IJAHP Article: Mallick, Sarkar, Das/A unified decision framework for inventory classification through graph theory International Journal of the Analytic Hierarchy Process 183 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.482 A UNIFIED DECISION FRAMEWORK FOR INVENTORY CLASSIFICATION THROUGH GRAPH THEORY Bivash Mallick Assistant Professor, Department of Industrial Engineering & Management Maulana Abul Kalam Azad University of Technology, West Bengal, India bivash.mallick@gmail.com Bijan Sarkar Professor, Department of Production Engineering, Jadavpur University, Kolkata, India bsarkar@production.jdvu.ac.in Santanu Das Professor and Head, Department of Mechanical Engineering Kalyani Govt. Engineering College, Kalyani, India sdas.me@gmail.com ABSTRACT Conventionally, a traditional ABC analysis based on a single criterion of annual consumption cost is employed in industry to facilitate classification of inventory items. However, other criteria may be important in inventory classification such as lead time, item criticality, storage cost, etc. Hence, for situations like this many multiple criteria decision-making methods are available and the Analytic Hierarchy Process (AHP) is a popular one. The present article demonstrates a new approach by integrating Graph Theory (GT) and the Analytic Hierarchy Process (AHP) as a decision analysis tool for multi-criteria inventory classification. In this paper, 47 disposable items used in a respiratory therapy unit of a hospital were considered for a case study. Output of this hybrid method shows more precise results than that of either traditional ABC or the AHP classification methods. As the proposed decision analysis tool is a simple, logical, systematic and consistent method, it may be recommended for application in diverse industries handling multi-criteria inventory classification systems. Keywords: ABC classification; graph theory; Analytic Hierarchy Process; inventory classification; graph theory-AHP integration; hybrid system 1. Introduction Inventories are resources of any kind that have an economic value. Appropriate inventory control is necessary because efficiency and cost of its operation are generally largely mailto:bivash.mallick@gmail.com mailto:bsarkar@production.jdvu.ac.in mailto:sdas.me@gmail.com IJAHP Article: Mallick, Sarkar, Das/A unified decision framework for inventory classification through graph theory International Journal of the Analytic Hierarchy Process 184 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.482 affected by both its excess and shortage. Inventory control is thus essential to determine the item(s) to indent (i.e., to order) along with its quantity, time to indent and the optimum stock to maintain so that purchasing and storage costs are reduced to a minimum (Mallick, Dutta, & Das, 2012).Hence, the planning and control of inventory attract substantial attention from the management in an organization. In a large organization, the inventory that must be maintained consists of a great variety of items. Statistics reveal that just a handful of items account for the bulk of the annual expenditure on materials. These few items, called ‘A’ items, therefore hold the key to the business. The other items, known as ‘B’ and ‘C’ items, are large in number, however their contribution is less. Traditional ABC analysis is performed based on the consumption values of inventory items. Consumption values are arranged in descending order. Cumulative consumption values are then converted corresponding to cumulative percentages. A, B and C classifications are then made based on the cumulative percentage figures. To classify the inventories, the break point percentages can be chosen by the management depending on the number of items that can be effectively managed under each category (Flores, Olson, & Dorai, 1992). A number of researchers have questioned the focus on the consumption value as a single criterion. There are many instances when other criteria, other than the annual consumption value, may be significant in deciding the importance of an inventory item (Cohen & Ernst, 1988). In these cases, multiple criteria decision-making methods are helpful. A few investigators have worked on multi-criteria inventory classification. Flores and Whybark introduced a multi-criteria inventory classification in 1986 and 1987 (Flores & Whybark, 1986, 1987). Though it included several criteria, e.g. obsolescence, lead times, substitutability, reparability, criticality and commonality, this approach became increasingly complicated if more than two criteria were considered to classify inventory items. Flores et al. (1992) applied the Analytic Hierarchy Process (AHP) for multiple criteria inventory classification. Guvenir & Erel (1998) applied Genetic Algorithm (GA) to solve the problem of multiple criteria inventory classification. Their proposed method called Genetic Algorithm for Multi-Criteria Inventory Classification (GAMIC) is within the framework of the AHP to deal with multi-criteria ABC analysis. On the other hand, Braglia et al. (2004) used the AHP models to solve various multi-attribute decision problems at different levels of the decision tree. They identified the best control strategy for the spare stocks by defining the inventory policy matrix linking different classes of spare parts with the possible inventory management policies. Ramanathan (2006) proposed a weighted linear optimisation model for multiple criteria ABC (MCABC) inventory classification. The performance score of each item was obtained using a Data Envelopment Analysis (DEA). The defect of his model was that an item with a high value in an unimportant criterion was inappropriately classified under class A. This drawback was rectified by Zhou & Fan (2007) who incorporated balancing features for multi-criteria inventory classification (MCIC) by using most favourable and least favourable scores for each item. In another work, Bhattacharya et al. (2007) IJAHP Article: Mallick, Sarkar, Das/A unified decision framework for inventory classification through graph theory International Journal of the Analytic Hierarchy Process 185 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.482 presented a distance-based multi-criteria consensus framework utilizing the concepts of the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) model. The technique took into account various conflicting criteria having different units of measurement, which were then ranked in categories A, B and C using TOPSIS. Cakir & Canbolat (2008) presented an inventory classification system based on FAHP, integrating the fuzzy concept with real inventory data and designed a decision support system. Torabi et al. (2012) applied a modified version of a common weight DEA-like model to ABC inventory classification in those cases where both quantitative and qualitative criteria existed, while Kabir & Hasin (2013) developed a multi-criteria inventory classification model through integration of Fuzzy Analytic Hierarchy Process (FAHP) and artificial neural network approach. Soylu & Akyol (2014) incorporated the preferences of the decision maker into the Multi-criteria inventory classification decision- making process in terms of reference items into each class. In a recent work, a new method of Evaluation based on Distance from Average Solution (EDAS) was introduced for multi-criteria inventory classification (MCIC) problems (Ghorabaee, 2015). A comparative analysis (involving seven sets of criteria weights and Spearman’s correlation coefficient) was also put forward to show the validity and stability of the proposed method in problems related to MCDM. Liu et al. (2016) mentioned the necessity of considering the non- compensation in the multiple criteria ABC analysis. The authors proposed a new classification approach based on the outranking model to cope with such a problem. Mallick et al. (2016) presented a multi- criteria inventory classification system by MOORA (Multi-Objective Optimization on the basis of Ratio Analysis) method for hospital inventory management. None of the multiple criteria decision-making methods mentioned above is foolproof and a single multi-criteria decision-making method doesn’t exist that can be applied to all decision problems with equal efficiency. Therefore, there is a need for a simple, systematic, logical and consistent method or tool to guide decision makers in making an optimal selection. Hence, a methodology was developed by integrating Graph Theory (GT) and the Analytic Hierarchy Process (AHP) for multi-criteria inventory classification to solve a typical hospital inventory management problem. The outcome of the present application of this hybrid method was also compared with certain previous researchers dealing with same problem. 2. The proposed methodology Graph theory is a logical and systematic approach that originated from combinatorial mathematics. The graph theory consists of the digraph representation, the matrix representation and the permanent function representation. The digraph is the visual representation of the variables and their interdependencies. The matrix converts the digraph into mathematical form and the permanent function is a mathematical representation that helps to determine the numerical index (Faisal, Banwet, & Shankar, 2007).Various researchers have addressed the application of the graph theory approach to engineering systems in a significant number of papers. IJAHP Article: Mallick, Sarkar, Das/A unified decision framework for inventory classification through graph theory International Journal of the Analytic Hierarchy Process 186 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.482 Rao (2007) in his book has demonstrated how this methodology could be effectively used for decision making in various situations in the manufacturing environment. The graph theory and matrix approach (GTMA) method has been employed to total quality management evaluation of an industry (Grover, Agrawal, & Khan, 2004), contractor ranking (Darvish, Yasaei, & Saeedi, 2009), non-traditional machining processes selection (Chakladar, Das, & Chakraborty, 2009), equipment selection (Paramasivam, Senthil, & Rajam Ramasamy, 2011), and assessing the vulnerability of supply chains (Wagner & Neshat, 2010). The Analytic Hierarchy Process (AHP) developed by Saaty (1980) has been successfully applied to multi-criteria inventory classification by Flores et al. (1992).The AHP has been used in a variety of business decision-making settings. A significant feature of the AHP is the consistency check of relative importance of the attributes. In view of this advantage, for multi-criteria inventory classification, a methodology combining graph theory and the AHP method is proposed. The AHP is applied for determination of the relative weights of the selected attributes and graph theory is applied for evaluation and ranking of the alternatives. This hybrid approach is similar to the one reported by Rao and Parnichkun (2009), Singh and Rao (2011) and Lanjewar et al. (2016). The proposed hybrid method is a simple, fast, logical, systematic and consistent method, that may be recommended for use as a decision making method for multi-criteria inventory classification. However, it can be used for any type of decision making situation. It enables more critical analysis than the common multi-attribute decision making methods since any number of quantitative and qualitative attributes can be considered. In the permanent procedure, even a small variation in the attribute leads to a significant difference in the overall performance value making it convenient to rank the alternatives in a descending order with a clear difference in the overall performance value. Moreover, the proposed procedure using the graphical representation not only enables the analysis of the alternatives, but also facilitates visualization of various criteria along with their interrelations. The steps for the proposed methodology are presented below after Rao (2007), Chakladar et al. (2009), and Darvish et al. (2009). Step1- Identify the inventory attributes or criterion for decision matrix. Step 2- Generate the decision matrix using raw inventory data. A typical multi-criteria decision making problem can be precisely presented with a decision matrix showing the performance of different alternatives with respect to various attributes or criteria as given in Equation1. D = [xij]i= 1,… ,m, j= 1,….,n (1) Where xij is the performance measure of i th alternative on jth criteria, m is the number of alternatives; n is the number of criteria. Information stored in a decision matrix is generally incommensurable, wherein performance ratings w.r.t. different criteria are IJAHP Article: Mallick, Sarkar, Das/A unified decision framework for inventory classification through graph theory International Journal of the Analytic Hierarchy Process 187 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.482 expressed after considering different units of measure. Therefore, it is desirable to transform data into comparable values, using the normalization procedure of Step 3. Step 3- Normalize the matrix calculation. The normalized values of a criterion assigned to the alternatives are calculated using the following formula: 𝐹𝑖 − 𝐹𝑚𝑖𝑛 𝐹𝑚𝑎𝑥 − 𝐹𝑚𝑖𝑛 where, Fi is the i th value of the factor, Fmax is the maximum value of the factor, and Fmin is the minimum value of the factor under transformation (Flores et al.,1992). There are also some other methods of normalizing which incorporate negative values into the AHP system (Millet & Schoner, 2005). Step 4- Construct the relative importance matrix A relative importance matrix (Equation 2) is constructed using a pair-wise comparison scale of absolute numbers from 1 to 9 as proposed by Saaty (1980). An element when self-compared is assigned a value of 1. Assuming that there are N number of criteria in a decision-making problem, the pair-wise comparison of i th criterion with respect to the j th one yields a square matrix, where aij = 1 when i = j and aji = 1/aij (aij is the comparative importance of i th criterion with respect to j th one). M = [aij] = [ a11 a12 … a1n a21... a22 … a2n... ai1 ai2 … ann ] (2) Step 5- Digraph model of the inventory attributes showing their interdependencies. A digraph consists of a set of nodes N= {ni} (i=1, 2, M) and a set of directed edges E= {eij}. A node ni represents i th selection criterion/attribute and edges represent the relative importance among the attributes. The total of nodes, M, is equal to the number of selection criteria considered. If a node i has relative importance over another node j in the selection process, a directed edge or arrow is drawn from node i to node j (eij). If j is having relative importance over i, then a directed edge or arrow is drawn from node j to node i(eji). The digraph depicts the graphical representation of the interdependence between various decision attributes taken two at a time and their relative importance for quick visual perception. Step 6- Develop other square matrix based on the digraph As the number of nodes and their interrelations increase, the digraph becomes complex. To address this problem, the digraph is usually represented in a matrix form. Matrix representation of a digraph gives one-to-one representation, taking all the attributes (Ai) and their relative importance (aij) into account. The matrix B for a digraph can be represented as Equation 3: IJAHP Article: Mallick, Sarkar, Das/A unified decision framework for inventory classification through graph theory International Journal of the Analytic Hierarchy Process 188 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.482 B = [ 𝐴𝑖 𝑎12 𝑎13 … … 𝑎21 𝐴2 𝑎23 … … 𝑎31 𝑎32 𝐴3 … … ⋮ ⋮ ⋮ ⋮ 𝑎1𝑀 𝑎2𝑀 𝑎3𝑀 ⋮ ⋮ 𝑎𝑀1 𝑎𝑀2 𝑎𝑀3 … … 𝐴𝑀 ] … (3) where Ai is the value of the i th attribute represented by node ni and aij is the relative importance of the i th attribute over the j th one represented by the edge eij. Step 7- Calculate the overall performance value using matrix method. The overall performance value of each inventory item can be found by calculating the permanent function value of matrix B, i.e. per (B) (Rao, 2007). The permanent of a matrix was introduced by Cauchy in 1812. At that time, while developing the theory of determinants, he also defined a certain subclass of symmetric function that later was named permanent by Muir. In the expression for the permanent of the matrix, as no negative sign appears, no information is lost. This permanent function is the determinant of a matrix, considering all the terms as positive. Step 8- Rank the inventory items based on overall performance value. In this step, all inventory items are ranked according to their overall performance value arranged in descending order. 3. Illustrative example The hybrid system, i.e. integrated graph theory and the Analytic Hierarchy Process, is applied in this paper for classifying inventory items using the data referred to in Flores et al. (1992) corresponding to hospital inventory management. Flores applied the Multiple Criteria ABC analysis method based on the Analytic Hierarchy Process (AHP) using data obtained from a traditional ABC inventory classification presented by Reid (1987) Now, in order to demonstrate and validate the proposed procedure, the multi-criteria inventory classification integrating Graph Theory (GT) and the Analytic Hierarchy Process and the various steps of the methodology, given in Section 2, are described below: Step 1- Four attributes are identified for the Multiple Criteria ABC analysis: (1) Average Unit Cost (AUC); (2) Annual Dollar Usage (ADU); (3) Critical Factor (CF): 1, 0.50, or 0.01 being assigned to each of the 47 disposable items, where values imply very critical, moderately critical, and non-critical respectively; (4) Lead Time (LT) (weeks) being the time that it takes to receive a replenishment order after it is placed, ranging from 1 to 7 weeks. IJAHP Article: Mallick, Sarkar, Das/A unified decision framework for inventory classification through graph theory International Journal of the Analytic Hierarchy Process 189 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.482 Step 2- Generation of decision matrix using raw inventory data based on the 47 disposable items referred to as S1 through S47 are taken from Flores et al.(1992) (Table 1). Table 1 Decision and normalised matrix Item Decision Matrix Normalised Matrix AUD ($) ADU ($) CF LT AUD ($) ADU ($) CF LT S1 49.92 5840.64 1 2 0.21866 1.00000 1.00000 0.16667 S2 210 5670.00 1 5 1.00000 0.97066 1.00000 0.66667 S3 23.76 5037.12 1 4 0.09098 0.86183 1.00000 0.50000 S4 27.73 4769.56 0.01 1 0.11036 0.81582 0.00000 0.00000 S5 57.98 3478.80 0.5 3 0.25800 0.59385 0.49495 0.33333 S6 31.24 2936.67 0.5 3 0.12749 0.50063 0.49495 0.33333 S7 28.2 2820.00 0.5 3 0.11265 0.48057 0.49495 0.33333 S8 55 2640.00 0.01 4 0.24346 0.44961 0.00000 0.50000 S9 73.44 2423.52 1 6 0.33346 0.41239 1.00000 0.83333 S10 160.5 2407.50 0.5 4 0.75840 0.40963 0.49495 0.50000 S11 5.12 1075.20 1 2 0.00000 0.18053 1.00000 0.16667 S12 20.87 1043.50 0.5 5 0.07687 0.17508 0.49495 0.66667 S13 86.5 1038.00 1 7 0.39721 0.17413 1.00000 1.00000 S14 110.4 883.20 0.5 5 0.51386 0.14751 0.49495 0.66667 S15 71.2 854.40 1 3 0.32253 0.14256 1.00000 0.33333 S16 45 810.00 0.5 3 0.19465 0.13492 0.49495 0.33333 S17 14.66 703.68 0.5 4 0.04656 0.11664 0.49495 0.50000 S18 49.5 594.00 0.5 6 0.21661 0.09778 0.49495 0.83333 S19 47.5 570.00 0.5 5 0.20685 0.09365 0.49495 0.66667 S20 58.45 467.60 0.5 4 0.26030 0.07604 0.49495 0.50000 S21 24.4 463.60 1 4 0.09410 0.07536 1.00000 0.50000 S22 65 455.00 0.5 4 0.29227 0.07388 0.49495 0.50000 S23 86.5 432.50 1 4 0.39721 0.07001 1.00000 0.50000 S24 33.2 398.40 1 3 0.13706 0.06415 1.00000 0.33333 S25 37.05 370.50 0.01 1 0.15585 0.05935 0.00000 0.00000 S26 33.84 338.40 0.01 3 0.14018 0.05383 0.00000 0.33333 S27 84.03 336.12 0.01 1 0.38515 0.05344 0.00000 0.00000 S28 78.4 313.60 0.01 6 0.35767 0.04956 0.00000 0.83333 S29 134.34 268.68 0.01 7 0.63071 0.04184 0.00000 1.00000 S30 56 224.00 0.01 1 0.24834 0.03415 0.00000 0.00000 IJAHP Article: Mallick, Sarkar, Das/A unified decision framework for inventory classification through graph theory International Journal of the Analytic Hierarchy Process 190 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.482 S31 72 216.00 0.5 5 0.32643 0.03278 0.49495 0.66667 S32 53.02 212.08 1 2 0.23380 0.03211 1.00000 0.16667 S33 49.48 197.92 0.01 5 0.21652 0.02967 0.00000 0.66667 S34 7.07 190.89 0.01 7 0.00952 0.02846 0.00000 1.00000 S35 60.6 181.80 0.01 3 0.27079 0.02690 0.00000 0.33333 S36 40.82 163.28 1 3 0.17425 0.02371 1.00000 0.33333 S37 30 150.00 0.01 5 0.12144 0.02143 0.00000 0.66667 S38 67.4 134.80 0.5 3 0.30398 0.01882 0.49495 0.33333 S39 59.6 119.20 0.01 5 0.26591 0.01613 0.00000 0.66667 S40 51.68 103.36 0.01 6 0.22725 0.01341 0.00000 0.83333 S41 19.8 79.20 0.01 2 0.07165 0.00925 0.00000 0.16667 S42 37.7 75.40 0.01 2 0.15902 0.00860 0.00000 0.16667 S43 29.89 59.78 0.01 5 0.12090 0.00592 0.00000 0.66667 S44 48.3 48.30 0.01 3 0.21076 0.00394 0.00000 0.33333 S45 34.4 34.40 0.01 7 0.14291 0.00155 0.00000 1.00000 S46 28.8 28.80 0.01 3 0.11558 0.00059 0.00000 0.33333 S47 8.46 25.38 0.01 5 0.01630 0.00000 0.00000 0.66667 Step 3- The quantitative values of the inventory attributes, which are given in Table 1, are normalized. It is to be noted that all four criteria considered in this study are positively related to the importance level of inventory items (Ramanathan, 2006). Step 4- Relative importance relation matrix (A1) as shown in Table 2 and weight (Wi) of each attribute for the above-mentioned criteria is kept the same as in Flores et al. (1992). Table2 Relative Importance Relation Matrix The weight (Wi) of each attribute is taken as: WAUC: 0.07872; WADU: 0.09161; WCF: 0.41969; and WLT: 0.40999 after Flores et al. (1992). Step 5- Digraph model of the inventory attributes showing their interdependencies is shown in Figure 1. As explained, the nodes in the digraph represent criteria. The interactions among criteria are represented by edges. Relative Importance Relation Matrix (A1) AUC ADU CF LT AUC 1 1 1/8 ¼ ADU 1 1 1/3 1/6 CF 8 3 1 1 LT 4 6 1 1 IJAHP Article: Mallick, Sarkar, Das/A unified decision framework for inventory classification through graph theory International Journal of the Analytic Hierarchy Process 191 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.482 Figure1. Digraph model of the inventory attributes Step 6- The attributes matrix for the above digraph is prepared as Equation 4. Attribute AUC ADU CF LT B = AUC ADU CF LT [ A1 1 8 4 1 A2 3 6 1 8⁄ 1 3⁄ A3 1 1 4⁄ 1 6⁄ 1 A4 ] (4) Step 7- The characteristic permanent function for Overall Performance Value can be written as Equation 5 (Rao, 2007): per(B) = ∏ Ai + ∑ ∑ ∑ ∑ (aijaji)AkAl 4 l=k+1 3 k=1 4 j=i+1 3 i=1 4 i=1 + ∑ ∑ ∑ ∑(aijajkaki+ aikakjaji)Al 4 l=1 4 k=j+1 3 j=i+1 2 i=1 + [∑ ∑ ∑ ∑ (aijaji)(aklalk) 4 l=j+2 3 k=i+1 4 j=i+1 1 i=1 + ∑ ∑ ∑ ∑ (aijajkaklali + ailalkakjaji 4 l=j+1 4 k=i+1 3 j=i+1 1 i=1 )] (5) Overall performance value is calculated using the values of Ai’s and aij’s for each inventory item. Ai’s are obtained from the normalized decision matrix and aij’s are obtained from relative importance matrix (Table 2). Step 8- According to graph theory and matrix method, classification of inventory items based on the overall performance value arranged in descending order are shown in Table 3. IJAHP Article: Mallick, Sarkar, Das/A unified decision framework for inventory classification through graph theory International Journal of the Analytic Hierarchy Process 192 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.482 In order to obtain comparison, the same distribution of 10 class A, 14 class B and 23 class C items was maintained. Inventory items with an overall performance value of 17.20748099 and above were classified as class A items, those with scores of 15.50751142 and below were classified as class C items, while those with scores between 17.20748099 and 15.50751142 were classified as class B items. The ABC classification using proposed method as shown in Table 3 is compared with the result of traditional ABC and the AHP classification methods as reported in Flores et al. (2002). Table 3 A comparison of ABC classification A comparison of ABC classification as determined by: Item Reid (1987) using only a single criterion. Flores et al. (1992) using AHP for multiple criteria inventory classification. This paper using the hybrid system i.e. integrated graph theory and AHP for multiple criteria inventory classification. Annual Dollar Usage Rank ABC AHP (Weighted Score) Rank ABC Overall performance value Rank ABC S1 5840.64 1 A 0.59684 7 A 18.98048509 5 A S2 5670 2 A 0.86066 2 A 26.10663828 1 A S3 5037.12 3 A 0.71080 4 A 19.5330094 4 A S4 4769.56 4 A 0.08342 42 C 13.82213779 37 C S5 3478.8 5 A 0.41910 24 B 16.68158993 11 B S6 2936.67 6 A 0.40029 26 C 15.83281893 16 B S7 2820 7 A 0.39728 27 C 15.70537438 19 B S8 2640 8 A 0.26535 37 C 15.1261434 25 C S9 2423.52 9 A 0.82538 3 A 20.43548102 2 A S10 2407.5 10 A 0.50995 13 B 18.72709566 6 A S11 1075.2 11 B 0.50456 17 B 14.91889907 27 C S12 1043.5 12 B 0.50314 18 B 15.78723561 17 B S13 1038 13 B 0.87690 1 A 20.33433735 3 A S14 883.2 14 B 0.53502 12 B 17.36928478 8 A S15 854.4 15 B 0.59480 8 A 16.77468681 10 A S16 810 16 B 0.37207 29 C 14.79540034 29 C S17 703.68 17 B 0.42707 22 B 14.8344986 28 C S18 594 18 B 0.57539 9 A 16.66968694 12 B S19 570 19 B 0.50592 16 B 15.97287062 15 B IJAHP Article: Mallick, Sarkar, Das/A unified decision framework for inventory classification through graph theory International Journal of the Analytic Hierarchy Process 193 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.482 S20 467.6 20 B 0.44018 21 B 15.45752996 22 B S21 463.6 21 B 0.63900 6 A 16.29263253 13 B S22 455 22 B 0.44250 20 B 15.56395933 21 B S23 432.5 23 B 0.66237 5 A 17.51474999 7 A S24 398.4 24 B 0.57302 10 A 15.71356327 18 B S25 370.5 25 C 0.01771 47 C 11.96390314 47 C S26 338.4 26 C 0.15263 40 C 12.98776323 41 C S27 336.12 27 C 0.03521 45 C 12.53370856 43 C S28 313.6 28 C 0.37435 28 C 15.32649641 23 B S29 268.68 29 C 0.46347 19 B 16.83859629 9 A S30 224 30 C 0.02268 46 C 12.13135833 46 C S31 216 31 C 0.50975 14 B 16.17631119 14 B S32 212.08 32 C 0.50937 15 B 15.24071987 24 B S33 197.92 33 C 0.29309 33 C 14.23044817 34 C S34 190.89 34 C 0.41335 25 C 14.59149575 32 C S35 181.8 35 C 0.16044 38 C 13.28125951 39 C S36 163.28 36 C 0.57224 11 B 15.69489064 20 B S37 150 37 C 0.28485 34 C 13.89944772 35 C S38 134.8 38 C 0.37004 30 C 14.7489373 31 C S39 119.2 39 C 0.29574 32 C 14.34183744 33 C S40 103.36 40 C 0.36078 31 C 14.75658732 30 C S41 79.2 41 C 0.07482 44 C 12.14000067 45 C S42 75.4 42 C 0.08164 43 C 12.37195808 44 C S43 59.78 43 C 0.28339 35 C 13.84673594 36 C S44 48.3 44 C 0.15362 39 C 13.0396684 40 C S45 34.4 45 C 0.42138 23 B 14.96412365 26 C S46 28.8 46 C 0.14582 41 C 12.75973946 42 C S47 25.38 47 C 0.27461 36 C 13.49604159 38 C IJAHP Article: Mallick, Sarkar, Das/A unified decision framework for inventory classification through graph theory International Journal of the Analytic Hierarchy Process 194 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.482 Table 4 A comparison of Annual dollar Usage percentage of class A, B and C type of items for integrated Graph theory-AHP,traditional ABC and the AHP Class of items % of items each category Annual Dollar usage percentage Integrated Graph theory-AHP (Proposed method for multiple criteria inventory classification.) Traditional ABC (Reid (1987) using only a single criterion.) Flores et al. (1992) using AHP for multiple criteria inventory classification A 21 54 74 43 B 30 23 19 30 C 49 23 7 27 Comparative analysis of Annual Dollar Usage percentage of A, B and C type of items of three types of ABC analyses is shown in Table 4. Table 4 illustrates that 54% of the Annual Dollar Usage for 21% of item (A type) was obtained by the proposed method. In accordance with the traditional ABC by Reid (1987),where only one criteria has been considered, 30% of item (B type) accounts for 19% of Annual Dollar Usage. In consonance with Flores et al. (1992), 49% of items (type C) are responsible for the 27% Annual Dollar Usage. Therefore, it can be stated that for any organization, both inventory cost-control as well as multi-criteria decision making can be done more effectively by applying the proposed method from a materials management point of view. The suggested hybrid method envisages the proper controlling of materials (47 items) in comparison to the other two methods. 4. Conclusions In the present investigation, the combination of graph theory and the Analytic Hierarchy Process was applied for multi-criteria inventory classification. It may be noted that, as per the author’s purview, this kind of hybrid system has not been used earlier to classify inventory items. A case study on a hospital inventory management system for 47 disposable items used in a respiratory therapy unit was considered. The results obtained using the proposed method were found to be more precise than that of solely the AHP classification method. Hence, this hybrid method being a simple, fast, logical, systematic and consistent method, may be recommended for use as a decision making method for multi-criteria inventory classification. It can also be used for any type of decision-making situation. It enables more critical analysis than the common multi-attribute decision making methods since any number of quantitative and qualitative attributes can be considered. In the IJAHP Article: Mallick, Sarkar, Das/A unified decision framework for inventory classification through graph theory International Journal of the Analytic Hierarchy Process 195 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.482 permanent procedure, even a small variation in the attribute leads to a significant difference in the overall performance value making it convenient to rank the alternatives in a descending order with clear difference in the overall performance value. Moreover, the proposed procedure using the graphical representation not only enables the analysis of the alternatives, but also facilitates visualization of various criteria along with their interrelations. 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