D
O INVENTORIES ENHANCE SUPPLY CHAIN
PERFORMANCE IN CASE OF DISRUPTIONS
?
R
OBINH
ANSMA(S2581922)
Master Thesis
prof. dr. D. P. (Dirk Pieter) van Donk
dr. O. A. (Onur) Kiliç
University of Groningen
Faculty of Economics and Business
Research master of Science in Economics and Business
Operations Research and Operations management
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A
BSTRACT:
It is commonly believed that redundant inventories reduce the impact of supply chaindisruptions. However, recent events illustrate that, like buildings, inventories are at risk of being destroyed by supply chain disruptions. To evaluate the impact of this on the common belief that inventories reduce the impact of supply chain disruptions; we study a single product problem with one supplier, two warehouses and customers, and propose ways in which inventory in combination with operational supply chain design can be used to adequately manage disruptions and enhance supply chain profitability.
K
EYWORDS:
Supply chain design; supply chain disruptions; Markov chain modeling; inventories;dynamic sourcing; adaptive inventory levels.
T
ABLE OFC
ONTENTS1 Introduction ... 3
2 Literature review ... 4
3 Methodology ... 6
4 Numerical analysis ... 14
5 Discussion and conclusion ... 20
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1
I
NTRODUCTIONIn today’s turbulent and uncertain environment, not only the physical structures of supply chains but also inventories and equipment are susceptible to disruptive events originating from factors, such as equipment failure, improper maintenance, human error or natural disasters. For example, in 2012 an earthquake in the north of Bologna (Italy) destroyed an estimated 400.000 cheese wheels (Giuseppe, 2012). In 2014 a fire at the warehouse of a cheese factory in Gerkesklooster (the Netherlands) destroyed 2.100 tons of cheese (Abdulla, 2014). Furthermore, in 2009 a forklift driver accidently damaged a warehouse in Moscow (Russia) filled with cognac and vodka destroying thousands of bottles (Mail Foreign Service, 2009).
Such disruptive events have severe negative consequences on the financial, market and operational performance of firms (van der Vegt et al., 2014; Wagner and Bode, 2006) as they disrupts the flow of goods or services in a supply chain (Ambulkar et al., 2015). Extant research suggests that inventory redundancy is an effective way to reduce the negative consequences of supply chain disruptions (Chen et al., 2011; Jüttner and Maklan, 2011; Schmitt et al., 2010; Schmitt and Singh, 2012). While this might be the case, most of these studies assume that inventories are placed at a location which is always available and exempt from disruptions (Kamalahmadi and Mellat, 2016; Snyder et al., 2016; Tukamuhabwa et al., 2015). However, supply chain disruptions are possible at any link of the supply chain (Qi et al., 2010) and not only the physical structures of supply chains but also the inventories within are at risk.
Failing to consider this may result in significant financial losses as inventories are valuable. Furthermore, inadequate consideration of the risks associated with holding inventory will most likely result in higher inventory levels, which further increases risk. We believe that it is important for supply chain managers and academics to be able to adequately account for the risk of holding inventories. As such we seek to determine optimal inventory levels for a supply chain where disruptions are possible at any link of the supply chain. In addition we explore ways in which costs savings can be achieved by altering the operational design of this supply chain. In realizing this we borrow ideas from inventory management, such as dynamic sourcing and adaptive inventory levels. We consider dynamic sourcing to entail having multiple potential modes of demand fulfillment (Berman et al., 2011; Fox, 2006) and adaptive inventory levels to entail using inventory levels tailored specifically to an idiosyncratic circumstance (Scarf, 1960).
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costs per time period. We solve the model using Markov Chains. Our results reveal that inventories in excess of customer demand could be a liability to supply chain performance. However, they can also be beneficial in the case of supply chain disruptions when combined with supply chain design changes. Recognizing that supply chain design changes can be both structural and operational (Carvalho et al., 2012; Garcia-Herreros et al., 2014), we solely focus on the latter.
The remainder of this paper is structured as follows. The next section will discuss the relevant literature, the modeling methods used and reflect upon the theoretical scope. This is followed by a discussion of our research methodology. Hereafter our numerical experiments and their results are presented. Finally we discuss these results and conclude upon fulfillment of the research aim.
2
L
ITERATURE REVIEWThe literature on supply chain disruptions can be divided into three areas. Studies belonging to the first area (e.g., Eisenstein, 2005; Xia et al., 2004) focus on supply disruptions. Supply disruptions always occur upstream and are events that impair the provision of goods in terms of quality and/or quantity to the focal entity (Kamalahmadi and Mellat, 2016). Studies in the next area (e.g., Azad et al., 2013; Peng et al., 2011; Shishebori et al., 2014) focus on location disruptions. Location disruptions always occur at the focal entity and are events that incapacitate, demolish a facility rendering it unusable (Kamalahmadi and Mellat, 2016). Studies in the last area focus on a combination of supply and location disruptions. Recognizing this, this study belongs to the latter area.
The issue with studies belonging to either the first or second area is that they assume that the supplier or focal location is exempt from disruptions (Kamalahmadi and Mellat, 2016; Snyder et al., 2016; Tukamuhabwa et al., 2015). By focusing on a single risk of supply chain disruption, they oversimplify the real-world (Qi et al., 2010) and fail to consider that improving one firms’ ability to cope with disruptions can decrease the supply chain’s overall ability to cope with disruptions (Garcia-Herreros et al., 2014; Schmitt and Singh, 2012). This may result in significant financial losses. Studies in the latter area of research consider a combination of supply and location disruptions to account for this issue. They are closely related to the well-known facility location problem and emphasize solution methods.
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warehouse and customer(s) are damaged. They assume that customers cannot be assigned to alternate warehouses if their closest warehouse is disrupted. Their solution method determines the number and locations of operational warehouses, their base-stock levels and customer assignment. Similar problems are studied by Chen et al. (2011) and DeCroix (2013). While these studies focus on determining the optimal inventory level when faced with supply and location disruptions, they provide little insight into the mechanisms and laws which underlie the use of inventory.
Our study is also related to the case-based literature on supply chain disruptions. Schmitt et al. (2010) considers a real-world three-tier supply chain in which supplier and focal locations fail at known times for known intervals. They experiment with a large number of different operational supply chain designs across multiple disruption risk scenarios. This allows them to generate insights into the mechanisms and laws underlying supply chain disruptions and its mitigation. The simulation findings indicate that the cost structure and the service level requirements of the production process affect the optimal location of inventory (i.e., further up- or downstream). However, because optimal solutions are not found they are unable to compare results across supply chain designs or risk scenario. Furthermore, they remain ambiguous about what constitutes a disruption and what effect is has on the supplier or focal location.
In light of the above, we feel that it is impossible for managers and academics to adequately account for the risk of holding inventories when faced with disruptions. This study aims to address this gap and uncover essential information on the risk of holding inventories. Furthermore, while one study considers dynamic sourcing we are the first to uncover if, how and how much it effects supply chain performance in case of supplier and location disruptions. Additionally, we explore if, how and how much adaptive inventory affect supply chain performance. By considering probabilistic occurrence of and recovery from disruptions, the disruption scenarios considered in this study are not very restrictive with regards to generalizability. That is, three categories of supply chain risks have been identified: (1) risks internal to the firm; (2) risks external to the firm, but internal to the supply chain; and (3) risks external to the supply chain (Christopher and Peck, 2004; Kleindorfer and Saad, 2005; Spiegler et al., 2012).
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3
M
ETHODOLOGY3.1 PROBLEM SETTING
The supply chain problem considered in this study is portrayed in Figure 3-1. It includes a single supplier, two warehouses and a set of customers. This setting was purposefully selected as it is the simplest supply chain problem possible given the aim of our study. In this regard, it was decided to include one supplier, however at least two warehouses where required to evaluate the effect of dynamic sourcing. With further regards to simplicity, the supply chain is centered on a single product because the mechanisms and laws associated with the use of inventory do not change by including more products. As a result of these considerations, our supply chain problem consists of fulfilling customer demand for a single product, which flows from the supplier through the warehouses to eight customers.
The warehouses satisfy customer demand from their on-hand inventory, which is replenished by the supplier. Any customer demand which cannot be fulfilled by the warehouses from their on-hand inventory is lost (i.e., no backorders). The supply chain is exposed to disruptions at the supplier and warehouses. Supplier disruptions render the facility unavailable for a period of time. During this time, it cannot receive orders from or send products to the warehouses. While warehouse disruptions also render the affected location(s) unavailable for a period of time, an additional assumption is that all inventories at the affected location are destroyed. In modeling disruptions, the present study, contrary to Schmitt and Singh (2012), Qi et al. (2010) and Garcia-Herreros et al. (2014), assumes that the occurrence of and recovery from disruptions is random. Such an approach should be preferred as it more closely reflect the uncertainty associated with real-world supply chain disruptions described by UN (2015).
Figure 3-1: Graphical representation supply chain
Supplier Warehouses Consumers
Focal
3.2 SUPPLY CHAIN DESIGNS
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supply chain performance. The supply chain designs we consider in our study are the traditional, the
dynamic and the dynamically adaptive. The traditional design closely reflects the examples mentioned
in the introduction. Both warehouses operate independently, each with a dedicated set of customers and a single inventory policy. As such, when a warehouse in this supply chain design is disrupted, its assigned customer demand is lost. In the dynamic design we allow demand of disrupted warehouses to be reassigned to operational warehouses. We think that having multiple potential modes of demand fulfillment is beneficial in the case of supply chain disruptions. For instance, when one warehouse is disrupted, its customers can be served by an operational warehouse which reduces lost sales. However, with a single inventory policy we need to balance performance in non-disrupted circumstances with performance under disrupted circumstance. It is also important to note that dynamic sourcing does require warehouse inventory levels to exceed its customer demand in order to reduce lost sales (see Berman et al., 2011). As such, we need to balance the benefits of reduced lost sales with the potential detrimental effect of additional inventory. In the dynamically adaptive design we use dynamic sourcing in tandem with adaptive inventory levels. We think that adaptive inventory levels in combination with dynamic sourcing are beneficial the case of supply chain disruptions. For example, without any disruptions the warehouses operate with the minimal inventory levels required to serve customer demand. However, when a warehouse is disrupted the operational warehouse increases its inventory level such that all can be served which reduces lost sales. We think that this should allow for better performance as it avoids the need to balance performance across disrupted and non-disrupted circumstances.
3.3 MODELING DISRUPTIONS
The model captures that the supplier and warehouses are prone to probabilistic disruption and recovery using a discrete Markov chain. While the use of continues-time would yield measurement improvements, this increase is minor as in real-world supply chains actions such as inventory review, reordering and fulfilling demand frequently occur at fixed intervals (Habermann et al., 2015). As such, discrete time should be preferred in our application because the minor (if any) measurement improvements do not justify the increased computational complexity. However, when more operational activities are incorporated, especially when these do not occur at fixed times, the use of a continuous time model should be preferred. In such a case, our model can relatively easy be converted to continuous time. The parameters used in our study are introduced in Table 3-1.
In our model, the state of the supply chain is denoted in triplets, with the first, second and third entity denoting the status of the supplier, warehouse 1, and warehouse 2. The status of the supplier/warehouse is denoted as 1: operational or 0: disrupted. Using this, Figure 3-1 depicts all one step transitions from state 1,1,1. Figure 3-1 also illustrates how the one step transition probabilities are calculated, for example from state 1,1,1 to state 0,0,0 has a probability of 𝜆𝑑𝑠𝜆
𝑑 𝑘λ
d
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Furthermore, we assume that the state of the supply chain only changes in-between time periods. In our model each time period is a day and each day has the following order of events: First operational warehouses review inventory levels and place orders at the supplier. Next, an operational supplier fulfills the orders placed by the warehouses. Subsequently, operational warehouses attempt to fulfill all customer demand, whilst adhering to their current supply chain design. Finally, as we transition from one day to the next, the supply chain transitions from state to state
Table 3-1: Model Parameters
𝜆𝑑𝑠 Probability of supplier transitioning from operational (1) to disrupted (0)
𝜆𝑜𝑠 Probability of supplier transitioning from disrupted (0) to operational (1)
𝜆𝑑𝑘 Probability of warehouse transitioning from operational (1) to disrupted (0)
𝜆𝑜𝑘 Probability of warehouse transitioning from disrupted (0) to operational (1)
𝑣𝑠𝑤 Expected initial inventory level at warehouse w in state s
𝑔𝜏𝑤 Initial inventory level at warehouse w after transition 𝜏
𝑝𝜏 Probability of state-transition 𝜏
𝑄𝑤 Base-stock level for warehouse w
𝑄𝑠𝑤 Base-stock level for warehouse w in state s
𝑛𝑤 Number of periods since warehouse w received its last order
𝑑𝑤 Demand faced by warehouse w
α Holding costs per item
β variable product ordering costs per item γ lost sales costs per item
𝜋𝑠 Steady state probability for state s
Page 9 of 25 3.4 INVENTORY POLICY AND CONTROL
For computational simplicity, we assume that the supplier has sufficient capacity to fulfill all orders from the warehouses, that lead times are zero and that customer demand is known and constant. In the absence of lead-times it has been shown that base-stock policies are optimal for supply chains facing disruptions (see Özekici and Parlar, 1999). As such, for each supply chain design we use a base-stock policy. For the traditional design, each warehouse uses a single base-stock level (𝑄𝑤). Moreover, the base-stock levels are allowed to differ across warehouses. During each period operational warehouses order products from the supplier to increase the inventory level to Qw. The customer demand which cannot be fulfilled because of disruptions or insufficient inventory it is lost.
For the dynamic design, each warehouses also uses a single base-stock level (𝑄𝑤). The base-stock levels can be different across warehouses. During each period, operational warehouses order products at the supplier to increase inventory up to 𝑄𝑤. Similar to the traditional design, the customer demand which cannot be fulfilled because of insufficient inventory it is lost. However, demand of disrupted warehouses can be reassigned to operational warehouses.
For the dynamically adaptive design, each warehouse uses multiple base-stock levels (𝑄𝑠𝑤). During each period, operational warehouses order products at the supplier to increase the inventory level up to Qsw. However, the supplier only fulfills these orders when operational. The base-stock level of a warehouse can be changed per state. Furthermore, demand of disrupted warehouses can be reassigned to operational warehouses.
3.5 OBJECTIVE FUNCTION
While it would be possible to employ a closed-form solution method, this is time consuming and there is no certainty that it would be possible to derive such a solution for all supply chain designs. Instead, we propose to model our supply chain problem using the concept of Markov processes. By modeling our problem as a Markov Chain we are able to derive an optimal solution for multiple designs whilst closely adhering to the real-world uncertainty associated with supply chain disruptions. That is, probabilistic occurrence of and recovery from supply chain disruptions.
The objective function that should be minimized in our model is the expected total costs of the supply chain. In order to find the minimal expected costs associated with each supply chain design, we need the expected initial inventory level at a warehouse in every state 𝑣𝑠𝑤= ∑ 𝑔
𝜏𝑤𝑝𝜏, 𝑤 = 1,2 𝑚
𝜏=1 ,
where (𝑔𝜏𝑤) is the inventory leftover at a warehouse (𝑤) after transition (𝜏) and (𝑝
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For the dynamic design if an order was the last event to occur at warehouse w, 𝑔𝜏𝑤= 𝑄𝑤− ∑𝑠∈𝜏𝑑𝑠𝑤where 𝑄𝑤 is the inventory level after receiving the last order and 𝑑𝑠𝑤 the demand served during each previous state from transition (𝜏).
For the dynamically adaptive design, if an order was the last event to occur at warehouse w, 𝑔𝜏𝑤= 𝑄𝑤− ∑𝑠∈𝜏𝑑𝑠𝑤 where 𝑄𝑤 is the inventory level after receiving the last order and 𝑑𝑠𝑤 the demand served during each previous state. It should be noted, that while it is possible to have an infinite number of states since the last order, we have to consider that if the number states in 𝜏 is equal to 𝑛 =𝑄
𝑑 then 𝑔𝜏 𝑤 = 0.
3.6 COSTS
3.6.1 TRADITIONAL DESIGN
In our model the objective is to minimize the expected total costs (TC) of the supply chain, which is the summation of the expected holding costs (HC), expected variable product purchasing costs (OC), and expected lost sales costs (LC). In our costs formulation α denotes the holding costs per item, β denotes the variable product ordering costs per item and γ denotes the lost sales costs per unit of demand. The objective function for the traditional design is to
minimize TC(𝑄1, 𝑄2) (1)
where
𝑇𝐶(𝑄1, 𝑄2) = 𝐻𝐶(𝑄1, 𝑄2) + 𝑂𝐶(𝑄1, 𝑄2) + 𝐿𝐶(𝑄1, 𝑄2) (2)
Note that the costs of products destroyed by disruptions are generated when products are ordered. LC are generated when demand cannot be fulfilled in the present day (i.e., no backorders). HC are generated when operational warehouses have inventory leftover after fulfilling demand. Given the traditional design, there are three different scenarios with regards to the HC. When the supplier is operational, operational warehouses order and receive products. As such, in states (1,1,1;1,1,0;1,0,1) the inventory leftover is 𝑄𝑤− 𝑑𝑤. When the supplier is disrupted, operational warehouses do not receive products. As such, in states (0,1,1;0,1,0;0,0,1) the inventory leftover at operational warehouses is 𝑣𝑠𝑤− 𝑑𝑤. Finally, when both warehouses are disrupted they do not hold any inventories. As such, in states (1,0,0;0,0,0) no inventory is leftover after fulfilling customer demand. Given this discussion, the HC for the traditional design can mathematically be expressed as
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Next we derive an expression for the OC using the expected inventory levels (𝑣𝑠𝑤). OC is generated by warehouses receiving products from the supplier. Given the traditional design, two different scenarios exist for the OC. When the supplier is operational, operational warehouses order and receive products. As such, in states (1,1,1;1,1,0;1,0,1) the products ordered per operational warehouse are 𝑄𝑤− 𝑣
𝑠𝑤. On the other hand, when the supplier or both warehouses are disrupted no products will be ordered. In light of the aforementioned, the OC for the traditional design can mathematically be expressed as 𝑂𝐶(𝑄1, 𝑄2) = 𝛽[∑ ( 2 𝑤=1 𝑄𝑤− 𝑣 1,1,1𝑤 )𝜋1,1,1+ (𝑄1− 𝑣1,1,01 )𝜋1,1,0+ (Q2− 𝑣1,0,12 )𝜋1,0,1] (4)
We now derive an expression for the LC. LC is generated when not all customer demand can be satisfied. Given the traditional design, three different scenarios exist for the LC. When the supplier is operational, warehouses receive products at the beginning of every period. As such, in states (1,1,1;1,1,0;1,0,1) lost sales only occur if 𝑄𝑤< 𝑑𝑤, which are in this case equal to 𝑑𝑤− 𝑄𝑤. When the supplier is disrupted warehouses cannot order any products. As such, in states (0,1,1;0,0,1;0,1,0) lost sales only occur if 𝑣𝑠𝑤 < 𝑑𝑤, which are in this case equal to 𝑑𝑤− 𝑣
𝑠𝑤. Finally, when both warehouses are disrupted no products can be sold to customers. As such, in states (1,0,0;0,0,0) lost sales are 𝑑𝑤. Given this discussion, the LC for the traditional design can mathematically be expressed as 𝐿𝐶(𝑄1, 𝑄2) = 𝛾[∑ max (𝑑𝑤 2 𝑤=1 − 𝑄𝑤, 0)𝜋 1,1,1+ {max(𝑑1− 𝑄1, 0) + 𝑑2}𝜋1,1,0 +{max(d2− 𝑄2, 0) + 𝑑1}𝜋 1,0,1+ ∑ max (𝑑𝑤− 𝑣0,1,1𝑤 , 0) 2 𝑤=1 𝜋0,1,1 +{max(𝑑1− 𝑣 0,1,0 1 , 0) + 𝑑2}𝜋 0,1,0+ {max(𝑑2− 𝑣0,0,12 , 0) + 𝑑1}𝜋0,0,1 + ∑ 𝑑𝑤 2 𝑤=1 𝜋1,0,0+ ∑ 𝑑𝑤 2 𝑤=1 𝜋0,0,0] (5) 3.6.2 DYNAMIC DESIGN
In this section we focus on the dynamic design. Similar to the traditional design, the objective function for the dynamic design is to
minimize TC(𝑄1, 𝑄2) (6)
where
𝑇𝐶(𝑄1, 𝑄2) = 𝐻𝐶(𝑄1, 𝑄2) + 𝑂𝐶(𝑄1, 𝑄2) + 𝐿𝐶(𝑄1, 𝑄2) (7)
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are 𝑄𝑤− 𝑑𝑤. When the supplier is operational and one warehouse is disrupted, the operational warehouse attempts to fulfill all customer demand using its inventor. As such, in states (1,1,0;1,0,1) the products leftover at the operational warehouse after fulfilling demand are 𝑄𝑤− ∑2 𝑑𝑤
𝑤=1 . On the other hand, when the supplier and one warehouse are disrupted, the operational warehouse attempts to fulfill all customer demand using its initial inventory level. As such, in states (0,1,0;0,0,1) the products leftover at the operational warehouse are 𝑣𝑠𝑤− ∑2𝑤=1𝑑𝑤. Similarly, when the supplier is disrupted but both warehouses are operational the warehouses attempt to fulfill demand from their initial inventory. As such, in state (0,1,1) the leftover inventory per operational warehouse is 𝑣𝑠𝑤− 𝑑𝑤. Finally, in the case where both warehouses are disrupted no inventories are held, there are no products leftover after fulfilling customer demand. As such, in states (1,0,0;0,0,0) there are no leftover inventories. Given this discussion, the HC for the dynamic design can mathematically be expressed as
𝐻𝐶(𝑄1, 𝑄2) = 𝛼[∑ max(𝑄𝑤− 𝑑𝑤, 0) 𝜋 1,1,1+ max(𝑄1− ∑ 𝑑𝑤 2 𝑤=1 , 0) 𝜋1,1,0 2 𝑤=1 + max(𝑄2− ∑ 𝑑𝑤 2 𝑤=1 , 0) 𝜋1,0,1+ max(∑{𝑣0,1,1𝑤 − 𝑑𝑤 2 𝑤=1 } , 0)𝜋0,1,1 + max(𝑣0,1,01 − ∑ 𝑑𝑤 2 𝑤=1 , 0) 𝜋0,1,0+ max(𝑣0,0,12 − ∑ 𝑑𝑤 2 𝑤=1 , 0) 𝜋0,0,1 (8)
Next we derive an expression for the OC for the dynamic design. As the reasoning and the expression are identical to the traditional design, we do not discuss it (again). The OC for the dynamic design can mathematically be expressed as 𝑂𝐶(𝑄1, 𝑄2) = 𝛽[∑ ( 2 𝑤=1 𝑄𝑤− 𝑣 1,1,1𝑤 )𝜋1,1,1+ (𝑄1− 𝑣1,1,01 )𝜋1,1,0+ (Q2− 𝑣1,0,12 )𝜋1,0,1] (9)
Page 13 of 25 𝐿𝐶(𝑄1, 𝑄2) = 𝛾[∑ max (𝑑𝑤 2 𝑤=1 − 𝑄𝑤, 0)𝜋 1,1,1+ max(∑ 𝑑𝑤 2 𝑤=1 − 𝑄1, 0) 𝜋 1,1,0 + max(∑ 𝑑𝑤− 𝑄2, 0)𝜋 1,0,1 2 𝑤=1 + max(∑{𝑑𝑤 2 𝑤=1 − 𝑣0,1,1𝑤 } , 0) 𝜋 0,1,1 + max(∑ 𝑑𝑤 2 𝑤=1 − 𝑣0,1,01 , 0) 𝜋0,1,0+ max(∑ 𝑑𝑤 2 𝑤=1 − 𝑣0,0,12 , 0) 𝜋0,0,1 + ∑ 𝑑𝑤 2 𝑤=1 𝜋1,0,0+ ∑ 𝑑𝑤 2 𝑤=1 𝜋0,0,0] (10)
3.6.3 DYNAMICALLY ADAPTIVE DESIGN
In this section we focus on the dynamically adaptive design. As the logic underlying the TC expression is the same as for the other supply chain designs, we do not (again) discuss it. For the dynamically adaptive design the objective function is to
minimize 𝑇𝐶(𝑄𝑠1, 𝑄𝑠2), 𝑠 = 1, … , 𝑚 (11)
where
𝑇𝐶(𝑄𝑠1, 𝑄𝑠2) = 𝐻𝐶(𝑄𝑠1, 𝑄𝑠2) + 𝑂𝐶(𝑄𝑠1, 𝑄𝑠2) + 𝐿𝐶(𝑄𝑠1, 𝑄𝑠2), 𝑠 = 1, … , 𝑚 (12)
Note that the difference between the objective function for the dynamically adaptive design and the other supply chain designs are the adaptive inventory levels. As visible from equation (11) and (12) the inventory level can change to allow for optimal performance in any state. We now derive the expression for the HC. Given the dynamically adaptive design, multiple different scenarios exist for the HC. The mechanisms underlying the HC for the dynamically adaptive design are identical to the HC for the dynamic design, with the only difference being the adaptive inventory levels. Considering this difference, the HC for the dynamically adaptive design can mathematically be expressed as
𝐻𝐶(𝑄𝑠1, 𝑄𝑠2) = 𝛼[∑ max(𝑄1,1,1𝑤 − 𝑑𝑤, 0) 𝜋1,1,1+ max(𝑄1,1,01 − ∑ 𝑑𝑤 2 𝑤=1 , 0) 𝜋1,1,0 2 𝑤=1 + max(𝑄1,0,12 − ∑ 𝑑𝑤 2 𝑤=1 , 0) 𝜋1,0,1+ max(∑{𝑣0,1,1𝑤 − 𝑑𝑤 2 𝑤=1 } , 0)𝜋0,1,1 + max(𝑣0,1,01 − ∑ 𝑑𝑤 2 𝑤=1 , 0) 𝜋0,1,0+ max(𝑣0,0,12 − ∑ 𝑑𝑤 2 𝑤=1 , 0) 𝜋0,0,1 (13)
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We now derive an expression for the LC for the dynamically adaptive design. Again, the expression and underlying reasoning is the same as the dynamic design with exception of the adaptive inventory levels. Accounting for this difference, the LC for the dynamically adaptive design can be mathematically expressed as 𝐿𝐶(𝑄𝑠1, 𝑄 𝑠2) = 𝛾[∑ max (𝑑𝑤 2 𝑤=1 − 𝑄1,1,1𝑤 , 0)𝜋 1,1,1+ max (∑ 𝑑𝑤− 𝑄1,1,01 , 0)𝜋1,1,0 2 𝑤=1 +max (∑ 𝑑𝑤− 𝑄 1,0,12 , 0)𝜋1,0,1+ max ( 2 𝑤=1 ∑{𝑑𝑤− 𝑣 0,1,1𝑤 2 𝑤=1 },0)𝜋0,1,1 + max(∑ 𝑑𝑤 2 𝑤=1 − 𝑣0,1,01 , 0) 𝜋0,1,0+ max(∑ 𝑑𝑤 2 𝑤=1 − 𝑣0,0,12 , 0) 𝜋0,0,1 + ∑ 𝑑𝑤 2 𝑤=1 𝜋1,0,0+ ∑ 𝑑𝑤 2 𝑤=1 𝜋0,0,0] (15)
4
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UMERICAL ANALYSISIn this section we discuss the numerical analysis. The numerical analysis is performed to evaluate the risks of holding inventories and explore if, how and how much incorporating both dynamic sourcing and adaptive inventory levels affects supply chain performance. We expect that the dynamically adaptive design will perform the best because it avoids the need to balance performance across disrupted and non-disrupted circumstances while allowing for demand of disrupted warehouses to be reassigned to operational warehouses. Furthermore, we expect that the traditional design will perform the worst because of high lost sales costs. However, it is important to know the extent of the differences in cost performance, as the dynamically adaptive and to a lesser extend dynamic designs, require more coordination among supply chain partners.
For the numerical analysis, each warehouse is assigned 50% of the total demand to balance the damages caused by disruptions. This also ensures that the benefit of dynamic sourcing is homogeneous across warehouses, which facilitates interpretation of the results as we do not need to account for differences in risk. Products are purchased by the warehouses at a cost of €1 per unit. Following Richardson (1996), the inventory carrying costs are set at twenty-percent of its purchase value. As the supply chain is centered on a consumer product, lost sales are costly. Saccani et al. (2007) argues that this is because lost sales costs are foregone profit and consumers often do not wait until the product becomes available again, instead they purchase another brand. To reflect this, we assume the lost sales to be €10 per unit of lost demand. The disruption- and recovery-rates (𝜆𝑑𝑠, 𝜆 𝑜 𝑠, 𝜆 𝑑 𝑘, 𝜆 𝑜
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While supply chains are frequently effected by disruptions (UN, 2015), the actual probability of a supply chain entity being disrupted is low (Urciuoli et al., 2014; Wagner and Bode, 2006; Zsidisin and Smith, 2005). In the absence of exact figures, we feel that the probabilities ranging from quarter-of-a-percent till four-percent are representative of most real-world environments. That is, environments ranging from the save and stable Benelux to hazard-prone areas such as the flood-prone Philippines, earthquake-prone Nepal, or regions that are characterized by unstable social and political situations such as the war-torn Middle-East with countries such as Iraq and Syria. When disrupted, supply chain entities need time to recover. While recovery from certain disruptions might take days, in other cases it might take weeks. In the absence of exact figures, we feel that recovery probabilities ranging from twenty-five-percent till sixty-percent are representative of most real-world situations. That is, situations where recovery is as simple as ordering new products to cases where recovery requires the affected party will find an alternative (temporary) storage location or rebuilt their demolished location. The model parameters described above are summarized in table 4-1.
Table 4-1: Parameters 𝛽 €1 𝛼 €0,20 𝑑𝑤 4 𝛾 €10 𝜆𝑑𝑠 0-0,04 𝜆𝑜𝑠 0,25-0,55 𝜆𝑑𝑘 0-0,04 𝜆𝑘𝑜 0,25-0,55
While 𝜆𝑑𝑠, 𝜆𝑜𝑠, 𝜆𝑑𝑘 and 𝜆𝑜𝑘 are changed in our experiments, this is done ceteris paribus. In other words, when changing one of these parameters, the others are kept constant at certain values. We refer to these constant values for the parameters as their baseline values. Because research by UN (2015) indicates that upstream disruptions occur more often than downstream disruptions, the probability of supplier disruptions is assumed higher than the probability of warehouse disruption. For 𝜆𝑑𝑠 we assume a baseline probability of two-percent and for 𝜆𝑑𝑘 we assume a baseline probability of one-percent. Research by Mehrjoo and Pasek (2016) suggests that in a supply chain upstream entities generally recover more quickly than downstream entities. As such, for 𝜆𝑜𝑠 we assume a baseline probability of forty-five-percent and for 𝜆𝑜𝑘 we assume a baseline probability of thirty-five-percent. For the readers’ convenience, the baseline probabilities are summarized in table 4-2.
Table 4-2: Baseline parameters
𝜆𝑑𝑠 0,02
𝜆𝑜𝑠 0,45
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The optimal base-stock level per warehouse (𝑄1, 𝑄2 for the traditional and dynamic design and 𝑄𝑠1, 𝑄𝑠2 for the dynamically adaptive design) that minimize the TC for the supply chain was obtained through enumeration. In our numerical analysis the supply chain is configured around a consumer product. Each time period is equivalent to one day and each day the supply chain faces a total customer demand of eight units.
4.1 NUMERICAL RESULTS
The next four sub-sections elaborates on the effect of disruption/recovery probability on the optimal base-stock levels and associated total costs (TC) for each the supply chain designs. Through enumeration the optimal base-stock levels are found using the Markov chain model, with the only one changing parameter. In each of these sections, the total costs and optimal base-stock levels are compared across the three different supply chain designs.
In our results we find that for a given design in a given experiment the base-stock levels are always the same across warehouses. As such, there is not benefit to reporting 𝑄1, 𝑄2 separately and we just report 𝑄. In tables 4-3, 4-4, 4-5 and 4-6 we find that 𝑄 is frequently equal to the demand. This is because of deterministic costs as holding costs are only charged over the inventory leftover after fulfilling customer demand.
From the tables we also observe that the base-stock levels of the dynamically adaptive design converge to those of the dynamic design when the disruption uncertainty is high. More specifically, because of the high uncertainty, having additional inventory to hedge against lost sales is more beneficial than reduced holding costs.
4.1.1 THE EFFECT OF WAREHOUSE DISRUPTION PROBABILITY
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the difference in TC between the dynamic and the dynamically adaptive design indicates that while the latter has higher LC, these costs are more than offset by a decrease in HC and OC. The base-stock levels for each supply chain design are given in table 4-3. While it would be possible for the dynamically adaptive design to use a different base-stock level for each state, table 4-3 indicates that this is not the case. Instead one base-stock level is found for state (1,1,1) and one for all states not (1,1,1). Observe also that for the traditional design the base-stock levels do change. This is because for the traditional design any inventory in excess of the customer demand per day is a liability for performance.
4.1.2 THE EFFECT OF WAREHOUSE RECOVERY PROBABILITY
This section elaborates on the effect of warehouse recovery probability (𝜆𝑜𝑘) on the optimal base-stock levels and associated total costs (TC) for each the supply chain designs. The results are presented in Figure 4-2 and the associated optimal base-stock levels are presented in Table 4-4. From Figure 4-2 it is noteworthy that the dynamic and dynamically adaptive designs are more robust to increases/decreases in the warehouse recovery probability. Specifically, the 35% increase in 𝜆𝑘𝑜 yields a 0,3% decrease in TC for the dynamic design, a 1% decrease in TC for the dynamically adaptive design and a 11% decrease in TC for the traditional design. Partly as a result of its increased sensitivity, the traditional design attains the highest TC. While visually there appears to be no difference in TC between the dynamic and dynamically adaptive design, numerical analysis reveals that there is. Specifically, it increases from €0,14 at 𝜆𝑜𝑘 = 25% to €0,25 at 𝜆
𝑜
𝑘= 60%. A breakdown of the difference in TC between supply chain designs into HC, OC and LC provides noteworthy insights. It reveals that the decreasing difference in TC between the traditional design and the dynamic and dynamically adaptive designs are as because of LC. On the other hand, the increasing difference in TC between the dynamic and dynamically adaptive design is as a result of HC. The optimal base-stock levels for each supply chain design are presented in table 4-4. Similar the previous section, we see that for the dynamically adaptive design two base-stock levels are used. Furthermore, for the traditional design base-stock levels do not increase beyond customer demand per day because this results in lower performance.
4.1.3 THE EFFECT OF SUPPLIER DISRUPTION PROBABILITY
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dynamically adaptive designs. Instead the dynamically adaptive design generates the largest increase in TC up until 𝜆𝑑𝑠 = 2,25% and the traditional design up until 𝜆
𝑑
𝑠 = 2,5%. It is as a result of the relative insensitivity of the dynamic design that the difference in TC between it and the traditional design initially increases while the difference between the dynamic and dynamically adaptive design decreases. A breakdown of the difference in TC between supply chain designs reveals that the initial (large) increase seen for both the traditional and dynamically adaptive design is as a result of increasing lost sales. As a result, optimal base-stock levels change for the traditional and dynamically adaptive design, see table 4-5. From this table it is noteworthy that for the dynamically adaptive design initially two different base-stock levels are optimal and later one base-stock level is optimal. Furthermore, optimal base-stock levels do increase for the traditional design indicating that for this specific circumstance it actually increases performance.
Table 4-3: Optimal base-stock warehouse disruption probability Base stock levels
Traditional Dynamic Dynamically adaptive 𝜆𝑑𝑘 𝑄 𝑄 𝑄1,1,1 𝑄
𝑠\(1,1,1)
0,00% 4 4 4 4
0,25-3,25% 4 8 4 8
3,50-4,00% 4 8 4 9
Table 4-4: Optimal base-stock warehouse recovery probability Base stock levels
Traditional Dynamic Dynamically adaptive 𝜆𝑘𝑜 𝑄 𝑄 𝑄1,1,1 𝑄𝑠\(1,1,1)
25% 4 4 4 9
30% 4 8 4 9
35-60% 4 8 4 8
Table 4-5: Optimal base-stock supplier disruption probability Base stock levels
Traditional Dynamic Dynamically adaptive 𝜆𝑑𝑠 𝑄 𝑄 𝑄1,1,1 𝑄𝑠\(1,1,1)
0,00-2,00% 4 8 4 8
2,25% 5 8 4 8
2,50% 7 8 8 8
2,75-4,00% 8 8 8 8
Table 4-6: Optimal base-stock supplier recovery probability Base stock levels
Traditional Dynamic Dynamically adaptive 𝜆𝑜𝑠 𝑄 𝑄 𝑄1,1,1 𝑄𝑠\(1,1,1)
25% 4 8 4 10
30-35% 4 8 4 9
40-60% 4 8 4 8
4.1.4 THE EFFECT OF SUPPLIER RECOVERY PROBABILITY
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Specifically, the 35% increase in 𝜆𝑜𝑠 yields a 11% decrease for the traditional design, a 22% decrease for the dynamically adaptive design and a 23% decrease for the dynamic design. As a result of the relative insensitivity of the traditional design, the difference in TC between it and the dynamic and dynamically adaptive design is increasing in 𝜆𝑜𝑠. On the other hand, the difference in TC between the dynamic design and the dynamically adaptive design is initially decreasing and remains constant thereafter. A breakdown of the TC differences into HC, OC and LC provides noteworthy insights. It reveals that the difference in TC between the traditional design and the dynamic and dynamically adaptive design is driven by LC and that the difference in TC between the dynamic and dynamically adaptive design is driven by HC and OC. These changes in TC are explained by the optimal-base stock levels depicted in Table 4-6. Finally, for the traditional design base-stock levels do not increase beyond customer demand per day because this results in lower performance.
€ 10 € 11 € 12 € 13 € 14 € 15 € 16 € 17 € 18 € 19 T o ta l Co st s
Disruption probability warehouse
Figure 4-1 € 11 € 12 € 12 € 13 € 13 € 14 € 14 T o ta l Co st s
Recovery probability warehouse
Figure 4-2 € 6 € 7 € 8 € 9 € 10 € 11 € 12 € 13 € 14 € 15 € 16 T o ta l Co st s
Supplier disruption probability
Figure 4-3 € 10 € 11 € 12 € 13 € 14 € 15 T o ta l Co st s
Supplier recovery probability
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5
D
ISCUSSION AND CONCLUSION5.1 CONCLUSIONS
The present study shows, using a variety of increasing risk scenarios that without any changes to the supply chain design, inventories in excess of customer demand present a substantial risk to supply chain performance when faced with supply and inventory disruptions. Specifically, for the traditional supply chain design we find that only in case of a large increase in the disruption probability at the supplier will inventories in excess of customer demand be beneficial. However, we also show that given any other change in the warehouse disruption probability, warehouse recovery probability or supplier recovery probability inventory levels in excess of customer demand yield sub-optimal performance. Furthermore, perhaps as a result of the absence of disruption coping mechanisms, the performance of the traditional supply chain are sensitive to changes in the disruption and recovery probability at the supplier and warehouses.
However, as the dynamic and dynamically adaptive supply chain designs have shown, increasing inventory levels at disruption prone locations should not be thoughtlessly rejected. For the dynamic design, inventory levels in excess of customer demand are a necessity and despite the higher inventory levels, it yields significantly lower costs than the traditional design under the same circumstances. Additionally, the dynamic supply chain design is better able to cope with disruptions, as is evident from the fact that the total costs are robust to changes in the disruption and recovery probability at the supplier and warehouse level. It is also important to note, that the optimal inventory levels for the dynamic design did not change throughout our experiments – making it an easy design to manage.
For the dynamically adaptive design, inventories in excess of customer demand are only carried when necessary. As a result, when faced with changing warehouse disruption and recovery probabilities, it yields lower costs than the dynamic supply chain design. However, when faced with changing supplier disruption and recovery probabilities the dynamically adaptive design only yields minor (if any) performance benefits over the dynamic design. Furthermore, for the dynamically adaptive design, the sensitivity of the total costs to changes in the disruption and recovery probabilities is nearly identical to that the dynamic supply chain design. However, there is a noteworthy exception: As a result higher inventory levels at the warehouses, the dynamic design is significantly more robust to changes in the supplier disruption probability.
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having additional inventory to hedge against lost sales is more beneficial than reduced holding costs. Furthermore, under the dynamically adaptive design attaining optimal performance given changes in the warehouse and supplier disruption/recovery probability requires numerous changes to the base-stock inventory levels and extensive coordination among the supply chain. Whereas under the dynamic design no changes needed to be made to attain optimal performance given similar changes in the warehouse and supplier disruption/recovery probability.
5.2 THEORETICAL AND MANAGERIAL IMPLICATIONS
By modeling distinctive disruptive events that have the potential to impair operational activities at the supplier and impair operational activities and destroy all on-hand inventories at the warehouses, this paper is among the first in the OR/MS literature that explicitly considers how tradeoffs with regard to downstream inventory influence supply chain performance. The present study extends prior research given that most of the existing literature on supply chain disruptions presupposes that increasing inventory levels at disruption prone locations is beneficial to supply chain performance.
While dynamic sourcing has been applied by earlier studies on supply and inventory disruptions, it was unknown if, how and how much this affects supply chain performance. By explicitly evaluating the effect of dynamic sourcing upon the total supply chain costs across a variety of increasing risk scenarios the present study addresses this gap. While dynamic sourcing was found effective in reducing total supply chain costs and reducing sensitivity to disruptions, we also established that its effectiveness can be improved through a combination with adaptive inventory levels. While extensively studied in the inventory management literature, the concept of adaptive inventory levels had not yet been applied in the context of supply and inventory disruptions.
In evaluating modeling methods suitable to evaluate the risk of holding inventories and compare the performance of multiple supply chain designs, we found that the approaches used by the extant literature where unsuitable. By modeling our problem as a Markov Chain we were able to adequately accounting for the risk of holding inventories and derive optimal inventory levels for a given supply chain design.
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chain. For managers without decentralized storage locations or without the option of implementing dynamic sourcing, it is advised to limit the amount of safety stock.
5.3 LIMITATIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH
In light of the exploratory nature of the model, as well as with regard to restricting complexity, lead-times are excluded. While this is common practice for studies in this area of research, it limits generalizability. Without lead-times the increased demand faced by warehouses in case of dynamic sourcing, is no problem as inventories are replenished immediately the following period. However, when orders arrive later, it might be necessary to account for the fact that the actions taken in this period can affect the supply chains’ ability to fulfill next periods’ demand. This is especially important when dynamic sourcing comes at an additional cost. As such, future research might find it worthwhile to evaluate how lead-times influence the effectiveness of dynamic sourcing with and without adaptive inventory levels.
As this study is the first to examine the risk of inventory individually and in combination with supply chain design changes, it is exploratory in nature. Further model based research is required to develop the comprehensive and explicit role of inventory and its interaction with both structural and operational supply chain design changes. The present model can be improved by including multiple suppliers, potentially with finite capacity or individual inventory policies. This is expected to have an effect on the use of inventory. Specifically, considering the increased supplier redundancy, increasing inventory levels without any supply chain design changes in response large increases in supplier disruption probability could become sub-optimal. Furthermore, as base-stock policies are rarely used in the real-world because ordering costs are non-linear, it will be interesting to examine the effect(s) of different inventory policies at the warehouses. In particular, it might be interesting to observe (cost) differences, if any, between the currently considered base-stock policy and, for example, an (s,S) policy.
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