An Organized Approach to Optimize Inventory Levels by Focusing on Lot Sizes

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An Organized Approach to Optimize Inventory

Levels by Focusing on Lot Sizes

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Abstract

Purpose – The aim of this study is to develop an organized approach to optimize inventory levels by

focusing on lot sizes, particularly at case company X. The practical problem is that most make-to-stock manufacturing companies produce an increasing variety of products. As a result, it becomes difficult to determine an optimal lot size for each product.

Methodology – A numerical analysis at case company X is conducted. First, the demand for products

during the year is calculated and the products are classified into 12 groups. Afterwards, the product characteristics of the product groups are put into a (Q, r) inventory model and the optimal lot sizes are determined.

Findings – For case company X, lot sizes have a large impact on the inventory value, especially for low

demand, high-value products. For these products, results indicate the theoretical yearly inventory costs can be decreased with €161.454 (-77%) and the theoretical overproduction value with €13.125 (-63%).

Research limitations and future work – The inventory model in this study assumed unlimited

production capacity. Future work could focus on the capacity constraint in combination with the inventory model to determine the optimal lot size.

Originality/value – This approach can be used for other make-to-stock companies, which produces a

high variety of products in fixed lot sizes, to optimize inventory levels.

Keywords: Deterministic Demand, Lot Sizing, Inventory Levels, (Q, r) inventory policy, Double ABC

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Content

2.2 Inventory control methods ... 5

2.3 Economic Order Quantity... 8

2.4 Inventory classification ... 10 3. Methodology ... 11 3.1 Case description ... 11 3.1.1 Problem description ... 12 3.2 Research design ... 14 3.3 Data collection ... 15

3.4 Reliability, validation & verification ... 15

3.5 Sensitivity analysis ... 16

4. Analysis ... 17

4.1 Demand investigation ... 17

4.2 Classification of key modules ... 19

4.3 Inventory costs ... 21

4.4 Inventory model ... 25

5. Results & interpretation ... 29

5.1 Inventory costs ... 29

5.2 Overproduction value ... 34

5.3 Sensitivity analysis ... 37

6. Conclusion, discussion & recommendations ... 39

6.1 Conclusion ... 39

6.2 Discussion ... 40

6.3 Recommendations ... 41

6.4 Limitations and further research ... 42

References ... 43

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1. Introduction

In today’s business environment, make-to-stock (MTS) companies produce a growing variety of products to satisfy diversified customer’s needs. This variety of products contributes to higher sales, by enabling better adaption to customer desires. Due to increasing product variety, more and more different subassemblies are produced on a shared resource. To avoid many setups, subassemblies and products are produced in large, fixed lot sizes and consequently flow through the supply chain. For example, subassemblies are only produced in full trolleys and finished products are only produced by full pallets because of practical reasons.

Lot sizes of products are kept on stock at the end of the final process. The lot size determines the volume flexibility of the company to meet the demand. The assumption can be made that larger lot sizes result in less volume flexibility to meet the demand. If for example the demand is 400 products and the fixed lot size is 1000 products, then there are 600 (1000-400) products left. If the demand is high enough, there is no problem because there is a high probability the products will be sold in the next period. However, when the product demand is low there is stock for many weeks and risk of obsolesce is high. This especially applies to the high-tech industry where product life cycles are getting shorter (Aytac & Wu, 2010). Furthermore, in high-tech industry products have high value and any decrease in inventory can save money. The inventory system with lot sizes is typically a (Q, r) inventory policy and this is widely applied in practice.

Harris (1913) developed the Economic Order Quantity (EOQ) formula. The aim of this formula is to minimize the ordering and holding cost to determine the optimal lot-size. However, this is under the assumption of an infinite time horizon and stable demand. Furthermore, this is based on zero lead time so without safety stock. With deterministic, seasonal demand it becomes harder to determine a fixed lot-size because of its seasonal variability. In this context, a variable lot lot-size is preferable, but with a (Q, r) inventory policy the lot size is fixed. In this study, the (Q, r) inventory policy is given and the aim is to develop an approach to determine the optimal lot size for case company X. The following research question has been formulated:

“How can inventory levels under deterministic seasonal demand be optimized by focusing on the lot sizes?”

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2. Theoretical background

The theoretical background starts with a general explanation of inventory management (section 2.1). Thereafter, the inventory control methods are discussed and the (Q, r) inventory method is explained in more detail (section 2.2). Section 2.3 gives the insights of the Economic Order Quantity (EOQ). Finally, the inventory classification is discussed (section 2.4).

2.1 Inventory management

Inventory management is the planning and controlling of inventories in order to meet the competitive priorities of the organization (Krajewski, Ritzman, & Malhotra, 2013). Effective management of inventories is not only essential for one organization but for the whole supply chain. Efficiency in the supply chain can only happen if the right amount of inventory is flowing through the supply chain – through suppliers, the firm, distribution centres, retailers, and customers (Krajewski, Ritzman, & Malhotra, 2013).

Inventory management is all about having the right inventory in the right quantity, in the right place, at the right time, and at the right cost. For efficient management of inventories, managers require to make decisions on two fundamental questions (Bushuev, Guiffrida, Jaber, & Khan, 2015):

1. How many products are we going to order? 2. When are we going to place the order?

The answers on these two questions are critical input for inventory management. There are different inventory control methods about how to deal with these two questions. Inventory control methods are discussed in the next section.

2.2 Inventory control methods

Inventory control methods differ from each other in how much to order and the moments to trigger an order.

The moment when an order can be triggered can be fixed or variable. A fixed reorder moment means that after each t periods an order is placed. Every time after t periods an order is placed, and it does not matter how much inventory there is. A variable reorder moment means that time does not trigger the order, but the inventory position does. If the reorder point is at or below the inventory position an order is placed.

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Fixed reorder moment Variable reorder moment Variable lot size (T, s) method (s,S) method

Fixed lot size (T, Q) method (Q, r) method

Table 2.1: Overview of inventory control methods

The (Q, r) method is explained in more detail because case company X controls the inventory with this method.

(Q, r) inventory control method

In this study, a (Q, r) inventory policy is tested for case company X. The Q and r variables are considered when aiming to minimize the total inventory cost (Hadley & Whitin, 1963). The main question of case company X is how much to order (Q), with the aim of minimizing the total cost. So, the focus is on lot size (Q) and not on the reorder point (r). However, these two decision variables interact with each other and will be shortly discussed.

Figure 2.1: Visualization of the (Q, r) inventory policy (Axsäter, 2015)

Q and r interact with each other in terms of inventory, production or order frequency, and customer service. It is important to recognize that two parameters generate two fundamentally different kinds of inventory (Hopp & Spearman, 2011). The lot size (Q) affects cycle stock. The cycle stock is the inventory between ordering cycles. The reorder point (r) covers the demand during lead time and affects the safety stock. The safety stock is the inventory that protects against uncertainties in supply and demand to prevent stockouts. The (Q, r) inventory policy is visualized in figure 2.1. The total inventory is the reorder level plus the order quantity minus the demand during lead time.

Most of the time only one lot size (Q) is needed to get above reorder point (r). However, it is also possible that the inventory position is sufficiently low that there are more lot sizes needed. Therefore, the considered policy is sometimes denoted as (nQ, r) policy (Axsäter, 2015). Still, with smaller lot sizes there are higher order costs involved. This is discussed in section 2.3.

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Effects of lot sizes

The focus of this study is to determine the effect of lot sizes on the inventory level and the number of orders by a variable, seasonal demand. To make it clear for the reader the effects of lot sizes on the inventory level and the number of orders is explained by an example. In this example is assumed the demand is stable, unlimited production capacity, and no lead time.

In the figure below three different lot sizes are visualized. The large lot size results in the highest average inventory, however, there is only one order needed. The medium lot size has a lower average inventory, but there are two orders needed and consequently, there are higher order costs. The small lot size has the lowest average inventory but the highest order costs. The optimal fixed lot size is a tradeoff between inventory level and number of orders needed to satisfy demand.

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2.3 Economic Order Quantity

Harris (1913) proposed the Economic Order Quantity (EOQ) model to deal with the lot-sizing problem. The objective of the EOQ model is to minimize the sum of the ordering and holding costs. However, in the EOQ model, there are assumptions that are not in line with this study. One of these assumptions is that the demand is constant over time. This is not the case for case company X. Nonetheless, the insight from Harris’s work can be used in this study. The fundamental insight from his work that there is a trade-off between lot size and level of inventory (Hopp & Spearman, 2011). Increasing the lot size increases the average inventory level, but reduces the frequency of ordering. By using setup cost to penalize frequent replenishments, an economic trade-off can be made. This is also visualized in the figures below. Holding costs increases proportionally as the lot sizes increases see figure 2.3. The larger the lot size is, the lower the influence on the setup and transport cost, see figure 2.4.

Figure 2.5 below combines the two previous cost functions with a green line as the sum of the two cost functions. The point at which the inventory costs are the lowest gives the optimal lot size. In figure 2.5, this point is indicated with the dots.

Figure 2.5: Inventory costs of lot sizes

Lot size

Inventory

costs

Figure 2.4: Holding costs of lot sizes Figure 2.3: Set up / transport cost of lot sizes

9 Next, there are some advantages and disadvantages of having a small lot size or large lot size. Advantages and disadvantages are summarized in the table below.

Small lot size Large lot size Advantages - Low capital;

- Low storage costs of product inventory;

- High flexibility if quantities change at suppliers or buyers.

- Lower administrative costs; - Fewer setups in production; because of large inventories; - Lower cost of shipping.

Disadvantages - The costs of frequent ordering; - High risk of setups in production

because of a small product inventory.

- High capital;

- Lower flexibility if quantities change at suppliers or buyers.

- High storage costs of product inventory.

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2.4 Inventory classification

ABC inventory classification systems are widely used by business firms to streamline the organization and management of inventories (Teunter, Babai, & Syntetos, 2010). This is needed because firms need to manage an increasing number of products in their inventory and this is also the case for company X in this study. The main purpose of classification is to simplify the task of inventory management by dividing products into different classes by their importance. By having different classes there can be set different inventory control parameters to each class based on their importance, such as service level and order quantity.

Originally, ABC analysis or Pareto analysis was designed for three classes: A, B, and C. The process of dividing products into classes is that the A class represent 20% of the products but account for 80% of the importance (Krajewski, Ritzman, & Malhotra, 2013). A-products should be extensively checked because these contribute most to the profit of the organization. The demand value is the most commonly applied criterion followed by demand volume (Teunter, Babai, & Syntetos, 2010). Demand value is the demand for a product multiplied by the price of a product.

Sometimes, it is insufficient by looking at one criterion at a time for managing inventories. Therefore, the ABC analysis can be extended to more classes, simply by dividing the ranked products into more groups (Teunter, Babai, & Syntetos, 2010). Various multi-criteria methodologies have been developed, such as weighted linear programming, analytic hierarchy process (AHP) and operations-related groups (ORG). However, these multi-criteria methodologies can become complex quickly. An alternative for using multi-criteria is a double ABC classification.

A double ABC classification considers two inventory criteria measures. It is visualized in figure 2.6 below. Going from a single to a double criterium result in an increase of products classes. Teunter et al (2010) mention the certainty of supply, the rate of obsolescence and lead time as possible inventory criteria. Other possible criteria are annual requests, order size requirements and demand distribution (Hadad & Keren, 2013).

AC AB AA BC BB BA CC CB CA Inventory criteria 1 In ve n to ry c ri te ri a 2

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3. Methodology

The methodology starts with a case description and a problem description. This is a description of the current situation of case company X and the motive of this study. Further, a detailed problem description is given. This is done in section 3.1. In the second section, the steps of this study are discussed. The aim of the research design is to structure the study (section 3.2). Data collection is stated in section 3.3. It discussed data that is used as input for this study. Reliability, validation and verification are discussed in section 3.4. The aim of this section is to ensure that the inventory model is sufficiently accurate. The last section is the sensitivity analysis. Here the parameters are described to test the robustness of the inventory model (section 3.5).

3.1 Case description

The case company of this study is company X. Company X consist of a Production Site, a Customization Centre and a Distribution Centre (DC), see figure 3.1 below. The production site of company X produces a technical, consumer product. To stay competitive in the market, company X is constantly striving for innovative products which leads to a large product portfolio. An increasing variety of products have to be produced and shipped through the supply chain. Further, company X is a make-to-stock company which means that production is based on forecasting. This is also known as a push supply chain. Each member in the supply chain has their own inventory. The Commercial Organization and the Planning Department communicate with each other about stock levels and demand on a weekly basis to determine how many products are needed for the coming period.

Figure 3.1: Overview of supply chain case company X

The production site delivers products, named key modules to the customization centre. The customization centre package the products for specific customers and accessories needed for the key module is added. After this, the key module is called a finished product. Afterwards, finished products are shipped to different DC’s and finally to the retailers.

Motive of this study

After production at the production site, key modules are stacked in quantities of 720 units on a pallet, and pallets are transported two at the time (a total of 1440) to the customization centre. At the

12 customization Centre, key modules are on stock, see figure 3.2. When there is demand for a key module, it is picked and packed for a specific retailer. However, most of the time not all 1440 key modules are needed. Only the demanded quantity is packed, and the remainder is still on stock.

The yearly demand for products of case company X is relatively stable. Every year around 10 million products are sold. However, product variety is increasing. Every year there are new key modules introduced and many “old” types are still in production. This means that the total demand per key module is lower. With the current lot size of 1440, there is a high probability that too many key modules of a certain type are produced. Consequently, the wrong types of key modules are kept on stock, while maybe other types are out of stock.

Figure 3.2: Focus of this study

3.1.1 Problem description

The problem of this study involves the limited production capacity, the seasonal demand pattern and the lot sizes of case company X. First, the effects of the limited production capacity on seasonal demand are explained. Thereafter, the impact of the current fixed lot size is elaborated on.

Demand and production capacity

Case company X faces a seasonal demand pattern for their products. At the end of the year, there is a significant higher demand compared to the beginning of the year. Further, at the end of the year, the demand is higher than the production capacity. Therefore, there is overproduction needed at the beginning of the year to fulfil the demand at the end of the year, see figure 3.3. In the first eight months, the production is higher than the demand. This is the building up inventory period. The last four months the demand is higher than the production capacity. The inventory is needed to fulfil the demand for the last 4 months. Currently, the yearly production is higher than the yearly demand because of the fixed lot size. This is explained in the next sections. For case company X it is preferable to end the year with a low inventory level because of fiscally reasons. In other words, overproduction has to be minimized, especially at the end of the year.

Figure 3.3: General demand and production pattern case company X

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Fixed lot sizes

Case company X works with fixed lot sizes. The issue of a fixed lot size is that an integer of that lot size must be produced. Figure 3.4 below shows the effect of the current lot size of case company X. If for example, the demand is 3000 units for a specific key module, then 3 lots of 1440 units (total of 4320 key modules) are needed to fulfil the demand. This way, there is overproduction of 1320 units (4320-3000) for this type of key module. This overproduction time could be used to produce another type of key module. Because of the fixed lot size of 1440, this is not possible. The fixed lot size of 1440 results in less volume flexibility in meeting the demand. Volume flexibility can be simply defined as the ability to alter production volumes (Upton, 1994). Consequently, with low volume flexibility, there is more overproduction and higher inventory levels than needed.

Figure 3.4: Effects of current lot size (1440)

In figure 3.5 the same demand pattern is shown as in figure 3.3. However, the production and inventory level of three different lot sizes are added. For all three lot sizes, there is overproduction in the first eight months because of the limited production capacity. This is the inventory building up period.

With a “large” lot size there is most overproduction due to less volume flexibility. Consequently, there is a high inventory level during the year and production capacity is high.

With a “medium” lot size there is less overproduction due to more volume flexibility. Consequently, there is a lower inventory level during the year. However, the production capacity is lower because with a “medium” lot size more production setups are needed. The question arises if there is enough production capacity with a “medium” lot size to fulfil the demand.

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Figure 3.5: Effects of lot sizes on inventory levels and production capacity

For case company X, the issue of fixed lot sizes can be divided into two perspectives, namely production perspective and market perspective. This study researches the impact of changing lot sizes on the inventory level at the Customization Centre (market perspective) and assumes that there is unlimited production capacity. Another study Mussche (2019), is carried out simultaneously with this study, focusing on the impact of lot sizes on the production capacity of case company X (production perspective). Mussche (2019) researched the impact of changing the lot size on the production capacity. To summarize, this study determines the impact of lot sizes from a market perspective whereas Mussche (2019) determines the impact from a production perspective. The optimal for case company X as a whole is to see what the differences are from both perspectives. At the end of this study, the insights of Mussche (2019) will be compared with this study.

3.2 Research design

This section contains the roadmap of this study.

Step 1 in this study is the demand investigation of key modules (section 4.1). Step two is the classification of the key modules (section 4.2). Key modules are the products under investigation. Step three is the determination of the inventory cost (section 4.3). Step four is to set up KPI’s and to build the inventory model (section 4.4). Step five are the outcomes of the model (section 5). The current situation is compared with the optimal situation. The steps are visualized in figure 3.6.

Figure 3.6: Roadmap of this study

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3.3 Data collection

Data can be divided into three categories, namely A-data, B-data, and C-data (Robinson, 2004). A-data is data derived directly from the company. B-data is not immediately available but collectable. The last one, C-data is not collectable and available at the company so must be assumed. For this study, all three data types are used.

The following A-data is used:

- The demand for finished products; - The lead time of key modules; - The current lot sizes of key modules;

- The cost price of key modules. Key modules are divided into different ranges and each range has a cost price;

The following B-data is used:

- Demand key modules; calculated based on demand for finished products;

- Demand value of key modules; total demand multiplied by cost price of a key module; - Product value low range; average cost price of a low range key modules;

- Product value medium range; average cost price of a medium range key modules; - Product value high range; average cost price of a high range key modules. The following C-data is assumed:

- Order cost per pallet from Production Site to the Customization Centre; - Holding costs per key module per year.

3.4 Reliability, validation & verification

Verification is the process of ensuring that the model design has been transformed into a computer model with enough accuracy, while validation is the process of ensuring that the model is sufficiently accurate for the purpose at hand (Robinson, 2004). For both, the aim is to ensure that the model is sufficiently accurate. This accuracy depends on the purpose of the model.

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3.5 Sensitivity analysis

Sensitivity analysis is important for model building, especially if there are different, uncertain inputs. In a sensitivity analysis, the consequences of changing model inputs are assessed (Robinson, 2004). The first step is changing input variables, run the model, and measure the output. If there is a significant difference, then the model is sensitive to the change of input. If there is a little change, then the response is insensitive to the change.

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4. Analysis

The first step in the analysis is the demand investigation of key modules. For case company X, demand for key modules is expressed in lot sizes of 1440. This demand cannot be used in this study because it is a rounded value of demand. Especially if demand is low because there is for many weeks stock and no order is placed. The calculation and analysis of the demand is done in section 4.1.

The second step is the classification of key modules. There are in total 116 key modules and these will be classified into different groups. The aim of this classification is to determine for each group an optimal lot size. This is helpful for case company X, because if a key module has the criterion of a group then the optimal lot size is already known. The classification is done in section 4.2.

The third step is determining inventory costs. This is done because a lot size influences inventory costs as described in the literature section 2.2. Determining inventory costs is done in section 4.3.

Step four is to set up KPI´s and to build the (Q, r) inventory model. The aim of this inventory model is to analyse inventory value and inventory costs by a change in lot size. The KPI’s and the inventory model are stated in section 4.4.

4.1 Demand investigation

This section describes the demand investigation for key modules. Weekly demand for key modules is based on weekly demand for finished products. The first step is to investigate which key module is used in which finished products. The second step is to select a yearly period to translate weekly demand for finished products into weekly demand for key modules. After this, the weekly demand of each key module is known.

The ratio between key module finished product

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Figure 4.1: Example schematic overview product structure case company X

Selection of demand period

The next step is to select a yearly period to translate weekly demand for finished products into weekly demand for key modules. A yearly period is needed because demand has a yearly, seasonal demand pattern. Four yearly demand forecasting periods of finished products are evaluated, see figure 4.2.

Figure 4.2: Four yearly demand forecasting periods finished products

All four periods have almost the same curve with relatively low demand at the beginning of the year and high demand at the end of the year. This study uses period 50.2018 – 49.2019 for determining the demand for key modules. This has a few arguments. First, 50.2018 – 49.2019 is the most recent period with all finished products. During the year there are new finished products introduced and these are not in older demand periods. Furthermore, each finished product has its own lifecycle. In the most recent period, this is most representative for the future. With this in mind, this period is chosen.

19 Next, the demand for finished products per week is translated to the demand of key modules per week. Figure 4.3 below shows this demand. The seasonal demand pattern looks the same as in figure 4.2. Around week 29 there is a peak in demand. This is due to a specific deal with a customer. This generally happens once or twice a year. The specific deals make the demand less stable. For case company X, higher volume flexibility is desirable in fulfilling the “unstable” demand.

Figure 4.3: Forecasted demand key modules

4.2 Classification of key modules

A double classification is used to classify key modules into different groups. The criteria are demand volume and product value. First is explained why these criteria are chosen. Thereafter, the demand volume criterion is executed on key modules. Afterwards, the product value criterion is conducted on key modules. Finally, these two criteria are combined in a matrix.

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Demand volume

The cumulative demand volume of key modules is shown in figure 4.4. There is not a traditional 20/80 ratio, whereby 20% of products account for 80% of demand. Therefore, one extra group is added to the traditional ABC analysis. Further, with more groups, the lot size can be determined more precisely because demand differences in a group are smaller. Five groups are considered as well but this has not a real impact on the lot size.

Figure 4.4: Cumulative demand volume key modules

Table 4.1 shows the results of the ABC demand classification. Four different groups are made; the first group are key modules with demand lower than 11.000 per year, the second group between 11.000 and 30.000, the third group between 30.000 and 90.000 and the fourth group higher than 90.000. Almost half of all key modules belong to low demand. Due to increasing product variety, it can be assumed that this group is even larger in the future.

Table 4.1: ABC demand classification

Product value

The other criterion in the classification is product value. As described before, key modules are divided into seven different ranges, and there are six different cost prices. Two ranges have the same cost price. Cost prices of ranges do not differ a lot so three groups are made. A low range with an average cost price of €1, a medium range with an average cost price of €2, and a high range with an average cost price of €3,75. These average cost prices are used to calculate the total inventory value for all key modules. An overview is given in table 4.2. Almost half of the key modules belong to the medium range.

Demand per year Key modules Cumulative demand

Low < 11.000 43% 5%

Medium low 11.000 > 30.000 25% 15%

Medium high 30.000 > 90.000 21% 30%

High > 90.000 11% 50%

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Table 4.2: Classification of product value

Classification matrix

The double classification is shown in the matrix in table 4.3. In total there are 116 key modules and each key module is classified in one of the 12 groups. However, there are no key modules in the high demand, high range group. The aim of this classification is to determine for each group a most optimal lot size.

Table 4.3: Double classification key modules

4.3 Inventory costs

This section describes the calculation of inventory costs. First order costs are explained, and thereafter holding costs.

4.3.1 Order costs

According to literature, it is hard to calculate order cost. Therefore, this study uses the fundamental insights from Harris (1913) to determine order cost. This is described in the literature section. The main idea is the smaller the lot size is, the larger the influence on setup and transport cost. First, the calculation of the order cost of the lot sizes is explained. Thereafter, the impact of lot sizes on order costs is elaborated on.

Order cost of lot sizes

Case company X uses pallets to deliver key modules. 12 boxes fit on a pallet and one box is 60 key modules. The minimum lot size in this study is 60 (= one box) and the maximal lot size is 1440 (two full pallets). In multiples of 60, lot sizes are tested in this study because these are full boxes. The order cost of one pallet is € 3,75 -.1 _{The following notations are used to calculate the order cost of lot sizes:}

𝑄 = 𝐿𝑜𝑡 𝑠𝑖𝑧𝑒

𝐵𝑄 = 𝐵𝑜𝑥𝑒𝑠 𝑓𝑜𝑟 𝑙𝑜𝑡 𝑠𝑖𝑧𝑒 𝑄,

1 _{This is ratio of the order costs of case company X}

Low range Medium range High range Cost price € 1,00 € 2,00 € 3,75

Key modules 27% 47% 26%

Product value

Low range Medium range High range

22 𝑃𝑄 = 𝑃𝑎𝑙𝑙𝑒𝑡𝑠 𝑓𝑜𝑟 𝑙𝑜𝑡 𝑠𝑖𝑧𝑒 𝑄, 𝐶𝑄 = 𝐶𝑜𝑠𝑡 𝑙𝑜𝑡 𝑠𝑖𝑧𝑒 𝑄, 𝑇𝑅𝑄 = 𝑇𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡 𝑟𝑎𝑡𝑖𝑜 𝑙𝑜𝑡 𝑠𝑖𝑧𝑒 𝑄, 𝑃𝑅𝑄 = 𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 𝑙𝑜𝑡 𝑠𝑖𝑧𝑒 𝑄, 𝐴𝑅𝑄 = 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑎𝑡𝑖𝑜 𝑙𝑜𝑡 𝑠𝑖𝑧𝑒 𝑄, 𝑠𝑄 = 𝑂𝑟𝑑𝑒𝑟 𝑐𝑜𝑠𝑡 𝑙𝑜𝑡 𝑠𝑖𝑧𝑒 𝑄, 𝑛 = { 720, 𝑖𝑓 𝑄 ≤ 720 1440, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

From a transport perspective, most efficient is the use of full pallets. Therefore, lot sizes of 720 and 1440 are set on 100% (= 1 or 2 pallets). Any decrease in lot size between 1440 and 720 and between 720 and 0 results in a higher order cost per product. This ratio is calculated as follows:

𝑇𝑅𝑄 = 𝑛

𝑄 (1)

From a production perspective, most efficient is the use of large lot sizes because the larger the lot size the fewer setups are needed. Furthermore, smaller lot sizes result in more internal transport at production. Currently, case company X uses a lot size of 1440, so this is set at 100%. Any decrease in lot size results in a higher cost per product. This ratio is calculated as follows:

𝑃𝑅𝑄 = 1440

𝑄 (2)

Next, the average ratio is a combination of the transport ratio and the production ratio. The average ratio is taken because it is assumed that both ratios have equal weights on the order cost. The ratio is used to calculate the cost of a full pallet of lot size Q. The average ratio of lot size Q is calculated as follows:

𝐴𝑅𝑄 =

𝑇𝑅𝑄 + 𝑃𝑅𝑄

2 (3)

Finally, the average order cost ratios are used to calculate the cost of lot size Q. The order cost of lot size Q is calculated as follows:

𝑠𝑄 = 𝐶𝑄 𝑇𝑅𝑄

23 The results of the formulas are shown in table 4.4. If the lot size is between 60 and 720 then the order cost is set on €5,63. If the lot size is between 780 and 1440 then the order cost is set on €7,50.

Table 4.4: Overview of order costs of lot sizes

Impact of lot sizes on order costs

The following notations are added to calculate the impact of lot sizes on order costs: 𝑆𝐹𝑃𝑄= 𝑂𝑟𝑑𝑒𝑟 𝑐𝑜𝑠𝑡 𝑓𝑢𝑙𝑙 𝑝𝑎𝑙𝑙𝑒𝑡 𝑙𝑜𝑡 𝑠𝑖𝑧𝑒 𝑄,

𝑆𝐹𝑃𝑛= 𝑂𝑟𝑑𝑒𝑟 𝑐𝑜𝑠𝑡 𝑓𝑢𝑙𝑙 𝑝𝑎𝑙𝑙𝑒𝑡 𝑙𝑜𝑡 𝑠𝑖𝑧𝑒 𝑛,

𝑆𝑅𝐹𝑃𝑄 = 𝑂𝑟𝑑𝑒𝑟 𝑐𝑜𝑠𝑡 𝑟𝑎𝑡𝑖𝑜 𝑓𝑢𝑙𝑙 𝑝𝑎𝑙𝑙𝑒𝑡 𝑙𝑜𝑡 𝑠𝑖𝑧𝑒 𝑄, First, the order cost of a full pallet of lot size Q is calculated. This is done as follows:

𝑆𝐹𝑃𝑄= 𝐶𝑄 ∗ 𝐴𝑅𝑄 (5)

Next, the ratio cost full pallet of lot size Q is calculated. This is done because this number shows the increase in order costs with a smaller lot size. This is done with the following formula:

24 The results of the formulas are shown in table 4.5.

Table 4.5: Impact of lot size on order costs

The figure below shows a graph of the Order Cost Ratio Full Pallet of lot size Q (SRFP_Q). Order cost decreases exponentially with the lot size. With other words, order cost increases exponentially if lot sizes are smaller. After lot size 720 there is a small change in the graph because two pallets are needed. From a transport perspective and production perspective, this is both less efficient.

Figure 4.5: Impact of lot sizes on order costs

25

4.3.2 Holding costs

From a literature point of view, holding cost per unit and time is often determined as a percentage of unit value (Axsäter, 2015). The level of this percentage really depends on financial risks. The dominating part of holding cost is capital costs, but there are also other parts that have to be taken into account, such as material handling, storage, damage and obsolescence, insurance, and taxes (Axsäter, 2015). In literature is often mentioned that holding costs, observed in industry range from 5 – 45% of a cost price. (Durlinger, 2012).

In conversation with different people from case company X holding costs are hard to determine but low at case company X. Therefore, there is calculated with 5% holding costs in this study.

4.4 Inventory model

This section explains the inventory model used in this research. The objective of the model is to balance holding costs and order costs by changing lot sizes. Outputs of the model are yearly holding costs, yearly ordering costs and the overproduction. First, the model with notations and formulas is discussed. Thereafter, the output of the model with notations and formulas is explained.

(Q, r) inventory model

The inventory model in this study is a single-item (Q, r) inventory model in which demand is variable and items are not perishable. This means that that unsold products in one period can be sold in the next period.

The model has the following assumptions: - Demand is deterministic;

- Lead time is constant; - No backlog;

- Unlimited production capacity.

The inventory model makes use of the following notations: 𝐷𝑖 = Demand arriving during week 𝑖, 𝑟𝑖 = Reorder point at week 𝑖,

𝑎𝑖 = Amount of lot sizes needed in week 𝑖, 𝑄𝑖 = Order quantity in week 𝑖,

26 Decision variable:

𝑄 = Fixed lot size

The model calculates the reorder point for a specific period as follows: 𝑟𝑖 = 𝐷𝑖+1+ 𝐷𝑖+2 + 𝐷𝑖+3 (7)

The reorder point for period i is summed demand of week i + 1, i + 2 and i + 3. Next two weeks of demand are chosen because the lead time is two periods for case company X. Furthermore, an extra week of demand is added because this is the safety stock in the model. Although, demand in the model is deterministic and no backlog is assumed in practice there is a safety stock. This affects the inventory level and eventually holding costs. Without safety stock in the model, lot sizes have a larger effect on holding costs. In conversation with different people of company X, one week of demand is added to the reorder point to represent safety stock in the model.

The model calculates for a week how many orders are needed by a given lot size. Amount of lot sizes needed in week i:

𝑎𝑖 = 𝑟𝑖− 𝐼𝑖

𝑄 (8)

The reorder point of week i minus inventory of week i divided by the fixed lot size. Then, the order quantity of week i is:

𝑄𝑖 = 𝑎𝑖 ∗ 𝑄 (9)

The orders needed in week i multiplied by the fixed lot size. Eventually, the inventory level of week i is calculated by the following formula:

𝐼𝑖= 𝐼𝑖−1+ 𝑄𝑖−𝐿 − 𝐷𝑖 (10)

The inventory level of week i is equal to the inventory level of the previous week i-1 plus ordered quantity in week i - L minus demand in week i. The inventory level calculates how many products are on stock and triggers an order. This is only the case if the inventory level is lower than the sum of demand for the next three weeks. This is also shown in formula 7. The inventory level in the model starts with the initial inventory. This level is set to cover the next three weeks of demand.

Output (Q, r) inventory model

The output of the model is total holding costs per year, total order cost per year and the total overproduction per year.

The following notations are added:

27 𝑘 = 5 years,

𝐶𝑝= Cost price of product 𝑃, 𝑃 = {€1, €2, €3,75},

ℎ = Holding costs,

𝑆𝑄 = Yearly order costs of lot size 𝑄, 𝑠𝑄 = Order costs of lot size 𝑄, 𝐼𝑄 = Inventory costs of lot size 𝑄, 𝑂𝑄 = Overproduction of lot size 𝑄.

In this study, 5 years (k = 5) is chosen because it gives a more reliable inventory level per year, amount of orders per year, and the overproduction per year. The yearly seasonal demand is five times copied and pasted in the model. More years are considered by the researcher, but this does not affect the output of the model. The holding costs h are 5%.

The yearly holding costs for lot size Q is calculated as follows:

𝐻𝑄 =

∑260𝑖=1𝐼𝑖

𝑘 ∗ 𝐶𝑝 ∗ ℎ 𝑝 ∈ 𝑃 (11)

The yearly holding costs of lot size Q is the summed inventory level of 260 weeks (= 5 years) divided by k, multiplied by a cost price of product P, multiplied by the holding costs. The holding costs are 5%. The order costs per year for lot size Q is calculated as follows:

𝑆𝑄 =

∑260𝑖=1𝑎𝑖

𝑘 ∗ 𝑠𝑄 (12)

The yearly order costs of lot size Q is the summed amount of orders of 260 weeks divided by k, multiplied by the order costs of lot size Q. The order costs of lot size Q are calculated in section 4.3.1. The inventory costs per year for lot size Q is calculated as follows:

𝐼𝑄 = 𝐻𝑄 + 𝑆𝑄 (13)

The yearly inventory cost of lot size Q is the sum of the yearly holding costs of lot size Q and the yearly order costs of lot size Q. The inventory costs are used to determine the most optimal lot size.

The overproduction per year for lot size Q is calculated as follows:

𝑂𝑄 =

∑260𝑖=1𝑄𝑖− ∑260𝑖=1𝐷𝑖

29

5. Results & interpretation

This section showed the outcomes of the study. First, the yearly inventory costs of different lot sizes are evaluated, and an optimal lot size is chosen. The aim of this section is to compare inventory costs of the current lot size with inventory costs of the optimal lot size (5.1). Thereafter, overproduction values of the current lot size and optimal lot size are evaluated. Overproduction value of the current lot size and optimal lot size are compared (section 5.2). Finally, outcomes of the sensitivity analysis are described. The aim of this section is to test the robustness of the model if some input data is changed (section 5.3).

5.1 Inventory costs

This section evaluates the inventory cost of different lot sizes. The inventory costs are holding costs and ordering costs. This section is structured as follows. First, the four key module demand groups are evaluated, and an optimal lot size is chosen. Further, inventory costs of the current lot size and inventory costs of the optimal lot size are compared.

This section ends with the classification matrix of key modules. This matrix gives an overview of the inventory costs savings.

Key module with low demand

Figure 5.1 illustrates the inventory costs of key modules with low demand. For all three ranges inventory cost decreases with a smaller lot size. The holding costs have a much larger impact on total cost compared to order cost, visualized in appendix B. The inventory costs of a high range key module decrease the fastest. This is due to the high value and holding cost have the largest impact on total costs. The lowest inventory cost point gives an optimal lot size. This is for low range 180, medium range 120, and for high range 60. Lot size 60 is the smallest lot in this study. However, the inventory costs of the high range still decrease with a lot of 60.

30 In figure 5.2 inventory costs of the current lot size and the optimal lot size are compared for all key modules with low demand. The highest cost savings can be realized in the high range.

Figure 5.2: Yearly inventory costs key modules with low demand

Key module with medium-low demand

Figure 5.3 illustrates the inventory costs of a key module with medium-low demand. For all three ranges inventory cost decreases at the start with a smaller lot size. However, the decrease is smaller compared to a key module with low demand. This is because holding cost have a less impact on total cost, visualized in appendix C. With other words, order costs have a larger impact because demand is higher. There are more orders needed to fulfil demand. The optimal lot sizes for low range, medium range and high range are 300, 300 and 180 subsequently.

31 In figure 5.3 inventory cost of the current lot size and the optimal lot size are compared for all key modules with medium-low demand. The differences are less compared to a key module with low demand.

Figure 5.4: Yearly inventory cost key modules with medium-low demand

Key module with medium-high demand

Figure 5.5 illustrates the inventory costs of a key module with medium-high demand. For all three ranges inventory cost decreases at the start with a smaller lot size. However, the decrease is less compared to a key module with a medium-low demand. This is because holding cost have a less impact on total cost, visualized in appendix D. With other words, order costs have a larger impact because demand is higher. There are more orders needed to fulfil the demand. The optimal lot sizes for low range, medium range and high range are 420, 300 and 240 subsequently.

32 In figure 5.6 inventory cost of the current lot size and the optimal lot size are compared for all key modules with high demand. The differences are less compared to a key module with medium-low demand.

Figure 5.6: Yearly inventory costs key modules with medium-high demand

Key module with high demand

Figure 5.7 illustrates the inventory costs of a key module with high demand. There are only two ranges because there is no high range key module with high demand. For the two ranges, inventory costs are almost the same with a smaller lot size. Order costs have a larger impact on total cost compared to a key module with medium-high demand. The demand is higher, and consequently, there are higher order costs with a smaller lot size, visualized in appendix E. The optimal lot sizes for low range and medium range are 660 and 540 subsequently.

33 In figure 5.8 inventory cost of the current lot size and the optimal lot size are compared for all key modules with high demand. The differences are less large compared to a key module with medium-high demand.

Figure 5.8: Yearly inventory costs key module with high demand

Classification matrix key modules

The classification matrix gives an overview of inventory costs differences between the current lot size and the optimal lot size of all key modules, see table 5.1. The difference between the inventory cost of the current lot size and an optimal lot size of one key module is multiplied by the number of key modules in a group.

The largest ratio differences are in low demand groups. Inventory cost can be decreased with approximately 77% for three groups. Most savings are possible in the low demand, high range group.

Table 5.1: Classification matrix yearly inventory costs savings

34

5.2 Overproduction value

This section illustrates the overproduction values of the current lot size and the optimal lot size. This section is structured as follows. First, for the four key module demand groups, the overproduction value of the current and optimal lot size is compared. Thereafter, a classification matrix of all key modules is given. This classification gives an overview of overproduction values savings if the optimal lot size is used.

Key module with low demand

Figure 5.9 illustrates overproduction values of the current and optimal lot size for all key modules with low demand. The largest differences are at the high range key module. These key modules have a higher value compared to medium and low range. Less overproduction in this range results in a lower inventory value compared to medium and low range.

Figure 5.9: Yearly overproduction value key modules with low demand

Key module with medium-low demand

Figure 5.10 shows the overproduction values of the current and optimal lot size for all key modules with medium-low demand. There is more overproduction compared to a key module with low demand. This is because there is a variable safety stock dependent on the level of demand. If demand is higher, there is a higher safety stock and consequently more overproduction.

35

Figure 5.10: Yearly overproduction value key modules with medium-low demand

Key module with medium-high demand

Figure 5.11 shows the overproduction values of the current and optimal lot size for all key modules with medium-high demand. Compared to a key module with medium-low demand there is a higher overproduction due to variable safety stock.

The largest differences are at the high range key modules. However, there is a smaller difference compared to a key module with medium-low demand.

36

Key module with high demand

Figure 5.12 illustrates overproduction values of the current and optimal lot size for all key modules with high demand. Compared to a key module with medium-high demand there is a higher overproduction due to variable safety stock. However, there is not much difference between overproduction value of the current and optimal lot size.

Figure 5.12: Yearly overproduction value key modules with high demand

Classification matrix key modules

The classification matrix gives an overview of differences in overproduction values between the current and optimal lot size for all key modules, see table 5.2. Differences between overproduction values of the current and optimal lot size of one key module are multiplied by the number of key modules in a group.

The largest ratio differences are in the low demand groups. Inventory value can be decreased by approximately 62% for three groups. Most savings are possible in low demand, high range group.

Table 5.2: Classification matrix overproduction savings

37

5.3 Sensitivity analysis

In this section, the sensitivity analysis is described for the inventory model. The input parameter holding cost is changed to examine the impact of this parameter. The sensitivity analysis is carried out on holding costs because these costs are assumed (C-data) and have the greatest impact on total cost, as illustrated in appendix B, C, D and E.

Holding costs are decreased from 5% to 2% and 1% of the cost price to examine the impact on the output of the model. The model output are inventory costs, and consequently the optimal lot size. These costs are decreased because the holding costs are expected to be low at case company X. Appendix F, G, H and I is inventory costs of the sensitivity analysis of a four key module demand groups.

Low demand

For key modules with low demand, a smaller lot size has the greatest impact on inventory costs, as stated in section 5.1. With 5% holding costs, inventory costs of low, medium and high ranges decrease with 76%, 77% and 77% subsequently, as stated in table 5.1. However, even with 1% holding costs, the inventory costs decrease with around 70% if the optimal lot size is used. The optimal lot sizes are 60 units larger if holding costs are 2% and 1%.

Medium-low demand

For key modules with medium-low demand inventory costs decrease with around 40% with 5% holding costs, see table 5.1. With 2% and 1% holding costs, the inventory costs decrease with around 33% and 27% if the optimal lot size is used. The optimal lot sizes are around 120 units larger if the holding costs are 2% and 1%.

Medium-high demand

For key modules with medium high demand inventory costs decrease with around 14% with 5% holding costs. With 2% and 1% holding costs, the inventory costs decrease with around 10% and 7% if the optimal lot size is used. The optimal lot sizes are around 180 units larger if the holding costs are 2% and 1%.

High demand

For key modules with high demand inventory costs decrease with around 4% with 5% holding costs. With 2% and 1% holding costs, the inventory costs decrease both with around 1% if the optimal lot size is used. With 1% holding cost, the optimal lot sizes are almost the same as in the current situation.

Optimal lot sizes sensitivity analysis

38 Changing holding cost has the largest impact on key modules with high demand. The optimal lot size is doubled from around 700 to 1400. Holding costs have less impact on key modules with low demand. The optimal lot size variates from 180 (5%) to 260 (1%).

39

6. Conclusion, discussion & recommendations

This section provides the conclusion and discussion of this study. First, a recap is given of this study (section 6.1). Thereafter, discussion points of this study are presented. Here the results and validity are discussed (section 6.2). Next, recommendations for case company X are provided. Practical implications are discussed in this section (6.3). Finally, limitations and further research is elaborated on (section 6.4).

6.1 Conclusion

This section provides the conclusion of this study. First, the research question is repeated, followed by the steps taken to answer the research question. This section ends with the results of this study and the outcomes of the sensitivity analysis.

As stated in the introduction, the research question is:

“How can inventory levels under deterministic seasonal demand be optimized by focusing on the lot sizes?”

Specifically, the aim of this study is to find an organized approach to optimize inventory levels by focussing on lot sizes at case company X.

Approach

A double ABC classification is executed to classify products. Criteria of this classification are volume demand and product value. Volume demand is divided into four groups and product value into three groups. So, in total there are 12 product groups with each their own characteristics. The 12 groups are examined in the (Q, r) inventory model in Excel. For each group, an optimal lot size is determined.

Results

The first insight is that almost half of the products belong to the low demand group. Due to increasing product variety, it can be assumed that this group is even larger in the future.

Most of the products of case company X have a lot size of 1440. Results indicate that a smaller lot size is preferable. The largest lot size is 660 for products with high demand and low value and the smallest lot size is 60 for products with low demand and high value. For products with low demand, the highest savings can be realised because 47% of products belong to this group. Especially, for products with high value. Yearly inventory costs can be decreased with €161.454 (-77%) and the overproduction value with €13.125 (-63%).

40 The approach is useful for make-to-stock companies which produce a variety of products in fixed lot sizes. Especially, if there are many products for which an optimal lot size has to be determined. These companies could use this organized approach to optimize inventory levels.

6.2 Discussion

This section presents a discussion of the results and validity of this study.

Discussion

First, the main finding of this study is that an approach needed to be developed to calculate optimal lot sizes. The approach was based on product characteristics demand volume and product value. Based on these criteria products are classified into 12 groups. The number of groups has been substantiated, but it will be possible to make more groups in the future.

For optimizations of inventories, product criteria demand volume and product value in this study are in line with the findings of Teunter, Syntetos & Babai (2017). However, a combination of both; demand value is often used as a criterion as well. This is considered by the researcher in this study, but this leads to sub-optimality with respect to cost efficiency. Product criteria are also in line with Pflitsch (2008), who argued that demand volume is more effective than demand value in reducing inventory costs. The products were tested in a (Q, r) inventory model to determine the optimal lot size. For products in the low demand group, a smaller lot size is preferable. Especially, if products have high value. This is in line with the expectations of the researcher. At the time of this study, some products had already a smaller lot size. Case company X assumed that this lot size could be smaller. Thus, results are in line with case company X.

Validity

41

6.3 Recommendations

The recommendations for case company X are as follows.

First, case company X produces in fixed lot sizes and they control the inventory with a (Q, r) inventory policy. Every period, an integer number of fixed lot sizes are needed to fulfil demand. This is detrimental to volume flexibility. However, case company X faces a seasonal demand pattern for their products. At the beginning of the year, there is a lower demand compared to the end of the year. From a market perspective, a variable lot size is preferable. There are small lots at the beginning of the year and larger at the end of the year. In this case, there is higher volume flexibility to meet demand more precisely and consequently, there is less overproduction. However, there can be assumed that this is not practical for production of case company X. Further research is needed about possibilities for a variable lot size in production.

Second, from a market perspective, results indicate that a smaller fixed lot size is preferable for all products. Especially, for products with low volume and high value. Even with 1% holding costs, the optimal lot size is 180 for this group. However, with a smaller lot size company X should order more often and the question arises if production can handle such small lot sizes. However, even if the lot size is halved (=720) there can be significant cost savings realized. The first results of Mussche (2019) indicate that if the lot size is halved, there is only 2% decrease in capacity due to setups. However, Mussche (2019) examined only one machine at case company X. More research is needed about the possibilties of smaller lot sizes in production.

42

6.4 Limitations and further research

This section describes which remarks can be made based on results and how other researchers can build on this study

In this study, there are several side notes based on results that can be made. In the (Q, r) inventory model demand is assumed to be deterministic. It will be interesting to research if there is a kind of uncertainty in demand. One way to include uncertainty is to build inventory model based on stochastic demand. Then there is change on a backlog, what is also the case in practice.

Case company X faces a seasonal demand for their products. Therefore, there is a variable reorder point to trigger orders. The sum of the next three periods of forecasted demand is the reorder point. In literature, there is not many researches done about variable reorder points. It will be interesting to examine the variable reorder point in more detail.

Next, lead time is assumed to be constant in this study. The assumption can be made that lead time changes if lot sizes changes because with a smaller lot size there is shorter production time. This affects the inventory level.

43

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Appendices

Appendix A: Overview (Q, r) inventory model in Excel

Appendix B: Inventory costs key module with low demand

This section illustrates the holding costs, order costs and total costs of key modules with low demand. The holding costs have the largest impact on the total costs.

46 The optimal lot size for a medium range key module with low demand is 120.

47

Appendix C: Inventory costs key module with medium low demand

This section illustrates the holding costs, order costs and total costs of key modules with medium low demand.

The optimal lot size for a low range key module with medium low demand is 300.

48 The optimal lot size for a high range key module with medium low demand is 180.

Appendix D: Inventory costs key module with medium high demand

This section illustrates the holding costs, order costs and total costs of key modules with medium high demand.

49 The optimal lot size for a medium range key module with medium high demand is 300.

50

Appendix E: Inventory costs key module with high demand

This section illustrates the holding costs, order costs and total costs of key modules with high demand.

The optimal lot size for a low range key module with high demand is 660.