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Multi-criteria inventory ABC classification using Gaussian Mixture Model
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Multi-criteria inventory ABC classification using Multi-criteria inventory ABC using Multi-criteria inventory ABC classification classification using Gaussian Mixture Model Multi-criteria inventory ABC classification using Gaussian Mixture Model Gaussian Mixture Model Gaussian Mixture Model
F.M. Zowid*. M.Z. Babai**. M. R. Douissa***. Yves Ducq**** F.M. Zowid*. M.Z. Babai**. M. R. Douissa***. Yves Ducq**** F.M. Zowid*. M.Z. Babai**. M. R. Douissa***. Yves Ducq**** F.M. Zowid*. M.Z. of Babai**. M. IMS, R. Douissa***. Ducq**** *Univ. Bordeaux, UMR 5218,Yves CNRS. *Univ. (e-mail: of Bordeaux, IMS, UMR 5218, CNRS. Talence, France, [email protected]). *Univ. of Bordeaux, IMS, UMR 5218, CNRS. Talence, France, (e-mail: [email protected]). *Univ. of Bordeaux, IMS, UMR 5218,France CNRS. **Kedge Business School, Talence, Talence, France, (e-mail: [email protected]). **Kedge Business School, Talence, France Talence, France, (e-mail: [email protected]). (e-mail: [email protected]). **Kedge Business School, Talence, France (e-mail: **Kedge School, France **** [email protected]). ofBusiness Bordeaux, IMS, Talence, UMR 5218, CNRS. (e-mail: [email protected]). **** Univ. of Bordeaux, IMS, UMR 5218, CNRS. (e-mail: [email protected]). Talence, France, (e-mail: [email protected]). **** Univ. of Bordeaux, IMS, UMR 5218, CNRS. Talence, France, (e-mail: [email protected]). **** Univ. of***University Bordeaux, IMS, UMR 5218, CNRS. of Tunis, Talence, France, (e-mail: [email protected]). ***University of Tunis, Talence, France, (e-mail: [email protected]). Tunis, Tunisia, (e-mail: [email protected]). ***University of Tunis, Tunis, Tunisia, (e-mail: [email protected]). ***University of Tunis, Tunis, Tunisia, (e-mail: [email protected]). Tunis, Tunisia, (e-mail: [email protected]). Abstract: ABC classification is a technique widely used by companies to deal with inventories Abstract: ABC is aof technique widely used units. by companies to dealABC with inventories consisting veryclassification large numbers distinct stock keeping Single-criterion Abstract: of ABC classification is a technique widely used by companies to deal withclassification inventories consisting of veryclassification large innumbers ofand distinct stock keeping units. Single-criterion ABC Abstract: ABC is a technique widely used by companies to deal with inventories methods are often used practice recently multi-criteria methods have attracted theclassification attention of consisting of very large numbers of distinct stock keeping units. Single-criterion ABC classification methods are often used innumbers practice and recently multi-criteria methods have attracted theclassification attention of consisting of very large of distinct stock keeping units. Single-criterion ABC academics and practitioners. Several models have been developed to deal with the multi-criteria ABC methods are often used in practice and recently multi-criteria methods have attracted the attention of academics and practitioners. Several models have been developed to deal with the multi-criteria ABC methods often used (MCIC). in practice methods have attracted the attention of inventory classification Toand therecently besthave of multi-criteria our very few researches have usedABC the academicsare and practitioners. Several models beenknowledge, developed to deal with the multi-criteria inventory classification (MCIC). To models the best of our knowledge, very fewwith researches havetheoretical used the academics andmachine practitioners. Several have beenMCIC developed to deal theattractive multi-criteria unsupervised learning methods address the problem despite their inventory classification (MCIC). To thetobest of our knowledge, very few researches have usedABC the unsupervised machine learning methods address the MCIC problem despite their attractive inventory classification (MCIC). To theto best oftheour knowledge, very few(GMM) researches havetheoretical used the and practical properties. Therefore, in this Gaussian mixture model is proposed to deal unsupervised machine learning methods topaper, address the MCIC problem despite their attractive theoretical and practical properties. Therefore, in this paper, the Gaussian mixture model (GMM) is proposed to deal unsupervised machine learning methods to address the MCIC problem despite their attractive theoretical with the multi-criteria inventory classification a simple optimization model that and practical properties.ABC Therefore, in this paper, the problem. Gaussian GMM mixtureismodel (GMM) is proposed to deal withbe theused multi-criteria ABC inventory classification problem. ismodel a simple optimization model that and properties. Therefore, in this paper, thecomputational Gaussian GMM mixture (GMM) is proposed toofdeal can for classification purposes with a low time. numerical investigation the withpractical the multi-criteria ABC inventory classification problem. GMM is aAsimple optimization model that can be used for classification purposes withisa presented low computational time. Asimple numerical investigation of that the with the multi-criteria ABC inventory classification problem. GMM is a optimization model cost-service inventory of the GMM model in this paper. The performance of the model is can be used for classification purposes with a low computational time. A numerical investigation of the cost-service inventory of the GMM model isa presented in thisMCIC paper. The performance of the model is can be used for classification purposes with low computational time. A numerical investigation of the also compared to some mathematical programming-based models. The numerical study is cost-service inventory of the GMM model is presented in this paper. The performance of the model is also compared to some mathematical programming-based MCIC models. The numerical study is cost-service of athe GMM model is presented in of this47 paper. The performance of the has model conducted byinventory means of theoretical dataset, consisting stockmodels. keeping units, which been also compared to some mathematical programming-based MCIC The numerical study is conducted by means of literature. amathematical theoretical dataset, consisting of 47 stockmodels. keeping units, whichcanhas been also compared to some programming-based MCIC The numerical study isa commonly used in the The numerical results show that the proposed model have conducted by means of a theoretical dataset, consisting of 47 stock keeping units, which has been commonly used in the literature. The numerical results show the model canhashave conducted by means of a theoretical dataset, consisting of 47 that stock keeping units, which promised along with the existing MCIC classification models inproposed the literature in terms ofbeen theaa commonlyperformance used in the literature. The numerical results show that the proposed model can have promised performance along with Copy-right the existing classification models the literature in terms of thea commonly used in the literature. The numerical the in model can have cost-service inventory efficiency. ©MCIC 2019results IFAC show that promised performance along with the existing MCIC classification models inproposed the literature in terms of the cost-service inventory efficiency. Copy-right ©MCIC 2019 classification IFAC promised performance along with the existing models in the literature in terms of the cost-service inventory efficiency. Copy-right © 2019 IFAC © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: inventory classification, multi-criteria inventory GMM, unsupervised clustering cost-service inventory efficiency. Copy-right © 2019 IFAC classification, Keywords: inventory classification, multi-criteria inventory classification, GMM, unsupervised clustering algorithm. Keywords: inventory classification, multi-criteria inventory classification, GMM, unsupervised clustering algorithm. inventory classification, multi-criteria inventory classification, GMM, unsupervised clustering Keywords: algorithm. algorithm. criterion to evaluate and classify the SKUs, which is 1. INTRODUCTION criterion to evaluate and multi-criteria classify the SKUs, is insufficient. Hence, the ABC which inventory criterion to evaluate and classify the SKUs, which is 1. INTRODUCTION insufficient. Hence, the multi-criteria ABC inventory 1. INTRODUCTION criterion to evaluate and classify the SKUs, which is classification, which can consider multiple criteria such as Inventory management is a vital function to meet market insufficient. Hence, the multi-criteria ABC inventory 1. INTRODUCTION classification, which can consider multiple criteria such as Inventory management is a vital function to meet market insufficient. Hence, the multi-criteria ABC inventory lead time, cost, price criticality, is acriteria contemporary demand. Amanagement lack inventory management control can classification, which canand consider multiple such as Inventory is aofvital function and to meet market time,developed cost, price criticality, is acriteria contemporary demand. lack inventory management and control can lead which can consider multiple such as Inventory is aof vital function to meet market approach by and (Flores and Whybark, 1986). In affect theA items lead time, of items, operation demand. Amanagement lack inventory ofavailability management and control can classification, lead time, cost, price and criticality, is a contemporary approach developed by (Flores and Whybark, 1986). In affect the items lead time, availability of items, operation lead time, cost, price and criticality, is a contemporary demand. A lack inventory of management and control can practice, it has also been recognized that there are other costs, the company’s profits, performance and any approach developed by (Flores and Whybark, 1986). In affect items lead time, financial availability of items, operation practice, it has also been recognized that there are other costs, company’s profits, financial performance and any developed and criteria Whybark, 1986). In affect itemsToday lead time,the availability of items,changes operation important and(Flores qualitative for classifying related efforts. with rapid and constant in approach practice, itquantitative has also by been recognized that there are other costs, the company’s profits, financial performance and any quantitative and qualitative criteria for classifying related efforts. Today with the rapiddifferent and constant changes in important practice, it has also been recognized that there are other costs, company’s profits, financial performance and any inventories that need to be considered such as “commonality, the business world, companies with sizes often cope related efforts. Today with the rapid and constant changes in important quantitative and qualitative criteria for classifying thatsubstitutability, need to be considered as “commonality, the business world, companies different sizes oftenmore. cope important qualitative criteria for classifying related efforts. Today withitems the with rapid and constant changes in inventories obsolescence, numbersuch of requests per year, with series of inventory in thousands or even inventoriesquantitative that need toand be considered such as “commonality, the business world, companies with different sizes often cope obsolescence, substitutability, number of requests per year, with series of inventory items in thousands or even more. inventories that need to be considered such as “commonality, the business world, companies with different sizes often cope scarcity, durability, repairability, order size requirement, Managers need sophisticated strategic-operational tactics. obsolescence, substitutability, number of requests per year, with series of inventory items in thousands or even more. durability, repairability, order size requirement, Managers sophisticated strategic-operational tactics. obsolescence, substitutability, number of requests per2006). year, with series need of ABC inventory itemsclassification in thousandsisortheeven more. stockability, and stock-out penalty cost” (Ramanathan, The traditional inventory most used scarcity, scarcity, durability, repairability, order size requirement, Managers need sophisticated strategic-operational tactics. stockability, and stock-out penalty cost” (Ramanathan, 2006). The traditional ABC inventory classification is the most used scarcity, durability, repairability, order size requirement, Managers need sophisticated strategic-operational tactics. The literatureand is stock-out abundant penalty with various studies to tackle the in companies. thisclassification method wasis introduced by stockability, cost” (Ramanathan, 2006). The traditional Historically, ABC inventory the most used literature is stock-out abundant with various studies to tackle the in companies. thisclassification method by The and cost” 2006). The traditional ABC is introduced the most used MCIC problem. Recently,penalty (Simha, et (Ramanathan, al., 2014) and few (Dickie, 1951) Historically, and inventory successfully used inwas General Electric The literature is abundant with various studies to tackle the in companies. Historically, this method was introduced by stockability, problem. Recently, (Simha, etmachine al., 2014) and type few (Dickie, 1951) and successfully used inwas General Electric The literature is utilized abundant with various to tackle the in companies. Historically, thiswith method introduced by researchers have unsupervised learning Company to facilitate the deal inventories. Typically, it MCIC MCIC problem. Recently, (Simha, et studies al., 2014) and few (Dickie, 1951) and successfully used in General Electric have utilized unsupervised machine learning type Company to that facilitate the deal withkeeping inventories. Typically, it researchers MCIC problem. Recently, (Simha, et al., 2014) and few (Dickie, 1951) and successfully used in General Electric to improve the inventory classification system from several is a method distributes stock units (SKUs) into Company to facilitate the deal with inventories. Typically, it researchers have utilized unsupervised machine learning type improve the inventory classification system from several is a method distributes stock units (SKUs) researchers have utilized unsupervised machine type Company to that facilitate withkeeping inventories. Typically, it to points of views. Examples of such methods are learning the K-means three classes based onthe thedeal importance of SKUs in whichinto A, to improve the inventory classification system from several is a method that distributes stock keeping units (SKUs) into points of views. Examples of such methods are the K-means three classes based on the importance of SKUs in which A, to improve the inventory classification system from several is a method that distributes stock keeping units (SKUs) into method (Keskin and Ozkan, 2013) and the Agglomerative very important, B, moderate important, C, least important. three classes based on the importance of SKUs in which A, points of views. Examples of such methods are the K-means and Ozkan, 2013) and the2016). Agglomerative very important, B, on moderate leastinimportant. points of(Keskin views. Examples of such methods the K-means three classes dollar based the importance of C, SKUs which Hierarchical Clustering method (Raja, et al., The usage is important, the single criterion used A, in method method (Keskin and Ozkan, 2013) and theareAgglomerative very annual important, B, moderate important, C, least important. Hierarchical Clustering method (Raja, et al., 2016). The annual dollar usage is the single criterion used in method (Keskin and Ozkan, 2013) and the Agglomerative very important, B, moderate important, C,criterion least important. To the best of our knowledge, research paper is the first evaluating alldollar SKUs for the is final Appropriate inventory Clustering method this (Raja, et al., 2016). The annual usage therank. single used in Hierarchical thethat best of our knowledge, this research paper is the first evaluating alldollar SKUs for the final Appropriate Hierarchical Clustering (Raja, et al., 2016). The annual usage is therank. single criterion used for in To work tomethod use the Gaussian mixture model review polices are then allocated to each SKUs’ inventory class To the best proposes of our knowledge, this research paper is the first evaluating all SKUs for the final rank. Appropriate inventory work that proposes to use the Gaussian mixture model review polices are then allocated to each SKUs’ class for To the best of our knowledge, this research paper is the first evaluating all SKUs for the final rank. Appropriate inventory (GMM)thatas proposes unsupervised classifier model to address the controlling purpose (Ernst and Cohen, 1990). Despite to use the Gaussian mixture model review polices are then allocated to each SKUs’ class the for work (GMM) as unsupervised classifier model to address the controlling purpose (Ernst and Cohen, 1990). Despite the work that proposes to use the Gaussian mixture model review polices are then allocated to each SKUs’ class for MCIC problem. The objective is to exploit the robustness simplicity purpose of the (Ernst implementation the Despite traditional the controlling and Cohen, of1990). the (GMM) as unsupervised classifier model to address and problem. The to objective isa to exploit robustness simplicity ofmethod, the implementation of1990). the Despite traditional (GMM) asofunsupervised classifier modelthe address and the controlling anda Cohen, the MCIC usefulness GMM provide competitive performance of classification it has total reliance a monoMCIC problem. The objective is to exploit theto robustness and simplicity purpose of the (Ernst implementation of theon traditional of GMM to provideisa to competitive performance of classification method, it has a total reliance on traditional a mono- usefulness MCIC problem. The objective exploit the robustness and simplicity of the implementation of the classification method, it has a total reliance on a mono- usefulness of GMM to provide a competitive performance of classification method, it has a total reliance on a mono- usefulness of GMM to provide a competitive performance of
2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2019, 2019 IFAC 1955Hosting by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic Copyright 2019 responsibility IFAC 1955Control. Copyright © 2019 IFAC 1955 10.1016/j.ifacol.2019.11.484 Copyright © 2019 IFAC 1955
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inventory cost and service when compared to other MCIC methods. An advantage of the GMM compared to others is the fact that the criteria are not weighted which means that the GMM algorithm does not involve subjectivity. In addition, an empirical cost-service inventory analysis of the GMM model is presented to show its performance when compared to four well known and used MCIC methods, namely: the R-method (Ramanathan, 2006); the ZF-method (Zhou and Fan, 2007); the Ng-method (Ng, 2007) and the Hmethod (Hadi-Vencheh, 2010). The rest of the paper is organized as follows. In the next section, the related research is described. A description of the classification model is provided in Section 3. The experimental investigation and results are discussed in Section 4. In the last section, conclusions and further research work are presented. 2. RELATED RESEARCH In the literature, a considerable research work has been developed to deal with the multiple criteria decision making (MCDM) approach that can be used in improving the MCIC problem. (Saaty, 1980) analytic hierarchy process (AHP) model has been developed and it has been then used by (Flores et al., 1992) to the MCIC. (Partovi and Burton, 1993) and other authors have applied the AHP in different ways to address the MCIC problem. Despite the merits of AHP, it has been criticized because it involves the subjectivity in a pairwise comparison to determine the criteria preferences and to rank inventory items. (Bhattacharya, et al., 2007) have designed an integrated model that includes a distance-based multi-criteria consensus framework based on the concepts of TOPSIS to improve the ABC classification. They have considered various important criteria (e.g. unit cost, LT, consumption rate, perishability of items and cost of storing items) for SKUs which are obtained from a pharmaceutical industry in India. The authors have tested the inventory cost and model performance using the ANOVA technique and they have shown that it contributes to a low average investment. (Chen, 2012) has presented an integrated model using DEA to generate criteria weights, which is based on an output maximizing multiplier DEA model with multiple outputs and a constant input and TOPSIS is then applied. Several classification models based on mathematical programming (MP) have been developed for the MCIC and to compute the performance of each inventory item under multiple criteria into a single score for ABC classification. In a similar manner, the concept of data envelopment analysis (DEA) technique, (Ramanathan, 2006) has developed a simple weighted linear optimization model, referred to as the R-model where it uses a weighted additive function that generates weighted scores. (Zhou and Fan, 2007) have proposed an extended work of R-model, which is known as ZF-model. The authors aimed to address the shortcomings and subjectivity of the R-model. Their model used two indices to reveal the most and least favorable weights along with a formulated control parameter that may show the preference of the decision maker on both indices. Other extended improvement research to the R-model is by (Ng, 2007) which has introduced a simple alternative model called Ng to obtain an optimal inventory item score which can
simply be solved without a linear optimizer. (Hadi-Vencheh, 2010) has presented a nonlinear optimization method that maintains the weights’ effects up to the final solution in providing a more reasonable and encompassing inventory item score. Note that both Ng and H models provide a descending ranking of criteria. The main drawback of the above mentioned of MP classification models is that they require high repeated optimizations when the inventory items are large in number to introduce. (Chen, 2011) has developed an optimization model in which provides a peer-estimation procedure depending on both R and ZF models to scoring inventory items. (Torabi, et al., 2012; Hatefi, et al., 2014; Chen, et al., 2008; Ladhari, et al., 2015) were among the important studies based on MP. Several artificial intelligence techniques have been employed in the MCIC problem. (Guvenir and Erel, 1998) have proposed a genetic algorithm (GA) to learn the criteria weights and found a solution for the cut-off points between the classes A-B and B-C. (Partovi and Anandarajan, 2002) have utilized an artificial neural network (ANN) along with GA and back-propagation to the MCIC problem by using real-world data obtained from a pharmaceutical company in the US. (Yu, 2011) has conducted a method of inventory classification which includes support vector machines (SVM), back-propagation neural networks (BNN) and knearest neighbor algorithm (KNN). Particle swarm optimization (PSO) has been applied by (Tsai and Yeh, 2008) in which the method finds the optimal number of inventory classes and generates items classifications. More recently, unsupervised learning models have been utilized to the MCIC. (Keskin and Ozkan, 2013) was the first work which has developed a fuzzy c-means clustering model for better inventory clustering under fuzzy circumstances. FCM has been used to determine the matching to any clusters and to show the membership degrees. (Simha, et al., 2014) have used self-organizing maps model (SOM) to the MCIC and obtained results compared to K-means, and fuzzy c-means using the known theoretical dataset in the literature. SOM outperformed the compared models in terms the inventory cost. The authors also stressed the use of unsupervised learning technique to the MCIC. (Lolli, et al., 2014) have developed a hybrid model consisting of AHP and the Kmeans clustering algorithm to improve the MCIC problem. The authors used K-means to create classes based on the ranking scores obtained from the AHP. (Raja, et al., 2016) have proposed the agglomerative hierarchical clustering (AHC) model by using ward’s method to minimize variance and using Euclidean distance for items measuring. They aimed to develop inventory police of storage and control based on the nature of spare parts (metal/non-metal) in a chemical company. 3. GAUSSIAN MIXTURE MODEL (GMM) The GMM is a probabilistic unsupervised classification model that represents a distribution data point within overall clusters. For example, when modeling the human beings weight data, for each gender, the weight is modeled as normal distribution within an average of about 65 kilograms. In this case, the weight data on each entry point are given
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with no corresponding data on gender. This example of a model following assumptions is known as a GMM. Applications of GMM include medical image segmentations, speech data extraction, predicting locations and tracking multiple objects. More detailed information of the procedures for classification can be found in the literature (e.g., Everit and Hand, 1981; Titterington, et al., 1985; Hastie and Tibshirani, 1996). 4. DISCUSSION AND NUMERICAL EXAMPLE In this section, the application of the GMM algorithm to the MCIC problem is explained and the SKUs are detailed. The results of classification and the validation of the model and results are provided. The dataset of SKUs is clustered based on the GMM algorithm where inventory items are allocated to different clusters according to their attributes. The GMM algorithm assigns each item to the belonging cluster centers by Gaussian distribution and by iteratively calculating the probability means, covariance and mixing coefficient in which items within cluster centers are as similar as possible and the differences between cluster centers as big as possible. By the GMM, colors of each datapoint (SKU) are used to determine which cluster it should belong to. The color was assigned to each SKU by its posterior probabilities’ values, which coincide with the (blue, green, and red) coloring function in the PYTHON [2, 1, and 0] correspond to clusters A, B, and C respectively. For example, if the GMM determines the posterior probabilities of item 1 as (0.89, 0.00, 0.01) then the item 1 should belong to cluster center A which will appear in blue. Items somewhere between two cluster centers are usually avoided since the EM steps are iteratively updated until convergence reached as a maximum likelihood. The dataset used for the experimental classification analysis contained 47 SKUs obtained from a Hospital Respiratory Therapy Unit (HRTU). Each inventory item measured with respect to three attributes namely average unit cost, annual dollar usage and lead time as depicted in Figures 1, 2, and 3. These attributes have been predefined in the literature. Due to the varied values of the dataset, a pre-processing procedure of a linear normalization used to convert all measurement into a 0–1 scale for all items on criteria. where i=1, …, M and j= 1, …, N is a benefit criterion
Fig. 1. Visualization of the data run through GMM (k=3) and membership degree of SKUs to clusters under pairs of attributes (ADU and AUC).
Fig. 2. Visualization of the data run through GMM (k=3) and membership degree of SKUs to clusters under pairs of attributes (LT and AUC).
(1)
is an input value, is normalized value, i is where number of alternatives, M represents total number of items, J represents criterion number and N which is total number of criteria. The result of normalization is showed in Table 1 with the original data. After the run of the GMM code in PYTHON, the membership of each item to the belonging cluster centers are generated. Figure 1, 2, and 3 indicate the belonging of each inventory item to cluster centers with respect to pairs of attributes. For example, the belonging of each item to cluster centers of A, B and C is the obtained measurements of pairs of attributes (AUC and ADU) as shown in Figure1.
Fig. 3. Visualization of the data run through GMM (k=3) and membership degree of SKUs to clusters under pairs of attributes (LT and ADU). The overall belonging of each item to the cluster centers with respect to the three attributes of (AUC, ADU and LT) is shown in Figure 4. For final ABC labelling, the cluster center with the maximum likelihood estimation is labelled.
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Fig. 4. Visualization of the data run through GMM (k=3) and membership degree of SKUs to all clusters under the three attributes (ADU, AUC, and LT). The final classification results are detailed in Table 1. The GMM model clusters the SKUs as follows, 10 items are grouped in cluster A (21.28%), 16 items are grouped in cluster B (34.04%) and 21 items are grouped in cluster C (44.68%) of the 47 SKUs. Overall similarity and dissimilarity of items classifications are 28 items (59.57%) and 19 items (40.43%) compared to GMM rank as shown in Table 1. Table 1. Results of classification based on GMM distribution.
The silhouette coefficient analysis is a known method used to validate the consistency results of the unsupervised learning model. It is used to validate how well the dataset used is clustered. The obtained consistency result is 0.36221. As it should be noted that this validation result is not compared to the MP models due to the differences in applications and validation processes. For comparing the cost-service inventory performance of the proposed GMM to four others MCIC models: ZF, R, Ng and H, the total holding inventory cost (safety stock inventory cost) is evaluated for all SKUs as well as the achieved fill rate when a predetermined service level is fixed to each class according to the methodology in (Teunter, et al., 2010; Babai, et al., 2015). The achieved fill rate is a factor to measure the fraction of routinely filling a customer order from on hand stock. Both total holding inventory cost and fill rate are calculated by assigning three fixed targets of cycle service levels as (99%, 95%, 90%), (95%, 90%, 85%) and (90%, 85%, 80%) to the classes A, B, and C respectively. Some notations and definitions are listed in Table 2. Table 2. Notations N N Di Ki hi σi Wi Qi Li FRi FRT CSLi Φ(.) G(x) CT
Number of items in the inventory systems. Number of item classes (A, B, C) Mean demand per item i Safety factor per item i Holding cost per item i Standard deviation of the demand of item i Fixed ordering cost per item i Quantity order per item i Lead time per item i Fill rate of item i Overall fill rate of inventory system evaluation Cycle service level of item i Standard normal distribution function Loss function of the standard normal distribution Total cost of the inventory system.
For calculating the total safety stock inventory cost, the equation is as follows: (2) The approximate fill rate of each ith item, is given as: (3) where,
-
[1-
(4)
The total achieved fill rate on the inventory is computed as: (5) The reader could refer to (Teunter, et al., 2010; Babai, et al., 2015) for more details and numerical examples of estimating the total cost and fill rate of inventory which used to the GMM in this research. The followed Table 3 is provided by illustrative samples of a numerical estimation the total inventory cost and achieved fill rate.
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efficient method. The results of the combined cost-service performance are depicted in Figure 5. All models are very close in curves with slight differences. At the same time, the GMM showed very competitive performance as compared to the other classification models within the three targets of CSL. The GMM efficiency curve is on the top of all other curves, which shows its higher efficiency compared to the four MCIC methods. GMM is followed by Ng in terms of efficiency. The results also show that the ZF method results in the lowest efficiency.
Table 3. Detailed calculation for the GMM
Fig. 5. Combined service-cost efficiency curves for the MCIC methods (under the normal distribution) of 47 SKUs 5. CONCLUSIONS
The numerical results illustrated in Table 4 notably show that there is an increase in the achieved fill rates when GMM is applied as compared to the other methods; however, this increase in the service is associated with an increase of the total holding inventory cost. Thus, a combined service-cost performance is conducted in order to have a better analysis and a fair comparison of the performance of the considered methods. Efficiency curves of the combined service-cost performance of the classification models are reflected in Figure 5.
In this paper, we have presented comprehensive results and analysis of the GMM clustering model, which is used to address the MCIC problem. The aim of our proposed unsupervised clustering model is not only to provide a classification of SKUs free of subjectivity processes, but also to provide a competitive total inventory cost with maximum service level. The results enable us to demonstrate that the proposed model can remain in a promised and competitive performance along with the other classification models within the same ABC distribution class. A future research opportunity would be to consider a larger dataset to validate our results and compare to relevant unsupervised machine learning applications. REFERENCES
Table 4. Numerical results Target CSL for Classes ABC Models GMM R ZF Ng H
1st Target of CSL
2nd Target of CSL
3rd Target of CSL
Aydin Keskin, G. and Ozkan, C. (2013) ‘Multiple Criteria ABC Analysis with FCM Clustering’, Journal of Industrial Engineering, 2013, pp. 1–7. doi: 10.1155/2013/827274.
99, 95, 90 % Cost FR 1032 0.992 947 0.988 964 0.987 1015 0.991 1003 0.990
95, 90, 85 % Cost FR 747 0.978 702 0.972 711 0.971 739 0.977 733 0.975
90, 85, 80 % Cost FR 587 0.959 557 0.952 563 0.951 582 0.957 578 0.956
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The curve of each model shows the achieved FRT as a function of the total inventory cost when CSL is varied. The curves can be interpreted as an indication that for a certain total holding inventory cost, the furthest curve from the xaxis (i.g., the highest achieved fill rate) means the more
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Multi-criteria inventory ABC classification using Multi-criteria inventory ABC using Multi-criteria inventory ABC classification classification using Gaussian Mixture Model Multi-criteria inventory ABC classification using Gaussian Mixture Model Gaussian Mixture Model Gaussian Mixture Model
F.M. Zowid*. M.Z. Babai**. M. R. Douissa***. Yves Ducq**** F.M. Zowid*. M.Z. Babai**. M. R. Douissa***. Yves Ducq**** F.M. Zowid*. M.Z. Babai**. M. R. Douissa***. Yves Ducq**** F.M. Zowid*. M.Z. of Babai**. M. IMS, R. Douissa***. Ducq**** *Univ. Bordeaux, UMR 5218,Yves CNRS. *Univ. (e-mail: of Bordeaux, IMS, UMR 5218, CNRS. Talence, France, [email protected]). *Univ. of Bordeaux, IMS, UMR 5218, CNRS. Talence, France, (e-mail: [email protected]). *Univ. of Bordeaux, IMS, UMR 5218,France CNRS. **Kedge Business School, Talence, Talence, France, (e-mail: [email protected]). **Kedge Business School, Talence, France Talence, France, (e-mail: [email protected]). (e-mail: [email protected]). **Kedge Business School, Talence, France (e-mail: **Kedge School, France **** [email protected]). ofBusiness Bordeaux, IMS, Talence, UMR 5218, CNRS. (e-mail: [email protected]). **** Univ. of Bordeaux, IMS, UMR 5218, CNRS. (e-mail: [email protected]). Talence, France, (e-mail: [email protected]). **** Univ. of Bordeaux, IMS, UMR 5218, CNRS. Talence, France, (e-mail: [email protected]). **** Univ. of***University Bordeaux, IMS, UMR 5218, CNRS. of Tunis, Talence, France, (e-mail: [email protected]). ***University of Tunis, Talence, France, (e-mail: [email protected]). Tunis, Tunisia, (e-mail: [email protected]). ***University of Tunis, Tunis, Tunisia, (e-mail: [email protected]). ***University of Tunis, Tunis, Tunisia, (e-mail: [email protected]). Tunis, Tunisia, (e-mail: [email protected]). Abstract: ABC classification is a technique widely used by companies to deal with inventories Abstract: ABC is aof technique widely used units. by companies to dealABC with inventories consisting veryclassification large numbers distinct stock keeping Single-criterion Abstract: of ABC classification is a technique widely used by companies to deal withclassification inventories consisting of veryclassification large innumbers ofand distinct stock keeping units. Single-criterion ABC Abstract: ABC is a technique widely used by companies to deal with inventories methods are often used practice recently multi-criteria methods have attracted theclassification attention of consisting of very large numbers of distinct stock keeping units. Single-criterion ABC classification methods are often used innumbers practice and recently multi-criteria methods have attracted theclassification attention of consisting of very large of distinct stock keeping units. Single-criterion ABC academics and practitioners. Several models have been developed to deal with the multi-criteria ABC methods are often used in practice and recently multi-criteria methods have attracted the attention of academics and practitioners. Several models have been developed to deal with the multi-criteria ABC methods often used (MCIC). in practice methods have attracted the attention of inventory classification Toand therecently besthave of multi-criteria our very few researches have usedABC the academicsare and practitioners. Several models beenknowledge, developed to deal with the multi-criteria inventory classification (MCIC). To models the best of our knowledge, very fewwith researches havetheoretical used the academics andmachine practitioners. Several have beenMCIC developed to deal theattractive multi-criteria unsupervised learning methods address the problem despite their inventory classification (MCIC). To thetobest of our knowledge, very few researches have usedABC the unsupervised machine learning methods address the MCIC problem despite their attractive inventory classification (MCIC). To theto best oftheour knowledge, very few(GMM) researches havetheoretical used the and practical properties. Therefore, in this Gaussian mixture model is proposed to deal unsupervised machine learning methods topaper, address the MCIC problem despite their attractive theoretical and practical properties. Therefore, in this paper, the Gaussian mixture model (GMM) is proposed to deal unsupervised machine learning methods to address the MCIC problem despite their attractive theoretical with the multi-criteria inventory classification a simple optimization model that and practical properties.ABC Therefore, in this paper, the problem. Gaussian GMM mixtureismodel (GMM) is proposed to deal withbe theused multi-criteria ABC inventory classification problem. ismodel a simple optimization model that and properties. Therefore, in this paper, thecomputational Gaussian GMM mixture (GMM) is proposed toofdeal can for classification purposes with a low time. numerical investigation the withpractical the multi-criteria ABC inventory classification problem. GMM is aAsimple optimization model that can be used for classification purposes withisa presented low computational time. Asimple numerical investigation of that the with the multi-criteria ABC inventory classification problem. GMM is a optimization model cost-service inventory of the GMM model in this paper. The performance of the model is can be used for classification purposes with a low computational time. A numerical investigation of the cost-service inventory of the GMM model isa presented in thisMCIC paper. The performance of the model is can be used for classification purposes with low computational time. A numerical investigation of the also compared to some mathematical programming-based models. The numerical study is cost-service inventory of the GMM model is presented in this paper. The performance of the model is also compared to some mathematical programming-based MCIC models. The numerical study is cost-service of athe GMM model is presented in of this47 paper. The performance of the has model conducted byinventory means of theoretical dataset, consisting stockmodels. keeping units, which been also compared to some mathematical programming-based MCIC The numerical study is conducted by means of literature. amathematical theoretical dataset, consisting of 47 stockmodels. keeping units, whichcanhas been also compared to some programming-based MCIC The numerical study isa commonly used in the The numerical results show that the proposed model have conducted by means of a theoretical dataset, consisting of 47 stock keeping units, which has been commonly used in the literature. The numerical results show the model canhashave conducted by means of a theoretical dataset, consisting of 47 that stock keeping units, which promised along with the existing MCIC classification models inproposed the literature in terms ofbeen theaa commonlyperformance used in the literature. The numerical results show that the proposed model can have promised performance along with Copy-right the existing classification models the literature in terms of thea commonly used in the literature. The numerical the in model can have cost-service inventory efficiency. ©MCIC 2019results IFAC show that promised performance along with the existing MCIC classification models inproposed the literature in terms of the cost-service inventory efficiency. Copy-right ©MCIC 2019 classification IFAC promised performance along with the existing models in the literature in terms of the cost-service inventory efficiency. Copy-right © 2019 IFAC © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: inventory classification, multi-criteria inventory GMM, unsupervised clustering cost-service inventory efficiency. Copy-right © 2019 IFAC classification, Keywords: inventory classification, multi-criteria inventory classification, GMM, unsupervised clustering algorithm. Keywords: inventory classification, multi-criteria inventory classification, GMM, unsupervised clustering algorithm. inventory classification, multi-criteria inventory classification, GMM, unsupervised clustering Keywords: algorithm. algorithm. criterion to evaluate and classify the SKUs, which is 1. INTRODUCTION criterion to evaluate and multi-criteria classify the SKUs, is insufficient. Hence, the ABC which inventory criterion to evaluate and classify the SKUs, which is 1. INTRODUCTION insufficient. Hence, the multi-criteria ABC inventory 1. INTRODUCTION criterion to evaluate and classify the SKUs, which is classification, which can consider multiple criteria such as Inventory management is a vital function to meet market insufficient. Hence, the multi-criteria ABC inventory 1. INTRODUCTION classification, which can consider multiple criteria such as Inventory management is a vital function to meet market insufficient. Hence, the multi-criteria ABC inventory lead time, cost, price criticality, is acriteria contemporary demand. Amanagement lack inventory management control can classification, which canand consider multiple such as Inventory is aofvital function and to meet market time,developed cost, price criticality, is acriteria contemporary demand. lack inventory management and control can lead which can consider multiple such as Inventory is aof vital function to meet market approach by and (Flores and Whybark, 1986). In affect theA items lead time, of items, operation demand. Amanagement lack inventory ofavailability management and control can classification, lead time, cost, price and criticality, is a contemporary approach developed by (Flores and Whybark, 1986). In affect the items lead time, availability of items, operation lead time, cost, price and criticality, is a contemporary demand. A lack inventory of management and control can practice, it has also been recognized that there are other costs, the company’s profits, performance and any approach developed by (Flores and Whybark, 1986). In affect items lead time, financial availability of items, operation practice, it has also been recognized that there are other costs, company’s profits, financial performance and any developed and criteria Whybark, 1986). In affect itemsToday lead time,the availability of items,changes operation important and(Flores qualitative for classifying related efforts. with rapid and constant in approach practice, itquantitative has also by been recognized that there are other costs, the company’s profits, financial performance and any quantitative and qualitative criteria for classifying related efforts. Today with the rapiddifferent and constant changes in important practice, it has also been recognized that there are other costs, company’s profits, financial performance and any inventories that need to be considered such as “commonality, the business world, companies with sizes often cope related efforts. Today with the rapid and constant changes in important quantitative and qualitative criteria for classifying thatsubstitutability, need to be considered as “commonality, the business world, companies different sizes oftenmore. cope important qualitative criteria for classifying related efforts. Today withitems the with rapid and constant changes in inventories obsolescence, numbersuch of requests per year, with series of inventory in thousands or even inventoriesquantitative that need toand be considered such as “commonality, the business world, companies with different sizes often cope obsolescence, substitutability, number of requests per year, with series of inventory items in thousands or even more. inventories that need to be considered such as “commonality, the business world, companies with different sizes often cope scarcity, durability, repairability, order size requirement, Managers need sophisticated strategic-operational tactics. obsolescence, substitutability, number of requests per year, with series of inventory items in thousands or even more. durability, repairability, order size requirement, Managers sophisticated strategic-operational tactics. obsolescence, substitutability, number of requests per2006). year, with series need of ABC inventory itemsclassification in thousandsisortheeven more. stockability, and stock-out penalty cost” (Ramanathan, The traditional inventory most used scarcity, scarcity, durability, repairability, order size requirement, Managers need sophisticated strategic-operational tactics. stockability, and stock-out penalty cost” (Ramanathan, 2006). The traditional ABC inventory classification is the most used scarcity, durability, repairability, order size requirement, Managers need sophisticated strategic-operational tactics. The literatureand is stock-out abundant penalty with various studies to tackle the in companies. thisclassification method wasis introduced by stockability, cost” (Ramanathan, 2006). The traditional Historically, ABC inventory the most used literature is stock-out abundant with various studies to tackle the in companies. thisclassification method by The and cost” 2006). The traditional ABC is introduced the most used MCIC problem. Recently,penalty (Simha, et (Ramanathan, al., 2014) and few (Dickie, 1951) Historically, and inventory successfully used inwas General Electric The literature is abundant with various studies to tackle the in companies. Historically, this method was introduced by stockability, problem. Recently, (Simha, etmachine al., 2014) and type few (Dickie, 1951) and successfully used inwas General Electric The literature is utilized abundant with various to tackle the in companies. Historically, thiswith method introduced by researchers have unsupervised learning Company to facilitate the deal inventories. Typically, it MCIC MCIC problem. Recently, (Simha, et studies al., 2014) and few (Dickie, 1951) and successfully used in General Electric have utilized unsupervised machine learning type Company to that facilitate the deal withkeeping inventories. Typically, it researchers MCIC problem. Recently, (Simha, et al., 2014) and few (Dickie, 1951) and successfully used in General Electric to improve the inventory classification system from several is a method distributes stock units (SKUs) into Company to facilitate the deal with inventories. Typically, it researchers have utilized unsupervised machine learning type improve the inventory classification system from several is a method distributes stock units (SKUs) researchers have utilized unsupervised machine type Company to that facilitate withkeeping inventories. Typically, it to points of views. Examples of such methods are learning the K-means three classes based onthe thedeal importance of SKUs in whichinto A, to improve the inventory classification system from several is a method that distributes stock keeping units (SKUs) into points of views. Examples of such methods are the K-means three classes based on the importance of SKUs in which A, to improve the inventory classification system from several is a method that distributes stock keeping units (SKUs) into method (Keskin and Ozkan, 2013) and the Agglomerative very important, B, moderate important, C, least important. three classes based on the importance of SKUs in which A, points of views. Examples of such methods are the K-means and Ozkan, 2013) and the2016). Agglomerative very important, B, on moderate leastinimportant. points of(Keskin views. Examples of such methods the K-means three classes dollar based the importance of C, SKUs which Hierarchical Clustering method (Raja, et al., The usage is important, the single criterion used A, in method method (Keskin and Ozkan, 2013) and theareAgglomerative very annual important, B, moderate important, C, least important. Hierarchical Clustering method (Raja, et al., 2016). The annual dollar usage is the single criterion used in method (Keskin and Ozkan, 2013) and the Agglomerative very important, B, moderate important, C,criterion least important. To the best of our knowledge, research paper is the first evaluating alldollar SKUs for the is final Appropriate inventory Clustering method this (Raja, et al., 2016). The annual usage therank. single used in Hierarchical thethat best of our knowledge, this research paper is the first evaluating alldollar SKUs for the final Appropriate Hierarchical Clustering (Raja, et al., 2016). The annual usage is therank. single criterion used for in To work tomethod use the Gaussian mixture model review polices are then allocated to each SKUs’ inventory class To the best proposes of our knowledge, this research paper is the first evaluating all SKUs for the final rank. Appropriate inventory work that proposes to use the Gaussian mixture model review polices are then allocated to each SKUs’ class for To the best of our knowledge, this research paper is the first evaluating all SKUs for the final rank. Appropriate inventory (GMM)thatas proposes unsupervised classifier model to address the controlling purpose (Ernst and Cohen, 1990). Despite to use the Gaussian mixture model review polices are then allocated to each SKUs’ class the for work (GMM) as unsupervised classifier model to address the controlling purpose (Ernst and Cohen, 1990). Despite the work that proposes to use the Gaussian mixture model review polices are then allocated to each SKUs’ class for MCIC problem. The objective is to exploit the robustness simplicity purpose of the (Ernst implementation the Despite traditional the controlling and Cohen, of1990). the (GMM) as unsupervised classifier model to address and problem. The to objective isa to exploit robustness simplicity ofmethod, the implementation of1990). the Despite traditional (GMM) asofunsupervised classifier modelthe address and the controlling anda Cohen, the MCIC usefulness GMM provide competitive performance of classification it has total reliance a monoMCIC problem. The objective is to exploit theto robustness and simplicity purpose of the (Ernst implementation of theon traditional of GMM to provideisa to competitive performance of classification method, it has a total reliance on traditional a mono- usefulness MCIC problem. The objective exploit the robustness and simplicity of the implementation of the classification method, it has a total reliance on a mono- usefulness of GMM to provide a competitive performance of classification method, it has a total reliance on a mono- usefulness of GMM to provide a competitive performance of
2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2019, 2019 IFAC 1955Hosting by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic Copyright 2019 responsibility IFAC 1955Control. Copyright © 2019 IFAC 1955 10.1016/j.ifacol.2019.11.484 Copyright © 2019 IFAC 1955
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inventory cost and service when compared to other MCIC methods. An advantage of the GMM compared to others is the fact that the criteria are not weighted which means that the GMM algorithm does not involve subjectivity. In addition, an empirical cost-service inventory analysis of the GMM model is presented to show its performance when compared to four well known and used MCIC methods, namely: the R-method (Ramanathan, 2006); the ZF-method (Zhou and Fan, 2007); the Ng-method (Ng, 2007) and the Hmethod (Hadi-Vencheh, 2010). The rest of the paper is organized as follows. In the next section, the related research is described. A description of the classification model is provided in Section 3. The experimental investigation and results are discussed in Section 4. In the last section, conclusions and further research work are presented. 2. RELATED RESEARCH In the literature, a considerable research work has been developed to deal with the multiple criteria decision making (MCDM) approach that can be used in improving the MCIC problem. (Saaty, 1980) analytic hierarchy process (AHP) model has been developed and it has been then used by (Flores et al., 1992) to the MCIC. (Partovi and Burton, 1993) and other authors have applied the AHP in different ways to address the MCIC problem. Despite the merits of AHP, it has been criticized because it involves the subjectivity in a pairwise comparison to determine the criteria preferences and to rank inventory items. (Bhattacharya, et al., 2007) have designed an integrated model that includes a distance-based multi-criteria consensus framework based on the concepts of TOPSIS to improve the ABC classification. They have considered various important criteria (e.g. unit cost, LT, consumption rate, perishability of items and cost of storing items) for SKUs which are obtained from a pharmaceutical industry in India. The authors have tested the inventory cost and model performance using the ANOVA technique and they have shown that it contributes to a low average investment. (Chen, 2012) has presented an integrated model using DEA to generate criteria weights, which is based on an output maximizing multiplier DEA model with multiple outputs and a constant input and TOPSIS is then applied. Several classification models based on mathematical programming (MP) have been developed for the MCIC and to compute the performance of each inventory item under multiple criteria into a single score for ABC classification. In a similar manner, the concept of data envelopment analysis (DEA) technique, (Ramanathan, 2006) has developed a simple weighted linear optimization model, referred to as the R-model where it uses a weighted additive function that generates weighted scores. (Zhou and Fan, 2007) have proposed an extended work of R-model, which is known as ZF-model. The authors aimed to address the shortcomings and subjectivity of the R-model. Their model used two indices to reveal the most and least favorable weights along with a formulated control parameter that may show the preference of the decision maker on both indices. Other extended improvement research to the R-model is by (Ng, 2007) which has introduced a simple alternative model called Ng to obtain an optimal inventory item score which can
simply be solved without a linear optimizer. (Hadi-Vencheh, 2010) has presented a nonlinear optimization method that maintains the weights’ effects up to the final solution in providing a more reasonable and encompassing inventory item score. Note that both Ng and H models provide a descending ranking of criteria. The main drawback of the above mentioned of MP classification models is that they require high repeated optimizations when the inventory items are large in number to introduce. (Chen, 2011) has developed an optimization model in which provides a peer-estimation procedure depending on both R and ZF models to scoring inventory items. (Torabi, et al., 2012; Hatefi, et al., 2014; Chen, et al., 2008; Ladhari, et al., 2015) were among the important studies based on MP. Several artificial intelligence techniques have been employed in the MCIC problem. (Guvenir and Erel, 1998) have proposed a genetic algorithm (GA) to learn the criteria weights and found a solution for the cut-off points between the classes A-B and B-C. (Partovi and Anandarajan, 2002) have utilized an artificial neural network (ANN) along with GA and back-propagation to the MCIC problem by using real-world data obtained from a pharmaceutical company in the US. (Yu, 2011) has conducted a method of inventory classification which includes support vector machines (SVM), back-propagation neural networks (BNN) and knearest neighbor algorithm (KNN). Particle swarm optimization (PSO) has been applied by (Tsai and Yeh, 2008) in which the method finds the optimal number of inventory classes and generates items classifications. More recently, unsupervised learning models have been utilized to the MCIC. (Keskin and Ozkan, 2013) was the first work which has developed a fuzzy c-means clustering model for better inventory clustering under fuzzy circumstances. FCM has been used to determine the matching to any clusters and to show the membership degrees. (Simha, et al., 2014) have used self-organizing maps model (SOM) to the MCIC and obtained results compared to K-means, and fuzzy c-means using the known theoretical dataset in the literature. SOM outperformed the compared models in terms the inventory cost. The authors also stressed the use of unsupervised learning technique to the MCIC. (Lolli, et al., 2014) have developed a hybrid model consisting of AHP and the Kmeans clustering algorithm to improve the MCIC problem. The authors used K-means to create classes based on the ranking scores obtained from the AHP. (Raja, et al., 2016) have proposed the agglomerative hierarchical clustering (AHC) model by using ward’s method to minimize variance and using Euclidean distance for items measuring. They aimed to develop inventory police of storage and control based on the nature of spare parts (metal/non-metal) in a chemical company. 3. GAUSSIAN MIXTURE MODEL (GMM) The GMM is a probabilistic unsupervised classification model that represents a distribution data point within overall clusters. For example, when modeling the human beings weight data, for each gender, the weight is modeled as normal distribution within an average of about 65 kilograms. In this case, the weight data on each entry point are given
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with no corresponding data on gender. This example of a model following assumptions is known as a GMM. Applications of GMM include medical image segmentations, speech data extraction, predicting locations and tracking multiple objects. More detailed information of the procedures for classification can be found in the literature (e.g., Everit and Hand, 1981; Titterington, et al., 1985; Hastie and Tibshirani, 1996). 4. DISCUSSION AND NUMERICAL EXAMPLE In this section, the application of the GMM algorithm to the MCIC problem is explained and the SKUs are detailed. The results of classification and the validation of the model and results are provided. The dataset of SKUs is clustered based on the GMM algorithm where inventory items are allocated to different clusters according to their attributes. The GMM algorithm assigns each item to the belonging cluster centers by Gaussian distribution and by iteratively calculating the probability means, covariance and mixing coefficient in which items within cluster centers are as similar as possible and the differences between cluster centers as big as possible. By the GMM, colors of each datapoint (SKU) are used to determine which cluster it should belong to. The color was assigned to each SKU by its posterior probabilities’ values, which coincide with the (blue, green, and red) coloring function in the PYTHON [2, 1, and 0] correspond to clusters A, B, and C respectively. For example, if the GMM determines the posterior probabilities of item 1 as (0.89, 0.00, 0.01) then the item 1 should belong to cluster center A which will appear in blue. Items somewhere between two cluster centers are usually avoided since the EM steps are iteratively updated until convergence reached as a maximum likelihood. The dataset used for the experimental classification analysis contained 47 SKUs obtained from a Hospital Respiratory Therapy Unit (HRTU). Each inventory item measured with respect to three attributes namely average unit cost, annual dollar usage and lead time as depicted in Figures 1, 2, and 3. These attributes have been predefined in the literature. Due to the varied values of the dataset, a pre-processing procedure of a linear normalization used to convert all measurement into a 0–1 scale for all items on criteria. where i=1, …, M and j= 1, …, N is a benefit criterion
Fig. 1. Visualization of the data run through GMM (k=3) and membership degree of SKUs to clusters under pairs of attributes (ADU and AUC).
Fig. 2. Visualization of the data run through GMM (k=3) and membership degree of SKUs to clusters under pairs of attributes (LT and AUC).
(1)
is an input value, is normalized value, i is where number of alternatives, M represents total number of items, J represents criterion number and N which is total number of criteria. The result of normalization is showed in Table 1 with the original data. After the run of the GMM code in PYTHON, the membership of each item to the belonging cluster centers are generated. Figure 1, 2, and 3 indicate the belonging of each inventory item to cluster centers with respect to pairs of attributes. For example, the belonging of each item to cluster centers of A, B and C is the obtained measurements of pairs of attributes (AUC and ADU) as shown in Figure1.
Fig. 3. Visualization of the data run through GMM (k=3) and membership degree of SKUs to clusters under pairs of attributes (LT and ADU). The overall belonging of each item to the cluster centers with respect to the three attributes of (AUC, ADU and LT) is shown in Figure 4. For final ABC labelling, the cluster center with the maximum likelihood estimation is labelled.
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Fig. 4. Visualization of the data run through GMM (k=3) and membership degree of SKUs to all clusters under the three attributes (ADU, AUC, and LT). The final classification results are detailed in Table 1. The GMM model clusters the SKUs as follows, 10 items are grouped in cluster A (21.28%), 16 items are grouped in cluster B (34.04%) and 21 items are grouped in cluster C (44.68%) of the 47 SKUs. Overall similarity and dissimilarity of items classifications are 28 items (59.57%) and 19 items (40.43%) compared to GMM rank as shown in Table 1. Table 1. Results of classification based on GMM distribution.
The silhouette coefficient analysis is a known method used to validate the consistency results of the unsupervised learning model. It is used to validate how well the dataset used is clustered. The obtained consistency result is 0.36221. As it should be noted that this validation result is not compared to the MP models due to the differences in applications and validation processes. For comparing the cost-service inventory performance of the proposed GMM to four others MCIC models: ZF, R, Ng and H, the total holding inventory cost (safety stock inventory cost) is evaluated for all SKUs as well as the achieved fill rate when a predetermined service level is fixed to each class according to the methodology in (Teunter, et al., 2010; Babai, et al., 2015). The achieved fill rate is a factor to measure the fraction of routinely filling a customer order from on hand stock. Both total holding inventory cost and fill rate are calculated by assigning three fixed targets of cycle service levels as (99%, 95%, 90%), (95%, 90%, 85%) and (90%, 85%, 80%) to the classes A, B, and C respectively. Some notations and definitions are listed in Table 2. Table 2. Notations N N Di Ki hi σi Wi Qi Li FRi FRT CSLi Φ(.) G(x) CT
Number of items in the inventory systems. Number of item classes (A, B, C) Mean demand per item i Safety factor per item i Holding cost per item i Standard deviation of the demand of item i Fixed ordering cost per item i Quantity order per item i Lead time per item i Fill rate of item i Overall fill rate of inventory system evaluation Cycle service level of item i Standard normal distribution function Loss function of the standard normal distribution Total cost of the inventory system.
For calculating the total safety stock inventory cost, the equation is as follows: (2) The approximate fill rate of each ith item, is given as: (3) where,
-
[1-
(4)
The total achieved fill rate on the inventory is computed as: (5) The reader could refer to (Teunter, et al., 2010; Babai, et al., 2015) for more details and numerical examples of estimating the total cost and fill rate of inventory which used to the GMM in this research. The followed Table 3 is provided by illustrative samples of a numerical estimation the total inventory cost and achieved fill rate.
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efficient method. The results of the combined cost-service performance are depicted in Figure 5. All models are very close in curves with slight differences. At the same time, the GMM showed very competitive performance as compared to the other classification models within the three targets of CSL. The GMM efficiency curve is on the top of all other curves, which shows its higher efficiency compared to the four MCIC methods. GMM is followed by Ng in terms of efficiency. The results also show that the ZF method results in the lowest efficiency.
Table 3. Detailed calculation for the GMM
Fig. 5. Combined service-cost efficiency curves for the MCIC methods (under the normal distribution) of 47 SKUs 5. CONCLUSIONS
The numerical results illustrated in Table 4 notably show that there is an increase in the achieved fill rates when GMM is applied as compared to the other methods; however, this increase in the service is associated with an increase of the total holding inventory cost. Thus, a combined service-cost performance is conducted in order to have a better analysis and a fair comparison of the performance of the considered methods. Efficiency curves of the combined service-cost performance of the classification models are reflected in Figure 5.
In this paper, we have presented comprehensive results and analysis of the GMM clustering model, which is used to address the MCIC problem. The aim of our proposed unsupervised clustering model is not only to provide a classification of SKUs free of subjectivity processes, but also to provide a competitive total inventory cost with maximum service level. The results enable us to demonstrate that the proposed model can remain in a promised and competitive performance along with the other classification models within the same ABC distribution class. A future research opportunity would be to consider a larger dataset to validate our results and compare to relevant unsupervised machine learning applications. REFERENCES
Table 4. Numerical results Target CSL for Classes ABC Models GMM R ZF Ng H
1st Target of CSL
2nd Target of CSL
3rd Target of CSL
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