The optimal pace of product updates

Available online at www.sciencedirect.com European Journal of Operational Research 192 (2009) 621–633 www.elsevier.com/locate/ejor O.R. Applications The optimal pace of product updates Cheryl T. Druehl a,* , Glen M. Schmidt b,1 , Gilvan C. Souza c,2 a Stonehill College, Easton, MA 02357, United States David Eccles School of Business, University of Utah Salt Lake City, UT 84112, United States R.H. Smith School of Business, The University of Maryland College Park, MD 20742, United States b c Received 20 July 2005; accepted 28 September 2007 Available online 13 October 2007 Abstract Some firms (e.g. Intel and Medtronics) use a time-pacing strategy for product development (PD), introducing new generations at regular intervals. If the firm adopts a fast pace (introducing frequently), it prematurely cannibalizes its old generation, incurring high development costs, while if it has a slow pace, it fails to capitalize on customer willingness-to-pay for improved technology. We develop a model to gain insight into which factors drive the pace. We consider PD cost, the diffusion rate (coefficients of innovation and imitation), the rate of margin decline, and the degree to which a new generation stimulates market growth. We find that a faster pace is generally associated with faster diffusion, a higher market growth rate and faster margin decay. Not so intuitively, we find that relatively minor differences in the development cost function can significantly impact the pace. Ó 2007 Elsevier B.V. All rights reserved. Keywords: OR in research and development; New product introduction; Diffusion; Time-pacing; Clockspeed 1. Introduction Consider a firm such as Intel that periodically updates its product line with a new generation of product. For example, since the introduction of the early 4004 microprocessor, Intel has typically introduced a new generation every three to four years as shown in Fig. 1.3 Each new technologically-advanced generation revitalizes the product line, initially commanding a relatively higher price. But then the price begins to decay as the new generation diffuses through the market, displacing the previous generation. As it ages, the new generation itself eventually becomes ripe for replacement. * Corresponding author. Tel.: +1 508 565 1909. E-mail addresses: [email protected] (C.T. Druehl), glen.schmidt@ business.utah.edu (G.M. Schmidt), [email protected] (G.C. Souza). 1 Tel.: +1 801 585 3160. 2 Tel.: +1 301 405 0628. 3 We recognize that the pace is not exactly constant. Market conditions have changed significantly since 1970 and the pace that was optimal then would not necessarily be optimal today. However, it appears that Intel does use some form of time pacing for planning purposes. 0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.09.043 Many electronic products are similarly updated at relatively regular intervals, including items such as computer components, printers and other peripherals, digital cameras, and cell phones. From a customer’s perspective, these products seem to improve at a somewhat regular pace, as new generations of products are repeatedly introduced. Eisenhardt and Brown (1998) refer to this type of product development (PD) strategy as time pacing. The pacemaker company Medtronics has benefited from a time-pacing strategy (Christensen, 1997): Uncertainty was reduced and requests for revisions to product features during the design process were essentially eliminated. This allowed the design and launch process to remain on schedule, even though the time-paced schedule did not specify the products to be developed, only the timetable. A notion similar to that of the pace of new product generations is that of industry clockspeed as discussed by Fine (1998) and by Mendelson and Pillai (1999): a fast pace contributes to a fast clockspeed. The pace at which a company introduces new product generations is an important but complex decision for the firm. For example, say Intel has just introduced a new 622 C.T. Druehl et al. / European Journal of Operational Research 192 (2009) 621–633 1990 1994 1998 2002 2006 Fig. 1. Intel’s new product development launch history (Intel, 2007). generation of microprocessor. It then needs to decide whether to introduce the next generation microprocessor next year, in two years, or at some point further down the road. If it goes for an early introduction, it may incur high costs (in PD, for example) and it may prematurely cannibalize sales of its previous generation. If it waits too long, it may fail to capitalize on customer willingness-topay for more advanced technology in addition to the possibility that competitors may (further) infiltrate the market (in other words, the margins on the existing product may have fallen significantly). Additionally, the existing product sales may decline due to market saturation. Thus it is unclear: How frequently should the firm introduce a new generation? Our contribution is to analyze the pacing decision considering a wide range of factors including the costs (both out-of-pocket and opportunity costs) of introducing the next generation either too early, or too late. Specifically, we focus on the interaction between: (1) the PD cost curve (considering both shape and scale parameters); (2) the new product diffusion characteristics (innovation and imitation parameters); (3) the growth in potential market size; and (4) the rate of decline in profit margin after a new generation’s introduction. We have chosen these factors because they directly or indirectly address the issues related to introducing too early or too late and discuss each factor briefly below. Due to the analytical complexity of the model, we proceed to study the optimal pace of product updates numerically. To simulate a broad array of environments, we consider five levels for each of the six parameters used in a full-factorial experiment – parameter levels are based on previous empirical work within the field as detailed in Section 4. Using an exhaustive numerical search, we find the optimal pace in each of the 15,625 runs, and statistically analyze the results to determine the effects of each parameter on pace and profits. The first parameter examined is PD cost, which we assume is convex. Others have similarly assumed that ‘‘crashing’’ is costly while extensively long development times also lead to inefficiencies in PD – see Morgan et al. (2001), Bayus (1997) and Graves (1989). Introducing too $800 1.8 GHz $600 2.0 GHz 1.4 GHz $400 1.6 GHz $200 $0 Oct-01 1986 Introduction Year Sep-01 1982 Aug-01 1978 Jul-01 1974 Jun-01 1970 May-01 4040 4004 Apr-01 80286 8086 Mar-01 80386 Feb-01 Generation Pentium 80486 Jan-01 Pentium 4 Pentium II early may cause excess PD costs due to more frequent introductions and possibly crashing costs. On the other hand, introducing too late may decrease PD costs overall, but may increase the PD costs per introduction. Although PD cost might typically represent less than 5% of the total product revenues over a generation’s life (Ulrich and Eppinger, 2004), our results show that seemingly relatively minor differences in the PD cost curve can significantly impact the optimal pace. Second, to model the diffusion process we follow the multi-generation model of Norton and Bass (1987), where we similarly assume the overall market size grows incrementally over time (or remains constant). To the best of our knowledge we are the first to consider how the coefficients related to the rate of diffusion (i.e. the Bass (1969) coefficients and the growth rate of the market between generations) impact the time-pacing decision. This is significant given the vast empirical evidence supporting the Bass model of diffusion (Norton and Bass, 1992). If a firm introduces too early, it may cannibalize the previous generation too quickly, not taking advantage of market growth. If it waits too long, sales may have slowed considerably as the product has already diffused through the market. We find the Bass coefficients of innovation and imitation are some of the most important determinants of the pace. Specifically, if there is not a sufficient base of customers of the innovator type, then the pace will be slow. But once this base of innovators exists, the pace will be increased by either innovators or imitators. Lastly, we assume the profit margin for a given generation of product declines exponentially over time – see Carrillo (2005), Krishnan et al. (1999), Bayus (1997), and Smith and Reinertsen (1991). Here introducing too early forgoes potential profit margin from the previous generation, while introducing too late results in sales of the previous generation at small margins. The rates of growth in market size and of decay in profit margin are effectively surrogates for such factors as the rate of technological change, the rate of change in customer willingness-to-pay, and the competitive nature of the market. For example, Dec-00 Pentium M Fig. 2. Price decay after introduction of an Intel microprocessor (Dataquest, 2003). C.T. Druehl et al. / European Journal of Operational Research 192 (2009) 621–633 the price of each subsequent upgrade of an Intel microprocessor decays over time as shown in Fig. 2. To formally model the margin decay one would also need to consider the pricing strategy and the cost trajectory, which may also be downward due to the learning curve. However, for simplicity, we directly assume an exponential decay in margin. We find that a faster margin decay increases the optimal pace. We review additional related literature in Section 2. We present the details of our model in Section 3, followed by presentation of our numerical studies in Section 4. We discuss implications of our results and conclude in Section 5. 2. Literature review An extant stream of literature examines the trade-offs between PD time, cost and product quality for a profitmaximizing firm. Several papers examine the initial time to market (i.e. for the first generation) in this context – see for example, Bhaskaran and Ramachandran (2006), Savin and Terweisch (2005), Klastorin and Tsai (2004), Bayus (1997), Bayus et al. (1997) and Kamien and Schwartz (1972). We focus below on related literature that looks at the entry timing of future generations. Wilson and Norton (1989) model a monopolist who offers one product, and then decides when to introduce a lower quality line extension in order to maximize profits. They model diffusion, but do not consider the PD time– cost trade-off. Cohen et al. (1996) consider the quality– cost–time trade-off in a two-generation model without diffusion. Morgan et al. (2001) extend the Cohen et al. study to multiple generations, but they also do not consider diffusion. They find that fixed costs are a stronger determinant of pace than unit costs. Recently, Krankel et al. (2006) examine the trade-off between demand diffusion and stochastic improvements in technology. Assuming a monopolist, they prove the optimality of a threshold introduction policy where the firm introduces the next generation if technology is above a certain level. Others have also studied pace, or clockspeed, as we do. In a game-theoretic model of a duopoly, Souza (2004) finds that increased manufacturing learning increases the firm’s pace of introductions. Souza et al. (2004) conclude that time pacing generally performs well, even if not always fully optimal. Both Souza (2004) and Souza et al. (2004) do not consider diffusion effects. Carrillo (2005) develops a firm-level model for determining optimal pace, assuming a single generation in the market at any time (in contrast, we explicitly consider diffusion and cannibalization effects with consecutive generations). Dhebar (1996) suggests companies establish an optimal pace of product improvement to prevent customers who are presented too frequently with upgrades from balking. (His use of pace differs from ours slightly, as we specify the pace of introduction, not necessarily the pace of improvements.) Other papers consider the sequence and timing of new product introductions for two or more products with differ- 623 ing quality levels as a way to alleviate cannibalism – see Moorthy and Png (1992), Chen and Yu (2002), Bhattacharya et al. (2003), Mallik and Chhajed (2006). Similarly, Dhebar (1994) finds the firm may limit the quality (or features) offered in each generation when faced with strategic customers in order to minimize consumer regret and the postponement of purchases by high-end consumers to the second period. Each of the papers discussed here analyzes one or two aspects of the introduction timing decision. We, in turn, have combined many aspects into one model to obtain a more complete picture. We consider multiple generations (Morgan et al., 2001; Souza, 2004; Souza et al., 2004; Carrillo, 2005), market growth (Bayus, 1997; Klastorin and Tsai, 2004; Carrillo, 2005), diffusion with cannibalization (Wilson and Norton, 1989), margin decay (Souza, 2004; Souza et al., 2004; Carrillo, 2005), a U-shaped PD cost curve (Bayus, 1997) and time pacing (Carrillo, 2005; Souza, 2004; Souza et al., 2004). To our knowledge our paper is the first to analyze these key trade-offs simultaneously, and the first to analyze diffusion of multiple generations when there is cannibalization across generations (this is a significant difference, as our results show diffusion to be a key factor in multi-generational production introduction). To do this, the model becomes complex to the point that analytical solutions are no longer possible, necessitating our numerical approach. 3. Model We seek insights into the factors that affect the pace of product updates of a durable good. Our model finds the optimal pace (i.e. rate) of new generation introductions, assuming the firm introduces product generational updates at constant time intervals. We summarize our notation in Table 1: time t takes on only integer values, denoting a time period (e.g. t = 10 might denote month 10), with the planning horizon involving T periods. The number of periods between product introductions is denoted by s. We call the inverse of s the pace (e.g. if the firm introduces every 24 months, then the pace is 0.5 introductions per year). The firm optimizes profits by setting the pace, with the optimal pace denoted by 1/s*. Pace influences the PD and launch costs as well as the sales of each generation. We assume without loss of generality that the first-generation product is introduced at the end of period t = 0, and denote the introduction of the ith generation as occurring at the end of period ti = (i  1)s. At t = 0, we assume the first generation product is ‘‘ready to go’’ and concern ourselves with the decision of when to introduce later updates to this product. We assume the firm only has two generations of product in the market at any one time (e.g. the first generation is dropped when the third is introduced). This assumption is supported, for example, by the observation that when Intel introduced the Pentium IV, shipments of Pentium II dropped to near zero (Dataquest, 2003). Thus generation i cannibalizes sales from generation 624 C.T. Druehl et al. / European Journal of Operational Research 192 (2009) 621–633 Table 1 Notation Symbol Description t T s 1/s* ti ri(t) r0 b D d a m1 mi Si(t) Fi(t) Yi(t) Zi(t) p q g P(s) Time period, t = 0, 1, 2, . . . , T Time horizon, in periods Number of periods between introduction of successive generations Optimal pace of introductions Time period in which generation i is introduced (at the end of the period); ti = (i  1)s (thus t1 = 0) Profit margin (price – cost) for generation i during period t; t P ti Profit margin in the period a generation is introduced Fractional decrease in profit margin per period Scale parameter for PD cost curve Shape parameter for PD cost curve Discount factor Gross potential market for generation 1 Incremental gross potential market for generation i relative to generation i  1, i > 1 Sales for generation i during time period t Fraction of generation i’s net potential market that will purchase in period t Gross potential market of generation i in period t Fraction of generation i’s gross potential market cannibalized by generation i + 1 in period t Coefficient of innovation in the Norton and Bass (1987) model Coefficient of imitation in the Norton and Bass (1987) model Per-period growth rate in the increase in gross potential market mi (growth accrues only when a new generation is introduced) Total discounted profit over all T periods i  1, but later has its sales cannibalized by generation i + 1. Diffusion of the new generation progresses either due to communication or because the new product is better in some way. For example, in microprocessors, the new chip has a faster processing speed and more transistors. See Schmidt and Druehl (2005). We apply the multi-generation Norton and Bass (1987) model to track the diffusion process (details are given later). We assume that unit profit margin ri(t) (selling price minus production cost) for each generation i is constant over a given period but decreases exponentially from period to period per the relationship ri(t) = r0 exp(b(t  ti)), where b denotes the rate of profit margin decrease per period. Exponential margin decay is considered a reasonable assumption given manufacturing learning (Yelle, 1979) on the cost side, and competition on the price side. Following Bayus (1997) and Smith and Reinertsen (1991), we combine exponential cost and price decay into a single variable for parsimony. Further support, in addition to the available empirical evidence, for our assumption of declining margins over time is found in Krishnan et al. (1999), where, under certain conditions, a monotonically decreasing price for a product that experiences learning effects is optimal. We assume that PD cost for introducing a generation in the market is a convex ‘‘U-shaped’’ function of its PD time: Rushing (crashing) a project is costly as it requires intense resource allocations, while lengthy PD times are also costly due to decreasing know-how, lack of motivation and high setup costs as people move to other projects. For a review on the justification (empirical and theoretical) for such a shape for the PD cost curve, the reader is referred to Bayus (1997) and Graves (1989). For example, Boehm (1981) finds an empirical curve similar to that proposed here when analyzing software development projects. We assume that some fixed number of new generations are under development at one time (e.g. Intel may have the Pentium III and IV under development when the Pentium II is introduced). A firm using a portfolio management technique such as aggregate portfolio planning (Wheelwright and Clark, 2003) to manage its product development will have at least the next generation in development (and possibly more). Due to limited PD resources in any firm, we assume that PD for a new generation starts when PD for some previous generation stops, i.e. when that earlier generation is introduced. This means PD time is equal to some multiple (say n) of s; for simplicity we scale the PD time (divide by n) such that PD time equals s and account for all PD cost of a given generation at the time of its introduction. We assume all generations face the same PD cost curve: we intend our model to apply to situations involving frequent, update-type projects (i.e., derivatives) as opposed to radically new products. More specifically, we assume that PD cost is a function of s according to the functional form D[1/[exp(ds)  1] + ds], where D is a scaling parameter, and d is a shape parameter that controls both the time of minimum cost, as well as the steepness of the curve (but all values of d yield the same PD cost minimum, thus D and d are independent). The scaling parameter D represents the size of the overall development project, which may vary by industry or by the scope of the of development project. On the other hand, the shape parameter indicates how costs will vary with the length of the project. A steep curve with a narrow bottom (high d) indicates a project where both ‘‘crashing’’ and delay of the PD program are increasingly costly; example curves are shown in Fig. 3. We also experimented with a quadratic form for the PD cost curve (used by Bayus, 1997), and reached the same directional results; we prefer the exponential–linear curve because it is defined by only C.T. Druehl et al. / European Journal of Operational Research 192 (2009) 621–633 PD Cost= D PD cost per generation Note that if t < ti then generation i has not yet been introduced, while if t = ti, then it is introduced only at the end of the period, so in either case its purchase fraction is zero for that period. If t > ti+2 then its purchase fraction is one. (But below we show that in this case generation i’s potential market is zero because the next generation has cannibalized all its potential.) If ti < t 6 ti+2, then there is a positive fraction of consumers who will purchase generation i at time t. Next consider Yi(t) which accounts for the cannibalization of some (or all) of older generation (i  1)’s potential market by generation i. For i > 1 define: 1 + dτ exp(dτ ) − 1 6D d .10 =0 0 d= .08 4D d= 6 0.0 .04 d=0 2D d = 0.02 0 20 40 60 625 80 Time between generations, τ Fig. 3. Product development cost curves. two parameters (where d defines the shape of the curve and D the scale), and because the PD cost approaches infinity as s approaches zero, and as s approaches infinity. This functional form we use thus not only follows the empirically-observed ‘‘U-shape’’ but also has some desirable mathematical properties. To our knowledge we are the first to introduce this desirable functional form for modeling product development cost. Finally, a cash flow F in period t is worth Feat at period zero to the firm. The firm’s objective is to maximize total discounted profit at period zero, where profit in each period is the sum of the net revenues (sales volume for each generation multiplied by its per-unit profit margin) minus the PD cost (if there is one). The Bass (1969) model has been applied in numerous settings and extended in many ways. See Mahajan et al. (1990) for a review of these extensions. We use the multigeneration Bass model of Norton and Bass (1987, 1992) to model demand, described as follows. The Bass (1969) and Norton and Bass (1987) models assume communication and ‘‘personality’’ drive the product diffusion. The purchases by innovators are a fraction p of the remaining population of non-buyers, while the purchases by imitators are a fraction q of the product of this remaining population and the fraction having already bought. In Norton and Bass (1987), each consumer who adopts buys exactly one unit per time period. Below we show how we allow for the number of consumers to grow with successive generations. The sales of generation i in period t, denoted by Si(t), is the product of three factors, Fi(t)Yi(t)Zi(t). Fi(t) denotes the fraction of generation i’s potential market that will purchase generation i in period t. Yi(t) denotes the potential market size after accounting for cannibalization from the previous generation i  1, and Zi(t) is the fraction of Yi(t) that is cannibalized by generation i + 1 in period t. First define: 8 0 if t 6 ti ; > < ðpþqÞðtt Þ i 1e F i ðtÞ  1þqpeðpþqÞðtti Þ if ti < t 6 tiþ2 ; ð1Þ > : 1 if t > tiþ2 : Y i ðtÞ  Y i1 ðtÞF i1 ðtÞ þ mi ; ð2Þ where the original potential market is m1 (for i = 1 we define Y1(t) = m1) and the incremental potential market each period thereafter is denoted mi, i > 1. In other words, Yi(t) represents all potential purchases of the previous generation, Yi1(t)Fi1(t), plus some incremental growth mi. The last factor is Zi(t), and it accounts for the fact that generation i may itself be partially (or fully) cannibalized by the newer generation i + 1. When a new generation is introduced, it cannibalizes the fraction Fi+1(t) of generation i’s gross potential market (by (1) and (2)). For generation i, this leaves the following fraction that has not been cannibalized:  1  F iþ1 ðtÞ if t 6 tiþ2 ; Z i ðtÞ  ð3Þ 0 otherwise: The product Yi(t)Zi(t) can be thought of as the net potential market. (Note that as claimed earlier, this is zero for t > ti+2.) The actual market or sales for generation i during time period t is thus: S i ðtÞ ¼ F i ðtÞY i ðtÞZ i ðtÞ; ð4Þ which can be interpreted as the fraction Fi(t) of the potential market Yi(t) that is not yet cannibalized Zi(t). We effectively assume that sales progress at the same rate over the duration of each period, with the rate given by (4). Assuming the firm is a monopolist, it will sell Si(t) in period t. If there is competition, a possible interpretation of our assumptions is that the firm’s market share is constant over the horizon (the total market is scaled up from Si(t)). Fig. 4 shows the impact of p on the diffusion curve shape. In this example, the first generation is introduced at time 0 and the second at t = 29, with roughly 40% growth in market size in the second generation. The sum p + q remains constant for all curves at 0.4. The curves show that as p decreases from 0.2 to 0.001, the diffusion process becomes slower. Specifically, when p = 0.2, the first generation sales immediately take off, while for p = 0.001, they do not begin to take off until around t = 10. The effect of q is similar: a larger q indicates a faster diffusion process. Thus in our model, a larger p + q indicates a faster diffusion process. To simulate the way in which a new generation of product often expands the market, we consider that the potential for a new generation to increase the market size 626 C.T. Druehl et al. / European Journal of Operational Research 192 (2009) 621–633 16 p=.1 q=.3 14 12 p=.1 q=.3 10 Sales p=.01 p=.001 q=.39 q=.399 p=.01 q=.39 Algorithm 1. Let P(s) denote total discounted profit over all periods. Let s* denote the optimal s, and let P(s*) denote the profit associated with s*. p=.001 q=.399 p=.2 q=.2 P 1. Set s* = 1 and Pðs Þ ¼ Tt¼0 r1 ðtÞeat m1 F 1 ðtÞ. 2. s = 1. 3. Find M, the number of generations over T, where M = dT/se. 4. For i = 2 to M, ti = (i-1)s; ti+1 = is. For t = ti to ti+1, compute Si1(t) and Si(t) by Eq. (4). p=.2 q=.2 8 6 4 2 0 0 10 20 30 40 50 60 Time (t) Fig. 4. Two-generation Bass model showing effect of p. grows at a rate of g P 0 per period, where g = 0 indicates no growth. Thus, the incremental gross potential market for the second-generation product, which is introduced at the end of period t2 = s, is m2 = m1((1 + g)s  1). This growth process continues throughout P the planning horizon; in general miþ1 ¼ ½ð1 þ gÞs  1 ik¼1 mk for i P 1. Norton and Bass (1992) fit empirical data to their model, showing that the potential market for each generation grows, although at a different rate for different product classes. We consider only non-negative growth. Note that in order for the potential market to grow there must be an introduction of a new generation: mi does not change during generation i’s lifetime. However, the longer the time between generations, the greater the expansion in the potential market (due to pent-up demand for higher performance, for example). We assume that the firm has the necessary production capacity and capabilities to serve the resulting market. Thus we do not focus on possible costs such as the costs of capacity expansion, training and inventory. For example, we assume that possible decisions regarding inventory will not affect the firm’s PD decisions. We also disregard organizational constraints. Although an organization’s capabilities may limit its ability to reach the optimal pace, knowing that 1/s* is faster than the organization can currently sustain is vital information in itself. Such a realization may be the necessary impetus for firm change. Thus in our model 1/s* depends on a complex set of factors: margin decay rate, sales level (which depends on the diffusion process), PD cost, and discount factor. Given the complexity and the fact that profit as a function of s is not concave (as we show later), we find s* numerically as described in Algorithm 1. Essentially, the algorithm computes the objective function for all T possible integer values of s, and selects the optimal among this set. We initialize the algorithm by calculating the profit for the case where the firm makes no product introductions during the horizon T (aside from the introduction at t = 0). 5. PðsÞ ¼ T M X X i¼1 eat S i ðtÞri ðtÞ  t¼0 M X D½1=ðexpðdsÞ i¼2  1Þ þ dseati : ð5Þ 6. If P(s) > P(s*) s*=s; P (s*)=P (s) s=s+1 if s = T stop; else Go to Step 3 else s=s+1 if s = T stop; else Go to Step 3 End End of Algorithm 1 Examining the profit expression (5), it is straightforward to see that profit for a given s decreases in b and D, and decreases in a as long as net revenues (i.e. profits without considering PD cost) exceed PD cost. However, profits may increase or decrease in d, depending on the value of s. 4. Numerical analysis In this section, we describe a numerical analysis to gain insights into the solution of Eq. (5). Without loss of generality, we fix m1 = 10 and r0 = 1: m1 and r0 are purely scale parameters. For ease of interpretation, a unit sale in our study can be thought of as, say, 1000 units. Further, we consider a long planning horizon T = 200 periods, where one period is equal to one month; thus the planning horizon corresponds to about 16 years. Finally, we set a discount rate at 0.5% per period (a = 0.005), which corresponds to an annual discount rate of 5.8%. We have performed a detailed sensitivity analysis on a (in more than 2000 scenarios) and concluded that it does not significantly impact the optimal time between product introductions, our decision variable, although it clearly impacts monotonically the total expected discounted profit over the planning horizon. 4.1. Parameter levels We ran a full-factorial experimental design for six model parameters. The parameters and levels examined are as 627 C.T. Druehl et al. / European Journal of Operational Research 192 (2009) 621–633 Factor Factor levels D d b p+q q/p g 10, 20, 30, 40, 50 0.02, 0.04, 0.06, 0.08, 0.10 0.01, 0.03, 0.05, 0.07, 0.09 0.1, 0.25, 0.4, 0.7, 1.0 1, 10, 40, 70, 100 0, 0.005, 0.008, 0.012, 0.015 shown in Table 2; all parameters are studied at five levels each. The five levels for each parameter range from a significantly low value (e.g. growth rate g = 0) to a significantly high value (e.g. g = 20% per year), with three intermediate levels in between the extremes. With a full-factorial setup, where the optimal solution is found for every combination of parameter values, our experimental design covers a very wide range of possible scenarios (56 = 15,625). Thus, our numerical study is extensive, and the insights derived from it are robust. The levels for the PD cost scaling parameter D range from 10 to 50; this corresponds to minimum PD costs from 16 to 80. To understand these numbers, consider that a minimum PD cost of 30 indicates a full quarter of earnings (net revenues; not sales) if the sales level is kept at its maximum for the first-generation product, without cannibalization from the second-generation product. The levels for the PD cost parameter d range from 0.02 to 0.10, which result in the PD cost curves previously shown in Fig. 3. The levels for the decay in profit margin with time range from 1% per month (b = 0.01), to 9% per month (or, from 11% to 66% per year). To justify these values, we consider the findings in Blackburn et al. (2004), who report value decay rates to be 1% per month for power tools (over a five-year life cycle) to 4–5% per month for personal computers (over an 18-month life cycle). Some PC component prices such as disk drives decay at rates up to 8% a month (Patterson, 2006). Regarding the Bass coefficients, we consider the sum p + q to be one parameter and the ratio q/p to be another (also used by Krishnan et al., 1999), due to the way the sum and ratio define diffusion as seen by Eq. (1). The levels for p + q and q/p capture a variety of diffusion processes encountered in practice and are inspired by the values found in the meta-analysis of diffusion studies performed by Sultan et al. (1990), covering a wide variety of product categories. We leave the values fixed over the time horizon as Norton and Bass (1992) find the values of p and q do not change from generation to generation in an empirical study. By considering different levels of diffusion coefficients, we are able to investigate different life-cycle patterns, as urged by Williams (1992), for example. Finally, the levels of market growth g range from a no growth scenario with g = 0 (stable market), to g = 0.015 per period (corresponding to 20% per year). Negative market growth (i.e. decline) is not considered. In this case, a pacing strategy would not be appropriate. Clearly, negative growth would increase the time between a firm’s introductions. However, the firm would need to assess whether it would be beneficial to incur the PD cost to introduce another generation, or just offer the current generation until the market declines to zero. We implement Algorithm 1 in MATLAB (1996). We first illustrate the profit function using two examples, followed by an analysis of results from the entire experiment. 4.2. Examples of profit maximization Consider a = 0.005 (discount factor of 5.8% per year), D = 19 (minimum PD cost equals 30), d = 0.02, b = 0.01 (1% profit margin decay per month), p + q = 0.4, q/ p = 10, and g = 0.005 (6.2% potential market growth per year). We subsequently refer to this as Example 1. Example 2 is similar to Example 1, except that p + q = 0.1 and g = 0. Fig. 5 plots P(s) for Example 1 and Fig. 6 for Example 2; note that both curves are not concave. Although P(s) appears to have a ‘‘unimodal’’ shape, it is not monotonically decreasing in s after the optimal s*. For example, for Fig. 6, P(50) = 590, which is the optimal solution, then P(s) decreases monotonically until P(66) = 560, but then P(67) = 570. For Example 1, s* = 29 months; thus, the optimal pace is 1/s* = (1/29) (12) = 0.41 introductions per year, and P(s*) = 1,304. For Example 2, s* = 50 months; thus, the optimal pace is 1/s* = (1/50) (12) = 0.24 introductions per year, and P(s*) = 590. Note that, as expected, if a firm introduces too frequently, negative profits accrue as in Example 2 because of excessive PD costs. Fig. 7 plots sales and undiscounted net revenues (i.e. profits without considering PD cost) as a function of time for all seven generations during the planning horizon T for Example 1 using s = s* = 29. After each generation is introduced, sales increase, level out, and then decrease as the next generation is introduced. The undiscounted net revenue per period increases due to diffusion (and consequently higher sales per period), despite the fact that each generation’s unit profit margin decreases exponentially with 1300 Total Profit, Π(τ) Table 2 Experimental design 1100 900 700 500 0 20 40 60 80 100 120 140 160 Time between generations, τ Fig. 5. Plot of total profit P(s) for Example 1. 180 200 628 C.T. Druehl et al. / European Journal of Operational Research 192 (2009) 621–633 10 800 Total Profit, Π(τ) 400 0 0 20 40 60 80 100 120 140 160 180 200 -400 Sales per period 9 S2(t) S1(t) S4(t) S3(t) 8 7 6 5 Net Revenues 4 3 2 1 -800 0 0 50 100 150 200 Time period t -1200 Time between generations, τ Fig. 8. Sales and undiscounted net revenues per period versus time for Example 2. Fig. 6. Plot of total profit P(s) for Example 2. 4.3. The main effects of the six parameters 25 S7(t) Sales / Net Revenues S6(t) 20 S5(t) Sales of gen 1 S2(t) 15 S1(t) 10 S4(t) S3(t) Net Revenues 5 0 0 50 100 150 200 Time period t Fig. 7. Sales and undiscounted net revenues per period versus time for Example 1. time. The undiscounted net revenue per period then peaks and afterward decreases with time, until a new generation is introduced again. This shape helps to explain why a firm would choose to introduce another generation: Its net revenues are increasing for only a short time after introduction. Then the margin decay effect dominates the sales increase from diffusion and net revenues fall. The gross potential market for the first generation is 10, and increases by a factor of 1.005^29 = 1.156 with each introduction or by 138% from time zero to when the seventh generation is introduced. This market growth also drives the firm to introduce more generations because of the assumption that market growth can only be captured by a new generation of product. Fig. 8 plots sales and undiscounted net revenues for Example 2 (s* = 50) as a function of time for the four generations during the planning horizon T. Due to a zero market growth rate g, market potential remains constant across generations. Net revenue for each generation initially increases (as sales increase due to diffusion). The firm chooses to introduce new product generations because then the net revenue for each generation decreases as margin decreases. We now focus on the results for the entire experimental design. The statistics for the optimal pace 1/s* are displayed in Table 3, for the 13,237 cells (85%) where it is optimal to introduce more than the first-generation product in the planning horizon; in 2388 cells (15%) it is not optimal to introduce the second-generation product (pace is not a meaningful term for these cells). The median value of 1/ s* is 0.71 introductions/year; this is equivalent to life cycles of approximately 17 months, which is similar to, for example, printers at 18 months (Blackburn et al., 2004). The minimum value of 1/s* is 0.12 introductions/year or a life cycle of 8.3 years, which is significantly lower than the planning horizon; this result reinforces our belief that the choice of planning horizon T is not critical to the insights our model generates, as long as T is ‘‘large enough’’. The maximum optimal pace is 2.4 introductions per year, or 5 months between introductions. We first analyze the 2388 cells where it is not optimal to introduce a second generation. To that end, we analyze the proportion in which each factor is observed at each of its five levels in those cells. If a factor’s influence on the decision of not introducing another generation is insignificant, then we should expect that the factor is observed at each of its five levels at roughly 20%; this is tested statistically. The results shown in Table 4 reveal that all factors play a significant role into the decision to not introduce the second generation, because they are observed at each of their five levels in proportions significantly different than 1/5, and there is a clear pattern. Specifically, the firm chooses to Table 3 Statistics for the optimal pace (in introductions per year) Statistic Value Minimum 25th percentile Median 75th percentile Maximum 0.12 0.48 0.71 1.00 2.40 629 C.T. Druehl et al. / European Journal of Operational Research 192 (2009) 621–633 D (%) d (%) b (%) p+q (%) q/p (%) g (%) Percent observed at lowest level Percent observed at second lowest level Percent observed at medium level Percent observed at second highest level Percent observed at highest level 6 12 5 60 6 45 12 15 8 20(ns) 14 22(ns) 19(ns) 19(ns) 18(ns) 10 23 16 27 24 29 5 27 10 36 30 40 4 30 7 All proportions are significantly different than 20% at p < 0.001 except when noted with ‘‘ns’’. 1.0 2200 Pace 0.8 1600 Profit 0.6 Profit Pace (introductions/yr) not introduce the second generation product at higher levels of D, higher levels of d, higher levels of b, lower levels of p + q, higher levels of q/p, and lower levels of g. In short, it is not optimal to introduce a second generation in the planning horizon in scenarios where products diffuse slowly, the profit margin decay is high, the potential market growth rate is low (a relatively mature market), and the PD cost is high and not ‘‘flat’’ around its minimum. We now consider the 85% of cells (13,237 cells) where it is optimal to introduce new generations in the planning horizon, that is, s* < T. Our main objective is to provide insights as to which factors are the most important in the 1000 0.4 0.0% 400 10.0% 5.0% 1.0 2200 Pace 0.8 1600 Profit 0.6 0.4 0.0% Margin decay, b (%/period) 0.4% Profit Statistic for factor decision regarding the frequency of introductions for future generations of products, or the pace. First, we provide a visual representation of the average impact of a factor on pace and profit. To that end, for each factor level (e.g. b = 0.005), we compute the average value of 1/s*, and the average value of total discounted profit over the planning horizon P(s*), across all respective experimental cells (where it is optimal to introduce). This shows the average impact of each factor level on the two performance measures of interest. The results are shown in Fig. 9a for the margin decay parameter b, Fig. 9b for the growth rate g, Fig. 10a for the PD cost scale parameter D, Fig. 10b for the PD cost shape parameter d, Fig. 11a for the sum of diffusion curve parameters p + q and Fig. 11b for the ratio of diffusion curve coefficients q/p. (We note that similar curves are obtained if one performs a one-at-a-time parameter sensitivity analysis for a base case.) The vertical scales for all figures are the same. We first note that all parameters (Fig. 9 through Fig. 11) have a strong impact on profit, but the PD cost curve parameters d and D (Fig. 10) to a lesser extent; this is due to the fact that PD costs represent in our experiments a relatively small fraction of the total profit over the long planning horizon. Fig. 9a shows that the higher the margin decay parameter b, the faster the optimal pace. Previous work suggests that fast clockspeed industries have shorter product life cycles, equivalently, a faster pace (Mendelson and Pillai, 1999; Souza et al., 2004). We confirm that as industry clock- Pace (introductions/yr) Table 4 Results of the 2388 experiments where it is not optimal to introduce the second generation 1000 0.8% 1.2% 400 1.6% Growth rate, g (%/period) 0.8 1600 Profit 0.6 1000 400 0.4 5 15 25 35 45 55 Scale parameter for PD cost, D 1.0 2200 Pace 0.8 1600 0.6 1000 Profit 0.4 0 0.03 0.06 Profit 2200 Pace Pace (introductions/yr) 1.0 Profit Pace (introductions/yr) Fig. 9. Optimal pace and profit versus (a) margin decay parameter b, and (b) growth rate g. 0.09 400 0.12 Shape parameter for PD cost, d Fig. 10. Optimal pace and profit versus (a) PD cost scale parameter D, and (b) PD cost shape parameter d. 630 0.8 1600 Profit 0.6 1000 0.4 400 0 0.25 0.5 0.75 1 1.0 2200 Pace 0.8 1600 Profit 0.6 Profit 2200 Pace Pace (introductions/yr) 1.0 Profit Pace (introductions/yr) C.T. Druehl et al. / European Journal of Operational Research 192 (2009) 621–633 1000 0.4 400 0 25 50 75 100 Sum of diffusion coefficients, p +q Ratio of diffusion coefficients, q / p Fig. 11. Optimal pace and profit versus (a) sum of diffusion coefficients p + q, and (b) ratio of diffusion coefficients q/p. speed (as measured by the margin decay rate b) increases, the pace 1/s* increases. Similar to Souza et al. (2004), we find that profits decrease when the margin decay is more rapid. When diffusion occurs quickly enough, it is beneficial for the firm to introduce another product to capture the high initial margin of each new generation. Fig. 9b shows that the pace 1/s* increases with a higher growth rate g. When the next generation is introduced, its potential market is determined by the growth parameter g. Our result can be interpreted as agreeing with innovation theory; for example, Abernathy and Utterback (1978) posit that in the beginning of the industry life cycle, a period of high demand growth, the focus is on product innovation, and the optimal pace of new products is faster than in the maturity stage of the industry life cycle when a dominant design has emerged. This latter stage is a period of much lower growth. We similarly see that in faster growth conditions, i.e. 20% per year, we predict a faster pace than a mature product with zero expected growth over time (i.e. g = 0.) Thus our results point to the need to recalibrate pace as the product category matures. The impact of the PD cost scale parameter D on optimal pace is intuitive (Fig. 10a) – as D increases, 1/s* decreases – a higher PD cost implies a longer time between generational updates. Fig. 10b shows that as d increases, optimal pace increases – a higher d means that PD cost increases more sharply as pace decreases, thus offering an incentive for the firm to not delay introduction of a generation for long. The diffusion parameters p + q and q/p (Fig. 11) have a very significant impact on 1/s* – a higher value of p + q (lower value of q/p) indicates a faster diffusion process, and consequently the optimal pace is faster. This is a very significant result, because it clearly demonstrates the importance of a product’s diffusion process in the timing of generational introductions. To the best of our knowledge, this is the first paper to show the importance of diffusion speed in the frequency of introducing new generations in a multi-generational product scenario. As diffusion nears completion and sales growth slows, the firm introduces another generation to increase sales, and therefore revenue. As the diffusion process speeds up, the introduction process does as well. However, note that the impact of the sum (p + q) is generally much stronger than the impact of the ratio q/p (as indicated by the relative slopes of the curves for pace). In particular, for ratios of q/p greater than ten or so, the pace does not change significantly with the ratio of q/p. This suggests that a long as there are enough innovators to ‘‘jump start’’ the diffusion process, it is to a large degree the sum of the number of innovators and imitators that drives pace. It is also interesting to compare the impact of each parameter on pace versus its impact on profit. In Fig. 9 we see that profit is impacted by margin decay and growth rate at least as much as is the pace, while in Fig. 10 we find that product development cost impacts the pace much more dramatically than it does the profit. An implication is that as a firm drives down product development costs it significantly increases the rate at which it develops new products, but the rate of profit generation increases more slowly. A similar impact could be inferred from marketing efforts – becoming more proficient at marketing efforts might yield an increase in the coefficients of innovation and imitation, which can dramatically increase the pace but less dramatically increase the profit (Fig. 11). To formalize our notion of impact of a factor on pace and profit, we regress 1/s* against each of the six experimental factors. The magnitude of the influence of a given factor on 1/s* is then measured by R2 (the percent of variance in pace explained by a given factor); a higher value of R2 implies a more influential factor. This approach is suggested by Wagner (1995) for performing sensitivity analysis in complex models; see also Downs et al. (2001) and Souza et al. (2004), who apply this method to a similar full-factorial experimental design. To confirm robustness of this approach, given that the relationship between pace and each of the factors is non-linear as shown in Fig. 9 through Fig. 11, we have also performed single non-linear regressions, one for each factor, using polynomial curves; we again compared the resulting R2 across regressions and our insights remained unchanged, as we discuss in the next paragraph. The results for each of the single linear regressions are displayed in Table 5. We first note that all regressions are highly significant at the p < 0.0001 level, thus all factors strongly influence 1/s*. The sign of the regression coefficient provides the direction of the relationship – a positive sign indicates that 1/s* increases as the factor increases. C.T. Druehl et al. / European Journal of Operational Research 192 (2009) 621–633 Table 5 Regression results for the optimal pace (introductions/year) for the 13,237 cells where it is optimal to introduce later generations (all pvalues < 0.0001) Importancea Factor Coefficient R2 (%) 5 3 4 1 2 6 b g D d p+q q/p 3.222 23.975 0.007 5.784 0.475 0.001 7.2 13.1 9.7 23.6 20.1 1.8 a This is a ranking of the factors based on the absolute value of R2. Table 6 Regression results for profit for the 13,237 cells where it is optimal to introduce later generations (all p-values < 0.0001) Importancea Factor Coefficient R2 (%) 2 1 6 5 3 4 b g D d p+q q/p 14650.1 122957.1 5.2 3399.2 953.2 3.6 15.6 36.4 0.5 0.9 8.5 1.7 a This is a ranking of the factors based on the absolute value of R2. The factors that most influence optimal pace are, in order of importance and with the sign of the coefficient given in parentheses: shape factor for PD cost curve d (+), sum of diffusion curve coefficients p + q (+), potential market growth rate g (+), scale factor for the PD cost curve D (), profit margin decay rate b (+), and ratio of diffusion curve coefficients q/p (). This ranking is based on the specific factor levels used in our experimental design, however, since each factor is varied over a wide range of parameter values (from low to high values based on previous research), we believe that our results are robust. If a nonlinear regression with a quadratic curve is used, then the R2 values for b, g, D, d, p + q, and q/p are 7.5%, 13.2%, 10.4%, 24.0%, 23.8%, and 2.2%, respectively; these values do not change noticeably if a polynomial fit of degree 3 is used instead. These R2 values are close to those reported in Table 5, confirming robustness of the approach. We now turn to the implications of these same factors on profit, given that we have optimized profit by choosing pace. Regression results are shown in Table 6. Again, all factors are important in explaining profit, but especially the potential market growth rate g, the margin decay rate b, and the sum of the diffusion coefficients (p + q). The product development cost curve parameters D and d have a relatively smaller impact on profit because they constitute a relatively small portion of profit throughout the planning horizon. We again verified the ranking with non-linear regression and found the results to be robust. 5. Discussion and summary We have examined how a firm might set its pace of updates to a product, as a function of a number of firm – and industry-specific factors. The intent of our model is 631 to lend insight into some of the trade-offs that a firm faces in making this decision. This is the first paper (to our knowledge) to find a significant link between the optimal pace of generations and the rate of product diffusion as determined by the Bass (1969) diffusion parameters. This is relevant given that the Bass (1969) and multi-generation Bass (Norton and Bass, 1987, 1992) models have been successfully fitted to a variety of product types such as electronics, pharmaceuticals, and consumer goods. Many new product introduction papers, such as Morgan et al. (2001) and Wilson and Norton (1989), only consider one or two generations and do not include the effect of diffusion. We have shown that the speed of diffusion is one of the key factors in setting pace. Diffusion speed may not alter decisions such as whether to introduce a high versus low quality product first, but certainly would affect the time between the introductions of two such products. We note that the pace that is optimal in one industry with a particular diffusion rate will not be the profit-maximizing pace for a firm in a different industry. Additionally, we find that as long as there are enough innovators to launch the diffusion process, a fast pace may be sustained with either innovators or imitators. For example, in the movie industry the optimal pace of introduction of movie sequels may not be dramatically different whether the movies are of the blockbuster variety, with extremely high coefficients of innovation, or of a ‘‘sleeper’’ variety where word-of-mouth from previous viewers is needed to generate excitement (assuming a similar total sum of innovation and imitation coefficients). In the case of movies, we note that calendar year and external events such as Oscar season also affect the optimal pace; these are not considered in our model. Other key drivers of pace are the shape of the PD cost curve, the market growth rate, and the rate of margin decline. The shape parameter of the PD cost curve determines the minimum cost PD time. One can imagine that the nature of the development project might determine this. However, a firm’s capabilities in PD would also play a role in setting this minimum and certainly in a firm’s being able to successfully complete the project near the time of the minimum cost. Thus a firm’s capabilities in PD are important, as a higher d (i.e. a lower time at which the PD cost is minimized) allows a firm to optimally choose a faster pace. We specifically find that subtle differences in the shape of the PD cost curve (which may impact PD cost by only 10% or so) can result in differences of more than 50% in the pace of new generations. This may help explain why product life cycles apparently have gotten shorter – even a relatively modest improvement in PD cost can significantly speed up the pace. A possible implication of our results is that a firm may achieve a significant strategic advantage by driving the market at a much faster pace, negatively impacting any competition. By considering different levels of potential market growth rates, we represent not only different classes of 632 C.T. Druehl et al. / European Journal of Operational Research 192 (2009) 621–633 products, but also different stages in the product life cycle. We find that for more mature industries associated with slower growth rates, the optimal pace of new product introduction will be slower. For new, high growth industries, we predict the pace of new product introduction will be faster. Some empirical studies suggest that industries with fast price decline, such as televisions and personal computers, tend to have more frequent product introductions, in other words a faster pace. For example, Mendelson and Pillai (1999) have used an observation of this type to develop a measure of an industry’s clockspeed. Our results agree with the empirical observation that price decline and pace of introductions go hand in hand, and generally differ from the result of Carrillo (2005) who finds that the pace decreases as margin decays more quickly. We find that when diffusion occurs quickly enough, it is beneficial for the firm to introduce another product to capture the high initial margin of each new generation. Our model leaves open the possibility for many extensions. Our model does not explicitly consider the gametheoretic nature of the relationship between firms or between the firm and its customers. However, to the extent that these factors might implicitly be built into the rate of margin decay or the diffusion parameters, our model could be applied to settings where these factors are at play. Given our finding that the optimal pace increases with faster margin decay, we might expect that competition hastens the margin decay and hence quickens the pace. We also do not explicitly take into account the firm’s ability to meet its launch date, which may negatively affect the firm’s financial performance (Hendricks and Singhal, 1997). Similarly, it would be of interest to formalize customer considerations regarding whether to make an immediate purchase or wait for an updated model (Dhebar, 1994, 1996). Here we might expect that decisions to delay purchase would reduce the diffusion coefficients such that the pace would slow, but this might in turn encourage customers not to delay their purchases such that we might find an equilibrium outcome. 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