.g] = (r - w)(1 - f)]+ [[lambda].sub.g.sup.2](1 - g)/2[k.sub.2] + b[[lambda].sub.g].

(9) - (12) constitute a system of differential equations in f, g, [[lambda].sub.f] and [[lambda].sub.g] with two initial conditions:

f(0) = [f.sub.0] and g(0) = [g.sub.0]

and two terminal conditions:

[[lambda].sub.f](T) = R(I-g(T)) and [[lambda].sub.g] (T) = R (1- f (T)).

As we cannot find a closed form solution for this problem, we use a numerical example to study the properties of the solution. The following parameters are used:

a = .8, b = .4, c = .1; r = 2, w = 6; [k.sub.1] = 3, [k.sub.2] = 1; [f.sub.0] =.6, [g.sub.0] = .4; R = 7, T = 5.

We obtain the complete numerical solution for u, v, f, g, q and all the components of the total quality costs. The procedure used to solve the problem numerically is discussed in Appendix B.

Figure 2 shows how the optimal prevention and appraisal efforts will be applied over time. The prevention effort committed begins at 0.493 and gradually drops to 0.182 when the end of planning horizon is reached. The appraisal effort committed is convex in time, which starts at 0.497 and ends with 0.682. Note that the optimal levels of efforts applied for prevention and appraisal is still between 0 and 1, which justifies the absence of any constraint on the level of efforts available. In this example, the optimal solution calls for extensive use of appraisal in exchange for less application of prevention.

[FIGURE 2 OMITTED]

The next figure (Figure 3) shows how the quality components change over time as a result of the prevention and appraisal efforts exerted. The default quality ends with 0.69 while the appraisal effectiveness improves to 0.56, leading to an improved ending outgoing quality of 0.86 (an improvement of over 13.16%).

[FIGURE 3 OMITTED]

Figure 4 shows that, on the cost side, the prevention cost declines over time because of the corresponding decrease in the usage of the prevention effort; the appraisal cost, however, increases toward the end of the planning horizon, leading to a slight increase in the internal failure cost from 0.32 to 0.35 while the external failure cost drops throughout from 1.44 to 0.82. This result is consistent with the empirical findings of Carr and Ponemon (1994) and implies that a quality improvement program with heavy reliance on the appraisal effort (see Figure 2) makes prevention less effective and leads to less use of it. The total quality cost begins with 2.74 and ends with 1.73 (an improvement of about 36.86%). Because of the way the quality evolution process is formulated, the increased use of the appraisal effort makes it more effective at catching the defective units before they are shipped to customers, leading to higher internal failure costs. In this case, the savings from the reductions in the prevention cost and the external failure cost are not large enough to cover the increases in the appraisal cost and the internal failure cost, so we observe an increase in the total quality cost near the end of the planning horizon.

[FIGURE 4 OMITTED]

We combine the quality and the quality cost information to show a largely complementary relationship between them in Figure 5. The data points are matched by time index. Note the kink near the end of the planning horizon (at approximately) as a result of the overtaking appraisal and the internal failure costs. The outgoing quality, on the other hand, improves throughout.

[FIGURE 5 OMITTED]

Finally, the failure costs are drawn against the outgoing quality in Figure 6 to demonstrate that the general movement of the failure cost curve corresponds to changes in the outgoing quality. Consistent with the trend of improved quality over time, the failure cost declines steadily over time from 1.76 to 1.17, showing an improvement of about 33.52%. In our example, it is easier to reduce failure costs than to improve quality over the planning horizon. This shows that changes in failure costs can be used as a surrogate to gauge the improvement in the outgoing quality, which may not be directly measurable or may not be measured at all. Of course, quality can also be measured as the number of failures or defects in a manufacturing setting. However, the magnitude of the failures is already taken into account in the measurement of failure costs, thus validating our approach here.

[FIGURE 6 OMITTED]

3.3 Model 2 (without decay for prevention)

To study the impact of the c factor (the decay rate for prevention) on Model 1, we set c = 0 in this model. Then the optimal control problem becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

(13) f'(t) = au(t)(1 - f(t)), f(0) = [f.sub.0] > 0, f(T) is free,

(14) g'(t) = v(t)(l - g(t)) - bg(t), g(0)=[g.sub.0] > 0, g(T)is free, u(t), v(t) [greater than or equal to] 0, t [member of] [0,T], T is fixed.

With the c factor set to zero, we effectively assume that the impact of the prevention effort will last forever, an unlikely situation, but it is somewhat mitigated by the fact that the planning horizon is fixed at T, a known parameter.

The derivation is similar to that of Model 1 and is omitted for brevity. As we still cannot solve the problem symbolically and obtain a closed form solution, the same set of parameters (without the c factor) is used in the numerical example.

Figure 7 shows how the optimal prevention and appraisal efforts should be applied over the planning horizon. Note that the level of the prevention effort used is again declining over time (from 0.64 to 0.09), while the convex curve of the appraisal effort starts at 0.43 and ends with 0.34.

[FIGURE 7 OMITTED]

The next figure (Figure 8) shows how the quality components change over time. Note that the default quality, the appraisal effectiveness and the outgoing quality start from the same values as before, at 0.6, 0.4 and 0.76, respectively. Without the decay factor, the impact of the prevention effort lasts as long as the planning horizon and the resulting default quality should be a non-decreasing one. In fact, it is increasing over time to 0.88 because of the non-zero level of the prevention effort applied. The appraisal effectiveness fluctuates a little and improves to 0.42 in the end. The ending outgoing shows an improvement of about 22.37%, to 0.93.

[FIGURE 8 OMITTED]

Figure 9 illustrates that, on the cost side, the prevention cost decreases over time from 1.25 to 0.02 while the appraisal cost fluctuates responding to the level of the appraisal effort used. The internal failure cost starts at 0.32 and gradually reduces to 0.1 in the end while the external failure cost starts at 1.44 and drops to 0.40. The total quality cost declines throughout the planning horizon, showing an improvement of about 80.25% (from 3.19 to 0.63).

[FIGURE 9 OMITTED]

The time-matched quality and quality cost information is shown in Figures 10. The smooth curve shows a nearly perfect complementary relationship. In addition, it shows results in terms of quality improvement (.93) and quality cost reduction (80.25%).

[FIGURE 10 OMITTED]

We next juxtapose the failure cost and the outgoing cost curves in Figure 11. Again, the trend in both curves is consistent with the practice of using failure cost reduction as a way to improve quality. Numerically, the failure cost declines over time from 1.76 to 0.5 (an improvement of about 71.6%) while the outgoing quality improves from 0.76 to 0.93 (an improvement of about 22.37%). So failure cost reduction is easier than quality improvement in our example.

[FIGURE 11 OMITTED]

3.4 Comparison of Models

We set up the numerical examples in such a way that the same set of parameters were used for both of the models we solved. The comparison is between Model 1 and Model 2 to study the impact of the decay factor for prevention on the overall performance of the proposed quality improvement program in our research. As is obvious, Model 2 (without the decay factor) outperforms Model 1 (with the decay factor) in terms of both of quality cost reduction (80.25% vs. 36.86%) and quality improvement (22.37% vs. 13.16%) over the planning horizon because the prevention effort shows no diminishing returns and is used more extensively. This demonstrates the importance of choosing the prevention measures with long-term impact such as automation, computer-assisted design and continuous education programs, among others, to mitigate the presence of any potential decay influence. However, setting the decay factor to zero is not as extreme as it seems because the quality improvement problem is studied within the framework of a finite planning horizon. The non-diminishing aspect of the prevention effort can only last as long as the finite horizon.

3.4 Comparison of Models

We set up the numerical examples in such a way that the same set of parameters were used for both of the models we solved. The comparison is between Model 1 and Model 2 to study the impact of the decay factor for prevention on the overall performance of the proposed quality improvement program in our research. As is obvious, Model 2 (without the decay factor) outperforms Model 1 (with the decay factor) in terms of both of quality cost reduction (80.25% vs. 36.86%) and quality improvement (22.37% vs. 13.16%) over the planning horizon because the prevention effort shows no diminishing returns and is used more extensively. This demonstrates the importance of choosing the prevention measures with long-term impact such as automation, computer-assisted design and continuous education programs, among others, to mitigate the presence of any potential decay influence. However, setting the decay factor to zero is not as extreme as it seems because the quality improvement problem is studied within the framework of a finite planning horizon. The non-diminishing aspect of the prevention effort can only last as long as the finite horizon.

A word of caution about the use of numerical examples is in order. Although we are able to obtain unique numerical solutions for the three models discussed so far, the validity of any conclusion drawn depends critically on the parameters adopted. It may very well be the case that different sets of parameters will lead to other unique solutions, the comparison of which can lead to a different relationship among models. So the results presented here are tentative at best and further research in this area is imperative. Table 1 summarizes the numerical examples discussed.

4. SUMMARY

In this paper, we study a quality improvement program with two versions of the quadratic cost structure for prevention and appraisal, one with the decay factor for prevention and one without. The advantage of the quadratic cost structure one lies in the concave objective function used, which guarantees an easier solution approach and an interior optimal solution. Unfortunately, no closed form solution is available and we have to make use of numerical examples to explore the properties of the variables and their interactions over time.

We find that a complementary relationship between quality and quality cost is reachable over time through the optimal quality improvement strategies proposed. It also shows the usefulness of the historical quality cost data in devising such strategies. In practice, the quality cost data is sometimes used in the quality improvement program to identify and eliminate those problem areas with the largest failure costs without specifically measuring whether the quality index is actually improved. Our results support the use of changes in failure costs to gauge the performance of the quality improvement program as reductions in failure costs are associated with improvement in quality measure over time. We also want to emphasize the monitoring role played by the ongoing measurement and reporting of quality cost data to ascertain that the optimal strategies for prevention and appraisal are carried out and the anticipated quality improvement and quality cost reduction can be achieved.

The decay factor in prevention plays an important role in encouraging the choice of longer-term prevention measures to reduce its impact, as evidenced by the superior performance of Model 2 (without the decay factor for prevention) relative to that of Model 1 (with the decay factor). As the use of the pure form of quadratic costs for prevention and appraisal is not expected in practice, more empirical research is needed to study whether cost structure plays a role in influencing the overall performance of a quality improvement program.

APPENDIX A (PROOF OF LEMMA)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[d.sup.2]H is strictly negative definite. We can conclude that the Hamiltonian function is strictly concave in the two control variables. Q.E.D.

APPENDIX B (PROCEDURE FOR SOLVING THE NUMERICAL EXAMPLES)

We use the mathematics package Maple V Release 4 to help solve the problem numerically as no symbolic solution is available. As in most software packages that deal with mathematics, Maple can handle either the initial value or the terminal value problems, but not both. However, our boundary conditions involve both, so instead we treat our problem as an initial value one and fine tune the initial values for [[lambda].sub.f] and [[lambda].sub.g] to meet their terminal value requirements.

That is, we use trial and error to fix the values of [[lambda].sub.f] and [[lambda].sub.g] (0) so that their terminal values satisfy the boundary conditions, or

[[lambda].sub.f] (T) = R(1 - g(T)) and [[lambda].sub.g] (T) = R(1 - f (T)).

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Chiaho Chang, Montclair State University, Montclair, New Jersey, USA

Dr. Chiaho Chang earned his Ph.D. at New York University in 1997. Currently he is an assistant professor of accounting at Montclair State University.

TABLE 1 (SUMMARY OF NUMERICAL EXAMPLES)

                                  Quality Costs

Model                             P      A      IF      EF     Total
Specifications

Quadratic Cost      t = 0         0.73   0.25   0.32    1.44     2.74
with decay (Model   t = 5         0.10   0.46   0.35    0.82     1.73
1)                  improvement                                36.86%

Quadratic Cost      t = 0         1.25   0.18   0.32    1.44     3.19
without decay       t = 5         0.02   0.11   0.10    0.40     0.63
(Model 2)           improvement                                80.25%

                                  Quality

Model                             f      g      q
Specifications

Quadratic Cost      t = 0         0.60   0.40    0.76
with decay (Model   t = 5         0.69   0.56    0.86
1)                  improvement                 13.16
                                                %
Quadratic Cost      t = 0         0.60   0.40    0.76
without decay       t = 5         0.88   0.42    0.93
(Model 2)           improvement                 22.37
                                                %

P - Prevention, A - Appraisal, IF - Internal Failure, EF - External
Failure

f - Default Quality, g - Appraisal Effectiveness, q - Outgoing Quality