Impact of lead time variability in supply chain risk management

Author’s Accepted Manuscript Impact of lead time variability in supply chain risk management Dia Bandaly, Ahmet Satir, Latha Shanker

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S0925-5273(16)30160-8 http://dx.doi.org/10.1016/j.ijpe.2016.07.014 PROECO6470

To appear in: Intern. Journal of Production Economics Received date: 13 April 2015 Revised date: 8 October 2015 Accepted date: 6 June 2016 Cite this article as: Dia Bandaly, Ahmet Satir and Latha Shanker, Impact of lead time variability in supply chain risk management, Intern. Journal of Production Economics, http://dx.doi.org/10.1016/j.ijpe.2016.07.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Impact of lead time variability in supply chain risk management Dia Bandalya*, Ahmet Satirb, Latha Shankerc a

Department of Information Technology and Operations Management, Adnan Kassar School of Business, Lebanese American University, P.O. Box 36, Byblos, Lebanon b

Department of Supply Chain and Business Technology Management, John Molson School of Business, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, QC, Canada H3G 1M8 c

Department of Finance, John Molson School of Business, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, QC, Canada H3G 1M8 *

corresponding author at: Department of Information Technology and Operations Management, Adnan Kassar School of Business, Lebanese American University, P.O. Box 36, Byblos, Lebanon; Tel: +961-9-547262 ext. 2504, fax: +961-9-546256; [email protected] Co-authros’ email addresses: Ahmet Satir: [email protected] Latha Shanker: [email protected] Abstract In this article, we study the impact of lead time variability on the performance of supply chain risk management in the beer industry. The stakeholders considered are the empty aluminum can supplier, the brewery and the distributor. The stochastic lead time model developed is an extension of the previously developed base model (with deterministic lead time) under which commodity price risk and demand uncertainty are managed via an integrated risk management approach using operational methods and financial derivatives. Simulation-based optimization is used to model and analyze such a complex system. We find that lead time variability does not always deteriorate the supply chain performance. Only high levels of lead time variability are found to justify more the need for a coordinated supply chain. The findings also reveal the need for using different hedging strategies to manage products flows across the supply chain under stochastic lead time versus the 1 of 39

deterministic case. Managerial insights for the supply chain studied are argued based on experimental design findings.

Keywords: beer supply chain; risk management; stochastic lead time; simulation-based optimization; opportunity cost 1. Introduction In this article, we study the impact of lead time variability on the supply chain risk management (SCRM) performance and how decisions change under lead time uncertainty. The setup of the problem described here is based on real-life data obtained during a case study by Kira, Satir, and Bandaly (2013) on risk management approaches used by a supply chain in the beer industry. The supply chain consisting of a can supplier, a brewery and a distribution center faces fluctuations in aluminum price and demand uncertainty. To mitigate these two risks, the supply chain stakeholders make decisions on operational and financial hedging positions, as well as, on inventory levels and products flows across the supply chain. The challenge is thus to determine: i) optimal quantities of aluminum sheets (used to produce the cans) to procure at different points in time (operational hedging), ii) optimal quantities of financial options to place (financial hedging), and iii) optimal inventory levels of empty cans and finished products to maintain across the supply chain. These decisions are to be made with the objective of minimizing the total opportunity cost of the supply chain. The benefits of integrating the above operational and financial risk management methods were highlighted by the authors in Bandaly, Satir and Shanker (2014). In that article, the effects of three principal factors on the expected total opportunity cost were studied, namely: i) level of risk the supply chain is willing to assume; ii) demand variability, and iii) volatility of aluminum prices. In this article, we extend the base model developed in the aforementioned article by introducing an important operational risk factor: lead time variability. Where lead time was assumed to be

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deterministic in the base model, stochastic lead time in the supply of aluminum cans to the brewery is used for the extended model here. We examine the effects of this factor on the supply chain performance and draw managerial insights. Because of the complexity of the problem and the stochastic nature of the main inputs, we use simulation-based optimization in order to find the optimal solutions. To support our hypothesis that centralized decision making is more beneficial than a decentralized one, we develop two models representing these two approaches and compare the results. We conduct an experimental design to study the effects of the above factors on the risk management performance. The main contribution of this article is of its focus on the effects of lead time variability on the performance of the supply chain studied, rather than an individual stakeholder of the supply chain. This study is also different from other works in terms of the type of decisions made (both operational and financial) to manage the risk and the performance measure (opportunity cost) used in the model. Our findings reveal a number of interesting points. For example, we find that reduction of lead time variability may not be necessary under certain conditions. This is a valuable insight for managers contemplating investing in such a reduction. We also observe how lead time variability might alter supply chain decisions in risk management. 2. Literature review and contribution of this article In this article, we study the effects of lead time variability (LTV) on supply chain risk management performance and on the decisions made to manage risks. We now review the related literature. We present our contribution at the end of each section. 2.1.

Deterministic lead time in supply chain literature While the impact of lead time has been widely studied in the operations management literature,

research incorporating variability of lead time is relatively sparse when the supply chain is the unit

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of analysis (Humair et al. 2013; Heydari 2014b). Research on the impact of lead time on supply chain performance is more common when lead time is treated as a deterministic parameter. Lee, Padmanabhan and Whang (1997) highlighted the contribution of long lead times in the creation of bullwhip effect along a supply chain. This work helped drawing lot of academic attention to the phenomenon studied (Isaksson and Seifert 2016, Wang and Disney 2016) and was followed by a stream of research on the bullwhip effect in supply chains. Some of the articles examine the explicit role of deterministic lead time, along with other factors, in this order amplification phenomenon (see for example, Chen et al. 2000, Agrawal, Sengupta, and Shanker 2009, Sadeghi 2015, Cannella, Bruccoleri and Framinan 2016). While the study of Sadeghi (2015) confirms the positive relationship between lead time and bullwhip effect, it provides new perspective on this effect. The author compares the bullwhip effects observed on two different products. The results show that when lead time increases the difference between those bullwhip effects gets larger, posing more challenges on managing the supply chain. A new perspective on the bullwhip effect is also highlighted by Cannella, Bruccoleri and Framinan (2016) who study the lead time effect in the specific case of closed-loop supply chain. The authors find that the reduction of remanufacturing lead time helps avoiding bullwhip effects. While the above articles use the bullwhip effect as supply chain performance measure in their analyses, others investigate the impact of lead time on supply chain inventory cost (for example, Liao and Chang 2010). In the aforementioned works, the effects of lead time on performance are studied for a given supply chain configuration. On the other hand, Hammami and Frein (2013) develop a model that determines the best supply chain configuration under lead time restrictions. Plant locations and suppliers selections are determined in a way that lead time is kept lower than a specified maximum limit. Li et al. (2012) propose a coordination mechanism between a buyer and supplier so that the former would order a quantity that optimizes

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the supply chain performance. Unlike other works, lead time here is a decision variable and its optimal length is determined as part of the solution. The common aspect between this study and the above is the study of the impact of lead time on supply chain performance. This study, however, considers a variable rather than a deterministic lead time. In our study we also examine the effects of lead time variability on the supply chain decisions. Similarly, Hlioui, Gharbi and Hajji (2015) study the effect of lead time on decisions made in the supply chain, but in the case when lead time is deterministic with random value. Using simulation model, the authors show how one quality control policy would be favored on another depending on the magnitude of lead time. 2.2.

Effects of lead time variability on supply chain performance The effects of uncertainties on supply chain performance are well documented in the literature.

While demand uncertainty is the factor most widely examined (Tang, 2006), managing the uncertainty stemming from lead time variability is also crucial in supply chain management. Unlike demand which is a factor external to the supply chain, lead time is an internal factor and its variability can be managed using different risk management approaches (Bandaly et al. 2012). Articles that study the impact of LTV on supply chain performance examine different measures such as: i) order variability (So and Zheng 2003; Chatfield et al. 2004; Heydari, Kazemzadeh and Chaharsooghi 2009, Chaharsooghi and Heydari 2010), ii) inventory level and cost (Chopra, Reinhardt and Dada 2004; Simchi-Levi and Zhao 2005; Acar, Kadipasaoglu and Schipperijn 2010; Chaharsooghi and Heydari 2010; Fang et al. 2013; Heydari 2014a) and iii) financial indicators (Christensen, Germain and Birou 2007). Similar to any other uncertainty, variability of lead time is expected to deteriorate supply chain performance. Simchi-Levi and Zhao (2005) demonstrate that the total inventory cost and the sum of stock levels across the stages of the supply chain drastically increase as LTV increases. Similarly, Chaharsooghi and Heydari (2010) also find a positive

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relationship between lead time variance and inventory positions. On the contrary, the authors find a diminishing effect of lead time variance on the bullwhip effect. Christensen, Germain and Birou (2007) report that reducing lead time variance is necessary to improve ROI, profit and profit growth. However, in some instances, the LTV impact can be negligible. Heydari, Kazemzadeh and Chaharsooghi (2009) find that an increase only in LTV alone may not have an impact on the bullwhip effect. Acar, Kadipasaoglu and Schipperijn (2010) find that the effect of lead time uncertainty is significant only on inventory carrying cost and not on other performance measures. In addition to the main effects of LTV, it is also important to observe the interaction effects with other factors. Chatfield et al. (2004) find that the impact of LTV ranges from ‘very little’ to ‘exponential’ under three different levels of information quality. The authors also report important interaction effect between LTV and demand information sharing. For the same LTV, the ratio of order variance at the factory to the variance of customer’s order drastically drops when demand information is shared. Chopra, Reinhardt and Dada (2004) report a threshold for the service level, below which decreasing lead time uncertainty increases the safety stock rather than lowering it. It is also important to study the effects of LTV under varying business environment. The findings of So and Zheng (2003) show that the effect of a variable lead time on the order variability increases exponentially when compounded with demand variability. In this article, we study the effects of LTV on the expected opportunity cost of the supply chain. Similar to the other works, we also highlight the interaction effects of LTV. However, contrary to the findings of So and Zheng (2003), our results reveal that the rate of deterioration of the supply chain performance due to an increase of LTV decreases under higher demand uncertainty. Similar to Chatfield et al. (2004), we also underline the interaction effect between LTV and a factor representing a supply chain initiative. While Chatfield et al. (2004) consider the initiative of information quality and information sharing

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across the supply chain, we consider the level of risk aversion of the supply chain. Our results show that lead time variability intensifies the impact of the risk aversion level on the expected opportunity cost. This same interaction effect also reveals an interesting observation. We find that the impact of LTV is not significant at some instances under low risk aversion level. 2.3.

Effects of lead time variability on operational supply chain decisions We review papers that examine the influence of lead time variability on the operational supply

chain decisions. Kouvelis and Tang (2012) examine how lead time variability would influence the decision of a retailer to ask for the shipment to be expedited. After observing the actual length of the first replenishment stage, the retailer may decide to ask the supplier to expedite his order to be received in a guaranteed period that is less than the expected value of the second stage. The authors develop a model to determine the optimal expediting policy. It is found that a more variable lead time would support a wider use of expediting. Heydari (2014b) studies the use of reducing lead time variability by a supplier, by using more reliable shipping means, as an incentive for the retailer to allow service level coordination. Hence, the retailer makes ordering decisions on the basis of a service level that maximizes the supply chain profit rather than its own profit. The results reveal that controlling LTV works well in coordinating the supply chain and results in increase profits for both the supplier and retailer. Sajadieh et al. (2009) incorporate a stochastic lead time in the joint economic lot sizing problem. The order quantity, reorder point and number of shipments are determined such that the total cost of both the buyer and vendor is minimized. The authors evaluate the performance of the joint model with a model where ordering decision is individual. They find that the joint ordering decision results in lower total cost and that savings achieved increase as LTV increases. To dampen the rise in inventory cost due to lead time variability, Heydari (2014a) proposes a coordination scheme based on which the supplier’s reorder point is determined in a

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manner that minimizes the supply chain cost. A coordination scheme is also used in the supply chain studied by Heydari, Mahmoodi and Taleizadeh (2016). In this paper, as variability of the upstream member’s lead time increases the probability of lead time aggregation increases, which forces the retailer to change his ordering policy to reduce his shortage costs. However, to ensure collaboration of upstream members, the retailer shares the cost savings and the supply chain as a whole benefits. Similar to Sajadieh et al. (2009), Li et al. (2012), Heydari (2014a), and Heydari, Mahmoodi and Taleizadeh (2016) the model developed in our study proposes a coordinated ordering scheme. However, our model is distinguished from the others as it entails two products. We determine the optimal flows of both the empty aluminum cans between the supplier and the brewery (upstream flow) and the beer filled cans between the brewery and the distribution center (downstream flow). We underline the changes in these two flows due to the variability of lead time. In this study, we make an important observation where variability of lead time has a higher impact on the upstream flow. Furthermore, our findings reveal changes in the risk hedging policies for the three levels of lead time variability experimented with. A comparative summary of the literature review summary on how lead time is incorporated into supply chain decision making is provided in Table 1. Table 1 is to be inserted here. 2.4.

Integrated operational and financial models As our model incorporates the integrated use of operational and financial instruments to manage

supply chain risks, the relevant literature also needs to be reviewed. As reported in Bandaly, Satir and Shanker (2014), the literature is rather sparse on models using interdisciplinary and integrated approaches in SCRM. For brevity, we refer the readers to the aforementioned article for detailed review on SCRM models. Our study differs from the reviewed articles on the types of risks which

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are addressed, as well as, on the risk management approach followed. In our model, we incorporate beer demand uncertainty, which leads to uncertainty in the demand for aluminum cans and aluminum sheets, as well as, commodity price risk in terms of aluminum price fluctuation. We manage these risks in an integrated and coordinated manner using inventory management techniques as well as, financial instruments such as options on aluminum futures. We present the problem setting and the model framework in the next section. We explain how lead time variability is incorporated and highlight the changes in the production plan and the product flows due to stochastic lead time. The model formulation is described in Section 4, where we explain the various components of the opportunity cost. Section 5 discusses the case of uncoordinated sequential decision making, where financial decisions are based on the operational decisions made. In Section 6, the simulation environment and the experimental results are discussed. The impact of lead time variability and managerial insights are provided. Concluding remarks are presented in Section 7. 3. The model 3.1.

Problem setting and model framework A brewery purchases aluminum cans from a can supplier, produces canned beer and then

transports it to a distribution center which maintains an inventory to meet retailers’ demand. The supply chain faces external risks which originate from both upstream and downstream. The can supplier faces aluminum price volatility causing fluctuations in packaging cost, while the distribution center faces uncertainty in beer demand causing either a shortage or a surplus in beer inventory. The supply chain also faces an internal risk that stems from uncertainty in the lead time in the supply of empty cans.

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The model developed aims to capture the benefits of integrating operational methods and financial instruments in managing the above risks. The model incorporates inventory levels of three items: canned beer at the distribution center, empty aluminum cans at the brewery and aluminum sheets at the can supplier. The model minimizes the expected total opportunity cost, E(TOC), of the supply chain as a whole, while maintaining the value at risk (VaR) of this cost within a predefined limit. The VaR limit is incorporated in the model as a constraint and its value depends on the level of risk aversion of the supply chain, to be collectively agreed upon by the supply chain members. Figure 1 presents the timeline of the supply chain activities pertinent to product flow. In the figure, ‘w’ is used to represent a week, ‘T’ is used to represent a time period that can span a number of weeks, and ‘t’ represents a point in time, that is, the beginning of a week. All decision variables and some parameters in the model are associated with inventory type and/or a point in time. For these variables and parameters, we use superscript i for the inventory type and subscript j for the point in time, where i = {a, b, c} denotes aluminum sheets, canned beer and empty cans, respectively, and j = {0, 1, …, 13}. Time t0 represents the current point in time at which the can supplier places the first order for aluminum sheets. This order quantity is necessary for operational hedging. . The notation used is provided in the appendix.

Figure 1. Timeline of supply chain activities.

3.2.

Supply chain risk management

3.2.1. Risk management using inventory and options on aluminum futures

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Faced with aluminum price variability and an uncertain demand for beer, the supply chain needs to make two strategic decisions on: i) the quantity of aluminum sheets to procure (Qa) and ii) the effective price to pay for the aluminum. At time t0, the can supplier purchases an initial quantity of aluminum

from the spot market at

the spot price of S0 per unit. This purchase is a hedge against future increases in the aluminum price. However, if the aluminum price were to decline in the future, then the supply chain would incur an opportunity cost, since by waiting to purchase aluminum on a later date, it could have done so at a lower price. At time t1, the can supplier purchases a second quantity of aluminum

from the spot

market at a spot price S1. The supply chain would incur an opportunity cost should the aluminum price increase. The purchase of aluminum in two batches reduces the total holding costs associated with holding aluminum sheets in inventory and allows time for the buyer to respond to price changes in the market place since time t0. To offset the opportunity costs associated with ), the can supplier buys at t0 a number Np (Nc) of European put (call) options on aluminum futures with a premium p0, an exercise price K and expiration date t1. The options are assumed to be at the money at purchase (F0 = K); F0 being the futures contract price at t0. It is also assumed that the delivery date of the underlying futures contract coincides with the options’ expiration date t1. The futures contract price F1 should be equal to S1, since the spot and futures price should converge on the futures contract’s delivery date. Table 2 summarizes the expected outcomes of the decisions regarding aluminum purchases and options. Decision Outcome at time t1 Decision at t0 If S1 < S0 (hence F1 < F0) If S1 > S0 (hence F1 > F0) Opportunity Cost ) Opportunity Cost ) Purchase at S0 Purchase Np Put options Options Payoff -Np(K-F1) Options Payoff 0 (1) ) Opportunity Cost ) Postpone the purchase of till t1 Opportunity Cost Purchase Nc Call options Options Payoff 0 (1) Options Payoff -Nc(F1-K) Table 2. Expected outcomes of decisions regarding aluminum purchases and options. (1) Options left unexercised

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3.2.2. Production plan and products flows process To manage the demand occurring over time span T1, the supply chain members maintain appropriate levels of the three inventory types in order to maximize the fill rate while minimizing holding costs. The lead times Lc and Lb are considered in scheduling production lots. Inventory flows are determined using pull logic with estimated beer demand as the starting point. Figure 2 illustrates information flow and product flows across the supply chain in order to fulfill the beer demand in week j. The figure also depicts the corresponding production quantities and the inventory levels at different stages of the supply chain.

pstream market

an supplier

rewery

istribution center

ownstream market

Information/Product Flow under Demand Uncertainty Information/Product Flow after Demand Realization

Figure 2. Typical production, information flow and product flows to fulfill demand in week j.

In anticipation of a weekly demand the brewery ships a quantity of beer at the beginning of the week. This shipment requires the production of a batch of beer which, in turn, requires the production of a batch of empty cans. The demand and production times are offset by the lead times. At each stage, the inventory is replenished at the rate of the inbound flow and depleted at the rate of outbound flow. The ending inventory at each stage is carried over to meet next week’s demand, subject to carrying cost. The actual demand is completely satisfied when inventory on hand is larger

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than the demand; otherwise a stockout occurs. Our model determines the optimal inventory levels by controlling the flows among the three inventory types of aluminum sheets, canned beer and empty cans. 3.3.

Incorporating the lead time factor

3.3.1. Variability of lead time We incorporate uncertainty in the lead time by considering a stochastic lead time with a discrete probability distribution. We consider three levels of lead time variability denoted as ‘high’, ‘moderate’ and ‘low’. The high variability corresponds to the case to which we assign probabilities of 0.464, 0.214, 0.179 and 0.143 to lead time durations of 3.5, 4, 4.5 and 5 weeks, respectively. In the case of moderate variability, the above probabilities are assigned to lead time durations of 3.625, 4, 4.375, and 4.75 weeks, respectively. For low variability, 3.75, 4, 4.25 and 4.5 weeks are assigned the same probabilities, respectively. Our choice for this probability distribution is mainly based on actual supply conditions observed when writing the case study on the beer company’s supply chain (Kira, Satir and Bandaly 2013). It is important to note here that the above lead time durations represent time of delivery from the time an order is placed. The durations include both production and shipment. For orders that are not completed on time, finished cans are held at the supplier’s facility until the next delivery time. This arrangement is necessary to ensure full-truck-load shipments. A load includes around 200,000 cans packed in pallets. While variable lead time is commonly described by normal distribution in the literature, (De Sensi, Longo, and Mirabelli 2008; Acar, Kadipasaoglu, and Schipperijn 2010; Osman and Demirli 2012; Hoque 2013, Fang et al. 2013, Heydari, Mahmoodi and Taleizadeh 2016), discrete probability distribution is also used to describe lead time variability (for example, Dolgui and Ould-Louly 2002; Hnaien, Delorme and Dolgui 2010). In our model, the average lead time at both levels is four weeks, which is the same

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deterministic duration used in the base model. This facilitates a fair comparison between the two models. The standard deviations of the discrete probability distributions with high, moderate and low variability are 0.551, 0.413 and 0.275 week, respectively. These values were derived from the variations in lead times of empty cans supply, based on insights we obtained from the aforementioned case study. 3.3.2. Modeling the stochastic lead time duration To introduce lead time variability into our model, we make a number of assumptions that are specific to the conceptual background of our model. These assumptions are necessary to justify our calculations of the relevant costs. First, we assume that the cans are produced in a continuous process with a production rate that is constant for each batch of cans ordered for the brewery’s consumption in a specific week. Second, the variability of the lead time duration stems from a change in this production rate (for example due to increased capacity allocation for another customer) and not from a delay in the production start time or a disruption in the production process. This assumption allows us to determine the quantity of cans produced at the end of the four weeks duration (the expected completion date) in a proportional manner as we will discuss in the following section. Third, we assume that shipments from the can supplier to the brewery are made once a week. This means that any unfinished portions of an order will be shipped with the next week’s order. Finally, we assume that no early shipments are allowed. This means that a batch of cans that is completed earlier than the expected delivery date will remain at the supplier’s premises until the agreed shipping date. 3.3.3. Impact of lead time variability on the SCM base process The introduction of stochastic lead time duration has an impact on the production plan and product flows process (hereafter referred to as SCM base process) discussed in Section 3.2.2 and

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depicted in Figure 2. In the SCM base process the quantity of aluminum cans shipped to the brewery (

) is equal to the quantity of a production lot that started four weeks earlier (

is, the planned production quantity ( of cans (

). That

) is actually produced and the planned shipment quantity

) is thus actually delivered. However, with stochastic lead time the actual produced

quantity of cans, (Pc)actual, may be less than the planned production quantity, Pc, caused by a lead time duration longer than four weeks. Consequently, the actual quantity of cans shipped to the warehouse may be less than the planned quantity (Qc). Under this new situation, the brewery places an order with the can supplier for a quantity of cans (

) that needs to be received at the beginning

of week wj. The can supplier starts producing this planned quantity, Pc, four weeks before the expected delivery time. In the event that the lead time duration, represented by X, is longer than the expected four weeks, only a proportion of the ordered quantity would be ready for shipment. This proportion is equal to Pc(4/X). The remaining balance that is still in production will be shipped with the batch produced for the next week wj+1. In the event that the lead time duration is shorter than four weeks, the supplier holds all the produced quantity and delivers it, as scheduled, at the beginning of week wj. 4. Model formulation The integrated risk management model solves for the decision variables (

,

, Nc, Np,

and

) in order to minimize the present value of expected total opportunity cost E(TOC) along the supply chain, while meeting, among others, the upper-bound constraint related to the value-at-risk of TOC (VaR). 4.1. Costs in the first time span (T0) The present value of opportunity cost associated with initial inventories at time t0 is given by:

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̃

)

(1)

where, r represents the weekly risk-free rate of return and f is an equivalence factor that converts aluminum tons into millions of cans. In (1) and all formulations that follow,

and

are the

weekly costs of carrying a quantity of inventory of type i associated with aluminum sheet quantities purchased at times t0 and t1 respectively. The second term captures the present value of the cost of carrying

over the time span from t0 to t1.

The opportunity cost (gain) associated with ̃

is given by:

)

(2)

The cost (gain) associated with the purchase of put options is given by: (

̃ )

(3)

The cost (gain) associated with the purchase of call options is given by: (̃

)

(4)

where, hop is the weekly holding cost associated with put and call options. The first two terms in each of (3) and (4) represent the premium paid for the options and the corresponding holding costs. The third term in (3) and (4) represents the present value of the payoff on the expiration date from the put and call options, respectively. 4.2. Costs in the second time span (T1) The present value of the stockout costs over an eight-week beer demand period is given by: ∑

(

)

)

(5)

Equation 6 represents the present value of the cost of carrying a surplus quantity of the corresponding inventory type. This surplus is determined by the weekly ending inventory.

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∑

)

) ́ )

∑

(

́

́ )

)

)

∑

(

́

(6)

where, u0 and u1 are the proportions of aluminum sheet quantities purchased at time t0 and t1, respectively. The unit inventory holding cost has two components,

and

, that are proportional

to the purchase price, S0 and S1, respectively. The contribution of each component is then weighted by u0 and u1.

As units of empty cans and canned beer move downstream, warehousing

requirements become more stringent and consequently unit holding costs increase. The model incorporates this increase in holding costs by setting ́

and ́

.

Equation (7) represents the present value of the holding cost associated with carrying the surplus quantity of beer during the production phase for the whole lead time period. In other words, this holding cost is the cost of insurance against uncertain demand. We only include the carrying cost corresponding to this surplus quantity, and not to the whole production lot, to be consistent with our definition of the opportunity cost. All the components of the opportunity cost penalize the supply chain for the deviations from ‘perfect’ decisions. Such decisions can be made only if ’perfect’ information on demand quantity and aluminum price is known a priori, which is not the case in reality. ∑

(

)

(

)

)

(7)

The cost of carrying surplus quantity of empty cans during the production phase is similar to the above, but this cost also includes an additional component that captures the uncertainty in lead time. As explained in Section 3.3.3, when lead time is longer than the planned 4 weeks only a proportion of the ordered quantity will be delivered and a balance quantity remains in production (BIP). Subsequently, the cost of carrying BIP is added to the total opportunity cost calculations. This cost is perceived as the cost of insurance against an uncertain lead time in supplying the cans. To

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determine this cost for every production week, we compute the actual quantity produced and the remaining balance quantity still in production. These quantities are computed as follows:

(P c )actual

~c c   P if L  4   c ~c  4 P ~c if L  4     L

 

(8)

The balance in production is then given by: BIP = Pc – (Pc)actual

(9)

Substituting the value of (Pc)actual in (9) by the relevant values from (8), the balance in production will be determined as follows. ~ 0 if Lc  4  BIP   c ~  c 4 P 1  L~c if L  4 





(10)

As

justified above, the cost of carrying this balance for four weeks is added to the cost of carrying the quantity produced in surplus. These two costs are calculated in (11). ∑

(

)

{(

4 ~c ) L

)

})

)

(11)

Equations (12) and (13) ensure that the final ending inventory is carried over to the next planning period. (12) (13) 4.3. Objective function The objective of our model is to optimize the risk management performance of the supply chain by minimizing the expected total opportunity cost E(TOC) along the supply chain, where the TOC is the summation of equations (1) through (7) and (11).

Min E(TOC)

(14)

4.4. Constraints 18 of 39

The following constraints are used in formulating the model. (15) (16) Constraints (15) and (16) ensure that the beginning aluminum sheets inventory in the second time period T1 equals the sum of the quantities of aluminum purchased at time t0 and t1. ̃

{

}

(17)

VaR  v

(18)

Constraint (18) captures the degree of risk aversion within the supply chain. The value of the upper bound v on the value at risk (VaR) of the total opportunity cost TOC is a function of the risk management policy to be collectively determined by the supply chain members. For lack of space, we omit in this paper the formulations of other constraints that: i) ensure transfer of inventories remaining at the end of one week to the next week, ii) ensure inventory flow conservation every week for the inventories of aluminum sheets, empty cans and beer, and iii) set upper limits for the decision variables due to operational and financial restrictions. 5. Sequential model The integrated model presented makes use of a coordinated decision-making approach based on which operational and financial risk management decisions are made simultaneously. Such a coordinated approach is not widely used by firms in practice. Instead, different functional areas make operational risk management decisions and financial risk management decisions independently. In order to compare the performances of coordinated and uncoordinated approaches, we present this latter approach as a sequential model that consists of two sub-models: i) the operational risk management sub-model and ii) the financial risk management sub-model. The operational sub-model is a replicate version of the integrated model with the exclusion of the

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financial variables and costs. Using the same problem parameters and probabilistic inputs used in the integrated model, the operational sub-model solves for all the decision variables in the integrated model, with the exception of the number of put and call options, Np and Nc. The optimal values of the decision variables obtained in the operational sub-model are then entered as fixed parameters in the financial risk management sub-model that solves for Np and Nc to minimize the expected total opportunity cost. The optimal values of the decision variables associated with the sequential model are the values optimized by the operational sub-model and then by the financial risk management sub-model. Hence, it is important to note that for the experimental design and statistical analyses that follow, the performance of the sequential model is measured by the expected total opportunity cost obtained by the financial risk management sub-model, the final stage in a two-stage solution procedure. 6. Experimental design, simulation environment and findings 6.1.

Experimental design The experimental design involves four factors: : i) value-at-risk (VAR), ii) standard deviation of

demand (SDD), iii) aluminum price volatility (APV) and iv) lead time variability (LTV). The first three factors are represented at two levels and the LTV factor is represented at three levels. Experimental design factors and levels are presented in Table 3. Treatments are created from all possible permutations of these factors. We find the optimal solutions for these treatments. To examine the impact of the lead time variability on the supply chain performance, we compare the results of the extended model with the results of the base model. We also compare the solutions in the extended model between the integrated approach and the sequential approach. These comparisons allow us to observe the effects of lead time variability on the operational hedging strategy and the product flows.

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Factor

Designation

Code

Value-at-risk VAR A Demand uncertainty SDD B Aluminum price volatility APV C Lead time variability LTV E Table 3. Experimental design factors and levels.

6.2.

L 1.5 3.8 (21.3 , 20.3) 0.275

Level M N/A N/A N/A 0.413

H 1.8 4.5 (28.8 , 27.4) 0.551

Units Million dollars Million cans % Weeks

Simulation environment The extent of the supply chain risk management system in our study, in terms of various

uncertainties and the nature of decision variables, renders the problem to be rather complex to be solved analytically. We, therefore, employ a simulation-based optimization technique to optimize the performance of the supply chain. To this end, @RISK, a risk management software which is part of the Decision Tools Suite provided by Palisade, was used. Using three levels for the LTV factor and two levels for each of the other three factors, we identify 24 treatment combinations for each of the three models (operational, financial and integrated) for a total of 72 model versions. To compare the effects of the various treatment combinations, we determine for each of these model versions the minimum expected total opportunity cost E(TOC). This cost is the response variable that we use to compare the effects of treatment combinations. We use @RISK to determine the values of the decision variables that minimize E(TOC) under the relevant constraints. Starting with initial values of the decision variables, the optimization involves running a large number of simulations. Each simulation consists of 10,000 iterations. In each iteration random values of the probabilistic inputs (S1, F1, Lc, and dj) are generated and used in the calculation of the expected total opportunity cost. The software uses genetic algorithms to find new solutions that improve the value of the objective function. Using the optimal solution found for the decision variables, we run a number of simulations as replications on each of the 72 model versions and record the values of E(TOC). These values

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represent the response variable in the replications for each treatment combination in the experimental design. The results are presented in Table 4. Table 4 is to be inserted here. 6.3.

Impact of lead time variability In this section we discuss the significant changes attributed to lead time variability in the

performance of the supply chain studied. 6.3.1. Increase in opportunity cost While it is expected that lead time variability (LTV) would increase the opportunity cost, our results reveal that this impact may not be significant under certain conditions. This observation is in line with the findings of Fang et al. (2013) who demonstrate an increasing marginal savings in inventory costs with a reduction of LTV, but, on the other hand, show that this would not be the case when lead time mean and lead time variance are correlated. It is important to note here that the task of reducing LTV calls for strong collaboration among the supply chain members. The brewery needs to share demand forecast information with the can supplier to facilitate the supplier’s production schedule. In turn, the can supplier needs to show commitment to comply with the fluctuating weekly quantities. It is evident that there is a cost associated with LTV reduction. Hence, the benefits of this reduction should outweigh the cost to justify actions of supply chain members. This argument complies with the findings presented in a number of papers in which the authors prove that reduction of lead time is not always beneficial (e.g. de Treville et al. 2014 and Li, Ye and Lin 2015). We compare the opportunity costs of the integrated model with lead time variability incorporated (extended model) with the opportunity costs in the corresponding treatments of the integrated model without lead time variability (base model). Table 5 shows the percentage increase

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in the expected total opportunity cost when lead time variability is incorporated. This increase, however, is statistically not significant in some instances. While in all the treatments with VAR equal to 1.5, except one, the increase in the expected opportunity cost is significant, this is not always the case when VAR is 1.8. Under this lower risk aversion level, only high or moderate lead time variability significantly increases the opportunity cost. Moreover, the findings show that, in the case of VAR 1.8, the increase in E(TOC), when lead time variability increases, is higher when demand uncertainty is lower. This could be explained by the fact that actions taken to manage high demand uncertainty have simultaneous effects on LTV, which is not the case for lower demand uncertainty. When lead time is deterministic, the brewery purchases more empty cans when demand uncertainty is higher (see Figure 3). While this additional quantity is used as a buffer against unexpected increase in demand, it also alleviates the shortage risk due to unexpected increase in lead time. VAR: 1.5 VAR: 1.8 LTV: M LTV: M LTV: L LTV: H LTV: L LTV: H 11.9%* 5.4% APV: L 5.3%* 14.5%* 3.8% 9.1%* SDD: 3.8 6.0%* 7.6%* APV: H 0.8% 11.2%* 4.0% 8.2%* 9.1%* 5.5%* APV: L 7.2%* 11.6%* 3.5% 5.9%* SDD: 4.5 10.5%* 5.2%* APV: H 5.4%* 11.1%* 3.0% 5.6%* Table 5. Percentage increase in E(TOC) in presence of lead time variability (extended vs base (no LTV) models). * Statistically significant at 0.05 significance level

6.3.2. Overall superiority of the integrated model over the sequential model In this section, we study the impact of lead time variability on the superiority of the integrated model over the sequential model. Table 6 depicts the percentage difference in the expected opportunity cost between the integrated model and the sequential model in the presence of lead time variability. It is important to note that the integrated model requires high collaboration among operational and financial departments of the supply chain members. Thus, significant benefits should accrue to justify such collaboration.

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VAR: 1.5 VAR: 1.8 LTV: M LTV: M LTV: L LTV: H LTV: L 3.0% 0.7% APV: L 4.3%* 2.8% 0.4% SDD: 3.8 1.6% 8.8%* APV: H 3.4%* 1.4% 8.8%* 0.9% 9.4%* APV: L 1.1% 0.0% 9.9%* SDD: 4.5 2.7%* 10.0%* APV: H 3.6%* 2.0%* 10.7%* Table 6. Percentage difference in E(TOC) with lead time variability (sequential vs integrated models). * Statistically significant at 0.05 significance level

LTV: H 1.6% 8.8%* 9.1%* 9.0%*

We compare the superiority of the integrated model for different treatments and for different lead time variability within the extended model. In most cases of lead time variability, the integrated model has a significant higher performance. Moreover, we notice that the degree of superiority changes with the level of lead time variability. The superiority of the integrated model is higher under lower lead time variability than it is under higher lead time variability. 6.3.3. Operational hedging strategy In this section, we study the impact of lead time variability on the operational hedging strategy. Table 7 shows the ratio (u0) of the quantity of aluminum sheets purchased at t0 over the total quantity purchased over the period T0. This ratio reflects the extent of hedging against aluminum price increases that the supply chain executes using the operational approach. The range of percentages in each cell of the table encompasses values of u0 at the two levels of APV for each treatment. VAR: 1.5 VAR: 1.8 Integrated Sequential Integrated Sequential LTV: L 26% 29% 7-10% 8-18% SDD: 3.8 LTV: M 31-34% 33-34% 9-11% 7-22% LTV: H 36-37% 34-36% 11% 8-26% LTV: L 34% 36-41% 11% 20-28% SDD: 4.5 LTV: M 35-39% 37-43% 11% 20-28% LTV: H 38-39% 37-43% 11% 20-28% Table 7. Ratio (u0) of aluminum sheets purchased at t0 to total purchased quantity.

First, we observe, within the extended model treatments, how the operational hedging strategy varies at the three levels of lead time variability. We note that higher lead time variability increases u0 for both the integrated and sequential models. However, this observation is valid mainly for the

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case of a high risk aversion level (VAR 1.5). Second, we compare the hedging strategy between treatments in the extended model and the corresponding treatments in the base model. We also observe that in the presence of lead time variability, only a more risk-averse supply chain uses more operational hedging. 6.3.4. Change in product flows across the supply chain An important change in the model performance in the presence of lead time variability is related to product flows across the supply chain. In the base model, the can supplier converts all the aluminum quantity purchased (Qa) into cans and ships them to the warehouse (Qc). The brewery fills all these cans with beer and sends them to the distribution center (Qb). That is, Qa = Qb = Qc. On the other hand, under a stochastic lead time, a larger quantity of aluminum is purchased and converted into cans. Such an action is expected to mitigate against shortages of aluminum cans due to delays in shipment of a proportion of the ordered quantity. However, a portion of these empty cans is left in the warehouse unused. That is, Qa = Qc > Qb. Such a situation is justified due to higher carrying cost of beer. Figure 3 depicts the product flows in the base and extended models under different treatments. In each treatment, the first column represents the quantities flowing in the base model, the second and third columns represent the flows in the extended model with low lead time variability, and the fourth and fifth columns represent the flows in the extended model with high lead time variability. In the base model Qa, Qb and Qc are equal and thus represented in a single column. In the extended model, Qa and Qc are equal and represented in single column, while Qb is represented in a separate column. To make the figure more readable and allow for easier comparison among the treatments, the columns pertinent to the moderate LTV are not shown. The latter do not add any new perspective to the below analysis.

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High LTV: Qb

High LTV: Qa = Qc

Low LTV: Qb

Low LTV: Qa = Qc

Base Model: Qa = Qb = Qc

High LTV: Qb

High LTV: Qa = Qc

Low LTV: Qb

Low LTV: Qa = Qc

Base Model: Qa = Qb = Qc

High LTV: Qb

High LTV: Qa = Qc

Low LTV: Qb

Low LTV: Qa = Qc

Base Model: Qa = Qb = Qc

High LTV: Qa = Qc

High LTV: Qb

175

Low LTV: Qb

180

Low LTV: Qa = Qc

Qty (million cans)

185

Base Model: Qa = Qb = Qc

190

170 165 160 T1 Lower Demand Uncertainty

T2 Higher Demand Uncertainty

Higher Risk Aversion Level

T3 Lower Demand Uncertainty

T4 Higher Demand Uncertainty

Lower Risk Aversion Level

Figure 3. Product flows in the base and extended models.

A number of observations can be made regarding product flows. i) When the risk aversion level is low, a lower product flow occurs across the supply chain. ii) Under the same risk aversion level, a larger flow is observed when the demand variability increases. iii) Higher lead time variability requires a larger quantity of empty cans to be dispatched to the warehouse (in Figure3, column #4 vs column #2). However, this variability has a lower impact on the quantity of beer moved to the distribution center (in Figure 3, column #3 and column #5 having the same volume). 6.3.5. Statistical analysis The previous analyses are based on optimization results of various versions of the models in which the factors are set at discrete values, as summarized in Table 3. Examining the change in the

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response variable, E(TOC), at other values of the factors would, however, provide more insights about the model behavior. For this purpose, we conduct a factorial analysis on the extended model using the experimental design software Design Expert®. The four factors are coded as follows: value at risk (A), demand uncertainty (B), aluminum price volatility (C) and lead time variability (E). In addition to these four factors, the model factor (coded as D) is incorporated in the analysis as a categorical factor with two values: integrated and sequential. The software generates a linear regression model that explains the variations in the response variable and can be used to predict the value of the response variable for any combination of the factors within their corresponding lower and upper levels. The software generates diagrams that depict these variations as well as interaction effects of the factors. Figures 4 to 6 depict examples illustrating interaction effects of two factors on the changes in E(TOC) while the other three factors are kept fixed. Considering the scope of this study, we focus on interaction effects involving LTV. As the interaction effect between LTV and APV is found not to be significant, we omit it from the following discussion. Figure 4 illustrates the decline in E(TOC) as VAR varies from $1.5 million to $1.8 million, when demand uncertainty and aluminum price volatility are fixed at the indicated levels in the integrated model. The rate of this decline is, however, a function of the lead time variability level. The decline is $ 78,000 at the low level of lead time variability and $ 118,000 at the high level of lead time variability. This figure allows us to conclude that, for the supply chain studied, lead time variability intensifies the impact of the risk aversion level on the expected opportunity cost. That is, with higher lead time variability, the improvement in the expected opportunity cost becomes more pronounced when the risk aversion level is lower.

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SDD: 3.8, APV: H, Integrated E: LTV

710

E (T O C )

678

646

High LTV 614

Low LTV

582

550 1.50

1.58

1.65

1.72

1.80

A: VAR

Figure 4. Interaction effect of VAR and LTV.

In contrast, an increase in lead time variability diminishes the effect of demand uncertainty on the expected opportunity cost. Figure 5 illustrates an example of such a reduction for the integrated model. Under a high risk aversion level and low aluminum price volatility, when the demand standard deviation increases from 3.8 million cans to 4.5 million cans, the increase in the expected opportunity cost is largely a function of lead time variability level. This increase is $ 180,000 at a low level of lead time variability and $ 159,000 at a high level of lead time variability. The milder slope of E(TOC) line when LTV is high renders the gap between the two lines smaller as SDD increases, as depicted in Figure 5.

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VAR: 1.5, APV: L, Integrated E: LTV

820

E (T O C )

776

High LTV

732

688

Low LTV

644

600 3.80

3.90

4.00

4.10

4.20

4.30

4.40

4.50

B: SDD

Figure 5. Interaction effect of SDD and LTV.

Finally, we observe the variations in E(TOC) as LTV changes between the lowest and highest levels. This is depicted in Figure 6. As expected, the opportunity cost decreases as LTV decreases. However, the rate of this decrease depends on the model used. We notice that the impact of the lead time variability on the expected opportunity cost is higher in the integrated model than in the sequential model. Under a high risk aversion level, coupled with a low demand uncertainty and high aluminum price volatility, the decrease in the expected opportunity cost (as the lead time variability decreases) is $ 64,000 in the integrated model and $ 47,000 in the sequential model.

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VAR: 1.5, SDD: 3.8, APV: H D: Model

710

E (T O C )

692

Sequential

674

656

Integrated 638

620 0.28

0.34

0.41

0.48

0.55

E: LTV

Figure 6. Interaction effect of LTV and model type.

7. Managerial Insights The above results provide practitioners with important insights that support their decision making process. We first underline implications on the general decision making approach. The superiority of the integrated model (centralized decisions) over the sequential model (de-centralized) assures practitioners that even in the presence of a risk which is traditionally managed by operational approaches, the integrated approach still proves to be superior. A supply chain would achieve higher benefits by reducing the variability of lead time when it adopts the centralized approach. Moreover, the results reveal that higher volumes of products would flow across the supply chain under lead time uncertainty. In our study, mitigation of lead time variability mainly takes place at the upstream side of the supply chain by increasing the quantities of empty cans at a much higher rate than the increase in the quantity of finished product by the brewery. More specific insights are drawn from results highlighting interaction effects of LTV with other factors. For example, when lead time is variable a supply chain could end up managing less volumes of product flow (figure 3) and would not significantly change its operational hedging

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policy (table 7) when the managers are less risk-averse. Moreover, the benefits of reducing the variability of lead time vary with the risk-taking behavior. Table 5 shows that for risk-averse managers, a reduction of lead time variability from a given level to a lower level yields a substantial improvement in performance. For risk takers, on the other hand, the change in performance resulting from a reduction in lead time variability is small and may not justify the efforts needed. It is very interesting to note here that, for these risk-takers, even the benefits of eliminating LTV may not be justified in cases of low variability. Another implication can be drawn from this same interaction effect. As depicted in figure 4, a manager willing to take higher risk is rewarded with higher performance improvement when variability of lead time is larger. Insights are also drawn from results emphasizing interaction effect of LTV with demand uncertainty. Observations in figure 5 allow us to conclude that it less compelling for the supply chain studied to work on reducing the lead time variability when operating under high demand uncertainty. This can be explained by the fact that the action of the supply chain in response to an increase in demand uncertainty would also mitigate the effects of lead time variability. When demand uncertainty increases, the supply chain increases the beer quantity in the distribution center. Such an increase necessitates a corresponding increase in the quantity of empty cans. The latter increase would contribute towards mitigating the higher variability in lead time of cans’ supply. 8. Conclusion In this article, a simulation-based optimization method is used to optimize the performance of a complex supply chain system. Through experimental design and statistical analyses of this system, we study the changes in the product flows due to variability in the lead time involved in supplying empty cans to the brewery. We also examine the impact of stochastic lead time on the expected opportunity cost and on the hedging decisions. In this extension of the base model previously

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developed in Bandaly et al. (2014), we modify the four-week fixed lead time to supply empty cans to the brewery to a stochastic lead time which follows discrete probability distributions with a mean lead time of four weeks. Based on experimental findings, we make a number of observations for the supply chain studied. While it is expected that lead time variability increases the opportunity cost, the results reveal that this increase may not be significant under certain conditions. For example, under a low risk aversion level, only high lead time variability would significantly increase the opportunity cost. The results also reveal that the increase in the expected opportunity cost is higher when demand uncertainty is lower. We explain this observation by the fact that actions taken to manage high demand uncertainty would simultaneously mitigate risk stemming from lead time variability. Given that the risk associated with a stochastic lead time is traditionally managed with operational tools, it is important to note that the integrated model is found to outperform the sequential model under lead time variability. The superiority of the integrated model is, however, not influenced by the lead time variability level when the supply chain is less risk averse As for the impact of lead time variability on the operational hedging decisions, we find that in both the integrated and the sequential models, a more risk averse supply chain would use operational hedging more as the lead time variability increases. This impact is not found to be significant in the case of being less risk averse. The results show substantial impact of lead time variability on the volume of products that flow across the supply chain. In the context of our study, the risk associated with lead time variability is mitigated mainly by an increase in the flow of empty cans, rather than an increase in the flow of the final product (beer). The potential positive impact of reducing lead time variability on the performance of a supply chain calls for initiatives to achieve such a reduction. As the contributions of the different members of the

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supply chain in these initiatives may not be of equal weight, the challenge would then be to align these efforts with the gained benefits. For example, our results show that mitigation of LTV necessitates a larger change of product flow on the upstream side than on the downstream. A future research may complement this finding by developing a supply chain coordination contract that quantifies the contributions of the different members and ensures proportional distribution of the benefits originating from such coordination. Acknowledgment The authors would like to thank the anonymous reviewer for his constructive and helpful comments. His valuable comments helped to improve the quality of this paper. References Acar, Y., Kadipasaoglu, S., Schipperijn, P., 2010. A Decision Support Framework for Global Supply Chain Modelling: An Assessment of the Impact of Demand, Supply and Lead-Time Uncertainties on Performance. International Journal of Production Research 48 (11), 3245-3268. Agrawal, S., Sengupta, R.N., Shanker, K., 2009. Impact of Information Sharing and Lead Time on Bullwhip Effect and on-Hand Inventory. European Journal of Operational Research 192 (2) 576593. Bandaly, D., Satir, A., Kahyaoglu, Y., Shanker, L., 2012. Supply chain risk management– II: A review of operational, financial and integrated approaches. Risk Management 14 (4), 249-271. Bandaly, D., Satir, A., Shanker, L., 2014. Integrated Supply Chain Risk Management via Operational Methods and Financial Instruments. International Journal of Production Research 52 (7), 2007-2025. Cannella, S., Bruccoleri, M., Framinan, J.M., 2016. Closed-loop supply chains: What reverse logistics factors influence performance? International Journal of Production Economics, 175, 35-49. Chaharsooghi, K.S., Heydari, J., 2010. LT variance or LT mean reduction in supply chain management:Which one has a higher impact on SC performance? International Journal of Production Economics, 124, 475-481. Chatfield, D.C., Kim, J.G., Harrison, T.P., Hayya, J.C., 2004. The Bullwhip Effect-Impact of Stochastic Lead Time, Information Quality, and Information Sharing: A Simulation Study. Production and Operations Management 13 (4), 340-353. Chen, F., Drezner, Z., Ryan, J.K., Simchi-Levi, D., 2000. Quantifying the Bullwhip Effect in a Simple Supply Chain: The Impact of Forecasting, Lead Times, and Information. Management Science 46 (3), 436-443. Chopra, S., Reinhardt, G., Dada, M., 2004. The Effect of Lead Time Uncertainty on Safety Stocks. Decision Sciences 35 (1), 1-24.

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Christensen, W.J., Germain, R.N., Birou, L., 2007. Variance Vs Average: Supply Chain Lead-Time as a Predictor of Financial Performance. Supply Chain Management 12 (5), 349-357. de Treville, S., Schürhoff, N., Trigeorgis, L., Avanzi, B., 2014. Optimal sourcing and lead-time reduction under evolutionary demand risk. Production and Operations Management, 23 (12), 21032117. De Sensi, G., Longo, F., Mirabelli, G., 2008. Inventory Policies Analysis Under Demand Patterns and Lead Times Constraints in a Real Supply Chain. International Journal of Production Research 46 (24), 6997-7016. Dolgui, A., Ould-Louly, M.A., 2002. A model for supply planning under lead time uncertainty. International Journal of Production Economics, 78 (2), 145-152. Fang, X., Zhang, C., Robb, D.J., Blackburn, J.D., 2013. Decision support for lead time and demand variability reduction. Omega, 41(2), 390-396. Hammami, R., Frein, Y., 2013. An Optimisation Model for the Design of Global Multi-Echelon Supply Chains Under Lead Time Constraints. International Journal of Production Research 51 (9), 2760-2775. Heydari, J., 2014a. Coordinating Supplier's Reorder Point: A Coordination Mechanism for Supply Chains with Long Supplier Lead Time. Computers & Operations Research 48, 89-101. Heydari, J., 2014b. Lead Time Variation Control using Reliable Shipment Equipment: An Incentive Scheme for Supply Chain Coordination. Transportation Research. Part E, Logistics & Transportation Review 63, 44-58. Heydari, J., Mahmoodi, M., Taleizadeh, A., 2016. Lead time aggregation: A three-echelon supply chain model. Transportation Research. Part E, Logistics & Transportation Review 89, 215-233. Heydari, J., Kazemzadeh, R.B., Chaharsooghi, K.S., 2009. A study of lead time variation impact on supply chain performance. International Journal of Advanced Manufacturing Technology 40, 1206– 1215. Hlioui, R., Gharbi, A., Hajji, A., 2015. Replenishment, production and quality control strategies in three-stage supply chain. International Journal of Production Economics 166, 90-102. Hnaien, F., Delorme, X., Dolgui, A., 2010. Multi-Objective Optimization for Inventory Control in Two-Level Assembly Systems Under Uncertainty of Lead Times. Computers & Operations Research 37 (11), 1835-1843. Hoque, M.A., 2013. A Vendor-Buyer Integrated Production-Inventory Model with Normal Distribution of Lead Time. International Journal of Production Economics 144 (2), 409-417. Humair, S., Ruark, J.D., Tomlin, B., Willems, S.P., 2013. Incorporating Stochastic Lead Times into the Guaranteed Service Model of Safety Stock Optimization. Interfaces 43 (5), 421-434. Isaksson, O.H.D., Seifert, R.W., 2016. Quantifying the bullwhip effect using two-echelon data: A cross-industry empirical investigation. International Journal of Production Economics 171, 311-320. Kira, D., Satir, A., Bandaly, D., 2013. Financial and Operational Risk Management at Molson Coors. In The Supply Chain Management Casebook: Comprehensive Coverage and Best Practices in SCM, edited by Munson, C., 50-67. Financial Time Press: New Jersey.

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Kouvelis, P., Tang, S.Y., 2012. On Optimal Expediting Policy for Supply Systems with Uncertain Lead-Times. Production and Operations Management 21 (2), 309-330. Li, Y., Ye, F., Lin, Q., 2015. Optimal lead time policy for short life cycle products under conditional value-at-risk criterion. Computers & Industrial Engineering 88, 354-365. Li, Y., Xu, X., Zhao, X., Yeung, J.H.Y., Ye, F., 2012. Supply Chain Coordination with Controllable Lead Time and Asymmetric Information. European Journal of Operational Research 217 (1), 108119. Liao, T.W., Chang, P.C., 2010. Impacts of Forecast, Inventory Policy, and Lead Time on Supply Chain Inventory - A Numerical Study. International Journal of Production Economics 128 (2), 527537. Lee, H.L., Padmanabhan, V., Whang, S., 1997. Information Distortion in a Supply Chain: The Bullwhip Effect. Management Science, 43 (4), 546-558. Osman, H., Demirli, K., 2012. Integrated Safety Stock Optimization for Multiple Sourced Stockpoints Facing Variable Demand and Lead Time. International Journal of Production Economics 135 (1), 299-307. Sadeghi, A., 2015. Providing a measure for bullwhip effect in a two-product supply chain with exponential smoothing forecasts. International Journal of Production Economics, 169, 44-54. Sajadieh, M.S., Jokar, M.R.A, Modarres, M., 2009. Developing a coordinated vendor–buyer model in two-stage supply chains with stochastic lead-times. Computers and Operations Research 36, 2484-2489. Simchi-Levi, D., Zhao, Y., 2005. Safety Stock Positioning in Supply Chains with Stochastic Lead Times. Manufacturing & Service Operations Management 7 (4), 295-318. So, K.C., Zheng, X., 2003. Impact of supplier's lead time and forecast demand updating on retailer's order quantity variability in a two-level supply chain. International Journal of Production Economics 86 (2), 169-179. Tang, C., 2006. Perspectives in supply chain risk management. International Journal of Production Economics 103 (2), 451-488. Wang, X., Disney, S.M., 2016. The bullwhip effect: Progress, trends and directions. European Journal of Operational Research, 250 (3), 691-701.

Appendix – Notation used in modeling

c0 dj

: level of inventory type i at the beginning of week wj : premium price at t0 of a call option : demand for beer during week wj (in millions of cans)

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f F0 F1 ́ hop K Lb Lc Nc Np p0

Qa

r s S0 S1 T tj v VaR wj

: level of inventory type i at the end of week wj : factor converting aluminum tons into millions of cans : price at time t0 of aluminum futures with delivery date that follows t1 : price at time t1 of aluminum futures with delivery date that follows t1 : weekly cost of carrying inventory of type i associated with aluminum sheet purchased at times tj ($/million cans) : weekly holding cost of inventory type i, as it moves downstream ($/million cans) : weekly holding cost of put and call options : exercise price of the put and call options : lead time to replenish beer inventory : lead time to replenish cans inventory : quantity of beer distributed in the market during week j : number of call options on aluminum futures with delivery date that follows t1 : number of put options on aluminum futures with delivery date that follows t1 : premium price at t0 of a put option : quantity of cans being filled and packed by the brewery during week wj (in millions) : quantity of cans being produced by the supplier during week wj (in millions) : total quantity of aluminum sheets purchased in period T0 (in tons) : quantity of aluminum sheets purchased at time t0 (in tons) : quantity of aluminum sheets purchased at time t1 (in tons) : quantity of beer cans shipped to the distribution center at time tj (in millions) : quantity of aluminum cans shipped by the can supplier at time tj (in millions) : weekly interest-free interest rate : stockout cost due to loss of beer sales ($/unit of unsatisfied demand) : aluminum spot price at time t0 : aluminum spot price at time t1 : period of time where T {T0, T1} : point of time where j = {0, 1, …, 13} : value set by the supply chain as a limit for the VaR : value at risk of the total expected opportunity cost : week starting at time j

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Sajadieh et al. (2009) Heydari et al. (2009) Acar et al. (2010) Chaharsooghi & Heydari (2010) Kouvelis & Tang (2012) Fang et al. (2013) Heydar (2014a) Heydari (2014b) Heydari, Mahmoodi & Taleizadeh (2016)

Simchi-Levi & Zhao (2005)

Chopra et al. (2004)

Chatfield et al. (2004)

So & Zheng (2003)

Cannella, Bruccoleri & Framinan (2016)

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Sadeghi (2015)

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Agrawal, Sengupta & Shanker (2009)

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This article

Christensen et al. (2007)

Lead time representatio V F F F F D C F F F V V V V n in the model* Supply chain performance measure Bullwhip effect        (order variations) Inventory levels     and/or costs Various cost    componen ts Opportuni ty costs (includes  inventory costs & input cost) Profit and/or other financial indicators Effects of lead time variability (LTV) on supply chain performance No significant effects Increasing (reducing) Deterministic lead time LTV deteriorate    s (improves ) supply chain V V V V V V V V V V

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performan ce LTV impact changes under different     levels of supply chain initiatives LTV impact changes under     varying business environme nt Effects of lead time variability (LTV) on supply chain decisions Supports order  expediting Justifies supply chain      coordinati Deterministic lead time on Changes inventory/ risk     hedging policies Table 1. A comparative summary of the literature review on how lead time is incorporated into supply chain decision making * fixed parameter (F); variable parameter (V); decision variable (D); constraint (C)

VA R

SD D

AP V

1.5

3.8

L

H

LT V 0.2 75 0.4 13 0.5 51 0.2 75 0.4 13 0.5 51

Sequential Model Operational Sub-model Financial Sub-model E(TO E(TO Np Nc C) C) O0 51. 123 S0 3,6 655.4 637.6 715 1 2 .1 1 55 O0 58. 115 S0 2,5 677.5 674.0 389 2 4 .1 2 27 O0 58. 115 S0 1,1 687.0 156 3 4 .1 3 680.9 54 O0 51. 123 S0 2,5 1,1 654.9 4 1 .1 4 655.7 38 13 O0 56. 117 S0 2,2 679.4 873 5 7 .2 5 672.8 30 O0 62. 112 S0 1,3 700.7 665 6 1 .6 6 701.8 27

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Integrated Model E(TO C) I0 1 I0 2 I0 3 I0 4 I0 5 I0 6

610.5 648.6 663.7 628.0 660.4 692.8

Np 45. 7 54. 3 62. 9 45. 9 59. 6 65. 9

129 .8 121 .0 113 .5 129 .8 116 .2 112 .0

3,9 81 3,9 79 3,7 84 2,1 52 2,6 75 2,8 30

Nc 872 690 94 983 576 433

0.2 O0 65. 114 S0 1,6 I0 60. 119 1,2 1,0 803.6 641 75 7 0 .6 7 798.6 76 7 790.2 5 .7 88 83 0.4 O0 67. 113 S0 1,3 I0 63. 116 1,4 813.5 432 838 13 8 1 .6 8 811.6 58 8 804.4 7 .8 53 0.5 O0 67. 113 S0 I0 68. 112 1,8 820.8 672 238 679 51 9 4 .9 9 822.3 9 822.8 4 .5 05 0.2 O1 73. 104 S1 1,1 I1 60. 120 1,2 1,0 827.3 386 H 75 0 6 .4 0 827.9 29 0 799.8 7 .2 38 93 0.4 O1 76. 102 S1 I1 70. 110 1,1 844.3 144 0 987 13 1 9 .5 1 849.7 1 838.4 5 .6 63 0.5 O1 77. 102 S1 I1 70. 110 1,1 854.9 3 0 857 51 2 9 .8 2 857.3 2 842.8 9 .9 37 0.2 O1 13. 161 S1 4,0 I1 13. 162 4,0 536.3 434 414 1.8 3.8 L 75 3 2 .5 3 510.7 00 3 508.9 1 .2 00 0.4 O1 12. 162 S1 4,0 I1 16. 160 4,0 540.9 529 360 13 4 9 .1 4 517.3 00 4 516.8 5 .0 00 0.5 O1 14. 162 S1 4,0 I1 19. 157 4,0 566.0 543.8 819 535.1 320 51 5 7 .4 5 00 5 0 .6 00 0.2 O1 34. 153 S1 3,0 1,1 I1 18. 157 3,2 1,3 610.2 H 75 6 4 .3 6 607.6 67 16 6 551.4 4 .7 57 50 0.4 O1 38. 138 S1 3,2 I1 18. 156 3,0 1,3 619.4 976 13 7 0 .0 7 614.0 83 7 568.2 9 .3 20 06 0.5 O1 45. 132 S1 3,8 I1 19. 155 2,6 1,0 649.7 665 51 8 4 .1 8 632.0 35 8 573.5 5 .6 54 90 0.2 O1 34. 142 S1 2,1 1,0 I1 19. 158 3,9 1,1 721.3 4.5 L 75 9 7 .8 9 719.7 70 84 9 650.5 4 .9 80 86 0.4 O2 35. 143 S2 2,3 1,0 I2 20. 159 3,9 1,1 726.4 13 0 4 .3 0 727.6 16 72 0 662.7 1 .1 85 80 0.5 O2 35. 143 S2 2,3 1,0 I2 20. 159 3,9 1,1 733.4 51 1 4 .3 1 731.5 76 61 1 664.5 3 .4 89 87 0.2 O2 49. 126 S2 1,6 I2 19. 159 1,7 1,3 760.1 541 H 75 2 6 .9 2 757.8 09 2 677.9 7 .1 15 62 0.4 O2 50. 127 S2 2,3 I2 20. 159 1,6 1,3 759.8 702 13 3 4 .6 3 760.1 89 3 692.5 6 .9 82 55 0.5 O2 50. 127 S2 2,8 I2 20. 159 1,6 1,3 770.7 784 51 4 3 .6 4 762.2 48 4 695.2 5 .8 45 63 Table 4. Optimization results. E(TOC): expected total opportunity cost (in thousands of dollars); : quantity of aluminum purchased at time t0 (in million cans); : quantity of aluminum purchased at time t1 (in million cans); Np: number of put options; Nc: number of call options 4.5

L

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