Warranty-based method for establishing reliability improvement targets.

1. Introduction

In order for product manufacturers to maintain their market positions in today's highly competitive global markets they must rely on continuous quality improvement programs as an essential part of business planning. Modifications in the physical design, production methods

and processes, and built-in redundancies to accommodate inevitable component failures without impacting operations are examples of improvements that are essential in keeping abreast of competitors. Consumers are well informed and demand high-quality products. Reliability, as the critical element of quality, is an essential concern in manufacturing and the primary target for continuous quality improvement. For this reason any approach to establish product quality improvement goals will generally include the allocation of reliability targets for the key elements of the manufactured products.

Reliability allocation is the process of translating the reliability requirements of the components in a system in order to achieve an overall system specification. This process can be quite difficult in the design of new systems where information on the operational and failure characteristics of the components is relatively unknown. Reliability allocation for the purpose of improving the quality of a product involves less uncertainty in the failure characteristics and relative impact of critical component failures, but still involves challenging decisions in selecting components and assigning improvement targets. The problem is to find, for a fixed t [greater than or equal to] 0, functions,

[g.sub.i][[R.sub.1](t), [R.sub.2](t) ..., [R.sub.n](t)], (1)

that assign the apportioned reliability improvement for the components i = 1,..., n that have current reliability [R.sub.i](t) so that an overall system reliability R(t):

g[[g.sub.1][[R.sub.1](t), [R.sub.2](t) ..., [R.sub.n](t)],..., [g.sub.n][[R.sub.1](t), [R.sub.2](t) ..., [R.sub.n](t)]] [greater than or equal to] R(t), (2)

is achieved. This improvement is typically relative to an overall reliability goal that is established through some continuous improvement analysis performed in the strategic business planning process. The components are assumed to be in a series configuration with independently distributed exponential failure times. An extensive literature exists on methods to solve reliability allocation problems under a range of conditions on the associated structure, and conditions on the variables. A review of these methods is provided in Kuo et al. (2001).

In developing process and product component reliability improvement goals, the cost or effort function used needs to include the quality measures of the product as well as the costs. This can be quite difficult due to the broad interpretation of quality and the many factors that influence consumer attitudes toward product quality. The purpose of this paper is to present a practical method to establish reliability goals, or targets for quality improvement strategies through reliability improvements of the components.

2. Overview and scope

In this paper, we present a practical method that is able to establish reliability goals, or targets for creating quality improvement strategies, through reliability improvements of the components.

We consider a multiple-component product system composed of n components and assume that the system is configured in such a way that the components can be considered independent and vital to the proper functioning of the product. We assume the product and production system are reviewed periodically through a formal continuous quality improvement program to accomplish quality improvements. These programs, often called Total Quality Management (TQM), are common among major manufacturers and are considered essential to maintain a competitive edge in global markets.

Setting reliability goals is an extensive process that engages all aspects of the manufacturing enterprise. The method proposed in this paper is based on the following concepts.

1. Quality is a multiple-attribute function of several features and characteristics that relate to the way a product is designed, developed, and utilized by customers (Thomas, 1997). Some of these elements are subjective and difficult to assess.

2. Reliability is a dominating characteristic of quality and therefore the major means for seeking improvement in product quality.

3. Warranty programs provide feedback on the relative quality and reliability among product components through the number and costs of claims that occur over time.

Observe that, in practice, setting reliability goals for a product involves the overall manufacturing enterprise. In fact, it is typically part of a business strategy that can involve other products and parts of the organization and always involves cost decisions and tradeoffs. Consequently, the TQM process starts with targets that are established for each unit or subsystem. These targets are strategic and are based on achieving the overall enterprise goals. However, they still require verification and further feasibility evaluations at the component level of analysis. For this reason, the method we propose is a two-stage process that can require multiple iterations between the strategic and components levels to finalize the reliability improvement goals.

In Section 3 we motivate the use of warranty costs in planning quality improvements for manufactured products. Most consumer products are covered by some type of warranty. From a total-system-quality perspective some elements of a product system are more sensitive to change than others and while it is difficulty to quantify the impact of these factors the overall aggregate effect is reflected in the warranty claims that accrue during warranty periods. In Section 4 we develop a procedure to establish the preliminary allocation of goals by allocating an overall product improvement objective among components based on the fraction of warranty expense contributed by the components. In Section 5 we present a method to find a more-refined allocation of reliability improvement targets that satisfies a given budget constraint on the improvement program. The problem is structured as a knapsack problem and a simple solution procedure is developed. A sample application of the method is presented in Section 6 and some concluding remarks are given in Section 7.

3. Quality, reliability, and warranty

Quality can be defined as a state of acceptance of a product or service that reflects the satisfaction that customers receive relative to given requirements. It refers to the value that is assigned by consumers to the way that a product is designed, developed, and used. Manufacturers are concerned with the efficiency and effectiveness of the processes, materials, and workmanship in producing products. By keeping waste to a minimum the costs can be kept under control, thus allowing producers to be more competitive. Many customers on the other hand tend to focus more on the features in a product, including its appearance and its ease of use. Others such as dealers or consumer groups focus more on the reputation of a product and manufacturer in providing products that last and perform at the specified levels.

Therefore, product quality is multi-dimensional and requires the specification of a set of attributes that characterize an item throughout its lifetime in order to be completely defined (Garvin, 1987). Thomas (1997, 2004) represents product quality as a vector-valued function of the six quality elements defined in Table 1 that we describe further next. Performance is a measure of the actual operational effectiveness of the item, indicating how well it functions within the system design. For example the performance of an automobile could be in terms of: (i) its fuel consumption in miles per gallon; (ii) its time to accelerate to 60 miles per hour; and (iii) the braking distance necessary to bring the vehicle to a complete stop. Durability is a measure of how long an item will last in absolute time or cycles of usage. Reliability is a probability statement about the chance of an item surviving over time. It can be expressed as a percentage or even as a subjective category class such as low, moderate, and high likelihood of not failing. Conformance is the degree to which standards are met in producing a product as measured through quality control testing and audits. For many years this was the accepted interpretation of product quality. There are many quantitative measures for assessing these first four elements of quality. The last two are more subjective and difficult to assess although they are critical in expressing quality. Aesthetics characterizes how an item appears in terms of its appropriate looks, smell, touch, and sound. Perceived quality is the overall image a product has among users and potential users. Brand names tend to have a higher image value than generic, based on experience in often different and unrelated product lines. This is an indicator of customer confidence.

The attributes of product quality relate to the way customers value the manner in which a product is designed, produced, and serviced. Those attributes that influence customer needs and preferences tend to be neglected in planning and design decisions. However, in reality it is the way customers accept and perceive the value of an item that determines its effectiveness as a product. Most consumer products have a hidden cost due to what customers interpret as poor service when in actuality the product is well constructed and operational but is not used properly due to inadequate user manuals or poor communications on how it should be used. This still translates to poor quality as it is perceived by the public.

Reliability is a vital element of quality since it has significant impact on the other elements, thereby influencing the way an item is valued by customers. Consequently, while the number of features that are necessary to describe the quality can vary with the type of product, an effective quality assessment must include some measure of the items reliability.

Ideally, we could find a function [pi]([Q.sub.1],..., [Q.sub.6]) that adequately relates the relative significance of these dimensions on some quantitative scale that would allow engineers and managers to diagnose and interpret marginal changes in particular dimensions. This is quite difficult in general due to the subjectivity and wide variation in customers' opinions. This function [pi] will also depend upon the choice of configuration and the dependencies of the modes of failure and warranty claims among the various physical components of the product system.

Warranty costs during the warranty period can provide an overall relative measure of quality. There are several types of warranties (see Blischke and Murthy (1997) and Thomas and Rao (1999)). One example is a free-replacement policy in which the customer receives full compensation for problems occurring during the warranty period. Another example is a pro rata warranty in which the cost of a failure is shared by the customer and manufacturer according to some weighting scheme. In all cases warranty claims impact the costs related to essentially all elements of [pi]. If quality is high then the warranty requirements are relatively low, and vice versa if quality is low. Since warranty expenditures serve as the penalty costs for imperfect quality to customers, they are often used as a relative measure for its quantification.

4. Method to establish reliability targets

The proposed method is for use in planning quality improvements by establishing reliability improvement goals through the allocation of an overall system goal among the product components. The procedure focuses on allocating the improvement effort where the impact on quality and cost effectiveness will be the greatest relative to customer and market preferences. To accomplish this we incorporate warranty costs as allocation criteria for assigning improvement effort. As with earlier methods we consider a series configuration of a product system consisting of n independent components. The system reliability is therefore the product of the component reliabilities. The method we propose works in two stages. In the first stage, we develop, at the system level, preliminary quality improvement targets for each component based on their failure rates and warranty costs. In the second stage, we derive, at the component level, technologically feasible quality improvement targets and determine which combination to select given a limited budget. We now focus on the determination of preliminary quality improvement targets while the derivation of actual improvement targets is given in Section 5. The additional notation for the subsequent development is as follows.

Let

i = 1,..., n represent the index for components.

[[beta].sub.i] = warranty burden for component i;

[gamma] = percentage improvement goal for the overall system, 0 [less than or equal to] [gamma] [less than or equal to] 1;

[lambda], [[lambda].sub.i] = failure rate of the existing system, and existing component i;

[[lambda].sup.0], [[lambda].sub.i.sup.0] = preliminary failure target for the system, and component i;

[lambda]*, [[lambda]*.sub.i] = technologically feasible failure rate goal for the system, and component i;

[M.sub.i](t) = expected number of component i failures in (0, t);

M(t) = expected number of system failures in (0, t);

[R.sub.i](t) = current component i reliability at age t;

R(t) = current system reliability at age t;

[R.sub.i.sup.0](t) = reliability target for component i at age t;

[R.sup.0](t) = reliability target for the system at age t;

[[rho].sub.i] = marginal warranty improvement opportunity rate for component i;

[T.sub.W] = length of the warranty period;

[c.sub.Wi] = warranty cost for component i;

[c.sub.Bi] = marginal cost to improve component i

For planning purposes we will use the length of the product warranty period, [T.sub.W], as a fixed time horizon and without loss of generality we will assume that the product is sold under a free-replacement warranty policy with an average cost to the manufacturer of [c.sub.Wi] dollars for each warranty event for component i. We assume that the length of the warranty is sufficiently small so that the resulting change in R(t) for multiple replacements under the warranty during [0, [T.sub.W]] can be neglected.

Starting with a total product perspective the objective is to determine the n component reliabilities, [R.sub.i.sup.0]([T.sub.W]) for accomplishing a 100[gamma], 0 < [gamma] < 1 percent aggregate improvement over the current component levels [R.sub.i]([T.sub.W]), i = 1,..., n. Thus, the product system goal is for:

[R.sup.0]([T.sub.W]) [greater than or equal to] (1 + [gamma])R([T.sub.W]). (3)

While the intent is to increase the reliability as much as possible, there will be cost limitations on the options that are available for making improvements. There are two types of costs that enter the decision: (i) the warranty cost for claims filed and restitution to customers that are not satisfied; and (ii) the cost for implementing improvements. The expected cost of component i failures during the warranty period [0, [T.sub.W]] is [c.sub.Wi][M.sub.i]([T.sub.W]) and for the product system:

C([T.sub.W]) = [n.summation over (j=1)] [c.sub.Wj][M.sub.j]([T.sub.W]), (4)

where [c.sub.Wi] is the average unit warranty cost for component i, and

M(t) = [[infinity].summation over (j=1)] [F.sup.(k)](t), (5)

is the renewal function with [F.sup.(k)] being the k-fold convolution of the distribution of the time of occurrence of claims with itself (see Ross (2000)). We now define the warranty burden for component i as:

[[beta].sub.i] = [[c.sub.Wi][M.sub.i]([T.sub.W])]/[[[summation].sub.j=1.sup.n] [c.sub.Wj][M.sub.j]([T.sub.W])], i = 1,..., n, (6)

which is the fraction of the total warranty cost that is contributed by component i. The rationale for this method is that since the relative quality of a product component is reflected through its warranty, then components that fail frequently or have high warranty expense or both will have a correspondingly high warranty burden. It is therefore argued that under the improvement initiative the goal should be to reduce failures at each component i in proportion to the relative fraction of warranty cost due to that component. Elements with higher values of [[beta].sub.i] should receive greater reduction targets than those having a smaller warranty burden. The reduction in the number of failures during warranty for component i, [M.sub.i]([T.sub.W]) - [M.sub.i.sup.0]([T.sub.W]) to the total number for the product should be at least [[beta].sub.i]. Therefore:

[[M.sub.i]([T.sub.W]) - [M.sub.i.sup.0]([T.sub.W])]/[M([T.sub.W]) - [M.sup.0]([T.sub.W])] [greater than or equal to] [[beta].sub.i], i = 1,..., n, (7)

from which it follows that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

We will consider two reliability allocation scenarios for determining component improvement targets: (i) the case of the system having a Constant Failure Rate (CFR); and (ii) the situation of an Increasing Failure Rate (IFR).

4.1. CFR Allocation

Suppose the products function during their useful life in [0, [T.sub.W]] with constant component failure rates [[lambda].sub.1],..., [[lambda].sub.n]. We wish to determine new failure rates [[lambda].sub.1.sup.0],..., [[lambda].sub.n.sup.0] so that the reliability of the improved system satisfies Equation (3). Note that the improved system reliability during warranty is then given by:

[R.sup.0]([T.sub.W]) = exp[.sup.(-[T.sub.W][[summation].sub.i=1.sup.n][[lambda].sub.i.sup.0])],

and it follows from Equation (3) that:

[n.summation over (i=1)] [[lambda].sub.i.sup.0] [less than or equal to] 1/[T.sub.W] ln (1/(1 + [gamma])R([T.sub.W])). (9)

The number of claims in [0, [T.sub.W]] is then Poisson distributed with the expected number of claims given by the renewal function:

[M.sub.i]([T.sub.W]) = [[lambda].sub.i][T.sub.W] for i = 1,..., n. (10)

Furthermore, from Equations (7) and (10) it follows that:

[[[lambda].sub.i] - [[lambda].sub.i.sup.0]]/[[lambda] - [[lambda].sup.0]] [greater than or equal to] [[beta].sub.i], i = 1,..., n, (11)

with

[lambda] = [n.summation over (i=1)][[lambda].sub.i] and [[lambda].sup.0] = [n.summation over (i=1)] [[lambda].sub.i.sup.0],

since the [T.sub.W] terms in the numerator and denominator of Equation (7) cancel out. We note that maximizing the reliability is equivalent to minimizing the failure rate and hence the allocated failure rates are given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

4.2. IFR allocation

We now consider the case where the time of occurrence of warranty claims is IFR but the product improvement goal will achieve a CFR system. So, for this more general case, in Equation (9) we want to accomplish the reliability improvement:

[R.sup.0]([T.sub.W]) = (1 + [gamma]) exp[.sup.(-[[integral].sub.0.sup.[T.sub.W]] [lambda](u)du)]. (13)

From Equation (11) we have that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

Suppose we have n = 3 components with [[lambda].sub.1](t) = [[lambda].sub.1],[[lambda].sub.2](t) = [[lambda].sub.2](t), and [[lambda].sub.3](t) = [[lambda].sub.3], for t [greater than or equal to] 0. It then follows that [M.sub.1](t) = [[lambda].sub.1]t and [M.sub.3](t) = [[lambda].sub.3]t. To compute [M.sub.2](t) we first determine the probability density function for the time of occurrence of warranty claims from:

[f.sub.2](t) = -[d/dt]R(t) = [[lambda].sub.2]t exp[.sup.([-[[lambda].sub.2][t.sup.2]]/2)] t [greater than or equal to] 0, (15)

which is the Erlang probability density function. It follows (see the Appendix) that the renewal function for this component is given by:

[M.sub.2](t) = [1/4]exp[.sup.(-2[[lambda].sub.2]t)] + [[[lambda].sub.2]/2]t - [1/4] t [greater than or equal to] 0. (16)

Therefore,

M([T.sub.W]) = [1/4]exp[.sup.(-2[[lambda].sub.2][T.sub.W])] + ([[lambda].sub.1] + [[[lambda].sub.2]/2] + [[lambda].sub.3]) [T.sub.W] - [1/4] t [greater than or equal to] 0. (17)

and substituting into Equation (6):

[[beta].sub.1] = [[c.sub.W1][[lambda].sub.1][T.sub.W]]/[[c.sub.W1][[lambda].sub.1][T.sub.W] + [c.sub.W2]((1/4) exp[.sup.(-2[[lambda].sub.2][T.sub.W])] + ([[lambda].sub.2]/2)[T.sub.W] - (1/4)) + [c.sub.W3][[lambda].sub.3][T.sub.W]], (18)

[[beta].sub.2] = [[c.sub.W2]((1/4) exp[.sup.(-2[[lambda].sub.2][T.sub.W])] + ([[lambda].sub.2]/2)[T.sub.W] - (1/4))]/[[c.sub.W1][[lambda].sub.1][T.sub.W] + [c.sub.W2]((1/4) exp[.sup.(-2[[lambda].sub.2][T.sub.W])] + ([[lambda].sub.2]/2)[T.sub.W] - (1/4)) + [c.sub.W3][[lambda].sub.3][T.sub.W]], (19)

and

[[beta].sub.3] = [[c.sub.W3][[lambda].sub.3][T.sub.W]]/[[c.sub.W1][[lambda].sub.1][T.sub.W] + [c.sub.W2]((1/4) exp[.sup.(-2[[lambda].sub.2][T.sub.W])] + ([[lambda].sub.2]/2)[T.sub.W] - (1/4)) + [c.sub.W3][[lambda].sub.3][T.sub.W]]. (20)

The target failure rates for improvement are then found by substituting the values from Equation (18) into Equation (20), and Equation (17) into Equation (14).

Example 1. For illustrative purposes suppose [[lambda].sub.1] = 0.1, [[lambda].sub.2] = 0.15, and [[lambda].sub.3] = 0.05 with [T.sub.W] = 1 year. The costs are [c.sub.W1] = c, [c.sub.W2] = 1.5c, and [c.sub.W3] = 0.5c units of cost for c > 0. The overall goal is to improve R([T.sub.W]) by 20%.

The current reliability for this product system is given by:

R([T.sub.W]) = exp[.sup.(-[[[lambda].sub.1][T.sub.W]+([[lambda].sub.2]/2)[T.sub.W.sup.2]+[[lambda].sub.3][T.sub.W]])],

or

R(1) = exp[.sup.(-1[0.1(1)+(0.15/2)(1)[.sup.2]+0.05(1)])] = 0.7985,

and from Equation (13) the overall reliability goal is [R.sup.0](1) = (1.2)(0.7985) = 0.9582. The mean number of warranty claims for the components are [M.sub.1](1) = 0.1(1) = 0.1, [M.sub.3](1) = 0.05(1) = 0.05, and from Equation (16)

[M.sub.2](1) = [1/4]exp[.sup.(-2(0.15)1)] + [0.15/2](1) - 1/4 = 0.0102,

thus giving a total mean of M(1) = 0.1602. Substituting these values into Equation (18):

[[beta].sub.1] = 0.1(1)/[0.1(1) + 1.5 (exp[.sup.(-2(0.15)1)]+(0.15/2)(1) - 1/4) + 0.5(0.05)1]

= 0.1027,

and from Equations (19) and (20), [[beta].sub.2] = 0.8716 and [[beta].sub.3] = 0.0257. Now for an improved CFR system with reliability [R.sup.0](1) = 0.9582, it follows from Equation (9) that the system failure rate is [[lambda].sup.0] = 0.0427.

Applying these values to Equation (14) leads to:

[M.sub.1.sup.0](1) = 0.1 - 0.1027[0.1602 - 0.0427] = 0.0162,

[M.sub.2.sup.0](1) = 0.0102, since [M.sub.2](1)/[[beta].sub.2] < M(1) - [M.sup.0](1); and [M.sub.3.sup.0](1) = 0.0291. The solution is therefore [[lambda].sub.1.sup.0] = 0.0162, [[lambda].sub.2.sup.0] = 0.0102, and [[lambda].sub.3.sup.0] = 0.0291. For these parameter values, and a 1-year warranty with [T.sub.W] = 1, the target improvements are to reduce the failures for components 1 and 3, accepting component 2 to remain as it is for the next period. This is due to the fact that even though component 2 has an IFR and the warranty burden is large, the expected number of failures during the 1 year period is relatively small compared to the other two components. However, if the warranty was extended an additional 90 days to [T.sub.W] = 1.25 then all of the components would be targeted for improvement.

5. Allocation with a limited budget

The method described in Section 4 provides guidelines on how to achieve an overall improvement level but most often they should not be taken as final. These are preliminary targets that require further analysis at the component level. Shifting the focus from the broader system level to the product subsystem and component levels can reveal other factors and constraints that can alter the final decision on targets. The target failure rates [[lambda].sub.1.sup.0],..., [[lambda].sub.n.sup.0] have to be further examined to determine if they are technologically feasible and if their cost is not prohibitive. At this level all technical details are considered and the various design and improvement options are evaluated to establish the best improvement alternatives for each component. Options such as upgrading modules through material selection, selecting different vendors, or adding redundant elements are evaluated with respect to their associated costs. Because budget constraints are always present, not all subsystems will necessarily be assigned a final improvement goal.

So for this analysis we assume that there is a budget limit B on the cost of implementing all selected improvements. For the sake of simplicity in the presentation, we also assume that for each component i, only one technologically and economically viable improvement option has been determined whose projected failure rate is [[lambda]*.sub.i] and marginal cost is [c.sub.Bi] for i = 1,..., n. These cost estimates include the costs for new devices, upgraded materials, and modifications of the component plus the costs for warranty expenditures, typically projected from sales and service data. Cases in which none or several improvement options are determined for some components can be handled with similar models. Note that, because it is typically not acceptable for a final component failure rate [[lambda]*.sub.i] derived after the various technical considerations and options to increase above its current level, we assume that [[lambda]*.sub.i] [less than or equal to] [[lambda].sub.i.sup.0]. We also assume that [c.sub.Bi] [less than or equal to] B for i = 1,..., n, since otherwise improvement option i cannot be afforded within the current budget limitations, and assume that [[summation].sub.i=1.sup.n] [c.sub.Bi] > B, since otherwise all the improvement options can be selected within the given budget and the solution is trivial. Under these assumptions the allocation can be formulated as the binary linear program:

z* = max [n.summation over (i=1)] [c.sub.Wi]([[lambda].sub.i] - [[lambda]*.sub.i])[x.sub.i],

Subject to

[n.summation over (i=1)] [c.sub.Bi][x.sub.i] [less than or equal to] B, (21)

[x.sub.i] [member of] {0, 1}, [for all]i = 1,..., n,

that maximizes the warranty cost savings over the warranty period. It is a form of the classical knapsack problem which is known to be NP-complete. However, because of its simple structure, various methods, both exact and approximate are available for its solution. Among the exact methods, branch-and-bound techniques and dynamic programming algorithms are the most commonly used (see Kellerer et al. (2004)). In particular, commercial optimization software such as CPLEX and X-PRESS will provide fast solutions to most instances of Equation (21). However, for this particular application where we are developing planning goals, we argue that it is sufficient to find a good approximate solution to Equation (21) since the data of the problem is itself approximate. Therefore, we next describe a simple algorithm that produces a good quality solution to Equation (21) without requiring heavy computing artillery or expensive optimization software. First we define the marginal warranty improvement opportunity rate by:

[[rho].sub.i] = [[c.sub.Wi]([[lambda].sub.i] - [[lambda]*.sub.i])]/[c.sub.Wi], [for all]i = 1,..., n, (22)

and assume without loss of generality that the improvement options are sorted in descending order of [rho], i.e.,

[[rho].sub.i] [greater than or equal to] [[rho].sub.j], [for all]i = 1,..., n, [for all]j = 1,..., n subject to i < j.

We define the "split option" as the option k for which:

[k-1.summation over (i=1)] [c.sub.Bi] [less than or equal to] B < [k.summation over (i=1)] [c.sub.Bi].

Then, we create two solutions [bar.x] and [~.x] where [bar.x.sub.k] = 0, [~.x.sub.k] = 1 and for j = 1,..., n, j [not equal to] k

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCTBLE IN ASCII].

We denote their objective values by [bar.z] and [~.z] respectively. Clearly these two solutions represent allocations that are feasible within the allowed budget. We select the one that has the largest objective value and call it [arc.x], i.e. [arc.x] = [bar.x] if [bar.z] > [~.z] and [arc.x] = [~.x] if [~.z] [greater than or equal to] [bar.z].

Proposition 1. [arc.z] [greater than or equal to] z*/2.

Proof. On the one hand z*, the optimal value of Equation (21) is no greater than the value of its linear programming relaxation. Therefore:

z* [less than or equal to] [z.sup.LP] = [k-1.summation over (i=1)] [C.sub.wi]([[lambda].sub.i] - [[lambda]*.sub.i]) + [[[c.sub.wk]([[lambda].sub.k] - [[lambda]*.sub.k])]/[c.sub.Bk]](B - [k-1.summation over (i=1)][c.sub.Bi])

[less than or equal to] [k.summation over (i=1)] [c.sub.wi]([[lambda].sub.i] - [[lambda]*.sub.i]). (23)

On the other hand, it follows from the definition of [arc.x] that:

[arc.z] = max{[bar.z], [~.z]} [greater than or equal to] [[bar.z]/2] + [[~.z]/2] [greater than or equal to] [[[summation].sub.i=1.sup.k-1] [c.sub.wi]([[lambda].sub.i] - [[lambda]*.sub.i])]/2 + [[c.sub.wk]([[lambda].sub.k] - [[lambda]*.sub.k])]/2 = [[[summation].sub.i=1.sup.k] [c.sub.wi]([[lambda].sub.i] - [[lambda]*.sub.i])]/2. (24)

Combining Equation (23) with Equation (24) we obtain [arc.z] [greater than or equal to] z [not equal to] 2, which proves the result. [black square]

Proposition 1 guarantees that the method described above never produces an improvement plan whose expected warranty cost reduction is less than half of that of the best possible warranty cost reduction plan. However, it will typically result in solutions whose quality is much better than the minimum performance guaranteed by Proposition 1, as we will observe in the example in Section 6. Note, however, that if needed, it is possible to produce an approximate solution whose guaranteed quality is as high as desired using the fully polynomial approximation schemes of Lawler (1979) and Magazine and Oguz (1981).

6. Sample application of the method

To illustrate an application of the warranty-based method to develop reliability improvement targets let us consider the gearbox for a motor vehicle described by Ivanovic (2000). The assembled unit consists of the following five subsystems that are considered to be in a series configuration.

Subsystem

1. box with components;

2. input and main shaft with internal gears, synchronizer assembly, and bearing;

3. counter and reverse idler shaft with internal gears, synchronizer assembly, and bearing;

4. selector forks with components;

5. gear selector mechanism.

In this example, we assume that the current gearboxes are produced with the subsystem failure rates and warranty costs as shown in Fig. 1.

We assume that these gearboxes are sold to a vehicle manufacturer with a 1-year free replacement warranty. The warranty cost for each component is obtained from the contribution of that component to the system warranty cost. The company would like to establish a reliability improvement goal as part of their continuous quality improvement program. Ideally, the management would like to improve the overall reliability by 20% while maintaining the costs of the improvements below $300 000.

[FIGURE 1 OMITTED]

6.1. Solution

We assume that all of the subsystems are functioning during their respective useful lives so the failure rates are all constant throughout the [T.sub.W] = 1 year warranty period. The system failure rate for the gearbox is then:

[lambda] = [5.summation over (i=1)][[lambda].sub.i] = 0.02737 + 0.07796 + 0.11093 + 0.07668 + 0.24207 = 0.53501,

and the current reliability is

R(1) = [e.sup.-0.53501(1)] = 0.58566.

So for an overall product reliability improvement of 20%, [gamma] = 0.20 in Equation (3) and hence:

R*(1) [greater than or equal to] (1.2)(0.58566) = 0.70279,

and the initial system failure rate goal is accordingly, from Equation (9):

[lambda]* [less than or equal to] (1/1) ln (1/0.70279) = 0.35270.

6.1.1. Preliminary targets

Applying the result of Equation (6) we determine the warranty burden for each subsystem. From Equation (4), the total expected warranty cost is:

w = [5.summation over (i=1)] [c.sub.Wi][[lambda].sub.i] = 7.37(0.02737) + 10.42(0.07796) + 9.58 x (0.11093) + 5.16(0.07668) + 8.23(0.24207)

= 4.4647,

Subsystem 1, the box with components, has a warranty burden of;

[[beta].sub.1] = (0.02737)7.37/4.4647 = 0.04518,

and similarly, [[beta].sub.2] = 0.18195, [[beta].sub.3] = 0.238 03, [[beta].sub.4] = 0.08862, and [[beta].sub.5] = 0.44622. Substituting these values into Equation (12) we have for subsystem 1 a target failure rate of:

[[lambda].sub.1.sup.0] [less than or equal to] 0.02737 - (0.04518)(0.53501 - 0.352 69) = 0.01913,

since [[lambda].sub.1]/[[beta].sub.1] > 0.182 32. The remaining target values follow with [[lambda].sub.2.sup.0] = 0.04479, [[lambda].sub.3.sup.0] = 0.06753, [[lambda].sub.4.sup.0] = 0.06052, and [[lambda].sub.5.sup.0] = 0.16072. These results are summarized in Table 2.

6.1.2. Final targets for a constrained budget

Now for the detailed analysis at the subsystem and component levels we start by exploring the options for pursuing the target levels [[lambda].sup.0] = ([[lambda].sub.1.sup.0],..., [[lambda].sub.5.sup.0]), and the respective costs [c.sub.B] = ([c.sub.B1],..., [c.sub.B5]), to implement the improvement changes for the final allocations. Suppose for illustrative purposes that [c.sub.B] = (75, 75, 80, 50, 150). The results for this bottom-up analysis are summarized in Table 3.

The optimal solution of Equation (21) can be easily found using commercial optimization software. It takes 0.2 seconds on a laptop for CPLEX 8.1 to find an optimal solution for the problem. The optimal solution consists in selecting improvement options 3, 4, and 5. We illustrate now how the simple method we described in Section 4 works and show that, in this case, it also produces an optimal solution for the problem. From Equation (23), the marginal warranty improvement opportunity rate for subsystem 1 is:

[[rho].sub.1] = [[c.sub.w1]([[lambda].sub.1] - [[lambda].sub.1.sup.0])]/[c.sub.B1] = [7.37(0.02737 - 0.019133)]/75

= 8.09 x [10.sup.-4],

and similarly, [[rho].sub.2] = 46.08 x [10.sup.-4], [[rho].sub.3] = 51.97 x [10.sup.-4], [[rho].sub.4] = 16.68 x [10.sup.-4], and [[rho].sub.5] = 44.63 x [10.sup.-4]. Now to determine the optimal allocation we find the ordering:

[[rho].sub.3] > [[rho].sub.2] > [[rho].sub.5] > [[rho].sub.4] > [[rho].sub.1].

Consecutively, for a budget B = $300 000, we have that:

[bar.x.sub.3] = 1, [bar.x.sub.2] = 1, [bar.x.sub.5] = 0, [bar.x.sub.4] = 1, [bar.x.sub.1] = 1,

whose objective value [bar.z] is 0.9055. We also have that:

[~.x.sub.3] = 1, [~.x.sub.2] = 0, [~.x.sub.5] = 1, [~.x.sub.4] = 1, [~.x.sub.1] = 0,

whose associate objective value [~.z] is 1.1687. Since [bar.z] < [~.z], we will use the solution [~.x] as the proposed improvement plan. Therefore, subsystems 3, 4 and 5 (the counter and reverse idler shaft subassembly, the selector forks, and the gear selector mechanism) will be assigned as failure rate reduction objectives. The subsystem objectives are therefore: [arc.[lambda].sub.1] = 0.02737, [arc.[lambda].sub.2] = 0.077 96, [arc.[lambda].sub.3] = 0.067 53, [arc.[lambda].sub.4] = 0.060 52, and [arc.[lambda].sub.5] = 0.160 72. The overall product failure rate objective is to achieve [arc.[lambda]] = 0.3941 or reliability R(1) = 0.67429, which will result in a 15.1% percent improvement in the reliability that is much less than the original target of [gamma] = 0.20. The implementation cost of the changes is:

[C.sub.B] = 80 + 50 + 150 = 280,

which is within the budgeted amount of B = $300, 000. The improvement gain through reduced warranty costs in general is given by:

C([T.sub.W], [lambda]) - C([T.sub.W], [arc.[lambda]]) = [n.summation over (i=1)] [c.sub.Wi]([[lambda].sub.i] - [arc.[lambda].sub.i])[T.sub.W]. (25)

For our example here:

C([T.sub.W], [lambda]) - C([T.sub.W], [arc.[lambda]]) = 9.58(0.11093 - 0.06753) + 5.16(0.07668 - 0.06052) + 8.23(0.24207 - 0.16072) = 1.1687.

Based on this analysis the allocation for reliability improvement is to target subsystems 3, 4, and 5, while holding the subsystems 1 and 2 at their current levels. Now suppose that rather than making the allocation using our suggested procedure, the decision is made to choose a different allocation, say [I.sub.B.sup.1] = {1, 2, 3, 4}. The failure rate improvement targets for this alternative would then be [[lambda].sub.1.sup.1] = [[lambda].sub.1.sup.0] = 0.019133, [[lambda].sub.2.sup.1] = [[lambda].sub.2.sup.0] = 0.04479, [[lambda].sub.3.sup.1] = [[lambda].sub.3.sup.0] = 0.06753, [[lambda].sub.4.sup.1] = [[lambda].sub.4.sup.0] = 0.06052 and [[lambda].sub.5.sup.1] = [[lambda].sub.5] = 0.16072. An advantage of this alternative is that it would allow an additional subsystem to show improvement, however, the failure rate for the overall product would be [[lambda].sup.1] = 0.6479 which is higher than [arc.[lambda]]. Moreover, the projected benefit through reduced warranty cost from Equation (25) is:

C([T.sub.W], [lambda]) - C([T.sub.W], [[lambda].sup.1]) = 7.37(0.02737 - 0.019133) + 10.42(0.07796 - 0.04479) + 9.58(0.11093 - 0.06753) + 5.16(0.07668 - 0.06052) = 0.9055,

which is less than the benefit for [arc.[lambda]].

7. Conclusions

The approach described in this paper is based on an alternative formulation to find an acceptable reliability allocation between components that will satisfy a given cost constraint. The proposed approach consists of two stages: (i) the top-down procedure described in Section 4 to establish preliminary goals; and (ii) a more detailed bottom-up analysis of the component options and allocation method of Section 5 that incorporates a budget constraint. The proposed method in Section 4, similar to the ARINC method (Von Alven, 1964), is an apportionment procedure that incorporates weighting factors to account for the relative importance of the components. Here the weighting factors are based on the warranty burden rates defined in Equation (6). We argue that this method has rational appeal since it incorporates the relative impact of warranty expenditures from claims filed as results of quality problems. We mention a variety of methods that can be used to solve Equation (21) and describe a simple-to-apply approximation algorithm that can be used to obtain quick provably good quality solutions to the problem. We argue that such an approximation is sufficient since the data of the problem is inevitably approximate.

Like most reliability allocation techniques the proposed method is based on an assumed series arrangement of independent components. The allocation algorithm in Section 4.1 is further based on the assumption that the time between component warranty claims is exponentially distributed. While this is a common assumption for reliability allocation, it might not hold well for some situations since while component failures times might reasonably be distributed exponentially, warranty claim events also occur without an actual failure. For example, a product with poor operating instructions can lead to erroneous claims of failure. However, for quality management purposes some organizations might choose to treat this as a failure anyway since it creates dissatisfaction among consumers. In Section 4.2 the case of IFR product warranty claims is considered. The reliability improvement strategy, for both cases of current CFR and IFR conditions is to develop goals aimed at an improved product system that has CFR conditions on product warranty claims. Note, however, that in example 1 the final optimum-improvement alternative was for the system to still be IFR.

For the method presented in this paper, we assumed that the reliability improvements will result in benefits of fewer warranty service requests thus resulting in reduced costs. An alternative approach would be to develop marginal service cost reduction goals with or in conjunction with reduced failures. This could be an area for future research.

As a first effort to incorporate a warranty burden factor for planning targets for reliability improvement, the method proposed in this paper considered only the expected value of the warranty cost of the components. A direction for future research is to generalize the proposed approach to take into account the variance of warranty costs. This generalization could be significant in situations where the variance is large.

In this paper we used the warranty period [0, [T.sub.W]] for the reliability improvement decision planning horizon. This is quite common in practice since this is the period that warranty costs are tracked and information beyond this period is difficult to obtain. Still the method is a rational means for developing strategic goals to improve product reliability, though it alone is not sufficient for developing overall quality improvement plans. Several elements of the quality vector in Table 1, such as durability and perceived quality are significantly influenced by product performance beyond the warranty period. Clearly, warranty information is an important source of valuable feedback that can be translated into continuous quality improvement programs.

Acknowledgement

The authors wish to express their gratitude to the Area Editor and referees for providing valuable improvements for this paper.

References

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Kuo, W, Prasad, V.R., Tillman, FA. and Hwang, C. (2001) Optimal Reliability Design, Cambridge Press, New York, NY.

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Appendix

Derivation of Equation (16)

The renewal function for the Erlang time-to-failure probability density function f(t) = [lambda]t [e.sup.[-[lambda][t.sup.2]]/2], t [greater than or equal to] 0 is given by:

M(t) = [1/4][e.sup.-2[lambda]t] + [[lambda]/2t] - [1/4[lambda]] t [greater than or equal to] 0. (16)

Proof. The Laplace transform for the renewal function is given by:

[bar.M](s) = [[bar.f](s)]/[s[1 - [bar.f](s)]], (A1)

where

[bar.f](s) = [[integral].sub.0.sup.[infinity]] [e.sup.-st]f(t)dt,

is the Laplace transform of f(t) with Re(s) > 0. Therefore, for the f(t) given in Equation (15):

[bar.f](s) = ([lambda]/[[lambda] + s])[.sup.2], (A2)

from which it follows that:

[bar.M](s) = [[lambda].sup.2]/[s(2[lambda] + s)]. (A3)

Taking a partial fraction expansion of Equation (A3):

[bar.M](s) = [lambda]/[2[s.sup.2]] - [1/4s] + [1/[4(2[lambda] + s)]],

which is easily inverted to Equation (16). [black square]

Biographies

Marlin U. Thomas is Dean, Graduate School of Engineering and Management at the Air Force Institute of Technology, and past Professor and Head of the School of Industrial Engineering at Purdue University. He received his BSE at the University of Michigan-Dearborn, and MSE and PhD at the University of Michigan. He has held other academic appointments at Lehigh University, Cleveland State University, University of Missouri-Columbia, University of Wisconsin-Milwaukee, and the Naval Postgraduate School. He has also served as a Program Director for the National Science Foundation; Manager, Reliability and Warranty Analysis, Chrysler Corporation; and Development Engineer, Owens-Illinois, Inc. He is past National Secretary of ORSA, Chairman of the Council of Industrial Engineering Academic Department Heads, and HE Past-President and member of the Board of Trustees. His research interests are in operations research with applications in reliability and contingency logistics. He is a Fellow of IIE, ASQ, and INFORMS. He is also a Captain, Civil Engineer Corps, U.S. Navy Reserve (Retired).

Jean-Philippe P. Richard is an Assistant Professor at the School of Industrial Engineering, Purdue University. He received a Bachelor's degree in Applied Mathematics Engineering from Universite Catholique de Lou-vain, Belgium. He holds a doctorate in Algorithms, Combinatorics and Optimization from the Georgia Institute of Technology. His research interests are in mathematical programming and more specifically in integer programming and discrete optimization. He is a member of MPS and INFORMS.

MARLIN U. THOMAS (1) and JEAN-PHILIPPE P. RICHARD (2)

(1) Graduate School of Engineering and Management, Air Force Institute of Technology, 2950 Hobson Way, Wright-Patterson AFB, OH 45433-7765, USA

E-mail: Marlin. Thomas@afit.edu

(2) School of Industrial Engineering, Purdue University, 315 N. Grant Street, West Lafayette, IN 47906, USA

E-mail: jprichar@purdue.edu

Received January 2005 and accepted January 2006

Table 1. Quality dimensions of a product

           Dimension          Definition

[Q.sub.1]  Performance        How the product functions relative to its
                                design
[Q.sub.2]  Durability         Ultimate usage before ultimate
                                deterioration
[Q.sub.3]  Reliability        Probability of failure having survived to
                                that point
[Q.sub.4]  Conformance        Degree to which design and engineering
                                standards are met
[Q.sub.5]  Aesthetics         How the product appears through sensory
                                skills
[Q.sub.6]  Perceived quality  Image of the product by potential users

Table 2. Preliminary analysis for the gearbox assembly example

                                       i
                        1         2         3        4

[[lambda].sub.i]        0.02737    0.07796  0.11093  0.07668
[c.sub.wi] ($/unit)     7.37      10.42     9.58     5.16
[[beta].sub.i]          0.04518    0.18195  0.23803  0.08862
[[lambda].sub.i.sup.0]  0.019133   0.04479  0.06753  0.06052

                              i
                        5        System

[[lambda].sub.i]        0.24207  0.53501
[c.sub.wi] ($/unit)     8.23
[[beta].sub.i]          0.44622
[[lambda].sub.i.sup.0]  0.16072  0.35269

Table 3. Bottom-up analysis for the gearbox assembly example

                                              i
                            1           2          3          4

[[lambda].sub.i]             0.027 37    0.077 96   0.110 93   0.076 68
[[lambda].sub.i.sup.0]       0.019 133   0.044 79   0.067 53   0.060 52
[c.sub.Bi] (k$)             75          75         80         50
[[rho].sub.i](x[10.sup.4])   8.09       46.08      51.97      16.68
<j>                          5           2          1          4
[arc.[lambda].sub.i]         0.027 37    0.077 96   0.067 53   0.076 68

                                 i
                            5           System

[[lambda].sub.i]              0.242 07  0.535 01
[[lambda].sub.i.sup.0]        0.160 72  0.3529
[c.sub.Bi] (k$)             150
[[rho].sub.i](x[10.sup.4])   44.63
<j>                           3
[arc.[lambda].sub.i]          0.160 72  0.3941