Product line design and positioning using add-on services

Int. J. Production Economics 163 (2015) 16–33

Contents lists available at ScienceDirect

Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

Product line design and positioning using add-on services Omkar D. Palsule-Desai a,n, Devanath Tirupati b, Janat Shah c a b c

Indian Institute of Management Indore, Block III, Ground Floor, Prabandh Shikhar, Rau Pithampur Road, Indore 453 331, India Indian Institute of Management Bangalore, Bangalore 560 076, India Indian Institute of Management Udaipur, Udaipur 313 001, India

art ic l e i nf o

a b s t r a c t

Article history: Received 3 March 2014 Accepted 3 February 2015 Available online 21 February 2015

In this paper, we consider a generic product line design and positioning problem in the context of variety creation using a core product and add-on services. While the functional output of the core product is identical across the products, product variety is created using add-on services that do not alter the functionality of the core product. The motivation for our study comes from emerging for-profit private healthcare service providers in India. We consider two specific scenarios – simultaneous and sequential design – and focus on obtaining insights into the implications of the core product on design and positioning of add-on services. We show that for the core product cost below a threshold the firm does not cover the entire market. For the cost beyond another threshold, it does not introduce any product. The product quality increases in both the core product cost and the consumer valuation of the product relative to the cost of quality; however, the quality based distinction decreases (increases) in the cost parameters (product valuation). In the two-product sequential product design scenario, we show that both products are active only under certain situations. We derive conditions under which the second product has quality higher (lower) than that of the first product. When the firm initially enters the market with a product that corresponds to single product design scenario, we show that the firm introduces a lower quality product and increases profit by increasing the market share. By comparing our results in the two scenarios, we show that the gap between positive (and similarly negative) implications of the two scenarios decreases as the core product cost increases with respect to the cost of quality. In a duopoly setting, we also illustrate how the design and positioning of an incumbent's product impacts a new entrant's product. Our model and results may be seen as building blocks for obtaining managerial insights. & 2015 Elsevier B.V. All rights reserved.

Keywords: Product line design Core product Add-on services

1. Introduction Increasing product variety is a popular and well recognized strategy in marketing to increase demand and market share (see, Kotler, 2002). Using empirical evidences, Kekre and Srinivasan (1990) found that broader product lines result in significant market share benefits and increase in firms' profitability. With rapidly evolving technologies, capabilities of firms to produce a variety of the same product are increasing. Over the years, managing product variety has also become a source of competitive advantage for firms (Meyer and Lehnerd, 1997). However, Ramdas and Sawhney (2001) and Gourville and Soman (2005) observe that increasing product variety does not guarantee increase in long run profits, and in fact, it can worsen firms' competitiveness. Researchers have studied the product line design problem extensively in a

n

Corresponding author. Tel.: þ 91 731 2439567; fax: þ 91 731 2439800. E-mail addresses: [email protected] (O.D. Palsule-Desai), [email protected] (D. Tirupati), [email protected] (J. Shah). http://dx.doi.org/10.1016/j.ijpe.2015.02.007 0925-5273/& 2015 Elsevier B.V. All rights reserved.

variety of settings in manufacturing industries (e.g., Kim and Chhajed, 2000; Lacourbe, 2012); however, related issues in services settings have not been addressed adequately. It has been observed in many service industries that firms develop a portfolio of product/service (henceforth, referred to as product only) using a core product and add-on services. In this case, the functional output of each product variant is delivered by the core product that is common across all variants, and a variety of the product is created using add-on services that do not alter the functionality of the core product. Consumers self-select a particular variant of the product based on the utility derived net of its price. Accordingly, the firm's strategy of offering a variety of the product has implications for its profitability; however, the functionality of each variant is identical.1 1 In the existing literature, product functionality is considered as one of the various dimensions implied by generic quality parameter used for vertically differentiated products (see, e.g., Desai, 2001). Thereby, better is the product functionality, more is the quality. However, in view of our motivating examples we specifically distinguish between product functionality and other dimensions of

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

The motivation for our study presented in this paper comes from emerging for-profit private healthcare service providers in India. The past two decades have witnessed emergence of two kinds of hospitals providing specialty care. On one hand, there are hospitals – such as Aravind Eye Care, Narayana Hrudayalaya – that were established to serve consumers at the bottom of the pyramid, and over the years they have introduced specialized services targeting elite sections of the market. On the other hand, other hospitals – such as Apollo Hospitals, Fortis Healthcare – that started as super-specialty healthcare providers eventually introduced products that could be afforded by the lower segment of the market. In these hospitals, the core product, e.g., cataract or cardiac surgery, is provided under different packages – such as premium, regular, economy – with add-on services that do not compromise on functionality of the core product. (The context is described further in Section 2.) Similar examples can be found in manufacturing industries in which firms offer service components as add-on to products manufactured. For instance, Dell Inc. offers laptops with either a limited warranty scheme or with an Accidental Damage Service cover. iPads and iPhones are available with various App Store Gift Cards. Similarly, many automakers offer a variety of car maintenance programs. While our motivation comes from specific examples in India, the model and structural results are valid in other service settings too. For example, medical tourism in countries such as Canada, Israel, UAE, Singapore, the UK, etc. (see, Gahlinger, 2008), books, telecommunication services, etc. The examples mentioned above can be characterized by a product line developed using a core product and add-on services. In this context, the firm's product line design problem focuses on determining the number of product variants and their positioning. Positioning of product variants implies relative quality and price levels of all variants. We model and analyze this problem for two specific scenarios: simultaneous and sequential design. In the former scenario, the quality and price levels for all products are determined jointly. In sequential design scenario, they are determined in sequence of the products, i.e., quality and price levels for the subsequent products are determined relative to the already existing products. Besides characterizing the optimal solution to the problem, we develop structural results to derive key managerial insights. Our results show that there exists a threshold level for the core product cost below which the optimal strategy for the firm is to cover the entire market. There also exists another threshold beyond which the firm does not produce any variety of the product. The optimal product quality level increases as both the core product cost and the consumer valuation of the product increase relative to the cost of quality. Some of the implications of our structural results are also quite unique. For example, in the multi-product scenario, we show that products in the product line are increasingly identical when the firm's cost parameters are larger. On the contrary, the products are increasingly distinguishable when the product valuation for the consumers is higher. In a sequential product design case with two products, we show that when the firm's product design and positioning decisions are arbitrary to begin with, there exist situations in which the firm introduces a new variety of quality level higher (similarly lower) than the existing variety. In this case, we also show that the two products are active in the market only under certain situations. In a special case of the sequential product design in which the firm's product design and positioning decisions are optimal at each stage, we show that the optimal strategy for the firm is to introduce a lower quality product following the first product and increase its profit by increasing the market share.

(footnote continued) quality. Thereby, the products in our model can be vertically differentiated on the basis of a generic quality parameter; however, they all deliver the same functional output.

17

The remainder of the paper is organized as follows. In Section 2, we describe our problem context, and in Section 3, we present an overview of the related literature. We develop the model in Section 4, and present our analyses and results in Sections 6 and 7. To bring out the implications of simultaneous and sequential design scenarios, we compare the firm's optimal decisions in both scenarios in Section 8. Section 9 reflects on a duopoly competitive setting based on our motivating examples. In Section 10, we mention key findings of the paper and conclude the paper. Proofs are relegated to appendix.

2. Problem description The motivation for our study comes from the product line design and positioning problems faced by some of the private sector hospitals in India. The past two decades have witnessed emergence of two kinds of hospitals providing specialty care. On one hand, there are hospitals such as Arvind Eye (AE) and Narayana Hridayalaya (NH) that were established with the objective of providing affordable and quality healthcare to the masses by primarily serving the consumers at the bottom of the pyramid (see, Shah and Murthy, 2004; Khanna et al., 2005). Khanna et al. (2005) describe this approach as “Wal-Martization of healthcare”. Over the years, in keeping with the reputation for their quality services and increasing popularity of medical tourism, AE and NH attained self-sustainability by exploring many avenues to attract consumers with high affordability from not only the western countries but also the Indian middle class that has been growing in its size, income levels and quality consciousness. These hospitals enhanced their market-reach by expanding the product portfolio without compromising on the quality of the basic services, i.e., surgeries. As described in Khanna et al. (2005), both AE and NH adopted a hybrid strategy to cater to both ends of the pyramid. In particular, they attracted consumers of high affordability by the virtue of reputation for their product quality and attracted consumers of low affordability via reasonable pricing. Unlike AE and NH, Apollo Hospitals (AH) and Fortis Healthcare (FH) established themselves for consumers at the high end of the economic pyramid (see, Oberhozer-Gee et al., 2007). They attracted people from everywhere to India for super-specialty healthcare by providing first-world healthcare at emergingmarket prices. However, due to capacity under-utilization in the existing systems and the scope for growing with the Indian middle class, alike AE and NH, both AH and FH expanded their product portfolio by introducing variants of the existing products. Accordingly, while the target consumer segments and the mission of the two kinds of hospitals are distinct, their strategies to attract consumers in the respective segments and the managerial problems arising out of this are identical as described below. Each of the above mentioned hospitals offers a menu of products that are identical in the basic medical treatment but distinct in supporting healthcare. The number of variants of supporting healthcare, their content (quality) and pricing decisions essentially depend on market characteristics and the hospital's cost structure. For example, at NH open heart surgery is offered under premium, regular and economy categories; and angioplasty is offered only under premium and regular categories (see Khanna et al., 2005). Nevertheless, the functional outcome of the surgery in each product variant (or simply, product) is the same which is evident from the fact that the surgery is performed by the same set of doctors. While a number of examples similar to our motivating example can be found in the service sector, manufacturing industries offering services as add-on to the manufactured products also demonstrate similar characteristics. For instance, the features considered in our problem are similar to those experienced by manufacturing firms

18

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

such as Dell Inc. in bundling warranties and service contracts with the product sales. A consumer interested in buying a laptop with required specifications – such as processor speed, RAM, screen size, etc. – first decides to purchase, say, from Dell, then decides on which of the warranty schemes and/or service contracts to add, if any, on top of the laptop purchase. In this case, a particular laptop meeting the requirements is the core product, and the type of warranty scheme and/or service contract selected as the add-on depends on the cost–benefit analysis. Similarly, add-on features such as App Store Gift Card reflect on a menu of bonuses offered by Apple Inc. on their core products such as iPad and iPhone. Audi, Jaguar, Land Rover and Lexus offer a menu of bonuses in the form of variety of car maintenance programs – such as Basic, Intermediate, Comprehensive – that the consumers choose from.2 In all these examples, the add-on services definitely impact the profitability of the seller, but they do not alter the functionality of the core products – laptop, iPad, iPhone, automobile, etc. In this paper, we consider the product line design problem from the hospital's (henceforth, firm) perspective and determine the number of variants to be offered, their design and relative positioning. A consumer interested in purchasing the core product selects a particular variety on the basis of net utility derived from the add-on service. Thereby, we particularly focus on obtaining insights into implications of the core product on design and positioning of add-on services. We consider two scenarios separately. In the first scenario, referred to as simultaneous product design, the firm upfront determines the number of products in the product line to be offered along with their design and relative positioning. In the second scenario, referred to as sequential product design, we consider the situation in which the firm expands its product line by adding one or more variants of the already existing products in the product line. In each scenario, the objective is to maximize the firm's profit which is appropriate for firms in the private sector.

3. Related literature Researchers have studied product line problem involving variety creation and variety management from various perspectives – such as economic (Lancaster, 1990), marketing (Mahajan and Wind, 1992; Eliashberg and Steinberg, 1993), consumer theory (Kahn, 1995), organizational behavior (Brown and Eisenhardt, 1995), etc. Ho and Tang (1998) provide a collection of research papers that address issues specific to management of product variety. The perspective that comes close to our work presented in this paper is at the interface of operations management and marketing. Krishnan and Ulrich (2001) and Ramdas (2003) survey the related literature. Krishnan and Ulrich (2001) study the literature in the areas of marketing, operations management and engineering design from the perspective of product development as a deliberate business process. Ramdas (2003) provides a framework for managerial decisions about product variety by focusing on functional interdependencies. An extensive and a comprehensive review of this literature is beyond the scope of this paper. Instead we mention a few studies that are perhaps most relevant to the work presented in this paper. These studies focus primarily on two characteristics of the product line design problem: (i) market segmentation with quality based product differentiation, and (ii) product line design using platform/modular products or common components. The problem of product line design in the presence of quality sensitive consumers has gained considerable visibility in the 2 Source: http://www.edmunds.com/car-buying/are-free-vehicle-maintenanceprograms-worth-it.html.

marketing and operations literature. The theory of market segmentation based on consumer self-selection has evolved using the framework provided by Mussa and Rosen (1978) and Moorthy (1984, 1988). The consumers are assumed to differ only in willingness to pay for product quality which is a single dimension of vertical product differentiation. The focus of these studies is on identifying the implications of demand conditions and product cannibalization on product line design. For instance, Moorthy (1984) shows that only the highest-valuation segment gets its preferred quality while the qualities of products aimed at other segments are distorted downwards. Using this framework, researchers have developed numerous models that focus on one or more important features of the problem and examine the impact of factors of interest on the firm's product line. To mention a few, marketing cost (Villas-Boas, 2004), scale economies (Krishnan and Gupta, 2001), channel considerations (Villas-Boas, 1998), competition (Desai, 2001), product architecture (Desai et al., 2001), production technology (Netessine and Taylor, 2007), costs, capacity and competition (Tang and Yin, 2010), reservation utility (Lacourbe, 2012), production batch (Yu, 2012), supplier competition (Altug and van Ryzin, 2013), risk-aversion (Xiao and Xu, 2014), green consumerism (Gu et al., 2015), etc. Studies present in the existing literature typically assess implications of a variety of costs in addition to managerial issues in product line design. For instance, Mussa and Rosen (1978), Moorthy (1984) and Katz (1984) focus on cannibalization issues while ignoring costs a firm would incur in carrying a wider product line. However, Dobson and Kalish (1988, 1993) and Krishnan et al. (1999) largely address these limitations by incorporating the product-level fixed and variable costs and by modeling the resource-sharing benefits of a firm's product line. Raman and Chhajed (1995) assume that the production costs are determined by product attributes and include costs related to common attributes across the products. Moorthy and Png (1992) consider quality based costs, but do not consider economies of scale or scope among the products. Kim and Chhajed (2000) study the effect of economies of scale in producing modular components, but do not consider any costs of modular components. Krishnan and Gupta (2001) explicitly model the costs of designing and producing common component. Desai et al. (2001) and Heese and Swaminathan (2006) analyze the impact of investments in design effort on manufacturing cost reduction. Our problem context is similar to that of the platform product based product line design problem (see, Krishnan et al., 1999; Krishnan and Ulrich, 2001; Ramdas, 2003). Mikkola and Gassmann (2003) in particular survey the literature on product modularity and component sharing. The platform product based product line design approach typifies a deliberate business process adopted in order to reduce product design and production costs. In this regard, the existing literature particularly focuses on exploring implications of product commonality for product line design. For instance, Desai et al. (2001) show that when the high-quality component is made common, the average quality of the products in the firm's product line increases. Heese and Swaminathan (2006) explicitly consider potential interdependencies between cost-reduction effort and quality decisions and show that it can be preferable to make those components common that, relative to their production cost, are attributed a higher importance by consumers. Kim and Chhajed (2000) and Krishnan and Gupta (2001) address product line design issues in the presence of both platform based product line and economies of scale, and determine situations that are appropriate for developing product lines using platform products. Adopting similar modeling techniques, Raghunathan (2000) and Chen and Seshadri (2007) provide insights into the development and versioning problem involving information and software goods. Zhang and Huang (2010) analyze a platform product based product line design problem using game theoretics solution concepts in a supply chain setting.

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

While there are similarities between our approach and the studies mentioned above, our focus and the problem context are different. First, as mentioned earlier, the product line design problem considering variety creation using add-on services has been rarely studied in the existing literature. Contrary to a product line in a manufacturing setting offering a variety of the product with distinct quality levels that also reflect distinct functional outcomes, add-on services create product variety with distinct quality levels but identical functional outcome of the products. Second, in view of the examples mentioned above, the problem of product line design is particularly important with respect to services as the firm's cost structure is variety specific that clearly has implications for its profitability but not for functional output of the product. Third, the existing studies on platform product based product line design typically assume that product quality is the (weighted) sum total of the quality of its components – core product and/or add-on components (see, e.g., Krishnan and Gupta, 2001; Desai et al., 2001). Moreover, they focus on examining the impact of the component quality on product line design, and likewise, ignore the impact of the core product on quality of add-on components. Our model presented in this paper particularly discusses implications of the core product on design and positioning of add-on services that create variety of the product. Fourth, the existing studies typically consider discrete consumer segments in the market, and assume that consumer valuations and the firm's cost parameters are such that supplying a variety of the product to an entire segment may be optimal. Likewise, they assume that the number of products in the product line and the supply quantity of each variety that is reflected in the size of the consumer segment both are given exogenously. Our model particularly obtains the optimal number of variants and size of each segment by optimally designing and positioning the products to maximize the firm's profit. In the existing studies considering sequential product introduction scenarios, the focus is on variety creation in advance while determining optimal timing and product introduction strategies. However, in view of our motivating examples, we consider both variety creation and product introduction as a sequential decision relative to the already existing variety in the market. Insights into such scenarios are clearly missing from the existing literature.

4. Model building In this section, we develop a model for the firm's product line design and positioning problem, and analyze it for two different scenarios. As described earlier, the firm's product line consists of a core product that is common to all variants that are distinguished by add-on services. We use a single generic quality parameter to model the product variants. Conceptually, the quality parameter may be seen as a composite of the several attributes that contribute to the additional features/service. It is modeled as a positive real number with higher value denoting superior product. As done by Shaked and Sutton (1982) and Hauser (1988), we can also assume that there exists an upper bound on the quality parameter, either due to technological limitations and/or market environment. However, such an upper bound is not essential for our model and the results would be valid even if such a bound is not present. Accordingly, it may be noted that our approach is similar to Moorthy (1984) and Desai et al. (2001). We model the total cost as an additive function with three components – a fixed cost for each variant, unit production cost of the core product, and the cost of quality for add-on services. Thus the total cost of providing x units of product with quality q is expressed as F þβx þ αq2 x. Here, F represents the fixed cost of introducing and managing the product variant of quality q. β denotes the unit variable cost of the core product and αq2 is the unit variable cost associated with the add-on quality component. (One may mildly interpret β as a reflection of quality level of the core product; higher the value of β is, higher is the quality of the

19

core product.) The additive nature of the cost function follows directly from the problem context where the costs associated with the core product and the quality features are quite distinct and clearly separable. We assume that both α and β are constants and positive. In the remainder of the paper, we refer to parameters α and β as the cost of quality and the core product cost, respectively. The assumption of constant unit cost for the core product may be appropriate for a mature product and given technology. However, this assumption may be relaxed for a new/innovative product that exhibits significant learning effects and/or scale economies. Anecdotal evidences from hospital administrators suggest that cost increases non-linearly with service quality level (see, Shah and Murthy, 2004). we model the marginal cost of the quality
Hence,  component αq2 as a quadratic function of the quality parameter. It may be noted that this approach is similar to studies in the marketing/economics literature (e.g., Mussa and Rosen, 1978; Moorthy, 1984, 1988; Kim and Chhajed, 2000, etc.) and the operations management literature (e.g., Desai et al., 2001; Heese and Swaminathan, 2006). On the demand side, we assume that there are M consumers in the market that are quality and price conscious and heterogeneous in terms of utility derived from a product. (In this paper, utility is also referred to as valuation of the product.) We model the gross utility of a consumer from a particular product of quality q as linear in quality, i.e, U ðu; qÞ ¼ uq. Here, the consumer heterogeneity is modeled through parameter u by assuming that u is uniformly distributed between a and b. Thus in our model, while consumers are heterogeneous and have individual preferences, they are identical in distribution. This modeling approach is similar to Mussa and Rosen (1978), Moorthy (1988), Mitra and Webster (2008), Lacourbe (2012), etc. It may be noted that our model is quite general and various other distributions of u may be used to capture market heterogeneity. For example, any non-uniform distribution of u – such as gamma – may be more realistic when the number of consumers at both ends of the pyramid are uneven. For instance, one would expect the number of consumers at the lower end to be significantly higher than that at the higher end. Nevertheless, uniform distribution provides analytical simplicity while capturing the fact that consumer valuations are not identical and chances of any two consumers being identical are zero. The net utility derived by a consumer is given by NUðu; q; pÞ ¼ uq  p, where p is the price of the product. We normalize the utility function by assuming that the net utility of any consumer is zero if the consumer does not purchase any of the products from the firm's product line. Accordingly, a consumer will buy a product only if the net utility from the product in greater than zero. We also assume that a consumer buys not more than one unit of the product of its choice. In order to focus on the firm's decision problem, this assumption is typical in the existing literature (see, Moorthy, 1984; Desai et al., 2001). Further, we assume that the variants of the product are substitutable, and in case of multiple variants with net utility greater than zero, the consumer would purchase the product which results in the highest net utility. Clearly, the firm's demand function is determined by the consumers' product selection as described below. A consumer with utility u from each quality unit observes the firm's product line specified as (i) the number of products, n, (ii) quality level, qi, and (iii) price, pi, i ¼ 1; 2; …; n. (In this paper, we use the subscripts such as i for product variant i. The subscripts are dropped and/or modified wherever they are evident from the context.) Thus when the firm offers exactly one product, the consumer purchases one unit of the product iff u 4 pq, and hence, the firm's total demand is  
 . Similarly, when the firm offers more than one product b pq b M a in its product line, then the consumer prefers to buy one unit of higher quality product i over lower product j, for i a j, iff  

 p p NU u; qi ; pi 4 NU u; qj ; pj , i.e., u 4 qi  qj , and NU u; qi ; pi 40, i.e., i

j

20

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

u 4 pqi . Here, the first condition is known as incentive compatibility, and i

the latter represents individual rationality of the consumer. Henceforth we assume, without loss of generality, that q1 r q2 r q3 r … r qn . It may be noted that product i is active such that it generate positive sales p p pi  1 . Otherwise, a consumer always prefers for the firm if qi þ 1  qi 4 pqi  q iþ1

i

i1

i

product ði þ 1Þ over product i. The demand for product i when it is   p p pi  1
M  . active is given by qi þ 1  qi  pqi  q ba iþ1

i

i

i1

We assume that the firm has complete and perfect information about the consumers' utility function, and as mentioned earlier, its objective is to maximize the total profit. In this case, the firm's ~ is given by product line design problem, Pn, ( ~ : π ¼ max Pn n;q ;p ;

i i i ¼ 1;2;…;n

   n 

 X  pi þ 1  pi pi  pi  1 M π n; qi ; pi ¼ pi  αq2i  β   Fi ba qi þ 1  qi qi  qi  1 i¼1

)

ð1Þ qi r qi þ 1 ;

s:t: ar

i ¼ 1; 2; …; n  1

ð2Þ

p1  p0 p2  p1 p  pn  1 r r⋯r n rb q1  q0 q2  q1 qn  qn  1

qi ; pi Z 0;

ð3Þ

i ¼ 1; 2; …; n

ð4Þ

where p0 ¼ 0 ¼ q0 , pn þ 1 ¼ b þ pn and qn þ 1 ¼ 1 þ qn . Constraint (2) essentially reflects the assumption that all n products are active in the market. Note that the fixed costs and the number of consumers in the market do not influence the firm's product quality and pricing decisions. However, both these factors are important in deciding whether to introduce a particular product in the market or not. For example, if the fixed cost for the particular product is relatively high and/or the number of consumers in the market is relatively small, then the firm's optimal strategy is to not introduce a product in the market. Therefore, for the analysis purpose, we ignore M and Fi, 8 i. For further analytical simplicity, we transform Problem ~ using the following substitution: Pn yi ¼

pi  pi  1 ; qi  qi  1

i ¼ 1; 2; …; n

ð5Þ

ð8Þ

qi Z0;

ð9Þ

i ¼ 1; 2; …; n

The firm's product line design problem may be formulated with alternate objective functions. For example, if the firm's objective is to maximize the total market share, then in Problem Pn above, (6) is to be redefined as follows:






  Pn : π ¼ max π n; qi ; yi ¼ b  y1 ¼ min π n; qi ; yi ¼ y1 n;q ;y ; n;q ;y i i i ¼ 1;2;…;n

i i i ¼ 1;2;…;n

However, in our model we use the firm's objective as shown in (6). For expositional clarity, we also use the following notations whenever it is evident from the context: β βα ¼ ; α

b bα ¼ ; α

and

aα ¼

a α

8 9 0 1 <
n i   X  X
= @ π n; qi ; yi ¼ yj qj  qj  1  αq2i  βA yi þ 1  yi : ; i i i ¼ 1;2;…;n

Pn : π ¼ max n;q ;y ;

i¼1

qi r qi þ 1 ;

5. Model analysis: single-product problem First, we analyze the firm's problem by characterizing the optimal solution for the single-product scenario that can be used as a building block in developing insights into two- and n-product problems. We particularly highlight the interplay between the core product cost and the cost of quality, and bring out its impact on the optimal product design and positioning. The firm's single-product problem, P1, is obtained by substituting n ¼1 in Problem Pn. (For notational simplicity, we drop the subscript 1 reflecting product index in the product line.)


  P1 : π ¼ max π ðq; yÞ ¼ yq  αq2  β ðb  yÞ s:t:

a r y r b;

qZ0

ð7Þ

qn ¼

yn 2α

Parameter

Description

i n qi pi u

Index (subscript) for product i in the firm's product line Number of products in the product line Quality of product i Price of product i

Utility derived by a consumer from unit quality level of any product in the product line; u  U a; b Fixed cost of producing product i Parameter reflecting the cost of quality for the firm Marginal cost of producing a unit of product of quality qi

yi yij π

ð12Þ

Proposition 1. The optimal solution to the firm's single-product problem is given by

Table 1 Notation.

Fi α αq2i β Xα

ð11Þ

ð6Þ

j¼1

i ¼ 1; 2; …n  1

ð10Þ

Note that βα represents the magnitude of the core product cost relative to the cost of quality. Similarly, bα represents the magnitude of consumer valuation of a product relative to the cost of quality. When βα (bα ) is higher/lower, we say that the core product cost (consumer valuation of a product) is higher/lower relative to the cost of quality. The complete notation used in this paper is summarized in Table 1.

q;y

(By definition, yn þ 1 ¼ b.) Here, yi can be interpreted as the incremental price per unit of the incremental quality of Product i over Product ði  1Þ. In this case, the transformed problem, Pn, is given by

s:t:

a r y1 r y2 r ⋯ r yn r b

Constant marginal cost of the core product

 X α ¼ X=α; X A a; b; β pi  pi  1 yi ¼ ; i ¼ 1; 2; …; n qi  qi  1 pi  pj yij ¼ ; iaj qi  qj Firm's profit

ð13Þ

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 99 8 8 <
ð14Þ

Corollary 1 provides a complete characterization of the optimal solution to the firm's single-product problem. Corollary 1. Define βmin ¼ α

aα ð3aα  2bα Þ ; 4

2

and

βmax ¼ α

bα 4

ð15Þ

The optimal solution to the firm's single-product problem is given by 8 aα > if βα rβmin > α > > 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > < 2 bα þ bα þ 12βα ð16Þ qn ¼ if βmin o βα o βmax α α > > 6 > > > > b > : α if βα Zβmax α 2 8 > a2 > > > > 2α > > > < bqn þαðqn Þ2 þ β pn ¼ > 2 > > > > > b2 > > : 2α

if βα r βmin α if βmin o βα o βmax α α

ð17Þ

if βα Z βmax α

In addition to characterizing the optimal solution to the singleproduct problem of the firm, Corollary 1 also brings out the impact of the firm's cost parameters on the volume of production. For instance, when the core product cost is below a threshold, i.e., βα r βmin α , the optimal strategy for the firm is to cover the entire market. On the contrary, when the core product cost is beyond a threshold, i.e., βα Z βmax , the optimal strategy is to not produce the α product at all. When βα r βmin α , both optimal price and quality of the product decrease in the cost of quality. However, it is interesting to note that the product quality and price are independent of the core product cost which can be explained as follows: given that the optimal strategy for the firm is to cover the entire market, the variable cost corresponding to the production of the core product is constant, and hence, it does not influence both price of the product and the consumer demand function. Thereby, β has no impact on product design and positioning. In this case, on the other hand, product quality level and price are influenced by the minimum utility of the product for the consumers, a. We note that the threshold βmin decreases in α implying that the range of β over α which it is optimal to cover the entire market decreases with the cost of quality. When βα Z βmax , the core product cost is too large for the firm α to recover the variable production cost, no matter what the choices of the product quality and price are. (Further considering fixed costs, the firm cannot recover the total cost of production.) Hence, it is not optimal for the firm to produce any product. Note that βmax is decreasing in α implying that the range for the core α product cost over which product is produced decreases in the cost of quality. (In keeping with our results, while analyzing the firm's problem in the following Sections, we assume that βα o βmax , or α 2 equivalently β o  b =4α.)  max , the optimal solution to the firm's When βα A βmin α ; βα single-product problem is in the interior implying that the firm covers the market partially. In this case, pn and qn are independent of a. We note that the optimal product quality increases in both bα and βα . The result implies that the product quality increases as the

21

core product cost and/or the consumer valuation of the product increases relative to the cost of quality. With the increasing βα , the input costs for the product increase by which the optimal price of the product also increases. The high valuation consumers would only purchase the pricey product. In this case, the firm can enhance the quality of the product while increasing these consumers' utility, and thereby, increasing its own profit. However, it is not surprising to note that the product quality decreases in the cost of quality, α. Consistent with the intuition, we note that price increases in the product quality which also implies that price increases with increase in the core product cost and the consumer valuation of the product relative to the cost of quality. However, it decreases with the cost of quality as the product quality also falls. The market share of the product, which is equal to ðb  yn Þ, deceases in α, βα and bα. It is consistent with the intuition that the market share of the product decreases in the firm's cost parameters. However, it is surprising to note that the market share decreases with relative increase in consumer valuation of the product. The result implies that the impact of relative increase in product valuation (for the consumers) on the price of the product is larger than that on the product quality. h i n It can be easily shown that pn ¼ bq þ αðqn Þ2 þ β = 2 and n

pn  αðqn Þ2  β ¼ bq  pn ¼ αðqn Þ2  β. It implies that the optimal price of the product is equal to the arithmetic average of the marginal production cost and the maximum net utility derived from the product. The optimal margin on the product is equal to the maximum net utility derived from the product. Moreover, this margin is equal to the difference between the quality based marginal cost and the core product cost. Consistent with the intuition, we observe that the product margin decreases with increasing cost parameters and increases with consumer valuation of the product.
n  We can also show that αðqn Þ2 ¼ pn =2 ¼ bq þ β = 3 which decreases in α and increases in both bα and βα . This implies that the quality based marginal cost of the product is decreasing in the cost of quality and it increases in the relative increase in the core product cost and the consumer valuation of the product. The result indicates that the product quality decreases at a rate much higher than the rate at which quality based marginal cost increases in α. However, it is not surprising to note that αðqn Þ2 increases in both bα and βα as the product quality also increases. It is also evident that the marginal cost for the product, β þ αðqn Þ2 , is decreasing in α and increasing in both bα and βα . 6. Simultaneous product design We begin this section by examining the two-product design problem of the firm that captures issues related to product cannibalization (see Moorthy, 1984). Later we generalize our structural results in the single- and two-product design scenarios to derive the optimal solution to the n-product problem that can be used to determine the optimal size of the product line. 6.1. Two-product problem When the firm's product line consists of multiple products, i.e., n Z 2, in addition to the firm's cost and demand functions, product cannibalization influences quality and pricing decisions (see, e.g., Frank et al., 1972; Mussa and Rosen, 1978; Moorthy, 1984). In this section, we develop key insights on product cannibalization in our problem context by analyzing the model for two products, Problem P2, that is obtained by substituting n ¼2 in Problem Pn. 





  P2 : π ¼ max π qi ; yi ¼ y1 q1 þ y2 q2 q1 αq22  β b  y2 q ;y ; i i i ¼ 1;2

22

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33



 þ y1 q1  αq21  β y2  y1 s:t:

q1 r q2 ;

a ry1 r y2 r b;

ð18Þ

q1 ; q2 Z 0

ð19Þ

Proposition 2. The optimal solution to the firm's two-product problem is given by yn q2 ¼ 2 ; 2α n

yn þyn2 b q1 ¼ 1 2α n

ð20Þ

that the market share of Product 1, which is equal to
n 
 y2  yn1 ¼ b  yn1 =2, decreases in the firm's cost parameters and the consumer valuation of the product. However, we do not find such monotonic trend in the optimal margins on the two products. 6.2. n-Product problem

A complete characterization of the optimal solution is given in Corollary 2 below.

In this section, we generalize our model for the two-product scenario and obtain the structural results that exhibit relative positioning of the products in the product line when it consists of multiple (more than two) products. This generalization also helps the firm in determining the optimal size of the product line. The optimal number of products in the product line is determined by the products that contribute negatively to the firm's profit function when the products are ranked in the increasing order of their quality levels.

Corollary 2. If βα r ð5aα  3bα Þð3aα  bα Þ=16, then the optimal solution to the firm's two-product problem is given by

Claim 1. For given y1 such that a r y1 r b, the optimal solution for the firm's n-product problem is given by

yn2 ¼

b þ yn1 ; 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 99 8 8 < <7b þ 2 b2 þ 60αβ == yn1 ¼ max a; min ;b : : ;; 15

qn2 ¼

aþb ; 4α

pn2 ¼

4a2  ab b ; 4α

qn1 ¼

3a  b 4α

2

pn1 ¼

3a2 ab 4α

ð21Þ

ð22Þ

qni ¼

yni þyni þ 1 b ; 2α

ð23Þ

yni ¼

ði 1Þb þ ðn  i þ 1Þyn1 ; n

2

If ð5aα  3bα Þð3aα  bα Þ=16 o βα o bα =4, then the optimal solution to the firm's two-product problem is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bα þ bα þ 60βα qn1 þ bα n n ð24Þ ; q1 ¼ q2 ¼ 10 3
2 n bq þ α qni þ β ; pni ¼ i 2

i ¼ 1; 2

ð25Þ

(From Corollary 1, recall that the firm does not produce any 2 product if βα Z bα =4.) In addition to characterizing the optimal solution to the firm's problem, Corollary 2 also reflects on the firm's strategy for market coverage. As in the single-product problem, we observe that there exists a bound on the core product cost relative to the cost of quality, βα , below which the optimal strategy for the firm is to cover the market entirely. In this case, the core product cost does not influence quality and prices of the two products, and the product characteristics depend on a, the minimum utility derived by a consumer from a unit quality product. On the other hand, when the core product cost relative to the cost of quality, βα , is beyond the threshold, the firm partially covers the market. In this case, we observe that the quality levels of both Product 1 and Product 2 decrease in α and increase in both bα and βα . The finding is identical to that in the firm's single-product problem. It can
 be shown that qn2  qn1 ¼ b yn1 =2α which is decreasing in both α and βα , and it is increasing in bα. The result implies that the quality based distinction between the two products decreases with the firm's cost parameters and it increases with the consumer valuation of the products. In other words, the products are increasingly identical with the increasing cost parameters, and they are increasingly distinguishable with the increasing product valuation for the consumers. As in the previous case, we observe that the optimal price of each product is equal to the arithmetic average of the marginal production cost and the maximum net utility derived from that product, and hence, the margin on each product is equal to the maximum net utility derived from the product. It is not surprising to note that the price of Product 2 is more than the price of Product 1. The same can be easily shown for the margins on the products as we obtain h i h i

2
2 
 pn2  α qn2  β  pn1  α qn1  β ¼ qn2  qn1 b yn1 =2. It may be noted that the market share of Product 2, which is
 equal to b  yn2 , decreases in α, βα , and bα. Similarly, we observe

i ¼ 1; 2; 3; …; n

ð26Þ

i ¼ 2; 3; …; n

ð27Þ

Further, when y1 a a, the optimal quality levels and prices of the products in the product line are given by qnn ¼

qn1 ¼

qn1 þ ðn  1Þbα ; ð2n  1Þ

bα þ

qnk ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 ffi 2 bα þ 4 4n2  1 βα 2ð2n þ 1Þ

;

qnk  1 þ qnk þ 1 ; 2 pnk ¼

k ¼ 2; 3; …; ðn  1Þ


2 n bqk þ α qnk þ β ; 2

k ¼ 1; 2; 3; …; n

ð28Þ

ð29Þ

In parallel with our results for the two-product design scenario, we develop structural results for n ¼ 3; 4; 5; …. For brevity of the paper, without reporting these results in detail, we conjecture that the optimal y1 is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 8
99 
 < < 2n2  1 b þ n b2 þ 4 4n2  1 αβ == n
 y1 ¼ max a; min ;b ð30Þ : : ;; 4n2  1 The insights obtained in this scenario are identical to those in the two-product design scenario. We also note that for n Z 3 the optimal quality level of any product is equal to the arithmetic average of the quality levels of the products on the either sides in the product line rank order. The optimal profit for the firm (as a function of n) is given by n
h
 i o

2  2 n2  1 b yn1 þ 3n2 yn1  4αβ b  yn1 π ðnÞ ¼ ð31Þ 12n2 α where yn1 is as given in (30). Due to analytical complexity, we obtain further insights into determining the optimal number of products in the product line using numerical examples illustrated in Fig. 1. We observe that, in the absence of fixed costs, the firm's profit is monotonically increasing in n, the number of products in the product line. The firm increases its profitability by offering as many variants of the product as possible—ideally one variety for each consumer. However, when fixed costs are associated with design and production of each variety, the number of variants offered depends on the marginal contribution of each variety in the firm's profit. From the figure we note that the marginal profit for the firm by adding one more product in the product line is lower if the product line is bigger vis-à-vis if it is smaller. We also observe that the marginal increase in the profit further decreases

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

1

Market Share

15

π(n)

14 13 12 11 10

23

1

2

3

4

5

6

0.5

0

7

1

2

3

n

4

5

6

7

n β=0.3

β=0.7

Fig. 1. Impact of number of products. Note: The parameter values chosen are a¼ 0, b ¼7, and α ¼ 1, βα A f0:3; 0:7g.

in the core product cost. The same is observed for the firm's market share. It is usual to expect that the fixed costs of expanding the product line increase with the size of the product line. Accordingly, one may provide reasonable justification to why the firms in our motivating examples characterized by high fixed costs and high core product costs typically offer two or three variants of the product. From (30), we can also show that as n-1, pffiffiffiffiffiffi b þ 2 αβ yn1 implying that the firm does not capture more than 2 half the market when the consumer heterogeneity is widespread, i.e., bZ 2a in particular. Recall that, in this section, we analyzed the firm's product line design problem when the number of products in the product line and their characteristics are determined simultaneously. In Section 7 below, we model and analyze the problem when the products are designed and positioned sequentially.

7. Sequential product design As observed in our motivating examples, firms do not always plan in advance the number of variants of the product to offer in the market. The success of the variants introduced initially leads to the firm offering more variants of the product subsequently. In this case, changing price and/or quality of the already existing variety may or may not be possible. Design and positioning of a new variant relative to the already existing variety is important for the firm from the profitability view point. In this section, we address issues related to the firm's product line design and positioning problem when a variety of the already existing product is designed and introduced in the market sequentially. The existing studies, e.g., Moorthy and Png (1992) and Bhattacharya et al. (2003), also address similar issues; however, as mentioned earlier, their focus is on creating product variety while determining optimal timing and product introduction strategy from the point of view of sequence of introducing product variety in the market. For instance, Moorthy and Png (1992) state that the high-quality product should be introduced first followed by the low-quality product. On the contrary, Bhattacharya et al. (2003) show that the low-quality product should be introduced first if the firm gains from technological evolution. Mallik and Chhajed (2006) establish the same result in the presence of learning effect by the market and the firm. In this section, we consider that the firm has a base product in the market (say Product 1) and examine the viability of adding the second product to enhance the firm's profit. The firm's decision involves design and positioning of Product 2 relative to Product 1. In particular, the questions that we address are: (i) When is it advisable for the firm to introduce the second product so that both products are active in market? and (ii) What are the optimal quality and price levels of Product 2 relative to those of Product 1? For many practical situations, changing quality levels of the existing products is a long term and strategic decision. On the contrary, price changes in a competitive scenario is commonplace.

Table 2 Product positioning and market share. Case

Quality level

Positioning

Segment size Product 1

I

q2 Z q1

II

q2 o q1

III

q2 Z q1

IV

q2 Z q1

V

q2 o q1

VI

q2 o q1

p1 p  p1 o 2 rb q1 q2  q1 p2 p1  p2 o rb q2 q1  q2 p2 p1 r q2 q1 p  p1 bo 2 q2  q1 p  p2 bo 1 q1  q2 p1  p2 p r 1 q1  q2 q1

p2  p1 p1  q2  q1 q1 p1  p2 b q1  q2 0 b

p1 q1

0 b

Product 2 p2  p1 q2  q1 p1  p2 p2  q1  q2 q2 p2 b q2 0 b

b p1 q1

p2 q2

0

Accordingly, we analyze two scenarios for the sequential product design problem in which: (i) both quality and price of Product 1 are fixed, and (ii) only quality of Product 1 is fixed. We assume that Product 1 is currently viable
in the market, i.e., p1 =q1 o b or y1 ob. 2 Moreover, p1 4pmin ¼ β þ α q1 . This follows from our results 1 presented in Section 6. Depending on the positioning of Product 2 relative to Product 1, six different cases that arise reflecting on active products in the market are shown in Table 2. The cases III–VI result in only one product being active in the market; the optimal solution in each of these cases corresponds to that in the single product design scenario as described in Proposition 1. Hence, in this section we analyze in detail our model for Cases I and II only that result in both Product 1 and Product 2 being active in the market. In order to maintain consistency with the model developed in Section 4, in this section we modify the notation yi as follows: yij ¼

pi  pj ; qi  qj

i a j and i; j ¼ 0; 1; 2

ð32Þ

From Table 2 and (32), we derive the firm's profit functions for Case I: q2 Z q1 and Case II: q2 o q1 . 8
 
> y q  αq22  β b  y20 > > 20 2
 


> > < y10 q1 þ y21 q2  q1  αq22  β b  y21


 π I q2 ; y21 ¼ > þ y10 q1  αq21  β y21  y10 > >  
>
> : y10  αq2  β b  y10 1

if y21 r y10 if y10 o y21 rb

ð33Þ

if b oy21

7.1. Quality and price fixed In this section, we analyze our model for the scenario in which both quality and price 1 are fixed, and the firm's
of Product  decision variables are q2 ; p2 . We characterize the optimal solution to the firm's product line design problem for the two separate cases: Case I: q2 Z q1 and Case II: q2 o q1 . Clearly, the optimal

24

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

8

 y q  αq22  β b  y20 > > > 20 2
 >  > >


> < y10 q1  y12 q1  q2  αq2 β y12  y10 q1  y12 q1  q2
 2 q2 π II q2 ; y12 ¼

 > > 2 > þ y q αq  β b  y > 10 1 12 1 > >

 > : y  αq2  β b  y 10

1

10

solution the firm's problem is the one that maximizes the profit in either of the two cases. 7.1.1. Case I: q2 Z q1 In this case, the firm's problem, PI, is given by


 P I : π In ¼ max π I q2 ; y21 q2 ;y21


 




¼ y10 q1 þ y21 q2  q1 αq22 β b  y21 þ y10 q1  αq21  β y21 y10

ð35Þ s:t:

q1 r q2 ;

y10 r y21 rb;

q2 ; y21 Z0

ð36Þ

Proposition 3. The optimal solution to Problem PI is given by 8
  > < max y ; 2 αq1 þb if 2αq1 r b 10 n 3 ð37Þ y21 ¼ > :ϕ otherwise 8 n < y21 q2 ¼ 2α : ϕ n

if 2αq1 r b

ð38Þ

otherwise

where ϕ signifies that the firm's problem is infeasible implying that it is not viable for the firm to introduce Product 2 with quality higher than that of Product 1, and both products are active in the market. From Proposition 3 we observe that the firm's decision to introduce the second product with quality higher than that of the first product is influenced by the quality level and price of the first product. It may also be noted that if q1 4 bα =2, then the firm does not introduce a higher quality product in the market. This result is consistent with an intuition that when the existing product is of high quality, it is not optimal for the firm to introduce a variant with higher quality so that both products are active in the market. 7.1.2. Case II: q2 o q1 In this case, the firm's problem, PII, is given by


 P II : π IIn ¼ max π II q2 ; y12 q2 ;y12


 


 y q  y12 q1 q2 ¼ y10 q1  y12 q1  q2  αq22  β y12  10 1 q2

 2 ð39Þ þ y10 q1 αq1  β b  y12

s:t:

q2 r q1 ;

y20 r y12 rb;

q2 ; y12 Z0

ð40Þ

Proposition 4. The optimal solution to Problem PII is given by 8
 n > < 2y10 þ αq2 q1  β if β r αq2 1 2q1 ð41Þ yn12 ¼ > :ϕ otherwise

qn2 ¼

8 > > > <

ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9
2

> > > :ϕ

if β r αq21 otherwise ð42Þ

if b ry12 ð34Þ if y10 oy12 r b if y12 ry10

where ϕ signifies that the firm's problem is infeasible implying that it is not viable for the firm to introduce Product 2 with quality lower than that of Product 1, and both products are active in the market. As in the previous case, we observe from Proposition 4 that the firm's decision to introduce the second product with quality lower than the first product is influenced by the quality level and price of the first product. In this case, if the core product cost is higher than the quality based marginal cost for Product 1, i.e., β 4 αq21 , then the firm does not introduce a lower quality product in the market. The result impliesp that ffiffiffiffiffi there exists a lower bound on the Product 1 quality level, βα , below which it is not optimal for the firm to introduce a variant with quality lower than that of the already existing variant so that both products are active in the market. As mentioned earlier, the optimal solution to the firm's sequential product design and positioning problem is obtained using Propositions 3 and 4. Recall that the results presented in these two propositions are for the cases q2 Z q1 and q2 o q1 wherein both products are active. Therefore, in order to determine the firm's optimal solution reflecting design and positioning of Product 2 relative to Product 1, we compare the firm's profit in these two cases using (33) and (34). We also compare the firm's profits in the single product design scenario as described in Proposition 1, and determine situations in which the optimal strategy for the firm is to have only one product active in the market. Likewise, we consider π In ; π IIn ; π n . The optimal strategy for the firm is such that qn2 4 q1 and both products are active in the market if π In 4 π IIn and π In 4 π n . Similarly, the optimal strategy is such that qn2 o q1 if π IIn 4 π In and π IIn 4 π n . On the other hand, the optimal strategy is to have only one product active in the market if π n 4 π In and π n 4 π IIn . In this case, product design and positioning is as described in Proposition 1, and this case corresponds to the Cases III–VI described in Table 2. We observe that comparing the firm's profit functions to obtain generalizable results analytically is quite difficult, and hence, we illustrate the scope of our approach to derive managerial insights through numerical examples as described in Fig. 2. We normalize α ¼ 1 and select b¼4. We consider q0 ¼1.7208 as a reference quality level that corresponds to the optimal solution to the firm's problem, i.e., q0 ¼ qn , when βα ¼ 2 in the single product design scenario. We select four Product 1 quality levels such that q1 ¼ xq0 , x A f0:7; 0:8; 1:1; 1:2g. We present our results in four different panels – one panel for each value of q1 chosen – in Fig. 2. In each panel, we

2 ¼ b =4α ¼ 4 in our numerical where βmax also select βα A 0; βmax α α example. We incorporate the optimal solutions described in Propositions 1, 3 and 4 in the firm's profit function in the respective scenarios, and develop each of the four panels. In particular, we determine and plot in the figure p1, for given q1 and βα , that



  maximizes the firm's profit, i.e., max π In qn2 ; pn2 ; π IIn qn2 ; pn2 ; π n . The regions separated by the curves shown in each panel provide ranges for the parameters q1 ; p1 and βα exhibiting a particular optimal strategy for the firm. We observe that over the region R1, the optimal strategy for the firm is such that qn2 o q1 and both Product 1 and Product 2 are active in the market. Similarly, the optimal strategy over the region R2 is such that qn2 4 q1 and both products are active. On the other hand, over the region R3, the firm maintains only one product active in the market wherein design and

4

R3

3

R2

2 pmin 1

1 0

0

1

1

2

R2

R1 2

q =0.7 q

pmin 1

0

0

3

0

1

βα

R1 R3

0

R2

min

p1

2 0

1

q1=1.1 q0 2

3

R3

6

3

p1

p1

8

4

2

q1=0.8 q0

βα

R3

6

25

R3

R3

4

p1

p1

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

R1

4 pmin 1

2 0

0

βα

1

2

q1=1.2 q0 3

βα

 , and q0 ¼ 1.7208. Fig. 2. Optimal strategy: quality and price of Product 1 are fixed. Note: The parameter values chosen are a ¼0, b¼ 4, α ¼ 1, βα A 0; βmax α

positioning of the active product is as described in Proposition 1. (These details are summarized in Table 3.) The curve separating the regions R2 and R3 corresponds to values of p1 and βα , for given q1,

 such that π In qn2 ; pn2 ¼ π n 4 π IIn qn2 ; pn2 . Similarly, the curve separating the regions R1 and R3 corresponds to values of p1 and βα such

 that π IIn qn2 ; pn2 ¼ π n 4 π In qn2 ; pn2 . The curve separating the regions R1 and R2 corresponds to values of p1 and βα such that

 π In qn2 ; pn2 ¼ π IIn qn2 ; pn2 4 π n . Recall that given q1 and βα when p1 o pmin 1 , Product 1 is not viable, and hence, the region below the curve designated as pmin is ignored. 1 From Fig. 2 and Table 3, we note that, given q1 and βα , both products are active when price of the existing product is relatively low. Similarly, given p1 and βα , when quality of the existing product is relatively low (high), the firm introduces a higher (lower) quality product. We also observe that, given p1 and q1, when the core product cost is low (high) relative to the cost of quality, the firm introduces a lower (higher) quality product; this result is consistent with our findings presented in Section 6. Likewise, we note that, given p1 ; q1 and βα , both products are active only under certain situations when price and quality of the first product are fixed. 7.2. Quality fixed In this section, we analyze the firm's problem for the situation in which the quality level of Product 1 is fixed, however, its price is optimally chosen while
introducing  Product 2. In this case, the firm's decision variables are q2 ; p2 ; p1 . It may be noted that the firm's problem in this scenario is a mathematical relaxation of the firm's problem for the scenario in which both price and quality of Product 1 are fixed. Therefore, in Proposition 5 below we present the results corresponding to the optimal decision on the price of Product 1 only. Proposition 5. Given the quality level of Product 1, its optimal price is such that
 bq þ αq21 þ β pn1 q1 ¼ 1 2

ð43Þ

From our results in Section 6, we note that, given the quality level of Product 1, the optimal price of the product is the same in both simultaneous and sequential product design scenarios. In parallel with the analysis presented in Section 7.1, using Propositions 3, 4 and 5 we obtain the optimal solution to the firm's sequential product design and positioning problem when only quality of the existing product is fixed. We also compare the

Table 3 Optimal strategy: quality and price of Product 1 are fixed. Region in Fig. 2

Number of active products

Optimal strategy

R1 R2 R3

Two Two One

qn2 o q1 qn2 4 q1 qn

firm's profit in the single product design scenario as described in Proposition 1 to determine situations in which the optimal strategy for the firm is to have only one product active in the market. Refer Figs. 3 and 4 in this regard. (The labels R1, R2 and R3 in Fig. 3 are as described in Table 3.). In Fig. 3, we select α ¼ 1 and b ¼ 5; 7. Using different values of b, we present our results in two different panels – one panel for each value of b chosen – in Fig. 3. In each panel, we choose βα A 0; βmax ¼ 6:25 when b ¼5 and βmax ¼ 12:25 when ; βmax α α α b¼7. We determine and plot in the figure values of q1, for given



βα , that maximize the firm's profit, i.e., max π In qn2 ; pn2 ; pn1 Þ; π IIn qn2 ; n n n p2 ; p1 Þ; π g. The regions separated by the curves shown in each panel provide ranges for the parameters q1 and βα exhibiting a particular optimal strategy for the firm. We observe that over the region R1, the optimal strategy for the firm is such that qn2 o q1 and both Product 1 and Product 2 are active in the market. Similarly, the optimal strategy over the region R2 is such that qn2 4 q1 and both products are active in the market. On the other hand, over the region R3, the firm maintains only one product active in the market wherein design and positioning of the active product is as described in Proposition 1. The curve separating the regions R1 and to values of q1 and βα for which
R2 corresponds 
π In qn2 ; pn2 ; pn1 ¼ π IIn qn2 ; pn2 ; pn1 4 π n . Similarly, the curve separating the
regions R2  and R3 corresponds
 to values of q1 and βα for which π In qn2 ; pn2 ; pn1 ¼ π n 4 π IIn qn2 ; pn2 ; pn1 . The curve designated as qmin 1 corresponds to the condition p1 o pmin for given βα . Likewise, we 1 note that, given q1 and βα , both products are active only under certain situations when quality of the first product is fixed. Fig. 4 describes the implications of q1 for the optimal quality level of Product 2 and the quality based product distinction denoted by qn2  q1 , i.e., qn2  q1 when q2 Z q1 and q1  qn2 when q2 o q1 . (The parameter values chosen are: α ¼ 1, β ¼ 2, and b¼ 5.) From the left panel of the figure it is not surprising to note that the optimal q2 is non-decreasing in q1. In this panel, the flat portion of the curve corresponds to the optimal strategy of the firm of having only one product active in the market. The first rising portion of the curve corresponds to the optimal

26

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

R3 1

4

R1

2 0

R1

4 2

bα = 5

R2 0

R3

6 q

q

1

6

2

4

0

6

R2

bα = 7

0

5

β

10 β

α

α

 . Fig. 3. Optimal strategy: quality of Product 1 is fixed. Note: The parameter values chosen are a ¼0, b A f5; 7g, α ¼ 1, and βα A 0; βmax α

3

2

2

*

q*

2

q2 ~ q1

3

1 0

0

1

2

3

4

5

q

1

1 0

0

1

2

3

4

5

q

1

Fig. 4. Sequential product design and positioning: quality of Product 1 is fixed. Note: the parameter values chosen are a ¼0, b ¼ 5, α ¼ 1, and β ¼ 2.

strategy of qn2 Z q1 , and it is observed when q1 is relatively low. The second rising portion of the curve, that is observed when q1 is relatively high, corresponds to the optimal strategy of qn2 o q1 . From the second panel in the figure, we note that the quality based distinction between the two products first decreases and then it increases in q1. Here, the falling portion of the curve corresponds to qn2 Z q1 , and the rising portion is for qn2 o q1 . 7.3. Special case: sequentially optimal product design and positioning In this section, we examine a special case of sequential product design in which the initial product offering corresponds to the results in Proposition 1 and Corollary 1 for the single-product design scenario. This scenario draws special attention because in the event of not having planned its products for the future, we expect a rational firm to design and position its products optimally at each instance. In this case, the analysis parallels that presented in Sections 7.1 and 7.2. For brevity of the paper, we present and discuss only the relevant results in this section. Proposition 6. Consider the sequential product design problem in which Product 1 is positioned as given in Proposition 1 and Corollary 1. The firm can increase its profit by introducing the second product in the market. Further, the optimal solution to the firm's problem is such that q2 o q1 . Proposition 6 essentially demonstrates the optimal strategy for the firm in sequential product design and positioning problem when each product is optimally designed and positioned relative to the already existing products in the market. In this case, the firm increases its profit by increasing its market share. The result implies that the benefits of increasing market share are more than the gain from extracting more consumer surplus by introducing a higher quality product.

8. Simultaneous versus sequential product design In this Section, we compare our results for the two-product design scenarios in which optimally designed products are offered in either simultaneous or sequential design settings. Our aim is to obtain insights into the strategic incentives that the firm has in designing the products either simultaneously or sequentially. To

evaluate the firm's incentives, we compare the firm's profit in the two-product simultaneous design scenario (Proposition 2 and Corollary 2) vis-à-vis that in the sequential design scenario (Propositions 4–6). For expositional purpose, we illustrate the firm's optimal decisions and their implications in the two scenarios in Fig. 5. In this figure, we exhibit the impact of the cost parameter βα on the quality levels of both high and low quality products separately in each of the two scenarios. We also illustrate the impact of the cost parameter on the average quality level of the two products in each scenario. To calculate the average quality of the firm's products in any scenario, we determine the market share based weighted average of quality level of each of the products; i.e., the average quality level of the firm's products is P given by AQ n ¼ 2i ¼ 1 MSni qni =MSn . Here, market shares for Product 1 and Product 2 in
the simultaneous product design are 
scenario  given by MS1nsim ¼ yn21  yn10 =ðb aÞ and MSn2sim ¼ b  yn21 =ðb  aÞ, respectively. Similarly, market shares for Product 1 and Product 2 in nseq the sequential product design

nscenario  are given by MS1 ¼ nseq n n b  y12 =ðb  aÞ and MS2 ¼ y12  y20 =ðb  aÞ, respectively. The total market share in a scenario is MSn ¼ MSn1 þ MSn2 . To avoid triviality, we focus only on the situations wherein yn1 4 a. From Fig. 5 we observe that the optimal quality levels of both high- and low-quality products in the sequential product design scenario are lower than those in the simultaneous design scenario. We also note that the optimal quality of Product 1, i.e., the highquality product introduced first, in the sequential design scenario is in between the quality levels of the two products offered in the simultaneous design scenario, and the quality of Product 2 – lowquality product that follows – is lower than that of the low-quality product offered in the simultaneous design scenario. We observe that the average quality level of the firm's products in the sequential scenario is lower than that in the simultaneous scenario. It implies that the firm's market share in the former scenario would be more than that in the latter. Nevertheless, the firm's optimal profit in the sequential scenario would be lower than that in the simultaneous scenario. (Our analysis also confirms these findings.). These findings together imply that, given a choice, the profit maximizing firm would prefer to design and introduce the products in the market simultaneously rather than sequentially. When the firm determines upfront to offer two products, it maximizes its profit by extracting more consumer surplus rather than by increasing its market share. On the other hand, when it offers products

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

Average Quality (AQ)

4

q*

x

3 2 1 0

0

5

10

4 3 2 1 0

0

5

β

q*sim

1

q*seq

2

10 βα

α

q*sim

27

1

q*seq

AQ

2

*sim

AQ

*seq

 . Superscripts “sim” and “seq” Fig. 5. Individual product quality and average quality level for firm. Note: the parameter values chosen are a ¼0, b ¼7, α ¼ 1, and βα A 0; βmax α imply simultaneous and sequential product design scenarios, respectively.

sequentially, the maximizes its profit by extracting as much consumer surplus as possible from the first product; it further increases its profit by increasing the market share by offering a lower quality product. However, the downside of this strategy is a drop in both profit and average quality of the products offered by the firm. Likewise, the sequential product design strategy benefits the firm via increased market share, however, the benefits accrue at the expense of falling quality levels of its product offering. The lower average quality level in the sequential product design vis-à-vis the simultaneous design implies that the implications of the former strategy as against the latter for product quality are severe compared to those for the firm's market share. Nevertheless, from Fig. 5 we observe that the gap between positive (and similarly negative) implications of the two scenarios decreases as the core product cost increases with respect to the cost of quality. It implies that both advantages and disadvantages of adopting the strategy of sequential design over simultaneous design decrease with the core product cost.

9. Implications for strategic competitor With the emergence of a number of private and state-sponsored hospitals in India in recent years, the well-known hospitals such as Arvind Eye and Narayana Hridayalaya have been struggling in retaining their doctors and medical-support staff. As described in Shah and Murthy (2004), access to the same group of expert medical professionals is easily available for any new hospital entering into this market. Moreover, due to technological liberalization, it is not difficult for new entrants to offer products in the market with the same quality level as that offered by the incumbents. In this regard, creating segments in the market in own interests by offering add-on services has been a wellrecognized strategy of new entrants. Analogous to the sequential product design problem faced by the firm describing our motivating examples, a new competitor entering into the market also faces the problem of designing and positioning of its product line relative to the already existing products of the incumbent firm. 
 8
y20 q2  αq22 β b y20 > > >

 >
 < y10 q1  αq21 β b y10 =2
 


π cI q2 ; y21 ¼ > y10 q1 þ y21 q2  q1  αq22  β b y21 > > > :0


 π cII q2 ; y12 ¼


 8
y20 q2  αq22  β b  y20 > > >  > > > < y q  y
q  q   αq2 β y 10 1

12

1

2

2



 > > > y10 q1  αq21  β b  y10 =2 > > > : 0

The existing literature provides two approaches to model competitive settings that are fundamentally distinct. On one hand, Chen and Seshadri (2007), Lacourbe (2012) and many others consider a non-strategic competitive scenario wherein the product line design problem is addressed for a single firm with distinct reservation utilities for various consumer segments in the market. On the other hand, Moorthy (1988), Desai (2001), Matsubayashi (2007), Tang and Yin (2010) and many others model this problem typically in a duopoly setting with one product for each competing firm. Likewise, the former approach focuses on competition across consumer segments, and the latter considers competition between firms. In view of the recent developments in regard to our motivating examples, we consider a situation of strategic competition between an incumbent firm and a new entrant in the market. We develop a model for a duopoly setting and provide insights into the implications of the already existing product of an incumbent firm for the product offered by the new entrant when the variety of the products offered by both firms are created using add-on services with the same core product.
 Consider that the incumbent firm
offers  a product q1 ; p1 , and the new entrant offers a product q2 ; p2 . We first present and analyze our model to address one particular scenario of competition in the context of our problem setting wherein the incumbent
 firm offers a generic product q1 ; p1 , and
thenew entrant offers its product by optimally determining q2 ; p2 . Nevertheless, for completeness we also discuss our results for two special scenarios: one in which the incumbent's product corresponds to the optimal solution to the single-product design scenario, i.e., q1 ¼ qn and p1 ¼ pn , as presented in Proposition 1 and Corollary 1; and two, the quality level of the incumbent's product is such that q1 ¼ qn , and it strategically determines p1 in the presence of the new entrant's product. We assume that the cost parameters of the new entrant are the same as those of the incumbent firm. As described in Section 7, the new entrant's profit function depending on whether (i) Case I: q2 Z q1 , or (ii) Case II: q2 o q1 is given by

if y21 o y10 if y21 ¼ y10

ð44Þ

if y10 o y21 rb if b o y21


 y10 q1  y12 q1  q2 12  q2

if b ry12 if y10 oy12 r b if y12 ¼ y10 if y12 oy10

ð45Þ

28

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

(Here, superscript ‘c’ reflects the competitor - new entrant.) From Eqs. (44) and (45), it may be noted that profits for both incumbent and new entrant are equal when the latter's product is identical to that of the former. The new entrant's problem is given as follows: Case I q2 Z q1 : In this case, the new entrant's problem, PcI, is given by ) (




  P cI : π cIn ¼ max π cI q2 ; y21 ¼ y10 q1 þ y21 q2 q1  αq22  β b y21 q2 ;y21

ð46Þ

ð47Þ q1 r q2 ; y10 r y21 rb; q2 ; y21 Z0 Case II q2 o q1 : In this case, the new entrant's problem, PcII, is given by 



P cII : π cIIn ¼ max π cII q2 ; y12 ¼ y10 q1  y12 q1  q2  αq22  β q2 ;y12
   y q  y12 q1  q2  y12  10 1 ð48Þ q2

s:t:

s:t:

q2 r q1 ;

y20 r y12 rb;

q2 ; y12 Z0

ð49Þ

The analysis of our model in this setting parallels that presented in Section 7. For brevity, we do not present the analysis in detail; rather we present only the important results providing insights into the implications of competition for the firms' product line. We illustrate the scope of our approach through numerical examples as described in Fig. 6. (The labels R1, R2 and R3 in Fig. 6 are as described in Table 3.). From Fig. 6, we note that, given q1 and βα , the products of both firms are active when price of the incumbent's product is relatively low. Similarly, given p1 and βα , when quality of the incumbent's product is relatively low (high), the new entrant offers a higher (lower) quality product. We also observe that, given p1 and q1, when the core product cost is low (high) relative to the cost of quality, the new entrant introduces a lower (higher) quality product. Likewise, our results in sequential product design in the duopoly setting are similar to those in the monopoly setting. What follows below is a discussion on two special scenarios with respect to the incumbent's product. Proposition 7. When the incumbent firm offers its product such that q1 ¼ qn and p1 ¼ pn , the optimal strategy for the new entrant is to offer a product that is identical to the incumbent firm's product, i.e., qn2 ¼ qn and pn2 ¼ pn . It may be noted in this case that the new entrant acts strategically and the incumbent firm is non-strategic as it does not respond to the new entrant's product offering by altering either quality or price of its product. Thereby, in the presence of a non-strategic incumbent firm offering the optimal product (as described in Proposition 1 and Corollary 1), the optimal strategy for the new entrant is to offer the identical product. By offering a higher quality product, loss due to lower market share surpasses the gain from extracting more consumer surplus. Similarly, by offering a lower quality product, loss due to lower consumer surplus surpasses the gain from more market share. By the entry of a competitor in the market, the average quality of the products

This result in the duopoly setting is analogous to that in the monopoly setting (sequential scenario). Therefore, we compare our results in both monopoly and duopoly settings to obtain further insights into the implications of competition for product quality levels, product profitability and both firms' market shares. The findings are illustrated in Figs. 7 and 8. From Fig. 7 we observe that the optimal quality of the new entrant's (low quality) product is less than that of the incumbent's low quality product in the monopoly setting. Since the high quality product is identical in both monopoly and duopoly settings, the average quality of the products supplied by both firms is less in the duopoly setting vis-àvis that in the monopoly setting. From Fig. 8, it is not surprising to note that profitability of both high and low quality products is higher in the monopoly setting vis-à-vis the duopoly setting, i.e., π inM 4 π ni D , i¼1,2. However, we observe that profitability of the lower quality product in the monopoly setting is more than the higher quality product in the duopoly setting, i.e., π 2nM 4 π 1nD , when the core product cost is higher. The result implies that the impact of (duopoly) competition is prominent for higher level of the core product cost vis-à-vis for lower level. Nevertheless, the market covered by both products – sum total of the market shares of Products 1 and 2 – is more in the duopoly setting than that in the monopoly setting implying that the competing firms increase their profit by increasing the market shares rather than by extracting consumer surplus.

10. Conclusions In this paper, we have considered a generic product line design and positioning problem in the context of variety creation using a core product and add-on services. While the functional output of the core product is identical across the products in the product line, product variety is created using add-on services that do not alter the

R3

3

pmin 1

1 0

R3

10

R2

2 0

Proposition 8. When the incumbent firm offers its product such that q1 ¼ qn , the optimal strategy for the new entrant is to offer a product with quality lower than that of the incumbent firm's product, i.e., qn2 o q1 .

p1

p1

4

offered by both firms does not alter from that when the incumbent is the only firm offering a product in the market. On the contrary, recall from Section 8 that when the incumbent offers the second product in addition to its already existing product, the average quality of the products available in the market decreases. It may also be noted that the market penetration for both firms together is equal to that of the monopoly firm with the single product; however, the market share for a monopoly firm offering two products is higher. These findings imply that the entry of the competitor harms the non-strategic incumbent, however, there are no implications for consumers. What follows below is a discussion on the optimal strategy for the new entrant in a scenario wherein the incumbent firm alters price of its product in response to the competitor's product offering. In particular, the incumbent firm's product is given as q1 ¼
qn , and  both incumbent firm and new entrant determine p1 and q2 ; p2 , respectively.

1

2

βα

pmin 1

R1

5

q1=0.7 q0 3

q =1.8 q 0

1

0

1

2

0

3

βα

 , Fig. 6. Optimal strategy for new entrant: quality and price of the incumbent firm's product are fixed. Note: The parameter values chosen are a ¼0, b ¼ 4, α ¼ 1, βα A 0; βmax α and q0 ¼1.7208.

4

q*

2

3 2 1 0

0

2

4

6 βα

8

q*D 2

10

12

Average Quality (AQ)

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

29

4 3 2 1 0

0

2

q*M 2

4

6 βα AQ*D

8

10

12

AQ*M

 . Superscripts “D” and “M” imply duopoly and Fig. 7. Implications of Duopoly Competition. Note: the parameter values chosen are a¼ 0, b ¼7, α ¼ 1, and βα A 0; βmax α monopoly competitive settings, respectively.

1 Market Coverage

*

πx

10

5

0

0

2

4

6

8

10

12

βα π*D 1

π*D 2

0.5

0

0

2

4

6

8

10

12

βα π*M 1

π*M 2

Duopoly

Monopoly

 . Superscripts “D” and “M” imply duopoly and Fig. 8. Implications of Duopoly Competition. Note: the parameter values chosen are a¼ 0, b ¼7, α ¼ 1, and βα A 0; βmax α monopoly competitive settings, respectively.

functionality of the core product. The motivation for our study comes from emerging for-profit private healthcare service providers in India. Over the years, this sector has witnessed emergence of hospitals – such as Aravind Eye Care, Narayana Hrudayalaya, Apollo Hospitals, Fortis Healthcare – that provide services to various segments of the market while the hospitals were established originally to serve either lower or elite consumers only. In this regard, we consider a firm's product line design and positioning problem that determines the number of products and their relative positioning in the product line. We consider two specific scenarios – simultaneous and sequential design – and focus on obtaining insights into the implications of the core product on design and positioning of add-on services. While the motivation for our work presented in this paper comes particularly from services sector, our model and the structural results can be applied to manufacturing settings in which firms offer service components as add-on to the products manufactured. Our structural results highlight the impact of the firm's cost parameters on design and positioning of each variety in the product line under both simultaneous and sequential design scenarios. The important results in a variety of situations considered within the framework of our motivating examples can be summarized as follows. Our main findings in the simultaneous design scenario are twofold – (i) For a threshold on the core product cost relative to the cost of quality below which the firm covers the entire market. On the contrary, beyond another threshold, the firm does not introduce any variety of the product in the market. (ii) The optimal product quality increases in the core product cost; however, products in the product line are increasingly identical. In the sequential product design scenario that is analyzed for the two-product scenario, we particularly derive conditions reflecting on relative design and positioning of the products. An important finding in this case is that both products are active in the market only under certain situations. We illustrate the scope of our approach to derive managerial insights through numerical examples. Depending on the quality level and price of the first product, there exist situations in

which the firm's optimal strategy is to offer both products in the market such that the quality level of the second product is higher (similarly, lower) than that of the first product. We obtain an interesting result in a special case of the sequential product design problem in which the firm's product design and positioning decisions are optimal at each stage. In this scenario, we particularly show that the optimal strategy for the firm is to introduce a lower quality product and increase its profit by increasing the market share. By comparing our results in both simultaneous and sequential design scenarios, we observe that the firm's profitability and average quality level of its products both are lower in the sequential design scenario vis-à-vis the simultaneous design scenario; however, the market share is higher. Nevertheless, both advantages and disadvantages of adopting the strategy of sequential design over simultaneous design decrease with the core product cost. In a duopoly setting, we also illustrate how the design and positioning of an incumbent's product impacts a new entrant's product. Our model and the results are developed on several simplifying assumptions, yet they provide an analytical framework and interesting results that are useful in incorporating additional problem features to develop sharper results. For instance, we ignored the firm's decisions on the core product quality level; it may be useful to examine implications of designing the core product while developing the firm's product line in which the core product impacts quality levels and the number of add-on services. We also assumed that the firm's product line is developed using a core product. The analysis can be extended to determine appropriateness of a core product based product line vis-àvis product variety with distinct serviceability and performance levels with a particular focus on services industry. Our model ignores the benefits of scale economies that are important in services industry considering repeatability of service delivery and learning effect. One may enrich the model and obtain interesting insights considering these two factors. The problem of product line design in services industry in an oligopoly setting has still not been addressed yet. Nevertheless, our model provides important building blocks and several structural results that can be used in future research in this area.

30

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

Acknowledgments We acknowledge the partial support provided by EADS-SMI Endowed Chair for Sourcing and Supply Management and the Supply Chain Management Center at the Indian Institute of Management Bangalore. We also sincerely thank the Editor and anonymous reviewers for their comments and suggestions to improve our previous versions of the manuscript.

Appendix A

Proof of Proposition 1. Consider any y for Problem P1 that satisfies a r y r b. From (11), we note that π ðq; yÞ is concave in q y . This is the for a given y with maximum occurring at qn ðyÞ ¼ 2α n same as (13). Substituting q ðyÞ in (11), we obtain
2  y  4αβ ðb  yÞ . πðyÞ ¼ maxq Z 0 π ðq; yÞ ¼ π ðqn ðyÞ; yÞ ¼ 4α Lemma 1. πðyÞ has the following properties: π 0 ðyÞ ¼

2yb  3y2 þ4αβ ; 4α

π ″ ðyÞ ¼

b 3y 2α

ð50Þ

b We observe that the function πðyÞ is convex in y for y r , and it 3 is concave otherwise. Further, πðyÞ attains a local minimum at pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 yn1 ¼ b  b3 þ 12αβ and a local maximum at yn2 ¼ b þ b3 þ 12αβ. Note b that yn1 r 0 and yn2 Z . 3 To prove the second part of the proposition, i.e., (14), consider the following two cases:

Case 1: 3b r a: In this case, πðyÞ is concave in the interval a; b . From Lemma

1, it follows that πðyÞ attains a maximum at max a; min yn2 ; b . This is the same as (14). Case 2: a o 3b: Consider the region a r yo 3b. Since πðyÞ is convex in this region and yn1 o 0, πðyÞ attains the maximum at 3b. Consider the region 3b ry r b. Since
 πðyÞ
is concave in this region and yn2 Z 3b, it follows that π yn2 Z π 3b and πðyÞ attains a

n2  2 maximum at min y ; b . Since a o yn and a ob, it follows that πðyÞ is maximum at max a; min yn2 ; b . □ Proof of Corollary 1. From (14), it is straightforward to note that
 yn ¼ a if β r 3a2 2ab =4α or equivalently, βα rβmin α . Similarly, 2 yn ¼ b if β Z b =4α or equivalently, βα Z βmax . We also note that α
 2 βmin o βmax as βmax  βmin ¼ bα =4  3a2α  2aα bα = 4 ¼ ðbα  aα Þð3aα α α α α þbα Þ=4 4 0. The rest is straightforward. □ Proof of Proposition 2. We prove the proposition in two stages. First, for any y1 and y2 that satisfy (19), we
show that the
solution  given in (20) is optimal. We substitute qn1 y1 ; y2 and qn2 y1 ; y2 in Problem P2 to obtain Problem P21 for which we show that the solution given in (21) is optimal. Stage 1: Consider any y1 and y2 for Problem P2 such that a
r y1 r y2 r b.  Now for given q1, we note that the function π q2 ; q1 ; y2 ; y1
given in (18) is concave in q2 with maximum y2 occurring at qn2 q1 ; y2 ; y1 ¼ 2α which is given in (20). Substituting
n q2 q1 ; y2 ; y1 in (18), we obtain



  π q1 ; y2 ; y1 ¼ max π q2 ; q1 ; y2 ; y1 ¼ π qn2 q1 ; y2 ; y1 ; q1 ; y2 ; y1 q2 Z 0

 ¼






 
y2 y1  y2 q1 þ 2  β b  y2 þ y1 q1  αq21  β y2  y1 4α

ð51Þ



 (20). Clearly, the solution qn1 y1 ; y2 and qn2 y1 ; y2 is also feasible as it satisfies (19) for y1 rb.

 Stage 2. Substituting qn1 y1 ; y2 and qn2 y1 ; y2 in (51) above, Problem P2 reduces to ( P21 : π ¼

max

a r y1 r y2 r b





)
 y  b b y1  y2  y21 þ y22  y1 y2 þ 4αβ π y2 ; y1 ¼ 1 4α


 Consider y1 such that y1 r b. Note that the solution yn2 y1 given
 in (21) satisfies y1 r y2 r b. Also, the function π y2 ; y1 given in
 (52) is concave in y2 with maximum occurring at yn2 y1 ¼ b þ2y1
 which is given in (21). Substituting yn2 y1 in (52) above, we obtain

 ðb  y1 Þ 5y21  2by1 þ b2  16αβ . □ π y1 ¼ 16α
 Lemma 2. π y1 has the following properties:   2 2
  15y1  14by1 þ 3b  16αβ
 7b 15y1 0 ; π ″ y1 ¼ π y1 ¼ 8α 16α ð53Þ
 7b The function π y1 is convex in y1 for y1 r 15 , and it is concave
 otherwise. Further, π y1 attains a local minimum at y1n1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð7b  2 b þ 60αβÞ=15 and a local maximum at yn12 ¼ ð7b þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 7b r y1n2 . 2 b þ 60αβÞ=15. Note that yn11 r 15 To prove the second part of the proposition, i.e., (21), consider the following two cases:


7b Case 1: 15 in the interval a; b . r a: In this case, π y1 is
concave  From Lemma

2, it follows that π y1 attains a maximum at max a; min y1n2 ; b . This is the same as (21).
 7b 7b Case 2: a o 15 : Consider the region 15 r y1 r b. Since
n2 π y1
7bis 7b n2 concave follows that π y1 Zπ 15
in  this region and y1 Z 15, it

n2 and π y1 attains a maximum ao at min y1n2 ; b . Since


y1 and  a o b, it follows that π y1 is maximum at max a; min yn12 ; b . Further, it is straightforward to note that min yn12 ; b satisfies 2 b y1 r b, and yn12 r b when β r 4α .
 7b Now, consider the region a r y o 15 . Since π y1 is convex in this region, it attains the maximum at the extremes, i.e., either at a or
 7b at 15 . Knowing that π y1 attains a local minimum at y1n1 , we consider the following two subcases:
 7b Subcase (i) yn11 o a: Clearly, π y1 attains the maximum at 15 for 7b

the region a; 15 , and hence, for the entire region a; b , the



 maximum is attained at max a; min yn12 ; b .
7b Subcase (ii) yn11 Za: In this case, if π 15 Z π ðaÞ, then from
 Subcase (i) above, π y1 attains the maximum at maxfa;




 min y1n2 ; b g for the region a; b . Otherwise, π y1 attains the 7b maximum at a for the region a; 15 . Now, it is sufficient to show

 that π yn12 Z π ðaÞ i.e., β r 15a2 þ whenever yn12 Z a, h  2 2 3b 14abÞ=16α. Note that π ðaÞ ¼ 5a2 2ab þb  16αβ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 2 2 ðb  aÞ=ð16αÞ and π y1n2 ¼  7b þ 2b b þ 60αβ  60αβÞð  4b qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 þ b þ60αβÞ =ð675αÞ. It can be observed that both π ðaÞ and
n2  π y1 are decreasing in β. Therefore, in order to complete the
 proof, it sufficient now to show that: (i) π yn12  π ðaÞ ¼ 0 for  
 2 β ¼ 15a2 þ3b  14ab =16α, and (ii) π yn12  π ðaÞ 4 0 for 2

β r b =4α. We obtain for β ¼


 15a2 þ3b  14ab
n2  ð5a  bÞðb  aÞ ¼ π ðaÞ; hence; π yn12  π ðaÞ ¼ 0 ; π y1 ¼ 2α 16α

for β ¼



 b ðb  aÞ ð5a þ 3bÞ ; π yn12 ¼ 0; π ðaÞ ¼  ; hence; π yn12  π ðaÞ 4α 16α

2






Note that the function π q1 ; y2 ; y1 is concave in q1 with
 y þ y2  b which given in maximum occurring at qn1 y1 ; y2 ¼ 1 2α

ð52Þ

2

2

2

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

   2ðαq1 þ bÞ Problem PI1 yn21 ¼ max 2αq1 ; y10 ; min ; b can be rewrit3

2

ðb  aÞ ð5a þ 3bÞ 40 ¼ 16α

□

ten as in (37) and (38). □

and thus the rest follows.

Proof of Corollary 2. From (21),   it is straightforward to note that 2 yn1 ¼ a if β r 15a2 þ 3b 14ab =16α or equivalently, βα r ð5aα  3bα Þð3aα  bα Þ=16. Now, the first part of the corollary clearly follows. Consider βα 4 ð5aα 3bα Þð3aα bα Þ=16. From (20)–(21), we 3yn1  b 4α

yn þ b

qn þ b

yn þ b

and qn2 ¼ 14α . Therefore, 1 3 α ¼ 14α ¼ qn2 . Conpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3yn1  b b þ 60αβ . Substituting yn1 in qn1 ¼ 4α , we obtain sider yn1 ¼ 7b þ 2 15

obtain qn1 ¼

n bq þ αðqn Þ þ β ¼ that pn1  1 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n 2 bq1 þ αðqn1 Þ þ β pn2  pn1 7b þ 2 b þ 60αβ n n n n ¼ 0 when y1 ¼ . Using y2 ¼ qn  qn y1 q1  2 15

2

(24).

and

In

addition,

bq þ αðqn Þ þ β p1 ¼ 1 2 1 , n

n

bq þ αðq2 Þ þ β 2 n

qn1 Þ  2 forward.

n 2

□

we

observe

2

we obtain

bq þ αðqn Þ þ β p2  2 2 2 2

n

n

y

n

n




2

1

( )






  ¼ max π II q2 ¼ π II yn12 q2 ; q2 P II1 : π IIn ¼ max π II q2 ¼ max π I q2 ; y12 q2 q2 y12 ðq2 Þ
    
  2y10 þ αq2 q1  β
αq1 q2  β ¼ max π II q2 ¼ y10 q1  q1  q2  αq22  β q2 2q1 2q2

 
 αq q  β þ y10 q1  αq21 β b y10  1 2 2q1 q2 r


 2 b  y10 q1 þ β ; αq1

q2 Z 0

ð57Þ

From (57), we observe that Problem PII1 is infeasible when
 β 4αq21 . Consider β r αq21 . The function π II q2 is such that 2 dπ II ðq2 Þ d π II ðq2 Þ α2 q q3  β2 ðαq q  βÞð  2αq22 þ αq1 q2 þ βÞ ¼ 1 2 , and ¼  12q23 . Clearly dq2 4q22 dq22 2
 for β r αq21 , π II q2 is concave in q2 with a local maximum at pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 αq þ ðαq1 Þ þ 8αβ , and hence, the optimal solution to Problem PII1 qn2 ¼ 1 4α pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi    2 αq1 þ ðαq1 Þ þ 8αβ 2ðb  y10 Þq1 þ β is qn2 ¼ max αqβ ; min ; ; q1 . Note that αq1 4α 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 αq þ 2ðb  y10 Þq1 þ β ðαq1 Þ þ 8αβ r q1 and αqβ r . Hence, for β r αq21 , αqβ r 1 4α αq 1

1

1

it is straightforward to note that the optimal solution is as shown in (41)–(42). □

q2 ;y21 ;y10

□


 




¼ y10 q1 þ y21 q2  q1  αq22  β b  y21 þ y10 q1  αq21  β y21  y10

( )






  ¼ max π I y21 ¼ π I qn2 y21 ; y21 P I1 : π In ¼ max π I y21 ¼ max π I q2 ; y21 y21 y21 q2 ðy21 Þ

 


 y2 ¼ max π I y21 ¼ y10 y21 q1 þ 21 β b  y21 y21 4α

 þ y10 q1  αq21  β y21  y10 y10 r y21 r b function

β rq2 ; αq1

q2 r q1 ;

ð56Þ

Proof of Proposition 5. We prove the proposition by considering the following two cases. Case I. When q2 Z q1 : In this case, the firm's problem PIII is given by


 P III : π In ¼ max π I q2 ; y21 ; y10

Proof of Proposition 3. For any y21 that satisfies (36), note that
 the function π I q2 ; y21 as given in (35) is concave in q2 with
 y
 maximum occurring at qn2 y21 ¼ 21 . Substituting qn2 y21 in 2α Problem PI, we obtain Problem PI1 as given below:

the

For any q2 that satisfies (40), we observe

obtain Problem PII1 as given below:

s:t:

ðy21  2αq1 Þð2αq1  3y21 þ 2bÞ , and 4α

2

2ðq1  q2 Þq1 . q2

12
 that π II q2 ; y12 is concave in y12 with maximum occurring at

 ð2y þ αq Þq  β yn12 q2 ¼ 10 2q 2 1 . Substituting yn12 q2 in Problem PII, we

¼ p1 þ y2 q2 

2ði  1Þb þ 2ðn  i þ 1Þyn1  ði  2Þb  ðn  i þ 2Þyn1  ib  ðn  iÞyn1 ¼0 2n

Here,

dy

¼

n

2yn  yni  1  yni þ 1 ¼ i 2

y21 Z 2αq1 ;

2

¼ 0 from (21) and qn1 þqn2 ¼ α1 . The rest is straight-

and the rest follows.

2

12

d π II ðq2 ;y12 Þ

1

  n 
 y þ yni þ 1  b yni  1 þ yni  b 2yni  b þ α qi þ qi  1 ¼ 2yni  b þ α i þ 2α 2α

s:t:


 Proof of Proposition 4. Note that the function π II q2 ; y12 as given II dπ ðq2 ;y12 Þ ðq  q Þ½2ðy10  y12 Þq1 þ αq1 q2  β ¼ 1 2 and in (39) is such that q dy

n

Proof of Claim 1. We prove the first part of the claim by showing that, for any y1 such that a r y1 r b, (26) and (27) simultaneously satisfy the first order optimality conditions, i.e., ∂π=∂qi ¼ 0, i¼
1; 2; …;  n and
∂π=∂yi¼ 0, i ¼ 2; 3; …; n. From (27), we obtain n yni  y1 ¼ ði 1Þ b  yn1 from which it straightforward to note that yni r yni þ 1 , i ¼ 2; 3; …; n. From (26), we also note that qni r qni þ 1 , i ¼ 1; 2; …; n, and hence, the solution (26)–(27) is feasible satisfying (7)–(8). The first order condition for optimal qi, i ¼ 1; 2; 3; …; n, is given as yi þ yi þ 1 ¼ b þ 2αqni is clearly satisfied by (26). Similarly, the first order condition for optimal yi, i ¼ 2; 3; …; n, is gives as
 2yni ¼ b þ α qi þ qi  1 . From (26) and (27) we note that

¼

31

ð58Þ ð59Þ
 For any y21 and y10 that satisfy (59), the function
π q2 ; y21  ; y10 y21 n is concave in q2 with maximum occurring at q2 y21 ; y10 ¼ 2α .
 Substituting qn2 y21 ; y10 in Problem PIII, we obtain Problem PIII 1 as s:t:


 π y21 d π I ðy21 Þ 2

2

dy21

¼

is

such

4αq1  3y21 þ b . 2α

that

y10 r y21 r b;

q2 ; y21 Z 0

I

(
 P III1 : π In ¼ max π I y21 ; y10 ¼ y21 ;y10


 max π I q2 ; y21 ; y10 q2 ðy21 ;y10 Þ



  ¼ π qn2 y21 ; y10 ; y21 ; y10

)


 ¼ max π I y21 ; y10 y21 ;y10

I

 



 y2 ¼ max π I y21 ; y10 ¼ y10  y21 q1 þ 21  β b  y21 þ y10 q1  αq21  β y21 ;y10 4α

ð54Þ


  y21  y10

ð55Þ

I

q1 r q2 ;

dπ I ðy21 Þ ¼ dy21


 Clearly, π I y21 is

convex in y21 for y21 r 4αq31 þ b, and concave otherwise. Further,
 2ðαq1 þ bÞ . We observe that for π I y21 attains a local maximum at 3 2ðαq1 þ bÞ r b. From (55), we also note that Problem 2αq1 r b, 2αq1 r 3 PI1 is infeasible when 2αq1 4b, and thus, the optimal solution to

s:t:

y21 Z 2αq1 ; For any y21

ð60Þ

y10 ry21 r b

ð61Þ
 that satisfies (59), the function π I y21 ; y10 is

concave in y10 with maximum occurring at yn10 ¼

bq1 þ αq21 þ β . 2q1

Note

that yn10 is independent of y21. Therefore, yn21 in Proposition 3 is also optimal for Problem PIII 1 . Clearly, by definition of y10, we obtain (43).

32

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

Case II. When q2 o q1 : Based on Proposition 4, the proof parallels that of Case I above, and hence, omitted. □ Proof of Proposition 6. To prove our results we show that π In Z π ðn ¼ 1Þ and π IIn Z π ðn ¼ 1Þ. From Proposition 1, we obtain   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 yn and y10 ¼ yn ¼ max a; min b þ b3 þ 12αβ; b . Conq1 ¼ qn ¼ 2α sider the following two cases. Case I. When q2 Z q1 : We note that q1 is increasing in β, and for  i 2 2 β ¼ b =4α, q1 ¼ b=2α. Therefore, for β A 0; b =4α , it is straightforward n o 2ðαq1 þ bÞ ; that 2αq1 rb. From Proposition 3, we obtain yn21 ¼ max y10 ; 3 2ðαq1 þ bÞ 2ðαq1 þ bÞ yn21 and qn2 ¼ 2α . We obtain yn21 ¼ as  y10 ¼ 3 3 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2ðαq1 þ bÞ bq1 þ αq21 þ β αq21 þ bq1  3β b þ b þ 12αβ  ¼ Z0 for q1 ¼ . Also 3 6α 2q 6q 1

1

yn21 r b.

Further, π In and π ðn ¼ 1Þ are such that 



π ðn ¼ 1Þ ¼ y10 q1  αq21  β b  y10 ¼ y10 q1  αq21 β yn21 y10



 þ y10 q1  αq21  β b  yn21 and π In ¼ y10 q1  αq21  βÞ yn21  y10 þ



  y10 q1 þ yn21 qn2  q1  αq2n2  β b yn21 . Further, we note that


 n
π In  π ðn ¼ 1Þ ¼ yn21 qn2  q1  α qn22  q21 b yn21 ¼ y21 α qn2 þ
n 

 q1 Þ q2 q1 b  yn21 is decreasing in β as q1, qn2 , qn2  q1 are decreasing and yn21 is increasing in β. Also, π In  π ðn ¼ 1Þ ¼ 0 for  i
n  2 2 Therefore, for β A 0; b =4α , as q2  q1 ¼ 0. β ¼ b =4α

π In  π ðn ¼ 1Þ Z 0. Case II. When q2 o q1 : The proof is similar to Case I above, and hence, omitted. Now we prove our results in Proposition 6 by showing that π IIn Zπ In  i

n  2 for β A 0; b =4α . We obtain π In ¼ y10 q1  αq21  β y21  y10 þ




 
y10 q1 þ yn21 qn2  q1  αq2n2  β b yn21 ¼ 14 bq1  αq21 β αqn2 
n 

β n2 n II n ¼ y10 q1  αq21  βÞðb q1 Þ þ bq2  αq2  β b αq1  αq2  and π h i


y q  yn ðq  qn Þ yn12 Þ þ y10 q1  yn12 q1  qn2  αqn22  β yn12  10 1 q12n 1 2 ¼ 2 

 
n β 1 2 n n2 . bq  αq  β b  αq  αq  αq  β αq þ bq 1 1 1  qn 2 1 2 2 4 2
 First consider π In . Note that bq1 αq21  β , i.e., margin on   Product 1, is nonnegative and decreasing in β. Also, αqn2  qβ , 1

i.e., market share of Product 1, is nonnegative and decreasing in β. 
 Therefore, bq1  αq21 β αqn2  qβ is nonnegative and decreasing 1
n 
 in β. Similarly, we can show that bq2  αq2n2  β b  αq1  αqn2 is nonnegative and decreasing in β (i.e., margin and market share of Product 2 is decreasing in β). Therefore, π In Z 0 and it is decreasing in β (i.e., profit is decreasing in β). Similarly, it can be shown that π IIn Z 0 and it is decreasing in β. Now, to prove Proposition 6, it is 2

sufficient to show that π IIn  π In Z 0 for β r b =4α. Note that for In

3

2

 π ¼ 11b =11; 664α Z 0 and for β ¼ b =4α, π IIn  π In ¼ 0.  i 2 Therefore, for β A 0; b =4α , π IIn  π In Z 0. □

β ¼ 0, π

II n

References Altug, M.S., van Ryzin, G., 2013. Product quality selection: contractual agreements and supplier competition in an assemble-to-order environment. Int. J. Prod. Econ. 141 (2), 626–638. Bhattacharya, S., Krishnan, V., Mahajan, V., 2003. Operationalizing technology improvements in product development decision-making. Eur. J. Oper. Res. 149, 102–130. Brown, S.L., Eisenhardt, K.M., 1995. Product development: past research, present findings, and future directions. Acad. Manag. Rev. 20 (4), 343–378. Chen, Y., Seshadri, S., 2007. Product development and pricing strategy for information goods under heterogeneous outside opportunities. Inf. Syst. Res. 18 (2), 150–172. Desai, P.S., 2001. Quality segmentation in spatial markets: when does cannibalization affect product line design?. Mark. Sci. 20 (3), 265–283.

Desai, P.S., Kekre, S., Radhakrishnan, S., Srinivasan, K., 2001. Product differentiation and commonality in design: balancing revenues and cost drivers. Manag. Sci. 47 (1), 37–51. Dobson, G., Kalish, S., 1988. Positioning and pricing a product line. Mark. Sci. 7 (2), 107–125. Dobson, G., Kalish, S., 1993. Heuristics for pricing and positioning a product-line using conjoint and cost data. Manag. Sci. 39 (2), 160–175. Eliashberg, J., Steinberg, R., 1993. Marketing-production joint decision-making. In: Eliashberg, J., Lilien, G.L. (Eds.), Handbooks in Operations Research & Management Science, vol. 5. Elsevier, Amsterdam, The Netherlands. Frank, R.E., Massy, W.F., Wind, Y., 1972. Market Segmentation. Prentice Hall, Englewood Cliffs, NJ. Gahlinger, P.M., 2008. The Medical Tourism Travel Guide: Your Complete Reference to Top-Quality, Low-Cost Dental, Cosmetic, Medical Care & Surgery Overseas. Sunrise River Press. Gourville, J.T., Soman, D., 2005. Overchoice and assortment type: When and Why variety backfires. Mark. Sci. 24 (3), 382–395. Gu, W., Chhajed, D., Petruzzi, N.C., Yalabik, B., 2015. Quality design and environmental implications of green consumerism in remanufacturing. Int. J. Prod. Econ. Forthcoming. http://dx.doi.org/10.1016/j.ijpe.2014.12.040. Hauser, J.R., 1988. Competitive price and positioning strategies. Mark. Sci. 7 (1), 76–91. Heese, H.S., Swaminathan, J.M., 2006. Product line design with component commonality and cost-reduction effort. Manuf. Serv. Oper. Manag. 8 (2), 206–219. Ho, T.H., Tang, C.S., 1998. Product Variety Management: Research Advances. Kluwer Academic Publishers, Boston, MA. Kahn, B.E., 1995. Consumer variety-seeking among goods and services. J. Retail. Consum. Serv. 2 (3), 139–148. Katz, M., 1984. Firm specific differentiation and competition among multiproduct firms. J. Bus. 57 (1), 149–166. Kekre, S., Srinivasan, K., 1990. Broader product line: a necessity to achieve success? Manag. Sci. 36 (10), 1216–1231. Khanna, T., Rangan, V.K., Manocaran, M., 2005. Narayana Hrudayalaya Heart Hospital: Cardiac Care for the Poor Harvard Business School Case N9-505078. Harvard University, Cambridge, MA. Kim, K., Chhajed, D., 2000. Commonality in product design: cost saving, valuation change and cannibalization. Eur. J. Oper. Res. 125 (3), 602–621. Kotler, P., 2002. Marketing Management, 11th ed. Prentice Hall, NJ. Krishnan, V., Gupta, S., 2001. Appropriateness and impact of platform-based product development. Manag. Sci. 47 (1), 52–68. Krishnan, V., Singh, R., Tirupati, D., 1999. A model-based approach for planning and developing a family of technology-based products. Manuf. Serv. Oper. Manag. 1 (2), 132–156. Krishnan, V., Ulrich, K.T., 2001. Product development decisions: a review of the literature. Manag. Sci. 47 (1), 1–21. Lacourbe, P., 2012. A model of product line design and introduction sequence with reservation utility. Eur. J. Oper. Res. 220 (2), 338–348. Lancaster, K.J., 1990. The economics of product variety: a survey. Mark. Sci. 9 (3), 189–206. Mahajan, V., Wind, J., 1992. New product models—practice, shortcomings and desired improvements. J. Prod. Innov. Manag. 9, 128–139. Mallik, S., Chhajed, D., 2006. Optimal temporal product introduction strategies under valuation changes and learning. Eur. J. Oper. Res. 172, 430–452. Matsubayashi, N., 2007. Price and quality competition: the effect of differentiation and vertical integration. Eur. J. Oper. Res. 180 (2), 907–921. Meyer, M., Lehnerd, A.P., 1997. The Power of Product Platforms. Free Press, New York. Mikkola, J.H., Gassmann, O., 2003. Managing modularity of product architecture: toward an integrated theory. IEEE Trans. Eng. Manag. 50 (2), 204–218. Mitra, S., Webster, S., 2008. Competition in remanufacturing and the effects of government subsidies. Int. J. Prod. Econ. 111, 287–298. Moorthy, K.S., 1984. Market segmentation, self-selection and product line design. Mark. Sci. 3 (4), 288–307. Moorthy, K.S., 1988. Product and price competition in duopoly. Mark. Sci. 7 (2), 141–168. Moorthy, K.S., Png, I.P.L., 1992. Market segmentation, cannibalization, and the timing of product introductions. Manag. Sci. 38 (3), 345–359. Mussa, M., Rosen, S., 1978. Monopoly and product quality. J. Econ. Theory 18, 301–317. Netessine, S., Taylor, T.A., 2007. Product line design and production technology. Mark. Sci. 26 (1), 101–117. Oberhozer-Gee, F., Khanna, T., Knoop, C.I., 2007. Apollo Hospitals—First-World Health Care at Emerging-Market Prices Harvard Business School Case 9-706440. Harvard University, Cambridge, MA. Raghunathan, S., 2000. Software editions: an application of segmentation theory to the packaged software market. J. Manag. Inf. Syst. 17 (1), 87–113. Raman, N., Chhajed, D., 1995. Simultaneous determination of product attributes and prices, and production process in product-line design. J. Oper. Manag. 12, 187–204. Ramdas, K., 2003. Managing product variety: an integrative review and research directions. Prod. Oper. Manag. 12 (1), 79–101. Ramdas, K., Sawhney, M., 2001. A cross-functional approach to designing multiple line extensions for assembled products. Manag. Sci. 47 (1), 22–36.

O.D. Palsule-Desai et al. / Int. J. Production Economics 163 (2015) 16–33

Shah, J., Murthy, L.S., 2004. Compassionate, High Quality Health Care at Low Cost: The Aravind Model. September, IIMB Management Review, Indian Institute of Management Bangalore, India. Shaked, A., Sutton, J., 1982. Relaxing price competition through price differentiation. Rev. Econ. Stud. 49 (1), 2–13. Tang, C.S., Yin, R., 2010. The implications of costs, capacity, and competition on product line selection. Eur. J. Oper. Res. 200 (2), 439–450. Villas-Boas, J.M., 1998. Product line design for a distribution channel. Mark. Sci. 17 (2), 156–169. Villas-Boas, J.M., 2004. Communication strategies and product line design. Mark. Sci. 23 (3), 304–316.

33

Xiao, T., Xu, T., 2014. Pricing and product line strategy in a supply chain with riskaverse players. Int. J. Prod. Econ. 156, 305–315. Yu, D.Z., 2012. Product variety and vertical differentiation in a batch production system. Int. J. Prod. Econ. 138, 314–328. Zhang, X., Huang, G.Q., 2010. Game-theoretic approach to simultaneous configuration of platform products and supply chains with one manufacturing firm and multiple cooperative suppliers. Int. J. Prod. Econ. 124 (1), 121–136.