Extending the strategic safety stock placement model to consider tactical production smoothing

European Journal of Operational Research 279 (2019) 429–448

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Production, Manufacturing, Transportation and Logistics

Extending the strategic safety stock placement model to consider tactical production smoothing Kunal Kumar a, Tarik Aouam a,b,∗ a b

Faculty of Economics and Business Administration, Ghent University, Tweekerkenstraat 2, Gent 9000, Belgium Université Internationale de Rabat, Rabat Business School - BearLab, Morocco

a r t i c l e

i n f o

Article history: Received 14 November 2018 Accepted 3 June 2019 Available online 8 June 2019 Keywords: Supply chain management Production smoothing Tactical planning model Safety stock placement Guaranteed service times

a b s t r a c t This paper extends strategic safety stock placement models under the guaranteed service approach (GSA) to incorporate tactical production smoothing. We propose a model to jointly optimize production capacity, production smoothing, and service times between all stages in the supply chain. Analysis of the model leads to several interesting findings. First, for certain service times, production smoothing is desirable to reduce both capacity and inventory costs. Second, inventory cost at a production stage is nonmonotonous in its net service time and consequently quoting a large service time may increase costs at both the production stage and its customers. Third, safety stocks can be pooled at downstream stages only when production is not smoothed, while production smoothing necessitates holding safety stocks. The formulated problem is solved using a dynamic program and numerical experiments are conducted based on a real-world instance from the literature. Through these experiments, we show that integrating production smoothing in GSA models results in considerable savings, especially when capacity costs are neither too high nor too low relative to holding costs. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Manufacturing firms are under an unprecedented pressure to improve service levels, reduce inventory costs and provide shorter lead times, in order to maintain or improve competitive advantage. Finding a balance between the conflicting pressures of delivering high service levels and reducing inventory costs lies at the heart of inventory management, which is a key driver behind the superior performance of the best-in-class companies (Aberdeen Group, 2015). As reported by the Aberdeen group, the use of (multiechelon) inventory optimization tools is an important enabler of the superior performance of these companies. The power of optimization models underlying these tools can be improved significantly by integrating decisions across different levels (strategic and tactical) and different functions such as inventory, production, and transportation (Armbruster & Kempf, 2012; Muriel & Simchi-Levi, 2003; Sarmiento & Nagi, 1999). In this paper, we extend strategic safety stock placement (SSP) models to incorporate tactical production smoothing. SSP models determine the optimal location and quantity of safety stocks that are strategic in nature because they decouple ∗

Corresponding author. E-mail addresses: [email protected] (K. Kumar), [email protected] (T. Aouam). https://doi.org/10.1016/j.ejor.2019.06.009 0377-2217/© 2019 Elsevier B.V. All rights reserved.

parts of a supply chain (Graves & Willems, 20 0 0). These strategic models do not capture the influence of tactical production decisions such as production smoothing. In fact, firms smooth production to reduce costs of overtime and idle capacity, and use safety stocks to insulate smoothed production plans from demand variability (Bray & Mendelson, 2015). As a result, production smoothing affects the amount of safety stocks needed to meet service levels (Bertrand, 1986; Graves, 1988). Several studies point to the prevalence of production smoothing in real-world supply chains. Mollick (2004) found evidence in support of production smoothing in the Japanese vehicle industry, where firms lay emphasis on techniques such as Heijunka (leveling). Cachon, Randall, and Schmidt (2007) showed that production smoothing is a common practice in U.S. industries. Bray and Mendelson (2015) studied the case of 20 automobile manufacturers across the U.S., Asia and Europe, and found that production for up to 75% of the 162 car models was smoothed by at least 5%. The prevalence of production smoothing in practice and its effect on supply chain safety stocks motivate us to analyze the interdependencies between strategic safety stock placement and tactical production smoothing. We consider a supply chain network that consists of production and logistics stages, both of which are potential locations for holding safety stocks. Each stage follows a periodic-review base-stock policy and promises its customers a guaranteed service time within which all orders are delivered with certainty, according to

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the guaranteed-service approach (GSA) (Graves & Willems, 20 0 0). If demand exceeds the on-hand inventory, a stage may resort to extraordinary measures to fill backorders. Production stages are controlled according to the tactical planning model (TPM) of Graves (1986) in order to smooth production levels. TPM is widely studied for its simplicity and analytical tractability (see Graves, 1988, Balakrishnan, Geunes, & Pangburn, 2004, Boute, Disney, Lambrecht, & Van Houdt, 2007, Zahraei & Teo, 2017). Boute and Van Mieghem (2015) demonstrated that the linear control rule of TPM closely follows the optimal dual base-stock policy. The proposed model builds on recent efforts to reflect production capacity and tactical decisions in supply chain safety stock models under the GSA. Graves and Schoenmeyr (2016) account for capacity using a modified base-stock ordering policy that restricts the maximum order quantity to the capacity. Kumar and Aouam (2018a,b) and Aouam and Kumar (2018) use queuing theory to model capacity through deterministic endogenous production lead times that are functions of tactical decisions. Kumar and Aouam (2018a,b) study the effect of batch sizing, while Aouam and Kumar (2018) analyze the impact of overtime and subcontracting. Following the suggestion of Graves (1988), the present work extends the SSP model to incorporate tactical production smoothing, while considering capacitated production stages. The proposed model jointly optimizes capacity, overtime and smoothing at the production stages, along with service times for each stage in the supply chain to minimize the total inventory and capacity costs. Unlike Kumar and Aouam (2018a,b) and Aouam and Kumar (2018), the present model considers stochastic lead times and does not use queuing theory. Analysis of a single production stage provides insights into the interplay between service times and production smoothing. In particular, we find three interesting insights. When service times are not taken into account, existing works on production smoothing find that more smoothing results in a more variable inventory and necessitates higher safety stocks (Bertrand, 1986; Disney & Towill, 2003; Parrish, 1987; Zahraei & Teo, 2017). When service times are considered, the established trade-off between smoothing and inventory holds true only when the net service time (outgoing minus incoming) at a stage is at most one review period. In cases where the net service time exceeds one review period, a certain extent of production smoothing yields not only less variable production but also less variable inventory, as compared to the production basestock policy (non-smoothed production). This may motivate manufacturers, even with highly flexible facilities, to smooth production as a measure to reduce both capacity and inventory costs. Second, we highlight the impact of the net service time on a production stage’s inventory costs. While existing GSA models consider inventory costs that are concave or monotonously decreasing functions of the net service time (Eruguz, Sahin, Jemai, & Dallery, 2016; Graves & Willems, 20 0 0), we find that the inventory cost at the stage is a non-concave and non-monotonic function of the net service time. In fact, upon increasing the net service time, inventory costs may first decrease and then increase because of a more variable inventory at large net service times. As a result, the optimal service time may lie in the interior of the feasible region and not necessarily at extreme points. This result is in agreement with Humair and Willems (2011), who simulated the inventory cost functions for stages with variable lead times and observed similar behavior. Humair, Ruark, Tomlin, and Willems (2013) also showed that inventory functions are non-monotonic and non-concave when lead times are stochastic, which is the case for 26 of the 38 supply chains studied by Willems (2008). Based on this property of inventory costs, we derive an upper bound on the optimal net service time as a function of the smoothing parameter. Finally, we show that all safety stocks can be pooled at a downstream stage when the production stage follows the pro-

duction base-stock policy (i.e. does not smooth production) and sets net service time to one period. However, for a smoothed production, safety stocks are always held at the production stage due to an ever-positive inventory variance. Humair et al. (2013) also found that stochastic lead times (or stochastic replenishment) tend to break the power of pooling. We conduct numerical studies on a real-world supply network from Zahraei and Teo (2017) to highlight the economical benefits of the integrated model. We compare our model to two existing approaches for safety stock placement and production smoothing, specifically, a single-echelon approach to production smoothing where service times are considered zero, and the sequential approach of the GSA models, where planned lead times are determined before the safety stock placement model is solved. We find that the integrated model results in lower smoothing than the single-echelon model and is able to leverage the benefits of inventory pooling through a better allocation of inventory in the network. Further, we demonstrate the economical value of using the integrated model as compared to the sequential approach used by the existing GSA models. The value of integration is especially high for moderate values of capacity costs relative to holding costs. The value can be zero when capacity is inexpensive. Lastly, we assess the suitability of the TPM policy in the GSA models by comparing it to the equal-weighted moving average (EMA) policy, which possesses similar characteristics as the TPM. More specifically, we highlight the role of service times on the better smoothing policy among TPM and EMA. TPM is preferred to EMA when the net service time at a stage is at most one review period. For net service times greater than one period, we find conditions on service times where one policy outperforms the other to achieve the same planned lead time. Numerical studies, however, indicate that for a wide range of capacity to holding cost ratios, the optimal cost for TPM policy is always lower than that of EMA, albeit with a gap lower than 0.37%. The remainder of this paper is organized as follows. Section 2 briefly reviews related literature. In Section 3, we present notations and assumptions, and formulate the integrated problem. Section 4 analyzes a production stage under the integrated model to study the impact of smoothing and service times on capacity requirements, inventory costs and safety stock placement. Section 5 lists multistage experiments and Section 6 provides a comparison between the TPM and EMA smoothing policies. We conclude in Section 7. 2. Literature review The present work lies within the broad domain of production and inventory planning in supply chains facing demand uncertainty. Readers may refer to Axsäter (2015), Stadtler (2015) and Kempf, Keskinocak, and Uzsoy (2011) for an overview of various topics under this theme, and to Swaminathan and Tayur (2003) and Mula, Poler, García-Sabater, and Lario (2006) for comprehensive reviews on production and inventory planning models. The following review concentrates on two special topics that are closely related to our work: strategic safety stock placement (SSP) under the guaranteed service approach (GSA) (Graves & Willems, 20 0 0) and production smoothing (Bray & Mendelson, 2015; Graves, 1988). GSA models follow Simpson Jr (1958), which studies the setting of safety stocks in serial networks under base-stock control. Each stage promises a fixed service time within which all orders up to a reasonable demand bound are fulfilled with certainty. The author showed that the optimal service times follow an extreme point property (all-or-nothing) due to concavity of cost functions. Graves and Willems (20 0 0) formalizes this framework to general network structures, and proposes a dynamic programming algo-

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rithm to efficiently solve the problem for spanning-tree networks. GSA models have been extended to consider non-stationary demand (Graves & Willems, 2008), stochastic lead times (Humair et al., 2013) and general cost functions (Humair & Willems, 2011). Eruguz et al. (2016) provides a comprehensive review on this topic. A majority of GSA models exclude tactical production planning decisions from their scope, and thus neglect the interplay between production and inventory planning. Tian, Willems, and Kempf (2011) was one of the earliest studies to jointly consider SSP and production planning problems. They propose an iterative procedure that first sets base-stock levels, which are then used to solve the capacitated lot-sizing problem. You and Grossmann (2011) treats an integrated process planning and SSP problem for the chemical process industry. In addition to determining the optimal production plan and safety stocks, their model also suggests optimal purchase of raw materials and sale of finished goods. Both these studies consider lead times as exogenous and independent of the production plan. Hua and Willems (2016) analytically characterized the impact of lead time and cost allocation on safety stock placement in a two-stage network. Lead times in their model are independent of the production plan and can be interpreted as design or control parameters. They showed that when the lead time at the downstream stage is sufficiently long, safety stocks can be pooled at the downstream stage. Their analysis, however, holds true only when the inventory costs at the stages are concave and monotonous functions of service times, which is not the case for many realworld supply chains (Humair & Willems, 2011). Graves and Schoenmeyr (2016) considered supply chains with fixed capacity at each stage and used a modified base-stock policy to limit the maximum order up to the stage capacity and recover the excess quantity in the upcoming periods. The modified basestock policy, in essence, smooths the production by clipping orders in excess of the fixed capacity. In comparison, our model smooths production without limiting the maximum quantity and relies on flexible capacity to process the excess workload. Another difference lies in our modeling of production lead times as stochastic and endogenous functions of capacity and the smoothing parameter. Recently, Kumar and Aouam (2018a,b) used queuing models to capture the impact of capacity and production planning on lead times. The authors studied the effect of batch sizing on lead times and safety stock placement in production networks, where production stages were modeled as G/G/1 workstations. They showed that batch sizing influences safety stock placement, and integrating the two decisions may result in lower costs. Similarly, Aouam and Kumar (2018) studied the effect of subcontracting and overtime on lead times and safety stocks by modeling production stages as G/M/1 queues. The present paper also captures the effect of production capacity and decisions on safety stock placement, however it is different from existing works in several ways. Firstly, instead of relying on FCFS rules and queuing models, the current work considers a flexible resource that is controlled using the TPM model. Secondly, previous models consider that production units do not hold inventory and quote production lead times to downstream inventory locations. In these models production decisions are optimized and production lead times are functions of these decisions. In the present work, production stages are potential safety stock holding locations and optimize planned lead times, which are considered as control parameters equal to the inverse of the smoothing parameters (see Graves, 1986 for discussion). Lastly, Kumar and Aouam (2018a,b) and Aouam and Kumar (2018) model production lead time as a deterministic time that an item spends at the production unit. Delayed items beyond the production lead time are expedited using special measures. In the current work, finished goods inventory supplied by a production unit observes a

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stochastic production cycle time. This supply variability is reflected through inventory shortfall and buffered using safety stocks similar to Humair et al. (2013). The second stream of literature pertains to production smoothing models. These models typically balance the cost of varying capacity against inventory requirements to derive optimal production plans (Silver, 1967). Graves (1986) proposed the tactical planning model (TPM), which uses a linear control rule and sets the production rate to process a fixed proportion of the work in queue. This rule effectively smooths production by spreading demand over a planned lead time. Graves (1988) modeled the link between production smoothing and safety stocks, and extended the model to multistage networks with guaranteed service times between stages. We build on this work in order to extend the safety stock placement problem to include production smoothing and present a dynamic program to solve the integrated problem that emerges. In addition, we conduct analysis to present new insights about the behavior of inventory costs as functions of smoothing parameters and service times. Boute et al. (2007) considered a two-echelon system with a make-to-order manufacturer and a retailer, and modeled the former as a PH/PH/1 queuing system. The orders to the manufacturer were smoothed based on the exponential smoothing policy. Instead of relying on a flexible capacity, they assumed a constant average production rate and modeled lead times to reflect congestion effects. Their network is equivalent to a production stage in our model, but with zero incoming and outgoing service times. Their analysis showed that, on the one hand, order smoothing makes inventory more variable, but on the other hand, it induces a shorter lead time that decreases safety stock requirements. Consequently, a certain extent of smoothing is desirable to reduce both capacity and inventory costs. In our work, we also find a similar behavior, where production smoothing may result in reduction of both capacity and inventory costs when the net service time is greater than one review period. The present work is also closely related to Zahraei and Teo (2017), who considered production smoothing in supply chains. They studied the impact of production smoothing on demand propagation, as orders to upstream stages were based on the smoothing policy at a downstream stage. Service between the stages was guaranteed using the expediting assumption of the GSA models; however, unlike GSA, all service times were considered as zero and the question of safety stock placement was not explored. On the contrary, our work builds on the GSA framework and considers a (unsmoothed) base-stock replenishment policy throughout the network. Production smoothing is considered at a stage level to optimally set the capacity and planned lead times. We allow positive service times between stages and focus on their joint effect with production smoothing on capacity, inventory costs, and safety stock placement. Balakrishnan et al. (2004) studied a supply chain with a supplier and one or several retailers. They analyzed and compared two classes of smoothing policies, namely moving-weighted average and exponential smoothing, and provided insights regarding the supply chain contexts that could benefit from these policies. In a similar essence, our work also compares these two policies, albeit in the context of production smoothing in supply networks under GSA and with non-negative service times. The literature on production smoothing emphasizes its impact on safety stocks; however, most of the studies have been conducted in the context of single-stage systems, or in the absence of service times. In contrast, GSA models tackle the strategic question of safety stock placement and neglect tactical production planning aspects. To the best of our knowledge, the present work is the first work to integrate safety stock placement and production smoothing problems and to analyze the joint impact of service times and

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production smoothing on network costs and safety stock placement. 3. Integrated safety stock placement and production smoothing In this section, we present the integrated safety stock placement and production smoothing (TP − SSP) problem. We begin by describing the notations and key assumptions needed to formulate the model. 3.1. Notations Sets M A P L MD MS

set of stages in the network set of arcs (i, j ) representing flow of material from stage i to j set of production stages (⊆ M ) set of logistics stages (⊂ M ) set of demand stages (⊂ M ) set of supply stages (⊂ M ) Parameters μj mean demand rate at j ∈ M (items per period) σj standard deviation of demand rate at j ∈ M (items per period) target service level at j ∈ M αj target safety factor at j ∈ M kj fixed outgoing service time at demand stages j ∈ M D (periods) sj fixed incoming service time at supply stages j ∈ M S (periods) si j fixed lead time at j ∈ L (periods) λj cost of nominal capacity at j ∈ P (per unit per period) c jf cost of overtime at j ∈ P (per unit) coj work-in-process (WIP) holding cost at j ∈ P (per unit per period) ωj inventory holding cost at j ∈ M (per item per period) hj cost of handling shortages at j ∈ M (per item per period) csj Decision variables nominal production capacity at j ∈ P (units per period) Kj production smoothing parameter ∈ (0, 1] at j ∈ P βj outgoing service time for j ∈ M (in periods) Sj incoming service time for j ∈ M (in periods) SI j Derived variables production quantity at j ∈ P in period t (units, random variable) X j (t ) WIP at j ∈ P in period t (units, random variable) Q j (t ) inventory shortfall at j ∈ P in period t (units, random variable) Y j (t ) base-stock level at j ∈ M (units) Bj planned production lead time at j ∈ P (periods) Tj Probability functions probability of an event A Pr (A ) φ uj (· ) probability density function (p.d.f.) of a variable U at j ∈ M uj (· ) cumulative distribution function (c.d.f.) of a variable U at j ∈ M

3.2. System description and assumptions First, we state the assumptions that follow from the GSA and then describe the additional assumptions needed for this study. 3.2.1. Assumptions from the GSA Supply chain network: We consider a supply chain with a spanning-tree structure that serves demand for a single end-item. The supply chain is modeled as a network where nodes (set M ) are the stages and arcs (set A ) represent the supplier-customer relationships. A stage is a major production or logistics activity center in the supply chain. Depending on the scope and granularity of the analysis, it may represent anything from a single step in manufacturing or distribution to an entire factory or a distribution facility (Graves & Willems, 2003). Each stage is a potential safety stock holding location. Further, the stages connecting to outside suppliers are termed as supply stages (M S ⊂ M ) and the stages serving external customers are called demand stages (M D ⊂ M ). Without loss of generality, we assume that the external customer demand occurs only at the demand stages. Demand and replenishment policy: We assume that the network operates under a periodic-review base-stock replenishment policy with a common review period. In each period, a demand stage j ∈ M D observes a random external demand while the internal

stages (non-demand stages) serve the demand from their downstream stages. All orders are placed at the same time and the external demand is transmitted through the entire supply chain. The demand di (t) seen by any internal stage i ∈ M \M D in period t can be calculated as



di (t ) =

ρi j d j (t ),

(1)

j:(i, j )∈A

where ρ ij is the number of items from stage i required to produce one item at stage j. We assume that the demand stream {dj (t)} at a stage j is independent and identically distributed (i.i.d.) with a mean μj and variance σ j2 , which can be derived from (1). Guaranteed-service approach: The network operates under the guaranteed service approach (GSA) and each stage quotes to its customers a deterministic service time within which it is able to meet all demand with certainty. For example, if a stage j quotes a service time Sj , then all orders from period t must be shipped by period t + S j . The incoming service time SIj seen by a stage j is greater than the maximum of the service times Si quoted by its suppliers, i.e., SIj ≥ Si , where (i, j ) ∈ A . Lastly, parameter sij represents the service time quoted by external suppliers to a supply stage j ∈ M S , whereas si is the maximum service time that a demand stage i ∈ M D can quote to an end-customer. 3.2.2. Additional assumptions Supply chain network: In this work, we distinguish between production and logistics stages based on their function. A production stage (∈ P) is an activity center that transforms goods from one form to another and may represent anything from a production unit, an assembly line to an entire factory. A production stage includes a production resource with a queue and a finished goods inventory to store processed items. A logistics stage (∈ L ) serves only a logistics function such as distribution or storage in the supply chain and may represent procurement of raw materials, distribution centers, warehouses or retailers. Both production and logistics stages are potential safety stock holding locations. In our setting, M = P ∪ L and P ∩ L = ∅. Base-stock level: To set the base-stock level Bj , we adopt the maximal approximation, which provides a distribution-free bound for any non-negative inventory shortfall distribution, based on Zipkin (20 0 0). Under this approximation, the base-stock level is given by

B j = E[Y j ] + k j



V ar (Y j ),

(2)

where Yj is the inventory shortfall and kj is a safety factor that provides a service level



⎛



⎞

1 − kj

1 α j = Pr Y j ≤ B j = 1 − ⎝  2

1 + k2j

⎠.

In cases where Yj can be approximated using a normal distribu  tion, α j = F k j , where F(·) is the standard normal c.d.f. The normal approximation is commonly used in GSA models (see Graves & Willems, 2003 and Eruguz et al., 2016). Special measures to handle backorders: There are two different approaches under the GSA to handle backorders and guarantee the service times. The original models by Simpson Jr (1958) and Graves and Willems (20 0 0) assume that the lead time demand for any end-item is bounded and all demand exceeding this bound is handled outside the supply chain. In contrast, the second approach allows an unbounded demand to propagate through the supply chain. A base-stock level is set to meet a target fraction of demand from stock, while the backorders are expedited from the pipeline inventory to satisfy 100% of the demand (Aouam & Kumar, 2018; Klosterhalfen, Dittmar, & Minner, 2013).

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Orders arriving from suppliers

433

Orders sent to customers Resource queue

Production resource

FG Inventory

Fig. 1. Production stage in a period t.

We follow the latter approach and consider unbounded demand; however, instead of expediting from the pipeline inventory we assume that shortages are supplied (or borrowed) from an external source. Furthermore, pipeline inventory corresponding to these shortages will eventually be processed and result in excess inventory. We assume that excess items are subsequently sold to a secondary market (or returned to the external source). Under this ‘borrow-return’ assumption, the smoothing model remains analytically tractable. Lee, So, and Tang (20 0 0) also make a similar assumption, which has been adopted in works such as Lutze and Özer (2008), Zhang and Zhao (2010) and Cheung, Song, and Zhang (2017). The ‘borrow-return’ assumption is plausible for industries with access to secondary markets, which are increasingly growing in several sectors to buy and sell excess inventory across the supply chains. Angelus and Özer (2017) reports that the secondary markets in the US were estimated at $424B in 2012, and are expected to grow 7%–8% a year. In electronics and high-end computing industries, these markets are facilitated by internet-based exchanges such as Converge, which allows the sale of excess components (Harrison, 2005; Lee & Whang, 2002). Recently, Cambridge Consultants developed an online trading platform to facilitate primary and secondary markets for the US pharmaceutical industry (Cambridge Consultants, 2018). For supply chains where the assumption of borrow-return is not plausible, we propose an alternate model to consider a demand bound in Appendix C. Please note that the forthcoming analysis of Sections 4 and 6 holds for the model with bounded demand. Capacity at production stages: These stages can be equipped with a nominal capacity of Kj units per period, which is treated as a decision variable in our model. Each extra unit of nominal caf pacity costs c j per period and includes capital, labor and overhead costs. In addition to the nominal capacity, the manufacturer can f use flexible capacity in the form of overtime at a cost of coj (> c j ) per unit. Production smoothing policy: Each production stage j ∈ P determines the production quantity Xj (t) based on the linear control rule of the tactical planning model (TPM) by Graves (1986). This policy sets the production quantity equal to a fixed proportion β j ∈ (0, 1] of the work in-queue, which dampens production variability and reduces capacity requirements. The production cycle time in stage j ∈ P, i.e., elapsed time between the release of an item into the production queue until it is processed and moved to the finished goods inventory, is random with a probability distribution that depends on the smoothing parameter β j (Hollywood, 20 0 0). We follow Graves (1986) in referring to the average production cycle time as planned lead time, which is given by T j = 1/β j . Note that while a planned lead time is related to a single produc-

tion stage, service times are quoted between production and logistics stages to denote the time within which a stage guarantees to satisfy orders. 3.3. Integrated problem formulation This section formulates the integrated problem TP − SSP, where the decision variables include the nominal capacity Kj , smoothing parameter β j for each production stage j ∈ P, and service times Sj and SIj for each stage j ∈ M . 3.3.1. Production smoothing model Consider a production stage as shown in Fig. 1. The stage plans production quantity Xj (t) for period t based on the linear control rule of the tactical planning model (TPM). Under this rule, a production target Xj (t) is set to clear a fixed proportion β j ∈ (0, 1] of the total work Qj (t) pending in queue at the beginning of period t, such that

X j (t ) = β j Q j (t ).

(3)

Graves (1986) showed that the linear control rule in (3) is equivalent to the exponential smoothing rule and can also be expressed as

X j (t ) = β j R j (t ) + (1 − β j )X j (t − 1 ) =

∞ 

 k β j 1 − β j R j (t − k ).

(4)

k=0

where, R j (t ) = d j (t − SI j ) is the quantity released to the queue at the beginning of each period t. We define demand dj (t) and all quantities derived from it to be zero for t < 0. The smoothing policy (4) is a generalization of the production base-stock policy, which is equivalent to the special case when β j = 1 and X j (t ) = R j (t ) = d j (t − SI j ). Based on Graves (1988), we can write the mean E[Xj ] and variance Var(Xj ) of production quantities as

E[X j ] = μ j ,

 V ar (X j ) =

(5)

βj σ 2. 2 − βj

(6)

It can be observed in (6) that under the production base-stock policy (β j = 1), the production quantities retain the variability of demand. For β j < 1, we have

βj

2−β j

< 1, which means that the pro-

duction is smoothed and its variance is lower than that of demand.

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3.3.2. Inventory model Inventory at a production stage: We define inventory shortfall (Yj (t)) at a production stage as the total inventory deficit in period t. Yj (t) is equivalent to the total quantity on-order, which includes both orders with suppliers and in-process at the stage itself, minus the outstanding orders to the customers. In any period t, the quantity on-order with the suppliers is SI j −1



OO j (t ) =

∀ j ∈ P,

d j (t − k ),

(7)

k=0

b where, k=a d (t − k ) = 0 for b < a. Further, the total work-inprocess (WIP) at the stage is

Q j (t ) =

∞  k=0

=

∞ 

R j (t − k ) −

∞ 

X j (t − k )

k=1



 m βj 1 − βj d j (t − SI j − k ),

k−1 

1−



∀ j ∈ P,

m=0

k=0

(8)

m where, Xj (t) follows from (4) and km−1 =0 β j (1 − β j ) = 0 for k = 0. Given an outgoing service time Sj , the total quantity of orders received from the customers but not yet delivered is BO j (t ) =

S j −1 d j (t − k ). Thus, the total inventory shortfall Yj (t) at the stage k=0 is written using (7) and (8) as

Y j (t ) = Q j (t ) + OO j (t ) − BO j (t ) =

∞ 



k−1 

1−



βj 1 − βj

SI j −1

d j (t − SI j − k )



SI j + λ j − S j ,

Assuming that the inventory shortfall

Y jl

d j (t − k ) −



d j (t − k ),

∀ j ∈ P,

(9)

k=0



SI j +

1

βj

− Sj

μ j,

∀j ∈ P

(10)

Var (Y j )

⎧  ⎪ ⎨ β j (21−β j ) + (SI j − S j ) σ j2 , =    S j −SI j  2 ⎪ 1 ⎩ + (S j − SI j ) − β2 1 − 1 − β j σj , β j (2−β j ) j

if S j ≤ SI j , if S j > SI j .

(11) It must be noted that the variance of inventory Var(Ij ) equals the variance of inventory shortfall Var(Yj ) since I j (t ) = B j − Y j (t ) for every period t, where Bj is not random. Base-stock level Bj is set as per (2) to ensure that the probability of Yj not exceeding Bj is equal to the service level α j . Then, the expected number of backorders at the stage is

E[(Y j − B j )+ ] =



∞ Bj



Yj − B j

 y  φ j Y j dY j ,

∀ j ∈ P,

(12)

where, φ j (· ) is the probability density function of inventory shortfall with mean E[Yj ] and variance Var(Yj ). The expected on-hand inventory can be written as y



Bj −∞

(B j − Y j )φ yj (Y j )dY j

= B j − E[Y j ] + E[(Y j − B j )+ ] = kj



V ar (Y j ) + E[(Y j − B j )+ ],

∀ j ∈ P.

∀j ∈ L.

(14)

is randomly distributed

l φ yj (· ),

with p.d.f. mean E[Yj ] and variance Var(Yj ), we can write the expected backorders and on-hand inventory as

E[(Y jl − B j )+ ] =





∞ Bj

l

(Y jl − B j )φ yj (Y jl )dY jl

SI j + λ j − S j +



∞ Bj



Y jl − B j

∀j ∈ L,

(15)

 yl  l  l φ j Y j dY j ∀ j ∈ L . (16)

S j −1

 k=0

E[OH j ] =

B j = (SI j + λ j − S j )μ j + k j σ j

E[OH j ] = k j σ j

and, its mean and variance can be written as

E[Y j ] =

E[Y jl ] = μ j (SI j + λ j − S j ) and V ar (Y jl ) = (SI j + λ j − S j )σ 2 , where SI j + λ j − S j is the net replenishment time. The base-stock level Bj to target a safety factor kj is

m=0

k=0

+

m



normal and maximal approximations. As a result, both expected expediting (12) and inventory holding (13) costs increase with inventory shortfall variance. We define the inventory variance  factor j = V ar (Y j )/σ j , which will be used as a proxy for a production stage’s inventory costs in Section 4. Inventory at logistics stages: A logistics stage j ∈ L involves a process that incurs a fixed lead time λj , which includes transportation, material handling, and the review period. Unlike production stages, processing workload is not smoothed and the released items are replenished in the inventory exactly after λj periods, i.e., X j (t ) = R j (t − λ j ). As a result, the inventory shortfall in any period t is Y jl (t ) = d j (t − SI j − λ j , t − S j ), where dj (a, b) is the cumulative demand within periods (a, b]. Based on Graves and Willems (20 0 0), the mean and variance of Y jl are written as

(13)

Zipkin (20 0 0) shows that the loss function E[(Y j − B j )+ ] is an increasing function of Var(Yj ) and independent of E[Yj ] for both

3.3.3. Expected total cost The expected total cost for the network includes the costs of capacity, holding work-in-process (WIP) inventory at production stages, and the costs of holding inventory and expediting shortages at all stages. Each of these cost elements are expressed below as functions of the decision variables Kj , β j , Sj and SIj . Expected capacity costs: Capacity costs at a production stage include the costs of nominal capacity and overtime. The marginal cost of adding one unit of nominal capacity per period at stage f j is assumed to be c j , which includes capital, labor and overhead costs. Therefore, the nominal capacity incurred at the manufacturer f is c j K j . In addition, overtime is used to produce all units exceeding the nominal capacity, and is given as

E[(X j − K j )+ ] =



∞ Kj



Xj − Kj

 x  φ j X j dX j ,

(17)

where φ xj is the p.d.f. for production requirements Xj , whose mean E[Xj ] and variance Var(Xj ) are given by (5) and (6), respectively. The expected cost of overtime is coj E[(X j − K j )+ ], which is an increasing function of Var(Xj ). Expected work-in process inventory cost: The expected WIP at a production stage can be derived from the (8) as

E[Q j ] =

1

βj

μ j.

(18)

The expected WIP holding cost is ωj E[Qj ], where ωj is the unit WIP holding cost per period. Expected inventory cost: The total inventory cost (E[ICj ]) at a production stage includes the cost of holding inventory (13) and expediting backorders (12) using special measures.

E[IC j ] = h j E[OH j ] + csj E[(Y j − B j )+ ]    = h j k j V ar (Y j ) + h j + csj

∞ Bj



Yj − B j

 y  φ j Y j dY j , j ∈ P , (19)

K. Kumar and T. Aouam / European Journal of Operational Research 279 (2019) 429–448

where hj is the unit cost of holding inventory and csj is the cost of expediting an item at a production stage j ∈ P. The inventory cost in (19) is an increasing function of Var(Yj ). Similarly, the inventory cost at a logistics stage can be written using (15) and (16) as

0 < βj ≤ 1

E[IC j ] = h j k j σ j

S j , SI j ∈ Z+



SI j + λ j − S j  ∞  l  l  + h j + csj Y j − B j φ yj Y jl dY jl ,



Bj

j ∈ L.

(20)

Expected total cost: The expected total cost E[TC] is the sum of capacity and WIP costs at the production stages and inventory costs at all stages.

E[T C] =





c jf K j + coj E[(X j − K j )+ ] + ω j

j∈P



+ hj + +



csj





 μj + h j k j V ar (Y j ) βj

E[(Y j − B j ) ] +

h jk jσ j





 s



SI j + λ j − S j + h j + c j E[(Y jl − B j )+ ] ,

j∈L

(21) where the E[(X j − K j )+ ] is given by (17) and is a function of Kj and β j . Terms Var(Yj ) and E[(Y j − B j )+ ] follow from (11) and (12), respectively, and are functions of β j , Sj and SIj . E[(Y jl − B j )+ ] is a function of Sj and SIj and is described in (15). Section 4 discusses the impact of the decision variables β j , Sj and SIj on these cost terms. Lastly, we can rewrite the total expected cost E[TC] for a normal approximation as

E[T C] =





c jf K j + coj E[(X j − K j )+ ]

j∈P

 + ω j E[Q j ] + H j (k j ) V ar (Y j )    + H j (k j )σ j SI j + λ j − S j

subject to: SI j +

1

βj

SI j + λ j − S j ≥ 0

E[T C]

− Sj ≥ 0

∀j ∈ L

(22)

(24)

(25)

∀(i, j ) ∈ A

(26)

Sj ≤ sj

∀j ∈ MD

(27)

∀j ∈ MS

∀j ∈ M

(31)

The objective function (23) minimizes the expected total cost as defined in (21). Constraints (24) and (25) ensure that the net lead time at all production and logistics stages is positive. Constraints (26) define the relationship between service times at consecutive stages. Outgoing service time at a demand stage j is at most sj periods (27) and the minimum incoming service time at a supply stage j is sij (28). Constraints (29) define β j in (0, 1], while nominal capacity Kj is a positive real number. Service times Sj and SIj are non-negative integers (31). 4. Analysis of a production stage This section analyzes a single production stage representing a manufacturer with an inventory holding location, as shown in Fig. 1. The objective of this analysis is to derive meaningful insights on the interaction between the manufacturer’s smoothing parameter β and service times S and SI. 4.1. Impact of smoothing parameter β On capacity requirements: The manufacturer smooths its production to reduce the overall capacity costs, where a lower β signifies a smoother production. Based on Boute and Van Mieghem (2015), for any given β , the optimal nominal capacity and total capacity costs are expressed in Lemma 1. cf co

), where x (·) is the c.d.f. of X with

mean μ and variance ( 2−β )σ 2 .

SI j ≥ Si

SI j ≥ si j

(30)

β

(23)

∀j ∈ P

∀j ∈ P

−1

3.3.4. Optimization model The integrated problem TP − SSP jointly determines the optimal capacity (Kj ), smoothing parameters (β j ), and service times (Sj and SIj ), and is expressed in (23)–(31).

min

(29)

duction stage is K ∗ = x (1 −

where, H j (k j ) = h j k j + (h j + csj )( f (k j ) − k j (1 − F (k j ))) and f(·) and F(·) are the standard normal p.d.f. and c.d.f. (Aouam & Kumar, 2018). For a given service level α j , Hj (kj ) is a constant and includes the cost of holding inventory and expediting backorders.

K j ,β j ,S j ,SI j

∀j ∈ P

Lemma 1. For any given β , the optimal nominal capacity at the pro-



j∈L

TP − SSP

Kj ≥ 0

435

(28)

Proof. See Appendix A for proof.



Corollary 1. When Xj is normally distributed, with f(·) and F(·) denoting the standard normal p.d.f. and c.d.f., we have: a) K ∗ = μ + z∗



β

2−β

σ, ∗

b) the minimum expected capacity cost is c f μ + co f (z ) ∗

c) where, z =

F −1 (1

−

cf co



β

2−β

σ,

) is the optimal capacity safety factor.

The expression for the optimal nominal capacity K∗ in Lemma 1 is analogous to the standard newsvendor’s solution. The optimal capacity K∗ balances the cost of owning idle capacity against the cost of not having sufficient capacity (overtime). For ease of exposition, we consider the production workload X to be normally distributed, i.e., X ∼ N (E[X ], V ar (X ) ). We let the ratio co /cf > 1 denote the rigidity of the capacity constraint, where co /cf → ∞ represents a hard capacity constraint. For facilities with rigid capacity such that δ = co/c f > 2, the safety factor z∗ is positive and capacity in excess of mean demand is needed to avoid additional overtime costs. When the production variability is high, as in the case of the production base-stock policy (β = 1), the manufacturer will require a large safety capacity. However, the capacity requirements can be reduced by making production less variable and setting β < 1. For facilities with a more flexible capacity (δ = co/c f < 2 ), the cost of idle capacity gets lower than the cost of overtime and z∗ becomes negative. In this case, a manufacturer using the production base-stock policy will keep the nominal capacity as low as possible to avoid high idle capacity costs. However, under smoothed production, the risks of having idle capacity

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K. Kumar and T. Aouam / European Journal of Operational Research 279 (2019) 429–448

Fig. 2. Performance of the smoothing policy relative to that of the base-stock policy for different δ .

are low, and the manufacturer chooses a higher nominal capacity K∗ as compared to that under the base-stock policy. This effect is shown in Fig. 2a that plots

K 1 −K β K1

, which is the reduction in opti-

mal nominal capacity (Kβ ) for a parameter

β relative to that of the production base-stock policy (K1 ). It can be seen that for δ > 2, the nominal capacity gets lower (or, reduction gets higher) in comparison to that in the production base-stock policy, since production is smoothed. While for δ =1.5 and 1.8 an opposite effect is observed. Production smoothing always leads to a reduction in overtime and the total capacity costs, irrespective of its effect on the optimal nominal capacity. Figs. 2b and 2c show this effect for cv = σ /μ = 1. For instance, at δ = 1.8, a smoothing policy with β = 0.5 will result in a nominal capacity that is 6.8% higher than the production base stock,yields a 42.3% reduction in overtime and a 17.5% reduction in total capacity costs. Therefore, production smoothing can help the manufacturer reduce its overall capacity costs. On inventory costs at the production stage: The inventory costs at the production stage include the costs of holding stocks and expediting shortages. For a given service level α , both these costs increase with the variance of  inventory shortfall Var(Y) and the inventory variance factor = V ar (Y )/σ . Traditional production smoothing models typically neglect service times and consider systems where S = SI = 0. For these systems, it is well-known that a

Fig. 3. Inventory variance factor =



Var (Y )/σ for S = SI = 0 as function of β .

smoother production (lower β ) results in a more variable inventory and higher inventory costs (Disney & Towill, 2003; Graves, 1999; Parrish, 1987). This behavior is demonstrated in Fig. 3, where is plotted as a function of β . We can notice that as the production is smoothed (β is reduced), inventory becomes more variable (right to left).

K. Kumar and T. Aouam / European Journal of Operational Research 279 (2019) 429–448

Fig. 4. Inventory variance factor =

437



Var (Y )/σ as function of β .

This behavior may not always hold true when positive service times are considered. As will be shown next, the behavior of Var(Y) with β depends on the net service time (NS = S − SI). Proposition 1. When the net service time NS = S − SI ≤ 1, inventory becomes more variable as production is smoothed, i.e., Var(Y) is a decreasing function of β . Proof. See Appendix A for proof.



Proposition 1 shows that Var(Y) is a decreasing function of β for net service times not exceeding one review period (NS < 1). This includes the systems with S = SI = 0. Fig. 4a plots the inventory variance factor for NS = −5, −3 and −1, and shows that the inventory becomes more variable as production is smoothed. The same effect can be seen in Fig. 4a for NS = 1. However, for stages with net service times exceeding one review period (NS > 1), inventory shortfall first decreases, reaches a minimum value and then increase as the production is smoothed. Fig. 4b shows this behavior for net service times NS = 3, 5 and 7. For instance, when NS = 5,

decreases from 2 at β = 1, to a minimum value of 1.03 at β = 0.2 and then increases to 2.49 at β = 0.05. This tendency of inventory shortfall variance implies that, for net service times exceeding one period, a certain extent of production smoothing is desirable to reduce both capacity and inventory costs. This behavior is characterized analytically next. This result is in contrast with the wellknown relationship between production smoothing and inventory and will be established analytically next. 4.2. Impact of the net service time On inventory costs: The net service time NS = S − SI is a key determinant of inventory shortfall and inventory costs at the warehouse. Proposition 2 describes the relationship between Var(Y) and NS. Proposition 2. For a given β , the following holds for a production stage: a. for NS = S − SI ≤ 1, Var(Y) is a convex and decreasing function of NS, and b. for NS = S − SI > 1, Var(Y) is convex in NS and its minima occurs at

 (β ) = max NS

⎧ ⎨

1,

⎩



ln

−β

2ln(1−β ) ln(1 − β )

Proof. See Appendix A for proof.

⎫ ⎬

.

⎭ 

Fig. 5. Var(Y)/σ 2 as a function of net service time NS for different β .

Fig. 5 illustrates the result of Proposition 2. For net service times less than one review period, inventory shortfall variance decreases linearly as NS is reduced from a high negative value to one. This effect occurs due to a decreasing quantity on-order with the supplier (OO(t)), which makes inventory less variable. Similarly, for net service times NS > 1, inventory shortfall variance first decreases, reaches a minimum value and then increases steadily. In this region, there are two simultaneous effects. On the one hand, inventory shortfall increases due to a higher amount of orders outstanding to the supplier (BO(t)), while on the other hand, there is a compensating effect due to the covariance of the quantity  ( β ), in queue (Q(t)) and the outstanding orders. For 1 ≤ NS ≤ NS the compensating effect is stronger than the former due to which the inventory becomes less variable upon increasing NS; while for S(β ) inventory shortfall variance increases. NS > N While most GSA models consider inventory costs that are concave and monotonically decreasing functions of NS (Eruguz et al., 2016; Graves & Willems, 20 0 0), we find that a production stage’s inventory costs are non-concave and non-monotonous in NS. As a result of the non-concavity and non-monotonicity of the cost function, the famous all-or-nothing property of the GSA models does not hold, and the optimal service time may lie at the interior of the feasible region (Humair & Willems, 2011). In Proposition 3, we  (β ) is an upper-bound to the optimal net service time show that NS NS∗ at a production stage.  (β ) is an upper bound to the optiProposition 3. For a given β , NS ∗ mal net service time NS at a production stage.

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K. Kumar and T. Aouam / European Journal of Operational Research 279 (2019) 429–448

Fig. 6. Total inventory cost at the manufacturer and retailer as a function of S, where the vertical dashed line represents Sˆ(β ). Instance: σ = 1, H2 (k2 ) = 1, H1 (k1 ) = 1.5, SI2 = 4 and λ1 = 2.

Proof. See Appendix A for proof.



 (β ) is a decreasing function of β . Corollary 2. NS To demonstrate, we consider a manufacturer ( j = 2) that supplies a single retailer ( j = 1), with the following instance: σ2 = σ1 = 1, H2 (k2 ) = 1, H1 (k1 ) = 1.5, SI2 = 4, S1 = 0 and λ1 = 2. Fig. 6 plots the total inventory cost at the manufacturer and retailer as a function of the service time S2 = S quoted by the manufacturer to retailer for different values of β2 = β = {0.1, 0.25, 0.5, 0.75}. The vertical dashed lines indicate upper bounds on service time,  (β ) − SI . It can be observed that the total inventory cost Sˆ(β ) = NS 2

is a non-monotonous (and non-concave) function of the service time S. Secondly, the service time S∗ that minimizes the total inventory cost may occur in the interior region and before Sˆ(β ). For example, in Fig. 6c, inventory costs are minimized at S∗ = 5, while Sˆ(0.75 ) = 5.47 periods. Lastly, we remark that the Sˆ(β ) is a decreasing function of β , or in other words, Sˆ(β ) increases as the production is smoothed (Corollary 2). In terms of computation, this means that a smaller β results in a larger search-space of the optimal S∗ . On safety stock placement: The safety stock placement problem addresses questions related to location and quantity of safety stocks. A production stage can completely pool its safety stocks at

K. Kumar and T. Aouam / European Journal of Operational Research 279 (2019) 429–448

439

Fig. 7. Supply network from Zahraei and Teo (2017).

the downstream stages only when its inventory shortfall variance is zero, whereas, when Var(Y) > 0, a certain amount of safety stock is required to meet the service levels. Proposition 4. A production stage does not carry safety stocks only when β = 1 and NS = 1; while for all other feasible values of β and NS, it is a safety stock holding location. Proof. See Appendix A for proof.



Proposition 4 shows that safety stocks can be pooled at downstream stages only when β = 1, which corresponds to the production base-stock policy. However, for a smoothed production, the inventory shortfall variance at the manufacturer is always positive and it must carry safety stocks. A similar effect was also observed in Humair et al. (2013), where the authors found that stochastic lead times (or equivalently stochastic replenishments) break the power of pooling. The manufacturer may, however, partially pool safety stocks at the downstream stages by quoting a service time 0 < S ≤ Sˆ. 5. Numerical study The intent of this section is to highlight the economical benefits of the integrated model TP − SSP over two existing approaches for solving production smoothing and safety stock placement problems. The first approach (TPP) resembles the existing production smoothing models and considers service times at each stage to be zero. As a result, every stage can be treated as a single-echelon that carries inventory and can be solved independently. The second approach (SSP) follows the original GSA models, where planned lead times from TPP are used as exogenous parameters to solve the safety stock placement problem that minimizes inventory holding and expediting costs defined in (19) and (20). We extend the dynamic programming algorithm of Graves and Willems (20 0 0) to solve the integrated model TP − SSP for spanning-tree networks. The details of the algorithm are provided in the Appendix B. 5.1. Test Instance We perform experiments on the 14-stage supply network from Zahraei and Teo (2017), shown in Fig. 7. The network represents a real-world supply chain and consists of four production stages, namely stages 4, 5, 10 and 11. The external demand occurs at stages 1, 2 and 3; the expected demand (standard deviation) at

these stages is 6500 (3100), 5300 (2300) and 11400 (5500), respectively, and follow normal distribution. Based on the original instance, it is assumed that the external demand is normally distributed. As a result, the random variables Xj and Yj can also be modeled as normally distributed. Table 1 lists the parameters for the network. The following modifications have been made to the instance: a) The lead times λj at logistics stages j ∈ L were uniformly distributed on {1, 2, 3}. f b) Cost of nominal capacity c j was set such that overtime is 50% f

more expensive than regular production, i.e., coj = 1.5c j , where the values for coj follow from the original instance. c) Nominal production capacity Kj is treated as a decision variable and set as per Lemma 1, while in the original instance it was a fixed parameter. d) WIP cost ωj for production stage j ∈ P was set as the average of the holding cost of raw materials and the finished goods as follows

ωj =

hj + 2

 i:(i, j )∈A

hi . 2

5.2. Comparison between TPP and TP–SSP We compare the single-echelon production smoothing model TPP with the integrated model TP − SSP to evaluate the impact of quoting positive service times between stages. Alternatively, the analysis herein studies the benefits of adopting a multi-echelon framework for production smoothing. Tables 2 and 3 provide a summary of the optimal solutions for TPP and TP − SSP models for the base instance. The integrated approach results in a 7.67% reduction in the total cost, the reason for which is two-fold. Firstly, the integrated model can reduce inventory shortfall variance at production stages by setting a net service time that in the region  , where Var(Y) is minimized (see Proposition 2). For 1 ≤ NS ≤ NS the base instance, the optimal NS∗ at stages 4, 10 and 11 equals 1 and results in lower inventory costs than NS = 0. Secondly, the integrated model can leverage the benefits of inventory pooling by setting β and NS such that all stocks are held downstream. For instance, the integrated model sets β = 1 and NS = 1 for stages 4, 5 and 11, which do not hold safety stocks. In TPP, stages 5 and 11 are safety stock holding locations as Var(Y) > 0. Further, we investigate the impact of the ratio of capacity to holding costs, which captures the most important trade-off in

440

K. Kumar and T. Aouam / European Journal of Operational Research 279 (2019) 429–448 Table 1 Parameters for the base instance. stage no.

λj

hj

csj

ωj

c jf

co

stage no.

λj

hj

csj

ωj

c jf

coj

1 2 3 4 5 6 7

1 2 1 0 0 2 3

0.50 0.49 0.51 0.47 0.48 0.15 0.28

8.46 5.30 7.22 4.00 4.20 3.54 12.90

– – – 0.43 0.43 – –

– – – 2.2 2.9 – –

– – – 3.3 4.35 – –

8 9 10 11 12 13 14

2 1 0 0 2 1 2

0.29 0.14 0.13 0.23 0.11 0.12 0.09

16.87 9.70 2.60 6.80 6.13 5.88 4.84

– – 0.11 0.21 – – –

– – 1.4 1.7 – – –

– – 2.1 2.55 – – –

Table 2 Optimal expected costs for the base instance. Metric

TPP

T P − SSP

% change

Inventory holding cost Inventory expediting cost WIP holding cost Nominal capacity cost Overtime cost Total cost

34594.30 8604.76 17797.43 99016.00 36178.10 196190.58

21850.86 6871.22 16298.28 97437.09 39756.27 182213.71

−58.32 −25.23 −9.20 −1.62 9.00 −7.67

Table 3 Optimal β for the base instance. Production stage

β∗ TPP

TP − SSP

4 5 10 11 Average β∗

1.00 0.84 0.77 0.93 0.89

1.00 1.00 0.89 1.00 0.97

Fig. 8. Savings as a function of ζ .

these models. We define a parameter ζ = coj /h j and vary it from 1 to 30 while fixing hj and setting c j = coj /1.5. A small value of ζ signifies a setting where wages are low and the flexibility to exceed normal working hours (overtime) is high, as is generally the case in an offshore location. A high ζ , on the other hand, represents markets such as U.S. or Western Europe where wages are high and labor working hours are relatively less flexible. Fig. 8 demonstrates the effect of ζ on absolute- and percentage- savings resulting from the use of TP − SSP model instead of TPP. The graph indicates that the savings are high when ζ is low, i.e., the multi-echelon smoothing approach is highly suitable for supply chains that are predominantly situated in markets with cheaper and flexible capacity. The benefits remain positive, if lower, for supply chains with production stages located in markets with high capacity costs (high ζ ). This trend can be explained based on Fig. 10, which shows the optimal smoothing parameters for different values of ζ . When ζ is f

Fig. 9. Average β ∗ as a function of ζ .

low ( < 6), both the models lead to a production base-stock policy and identical capacity costs at all the stages. The savings (approx. 14,500) for these ζ -values are a result of multi-echelon inventory optimization, where TP − SSP is able to effectively allocate inventory across the network. This represents the maximum savings potential for the network. As capacity gets expensive (ζ increases), TP − SSP leads to smoothing parameters that are less than one and breaks the power pooling, which diminishes the extent of savings. From a managerial perspective, the analysis above shows that cost savings can be realized when production smoothing decisions are optimized together with the supply chain safety stocks, as opposed to local optimization of production smoothing. The integrated decision-making results in lower production smoothing, which increases the capacity costs but decreases the inventory holding and expediting costs. Therefore, the integrated model is recommended for supply chains where capacity costs are low as compared to inventory holding and expediting costs. 5.3. Value of integration This section compares the solution costs of TP − SSP with the sequential approach of SSP. We measure value of integration (VOI) as the percentage savings achieved by the integrated model over the optimal solution of the sequential approach. For the base instance, VOI is measured at 3.89%, i.e., the integrated model results in a 3.89% lower cost as compared to the sequential approach used in the existing GSA studies. We find that the ratio of overtime to holding costs, ζ , is an important influencer of the value of integration. Fig. 11 shows this effect, where the VOI is zero for ζ ≤ 5, which then increases to more than 5.5% at ζ = 11 and then declines steadily as capacity gets expensive. The VOI occurs due to the sub-optimal setting of planned lead times (or, smoothing parameters) in the first step of the sequential approach, and can be explained using Fig. 9. For ζ ≤ 5, both the methods result in the the production base-stock policy and T = 1 at all stages. In this case, the safety stock placement solutions under both these methods are also identical. However,

K. Kumar and T. Aouam / European Journal of Operational Research 279 (2019) 429–448

441

Fig. 10. β ∗j for TPP and TP − SSP.

these two policies may result in cost savings. These savings may be negligible when capacity costs are low; while, it is worthwhile to jointly optimize these decisions when capacity cost is high relative to the holding cost. 6. Effect of service times on the suitability of the TPM

Fig. 11. Value of integration as a function of ζ .

as capacity becomes expensive, the smoothing solutions under the sequential and integrated approaches begin to differ, which results in a positive value of integration. As ζ increases further, β -values start to become similar, which reduces the VOI. This study emphasizes the importance for managers to integrate production smoothing and safety stock placement decisions. Traditionally, safety stock placement is determined assuming the planned lead times as inputs, which are themselves the output of production smoothing models. We show that jointly determining

The preceding sections analyzed a model based on the linear control rule of the tactical planning model (TPM), which is one of the most commonly used smoothing rules in literature. However, service times at a stage may influence the suitability of TPM in the integrated model. To demonstrate the effect of service times, we compare TPM to another policy that possesses similar characteristics. In this section, we focus on the equal-weighted moving average (EMA) policy, which like similar to TPM, is simple, descriptive and can be defined using a single policy parameter. Balakrishnan et al. (2004) showed that the EMA results in the highest degree of smoothing among all the moving average policies. The goal here is to compare similar to TPM and EMA in the context of safety stock placement problem and identify the role of service times on the choice of the better smoothing policy among the two. 6.1. Safety stock placement with EMA policy The EMA policy considers a time-window Wj and sets the production quantity Xj (t) to an equal-weighted combination of the last Wj released quantities. Under the production base-stock policy, W j = 1 and all quantities Rj (t) released at the beginning of a

442

K. Kumar and T. Aouam / European Journal of Operational Research 279 (2019) 429–448

period are produced in the same period. However, for Wj ≥ 1, an equal and fixed fraction γ j = 1/W j of Rj (t) is produced from periods t until t + W j − 1. The production quantity in a period t under EMA is described as: W j −1



X j (t ) = γ j

R j (t − k )

k=0 W j −1



= γj

d j (t − k − SI j )

(32)

k=0

Consequently, the state variables for the EMA policy are written as functions of γ j as in Eqs. (33)–(39), where the superscript  refers to EMA policy. Hereafter, we use superscript † to refer to the TPM variables.

T j =

Fig. 12. Var(Y )/σ 2 as a function of net service time NS for different γ .

γj + 1 2γ j

E[X j ] =

(33)

μj

(34)



V ar (X j ) = γ j σ j2

 E[Q j ] =

1 + γj

 SI j +

 V ar (Y j ) =



(36)

2 + γj

1 + γj 2γ j

1 + γj



occurs at NS j (γ j ) = T j =

1+γ j 2γ j .

Proof. See Appendix A for proof.



6γ j

 E[Y j ] =

(35)

1 + γj μj 2γ j 

V ar (Q j ) =

a) for NS j = S j − SI j ≤ 1, V ar (Y j ) is a convex and decreasing function of NSj , and b) for NS j = S j − SI j > 1, V ar (Y j ) is convex in NSj and its minima

σ j2

− Sj

2 + γj

6γ j



(37)

μj

(38)

  + S j − SI j  + [S j − SI j ]+

  2 × S j − SI j − 1 γ j − 2 σ j 

(39)



For net replenishment time to be positive, it is required that SI j + T j − S j ≥ 0 or NS j ≤ T j . Therefore, it follows from Proposition 5 that the inventory shortfall variance under the EMA policy, unlike TPM, is always a decreasing function of the net service time. Fig. 12 shows the behavior of V ar (Y j )/σ j2 as a function of NSj for different γ j values. 6.2. Comparison of TPM and EMA policies This section compares the optimal costs under the TPM and EMA policies, and presents conditions under which one policy dominates the other for a target planned lead time Tj . Further, we aim to highlight the influence of net service time NS j = S j − SI j on the choice of the better policy.

Following an analysis similar to that of the TPM policy, we find the hereunder properties for the EMA policy.

Proposition 6. For service times such that NS j = S j − SI j ≤ 1, the optimal TPM policy always results in lower costs than the optimal EMA policy.

Lemma 2. For any given γ j , the optimal nominal capacity at a pro-

Proof. See Appendix A for proof.

∗

x−1

duction stage j ∈ P under the EMA is K j =  j −1

(1 −

f

cj

coj

), where

xj (· ) is the inverse c.d.f. of Xj with mean μj and variance γ j σ j2 . Proof. See Appendix A for proof.



Corollary 3. When Xj is normally distributed, with f(·) and F(·) being the standard normal p.d.f. and c.d.f., we have: ∗

a) K j = μ j + z∗j



γ jσ j,



b) the minimum expected capacity cost is c j μ j + coj f (z∗j ) γ j σ j , f

c) where, z∗j = F −1 (1 −

f

cj

coj

) is the optimal capacity safety factor.

Under EMA, Lemma 2 describes the optimal nominal capacity ∗ K j and the optimal capacity costs as a function of the smoothing parameter γ j . A smoother production with γ j < 1 leads to lower ∗ (higher) nominal capacity K j as compared to the production basestock policy when z∗j is positive (negative). Proposition 5. For a given γ j , the following holds for a production stage j under the EMA policy :



Proposition 6 highlights the superiority of the TPM over the EMA policy when the net service time is at most one review period. The result includes the case of traditional production smoothing models, where the service times are generally considered zero. In such systems, the TPM is a superior policy as compared to the EMA. Further, we note that manufacturers operating under a pure make-to-stock strategy, i.e., S j = 0, will be better off using the TPM policy. When the net service time exceeds one review period (NSj > 1), the choice of the better smoothing policy is not trivial and depends on the planned lead time and the net service time. Next, we compare EMA and TPM policies with the same planned lead times. For a planned lead time Tj , we consider the EMA policy with γ j = 1/(2T j − 1 ) and TPM policy with β j = 1/T j . Further, we note that it can be easily shown that for the same planned lead time, both the TPM and EMA policies result in the same degree of production smoothing and incur identical costs of WIP and capacity. The two policies differ only in terms of their inventory shortfall variances (and consequently inventory costs) at the manufacturer. Therefore, it would be sufficient to compare inventory shortfall variances of the two policies, for a given net service time NSj to

K. Kumar and T. Aouam / European Journal of Operational Research 279 (2019) 429–448

443

Fig. 13. /σ 2 as a function of NS for given T.

determine the better policy. By defining  j = V ar (Y j ) − V ar (Y j ), a positive j signifies the dominance of the TPM over EMA; while a negative value of  denotes otherwise. †

Proposition 7. For any lead time Tj ,  j = V ar (Y j ) − V ar (Y j ) is a non-increasing function of the net service time NSj . †

Proof. See Appendix A for proof.



From Proposition 4, it can be inferred that j > 0 for NSj ≤ 1. However, as NSj is increased, j may decrease and eventually become negative (Proposition 7), at which point EMA starts to dominate TPM. This effect is shown in Fig. 13, which plots  j /σ j2 as a function of NSj for different lead times. In fact, the plots can be divided into five distinct subregions. a. In the first subregion, NSj ≤ 1 and j has a constant positive value. As shown in Proposition 4, TPM dominates in this region. b. The second subregion corresponds to 1 < NS j ≤ NS j (1/T j ), where NS j (1/T j ) is the net service time that minimizes the shortfall variance under TPM and was derived in Proposition 2. Here, j decreases as NSj is increased; however it retains a positive value and the TPM dominates the EMA policy. c. The third subregion lies between NS j (1/T j ) and NS0j , where NS0j is the net service time at which j becomes zero. In this region, j is positive and TPM results in lower costs as compared to the EMA. Although TPM dominates in this region, no optimal net service time NS∗j for TPM such that NS j (1/T j ) < NS∗j < NS0j would be selected. This follows from Proposition 1, where it was established that NS j (1/T j ) is an upper bound on the optimal net service time. d. For N S0j ≤ N S j ≤ T j , j is negative and EMA leads to lower inventory shortfall variance and lower costs as compared to the TPM policy. e. Both EMA and TPM policies lead to infeasible service times with NSj > Tj . The observations in Fig. 13 are formally stated in Proposition 8 for which the mathematical proof is relegated to the Appendix. Proposition 8. For a given lead time Tj , let NS0j be the service time difference at which  j = 0. Then,

Fig. 14. Percentage savings from using the TPM over EMA policy.

a) the TPM policy dominates for N S j ≤ N S j (1/T j ), b) the EMA policy dominates for N S0j < N S j ≤ T j , and

c) no optimal solution will occur for NS j (1/T j ) < NS j ≤ NS0j .

Proof. See Appendix A for proof.



A numerical comparison. We compared the performance of both the smoothing policies on the 14-stage instance described above. Fig. 14 shows percentage difference between the optimal costs under EMA and TPM policies, where a positive difference signifies the dominance of TPM. Despite the result of Proposition 8, we find that the TPM policy outperforms EMA for all realistic values of ζ ≥ 8, while for ζ < 8, both the policies result in similar costs. However, it must be noted that gap between the optimal costs of these policies is very low, less than 0.37% for the studied cases. Similar observations were found when we varied the holding costs and fixed lead times through the network. Our study indicates that the choice of TPM as a smoothing rule is robust to changes in policy parameters. It is most likely to result in lower costs than the EMA, even though the difference in costs may not be significant. This result resonates with the study of Balakrishnan et al. (2004), who also found that the TPM (or exponential soothing) results in lower costs when compared with other moving average policies with time-windows.

444

K. Kumar and T. Aouam / European Journal of Operational Research 279 (2019) 429–448

7. Conclusion

Acknowledgment

On the one hand, GSA models typically do not reflect the effects of tactical production decisions, such as production smoothing, on lead times. On the other hand, tactical production smoothing lay great emphasis on the use of safety stocks to insulate production from demand variability but do not address the questions of safety stock placement. In the current work, we fill this research gap and formulate a model to integrate strategic safety stock placement with tactical production smoothing under the exponential smoothing rule. The integrated problem minimizes the costs of capacity and holding WIP at each production stage, along with inventory holding and expediting cost in the supply chain. Our analysis focuses on studying the combined effect of the smoothing parameter and guaranteed service times on safety stocks and total costs. A key feature of traditional smoothing models is the trade-off between production smoothing and safety stocks, where dampening of production results in a more variable inventory and higher safety stocks. We found that this relationship holds only when the net service time at a stage (outgoing minus incoming) is less than one review period. When the net service time is more than one review period, we showed that a certain degree of smoothing results in less variable inventory and lower holding and expediting costs. Secondly, the inventory costs for the integrated model are non-monotonous and non-concave functions of the net service times, as opposed to traditional GSA models, where inventory costs are generally considered to be concave and decreasing in the net service time. Furthermore, we derived an upper bound on the optimal net service time. Thirdly, we showed that production smoothing at a stage directly impacts safety stock placement. We find that all safety stocks at a stage are pooled at the downstream stages only under the production base-stock policy, while for smoothed production safety stocks must be held at the production stage. This effect occurs due to a positive inventory shortfall variance at the stage when the production is smoothed. Lastly, similar to Balakrishnan et al. (2004) and Dejonckheere, Disney, Lambrecht, and Towill (2003), we compared the TPM policy with another commonly used smoothing rule - the equal-weighted moving average (EMA) policy. Our study is the first to highlight the impact of service times on the choice of smoothing policy. Our analysis showed that for net service times less than one review period, TPM results in lower costs as compared to the EMA. Further, for a given planned lead time, we characterize the conditions on service times where one of the two policies outperforms the other. Numerical results however favored TPM for optimal planned lead times. The present work extended the dynamic programming algorithm from Graves and Willems (20 0 0) to solve for spanning tree structures. This work can be extended to consider general acyclic structures and develop efficient solution algorithms for these structures. One possibility could to be incorporate the algorithm from this paper into the branch-and-bound schema of Humair and Willems (2011). Other possibilities may include development of metaheuristic procedures. Further, it will be interesting to consider smoothed ordering policy as in Zahraei and Teo (2017) and study the effect of service times on variability propagation through the network. The assumption of common review periods can also be relaxed to consider non-nested review period as in Bossert and Willems (2007). However, in this case, we expect cost functions that are irregular as shown in Humair and Willems (2011). In addition, an interesting extension is to consider correlated demand across periods and study the impact of smoothing and safety stock placement.

We acknowledge the support provided by the “Bijzonder Onderzoeksfonds” (BOF) for the project with contract number BOF.STA.2015.0 0 06.01 Appendix A. Proofs for propositions and lemmas Proof of Lemma 1. Following (21), we write the expected cost at a production stage as a function of K as:

T C (K ) = c f K + coE[(X − K )+ ] + ωE[Q] + E[IC]   ∂ T C (K ) ∂  ∞ = c f + co (X − K )φ x (X )dX ∂K ∂K K  ∞ f o =c −c φ x (X )dX K

= c f − co (1 − x (K ) )

∂ 2 T C (K ) = co φ x (k ) ≥ 0 ∂ K2 2 Since ∂ ∂TKC2(K ) ≥ 0, we can find the optimal K∗ by setting ∂ T∂CK(K ) = 0 as follows.



x−1

K = ∗

cf 1− o c



.

For a normal distribution, we have: K ∗ = μ + z∗ z∗ = F −1 (1 −

cf co



β

2−β

σ , where

) and F(·) is the standard normal c.d.f. 

Proof of Proposition 1. Using (11), we can write the inventory shortfall variance for all net service times NS ≤ 0 and NS = 1 as:



V ar (Y )|NS≤1 =

1 − NS β (2 − β )

σ2

The first term is a decreasing function of β , while the second term is independent of β . Therefore, Var(Y) is a decreasing function of β for all NS ≤ 1 (and NS ∈ Z).  Proof of Proposition 2. Using (11), we write Var(Y) for all NS ≤ 1 as:



V ar (Y ) =

1 − NS β (2 − β )

σ2

which is a linearly decreasing function of NS. For the case where NS > 1, we have



V ar (Y ) =

σ2

 2 1 NS + NS − 1 − (1 − β ) β (2 − β ) β

 ∂ V ar (Y )  2 NS = 1 + (1 − β ) ln(1 − β ) σ 2 ∂ NS β

Equating the above expression to zero, we get an extreme point



 (β ) = ln − NS

β

2ln(1 − β )



/ln(1 − β )

Next, we prove the convexity of Var(Y) in NS.

∂ 2V arY 2 NS 2 = (1 − β ) (ln(1 − β ) ) σ 2 > 0 β ∂ NS2 This shows that Var(Y) is convex and decreasing for NS > 0 with a  (β ). However, for β ≥ 0.715, we have 0 < NS  ( β ) < 1, minima at NS which is undefined as NS ∈ Z. Therefore, the minimum feasible  ( β )}.  value of Var(Y) occurs at max{1, NS

K. Kumar and T. Aouam / European Journal of Operational Research 279 (2019) 429–448

Proof of Proposition 3. Consider a network with an optimal solution such that there exist at least one node j ∈ P such that the  (β ). Let the optimal cost be E[T C ∗ ] = optimal NS∗j = S∗j − SI∗j > NS j j ∗ ∗ ∗ costd + cost j + costu , where terms represent the cost of downstream subnetwork, the node and the upstream subnetwork, reS , we can either set S = ! spectively. Now, to set N S j = N S j < S∗j j j ! > SI∗ (option 2). The first option will re(option 1) or SI = SI j

j

j

duce cost ∗j (Proposition 2), without necessarily affecting costd∗ , as the incoming service time at the downstream stages can remain the same (constraints 26). Therefore, option 1, will result in at least one solution that is lower than NS∗j . Similarly, we can increase SIj ! without changing cost ∗ . Therefore, NS∗ is not an optimal soto SI j

u

j

 is an lution and through reduction ad absurdum, we show that NS j upper bound on the optimal net service time at node j. 

Proof of Proposition 6. For any NS, consider the optimal EMA policy such that it leads to a lead time T, such that γ = 2T1−1 . Now, consider a TPM policy with the same lead time T such that β = 1/T . Under these parameters, both policies result in identical WIP and overtime costs; the only difference lies in Var(Y) and the inventory costs at the node. We define  = V ar (Y  ) − V ar (Y † ) as the difference between inventory shortfall variances of EMA and TPM. A positive  signify that TPM is better, while a negative  would denote that EMA is better. For the case NS ≤ 0, we have

1  (1 + γ ) (2 + γ ) = − NS − + NS 6γ β (2 − β ) σ2 T2 ( T ) ( 4T − 1 ) =

− 6T − 3 2T − 1 T (T − 1 ) = ≥ 0, ∀ T ≥1 6T − 3

Proof of Proposition 4. First, we consider the case where NS ≤ 0. In this case, using (10), we can write Var(Y) as



V ar (Y ) =

1 − NS β (2 − β )

σ2

where, both first and the second terms are positive for all β and NS ≤ 0, and Var(Y) is always positive. When NS = 1, we have



V ar (Y ) =

1 −1 β (2 − β )

Now, consider the case where NS = 1.

1  (1 + γ ) (2 + γ ) = + 1 + (−2 ) − 6γ β (2 − β ) σ2

σ2

V ar (Y ) =

 2 2 1 NS + NS − 1 − (1 − β ) σ β (2 − β ) β

6T − 3



V ar (Y )|NS=NS =

1

β (2 − β )



−β

2ln(1−β ) ln(1 − β )

+

−

2

β

 1+

 = V ar (Y  ) − V ar (Y † ) =

β



Proof of Proposition 5. When NSj ≤ 1, we write V ar (Y j ) given by (39) as

V ar (Y j ) =

1 + γj



2 + γj

6γ j





− NS j

σ2

which is a linearly decreasing function of NSj . For the case where NSj > 1, we have



V ar (Y j ) =

     2 1 + γj 2 + γj + NS j + NS j (NS j − 1 )γ j − 2 σ j 6γ j

∂ V ar (Y j )     = γ j 2NS j − 1 − 1 σ j2 ∂ NS j ∂ 2V ar (Y j ) = 2γ j σ j2 ∂ NS2j Therefore, V ar (Y j ) for EMA is convex in NSj > 0, and the minimizer

∂ Var (Y  ) of V ar (Y j ) can be found by setting ∂ NS j = 0, which is given as j   = 1+γ j = T  .  NS j 2γ j j

T2 − T σ2 3 ( 2T − 1 )

(A.1)

In (A.1),  is independent of NS. Next, we consider the case where 1 < NS ≤ T. On setting γ = 2T1−1 and β = T1 , we get

   T ( 4T − 1 ) 1 = + N S N S − 1 − 2 ( ) 6T − 3 2T − 1 σ2   NS 2

2ln(1 − β )

which is positive for all β > 0.834 or T < 1.19, where T = 1/β . For net lead time to be positive (constraints (24)), we have NS ≤ T and there exist no integer that is NS < 1.19 and NS > 1. Therefore, for all feasible values of β and NS, we show that V ar (Y ) = 0 only when β = 1 and NS = 1. 



 Proof of Proposition 7. We first consider the case where NS ≤ 1:

 , is given as and its minima, which occurs at NS

ln

2

(1 − (1 − β ) ) β 1 (1 + γ ) (2 + γ ) = − 6γ β (2 − β ) T (T − 1 ) = ≥ 0, ∀ T ≥1 −1+

which has a unique solution β = 1, which corresponds to the basestock policy. In the last case which is NS > 1, we have



445

−

T 1 + 2T 1 − 1 − 2T − 1 T

    4T − NS − 1 1 NS 1 ∂ NS = − + − 2 T 1 − ln 1 − 2T − 1 2T − 1 T T σ 2 ∂ NS 1

Now for  to be decreasing in NS, the following must be true:

−



4T − NS − 1 1 NS + − 2T 1 − 2T − 1 2T − 1 T ⇔



1 2NS + 1 − 4T < 2T 1 − 2T − 1 T

For 1 < NS ≤ T and 0 ≤ 1 −



1−

 1 NS T



≤ 1−

⇔ 2T 1 −





 1 T

  1 NS

ln 1 −

T

because ln 1 −

1 T



1 T

1 T

NS 

ln 1 −

NS 

ln 1 −

1 T



1 T



<0 (A.2)

≤ 1, the following result is true:





≥ 2T 1 −

 

1 1 ln 1 − T T



< 0. Now, using this result in (A.2), we get



 





1 1 2NS + 1 − 4T < 2T 1 − ln 1 − 2T − 1 T T





1 NS 1 ln 1 − T T 2NS + 1 − 4T ⇔ 2T − 1 ≤ 2T 1 −



446

K. Kumar and T. Aouam / European Journal of Operational Research 279 (2019) 429–448



"

 

#

1 1 ln 1 − T T T     1 1 ≤ 2T 1 − ln 1 − T T  1 2NS + 1 − 4T ⇔ < 2T − 2T − 1 e " 2NS + 1 − 4T # 2NS + 1 − 4T ⇔ ≤ max NS 2T − 1 2T − 1  1 < 2T − e  1 1 − 2T ⇔ < 2T − 2T − 1 e  2  2 ⇔ −1 < min {T } − ≤T − T e e 2 ⇔ −1 < − e < 2T min

1−

Thus, we prove that the difference  is strictly decreasing in NS for 1 < NS ≤ T. For NS ≤ 1,  is constant. Therefore,  is a nonincreasing function of NS ≤ T, where NS is defined as an integer.  S(1/T ) = Proof of Proposition 8. We compute  at N S = N ln( −1/T ) 2ln(1−1/T ) and it can be verified that |NS=NS(1/T ) > 0. Since  ln(1−1/T ) is non-decreasing in NS (Proposition 6), it can also be concluded S(1/T ) and TPM dominates over EMA. Anthat  > 0 for N S ≤ N  (1/T ) < NS0 . Now, in the other implication of this result is that NS 0  region NS(1/T ) < NS ≤ NS ,  > 0 and TPM is better than EMA. However, from Proposition 1, we know that a solution with lower S(1/T ). Therefore, no optimal solution will costs exists in N S ≤ N  (1/T ) < NS ≤ NS0 . occur in the region NS Further, we compute  at NS = T and it can be easily verified that |NS=T < 0. This also implies that NS0 < T. In the range NS0 < NS ≤ T, EMA dominates the TPM policy. Further, NS > T does not exist as the net lead time becomes negative.  Appendix B. Dynamic programming algorithm We adapt the dynamic programming algorithm (DP-GW) from Graves and Willems (20 0 0) to solve the integrated problem TP − SSP for supply chains with a spanning-tree structure. DP-GW decomposes the problem into |M | subproblems, each representing a stage in the network. The order of recursion over these subproblems is defined as per the labeling procedure of DP-GW, which assigns a label m to each stage. Stages are labeled such that the minimum cost function at a stage can be written as function of a single-state variable (either S or SI). A parent node p(m) is the node adjacent to m with a higher label. The subset Nm is the set of nodes with label less than m and through which there exists a path to m, and is written as

Nm = {m} +

$

l
Nl +

$

Nn

n
The minimum cost functions are defined to find the minimum cost in Nm as function of a single state variable, either S or SI. In view of our study, we distinguish between the minimum cost functions at production and logistics stages. For production stages, we p p denote the minimum cost function as fm (S ) and gm (SI ), whereas l (S ) and for logistics nodes the minimum cost functions are fm l gm (SI ). Further, if p(m) lies downstream of m, S is the state varip l (S )) is solved for able and the minimum cost function fm (S ) (or fm a given S. When p(m) lies upstream of m, SI is the state-variable p and gm (SI ) (or glm (SI )) is solved for a given SI. As a first step to defining the minimum cost function, we write the total operating costs cm (·) in the subgraph Nm as function of

the decision variables at production stages in (B.1) and for logistics stages in (B.2). ∗ o ∗ + cm (S, SI, β ) = cmf Km + cm [(Xm − Km ) ] + ωm E[Qm ]

+ hm km  +



s V ar (Ym ) + (hm + cm )E[(Ym − Bm )+ ]

min fn♦ (x )

(n,m )∈A

0≤x≤SI

n
+



min g♦ o ( y ),

(m,o)∈A

(S≤y≤Lo )

∀m ∈ P

(B.1)

o


cm (S, SI ) = hm km SIm + λm − Sm + E[(Yml − Bm )+ ]  + min fn♦ (x ) (n,m )∈A

0≤x≤SI

n
+



(m,o)∈A

min g♦ o ( y ),

(S≤y≤Lo )

∀m ∈ L

(B.2)

o
where, ♦ = { p, l } and Lm is the maximum service time at stage m. ∗ follows from Lemma 1. The second last terms Further, in (B.1) Km in (B.1) and (B.2) correspond to the minimum operating costs in the subnetwork downstream of m, assuming a maximum outgoing service time SI for the downstream stages. Similarly, the last terms in (B.1) and (B.2) represent the minimum operating costs in the subnetwork upstream of m, assuming a minimum incoming service time S for the upstream stages. We solve the minimum cost function below to minimize the total operating costs in the subnetwork Nm .

fmp (S ) = minSI,β {cm (S, SI, β )} s. t. 0 < β ≤ 1  (β ) S − SI ≤ NS S, SI ∈ Z+ fml (S ) = minSI {cm (S, SI )} s. t. S − SI ≤ λm (β ) S, SI ∈ Z+

p gm (SI ) = minS,β {cm (S, SI, β )} s. t. 0 < β ≤ 1  (β ) S − SI ≤ NS S, SI ∈ Z+

glm (SI ) = minS {cm (S, SI )} s. t. S − SI ≤ λm (β ) S, SI ∈ Z+

The dynamic programming algorithm for TP − SSP is then written as follows: a) For m := 1 to |M | − 1 ♦ i) If p(m) is upstream to m, evaluate fm (S ) for S = 0, . . . , Lm , where ♦ = { p, l }. ii) If p(m) is downstream to m, evaluate g♦ m (SI ) for SI = 0, . . . , Lm , where ♦ = { p, l }. b) For j := |M | evaluate g♦ m (SI ) for SI = 0, . . . , Lm . c) Minimize g♦ ( SI ) for SI = 0, . . . , L|M | to find the optimal cost |M | of the solution. Standard backtracking procedure for dynamic programs can then be used to find the optimal service times and smoothing parameters.

Appendix C. Incorporating demand bound In this paper, we considered an unbounded demand to flow through the network and assumed that demand exceeding onhand inventory is satisfied based on the ‘borrow-return’ policy of Lee et al. (20 0 0). Alternatively, we can adopt the boundeddemand assumption of the original GSA framework as in Graves and Willems (20 0 0). Under this assumption, the lead time demand or inventory shortfall is considered to be bounded, and it is only this bounded stream that is allowed to propagate through the supply chain. Items in excess of the demand bound are handled outside the network. −1 Specifically, we assume that dj (t) is bounded by D¯ j = dj (α ), where dj (· ) is the c.d.f. of demand and α is the target fraction

K. Kumar and T. Aouam / European Journal of Operational Research 279 (2019) 429–448

of demand to be met from stock. Further, the inventory shortfall y−1 is assumed to be bounded by the function Y¯ =  (α ) = E[Y j ] +



j

k j V ar (Y j ), where kj is a safety factor corresponding to α . In this case, the minimum base-stock level to guarantee the demand is B j = Y¯ , which is the same as (2). Based on Graves and Willems (20 0 0), the expected on-hand inventory at any stage j is

E[OH j ] = k j



V ar (Y j ).





D¯ j

Kj

Xj − Kj

  x φ j (X j )dX j +



∞

D¯ j

D¯ j − K j

 x φ j (X j )dX j , (C.1)

while, it can be easily shown that it is never optimal to have K j > D¯ j . The total cost for the network is then written as follows:

E[T C] =





c jf K j + coj E[(X j − K j )+ ] + ω j E[Q j ] + k j

j∈P

+



k jσ j





SI j + λ j − S j



V ar (Y j )

(C.2)

j∈L

We note that for a given Bj , the production smoothing and inventory models from Section 3.3 remains the same for the case of bounded demand. An optimization model TP-SSP-BD can be written with an objective function (C.2) and constraints (24)–(31) to jointly optimize production smoothing and safety stock placement when the demand is bounded. Further, key results proposed in this paper (Propositions (1)–(8) and Corollary (2)) extend to the bounded-demand model TP-SSP-BD, while Lemmas 1 and 2 can be rewritten as below. Lemma 3. For any given β , the optimal nominal capacity at the production stage is

%



−1 K = min D¯ , x 1−

∗

cf co

&

,

−1

where x (· ) is the inverse c.d.f. of X with mean μ and variance β σ 2. 2−β Proof. Following (C.1), we write the expected capacity cost C(K) at a production stage for any K ≤ D¯ as

C (K ) = c K + c f



D¯

o K

 

∂ C (K ) = c f + co − ∂K 

= c f − co

∞

K

(X − K )φ (X )dX +

D¯ K



x

φ (X )dX − x



∞

D¯

∞ D¯



(D¯ − K )φ x (X )dX



φ (X )dX x

φ x (X )dX

= c f − co (1 − x (K ) )

∂ 2C (K ) = co φ x (k ) ≥ 0 ∂ K2 From above, it follows that:

%



−1 K ∗ = min D¯ , x 1−

cf co

& 

Lemma 4. For any given γ j , the optimal nominal capacity at a production stage j ∈ P under the EMA is

'



x−1 j

K j = min D¯ j ,  ∗

−1

where xj (· ) is the inverse c.d.f. of Xj with mean μj and variance

γ j σ j2 .

Lastly, the dynamic programming algorithm from Appendix B can be used to solve TP-SSP-BD, where K∗ is given by Lemma 3 for TPM. References

For any given nominal capacity K j ≤ D¯ j , the expected overtime cost is given as

E[(X − K )+ ] =

447

1−

c jf coj

(

,

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