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Journal of Operations Management

journal homepage: www.elsevier.com/locate/jom

Valuing supply-chain responsiveness under demand jumps

Işık Biçera, Verena Hagspielb, Suzanne de Trevillec,∗ a Erasmus University, Rotterdam School of Management, PO Box 1738, 3000 DR, Rotterdam, the Netherlands b Norwegian University of Science and Technology, Department of Industrial Economics and Technology Management, 7491, Trondheim, Norway c University of Lausanne, Faculty of Economics and Business, Swiss Finance Institute, 1015, Lausanne, Switzerland

A B S T R A C T

As the time between the decision about what to produce and the moment when demand is observed (the decision lead time) increases, the demand forecast becomes more uncertain. Uncertainty can increase gradually in deci- sion lead time, or can increase as a dramatic change in median demand. Whether the forecast evolves gradually or in jumps has important implications for the value of responsiveness, which we model as the cost premium worth paying to reduce the decision lead time (the justified cost premium). Demand uncertainty arising from jumps rather than from constant volatility increases the justified cost premium when an average jump increases median demand, but decreases the justified cost premium when an average jump decreases median demand. We fit our model to two data sets, first publicly available demand data from Reebok, then point-of-sale data from a supermarket chain. Finally, we present two special cases of the model, one covering a sudden loss of demand, and the other a one-time adjustment to median demand.

1. Introduction

Postponing an order quantity decision until demand is known—thus reducing the decision lead time to zero—eliminates demand-risk ex- posure.1 Conversely, demand-risk exposure tends to increase in the decision lead time, resulting in stockouts or overstocks that generate mismatch costs. The ability to postpone the decision about what to order so that the order quantity can be based on better demand in- formation can be conceptualized as a real option (de Treville and Trigeorgis, 2010), and that option's value can be estimated using quantitative-finance methods. Being able to quantify the value of re- ducing demand-uncertainty exposure that arises from an increase in the decision lead time transforms time into a decision variable.

The first step in estimating option value is to specify the forecast- evolution process: how demand uncertainty increases in decision lead time.2 The simplest case is the random-walk assumption that underlies the Black-Scholes option-pricing model (Black and Scholes, 1973). Each instant that the decision lead time increases, demand uncertainty in- creases by a minute amount following a geometric Brownian motion. When this constant-instantaneous-volatility process holds, demand is lognormally distributed with volatility increasing in the square root of

the decision lead time. This assumption underlies the Cost-Differential Frontier decision tool proposed by de Treville et al. (2014b) that esti- mates the cost differential that must be offered by a long-lead-time supplier to compensate for the increase in demand-uncertainty ex- posure resulting from an increase in decision lead time.3

In practice, changes in demand may occur suddenly as a change in median demand (jump) rather than as an instantaneous increase in volatility. Demand is frequently subject to jumps: The World Economic Forum in its 2012 report on supply-chain risk attributed 44% of supply- chain disruptions to demand shocks (World Economic Forum, 2012).4

In finance, the limitations of the Black-Scholes model are well known, but the model is generally used as a reasonable approximation (e.g., Bakshi et al., 1997). When the true forecast-evolution process is subject to jumps but the mismatch cost is estimated assuming that all demand uncertainty emerges from a constant-volatility process, how bad is the error? Does the constant-volatility version of the model give a good enough approximation of the mismatch cost for practical purposes, or does the error impact decision making enough to warrant the use of a more complex model?

To address this question we extend the Cost-Differential Frontier decision tool to include jumps following the classic model proposed by

https://doi.org/10.1016/j.jom.2018.06.002 Received 4 October 2017; Received in revised form 20 May 2018; Accepted 5 June 2018

∗ Corresponding author. E-mail address: [email protected] (S. de Treville).

1 A decision lead time of zero means that the production decision can be postponed enough to permit working with firm demand data. Even with a decision lead time of zero, the delivery lead time may well be positive. It need only be short enough that the production can be decided after demand is observed.

2 For a stock option, decision lead time translates into the time until the stock price is known. 3 The tool also calculates the cost premium worth paying to reduce decision lead time. 4 A sudden shift in median demand represents a common type of demand shock, see https://www.investopedia.com/terms/d/demandshock.asp.

Journal of Operations Management 61 (2018) 46–67

Available online 02 August 2018 0272-6963/ © 2018 Elsevier B.V. All rights reserved.

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Merton (1976). We use publicly available demand data from Reebok to gain insight into how the choice of model impacts supply-chain decision making. Parsons (2004) studied the cost of demand-risk exposure faced by Reebok in the context of the exclusive license held during the period 2000–2010 to produce replica jerseys with the National Football League (NFL, see also Graves and Parsons, 2005).5 Available published Reebok data includes the mean and standard deviation of annual de- mand for replica jerseys for New England Patriots fans; price, cost, and residual value; and a qualitative description of the many types of de- mand jumps observed by Reebok. Parsons (2004, pp. 74–75) concluded his analysis of Reebok data by proposing that “perhaps the single greatest opportunity for Reebok is to improve its ability to respond to shifts in demand through shorter lead times.” This data is used by Parsons (2004); Graves and Parsons (2005); Parsons and Graves (2005) to demonstrate the value of postponement. Cattani et al. (2008) cited this work as exemplifying the importance of analyzing the value of responsiveness (see also Uppala, 2016). The decision by the authors of the Reebok study to make their data and analysis publicly available made it possible for us to build directly on their work and use our model to extend their analysis.

Our first result is that the impact of jumps on the cost premium worth paying to reduce decision lead time depends on whether a jump is expected to increase or decrease median demand. If a jump is ex- pected to increase median demand, then treating demand uncertainty as though it came from a constant-volatility process results in an un- derstatement of the justified cost premium. If, however, a jump is ex- pected to reduce median demand, then assuming a constant-volatility process will lead to an overstatement of that cost premium. This result arises from how the jump changes the skewness of the marginal de- mand density. Jumps that are expected to increase median demand will increase skewness as long as they occur relatively rarely.6 The resulting increase in right-tail weight increases the value of the option to post- pone the production commitment. A jump that reduces median demand reduces skewness, making the postponement option less valuable. Managers with whom we have reviewed this result have found it counterintuitive, as they experience more concern about being stuck with excess inventory if a negative jump occurs than about stocking out following a positive jump.

In order to make the analysis as useful as possible to practitioners, we explore two special cases of jumps that are frequently encountered in practice. The first special case models the risk that demand would be completely lost. In the Reebok case this corresponds to a change of team jersey that reduces demand for the old model to zero. We show that adding any reasonable risk of demand loss to a constant-volatility process substantially increases the justified cost premium. The second special case models a one-time update of median demand such as occurs when decision makers obtain early-sales data, which we use to quantify the impact of a possible Super-Bowl win on the cost premium worth paying to reduce decision lead time. These results are not surprising in their direction, but they are striking in their magnitude. When the jumps that everyone knows to exist are explicitly considered in setting the decision lead time, the company is likely to much more aggressively reduce decision lead time.

A question that arose during the research project was whether de- mand jumps are experienced in supply chains. To address this question, we randomly selected two products from a supermarket chain and analyzed 100 observations of daily demand from point-of-sale data, then counted how many observations had standardized residuals more than three standard deviations from zero. The first product had four such outliers, indicative of demand jumps, and the second had none. We then considered what would happen if we forced a jump model on a

product where it seemed like a constant-volatility assumption would suffice. By moving the threshold defining outliers from the usual three down to 2.52 standard deviations, the number of outliers for the second product increased from zero to four. Interestingly, treating these four points as jumps rather than normal variation for the second product substantially increased the cost-premium frontier. Which representa- tion is correct? Our model cannot say. But, the fact that the option value of responsiveness is quite sensitive to when outliers are assumed to represent demand jumps indicates that this is an area that managers should be pondering.

2. Literature review

When we fit a normal or lognormal distribution to demand data, this is consistent with assuming a constant instantaneous-volatility process: The increase in demand uncertainty as the decision lead time increases can be modeled by a Brownian motion. Assuming that demand follows a normal (lognormal) distribution implies an underlying process that follows an arithmetic (geometric) Brownian motion. The standard de- viation of demand (log demand) increases with the square root of the decision lead time. Modeling the evolution of a demand forecast as a constant instantaneous-volatility process is not new in supply-chain research. Hausman (1969) demonstrated that a forecast may plausibly evolve according to a geometric Brownian motion. This research formed the basis for the Martingale Model of Forecast Evolution (e.g., Heath and Jackson, 1994; Milner and Kouvelis, 2005). Oh and Özer (2013) emphasized the importance of accurately capturing the forecast- evolution process when deciding about investing in lead-time reduc- tion, giving as an example the case where manufacturer and supplier have asymmetric information. de Treville et al. (2014b) demonstrated the use of a constant-volatility process to transform decision lead time into a decision variable, showing that an apparently compelling cost differential offered by a long-lead-time supplier may result in an in- crease in mismatch cost that eliminates the cost advantage when vo- latility is high and the residual value of the item being acquired is low. Including the value of responsiveness into decision making will fre- quently change the production-location decision. A summary of the calculations that underlie the Cost-Differential Frontier is given in Appendix E. The literature on the value of lead time is summarized in de Treville et al. (2014b) and de Treville et al. (2014a), so we refer readers to those papers. It is generally agreed that lead-time reduction makes companies better able to respond to demand uncertainty, whe- ther from general randomness or demand shocks, and the increased responsiveness reduces the supply-demand mismatch cost (Fisher and Raman, 1996; Iyer and Bergen, 1997; Milner and Kouvelis, 2005; Lutze and Özer, 2008). de Treville et al. (2014a) applied the insights arising from consideration of forecast evolution in decision making in three industrial settings: one with constant instantaneous volatility, one with stochastic instantaneous volatility due to a bullwhip effect, and one with a simple jump-diffusion process in which there was a risk of de- mand suddenly dropping to zero. The demand forecasts considered in de Treville et al. (2014b) and here are assumed to eventually converge to the true value of demand. They are also assumed to be unbiased, so that the expected value of any given forecast update is zero (this is explained in further detail in de Treville et al., 2014a). de Treville et al. (2014b) considered instantaneous volatility that was both constant and stochastic, showing that stochasticity in the instantaneous volatility increases the value of lead time, especially when comparing longer decision lead times. Thus, demand risk can be a source of profit for a responsive firm. Similar results are emerging in the field of marketing concerning the use of demand clumpiness to increase profit (e.g., Zhang et al., 2014).

The ability to extract value from responsiveness has turned out to be a key piece of the puzzle to policy makers that recognize the importance of manufacturing to the local economy, and that are seeking to identify what kind of manufacturing has the best chance of being competitive in

5 The license was won by Nike in 2010. 6 As jumps become less rare, their impact on the marginal density moves from the tail

to the body.

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a high-cost economy.7

Demand jumps are described as a source of operational inefficiency in the economics and the operations management literature (Lorenzoni, 2009; Albuquerque and Bronnenberg, 2012; Kesavan and Kushwaha, 2014; Kesavan et al., 2016). Empirical research supports the idea that shorter lead times reduce the negative consequences of jumps (Kesavan et al., 2016; Tokar et al., 2014). Our models and analysis contribute to this literature in making it possible to quantify the value of respon- siveness when demand uncertainty includes jumps.

3. The impact of jumps on the justified cost premium

3.1. Model presentation and application to Reebok demand data

In this section we present a model that estimates the cost premium worth paying to reduce the relative decision lead time when jumps are added to a constant-volatility forecast-evolution process. Not only does demand uncertainty incorporate the variance of log demand that in- creases linearly in lead time, but median demand might shift up or down.

Fig. 1 compares sample paths from three processes that have the same constant instantaneous volatility, and that begin from a normal- ized demand forecast. The dashed (middle) line illustrates the case where noise is added in very small amounts each instant, captured by geometric Brownian motion without drift. The solid (top) line combines geometric Brownian motion with a jump-diffusion process, with an average jump increasing median demand by 60%. The dotted (bottom) line combines geometric Brownian motion with the risk of complete loss of demand. Our objective is to quantify how the form of the de- mand forecast-evolution process affects the value of lead time.

At a decision lead time t 0= , the demand forecast is equal to actual demand. The constant instantaneous volatility σ is 0.8 for the three paths. The solid (top) line adds jumps that change median demand at the previous instant by an average of 60%. The dashed (middle) line denotes addition of noise to the forecast based solely on the geometric Brownian motion. The dotted (bottom) line adds the risk that at any moment demand may drop to 0.

As the decision lead time increases, the dynamics illustrated in Fig. 1 mean that the demand uncertainty also increases. Our model (the development of which is described in detail in Appendices A and B) estimates the per-unit cost premium relative to the cost offered by the long-lead-time supplier that the decision maker should be willing to pay to reduce the decision lead time.

We fit the model to publicly available demand data for Reebok re- plica NFL jerseys for the New England Patriots (Parsons, 2004; Parsons and Graves, 2005; Graves and Parsons, 2005) to illustrate the impact of assumptions about the underlying forecast-evolution process on the cost premium worth paying to reduce decision lead time. Jerseys were produced by an Asian contract manufacturer, with the production de- cision made well before demand was known. Demand for jerseys evolved over the year based on team and player performance, also on player trades. As the Super Bowl approached, demand increased for teams that made it to the playoffs, with a dramatic increase in jersey demand for the winning team. We first analyzed this data using our models in early 2014, at which point the New England Patriots were estimated by various betting websites to have a 10% chance of winning the 2015 Super Bowl. That probability remained fairly constant throughout the fall of 2014. The Patriots won the 2015 and 2017 Super

Bowls, then made it to the 2018 Super Bowl, only to lose to the Phi- ladelphia Eagles—dramatically bringing our story to life. Winning a Super Bowl is expected to increase expected jersey demand several fold: Reaping the profit from being able to meet such a demand peak requires capacity rather than inventory. We calculate the justified cost premium arising from a possible Super Bowl win in Section 5.

Our objective in this section is to consider jumps that are less spectacular, such as occur during the season when teams perform above or below average or when players are traded. The probability of a jump occurring over a given relative lead time is captured by a Poisson process with jump intensity λ, following standard practice in quanti- tative finance (Merton, 1976). For an order placed at time t [0,1]∈ where demand will be known at time t 1= , the probability of a jump occurring is e1 λ t(1 )− − − . For an order placed at time t 0= —representing a full relative lead time—and a jump intensity of λ 0.2= the probability of a jump occurring at some point is

e1 0.180.2− =− . If the order can be postponed until t 0.4= , the prob- ability of a jump occurring between when the order is placed and when demand is observed is reduced to 0.11. We refer the reader to Appen- dices 7 and 8 for a full description of the mathematical formulation of the general model.

The results of our analysis are shown in Fig. 2. The selling price for a blank jersey is $21.60.8 The residual value for a blank jersey—one that has not yet been printed with a player's name—left over after the season is $8.46. The cost of purchasing a blank jersey from an offshore contract manufacturer is $9.50. These values yield a critical fractile of 92.1%, which corresponds to a fill rate of 98.8% if log demand is normally distributed (de Treville et al., 2014a).

The bottom curve in Fig. 2 shows the justified cost premium in the absence of jumps. We calibrate the volatility parameter by fitting a lognormal density to the demand data given in the case, assuming that the demand variation captured in case data has emerged from a con- stant instantaneous-volatility process. This yields a volatility parameter σ of around 0.22. We make the simplifying assumption that the demand uncertainty observed in the published demand data arises from vola- tility rather than jumps. Although the case describes the kinds of jumps that are typically experienced in the industry, no jumps are mentioned with respect to published data concerning the New England Patriots. If this assumption is incorrect and the demand uncertainty in the pub- lished data does arise to some extent from jumps, then model para- meters should be adjusted to reduce the constant volatility parameter and increase the expected jump impact. The cost premium worth

Fig. 1. Evolution of demand forecasts under a constant-volatility (geometric Brownian motion), a jump-diffusion, and a demand-loss model.

7 A decision tool called the Cost-Differential Frontier emerged from this analysis. It addresses the question: What cost differential does a supplier have to offer to compensate for the increase in demand-volatility exposure arising from an increase in decision lead time? The Cost-Differential Frontier has been made available since 2014 as part of the U.S. Department of Commerce toolbox to aid decision makers in valuing lead time (acetool.commerce.gov/inventory), and it has also been made available by the Swiss State Secretariat for Economic Affairs (https://goo.gl/rjvQEO).

8 The player name could be added by the Asian contract manufacturer, or the blank jersey could be printed locally. We are only considering the demand-risk exposure for blank jerseys, and have adjusted the selling price accordingly: The published data gives the selling price for a printed jersey as $24.00. We remove the cost of local printing of $2.40 to obtain our estimate of the unit price for a blank jersey of $21.60.

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paying to eliminate demand-volatility exposure for these parameter values is less than 6%.

We set the selling price for a jersey to $21.60, the unit cost to $9.50, and the unit residual value to $8.46, yielding a critical fractile of 92.1%. The constant instantaneous volatility σ is set to 0.22, the jump intensity λ to 0.2, and σ to 0.43 for the modified constant-volatility model. The bottom curve shows the cost-premium frontier arising solely from constant volatility. Adding a jump to a constant-volatility process pu- shes up the cost-premium frontier because of the increase in demand uncertainty exposure. Positive jumps—shown in the top curve—are produced by a process Yt that follows a lognormal distribution with parameters τ 0= and ς 0.83= , so that a median jump leaves median demand unchanged, and the expected impact of a jump that occurs is

Y( ) 1.4.t = The second curve from the top shows the cost-premium frontier with the same demand uncertainty as under positive jumps (shown in the top curve) but coming instead from a constant-volatility process. The third curve from the top shows the cost-premium frontier arising from negative jumps, where the expected impact of a jump is a 40% reduction in median demand, in contrast to the 40% increase re- presented in the top curve. The negative jump process follows a log- normal distribution with τ 0.64= − and ς 0.51= , which parameters were chosen by trial and error so that the negative jump curve re- presents the same demand uncertainty as the curves with the positive jump and the modified model.

To this constant-volatility process we add jumps, following the qualitative description of jumps given in the published company in- formation. Jumps increase demand uncertainty, so push up the cost- premium frontier. We consider three scenarios designed to have the same overall demand uncertainty: 1) jumps that are expected to in- crease expected demand when they occur, 2) jumps that are expected to reduce expected demand when they occur, and 3) demand uncertainty modeled as though coming from a constant-volatility process.

Again following standard practice in quantitative finance we assume that the jump follows a lognormal distribution (Merton, 1976). We have programmed the model into a decision tool in R that is available at https://r.erim.eur.nl/r-apps/bicer/ so that interested readers can test their own parameter values. We begin by considering the case where the location parameter is 0. This yields a median jump size of 1, so that a jump is equally likely to increase or decrease median demand. The result is shown in Fig. 2 in the top curve, which describes a jump that is expected to increase median demand by 40%, which corresponds to a jump volatility of 0.83.9 For these parameter values the cost premium worth paying to eliminate demand uncertainty exceeds 15%.

Our next question is how the estimate of the justified cost premium

would differ were we to correctly estimate the amount of demand un- certainty implied by the combination of constant volatility and the jump expected to increase median demand by 40% when it occurs, but incorrectly model it as coming from a constant-volatility process. Would this be close enough, or does the difference justify explicitly considering the two forecast-evolution processes? To answer this question we simulate demand values from the jump demand process and fit a lognormal distribution, which has a volatility of σ 0.43= : A decision maker who interprets demand uncertainty from jumps as coming from a constant-volatility process will base her analysis on a volatility parameter of 0.43. The result is shown in the second curve from the top in Fig. 2. If demand uncertainty from jumps is taken into consideration but treated as coming from a constant-volatility process, the cost premium worth paying drops from 15% to around 10%. This simple illustration demonstrates the potential impact of correctly spe- cifying the forecast-evolution process on the production-location deci- sion: A 15% cost premium might suffice to make local production of blank jerseys feasible, but 10% might fall short.

Now let's consider a jump that is expected to reduce rather than increase demand by 40%.10 The result is shown in the second curve from the bottom in Fig. 2. The cost premium worth paying to elim- inate demand uncertainty drops to around 7%, considerably below that for either positive jumps or the same demand uncertainty coming from a constant instantaneous process. The constant-volatility ap- proximation thus overestimates the justified cost premium for nega- tive jumps and underestimates it for positive jumps. This insight has the intriguing managerial implication that managers should be more willing to invest in reducing decision lead time in the face of positive than negative jumps. We have shared this result with several man- agers, all of whom have found it counterintuitive and contrary to how they manage risk. The reaction across the board has been that these managers find more worrisome—thus would be willing to pay a higher cost premium to avoid—jumps that reduce median demand by 40% when they occur than those that instead increase median de- mand by the same amount.

We lack an explicit form for the demand density arising from the jump-diffusion model. We can, however, use asymptotic analysis to gain initial understanding of how the cost-premium frontier differs when estimated under the modified rather than the jump model. For very small values of λ, τ, and ς we can approximate the density for the jump model using an Edgeworth expansion. Although the parameter values for which the Edgeworth series provides a reasonable approx- imation are much smaller than what we observe in practice, this asymptotic analysis allows us to formulate Propositions 1–3. These propositions are proved in Appendix A for the values for which the Edgeworth approximation holds. We then use numerical analysis to demonstrate that the propositions continue to hold for parameter values commonly encountered in a realistic supply-chain setting.

Proposition 1. The cost premium worth paying to reduce decision lead time will be underestimated (overestimated) when demand uncertainty is approximated by a constant-volatility model and true demand uncertainty includes jumps that increase (decrease) the skewness of the marginal demand density.

Proposition 2. The skewness of the marginal demand density of the jump model increases in the median jump eτ. For τ 0< the resulting decrease in skewness offsets the increase resulting from jump volatility, and may result in the skewness of the marginal demand density of the jump model being lower than that of the constant-volatility approximation.

Proposition 3. The skewness of the marginal demand density of the jump model increases in the jump volatility ς both absolutely and relative to the

Fig. 2. Impact of jumps on the cost-premium frontier.

9 The mean of a lognormal density with location parameter τ and scale parameter ς is eτ ς

2/2+ . To consider a jump that increases median demand by 40%, we set τ σlog (1.4) /22= + . In Fig. 2 we assume that a median jump does not change median de-

mand, that is, the median jump size is 1 and τ 0= . This allows us to solve for ς 0.83= .

10 We hold total demand uncertainty constant at σ 0.43= . The expected jump si- ze—reducing median demand by 40%–is 0.6. We used trial and error to find values of τ and ς such that σ 0.43= and τ ςlog (0.6) /22= + , which yielded τ 0.64= − and ς 0.51= .

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constant-volatility approximation for τ 0≥ as long as jumps are relatively infrequent.

The residual value for a blank jersey not sold during the demand period is described as 89% of the acquisition cost, which results in the relatively high target service level of 92%. Even though the right tail of the marginal demand density for the full decision lead time under po- sitive jumps is heavy, only 8% of the density lies to the right of what the order quantity should cover. A lower residual value reduces the critical fractile. In Fig. 3 we vary the residual value from 0% to 50% to (the actual) 89%–99% of acquisition cost and derive the cost-premium frontier assuming positive jumps. From the figure we observe that as the critical fractile decreases with the reduction in residual value, the cost premium worth paying to reduce the decision lead time sub- stantially increases.

We next consider the impact of changes in the critical fractile due to changes in price or cost rather than residual value. In Fig. 4 we explore how the justified cost premium to eliminate decision lead time varies as the critical fractile increases from 0.5 to 1.0 when a jump is expected to increase median demand by 40%. In each curve, we vary one element of the critical fractile and hold the other two constant: price (solid curve), cost (dashed curve), and residual value (dashed and dotted curve). The solid curve shows that the justified cost premium for a full lead-time reduction increases when the critical fractile increases because of an increase in price. The dashed line shows that the justified cost premium for an elimination of decision lead time first increases, then decreases, as the critical fractile increases because of a reduction in cost. The dashed and dotted curve shows that the justified cost premium for an elimination of decision lead time decreases as the critical fractile in- creases because of an increase in residual value.

In this figure, we explore the impact of residual value on the justi- fied cost premium assuming positive jumps (the top curve that we saw

in Fig. 2, the same relationship holds for negative jumps). We hold constant the original Reebok parameter values for price and cost, and vary the residual value of an unsold unit of product as a percent of acquisition cost from 0 to 99%. The second curve from the bottom corresponds to the top curve in Fig. 2, with the actual residual value 89% of acquisition cost. As the residual value decreases, the cost pre- mium worth paying to reduce the decision lead time increases.

The justified cost premium for a full elimination of decision lead time varies both in the critical fractile, and in the type of mismatch risk that dominates in the critical-fractile calculation. This figure shows the justified cost premium for a complete elimination of decision lead time for a jump that is expected to increase median demand by 40%. In each curve we hold constant two parameter values and vary the third to generate critical fractiles that vary from 0.5 to 1.0. The solid curve represents the case where price is varied between $10.54 and $100, and the unit cost of purchasing from the long-lead-time supplier and the residual value are fixed: The justified cost premium increases with the critical fractile. The dashed curve represents the case where the unit cost of purchasing from the long-lead-time supplier is varied between $8.50 and $15.03, and the price and residual value are fixed: The jus- tified cost premium increases then decreases with the critical fractile. The dashed-and-dotted curve represents the case where the residual value is varied between -$2.60 and $9.40, and the price and the unit cost of purchasing from a long-lead-time supplier are fixed: The justi- fied cost premium decreases with the critical fractile. Note that redu- cing the critical fractile to 0.5 via the residual value implies a negative residual value, such that the company must pay to eliminate leftover inventory.

3.2. Estimating general-model parameter values from time-series data

In this section, we fit the general model to time-series data. We had access to point-of-sale data for 4000 products from a large supermarket retail chain, from which we chose two products at random and con- sidered daily sales over 100 days. Product 1 had four standardized re- siduals that were over three standard deviations from zero (4%). Product 2 had no standardized residuals more than three standard de- viations from zero, but four that were at least 2.52 standard deviations from zero: The decision maker would need to use judgment in deciding whether these demand values arose from white noise in a constant- volatility process or indicated a jump. For both products, we fit a SARIMA model to the demand data, replaced the four outliers (potential outliers) by SARIMA forecast values, then used the outliers to estimate the parameters of a jump model. The next step was to compare the cost- premium frontier under the two assumptions with critical fractiles of 99.5% and 70%. For both products, the cost-premium frontier was considerably higher when outliers were treated as jumps rather than white noise when the critical fractile was very high. When the critical fractile was a more modest 70%, counting outliers as jumps had little impact on the frontier. In fact, because the revised SARIMA model provided a more accurate forecast for Product 1, the justified cost premium decreased slightly for the 70% critical fractile. The detailed calculations are provided in Appendix F.

Analysis of these two products suggests that when the service level is high, the value of responsiveness depends critically on how demand values that approach three-standard-deviation limits are interpreted. We thus propose:

Proposition 4. For very high service levels, the threshold used to define outliers substantially impacts the premium worth paying to reduce the decision lead time.

Our approach contrasts with more traditional treatment of outliers in demand data. That outliers arising from demand shocks can lead to an overstatement of underlying volatility is well understood. In demand forecasting, techniques such as winsorization have historically been used in practice to avoid such overestimation of volatility.

Fig. 3. Impact of residual value on the cost-premium frontier.

Fig. 4. Impact of critical fractile on the justified cost premium.

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Winsorization involves defining some maximum percentile (e.g., 95%), then replacing the demand value by the value at that threshold. Such approaches focus on the body of the marginal demand density, and do not make use of information provided by outliers. Our results suggest that—especially for high critical fractiles—the information provided by outliers may be of value.

4. Special case 1: the demand-loss model

In this section we develop the first special case of a jump-diffusion model in which there is a risk that demand would fall to zero. We continue our analysis of the Reebok case and explore the situation where a team suddenly changes its jersey, reducing the value of stock of the old jersey held by Reebok to zero.11

The unit price received for a blank jersey is $21.60, the unit cost to $9.50, and the unit residual value is $8.46. The constant instantaneous volatility without jumps σ 0.22= . We vary λ from 0% to 20%.

We now apply our model to Reebok's risk of jerseys losing their value following a jersey change. As in the previous section, demand volatility in the absence of jumps is σ 0.22= , the price received for the blank jersey is $21.60, the cost for production in Asia is $9.50, and the residual value is $8.46.

There are 32 NFL teams. Uniform changes are permitted every 5 years, suggesting a maximum jump intensity of 20%. In recent years, one to three teams have changed their uniform each year (http://goo. gl/GbRvfk), suggesting that jump intensities of 3%–10% are worth in- corporating into analysis of demand uncertainty. We vary the prob- ability of a change of jersey λ from 0% (no risk of a jersey change, constant instantaneous volatility only) to 20%.

Fig. 5 illustrates how the cost premium that Reebok should be willing to pay to avoid demand certainty increases in the probability of a sudden change in team jersey. Here, we are not comparing the jump model to a modified model as we did in the general case. Rather, we are demonstrating the understatement of the justified cost premium for reducing decision lead time that occurs when the risk of a complete loss of demand is not included in decision making. As the probability in- creases to 10% or 20%, the cost premium justified by the mismatch cost increases enough to make local production competitive. The R tool that is available at https://r.erim.eur.nl/r-apps/bicer/ permits exploration of different jump intensity and constant-volatility combinations.

5. Special case 2: making a single adjustment to the forecast

In this section, we consider a single adjustment to the forecast, such as occurs when early sales are used to update a demand forecast, as discussed by authors such as Fisher and Raman (1996) in considering how to allocate production between reactive and speculative capacity, and exemplified in the classic Sport Obermeyer case (Fisher and Raman, 1996; Hammond and Raman, 1995). de Treville et al. (2014b) extended analysis of the Sport Obermeyer case to consider lead time as a decision variable, modeling the observation of early sales either as a reduction of stochasticity in the instantaneous volatility, or as a one-time reduction in the right-tail heaviness of the marginal demand density. Here we consider a third alternative, where median demand is adjusted to take into consideration the updated information by being multiplied by a factor that follows an independent lognormal distribution Y with parameters τ and ς. The median value of the factor is eτ and the ex- pected value is eτ ς /22+ . Demand uncertainty is thus composed of 1) the constant instantaneous volatility σ that holds after the forecast is ad- justed to take early sales into consideration, 2) the median eτ of the factor by which the forecast is multiplied based on the early sales, and 3) the volatility ς of the factor.

This special case of the model allows us to estimate the justified cost premium taking into consideration the possibility that a team would win the Super Bowl. Fig. 6 shows the cost premium that Reebok should be willing to pay to postpone their production commitment to in- corporate the demand uncertainty arising from a possible Super-Bowl win. The dashed line shows the calculation assuming that all demand uncertainty comes from a constant-volatility model, and the solid line separates out the demand uncertainty coming from the Super Bowl. As the Super Bowl date approaches, some uncertainty will be resolved, but—as was observed in 2017—much remains until the last second of the game. Therefore, we will simplify the model to combine a constant- volatility process with a single update at a relative lead time of 0.05, just before the end of the demand period. The constant volatility is again 22%, the price remains at $21.60, the long-lead-time unit cost at $9.50, and the unit residual value at $8.46.

We follow betting markets in assigning the probability of the New England Patriots winning the Super Bowl at 10% at the time when an order would need to be placed to the distant supplier. If a Super Bowl win is expected to increase demand by 200% and the probability of winning is estimated to be 10%, then the possible jump increases ex- pected demand by 20%. The standard deviation changes by a factor of

(0.1 3 0.9 1 ) 1.2 0.62 2 2× + × − = , so the coefficient of variation is 0.5, which corresponds to a volatility of log (0.5 1) 0.472 + = , justifying a cost premium of between 12 and 15%. The available R tool that lets readers experiment with their own parameter values is available at https://r.erim.eur.nl/r-apps/bicer/.

In our experience, managers are typically not able to estimate what the impact of a Super-Bowl win will be on demand, but are able to answer questions like, “Do you expect that a Super-Bowl win would at least double demand?” The model makes it possible to identify the demand impact that suffices to make a given cost premium required for

Fig. 5. Impact of the risk of complete demand loss on the value of lead-time reduction.

Fig. 6. The justified cost premium given a possible Super-Bowl win.

11 The theoretical results developed in this section have been applied to tender-loss risk (as described in de Treville et al., 2014a, which cites an early version of this paper).

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51

local responsive production worth paying, making it easier for man- agers to quantify their intuition.

The figure compares the cost differential worth paying to postpone the production decision enough to observe who wins the Super Bowl under the jump and constant-volatility assumptions. The probability of winning the Super Bowl at the time that an offshore order would need to be placed is set to 10%, and demand for jerseys is expected to triple for the winning team. If demand uncertainty is assumed to come from a constant-volatility process, then the cost premium worth paying to eliminate decision lead time is less than 6%. When correctly modeled as coming from a jump process, the cost premium worth paying to reduce decision lead time enough to know who won the Super Bowl more than doubles, but the premium to reduce decision lead time less than what is required to observe the Super Bowl is close to zero.

6. Conclusion

Although it is not traditional to include forecast evolution in cal- culating the supply-demand mismatch cost, research is increasingly demonstrating the importance of including this dynamic in supply- chain analysis. We contribute to this stream of research by exploring the impact of jumps on the cost premium worth paying to reduce decision lead time. We ask: If forecast uncertainty evolves according to a jump process but we make the simplifying assumption that the forecast evolves according to a constant volatility, how much will our decision be affected? Option pricing has been used for many decades in finance: Although the constant-volatility assumption is often violated and tools have been developed that allow an accurate portrayal of the evolution process, results from models that assume a constant volatility are often good enough for practical purposes. In the supply-chain domain, does the same hold? Is correct incorporation of the forecast-evolution pro- cess likely to change key decisions about how much or where to pro- duce, or whether to buffer with inventory or capacity?

We fit our general model to two data sets, first to published demand data for Reebok NFL replica jerseys, then to time-series data from a

supermarket chain. We then model and explore using the Reebok data the risk that demand would drop to zero at some point during the de- cision lead time, then a one-time adjustment to median demand at an expected point in time. The empirical analysis demonstrates that con- sideration of jumps has a substantial impact on how much the company should be willing to invest in responsiveness when a jump is expected to increase median demand, but little impact when a jump is expected to reduce median demand. This in turn has far-reaching implications for the production-location decision.

The field of operations and supply-chain management has access to a wide variety of analytical models. Many of these models have the potential to be useful to decision makers, but are not used because implementation is left to the users. And, many of these models have the potential to lead decision makers into error: It is only in observing what happens when we apply these models to a real problem that we see potential inconsistencies. Neglecting model implementation in research thus results in models that are less used and useful than they could be, and a sharper-than-needed divide between practice and theory. Our primary contribution lies neither in the analytical model nor in its ability to generalize empirically, but rather in the interaction between model development and its impact on decision making in a typical application. For this reason, we focus in the body of the paper on the research question that lies at this intersection: Does the valuation of responsiveness change enough when using a model that accurately captures jumps in the forecast-evolution process to compensate for in- creased model complexity? The model that we have developed to ad- dress that question is made available for users through an online cal- culator at https://r.erim.eur.nl/r-apps/bicer/. Model development is described in detail in the appendices, but is not given center stage as would be typical were the model to be the primary contribution. Similarly, although the two cases that we examine are realistic, we make no claims about the prevalence of jumps: Our focus is on how decision making changes when our model is used in an empirical setting where demand jumps are considered to be possible.

Appendix A. Development of the general model

The general model combines a compound Poisson and constant-volatility process to explore how jumps affect the mismatch cost, following the jump-diffusion option-pricing model developed by Merton (1976). We normalize lead time using the longest lead time under consideration: Actual demand is realized at t 1= and the supplier with the longest decision lead time requires an order-quantity decision at relative lead time t 0= .

We normalize the demand forecast using median demand. The instantaneous volatility (assumed here to be constant) is denoted by σ. The standard Brownian motion process is denoted by Zt. The jump intensity λ is captured by a Poisson process Nt. Consistent with jump models from quantitative finance (Merton, 1976), we assume the jump Yt to follow a lognormal distribution with parameters τ ς( , )2 . A jump at time t changes the demand forecast from Dt to D Yt t, where Y e( )t τ ς /2

2= + . We offset the impact of jumps on the drift rate by adding the term λ Y λ e(1 ( ) ) ( 1)t τ ς /2

2− − = − −+ . Demand follows the process

( )dD λ e D dt σD dZ Y D dN1 ( 1) ,t τ ς t t t t t t/22= − − + + −+ (1) Applying Ito's lemma, the log-demand process can be formulated as follows:

( ) ( )

d D λ e σ dt σdZ Y dN

D D λ e σ t σZ N Y

(log ( ) ) ( 1) log ( ) ,

log ( ) log ( ) ( 1) log ( ).

t τ ς

t t t

t τ ς

t i t

i

/2 1 2

2

0 /2 1

2 2

1

2

2

= − − − + +

= + − − − + + ∑

+

+ = (2)

The demand processes Dt S and Dt

J have the same expected demand ( D D( ) ( )J S1 1=  ) and the same instantaneous volatility: Dt S evolves according

to a constant-volatility process, and Dt J adds a jump-diffusion process to Dt

S.

dD σD dZ ,t S

t S

t= (3)

dD λkD dt σD dZ Y D dN( 1) .t J

t J

t J

t t t J

t= − + + − (4)

We simulate demand values from the jump process and fit a lognormal density to those values, defining a modified process Dt M.

dD σD dZ ,t M

t M

t= (5)

with modified volatility σ σ> and where D M1 and D J

1 have the same mean and variance.

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52

We lack an explicit form for the demand density f (·) arising from the general jump-diffusion model given in Equation (4). When λ 02 ≈ and τ and ς are small, f (·) is close enough to a lognormal that it can be approximated using an Edgeworth expansion, as described in Jarrow and Rudd (1982). This makes it possible to prove Propositions 1 to 3 for the Edgeworth case. Numerical analysis using larger values suggests that the same re- lationships continue to hold.

Proposition 1: The cost premium worth paying to reduce decision lead time will be underestimated (overestimated) when demand uncertainty is approximated by a constant-volatility model and true demand uncertainty includes jumps that increase (decrease) the skewness of the marginal demand density.

Proof.

f D φ D κ f κ φ φ D

D κ f κ φ φ D

D o D( ) ( )

( ( ) ( ) ) 2!

( ) ( ( ) ( ) ) 3!

( ) ( ),2 2

2

2 3 3

3

3 3= +

− ∂ ∂

− − ∂

∂ +

(6)

where φ D( ) is a lognormal density used as the approximating density of demand, and o D( )3 is the third-order approximation error (Jarrow and Rudd, 1982). The term κ h( )i , the i'th cumulant for a given probability distribution h f φ{ , }∈ , is given by:

κ h Dh D dD D( ) ( ) ( ),1 ∫= = −∞

+∞  (7)

κ h D D h D dD( ) ( ( ) ) ( ) ,2 2∫= − −∞

+∞  (8)

κ h D D h D dD( ) ( ( ) ) ( ) ,3 3∫= − −∞

+∞  (9)

The expected mismatch cost at time t is

MC Q p c D V Q

p s D Q f D dD c s Q D

( ) ( ) ( ) ( ),

( ) ( ) ( ) ( ) ( ( ) ) ,

t t

Q∫⎜ ⎟

= − −

=⎛ ⎝

− − + − − ⎞ ⎠

∞



 (10)

where V Q( )t is given by Equation (28). Combining Equations (6) and (10), the third-order approximation of the mismatch cost is formulated as

MC Q MC Q e p s D Q dD

D Q dD o Q

( ) ( ) ( ) ( )

( ) ( ),

f φ r t

Q κ f κ φ φ D

D

Q κ f κ φ φ D

D

(1 ) ( ( ) ( ) ) 2 !

( ) 1

( ( ) ( ) ) 3 !

( ) 3

2 2 2

2

3 3 3

3

∫

∫

= + − ⎧ ⎨⎩

−

− − ⎫ ⎬⎭

+

− − ∞ − ∂ ∂

∞ − ∂ ∂

(11)

where MC Q( )f is the true value of the mismatch cost, and MC Q( )φ the mismatch-cost value based on the assumption that demand has a lognormal distribution. As Dt

J and Dt M are constructed to have the same variance, then κ f κ φ( ) ( )2 2= , and the second term drops out of the equation.

Jarrow and Rudd (1982) showed that for a lognormal distribution:

D Q φ D D

dD φ x

x j( )

( ) ( ) , for 2.

Q

j

j

j

j x Q

2

2∫ − ∂

∂ =

∂ ∂

≥ ∞ −

− = (12)

Combining Equations (11) and (12) we obtain:

MC Q MC Q e p s κ f κ φ φ x

x o Q( ) ( ) ( )

( ( ) ( ) ) 3!

( ) ( ).f φ r t

x Q

(1 ) 3 3 3= + − ⎧ ⎨⎩

− − ∂

∂ ⎫ ⎬⎭

+− − = (13)

From Equation (13) we see that the mismatch cost from the jump model exceeds that of the modified model when:

φ Q D ( )

0, and 1

∂ ∂

< (14)

κ f κ φ( ) ( ).3 3> (15)

The first derivative of a lognormal density with parameters η and γ is

φ x x Q πγ

e Q η γ

( ) 1 2

log ( ) 1 , so that

x Q

Q η γ

2 2

(log ( ) ) 2

2

2

2 ⎜ ⎟

∂ ∂

= ⎛ ⎝

− −

− ⎞ ⎠=

− −

(16)

φ Q D

Q e ( )

0 .η γ 1

2∂ ∂

< ⇔ > − (17)

As γ σ= , the first derivative of the lognormal density is negative for an order quantity Q e σ 2= − times median demand, corresponding to a service level of σΦ ( )− . As this is below median demand, we can infer that the first condition holds whenever an effort is made to match supply to demand.

The second condition (15) states that the mismatch cost is higher for the jump model when the third cumulant—the skewness—is greater for the jump than the lognormal model (κ f κ φ( ) ( )3 > ), thus completing the Proof based on asymptotic analysis.

That making a constant-volatility assumption—using the modified model—understates the justified cost premium for positive jumps and un- derstates it for negative jumps stems from the impact of the two jump types on skewness. A positive jump increases the skewness of the demand density, and a negative jump decreases it. More generally, the skewness of the demand density is increasing in τ. We thus propose:

Proposition 2: The skewness of the marginal demand density of the jump model increases in the median jump eτ . For τ 0< the resulting decrease

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53

in skewness offsets the increase resulting from jump volatility, and may result in the skewness of the marginal demand density of the jump model being lower than that of the modified model.

Proof. The second and third cumulants of the modified model are formulated as:

κ φ τ e( ) 1,σ2 2= − (18)

( )( )κ φ τ e e( ) 2 1 .σ σ3 22 2= + − (19) The marginal density for the jump model has the same variance as that of the modified model such that:

κ φ τ κ f τ( ) ( ),2 2= (20)

( )e λe e e1 .σ τ ς ς σ22 2 2 2= − ++ (21) Integrating Equation (21) into Equation (19) yields:

( )( ) ( )( )κ φ τ e e λe e e( ) 2 1 3 1 1σ σ τ ς σ ς3 2 2 22 2 2 2 2= + − + − −+ (22) for λ 02 ≈ . The relative skewness equates to:

( )( ) ( )( )κ f τ κ φ τ λe e e λe e e( ) ( ) 2 1 3 1 1 ,τ ς ς ς τ ς σ ς3 3 3 (3/2) 2 2 22 2 2 2 2 2− = + − − − −+ + (23)

( )( ( )( ) ( ))λe e e e e e1 2 1 3 1 .τ ς ς τ ς ς ς σ2 /2 2 2 2 2 2 2 2= − + − − −+ + (24)

For τ 0≪ , the first part in the parenthesis of Equation (24) reduces to zero, making the relative skewness negative. This result together with Equation (26) completes the Proof of Proposition 2.

We use numerical analysis to explore whether Propositions 1 and 2—proved by asymptotic analysis—continue to hold for parameter values like those that we observe in practice. In Table 1 we use the cost parameters from the Reebok case (p $21.60= , c $9.50= , s $8.46= , yielding a critical fractile of 92.1%). The first nine runs assume a jump intensity of 0.05, rising to 0.2 for the second nine runs. We vary the jump volatility ς from 0.05 to 0.8. From Table 1 we see that the results continue to hold for the parameter values tested. The modified model overstates the true justified cost premium CPtrue for negative jumps, and understates it for positive jumps. From these results, it is also useful to observe that the Edgeworth ap- proximation is reasonably accurate for small values of λ and ς, but is inaccurate as ς increases even though λ is limited to 0.05. In finance the Edgeworth approximation is often close enough for use in estimation, but our numerical analysis demonstrates that this does not hold for parameter values that are typical in the supply-chain context.

When jumps are relatively infrequent (λ 0.3≤ ), increases in ς lead to increases both absolutely and relative to the modified model. As jumps become more frequent (λ 0.5≥ ), they impact the body of the demand density as well as the tail, so their impact is better captured by the modified model. We thus propose:

Proposition 3: The skewness of the marginal demand density of the jump model increases in the jump volatility ς—absolutely and relative to the modified model—for τ 0≥ as long as the jump intensity is such that jumps are relatively infrequent.

Proof. The second and third cumulants for the jump-diffusion model are formulated as:

( )κ f τ λe e e( ) 1 1,τ ς ς σ2 2 2 2 2= − + −+ (25)

( )( ) ( )( )κ f τ e e λe e e( ) 2 1 2 1 .σ σ τ ς ς ς3 2 3 (3/2) 22 2 2 2 2= + − + + −+ (26)

Table 1 The actual justified cost premium is compared to the estimated justified cost premium first using the Edgeworth approximation and then using the modified model. The price, cost, and salvage values are taken from the Reebok case. We hold λ constant at 0.05, and σ constant at 0.22. The first nine lines of the table represent negative jumps with τ 0.64= − , and the bottom nine lines represent positive jumps with τ 0= . The results are consistent with Propositions 1 and 2: The justified cost premium is understated by the modified model for positive jumps, and overstated by the modified model for negative jumps. The results in the table further illustrate that the Edgeworth approximation does not provide a good estimate of the justified cost premium for larger values of ς, even though λ is relatively small in this example.

Run σ λ τ ς Y[ ] 1− CPtrue CPedgeworth σ CP

1 0.22 0.05 -0.64 0.05 -0.472 5.52% 5.22% 0.2602 6.33% 2 0.22 0.05 -0.64 0.1 -0.471 5.52% 5.22% 0.2638 6.44% 3 0.22 0.05 -0.64 0.2 -0.462 5.51% 5.24% 0.2664 6.51% 4 0.22 0.05 -0.64 0.3 -0.449 5.51% 5.28% 0.2707 6.63% 5 0.22 0.05 -0.64 0.4 -0.429 5.52% 5.37% 0.2796 6.89% 6 0.22 0.05 -0.64 0.5 -0.403 5.57% 5.58% 0.2834 7.00% 7 0.22 0.05 -0.64 0.6 -0.369 5.67% 6.04% 0.2932 7.29% 8 0.22 0.05 -0.64 0.7 -0.326 5.84% 7.15% 0.3046 7.63% 9 0.22 0.05 -0.64 0.8 -0.274 6.08% 9.91% 0.3177 8.03% 10 0.22 0.05 0 0.05 0.0013 5.22% 5.22% 0.2202 5.22% 11 0.22 0.05 0 0.1 0.005 5.25% 5.23% 0.2205 5.23% 12 0.22 0.05 0 0.2 0.0202 5.36% 5.31% 0.2243 5.30% 13 0.22 0.05 0 0.3 0.046 5.56% 5.50% 0.2298 5.48% 14 0.22 0.05 0 0.4 0.0833 5.86% 5.96% 0.2388 5.73% 15 0.22 0.05 0 0.5 0.1331 6.24% 7.10% 0.2475 5.98% 16 0.22 0.05 0 0.6 0.1972 6.70% 9.87% 0.2576 6.26% 17 0.22 0.05 0 0.7 0.2776 7.27% 16.80% 0.2695 6.60% 18 0.22 0.05 0 0.8 0.3771 7.94% 34.60% 0.2834 7.00%

I. Biçer et al. Journal of Operations Management 61 (2018) 46–67

54

Equation (26) completes the Proof of Proposition 3. For our numerical analysis of Proposition 3, we use a higher critical fractile (p $100.00= , c $1.00= , and s $0.00= to yield a critical fractile of

99%) to better explore the region under the right tail. The results are shown in Table 2. This table only considers positive jumps, as negative jumps reduce skewness. We have removed the column with the Edgeworth approximation because our focus here is on parameters values that go outside of its useful range. We have added a column with the ratio of the true justified cost premium to that estimated using the modified model. The first nine runs have a jump intensity λ 0.05= , increasing to 0.2 and then 0.5 for the next two sets of nine runs. As predicted by our asymptotic analysis, the ratio of the true justified cost premium to that of the modified model increases in ς for all values of λ. As we increase λ from 0.2 to 0.5, we see that CP

CP true decreases.

Appendix B. Derivation of the newsvendor profit-maximizing order quantity and expected profit by Fourier transform

Because the demand density arising from the general model is too complex to be presented in an explicit form, in this appendix we demonstrate the use of a Fourier transform to calculate the newsvendor profit-maximizing order quantity (e.g., Carr and Madan, 1999; Lee, 2004; Bakshi and Madan, 2000). The Fourier transform is widely used in finance to derive option-pricing formulas when asset prices follow complex, non-Gaussian processes. Applying the Fourier transform, an integral over a probability distribution is first transformed from probability space to Fourier space. The explicit form of the integral is then obtained by the inverse Fourier theorem.

Let Dt be a random variable denoting demand at time t, p the retail price of the product per unit, c the cost of the product per unit, and s the salvage value per unit. We denote by f (·) the probability density function of demand. We normalize the total time length in that demand forecasts evolve over t [0,1]∈ and actual demand is realized at time t 1= . The newsvendor profit function for an order quantity Q is given by:

π D Q p c Q p s E Q D( , ) ( ) ( ) [max ( , 0) ].1 1= − − − − (27)

The expected profit for the demand period is

V Q p c Q p s Q D f D D dD( ) ( ) ( ) ( ) ( ) , Q

t 0

∫= − − − − (28) where f D D( )t is the density function of demand conditional on Dt.

We recall that demand follows the process:

( )dD λ e D dt σD dZ Y D dN1 ( 1) ,t τ ς t t t t t t/22= − − + + −+ (29) The log-demand process is then:

( ) ( )

d D λ e σ dt σdZ Y dN

D D λ e σ t σZ N Y

(log ( ) ) ( 1) log ( ) ,

log ( ) log ( ) ( 1) log ( ).

t τ ς

t t t

t τ ς

t i t

i

/2 1 2

2

0 /2 1

2 2

1

2

2

= − − − + +

= + − − − + + ∑

+

+ = (30)

Table 2 The actual justified cost premium is compared to the estimated justified cost premium using the modified model. The price, cost, and salvage values are p $100= , c $1= , and s $0= to yield a critical fractile of 99%. We hold τ constant at 0, and σ constant at 0.22. The first nine lines of the table represent low jump intensity with λ 0.05= ; the next nine lines represent moderate jump intensity with λ 0.2= ; and the last nine lines represent high jump intensity with λ 0.5= . The results are consistent with Proposition 3. The actual justified cost premium relative to the modified model increases in jump volatility.

Run σ λ τ ς Y[ ] 1− CPtrue σ CP CPtrue CP

1 0.22 0.05 0 0.05 0.0013 76.00% 0.2202 75.86% 1.002 2 0.22 0.05 0 0.1 0.005 76.49% 0.2205 75.99% 1.006 3 0.22 0.05 0 0.2 0.0202 79.65% 0.2243 76.20% 1.045 4 0.22 0.05 0 0.3 0.046 87.84% 0.2298 79.07% 1.11 5 0.22 0.05 0 0.4 0.0833 102.60% 0.2388 84.19% 1.22 6 0.22 0.05 0 0.5 0.1331 124.54% 0.2475 88.16% 1.41 7 0.22 0.05 0 0.6 0.1972 154.02% 0.2576 92.85% 1.66 8 0.22 0.05 0 0.7 0.2776 191.61% 0.2695 98.50% 1.95 9 0.22 0.05 0 0.8 0.3771 238.19% 0.2834 105.29% 2.26 10 0.22 0.2 0 0.05 0.0013 76.39% 0.221 76.31% 1.001 11 0.22 0.2 0 0.1 0.005 78.31% 0.225 78.05% 1.003 12 0.22 0.2 0 0.2 0.0202 89.86% 0.237 83.38% 1.078 13 0.22 0.2 0 0.3 0.046 116.19% 0.257 92.57% 1.25 14 0.22 0.2 0 0.4 0.0833 157.83% 0.283 105.09% 1.50 15 0.22 0.2 0 0.5 0.1331 213.42% 0.314 120.92% 1.76 16 0.22 0.2 0 0.6 0.1972 283.85% 0.347 138.88% 2.04 17 0.22 0.2 0 0.7 0.2776 372.07% 0.382 159.27% 2.34 18 0.22 0.2 0 0.8 0.3771 482.18% 0.416 180.44% 2.67 19 0.22 0.5 0 0.05 0.0013 77.17% 0.221 76.30% 1.011 20 0.22 0.5 0 0.1 0.005 81.84% 0.234 82.03% 0.997 21 0.22 0.5 0 0.2 0.0202 106.81% 0.261 94.45% 1.13 22 0.22 0.5 0 0.3 0.046 154.36% 0.301 114.16% 1.35 23 0.22 0.5 0 0.4 0.0833 222.24% 0.350 140.58% 1.58 24 0.22 0.5 0 0.5 0.1331 312.51% 0.401 170.93% 1.83 25 0.22 0.5 0 0.6 0.1972 430.42% 0.475 220.59% 1.95 26 0.22 0.5 0 0.7 0.2776 583.09% 0.551 279.20% 2.08 27 0.22 0.5 0 0.8 0.3771 779.47% 0.613 333.24% 2.34

I. Biçer et al. Journal of Operations Management 61 (2018) 46–67

55

We rewrite Dlog ( )t as the sum of four parts by defining Dlog ( )0=A , ( )λk σ t12 2= − −� , σZt=C , and Ylog ( )iN i1t= ∑ =D . The characteristic function of Dlog ( )t is created from these four parts as follows:

ϕ ω ϕ ω ϕ ω ϕ ω ϕ ω( ) ( ) ( ) ( ) ( ),Dlog ( )t = × × ×A C D� (31)

with

( )ϕ ω D ϕ ω e ϕ ω e ϕ ω e( ) , ( ) , ( ) , ( ) ,( )iω iω λk σ t σ tω λt ϕ ω0 1 2 2 ( ) 1y

2 2 2 ͠= = = =− − − −A C D�

where ϕ ω( )y͠ is the characteristic function of Ylog ( )i , and ϕ ω e( )y iωτ

͠

ς ω2 2 2=

⎜ ⎟ ⎛ ⎝

− ⎞ ⎠.

Theorem 1. If demand forecasts follow a process with a characteristic function ϕ (·), the optimal order quantity Q* satisfies the equation:

βQ e

π Re e

ϕ ω α i iω α

dω ( ( 1) ) ( 1)

, α Q

iω Q* log ( )

0 log ( )

* *∫ ⎜ ⎟= ⎛

⎝

− + − + +

⎞ ⎠

− +∞ −

(32)

where β p c p s

= − −

is the newsvendor critical fractile, Re (·) is an operator that gives the real part of a complex number, i 1= − , and α is a damping factor required to permit square integrability as described in Carr and Madan (1999) (as explained below).

To apply the Fourier transform to the expected net present value of the profit, we rewrite Equation (28) by change of variables:

( )V q e p c e p s e e e f y y dy( ) ( ) ( ) ( ) ,t r t q r t q q y

t (1 ) (1 )

1 11∫= − − − −− − − − −∞ (33)

where y Dlog ( )1 1= , and q Qlog ( )= . Let P q( ) denote the integral in Equation (33).

( )P q e e f y y dy( ) ( ) . q

q y t1 11∫= −−∞ (34)

The term P q( ). in Equation (34) tends to + ∞ as q → +∞, thus P q( ) is not square-integrable. Carr and Madan (1999) note that the sufficient condition for square integration is

P q dq( ) ) .2∫ < ∞ −∞

∞

(35)

and suggest to work instead with

p q e P q( ) ( ),αq= (36)

where α 1< − serves as a damping factor that makes p q( ) square integrable. The Fourier transform of p q( ) is:

( )ψ ω e e e e f y y dy dq( ) ( ) ,iωq αq q

q y t1 11∫ ∫= −−∞

+∞

−∞ (37)

ϕ ω α i α α ω iω α

( ( 1) ) (2 1)

, 2 2

= − +

+ − + + (38)

where ϕ (·) is the characteristic function of the random variable y1. Using the inverse Fourier transform, Equation (34) becomes

P q e

π e ψ ω dω( )

2 ( ) ,

αq iωq∫=

−

−∞

+∞ −

(39)

where ψ ω( ) is given by Equation (38). Since P q( ) is a real number without any imaginary part (Carr and Madan, 1999), Equation (39) can be rewritten as

P q e

π Re e ψ ω dω

e π

Re e ψ ω dω( ) 2

( ( ) ) ( ( ) ) . αq

iωq αq

iωq 0

∫ ∫= = −

−∞

+∞ −

− +∞ −

(40)

The expected net value of the profit, given by Equation (33), in terms of q Qlog ( )= becomes

V q e p c e p s e e π

Re e ψ ω dω( ) ( ) ( ) ( ) .t r t q r t αq

iωq(1 ) (1 ) 0

∫= − − − ⎛ ⎝ ⎜

⎛

⎝ ⎜

⎞

⎠ ⎟

⎞

⎠ ⎟

− − − − − +∞ −

(41)

Equation (41) gives the expected profit function in terms of the characteristic function of demand. Let q* denote the value of q that maximizes V q( )t :

q V qargmax ( ) q

t * =

(42)

dV q dq

p c e p s dP q

dq ( )

( ) ( ) ( )

0,t q= − − − = (43)

βe e

π Re e

ϕ ω α i iω α

dω ( ( 1) ) ( 1)

.q αq

iωq 0

* *

*∫ ⎜ ⎟= ⎛ ⎝

− + − + +

⎞ ⎠

− +∞ −

(44)

The profit-maximizing order quantity for a full lead time is:

β e e

π Re e

ϕ ω α i iω α

dω ( ( 1) ) ( 1)

,L q

αq iωq

0 0 * 0

*

0 *∫ ⎜ ⎟= ⎛

⎝

− + − + +

⎞ ⎠

− +∞ −

(45)

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This yields the full-lead-time expected profit assuming a profit-maximizing order quantity:

( )V q c p c e p s e π

Re e ψ ω dω( ) ( ) ( ) ( ) ,L L q αq

iωq 0 0

* 0

0 * 0

*

0 *∫= − − − ⎛

⎝ ⎜

⎞

⎠ ⎟

− +∞ −

(46)

For lead time T t− , we iterate to find the unit cost that yields an equivalent maximum profit:

c c V q c V q c{ ( ) ( ) }.t t t L *

0 0 *= = (47)

The cost premium CP worth paying to obtain the reduced demand-risk exposure resulting from reducing the lead time from T to T t− is then:

CP c c/ 1.t L= − (48)

Appendix C. Development of the demand-loss model

Demand forecasting here has two components: a jump-to-zero component where demand or sales is lost, and constant volatility otherwise. We represent the demand-loss process using the stochastic differential equation given by Equation (49).12 The demand forecast at time t is denoted by Dt, σ the instantaneous volatility (assumed to be constant), Zt the standard Brownian motion process, and Nt a Poisson process with intensity λ. We again assume that the supplier with the longest lead time requires that the order be placed at t 0= , with actual demand realized at t 1= .

dD λD dt σD dZ D dN .t t t t t t= + − (49)

The drift rate in Equation (49) is adjusted by adding λ to offset the effects of the jumps on the drift, thereby making the absolute drift rate equal to zero. Under Equation (49), a jump reduces demand to zero, and its value remains at zero after the jump. If no jump occurs, demand forecasts follow a geometric Brownian motion with a drift rate of λ and an instantaneous volatility of σ. The log-demand process can be formulated as

( ) ( )

d D λ σ dt σdZ dN

D D λ σ t σZ t t

(log ( ) ) log (0) ,

log ( ) log ( ) if ˆ,

log (0) otherwise,

t t t

t t

1 2

2

0 1 2

2

= − + +

= ⎧ ⎨ ⎩

+ − + < ⎫ ⎬ ⎭ (50)

where t̂ is the time of the first jump. The density function of demand conditional on the demand forecast at time t can be written as follows:

f D D δ D( 0) ( ),t1 1= = (51)

f D D e D

e e D ( 0)

1 , 0

, 0 t

λ t

λ t D σ t π

1

(1 ) 1

(1 ) 1 (1 ) 2 1

D Dt λ σ t σ t

1

(log ( 1) log ( ) ( 2/2) (1 ) ) 2

2 2 (1 ) > =

⎧

⎨ ⎪

⎩ ⎪

− =

>

− −

− − −

− − − − −

− (52)

where δ D( )1 is the Dirac delta function:

δ D D

( ) , if 0,

0, otherwise.1 1= ⎧

⎨⎩

∞ = (53)

Equation (51) implies that D D 0t1 = = if a jump occurs between 0 and t. Otherwise, the point density of demand at D 01 = is equal to the probability of observing a jump between t and 1. If there is no jump occurring between t and 1, the value of demand would be larger than 0 (D 01 > ), and the demand follows a lognormal distribution as given in Equation (52).

Appendix D. Development of the early-sales model

We combine the geometric Brownian motion process with a single jump at time tES as follows:

( )dD e D dt σD dZ Y D d1 ( 1) Θ ,t τ ς t t t t t/22= − + + −+ (54) Where Θt is a deterministic time, discrete-state process:

t tΘ 1 if , 0 otherwise.

t ES= ⎧

⎨⎩

= (55)

In Equation (54), we add a drift term with a rate of e1 τ ς /22− + to offset the effect of the jump on the drift. Demand forecasts now follow a geometric Brownian motion process with a drift rate of e1 τ ς /22− + and an instantaneous volatility of σ when t tES< . At t tES= , the demand forecast jumps from Dt to e Dτ ς t/2

2+ , and then follows the geometric Brownian motion with the parameters e1 τ ς /22− + and σ. The log-demand process can be written as:

( )D D e σ t σZ τ ςlog ( ) log ( ) 1 1 2

(1 ) ( /2) 1 ,t τ ς t t t1 /2 2 {1 } 2 { }ES 2= + ⎛

⎝ − − ⎞

⎠ − + + ++ − < (56)

where t [0,1]∈ , and t1 ( )t{ }ES is an indicator function such that

12 This equation is used in the quantitative finance literature to model the default risk of assets (for a complete review of this jump-to-default model, see Gatheral, 2006).

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t t1 1 if , 0 otherwise.

t t ES

{ }ES = ⎧ ⎨⎩

< <

The distribution of a demand forecast made at time t [0,1]∈ and realized at time t 1= will be

D D D e σ t σ t ςlog ( ) (log ( ) (1 ) /2) (1 ), (1 ) 1 )t t τ ς t t1 /2 2 2 2 { }ES 2∼ + − − − − ++ <N (57)

From Equation (57) we can see that the log variance of the conditional distribution of demand before observing the early sales (i.e., t tES< ) is simply the weighted average of the squared jump and constant volatilities, and the median is the product of the median under constant volatility times the expected value of the jump factor.

Appendix E. Review of the cost-differential frontier

Here we review the calculation of the cost differential that underlies the constant instantaneous-volatility version of the Cost-Differential Frontier (de Treville et al., 2014b; a). In its simplest form, the Cost-Differential Frontier represents evolution of demand forecasts from time t 0= to t T= by a geometric Brownian motion. Actual demand is realized at time t T= . The total time period T is normalized by the longest lead time under con- sideration, which is equivalent to setting T 1= . The volatility parameter σ is then calibrated using the volatility for the full lead time.

Let Dt denote the initial demand forecast at time t, σ the constant instantaneous volatility, μ the rate at which median demand approaches average demand as t approaches 1, and Z( )t t 0≥ a standard Brownian motion. Then,

dD μD dt σD dZ .t t t t= + (58)

The demand forecast at time t [0,1]∈ follows a lognormal distribution with the following parameters:

D D D μ σ t σ tlog ( ) (log ( ) ( /2) (1 ), (1 ) ).t t1 2 2∼ + − − −N (59)

The expected value of the newsvendor profit for an order quantity Q is given by:

E π Q D p c Q p s Q D f D D dD[ ( ) ] ( ) ( ) ( ) ( ) ,t Q

t 0

1 1 1∫= − − − − (60) where f ( )⋅ ⋅ denotes the conditional demand density that follows a lognormal distribution given by Equation (59) for a geometric Brownian motion process with a drift of μ and a volatility of σ. The optimal value of Q maximizing Equation (60) is found by the critical-fractile approach:

Q F p c p s

,* 1 ⎜ ⎟= ⎛ ⎝

− −

⎞ ⎠

−

(61)

z Q D μ σ t

σ t log ( / ) ( /2) (1 )

1 .t*

* 2 =

− − − − (62)

Then,

E π Q D p s D f D D dD

p s D e z σ t

[ ( ) ] ( ) ( ) ,

( ) Φ ( 1 ). t

Q t

t μ t

* 0 1 1 1

(1 ) *

∫= − = − − −− (63)

Now, we consider a situation in which the decision maker places the profit-maximizing order quantity when t 0= (i.e., full lead time) at a unit cost of cl. Thus,

E π Q D p s D e z σ[ ( ) ] ( ) Φ ( ).l μ l *

0 0 *= − − (64)

If the decision maker places the profit-maximizing order quantity when t 0> (i.e., with a shorter lead time) at a unit cost c cs l> , the expected profit becomes

E π Q D p s D e z σ t[ ( ) ] ( ) Φ ( 1 ).s t t μ t s * (1 ) *= − − −− (65)

It follows from the equality of Equations (64) and (65):

z z σ t(1 1 ).s l * *= − − − (66)

As time t increases from 0 (the order is placed the full lead time before knowing demand) to 1 (the order is placed with full knowledge of demand), the volatility of demand will decrease from σ to 0. Expected demand is D e μ0 for all values of t because the process is a martingale. Median demand is D e μ σ0 /2

2− when t 0= , then increases to the mean as t approaches 1. By normalizing the median of the initial demand forecast density to one by setting D 10 = and μ σ /22= we are able to work with expected

demand, expected sales, the order quantity, and expected leftover inventory as multiples of median demand. Median demand approaches expected demand as t approaches 1 at a rate of σ /22 , so median demand at t will be eσ t/22 .

The quantity ordered at t 0= (i.e., with a full lead time) is set to achieve a target service level θ, corresponding to z θΦ ( )1= − multiplicative standard deviations above the median, where Φ ( )⋅ is the standard normal distribution function and Φ ( )1 ⋅− is the inverse of Φ ( )⋅ .

The order quantity as a multiple of the median demand forecast at t 0= then becomes:

Q e ,σz=

The fill rate will be (for a derivation of the fill rate, see de Treville et al., 2014a):

z σ z σ e zΩ ( , ) Φ ( ) (1 Φ ( ) ).zσ σ 2 2

= − + −− (67)

Expected demand as a multiple of median demand at t 0= will be:

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58

e[Demand] .σ /22= (68)

Expected sales will be:

e z σ[Sales] Ω ( , ).σ /22= (69)

Expected leftover inventory will be the difference between the order quantity and expected sales:

e e z σ[Leftover inventory] Ω ( , ).zσ σ /22= − (70)

If an order is placed based on exact information as to the quantity demanded, the expected profit per unit of expected demand π( )t t 1= will be:

π p c( ) ,t t s1 = −= (71)

where p denotes the retail price, and cs is the short-lead-time cost of the product per unit. We then solve for the breakeven long-lead-time cost cl:

( )p c p c z σ c s e z σ( ) Ω ( , ) ( ) Ω ( , ) ,s l l zσ σ /22− = − − − −− (72) where s denotes the salvage value per unit. The cost differential that must be offered by the long-lead-time supplier to compensate for the mismatch cost is c c1 /l s− . Similarly, the cost premium a retailer is willing to pay at most to reduce lead time is c c/ 1s l − . By definition, both cost differential and cost premium are positively correlated with the mismatch cost:

p c c s[Mismatch cost] ( ) [Lost sales] ( ) [Leftover inventory],l l= − + −  

p c c s( ) ( [Demand] [Sales] ) ( ) [Leftover inventory],l l= − − + −  

( )p c e z σ c s e e z σ( ) (1 Ω ( , ) ) ( ) Ω ( , ) .l σ l zσ σ/2 /22 2= − − + − − (73)

Appendix F. Fitting a jump model from time-series data

In this section we apply prescriptive analytics to raw data to explore the kinds of jumps that are routinely encountered in analyzing time-series data. We randomly chose two products from a set of 4000 products sold by a large supermarket chain for which we have point-of-sale data. The first product exhibited behavior strongly suggestive of jumps, while the other was consistent with a constant-volatility assumption. We use the first product to illustrate how one estimates jump-model parameters from time-series demand data, then use the second to explore the implications of force fitting a jump model in the absence of obvious jumps. We only considered these two products. In future research, it would be interesting to observe the jumps that occur over the entire range of products for this data set and compare those values to other product types to gain an idea of the range of jumps that one might expect to encounter across a wide variety of demand settings.

For the two products, we considered demand over 100 days. We followed the protocol recommended by Shumway and Stoffer (2011, following the protocol presented in Tables 3.1 and 3.3) to fit a model that combines seasonal (S), autoregressive (AR), integrated (I), and moving average (MA) components. The resulting SARIMA model is parameterized as ARIMA a b c x y z( , , ) ( , , )n. The seasonality n is estimated by observing cycles that show up in the data. The first parameter values a and x specify the number of lags included in the autoregressive component (AR), first for the original values and then for those values after a lag of n periods, referred to as the seasonality. These autoregressive parameter values are estimated by observing when partial autocorrelations disappear. The second parameter values b and y specify the differencing (I) required to compensate for any trend in the data: When time-series data shows a trend, differencing the data points in sequence yields a stationary process ( x x xΔ t t t 1= − − ). Differencing can be applied several times for non-linear trends (denoted by xΔd t for d applications of differencing). The third parameter values c and z specify the number of lags included in the moving average (MA) component.

This SARIMA model allows us to replace extreme values of demand by a forecast. The extreme values are then used to fit the jump portion of the model.

F.1. Product 1

The top panel (panel a) shows autocorrelations of the natural logarithm of daily sales. The weekly autocorrelations suggest a seasonality effect for a lag of 7 days. The daily autocorrelations do not show a consistent pattern. The bottom panel (panel b) shows the partial autocorrelations. Error bounds are set to + n2/ 0.2= + , corresponding to a 95% confidence interval for both panels. The weekly partial autocorrelations cut off after one lag: A partial autocorrelation that exceeds the lower error bound on day 8, for example, is followed by one that is close to zero on day 9, so that a x 1= = . The time-series data shows neither a trend, nor a moving average, and has a seasonality of 7 days, so we fit a SARIMA model of type ARIMA (1,0,0) (1,0,0)7.

In this figure we analyze the goodness of fit of the SARIMA model of type ARIMA(1,0,0) (1,0,0)7. The top panel (panel a) shows the standardized residuals. The middle-left panel (panel b) shows the autocorrelation values for these residuals. The error bounds given in panel b correspond to a 95% confidence interval: The majority of autocorrelation values lie within these bounds, suggesting that the time-series model is a good fit with the sales data and that it explains all the autocorrelation in the original data. The middle-right panel (panel c) shows the Q-Q plot of residuals, which suggests that residuals follow a normal distribution for the kurtosis to the far right. The bottom panel (panel d) shows the p-values for the Ljung-Box test statistics. The null hypothesis of this test is that the residuals are not correlated for most lags. Panel d shows that the null hypothesis cannot be rejected.

Fig. 7 shows the autocorrelation (panel a) and partial autocorrelation (panel b) for the natural log of daily sales over 100 days for Product 1. The autocorrelation plot of the log of the daily sales shows continuous decay for daily and weekly autocorrelations. The weekly partial autocorrelations disappear after one lag, suggesting an AR1 model type. There does not appear to be a moving-average effect. These graphs suggest clear weekly seasonality, so we capture the seasonality (S) component of the SARIMA model by setting n 7= . As our time-series data does not show a trend we set b y 0= = . The resulting SARIMA model is of type ARIMA(1,0,0) (1,0,0)7.

Fig. 8 shows the standard output from the R package made available by Shumway and Stoffer (2011). The standardized residuals are shown in

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panel a. From panel b of Fig. 8 we see that most autocorrelation values fall within 95% confidence intervals for a white-noise process, suggesting that the SARIMA model of type ARIMA (1,0,0) (1,0,0)7 provides a good fit to the data.

13

The top panel (panel a) shows outliers that are more than three standard deviations from zero. The bottom panel (panel b) shows the impact over time of these outliers on the standardized residuals. The outlier that occurs right before day 60, for example, affects subsequent residuals for almost a week. Thus, interpreting an outlier as white noise may well reduce the forecasting ability of the model.

The top panel (panel a) shows autocorrelations of the natural logarithm of daily sales for the adjusted model. The daily autocorrelations suggest seasonality effect of lag 7. The weekly autocorrelations decay continuously, whereas the daily autocorrelations cut off after lag 4. The bottom panel shows partial autocorrelations. The weekly partial autocorrelations cut off after one lag. We also observe that the daily partial autocorrelations do not show a regular pattern. Therefore, we selected a SARIMA model of type ARIMA(0,0,4) (1,0,0)7.

Although the model fit based on assuming that all variation in daily demand comes from a constant-volatility paradigm, the Normal Q-Q plot shown in panel c of Fig. 8 shows kurtosis. This suggests that the demand data has some extreme values. The standardized residuals are shown in Fig. 9. Four of 100 residuals were more than three standard deviations from zero, which corresponds to λ 0.04= in our jump model. To explore the impact of these extremes on development of the cost-premium frontier, we fit a revised SARIMA model based on the assumption that such outliers represent jumps rather than normal volatility. The revised autocorrelation plot shown in Fig. 10 (panel a) shows a cutoff point after 4 lags for daily sales, suggesting an MA4 model. The revised partial autocorrelation plot shown in Fig. 10 (panel b) no longer shows a cutoff point for daily sales, and cuts off after one lag for weekly sales. The type of SARIMA model thus changes from ARIMA (1,0,0) (1,0,0)7 to ARIMA (0,0,4) (1,0,0)7 with the new interpretation of the four outliers. The SARIMA model based on the adjusted data performs better than the original SARIMA model. We thus note that the outliers had hidden a strong 4-day moving-average effect. The revised SARIMA model performed better in forecasting daily demand than did the original model that included the extremes.

In this figure we analyze the goodness of fit of the SARIMA model of type ARIMA(0,0,4) (1,0,0)7 to the natural logarithm of daily sales data after the outliers were replaced by their forecasts. The top panel (panel a) shows the standardized residuals. The middle-left panel (panel b) shows the autocorrelation values for these residuals. The error bounds given in panel b correspond to a 95% confidence interval: The majority of auto- correlation values lie within these bounds, suggesting that the time-series model is a good fit with the sales data and that it explains all the autocorrelation in the original data. The middle-right panel (panel c) shows the Q-Q plot of residuals, which suggests that residuals follow a normal distribution. The bottom panel (panel d) shows the p-values for the Ljung-Box test statistics. The null hypothesis of this test is that the residuals are not correlated for most lags. The p values are less than 0.05 so that we can reject the null hypothesis. However, we saw from panel c that the selected model fits well to the data.

After replacing each outlier by the SARIMA forecast for that day, we analyzed the adjusted time-series data as shown in Fig. 11. Note that this eliminates the kurtosis. The demand volatility was re-estimated using the adjusted data: Modeling the outliers as jumps rather than as white noise reduced the estimate of volatility from 0.169 to 0.11. The four outliers were used to estimate the parameters of the jump model, with τ 0.28= and ς 0.54= . The parameters for the continuous volatility and jump versions of the SARIMA model are summarized in Table 3.

Fig. 7. Autocorrelation and partial autocorrelation plots for the natural logarithm of daily sales for Product 1.

13 As usually occurs, the Ljung-Box test p values—applied to data cleansed from seasonality and AR1 effects as shown in Fig. 8, panel d—are too high to permit rejecting the null hypothesis that residuals are independent for most lags.

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In this table, we summarize the parameters of the constant-volatility and jump models. We used the standardized residuals given in Fig. 8 to estimate the σ value of the constant-volatility model. Then, we used the outliers given in Fig. 9 and the standardized residuals given in Fig. 11 to estimate the parameters of the jump model.

In Fig. 12a, we present the cost-premium frontiers for a jump and a constant-volatility forecasting model using the parameters given in Table 3 assuming a high critical fractile (99.5%). Whether outliers are assumed to represent jumps or white noise has a substantial impact on the value of responsiveness: The premium worth paying to postpone the production enough to obtain full demand information is around 23% in the case of jumps, and just over 5% in the case of a constant-volatility assumption.

Fig. 12b shows the cost-premium frontiers for a jump and a constant-volatility forecasting model using the parameters given in Table 3 assuming a lower critical fractile (70%). Paradoxically, modeling the outliers as jumps reduces the justified cost premium when the target service level is lower: That the adjusted SARIMA model provides an improved forecast combines with the less-ambitious service level to reduce the premium that a decision maker should be willing to pay to reduce the decision lead time.

This example demonstrates how to estimate jump parameter values from time-series data in a case where a SARIMA model is an appropriate choice. It also shows that the extra step of fitting the jump model provides useful information to consider in estimating the value of responsiveness. This was the first product that we examined, chosen randomly from 4000 products available in our data base, which suggests that extreme values may be routinely encountered.

Panel a shows the cost-premium frontier for a jump and a constant-volatility forecasting model using the parameters given in Table 3 and assuming a high critical fractile of 99.5%, obtained by normalizing the unit price to 100, then setting the cost of the unresponsive supplier to 5% of the unit price, and the residual value to 4.5% of the unit price. The analysis in panel a shows that for this example, Whether outliers are assumed to represent jumps or white noise has a substantial impact on the value of responsiveness: The premium worth paying to postpone the production enough to obtain full demand information is around 23% in the case of jumps, and just over 5% in the case of a constant-volatility assumption. Other combinations of price, cost, and residual that yield the same critical fractile result in a very similar cost-premium frontier. Panel b shows the cost- premium frontier for a jump and a constant-volatility forecasting model using the parameters given in Table 3, but now assuming a lower critical fractile of 70%, obtained by again normalizing the unit price to 100, then setting the cost of the unresponsive supplier to 30% of the unit price, and the residual value to 0. Here we see that the cost-premium frontiers are very close to each other, so treating the outliers as jumps or as white noise makes little difference. Other price, cost, and residual value combinations again result in very similar cost-premium frontiers. In fact, because the adjusted SARIMA model forecasts better than the original model, accounting for jumps actually slightly lowers the cost-premium frontier when the service-level target is modest.

Fig. 8. Fitting a time-series model for Product 1.

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Fig. 9. Outliers of the standardized residuals.

Fig. 10. Autocorrelation and partial autocorrelation plots after correcting for the outliers for Product 1.

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Fig. 11. Fitting a time-series model after the outliers were replaced by the forecast.

Table 3 Estimation of constant volatility and jump parameters for Product 1.

Model σ λ τ ς SARIMA parameters

Constant Volatility 0.169 N/A N/A N/A ARIMA(1,0,0) (1,0,0)7 Jump diffusion 0.11 0.04 0.28 0.54 ARIMA(0,0,4) (1,0,0)7

Fig. 12. Cost-premium frontiers for Product 1, assuming high and moderate critical fractiles.

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F.2. Product 2

As with Product 1, we fit a SARIMA model to the natural logarithm of daily sales over 100 days. The autocorrelation plot shown in Fig. 13, panel a indicates that daily and weekly autocorrelations decay continuously: The moving-average effect does not appear to be significant. This panel also shows clear weekly seasonality. The partial autocorrelations (Fig. 13, panel b) disappear after one lag, consistent with a model of type AR1. Thus, we again fit a SARIMA model of type ARIMA (1,0,0) (1,0,0)7.

The top panel (panel a) shows autocorrelations of the natural logarithm of daily sales. The weekly autocorrelations suggest a seasonality effect for a lag of 7 days. The daily autocorrelations do not show a consistent pattern. The bottom panel (panel b) shows partial autocorrelations. The weekly partial autocorrelations cut off after one lag. We also observe a strong partial autocorrelation on day 8, and it cuts off on day 9. Therefore, we selected the SARIMA model of type ARIMA (1,0,0) (1,0,0)7. In both panels, error bounds correspond to a 95% confidence interval.

Fig. 14, panel b shows that the autocorrelation of residuals is low for different lags.14 Given that the autocorrelation values for the first 20 lags are also very low the SARIMA model of type ARIMA (1,0,0) (1,0,0)7 provides a good fit to the Product 2 data. All of the residuals lie within three standard deviations of zero and the Q-Q plot does not suggest kurtosis. In contrast to the first product analyzed, this demand data does not suggest jumps.

What happens to the cost-premium frontier when an enthusiastic user is determined to fit a jump model even though the demand data is not suggestive of jumps? We explored this scenario by reducing the threshold for defining outliers from three to 2.52 standard deviations, which was the number of standard deviations that resulted in exactly four outliers, so that λ 0.04= as for the first product analyzed. The revised threshold and resulting outliers are shown in Fig. 15. We then used these outliers—replaced by the SARIMA forecast—to fit a jump model as described for Product 1. The revised time-series model is developed as shown in Fig. 16. The parameter values for both versions of the SARIMA model are given in Table 4. The form of the SARIMA model remained ARIMA(1,0,0) (1,0,0)7 whether the four outliers were treated as jumps or as white noise.

Fig. 17a shows the cost-premium frontiers for a jump and a constant-volatility forecasting model using the parameters given in Table 4 assuming a high critical fractile (99.5%). As in Fig. 12a, whether outliers are assumed to represent jumps or white noise has a substantial impact on the value of responsiveness. Interpretation thus calls for judgment on the part of decision makers as to whether the outliers should be interpreted as emerging from a constant-volatility process or as evidence of a jump when the critical fractile is very high.

In this table, we summarize the parameters of the constant-volatility and jump models. We used the standardized residuals given in Fig. 14 to estimate the σ value of the constant-volatility model. Then, we used the standardized residuals given in Fig. 16 and the outliers given in Fig. 15 to estimate the parameters of the jump model.

In Fig. 17b, we show the cost-premium frontiers for a jump and a constant-volatility forecasting model using the parameters given in Table 4 assuming a lower critical fractile (70%). The cost-premium frontiers are very close to each other.

In this figure we analyze the goodness of fit of the SARIMA model of type ARIMA(1,0,0) (1,0,0)7. The top panel (panel a) shows the standardized residuals. The middle-left panel (panel b) shows the autocorrelation values for these residuals. The error bounds given in panel b correspond to a 95% confidence interval: All of the autocorrelations lie within these bounds, suggesting that the time-series model is a good fit with the sales data and that it explains all the autocorrelation in the original data. The middle-right panel (panel c) shows the Q-Q plot of residuals, which suggests that

Fig. 13. Autocorrelation and partial autocorrelation plots for Product 2.

14 As with the first product and as usually occurs, the Ljung-Box test statistics do not let us reject the null hypothesis that the residuals are not correlated for most lags.

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Fig. 14. Fitting a time-series model to daily sales of Product 2.

Fig. 15. Outliers of the standardized residuals for Product 2.

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Fig. 16. Fitting a time-series model after the outliers were replaced by the forecast for Product 2.

Table 4 Estimation of constant-volatility and jump parameters for Product 2.

Model σ λ τ ς SARIMA parameters

Constant volatility 0.123 N/A N/A N/A ARIMA(1,0,0) (1,0,0)7 Jump diffusion 0.111 0.04 0.14 0.248 ARIMA(1,0,0) (1,0,0)7

Fig. 17. Cost-premium frontiers for Product 2.

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residuals follow a normal distribution. The bottom panel (panel d) shows the p-values for the Ljung-Box test statistics. The null hypothesis of this test is that the residuals are independent for most lags. The p values are too high to allow us to reject the null hypothesis.

The top panel (panel a) shows outliers that are more than 2.52 standard deviations from zero. The bottom panel (panel b) shows the impact over time of these outliers on the standardized residuals. The outlier that occurs on day 50, for example, slightly affects subsequent residuals for almost a week.

In this figure we analyze the goodness of fit of the SARIMA model of type ARIMA(1,0,0) (1,0,0)7 to the natural logarithm of daily sales data after the outliers were replaced by their forecasts. The top panel (panel a) shows the standardized residuals. The middle-left panel (panel b) shows the autocorrelation values for these residuals. The error bounds given in panel b correspond to a 95% confidence interval: All of autocorrelation values lie within these bounds, suggesting that the time-series model is a good fit with the sales data and that it explains all the autocorrelation in the original data. The middle-right panel (panel c) shows the Q-Q plot of residuals, which suggests that residuals follow a normal distribution. The bottom panel (panel d) shows the p-values for the Ljung-Box test statistics. The null hypothesis of this test is that the residuals are not correlated for most lags. The p values are too high to allow us to reject the null hypothesis. The results are not significantly different from those given in Fig. 14.

In panel a, we plot the cost-premium frontier for a jump and a constant-volatility forecasting model using the parameters given in Table 4 assuming a high critical fractile (99.5%). We normalize the unit price to 100. The cost of the unresponsive supplier is 5% of the unit price, and the residual value is 4.5% of the unit price. Whether outliers are assumed to represent jumps or white noise has a substantial impact on the value of responsiveness: The premium worth paying to postpone the production enough to obtain full demand information is around 18% in the case of jumps, and just around 4% in the case of a constant-volatility assumption. Other combinations of price, cost, and residual that yield the same critical fractile result in a very similar cost-premium frontier. In panel b, we plot the cost-premium frontier for a jump and a constant-volatility forecasting model using the parameters given in Table 4 assuming a lower critical fractile (70%). The unit price is normalized to 100, the cost of the un- responsive supplier is 30% of the unit price, and the residual value is 0. We see that the cost-premium frontiers are very close to each other. Other price, cost, and residual value combinations again result in very similar cost-premium frontiers.

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