Evolutionary Optimization of Spare Parts Inventory Policies: a Life Cycle Costing Perspective

Control 9th IFAC Conference on Manufacturing Modelling, Management and Berlin, Germany, August 28-30, 2019 Available online at www.sciencedirect.com Control 9th IFAC IFAC Conference Conference on on Manufacturing Manufacturing Modelling, Management and 9th Modelling, Management and Berlin, Germany, August 28-30, 2019 Control Control Berlin, Berlin, Germany, Germany, August August 28-30, 28-30, 2019 2019

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IFAC PapersOnLine 52-13 (2019) 2243–2248 Evolutionary Optimization of Spare Parts Inventory Policies: a Life Cycle Evolutionary OptimizationCosting of SparePerspective Parts Inventory Policies: a Life Cycle Evolutionary of Parts Evolutionary Optimization OptimizationCosting of Spare SparePerspective Parts Inventory Inventory Policies: Policies: aa Life Life Cycle Cycle Orlando Durán *, Arturo Carrasco* Costing Perspective Costing Perspective Paulo Sérgio Afonso**, Paulo Andrés Durán***

Orlando Durán *, Arturo Carrasco* PauloOrlando Sérgio Afonso**, Durán*** Durán Arturo Carrasco* Orlando Durán *, *, Paulo ArturoAndrés Carrasco* Paulo Sérgio Afonso**, Paulo Andrés Durán*** Paulo Sérgio Afonso**, Paulo Andrés Durán*** * Pontificia Universidad Católica de Valparaíso, Valparaíso, 2340025, Chile (e-mail: [email protected], * Pontificia Universidad Católica [email protected]) Valparaíso, Valparaíso, 2340025, Chile ***Universidade Do Minho,Católica Campusde deValparaíso, Azurém, 4800 - 058 Guimarães, Portugal Universidad (e-mail: [email protected], * Pontificia Pontificia Universidad Católica [email protected]) Valparaíso, Valparaíso, Valparaíso, 2340025, 2340025, Chile Chile (e-mail: [email protected]) (e-mail: [email protected], [email protected]) ** Universidade Do Minho, Campus de Azurém, 4800 - 058 Guimarães, Portugal (e-mail: [email protected], [email protected]) *** INACAP Universidad Tecnológica de Chile, Valparaiso. ** Minho, Campus de (e-mail: [email protected]) ** Universidade Universidade Do Do (e-mail: Minho, Campus de Azurém, Azurém, 4800 4800 -- 058 058 Guimarães, Guimarães, Portugal Portugal [email protected]) [email protected]) *** INACAP(e-mail: Universidad Tecnológica de Chile, Valparaiso. (e-mail: [email protected]) *** Universidad Tecnológica (e-mail: [email protected]) *** INACAP INACAP Universidad Tecnológica de de Chile, Chile, Valparaiso. Valparaiso. (e-mail: [email protected]) (e-mail: [email protected]) Abstract: This paper proposes the development of an optimization model for spare parts management during theThis life cycle a physical The mainofcharacteristic of the proposed modelparts is that it is based Abstract: paperofproposes theasset. development an optimization model for spare management on the principles of Activity Based Costing (ABC). The aim of this model is to allow decision to during the life cycle of a physical asset. The main characteristic of the proposed model is that itmakers is based Abstract: This This paper paper proposes proposes the the development development of of an an optimization optimization model model for for spare spare parts parts management management Abstract: perform an life optimal inventory policyasset. and the correspondent parameter definitions for a series of spare parts. on the principles of Activity Based Costing (ABC). The aim of this model is to allow decision makers to during the cycle of a physical The main characteristic of the proposed model is that it is based during the life cycle of athrough physical The main characteristic ofhypothetical the proposed model isisthat it is based The model is optimized theasset. use ofthe a Genetic Algorithm. A case study presented and on the principles of Activity Based Costing (ABC). The aim of this model is to allow decision makers to perform an optimal inventory policy and correspondent parameter definitions for a series of spare parts. on the principles of Activity Based Costing (ABC). The aim of this model is to allow decision makers to discussed. perform an optimal inventory policy and correspondent parameter definitions for aa series spare The model optimized through the use a Genetic Algorithm. A hypothetical is of presented and perform an is optimal inventory policy andofthe the correspondent parameter definitionscase for study series of spare parts. parts. The model is optimized through the use of a Genetic Algorithm. A hypothetical case study is presented discussed. © 2019, IFAC (International Federation Control) Hosting by Elseviercase Ltd. All rights reserved.and The model is optimized the useof ofAutomatic a Genetic Algorithm. A hypothetical study is presented and Keywords: Spare Parts through Management; Inventory Control; Genetic Algorithms; Optimization discussed. discussed. Keywords: Spare Parts Management; Inventory Control; Genetic Algorithms; Optimization

Keywords: Genetic Optimization methodology for the selection of spare parts management Keywords: Spare Spare Parts Parts Management; Management; Inventory Inventory Control; Control; Genetic Algorithms; Algorithms; Optimization 1. INTRODUCTION policies based on the criticality level. Sanchez-Partida et al. methodology for the selection of spare parts management 1. INTRODUCTION (2012) suggest a methodology based on the ABC classification Life cycle costs have been studied extensively in several methodology policies basedfor on the criticality level. Sanchez-Partida et al. selection of spare parts management methodology the selection the of spare parts of management 1. INTRODUCTION of spare partsfor for determining parameters inventory 1. INTRODUCTION engineering areas. Several models have been developed to (2012) suggest a methodology based on the ABC classification policies based on the criticality level. Sanchez-Partida Life cycle costs have been studied extensively in several policies based on the criticality level. Sanchez-Partida et et al. al. control inparts spare parts management. All ABC of these authors estimate the costs of products and production systems. Among (2012) suggest a methodology based on the classification of spare for determining the parameters of inventory engineering areas. Several models have been developed to Life cycle costs have been studied extensively in several (2012) suggest a methodology based on the ABC classification Lifemodels cycle costs have gained been studied extensively inwe several addressed the problem ofmanagement. definingthe inventory policies and their the that theproduction most may spare parts All of these authors of for determining of estimate the costs ofSeveral productsmodels and systems. Among engineering areas.have Several models havereputation, been developed to control of spare spareinparts parts for determining the parameters parameters of inventory inventory engineering areas. have been developed to parameters as a single period problem in combination with highlight Woodward´s (1993, 1997) models. problem defining inventory policies their control in spare parts All these authors the models that have gainedand theproduction most reputation, may addressed estimate the of systems. Among control rules inthe spare partsofmanagement. management. All of of theseand authors estimate the costs costs of products products and production systems.we Among simple of thumb. parametersthe asproblem a singleof problem in combination with addressed defining inventory policies highlight Woodward´s (1993, models. the models that gained the most reputation, we the problem ofperiod defining inventory policies and and their their the models that have have gained 1997) themaintenance most reputation, we may may addressed Despite its great importance in and operation simple rules of thumb. parameters as a single period problem in combination with highlight Woodward´s (1993, 1997) models. parameters as a single period problem in combination values for lot sizes, replenishment frequencieswith and highlight Woodward´s (1993, 1997) models. costs (OPEX), which can represent up to 40% and of total costs, Different Despite its great importance in maintenance operation simple rules of thumb. simple rules of thumb. ordering levels can significantly affect long-term management until now no model has incorporated spare parts management Different values for lot sizes, replenishment frequencies and costs (OPEX), which can represent up to 40% and of total costs, Despite its great importance in maintenance operation Despite itscycle great importance infor maintenance and operation costs. However, the effect inreplenishment long term of frequencies such inventory in the life cost calculation high value equipment. ordering levels can significantly affect long-term management Different values for lot sizes, and until no model incorporated spare partsof costs (OPEX), which can represent to total values for lot sizes, replenishment frequencies and costs now (OPEX), whichhas can represent up up to 40% 40% ofmanagement total costs, costs, Different policies and parameters hasinbeen scarcely studied (Duran costs. However, the effect long term of such inventory ordering levels can significantly affect long-term management in the life cycle cost calculation for high value equipment. until now no model has incorporated spare parts management ordering levels can significantly affect long-term management until spare now noparts model has incorporated spare management The management process is parts activated when a 2016a). Recently, Duran et (2019) proposed a model for costs. However, the in long term of such policies and parameters hasal. (Duran in the life cycle cost calculation for high value equipment. costs. However, the effect effect inbeen longscarcely term of studied such inventory inventory in the life cycle cost calculation for high value equipment. failure occurs or when units are replaced preventively. When costing logistics activities considering a multi-period approach The spare parts management process is activated when a policies 2016a). Recently, Duran et (2019) proposed a model for and has scarcely studied (Duran policies 2019). and parameters parameters hasal.been been scarcely studied (Duran this replacement a number of logistics activitieswhen related failure occurs or occurs, when units areprocess replaced Whenaa (Duran, The spare parts management process ispreventively. activated 2016a). Recently, Duran et al. (2019) proposed a model costing logistics activities considering a multi-period approach The spare parts management is activated when 2016a). Recently, Duran et al. (2019) proposed a model for for to management of spare parts are carried out (Cavalieri et (Duran, thisthe replacement occurs, a number of logistics activities related failure occurs or when units are replaced preventively. When 2019). costing logistics considering aa multi-period approach failure occurs or when units are replaced preventively. When costingterm logistics activities considering multi-period approach Long spareactivities parts planning involves the forecasting of al., 2008). Huiskonen (2001) provided a useful basis for to the management of spare parts are carried out (Cavalieri et (Duran, 2019). this replacement occurs, aa number of activities related this replacement occurs, number of logistics logistics activities related (Duran, 2019). future demand. That estimation constitutes oneforecasting of the most mapping spare parts logistics activities. Some activities are Long term spare parts planning involves the of al., 2008). Huiskonen (2001) provided a useful basis for to the management of parts are out (Cavalieri et to the management of spare spare parts are carried carried out (Cavalieri et complex aspects of life cycle cost models (Syntetos, 2012). Hu carried out as a consequence of purchasing or ordering one or Long term spare parts planning involves the forecasting of future demand. That estimation constitutes one of the most mapping spare parts logistics activities. Some activities al., 2008). Huiskonen (2001) provided aa useful basis for Long term spare parts planning involves the for forecasting of al., 2008). Huiskonen (2001) provided useful basis are for et al. (2015) modelled a forecasting method a2012). complex more units of a given spare part. Others relate mainly to complex aspects of life cycle cost models (Syntetos, Hu future demand. That estimation constitutes one of the most carried out as a consequence of purchasing or ordering one or mapping spare parts logistics activities. Some activities are future demand. That estimation constitutes one of the most mapping spare the parts logistics to activities. Some activities are preventive maintenance policy. In particular, this is a major shipping from warehouse the Others repair site. And finally, complex aspects of cycle cost models (Syntetos, Hu al. (2015) modelled a forecasting method for a2012). complex more units ofaa aconsequence given spareof part. relate mainly to et carried out as purchasing or ordering one or complex aspects of life life cycle cost models (Syntetos, 2012). Hu carried out as consequence of purchasing or ordering one or difficulty since spare parts with very complex demand patterns there are a number of activities that are devoted to the preventive maintenance policy. In particular, this is a major et al. (2015) modelled a forecasting method for a complex shipping from the warehouse to the repair site. And finally, more units of a given spare part. Others relate mainly to et al. (2015) modelled a forecasting method for a complex more units ofofaexisting given spare part. Others relate mainly to (Baykasoglu and Kaplanoglu, 2008; Boylan and Syntetos, management thearewarehouse. In the difficulty since spare partspolicy. with very complex demand maintenance In this is aa major there arefrom a number of stocks activities devoted to the preventive shipping from the warehouse warehouse towithin thethat repair site. And finally, finally, preventive maintenance policy. In particular, particular, thisincorporated is patterns major shipping the to the repair site. And Tosince meet thisKaplanoglu, challenge, some models have specialized literature, such activities have been treated in a 2010). (Baykasoglu and 2008; Boylan and Syntetos, difficulty spare parts with very complex demand patterns management of existing stocks within the warehouse. In the there are a number of activities that are devoted to difficulty since spare parts with very complex demand patterns there are a number of activities that are devoted to the Weibull reliability function. Such a function models the very generalliterature, way and such with activities approximations that treated can lead to the 2010). To meet thisKaplanoglu, challenge, some models haveand incorporated (Baykasoglu and 2008; Boylan Syntetos, specialized been management of the In and Kaplanoglu, 2008; Boylan and Syntetos, management of existing existing stocks stocks within withinhave the warehouse. warehouse. Ininthe thea (Baykasoglu behavior of the failure rate over time in a meaningful and erroneous decisions. the Weibull reliability function. Such a function models the 2010). To To meet meet this this challenge, challenge, some some models models have have incorporated incorporated very generalliterature, way and such with activities approximations that treated can leadin specialized have specialized literature, such activities have been been treated in toaa 2010). representative form. behavior of the failure rate over time in a meaningful and the Weibull reliability function. Such a function models the erroneous decisions. very general way and with approximations that to very general way andparts withmanagement approximations that can can lead to the Weibull reliability function. Such a function models the Traditionally, spare literature haslead been behavior of the failure rate over time in a meaningful and representative form. erroneous behavior the out failure rate over time in aprocess, meaningful and In order toofcarry the spare management resources erroneous decisions. focused ondecisions. defining and their parameters in spare Traditionally, sparepolicies parts management literature has parts been representative form. representative form. are needed. In the "activity-based-costing" method resources inventory Aisyati al. (2013) proposed a model In order to carry out the spare management process, resources Traditionally, spare parts management literature has been focused onmanagement. defining andettheir parameters in spare parts Traditionally, sparepolicies parts management literature hasaircraft been costs are firstly allocated to the activities and in a second step, to determine ordering quantity and reorder point for are needed. In the method resources In to out the inventory Aisyati al. (2013) proposed a model focused on defining policies and parameters in spare parts In order order to carry carry out"activity-based-costing" the spare spare management management process, process, resources focused onmanagement. defining policies andettheir their parameters inproposed spare partsa costs consumable spares parts. Miranda et al. (2014) are firstly allocated to the activities and in a second step, are needed. In the "activity-based-costing" resources to determine ordering quantity and reorder point for aircraft inventory management. management. Aisyati Aisyati et et al. al. (2013) (2013) proposed proposed aa model model are needed. In the "activity-based-costing" method method resources inventory costs are firstly allocated to the activities and in a second step, consumable spares parts. Miranda et al. (2014) proposed a to determine ordering quantity and reorder point for aircraft to determine ordering quantity and reorder point for aircraft costs are firstly allocated to the activities and in a second step, consumable spares parts. Miranda et al. (2014) proposed a Copyright © 2019 IFAC 2293 consumable spares parts. MirandaFederation et al. (2014) proposed a Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2019, IFAC (International of Automatic Control) Copyright 2019 responsibility IFAC 2293Control. Peer review©under of International Federation of Automatic 10.1016/j.ifacol.2019.11.539 Copyright © 2019 IFAC 2293 Copyright © 2019 IFAC 2293

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the costs of activities are allocated to the relevant cost objects. (Afonso and Paisana, 2009).

management costs with the total costs of ownership of high valued physical assets.

With this approach, it is possible to increase the level of accuracy in the computation of the costs of all the logistics activities carried out in the warehouse. It is worth mentioning that these costs depend, to a large extent on the levels of demand that each class of spare parts has along the analysis horizon.

The activity-based costing method maps production and business processes, and these processes into activities. An activity represents what the organization does, the time it spends doing it, and the product obtained. Activity based costing identifies the activities performed and determines the resources consumed by each one of that activities. In other words, an activity-based costing model allocates the resources’ costs to a series of activities which are carried out to produce/handle the relevant cost objects (Almeida and Cunha, 2017). In a second phase, the activity costs are allocated to the cost objects (Figure 1). In this case, the cost objects are spare and/or service parts.

This paper explores the integration of an ABC approach with a spare parts management costing model to develop an evolutionary optimization methodology based on a life cycle perspective. Once the costing model is defined, a genetic algorithm approach for optimizing the long-term inventory control decisions can be proposed. Specifically, the proposed model and the optimization technique are focused on the optimization of the main parameters of an inventory policy. The global cost function is defined in terms of direct costs, holding costs and logistics costs. Given the large number of variables in this type of models, we have incorporated genetic algorithms as a solution mechanism. With them we seek to determine the inventory policy for a significant group of spare parts and, even more important, to obtain the optimal parameters of such policy, which as a whole, minimize long-term management costs. The paper is organized as follows. Section 2 is focused on discussing the theoretical background on spare parts management costs and life cycle costing. Section 3 presents the proposed model. Next, in section 4, the main results of the optimization process are presented and discussed. Finally, in section 5 the main conclusions and final remarks of along with the future research are proposed. 2. THEORETICAL BACKGROUND 2.1 Activity Based Costing According to Cooper (1990), costs are divided into two groups: direct and indirect, depending on how viable it is to allocate such cost to the corresponding cost objects. Indirect costs are thus allocated to cost objects, using allocation bases such as production volume, direct labor or the amount of raw material used. Much has been said about the need to seek new and more precise approaches to allocating these costs, as traditional methods are not considered entirely accurate. By applying a correct method of cost allocation, managers will have useful information for decision making and cost optimization (Cooper, 1988; Cooper and Kaplan, 1992). If indirect costs are incorrectly allocated, there will be an uneven distribution of the indirect costs and consequently cost computation will result inaccurate (Lips, 2017. Regarding spare parts logistics, this inequality can affect decisions such as: stock or non-stock specific spare parts, the definition of lot sizes, the application of an appropriate inventory policy, among others (Driessen et al. 2015), affecting the long-term profitability of the organization (Du Toit, 2014). Duran et al. (2016b) highlighted the importance of linking the spare parts

Figure 1. The Generic Allocation mechanism in ABC. Using the correct cost drivers for each phase, it is possible, with a high level of precision, to calculate the amount of resources that are consumed by each cost object (Badad and Balachandran, 1993; Homburg, 2001). Afonso and Paisana (2009) proposed a matrix ABC model for implementing the ABC method. In such approach, resources, activities, cost objects and costs drivers are related using matrixes and/or vectors. In the first allocation phase, we perform the calculation of the cost per activity. The resource-activity matrix, which contains rij terms each one representing the proportion of the resource driver j that is related to activity i, is multiplied by the resources vector whose terms rj represent the total amount of the resource j spent during the period under analysis. That multiplication results in the so-called activities vector, where the term ai represents the amount of cost allocated to activity i. [rij][rj] = [ai]

(1)

Note that in the resource-activity matrix, its elements represent the proportion of the resource driver j that is related to activity i. That proportion is obtained as the ratio between the resource driver j related to activity i (rij) and the total amount of the resource driver j (rj). Then, the cost allocated to each activity will be obtained by:

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𝑎𝑎" = $ r"& ∗ 𝑟𝑟&

be represented by the following equation (Duffuaa and Raouf, 2015):

(2)

& ∈ +

2245

(7)

In the second allocation phase, the cost calculation per cost object will be carried out by multiplying the activity-product matrix (aki) by the vector column of the activities costs (ai) obtaining the so-called cost object vector (pk): [aki] [ai] = [pk]

(3)

Considering the activity-product matrix, each element aki is the proportion of the activity driver related to product k. That proportion is obtained by computing the ratio between the activity-driver i, related to product or cost object (e.g., spare part) k (aki) and the total amount of the activity-driver i (ai). Then, the cost allocated to each activity will be obtained by: +

(4)

𝑎𝑎𝑖𝑖 = $ 𝑎𝑎-" ∙ 𝑎𝑎" "/0

In summary, the product or in more general terms the cost object vector can be obtained in just one step as shown below: (5)

[akj] [rij][rj] = [pk]

In this research cost objects/products are spare parts. In generic terms, spare parts management is based on a set of logistics activities that consume a series of resources. Once the activity costs have been calculated, these costs are allocated to each of the spare parts considered in the model. Through this model it is possible to measure the effect of cost variation on the use of different inventory policies and compare the effects they generate in the long term, i.e., considering its entire life cycle.

Where β is the shape parameter and characterizes the failure pattern. The higher value of β, the greater the failure probability in a given period of time. The time to is the location parameter and provides an estimate of the earliest time at which a failure may be observed. Also, it can represent the beginning of the deterioration process of the equipment. Finally, η is the scale parameter and represents the characteristic life of the equipment. This parameter corresponds to the time in which 63.2% of the failures are expected to occur. Therefore, knowing both Weibull parameters, it is possible to estimate the failure rate for a given component in a time t. The equation (8) shows the relationship between β, λ and η: 4

6 480

l(t) = 5 7 h h

(8)

The most common way to represent the lifecycle phases of an asset or group of physical assets is the bathtub curve (Figure 2). This curve shows the failure rate as a function of the operating time of the asset. It consists of three phases, namely: •

Infant mortality, with λ(t) decreasing over time.

•

The useful life phase, where λ(t) is constant.

• The wear out phase, where λ(t) increases in time, up to the moment where it is decided to discard the equipment or deactivate the installation.

Few works have been focused on the combined use of ABC and Life-cycle costing (LCC). Emblemsvag (2001; 2003) suggested this approach and suggested the use of Montecarlo simulations to represent with more accuracy the projected values of the cash flows. 2.2 Spare Parts Management Costs The value of the global cost of spare parts inventory management along the entire life cycle, can be modelled as follows: GC = DC + HC + LC

Figure 2. Bathtub curve of a physical asset.

(6) Particularly, as the Weibull parameters of a component vary in time, the failure rate will be affected and consequently the demand for a new spare part unit will fluctuate (Ebeling, 2004). As a result, also some logistics activities related to these parts will have their execution rates altered, along with the respective changes in the related costs.

Where DC = Net present value of Direct Costs HC = Net present value of Holding Costs LC = Net present value of Logistics Costs

3. OPTIMIZATION MODEL

2.3. Weibull-based Reliability Function The Weibull statistical distribution is one of the most used function to represent the behavior of the reliability throughout the life of a physical asset. The reliability of a component can

This section is dedicated to the presentation of a life cycle cost optimization model for spare parts management. As already mentioned, the logistics costs will be computed using a matrix

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activity-based costing model. Along with that, Weibull's reliability function is used to model the demand for each spare part in the life cycle periods. 3.1 Model Structure Regarding the continuous review inventory policy for an individual spare part (v) and a given period of the life cycle, the two first terms of equation 6 can be expressed as (Roach, 2005): 𝐺𝐺𝐺𝐺𝐺𝐺 = 𝐶𝐶< ⋅ 𝜆𝜆< +

@A ⋅BA C

+ 𝐿𝐿𝐿𝐿<

(9)

periods y of the entire life cycle that minimize the global costs considering that there is a budget By for each year available to invest in such items. For this purpose, the average stock level of each type of spare part is valued at its unitary cost. The sum of these costs must not exceed the available budget for each period (By). Therefore, the optimization problem can be expressed as:

𝐺𝐺𝐺𝐺

Min s.t.:

Parameters and sets are the following:

O<

6

$$

Cv = Unitary cost of item v

I/0 -/0

λ(v) = Demand of item v

(𝑐𝑐-I ⋅ 𝑄𝑄-I ) ≤ 𝐵𝐵" 2

Qv = Replenishment lot size of item v

3.2 Solving Strategy

Hv = Capital costs of item v (Cv ∙ td)

In order to validate and evaluate the model and the mechanism of solution through genetic algorithms, a series of experiments were made. We used an hypothetical spare parts distribution center. Such distribution center receives several units of different classes of spare parts, keep them in its interior and when requested provide the requested units to its customers. The following resources are consumed by logistics activities which are performed within the distribution center:

Regarding the LCv costs, we propose here an ABC-based methodology to compute such costs. In this category we have considered those cost elements called as inventory service costs, space and warehousing costs, ordering costs and so on (Azzi et al., 2014). Therefore, LCv can be represented as: 𝐿𝐿𝐿𝐿< = p<

• • • • • • • •

Where pv represents the proportion of the logistics costs corresponding to the spare part v, obtained by the activitybased costing model (third term of equation 9). Last but not least, as it was aforementioned, the demand λ(v) is modelled using the Weibull distribution. In summary, the life cycle costs of spare parts management for a period of cv years, T types of spare parts, M categories of resources and N activities will be given by the net present value of the sum of the global cost of the cv periods of the life cycle: 𝐺𝐺𝐺𝐺

O<

=$

I/0 N

For the logistics activities, those listed by (Varila et al., 2005) were considered: • • • • • • • • • • •

N

(𝐻𝐻-I ⋅ 𝑄𝑄-I ) 1 J$ 𝐶𝐶-I ⋅ 𝜆𝜆-I + I 2 (1 + 𝑡𝑡𝑡𝑡I ) Q

P

-/0

+ $ $ $ 𝑎𝑎-" I ⋅ 𝑟𝑟 "& I ⋅ 𝑟𝑟 & I R -/0 "/0 &/0

It is worth noting that we have incorporated two variables oriented to the evaluation of costs under the life cycle approach. We refer to the variable y which identifies the specific period within the life cycle and the capitalization factor, (1+ tdy) y, which in turn, is affected by the expected discount rate for that period tdy. If we consider the following decision variables:

𝑄𝑄𝑘𝑘𝑘𝑘: Lot size of spare part 𝑘𝑘 in period 𝑦𝑦.

The optimization problem can be translated into finding the lot sizes for each one of the spare parts (𝑄𝑄𝑘𝑘𝑘𝑘), in each one of the

Labor Supervisor Handling Equipment Vehicles IT infrastructure Buildings Energy Other overheads

Receiving Spare parts Storage Put-away Dispatching Picking Handling, maintaining Packing, marking, etc. Shipping Purchasing Dispatching Reporting

The initial experiments consider one or few types of spare parts and a small number of periods for the life cycle. In those experiments it was possible to carry out an comprehensive search to make comparisons with the results obtained through the evolutionary approach. Later, additional difficult was established, with a larger number of spare parts and longer life cycles. In addition, a limit was added to the amount to be invested in each period of the life cycle in order to make the

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experiment more realistic. The summary of the experiments is shown in Table 1. Table 1. Experiments parameters. Number of Spare Parts

Number of Periods

1 27 81

6 9 27

Several genetic algorithms, with differentiated structures and strategies, were tested. Table 2 shows the set of configurations that were tested with the experiments defined before. Table 2. Genetic algorithm configurations.

As it was commented before, when possible, the results were compared with those obtained through exhaustive searches. Based on that, the configuration with the best results is the one with the following characteristics: • • • •

Population size: 10 individuals Selection mechanism: Tournament Double point crossover with 20% of the population Ending criteria: number of generations (100) or 10 seconds of execution time.

The model and the genetic algorithms were implemented using MATLAB. 4. RESULTS Table 3 provides a comparison of the results in terms of Net present Value (NPV) obtained in the last two experiments. That is, with and without budgetary restriction. These experiments were conducted ten times each, obtaining a minimum dispersion of the results (standard deviation of approximately 0.15%). It can be seen that the overall cost differences between the two cases are minimal and do not exceed 0.5%. Another aspect that should be highlighted is the reduced resolution time. These experiments were carried out in about 20 minutes. Table 3. Main results of the two last study cases. CONTINUOUS REVIEW NPV GC

Exec. Time

[$]

[sec]

Without restriction

$ 46,453,533,474

1943

With restriction

$ 46,685,566,475

892

$ 232,033,001

-1051

Variation ($) Variation [%]

0.50%

-54%

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5. CONCLUSIONS AND FURTHER RESEARCH An optimization model based on activity-based life cycle costing is proposed here. This model incorporates aspects related to the reliability of spare parts (in any number) expressed through the respective Weibull distributions. Besides, the activity-based costing model was used to calculate the cost of the logistics activities. For optimizing decision making, a genetic algorithm was used. For illustrating the application and usefulness of the proposed strategy, a hypothetical case study was presented. The results obtained allow to guarantee the potential of the mentioned evolutionary technique to solve the problem of minimizing the global life cycle cost for a given inventory policy. This also allow the optimization of spare parts management considering a large number of different spare parts in a joint and in an effective way. After the analysis, the hypothesis that the inventory policy, and its main parameters, affect mainly the inventory average levels and the number of executions of logistics activities is proved as correct. The case study shown validates the potential use of the proposed methodology in real situations and allows defining the parameters of the inventory policy, preferably where there are a large number of components and it is desired to project costs in the long term (multi-period). On the other hand, this model can be useful with other types of inventory policies, because it does not need to be reconfigured to address different inventory policies and their respective parameters. This is because the decision parameters are relative to lot sizes and the number of times the activities are performed. Note that the latter, number of times the activities are executed, also depends on the lot sizes. Future developments point to the application of the technique to a real-life case, and the purification of the genetic algorithm to achieve better performances. ACKNOWLEDGEMENT Authors would like to acknowledge the funding of Fondecyt Regular n° 1170915: “Design and optimization of a life cycle based critical spare parts management system”. REFERENCES Afonso, P. S., & Paisana, A. M. (2009). An algorithm for activity-based costing based on matrix multiplication. In Industrial Engineering and Engineering Management, 2009. IEEM 2009. IEEE International Conference on (pp. 920-924). IEEE. Almeida, A., & Cunha, J. (2017). The implementation of an Activity-Based Costing (ABC) system in a manufacturing company. Procedia Manufacturing, 13, 932-939. doi.org/10.1016/j.promfg.2017.09.162 Anna Azzi, Daria Battini, Maurizio Faccio, Alessandro Persona, Fabio Sgarbossa, (2014) "Inventory holding costs measurement: a multi-case study", The International

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