Product portfolio planning with customer-engineering interaction.

Nomenclature

[a.sub.k] = product attribute;

A = set of product attributes;

[a*.sub.kl] = attribute level;

[A*.sub.k] = set of attribute levels for an attribute;

[z.sub.j] = product profile;

Z = set of products;

[LAMBDA] = product

[C.sub.j] = engineering cost of a product;

[P.sub.j] = price of a product;

[s.sub.i] = market segment;

[Q.sub.i] = size of a market segment;

[U.sub.ij] = customer-perceived utility of a product;

[P.sub.ij] = probability of a customer choosing a product;

[C.sub.j.sup.V] = variable cost per product;

[C.sub.j.sup.F] = fixed cost per product;

E[V] = expected value of shared surplus;

[u.sub.ikl] = part-worth utility of segment [s.sub.i] for the lth level of attribute [a.sub.k];

[w.sub.jk] = utility weights among attributes;

[[pi].sub.j] = constant of composite utility for a product;

[[epsilon].sub.ij] = error term for a segment-product pair;

[mu] = scaling parameter of conditional multinomial logit choice rule;

PC[I.sub.j] = process capability index corresponding to a product;

LS[L.sup.T] = lower specification limit of cycle time estimation;

[[mu].sub.j.sup.T] = mean value of the estimated cycle time corresponding to a product;

[[sigma].sub.j.sup.T] = standard deviation of the estimated cycle time corresponding to a product;

[beta] = constant indicating the average dollar cost per variation of process capabilities;

[[zeta].sub.jk]/[[omega].sub.j] = regression coefficients;

[[mu].sub.kl.sup.t] = mean value of the part-worth standard time for attribute level [a*.sub.kl];

[[sigma].sub.kl.sup.t] = standard deviation of the part-worth standard time for attribute level [a*.sub.kl];

[x.sub.jkl] = binary variable indicating the choice of an attribute level in a product;

[y.sub.j] = binary variable indicating the decision of offering of a product.

1. Introduction

In order to compete in the marketplace manufacturers are looking to expand their product lines and differentiate between their product offerings in the intuitively appealing belief that a large product mixture may stimulate sales and hence increase revenues (Ho and Tang, 1998). Whereas a strategy based on a large product collection may offer an effective means for a company to differentiate itself from its competitors, it unavoidably leads to a high complexity and high costs for product fulfillment (Child et al., 1991). Moreover, producing a wide variety of products and then letting the customers choose the appropriate product may lead to the customer becoming overwhelmed by the large product range or frustrated by the complexity involved in making a choice (Huffman and Kahn, 1998). Therefore, it becomes imperative for the manufacturer to determine how to offer the "right" product variety to the target market. Such decisions on the optimal amount of product mix adhere to the general wisdom as suggested in the Boston Consulting Group's notion of product portfolio strategy (Henderson, 1970). While representing the spectrum of a company's product offerings, the product portfolio must be carefully set up, planned and managed so as to match those customer needs in the target market (Warren, 1983).

Product portfolio planning has been traditionally dealt with in the management and marketing fields with the focus being on portfolio optimization based on customer preferences. The objective is to maximize profit, share of choices, or sales (Urban and Hauser, 1993). Consequently, the measurement of customer preferences among multi-attribute alternatives has been a primary concern in marketing research. Among the many developed methods conjoint analysis has turned out to be one of the most popular preference-based techniques for identifying and evaluating new product concepts (Green and Krieger, 1985, 1996). A number of conjoint-based models have been developed with a particular focus on mathematical programming techniques for optimal product line design (for example: Dobson and Kalish (1993)). These models seek to determine optimal product concepts using customers' idiosyncratic or segment-level part-worth (i.e., customer-perceived value of a particular level of an attribute) preference functions that are estimated within a conjoint framework (Steiner and Hruschka, 2002). Whereas many methods excel in determining optimal or near-optimal product designs from conjoint data, traditional conjoint analysis is limited in that it only considers input from the customers rather than analyzing distinct conjoint data from both customers and engineering concerns (Tarasewich and Nair, 2001).

In the engineering community, product portfolio decisions have been extensively studied with a primary focus on the costs and flexibility issues associated with product variety and mix (for example: Lancaster (1990)). However, the effect of product lines on the profit side of the equation has been seldom considered (Yano and Dobson, 1998). Few industries have developed an effective set of analyses to simultaneously manage the profit due to product variety and the costs due to complexity in product portfolio decision-making (Otto et al., 2003). It is imperative to take into account the combined effects of multiple product offerings on both the profit and engineering costs (Krishnan and Ulrich, 2001). Therefore, product portfolio planning should be positioned at the crossroad of engineering and marketing, where the interaction between the customer and engineering concerns is the linchpin (Markus and Vancza, 1998). In particular, portfolio decisions with customer-engineering interactions need to address the tradeoffs between the economies of scope in profit from the customers and markets and diseconomies of scope in design, production, and distribution at the backend of product fulfillment (Yano and Dobson, 1998). Moreover, achieving a synergy of engineering concerns among products in portfolio planning is deemed to be increasingly beneficial given the efforts in many industries to improve the coordination of design and manufacturing activities across product families and platforms (Morgan et al., 2001).

Towards this end, this paper examines the benefits of integrating marketing implications of product portfolio with engineering implications. A comprehensive methodology for product portfolio planning is developed that aims at leveraging both customer and engineering concerns.

2. Critical review of related work

Most of the literature on product line design tackles the optimal selection of products by maximizing the surplus: the margin between the customer-perceived utility and the price of the product (Kaul and Rao, 1995). Other objectives widely used in selecting products among a large set of potential products include: (i) maximization of profit (Monroe et al., 1976); (ii) net present value (Li and Azarm, 2002); (iii) a seller's welfare (McBride and Zufryden, 1988); (iv) market share (Kohli and Krishnamurti, 1987); and (v) share of choices (Balakrishnan and Jacob, 1996) within a target market.

Whereas numerous papers in the marketing literature deal with the selection problem using various objectives originated from the profit, few of them explicitly model the costs of manufacturing and engineering design (Yano and Dobson, 1998). Dobson and Kalish (1993) have introduced per-product fixed costs. Recent product line design models allow for more complex cost structures. Raman and Chhajed (1995) have observed that, in addition to choosing which products to produce, one must also choose the process by which these products are manufactured. Dobson and Yano (1994) have allowed for complex interactions by admitting per-product fixed costs, resources that can be shared by multiple products, as well as technology choices for each. Morgan et al. (2001) have examined the benefits of integrating the marketing implications of product mix with more detailed manufacturing cost implications, which sheds light on the impact of alternative manufacturing environment characteristics on the composition of the optimal product line.

Product line design involves two basic issues: (i) generation of a set of feasible product alternatives; and (ii) subsequent selection of promising products from this reference set to construct a product line (Li and Azarm, 2002). Along this line, existing approaches to product line design can be classified into two types (Steiner and Hruschka, 2002). One-step approaches aim at constructing product lines directly from part-worth preference and cost/return functions. On the other hand, two-step approaches first reduce the total set of feasible product profiles to a smaller set, and then select promising products from this smaller set to constitute a product line. Most of the literature follows the two-step approach and emphasizes the maximization of profit contributions in the second step (McBride and Zufryden, 1988; Dobson and Kalish, 1993). The determination of a product line from a reference set of products is thereby limited to partial models due to the underlying assumption that the reference set is given a priori. Following the two-step approach, Green and Krieger (1989) have introduced several heuristic procedures and considered how to generate a reference set appropriately. On the other hand, Nair et al. (1995) have adopted the one-step approach, in which product lines are constructed directly from part-worth data rather than by enumerating potential product designs. In general, the one-step approach is more preferable, since the intermediate step of enumerating utilities and profits of a huge number of reference set items can be eliminated (Steiner and Hruschka, 2002). Only when the reference set contains a small number of product profiles can the two-step approach work well. As a result, few papers in the marketing literature allow a large number of attributes to describe a product (Yano and Dobson, 1998).

Whereas traditional design focuses more on the designers' perspective (Tarasewich and Nair, 2001), the measurement of customer preferences in terms of expected utilities is the primary concern of optimal product design (Krishnan and Ulrich, 2001) or decision-based design (Hazelrigg, 1998). In typical preference-based product design, conjoint analysis (Green and Krieger, 1985) has proven to be an effective means to estimate individual-level part-worth utilities associated with individual product attributes. The conjoint-based search for optimal product designs always results in combinatorial optimization problems because typically discrete attributes are used in conjoint analysis (Kaul and Rao, 1995; Nair et al., 1995).

Product positioning involves decisions about abstract perceptual attributes and customer heterogeneity (Kaul and Rao, 1995). To optimize a new product's positioning, Shocker and Srinivasan (1979) have proposed a framework using joint space models of customer perceptions and preferences. Using a joint mapping of ideal points and product locations, a manager can model customers' choices of existing products, predict their responses to new products, and hence identify optimal new product concepts (Sudharshan et al., 1987). A number of multi-dimensional scaling-based algorithms have been developed that depend on the number of ideal points (individuals or segments) in the joint space (Kaul and Rao, 1995). Consequently, as the number of ideal points increases, so does the complexity of the optimization problem. Genetic algorithms have been shown to outperform most existing optimal positioning algorithms in dealing with the choice set size heterogeneity between a customer's decision setting and variations in the size of the individual's choice sets (Balakrishnan and Jacob, 1996).

On the other hand, many algorithms have been formulated with the intent of improving the realism of the customer choice setting. Deterministic first-choice models assume that customers choose the offered product that is closest to the ideal point. Probabilistic choice settings postulate a customer's propensity to buy a particular product based on a weighted distance between the ideal point and the offered product. Discrete choice analysis is widely used to identify patterns in choices that customers make among competing products (Ben-Akiva and Lerman, 1985). It allows for the examination of the interaction between market shares and product features, price, service, and promotion with respect to different classes of customers. Sudharshan et al. (1987) observed that a probabilistic choice model tends to provide better solutions and larger share projections for new product positioning.

3. Problem description

This research addresses the product portfolio planning problem with the goal of maximizing an expected surplus from both the customer and engineering perspectives. More specifically, we consider a scenario in which a large set of product attributes, A [equivalent to] {[a.sub.k]|k = 1,..., K}, have been identified (a few methods are available, for example: Jiao and Zhang (2005)), given that the firm has the capabilities (both design and production) to produce all these attributes. Each attribute, [for all][a.sub.k] [member of] A, possesses a few levels, either discrete or continuous, i.e., [A*.sub.k] [equivalent to] {[a*.sub.kl]|l = 1,..., [L.sub.k]}. One advantage of using discrete levels is that it does not presume linearity with respect to the continuous variables (Train, 2003).

A set of potential product profiles, Z [equivalent to] {[z.sub.j]|j = 1,..., J}, are generated by choosing one of the levels for certain attributes, subject to satisfying certain configuration constraints. That is, a product assumes certain attribute levels that correspond to a subset of A. Each product, [for all][z.sub.j] [member of] Z, is defined as a vector of specific attribute levels, i.e.,

[z.sub.j] = [[a*.sub.kl.sub.j]][.sub.K],

where any [a*.sub.kl.sub.j] = [empty set] indicates that product [z.sub.j] does not contain attribute [a.sub.k]; and any [a*.sub.kl.sub.j] [not equal to] [empty set] represents an element of the set of attribute levels that can be assumed by product [z.sub.j], i.e., {[a*.sub.kl.sub.j]}[.sub.K] [member of] {[A*.sub.1] X [A*.sub.2] X ... X [A*.sub.K]}.

A product portfolio, [LAMBDA], is a set consisting of a few selected product profiles, i.e., [LAMBDA] [equivalent to] {[z.sub.j]|j = 1,..., [J.sup.[dagger]]} [??] Z, [there exists][J.sup.[dagger]] [member of] {l,..., J}, denotes the number of products contained in the product portfolio.

Every product is associated with certain engineering costs, denoted as {[C.sub.j]}[.sub.J]. The manufacturer must make decisions about which products to select and how about their respective prices, {[p.sub.j]}[.sub.J]. As for portfolio decisions, the manufacturer must also determine what combinations of attributes and their levels should be introduced to offer the potential products, or should be discarded from product offerings. This is different from traditional product line design, which solely involves the selection of products and leaves the sets of attributes and their levels intact, and assumes that the products are generated a priori by enumerating all possible attribute levels. In this sense, this research adopts a one-step approach to the optimal product line design problem, which excels in simultaneously optimizing product generation and selection when faced by a large number of combinations of attributes and their levels (Steiner and Hruschka, 2002).

There are multiple market segments, S [equivalent to] {[s.sub.i]|i = 1,..., I}, each containing homogeneous customers, with a size, [Q.sub.i]. The customer-engineering interaction is embodied in the decisions associated with customers' choices of different products. Various customer preferences on diverse products are represented by respective utilities, {[U.sub.ij]}[.sub.I X J]. Product demands or market shares, {[P.sub.ij]}[.sub.I X J], are described by the probabilities of customers' choosing products, denoted as customer or segment-product pairs, {([s.sub.i], [z.sub.j])}[.sub.I X J] [member of] S X Z.

Customers choose a product based on the surplus buyer rule (Kaul and Rao, 1995). They have the option of not buying any products (if none of them produces a positive surplus) or buying competitors' products. We assume that competitors do not respond to the manufacturer's moves, meaning that, in the short run, the competition does not react by introducing new products. This is supported by the findings of Robinson (1988). As a result, competitive reactions appear implicitly in the customer utilities, which are influenced by the attributes and prices of competing products. In addition, we assume that neither price nor supply discrimination is allowed. That is, each offered product bears the same price for all segments and each segment can buy any of the products offered (Yano and Dobson, 1998). Moreover, we assume that customers can access complete information regarding the available products and their prices. The growing presence of electronic commerce for business-to-business and business-to-customer sales is also expanding the availability of product and price information.

4. Fundamental issues

4.1. Objective function

In the customer's-preference- or seller's-value-focused approaches, the objective functions used to solve the selection problem are formulated by measuring the consumer surplus, i.e., the amount that customers benefit by being able to purchase a product for a price that is less than that they would be willing to pay. The idea behind this approach is that the expected revenue (utility minus price) comes from the gain between customer preferences (utilities indicating the dollar value that they would be willing to pay) and the actual price they would pay, whilst the price implies all related costs. A general form is given as the following (see for example: Green and Krieger (1985)):

Max [I.summation over (i=1)] [J.summation over (j=1)] ([U.sub.ij] - [p.sub.j])[P.sub.ij][Q.sub.i], (1)

where the modeling of customer choices ([P.sub.ij]) based on customer preferences (utilities) is most important (for example, discrete choice analysis as discussed in Ben-Akiva and Lerman (1985)).

When engineering concerns are of importance then the selection problem is approached by measuring the producer surplus, i.e., the amount that producers benefit by selling at a market price that is higher than that they would be willing to sell for. The principle is to measure the expected profit (price minus cost) based on the margin between the actual price they would receive and the cost (indicating the dollar value they would be willing to sell for), and the price implies customer preferences. A general form is given as the following (see for example: Yano and Dobson (1998)):

Max [I.summation over (i=1)] [J.summation over (j=1)] ([p.sub.j] - [C.sub.j.sup.V])[P.sub.ij][Q.sub.i] - [C.sub.j.sup.F], (2)

where [P.sub.ij] represents the probability of a produced product that can be sold to a market (i.e., product demand), and [C.sub.j.sup.V] and [C.sub.j.sup.F] indicate the variable cost and allocated fixed cost per product (with respect to [z.sub.j]), respectively.

In practice, either the consumer or producer surplus-based optimization approach encounters difficulties when dealing with pricing or cost accounting. As a matter of fact, price competition is one of the most complicated topics in marketing research, and a number of approximations have to be assumed such as price equilibrium, monopolistic producers, oligopoly, market mavenism, etc. (Choi and DeSarbo, 1994). The difficulty in cost estimation lies in its reliance on a detailed knowledge of product design and process plans (Jiao and Tseng, 1999). A complete description of product design, however, is rarely available at the portfolio planning phase, nor does there exist any well-defined relationship, at the early design stage, between various attribute levels and the cost figures for their manufacture. More difficult is the allocation of variable and fixed costs among products (Dobson and Kalish, 1993), although a linear-additive fixed cost function is always employed (Moore et al., 1999).

Considering the customer-engineering interaction in product portfolio planning, the above economic surpluses should be leveraged from both the customer and engineering perspectives. This research proposes to use a shared surplus to leverage both the customer and engineering concerns. Then the objective function can be formulated as:

Max E[V] = [I.summation over (i=1)][J.summation over (j=1)] [[U.sub.ij]/[C.sub.j]] [P.sub.ij][Q.sub.i][y.sub.j], (3)

where E[*] denotes the expected value of the shared surplus, V, which is defined as the utility per cost, modified by the probabilistic choice model, {[P.sub.ij]}[.sub.I X J], and the market size, {[Q.sub.i]}[.sub.I], [C.sub.j] indicates the cost of offering product [z.sub.j], and [y.sub.j] is a binary variable such that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The underpinning principle of the shared surplus coincides with the implications of customer values in marketing, i.e., a customer's expectations of product quality in relation to the actual amount paid for it. It is often expressed as the ratio of the customer-perceived utility and the costs to produce it (Zeithaml, 1988). In addition, introduction of the shared surplus contributes to the maintenance of a consistent measure for the relative comparison of various alternatives on a common ground, while avoiding the intricate pricing and cost estimation problems. This is consistent with the findings reported by Choi and DeSarbo (1994) in that "exact cost estimates are not necessary as long as the relative magnitudes are in order". Furthermore, the incorporation of a choice model into customer values enables the modeling of customer decision-making when facing similar product offerings from competitors or even competing products from the same brand. In practice, customer-perceived value of a product tends to decrease if there are counterparts, whereas a premium value can be expected for a unique product owing to limited choices for the customer.

4.2. Conjoint analysis and customer preference

Following the part-worth model, the utility of the ith segment for the jth product, [U.sub.ij], is assumed to be a linear function of the part-worth preferences (utilities) of the attribute levels of product [z.sub.j], i.e.,

[U.sub.ij] = [K.summation over (k=1)] [[L.sub.k].summation over (l=1)] ([w.sub.jk][u.sub.ikl][x.sub.jkl] + [[pi].sub.j]) + [[epsilon].sub.ij], (4)

where [u.sub.ikl] is the part-worth utility of segment [s.sub.i] for the lth level of attribute [a.sub.k] (i.e., [a*.sub.kl]) individually, [w.sub.jk] is the utility weights among attributes, {[a.sub.k]}[.sub.K], contained in product [z.sub.j], [[pi].sub.j] is a constant associated with the derivation of a composite utility from part-worth utilities with respect to product [z.sub.j], [[epsilon].sub.ij] is an error term for each segment-product pair, and [x.sub.jkl] is a binary variable such that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

There are a number of methods available to estimate the regression utility weights, {[w.sub.jk]}[.sub.J X K], and the constant, {[[pi].sub.j]}[.sub.J], given a set of observed choice data. These include full-profile conjoint analysis, adaptive conjoint analysis, hybrid conjoint analysis, and experimental choice analysis, or choice-based conjoint analysis (see: http://www.sawtoothsoftware.com). In addition, a great deal of marketing research has been devoted to recovering model parameters through latent classes, such as using finite mixtures, hierarchical Bayes methods, the maximum likelihood formulation, and the least-squares method (Lilien et al., 1992).

In the above formulation, we model customer behavior at the segment-level, although one could also assume individual-level part-worth utilities without loss of generality. As observed by Wittink and Cattin (1989), market segmentation ranks among the primary purposes of suppliers in conjoint studies. If segmentation issues are of particular interest, individual-level part-worth estimations might further be clustered to form market segments (post hoc segmentation). Moreover, a number of procedures for simultaneously performing market segmentation and calibrating segment-level part-worth utilities in conjoint analysis have been developed in recent years. Such methods for simultaneous segmentation and estimation have been proposed for both the traditional conjoint analysis and the choice-based conjoint analysis (Wedel and Kamakura, 1998).

4.3. Choice model and product demand

Probabilistic choice rules can provide more realistic representations of the customer decision-making process (Sudharshan et al., 1987). Some probabilistic choice rules can offer flexibility in calibrating actual choice behavior such as the option of mimicking the first choice rule (Kaul and Rao, 1995). In general, there are two types of probabilistic choice rules (Ben-Akiva and Lerman, 1985): (i) the generalized (or powered) Bradley-Terry-Luce (BTL) share-of-utility rule; and (ii) the conditional MultiNomial Logit (MNL) choice rule. With the assumption of independently and identically distributed error terms, the logit choice rule suggests itself to be a discrete choice model (Ben-Akiva and Lerman, 1985). Discrete choice models are best suited to estimate customer preferences directly from choice data (Green and Krieger, 1996), i.e., the case of product portfolio planning, in which customers' choices are directed to the attribute levels that constitute products. Moreover, with discrete choice models, preference estimation and model calibration are performed simultaneously and tests for statistical inferences about a particular model and its parameters are available (Ben-Akiva and Lerman, 1985). Therefore, this research employs the logit choice rule to model product demands.

Under the MNL model, the choice probability, [P.sub.ij], that a customer or a segment, [there exists][s.sub.i] [member of] S, chooses a product, [there exists][z.sub.j] [member of] Z, with N competing products, is defined as the following:

[P.sub.ij] = [exp([mu][U.sub.ij])]/[[[summation].sub.n=1.sup.N] exp([mu][U.sub.in])], (5)

where [mu] is a scaling parameter. As [mu] [right arrow] [infinity], the logit behaves like a deterministic model, whereas it becomes a uniform distribution as [mu] [right arrow] 0. Therefore, as with the BTL model, calibration on actual market shares can be carried out subsequently to elaborate preference estimation by post hoc optimization with respect to [mu] (Train, 2003).

Based on a customer survey, the response rate (i.e., how often each product alternative is chosen) can be depicted as a probability density distribution. The demand for a particular product is the summation of the choice frequency of each respondent, [for all][s.sub.i] [member of] S, adjusted for the ratio of respondent sample size versus the size of the market population (Train, 2003). The accuracy of demand estimates can be improved by identifying unique customer utility functions per market segment, or class of customers to capture systematic preference variations (Ben-Akiva and Lerman, 1985). Estimates of future demand can also be facilitated using pattern-based or correlation-based forecasting of existing products. Forecasts of economic growth and the estimated change of the socioeconomic and demographic background of the market populations help to refine these estimates (Lilien et al., 1992).

4.4. Dealing with engineering costs

The premise of existing profit-maximizing approaches is to assume that costs can be estimated, provided that the manufacturer has established an operating cost accounting system (Dobson and Kalish, 1993). As discussed in Section 2, cost estimation, however, is deemed to be very difficult, especially at the portfolio planning phase. Furthermore, traditional cost accounting by allocating fixed costs and variable costs across multiple products (e.g., Equation (2)) may produce distorted cost-carrying figures due to possible sunk costs associated with investment into product and process platforms. It is quite common in mass customization that design and manufacturing admit resources (and thus the related costs) to be shared among multiple products in a reconfigurable fashion, as well as per-product fixed costs (Moore et al., 1999). In fact, Yano and Dobson (1998) have observed a number of industrial settings in which a wide range of products are produced with very little incremental costs per se, or very high development costs are shared across broad product families, or fixed costs and variable costs change dramatically with product variety. They have pointed out that "the accounting systems, whether traditional or activity-based, do not support the separation of various cost element".

Furthermore, the cost advantages in mass customization lie in the achievement of mass production efficiency. Rather than the absolute amount of dollar costs, what is important to justify optimal product offerings is the magnitudes of the deviations from existing product and process platforms due to design changes and process variations in relation to product variety. Therefore, Jiao and Tseng (2004) have proposed to model the cost consequences of providing variety by varying the impacts on process capabilities. The process capability index lends itself as an instrument for handling the sunk costs that are related to the product families and shared resources.

To circumvent the difficulties inherent in estimating accurate cost figures, this research adopts a pragmatic costing approach that is based on the standard time estimation technique developed by Jiao and Tseng (1999). The idea is to allocate costs to those established time standards from previously reported work and time studies and thereby circumvent the tedious tasks of identifying various cost drivers and cost-related activities. The key is to develop mapping relationships between different attribute levels and their expected consumptions of standard times within legacy process capabilities. These part-worth standard time accounting relationships are built into the product and process platforms (Jiao et al., 2003). Any product configured from available attribute levels is justified based on its expected cycle time. This expected cycle time is accounted for by the aggregation of the part-worth standard times. The rationale is particularly applicable to portfolio planning, where "the optimal product profiles are not as sensitive to absolute dollar costs as they are to the relative magnitudes of cost levels" (Choi and DeSarbo, 1994).

The expected cycle time can be used as a performance indicator of variations in process capabilities (Jiao and Tseng, 2004). The characteristic for the cycle time is of "the smaller the better" type. The cycle time demonstrates the distinctions between variables that differ as a result of random error and are often well described by a normal distribution. Hence, the one-side specification limit process capability index, PCI, can be formulated as:

PCI = [[[mu].sup.T] - LS[L.sup.T]]/[3[[sigma].sup.T]], (6)

where LS[L.sup.T], [[mu].sup.T], and [[sigma].sup.T] are the lower specification limit, the mean and the standard deviation of the estimated cycle time, respectively. Variations in the cycle time are characterized by [[mu].sup.T] and [[sigma].sup.T], reflecting the compound effect of multiple products on production in terms of process variations. The LS[L.sup.T] can be determined ex ante based on the best case analysis of a given process platform, in which standard routings can be reconfigured to accommodate various products derived from the corresponding product platform (Jiao et al., 2003).

The value of PCI falls into (0, [infinity]), where a large value suggests that the related production process is easy to implement (as it involves little deviation from existing platforms), and a small value a difficult one. Since there exist close correlations between costs and the cycle time, the PCI can indicate how expensive a product can be expected to be if produced within the existing capabilities. Introducing a penalty function, the cost function, [C.sub.j], corresponding to product [z.sub.j], can be formulated based on the respective process capability index, PC[I.sub.j], that is:

[C.sub.j] = [beta] exp(1/PC[I.sub.j]) = [beta] exp ([3[[sigma].sub.j.sup.T]]/[[[mu].sub.j.sup.T] - LS[L.sup.T]]), (7)

where [beta] is a constant indicating the average dollar cost per variation of process capabilities. LS[L.sup.T] denotes the baseline of cycle times for all product variants to be produced within the process platform, [[mu].sub.j.sup.T] and [[sigma].sub.j.sup.T] are the mean and the standard deviation of the estimated cycle time for product [z.sub.j], respectively.

The estimated cycle time for product [z.sub.j], ([[mu].sub.j.sup.T], [[sigma].sub.j.sup.T]), is assumed to be a linear function of the part-worth standard times of the attribute levels assumed by product [z.sub.j], i.e.,

[[mu].sub.j.sup.T] = [K.summation over (k=1)] [[L.sub.k].summation over (l=1)] ([[zeta].sub.jk][[mu].sub.kl.sup.t][x.sub.jkl] + [[omega].sub.j]), (8a)

[[sigma].sub.j.sup.T] = [square root of ([K.summation over (k=1)] [[L.sub.k].summation over (l=1)] ([[sigma].sub.kl.sup.t][x.sub.jkl])[.sup.2])], (8b)

where [[zeta].sub.jk] and [[omega].sub.j] are regression coefficients, [x.sub.jkl] possesses the same meaning as that in Equation (4), and [[mu].sub.kl.sup.t] and [[sigma].sub.kl.sup.t] are the mean and the standard deviation of the part-worth standard time associated with the lth level of attribute [a.sub.k], respectively.

The meaning of [beta] is consistent with that of the dollar loss per deviation constant widely used in Taguchi's loss functions. It can be determined ex ante based on the analysis of existing product and process platforms. Such a cost function produces a relative measure, instead of actual dollar figures, for evaluating the extent of process variations among multiple products. Modeling the economic latitude of product portfolio planning through the cycle time performance and the impact on process capabilities can alleviate the difficulties in traditional cost estimation which is tedious and less accurate.

5. Model development

Surplus-based optimization models assume that customers only choose a product with a positive surplus rather than the lowest price. Otherwise, the price of each offered product becomes a decision variable, making the problem nonlinear (Yano and Dobson, 1998). To avoid explicitly modeling the price, the general practice is to treat the price as a separate attribute that can be chosen from a limited number of values for each product (Nair et al., 1995; Moore et al., 1999). With the addition of the price as one attribute, the attribute set becomes A [equivalent to] {[a.sub.k]}[.sub.K+1], where [a.sub.K+1] represents the price possessing a few levels, i.e., [A*.sub.K+1] [equivalent to] {[a*.sub.(K+1)l]|l = 1,..., [L.sub.K+1]}. Let p = [[a*.sub.(K+1)l],...,[a*.sub.(K+1)[L.sub.K+1]]] be the vector of feasible price levels. Further let [x.sub.j(K+1)l] be a binary vector of length [L.sub.K+1] that indicates the presence or absence of the lth price level with respect to product [z.sub.j]. Then [p.sub.j] = p [cross product] [x.sub.j(K+1)l] suggests the price assigned to product [z.sub.j].

Combining Equations (3)-(5) and (7), the product portfolio planning problem can be formulated as a mixed integer program, as below:

Max E[V] = [I.summation over (i=1)][J.summation over (j=1)] [U.sub.ij]/[beta] exp ([3[[sigma].sub.j.sup.T]]/[[[mu].sub.j.sup.T] - LS[L.sup.T]]) X exp([mu][U.sub.ij]) / [N.summation over (n=1)] exp([mu][U.sub.in])[Q.sub.i][y.sub.j], (9a)

subject to

[U.sub.ij] = [(K+1).summation over (k=1)][[L.sub.k].summation over (l=1)] ([w.sub.jk][u.sub.ikl][x.sub.jkl] + [[pi].sub.j]) + [[epsilon].sub.ij], [for all]i [member of] {1,...,I}, [for all]j [member of] {1,...,J}, (9b)

[[L.sub.k].summation over (l=1)][x.sub.jkl] = 1, [for all]j [member of] {1,...,J}, [for all]k [member of] {1,..., K + 1}, (9c)

[(K+1).summation over (k=1)] [[L.sub.k].summation over (l=1)] |[x.sub.jkl] - [x.sub.j'kl]| > 0, [for all]j, j' [member of] {1,...,J}, j [not equal to] j', (9d)

[J.summation over (j=1)] [y.sub.j] [less than or equal to] [J.sup.[dagger]], [for all][J.sup.[dagger]] [member of] {1,...,J}, (9e)

[x.sub.jkl], [y.sub.j] [member of] {0, 1}, [for all]j [member of] {1,...,J}, [for all]k [member of] {1,..., K + 1}, [for all]l [member of] {1,..., [L.sub.k]}. (9f)

Equation (9a) maximizes the expected shared surplus by offering a product portfolio consisting of products, {[z.sub.j]}[.sub.J], to customer segments, {[s.sub.i]}[.sub.I], each with size [Q.sub.i], The market potentials, {[Q.sub.i]}[.sub.I], can be given exogenously at the outset or estimated through a variety of techniques based on historical data or test markets (Lilien et al., 1992). Equation (9b) refers to the conjoint analysis; it ensures that the composite utility of segment [s.sub.i] for product [z.sub.j] can be constructed from the part-worth utilities of the individual attribute levels, {[A*.sub.k]}[.sub.K+1]. Equation (9c) suggests an exclusiveness condition; it enforces that exactly one and only one level of each attribute can be chosen for each product. Equation (9d) denotes a divergence condition; it requires that the offered products must pairwise differ in at least one attribute level. Equation (9e) is a capacity condition; it limits the maximum number of products that can be chosen by each segment. It can be in the form of an inequality or an equality. In the case of an inequality constraint, [J.sup.[dagger]] is the upper bound on the number of products that the manufacturer wants to introduce into a product portfolio, whereas, with an equality constraint, [J.sup.[dagger]] is the exact number of products contained in a product portfolio. Equation (9f) represents the binary restriction with regard to the decision variables of the optimization problem.

In the above mathematical program, there are two types of decision variables involved, i.e., [x.sub.jkl] and [y.sub.j], representing two layers of decision-making in the portfolio planning, respectively. The first layer is the selection of attributes and their levels for different products (i.e., product generation) and the second one is to decide which products to offer (i.e., product selection). Both types of decisions depend on simultaneously satisfying the target segments. The manufacturer's decisions about what (i.e., layer I decision-making) and which (i.e., layer II decision-making) products to offer to the target segments are implied in various instances of {[x.sub.jkl]|[for all]j, k, l} and {[y.sub.j]|[for all]j}, respectively. As a result, an optimal product portfolio, [[LAMBDA].sup.[dagger]] [equivalent to] {[z.sub.j.sup.[dagger]]|j = 1,..., [J.sup.[dagger]]} is created as a combination of selected products corresponding to {[y.sub.j]|[for all]j}, where each selected product, [z.sub.j.sup.[dagger]], contains a few selected attributes and the associated levels corresponding to {[x.sub.jkl]|[for all]j, k, l}. The framework and solution procedures for product portfolio planning are schematically shown in Fig. 1, where a heuristic genetic algorithm solver is developed to solve the mixed-integer optimization problem.

The conjoint-based search for an optimal product portfolio always results in combinatorial optimization problems because in general discrete attributes are used in conjoint analysis (Kaul and Rao, 1995). Nearly all of these problems are known to be mathematically intractable or NP-hard, and thus heuristic solution procedures are normally proposed to solve the various problem types (Nair et al., 1995). Genetic Algorithms (GAs) have been shown to excel in solving combinatorial optimization problems as compared to traditional calculus-based and approximation optimization techniques (Steiner and Hruschka, 2002). The GA approach adopts a probabilistic search technique that is based on the principles of natural selection, i.e., the survival of the fittest, and merely uses objective function information. Thus it is easily adjustable to different objectives with little algorithmic modification (Holland, 1992). An important feature of GAs is that they allow product profiles to be constructed directly from attribute-level part-worth data (Kohli and Sukumar, 1990). This is particularly useful for reference set enumeration if the number of attributes and their levels is large and most multi-attribute products that can be represented by different attribute-level combinations are economically and technologically feasible (Nair et al., 1995). Hence, a GA approach is employed in this research to solve the mixed integer program of Equations (9a)-(9f). Our aim is to produce acceptable solutions for the combinatorial optimization problem involving a wide variety of attribute configurations and their levels as well as product profiles in portfolio planning.

6. Case study

6.1. Problem illustration

The proposed framework has been applied to a notebook computer portfolio planning problem for a world-leading computer manufacturing company. The company had conducted extensive market studies and competition analyses and projected the trends of technology development in the business sector concerned. Based on existing technologies, product offerings of notebook computers manifest themselves through various instances of a number of functional attributes. For illustrative simplicity, a set of key attributes and available attribute levels are listed in Table 1. Among them, "price" is treated as one of the attributes to be assumed by a product. Every notebook computer is thus described as a viable configuration of available attribute levels.

It is interesting to observe the importance of product portfolio planning in this case problem. Taking the "processor" attribute as an example, existing microelectronics technologies have made it possible to achieve a CPU performance that ranges from Centrino 1.4 GHz up to Centrino 2.0 GHz. As a matter of fact, one of the two existing competitors to our company does offer products with an extensive portfolio, including Centrino 1.4, 1.5, 1.6, 1.7, 1.8 and 2.0 GHz. On the other hand, the other competitor only offers Centrino 1.4, 1.8 and 2.0 GHz. It hence becomes imperative to isolate the correct products for the company's product portfolio, regardless of the fact that all these attributes levels are technologically feasible. The question is can the granularity of the product offerings leverage the resulting costs and complexity with respect to the company's engineering capabilities.

[FIGURE 1 OMITTED]

6.2. Customer preference

Conjoint analysis starts with the construction of product profiles. Given all attributes and their possible levels as shown in Table 1, a total number of 9 X [4.sup.2] X [3.sup.3] X [2.sup.2] X 5 = 77 760 possible combinations may be constructed. To overcome such an explosion of configurations with enumeration, orthogonal product profiles are always used in practice (Wittink and Cattin, 1989). Using the Taguchi Orthogonal Array Selector provided in the SPSS software (www.spss.com), a total number of 81 orthogonal product profiles were generated. With these profiles, a fractional factorial experiment was designed to explore customer preferences, as shown in Table 2. In Table 2, columns 2-10 indicate the specification of offerings that are involved in the profiles and column 11 collects the preferences given by the customers.

A total number of 30 customers were selected to act as the respondents. Each respondent was asked to evaluate all 81 profiles one by one by giving a mark based on a nine-point scale, where "nine" means the customer prefers a product most and "one" the least. This results in 30 X 81 groups of data. Based on these data, clustering analysis was run to find customer segments based on the similarity among customer preferences. Three customer segments were formed: [s.sub.1], [s.sub.2] and [s.sub.3], suggesting home users, regular users, and professional/business users, respectively. These segments, [s.sub.1], [s.sub.2] and [s.sub.3], divide the 30 respondents into three respective groups: (i) customers 1, 2, 7, 8, 11, 12, 14, 15, 24 and 29; (ii) customers 3, 4, 5, 9, 10, 13, 17, 19, 20, 23, 26 and 30; and (iii) customers 6, 16, 18, 21, 22, 25, 27 and 28.

For each respondent in a segment, 81 regression equations were obtained by interpreting their original choice data as a binary instance of each part-worth utility. Each regression corresponds to a product profile and indicates the composition of their original preference in terms of part-worth utilities according to Equation (4). With these 81 equations, the part-worth utilities for a respondent can be derived. A segment-level utility is obtained for each attribute level by averaging the part-worth utility results of all respondents belonging to the same segment. Columns 2-4 in Table 3 show the part-worth utilities of three segments with respect to every attribute level.

6.3. Engineering cost

Table 3 also shows the part-worth standard times for all attribute levels. The company fulfills customer orders through assembly-to-order production while importing all components and parts via global sourcing. The part-worth standard time of each attribute level is established based on time and motion studies of the related assembly and testing operations. With assembly-to-order production, the company has identified and established standard routings as basic constructs of its process platform. Based on empirical studies, costing parameters are known to be LS[L.sup.T] = 45 seconds and [beta] = 0.004.

6.4. GA solution

To determine an optimal notebook computer portfolio for the target three segments, the GA procedure was applied to search for a maximum of the expected shared surplus among all attribute, product and portfolio alternatives. We assumed that each portfolio consists of a maximum number of five products. Then the chromosome string comprises 9 X 5 = 45 genes. Each substring is as long as nine genes and represents a product that constitutes the portfolio. For every generation, a population size of M = 20 was maintained, meaning that only the top 20 fit product portfolios were kept for reproduction.

In addition, it is not uncommon that in the notebook computer business most manufacturers directly order components and parts from their suppliers. This means that all the companies possess similar technological capabilities to provide the attributes and levels listed in Table 1. In fact, the ability to create these attributes and levels depends on the global semiconductor manufacturers rather than the notebook computer manufacturers themselves. Therefore, we assume that the competitors of the company under our study offer the same product attributes and levels. As a result, the status-quo product alternatives in the current generation are used as the pool of competing products for the choice model in Equation (5).

6.5. Results

The results of the GA solution are presented in Fig. 2(a-c). As shown in Fig. 2(a), the fitness value is continuously improved generation by generation using the reproduction process. Certain local optima (e.g., around 100 generations) are successfully overcome. The saturation period (350-500 generations) is quite short, indicating that the GA search is efficient. This proves that the moving average rule is a reasonable measure of convergence. It helps avoid the possible problem that the GA procedure may unnecessarily run for upto 1000 generations. Upon termination at the 495 generation, the GA solver returns the optimal result, which achieves an expected shared surplus of 802 000 and an unbalance index of 200.0, as shown in Table 4.

As shown in Table 4, the optimal product portfolio consists of two products, [z.sub.1.sup.1] and [z.sub.2.sup.1]. From the specifications of attribute levels, we can see they basically represent the low-end and high-end notebook computers, respectively. With such a two-product portfolio, all home, regular and professional/business users can be served with an optimistic expectation of maximizing the shared surplus. Whereas the low-end notebook computer [z.sub.1.sup.1] includes all available attributes, the high-end notebook computer [z.sub.2.sup.1] does not contain the "software" attribute. This may be due to the fact that most professionals prefer to install software authorized by their business organizations for the purpose of, for example, systems maintenance and technical support.

[FIGURE 2 OMITTED]

[TABLE 4 OMITTED]

6.6. Performance evaluation

Figure 2(b) compares the results of utility with choice probability, [[summation].sub.i=1.sup.3] [[summation].sub.j=1.sup.5]([U.sub.ij][P.sub.ij]), among generations. It is interesting to observe that the distribution of utility with choice probability does not tally with that of the fitness shown in Fig. 2(a). The optimal solution (i.e., the last generation) does not produce the best utility performance. On the other hand, a number of high utility achievements do not correspond to a high fitness. Likewise, as shown in Fig. 2(c), the distribution of cost performance among generations disorders the pattern of fitness distribution shown in Fig. 2(a). This may be explained by the fact that a high utility achievement is usually accompanied with high incurred costs. Therefore, the shared surplus is a more reasonable fitness measure to leverage both customer and engineering concerns than either utility or cost alone.

Figure 3 compares the achievements, in terms of the normalized shared surplus, cost, and utility with choice probability, of 20 product portfolios in the 495 generation that returns the optimal solution. It is interesting to see that the peak of utility achievement (portfolio 8) does not result in the best fitness since its cost is estimated to be high. On the other hand, the minimum cost (portfolio 4) does not result in the best shared surplus since its utility performance is moderate. It is also interesting to observe that the worst fitness (portfolio 20) performs with neither the lowest utility achievement nor the highest cost figure. The best portfolio (1) results from a leverage of both utility and cost performances.

[FIGURE 3 OMITTED]

7. Conclusions

Product portfolio planning differs from the conventional product line design problem in that it must not only optimize a mix of products but also simultaneously optimize the configurations of individual products in terms of specific attributes. This research allows products to be constructed directly from attribute-level part-worth utilities and costs. The shared surplus model accounts for both diverse customer preferences across market segments, and engineering costs that vary with the composition of a product portfolio. By integrating marketing inputs with detailed cost information attained through coordinated product and process platforms, the model captures the tradeoffs between the benefits derived from providing variety to the marketplace, and the cost savings that can be realized by selecting a mix of products that can be produced efficiently within a company's manufacturing capabilities.

To model customers' choices, the paper employs segment-specific conjoint models of the conditional MNL type. It would also be possible to start with part-worth data estimated at the individual-level and to incorporate other probabilistic choice models. Nevertheless, the logit choice model implies a property of independence from irrelevant alternatives, which applies to an individual, rather than the whole population. This suggests a possible way to improve the predictive quality of a logit model by incorporating more socioeconomic and demographic attributes. In addition, the logit assumes that the values of the scaling parameter [mu] are equal, indicating that the shape of the disturbance of every attribute is constant. Such a simplistic treatment, however, is a limitation on the choice model used in measuring the expected shared surplus.

Although uncertainty about customer utilities could be incorporated into the model, it would be interesting to also allow for uncertainty in demands and costs. Other interesting avenues for future research include allowing for sequential entry strategies. time-varying utilities and changing customer behaviors. Another fruitful direction would be to model active competition. Most competitors eventually react to new entries with changes in the prices. Therefore, different competitive scenarios and dynamics should be analyzed. This may be verified by explicitly modeling competitive reactions within a game theoretic framework or by deriving competitive strategies in conjoint analysis under the Nash equilibrium concept.

Acknowledgements

This research was supported by Singapore NTU-Gintic Collaborative Research Project (U01-A-130B). The authors would like to express their sincere thanks to Professors Mitchell M. Tseng, Martin Helander and Halimahtun M. Khalid, and their colleagues at the Global Manufacturing & Logistics Forum, Nanyang Technological University for valuable discussions. The authors would also like to thank the anonymous referees and the Editor for their constructive comments.

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Biographies

Jianxin Jiao is an Assistant Professor of Systems and Engineering Management, School of Mechanical and Production Engineering, Nanyang Technological University, Singapore. He is the convener and coordinator of Nanyang Global Manufacturing & Logistics Forum. He received a Ph.D. from the Department of Industrial Engineering and Engineering Management, Hong Kong University of Science & Technology. He holds a Bachelor's degree in Mechanical Engineering from the Tianjin University of Science & Technology in China, and a Master's degree in Mechanical Engineering from Tianjin University in China. He has been a lecturer in the Department of Management at Tianjin University. His research interests include mass customization, design theory & methodology, reconfigurable manufacturing systems, engineering logistics, and intelligent systems.

Yiyang Zhang is a Ph.D. candidate in the School of Mechanical and Production Engineering at Nanyang Technological University, Singapore. She received her BBA and MBA degrees from the School of Management at Northeastern University, China, in 1999 and 2002, respectively. Her current research interests are customer decision-making processes, product portfolio planning and customer requirement management.

Contributed by the Engineering Design Department

JIANXIN JIAO* and YIYANG ZHANG

School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore, 639798

E-mail: jiao@pmail.ntu.edu.sg

Received April 2004 and accepted October 2004

*Corresponding author

Table 1. List of attributes and their feasible levels for notebook
computers

           Attribute                  Attribute levels
[a.sub.k]  Description   [a*.sub.kl]  Code  Description

[a.sub.1]  Processor     [a*.sub.11]  A1-1  Pentium 2.4 GHz
                         ...          ...   ...
                         [a*.sub.19]  A1-9  Centrino 2.0 GHz
[a.sub.2]  Display       [a*.sub.21]  A2-1  12.1" TFT XGA
                         [a*.sub.22]  A2-2  14.1" TFT SXGA
                         [a*.sub.23]  A2-3  15.4" TFT XGA/UXGA
[a.sub.3]  Memory        [a*.sub.31]  A3-1  128 MB DDR SDRAM
                         ...          ...   ...
                         [a*.sub.34]  A3-4  1 GB DDR SDRAM
[a.sub.4]  Hard disk     [a*.sub.41]  A4-1  40 GB
                         ...          ...   ...
                         [a*.sub.44]  A4-4  120 GB
[a.sub.5]  Disk drive    [a*.sub.51]  A5-1  CD-ROM
                         [a*.sub.52]  A5-2  CD-RW
                         [a*.sub.53]  A5-3  DVD/CD-RW combo
[a.sub.6]  Weight        [a*.sub.61]  A6-1  Low (below 2.0 Kg with
                                              battery)
                         [a*.sub.62]  A6-2  Moderate (2.0-2.8 Kg with
                                              battery)
                         [a*.sub.63]  A6-3  High (2.8 Kg above with
                                              battery)
[a.sub.7]  Battery life  [a*.sub.71]  A7-1  Regular (around 6 hours)
                         [a*.sub.72]  A7-2  Long (7.5 hours above)
[a.sub.8]  Software      [a*.sub.81]  A8-1  Multimedia package
                         [a*.sub.82]  A8-2  Office package
[a.sub.9]  Price         [a*.sub.91]  A9-1  Less than $800
                         ...          ...   ...
                         [a*.sub.95]  A9-5  $2.5K above

Table 2. Response surface experiment design

                                    Conjoint test
                    Display           Hard  Disk             Battery
Profile  Processor  (inches)  Memory  disk  drive  Weight    life

 1       C-1.6      14.1      256      60   CD-R   Low       Regular
 2       C-2.0      14.1      256      80   CD-RW  Low       Regular
 3       P-2.4      12.1      128      60   CD-RW  Moderate  Long
 4       C-1.7      12.1      128      40   Combo  Low       Regular
...      ...        ...       ...     ...   ...    ...       ...
79       C-1.7      15.4      256      80   CD-R   Moderate  Regular
80       C-1.5      15.4        1     120   Combo  Low       Regular
81       C-1.5      14.1      128      80   CD-RW  High      Regular

             Conjoint test             Preference scale
                     Price ($)         least    most
Profile  Software    (X 10%[.sup.8]    1        9

 1       Multimedia  <0.8                    9
 2       Multimedia   1.8-2.5                5
 3       Office       0.8-1.3                7
 4       Multimedia   1.3-1.8                4
...      ...          ...                    ...
79       Multimedia   1.8-2.5                3
80       Multimedia   0.8-1.3                8
81       Office       1.3-1.8                4

Table 3. Part-worth utilities and part-worth standard times

              Part-worth utility
Attribute     (customer segment)
Level      [s.sub.1]  [s.sub.2]  [s.sub.3]

A1-1        0.75       0.65       0.62
...        ...        ...        ...
A1-9        0.84       0.85       1.22
A2-1        1.18       1.05       0.75
A2-2        1.21       1.47       1.18
A2-3        1.25       1.49       1.38
A3-1        1.02       0.5        0.4
...        ...        ...        ...
A3-4        1.14       1.18       1.11
A4-1        1.33       0.97       0.63
...        ...        ...        ...
A4-4        1.56       1.19       1.22
A5-1        0.86       0.93       0.78
A5-2        0.88       1.11       0.82
A5-3        0.92       1.35       0.83
A6-1        0.7        0.2        0.3
A6-2        0.9        0.7        0.8
A6-3        1.1        0.9        0.9
A7-1        0.7        0.6        0.3
A7-2        0.8        0.9        1.2
A8-1        1.2        1.1        1.2
A8-2        0.5        0.8        1.0
A9-1        0          0          0
...        ...        ...        ...
A9-5       -3.5       -3.3       -0.95

                            Part-worth
                        standard time (assembly)
Attribute                & testing operations)
level      [[mu].sup.t] (second)     [[sigma].sup.t] (second)

A1-1       497                              9.5
...        ...                             ...
A1-9       637                             24
A2-1       739                             35
A2-2       819                             37
A2-3       836                             39
A3-1       659                             24.5
...        ...                             ...
A3-4       756                             36
A4-1       641                             26
...        ...                             ...
A4-4       865                             40
A5-1       293                              4.4
A5-2       321                              5.1
A5-3       368                              5.5
A6-1       215                              3.8
A6-2       256                              4.0
A6-3       285                              4.1
A7-1       125                              1.6
A7-2       458                             19.1
A8-1       115                              1.55
A8-2        68                              0.95
A9-1       N.A.                            N.A.
...
A9-5