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MANUFACTURING & SERVICE OPERATIONS MANAGEMENT

Vol. 14, No. 3, Summer 2012, pp. 472–484 ISSN 1523-4614 (print) — ISSN 1526-5498 (online)

http://dx.doi.org/10.1287/msom.1120.0387 © 2012 INFORMS

INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/.

A Multiordering Newsvendor Model with Dynamic Forecast Evolution Tong Wang NUS Business School, National University of Singapore, Singapore 119245, [email protected]

Atalay Atasu College of Management, Georgia Institute of Technology, Atlanta, Georgia 30308, [email protected]

Mümin Kurtulu¸s Owen Graduate School of Management, Vanderbilt University, Nashville, Tennessee 37203, [email protected]

W

e consider a newsvendor who dynamically updates her forecast of the market demand over a finite planning horizon. The forecast evolves according to the martingale model of forecast evolution (MMFE). The newsvendor can place multiple orders with increasing ordering cost over time to satisfy demand that realizes at the end of the planning horizon. In this context, we explore the trade-off between improving demand forecast and increasing ordering cost. We show that the optimal ordering policy is a state-dependent base-stock policy and analytically characterize that the base-stock level depends on the information state in a linear (loglinear) fashion for additive (multiplicative) MMFE. We also study a benchmark model where the newsvendor is restricted to order only once. By comparing the multiordering and single-ordering models, we quantify the impact of the multiordering strategy on the newsvendor’s expected profit and risk exposure. Key words: newsvendor; MMFE; forecast evolution; dynamic ordering History: Received: April 2, 2010; accepted: January 27, 2012. Published online in Articles in Advance May 4, 2012.

1.

Introduction

early and late orders may help achieve better operational performance. This research studies a multiordering strategy for a firm that sells a seasonal product. We formulate a dynamic newsvendor model. The newsvendor has multiple opportunities to place orders at different times before the demand is realized. Multiple ordering options can be found in several practical contexts: it can be that the newsvendor sources from multiple suppliers that require different lead times and costs (Yan et al. 2003); or that the newsvendor produces inhouse using multiple technologies with different costs and lead times (Donohue 2000); or that the newsvendor orders from a single supplier offering a menu of advance purchase discounts (Tang et al. 2004). Early orders are cheaper but are exposed to higher demand uncertainty, and late orders incur cost premiums. To reduce demand uncertainty, the newsvendor constantly updates her demand forecast based on market information observed over time, before demand is realized. The information can be expert estimates (Fisher and Raman 1996), market research reports (Donohue 2000), retail test results (Fisher and Rajaram 2000), etc., which are valuable in forecasting the final demand. To model such demand information and

Creative businesses, such as those in toy and fashion industries, have difficulty matching supply and demand due to high demand uncertainty, long supply lead times, and short selling seasons. Firms commit to orders well in advance of the selling season (e.g., up to nine months in the apparel industry) in face of high uncertainty as initial demand forecasts available for planning are highly inaccurate due to the unpredictable nature of customer preferences and constantly changing market trends (Fisher et al. 1994, Boyaci and Özer 2010, Wang and Tomlin 2009). At the same time, new information about demand gradually becomes available and the uncertainty resolves as the selling season approaches. To take advantage of more accurate demand information, firms may need to postpone their order commitment until closer to the selling season. This requires the use of more flexible production technologies and facilities or more responsive suppliers, which are often more expensive. As such, many apparel (Agrawal et al. 2002) and toy (Wong et al. 2005) companies respond to constantly evolving market information by ordering from a portfolio of sources that differ in lead times and costs. In this context, an optimally designed portfolio of 472

Wang, Atasu, and Kurtulu¸s: Multiordering Newsvendor with Dynamic Forecast Evolution

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Manufacturing & Service Operations Management 14(3), pp. 472–484, © 2012 INFORMS

forecast evolution process, we adopt the martingale model of forecast evolution (MMFE) developed by Hausman (1969), Graves et al. (1986), and Heath and Jackson (1994), which is also a special case of the generalized MMFE by Oh and Özer (2012) for a single forecaster. Observing an evolving demand forecast, the newsvendor dynamically decides on the order quantity at each possible ordering opportunity. In this setting, we explore the trade-off between increasing ordering cost and improving demand information. We provide a complete characterization of the optimal ordering policy. We show that the optimal policy is a state-dependent base-stock policy, where the state represents observed market information. Moreover, with additive forecast adjustments, there is a simple linear relationship between the optimal basestock level and the information state: the base-stock level is the sum of updated demand forecast and a safety stock term. Similarly, a log-linear relationship holds in the multiplicative case: logarithm of the base-stock level is a linear function of information state. We demonstrate how the safety stock term captures the trade-off between ordering too early and too late, in addition to the classic single-period newsvendor trade-off of ordering too much versus too little. We also show that the safety stock term is independent of the forecast evolution. It is characterized by a series of one-dimensional recursive equations, which can be solved off-line. This finding substantially simplifies the calculation of the optimal base-stock level and can be generalized for scenarios with fixed ordering costs and cancelations (see the online supplement, available at http://msom.journal.informs.org/, for details). Next, to illustrate the benefits of the multiordering strategy, we consider a benchmark model where the newsvendor is restricted to place a single order. We consider two scenarios. In the static singleordering model, the newsvendor chooses the timing of her order at the beginning of the planning horizon, whereas in the dynamic single-ordering model, the newsvendor decides on her order timing dynamically, based on observed demand information. We find that the newsvendor does not necessarily benefit from making the timing decision dynamically under the single-ordering benchmark. The static and dynamic single-ordering strategies result in the same expected profit under multiplicative MMFE, and the profit difference is marginal under additive MMFE. Finally, we numerically quantify the benefits of the dynamic multiordering strategy by comparing it with the single-ordering benchmark. As expected, the multiordering strategy always yields a higher expected profit. The analysis also suggests that the multiordering strategy is more valuable in boosting profit when (i) demand is highly uncertain, (ii) ordering

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options differ significantly in their designated order timing, and/or (iii) the ordering options lead to similar expected profits if they are the only source that can be ordered from. Our analysis also illustrates that the profit variability can sometimes be higher under the multiordering strategy, which may not be desirable. A detailed investigation of this phenomenon, however, reveals that the increase in profit variability is driven by the up-side risk, i.e., variability of outcomes with better-than-expected profits. The down-side risk, i.e., outcomes below expected profit, can be mitigated by multiple orders. The rest of this paper is organized as follows. We review the related literature in §2. Model setup and assumptions are described in §3. In §4, we analyze the optimal policies for multiordering and singleordering strategies. A numerical comparison between the two strategies is conducted in §5. In §6, we conclude with a summary of our results. All proofs are provided in the appendix.

2.

Literature Review

Our research contributes to the literature that incorporates forecast updating into inventory control decisions. This literature can be broadly categorized into two main streams. The first stream studies newsvendor models (i.e., short selling seasons). A series of papers in this stream focus on generating managerial insights by studying stylized models with up to two ordering periods (opportunities). Fisher and Raman (1996) consider a multiproduct capacitated production problem with demand learning from the actual sales in the first period. Gurnani and Tang (1999) analyze the tradeoff between demand uncertainty and purchasing cost uncertainty: although demand uncertainty can be reduced by forecast updating, the purchasing cost becomes uncertain in the second period. Donohue (2000) examines contracting and coordination issues when there is a cheaper production mode and a more expensive but quicker production mode with forecast updating. Özer et al. (2007) study a dual sourcing contract with a forecast update in a similar setting. Erhun et al. (2008) analyze the benefits of information updating and the strategic interactions between a buyer and a seller in a two period model with dynamic pricing and procurement. Yan et al. (2003) focus on the tradeoff between increasing cost and demand uncertainty in a dual-supplier system and present an industry application of the model. Huang et al. (2005) study a problem with costly order adjustment in the second period after obtaining an improved forecast. Our first contribution to this stream of literature is extending these models by (i) modeling a multiperiod forecast evolution by the general MMFE, (ii) allowing for

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multiple ordering opportunities, and (iii) providing an explicit analytical characterization of the optimal order quantity. A number of other papers in this stream consider multiple ordering opportunities. Wang and Tomlin (2009) study a newsvendor that dynamically decides when to place a single order on a continuous time line. They assume a stochastic delivery lead time and model the demand forecast by a multiplicative Markovian forecast revision, which is equivalent to our multiplicative MMFE. They focus on the trade-off between improving demand information and increasing risk of not receiving the delivery on time (the ordering cost is constant in their model). Oh and Özer (2012) propose a generalized MMFE model, which allows asymmetric information to be combined into a forecast evolution, and Boyaci and Özer (2010) study a manufacturer’s optimal timing of capacity commitment in a general setting with advance sales information and price dependent demand. In our dynamic single-ordering model, we study a similar optimal stopping problem as in the aforementioned papers but with different trade-offs. In addition, our main focus is on managing a portfolio of multiple ordering options. Another relevant paper in this stream is by Song and Zipken (2009), who study a problem where the initially ordered inventory can be sold off at multiple points in time when partial demand information is revealed. On the contrary, we assume that inventory can be built-up as demand information is revealed over time, and we focus on characterizing the optimal policy. The second stream is concerned with multiperiod inventory models (i.e., long selling horizons with multiple periods) under forecast evolution (Güllü 1997, Graves et al. 1998, Aviv 2001, Toktay and Wein 2001, Iida and Zipkin 2006, Lu et al. 2006, to name a few). This literature incorporates MMFE into inventory/production problems. Our research differs from this stream by incorporating MMFE into a simpler dynamic newsvendor model that only allows demand in the last period. Yet the newsvendor can order in multiple periods in response to the forecast evolution, which preserves the dynamic flavor of the ordering policy. This allows us to explicitly characterize the optimal state-dependent base-stock policy. This characterization is of theoretical significance as it demonstrates the cost-information trade-off in an intuitive and transparent way, and it is of practical interest as the obtained optimal policy is easy to calculate and implement.

3.

Model Setup

Throughout the paper, we refer to a period as a point on the time line; a period is equivalent to a time

epoch. The planning horizon consists of N +1 periods, from 1 to N + 1. The first N periods are ordering periods, representing N ordering options available to the newsvendor at different times. Sales take place instantaneously at the end, in period N + 1, which is also referred to as the selling season. The retail price, denoted by r, is exogenously given. There is no salvage value for unsold inventory, no penalty for unsatisfied demand, and no discounting. Before the selling season, the newsvendor can order the product in periods 1 to N . Let cn be the cost of ordering one unit in period n. We assume 0 < c1 < c2 < · · · < cN < r, i.e., it is cheaper to order earlier. Note that otherwise, if there exists ci ≥ ci+1 or ci ≥ r, period i can be eliminated as the newsvendor would never order in that period. We do not explicitly consider inventory holding costs that might be incurred for the goods ordered; they can be easily embedded into ordering costs. We also note that although we assume no fixed ordering costs and cancelations in our model, our results can be extended to include these under certain conditions (see the online supplement). Market demand D is a random variable and realizes in period N + 1. As market signals unveil gradually over the planning horizon, the demand forecast improves over time. Let Dn be the forecast of demand D in period n. We assume that the initial forecast D1 is given, the final forecast DN +1 is simply the realization of demand D. We model the forecast process 8Dn 1 n = 11 0 0 0 1 N + 19 by the MMFE. We adopt a special case of the MMFE setup from Oh and Özer (2012) who provide a generalized model for multiple forecasters. We also refer the reader to Heath and Jackson (1994), Toktay (1998), and Chen and Lee (2009) for a discussion on the applicability of MMFE in operational problems. We consider both additive and multiplicative MMFE that differ in how forecasts are adjusted. In the additive model (a-MMFE), the demand forecast in period n = 21 0 0 0 1 N + 1 is given by Dn = D1 + …2 + 0 0 0 + …n , where …i represents forecast adjustments during period i = 21 0 0 0 1 n. The updates are independent and normally distributed with mean 0 and variance ‘i2 . Let Œ = D1 be the expected demand in period 1 and In = …2 + · · · + …n be the cumulative forecast adjustment up to period n for the a-MMFE case. In the multiplicative model (m-MMFE), the forecast adjustments, i.e., ratios of successive forecasts, are Dn /Dn−1 = exp4…n 5, where …n is normally distributed with mean −‘n2 /2 and variance ‘n2 . Let Œ = P log D1 − Ni=2+1 ‘i2 /2 be expectation of the logarithm of P demand in period 1 and In = ni=2 4…i + ‘i2 /25 be the mean-adjusted cumulative forecast adjustment. Then, the estimate of D after observing In under a-MMFE (respectively, m-MMFE) is normal (log-normal) with

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Manufacturing & Service Operations Management 14(3), pp. 472–484, © 2012 INFORMS

P +1 2 parameters 4Œ + In 1 ‘˜ n2 5, where ‘˜ n2 = Ni=n+1 ‘i represents the residual uncertainty after period n. Both 8Dn 9 and 8In 9 are sufficient to represent up-to-date information and describe demand forecast evolution. Accordingly, when we formulate the dynamic program (DP), we use In instead of Dn as the information state in period n. This enables us to represent the additive and the multiplicative cases in the same formulas.

4.

Analysis of the Ordering Strategies

In this section, we first analyze the dynamic multiordering strategy in §4.1 and establish the structure of the optimal ordering policy. Then, in §4.2, we study a single-ordering strategy, which is used as a benchmark to illustrate the value of placing multiple orders. 4.1. The Multiordering Strategy With this strategy, the newsvendor can order multiple times and replenish the stock level in response to the most up-to-date demand forecast. The newsvendor can spread her orders over the planning horizon to take advantage of lower ordering costs with early orders and more accurate demand forecasts with late orders. We formulate the discrete-time dynamic program as follows. The decision periods are n = 11 0 0 0 1 N . The state variables are 4xn−1 1 In 5, where xn−1 is the inventory state tracking the total number of units ordered before period n (with x0 = 0), and In is the information state representing the up-to-date information about forecast adjustments (with I1 = 0). The decision in period n is Qn , a nonnegative order quantity. The sequence of events in an ordering period are (1) observe In , (2) update demand forecast, (3) review inventory position xn−1 , (4) order Qn and incur cost cn Qn , and (5) raise inventory position to xn = xn−1 + Qn . The events in the selling season are (1) demand D is realized, and (2) revenue r · min8D1 xN 9 is collected. Let Vn be the profit-to-go function at the beginning of period n after observing In . Then, Vn 4xn−1 1 In 5 = max 8EIn+1 — In 6Vn+1 4xn 1 In+1 57 − cn 4xn − xn−1 591 xn ≥xn−1

n = 11 0 0 0 1 N − 13 (1)

Gn 4xn 1 In 5 = EIn+1 — In 6Vn+1 4xn 1 In+1 57 − cn xn 1 n = 11 0 0 0 1 N − 13 GN 4xN 1 IN 5 = ED — IN 6r min4D1 xN 57 − cN xN 0

= max 8ED — IN 6r min4D1 xN 57 − cN 4xN − xN −1 590 (2) xN ≥xN −1

M

The total expected profit is ç = V1 401 05. We can rewrite (1) and (2) as Vn 4xn−1 1 In 5 = max 8Gn 4xn 1 In 59 + cn xn−1 1 xn ≥xn−1

n = 11 0 0 0 1 N 3

(3)

(5)

Intuitively, we can see that if the observed information state In is large, this implies a high demand in the selling season, and the expected profit-to-go Vn will increase. (Throughout the paper we use increasing and decreasing in a weak sense, i.e., increasing means nondecreasing.) Also for the ordered inventory xn−1 , if it increases, the profit-to-go will also increase, but the marginal benefit of an additional unit of inventory diminishes. Proposition 1. The optimal ordering policy is a statedependent base-stock policy. In each period n, n = 11 0 0 0 1 N , there exists an optimal base-stock level Sn 4In 5 = arg maxx 8Gn 4x1 In 59, which is a function of the current information state In . Proposition 1 states that the newsvendor should raise the inventory position up to the threshold Sn 4In 5 if the inventory position is below this threshold. This threshold depends on the updated forecast through In and is given by a function Sn 4In 5. This is an established result, and is consistent with the policy structures that can be found in both streams of research reviewed in §2. This result extends the two-period model in Özer et al. (2007) to multiple periods under MMFE, and is similar to the policy structures in the multiperiod inventory models in Burnetas and Gilbert (2001) and Iida and Zipkin (2006). However, although one can solve the two-dimensional dynamic program numerically and find the optimal base-stock level in each period for any given In , this could be computationally intensive. In Proposition 2, we analytically show that Sn 4In 5 is a linear (log-linear) function of In for a-MMFE (m-MMFE), and further characterize the linear (log-linear) relationship. This result simplifies the calculation of the base-stock levels. Proposition 2. Under a-MMFE, the optimal basestock level for period n, Sn 4In 5, n = 11 0 0 0 1 N , is a linear function of In , and can be written as Sn 4In 5 = Œ + In + bn . The constant bn is the solution to gn 4yn 5 = 0, where the function gn 4 · 5 is given by Z 4yn −bn+1 5/‘n+1 gn 4yn 5 = gn+1 4yn −‘n+1 †5dê4†5+cn+1 −cn 3 −ˆ

VN 4xN −1 1 IN 5

(4)

(6) ¯ gN 4yN 5 = r ê



yN ‘N +1

 − cN 0

(7)

Here ê4 · 5 is the standard normal cumulative distribution ¯ function (c.d.f.), and ê4x5 = 1 − ê4x5. Under m-MMFE, the optimal base-stock level Sn 4In 5 is a log-linear function of In , and can be written as Sn 4In 5 = exp4Œ + In + bn 5, where bn is also defined by (6) and (7).

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The detailed proof of the above proposition is provided in the appendix. Here we sketch the proof (for a-MMFE) and discuss the key findings. Because by definition, Sn 4In 5 maximizes Gn 4·1 In 5, we check the first-order derivative of Gn 4·1 In 5, which is of the following recursive form (n = 11 0 0 0 1 N − 1): ¡Gn 4xn 1 In 5 ¡xn Z …˜n+1 ¡G 4x 1 I + … 5 n+1 n+1 n n+1 = dFn+1 4…n+1 5 xn+1 =xn ¡xn+1 −ˆ + cn+1 − cn 0

(8)

2 Here …n+1 = In+1 − In , which follows N401 ‘n+1 5, Fn+1 is the c.d.f. of …n+1 , and …˜n+1 (as a function of xn and In ) is the threshold such that in period n + 1, the orderup-to level Sn+1 4In + …n+1 5 is less than xn if …n+1 ≤ …˜n+1 , or equivalently, nothing should be ordered in n + 1 if the forecast adjustment …n+1 is below the threshold. We note that the problem in the last period reduces to the standard newsvendor problem, and Equation (7) is the first-order condition of the standard problem. Expanding the recursive Equation (8), we can write

¡Gn 4xn 1 In 5 = −cn + cn+1 Pn+1 + · · · + cN PN + rP 1 ¡xn

(9)

where Pi , i = n + 11 0 0 0 1 N is the probability that after period n, the next order is placed in period i; and P is the probability that no order is placed in all the remaining periods n + 11 0 0 0 1 N and the realized demand turns out to be larger than xn . These probabilities depend on xn and In , and computing these probabilities involves integrals and is nontrivial. Nevertheless, Equation (9) can be interpreted as being equivalent to the first-order condition in the traditional newsvendor model. In particular, if the newsvendor raises xn by one unit, then she (i) incurs an immediate ordering cost cn ; (ii) saves cn+1 if that unit was to be ordered in n + 1 (with probability Pn+1 ), or cn+2 if the unit was to be ordered in n + 2 (with probability Pn+2 ), and so on; and (iii) collects additional revenue r if no order is placed in n + 11 0 0 0 1 N and the realized demand is larger than xn (with probability P ). Note that (i) and (iii) correspond to the standard overage and underage costs in the traditional newsvendor model, and in our model, we have an extra underage cost (ii), which is the loss due to postponing orders. The optimal base-stock level is chosen to balance the overage cost (i) and the expected underage cost (ii) and (iii). Although Gn is defined on the two-dimensional state space 4xn 1 In 5, it turns out (see the proof in the appendix) that the partial derivative ¡Gn /¡xn (or say probabilities Pi and P in (9)) depends only on yn = xn − Œ − In and can be written as gn 4yn 5 defined in (6) and

(7). Let bn be the optimal value of yn such that the partial derivative is equal to 0, which exists and is unique. Then yn∗ = bn implies xn∗ − Œ − In = bn . Thus, the optimal base-stock level Sn 4In 5 = Œ + In + bn is linear in In . The significance of Proposition 2 is twofold. First, from a computational perspective, it shows that the impact of forecast adjustments on the base-stock level is all captured by the term In in a linear fashion, whereas bn is static and is independent of In and the forecast evolution. It is solely determined by a set of one-dimensional recursive Equations (6) and (7), which can be solved off-line. In each period, after observing In , the newsvendor can find the optimal base-stock level by calculating Œ + In + bn , instead of inputting In into the model and re-solving the twodimensional DP. This substantially simplifies the process of searching for the optimal base-stock level. We note that similar forms of linear state dependence has been assumed by a number of papers in the multiperiod inventory models literature. For instance, Chen and Lee (2009) define an affine and stationary base-stock policy for their infinite horizon problem; Schoenmeyr and Graves (2009) assume linearity when constructing the forecast-based ordering policy, and Toktay and Wein (2001) restrict their analysis to linear policies motivated by material requirements planning logic. Proposition 2, however, not only proves the linearity of the base-stock level for our setting with a finite horizon and increasing costs, but also explicitly characterizes the slope and intercept. Second, this result extends the trade-off between overage and underage costs in the traditional news vendor model. Recall that in the traditional newsvendor model with normal demand, the optimal order quantity is mean demand plus a safety stock term. The safety stock level depends on demand uncertainty and overage/underage costs (see (10) in §4.2.1). Proposition 2 shows a similar relationship: in each period n, the optimal base-stock level can also be written as the last-updated demand forecast Œ + In plus a “safety stock” term bn . The meaning of bn here is richer as it captures not only the impact of future demand uncertainty and overage/underage concerns but also the intertemporal trade-off between ordering too early and too late (see Wang and Tomlin 2009 for a similar discussion in a single-ordering setting). Corollary 1 summarizes some properties of the safety stock term bn . Everything else being equal, the newsvendor should stock more if it is cheaper to buy in period n, or if it is more expensive to buy later, or if the selling price is higher. Furthermore, the optimal safety stock level is always lower than the myopic safety stock level (bˆ n ) that neglects future ordering opportunities (or equivalently, the safety stock level for the single order model in the following section defined by Equation (10)).

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Corollary 1. The safety stock term bn is independent of c1 1 0 0 0 1 cn−1 , decreasing in cn , and increasing in cn+1 1 0 0 0 1 cN , and r, for all n = 11 0 0 0 1 N . Furthermore, let bˆ n be the myopic safety stock level when future ordering opportunities are neglected, then bn ≤ bˆ n . 4.2. The Single-Ordering Strategy To evaluate the benefits the of multiordering strategy, we next consider a benchmark model where the newsvendor is restricted to place a single order. She can still acquire market information and update her forecast as before. In this context, there are two decisions for the newsvendor to make: (1) timing— when to place the single order, and (2) quantity—how much to order (for similar discussions, see Milner and Kouvelis 2005, Oh and Özer 2012, Boyaci and Özer 2010). Depending on how the timing decision is made, there are two possible models: static and dynamic. 4.2.1. The Static Policy. The newsvendor chooses the time to order statically at the beginning of the planning horizon. The quantity decision is delayed until the chosen period, i.e., if she is to order in period n, she waits until period n to decide the order quantity. By then, she will have observed In and updated the demand forecast accordingly. The optimal policy is as follows. Proposition 3. If the newsvendor decides to order in period n, where Z‚n = ê −1 41 − cn /r5, the optimal order quantity and the corresponding expected profit are given by  Œ + In + ‘˜ n Z‚n 4a-MMF E51 Sn∗ 4In 5 =  exp4Œ + In + ‘˜ n Z‚n 5 4m-MMF E51

ç∗n 4In 5 =

n = 11 0 0 0 1 N 3 (10)   4r − cn 54Œ + In 5 − r ‘˜ n ”4Z‚n 5      4a-MMF E51   ¯ ‘˜ n − Z‚ 5 r exp4Œ + In + ‘˜ n2 /25ê4   n   4m-MMF E51 n = 11 0 0 0 1 N 0

(11)

Let ç∗N +1 4 5 = 0 be a dummy option standing for not ordering in any period. The optimal profit of the static single-ordering model is çSs = maxn=110001N +1 E6ç∗n 4In 57, and the optimal timing is n∗ = arg maxn=110001N +1 · E6ç∗n 4In 57. 4.2.2. The Dynamic Policy. Under this policy, the time to order is contingent on the observed information. The newsvendor observes the forecast evolution process and decides on the timing and quantity of the single order dynamically. In other words, the newsvendor not only enjoys quantity flexibility but also timing flexibility. This is an optimal stopping

problem. In any period n (if she has not ordered yet), after observing In , the newsvendor faces two options: to wait or to order. If she chooses to wait, nothing happens in n and she moves on to the next period. The payoff will be the expected profit if the order is placed later, conditional on In . If she chooses to order, then she determines the order quantity according to (10) and collects her profit as in (11), both of which are functions of In . Let Wn 4In 5 be the optimal profit-to-go given that the observed information is In and that the newsvendor has not yet ordered in periods 11 21 0 0 0 1 n − 1. We can write the DP as Wn 4In 5 = max8E6Wn+1 4In+1 5 — In 71 ç∗n 4In 591 n = 11 21 0 0 0 1 N 3 WN +1 4IN +1 5 = 00

(12) (13)

As before, period N + 1 is the dummy option of not ordering anything. The optimal profit of the dynamic single-ordering model çSd = W1 405. Proposition 4 characterizes the optimal policy for the a-MMFE case and establishes the equivalence of the expected profits in static and dynamic single-ordering models for the m-MMFE. Proposition 4. For a-MMFE, the optimal order policy is a threshold policy. In each period n, there exists a threshold I˜n such that the newsvendor should wait if In < I˜n , otherwise she should order the quantity given in (10). The resulting expected profit is larger than that of the static single-ordering strategy, i.e., çSd ≥ çSs . For m-MMFE, the order timing decision is independent of In , so the dynamic single-ordering strategy reduces to the static strategy and leads to the same expected profit, i.e., çSd = çSs . The dynamic single-ordering problem is of similar flavor to Wang and Tomlin (2009) and Boyaci and Özer (2010), and is a special case of Oh and Özer (2012), who study an optimal stopping problem with a generalized MMFE model that allows asymmetric demand information from multiple parties to be combined into a forecast evolution. The threshold policy obtained in Proposition 4 is a special case of the control band policies studied in Oh and Özer (2012) and Boyaci and Özer (2010) as well. The second part of Proposition 4 shows that the optimal policy boils down to a state-independent policy and that the static and dynamic single-order strategies are equivalent under m-MMFE. Oh and Özer (2012) also show, under different conditions, the optimality of static ordering strategy for their capacity planning problem. We note that although one would expect additional benefits from timing flexibility in the dynamic singleordering model, in our setting it is only true for

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the a-MMFE case but not for m-MMFE. Three key observations help us build an intuitive explanation of this result in hindsight. First, although the demand forecast evolves randomly, the change of the key trade-off between decreasing uncertainty and increasing ordering cost is deterministic. Second, because of the martingale nature of the demand process, the expectation of the future is equal to the current state. Third, which is specific for m-MMFE, the effect of forecast evolution (the term exp4Œ + In + ‘˜ n2 /25 in Equation (11)) and the effect of cost-uncertainty trade¯ ‘˜ n − Z‚ 5 in Equation (11)) are nicely off (the term ê4 n decoupled such that the optimal timing decision is independent of the forecast evolution.

5.

Numerical Comparison of the Ordering Strategies

In this section, we conduct numerical experiments to investigate the benefits of dynamic decision making and multiple orders similarly to Erhun et al. (2008). We provide insights on how uncertainty reduction, time span, and cost differentials between procurement options affect the benefits of multiple orders. We run two sets of experiments for a-MMFE and m-MMFE, respectively. We consider a continuous time line starting at time t = 0 and ending at t = 1. The selling season (period N + 1) takes place at t = 1. There are three ordering opportunities (N = 3), and ordering period n, n = 11 0 0 0 1 3, is located at tn on the time line. We fix t1 = 0, i.e., the first ordering period is at time 0. The other periods are evenly spaced within the range from 0 to T , i.e., t2 = T /2, t3 = T . The last possible ordering point, T , takes values 0011 0021 0 0 0 1 009. Ordering cost in period n is given by cn = 1+4n−15‹, where ‹ captures the magnitude of cost difference between order options and takes values 00021 00041 0 0 0 1 002. The retail price is fixed at r = 2. Initial demand information Œ is fixed at 1, and ‘, the parameter capturing overall market uncertainty, takes values 00051 0011 0 0 0 1 003 for a-MMFE and 0011 0021 0 0 0 1 006 for m-MMFE. We let the uncertainty diminish linearly over time, i.e., ‘n2 = 4tn − tn−1 5‘ 2 , and residual uncertainty is given by ‘˜ n2 = 41 − tn 5‘ 2 . Under a-MMFE (m-MMFE), this linear reduction in uncertainty is equivalent to assuming that forecast evolves according to a Brownian (geometric Table 1

Brownian) motion. With this parameter set, we have 540 scenarios for both models. For each scenario, we first calculate the optimal ordering policies by applying the results from the previous section, then simulate 10,000,000 sample paths of forecast evolution, evaluate performance of the optimal policies obtained from the single-ordering and multiordering strategies under each sample path, and summarize the differences. 5.1. Comparison of Expected Profits We first compare the expected profits under static and dynamic single-ordering strategies, çSs and çSd , under a-MMFE. Define the profit gap between the two strategies as 4çSd − çSs 5/çSs × 100%. The gap represents the benefit of dynamic order timing in the single-ordering model. Our numerical result suggests that the benefit is marginal: among the 540 scenarios, the gap ranges from 0% to 0.084%, with an average being less than 0.001%. This observation, combined with the m-MMFE result in Proposition 4, leads to an important insight: being dynamic and being able to incorporate real-time forecast evolution into the order timing decision does not help much if the newsvendor is restricted to order only once. On the other hand, the newsvendor can obtain significant benefits when a dynamic policy is coupled with the flexibility of placing multiple orders, that is, when she adopts the dynamic multiordering strategy. Hence, we next compare the expected profits under single-ordering and multiordering strategies. Hereafter, single-ordering strategy means the static single-ordering strategy, unless explicitly noted. Define the profit gap between the two strategies as ãç = 4çM − çSs 5/çSs × 100%. This gap is always nonnegative as the multiordering strategy is at least as good as the static single-ordering strategy (Table 1 reports the minimum, maximum, and average of ãç among the scenarios studied). We are also interested in understanding how the profit gap is affected by our model parameters: ‘, T , and ‹. These parameters capture the magnitude of market uncertainty, time span of ordering options, and cost differences, respectively. Figure 1 illustrates the average profit gaps subtotaled with regard to the three parameters for a-MMFE (with the graphs being very similar for m-MMFE). For example, among a total of 540 scenarios of a-MMFE,

Statistics of the Gaps in Expected Profit, Vairance, Semivariances, and Coefficient of Variation a-MMFE

Min Max Average

m-MMFE

ãç (%)

ãVar (%)

ãCV (%)

ãVar− (%)

ãVar+ (%)

ãç (%)

ãVar (%)

ãCV (%)

ãVar− (%)

ãVar+ (%)

0001 13013 2043

−23049 45032 −7076

−20067 6056 −6026

−44087 68076 −13030

−0011 37087 8097

0002 27080 4090

−15009 44017 2039

−11081 0008 −3059

−40032 71052 −7060

0013 180022 28081

Wang, Atasu, and Kurtulu¸s: Multiordering Newsvendor with Dynamic Forecast Evolution

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Manufacturing & Service Operations Management 14(3), pp. 472–484, © 2012 INFORMS

∆Π (%)

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Figure 1

Comparison of the Expected Profits Between the Multiordering and Static Single-Ordering Strategies: a-MMFE

5

5

5

4

4

4

3

3

3

2

2

2

1

1

1

0 0.05

0.10

0.15

0.20

0.25

0.30

0 0.1

0.3




60 are with T = 001, 60 are with T = 002, and so on. Profit gaps are averaged within the nine subcategories for the nine values of T , and the average gaps are plotted in the second graph of Figure 1. The plots in Figure 1 reveal several interesting patterns. First, the profit gap is larger when market demand is more uncertain. This is intuitive as the multiordering strategy provides extra flexibility for the newsvendor to respond to unexpected market conditions. Second, the profit gap is increasing in T . When T is large, ordering options are spread out on the time line. Having a portfolio of spread out ordering options helps the newsvendor take advantage of the low-cost of early orders and low-uncertainty of late orders. Finally, for parameter ‹, the profit gap appears to be first increasing and then decreasing. When ‹ is very low (high), the cost increases slowly (quickly), thus it will be optimal for the newsvendor to order all at once in the very last (first) period. Therefore, toward the two extremes, benefits of multiordering strategy diminish. A more interesting question is at what value of ‹ the newsvendor can expect the most benefit from the multiordering strategy. A further detailed numerical analysis (available from the authors) reveals that benefits from the multiordering strategy (ãç ) peaks when ‹ is at a moderate level such that ordering solely from any of the three options would lead to roughly the same expected profit. In other words, when ordering options are similar in terms of their expected profits when considered individually, dynamic ordering from the portfolio is most valuable. It can also be shown that the benefit of multiordering strategy is increasing and concave in the number of ordering opportunities N : the marginal benefit of one more ordering opportunity is diminishing. 5.2. Comparison of Profit Variability Next, we study the profit variability gap, which is defined as ãVar4ç5 = 4Var4çM 5−Var4çSs 55/Var4çSs 5×100%0

0.5

0.7

0.9

0 0.02

0.06

0.10

0.14

0.18 0.20



T

Although there are scenarios with negative gaps of profit variability (i.e., the multiordering strategy reduces profit variability), there are also scenarios with a positive profit variability gap, suggesting that multiordering strategy can accentuate profit variability. Even in terms of coefficient of variation (reported under the column ãCV in Table 1), results are mixed (though on average, the CV under multiordering is lower). This may not sound appealing at first. One possible explanation is that compared to the static single-ordering strategy, the multiordering strategy is capable of adapting to market changes better, so its profit is more closely pegged to market condition and fluctuates more as the market demand varies. To understand this, we investigate the corresponding semi-variances (Markowitz 1959, Jin et al. 2006). Clearly, for the newsvendor’s profit considered in this paper, high upside semivariance is preferable whereas high downside semivariance is not. An examination of semivariances (also reported in Table 1 under columns ãVar− and ãVar+ ) reveals that on average, the multiordering strategy helps boost the more preferable upside semivariance, which reflects a superior match of supply and demand, and mitigate the downside semivariance, which is a more accurate measure of the newsvendor’s risk exposure. The fact that the multiordering strategy results in higher upside semivariances and lower downside semivariance on the average suggests that the multiordering strategy can also reduce the newsvendor’s risk exposure.

6.

Summary of Results and Conclusion

The analytical model presented in this paper extends the traditional newsvendor model by incorporating dynamic forecast evolution (the MMFE model) and multiple ordering opportunities with increasing

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480

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ordering costs to solve a practical operational problem. Our main contribution is identifying the optimal ordering policy and its structural properties when firms face short selling seasons and multiple ordering opportunities from a variety of sources with different costs and order timing requirements. The significance of our work lies in the analytical characterization of the linear/log-linear structure of the state-dependent base-stock policy, which allows for a simple way to find and implement the optimal ordering policy in a general setting. From a practical point of view, the explicit analytical form of the optimal policy we identify makes it easy for it to be built into decision support systems. We also note that these structural results can be extended to allow order cancelations and fixed ordering costs under certain conditions (see the online supplement). This analysis also helps improve our understanding of the cost-information trade-off in our context. Although the forecast evolves stochastically, the cost increases and the uncertainty diminishes in a deterministic manner. We demonstrate that a safety stock term, which is independent of the forecast evolution, is sufficient to capture the trade-off between ordering too early and too late, in addition to the classic singleperiod newsvendor trade-off of ordering too much versus too little. The single-ordering model sheds light on when dynamic order timing based on forecast evolution is valuable. We find that a dynamic order timing decision does not necessarily increase the expected profit if the newsvendor orders only once. Making the single-order timing choice statically at the beginning of the horizon without being contingent on the particular realization of forecast evolution leads to a negligible loss (if any) in expected profit relative to the dynamic single-ordering strategy. However, when a portfolio of ordering options are available (as in our multiordering model), dynamic ordering based on forecast evolution is significantly more valuable in boosting profit. The numerical analysis quantifies the value of the multiordering strategy relative to the singleordering strategy. Understanding the magnitude and the drivers of the profit difference between the two models helps us identify the operating conditions where the multiordering strategy can be most valuable. In particular, we find that multiple orders are most valuable when the market is highly uncertain, the ordering options are widely spread on the time line, and when no option clearly dominates others if they are considered individually. Our analysis on profit variability also leads to an important insight from a risk management perspective. Although the multiordering strategy may seem riskier (as it is likely to augment overall profit variability), this risk may

not necessarily be bad. The analysis on semivariances reveals that the multiordering strategy helps reduce down-side risk and boost up-side variability. Electronic Companion An electronic companion to this paper is available as part of the online version that can be found at http://msom.journal .informs.org/.

Acknowledgments The authors thank Nils Rudi, Beril Toktay, and Ilia Tsetlin for comments on an earlier version of this paper. The authors also thank Stephen Graves, the anonymous associate editor, and three reviewers for their constructive suggestions. The first author’s research was supported in part by National University of Singapore’s academic research [Grants R-314-000-076-133, R-314-000-084-112].

Appendix. Omitted Proofs Proof of Proposition 1. First we show that Gn 4xn 1 In 5 is increasing in In and concave in xn ; Vn 4xn−1 1 In 5 is increasing in In and xn−1 and concave in xn−1 for all n = 11 0 0 0 1 N . Note the fact that the demand estimate D — In is stochastically increasing in In . For a-MMFE, it is straightforward as D — In is N4Œ + In 1 ‘˜ n2 5, whereas for m-MMFE, D — In ∼ LogN4Œ + In 1 ‘˜ n2 5. The following result in Levy (1973) is sufficient to complete the proof of this part: Let X1 and X2 be two log-normally distributed random variables with parameters 4Œ1 1 ‘12 5 and 4Œ2 1 ‘22 5, respectively. Random variable X1 stochastically dominates X2 in a first-order sense if and only if Œ1 > Œ2 and ‘12 = ‘22 . We can then prove the monotonicities of Gn and Vn with regard to In . Consider GN . Demand estimate D — IN is stochastically increasing in IN because of the abovementioned result. Because min4D1 xN 5 is an increasing function of D, ED — IN 6r min4D1 xN 57 is increasing in IN , so is GN . Then it follows that VN is increasing in IN . Now suppose Vn+1 is increasing in In+1 , then Gn is increasing in In because In+1 — In is stochastically increasing in In . In the end we have that Vn is increasing in In , which completes the induction argument. Next, we prove the properties with regard to x by induction. It is obvious that GN 4·1 IN 5 is concave. Moreover, ¡GN /¡xN goes from r − cN to −cN . Letting SN 4IN 5 be the global maximizer of GN 4·1 IN 5, we know that VN 4xN −1 1 IN 5 = max 8GN 4xN 1 IN 59 + cN xN −1 xN ≥xN −1

=

 GN 4SN 4IN 51 IN 5 + cN xN −1

if xN −1 < SN 4IN 51

 GN 4xN −1 1 IN 5 + cN xN −1

if xN −1 ≥ SN 4IN 5

is continuous, increasing, and concave in xN −1 for any IN . Now it is straightforward to show GN −1 4·1 IN −1 5, which is equal to E6VN 4 · 1 · 57 plus a linear term, is concave, and the derivative goes from cN − cN −1 to −cN −1 monotonically. Applying the argument to periods N − 11 N − 21 0 0 0 1 1, we are able to show that all the Gn 4·1 In 5’s are concave and all the Vn 4·1 In 5’s are increasing and concave. Finally, a base-stock policy is optimal because function Gn , n = 11 0 0 0 1 N is concave in xn for any given In and has

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an internal maximum. The optimal base-stock level Sn 4In 5 is given by Sn 4In 5 = arg maxx 8Gn 4x1 In 59. Proof of Proposition 2. The a-MMFE Case. The proof is by induction. First, in period N with IN being observed, the problem reduces to single-period newsvendor problem with overage cost cN , underage cost r − cN and the demand being normal with mean and variance 4Œ + IN 1 ‘N2 +1 5. The optimal stock level is SN = Œ+IN +‘N +1 Z‚N , where ‚N = 4r −cN 5/r is the critical fractile and Z‚ = ê −1 4‚5 is the associated z-value. Leting bN = ‘N +1 Z‚N , we can write SN = Œ + IN + bN . More¯ N − Œ − IN 5/ over, ¡GN 4xN 1 IN 5/¡xN = r F¯N 4xN 5 − cN = r ê44x ‘N +1 5 − cN , where FN is the normal c.d.f. of D — IN and ê is the c.d.f. of standard normal random variable. This implies 0 that the first-order derivative is actually a function of yN = xN − Œ − IN . So, it can be written as gN 4yN 5 = gN 4xN − Œ − IN 5 = ¡GN 4xN 1 IN 5/¡xN , and, by definition, gN 4bN 5 = 0. Suppose in period n + 1, Sn+1 4In+1 5 = Œ + In+1 + bn+1 , and ¡Gn+1 4xn+1 1 In+1 5/¡xn+1 only depends on xn+1 − Œ − In+1 . Similarly, we define gn+1 4yn+1 5 = gn+1 4xn+1 − Œ − In+1 5 = ¡Gn+1 4xn+1 1 In+1 5/¡xn+1 , and gn+1 4bn+1 5 = 0. Now consider period n. We need to show what holds in n + 1 also holds in n. Recall that Vn+1 4xn 1 In+1 5 = max 8Gn+1 4xn+1 1 In+1 59 + cn+1 xn xn+1 ≥xn

=

 G∗n+1 4In+1 5 + cn+1 xn

if xn < Sn+1 4In+1 51

 Gn+1 4xn 1 In+1 5 + cn+1 xn

if xn ≥ Sn+1 4In+1 51

where G∗n+1 4In+1 5 = Gn+1 4Sn+1 4In+1 51 In+1 5. Note 2 In+1 − In ∼ N401 ‘n+1 5. Substitute into Gn 4xn 1 In 5,

that …n+1 =

Gn 4xn 1 In 5 = EIn+1 — In 6Vn+1 4xn 1 In+1 57 − cn xn = E…n+1 6Vn+1 4xn 1 In + …n+1 57 − cn xn Z ˆ = G∗n+1 4In + …n+1 5 dFn+1 4…n+1 5

Let bn be the solution to gn 4yn 5 = 0 (such a bn must exist because it can be checked recursively that gn 4yn 5, n = 11 0 0 0 1 N − 1, goes from cn+1 − cn to −cn continuously and monotonically as yn goes from −ˆ to ˆ), then, by definition, the optimal base-stock level is such that Sn − Œ − In = yn∗ = bn , or equivalently Sn = Œ + In + bn , which completes the induction argument. The m-MMFE Case. Proof for the m-MMFE case follows the same logic. Below we highlight the differences. In period N with IN being observed, the demand is lognormal with parameters 4Œ + IN 1 ‘N2 +1 5. Therefore, the optimal stock level is SN = exp4Œ + IN + bN 5, and   ¡GN 4xN 1 IN 5 ¯ log xN − Œ − IN − cN 0 = r F¯N 4xN 5 − cN = r ê ¡xN ‘N +1 0 The first-order derivative is actually a function of yN = log xN − Œ − IN . So it can be written as gN 4yN 5 = gN 4log xN − Œ − IN 5 =

Suppose in period n + 1, Sn+1 4In+1 5 = exp4Œ + In+1 + bn+1 5, and ¡Gn+1 4xn+1 1 In+1 5/¡xn+1 only depends on log xn+1 − Œ − 2 In+1 . Let „n+1 = In+1 − In = …n+1 + ‘n+1 /2, which follows 2 N401 ‘n+1 5. Also let „˜ n+1 = log xn −Œ−In −bn+1 be the threshold such that Sn+1 4In + „˜ n+1 5 = xn . Then ¡Gn 4xn 1 In 5 ¡xn Z „˜ n+1 ¡G 4x 1 I + „ 5 n+1 n n n+1 = dFn+1 4„n+1 5 + cn+1 − cn ¡xn −ˆ Z log xn −Œ−In −bn+1 = gn+1 4log xn − Œ − In − „n+1 5 dFn+1 4„n+1 5 −ˆ

…˜n+1

+

Z

…˜n+1 −ˆ

= Gn+1 4xn 1 In + …n+1 5 dFn+1 4…n+1 5

+ 4cn+1 − cn 5xn 1 where …˜n+1 is defined such that Sn+1 4In + …˜n+1 5 = xn (so it is a function of xn and In , or more precisely, …˜n+1 = xn − Œ − In − bn+1 ), and Fn+1 is the c.d.f. of …n+1 . Since Sn 4In 5 maximizes Gn 4·1 In 5, we need to check the first-order derivative: ¡Gn 4xn 1 In 5 ¡xn Z …˜n+1 ¡G 4x 1 I + … 5 n+1 n+1 n n+1 = xn+1 =xn dFn+1 4…n+1 5 + cn+1 − cn ¡xn+1 −ˆ Z xn −Œ−In −bn+1 = gn+1 4xn −Œ−In −…n+1 5 dFn+1 4…n+1 5+cn+1 −cn −ˆ

=

Z

4xn −Œ−In −bn+1 5/‘n+1 −ˆ

gn+1 4xn −Œ−In −‘n+1 †5 dê4†5+cn+1 −cn 0

Now it is clear that the derivative depends only on xn − Œ − In . We can write the derivative as a function of yn = xn − Œ − In , i.e., Z 4yn −bn+1 5/‘n+1 gn 4yn 5 = gn+1 4yn − ‘n+1 †5 dê4†5 + cn+1 − cn 0 −ˆ

¡GN 4xN 1 IN 5 0 ¡xN

+ cn+1 − cn Z 4logxn −Œ−In −bn+1 5/‘n+1 −ˆ

gn+1 4logxn −Œ−In −‘n+1 †5 dê4†5

+ cn+1 − cn 0 Now it is clear that the derivative depends only on yn = log xn − Œ − In . Let bn be the solution to gn 4yn 5 = 0, then, by definition, the optimal base-stock level is such that log Sn − Œ − In = yn∗ = bn , or equivalently Sn = exp4Œ + In + bn 5, which completes the induction argument. Proof of Corollary 1. We first prove the monotonicity by induction. Consider period N . By definition, gN 4bN 5 = ¯ N /‘N +1 5 − cN = 0. It is easy to see that gN is indepenr ê4b dent of c1 1 0 0 0 1 cN −1 . In addition, we can verify     ¡gN r bN ¡gN bN ¯ =− ” < 01 =ê > 01 and ¡bN ‘N +1 ‘N +1 ¡r ‘N +1 ¡gN = −1 < 00 ¡cN By the implicit function theorem, we have ¡bN ¡g . ¡gN =− N >0 ¡r ¡r ¡bN

and

¡bN ¡g . ¡gN =− N < 00 ¡cN ¡cN ¡bN

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Now suppose in period n + 1, gn+1 is independent of c1 1 0 0 0 1 cn and ¡gn+1 /¡bn+1 < 0, ¡gn+1 /¡r > 0, ¡gn+1 / ¡cn+1 = −1. Then in period n, Z 4bn −bn+1 5/‘n+1 gn 4bn 5 = gn+1 4bn − ‘n+1 †5 dê4†5 + cn+1 − cn = 00

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−ˆ

It is clear that gn is independent of c1 1 0 0 0 1 cn−1 and ¡gn Z 4bn −bn+1 5/‘n+1 ¡ = g 4b − ‘n+1 †5 dê4†5 < 03 ¡bn ¡bn n+1 n −ˆ ¡gn Z 4bn −bn+1 5/‘n+1 ¡ = g 4b − ‘n+1 †5 dê4†5 > 03 ¡r ¡r n+1 n −ˆ ¡gn = −1 < 03 ¡cn Z 4bn −bn+1 5/‘n+1 ¡ ¡gn = g 4b − ‘n+1 †5 dê4†5 + 1 > 00 ¡cn+1 ¡cn+1 n+1 n −ˆ In addition, the last inequality implies, for k = 21 0 0 0 1 N − n, Z 4bn −bn+1 5/‘n+1 ¡ ¡gn = g 4b − ‘n+1 †5 dê4†5 > 00 ¡cn+k ¡cn+k n+1 n −ˆ We can then apply the implicit function theorem and complete the proof. With the result above, the proof of bn ≤ bˆ n follows directly. In any period n with In being observed, bn is determined by (6) and (7). The myopic safety stock level bˆ n is essentially the optimal bn obtained with parameters cn+1 1 cn+2 1 0 0 0 1 cN all being augmented to be equal to r—hence, future ordering options have no value. From the monotonicity result above, we know that bn is increasing in cn+1 1 cn+2 1 0 0 0 1 cN ; hence, bn ≤ bˆ n . Proof of Proposition 3. For a given order period n, the newsvendor decides on the order quantity after observing In to maximize the expected profit çn 4Sn — In 5 = ED — In 6r min4D1 Sn 57 − cn Sn , where the demand estimate D — In follows N4Œ + In 1 ‘˜ n2 5 for a-MMFE (or LogN4Œ + In 1 ‘˜ n2 5 for m-MMFE). This is a typical newsvendor problem. The optimal order quantity is given by  Œ + In + ‘˜ n Z‚n (a-MMFE)1 Sn∗ 4In 5 =  exp4Œ + In + ‘˜ n Z‚n 5 (m-MMFE)0 Substituting the optimal order quantity back into the profit function, we have the newsvendor’s optimal profit if the order is made in period n and the realized forecast adjustment is In :  4r − cn 54Œ + In 5 − r ‘˜ n ”4Z‚n 5 (a-MMFE)1 ç∗n 4In 5 =  ¯ ‘˜ n − Z‚ 5 (m-MMFE)0 r exp4Œ + In + ‘˜ n2 /25ê4 n Taking the expectation with respect to In , we have the expected profit at the beginning of the planning horizon if the newsvendor orders in n. Denote this expected profit by ç∗n : ç∗n = E6ç∗n 4In 57  4r − cn 5Œ − r ‘˜ n ”4Z‚n 5 (a-MMFE)1 =  ¯ ‘˜ n − Z‚ 5 (m-MMFE)0 r exp4Œ + ‘˜ 12 /25ê4 n

Let ç∗N +1 = 0 be a dummy option standing for not ordering in any period. Then, the optimal profit of the static singleordering model çSs = maxn=110001N +1 ç∗n . Proof of Proposition 4. The a-MMFE Case. In any period n, the newsvendor needs to decide whether to wait, which leads to a profit E6Wn+1 4In+1 5 — In 7 (the first component in max89 of Wn 4In 5), or to order, which leads to a profit ç∗n 4In 5 (the second component in max89). According to (11), ç∗n 4In 5 is linearly increasing in In with a strictly positive slope r − cn . We first show by induction that the following properties hold for n = 11 0 0 0 1 N + 1, (1) Wn 4 5 ≥ 0, (2) Wn 4 5 is increasing and convex, (3) Wn0 4 5 ≤ r − cn , and (4) limIn →−ˆ Wn 4In 5 = 0. In period N + 1, WN +1 4 5 = 0, so the properties are trivially true. Suppose they are also true in period n + 1, then in period n, Wn 4In 5 = max8E6Wn+1 4In + …n+1 571 ç∗n 4In 59 = max8E6Wn+1 4In + …n+1 571 4r − cn 54Œ + In 5 − r ‘˜ n ”4Z‚n 590 Inside the max89, the second component is linearly increasing in In with a strictly positive slope r − cn . Regarding the first component, we know that because of the induction assumptions, Wn+1 4 5 is nonnegative, increasing and convex, 0 and Wn+1 4 5 ≤ r − cn+1 . So the expectation E6Wn+1 4In + …n+1 57 is also nonnegative and increasing and convex in In with slope being less than or equal to r − cn+1 . It is then straightforward to verify that Wn 4In 5, the maximum of the two components, is (1) nonnegative, (2) increasing and convex, and (3) its slope is less than or equal to r − cn . When In → −ˆ, the first component goes to 0 and the second component goes to −ˆ. So property (4) also holds. This completes the induction argument. Having the four properties, we can study the optimal stopping policy by investigating the difference between the first and second component. Let I˜n be the threshold such that the two components are equal, i.e., E6Wn+1 4In+1 5 — I˜n 7 − ç∗n 4I˜n 5 = 0. By property (4), as In → −ˆ, the difference E6Wn+1 4In+1 5 — In 7 − ç∗n 4In 5 goes to infinity, implying the newsvendor will prefer waiting in extremely bad market condition. Moreover, ¡8E6Wn+1 4In+1 5 — In 7 − ç∗n 4In 59 ¡In =

¡E6Wn+1 4In+1 5 — In 7 ¡ç∗n 4In 5 − ≤ 4r − cn+1 5 − 4r − cn 5 ¡In ¡In

= cn − cn+1 < 01 suggesting that the difference is strictly decreasing in In , which assures the existence and uniqueness of threshold I˜n in period n. It is optimal to order if In ≥ I˜n and wait otherwise. Next, we prove çSd ≥ çSs . Again, we show by induction that Wn 4In 5 ≥ maxi=n1 0001 N +1 ç∗i 4In 5. It is trivially true for

Wang, Atasu, and Kurtulu¸s: Multiordering Newsvendor with Dynamic Forecast Evolution Manufacturing & Service Operations Management 14(3), pp. 472–484, © 2012 INFORMS

N + 1. Suppose it is also true in some period n + 1, then in period n, Wn 4In 5 = max8E6Wn+1 4In+1 5 — In 71 ç∗n 4In 59 n h i o ≥ max E max ç∗i 4In+1 5 — In 1 ç∗n 4In 5

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i=n+11 0001 N +1

≥ max

n

= max

n

=

max

E6ç∗i 4In+1 5 — In 71 ç∗n 4In 5

max

o ç∗i 4In 51 ç∗n 4In 5

i=n+11 0001 N +1

i=n+11 0001 N +1

max

i=n1 0001 N +1

o

ç∗i 4In 50

Note the first inequality is due to the induction assumption and the second inequality is due to Jensen’s inequality because maxi=n1 0001 N +1 ç∗i 4In 5 is a convex function of In . This completes the induction argument. Then it follows that at the beginning of the horizon, the expected profit under the dynamic single-ordering strategy çSd = W1 405 ≥ maxi=11 0001 N +1 ç∗i 405 = maxi=11 0001 N +1 ç∗i = çSs . The m-MMFE Case. We prove by induction that the expected profit-to-go can be written as ¯ ‘˜ i − Z‚ 50 Wn 4In 5 = r exp4Œ + In + ‘˜ n2 /25 max ê4 i i=n1 0001 N

ç∗N 4IN 5

In period N , because is always positive and WN +1 = 0, WN 4IN 5 = max8E6WN +1 71 ç∗N 4IN 59 = ç∗N 4IN 5. So the statement above is true. Now suppose it is also true for period n, then Wn−1 4In−1 5 = max8E6Wn 4In 5 — In−1 71 ç∗n−1 4In−1 59 n h i ¯ ‘˜ i − Z‚ 5 — In−1 1 = max E r exp4Œ + In + ‘˜ n2 /25 max ê4 i i=n1 0001 N

ç∗n−1 4In−1 5

o

n 2 ¯ ‘˜ i − Z‚ 51 = max r exp4Œ + In−1 + ‘˜ n−1 /25 max ê4 i i=n1 0001 N

r

2 ¯ ‘˜ n−1 exp4Œ + In−1 + ‘˜ n−1 /25ê4

− Z‚n−1 5

o

2 = r exp4Œ + In−1 + ‘˜ n−1 /25 n o ¯ ‘˜ i − Z‚ 51 ê4 ¯ ‘˜ n−1 − Z‚ 5 · max max ê4 i n−1 i=n1 0001 N

2 = r exp4Œ + In−1 + ‘˜ n−1 /25

max

i=n−11 0001 N

¯ ‘˜ i − Z‚ 50 ê4 i

¯ 5 Note that the third equality holds because maxi=n10001N ê4 is independent of the value of In . It follows that at the beginning of the horizon, the expected profit under the dynamic single-ordering strategy çSd = W1 405 = ¯ ‘˜ i −Z‚ 5 = maxi=11 0001 N ç∗ = çSs . r exp4Œ+ ‘˜ 12 /25maxi=11 0001 N ê4 i i

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