vailable online at www.ispacs.com/conferences/cjac Volume, Year 202, rticle ID cjac-00-028, Pages 22-25 doi:0.5899/202/cjac-00-028 onference rticle The First Regional onference on the dvanced Mathematics and Its pplications 29 FE- MR 202, Mobarakeh ranch, Islamic zad University, Mobarakeh, Iran. inventory classification withmultiple-criteria using weighted non-linear programming Z. Eslaminasab *, T. Dokoohaki Department of Mathematics, Islamic zad University, Khorasgan ranch, Isfahan opyright 202 Z. Eslaminasab and T. Dokoohaki. This is an open access article distributed under the reative ommons ttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. bstract: To have an efficient control of a huge amount of inventory items, traditional approach is toclassify the inventory into different groups. Different inventory control policies can thenapplied to different groups. The well-known classification is simple to understandand easy to use. In proposed method, the performance matrix is normalized first,then theideal alternative is introduced. The appropriate weights are obtained into multiple criteriaby a non-linear programming. In next stage, alternatives are evaluated based on Distanceased pproach (D)method and classified. Finally, the proposed method is comparedwith another methods by an example. Keywords: inventory classification; Distance ased pproach; Multiple criteria analysis. Introduction Inventory classification using analysis is one of the most widely employed technique in organization. Different inventory policies can the applied to different groups. analysis is a well-known and practical classification based on the pareto principle.for example, group inventory items are those making about 70% of company s business but only taking up 0% of inventory. They are critical to the functioning of the com-pany.group inventory items are those presenting about 20% of company s business and taking about 20% of inventory. Group items are those representing 0% of company business but taking up about 70% of inventory. analysis is simple to understand and easy to use.however, traditional analysis is based on only single measurement such as annual dollar usage.it has been recognized that other criteria,such as inventory cost, part criticality, lead time, commonality, obsolescence, substitutability, number of requests for the item in a year, scarcity, durability, * orresponding author. Email addresses: z.esb.math@gmail.com, dokoohakitahereh@yahoo.com
acta computare Volume, Year 202, 22-25 23 substitutability, reparability, order size requirement, stock-ability, demand distribution, and stock-out penalty cost are also important in inventory classification. Research literature on decision tools for multi-criteria inventory classification (MI) has been developed in past 20 years. omplex computational tools are needed for multi-criteria classification. Flores et al. [2] provide a matrix-based methodology. joint criteria matrix is developed in the case of two criteria. However, the methodology is relatively difficult to use when more criteria have to be considered. Several multiple criteria decision-making (MDM) tools have also been employed for the purpose. ohen and Ernst [3] and Ernst and ohen [] have used cluster analysis to group similar items. The analytic hierarchy process (HP) [5] has been employed in many MI studies [2, 6, 7]. When HP is used, the general idea is to derive a single scalar measure of importance of inventory items by subjectively rating the criteria and/or the inventory items [2, 8]. The single most important issue associated with HP- based studies is the subjectivity involved in the analysis. Heuristic approaches based on artificial intelligence, such as genetic algorithms [8] and artificial neural networks [9], have also been applied to address the MI problem. learly, these approaches are heuris- tics and need not provide optimal solutions at all environments. very recent attempt by ramanathan [0] is to develop a weighted linear optimization model to the problem. The basic concept of model in [0] is closely similar to the concept of data envelopment analysis (DE). The model first converts all criteria measurement into a scalar which is a weighted sum of measures under individual criteria. To avoid the subjectively on the weight assignment, the weights are generated by a DE-like linea optimization. The clas- sification is then performed by grouping the items based on the scores generated. however, a linear optimization is required for each item. The processing time can be very long when the number of inventory items is large in scale of thousands of items in inventory. Zhu and fan []present an extended version of the Ramanathan s model by incorporating some balancing features for MI. Zhu and fan model, here after ZF-model, uses two sets of weights that are most favorable and least favorable for each item. Ng [2] propose a simple model for MI. The model converts all criteria measures of an inventory item into a scalar score. With proper transformation, Ng obtains the scores of inventory items without a linear optimizer. The Ng-model is flexible as it could easily integrate additional information from decision makers for inventory classification. ut, Ng-model leads to a situation where the score of each item is independent of the weights obtained from the model. That is, the weights do not have any role for determining total score of each item and this may lead to a situation where an item is inappropriately classified.in this article the performance matrix is normalized based on desirable and un desirable criteria, at first.the appropriate weights are obtained into multiple criteria by a non-linear programming. In next stage, alternatives are evaluated based on Distance ased pproach (D) method. The mathematical formulation is presented in Section 2. n illustration is provided in Section 3 with comparisons to the result from those in the literature. Short conclusions are given in Section. 2. The proposed model onsider an inventory with n items and these items are to be classified based on m riteria. In particular, let the performance of ith inventory item in terms of the jth criteria
2 acta computare Volume, Year 202, 22-25 e denoted as yij (i =, 2,..., n; j =, 2,...,m). We evaluate an item m (m =, 2,...,M) by converting multiple measures under all criteria into a single score. common scale for all measures is also an important issue. particular criterion measure, in a large scale, may always dominate the score. For this, we propose normalizing all measures yij into a 0- scale. We denote all transformed measures as xij. In order to transform the performance ratings, the performance ratings are normalized into the range of [0, ] by the following equations []. (i) The larger the better type: * + * + * + (2.) (ii) The smaller the better type: * + * + * + Let Xj * = max {xij ; i=, 2,..., n} denote the largest value that appears in the jth criterion. Then I = ( X *, X2 *,, Xm * ) is empirically the ideal item because it has the best performance in every criterion. consider the following model : * + 2 2 where Wj (j =, 2,...,m) is the importance associated with the jth criterion and c obtained from the following model : 2 where is a small positive quantity imposed to restrict any criterion from being ignored. when we have n items and m criterion, the whole set of items can be represented by the following matrix [3] :
acta computare Volume, Year 202, 22-25 25 2 2 22 2 [ 2 ] that xij (i =, 2,..., n; j =, 2,...,m) denote the performance of ith inventory item in terms of the jth criteria. Thus in this matrix, a vector in an n-dimensional space represents every inventory items. In order to ease process and in the same time to eliminate the influence of different units of measurement, the matrix is standardized using z formula as : ( ( ) ) where i =,2,..., n. m = Number of different criterion. n = Number of inventory items. xij = Indicator value for inventory item i for criteria j. sj = Standard deviation of criterion j. IN this study the optimal state of the objective is represented by the optimum inventory item, the OPTIML. The vector I = ( X *, X2 *,, Xm * ) is a set of optimum simultaneous criterion. The vector I is called the optimal point. For practical purpose the optimal good value for criterion is defined as the best value within the range of criteria. The standard matrix is given as : [ ] where
26 acta computare Volume, Year 202, 22-25 The next step is to obtain the difference or distance from each item to the optimal point by subtracting each element of optimal by correspondence element in the item set. This results in another interim matrix: [ ] where * + The next step is to introduce the aggregated performance weights for each selection criteria. If the aggregated performance weight for any selection criteria j is denoted by wj then this will be result in another interim matrix as given: [ ] Finally the Euclidean omposite Distance, D, between each item inventory to the optimal state, is derived from the following formula: * + where the weights are obtained from (2.3) and (2.) nd the items are ranked based on the minimal distance to the optimal point. 3.Illustrative example Following [2, ] let us consider three criteria: nnual Dollar Usage (DU), verage Unit ost (U) and Lead Time (LT) for inventory classification. ll the criteria are positive related to the score of the inventory items. n inventory with 7 items and measurement of performance under each of the criteria considered are shown in Table. This table also shows the maximal and minimal measures under each criteria as well as transformed
acta computare Volume, Year 202, 22-25 27 measures in a scale of 0- as suggested in Section 2. For comparison purpose, we maintain the same distribution of class, and items as in literature studies [0, ], i.e. 0 class, class and 23 class. Table 2 shows the classification based on our proposed model.the classification with the three criteria by ZF-model, Ng-model and traditional analysis using annual dollar usage are listed in this table as well. s shown in Table 2, the classification using our approach provides different results compared with the other methods. omparing to traditional analysis based on only annual dollar usage, only 25 out of 7 items are kept in the same classes when classification using proposed model with multi-criteria. seven out of the ten class items in traditional classification is still classified as class items when multiple criteria is considered in proposed model. The other two (S2, S9 and S0) are re-classified as class using our model. For the class items are matched in both models, and for class, 6 out of 23 items are classified as class items in both models.
28 acta computare Volume, Year 202, 22-25 Table Source and transformed measures of items under criteria Item DU U LT DU (transformed) U (transformed) LT (transformed) S 580.6 9.92 2.0000 0.783 0.8333 S2 5670 20 5 0.9707 0.0000 0.3333 S3 5037.2 23.76 0.869 0.9090 0.5000 S 769.56 27.73 0.859 0.8896.0000 S5 378.8 57.98 3 0.5939 0.79 0.6666 S6 2936.67 3.2 3 0.5007 0.8725 0.6666 S7 2820 28.2 3 0.806 0.8873 0.6666 S8 260 55 0.97 0.7565 0.5000 S9 223.52 73. 6 0.2 0.6665 0.666 S0 207.5 60.5 0.097 0.26 0.5000 S 075.2 5.2 2 0.806.0000 0.8333 S2 03.5 20.87 5 0.75 0.923 0.3333 S3 038 86.5 7 0.72 0.6027 0.0000 S 883.2 0. 5 0.76 0.86 0.3333 S5 85. 7.2 3 0.26 0.677 0.6666 S6 80 5 3 0.350 0.8053 0.6666 S7 703.68.66 0.67 0.953 0.5000 S8 59 9.5 6 0.0978 0.7833 0.666 S9 570 7.5 5 0.0937 0.793 0.3333 S20 67.6 58.5 0.076 0.7397 0.5000 S2 63.6 2. 0.075 0.9058 0.5000 S22 55 65 0.0739 0.7077 0.5000 S23 32.5 86.5 0.070 0.6027 0.5000 S2 398. 33.2 3 0.062 0.8629 0.6666 S25 370.5 37.05 0.059 0.8.0000 S26 338. 33.8 3 0.0539 0.8598 0.6666 S27 336.2 8.03 0.0535 0.68.0000 S28 33.6 78. 6 0.096 0.623 0.666 S29 268.68 3.3 7 0.09 0.3692 0.0000 S30 22 56 0.032 0.755.0000 S3 26 72 5 0.0328 0.6735 0.3333 S32 22.08 53.02 2 0.0322 0.7662 0.8333 S33 97.92 9.8 5 0.0297 0.783 0.3333 S3 90.89 7.07 7 0.0285 0.990 0.0000 S35 8.8 60.6 3 0.0269 0.7292 0.6666 S36 63.28 0. 3 0.0238 0.8257 0.6666 S37 50 30 5 0.025 0.8785 0.3333 S38 3.8 67. 3 0.089 0.6960 0.6666 S39 9.2 59.6 5 0.062 0.730 0.3333 S0 03.36 5.68 6 0.035 0.7727 0.666 S 79.2 9.8 2 0.0093 0.9283 0.8333 S2 75. 37.7 2 0.0087 0.809 0.8333 S3 59.78 29.89 5 0.0060 0.8790 0.3333 S 8.3 8.3 3 0.000 0.7892 0.6666 S5 3. 3. 7 0.006 0.8570 0.0000 S6 28.8 28.8 3 0.0006 0.88 0.6666 S7 25.38 8.6 5 0.0000 0.9836 0.3333
acta computare Volume, Year 202, 22-25 29 Table classifications by different models Item DU U LT si classification proposed model ZF H. Ng Traditional S S 580.6 5037.2 9.92 23.76 2.0000 0.992 S3 769.56 27.73 0.96 S6 2936.67 3.2 3 0.702 S7 2820 28.2 3 0.698 S5 S8 378.8 260 57.98 55 3 0.678 0.5802 S 075.2 5.2 2 0.565 S6 03.5 20.87 5 0.50 S25 223.52 73. 6 0.873 S2 S7 703.68 90.89.66 7.07 7 0.809 0.35 S 25.38 8.6 5 0.78 S26 63.6 2. 0.60 S5 S30 5670 80 0 5 5 3 0.08 0.3862 S2 79.2 9.8 2 0.3809 S32 398. 33.2 3 0.3765 S2 338. 33.8 3 0.3668 S2 S36 370.5 50 37.05 30 5 0.3593 0.3567 S6 570 7.5 5 0.37 S 59.78 29.89 5 0.360 S35 28.8 28.8 3 0.358 S9 S20 59 3. 9.5 3. 6 7 0.330 0.3397 S38 75. 37.7 2 0.3200 S22 63.28 0.82 3 0.399 S9 85. 7.2 3 0.298 S37 S23 67.6 97.92 58.5 9.8 5 0.2953 0.2933 S7 22.08 53.02 2 0.2826 S3 8.3 8.3 3 0.2788 S33 03.36 5.68 6 0.2737 S39 S8 22 55 56 65 0.2733 0.2703 S3 038 86.5 7 0.2669 S 8.8 60.6 3 0.257 S0 S0 9.2 3.8 59.6 67. 5 3 0.27 0.227 S28 26 72 5 0.25 S3 33.6 78. 6 0.209 S3 32.5 86.5 0.909 S27 S5 336.2 207.5 8.03 60.5 0.876 0.753 S2 883.2 0. 5 0.62 S29 268.68 3.3 7 0.523
250 acta computare Volume, Year 202, 22-25.onclusion In this paper, a simple approach for inventory classification is proposed. weighted nonlinear programming model has been proposed and illustrated in this paper for classifying inventory items in the presence of multiple criteria. It is a very simple model that can be easily understood by inventory managers. References [].H. heng, Evaluating weapon systems using ranking fuzzy numbers, Fuzzy Sets and Systems, 07, 25-35, (999). http://dx.doi.org/0.06/s065-0(97)0038-5 [2].E. Flores, D.L. Olson and V.K. Dorai, Management of multicriteria inventory classification, Mathematical and omputer Modelling 6 7-82 (992). http://dx.doi.org/0.06/0895-777(92)9002- [3] M.. ohen, R. Ernst, Multi-item classification and generic inventory stock control policies, Production and Inventory Management Journal 29 (988). [] R. Ernst, M.. ohen, Operations related groups (ORGs): a clustering procedure for production/inventory systems, Journal of Operations Management, 9, 57-598, (990). http://dx.doi.org/0.06/0272-6963(90)9000- [5] T.L. Saaty, The analytic hierarchy process, McGraw-Hill: New York; (980). [6] F.Y. Partovi, J. urton, Using the analytic hierarchy process for analysis, International Journal of Production and Operations Management, 3, 29-, (993). http://dx.doi.org/0.08/03579300369 [7] F.Y. Partovi and W.E. Hopton, The analytic hierarchy process as applied to two types of inventory problems, Production and Inventory Management Journal, 35, 3-9, (993). [8] H.. Guvenir, E. Erel, Multicriteria inventory classification using a genetic algorithm, European Journal of Operational Research, 05, 29-37, (998). http://dx.doi.org/0.06/s0377-227(97)00039-8 [9] F.Y. Partovi, M. nandarajan, lassifying inventory using an artificial neural network approach, omputers and Industrial Engineering,, 389-0, (2002). http://dx.doi.org/0.06/s0360-8352(0)0006-x [0] R. Ramanathan, inventory classification with multiple-criteria using weighted linear optimization, omputers & Operations Research, 33, 695-700, (2006). http://dx.doi.org/0.06/j.cor.200.07.0
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