A Multi-Attribute Utility Theory-Based Approach to Product Line

Proceedings of IDETC/CIE 2006 ASME 2006 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference September 10-13, 2006, Philadelphia, Pennsylvania, USA

DETC2006-99506 A MULTI-ATTRIBUTE UTILITY THEORY-BASED APPROACH TO PRODUCT LINE CONSOLIDATION AND SELECTION 1

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Henri J. Thevenot , Elizabeth D. Steva , Gül E. Okudan , and Timothy W. Simpson The Pennsylvania State University University Park, PA 16802, USA 1234

single platform, while maintaining the competitiveness of the product line. One recognized approach to product family design is a bottom-up (reactive design) approach, wherein a company redesigns and/or consolidates a group of distinct products to standardize components and thus reduce costs [2]. For instance, Black & Decker redesigned their products to reduce variety in their motors [3], and Lutron Electronics Inc. redesigned its product line of lighting controls around 15-20 standard components that can be configured into more than 100 models specified by the customers [4]. Similar situations can be found when several companies merge, seeking to reduce product proliferation by redesigning or consolidating one or more product lines. As more manufacturing companies seek to benchmark, redesign and consolidate their product lines, there is an increasing need for more systematic and consistent approaches to help them do so. While literature in engineering design provides an extensive range of concept selection methods (CSM) and their applications, none specifically address the problem of product family consolidation. An extensive review of existing CSMs can be found in Refs. [5,6], and some of the more common CSMs found in the literature are described next. There are several ways to classify CSMs, and numerical / nonnumerical [5] as well as formulation-based [6] classifications have been proposed. Using the former classification, nonnumerical CSMs include those from Pugh [7], Otto [8], Ulrich and Eppinger [9], and Kuppuraju, et al. [10]. One major drawback usually found in these approaches is the inability to weight the importance of concepts relative to one another. Meanwhile, numerical CSMs have also been more extensively reported in the literature. For example, Quality Function Deployment (QFD) [11] can be seen as an extension of Pugh’s Concept Selection Method, which takes interactions between concepts into account. Finally, in the Analytic Hierarchy Process (AHP), the criteria for the concepts are ranked against all criteria from the other concepts [12,13].

ABSTRACT Concept selection, an important step in the product design process, can also be used when redesigning or consolidating an existing product line for the purpose of effectively selecting a set of products that satisfies customer demands while increasing profits. If product line consolidation is not done properly, it can lead to unsuccessful platforms and ineffective family architectures that yield products that fail to satisfy customer demands. Accordingly, this paper presents a method that facilitates the consolidation of an existing product line using multi-attribute utility theory. The method helps select the subset of products to manufacture based on criteria at the (1) product level (e.g., costs and profits) and (2) product family-level (e.g., component reuse, variety, commonality, market coverage). The application of the proposed method is demonstrated using an existing line of staplers. The resulting decision, namely, the subset of products recommended by the proposed method, would substantially increase profits while minimizing product proliferation and still maintain good market coverage if the decision were implemented. Keywords: Product family design, product line consolidation, selection, multi-attribute utility theory. 1. INTRODUCTION Product platform and product family design is an approach many companies have taken in today’s highly competitive and increasingly global market economy to reduce product development and production costs and to diversify their product offerings [1]. A key strategy in creating product families is to identify the products that can be produced effectively from a 1 2 3 4

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Research Assistant, Industrial & Manufacturing Engineering Graduate Research Assistant, Mechanical & Nuclear Engineering Assistant Professor, Engineering Design Program Professor of Mechanical and Industrial Engineering and Engineering Design, Member ASME, and Corresponding Author: 329 Leonhard Building, Penn State University, University Park, PA 16802 USA, Phone/fax: (814) 863-7136/4745, Email: [email protected]

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account. In this paper, the proposed method facilitates the consolidation of existing product lines using MAUT. The method recommends a subset of products to manufacture based on criteria at the (1) product level (e.g., costs and profits) and (2) product family level (e.g., component reuse, variety, commonality, market coverage). As an example application, an existing line of staplers is analyzed. The remainder of this paper progresses as follows: Section 2 describes the proposed method, Section 3 presents the products being analyzed, and Section 4 defines the attributes and their specific utilities. The implementation is found in Section 5, and Section 6 discusses the results. Section 7 gives closing remarks and future work.

The importance of decision-making in engineering design (e.g., Ref. [14]), and the importance of selecting an appropriate decision-making tool in design applications (e.g., Refs. [15,16]) have been discussed extensively in the literature lately. Despite this fact, however, inclusion of appropriate decision-making tools for complex engineering design problems is rare, especially for multi-attribute problems under risk and uncertainty considerations. The importance of uncertainty and risk considerations has been put forth by Hazelrigg [14]. He argues that outcomes of a design decision cannot be determined with certainty. Accordingly, uncertainty and risk should be integrated into design decision-making. Strengthening this argument for the multi-attribute decision making situations, Thurston and Locascio [17] explain the inadequacy of pair-wise comparisons under uncertainty and risk, and point out the need for better decision-making methods: “…once the designer acknowledges that multiple attributes are considered in evaluating the relative worth of design alternatives …the notion that one attribute is ‘more important’ than another loses its meaning and becomes inappropriate. A better method is needed which accurately reflects the fact that willingness to make tradeoffs depends on the current attribute levels” (p.50). One method that integrates uncertainty along with the willingness to make tradeoffs at various attribute levels is multiattribute utility theory (MAUT) [18]. MAUT involves a single decision-maker who chooses among a number of alternatives on the basis of two or more criteria or attributes. The alternatives involve risks and uncertainties. The decision-maker seeks to maximize a utility function that depends on these attributes [18]. MAUT integrates a large body of mathematical theory for utility models and a wide range of practical assessment techniques that together assist in the decision problem to rank alternatives or make a choice. Applications of MAUT in the public sector are numerous since most public sector problems involve multiple conflicting objectives, such as in public health care systems, environmental policy, regulatory issues, site selection, energy or public policy [19]. Considerably fewer applications of MAUT to corporate problems appeared in the decision analysis and operations research literature by the mid 1990s [19]. The engineering design field is no exception; however, documented examples have been increasing given the emphasis of Decision-Based Design recently in the field. Among the handful of engineering design-related applications of MAUT applications for concept selection are the following. Keefer and Kirkwood [20] discuss an application to optimally allocate an operating budget for project engineering. Thurston and Locascio [17] apply MAUT to a turnbuckle design problem. Gurnani and Lewis [21] apply MAUT for an airplane selection problem. Finally, Olewnik, et al. [22] use MAUT to support decision-making for flexible systems design. This paper not only responds to the need for documenting MAUT applications in engineering design but also contributes to the product family design literature by introducing a method with which situations of marginal returns on the objective function with improvement in an attribute level are taken into

2. DECISION PROBLEM The method in Figure 1 is used to consolidate a product line based on product family concept selection. The method follows the optimal decision process common to rational choice analysis [23]: identify options, identify ways to evaluate each option, weight each evaluation dimension, do the rating, and pick the option with the highest score. In the proposed method, we utilize MAUT to evaluate each option based on a set of attributes, which are defined as part of the method.

Figure 1. Proposed method for product family concept selection As shown in Figure 1, the first step is to identify the products in the product line. From this set of products, the concepts under consideration are the different possible combinations of products that can be produced. The second step is to identify the attributes related to the product line being redesigned. This is accomplished by choosing a variety of attributes that encompass the design problem. Once the

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return for a company (lifecycle volume of about 2 million units, a 5 year life cycle selling 400,000 units per year, information obtained from the stapler manufacturer). Finally, the market coverage attribute expresses the robustness of the business strategy to target different groups of consumers [26]. The three attributes are explained in detail in Section 4. General interactive relationships are known among the different parameters. For instance, as the commonality index for a family increases, the market coverage may decrease, and as market coverage decreases, profit may also decrease. The interactive nature of the parameters prescribes the use of the multiplicative form of the utility function. Additional interactive parameters (e.g., functional similarity) can be added or substituted with these given attributes based upon the type of analysis and development stage of the products. After defining the attributes, the data to evaluate the concepts with respect to the attributes is collected (third step of the method). Limited data was available for these products; hence, dissection was conducted to gather the necessary data. The dissection was performed in the Mechanical Engineering Department at Bucknell University as part of a summer Research Experience for Undergraduate Program (see Refs. [27,28] for more detail). To ensure consistency in the dissection, each product within the family was dissected to the lowest possible level, i.e., each component could not be dissected any further. Three of the dissected products are shown in Figure 2, and details on the dissection method can be found in Ref. [29]. The fourth stapler, still at the prototype stage, could not be dissected and is not shown in the figure.

attributes are defined the necessary data is gathered using appropriate product and market analysis techniques. The utility functions for each attribute are assessed separately, and the parameters for the assessment are obtained from product and market analyses and corporate strategy review. The attribute utilities are then aggregated using MAUT, and the concepts are ranked based on their utility value. Each step is detailed in the sections that follow using a family of staplers as an example. 3. INTRODUCTION TO THE FAMILY OF STAPLERS This work uses a case study involving a small startup company producing a line of innovative staplers. Currently, the company is producing three different kinds of staplers, but they would like to introduce a fourth model to increase their market coverage; however, introduction of a new product will result in higher inventory costs, etc. Hence, the goal in this work is to determine the best set of staplers to manufacture in order to increase profits while minimizing product proliferation and still maintaining good market coverage. While this case study is based on real data and discussions with the company, the actual implementation of the decision resulting from the proposed method was not implemented. Hence, the example is only described for illustrative purpose. As shown in Figure 1, the first step in the method is to identify the set of concepts. The concepts analyzed consist of all different possible combinations of the four staplers shown in Table 1. The goal can be achieved by creating and maximizing a multi-attributed utility function involving the following attributes identified in the second step of the method: 1) Commonality Index (for this case study, we use the Product Line Commonality Index, PCI [24]) 2) Profit Over 5-Years 3) Market Coverage The attributes are chosen so that they reflect the tradeoffs between design, production, and marketing at the product and product family levels. The attributes are also selected so that they satisfy the preferential and utility independence conditions required for the multiplicative utility formulation [18]. In this example, only a limited set of attributes is used for ease of understanding; however, the same method can be used with more attributes to get even more comprehensive results. Table 1. The stapler family Model

1 (500)

2 (1000)

3 (2000)

4 (3000)

Capacity

2-15 sheets

2-20 sheets

2-60 sheets

2-100 sheets

Figure 2. Components in dissected staplers The stapler data is stored in a spreadsheet format, see Refs. [30,31] for more details. For each product, the components and the following associated attributes are captured: size, geometry, material, manufacturing process, assembly, fastening scheme, and cost. Regarding the costs, the components are either manufactured in-house or purchased. Hence, for the manufactured components, the initial set-up costs, the volume of the component and the material cost information is recorded;

Picture

The first attribute, commonality index, is used to quantify the “fitness” of a product family in relation to a number of components that the products in the family share [25]. The second attribute, profit over 5-years, is a numerical measure of the desirability of products in the marketplace and the potential

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for the purchased components, the purchasing price and possible bulk discount price are recorded. These numbers were verified with the stapler manufacturer to be reasonable estimates of their production costs. The production level for each variant is determined automatically. The size and geometry factor affects the production level, and hence the initial cost (setup price/total production for this component); the material factor, associated with the component volume, gives an estimate of the overall material cost.

= Ratio of the greatest number of models that share component i with identical materials and manufacturing processes to the greatest possible number of models that could have shared component i with identical materials and manufacturing processes (ni). f3i = Assembly and fastening schemes factor for component i. = Ratio of the greatest number of models that share component i with identical assembly and fastening schemes to the greatest possible number of models that could have shared component i with identical assembly and fastening schemes (ni).

4. DEFINITIONS OF ATTRIBUTES AND ATTRIBUTESPECIFIC UTILITIES This section provides the details regarding the three attributes used in this paper for product line consolidation (step four of the method): (1) the Product Line Commonality Index (PCI), (2) the Profit over a 5-year period, and (3) the Market Coverage. The attributes are chosen so that they reflect the tradeoffs between design, production, and marketing at the product and product family levels. Each is explained in detail in the following sections.

By substituting the values of CCIi, MinCCIi, and MaxCCIi, the following formula is obtained for the PCI: P P P 1 1 PCI = 100 * (∑ ni * f1i * f 2 i * f 3i − ∑ 2 ) /( P * N − ∑ 2 ) (2) i =1 ni i =1 i =1 ni PCI varies continuously between 0 and 100. When PCI = 0, either none of the non-differentiating components are shared across models, or if they are shared, their size/shapes, materials/manufacturing processes, and assembly processes are all different. When PCI = 100, it indicates that all the nonunique components are shared across models and that they are of identical size and shape, made using the same material and manufacturing process, and assembled in the same manner. The PCI values for all possible combinations of stapler alternatives are listed in Table 2. The PCI is computed automatically using the data described previously.

4.1 Commonality Index: PCI The Product Line Commonality Index, PCI [24], is one of a group of metrics called commonality indices that measure the “fitness” of a product family based on the performance of the family in a number of areas. Commonality indices are intended to provide valuable information about the degree of commonality in the family and help reduce costs [25]. Contrary to the indices that only measure the percentage of components that are common across a product family (and hence penalizing families with a broader feature mix), the PCI measures and penalizes the non-unique components, given the product mix. The PCI is given by [24]: P

P

P

P

i =1

i =1

i =1

i =1

PCI = 100 * ( ∑ CCI i − ∑ MinCCI i ) /( ∑ MaxCCI i − ∑ MinCCI i )

Table 2. PCI values for the different family alternatives

(1)

where: CCIi = Component Commonality Index for component i. = ni * f1i * f2i * f3i MaxCCIi = Maximum possible Component Commonality Index for component I = N MinCCIi = Minimum possible Component Commonality Index for component i. = ni * 1/ni * 1/ni * 1/ni = = 1/ni2 P = Total number of non differentiating components that can potentially be standardized across models. N = Number of products in the product family. ni = Number of products in the product family that have component i. f1i = Size and shape factor for component i. = Ratio of the greatest number of models that share component i with identical size and shape to the greatest possible number of models that could have shared component i with identical size and shape (ni). f2i = Materials and manufacturing processes factor for component i.

The nature of PCI lends itself to being assessed as a risk averse attribute. It is initially easy to increase PCI from 0 to 50, for example, by substituting variant components with common ones among the different products in the family. It is rather difficult, however, to increase the PCI from 95 to 100, since a PCI of 100 means that all shared components of the products are identical. Therefore, the utility of PCI can be represented as a monotonically increasing exponential function with a positive risk tolerance, RT, according to the following relationship [32]:  − x1   

(3)

U1 ( x1 ) = A − B ∗ e RT  where:

A=

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e  e 

 − Min ( x1 )    RT  

 − Min ( x1 )    RT  

−e

 − Max ( x1 )    RT  

  

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B=

RT Min(x1) Max(x1)

than or equal to the expected value from the lottery, the utility function takes the form of a risk neutral function, the graph of which is a positively sloped linear function through the maximum and minimum utility points. The CE cannot be less than the lowest PCI value in the set, because the decision-maker would not be indifferent to the decision expressed as a lottery in Figure 3a and Figure 3b. By this assessment, the certainty equivalent chosen for the staplers is PCI = 48.0. This means that the company values a PCI of 48.0 just as much as the 50% chance of having a PCI of 36.5 or 63.3. The risk tolerance (RT) can be derived from the certainty equivalent by finding the utility function with the form of Eq. 3 that satisfies the following relationship with CE1 = 48.0: U1(CE1) = .5U1(Max(x1))+.5U1(Min(x1)) (6) Solving for RT,

1 ( x1 )   − Max ( x1 )    −Min    e RT  − e RT    

(5)

= Risk tolerance for the PCI, = Minimum value of PCI across all alternatives, = Maximum value of PCI across all alternatives.

The risk tolerance of the PCI attribute is assessed by determining the best and worst values of PCI in the data set of potential families and using the concept of the certainty equivalent (CE). The CE is the value of the attribute, in this case PCI, for which the decision-maker is indifferent between the CE and a lottery between the best and worst PCI. In short, it is a measure of how risk averse or risk prone the decisionmaker, or product family designer, is toward the attribute. A higher CE for PCI, which is being maximized and is risk averse, indicates a less risk averse attitude. Likewise, a lower CE indicates a more risk averse attitude. The lottery is referred to as a von Neumann-Morgenstern lottery and is built upon a multi-axiomatic formulation. The family with the lowest utility has a PCI of 36.5, and the one with the highest utility among the alternatives has a value of 63.3. These two points correspond to utilities of 0 and 1, respectively. A lottery is constructed with the best and worst alternative as shown in Figure 2a, where each alternative has a 50% probability of being selected.

p =0.50

or

(7)

Equations 4, 5 and 7, are solved iteratively in Excel to yield RT1=0.1476. The overall utility function for PCI is given in Eq. 8. The graph of utility data for the family of staplers is shown in Figure 4 based upon the exponential utility function (see Appendix for the complete set of data).

U 1 ( x1 ) = 2.301 − 5.01∗ e

 − x1     46.93 

(8))

PCI = 63.3

CE = ? When p = 1.00

EV(CE) = 0.5*(36.5)+0.5*(63.3) = 49.9 (a)

− CE1  − 0.5U1 ( Max( x1 )) − 0.5U1 ( Min( x1 )) + A  ln  B  

PCI = 36.5

CE = p =0.50

RT1 =

Figure 4. PCI utility for the families of staplers

(b)

Examining Figure 4, the graph of PCI utility is monotonically increasing, reflecting the risk averse nature of this attribute. The data points, representing the families, range in utility from 0 to 1, with most family alternatives clustered with low or high utility. The data indicates that all 3-product families have utilities on the low end of the scale. This is expected because, in general, it is easier to make two products common than three products.

Figure 3. Choosing certainty equivalent by a) expected value of lottery or b) value assessment The decision problem depicted in Figure 3 can be written for what value of the CE in Figure 3b is the decision-maker indifferent between the “sure thing” in Figure 3b and the lottery in Figure 3a. For the family of staplers, the company is interested more in obtaining quick profits to fuel growth than in creating a long-lasting product platform and is therefore less risk averse. The CE for a risk averse attribute that is being maximized must be less than the expected value of CE from the lottery (see Figure 3a) but greater than the minimum consequence value; the closer the CE to the expected value the less risk averse is the decision-maker (for more details, refer to the risk premium discussed by Pratt [33]). If the CE is greater

4.2 Profit Over 5-years The profit over 5-years is determined from the production estimates of each stapler over its lifetime (see Table 3). When more units are produced and sold, profits for that product increase. The production estimates therefore are forecast from data gathered from the composition of the market and from discussions with engineers from the stapler manufacturer.

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Table 3. Products and production volume for staplers Product 1 (500) 2 (1000) 3 (2000) 4 (3000)

Quantity 2,000,000 3,750,000 2,500,000 1,250,000

The profits values are calculated by subtracting the cost per unit from the wholesale price per unit and multiplying by the number of units produced. By aggregating profits for each potential family, the total profits for the family are generated in Table 4. For example, the second set of products contains staplers 1, 2 and 3. After a 5-year period, the profit generated from stapler 1 is $3,881,896, from stapler 2 is $13,245,118, and from stapler 3 is $19,153,504, for a total profit of $36,280,518. Although the fourth model of stapler is not yet in production, the production forecasts and costs are estimated based on the data for the other models.

Figure 5. Utility of profit over 5-years for stapler families 4.3 Market Coverage To sustain competitiveness, companies wish to achieve good market coverage [26]. One way to accomplish this is to develop one product for each market segment, which may result in higher development and manufacturing costs. Therefore, it is generally seen as advantageous to maximize market coverage with as few products as possible. For staplers, the market coverage is determined based on one factor, namely, the number of sheets being stapled. After analyzing a variety of existing staplers in the market, the following segments are defined based on the maximum number of sheets that can be stapled: a. less than 20 sheets, b. between 20 and 40 sheets, c. between 40 and 80 sheets, and d. more than 80 sheets. For these four market segments, the market coverage is defined as being proportional to the actual sales number for the staplers analyzed, as shown in Figure 6.

Table 4. Total profit over 5-years for stapler families (in $)

Considering the attitude of a small startup company producing a line of staplers, we assume that the anticipated profit strategy of the company is risk prone; therefore, the utility function is monotonically increasing with a negative risk tolerance in the exponential equation. The utility function for profit over 5-years, U2, has the form given by Eq. 3. As with PCI, the risk tolerance of profit, RT2, can be calculated by finding the certainty equivalent for profit. This time the decision-maker finds the indifference point between a lottery of the lowest and highest profit values ($17 million and $49 million, respectively) and the CE in a von NeumannMorgenstern lottery. Based upon the startup company’s aggressive pursuit of profit in the first few years of business, the risk attitude is assessed as more risk prone. This indicates a higher CE value, which is even further away from the expected value of the lottery. The CE is assessed to be indifferent with the lottery when the profit over 5-years has a value of $42 million. Equations. 4, 5 and 7, are solved iteratively in Excel with CE2 = $42M to yield RT2= -10.54e6. The overall utility function for profit over 5-years is given in Eq. 9. The graph of utility data for the family of staplers is shown in Figure 5 based upon the exponential utility function for profit, and the tabulated results can be found in Appendix.  

− x2

 

U 2 ( x2 ) = −0.0521 − ( −0.0103) ∗ e −10.54e 6 

Figure 6. Market coverage The actual sales numbers were obtained from the stapler manufacturer. One can see that the primary market is the first segment (less than 20 sheets), while the fourth segment (80 sheets and over) is much smaller. This is explained by the fact that small staplers are needed by most individuals and can be found virtually on any desk, while more “heavy-duty” staplers are shared between several people, as their use is less frequent.

(9)

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For the product line analyzed, the market segments are shown in Figure 7. One can see that staplers 1 and 2 (500 and 1000 models) compete in the same market segment, which can lead to “cannibalization” [34], as seen in the results in Section 6. The values for the market coverage for all possible sets of products are listed in Table 5.

tolerance can be found in the same manner as the other attributes. The iterative solution of Eqs. 4, 5 and 7 gives RT3= 0.2176. Equation 10 gives the utility function for market coverage, substituting the values for A3, B3 and RT3 into Eq. 3. Figure 8 shows the graph of the market coverage utility function. The complete data set can be found in Appendix.

U 3 ( x3 ) = −0.1117 − (−0.0281) ∗ e

 − x3     −0.2176 

(10)

The graph and the data show that the families of products span the range of utilities fairly evenly. The two 3-product alternatives, which span three different market segments, have the highest utility. Families with products in the same market segment have much lower utilities.

Figure 7. Market segmentation grid for the staplers Figure 8. Market coverage utility for families of staplers

Table 5. Percentage of market coverage by family

5. ATTRIBUTE TRADEOFFS & AGGREGATED UTILITY The fifth step in the proposed method is determining the tradeoffs between each attribute and collecting the attributespecific utilities into one aggregated utility function. As discussed previously, the relevant attributes interact with each other, but they satisfy preferential and utility independence conditions. Therefore, the overall utility is represented by the multiplicative form of the utility function rather than a simple weighted average. The multiplicative form is given as a product of the attribute utilities as given by Keeney and Raiffa [18]:

The anticipated strategy of the company toward market coverage is risk prone. As market coverage increases, the number of potential customers also increases. More customers means more profit and higher utility; so, utility is monotonically increasing with a negative risk tolerance. A small amount of market coverage is not desirable. It becomes more difficult to increase market coverage at the higher end. We assume that an increase from 80 to 85% is more desirable than an increase from 10 to 15%, reflecting the risk prone nature of the attribute. The utility function for market coverage, U3, has the form given by Eq. 3. As with PCI and profit, the risk tolerance of market coverage, RT3, is calculated by finding the certainty equivalent (CE) for market coverage. The von NeumannMorgenstern lottery is once again constructed between a lottery of the lowest and highest market coverage values and the certainty equivalent, CE3. The indifference point between the alternatives of the lottery is chosen based upon the degree of the attitude toward risk for the market coverage attribute. It is assumed that a small startup company is only moderately risk prone with regard to percent of market coverage. The CE value is therefore closer to the expected value of the lottery of the lowest and highest market coverage values. The CE is assessed to be an indifferent alternative when CE3 = 67%. The risk

U ( x) =

1 K

n  ( Kk iU i ( xi ) + 1) − 1 ∏ i =1 

(11)

where: U(x) = the total utility of design alternative x xi = the performance level of attribute i Ui(xi) = the single attribute utility for attribute i i = 1, 2, 3, …, n attributes ki = attribute-scaling parameter for attribute i K = normalizing constant The normalizing constant, K, is itself derived from the attributescaling parameters by the following relationship: n

1 + K = ∏(1 + Kk i ) i =1

(12)

It is apparent from Eqs. 11 and 12 that the attribute-scaling parameters must be assessed before determining the overall utility of the family alternatives. This is achieved by following

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achieved when PCI=55 and MC=60%, generating Eq. 14. The utilities are then evaluated using the attribute-specific utility functions in Eqs. 8 and 10 to obtain a relationship between k1 and k3 as seen in Eqs. 15-17.

a step-by-step process of evaluating the tradeoffs between the attributes, similar to the hypothetical equivalence method [35]. First, the attributes must be ranked in order of importance. This ranking does not reflect the magnitude of importance, only their order. The three stapler family attributes are assessed as follows: first, the Profit over 5-years P (corresponding scaling parameter k2), then the Market coverage MC (corresponding scaling parameter k3) and finally the Commonality Index PCI (with a corresponding scaling parameter k1). The ranking is chosen based upon the business strategy of the stapler manufacturer. As a small startup company, returning a profit quickly is of most importance. In order to broaden the appeal of the products, market coverage is second most important. Finally, commonality among the products is a favorable goal to reduce development costs but is not essential to succeed. Determining the ki values begins by finding the indifference point in the following decision problem involving the most important attribute, profit. The decision problem consists of comparing the two choices shown in Figure 9a and Figure 9b. The goal is to find the value of probability, p, such that the decision-maker is indifferent between the two choices shown in Figure 9a and Figure 9b. Figure 9a shows a lottery between the best alternative, where the values of all attributes—profit, market coverage, and PCI—are all at their highest values, and the worst alternative with all attributes at their lowest values. The expected value of utility for the choice in Figure 9a is the probability, p. The choice in Figure 9b is the alternative with profit at its highest value and the other two attributes at their lowest values.

p

(14) (15) (16) (17)

A similar decision problem is constructed between profit and market coverage. The choice is between the lowest value of profit and unknown market coverage, and an unknown value of profit and the lowest value of market coverage (see Eq. 18). Substituting appropriate indifference values into Eq. 18, MC=80% and P=$45 million, k3 is determined to be 0.4433: u($17M, MC) ~ u(P, 30%) k2*u2($17M)+k3*u3(80%)=k2*u2($45M)+k3*u3(30%) 0.65(0)+k3(1.0)=0.65(0.682)+k3(0) k3=0.65(0.682)=0.4433

(18) (19) (20) (21)

Then k1 is determined from Eqs. 17 and 21 to be k1= 0.1856. The three ki values are then used to find K by using the product relationship given by Eq. 12. Solving for K yields K=0.60824. All of the scaling factors are substituted into the multiplicative equation, Eq. 11. Each of the family alternatives is evaluated in this multiplicative equation, and the resulting rankings are listed in Table 6. Table 6. Aggregated utility and ranking for stapler families

U($59M, 80%, 83.3)= 1 or

1- p

u(55, 30%) = u(36.5, 60%) k1*u1(55)+k3*u3(30%)=k1*u1(36.5)+k3*u3(60%) (0.793)k1+0=0+(0.332)k3 0.793k1=0.332k3

U($49M, 30%, 36.5)

U($17M, 30%, 36.5)= 0

EV = p*(1.0)+(1-p)*(0.0) (a)

(b)

6. DISCUSSION OF RESULTS The aggregated utility and the resulting ranking for the stapler families (final step in the method) are shown in Table 8, from which the following remarks can be made. (1) The best subset of products to manufacture is the set 2-3-4. This set maximizes profits, and market coverage, although commonality between the three products could be increased. (2) In general, the sets containing three products rank higher than the sets having only two products. This can be understood by the fact that producing only two staplers does not provide enough market coverage, and profit can easily be increased by offering a third stapler. (3) In general, the sets containing both staplers 1 and 2 rank lower than the ones containing at most either stapler 1 or stapler 2. This is due to the fact that these two staplers,

Figure 9. Profit scaling factor decision problem: (a) lottery and (b) value assessment Then the decision problem in Figure 9 is evaluated based on the company’s business strategy, yielding an indifference point when p=0.65. This implies that the value of the profit scaling factor k2 is also 0.65. Additional decision problems are constructed to find k3 for market coverage and k1 for PCI. First market coverage and PCI are compared, while holding profit at a constant level. A decision problem is constructed to make the following choices indifferent: u(PCI, 30%) ~ u(36.5, MC) (13) The first choice contains an unknown value of PCI and the lowest value of market coverage. The choice is compared to the lowest value of PCI and an unknown value of market coverage. An indifference point for the stapler families is

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[4] Pessina, M. W. and Renner, J. R., 1998, "Mass Customization at Lutron Electronics - A Total Company Process," Agility & Global Competition, 2(2), pp. 50-57. [5] Mattson, C. A. and Messac, A., 2005, "Pareto Frontier Based Concept Selection Under Uncertainty, with Visualization," Optimization and Engineering - Special Issue on Multidisciplinary Design Optimization, 6(1), pp. 85-115. [6] Okudan, G. E. and Shiraiker, R., 2006, "A Multi-Stage Problem Formulation for Concept Selection for Improved Product Design," Portland International Conference on Engineering and Technology Management, Istanbul, Turkey, PICMET. [7] Pugh, S., 1981, "Concept Selection - A Method that Works," International Conference on Engineering Design, Roma, Italy. [8] Otto, K. N., 1995, "Measurement Analysis of Product Design Methods," Research in Engineering Design, 7(2), pp. 86-101. [9] Ulrich, K. T. and Eppinger, S. D., 2000, "Chapter 7: Concept Selection," Product Design and Development - Second Edition, McGraw-Hill, Boston, MA, pp. 138-159. [10] Kuppuraju, N., Ittimakin, P. and Mistree, F., 1985, "Design through Selection: A Method that Works," Design Studies, 6(3), pp. 91-106. [11] Sullivan, L. P., 1986, "Quality Function Development," Quality Progress, 19(6), pp. 39-50. [12] King, A. M. and Sivaloganathan, S., 1999, "Development of a Methodology for Concept Selection in Flexible Design Strategies," Journal of Engineering Design, 10(4), pp. 329-349. [13] Saaty, T. L., 1990, "How to Make a Decision: The Analytical Hierarchy Process," European Journal of Operational Research, 48(1), pp. 9-26. [14] Hazelrigg, G. A., 1998, "A Framework for Decision-Based Engineering Design," ASME Journal of Mechanical Design, 120(4), pp. 653-658. [15] Dym, C. L., Wood, W. H. and Scott, M. J., 2002, "Rank Ordering Engineering Designs: Pairwise Comparison Charts and Borda Counts," Research in Engineering Design, 13(4), pp. 236-242. [16] Saari, D. G., 2000, "Mathematical Structure of Voting Paradoxes I: Pairwise Votes," Economic Theory, 15 (1), pp. 153. [17] Thurston, D. L. and Locascio, A., 1994, "Decision Theory for Design Economics," The Engineering Economist, 40(1), pp. 41-72. [18] Keeney, R. L. and Raiffa, H., 1993, Decisions with Multiple Objectives: Preferences and Value Tradeoffs, Cambridge University Press, Cambridge, UK. [19] Walls, M. R., 1995, "Integrating Business Strategy and Capital Allocation: An Application Of Multi-Objective Decision Making," The Engineering Economist, 40(3), pp. 247-266. [20] Keefer, D. L. and Kirkwood, C. W., 1978, "A Multiobjective Decision Analysis: Budget Planning for Product Engineering," Operational Research Quarterly, 29(5), pp. 435442.

rather than targeting different markets, compete in the same market, decreasing profits due to cannibalization. The aggregated utility and ranking for the alternative stapler families enable the ranking of different production scenarios, and one can see that the recommended set to produce (staplers 2-3-4) differs from the current set of staplers produced (1-2-3). Implementing these changes could result in a dramatic increase of profits, from $36M to more than $48M (+33%). While the results provided by the proposed method can appear trivial on a simple example such as the one described in this paper, the proposed method is systematic and can be applied on much larger-scale problems to help companies where decisions on product line consolidation are much less clear cut. 7. CLOSING REMARKS In this paper we proposed a method to consolidate a product line using multi-attribute utility theory. An existing line of staplers was analyzed, and the subset of products recommended by the method would substantially increase profits (+33%) while minimizing product proliferation if implemented. The proposed product family has the same number of products as the current product line but includes a high-end stapler in place of a low-end model. Three attributes were considered: (1) the Product Line Commonality Index, (2) profit over 5-years, and (3) percent market coverage. While this work shows promising results, only a limited breadth of aspects that are relevant in making decisions for product line consolidation are considered. The proposed method can be easily extended, however, to include more parameters, such as the engineering cost to develop a new product. Suggested future work includes considering more criteria to give more accurate aggregate utilities and to obtain a more comprehensive ranking of the different sets of products. ACKNOWLEDGMENTS The authors acknowledge Mike Haile and Fabrice Alizon from Bucknell University for the dissection of the staplers. This work was funded by the National Science Foundation under Grant Nos. IIS-0325402 and DMI-0133923. Any opinions, findings, and conclusions or recommendations presented in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation. REFERENCES [1] Simpson, T. W., Siddique, S. and Jiao, J., Eds., 2005, Product Platform and Product Family Design: Methods and Applications, Springer, New York. [2] Simpson, T. W., 2004, "Product Platform Design and Customization: Status and Promise," Artificial Intelligence for Engineering Design, Analysis and Manufacturing, 18(1), pp. 320. [3] Meyer, M. H. and Lehnerd, A. P., 1997, The Power of Product Platforms: Building Value and Cost Leadership, The Free Press, New York.

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[21] Gurnani, A. P. and Lewis, K., 2005, "Robust Multiattribute Decision Making Under Risk and Uncertainty in Engineering Design," Engineering Optimization, pp. 813-830. [22] Olewnik, A. and Lewis, K., 2006, "A Decision Support Framework for Flexible System Design," Journal of Engineering Design, 17(1), pp. 75-97. [23] Soelberg, P. O., 1967, "Unprogrammed Decision Making," Industrial Management Review, 8(1), pp. 19-29. [24] Kota, S., Sethuraman, K. and Miller, R., 2000, "A Metric for Evaluating Design Commonality in Product Families," Journal of Mechanical Design, 122(4), pp. 403-410. [25] Thevenot, H. J. and Simpson, T. W., 2006, "Commonality Indices for Product Family Design: A Detailed Comparison," Journal of Engineering Design, 17(2), pp. 99-119. [26] Porter, M. E., 1985, Competitive Advantage: Creating and Sustaining Superior Performance, Free Press, New York. [27] Sahin, A., Terpenny, J. P., Shooter, S., Stone, R. B. and Simpson, T. W., 2006, "A Method to Evaluate Relative Instructional Efficiencies of Design Activities for Product Platform Planning," ASEE Annual Conference & Exposition, Chicago, IL, ASEE, in press. [28] Lukman, H., Shooter, S., Terpenny, J. P., Simpson, T. W., Stone, R. B. and Kumara, S. R. T., 2006, "An Inter-University Collaborative Undergraduate Research/Learning Experience for Product Platform Planning: Year 2," ASEE Annual Conference & Exposition, Chicago, IL, ASEE, in press. [29] Simpson, T. W. and Thevenot, H. J., 2005, "Using Product Dissection to Integrate Product Family Design Research into the Classroom and Improve Students' Understanding of Platform Commonality," International Journal of Engineering Education, in press. [30] Thevenot, H. J., 2006, "A Method For Product Family Redesign Based on Component Commonality Analysis," Ph.D. Dissertation, Industrial & Manufacturing Engineering, The Pennsylvania State University, University Park, PA. [31] Thevenot, H. J. and Simpson, T. W., 2006, "A Comprehensive Metric for Evaluating Component Commonality in a Product Family," ASME 2006 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Philadelphia, PA, ASME, Paper No. DETC2006/DAC-99268. [32] Kirkwood, C. W., 1997, Strategic Decision Making: Multiobjective Decision Analysis with Spreadsheets, Wadsworth Publishing Company, Belmont, CA. [33] Pratt, J. W., 1964, "Risk Aversion in the Small and in the Large," Econometrica, 32, pp. 122-136. [34] Desai, P., Kekre, S., Radhakrishnan, S. and Srinivasan, K., 2001, "Product Differentiation and Commonality in Design: Balancing Revenue and Cost Drivers," Management Science, 47(1), pp. 37-51. [35] Simpson, T. W., Seepersad, C. C. and Mistree, F., 2001, "Balancing Commonality and Performance within the Concurrent Design of Multiple Products in a Product Family," Concurrent Engineering: Research and Applications, 9(3), pp. 177-190.

APPENDIX: ATTRIBUTE-SPECIFIC UTILITY DATA

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