Fuzzy Continuous Review Inventory Model Using ABC Multi-Criteria Classification Approach: a Single Case Study

. Inventory is considered as the most expensive, yet important,to any companies. It representsapproximately 50% of the total investment. Inventory cost has become one of the majorcontributorsto inefficiency, therefore it should be managed effectively. This study aims to propose an alternative inventory model, by using ABC multi-criteria classification approach to minimize total cost. By combining FANP (Fuzzy Analytical Network Process) and TOPSIS (Technique of Order Preferences by Similarity to the Ideal Solution), the ABC multi-criteria classification approach identified 12 items of 69 inventory items as “outstanding important class” that contributed to 80% total inventory cost. This finding is then used as the basis to determine the proposed continuous review inventory model.This study found that by using fuzzy trapezoidal cost, the inventory turnover ratio can be increased, and inventory cost can be decreased by 78% for each item in “class A” inventory.


Invento
inventory classification is the optimal number and it is familiar to managers. The number of inventory items does not affect the determination or a determination of the number of classes. Rezaei et al. (2010), Torabi et al. (2012), Keskin et al. (2013) classify the raw materials amounting to an average of 50 items into three classes; Kabir et al. (2012) classifies 315 raw materials of construction industry into three classes; and Kartal et al. (2012) classifies 715 raw materials of automotive industry into three classes as well. Therefore, this study classified the inventory items into three classes: A (outstandingly important), B (of average importance), and C (relatively unimportant).
Determination of the borders between the classes affected by the classification criteria and classification techniques (Kampen et al., 2012). The traditional ABC classification has generally been bases on just one criterion, the annual dollar usage. However, using single criteria are irrelevant in real life. Yu (2010) and Keskin et al. (2013) revealed that using the annual dollar usage criterion only might create problems of significant financial loss because there are other important criteria that should be considering such as lead-time, criticality, durability, and so on. Thus, to get the inventory classification criteria, which are relevant to the subject of this research, the study gives authority on managers to choose their own criteria. To assist the selection criteria process, the master list of criteria that have been used in academic researchers are provided.
Multi-criteria inventory classification is a part of Multi-criteria Decision Making (MCDM) problems. Kampen et al. (2012) distinguished this MCDM technique into two types based on the knowledge source: the statistical and judgmental techniques. Statistical techniques knowledge sources are based on data of a number of inventory items characteristics. Yu (2010) and Fernandez et al. (2011) used the statistical techniques in classify inventory items. They used metaheuristic approach. The advantage of statistical techniques is the classification result spared from subjectivity. However, these techniques have a high level of complexity. The application of these techniques could be cumbersome for inventory managers especially there is no participation of the manager in it (Rezaei et al., 2010;Kampen et al., 2012).
In contrast to statistical techniques, the judgemental techniques involve the opinions of manager especially in the determination of criteria weights. There are some of the judgemental techniques proposed by previous researchers such as Technique of Order Preferences by Similarity to The Ideal Solution (TOPSIS)by Bhattacharya et al. (2007), Fuzzy Analytic Hierarchy Process (FAHP) by Kabir et al. (2012), combination of Fuzzy Delphi and FAHP by Kabir et al. (2013), Simple Additive Weighting (SAW) by Kartal et al. (2013), and Fuzzy Analytic Network Process (FANP) by Kiris (2013).
The main advantage of TOPSIS and SAW technique is that these techniques are practical and suitable for a relatively large amount of inventory. However, these two techniques do not have provisions to determine the weights of criteria. Compared to SAW, TOPSIS has advantages in determining the composite priority weight of alternatives that takes into account the closest distance from the positive ideal solution. The main advantage of AHP is this technique has consistency consideration in determining the weights of each criterion in which it can cover up the weakness of TOPSIS. However, the FAHP is still having an element of subjectivity and assumption that each criterion is independent. Therefore, the extended version of FAHP technique, namely FANP can cover up the weakness of FAHP because this technique considers the dependency factor among criteria and it produces a more stable weight than FAHP. Based on the advantages and disadvantages of the above techniques, this study proposes a combination of FANP and TOPSIS to classify the inventory items.

Inventory Model
There are two fundamental decisions that should be determined when designing an inventory model. They are when should an order be placed and how much should be ordered (Nahmias, 2004: 193;Balakrishnan et al., 2011: 12-3 -12-4). The complexity of the resulting inventory model depends upon the assumptions one makes about the various variables of the system. The variables are demand, lead-time, excess demand, inventory costs, and review time (Nahmias, 2004: 193-197).The determination of the characteristics of these variables needs to adjust to the research subject because the proposed model is designing not only to describe the situation of a system but also provides the best answer to the inventory problem.
Demand variable is distinguished into two types: known demand (deterministic model) and uncertain (stochastic model) (Nahmias, 2004:196). According to the condition of the subject of research, this study uses uncertain demand model. It means that the exact number of future demand cannot be predicted at the beginning. The uncertain demand variable is influenced by lead-time variable (Nahmias, 2004:197). In this case, although the future demand cannot be predicted at the beginning, one's past experience can provide useful information for planning. The random demand on the past can be used to estimate its lead-time demand probability distribution. Previous researchers generally assumed the lead time demand distribution is uniform for all inventory items, such as entirely normal distribution (Silver et al., 2011, Joshi et al., 2011, Zheng et al., 2011Sadi-Nezhad et al., 2011) or uniform and exponential distribution (Taleizaideh et al., 2013) without statistical testing, whereas the different type of distribution affects the value of decision variables. For this reason, the study proposes to examine the distribution type of lead-time demand before formulating the inventory model.
Another important characteristic that determines the inventory model formulation is how the system reacts to excess demand (Nahmias, 2004: 197). In accordance with the subject of research, this study used to apply backorder systems in which the system will satisfy the customer's need in the future.
Determination of inventory models, especially on what variable to be decided is influenced by a review or the review time variable.
Review time variables distinguished into two types, continuous review and periodic review (Sipper et al., 1997: 211;Nahmias, 2004: 244). Determining the review time variable should be adjusted to the importance of inventory items on company performance (Nahmias, 2004: 276;Motadel et al., 2012). Previous researchers researchers that considered the importance of inventory item in the review type determination are still limited. Aisyati et al. (2013) for example, who designed the continuous review inventory model for class A and B items. Continuous review inventory model considered suitable for class A and B items that have high costs because this model gives the amount of safety stock that is smaller than the periodic review models. In addition, based on the characteristic of continuous review model where fixed quantity ordering, the supplier are easy to predict the order quantity. For these reasons, in accordance with the objective of this study, we propose to use this continuous review inventory model.
Variable that becomes optimization criteria in designing an inventory model is inventory cost. In real life, estimating the exact value of cost variables may not be possible (Sadi-Nezhad et al., 2011;Prasath et al., 2012;Jaggi et al., 2012;Dutta et al., 2012). Inspired by the concept of fuzzy sense, this research proposes to adopt this concept in the cost variable. In the fuzzy sense, two main things   One such dependence matrix of the pair wise comparisonis is shown in Table 3. It shows the result of criticality criteria as the controlling criterion over other criterions. The semi-structured interview question is "What criterion is more influential to the criticality criterion: cost or procurement process? How the level of influence of this criterion compared to the other criterion with respect to the criticality?" (1,1,1) (5/2,3,7/2) 0.75 Procurement process (C4) (2/7,1/3,2/5) (1,1,1) 0.25 λ = 0.99 Global weights of criteria are computed by multiplying the dependence matrix of the criteria and the local weights of criteria. Global weights of sub-criteria are calculated by multiplying the local weights of subcriteria and the global weights of the related criteria. The global weights are shown in Table 4. Global weights of sub-criteria and data for each inventory item based on the subcriteria were calculated by using TOPSIS to obtain composite priority weight for each inventory item. The inventory items are then classified using traditional ABC classification technique. After careful consideration, the authors of this paper and the management of the company decided that the borders between classes are determined based on Pareto Principle. Therefore, the borders between classes were derived from the following basis. Class A involves 80% of the cumulative composite priority weights. Class B involves 15% of the cumulative composite priority weights while 5% of total composite priority weights belong to class C. Table 5 shows the classification of 69 inventory items of the company.

The Proposed Inventory Model
Based on the inventory classification in Table 5, the 12 Class A items will be designed for their inventory model. Using Arena Input Analyzer software yields various types of lead time demand distribution of the items: normal distribution for item S-54, S-8, S-14, S-15 and S-39; uniform distribution for item S-1, S-42, S-62 and S-55; lognormal distribution for item S-3 and S-28; and exponential distribution for item S-2.
Decision support system was designed to support decision makers in using the proposed model efficiently. It was designed using Microsoft Excel-VBA software that integrates Microsoft Visual Basic and Microsoft Excel. By translating the fuzzy formulation of continuous review inventory model in Visual Basic programming code, the decision support system was successfully constructed. Comparison of Inventory Turnover Ratio (ITR) and inventory total cost between the existing and proposed model is shown in Table 7.

Discussion
The fuzzy continuous review inventory model using multi-criteria ABC classification approaches is presented in this study to the answer the research question. In classifying the 69 inventory items, ten relevant subcriteria to the subject of research had been selected. Using the FANP techniques, this research found that annual demand subcriterion has the highest importance weight in the inventory. It means that the use of dollar usage in traditional ABC classification proved to be irrelevant in real life. This research also found pack size sub-criterion as a new criterion in the ABC multi-criteria classification research.
The ABC classification based on ten subcriteria in this research was constructed three inventory classes. The result of this research (as shown in Table 5) by using Pareto principle shows that among 69 items, 12 items (17% of all items) are identified as class A or outstandingly important group, 22 items as class B (32% of all items) as class B or average important group, and the remaining 35 items as class C or relatively unimportant group as a basis for a control scheme. In terms of the number of items per class, this research shows that inventory items that have the highest priority weight will go into class A while inventory items that have the lowest weight will go into the class C and it contains 51% of all items.
These results provide a recommendation for a manager or the manager to start implementing the classification system on their company, so attentions to each inventory item given proportionally. Fuzzy continuous review inventory model has also presented in this research. Before design the model, it was found that from 12 class A items, four items have lead time demand distribution that shaped uniform, five items have normal distribution, two items have lognormal distribution, and an item has exponential distribution. The various types of lead-time demand distributions make this research relevant to the real life situations.
This research also proved that the previous researchers that generally used "assumptions" in determining the type of distribution is a less relevant method to the real life situations. For example, Godwin et al. (2013) who designed a continuous review inventory model in a company in Nigeria assumes all of the inventory items had uniform lead-time demand distribution, or Sadi-Nezhad et al. (2011) who designed the periodic and continuous review inventory model on transformer manufacturing in Iran assume that the lead-time demand distribution is entirely normal. The assumption of lead-time demand distribution is becoming irrelevant because the differences in the distribution type will affect the mathematical formulation of the expected number of shortage and the value of decision variables.
In developing a practice model for the manager, a decision support system using Microsoft Excel-VBAis presented as the main result of fuzzy continuous review inventory model in this research (as shown in Figure 3). By using this decision support system, the manager can make rapid and accurate decisions. The value of decision variables: optimal order quantity (Q) reorder points (R), inventory total cost have been determined (as shown in Table 6). This research found that reorder point value of each inventory item has a higher value than expected lead-time demand. This means that safety stocks were prepared in this proposed model.
Comparison of Inventory Turnover Ratio (ITR) and inventory total cost between the existing and proposed model have been provided (as shown in Table 7). ITR comparison shows that the proposed model has a higher ratio than the existing model for each Class A item. According to Rao et al. (2009) and Bahagia (2006: 42), incremental of ITR showed an improvement on inventory management in reducing the overstocking level because of the increase in inventory turnover per cycle. Inventory total cost comparison shows that the proposed model has a lower total cost than the existing model for each Class A item. This result also shows that the proposed continuous review inventory model gives average savings of 77.7% compared to an existing model that was used by the company.
Finally, the ABC multi-criteria classification approach to design appropriate inventory model was supported by Aisyati et al. (2013) research. Aisyati et al. (2013) who used a continuous review inventory model for class A and B found that there are several items show that existing model performs better than continuous review model or it can be said that the percentage of saving is negative. In their analysis, Aisyati et al. (2013) explained that the continuous review model might be failing to result in or result from better inventory model since the demand of the items is too lumpy or it has pattern of Poisson lead-time demand distribution. Aisyati et al. (2013) also recommended that this Poisson demand could be managed by periodic review inventory model. This finding is caused by the classification technique. Aisyati et al. (2013) used traditional ABC classification in their research. It means that the class A items are high-value inventory based on the dollar usage only. Consequently, although the demand of a high-cost item is too lumpy, it will be classified as class A. Therefore, the findings of this research, the positive percentage of saving of each class an items prove that the design of the inventory model using ABC multi-criteria classification approach would be more effective in saving the inventory cost than using traditional ABC classification.

Conclusion
Based on the study, it can be concluded that Combination of Fuzzy Analytic Network Process (FANP) and Technique of Order Preferences by Similarity to the Ideal Solution (TOPSIS) in ABC multi-criteria classification techniques identified 12 items of 69 inventory items as class A (outstandingly important class. They contribute to 80% total inventory cost. Furthermore, the appropriate inventory model for class A items is fuzzy continuous review inventory model using trapezoidal fuzzy numbers and the statistical testing of lead time demand distribution. This proposed inventory model increase the inventory turn ratio and reduce the inventory total cost with average savings of 78% for each inventory item. These findings show that the proposed model is feasible to be implemented in the company.

Future Research
Future research can look at designing the appropriate inventory control model for more than one company or more than one class item. An inventory control model that also consider several factors such the decay factor, partial delivery from suppliers, and warehouse capacity constraint can also be designed in future research. Future research can also look to design an inventory-model or the inventory-model decision support system using demand-forecasting approach.

Contribution
This researched expected to contribute to the development of inventory management in real situations. The combination of FANP, TOPSIS, and traditional ABC classification technique found as an effective combination technique to classify the inventory items, especially a relatively large amount of inventory. a previous or the previous study, the graded mean integral representation defuzzification technique and trapezoidal membership function were just applied the indeterministic or an indeterministic model. Therefore, applying this defuzzification technique and membership function is the contribution of this research in stochastic inventory model environment. This research shows that the lead-time demand distribution is not always normally distributed and it needs to be testing before design the mathematical formulation of inventory model. Finally, the research that proposes a fuzzy continuous review inventory model with various types of lead-time demand distribution can also contribute to the mathematical formulation of inventory control models.
The other contribution of this research is that the inventory classification result gives a suggestion for the manager to develop an inventory control policy based on the importance level of the items. By implementing the proposed inventory model in this research, the enterprise should be able to reduce their overstocking level and inventory total cost. This research also provided an inventory model decision support system for the company so managers can make rapid and accurate decisions.