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Optimal Dynamic Order Scheduling under Capacity Constraints Given Demand-Forecast Evolution

Is�ık Bic�er* Rotterdam School of Management, Erasmus University, Rotterdam, 3000 DR, The Netherlands, bicer@rsm.nl

Ralf W. Seifert Ecole Polytechnique F�ed�erale de Lausanne, CH-1015, Lausanne, Switzerland

IMD, Chemin de Bellerive 23, P.O. Box 915, CH-1001, Lausanne, Switzerland, ralf.seifert@epfl.ch

W e consider a manufacturer without any frozen periods in production schedules so that it can dynamically update the schedules as the demand forecast evolves over time until the realization of actual demand. The manufacturer

has a fixed production capacity in each production period, which impacts the time to start production as well as the pro- duction schedules. We develop a dynamic optimization model to analyze the optimal production schedules under capac- ity constraint and demand-forecast updating. To model the evolution of demand forecasts, we use both additive and multiplicative versions of the martingale model of forecast evolution. We first derive expressions for the optimal base stock levels for a single-product model. We find that manufacturers located near their market bases can realize most of their potential profits (i.e., profit made when the capacity is unlimited) by building a very limited amount of capacity. For moderate demand uncertainty, we also show that it is almost impossible for manufacturers to compensate for the increase in supply–demand mismatches resulting from long delivery lead times by increasing capacity, making lead-time reduction a better alternative than capacity expansion. We then extend the model to a multi-product case and derive expressions for the optimal production quantities for each product given a shared capacity constraint. Using a two-product model, we show that the manufacturer should utilize its capacity more in earlier periods when the demand for both products is more positively correlated.

Key words: forecast evolution; dynamic scheduling; production capacity; production postponement History: Received: June 2016; Accepted: July 2017 by Panos Kouvelis, after 2 revisions.

1. Introduction

The purchasing behavior of customers has been changing from buying standard products to ordering customized and niche items (Anderson 2009, Brynjolf- sson et al. 2011). This trend has led to product prolif- eration in almost all industries. Some manufacturers have adapted to this change by replacing cost-based manufacturing strategies with time-based manufac- turing. Time-based manufacturing considers produc- tion lead time as an important resource and focuses on cutting the lead time (Stalk 1990, Suri 1998, 2010). It allows manufacturers to place production orders close to the realization of actual demand, thereby sig- nificantly reducing supply–demand mismatches. With the help of quick changeover systems, the inter- net of things (IoT) and 3D printing technologies, some companies successfully implemented time-based manufacturing by eliminating frozen periods in pro- duction schedules and reducing batch sizes (Ghe- mawat and Nueno 2003, Wainwright 2015). However, the ability to implement time-based manufacturing may be restricted by capacity limitations. Even

manufacturers with very short throughput times could end up with very long production lead times if their customer demand far exceeds their production capacity. For example, a manufacturer with very lim- ited production capacity needs to start production well in advance of the realization of actual demand (e.g., to support seasonal sales or a new product launch on a global scale), especially when the cost of lost sales is relatively high (Avanzi et al. 2013, Bayus et al. 2003, Fisher 1997). As the time between a pro- duction order and delivery to a customer increases, the order quantity has to be determined based on more inaccurate demand forecasts, causing an increase in supply–demand mismatches. Therefore, capacity problems can be regarded as a major obstacle to fully benefiting from time-based manufacturing strategies. In a multi-product setting, time-based manufactur-

ing makes it possible to balance capacity and total demand by reserving the capacity during off-peak periods (i.e., speculative capacity) for products with predictable demand while using the capacity during peak periods (i.e., reactive capacity) for products with

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highly uncertain demand (Cattani et al. 2008, 2010, Fisher and Hammond 1994, Fisher et al. 1997). Although some fashion retailers successfully imple- mented this strategy and achieved a significant reduc- tion in supply–demand mismatches (Fisher et al. 1997), it might in fact be optimal to produce a propor- tion of standard (innovative) products with predictable (uncertain) demand using the reactive (speculative) capacity. In this study, we develop a dynamic programming model and provide analytical expressions for optimal production schedules under capacity constraints and demand-forecast evolution. We consider a manufacturer that has fixed capacity

in each period to produce seasonal products with evolving demand forecasts. The manufacturer aims to optimally schedule its production orders before the selling season and satisfy market demand during the selling season. We address the following questions:

1. Would it be possible for a manufacturer to realize most of the potential profit (i.e., profit made when the capacity is unlimited) by build- ing a limited amount of capacity?

2. How much extra capacity should a manufac- turer with a long delivery lead time reserve to compensate for the negative impact of mis- match losses due to long lead times?

3. How can manufacturers optimally allocate their production capacity between different products with evolving demand forecasts?

We use both additive and multiplicative versions of the martingale forecast-evolution process to model the evolution of demand forecasts. We first develop a stochastic dynamic optimization model and provide expressions for optimal production schedules for a single-product problem under a capacity constraint. We find that manufacturers located near their market bases can realize most of their potential profits by building a very limited amount of capacity. For mod- erate demand uncertainty, we also show that manu- facturers with moderate delivery lead times should significantly increase their production capacity to compensate for the increase in mismatch losses result- ing from the lead times. For manufacturers with long delivery lead times, however, it is almost impossible to compensate for the increase in the resulting mis- match losses by increasing capacity. This means that lead-time reduction is a better alternative than capac- ity expansion for those manufacturers. In practice, though, capacity expansion is considered more effec- tive than lead-time reduction in reducing mismatch losses when rationalizing offshore-sourcing decisions of companies (Atkinson and Ezell 2012, Girotra and Netessine 2012, Tongarlak et al. 2017). Therefore, our finding can be considered as counterintuitive, poten- tially providing decision makers with additional

insights regarding optimal production strategies. Building on our results for the single-product model, we develop a simple typology proposing effective production strategies for different product types. We extend our results to a multi-product problem

and derive expressions for the optimal schedules under a shared capacity constraint. We apply our derivations to a two-product model and show that the more positively correlated the demand for the prod- ucts, the more utilized the capacity in earlier periods. We interpret this result as indicating that the proba- bility of observing demand peaks for all the products is relatively high under positive correlation. There- fore, it is optimal to utilize the capacity more in earlier periods under positive correlation in order to mitigate potential stock-out risks. Under negative correlation, the changes in demand forecasts are expected to be in opposite directions, allowing the manufacturer to profitably postpone production and to derive more benefits from accurate demand forecasts. The remainder of this study is organized as follows.

After reviewing the related literature in the next sec- tion, we describe our model in section 3. We analyze the single-product problem in section 4 and the multi- product problem in section 5. Finally, section 6 pre- sents our concluding remarks.

2. Literature Review

Our work relates to studies that integrate the forecast- updating process into the operational decision-mak- ing process. In the related literature, the martingale model of forecast evolution (MMFE) has been exten- sively used to model the forecast-updating process. The MMFE dates back to Hausman (1969), who used agricultural data and found that the multiplicative version of MMFE (m-MMFE) is empirically well sup- ported. Graves et al. (1986, 1998) used the additive version of MMFE (a-MMFE) to dynamically optimize production planning in single and multi-stage sys- tems. Heath and Jackson (1994) provided a structural overview of the MMFE, demonstrating that the mar- ginal demand distribution follows a normal distribu- tion under the a-MMFE, whereas it has a lognormal distribution under the m-MMFE. Aviv (2001) devel- oped a unified forecast-evolution model using corre- lated martingale models. He analyzed the benefits of the unified forecast-evolution approach in a two-eche- lon supply chain and stressed the importance of col- laborative forecasting. Aviv (2007) studied a similar problem, except that the manufacturer dynamically updates production plans, and deviations from the initial plan result in increased production costs. Intu- itively, he demonstrated that the benefits of collabora- tive forecasting increase as the agility of the manufacturer improves. De Treville et al. (2014a)

studied the value of lead-time reduction when demand forecasts evolve according to an m-MMFE model. They developed an analytical model to calcu- late how much cheaper a long-lead-time supplier should be to compensate for the increase in mismatch losses resulting from long lead times. The required cost differential as lead time increases from 0 to the full lead time T becomes an indifference, or cost-differ- ential frontier. The cost-differential frontier was tested empirically in a variety of supply chains in de Treville et al. (2014b); it has become part of the U.S. Depart- ment of Commerce toolbox to aid decision makers in valuing lead time (acetool.commerce.gov/inventory). We contribute to this literature stream by imposing a capacity constraint, so production cannot necessarily be postponed to the last period before the shipment of all products produced. Therefore, we capture the dynamics of forecast evolution during a sequence of production periods before the shipment. The evolution of demand forecasts has been incor-

porated in analytical models to determine optimal decision timing under demand uncertainty. Milner and Kouvelis (2005) considered a single-period inven- tory control problem with two ordering opportuni- ties, where the time of placing the second order is an endogenous decision variable. They determined a boundary value of the reorder point and then found the optimal time for the second order by iteration. Wang and Tomlin (2009) studied the ordering prob- lem of a buyer under demand and lead-time uncer- tainty. In particular, they analyzed the trade-off between ordering early to reduce supply risk and ordering late to reduce the demand risk. They formu- lated the problem as a Markov Decision Process and used a dynamic programming approach to determine the optimal order time. Boyaci and €Ozer (2010) ana- lyzed the investment decision of a manufacturing firm that has to determine production capacity in the face of demand uncertainty. The firm can sell a pro- portion of its products in advance at a discounted price to obtain valuable early-sales data, thereby reducing the risk of unused capacity. The authors developed a model, using a dynamic programming approach to determine the optimal time for the capac- ity investment. They also extended the model by incorporating a dynamic pricing policy for both advance sales and regular sales periods. €Ozer and Wei (2004) studied an inventory control problem under capacity constraint, where the manufacturer could sell a proportion of total demand in advance and obtain advance demand information. In their model, advance orders are not correlated with regular sales, and both sales and production can occur in each period. They developed a dynamic programming model and solved the problem by backward induc- tion. Hausman and Peterson (1972) considered a

multi-period ordering problem, similar to ours, under a capacity constraint and forecast updates. They formulated the problem as a dynamic pro- gramming model and provided a solution proce- dure to numerically solve the problem. They also developed three heuristic methods for a multi-pro- duct case because numerical solutions based on the dynamic programming model are not computation- ally feasible for a multi-product case. Bitran et al. (1986) studied a similar problem with setup costs and forecast updates. In their model, there is no setup cost if a changeover occurs from one product to another within the same product family; how- ever, there is a high setup cost associated with a changeover from one product family to another. They restricted the problem such that each product family is set up once over the planning horizon and found an approximate solution of the periodic order quantities based on a deterministic, mixed integer programming (MIP) model. More recently, Wang et al. (2012) studied a single-product, unca- pacitated newsvendor model with the option of placing multiple orders over time. They assumed that the ordering cost increases over time, trading off improved demand forecasts against increasing ordering cost. They formulated a dynamic pro- gramming model and derived an expression of the marginal value of producing one additional unit in each period (Wang et al. 2012, equation (9)). Using this expression, they found the optimal results for their multi-period newsvendor problem. We extend the Wang et al. model by adding a periodic capac- ity constraint to the multi-period newsvendor prob- lem. We also analyze a multi-product case and provide analytical expressions for the optimal base- stock level for each product in each period.

3. Model Description

We consider a newsvendor model with T + 1 plan- ning periods. The first T periods are production peri- ods, where the newsvendor has an option to place a production order in each of these periods. The last period, T + 1, is the unique selling period. All actual demand is realized in period T + 1. The capacity con- straint per production period is fixed to K. In each period, the demand forecast is updated according to the MMFE. Thus, demand uncertainty reduces gradu- ally over time until the realization of the actual demand. We use Dt to denote the demand forecast in period t 2 {1, 2, . . ., T + 1}. The last demand forecast in period T + 1 (i.e., DT+1) is equal to the actual demand. We denote by ðX; F; PÞ a filtered probability space

on which demand forecasts follow additive or multi- plicative versions of the MMFE. Thus, demand

process on ðX; F; PÞ is adapted by filtration ðF tÞt � 0 � F . In the additive model (a-MMFE), the demand forecasts in period t 2 {2, . . ., T+1} are given by Dt = D1 + ɛ2 + ɛ3 + ⋯ + ɛt, where ɛt follows a nor- mal distribution:

et � N ð0; r2Þ: ð1Þ Let l be the expected demand in the first period. Then, the demand distribution conditional on the forecast adjustments until period t is:

DTþ1jAt � N ðl þ At; ðT þ 1 � tÞr2Þ; ð2Þ where At = ɛ2 + ɛ3 + ⋯ + ɛt represents the cumula- tive forecast adjustments in the a-MMFE. In the multiplicative model (m-MMFE), the

demand forecasts in period t 2 {2, . . ., T + 1} are given by Dt = D1 exp (ɛ2 + ɛ3 + ⋯ + ɛt), where ɛt follows a normal distribution:

et � N ð�r2=2; r2Þ: ð3Þ Let l = ln(D1) � Tr2/2 be the expectation of the logarithm of demand in the first period. Then, the demand distribution conditional on the information observed in period t is:

lnðDTþ1ÞjAt � N ðl þ At; ðT þ 1 � tÞr2Þ; ð4Þ where At ¼

Pt i¼2ðei þ r2=2Þ represents the cumula-

tive forecast adjustments in the m-MMFE. The main difference between a-MMFE and m-

MMFE is that forecast errors depend on the value of the initial forecast for the m-MMFE, whereas those errors are independent of the forecast for the a-MMFE (Heath and Jackson 1994, p. 22). This makes the value of observing forecast updates higher in the m-MMFE than in the a-MMFE. The a-MMFE, where demand follows the normal distribution given in Equation (2), provides a good fit to empirical data when demand uncertainty is low, whereas the m-MMFE fits better when the uncertainty is high (Gallego et al. 2007). Fit- ting an a-MMFE to a highly uncertain demand pro- cess also causes excessive ordering problems as discussed by Gallego et al. (2007, table 1). The decision periods are t 2 {1, 2, . . ., T}. The state

variables are (xt�1, At), where xt�1 is the inventory level at the beginning of period t before placing an order, and At is the cumulative forecast adjustments given above. The sequence of events in each period t 2 {1, 2, . . ., T} occurs as follows:

1. At is observed, 2. The demand forecast is updated, 3. Inventory position xt�1 is reviewed, 4. The decision maker decides an order quantity

Qt and incurs a cost of cQt,

5. Inventory position is updated as xt = xt�1 + Qt.

4. Single-Product Model

In the selling period (i.e., period T + 1), the actual demand D is realized and total revenue p minðD; xTÞ is collected. We formulate our problem as a dynamic programming model with the following Bellman equation at the beginning of period t < T:

Vtðxt�1;AtÞ ¼ max xt�1 �xt �xt�1þK

n EAtþ1jAt

h Vtþ1ðxt;Atþ1Þ

�cðxt �xt�1Þ o ; ð5Þ

where the inventory position xt�1 and the cumula- tive forecast adjustments are the state variables, and the decision variable is the order quantity Qt. We specify an auxiliary function Gt(xt, At) such that

Gtðxt; AtÞ ¼ EAtþ1jAt h Vtþ1ðxt; Atþ1Þ

i � cxt; ð6Þ

Vtðxt�1; AtÞ ¼ max xt�1 � xt � xt�1þK

fGtðxt; AtÞg þ cxt�1: ð7Þ

Then, we develop the following theorem that char- acterizes the optimal scheduling policy.

THEOREM 1. The optimal policy is a state-dependent base- stock policy, which depends on the time a production order is placed and on the cumulative forecast adjustments.

The proof of Theorem 1 is given in Appendix A, which includes expressions for the optimal base-stock levels. We use St(At) to denote the optimal base-stock levels for t 2 {1, . . ., T}. The order quantities in each period are then found by the following:

Qt ¼ 0 if StðAtÞ\xt�1; StðAtÞ�xt�1 if xt�1 �StðAtÞ\xt�1 þK; K if xt�1 þK�StðAtÞ:

8< : ð8Þ

The results given in Theorem 1 can be extended to analyze the case in which a supplier offers advance ordering discounts if the buyer places order quantities well in advance of the realization of actual demand. In Appendix B, we provide expressions that make it possible to find the optimal order quantities in each period under a capacity constraint and with advance ordering discounts. We develop the following corollary that shows the

impact of capacity constraint on the optimal base- stock level.

COROLLARY 1. Let x ¼ fx1; x2; . . .; xTg and x0 ¼ fx01; x02; . . .; x

0 Tg be the vectors of optimal inventory position

levels for an identical product under two different scenar- ios—that is, with a periodic capacity of K and K0, respec- tively. The vector x weakly majorizes x0 when K0 > K.

The proof is given in Appendix C. The corollary implies that the vector of inventory positions becomes more concentrated toward the last production period as capacity is increased (i.e., xt=xT is weakly decreas- ing in K for t 2 {1, 2, . . ., T � 1}). This result pro- vides useful insights regarding the ordering policy of companies. As K increases, it is optimal to postpone production to benefit from accurate demand forecasts close to the realization of market demand. As K decreases, it is optimal to start production earlier in order to mitigate stock-out risk, trading off supply shortages due to order postponement and capacity constraint against improved demand forecasts. This result is intuitive and in line with the findings of Fisher and Raman (1996), in which the authors ana- lyzed the value of early sales in a two-stage produc- tion system. They showed that it is optimal to postpone most of the production after receiving early- order information if the production capacity in the second stage (i.e., after receiving early-order informa- tion) is high. Our model can be considered a multi- stage extension of the Fisher and Raman (1996), model, with information updates occurring at the beginning of each period and thereby allowing us to better analyze the value of production capacity and the dynamics between production capacity and deliv- ery lead time.

COROLLARY 2. The expected profit at the beginning of the first period (i.e., V1(x0, A1|K)) is a non-decreasing

concave function of K with @V1ðx0; A1jKÞ

�� K!0

¼ ðp � cÞT and

@V1ðx0; A1jKÞ @K

�� K!þ1

¼ 0.

The proof is given in Appendix D. Corollary 2 reflects a pattern of diminishing returns from addi- tional capacity. Therefore, manufacturers can realize most of the potential profit (i.e., profit made when the capacity is unlimited) by building a limited amount of capacity. The marginal return from addi- tional capacity becomes zero when the capacity is excessive. As the number of production periods increases, so does the value of @V1(x0, A1|K)/@K|K? 0. Therefore, the ability of manufacturers to realize most of the potential profit by building a limited amount of capacity increases with the number of production periods.

4.1. Valuing Production Capacity In Figure 1, we present an example to demonstrate the impact of capacity constraint on the production schedule and the expected profit for the a-MMFE. We consider a product with a mean demand of 100 units and a coefficient of variation (CV) of 0.7. We assume that there are 10 periods over the planning horizon. Therefore, the parameters of the a-MMFE are l = 100 and r = 21.11. The price of the product is $10 per unit, and it costs the manufacturer $2 per unit to produce the product. The salvage value is set to zero. Figure 1a displays the impact of capacity on the time to start production. The x-axis represents capacity (K) and the y-axis represents the period when production starts. An increase in capacity allows the manufacturer to postpone production to later periods, which in turn reduces the production lead time. Figure 1b shows the impact of capacity on the expected profit. By increasing capacity and postponing production, the manufacturer is better able to match the end demand with the supply. This helps to increase the expected profit. As capacity increases, the expected profit also increases at a decreasing rate. Therefore, the

(a) (b)

Figure 1 Analysis of Single-Product Model for the a-MMFE. (a) Impact of Capacity Constraint on the Production Schedule; (b) Impact of Capacity Constraint on the Expected Profit [Color figure can be viewed at wileyonlinelibrary.com]

manufacturer can realize most of the profits by build- ing up a limited amount of capacity. In Figure 2, we demonstrate the results for the

m-MMFE. We assume the same values of cost param- eters, mean demand, and CV as in Figure 1, and esti- mate the parameters of the m-MMFE as l = 4.42 and r = 0.19. In Figure 2a, we show the impact of capacity on the time to start production for the m-MMFE. In Figure 2b, we indicate that the expected profit increases at a decreasing rate as capacity increases. Similar to the results for the a-MMFE given in Fig- ure 1, a significant increase in profits can be achieved by building up a limited amount of capacity. Although optimal inventory levels under a normal and a lognormal distribution with the same first- and second-order moments may be significantly different (Gallego et al. 2007), we obtain similar results in Fig- ures 1 and 2. This difference is due to the capacity constraint in our model. When the capacity is too restricted, it is expected to highly utilize the overall capacity for both models, thereby reducing the devia- tion between the results under normal and lognormal distribution. When the capacity is sufficiently large, however, the manufacturer postpones production and faces low demand uncertainty. The reduction in demand uncertainty helps minimize the gap between lognormal and normal distribution results. In Figures 1 and 2, we consider a product with a

high profit margin and a high coefficient of variation (CV). In Table 1, we provide the optimal period to start production and total expected profit for different profit margins and CV values. In particular, we focus on four different product types: (i) high margin with high demand uncertainty, (ii) high margin with low demand uncertainty, (iii) low margin with high demand uncertainty, and (iv) low margin with low demand uncertainty. As observed in the table, most of the potential profits can be realized by a limited

amount of capacity (e.g., when K = 15) for each com- bination of profit margin and CV value. In Table 1, we also present the safety capacity val-

ues, which gives the excess capacity starting from the point when production starts. Suppose that produc- tion starts in period t = 5 and K = 25. Thus, produc- tion takes place in the last six periods, and total capacity starting from the point the production is started becomes 6 9 25 = 150. Then, we compute the safety capacity by subtracting the mean demand from the total capacity (i.e., 150 � 100). The results in the table show that it is possible to realize most of the potential profit when the safety capacity is not nega- tive. When a manufacturer reserves 20%–50% safety capacity for its seasonal products, it can successfully reduce supply–demand mismatches while balancing capacity throughout the year.

4.2. Comparison between Lead-Time Reduction and Capacity Expansion Manufacturers often face a trade-off between produc- tion capacity and delivery lead time when they have to determine the location of a new production facility. Building a facility close to the market is expected to be costlier than building one far away from the mar- ket. Thus, manufacturers that are willing to build a production facility should often choose between a high-capacity facility far away from the market and a low or moderate-capacity facility close to the market. We now consider a manufacturer that has to ship

products at the end of the (T � L)th period due to a delivery lead time of L periods. This prevents the manufacturer from benefiting from improved demand forecasts in the last L periods. To compensate for the negative effect of long delivery lead times, the manufacturer may increase its production capacity and postpone the start of production. Starting from the (T � L)th period and going backward, our

(a) (b)

Figure 2 Analysis of Single-Product Model for the m-MMFE. (a) Impact of Capacity Constraint on the Production Schedule; (b) Impact of Capacity Constraint on the Expected Profit [Color figure can be viewed at wileyonlinelibrary.com]

analytical model makes it possible to find the optimal production schedule for a delivery lead time of L peri- ods. In Figure 3, we demonstrate the required percent-

age capacity increase that should be offered by the manufacturer to compensate for the increase in mis- match losses resulting from a delivery lead time of L periods. We assume that the price of the product is $10 per unit, the unit cost $2, and the salvage value is equal to zero. There are 10 periods over the planning horizon, and the production capacity per period is equal to 15 units (K = 15). We first calculate the expected profit for L = 0 and then find the required percentage capacity increase for L > 0 to generate the

same expected profit. Therefore, our results can be considered as iso-profit curves for different pairs of capacity and lead time values. Figure 3a shows the results for a-MMFE, for which we use the same demand parameters as in Figure 1. Figure 3b shows the results for the m-MMFE, for which we use the same demand parameters as in Figure 2. For the a- MMFE model, the percentage capacity increase is a convex increasing function of L with a cutoff point of six periods of lead time. The manufacturer has to increase capacity by more than 100% to compensate for the increase in mismatch losses when L = 4. When L > 6, it is not possible for the manufacturer to com- pensate for the increase in mismatch losses by

Table 1 Optimal Results for Alternative Parameter Values

Cost parameters Demand parameters Results

Price Cost Margin Model l r CV Capacity (K) Start of

production Safety capacity Total profit

$10 $2 80% a-MMFE 100 21.1 0.7 5 1 �50 $344.39 $10 $2 80% a-MMFE 100 21.1 0.7 15 1 50 $579.04 $10 $2 80% a-MMFE 100 21.1 0.7 25 1 150 $628.69 $10 $2 80% a-MMFE 100 21.1 0.7 50 6 150 $685.72 $10 $2 80% a-MMFE 100 21.1 0.7 100 8 200 $713.04 $10 $2 80% a-MMFE 100 9.05 0.3 5 1 �50 $399.80 $10 $2 80% a-MMFE 100 9.05 0.3 15 1 50 $747.75 $10 $2 80% a-MMFE 100 9.05 0.3 25 2 125 $765.13 $10 $2 80% a-MMFE 100 9.05 0.3 50 6 150 $773.33 $10 $2 80% a-MMFE 100 9.05 0.3 100 8 200 $773.40 $10 $8 20% a-MMFE 100 21.1 0.7 5 1 �50 $40.62 $10 $8 20% a-MMFE 100 21.1 0.7 15 1 50 $83.51 $10 $8 20% a-MMFE 100 21.1 0.7 25 4 75 $93.34 $10 $8 20% a-MMFE 100 21.1 0.7 50 7 100 $100.60 $10 $8 20% a-MMFE 100 21.1 0.7 100 9 100 $104.65 $10 $8 20% a-MMFE 100 9.05 0.3 5 1 �50 $99.80 $10 $8 20% a-MMFE 100 9.05 0.3 15 1 50 $159.88 $10 $8 20% a-MMFE 100 9.05 0.3 25 3 100 $166.18 $10 $8 20% a-MMFE 100 9.05 0.3 50 7 100 $171.95 $10 $8 20% a-MMFE 100 9.05 0.3 100 9 100 $173.41 $10 $2 80% m-MMFE 4.42 0.19 0.7 5 1 �50 $390.79 $10 $2 80% m-MMFE 4.42 0.19 0.7 15 1 50 $572.50 $10 $2 80% m-MMFE 4.42 0.19 0.7 25 1 150 $626.92 $10 $2 80% m-MMFE 4.42 0.19 0.7 50 5 200 $686.65 $10 $2 80% m-MMFE 4.42 0.19 0.7 100 7 300 $716.28 $10 $2 80% m-MMFE 4.57 0.089 0.3 5 1 �50 $399.09 $10 $2 80% m-MMFE 4.57 0.089 0.3 15 1 50 $739.10 $10 $2 80% m-MMFE 4.57 0.089 0.3 25 1 150 $757.64 $10 $2 80% m-MMFE 4.57 0.089 0.3 50 4 250 $768.07 $10 $2 80% m-MMFE 4.57 0.089 0.3 100 7 300 $768.10 $10 $8 20% m-MMFE 4.42 0.19 0.7 5 1 �50 $79.04 $10 $8 20% m-MMFE 4.42 0.19 0.7 15 1 50 $110.27 $10 $8 20% m-MMFE 4.42 0.19 0.7 25 4 75 $120.14 $10 $8 20% m-MMFE 4.42 0.19 0.7 50 7 100 $129.48 $10 $8 20% m-MMFE 4.42 0.19 0.7 100 9 100 $134.70 $10 $8 20% m-MMFE 4.57 0.089 0.3 5 1 �50 $99.96 $10 $8 20% m-MMFE 4.57 0.089 0.3 15 1 50 $151.02 $10 $8 20% m-MMFE 4.57 0.089 0.3 25 1 150 $159.11 $10 $8 20% m-MMFE 4.57 0.089 0.3 50 5 200 $167.10 $10 $8 20% m-MMFE 4.57 0.089 0.3 100 8 200 $169.17

increasing capacity (even for K ? +∞). For the m- MMFE model, the cutoff point is four periods of lead time. Thus, it is not possible for the manufacturer to compensate for the increase in mismatch losses, even for K ? +∞, when L > 4. The difference between the cutoff points of a-MMFE and m-MMFE results from the fact that the value of observing forecast updates is higher in the m-MMFE than in the a-MMFE (Heath and Jackson 1994, p. 22). In Table 2, we show the pairs of different lead time

and capacity values that give the same expected profit (which we refer to as iso-profit pairs). We focus on four different cases: (i) high margin with high demand uncertainty, (ii) high margin with low demand uncer- tainty, (iii) low margin with high demand uncertainty, and (iv) low margin with low demand uncertainty. For each set of cost and demand parameters, we calcu- late the expected profit for three different capacity val- ues (i.e., K = {15, 25, 50}) with zero lead time. Then, we incrementally increase the lead time and find the capacity that gives the same expected profit. The results shown in Figure 3 correspond to the 1st and 13th rows in the table. As shown in Figure 3 and Table 2, it is almost impossible for the manufacturer to compensate for the increase in mismatch losses due to the lead time by increasing capacity when the capacity is greater than or equal to 15 units (i.e., 15% of total expected demand) per period. Our results provide useful insights into the compar-

ison between the value of lead-time reduction and capacity expansion. For a moderate value of produc- tion capacity, we show that when the delivery lead time is relatively long, it is almost impossible for the manufacturer to compensate for the increase in mis- match losses due to the lead time by increasing capac- ity. In practice, however, capacity expansion is considered more effective than lead-time reduction in reducing mismatch losses when companies rational- ize their decisions on long-distance sourcing

(Atkinson and Ezell 2012, Girotra and Netessine 2012, Tongarlak et al. 2017). For example, Tongarlak et al. (2017) commented that retail stores are not willing to pay more to capacity-constrained local suppliers of fresh products for their responsiveness than to large long-distance farms, because they value capacity more than responsiveness. Therefore, our results in this subsection can be considered as counterintuitive in practice. Capacity-constrained local manufacturers that improve their demand forecasting can increase the value of forecast updates toward the end of pro- duction periods. If they are also able to integrate the updated demand forecasts into their production schedules (e.g., by reducing frozen periods in their Master Production Schedules), they can compete with large long-distance manufacturers. The difference between our findings and common

practice regarding the value of lead-time reduction rel- ative to the value of production capacity results from the underlying assumption of the MMFE that useful demand information that would facilitate accurate forecasts increases continuously over time until the real- ization of actual demand. Although this assumption is relevant to different contexts, there may be some cases for which demand information transfer to manufactur- ers is not possible or feasible (de Treville et al. 2004, p. 622). The unavailability of information transfer makes lead-time reduction worthless, favoring capacity increase over lead-time reduction.

4.3. Comparison between Dynamic Scheduling and Forecast Improvement Throughout the study, we use two standard versions of the MMFE to model the forecast-evolution process, where demand forecasts are ideally obtained by judg- mental forecasting (i.e., experts aided with statistical tools give point estimates of demand for each product in a product portfolio). It has been well established in the literature that accuracy of judgmental forecasts

(a) (b)

Figure 3 Justified Capacity Increase for Offshore Manufacturing to Compensate for the Increase in the Mismatch Cost Resulting from the Shipment Lead Time. (a) Additive Model; (b) Multiplicative Model [Color figure can be viewed at wileyonlinelibrary.com]

decreases with the forecast horizon (Lawrence et al. 1986), thus providing support for the MMFE. Heath and Jackson (1994) also used the MMFE to capture the dynamics of the judgmental forecasting-updating process. Judgmental forecasting is in general more effective than statistical forecasting, especially when unusual events (e.g., promotion and demand shocks) are likely to occur (Bunn and Wright 1991, Gaur et al. 2007). Therefore, an intrinsic assumption in this study is that manufacturers that have integrated dynamic scheduling with the forecast-updating process should already have progressed in demand forecasting. Thus, there is no room for improvement in demand forecasting. In this subsection, we relax this assumption to

make a comparison between dynamic scheduling and static scheduling with more accurate demand fore- casts. In static scheduling policy, production sched- ules are determined and frozen at the very beginning. Therefore, the manufacturer does not benefit from the resolution of demand uncertainty as the selling sea- son approaches. To compensate for that, the manufac- turer puts some effort into improving the forecast accuracy at the very beginning. In Figure 4, we pre- sent the required percentage reduction in the coeffi- cient of variation that makes the expected profit for static policy equal to that of dynamic policy. We use the same parameter values for a-MMFE and m-MMFE as in Figures 1 and 2, respectively. The figure shows

that manufacturers with a static scheduling policy can easily generate the same profit as those with a dynamic policy by slightly increasing the forecast accuracy when the capacity is very limited. As the capacity increases, so does the required percentage reduction in CV, making the dynamic policy more advantageous than the static policy.

4.4. Managerial Insights In Figure 5, we propose a typology showing effective production strategies for different product and pro- duction alternatives. When the production flexibility is high, manufacturers can follow a decentralized manufacturing strategy with short delivery lead times

Table 2 Iso-Profit Pairs of Capacity and Lead Time Values

Cost parameters Demand parameters Results

Price Cost Margin Model l r CV Capacity (K) Iso-profit pairs (L, K) Cutoff Profit

$10 $2 80% a-MMFE 100 21.1 0.7 15 (1, 16), (2, 20), (3, 25), (4, 32), (5, 47), (6, 85) 6 $579.05 $10 $2 80% a-MMFE 100 21.1 0.7 25 (1, 33), (2, 48), (3, 92) 3 $628.70 $10 $2 80% a-MMFE 100 21.1 0.7 50 (1, 154) 1 $685.71 $10 $2 80% a-MMFE 100 9.05 0.3 15 (1, 20), (2, 34), (3, 58), (4, 118) 4 $747.75 $10 $2 80% a-MMFE 100 9.05 0.3 25 (1, 68) 1 $765.13 $10 $2 80% a-MMFE 100 9.05 0.3 50 N/A 0 $773.33 $10 $8 20% a-MMFE 100 21.1 0.7 15 N/A 0 $83.51 $10 $8 20% a-MMFE 100 21.1 0.7 25 N/A 0 $93.34 $10 $8 20% a-MMFE 100 21.1 0.7 50 N/A 0 $100.60 $10 $8 20% a-MMFE 100 9.05 0.3 15 (1, 22), (2, 98) 2 $159.88 $10 $8 20% a-MMFE 100 9.05 0.3 25 (1, 95) 1 $166.18 $10 $8 20% a-MMFE 100 9.05 0.3 50 N/A 0 $171.95 $10 $2 80% m-MMFE 4.42 0.19 0.7 15 (1, 19), (2, 25), (3, 38), (4, 61) 4 $575.20 $10 $2 80% m-MMFE 4.42 0.19 0.7 25 (1, 39), (2, 61) 2 $634.43 $10 $2 80% m-MMFE 4.42 0.19 0.7 50 N/A 0 $696.47 $10 $2 80% m-MMFE 4.57 0.089 0.3 15 (1, 21), (2, 35), (3, 62) 3 $739.10 $10 $2 80% m-MMFE 4.57 0.089 0.3 25 (1, 42) 1 $757.64 $10 $2 80% m-MMFE 4.57 0.089 0.3 50 N/A 0 $768.07 $10 $8 20% m-MMFE 4.42 0.19 0.7 15 N/A 0 $110.27 $10 $8 20% m-MMFE 4.42 0.19 0.7 25 N/A 0 $120.14 $10 $8 20% m-MMFE 4.42 0.19 0.7 50 N/A 0 $129.48 $10 $8 20% m-MMFE 4.57 0.089 0.3 15 (1, 23), (2, 62) 2 $151.02 $10 $8 20% m-MMFE 4.57 0.089 0.3 25 (1, 42) 1 $159.11 $10 $8 20% m-MMFE 4.57 0.089 0.3 50 N/A 0 $167.10

Figure 4 Required Forecast Improvement for Static Scheduling Policy [Color figure can be viewed at wileyonlinelibrary.com]

and moderate production capacity (i.e., Strategies 1 and 2 in the figure). These types of manufacturers benefit from integrating forecast updates into produc- tion schedules. This integration allows manufacturers to significantly reduce supply–demand mismatches even for products with high demand uncertainty. When the demand uncertainty is relatively low, high flexibility also makes it possible for manufacturers to increase product variety by offering more customized products. This causes an increase in demand uncer- tainty, resulting in a move from Strategy 1 to 2. Manu- facturers of consumer packaged goods (e.g., Nestl�e, P&G and Unilever) may optimally follow a decentral- ized production policy. These manufacturers can easily have access to sales data on their products in retail stores through vendor-managed inventory sys- tems or by observing point-of-sale data. If they also invest in production flexibility that enables the dynamic adjustment of production schedules in response to changes in demand forecast, having a production facility close to the market helps them to increase their profits. Following such a strategy, Nestl�e operates in 47 states of the United States with 25 production facilities and 43 distribution centers and generated $9.7 billion in sales in 2014 (http:// www.nestleusa.com/about-us/key-figures). Recent research on the sourcing decisions of global firms has also revealed that consumer goods manufacturers tend to restructure their manufacturing footprint to be in close proximity to different markets (Cohen et al. 2016, pp. 14 and 17). Manufacturers of fashion apparel items can also successfully follow a decentral- ized production strategy for their seasonal products with high demand uncertainty (i.e., Strategy 2). These manufacturers can obtain useful demand information through fashion shows or by conducting market anal- ysis before the selling season (Fisher and Raman 1996). Then, they balance their capacity throughout the year by dynamically integrating forecast updates into production schedules. Owing to the short deliv- ery lead times, decentralized production networks

also allow them to react to demand fluctuations by performing in-season replenishments (Gallien et al. 2015). Manufacturers of products with long processing

times (e.g., agricultural goods and commodity items) cannot easily change their production schedules. These manufacturers should not invest in the accumulation of demand information over time until the realization of actual demand because they cannot benefit from accurate demand forecasts due to frozen schedules. They can cost-effectively follow a centralized produc- tion strategy for their low-demand-uncertainty prod- ucts (i.e., Strategy 3). When demand uncertainty is high, they should put some effort into improving pro- duction flexibility (i.e., moving from Strategy 4 to 2) or into improving demand forecasts to reduce demand uncertainty (i.e., moving from Strategy 4 to 3). If none of them is possible, manufacturers may change the price in response to demand fluctuations. However, we caution that dynamic pricing may cause price volatility as observed in commodity markets. In the automotive industry, production schedules

are highly inflexible due to frozen production periods. If the automotive industry does not improve produc- tion flexibility by reducing the length of frozen peri- ods, it may find itself in region 3 (i.e., Strategy 3). Although lean production principles, mainly applied to the automotive industry, emphasize the impor- tance of being in close proximity to suppliers and customers (Womack et al. 2007, ch. 6–7), auto manu- facturers pay a disproportionate amount of attention to improving their productivity (Womack et al. 2007, ch. 4) to keep costs down, leading to high utilization of factories. Thus, auto manufacturers cannot easily reduce their production volume, and they smooth demand through marketing activities (Womack et al. 2007, pp. 190–191). Under these conditions, auto man- ufacturers do not need to be in close proximity to cus- tomers, as evidenced by Cohen et al. (2016, p. 14) who empirically found that auto manufacturers tend to offshore their production activities.

5. Multi-Product Model

In this section, we consider N products and analyze the optimal order quantities in periods t 2 {1, 2, . . ., T} under a single shared capacity con- straint. We use the subscript and superscript j to denote the parameter values for the product j 2 {1, 2, . . ., N}. Each product can have different cost and demand parameters. Demand for the prod- ucts can also be correlated; the only restriction is that all products should have the same type of forecast- evolution process, either a-MMFE or m-MMFE. We denote by et ¼ ðe1t ; e2t ; . . .; eNt Þ the vector of forecast adjustments in period t. For the a-MMFE, ɛt follows a

Demand uncertainty

Low High

Pr od

uc ti

on fl

ex ib

ili ty

Lo w

H ig

Strategy 1 • Decentralized production • Room for product proliferation • Moderate capacity & short delivery lead

time • Full demand information transfer

Strategy 3 • Centralized production • High capacity & long delivery lead time • No demand information transfer

Strategy 2 • Decentralized production • Moderate capacity & short delivery lead

time • Full demand information transfer

Strategy 4 • High mismatch risk • Needs either production flexibility or

product standardization

Figure 5 Production Decision Typology

http://www.nestleusa.com/about-us/key-figures

multivariate normal distribution for t 2 {1, . . ., T} such that:

et � N ðM; RÞ; ð9Þ

where M is an N-dimensional vector with the entries: Mi = 0 for i 2 {1, . . ., N}; and Σ is an N 9 N covariance matrix such that Σi,j = q(i,j)rirj for i 2 {1, . . ., N} and i 2 {1, . . ., N}, and q(i,j) is the correlation between the forecast adjustments of pro- duct i and product j. For the m-MMFE, et � N ðM; RÞ, where M is an

N-dimensional vector such that Mi ¼ �r2i =2 for i 2 {1, . . ., N}; and Σ is an N 9 N covariance matrix with the entries: Σi,j = q(i,j)rirj for i 2 {1, . . ., N} and i 2 {1, . . ., N}. In each period, the total order quantity is restricted

to be less than or equal to the total capacity K:

X j2f1;2;...;Ng

x j t �

X j2f1;2;...;Ng

x j t�1 � K: ð10Þ

For the uncapacitated problem, the optimal value of bt satisfies the following equations for each product (Wang et al. 2012):

g j tðy

j tÞ ¼

Zðyjt�bjtþ1Þ=rj

�1 gtþ1ðyjt � rjdjÞ/ðdjÞddj ¼ 0; ð11Þ

for t 2 {1, 2, . . ., T � 1}, and

g j Tðy

j TÞ ¼ pjU

� yjT rj

� � cj ¼ 0: ð12Þ

We recall that y j t ¼ x

j t � lj � A

j t under the a-MMFE

and y j t ¼ lnðx

j tÞ � lj � A

j t under the m-MMFE for

t 2 {1, 2, . . ., T}. Equations (11) and (12) are recur- sive expressions that make it possible to find opti- mal base-stock levels in each period for the single- product model without any capacity constraint. These equations are equivalent to equations (6) and (7) of Wang et al. (2012). They can be interpreted as the marginal value of producing one unit in each period. We use these equations to find optimal base- stock levels for each product in each period under a fixed capacity constraint.

THEOREM 2. The optimal solution in each period t 2 {1, 2, . . ., T} is guaranteed if either

g1t ðy1t Þ ¼ � � � ¼ g j tðy

j tÞ ¼ � � � ¼ gNt ðyNt Þ ¼ 0; ð13Þ

X j2f1;2;...;Ng

x j t �

X j2f1;2;...;Ng

x j t�1\K; ð14Þ

g1t ðy1t Þ ¼ � � � ¼ g j tðy

j tÞ ¼ � � � ¼ gNt ðyNt Þ � 0; ð15Þ

X j2f1;2;...;Ng

x j t �

X j2f1;2;...;Ng

x j t�1 ¼ K; ð16Þ

are satisfied.

The proof is given in Appendix E. If the capacity constraint (i.e., Equation (10)) is not binding, the opti- mal order quantity for each product will be set such that the marginal value of producing one additional unit is equal to zero (i.e., Equation (13) holds). Other- wise, the optimality is guaranteed when Equations (15) and (16) are satisfied. However, it may not be possible to obtain Equation (15) in some cases, espe- cially when the capacity in each period is very lim- ited. In those cases, the optimal solution is obtained when maxfg1t ðy1t Þ; . . .; g

j tðy

j tÞ; . . .; gNt ðyNt Þg is mini-

mized. This objective function guarantees that the limited capacity is allocated to products with the highest marginal value of producing one additional unit. Therefore, we develop a mathematical model to find the optimal solution in each period:

Minimize: maxfg1t ðy1t Þ; . . .; g j tðy

j tÞ; . . .; gNt ðyNt Þg

subject to: ð17Þ

X j2f1;2;...;Ng

x j t �

X j2f1;2;...;Ng

x j t�1 � K; ð18Þ

g j tðy

j tÞ � 0; 8j 2 f1; . . .; Ng: ð19Þ

We determine the marginal value of producing one additional unit for each product in the absence of any capacity constraints as follows:

g j tðy

j tÞ¼

@G j tðx

j t;I

j tÞ

@x j t

¼pjPðDjTþ1[x j tÞP

j tðW

j tjW

j t �F½tþ1;T�Þ

�cjPjtðW j tjW

j t �F ½tþ1;T�Þ;

¼ � pjPðDjTþ1[x

j tÞ�cj

� P

j tðW

j tjW

j t �F½tþ1;T�Þ

ð20Þ

where P j tðW

j tjW

j t � F ½tþ1;T�Þ is the probability that pro-

duct j will not be ordered in the remaining periods. The probability PðDjTþ1 [ x

j tÞ can be easily calcu-

lated based on the forecast-evolution process and the cumulative forecast adjustments in period t. However, the second probability value (i.e., P

j tðW

j tjW

j t � F½tþ1;T�Þ) cannot easily be determined

because it depends on the capacity constraint and demand densities of other products. We estimate this probability by:

P j tðW

j tjW

j t �F ½tþ1;T�Þ¼PðD

j Tþ1 �x

j tjAtÞ

þPðDjTþ1 [x j t;D

J=j Tþ1 [ðT�tÞKjAtÞ;

ð21Þ where D

J=j Tþ1 denotes total demand for all other pro-

ducts. Equation (21) implies that the probability of not ordering product j in the remaining periods is the sum of the probability that demand for product j will be less than the available inventory at the end of period t and the probability that total cumulative capacity in the remaining periods is not sufficient to satisfy total demand for other products when demand for product j is higher than the available inventory. By using Equations (20) and (21) in our mathematical model (17)–(18), we can solve the dynamic scheduling problem in a computationally efficient way.

COROLLARY 3. Let x1 ¼ fx11; x12; . . .; x1Tg and x2 ¼ fx21; x22; . . .; x2Tg be the vectors of optimal inventory position levels for two different products that share the same capacity of K units in the periods t 2 {1, . . ., T}. E½x1t þ x2t � weakly increases (decreases) if demand for the products becomes more positively (negatively) correlated.

The proof is straightforward from Equation (21). As demand for the products becomes more positively (negatively) correlated, the term P

j tðW

j tjW

j t � F½tþ1;T�Þ

weakly increases (decreases) due to an increase (de- crease) in the value of the last term of the right-hand side of Equation (21). This leads to an increase (de- crease) in the g

j tðy

j tÞ values because non-negativity of

the term pjPðDjTþ1 [ x j tÞ � cj in Equation (20) is guar-

anteed by the constraint (19). If the total capacity in period t is not fully utilized, an increase in g

j tðy

j tÞ val-

ues results in higher capacity utilization. Therefore, the expected value of the sum of the inventory posi- tions (i.e., E½x1t þ x2t �) weakly increases (decreases) if demand for the products is more positively (nega- tively) correlated. Corollary 3 implies that the sum of inventory positions is less (more) concentrated toward the last production period under more posi- tively (negatively) correlated demand. Therefore, it is optimal to highly utilize production capacity in earlier periods under positive correlation of demand. This allows the manufacturer to reduce potential stockout costs when demand for both products turns out to be high, which has a higher probability under positive correlation. Under negative correlation of demand, it is less likely that both products are highly in demand at the same time. Then, the manufacturer can opti- mally use production capacity in later periods to sat- isfy market demand. Our min-max model given by Equations (17)–(19)

is similar to the inventory allocation rule in Jackson

(1988), in which the author considered an inventory allocation problem from a central warehouse to N different retailers. The objective in Jackson (1988) is to allocate scarce centralized inventory among retailers to satisfy demand at retailers until the cen- tral warehouse is replenished. He aggregated total demand at retailers until the next replenishment of the warehouse and set the sum of all the retailers’ base-stock levels to the total warehouse inventory. In his model, there is no constraint on tranship- ment capacity, so that all centralized inventory can be delivered to retailers in a single period. Thus, our problem is inherently different from that of Jackson (1988). It might look analogous to the clas- sic multi-period inventory-allocation problem from a central warehouse to N different retailers if we conceptualize the shared capacity and N different products in our problem as warehouse inventory and N retailers in the inventory-allocation problem, respectively. However, our problem departs from the inventory-allocation problem in four different ways. First, sales occur in each period in the inven- tory-allocation problem and hence the objective is to balance the inventory levels in retail stores (Agrawal et al. 2004, G€ull€u 1997, McGavin et al. 1997), whereas sales occur at the end of the last period in our problem. Second, the optimal solu- tion is myopic in the inventory-allocation problem, since it does not depend on future demand esti- mates (Federgruen and Zipkin 1984), whereas it is not myopic in our problem. We optimize the pro- duction orders depending on the evolution of demand forecasts and remaining capacity until the realization of actual demand. Third, it is common in the literature on multi-period inventory-alloca- tion problems to assume that no inventory is kept in the central warehouse at the end of each period (G€ull€u 1997), whereas we do not impose any con- straint of fully utilizing the capacity. Thus, it is possible to have buffer capacity in our model which can be idle or utilized depending on the evolution of demand forecasts. We also note that although Jackson (1988) does not impose such a constraint, his model is structurally very different from and not directly related to ours. Finally, it is possible to rebalance inventory (i.e., transporting some inventory from one retailer to another) in the inventory-allocation problem (Agrawal et al. 2004, G€ull€u 1997), whereas it is not possible to pool the demand for different products in our model. In Figures 6 and 7, we provide examples to comple-

ment our analytical results. We consider a two-pro- duct scheduling problem under a shared capacity constraint of K = 30. As in the previous section, we set T = 10. Product 1 has high demand uncertainty and a high profit margin, for which we set p1 = $10

per unit, c1 = $2 per unit, the expected value of demand to 100 units, and the CV to 0.7. Product 2 has low demand uncertainty and a low profit margin with the parameter values: p2 = $10 per unit, c2 = $8 per unit, an expected value of demand of 100 units, and a CV of 0.3. The parameters of both the additive and the multiplicative processes corresponding to these mean demand and CV values are given in Tables 1 and 2. In Figure 6, we provide the expected production

quantities for the a-MMFE under three different correlation structures: Figure 6a for perfect positive correlation; Figure 6b for the case of no correlation; and Figure 6c for perfect negative correlation. Fig- ure 6 shows that the more positively correlated the demand for the products, the more utilized the capacity in earlier periods. For the perfect negative correlation shown in Figure 6c, production shifts toward the later periods. These results are in line with Corollary 3 and interpreted as follows. The probability that both products have a surge in

demand is relatively high under positive correla- tion, increasing the risk of having high lost sales at the end. Thus, it is optimal to utilize the capacity more in earlier periods under positive correlation in order to mitigate potential stock-out risks. Under negative correlation, the changes in demand fore- casts are expected to be in opposite directions. Therefore, it is optimal to postpone production to derive more benefit from accurate demand fore- casts. In Figure 7, we provide the expected production

quantities for the m-MMFE under three different correlation structures: Figure 7a for perfect positive correlation; Figure 7b for the case of no correlation; and Figure 7c for perfect negative correlation. The results are very similar to those of the additive model given in Figure 6, which is also in line with Corollary 3. As previously discussed in section 4.1, the similarity of the results between a-MMFE and m- MMFE results from the capacity constraint in our model.

(a)

(b)

(c)

Figure 6 Expected Production Quantities for the Additive Model. (a) Perfect Positive Correlation; (b) No Correlation; (c) Perfect Negative Correlation

(a)

(b)

(c)

Figure 7 Expected Production Quantities for the Multiplicative Model. (a) Perfect Positive Correlation; (b) No Correlation; (c) Perfect Negative Correlation

Figures 6 and 7 depict that the utilized capacity is allocated more to Product 1 in earlier periods but to Product 2 in later periods when demand for the prod- ucts is positively correlated. High stock-out risk under positive correlation leads to an increase in the production quantity of Product 1 because this product has a higher profit margin and hence a higher cost of lost sales. The capacity in later periods is underuti- lized on average due to the excessive production in earlier periods. Under negative correlation, the uti- lized capacity is allocated more to Product 2 in earlier periods, but more to Product 1 in later periods. This is because precious capacity in later periods, once more accurate demand forecasts are available, is allocated more to Product 1 to reduce the lost-sales costs of this product, which are much higher than those for Pro- duct 2.

6. Conclusion

In this study, we consider the production scheduling problem of a manufacturer under the evolution of demand forecasts and capacity constraint. The man- ufacturer may need to start production well in advance of the realization of customer demand in order to satisfy demand peaks. As demand forecasts evolve over time, production schedules can be updated. We use both additive and multiplicative versions of the martingale forecast-evolution process to model the evolution of demand forecasts. We derive expressions for the optimal production sched- ules for a single-product problem. We find that man- ufacturers with zero delivery lead time can realize most of their potential profits by building a very lim- ited amount of capacity. As the delivery lead time increases, manufacturers should excessively increase their production capacity to compensate for the increase in mismatch losses resulting from the lead times, making lead time reduction a better alterna- tive than capacity expansion. We then extend our results to a multi-product problem and provide expressions for the optimal schedules under a shared capacity constraint. We apply our derivations to a two-product model and show that the more posi- tively correlated the demand for the products, the more utilized the capacity in earlier periods. Our analytical model can be extended to deter-

mine the optimal ordering policy in a multi-sup- plier model, whereby a retailer can purchase products from different suppliers and sell them to a market with uncertain demand. The suppliers may also differ in terms of delivery lead time, wholesale price and production capacity. The retai- ler then has to place periodic orders with each sup- plier to profitably bring its inventory to a desired level before the realization of market demand. By

dynamically adjusting its periodic orders, the retai- ler would be able to maximize its profit by mini- mizing mismatch costs under the evolution of demand forecasts. We suggest that the application of dynamic programming formulation to a multi- supplier problem would be an interesting topic for future research.

Acknowledgments

We are grateful to Panos Kouvelis, the Senior Editor, and the reviewers for their constructive comments on the earlier versions of the paper. Any errors or omissions are our responsibility.

Appendix A. Proof of Theorem 1

In period T, we have the following:

GTðxT; ATÞ ¼ EDjAT h p minðD; xTÞ

i � cxT; ðA1Þ

VTðxT�1; ATÞ ¼ max xT�1 � xT � xT�1þK

fGTðxT; ATÞg þ cxT�1: ðA2Þ

Thus, the ordering decision in period T can be con- sidered as a newsvendor problem:

GTðxT; ATÞ ¼ p ZxT

DfðDjATÞdD þ pxTð1 � FðxTjATÞÞ

� cxT; ðA3Þ

@GT @xT

¼ pð1 � FðxTjATÞÞ � c: ðA4Þ

where f(�|�) and F(�|�) denote conditional demand density and distribution, respectively, which are given by Equations (2) and (4) for a-MMFE and m- MMFE. With p > c, GT(�, AT) is a concave function for any

given AT. For STðATÞ ¼ fxTj@GT=@xT ¼ 0g, VTðxT�1;ATÞ

¼ GTðxT�1;ATÞþcxT�1 ifxT�1[STðATÞ;

GTðSTðATÞ;ATÞþcxT�1 ifxT�1þK[STðATÞ�xT�1; GTðxT�1þK;ATÞþcxT�1 ifSTðATÞ�xT�1þK:

8>< >:

ðA5Þ Thus, VT(�, AT) is a non-decreasing concave function. Then,

GT�1ðxT�1; AT�1Þ ¼ EATjAT�1 h VTðxT�1; ATÞ

i � cxT�1;

ðA6Þ

which is a concave function because VT(�, AT) is con- cave. Similar to Wang et al. (2012), Gt(xt, At) is a concave function for t 2 {1, 2, . . ., T � 1} (by induc- tion), and the optimal policy is a base-stock policy:

StðAtÞ ¼ arg max x

fGtðxt; AtÞg; 8t 2 f1; 2; . . .; Tg: ðA7Þ

A.1. Derivation of St(At) Values for the Additive Model In the last production period, conditional demand density follows a normal distribution such that:

DjAT � N ðl þ AT; r2Þ: ðA8Þ

The optimal base-stock level is found by:

@GT @xT

¼ gTðxT � l � ATÞ ¼ pU � xT � l � AT

� � c ¼ 0;

ðA9Þ

where (�) = 1 � Φ(�). Thus, STðATÞ ¼ l þ AT þ bT; ðA10Þ

where bT ¼ rZaT , ZaT ¼ U�1ðaTÞ, and aT ¼ ðp � cÞ= p. We define a new variable such that:

yt ¼ xt � l � At; ðA11Þ for t 2 {1, . . ., T}. Suppose in period t + 1,

Vtþ1ðxt;Atþ1Þ

¼ Gtþ1ðxt;Atþ1Þþcxt if xt [Stþ1ðAtþ1Þ; G tþ1ðAtþ1Þþcxt if xt þK[Stþ1ðAtþ1Þ�xt;

Gtþ1ðxt þK;Atþ1Þþcxt if Stþ1ðAtþ1Þ�xt þK;

8>< >:

ðA12Þ where G tþ1ðAtþ1Þ ¼ Gtþ1ðStþ1ðAtþ1Þ; Atþ1Þ. Using Equation (A12), we formulate Gt(xt,At) as follows:

Gtðxt; AtÞ ¼EAtþ1jAt h Vtþ1ðxt; Atþ1Þ

i � cxt;

¼ Zetþ1

�1 Gtþ1ðxt; At þ etþ1Þfðetþ1Þdetþ1

þ Zetþ1

etþ1

G tþ1ðAt þ etþ1Þfðetþ1Þdetþ1

ðA13Þ

þ Zþ1

etþ1

Gtþ1ðxt þ K; At þ etþ1Þfðetþ1Þdetþ1; ðA14Þ

where etþ1 and etþ1 satisfy:

Stþ1ðAt þ etþ1Þ ¼ xt; ðA15Þ Stþ1ðAt þ etþ1Þ ¼ xt þ K: ðA16Þ

Then,

@Gt @xt

¼ Zetþ1

�1 gtþ1ðxt � l � At � etþ1Þfðetþ1Þdetþ1

þ Zþ1

etþ1

gtþ1ðxt þ K � l � At � etþ1Þfðetþ1Þdetþ1;

ðA17Þ

gtðytÞ ¼ Zðyt�btþ1Þ=r

�1 gtþ1ðyt �rdÞ/ðdÞdd

þ Zþ1

ðytþK�btþ1Þ=r

gtþ1ðyt þK �rdÞ/ðdÞdd: ðA18Þ

A.2. Derivation of St(At) Values for the Multiplicative Model In the last production period, conditional demand density follows a lognormal distribution such that:

DjAT � ln �N ðl þ AT; r2Þ: ðA19Þ

The optimal base-stock level is found by:

@GT @xT

¼ gðlnðxTÞ � l � ATÞ

¼ pU � lnðxTÞ � l � AT

� � c: ðA20Þ

Thus,

STðATÞ ¼ expðl þ AT þ bTÞ; ðA21Þ where bT ¼ rZaT , ZaT ¼ U�1ðaTÞ, and aT ¼ ðp � cÞ =p. We define two new variables such that:

yt ¼ lnðxtÞ � l � At; ðA22Þ ut ¼ lnðxt þ KÞ � l � At; ðA23Þ

for t 2 {1, . . ., T}. Suppose in period t + 1,

Vtþ1ðxt;Atþ1Þ

¼ Gtþ1ðxt;Atþ1Þþcxt if xt [Stþ1ðAtþ1Þ; G tþ1ðAtþ1Þþcxt if xt þK[Stþ1ðAtþ1Þ�xt;

Gtþ1ðxt þK;Atþ1Þþcxt if Stþ1ðAtþ1Þ�xt þK;

8>< >:

ðA24Þ

where G tþ1ðAtþ1Þ ¼ Gtþ1ðStþ1ðAtþ1Þ; Atþ1Þ. Let xt+1 = At+1 � At = ɛt+1 + r2/2. Then, we obtain the following result:

Gtðxt; AtÞ ¼ EAtþ1jAt h Vtþ1ðxt; Atþ1Þ

� cxt; ðA25Þ

¼ Zxtþ1

�1 Gtþ1ðxt; At þ xtþ1Þfðxtþ1Þdxtþ1

þ Zxtþ1

xtþ1

G tþ1ðAt þ xtþ1Þfðxtþ1Þdxtþ1

þ Zþ1

xtþ1

Gtþ1ðxt þ K; At þ xtþ1Þfðxtþ1Þdxtþ1;

ðA26Þ

where xtþ1 and xtþ1 satisfy:

Stþ1ðAt þ xtþ1Þ ¼ xt; ðA27Þ Stþ1ðAt þ xtþ1Þ ¼ xt þ K: ðA28Þ

Then,

@Gt @xt

¼ Zxtþ1

�1 gtþ1ðlnðxtÞ � l � At � xtþ1Þfðxtþ1Þdxtþ1

þ Zþ1

xtþ1

gtþ1ðlnðxt þ KÞ � l � At

� xtþ1Þfðxtþ1Þdxtþ1; ðA29Þ

gtðytÞ ¼ Zðyt�btþ1Þ=r

�1 gtþ1ðyt � rdÞ/ðdÞdd

þ Zþ1

ðut�btþ1Þ=r

gtþ1ðut � rdÞ/ðdÞdd: ðA30Þ

A.3. Derivation of gt(yt) Values For period T, we have

gTðyTÞ ¼ pU � yT r

� � c;

where yT ¼ xT � l � AT and UðyT=rÞ ¼ PðDTþ1 [ xTjATÞ. From Equation (A17) or (A29),

gT�1ðyT�1Þ

¼ ZeT

�1

@GTðxT; AT�1 þ eTÞ @xT

�� xT¼xT�1

fðeTÞdeT

þ Zþ1

@GTðxT; AT�1 þ eTÞ @xT

�� xT¼xT�1þK

fðeTÞdeT;

¼ ZeT

�1

� pPðDTþ1 [ xT�1jAT�1 þ eTÞ � c

� fðeTÞdeT

þ Zþ1

� pPðDTþ1 [ xT�1 þ KjAT�1

þ eTÞ � c � fðeTÞdeT;

¼ pPðDTþ1 [ xT�1; eT [ eTjAT�1Þ � cPðeT [ eTjAT�1Þ þ pPðDTþ1 [ xT�1 þ K; eT [ eTjAT�1Þ � cPðeT [ eTjAT�1Þ:

ðA31Þ Then,

gT�2ðyT�2Þ

¼ ZeT�1

�1

@GT�1ðxT�1;AT�2þeT�1Þ @xT�1

�� xT�1¼xT�2

fðeT�1ÞdeT�1

þ Zþ1

eT�1

@GT�1ðxT�1;AT�2þeT�1Þ @xT�1

�� xT�1¼xT�2þK

fðeT�1ÞdeT�1:

ðA32Þ

Combining Equation (A31) into (A32), we obtain

gT�2ðyT�2Þ ¼ pPðDTþ1[xT�2;eT[eT;eT�1[eT�1jAT�2Þ �cPðeT[eT;eT�1[eT�1jAT�2Þ þpPðDTþ1[xT�2þ2K;eT[eT;eT�1[eT�1jAT�2Þ �cPðeT[eT;eT�1[eT�1jAT�2Þ þpPðDTþ1[xT�2þK;eT[eT;eT�1[eT�1jAT�2Þ �cPðeT[eT;eT�1[eT�1jAT�2Þ þpPðDTþ1[xT�2þK;eT[eT;eT�1[eT�1jAT�2Þ �cPðeT[eT;eT�1[eT�1jAT�2Þ:

The probability values PðeT [ eT; eT�1 [ eT�1jAT�2Þ and PðeT [ eT; eT�1 [ eT�1jAT�2Þ are the probabil- ities of utilizing the capacity fully in one period and not producing anything in the next period and vice versa. We assume that they are equal to zero

because neither additive nor multiplicative models allow such steep changes in demand forecasts (in quantitative finance, an additional jump term is added to geometric Brownian motion to capture sudden increases or decreases in stock price, which is beyond the scope of this study). We also assume that ϒt and Ψt are subsets of the set of all events in the time domain [t + 1, T] (i.e., (F½tþ1;T�), such that ϒt and Ψt denote the events of no capacity utiliza- tion and full capacity utilization in the remaining periods {t + 1, . . ., T}, respectively. Then, the last expression is rewritten as follows:

gT�2ðyT�2Þ ¼ pPðDTþ1[xT�2;!T�2jAT�2;!T�2�F ½T�1;T�Þ �cPð!T�2jAT�2;!T�2�F ½T�1;T�Þ þpPðDTþ1[xT�2þ2K;WT�2jAT�2;WT�2 �F½T�1;T�Þ�cPðWT�2jAT�2;WT�2�F½T�1;T�Þ:

Following the same logic, we get the following result for period t 2 {1, . . ., T � 1}.

gtðytÞ ¼ pPðDTþ1 [ xt; !tjAt; !t � F ½tþ1;T�Þ � cPð!tjAt; !t � F½tþ1;T�Þ þ pPðDTþ1 [ xt þ ðT � tÞK; WtjAt; Wt � F ½tþ1;T�Þ � cPðWtjAt; Wt � F½tþ1;T�Þ: ðA33Þ

The probabilities of ϒt and Ψt are calculated by

!t ¼Pðetþ1 [ etþ1; etþ2 [ etþ2; . . .; eT [ eTÞ; Wt ¼Pðetþ1 [ etþ1; etþ2 [ etþ2; . . .; eT [ eTÞ:

The terms fetþ1; etþ2; . . .; eTg correspond to the fore- cast adjustments that make the optimal inventory level equal to xt in periods {t+1, t+2, . . ., T}, respec- tively. Similarly, the terms fetþ1; etþ2; . . .; eTg corre- spond to the points that make the optimal order-up- to level equal to {xt+K, xt+2K, . . ., xt + (T�t)K} in periods {t+1, t+2, . . ., T}, respectively. To compute these forecast adjustment boundaries, we use back- ward induction and generate a (T � t + 1) 9 (T � t + 1) matrix only with lower triangular ele- ments such that the (i, j)th element in the matrix gives us the ɛt+i�1 that makes the optimal inventory level equal to xt + (t + j � 1)K. Starting from the last period, we first calculate (T � t + 1) different ɛT val- ues that make the optimal order-up-to level in the last period equal to {xt, xt + K, xt + 2K, . . ., xt + (T � t)K}. We use the last two ɛT values to cal- culate the ɛT�1 value that makes the optimal order- up-to level in period T � 1 equal to xt + (T � 1 � t) K, which is also equal to eT�1. Similarly, we need the last three ɛT values to calculate eT�2, and so on. Finally, the first column of the matrix provides the e

values, and the diagonal elements give us the e val- ues for all the remaining periods. Equation (A33) can be interpreted as the marginal

value of producing one additional unit. The optimal base-stock level corresponds to the point at which the marginal value of producing one more unit is equal to zero. As K ? +∞, full capacity utilization is not pos- sible (i.e., PðWtjAt; Wt � F½tþ1;T�Þ ¼ 0), and Equation (A33) reduces to equation (9) of Wang et al. (2012). Then, the optimal policy is not to order until the last period and to bring the inventory level to ST(AT) in the last period.

Appendix B. An Extension to the Single-Product Model

We now consider the case in which the supplier offers early purchase discounts if the buyer places produc- tion orders well in advance of the realization of the actual demand. We use ct to denote the transfer price per unit in period t 2 {1, . . ., T}, where c1 < c2 < ⋯ < ct. This payment structure is similar to that of Wang et al. (2012). However, we also impose a capac- ity constraint on our model. It directly follows from Theorem 1:

(A) Under the a-MMFE, the optimal base-stock level is found by:

StðAtÞ ¼ l þ At þ bt: ðB1Þ

The safety stock term bt has a boundary value bT ¼ rZaT , where aT ¼ ðp � cÞ=p and ZaT ¼ U�1ðaTÞ. In periods t 2 {1, 2, . . ., T � 1}, bt satisfies the following equations:

gtðytÞ ¼ Zðyt�btþ1Þ=r

�1 gtþ1ðyt � rdÞ/ðdÞdd

þ Zþ1

ðytþK�btþ1Þ=r

gtþ1ðyt þ K � rdÞ/ðdÞdd

þ ctþ1 � ct ¼ 0;

ðB2Þ

gTðyTÞ ¼ pU � yT r

� � cT ¼ 0; ðB3Þ

where yt = xt � l � At. (B) Under the m-MMFE, the optimal base-stock

level is found by:

StðAtÞ ¼ expðl þ At þ btÞ: ðB4Þ The safety stock term bt has a boundary value bT ¼ rZaT , where aT ¼ ðp � cÞ=p and ZaT ¼ U�1ðaTÞ. In periods t 2 {1, 2, . . ., T � 1}, bt satisfies the following equations:

gtðytÞ ¼ Zðyt�btþ1Þ=r

�1 gtþ1ðyt �rdÞ/ðdÞdd

þ Zþ1

ðut�btþ1Þ=r

gtþ1ðut �rdÞ/ðdÞddþctþ1 �ct ¼ 0;

ðB5Þ

gTðyTÞ ¼ pU � yT r

� � cT ¼ 0; ðB6Þ

where yt = ln (xt) � l � At and ut = ln (xt + K) � l � At.

Appendix C. Proof of Corollary 1

The base-stock level in the last period is given by the newsvendor solution. We start from the last period and set STðATÞ ¼ xT ¼ x0T. For the periods t 2 {1, 2, . . ., T � 1}, it follows from Equation (A18) that

gtðbtÞ ¼ Zðbt�btþ1Þ=r

�1 gtþ1ðbt � rdÞ/ðdÞdd

þ Zþ1

ðbtþK�btþ1Þ=r

gtþ1ðbt þ K � rdÞ/ðdÞdd

¼ 0;

for the a-MMFE. At the optimal solution,

Zðbt�btþ1Þ=r

�1 gtþ1ðbt �rdÞ/ðdÞdd

¼ � Zþ1

ðbtþK�btþ1Þ=r

gtþ1ðbt þK �rdÞ/ðdÞdd;

Zðbt�btþ1Þ=r

�1

@gtþ1ðbt �rdÞ @bt

/ðdÞdd

¼ � Zþ1

ðbtþK�btþ1Þ=r

@gtþ1ðbt þK �rdÞ @bt

� 1þ@K=@bt

� /ðdÞdd:

It follows from the concavity of Gt(�, At) that @gt(yt)/ @yt < 0 for t 2 {1, 2, . . ., T}. Therefore, the last expression holds when @K/@bt < �1 for t 2 {1, 2, . . ., T � 1}, meaning that the optimal safety stock level decreases in K. The optimal inven- tory position for t 2 {1, 2, . . ., T � 1} is:

xt ¼ l þ At þ bt: ðC1Þ

Combining this expression with @K/@bt < �1 yields the result that xt decreases in K for t 2 {1, 2, . . ., T � 1}, completing the proof for the a- MMFE. For the m-MMFE, the term xt can be written as:

xt ¼ expðl þ At þ btÞ: ðC2Þ

Then, following the same steps as in the a-MMFE given above, it is straightforward that Corollary 1 also holds for the m-MMFE.

Appendix D. Proof of Corollary 2

It follows from Equation (A5) that VT(xT�1, AT|K) is a non-decreasing concave function of K with

@VTðxT�1; ATjKÞ @K

¼ 0; ðD1Þ

for xT�1 + K > ST(AT). Then,

GT�1ðxT�1; AT�1jKÞ ¼ EATjAT�1½VTðxT�1; ATjKÞ� � cxT�1;

ðD2Þ

which is also a non-decreasing concave function of K. By induction, we have V1(x0, A1|K) as a non- decreasing concave function. When K ? 0, Equation (A5) can be reformulated

such that:

VTðxT�1; ATjKÞ ¼ GTðxT�1 þ K; ATjKÞ þ cxT�1; ðD3Þ

and

V1ðx0; A1jKÞ ¼ EATjA1½GTðTK; ATjKÞ: ðD4Þ

Therefore, @V1ðx0; A1jKÞ

�� K!0

¼ ðp � cÞT. When K ? +∞, our model reduces to the form of

Wang et al. (2012) and V1(x0, A1|K) becomes inde-

pendent of K. Thus, @V1ðx0; A1jKÞ

�� K!þ1

¼ 0.

Appendix E. Proof of Theorem 2

We consider our model as an extension of the newsvendor network, originally developed by Van Mieghem (1998) and Harrison and Van Mieghem (1999). In these papers, the authors assumed that a newsvendor decision maker determines a capacity vector before observing demand and a production vector after observing demand. They calculated the expected shadow price for capacity constraints and proved that the optimal solution exists at the point where the expected shadow price is equal to the unit

capacity cost. Our model is slightly different in that we assume that order quantities under a total capacity constraint are determined before observing demand. We solve the dual problem of our newsvendor network to derive some analytical results that give the optimal solution. In the following, we present the optimality conditions for our multi-product model. In period T, the primal problem for two products

with a limited capacity of K is formulated as follows:

Maximize x j T ;8j2f1;...;Ng

z ¼ E � X

j2f1;...;Ng pjJ j � cjxjT

� ðE1Þ

subject to:

J j � xjT � 0; 8j 2 f1; . . .; Ng; ðE2Þ

J j � DjTþ1; 8j 2 f1; . . .; Ng; ðE3Þ X

j2f1;...;Ng x j T � K þ

X j2f1;...;Ng

x j T�1: ðE4Þ

Then, the dual problem is:

Minimize �j;bj;c

w ¼ E � X

j2f1;...;Ng bjD

j Tþ1

þ cðK þ X

j2f1;...;Ng x j T�1Þ

� ðE5Þ

subject to:

Eð�j þ bjÞ � pj; 8j 2 f1; . . .; Ng; ðE6Þ

Eð�j � cÞ � cj; 8j 2 f1; . . .; Ng: ðE7Þ

The values of kj and bj for each j 2 {1, . . ., N} are found by the parametric analysis:

1. If D j Tþ1 \ x

j T, kj = 0 and bj = pj for j 2

{1, . . ., N}. 2. If D

j Tþ1 [ x

j T, kj = pj and bj = 0 for j 2

{1, . . ., N}.

Combining these results with the constraint (E.7), we obtain:

pjð1 � FjTþ1ðx j TÞÞ � c � cj; j 2 f1; . . .; Ng: ðE8Þ

We set a value for the dual variable c such that:

c ¼ ðp1 � c1Þ � p1F1Tþ1ðx1TÞ ¼ � � � ¼ ðpN � cNÞ � pNFNTþ1ðxNT Þ: ðE9Þ

Therefore,

c ¼ g1Tðy1TÞ ¼ � � � ¼ g j Tðy

j TÞ ¼ � � � ¼ gNT ðyNT Þ � 0: ðE10Þ

Then, the objective function value for the dual prob- lem becomes:

z ¼ X

j2f1;...;Ng

h ðpj � cjÞxjT � pj

ZxjT

ðxjT � D j Tþ1ÞfjðD

j Tþ1ÞdD

j Tþ1

i :

ðE11Þ

Equation (E11) is equivalent to the solution of the pri- mal problem for a feasible pair of x

j T and x

k T values. It

follows from the strong theorem of duality that the optimal solution satisfies Equation (E9). Therefore, we have the following conditions of optimality:

c ¼ g1Tðy1TÞ ¼ � � � ¼ g j Tðy

j TÞ ¼ � � � ¼ gNT ðyNT Þ; ðE12Þ

X j2f1;...;Ng

x j T � K þ

X j2f1;...;Ng

x j T�1: ðE13Þ

If the capacity constraint (E4) is not binding, the dual variable c becomes zero. In this case, the opti- mal solution reduces to the solution of N indepen- dent newsvendor problems in the last period. Otherwise, the optimal solution exists at the point where the marginal value of producing one unit is the same for all products. In period t 2 {1, 2, . . ., T � 1}, the optimization

problem is written as follows:

Maximize x j t ;j2f1;...;Ng

z ¼ X

j2f1;...;Ng G

j tðx

j t; A

j tÞ ðE14Þ

subject to: X j2f1;...;Ng

x j t � K þ

X j2f1;...;Ng

x j t�1:

ðE15Þ

Then, the dual problem is:

Minimize �j;bj;c

w ¼ cðK þ X

j2f1;...;Ng x j t�1Þ ðE16Þ

subject to:

c � gjtðy j tÞ; j 2 f1; . . .; Ng:

ðE17Þ

Then, the optimal solution in each period t 2 {1, . . ., T} satisfies the following equations:

c ¼ g1t ðy1t Þ ¼ � � � ¼ g j tðy

j tÞ ¼ � � � ¼ gNt ðyNt Þ � 0; ðE18Þ

X j2f1;2;...;Ng

x j t � K þ

X j2f1;2;...;Ng

x j t�1: ðE19Þ

If the capacity constraint (E15) is not binding, the dual variable c becomes zero. In this case, the opti- mal solution reduces to the solution of N uncapaci- tated problems (see proposition 1 in Wang et al. 2012).

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