The effect of more accurate input on the repair
kit solution using sophisticated demand
forecasting techniques
M. Dijkhuizen
Master’s Thesis Econometrics, Operations Research and Actuarial Studies Supervisor:
The effect of more accurate input on the repair kit
solution using sophisticated demand forecasting
techniques
M. Dijkhuizen
January 7, 2019
Abstract
Contents
1 Introduction 5
2 Literature review 6
2.1 The repair kit problem . . . 6
2.1.1 Single-period, multiple-item inventory model . . . 7
2.1.2 Multiple-period, multiple-item inventory model . . . 8
2.2 Demand forecasting, classification and aggregation . . . 10
2.2.1 Demand forecasting methods . . . 11
2.2.2 Forecasting performance measures . . . 12
2.2.3 Demand classification . . . 14
2.2.4 Aggregation . . . 17
2.2.5 Post-processing . . . 18
2.3 Contribution to the existing literature . . . 19
3 Case study description 19 3.1 Problem formulation . . . 19
3.2 Data description . . . 20
3.3 Temporal aggregation . . . 21
4 Model and analysis 23 4.1 Demand forecasting . . . 23
4.1.1 Demand forecasting methods . . . 24
4.1.2 Forecasting performance measures . . . 25
4.1.3 Demand classification . . . 26
4.2 The repair kit problem . . . 27
5 Results 30 5.1 Demand forecasting . . . 30
5.1.1 Sophisticated demand forecasting techniques . . . 30
5.1.2 Demand forecasting for the benchmark . . . 33
5.1.3 Comparison of the performance of the sophisticated demand forecasting techniques and the benchmark . . . 35
5.2 Repair kit solutions . . . 37
5.2.1 Repair kit solutions from sophisticated demand forecasting techniques . . . 37
5.2.2 Repair kit solution from the benchmark . . . 39 5.2.3 The effect of more accurate input on the repair kit solution . 41
1
Introduction
In practice many organizations provide the service of maintaining various types of products for their customers. This is done by repairmen who often visit mul-tiple customers on a daily basis. The customers do not have any spare parts on location and the repairmen, who aim to get a high service level, carry a large amount of spare part types with them. In most cases, the repairman’s inventory get replenished at the end of the day. Failing to have the spare parts available that are needed for the repair leads to a revisit with additional return-to-fit costs and customer dissatisfaction. However, carrying too many types of spare parts leads to higher (holding) costs and loss of overview for the repairman as to what is available in his inventory. The repair kit problem is concerned with finding the optimal set and optimal quantities for the various types of spare parts. Such a set is called the repair kit.
The composition of the repair kit balances two criteria: the service level and the (holding) costs. Including more spare parts in the repair kit increases the service level (measured by the first fix rate) but also increases the holding cost. In deciding which spare parts to include in the repair kit, spare part usage is essential. A complicating factor is that the expected usage or probability that a spare part is needed is often unknown and needs to be estimated.
Previous authors have assumed complete data with known fail rates or simu-lated demand data from a probability distribution and subsequently expanded the repair kit problem to be a better representation of reality by making additional or more restricting assumptions. This paper will focus on how the repair kit solution improves with better input. That is, fail rates based on more accurate demand forecasts. Thus, this paper can be seen as having two parts. The first part is concerned with (aggregate) demand forecasting and how well the individual de-mand for the various types of spare parts can be forecast. And the second part of the paper applies this to the case of the repair kit problem and how the model improves with better data.
For this purpose, and the primary motivation to write this paper, real data from a Dutch-based company is used. The dataset contains extensive information on spare part usage from a considerable amount of repairmen. Applying the demand forecasting techniques as proposed in this paper in combination with the repair kit problem will provide new insights for both the literature and the company.
items one by one based on certain proposed assumptions until a certain service level is achieved.
Furthermore, (aggregate) demand forecasting and what has been done in the literature in this field of research will be discussed. Usage data for spare parts is often found to be lumpy or slow-moving which means that standard forecasting techniques are usually less effective. In the existing literature techniques are dis-cussed which are more effective for these kind of datasets and the usefulness of aggregation is discussed.
The rest of the paper is organized as follow. First, an overview will be given in section 2 of the existing literature on (aggregate) demand forecasting, the repair kit problem and the various variations. Secondly, the problem formulation and dataset will be discussed in section 3. Then, the model and the results are presented in section 4 and section 5, respectively. And lastly, the conclusion of this paper is presented in section 6.
2
Literature review
First, the repair kit problem and the contributions to the literature over the years will be discussed. Various authors have introduced variants and additions to the repair kit problem in an attempt to make it more efficient and realistic. Even though relatively few authors are working in this field of research, considerable progress has been made in improving the repair kit problem.
There is much more literature available on the broad subject of demand casting. A selection of the most recent and prominent papers on demand fore-casting techniques for smooth and intermittent demand will be discussed. This includes demand forecasting methods, demand classification methods, temporal and cross-sectional aggregation, and demand forecasting performance measures.
Lastly, the contribution of this paper to the existing literature on the repair kit problem will be discussed.
2.1
The repair kit problem
one part of each spare part type to be used on a given job. Moreover, additions such as a space constraint and the introduction of lead-times are discussed. 2.1.1 Single-period, multiple-item inventory model
Smith, Chambers, and Shlifer (1980), Graves (1982), and Brumelle and Granot (1993) are among the first authors who discuss the repair kit problem and analyse the single-period, multiple-item inventory model. In addition, Smith et al. (1980) and Graves (1982) assume failures of various spare part types to be stochastically independent and at most one part of each spare part type is assumed to be used on a given job (single-unit). In contrast, Brumelle and Granot (1993) consider dependence of the failures of various spare part types. Furthermore, Brumelle and Granot (1993) do not make the assumption that at most one part of each spare part type is used on a given job and their model is thus considered multiple-unit. Smith et al. (1980) are the first to present a formulation which yields an optimal policy for this inventory model using an optimal marginal analysis procedure. Furthermore, Smith et al. (1980) assume a cost model which minimizes the holding cost for spare parts and the return-to-fit cost if a job could not be completed. Smith et al. (1980) discuss standard topics in inventory theory in which a part-fill rate is considered. The part-fill rate measures the fraction of spare part types available during a repair. This is more simplistic than the job-fill rate which considers the fraction of jobs completed without needing a return-to-fit visit from the repairman. Smith et al. (1980) note that a job-fill rate is more appropriate in many inventory applications and incorporate it in the proposed inventory model. Furthermore, Smith et al. (1980) propose a solution method which takes into account the fact that the various types of spare parts are ranked by their holding cost divided by their usage frequency.
ratio of the increase in holding cost and the increase in fill rate. Graves (1982) note that for small values of the spare part type fail rates this is nearly equivalent to the ratio which was proposed by Smith et al. (1980). To see if the repair kit problem improves with better data, at least at first, a simple model is considered. Since the binary knapsack problem by Graves (1982) is the most complete single-period model without any additional restrictions, this model will be considered in the following sections of this paper.
Brumelle and Granot (1993) note that the parametric knapsack problem as discussed by Smith et al. (1980) and Graves (1982) may have an exponential number of optimal kits. Brumelle and Granot (1993) note that by parametrically varying the job completion rate all Pareto optimal kits can be generated for the holding cost and cost of failing to complete a job. However, generating all Pareto kits may be computationally very difficult. Brumelle and Granot (1993) use the theory of lattice programming to show that a monotone sequence of optimal kits can be selected for a number of different parametrizations of repair kit models and discuss the implications for the repair kit models proposed by Smith et al. (1980) and Graves (1982).
2.1.2 Multiple-period, multiple-item inventory model
Heeremans and Gelders (1995), Teunter (2006), Bijvank, Koole, and Vis (2010), Gorman (2016), and Prak, Saccani, Syntetos, Teunter, and Visitin (2017) consider the repair kit problem and analyse the multiple-period, multiple-item inventory model. Additionally, Heeremans and Gelders (1995), Teunter (2006), Bijvank et al. (2010), Gorman (2016), and Prak et al. (2017) assume that the failures of various spare part types are stochastically independent. In addition, Teunter (2006) also consider the case where the failures of various spare part types are stochastically dependent. Only Heeremans and Gelders (1995) assume throughout their paper that at most one part of each type is used on a given job. However, since multiple periods are considered, the repair kit is not replenished after each job. Hence, for j periods a repairman may now need up to j parts of each spare part type.
model on a case study with a real-life dataset. Heeremans and Gelders (1995) conclude that the determination of a standard inventory for all technicians is im-possible, hence personal inventories are considered. The probability of needing a part during a random visit is derived from historical data. For this, Heeremans and Gelders (1995) used the Expected Weighted Moving Average forecasting tech-nique. The probabilities of needing j units of spare part type i during m visits and the probabilities that no more than j units of spare part type i are necessary during m visits are calculated. Finally, a knapsack heuristic was used and Heere-mans and Gelders (1995) conclude that a considerable improvement in customer service and inventory investment was achieved compared to the benchmark.
Teunter (2006) further analyses the multiple-period, multiple-item inventory model. Teunter (2006) notes that the integer (binary) linear program of Heeremans and Gelders (1995) is complex and not insightful and presents two new heuristics to solve the model. Both heuristics can be used to solve the service model and the cost model. The first heuristic is a reformulation of the knapsack heuristic proposed by Heeremans and Gelders (1995) with the addition of a job-fill rate and determines the exact optimal solution in most cases. Teunter (2006) notes that job-fill rates are difficult to calculate and that the first heuristic can only be applied with ease if the number of jobs in a tour is fixed, at most one part of each spare part type is needed on a job, and demand for parts of different spare part types are independent. Therefore, a second heuristic is proposed for which failures of various parts can be assumed to be dependent and several spare parts of each spare part type may be used on a given job. Furthermore, the heuristic uses the part-fill rate and only needs tour demand data instead of job demand data. Teunter (2006) argues that this heuristic is more appropriate in practice and that the cost-model should be considered since the solution may not satisfy the service requirement if the job-fill rate is overestimated.
is presented using a greedy marginal analysis procedure which immediately stops when the job-fill rate is obtained. Bijvank et al. (2010) note that it is of importance to first determine the order of the number of units to add to the repair kit for each spare part type. Furthermore, an algorithm is proposed which adds units that immediately contribute (relatively) the most to the repair kit, contrary to adding units to the repair kit based on the potential of each spare part type to increase the service level.
Similar to Heeremans and Gelders (1995), Gorman (2016) considers the repair kit problem with space (or capacity) constraints. However, Gorman (2016) pro-poses an inventory model with a cost model and a part-fill rate. Gorman (2016) notes that the part-fill rate is a reasonable approximation when the number of parts on each job is low. Obviously, complex calculations of a job-fill rate is unnecessary if only a single unit of a spare part type is needed for a repair. Furthermore, a stochastic tour length is considered, but with a known time interval between replenishments. Gorman (2016) introduces the Net Benefit per unit Volume as a way to measure and balance the costs, increase in service level, and space of var-ious spare parts. Gorman (2016) states that this is a more appropriate objective function than simply the service level when the repairman’s time and customer service are valued heavily relative to inventory holding cost. However, Gorman (2016) notes as well that a service model naturally minimizes the number and amount of space of the various spare part types in the repair kit.
Prak et al. (2017) propose an inventory model with positive replenishment lead times and fixed ordering costs. Prak et al. (2017) note that zero replenishment lead times are difficult to justify for daily tours in practice. Furthermore, Prak et al. (2017) introduce ordering costs with corresponding policies for reordering and order-up-to levels and incorporate this in the repair kit problem. In addition, Prak et al. (2017) compute a job-fill rate based on the reorder levels and order-up-to levels for the various spare part types. Prak et al. (2017) propose a heuristic order-up-to set the reorder levels and order-up-to levels for every item using a greedy marginal analysis procedure. Moreover, Prak et al. (2017) present a simpler heuristic which assumes a zero lead time. Finally, Prak et al. (2017) discuss that a critical mass of items needs to be added to the repair kit for a significant increase in the service level and the importance of the order in which items are added to the repair kit, which depends on the item price and the item’s demand distribution.
provide a literature study on demand forecasting specifically in the area of spare parts management.
In this paper a framework similar to the forecast support system, as stated by Boylan and Syntetos (2010), is applied. This consists of a demand classifica-tion (pre-processing) followed by the corresponding demand forecasting method using the appropriate forecasting performance measures (processing). Further-more, this can be followed up by manual adjustments made by the practitioner (post-processing).
Since demand classification depends and refers to several of the possible de-mand forecasting methods, this section starts with an overview of the literature on demand forecasting methods. Several well known demand forecasting methods are discussed with their corresponding advantages and disadvantages. Subsequently, to be able to properly compare different forecasting methods or smoothing pa-rameter values for a certain demand forecasting method, forecasting performance measures are discussed. Several error metrics are considered and their usefulness as a performance measure is discussed. Furthermore, having considered demand forecasting methods and performance measures, demand classification is discussed. Several demand classification schemes are discussed and a choice will be made re-garding the demand classification scheme that is to be used in this paper. More-over, demand aggregation is discussed in order to make the demand data more manageable. As will be discussed, temporal aggregation can be applied which will reduce the intermittence of the demand data and thus may lead to a different demand classification. Finally, post-processing and the gap between research and practice regarding the field of spare parts management is discussed. Highlight-ing the fact that in practice forecastHighlight-ing methods are used that are often deemed suboptimal in the literature.
2.2.1 Demand forecasting methods
A considerable amount of different forecasting methods and derivatives exist and are discussed in the literature.
One of the most widely known forecasting methods is the (single) exponential smoothing method (SES) as discussed by e.g. Syntetos et al. (2009) and Boylan and Syntetos (2010). Single exponential smoothing uses a smoothing parameter and updates the demand forecast partly on the last demand occurrence and partly on the previous demand forecast.
non-zero demand forecast and the inter-demand interval forecast. Hence, if the inter-demand interval forecast is one, Croston’s method is equivalent to single exponential smoothing. Furthermore, the demand intervals are assumed to be independent and identically distributed.
Syntetos and Boylan (2001) showed that Croston’s method is biased due to the ratio in the final computation and propose their approximation, the Syntetos-Boylan Approximation (SBA), in Syntetos and Syntetos-Boylan (2005). The Syntetos-Boylan Approximation corrects the final computation by introducing a bias cor-rection which includes the smoothing parameter for the inter-demand intervals. In several papers the Syntetos-Boylan Approximation has since then been discussed (e.g. in Teunter and Sani (2009)) and is considered an appropriate forecasting method in case of intermittent demand. Eaves and Kingsman (2004) and Syn-tetos, Keyes, and Babai (2009) praise the Syntetos-Boylan Approximation for its robust performance regarding forecasting and inventory control.
However, Croston’s method and the Syntetos-Boylan approximation both lack the ability to deal with obsolescence and Teunter, Syntetos, and Babai (2011) pro-pose a derivative of Croston’s method with a demand probability which updates every period instead of a demand interval which updates only after a non-zero demand occurrence, which may be less biased and more accurate as discussed by Romeijnders, Teunter, and van Jaarsveld (2012). Hence, the Teunter-Syntetos-Babai method (TSB) works especially well when the time series has a large amount of zero-demands, with the possibility that the product has been or will be discon-tinued.
As single exponential smoothing, Croston’s method, the Syntetos-Boylan Ap-proximation, and the Teunter-Syntetos-Babai method have been widely accepted in the literature (e.g. Syntetos et al. (2009), Boylan and Syntetos (2010), Synte-tos, Boylan, and Croston (2005), Petropoulos and Kourentzes (2015)), these will be the demand forecasting methods that will be considered throughout this paper. The methods themselves will be discussed in greater detail in section 4.
2.2.2 Forecasting performance measures
Furthermore, during demand forecasting the forecasting methods should be tested on accuracy and efficiency. Among other things, this gives the opportunity to compare different smoothing parameter values for a certain demand forecasting method and possibly finding the optimal one within the set of possible smoothing parameter values. Moreover, it is possible to compare different demand forecasting methods in forecasting accuracy. This is done by considering error metrics, in the literature also known as optimisation criteria, cost functions or loss functions.
propose the use of traditional measures of accuracy, namely the mean absolute deviation (MAD), root mean square error (RMSE), and mean absolute percentage error (MAPE), and compare the values of these error metrics for various forecast-ing techniques and demand aggregation levels. Eaves and Kforecast-ingsman (2004) notes the inability of the mean absolute percentage error to deal with zero-demand oc-currences and show that these traditional error metrics are not ideal when dealing with slow-moving or intermittent demand due to the periods with zero demand. Eaves and Kingsman (2004) propose the use of implied stock-holdings and men-tion the relative geometric root mean square error (RGRMSE) as an alternative. Gardner (2006) review the state of exponential smoothing and note that tradition-ally error metrics are based on squared errors and absolute error. With the mean squared error as the most widely used error metric and the mean absolute error, also known as the mean absolute deviation, as an alternative. Gardner (2006) note how the mean squared error is especially influenced by extreme values. Syntetos et al. (2009) discuss demand categorization and use the mean absolute deviation as the error metric in finding optimal parameter values. In conclusion, the traditional error metrics are given by
M SE = 1 n n X i=1 (yi− ˆyi)2 RM SE = v u u t 1 n n X i=1 (yi− ˆyi)2 M AE = 1 n n X i=1 | yi− ˆyi | M AP E = 1 n n X i=1 | yi− ˆyi | yi × 100
Furthermore, Nikolopoulos, Syntetos, Boylan, Petropoulos, and Assimakopou-los (2011) discuss that forecasting methods should not only be evaluated based on forecasting accuracy but also on the implied inventory control decisions. Kourentzes (2014) mentions the periods in stock (PIS) error metric as introduced by Wallstr¨om and Segerstedt (2010). P IS = − n X i=1 i X j=1 (yj − ˆyj)
use of these error metrics, although there is also evidence for good performance of the traditionally used error metrics. Forecasting methods using error metrics based on squared errors or absolute errors produce forecasts biased towards the zero-demand due to the many zero-demands in the case of intermittent demand. Furthermore, the final demand forecast of Croston’s method and its derivatives is a demand rate (of the forecast for non-zero demand occurrences and the fore-cast for the inter-demand intervals) rather than an expected demand. For these reasons Kourentzes (2014) propose two new error metrics for these demand fore-casting methods acknowledging the demand rate. Based on the cumulative mean, Kourentzes (2014) propose the mean absolute rate (MAR) and the mean squared rate (MSR). Kourentzes (2014) notes superior performance of the two new error metrics in the case of intermittent demand and highlight the mean absolute rate as the most effective error metric. The error metrics considered in the case of intermittent demand are given by
rj = ˆyj − 1 j j X i=1 yi M AR = n X j=1 | rj | M SR = n X j=1 rj2
Throughout this paper, the mean absolute rate will be the error metric con-sidered in the case of intermittent demand. For time series with demand patterns that can be considered smooth, one of the traditional error metrics shall be con-sidered. In this paper the mean squared error shall be used as the error metric in the case of such demand patterns.
2.2.3 Demand classification
Regarding pre-processing and thus the demand classification, several authors have discussed and developed demand classification schemes in the literature.
Based on Johnston and Boylan (1996), Syntetos et al. (2005) proposed a 2 by 2 classification scheme based on the average inter-demand interval and the squared coefficient of variation of the non-zero demand occurrences. Syntetos et al. (2005) consider the following demand patterns: smooth, intermittent, erratic, and lumpy. For smooth demand patterns they propose the use of Croston’s method as the demand forecasting method and for the three other possible demand patterns the Syntetos-Boylan Approximation as the demand forecasting methods. These de-mand forecasting methods were discussed in greater detail in a previous subsection. For a graphical representation, see figure 1. Using the theoretical mean squared error (MSE) of the demand forecasting methods, Syntetos et al. (2005) find values for the average inter-demand interval and the squared coefficient of variation of the non-zero demand occurrences for which to properly distinguish between the four demand patterns.
Figure 1: Graphical representation of the Syntetos-Boylan-Croston classification Kostenko and Hyndman (2006) evaluate the classification scheme of Synte-tos et al. (2005) and propose an improved classification scheme using different threshold values for the average inter-demand interval and the squared coefficient of variation of the non-zero demand occurrences. Heinecke, Syntetos, and Wang (2013) empirically test the different threshold values proposed by Syntetos et al. (2005) and Kostenko and Hyndman (2006) and conclude that the threshold values of Kostenko and Hyndman (2006) are indeed superior.
methods are equal when the inter-demand interval forecast is equal to one. As mentioned in the discussion on demand forecasting methods, the Syntetos-Boylan Approximation introduces a damping factor based on the smoothing parameter of the inter-demand interval forecast in an effort to reduce the bias of Cros-ton’s method. The Syntetos-Boylan Approximation is equal to CrosCros-ton’s method only when the smoothing parameter of the inter-demand interval is zero. Hence, the Syntetos-Boylan Approximation is equal to single exponential smoothing only when the smoothing parameter of the demand interval is zero and the inter-demand interval forecast is equal to one. In contrast to Petropoulos and Kourentzes (2015), the demand classification schemes by Syntetos et al. (2005) and Kostenko and Hyndman (2006) advise the use of the Syntetos-Boylan Approximation when the squared coefficient of variation of the non-zero demand occurrences is larger than 12, regardless of the average inter-demand interval (being equal to one). How-ever, using single exponential smoothing as the demand forecasting method in the case where the demand data has exclusively non-zero demand occurrences is often preferred in the literature (Boylan, Syntetos, and Karakostas (2008)). Hence, Petropoulos and Kourentzes (2015) propose the addition of single exponential smoothing to the classification scheme when the average inter-demand interval is equal to one regardless of the coefficient of variation of the non-zero demand occurrences. In order to confirm the usefulness of the addition of single exponen-tial smoothing in the demand classification scheme, Petropoulos and Kourentzes (2015) provide empirical evidence of improvements in various forecasting perfor-mances measures for a range of aggregation levels. Syntetos, Babai, Kolassa, and Nikolopoulos (2016) mention the results of Petropoulos and Kourentzes (2015) and note the usefulness of the methods that are introduced.
Figure 2: Graphical representation of the Petropoulos-Kourentzes (approximation) classification
2.2.4 Aggregation
In addition to demand classification, aggregation of the demand data can be used in order to make the demand data more manageable. In the literature, Nikolopoulos et al. (2011) are credited with being the first to present empirical re-sults of aggregation in the case of intermittent demand introducing the aggregate-disaggregate intermittent demand approach (ADIDA). Spithourakis, Petropou-los, Babai, NikolopouPetropou-los, and Assimakopoulos (2011) apply the ADIDA on non-intermittent demand and are positive regarding the framework and forecasting accuracy. Syntetos et al. (2016) briefly discuss experience-driven heuristics and note that methods such as aggregation (both across time and across time series) are hard to outperform. Furthermore, they note that the time buckets in which data is collected are often inconsistent in practice, making it difficult to find an ap-propriate time bucket length across the demand data. Aggregation can be across time, which is called temporal aggregation, or across time series, which is called cross-sectional aggregation.
process. Syntetos et al. (2016) note that disaggregation makes strong and possibly incorrect assumptions on the individual time series such as the individual time series following the same pattern as the aggregated demand data. On the other hand, Nikolopoulos et al. (2011) note that aggregation can also help identify the underlying series’ characteristics such as a trend or seasonality that are otherwise may be lost due to the number of observations with zero demand.
Using cross-sectional aggregation, demand is aggregated for a selection (or possibly all) of the time series. In the literature, the top-down and bottom-up approaches and their effectiveness are often discussed as mentioned in the compre-hensive review of the literature by Syntetos et al. (2016). The bottom-up approach applies forecasting methods on the individual time series and aggregates the re-sults to be applied for a larger group of time series. In contrast, the top-down approach uses forecasting methods on the aggregated data and disaggregates the forecasts to be applicable on the individual time series. Rostami-Tabar, Babai, Ducq, and Syntetos (2015) further explore cross-sectional forecasting and compare the top-down and bottom-up approaches under certain conditions. Carson, Ce-nesizoglu, and Parker (2011) compare the forecasting of air travel demand using exogenous macroeconomic variables versus using the aggregation of airport specific forecasts. Syntetos et al. (2016) note that cross-sectional aggregation can provide useful information otherwise lost due to limited data at the individual level. Using cross-sectional aggregation for a certain group means more time series are avail-able to extract knowledge from. This can be used, for example, to find a seasonal pattern in the demand when aggregation a family of products.
Using aggregation on demand data, a decision must be made regarding the aggregation level. Nikolopoulos et al. (2011) note that an optimum aggregation level can be determined empirically for each demand series. In case of temporal aggregation, the choice is the size of the time bucket which is considered for the time series. Similarly, for cross-sectional aggregation, the groups of time series which are considered are arbitrary.
However, aggregation methods are often considered computationally costly compared to the more traditional forecasting methods. Furthermore, the com-bination of temporal aggregation and cross-sectional aggregation has been cited as an interesting new field in the literature.
2.2.5 Post-processing
Bacchetti and Saccani (2012) and Syntetos et al. (2016) discuss the existence of a gap between research and practice regarding the field of spare parts management, specifically spare parts classification and demand forecasting for stock control, and provide a critical review of the existing literature on these topics. They note that relatively simple forecasting methods such as the moving average and single expo-nential smoothing are the most used forecasting methods in practice while often deemed as not optimal in the literature. There are multiple reasons for the exis-tence of a gap between theory and practice. Bacchetti and Saccani (2012) mention that recent forecasting methods might simply still be unknown. Furthermore, they might be complex and costly to implement in practice.
2.3
Contribution to the existing literature
The main contribution of this paper to the existing literature on the repair kit problem will be to empirically test the effect of having more accurate input on the repair kit solution. The repair kit solution depends on both the service level and (holding) costs. Since the estimations on spare part usage affects both, the estimation of the expected spare part usage (or probability that a spare part is needed) is crucial. In this paper it is taken into consideration that the data in practice is often much more involved than assumed in the repair kit problem literature. In the field of spare parts management, demand data with difficulties such as intermittence or corrections are not uncommon. In this paper the effect of demand forecasting on the repair kit solution is investigated.
3
Case study description
3.1
Problem formulation
The data for the case study which will be discussed in this paper is from a Dutch-based company. The company is a wholesaler that offers their customers and customer service repairmen services in addition to providing them with the re-quested spare parts. The newest addition to these services is an innovative service using Radio-Frequency Identification (RFID) tags and scans to optimize the car stock of their repairmen. The company wants to make use of this new feature, the RFID-scan, that can periodically scan the inventory of the repairman’s car to provide more insight in the repairman’s inventory control. For this purpose, the company has bought over 3 million RFID-tags.
results from subsequent scans. If a customer agrees to be part of the RFID-program, their repairmen are obliged to buy at least 95% of their various types of spare parts at the company. This could give the company a much clearer picture on spare part usage than in the previous situation. Currently, the company decides on their repairmen’s starting inventory based primarily on the total quantity of spare parts that the company supplies its entire set of customers with. Further-more, several arbitrary decisions are made depending largely on the specific input and the experience of the repairman.
3.2
Data description
The data used in this paper is collected from the customers of the company and the corresponding repairmen which make use of these new RFID features. The dataset spans the entire year of 2017. Among other things, information is available on the customers, the repairmen, the products, the dates, and the usage or deliveries.
Two problems are immediately identified. First, several identical products have different product codes in the dataset. This is mostly due to administration adding numbers to the product codes in the case of redistribution of spare parts within the company. Hence, these numbers are removed from the product codes such that the product codes are consistent for the various types of spare parts. Secondly, several products are not considered to be RFID-items and can be removed from the dataset and disregarded in the advice on the initial inventory stock. This mostly includes basic items such as glue. The company has provided a list of product codes for all RFID-items and the dataset can be corrected accordingly.
Furthermore, the company considers two groups of customers. The first group consist of the regular customers of the company. For the regular customers, the company has data available on the deliveries of the company to the repairmen, but not the actual spare part usage and not on the actual moment of the repair. However, usage data can be derived using available information from a yearly bus scan at the end of 2017 which results in usage data in the form of one value per spare part type per repairman. The second group of customers consists customers that have a special agreement with the company. From now on these customers will be called the comfort partners. These comfort partners frequently receive spare parts from the company. However, while the spare parts are in the inventory of the repairman, the spare parts are still considered property of the company. In theory, the company charges the repairmen for the used spare parts when the repairman reports the spare part usage on the moment of a repair. Hence, this would imply that usage data could be considered and on a daily level. Since frequent usage data is preferred in the following sections of this paper, the data from the comfort partners will be the main dataset considered in this paper.
partners does not match the usage data on a daily level. First, from the data it can be seen that in practice the repairman frequently does not report usage of the spare part immediately on the moment of a repair. Hence, repairmen occasionally have periods where there is no recorded demand for any of the various types of spare parts. Secondly, a considerable amount of repairmen still occasionally purchase spare parts on their own while these parts should be supplied by the company.
Fortunately, comfort partners are allowed to have regular bus scans, and based on those the usage data is usually corrected within one week after a scan occurred. For unreported usage of spare parts, this often results in certain days for which a lot of usage is falsely reported as usage on that specific day. Furthermore, the spare parts which were bought by repairmen themselves are considered property of the company due to the agreement with comfort partners. The regular bus scans for comfort partners detect these purchases and correct the “administrative inventory” with negative values. These negative values represent positive values and are corrected correspondingly in the dataset.
Finally, several of the comfort partner’s repairmen in the dataset have recorded usage data only around specific days such as around the Christmas holidays or have only a few dates available because they have either quit or have started somewhere during 2017. This relatively small group of repairmen are considered outliers and shall be excluded from the dataset.
3.3
Temporal aggregation
The conclusion that can be drawn from the previous section is that the obtained data of the company’s comfort partners can not be properly used as usage data on a daily basis. This is mainly due to the fact that the corrections in the dataset for the unreported usage of spare parts lead to certain days for which a lot of usage is falsely reported as usage on that specific day. Hence, using the data in this format as a basis for demand forecasting would lead to poor conclusions. Thus the question rises on how to properly deal with the previously stated issues with the dataset.
Since the corrections in the dataset are frequent but irregular, temporal ag-gregation is applied on the various types of spare parts of the repairmen in order to deal with the usage spikes on certain days. Hence, a decision must be made regarding the time bucket length for the temporal aggregation. As was discussed in the literature review, Syntetos et al. (2016) note that finding an appropriate time bucket length across the dataset is difficult as the time buckets in which data are collected is often inconsistent in practice.
for each repairman in the dataset the periods where no usage data for any of the various spare part types were recorded will be analysed. This includes, for all repairmen, the maximum period in which the repairman did not record spare part usage as observed in the dataset as presented in figure 3. Furthermore, after inves-tigating several time frames, the amount of times the repairman did not recorded spare part usage for over a month and for over two months are presented in figure 4a and figure 4b.
(a) For all repairmen in the dataset (b) Excluding repairmen with over 150 days of no recorded usage
Figure 3: The maximum period of no recorded usage
(a) No recorded usage for over a month (b) No recorded usage for over two months
Figure 4: Counting the number of times there was no recorded usage data To make the time buckets easily interpretable, a week, two weeks, a month, and two months were considered as time buckets. With the knowledge gained from figure 3b, figure 4a and figure 4b are presented. Figure 4a shows that a considerable amount of repairmen have a period of over 30 days where no usage data was collected. Furthermore, figure 4b shows that very few repairmen have a period of over 60 days of no usage data collected.
Hence, using temporal aggregation with a time bucket of two months will be used to appropriately interpret the usage data from the repairmen from the com-pany’s comfort partners.
4
Model and analysis
4.1
Demand forecasting
In this section the demand forecasting methods, forecasting performance measures, and demand classification are explained in great detail.
The computations will be applied to the datasets obtained from the company’s comfort partners, aggregated with time buckets of two months. Hence, demand classification is applied for each product of each repairman in the dataset to find the most appropriate demand forecasting method. Then, this demand forecasting method is used with an appropriate error metric to make an accurate forecast for one period ahead.
4.1.1 Demand forecasting methods
The following four demand forecasting methods are considered: single exponential smoothing (SES), Croston’s Method (CRO), the Syntetos-Boylan Approximation (SBA) and the Teunter-Syntetos-Babai method (TSB). In section 2, the advantages and disadvantages of these demand forecasting methods were discussed. In this section, these four demand forecasting methods will be explained in greater detail. Single exponential smoothing uses a smoothing parameter α which updates the demand forecast partly on the last demand occurrence and partly on the previous demand forecast. Typical values for the smoothing parameter α are between 0.05 and 0.3 Let yt be a demand occurrence at time t and ˆyt be the demand forecast
for time t. Let 0 < α < 1. This gives the following model for single exponential smoothing
ˆ
yt= αyt−1+ (1 − α)ˆyt−1
Croston’s method updates both the non-zero demand occurrences and inter-demand intervals using single exponential smoothing. Let yt be a demand
occur-rence at time t and ˆyt be the demand forecast for time t. Furthermore, let pt be
the inter-demand interval at time t, zt be the non-zero demand occurrence at time
t. Let 0 < α < 1 and 0 < β < 1. This gives the following model for Croston’s method If yt= 0, ˆ pt = ˆpt−1 ˆ zt = ˆzt−1 If yt> 0, ˆ pt= αpt−1+ (1 − α)ˆpt−1 ˆ zt= βzt−1+ (1 − β)ˆzt−1
where the smoothing parameter α is used to update the inter-demand interval and the smoothing parameter β is used to update the non-zero demand occur-rences. Smoothing parameter α need not be different from smoothing parameter β. The smoothing parameter for the non-zero demand and demand intervals will be optimised separately. Finally,
ˆ yt = ˆ zt ˆ pt
were in Croston’s method, this gives the following model for the Syntetos-Boylan Approximation ˆ yt = 1 − α 2 zˆt ˆ pt
where α is the smoothing parameter for the inter-demand intervals.
The Teunter-Syntetos-Babai method uses a demand probability instead of an inter-demand interval. Let yt, ˆyt, zt, and ˆzt be as before. Let dt be the demand
probability at time t and ˆdt the demand probability forecast at time t. Let 0 <
α < 1 and 0 < β < 1. This gives the following model. If dt= 0, ˆ dt = ˆdt−1+ α(dt− ˆdt−1) = αdt+ (1 − α) ˆdt−1 = (1 − α) ˆdt−1 ˆ zt = ˆzt−1 If dt= 1, ˆ dt = ˆdt−1+ α(dt− ˆdt−1) = αdt+ (1 − α) ˆdt−1 = α + (1 − α) ˆdt−1 ˆ zt = βzt+ (1 − β)ˆzt−1 Finally, ˆ yt = ˆdtzˆt
In the computations above, the smoothing parameter α is used to update the demand probability and the smoothing parameter β is used to update the non-zero demand occurrences. Smoothing parameter α need not be different from smoothing parameter β, although Teunter et al. (2011) argue that α < β.
4.1.2 Forecasting performance measures
For i = 1, 2, . . . , n, let yi be a demand occurrence at time i and ˆyi the demand
forecast at time i. Then, the mean squared error is given by M SE = 1 n n X i=1 (yi− ˆyi)2
Furthermore, let the cumulative mean of a time series be given by
rj = ˆyj− 1 j j X i=1 yi
Then the mean absolute rate is given by M AR = n X j=1 | rj | = n X j=1 | ˆyj− 1 j j X i=1 yi | 4.1.3 Demand classification
As was discussed in section 2, demand classification depends on the average inter-demand interval (p) and the squared coefficient of variation of the non-zero de-mand occurrences for the time series (CV2). Obviously, the average inter-demand
interval is the sum of the demand intervals divided by the number of inter-demand intervals. Furthermore, the squared coefficient of variation of the non-zero demands, in the literature often denoted as v, is given by
v = CV2 = σ µ
2
where σ is the standard deviation of the non-zero demands and µ is the mean of the non-zero demands.
Using the mean squared error, the Syntetos-Boylan Approximation was shown to be superior to Croston’s method when
v > 4p(2 − p) − α(4 − α) − p(p − 1)(4 − α)(2 − α) p(4 − α)(2p − α)
Using limiting values, this was simplified to the following approximation v > 2 − 3
The minor addition to this existing classification scheme is that if there is only a single demand occurrence and no proper inter-demand interval, then the Teunter-Syntetos-Babai method is preferred. Furthermore, as discussed in section 2, single exponential smoothing is applied when p ≤ 1. In conclusion, the demand classifi-cation method which will be applied in this paper is given by
The preferred demand forecasting method is TSB, if yt> 0 only once SES, if p ≤ 1 CRO, if 1 < p < 43 and v < 12 SBA, if 1 < p < 43 and v ≥ 12 SBA, if p ≥ 43
4.2
The repair kit problem
As discussed in section 2, a binary knapsack problem and a greedy marginal anal-ysis are considered in this paper. In this section, the details of the model are given.
In order to compute the repair kit solution for any repairman, the fail rates for the various types of spare parts for the repairman must be computed first. Let there be n types of spare parts. Let pi, i = 1, 2, . . . , n, be the probability that a
spare part of type i has failed and needs to be replaced at the time of the service visit. This probability is estimated based on the bimonthly demand forecast ˆyi,t of
spare part type i for the repairman divided by the expected number of jobs in the same period. Let Ji be the expected number of bimonthly jobs for repairman i. As
mentioned in section 3, there is no exact information in the dataset on the number of jobs for each repairman. In consultation with the management, an estimate of 8 jobs per day for an average of 18 days per month is considered as an upper bound for the expected number of bimonthly jobs for a repairman. Resulting in an upper bound of 288 jobs in a time span of 2 months. Let Ujob = 288 be this upper bound. The actual number of reported demand usages of the repairman in the non-aggregated dataset is used as a correction for a more accurate estimated number of bimonthly jobs for the repairman. Let Ni be the actual number of reported demand
usages of repairman i in the non-aggregated dataset and let M = max {Ni} for
Hence, Ji = Ni M Ujob = U job M Ni = 288 1107 Ni
Then, the fail rates for the various types of spare parts of each repairman are computed as the one-period-ahead forecast divided by the corrected estimated number of bimonthly jobs. Should the fail rate of any of the various types of spare parts, for any of the repairmen, exceed the value of 1, it is set to 1 instead. That is, the spare part type has a 100% fail rate per job. However, this only occurred once, for one spare part type of a single repairman in the dataset. Thus,
pi = max
ˆyi,t
Ji
, 1
Furthermore, each repairman should only experience failures of a subset of the total amount of possible spare parts. This depends on the types of boilers the repairman has in its routes which, in practice, is frequently unknown. In order to produce the most realistic repair kit solutions possible, it is assumed that for each repairman it holds that the unknown (sub-)set of possible types of spare parts consists of the spare part types that failed at any point during the time frame of the dataset. In this case, at any point during the year 2017. Hence, it is assumed that each repairman only services types of boilers which contain spare part types the repairman used at least once at some point during the time frame of the dataset. Let hi, i = 1, 2, . . . , n, be the annual holding cost for spare part
variable for the inclusion of spare part type i in the repair kit. Then, min n X i=1 hixi s.t. n Y i=1 (1 − pi)1−xi ≥ γ xi = {0, 1}, i = 1, 2, . . . , n
The job-fill rate can be rewritten as
n
P
i=1
(1 − xi) log(1 − pi) ≥ log(γ).
Using this, the model can be formulated as a binary knapsack problem. max n X i=1 −hixi s.t. n X i=1 xilog(1 − pi) ≤ −β xi = {0, 1}, i = 1, 2, . . . , n where −β = n P i=1 log(1 − pi) − log(γ).
The greedy marginal analysis procedure ranks the various types of spare parts in non-decreasing order according to the ratio
hi
− log(1 − pi)
Hence, if hi is estimated at some percentage of the cost price for all parts, the ratio
and thus the ranking is not affected by the percentage value previously chosen for hi. After ranking the spare part types in non-decreasing order in terms of the
knapsack ratio, the job-fill rate and total holding costs are computed at every spare part type assuming the previous spare part types were added to the repair kit solution. For each repairman in the dataset, the spare part types are added one by one to the repair kit until
n
X
i=1
(1 − xi) log(1 − pi) ≥ log(γ)
5
Results
In this section the results from the demand classification, the demand forecast-ing methods, and the repair kit problem are discussed. Furthermore, a demand forecasting method is introduced which will be used as the benchmark and to investigate the implication of more accurate input on the repair kit solution.
5.1
Demand forecasting
First, the results from the demand classification are discussed. Secondly, the fore-casts from the corresponding demand forecasting methods are briefly examined using the mean squared error as a performance measure. The combination of the demand classifications and the corresponding demand forecasting methods, with appropriate forecasting performance measures, are from now on called the sophisticated demand forecasting techniques and are discussed in section 5.1.1. Furthermore, the demand forecasting method which will be used as the bench-mark is introduced and the corresponding forecasts are similarly examined using the mean squared error as a performance measure in section 5.1.2. Although the mean squared error provides little information in itself, it provides a means to compare different one-period ahead forecasts as will be explained in more detail in section 5.1.3 when the mean squared errors obtained from the sophisticated demand forecasting techniques are compared to the mean squared errors obtained from the benchmark for every spare part type of every repairman. Hence, the percentage of spare part types which are predicted more accurately using sophisti-cated demand forecasting techniques can be computed for every repairman in the dataset.
5.1.1 Sophisticated demand forecasting techniques
approximation or the Teunter-Syntetos-Babai method, respectively.
(a) Single exponential smoothing. (b) Croston’s method.
Figure 5: Histograms of the two demand forecasting methods which were classified relatively infrequently for repairmen in the dataset.
(a) The Syntetos-Boylan approximation. (b) The Teunter-Syntetos-Babai method.
Figure 6: Histograms of the two demand forecasting methods which were classified relatively frequently for repairmen in the dataset.
repairman in the dataset is 18.99%. For Croston’s method this is 14.91%, for the Syntetos-Boylan approximation this is 75.00% and for the Teunter-Syntetos-Babai method this is 90.62% of the spare part types of a repairman.
SES CRO SBA TSB
Minimum of any repairman 0.00 0.00 9.38 13.89
Mean across repairmen 6.24 2.23 41.68 49.85
Maximum of any repairman 18.99 14.91 75.00 90.62
Table 1: Demand classification results in percentage values for all repairmen in the dataset.
Figure 7: Histogram of the average mean squared error of bimonthly spare part type usage forecasts per repairman for the sophisticated demand forecasting tech-niques.
Average mean squared error across all spare part types
Standard deviation of the mean squared er-rors across all spare part types
Lowest for any repairman 1.15 1.38
The median for repairmen 13.73 64.24
The mean for repairmen 33.00 214.14
Highest for any repairman 257.94 2538.69
Table 2: Further details on the mean squared errors obtained for all repairmen in the dataset.
5.1.2 Demand forecasting for the benchmark
the benchmark. That is, the benchmark forecast equals ˆ yi,t = 1 t − 1 t−1 X j=1 yi,j , t = {2, . . . , 7}
Similarly to section 5.1.1, using the benchmark a one-period (of two months) ahead forecast is produced for each period, except the first, using the previous available period or periods. Hence, six benchmark forecasts are produced for March-April of 2017 until January-February of 2018. The mean squared error obtained from the five forecasts in 2017 is used as an performance measure of the forecast prediction and a mean squared error is produced for every spare part type of every repairman. A histogram is presented in figure 8 displaying the average mean squared error obtained for the benchmark across the spare part types for each repairman. Furthermore, additional information on the mean squared errors for all the repairmen in the dataset and are presented in table 3.
Average mean squared error across all spare part types
Standard deviation of the mean squared er-rors across all spare part types
Lowest for any repairman 2.06 2.09
The median for repairmen 17.77 71.82
The mean for repairmen 37.80 228.34
Highest for any repairman 271.14 2507.85
Table 3: Further details on the mean squared errors obtained for all repairmen in the dataset.
5.1.3 Comparison of the performance of the sophisticated demand forecasting techniques and the benchmark
In order to empirically test the claim of having more accurate input using the demand classification, the demand forecasting methods and the forecasting per-formance measures used in this paper, the mean squared errors obtained from the sophisticated demand forecasting techniques are compared to the mean squared errors obtained from the forecasts obtained from the benchmark which uses aver-age values. The results obtained from sections 5.1.1 and 5.1.2, taking the averaver-age mean squared errors across spare part types for each repairman, are combined in table 4.
Mean squared error for sophis-ticated demand forecasting tech-niques
Mean squared error for the bench-mark with average value forecasts
Minimum 1.15 2.06
Median 13.73 17.77
Mean 33.00 37.80
Maximum 257.94 271.14
Table 4: Comparing the forecasting results of the sophisticated demand forecasting techniques with average values as forecasts.
for which the mean squared error obtained from the sophisticated demand forecast-ing techniques is lower than the mean squared error obtained from the forecasts obtained from the benchmark is computed. The results are given in figure 9 and further details of these results are given in table 5. There are no repairmen in the dataset for which it holds that the majority of the spare part types are more accurately predicted using the forecasts obtained from the benchmark than with the sophisticated demand forecasting techniques. In fact, for all repairmen in the dataset the vast majority of the spare part types is more accurately predicted using the sophisticated demand forecasting techniques as can be seen in figure 9. Fur-thermore, for a considerable amount of repairmen in the dataset it holds that 100% of the spare part types are more accurately predicted using sophisticated demand forecasting techniques than using the forecasts obtained from the benchmark using average values.
Figure 9: The percentage of spare part types per repairman which are predicted more accurately using sophisticated demand forecasting techniques.
the benchmark forecasts than the sophisticated demand forecasting techniques is 5.69%.
Percentage of spare part types
Minimum for any repairman. 94.31
Mean across the repairmen. 98.75
Maximum for any repairman. 100.00
Table 5: The percentage of spare part types for which the mean squared error obtained from the sophisticated demand forecasting techniques is lower than the mean squared error obtained from the benchmark.
5.2
Repair kit solutions
First, the repair kit solutions acquired from using the fail rates which were ob-tained from the sophisticated demand forecasting techniques as input are discussed in section 5.2.1. These repair kit solutions are from now on called the sophisti-cated repair kit solutions. The resulting amount of spare part types, corresponding jfill rate and holding costs are presented. Secondly, the repair kit solutions ob-tained from using the fail rates which were obob-tained from the benchmark demand forecasting method as input are discussed in section 5.2.2 and similarly the effect on the amount of spare part types and the corresponding job-fill rate and holding costs is discussed. These repair kit solutions are from now on called the bench-mark repair kit solutions. As was the case with demand forecasting, these results do not mean much yet on their own and will become especially useful when used to compare the two models with different inputs. Hence, in section 5.2.3 the effect of more accurate input on the repair kit solution is examined and discussed. By considering the spare part types obtained from the benchmark repair kit solution in combination with the one-period ahead forecasts resulting from sophisticated demand forecasting methods and comparing it to the sophisticated repair kit solu-tion of secsolu-tion 5.2.1, the effect of more accurate input on the amount of spare part types, corresponding job-fill rate and holding costs of the repair kit solution can be investigated. The benchmark repair kit solutions but with fail rates from the sophisticated demand forecasting techniques are from now on called the realistic benchmark repair kit solutions.
5.2.1 Repair kit solutions from sophisticated demand forecasting tech-niques
discussed. To prevent repetition, this is assumed to be the case throughout section 5.2.1 and the results are called the sophisticated repair kit solutions. The resulting amount of spare part types, corresponding job-fill rate and holding costs of the repair kit solution will be presented. As was discussed in section 4, applying the greedy marginal analysis procedure on the repair kit problem for each repairman results in a repair kit solution for each repairman in the dataset. Each repair kit solution has a possibly unique set of spare part types, corresponding holding costs, and with a corresponding job-fill rate of at least γ. Obviously, a range of values for γ can be explored. However, in consultation with the management of the company, a job-fill rate of at least 95% will be assumed in the repair kit problem. The resulting amount of spare part types and holding costs of the repair kit solutions for all repairmen in the dataset are given in figure 10a and figure 10b respectively.
(a) The amount of spare part types in the so-phisticated repair kit solution.
(b) The holding cost of the sophisticated repair kit solution.
Figure 10: The sophisticated repair kit solutions per repairman.
Where the average amount of spare part types in a sophisticated repair kit solution for the repairmen in the dataset is 85.72. The lowest amount of spare part types in a sophisticated repair kit solution for any repairman in the dataset is 17, where the highest amount of spare part types in a sophisticated repair kit solution is equal to 170. Furthermore, the average holding costs of the sophisticated repair kit solutions for the repairmen in the dataset is 73.39. The lowest holding costs for a sophisticated repair kit solution of any repairman in the dataset is 19.77, while the highest holding cost is equal to 186.08. Moreover, the job-fill rates of the sophisticated repair kit solutions range from 95.00% to 96.63% with an average of 95.28% and a median of 95.21%. That the job-fill rate exceeds the specified value for γ is unsurprising since the heuristic adds spare part types one by one, resulting in a possibly higher job-fill rate than necessary.
solution with a “100% job-fill rate”. The amount of spare part types and the corresponding holding costs in the sophisticated repair kit solutions are compared to the case where all spare part types are added to the repair kit are given in figure 11a and figure 11b respectively.
(a) The percentage of spare part types in the sophisticated repair kit solution.
(b) The relative holding cost of the sophisti-cated repair kit solution.
Figure 11: Sophisticated repair kit solution per repairman compared to a full repair kit.
On average a repairman only needs 80.79% of the set of possible spare part types to achieve a job-fill rate equal to γ. Furthermore, on average the holding costs for a repairman are only 46.06% of the total holding cost for using all possible spare part types in the repair kit. Hence, using this repair kit problem with sophisticated demand forecasting techniques a repair kit solution can be found for which, on average for all repairmen, the amount of spare part types can be reduced by 19.21% which results in the corresponding holding costs being 53.94% lower, while still achieving a job-fill rate of at least γ.
5.2.2 Repair kit solution from the benchmark
the fail rates obtained from the benchmark, which were shown to be less accurate forecasts based on the mean squared error. The resulting amount of spare part types and holding costs of the benchmark repair kit solution for all repairmen in the dataset are given in figure 12a and figure 12b respectively.
(a) The amount of spare part types in the benchmark repair kit solution.
(b) The holding cost of the benchmark repair kit solution.
Figure 12: The benchmark repair kit solutions per repairman.
Where the average amount of spare part types in a benchmark repair kit so-lution for the repairmen in the dataset is 85.18. The lowest amount of spare part types in a benchmark repair kit solution for any repairman in the dataset is 17, where the highest amount of spare part types in a benchmark repair kit solution is equal to 169. Furthermore, the average holding costs of the benchmark repair kit solutions for the repairmen in the dataset is 70.38. The lowest holding costs for a benchmark repair kit solution of any repairman in the dataset is 19.71, while the highest holding cost is equal to 186.42. Moreover, the presumed job-fill rates of the benchmark repair kit solutions range from 95.01% to 97.75% with an average of 95.31%.
(a) The percentage of spare part types in the benchmark repair kit solution.
(b) The relative holding cost of the benchmark repair kit solution.
Figure 13: The benchmark repair kit solution per repairman compared to a full repair kit.
Mean amount of spare part types in the repair kit solution per re-pairman
Mean job-fill rate of the re-pair kit solution per repairman
Mean holding
costs of the re-pair kit solution per repairman Sophisticated
repair kit solution 85.72 95.28 73.39
Benchmark repair
kit solution 85.18 95.31 70.38
Table 6: The sophisticated repair kit solution compared to the benchmark repair kit solution.
Assuming that the sophisticated demand forecasting techniques provide the most realistic fail rates, the benchmark repair kit solutions are re-evaluated and the job-fill rates recalculated with the more realistic fail rates as obtained from the sophisticated demand forecasting techniques. Hence, by considering the spare part types from the benchmark repair kit solution in combination with the one-period ahead forecasts resulting from sophisticated demand forecasting methods, more realistic job-fill rates can be computed for the benchmark repair kit solution. These repair kit solutions are from now on called the realistic benchmark repair kit solutions. In short, the fail rates and knapsack ratios are calculated as was the case in the sophisticated repair kit solution. However, when adding the spare part types to the knapsack, the spare part types obtained from the benchmark repair kit solution are the only spare part types which are added to the knapsack. A repair kit solution is presented and the corresponding amount of spare part types, the obtained job-fill rate (which need not be equal to γ), and holding costs are presented. Furthermore, the realistic benchmark repair kit solutions are compared to the sophisticated repair kit solutions. Obviously, the amount of spare part types and corresponding holding costs are identical to the benchmark repair kit solution. However, the obtained job-fill rate is worth investigating.
Achieved job-fill rate
Minimum 93.92
Median 94.88
Mean 94.96
Maximum 97.30
Table 7: The job-fill rates as achieved by the realistic benchmark repair kit solu-tions for all repairmen.
(a) The amount of spare part types in the sophisticated repair kit solution minus the amount of spare part types of the realistic benchmark repair kit solution.
(b) The holding cost of the sophisticated re-pair kit solution minus the holding cost of the realistic benchmark repair kit solution.
Job-fill rate of the sophisticated repair kit solu-tion minus the job-fill rate of the realistic bench-mark repair kit solution
Minimum for any repairman -1.04 Mean across repairmen 0.32 Maximum for any repairman 1.49
Table 8: The achieved job-fill rate of the sophisticated repair kit solution compared to the realistic benchmark repair kit solution.
Amount of spare part types in the sophisti-cated repair kit solu-tion minus the amount of spare part types in the realistic bench-mark solution
Holding cost of the sophisticated repair kit solution minus the holding cost of the realistic benchmark repair kit solution
Minimum for any repairman -6.00 -4.65
Mean across repairmen 0.53 3.01
Maximum for any repairman 6.00 14.23
Table 9: Differences in the amount of spare part types and the holding costs for the sophisticated repair kit solution compared to the realistic benchmark repair kit solution.
6
Conclusion
This paper empirically tests the effect of having more accurate input on the repair kit solutions and finds that for repair kit solutions with more accurate input the achieved mean job-fill rate was 0.32% higher than the achieved mean job-fill rate of the repair kit solutions with less accurate input. These results were found using spare part usage data obtained from a Dutch-based company. First, by proving the existence of more accurate input data. And subsequently, by testing the effect of this more accurate input on the repair kit solutions.
the sophisticated demand forecasting techniques certainly produce more accurate forecasts, the one-period (of two months) ahead forecasts were transformed into fail rates using an estimated number of bimonthly jobs for all products of every repairman. Hence, the presence of better input data.
Using a single-period, multiple-item repair kit problem, to be solved using a marginal analysis (greedy) procedure, knapsack ratios are computed and then sorted in non-decreasing order. Depending on the fail rates, from either the sophis-ticated demand forecasting techniques or the benchmark, and the specified job-fill rate, the repair kit solution is constructed. The repair kit solution consists of the spare part types and the holding costs with the presumably achieved job-fill rate. Assuming that the sophisticated demand forecasting techniques provide more real-istic fail rates than the benchmark, the repair kit solutions from the sophreal-isticated demand forecasting techniques and the re-evaluated repair kit solutions from the benchmark are compared and as such the effect of more accurate input on the repair kit solution is examined.
As such a noticeable effect of more accurate input on the repair kit solution was observed. This is especially due to the fact that less-accurate input data leads to inaccurate job-fill rates. For the dataset used in the case study of this paper, the job-fill rates of the benchmark repair kit solutions were in general too optimistic. In practice, this would result in more revisits than expected with additional return-to-fit costs and customer dissatisfaction.
References
Bacchetti, A. and N. Saccani (2012, December). Spare parts classification and demand forecasting for stock control: Investigating the gap between research and practice. Omega 40 (6), 722–737.
Bijvank, M., G. Koole, and I.F.A. Vis (2010, July). Optimising a general re-pair kit problem with a service constraint. European Journal of Operational Research 204 (1), 76–85.
Boylan, J.E. and A.A. Syntetos (2010, July). Spare parts management: a review of forecasting research and extensions. IMA Journal of Management Mathemat-ics 21 (3), 227–237.
Boylan, J.E., A.A. Syntetos, and G.C. Karakostas (2008). Classification for fore-casting and stock control a case study. Journal of the Operational Research Society 59 (4), 473–481.
Brumelle, S. and D. Granot (1993, September-October). The repair kit problem revisited. Operations Research 41 (5), 994–1006.
Carson, R.T., T. Cenesizoglu, and R. Parker (2011, July-September). Forecast-ing (aggregate) demand for us commercial air travel. International Journal of Forecasting 27 (3), 923–941.
Croston, J.D. (1972, September). Forecasting and stock control for intermittent demands. The Journal of the Operational Research Society 23 (3), 289–303. Eaves, A.H.C. and B.G. Kingsman (2004, April). Forecasting for the ordering
and stock-holding of spare parts. The Journal of the Operational Research So-ciety 55 (4), 431–437.
Fildes, R., K. Nikolopoulos, S.F. Crone, and A.A. Syntetos (2008, September). Forecasting and operational research: a review. Journal of the Operational Research Society 59 (9), 1150–1172.
Fox, B. (1966). Supply chain forecasting theory practice their gap and the future. Management Science 13 (3), 210–216.
Gardner, E.S. (2006, October). Exponential smoothing the state of the art part 2. International Journal of Forecasting 22 (4), 637–666.
