Managing products subject to obsolescence in the context of nonstationary demand and quantity discounts: a dynamic inventory control policy.

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Managing products subject to obsolescence in the context of

nonstationary demand and quantity discounts: a dynamic inventory

control policy.

MSc Technology & Operations Management MSc Supply Chain Management Faculty of Economics and Business

University of Groningen July 10, 2017 L.W. van Linge s2165813 l.w.van.linge@student.rug.nl Peizerweg 68-22 9726JM Groningen

Supervisor: dr. N.D. van Foreest Second-assessor: prof. dr. J. Wijngaard

Acknowledgements: I would like to thank my supervisor, dr. N.D. van Foreest of the Faculty of

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Abstract

Shop4 sells smartphone covers through their online web shop. Due to the introduction of new smartphone models, the demand for old covers diminishes. This often results in obsolete inventory. Therefore, the company wants to adjust the inventory control policy to avoid high obsolescence and reduce inventory costs. In addition to e-commerce, this problem also exists in other industries, such as the apparel business. This thesis proposes a dynamic inventory control policy based on the (Q, r) model, in order to deal with nonstationary demand and quantity discounts while avoiding high obsolescence. The results show that the best performing policy, adjusts the order size and reorder level (Q, r) when the reorder level is reached. The differences in the total profit of the policies are in general small. But the results should be considered in light of applying the policy to the whole assortment. As a decrease in inventory costs of a large amount of stock keeping units (SKUs) could have a significant impact on total profit.

Keywords: obsolete inventory, excess inventory, nonstationary demand, quantity discounts,

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1. Introduction

Shop4 sells smartphone covers through their online web shop. Due to the introduction of new smartphone models, the demand for old covers eventually plummets and suppliers stop producing these covers. Therefore, the case company wants to adjust their order policy for older smartphone covers in order to prevent high obsolescence and reduce inventory cost. However, nonstationary demand causes difficulties in determining when to change their order policy. Thus, the questions arise: (i) Should the company change the order policy? If so, (ii) when should the order policy be changed? Pinçe and Dekker (2011) state that the timing of the control policy change primarily determines the tradeoff between backordering penalties and obsolescence costs. However, for the case company there are no backorders, as insufficient inventory leads to lost sales. In addition, obsolete inventory can be salvaged, reducing the cost of obsolescence. Furthermore, the company would like to postpone adjustment of the order size, in order to benefit from higher quantity discounts on the purchase price as long as possible. Suppliers offer discounts to entice buyers to purchase more and to achieve economies of scale for transportation and processing costs (Benton & Park, 1996). But, large orders also result in increased inventory, and hence a higher risk of obsolete inventory.

Bakker, Riezebos and Teunter (2012, p.275) state that: “Items are subject to obsolescence if they lose their value over time because of rapid changes of technology or the introduction of a new product by a competitor, or because they go out of fashion.” Hadley and Whitin (1962) are one of the first to consider obsolescence by analyzing a finite horizon periodic review inventory system, in which the mean demand varies per period and a finite number of possible obsolescence dates exist. Song and Zipkin (1996) employed a Markovian model to represent the processes leading to obsolescence, and found that obsoleteness has a substantial effect on inventory costs and that this cannot be prevented by simple parameter adjustments. More recently, Pinçe and Dekker (2011) countered obsolescence by adapting to lower demand through changing the control policy in advance and letting the demand take away the excess inventory.

Just like the smartphone accessories market, the apparel business also runs the risk of obsolete inventory if too much of a product is bought. Retail companies could thus benefit from a method which determines when the order policy should be changed to deal with nonstationary demand and subsequently reduce or even prevent inventory obsolescence. Therefore, this thesis investigates various policies which determine when to change the order policy, for products subject to obsolescence, in the context of nonstationary demand. The policies adjust the order size and reorder level to the diminishing demand and are tested by means simulation. The simulation model acts a representation of the inventory system of the case company. In addition, a forecast is made to provide a total of four years of data for the selected items. The results show that obsolete inventory, in the context of nonstationary demand, can be reduced by applying a dynamic order policy.

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2. Theoretical background

This section discusses the literature relevant to the context of the case company, namely: inventory obsolescence, nonstationary demand and quantity discounts. The structure is as follows: first the definitions of the subjects are given after which associated methods are reviewed. By discussing the relevant theory, the differences with regard to the context of the case company can be determined. Subsequently, the research question is presented. After which the methodology follows.

2.1 Obsolete inventory

In order to stress the impact of obsolete inventory, the concept itself has to be explained. Various definitions and explanations of obsolete inventory exist. Masters (1991) describes the risk of obsolescence as the chance that the anticipated demand will fail to materialize, rendering the held inventory useless. In addition, he emphasizes that optimal stocking decisions must take into account potential obsolescence. Song and Zipkin (1996) state that in general obsolescence is the result of deteriorating demand, which relates to the state of demand and the chance of a future drop in demand. A recent and more extensive definition is given by Bakker, Riezebos and Teunter (2012, p.275) who state that: “Items are subject to obsolescence if they lose their value over time because of rapid changes of technology or the introduction of a new product by a competitor, or because they go out of fashion.” This implies that if an item becomes obsolete it impacts the profit of businesses through increased inventory cost. The money invested in obsolete inventory becomes sunk costs, but can sometimes be partly earned back through a salvage revenue. This so-called salvage revenue can be obtained by disposing the obsolete inventory (Willoughby, 2010), through selling it to a third party, such as a wholesaler or a recycling company. Nevertheless, obsolete inventory can reduce profits by up to 1% (Cattani & Souza, 2003). Pay (2010, p.1) even states that: “Obsolete inventory is one of the largest components of inventory cost and often is larger and more costly than executives are willing to admit.” This is presumably the case for companies with a large amount of stock keeping units (SKUs), as more items increase the risk of potential obsolescence. Furthermore, the risk of obsolescence exists in virtually every supply chain in which products are manufactured, transported and sold to customers. Considering that the reason of obsolescence ultimately lies in the uncertainty in both supply and demand (Pay, 2010). In addition to the aforementioned negative consequences of obsolete inventory, businesses should also consider the topic from environmental perspective. Especially, in a time when green supply chain management has become increasingly more important.

2.2 Methods dealing with obsolete inventory

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receive a salvage revenue. Older research on decreasing the cost of obsolescence determines the economic retention quantity, by calculating the amount of inventory to dispose of (Rosenfield, 1989; Tersine & Toelle, 1984). Other research has focused on the preceding step, namely preventing obsolescence. Hadley and Whitin (1962) are one of the first to consider obsolescence by analyzing a finite horizon periodic review inventory system, in which the mean demand varies per period and a finite number of possible obsolescence dates exist. They solve a dynamic programming problem which minimizes the cost of carrying inventory, stock outs, and obsolescence in order to determine procurement quantities. Song and Zipkin (1996) employ a Markovian model to represent the processes leading to obsolescence, and found that obsoleteness has a substantial effect on inventory costs and that this cannot be prevented by simple parameter adjustments. Pinçe and Dekker (2011) consider a continuous review inventory system of a slow moving item. They propose to adjust the control policy to the diminishing demand and find that the advance policy change results in significant cost savings. In addition, they state that the timing of the control policy change primarily determines the tradeoff between backordering penalties and obsolescence costs. Van Jaarsveld and Dekker (2011) propose a method that estimates the obsolescence risk of service parts by observing demand data. Which is subsequently used to enhance the inventory control for those parts. They state the method can be adopted by companies with sufficient data for a sufficient number of parts, and products with long life cycles. Yet, it should be noted that except businesses applying a make-to-order strategy, no method can prevent obsolescence completely without risking some lost sales. However, unless one could accurately predict the future. Therefore, the cost of lost sales should be considered when applying these methods.

2.3 Nonstationary demand

Treharne and Sox (2002, p.607) state that: “Most inventory problems exist in situations in which demand is both nonstationary and partially observed.” Neal and Willems (2009, p.388) state that: “It is the rule rather than the exception in most industries today.” Nonstationary relates to the change of the probability distribution of demand over time (Neale & Willems, 2009; Treharne & Sox, 2002), caused by short product life cycles, seasonality, customer buying patterns, economic conditions and other factors (Neale & Willems, 2009; Shang 2012). Not only does the distribution of demand change over the life cycle, the uncertainty does also (Neale & Willems, 2009). Furthermore, the distribution is never directly observed, but only partially through the actual demand values (Treharne & Sox, 2002). This implies that forecasts made in the context of nonstationary demand are unreliable, impacting the chance that the demand will materialize and subsequently increasing the risk of obsolete inventory (Masters, 1991). Appendix A provides an exemplary illustration of nonstationary demand over the life cycle of a product.

2.4 Methods dealing with nonstationary demand

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inventory problem under nonstationary demand. The demand is only partially observed and the distribution is represented by the state of a Markov chain. They find that their practical policies almost always provide better solutions than policies commonly used in practice. Neale and Willems (2009) address managing inventory in a supply chain facing stochastic, nonstationary demand. They show how inventory levels should adapt to changes in demand at a single stage and how nonstationary demand propagates in a supply chain. This allows them to link stages in the supply chain and subsequently apply a multi echelon optimization algorithm designed originally for stationary demand.

2.5 Quantity discounts

Suppliers may offer a discount on the purchase price if a certain quantity of an item is purchased (Taleizadeh & Pentico, 2014). In doing so the buyers are enticed to purchase more while the supplier achieves economies of scale for transportation and processing costs (Benton & Park, 1996). However, several types of quantity discounts are used in practice and have been discussed in literature. Three main types can be distinguished, namely: all-units discounts, incremental discounts and standard-quantity discounts. If a discount is offered for all purchased units, this is known as all-units discounts, in contrary to incremental discounts, where the discount only applies to the units above a certain amount (Katehakis & Smit, 2012; Taleizadeh & Pentico, 2014). Furthermore, standard-quantity discounts relate discounts to a combination of different standard quantities, e.g. units, boxes and cases, and the discount depends on the specific price of the standard quantity and the cost per unit is lower for larger standard quantities (Taleizadeh & Pentico, 2014). The type of discount offered by suppliers has implications for determining the total purchase cost. The supplier has defined ranges and subsequent discounts, mostly expressed in percentages. In all-units discounts the total cost of an order is obtained through the specific range within which the order quantity falls (Taleizadeh & Pentico, 2014). This implies that the total quantity determines the percentage discount applied to the order. Whereas in incremental discounts the cost is obtained through multiple ranges within which certain quantities fall (Taleizadeh & Pentico, 2014). Hence, the total purchase cost can be calculated by applying the discounts of specific ranges to specific quantities and summing the total. Standard-quantity discounts differ from the previous, because the price depends on the kind of standard quantity instead of a discount based on the total amount of items ordered. Appendix B provides examples on how the type of discount effects the calculation of the total purchase cost.

2.6 Methods dealing with quantity discounts

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Teng (2002) also provided the optimal order policy for this situation and presented an easy-to-use algorithm to find the optimal order quantity and replenishment time. Katehakis and Smit (2012) compute optimal (Q, r) replenishment policies under quantity discounts, and provide efficient algorithms for when there is a multi-breakpoint discount pricing structure.

Table 1: Summary of factors addressed by literature

2.7 Relating theory to the context of the case company

This research aims to identify if and when businesses should adapt their inventory control policy for products subject to obsolescence, in the context of nonstationary demand and quantity discounts. Literature mostly focusses on the individual elements of this particular context (see table 1). Whereas this thesis considers these factors jointly by answering the following research question:

If and when to adjust the order policy for products subject to obsolescence, in the context of nonstationary demand and quantity discounts?

To the best of the knowledge of the author, no previous research has combined the factors obsolete inventory, nonstationary demand and quantity discounts. However, it is shown in the previous sections that all these factors separately affect the management of inventory. Considering these factors jointly in the development of inventory control policies could lead to increased cost savings. As the whole might be greater than the sum of its parts. Therefore, the methodology developed in this research considers the influence of these factors jointly.

Authors Obsolete inventory Nonstatio nary demand Quantity discounts

Brounen (2017); Hadley and Whitin (1962); Pinçe and Dekker (2011); Song and Zikpin (1993, 1996); Van Jaarsveld and Dekker (2011)

X X

Rosenfield (1989); Tersine and Toelle (1984) X

Graves (1999); Hadley and Whitin (1961); Sethi and Cheng (1995); Treharne and Sox (2002); Neale and Willems (2009)

X

Burwell et. al (1997); Weng (1995) X X

Carlson et. al (1996); Chang (2002); Chung et. al (1996); Gafaar and Choueiki (2000); Katehakis and Smit (2002); Lin and Kroll (1997); Ouyang et al. (2002); Tersine et al (1995)

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3. Methodology

This section elaborates on the problem approach in general, after which the different elements of this approach are discussed in more detail. The case company wants to avoid inventory obsolescence of their older smartphone covers. Quantitative research is required to test the performance of different policies which adjust the order policy by a certain criterion. To this end, a simulation model is a suitable research method. In order to use simulation a forecast is needed. Section 3.1 clarifies how the forecast is made and which items are chosen. Section 3.2 then explains why simulation is chosen in this context and how the model is made. Finally, section 3.3 elaborates on the chosen policies and the key performance indicators (KPIs) used to compare the policies.

3.1 Demand forecast

The case company has provided historical demand data of the past three years. The selection of items subject to obsolescence is based on Van Jaarsveld and Dekker (2011). They estimate the obsolescence risk by observing demand data. The case company assigns a TUS label to items which show a decline in demand. Customers can still order these products, but the case company does not replenish any inventory. This is done to let the inventory level drop, if demand increases the company can choose to replenish the inventory again. Similarly, US labels are assigned to items that which the case company cannot order anymore. Customers can also still order these products, but the company will not replenish inventory for these products. These are the kind of items for which the case company could benefit from a change in order policy to avoid inventory obsolescence. The labels thus provide a good indication of suitable items. Table 2, shows the characteristics of the selected items1_.

Category Item 1 Item 2 Item 3 Item 4

US X X

TUS X X

Inbound date 04-07-2014 04-07-2014 01-04-2015 15-07-2015

Purchase price €0,37 €0,31 €0,29 €0,31

Selling price €3,16 €3,30 €3,17 €3,30

Table 2: Characteristics of selected items

Due to time constraints of this research four items are chosen, two items with the label TUS and two items with the label US. Around two years of historical data is available for the selected items. This data is extracted from the online portal of the case company called Shop4raad. The portal provides the raw data in the form of a .csv file and the contents of these files are copied to an Excel file in which the forecast is carried out. The data is then analyzed in order to make a proper forecast, i.e. one that resembles the demand patterns in the historical data. The items show different declines in demand which need to be accounted for in the forecasting procedure.

The forecasting procedure consists of assigning probabilities to the amount of orders on any day. This is done by dividing the frequency of having X-orders per day, by the total amount of days that the item was in assortment. The decline in demand is accounted for by letting the probabilities decrease over time. A random number between zero and one is then drawn to represent the forecasted demand on that day. This way demand can be forecasted per day, which is necessary for the simulation (see section 3.2). Furthermore, as the length of the historical data varies per item, it allows for having the same horizon in terms of total days. A total of four years is chosen because it is assumed that smartphone covers last about four years before becoming obsolete, due to the introduction of new

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smartphone models. Therefore, the amount of days that need to be forecasted in order to reach a total of four years is calculated for each item. Per item five different demand sequences are forecasted this way, to serve as input for the simulation model. The outcomes of the simulation model are subsequently averaged, this reduces the effect of potential outliers caused by the forecasting procedure. Hence, increasing the reliability of the results.

3.2 Simulation

Simulation models are capable of incorporating variability and interconnectedness. Making it possible to predict system performance, compare alternative system designs and to determine the effect of alternative policies on system performance (Robinson, 2014). In addition, simulation has the benefits of face validity (Robinson, 2014), making it possible to validate the model by observing the behavior when variables and parameters are changed. In this research simulation is used to simulate the inventory system of the case company, in order to test different order policies. A simulation model has various prerequisites for it to be valid and reliable. In this context the model has to be an accurate representation of the inventory system of the case company. This means that parameters and variables relevant to the case company have to be correctly modelled and incorporated. In addition, it has to be made in such a way that one would be able to reproduce it and get roughly the same results. For more information on simulation and its advantages see Robinson (2014).

The simulation of the inventory system of the case company is carried out in Excel 2016. The parameters and variables required for the simulation are depicted in table 3. Variables such as the demand, inventory level, and lost sales are shown per day. The inventory level is tracked threefold: at the beginning of the day, at the end of the day, and the inventory level with outstanding orders. Other inventory related variables, such as the reorder level and order size, are implemented in such a way that they can easily be adjusted. In addition, the simulation will automatically issue an order when the inventory level reaches the reorder point. The issued order is received after the lead time of 14 days. Financial parameters are used to subsequently quantify the revenue, purchase cost, holding cost, order cost and profit per day.

By tracking the variables daily, more insights into the results can be obtained. Specifically, it can be observed when an order policy leads to lost sales. Or when the reorder level is reached and an order is issued. In addition, it allows for validating the simulation model by observing its behavior. If for instance an order is issued, this order should be received 14 days later. This can then be visually verified. Appendix C provides more details on the simulation, and the variables and parameters. The next section explains the policies conceived in this research. Furthermore, the KPIs used are elaborated upon.

Inventory related Financial

Inventory level Purchase price

Order size (Q) Selling price

Reorder level (r) Order cost

Lead time Holding cost

Lost sales Quantity discounts

Overage (obsolete inventory) Salvage revenue

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3.3 Policies and KPIs

Literature describes various inventory control models, such as the EOQ model, the news vendor model and the base stock model (e.g. Silver, Pyke, & Peterson, 1998; Hopp & Spearman, 2008). In this research the (Q, r) model is used as a basis for inventory control, due to its inherent practicality and ability to provide powerful insights (Hopp & Spearman, 2008). In addition, it was one of the earliest attempts to model uncertainty in the demand process, and it is a synthesis of the EOQ model and the base stock model, thereby providing both cycle stock and safety stock (Hopp & Spearman, 2008). Cycle stock is normally used to avoid excessive replenishments costs, but for the case company the order cost is a fixed amount per item. Therefore, the purpose of cycle stock in this context is to gain quantity discounts. Whereas the safety stock is used to guard against uncertainty and potential lost sales. Literature does not provide consensus on when the order policy should be changed. However, this decision could be based on various criteria, such as the time that an item has been in assortment, the past demand or the inventory level. Yet, it remains unclear which criteria would work best for the case company. Therefore, this research uses multiple criteria to determine the timing of changing the order policy. The criteria lead to a specific date on which the order size and reorder level (Q, r) are adjusted. Since there is no base case to compare with the policies developed in this thesis, a ‘do nothing’ policy is used for comparison. The policies are depicted in the table below.

# Description Criterion Order policy

0 ‘Do nothing’ N/A Q1, r1

1 After two years Time in assortment Q1, r1  Q2, r2 2 After 4 periods of demand ≤ 2 Past demand Q1, r1  Q2, r2

3 After reorder level is reached Inventory level Q1, r1  Q2, r2  Q3, r3 Table 4: Policies

The ‘do nothing’ policy applies a fixed Q and r throughout the finite horizon of four years. The other policies base the timing of changing Q and r on the criteria shown above. The rationale behind policy 1 is that on average smartphone covers last about four years before becoming obsolete. Therefore, it is expected that after two years demand will stagnate or even decline, hence it would make sense to evaluate the order policy after two years. The decline in demand can also be used as a trigger to determine the timing of changing the order policy. Policy 2 makes use of this concept by observing the demand periodically. Periods of two weeks are chosen, as the case company keeps an overview of their demand per two weeks per item. Due to the erratic nature of nonstationary demand, a boundary of 4 periods of demand ≤ 2 is chosen in order to find a balance between changing the order policy too soon or too late.

Policy 3 is based on Graves (1999) and Pinçe and Dekker (2011), by combining the idea of an adaptive policy to deal with nonstationary demand with a control policy change to prevent obsolescence. The policy is modelled in such a way that Q and r are automatically adjusted when the reorder level is reached. The following example illustrates how this works: The inventory level on day 1 of the simulation consists of Q1 + r1. If r1 is reached, Q2 is ordered and r1 is adjusted to r2. Subsequently, if r2

is reached, Q3 is ordered and r2 is adjusted to r3. Q3 and r3 are then used until the end of the horizon.

The advantage of this policy lies in the automated nature of adjusting Q and r. In addition, Q1 is only

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Figure 1: illustration of policy 3

The main KPI used in this research is the total profit, which consists of the revenue minus the purchase cost, holding cost, order cost and the cost of lost sales. It is assumed that overage (obsolete inventory) is sold for a salvage revenue, which is also included in the total profit. The total profit is chosen as the main KPI because it incorporates all the benefits and costs throughout an items life cycle. It is therefore, the most determinative factor of performance. In addition, the performance is also quantified by the service level and overage level. As both are considered while developing the policies. Moreover, to explain any observed differences between the policies, the performance might be dissected into the individual cost components (e.g. purchase cost, holding cost, etc.). The KPIs are depicted in table 5. All the KPIs are expressed in percentages. The total profit of the best performing policy is set to 100%, after which the performance (expressed in percentages) of the other policies can be determined. Furthermore, the service level is determined by one minus the amount of lost sales divided by the total demand, while the overage level is determined by dividing the amount of obsolete inventory by the total demand.

Table 5: Key performance indicators

jan feb mrt apr mei jun jul aug sep okt nov dec

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3.4 Scenarios

This research makes use of multiple scenarios to determine which inventory control policy works best under different circumstances. The policies are based on the (Q, r) model and by adjusting Q and r between scenarios more insights into the performance of the policies can be obtained. Moreover, increasing the generalizability of the results.

All scenarios start on day 1 and at this moment in time demand data is not available yet, hence an exact calculation of r is impossible. Therefore, the first reorder level of each policy is based on the lead time demand, which is assumed to be a maximum of 1 unit per day. This method takes into account the relatively high cost of lost sales for the case company, since it can be expected that the demand will be smaller than 15 units during the lead time of 14 days. The second reorder level, which is applied on the dates assigned by the policies, is calculated by using the backorder cost model as explained in Hopp and Spearman (2008). The backorder cost model takes into account the cost of backorders versus the cost of carrying inventory to determine the critical fractile. However, for the case company backorders are essentially non-existent, since unavailability of their products mostly results in lost sales. Therefore, the cost of backorders is equal to the average cost of a lost sale.

In addition, this method also assumes that demand can be approximated by a continuous distribution (Hopp & Spearman, 2008). The approximation is then used to calculate the average demand during the lead time. For the case company this is not a viable method due to the nonstationary demand. Instead the demand of the past quarter is used to intercept and stress changes in demand, in the calculation of r2. A reliable calculation of Q is not available, because formulas assume that the demand

is stable and non-fluctuating over time. Therefore, Q will be based on the trade-off between quantity discounts and obsolescence. A larger order size means that a higher discount on the purchase price is acquired, whereas a smaller order size decreases the risk of inventory obsolescence. R2 is calculated

for each policy, using the backorder cost model. The average of all policies is then taken, to be able to relate performance differences to the timing of changing the order policy instead of different values of r. The first scenario will be based on avoiding obsolete inventory, the second scenario on quantity discounts and the third scenario on a combination of both (see table 6).

# Description Order policy

1 Minimize obsolete inventory Q1=30, r1=15  Q2=20, r2=10  Q3=10, r3=5 2 Maximize quantity discounts Q1=100, r1=10  Q2=50, r2=10  Q3=10, r3=5 3 Combination of both Q1=100, r1=10  Q2=20, r2=10  Q3=10, r3=5

Table 6: Description of scenario’s

The first scenario represents a modest situation, where Q and r are set to minimize obsolete inventory, instead of maximizing quantity discounts on the purchase price. The scenario starts with Q1=30 and

r1=15, then on the dates assigned by the policies Q and r are adjusted to 20 and 10 respectively.

However, policy 3 changes Q and r once more, to 10 and 5 respectively. In the second scenario, Q1 is

increased to 100 pieces in order to get the highest discount on the purchase price of 11%. R2 is

subsequently set to 10, because a larger order size results in reaching the reorder level in a later stadium. Therefore, it is likely that the demand during the lead time is smaller compared to the first scenario, and hence a smaller reorder level can be used. Q2 is then set to 50 to get the second highest

discount of 10%. The third scenario is a combination of maximizing quantity discounts and minimizing obsolete inventory, by starting with Q1=100 and then adjusting to Q2=20, as in the first scenario. It

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4. Experiments

In this section the simulation of the inventory system is used to test the policies conceived in section 3.3. The forecast, as explained in section 3.1, is used to generate five demand sequences per item. So for each policy five demand sequences are tested and the average of these sequences is then used to determine the performance. Yet before testing the policies, the validity and reliability of the simulation model is determined in section 4.1. After which the results of the individual scenarios are presented. The scenarios are then compared in section 4.3. Subsequently, a sensitivity analysis is performed to establish the robustness of the policies.

4.1 Validity and reliability

One of the advantages of simulation, mentioned in section 3.2, is face validity. Face validity is now used to validate the simulation model. As the model is a representation of a real life inventory system it should behave accordingly. This means that the validity can be determined by observing the behavior of the model through altering the variables and parameters given in section 3.2. If for example the reorder level is increased, the model should order more frequently. Similarly, if the order size is decreased, the model should also order more frequently and vice versa. The lead time can then be adjusted to see if orders are coming in on time. Thus, by changing the variables and parameters the validity can be determined. In addition, manual calculations are performed to assure the correctness of the key performance indicators.

The reliability of the simulation model is assured by repeating the above mentioned adjustments for multiple items and then comparing the outcomes. If for instance an adjustment to the reorder level would lead to lost sales for a product with low demand, then this should also occur for a product with high demand. Therefore, it is known that under similar circumstances the model should provide similar outcomes, every time. Reliability is further assured by using the simulation software in an appropriate manner. Meaning that if one would repeat the method used in this research, one would likely set up the simulation in a similar fashion. Thereby, increasing the likelihood of getting the same results. Specifics on how the validity and reliability has been determined can be found in Appendix E.

4.2 Results

After forecasting five demand sequences per item, the different policies can be tested. In doing so it will become clear whether it is worth the time to use a policy to determine when to adjust the order policy. In addition, it clarifies whether the case company could improve by using one of the policies conceived in this research. But first the assigned dates of the individual policies are explained. The following table shows the inbound date of the selected items and subsequently the dates assigned by the policies.

Table 7: Dates per policy per item

Each policy has a certain criterion, which leads to changing the order policy at a specific moment in time. It can be observed that in general policy 2 leads to an early assignment. This is caused by periods

Item 1 Item 2 Item 3 Item 4

Inbound date 04-07-2014 04-07-2014 01-04-2015 15-07-2015

Policy 0: ‘Do nothing’ N/A N/A N/A N/A

Policy 1: After two years 04-07-2016 04-07-2016 01-04-2017 15-07-2017 Policy 2: After 4 periods of demand ≤ 2 15-03-2015 03-01-2016 10-06-2015 14-09-2015

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of low demand early in the item’s life. It should be noted that the dates of policy 0 and policy 3 are not listed in table 7, because policy 0 does not adjust the order policy. Whereas policy 3 adjusts the order policy when the reorder level is reached, which differs per demand sequence and values of Q and r. The results are now given per scenario, after which the scenarios are compared.

4.2.1 Scenario 1: Minimize obsolete inventory

The first scenario represents a modest situation, in which Q and r are set to avoid obsolete inventory. Specifics on the scenarios are listed in section 3.4. Figure 2 shows a comparison between the total profit of the policies per item, expressed in percentages. Policy 3 leads to the highest performance, the differences are in general small (less than 10%). But policy 3 performs well for all items and is on average 3% better. In order to explain the observed differences, the service and overage level of each policy are listed in table 8.

Figure 2: Total profit per policy scenario 1

It can be observed that although policy 0, 1 and 2 have a service level of 100% for all items, the performance in terms of total profit is degraded by the amount of overage, which is significantly larger for policy 0, 1 and 2. Contrary to policy 3, which has the lowest amount of overage and the highest performance, even though this policy leads to some lost sales for item 2 and 4.

Service level Overage level

Policy Item 1 Item 2 Item 3 Item 4 Item 1 Item 2 Item 3 Item 4 Policy 0: ‘Do nothing’ 100% 100% 100% 100% 18,6% 17,4% 51,8% 19,5% Policy 1: After two

years

100% 100% 100% 100% 13,6% 8,8% 27,3% 14,7%

Policy 2: After 4 periods of demand ≤ 2

100% 100% 100% 100% 13,6% 9,9% 30,4% 13,6%

Policy 3: After reorder level is reached

100% 98,9% 100% 98,8% 7,3% 7,2% 18,1% 7,0%

Table 8: Service and overage level scenario 1

80,0% 82,0% 84,0% 86,0% 88,0% 90,0% 92,0% 94,0% 96,0% 98,0% 100,0% 1 2 3 4 To ta l p ro fit p erf o rm an ce Item

Comparison of policies

Policy 0: 'Do nothing' Policy 1: After two years

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By plotting the cumulative demand, inventory position and holding cost, the effect of the different policies can be visualized. This is done for item 3 and 2, because item 3 has the lowest overall demand and item 2 has the highest overall demand. Hence, providing a better picture on how the policies react to different demand patterns. It should be noted that the graphs are derived from the first demand sequence of each item. The gap between the inventory position (the red line) and the demand (the blue line) at the end of the horizon represents the amount of overage. Figure 3 to 6 show the policies for item 3.

Figure 3: Illustration of policy 0 for item 3

0 10 20 30 40 50 60 70 0 15 30 45 60 75 90 105 2-4-2015 2-4-2016 2-4-2017 2-4-2018 2-4-2019 H o ld in g co st (in eu ro 's ) Qu an tit y (in u n its ) Date Policy 0

Demand Inventory position Holding cost

Q1 Q1 0 10 20 30 40 50 60 70 0 15 30 45 60 75 90 105 2-4-2015 2-4-2016 2-4-2017 2-4-2018 2-4-2019 H o ld in g co st (in eu ro 's ) Qu an tit y (in u n its ) Date Policy 1

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It can be seen that policy 0 leads to the highest amount of inventory for item 3 (see figure 3). This happens because Q and r are fixed and the reorder level is reached 8 months before the end of the horizon, while the remaining demand is not large enough to justify this second order. Policy 1 adjusts Q and r after two years and because the demand is not large enough to reach r2, a second order is not

issued. Hence, resulting in a lower amount of overage compared to policy 0. Policy 2 adjusts Q and r after two months, leading to more overage in comparison to policy 1, because the smaller order size (Q2) results in a second order as now r2 is reached. Despite this, policy 2 outperforms policy 1, because

policy 1 carries a higher of amount inventory for a longer period of time. Resulting in higher inventory

Q2 0 10 20 30 40 50 60 70 0 15 30 45 60 75 90 105 2-4-2015 2-4-2016 2-4-2017 2-4-2018 2-4-2019 H o ld in g co st (in eu ro 's ) Qu an tit y (in u n its ) Date Policy 3

Q₂

Q3

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holding cost (25% higher). For item 3, policy 3 is the only policy in scenario 1 that issues multiple order sizes. This happens because Q2 is automatically ordered when r1 is reached and subsequently Q3 is

ordered when r2 is reached. Whereas for the other policies the use of Q1 and Q2 also depends on the

timing of adjusting Q and r. Meaning that if Q and r are adjusted too late, the inventory level will not reach r2,because an order of Q1 has already been issued. Similarly, if Q and r are adjusted to early, Q2

will be ordered before r1 was ever reached. Figure 7 to 10 show the policies for item 2.

0 8 15 23 30 38 45 53 60 0 30 60 90 120 150 180 210 240 7-7-2014 7-7-2015 7-7-2016 7-7-2017 7-7-2018 H o ld in g co st (in eu ro 's ) Qu an tit y (in u n its ) Date Policy 0

Q1 0 8 15 23 30 38 45 53 60 0 30 60 90 120 150 180 210 240 7-7-2014 7-7-2015 7-7-2016 7-7-2017 7-7-2018 H o ld in g co st (in eu ro 's ) Qu an tit y (in u n its ) Date Policy 1

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The policies for item 2 provide a different picture, since the demand for this item is much larger, resulting in multiple orders throughout the horizon. Policy 0 leads again to the highest amount of overage, which is caused by an order near the end of the horizon. Because policy 0 does not adjust Q and r, an order near the end of the horizon always leads to overage, since by then demand has stagnated and a smaller order size would have been sufficient. Policy 1 prevents this by adjusting the r before the end of the horizon. In general, the gaps between the inventory position (the red line) and demand (the blue line) are much smaller for this item. Therefore, it can be stated that higher demand leads to better performance for all the policies in scenario 1.

Demand Inventory position Holding cost cum

Q2

Q₃

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To further substantiate this statement, the holding cost of the policies for item 3 and 2 are compared in figure 11 and 12 respectively. Although item 3 has substantially less demand (34% less), the total holding cost at the end of the horizon is relatively larger. This is caused by carrying the inventory for much longer periods of time, which can also be seen from the staircase shape of the inventory position. Whereas the policies for item 3 only have 1 or 2 stairs, the policies for item 2 have considerably more, which implies that the inventory is carried for shorter periods of time. Hence, resulting in lower inventory costs, especially for policy 3. Furthermore, the order costs are a fixed amount per item. Therefore, the frequency of ordering does not impact the total profit through increased order costs. In addition, making the trade-off between order cost and holding cost non-existent. Moreover, leading to the realization that the decision of how much and how often to order should depend on the discount acquired for a certain quantity versus the holding cost of carrying this quantity and the associated risk of obsolescence.

Figure 11: Holding cost of the policies for item 3

Figure 12: Holding cost of the policies for item 2

0,00 10,00 20,00 30,00 40,00 50,00 60,00 70,00 2-4-2015 2-4-2016 2-4-2017 2-4-2018 2-4-2019

Holding cost item 3

Policy 0 Policy 1 Policy 2 Policy 3

0,00 10,00 20,00 30,00 40,00 50,00 60,00 7-7-2014 7-7-2015 7-7-2016 7-7-2017 7-7-2018

Holding cost item 2

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4.2.2 Scenario 2: Maximize quantity discounts

The second scenario represents a situation in which Q and r are set to maximize the use of quantity discounts. Q1=100 and Q2=50 provide the highest and second highest discount on the purchase price.

Figure 13 shows the total profit of the policies in scenario 2. Remarkably, the policies for item 3 lead to same performance. This is caused by the low demand for this item, since the average demand of five sequences is lower than 100, resulting in only one order for each policy. Still, policy 3 performs on average 5% better in this scenario.

Table 9 supports this statement as the overage level for item 3 is the same for all policies. However, it cannot be stated that in general the same overage level leads to the same performance. Because, policy 0, 1 and 2 have the same overage level for item 4, yet policy 2 performs better. This is caused by a significantly lower holding cost of policy 2, which is on average 28,7% lower compared to policy 0 and 1. Therefore, it can be stated that the timing of changing Q and r is better for policy 2. Nevertheless, policy 3 has the best performance in this scenario. Figure 14 to 21 now show the policies for item 1 and 2, since the policies for item 3 lead to the same performance in this scenario. It should be noted that for item 1 the graphs are based on demand sequence 1 and for item 2 on demand sequence 2, because the first sequence of item 2 lead to the same graphs for three policies.

years

100% 100% 100% 100% 31,7% 17,9% 68,7% 24,9% Policy 2: After 4

periods of demand ≤ 2

100% 100% 100% 100% 25,5% 17,9% 68,7% 24,9% Policy 3: After reorder

level is reached

100% 100% 100% 100% 6,6% 4,0% 68,7% 7,0%

Comparison of policies

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It can be seen that policy 0 and 1 lead to the same performance for item 1. This occurs because policy 1 adjusts Q and r too late, allowing the inventory level to reach r1 and issuing an order of Q1=100.

Whereas policy 2 adjusts Q and r before r1 is reached, thereby only issuing orders of size Q2. This

results in much lower holding cost, as the inventory is carried for shorter periods of time.

Q2

Q₂

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Q1

Q2

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For item 2 policy 1 and 2 lead to the same performance. Even though the dates of adjusting Q and r differ, the demand pattern causes both policies to behave in the same manner. After observing the graphs of scenario 2 it can be stated that the policies acquiring higher quantity discounts do not lead to higher performance in terms of total profit. Leading to the conclusion that quantity discounts in this scenario do not outweigh the increased inventory holding cost, caused by issuing larger orders and subsequently carrying more inventory for longer periods of time.

Q2

Q₂

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4.2.3 Scenario 3: Combination of both

The third scenario represents a situation in which Q and r are set to combine the use of quantity discounts and avoid obsolete inventory. Q1=100 provides the highest discount on the purchase price

and Q2=20 is used to avoid high obsolete inventory at the end of the horizon. The total profit of the

policies is shown in figure 22. The policies for item 3 lead again to the same performance, because Q1

is still larger than the average demand of five sequences for this item. But on average policy 3 performs 5% better in this scenario.

This is caused by a much lower amount of overage (see table 10). Figure 23 to 26 show the policies for item 1 and 2 in this scenario. Again, demand sequence 1 is used for item 1 and sequence 2 for item 2, since the first sequence of item 2 lead to the same graphs for three policies. Noticeably, policy 0 and 1 for item 1 and 4, and policy 1 and 2 for item 2 leads to same performance. By observing figure 20 to 23 this behavior can be explained.

years

100% 100% 100% 100% 31,7% 14,7% 68,7% 24,9% Policy 2: After 4

periods of demand ≤ 2

100% 100% 100% 100% 14,2% 14,7% 68,7% 13,0% Policy 3: After reorder

level is reached

100% 100% 100% 100% 6,6% 4,0% 68,7% 5,8%

Comparison of policies

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Q₁ 0 20 40 60 80 100 120 140 0 30 60 90 120 150 180 210 7-7-2014 7-7-2015 7-7-2016 7-7-2017 7-7-2018 H o ld in g co st (in eu ro 's ) Qu an tit y (in u n its ) Date Policy 1

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As in the second scenario, r1 of policy 1 is reached before r has been adjusted. This causes the inventory

system to issue another order of Q1=100, after which the inventory level never reaches r2. Hence,

resulting in the same performance as policy 0, which holds Q and r fixed over time. For policy 2 it can be seen that Q and r are adjusted before r1 was reached, resulting in only orders of Q2=20.

Consequently, leading to much lower holding cost in comparison to policy 0 and 1. In addition, also resulting in less overage. Figure 27 to 31 show the policies for item 2 in this scenario.

Q2

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Figure 28: Illustration of policy 1 for it em 2

Q1

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Policy 1 and 2 show the same behavior as in the second scenario, resulting in the same performance of both policies. As in the previous scenario’s, policy 3 leads to the highest performance. The holding cost of policy 3 is much lower compared to the other policies, which is caused by the inventory position closely following the demand pattern. Even though this policy benefits less from quantity discounts, the lower holding cost and amount of overage result in higher performance. To further substantiate these statements, a comparison of the policies between the scenarios is performed in the next section.

Q2 Q1 0 16 31 47 63 78 94 109 125 0 40 80 120 160 200 240 280 320 7-7-2014 7-7-2015 7-7-2016 7-7-2017 7-7-2018 H o ld in g co st (in eu ro 's ) Qu an tit y (in u n its ) Date Policy 3

Q2

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4.3 Comparison of scenarios

This section compares the results of the three scenarios in order to select the best performing policy. After which a sensitivity analysis is carried out in the next section. Figure 31 shows a comparison of the policies between scenarios. The best performing policy in scenario 1 (policy 3) has been set to 100%, and the scenarios are compared to this policy.

Figure 31: Comparison of policies between scenarios

In general, policy 3 leads to the highest total profit, while scenario 1 results in the best performance for all policies. Except for policy 3, which performs better in scenario 2 and 3 for item 2 and 4 (see figure 31). To explain the observed differences, the best performing policy (policy 3) is now compared with the second best performing policy (policy 2) in figure 32 and 33.

Figure 32: Comparison of policy 2 and 3 for item 1

70,0% 72,0% 74,0% 76,0% 78,0% 80,0% 82,0% 84,0% 86,0% 88,0% 90,0% 92,0% 94,0% 96,0% 98,0% 100,0% 102,0% P0 P1 P2 P3 P0 P1 P2 P3 P0 P1 P2 P3 P0 P1 P2 P3

Item 1 Item 2 Item 3 Item 4

To ta l p ro fit p erf o rm an ce

Comparison of policies between scenarios

Scen 1 Scen 2 Scen 3

0,00 15,00 30,00 45,00 60,00 75,00 90,00 105,00 120,00 135,00 150,00 165,00 180,00 195,00 210,00

Scen 1 Scen 2 Scen 3 Scen 1 Scen 2 Scen 3

Policy 2 Policy 3 In ve n to ry co sts (e u ro 's )

Comparison policy 2 and 3 for item 1

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Figure 33: Comparison policy 2 and 3 for item 2

It can be observed that the acquired discount (expressed in euro’s) is relatively low for both policies. The second scenario, in which quantity discounts are maximized, does lead to the highest discounts. Yet, the differences between scenarios are small. In addition, policy 2 acquires higher discounts than policy 3, but the amount of overage (expressed in euro’s) is also larger. This means that policy 2 builds up more inventory throughout the horizon of four years. Consequently, resulting in higher purchase cost, order cost and inventory holding cost. Appendix F shows the development of the inventory costs and quantity discounts of policy 2. Figure 32 and 33 also indicate that quantity discounts are associated with higher holding cost, since the holding cost is much larger in the second and third scenario. It can be concluded that policy 3 results in the highest total profit and subsequently the sensitivity of the results of policy 3 are tested in the next section. Figure 34 illustrates the performance of policy 3 for all items and in all scenarios.

Figure 34: Comparison of scenarios policy 3

0,00 15,00 30,00 45,00 60,00 75,00 90,00 105,00 120,00 135,00 150,00 165,00 180,00 195,00

Scen 1 Scen 2 Scen 3 Scen 1 Scen 2 Scen 3

Policy 2 Policy 3 In ve n to ry co sts (in eu ro 's )

Comparison policy 2 and 3 for item 2

Purchase cost Order cost Holding cost Discount Overage

70% 72% 74% 76% 78% 80% 82% 84% 86% 88% 90% 92% 94% 96% 98% 100% 102% 1 2 3 4 To ta l p ro fit p erf o rm an ce Item

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4.4 Sensitivity analysis

Policy 3 leads to the highest total profit performance, but what happens if this policy is applied in another context? In order to investigate the robustness of the policy, the relationship between input- and output variables can be tested by means of a sensitivity analysis (Pannell, 1997). In the previous section it became apparent that the most prominent factor seems to be holding cost, as the holding cost for policy 3 is much lower. In addition, the influence of quantity discounts on total profit seems to be insignificant, since policies and scenarios leading to higher discounts performed less.

Moreover, purchasing larger quantities lead to higher total purchase cost, order cost, holding cost and overage cost. The sensitivity analysis therefore consists of using different values of holding cost, discounts and salvage revenue, to account for the consequences of buying larger quantities. Whereas higher discounts reduce the purchase cost, a lower holding cost presumably makes quantity discount more attractive. In addition, a higher salvage revenue decreases the cost of obsolete inventory. The said parameters are changed and tested for item 2 and 3.

4.4.1 Holding cost

The holding cost of the case company is €0,52 per year, which is relatively low. Still, it is interesting to see whether the performance of the policies would differ under even lower holding cost. Especially considering that the total profit of policy 0, 1 and 2 is degraded by high total holding cost (see figure 11 and 12). Figure 35 shows the total profit performance of the policies for item 1 under different values of holding cost. It can be seen that even with much lower holding costs, policy 3 outperforms the other policies in every scenario.

Figure 35: Sensitivity of total profit to holding cost for item 1

85,0% 86,0% 87,0% 88,0% 89,0% 90,0% 91,0% 92,0% 93,0% 94,0% 95,0% 96,0% 97,0% 98,0% 99,0% 100,0% P0 P1 P2 P3 P0 P1 P2 P3 P0 P1 P2 P3 P0 P1 P2 P3

Holding cost /8 Holding cost /4 Holding cost /2 Holding cost /1

To ta l p ro fit p erf o rm an ce Item 1

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However, the performance of the policies for item 2 present a different picture (see figure 36). Noticeably, policy 3 performs less in scenario 1 when the holding cost is decreased. Instead policy 2 leads to the highest performance when the holding cost is halved, and policy 1 when the holding cost is decreased even more. Figure 33 does not provide an explanation for why this is the case, because policy 2 performs worse in every aspect and even with a decrease in holding cost, the holding cost is still higher compared to policy 3. But, table 8 provides an explanation, in the first scenario only policy 3 lead to lost sales. Since the cost of lost sales is the lost profit, the performance of policy 3 is degraded by the lost profit. The lost profit thus outweighs the amount saved by lost sales (the saved purchase- and holding cost), when the holding cost is decreased. Therefore, policy 1 and 2 outperform policy 3 in scenario 1, for item 2, as these policies do not result in any lost sales. Yet, in all other scenarios and in the context of the case company policy 3 performs best.

Figure 36: Sensitivity of total profit to holding cost for item 2

Holding cost /8 Holding cost /4 Holding cost /2 Holding cost /1

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4.4.2 Quantity discounts

The case company receives all-units discounts at various quantities. It is now investigated whether the policies, which order larger quantities more frequently, would perform better if the granted discounts were higher. Figure 37 shows the total profit performance of the policies for item 1 when the received discounts are increased. It is shown that also with higher discounts, policy 3 outperforms the other policies in every scenario. In addition, it can be seen that the increase in total profit is only marginal.

Figure 37: Sensitivity of total profit to quantity discounts for item 1

Figure 38 shows more or less the same behavior of the policies for item 2. Except that for this item scenario 1, for policy 0, 1 and 2, always outperforms the other scenarios. Policy 3 outperforms all other policies in all scenarios.

Figure 38: Sensitivity of total profit to quantity discounts for item 2

Discounts x1 Discounts x1,5 Discounts x2 Discounts x2,5

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4.4.3 Salvage revenue

The salvage revenue in this research was assumed to be 50% of the purchase price without discounts. However, the performance of the policies that result in obsolete inventory might improve when the salvage revenue is increased. Figure 39 shows that the increase in total profit is in very small for item 1. The performance of policy 2 in scenario 2 improves the most, by just 2%.

Figure 39: Sensitivity of total profit to salvage revenue for item 1

The increase in total profit is even smaller for item 2 (see figure 40). For both items the increase in total profit is very small. Thus, the sensitivity of the total profit to salvage revenue is insignificant. Whereas the sensitivity to the holding cost is much higher.

Figure 40: Sensitivity of total profit to salvage revenue for item 2

Salvage rev 50% Salvage rev 75% Salvage rev 100% Salvage rev 125%

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5. Discussion

In the experiments section four policies were tested to determine if and when to adjust the order policy for products subject to obsolescence, in the context of nonstationary demand and quantity discounts. Four items were selected based on Van Jaarsveld and Dekker (2011), by estimating the obsolescence risk from demand data of the case company. It became apparent that items with the label TUS or US were subject to potential obsolescence, as the case company used these labels for items with high inventory levels. A forecast was then made for the selected items, based on the probability distribution of the historical demand. The change of the distribution was taken into account by letting the probabilities decrease over time. This decrease was assumed to be linear. However, one could argue that this does not represent reality, as it is unknown whether such a decrease is for instance linear or exponential. It could also be that the decrease has stagnated and does not decrease any further in the future. Although this is a valid argument, this unpredictability is inherent to nonstationary demand.

The policies conceived in this research used different criteria to determine the timing of adjusting the inventory control policy. Except for policy 0 that represented a base case in which the order policy was fixed throughout the horizon of four years. Policy 1 adjusted the order policy after two years and policy 2 after four periods of demand less than or equal to two. Policy 1 was based on the assumption that smartphone covers last about four years before becoming obsolete, due to the introduction of new smartphone models. Therefore, it was argued that reviewing and adjusting the policy after two years seemed appropriate, as it was expected that by then demand had declined. However, this might not suit items with shorter or longer life cycles. Policy 2 used the past demand as a trigger itself, by observing the demand periodically. Periods of two weeks were chosen, as the case company keeps an overview of their demand per two weeks per item. Policy 3 was based on Graves (1999) and Pinçe and Dekker (2011), by combining the idea of an adaptive policy to deal with nonstationary demand with a control policy change to prevent obsolescence. This was accomplished by modeling the policy in such a way that the order size and reorder level were automatically adjusted when the reorder level was reached.

Multiple scenarios were investigated to see whether the performance of the policies would vary under different circumstances. The first scenario represented a situation in which quantity discounts were maximized. The second scenario focused on avoiding obsolete inventory and the third scenario on a combination of both. However, more scenarios could be thought of to represent reality, which could potentially further increase the generalizability of the results. Nevertheless, policy 3 outperformed the other policies in all scenarios. In addition, the second and third scenario lead to even better performance for item 2 and 4. For item 3 all policies lead to the same performance in scenario 2 and 3. The low demand for this item, caused Q1 plus r1 to immediately result in obsolete inventory. A

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After analyzing the scenarios, it seemed that the holding cost was the most prominent factor influencing the total profit performance of the policies. Subsequently, a sensitivity analysis was performed in order to test the relationship between input- and output variables (Pannell, 1997), and determine the sensitivity of the results to the parameters of the case company. The sensitivity analysis showed that when the holding cost was decreased the differences in performance of the policies became smaller. Still, the holding cost of the case company is relatively low and it is questionable whether other companies would charge lower costs.

Pinçe and Dekker (2011) stated that the timing of adjusting the control policy primarily determines the tradeoff between backordering penalties and obsolescence costs. However, in the context of the case company insufficient inventory leads to lost sales, making backorders nonexistent. In addition, the cost of lost sales is relatively high and the reorder levels were set to avoid lost sales as much as possible. Therefore, it was found that in this particular context, the timing primarily determines the tradeoff between quantity discounts, holding costs and obsolescence costs. Since the policies that changed the order policy later in the horizon, ordered larger quantities more frequently. Hence, obtaining a higher total discount, but also resulting in higher holding- and obsolescence costs. In the sensitivity analysis it was found that the amount saved through higher discounts never exceeded the costs of ordering larger quantities more frequently. Even if the discounts granted were multiplied by 2,5, the holding cost divided by 8 or the salvage revenue increased to 125%, the benefits from quantity discounts could not justify the impact on inventory costs. That is not to say that quantity discounts should not be considered while developing order policies. But, to stress the possible consequences of ordering larger quantities more frequently just to obtain higher discounts.

Managing products subject to obsolescence in the context of nonstationary demand and quantity discounts: a dynamic inventory control policy.