Monte Carlo Simulation for Sparepart Inventory

Monte Carlo Simulation for Sparepart Inventory I Nyoman Pujawan 1, Niniet Indah Arvitrida 2, and Bathamas P. Asihanto Department of Industrial Engineering Sepuluh Nopember Institute of Technology Surabaya, Indonesia Email: pujawan@ie.its.ac.id 1 niniet@ie.its.ac.id 2 Abstract - Most inventory models are based on assumptions that demand follows a certain theoretical distribution (such as Normal or Poisson distribution). For items with relatively fast moving, such an assumption may hold true. However, for slow moving items such as spare parts, we rarely found that demand follows such theoretical distribution. Consequently, using standard inventory control models to set inventory parameters for spare parts would result in either low service level, high inventory level, or both. In this study we use Monte Carlo simulation to determine inventory control parameters for spare parts. We employ a well known periodic review base-stock inventory control models where the replenishment decisions are governed by maximum and minimum inventory levels. The initial solutions were generated using standard inventory control models. Monte Carlo Simulation is then used to evaluate the parameters against other neighboring values in terms of inventory cost and service level. We apply the procedure for cabin spare parts in an aircraft maintenance facility. Keywords: inventory management, spare parts, simulation. 1. INTRODUCTION Managing spare parts inventory has always been challenging for a number of (conflicting) reasons. First, demand for spare parts is often intermittent or lumpy making it difficult to forecast. Second, the required service level is high for most spare parts as unavailability would result in costly consequences such as lengthy facility breakdown or high costs associated with urgent purchase. Third, the price for a spare part could be very high making it less favorable to hold inventories. For many companies, inventory investment for spare parts is enormously large. An informal discussion with a gas exploration company in Indonesia reveals that the spare parts inventory value in this company is about US$ 47 million. If the annual inventory holding cost is about 30% then the spare parts cost this company about US$12 million per year. In addition, the turnover ratio of spare parts is usually very low (usually between 0.5 and 1 which means that on average one spare part is consumed every one to two years. Despite that managing spare parts inventory is critical to most companies, studies addressing this issue is fairly limited, in contrast to inventory for raw materials or finished goods that have received much attention from academics. One of the reasons why inventory management for spare parts received only little attention is because the demand pattern for spare parts is slow moving, often intermittent or lumpy (Teunter et al, 2010). As a consequence, they are difficult to forecast. Lumpy demand is characterized by many periods with zero demand (Pujawan & Kingsman, 2003). The inventory control models that have been developed for fast moving items are hardly ever appropriate for spare parts. There are a number of different stream of papers about spare parts inventory. One of the streams is dealing with forecasting. Croston (1972) is among the early papers proposing a method for forecasting demand of spare parts inventory. This model basically decomposes demand pattern into two components, i.e., the magnitude of positive demand and time between demands. This method is further discussed or developed by other authors such as Regattieri et al (2005) and Syntetos & Boylan (2005). The other group of authors deal with optimum setting of replenishment parameters such as reorder point and maximum inventory. The most popular method appears to be the (s, S) or its periodic review version (R, s, S). This model works as follows. In a review period, when inventory falls to less than or equal to s then place an order to bring the inventory level back to S. The determination of : Corresponding Author

optimum s and S values has been discussed by many authors, but most of them have assumed that demand follows Normal or Poisson distribution. Due to its demand being intermittentt or lumpy, the demand distribution for spare parts hardly follows any theoretical distribution. Consequently, using standard inventory control models to set inventory parameters for spare parts would result in either low service level, high inventory level, or both. In this study we use Monte Carlo simulation to determine inventory control parameters for spare parts. By using simulation, we can capture various demand characteristics including those which do not follow any theoretical distribution. According to Regattieri et al. (2005), the demand pattern of a material is characterized by two attributes, i.e., the average demand internal (ADI) which is the average time between occurrence of two successive demands and the coefficient of variation of the positive demand (CV). Large values of both ADI and CV indicated high demand uncertainty and hence, large inventory investment is required to achieve an acceptable service level. As shown by figure 1, most items have CV larger than 50% and ADI more than 1.5, meaning that the degree of lumpiness of those items is high. 2. CASE STUDY We do the case study in PT. GMF Aeroasia (will be referred to as GMF in the rest of this paper), an aircraft maintenance facility, a sister company of Garuda Indonesia. The GMF manages the maintenance, repair, and overhaul of aircraft from various airline companies. A critical activity within the maintenance, repair, and overhaul is the management of spare parts inventory. With the increasing variety of aircrafts and the fact that each aircraft consists of a very large number of components, the management of spare parts inventory is becoming so critical to achieve an efficient operations as well as high service level. In general, there are two major classification of spare parts related to an aircraft. The first is spare parts that are required for safety flight such as break, wheel, and many others. If any of these components does not working well, the aircraft can not or is not allowed to fly because it will pose high safety risks. The second classification is related to cabin materials. If any of the cabin materials is not functioning well, the aircraft is still able to fly, but it will compensate the convenience of the passengers. In this study we will focus on seat materials for B737-800 as one class of cabin materials. In general the cabin materials can be classified into six groups, namely components related to seat (such as hydro-lock, arm rest, in arm table), in flight entertainment (such as seat electronic box, video monitor, and harness), lavatory (such as portable water tank, toilet assembly), lighting, luggage bin and galley (such as oven, coffee maker, and boiler). The current situation in GMF shows that many of the cabin materials are on what they refer to as hot item list or HIL, meaning that those items are not available when needed. For B 737-800 alone, at the time this observation is made, there were 45 items on HIL status. Among those on the HIL status, the majority are for seat components. There are in total 94 items that belong to seat components of B737-800. Like other spare parts in general, we found that the demand pattern is lumpy for most of the seat materials. Figure 1 Demand pattern classification 2.1 The Use of Analytical Models Most inventory control models have been developed based on an assumption that demand follows a certain theoretical distribution. In particular, majority of the models have assumed that demand follows either normal or Poisson distribution. In reality, when demand is lumpy, such an assumption is hardly ever holds true. Consequently, the use of analytical models for lumpy demand often ends up with inappropriate inventory parameters. In this study we have used a periodic base-stock policy called (R, s, S) where R stands for interval between review periods, s is reorder point and S is maximum inventory position (Silver et al, 1998). The model reads as follows. During a review period, if inventory position falls below s then we have to order to bring inventory position back to S. Inventory position is on hand inventory plus on order minus backorder. Determination of optimum R, s, and S is not an easy task. All of these three parameters will affect both inventory costs and service level. One of the analytical methods often used for determining those parameters is called power approximation. However, this approximation does not always provide good results, especially when demand is lumpy. We have tried to use the power approximation to set inventory parameters for a couple of

spare parts. We then simulate the performance of those parameters to see if they can give good performance in terms of service level. One of our observations is that this approximation fails to achieve the intended service level for almost all situations. As shown in figure 2, the actual service level is always below the theoretical service level. Figure 2 - The actual service level (vertical) is always lower than the intended service level (horizontal) 2.2 The Use of Monte Carlo Simulation We have used the Monte Carlo simulation to evaluate the performance of the parameters from power approximation. As we believe that those parameters are rarely the best ones, we also evaluate other set of parameters close to those recommended by the power approximation. Since the demand for parameters is mostly slow moving, the choice of s and S values is relatively limited, making the simulation quite effective for finding the best set of parameters. The simulation procedure can be summarized as follows. First, based on historical data of 28 months, we model the empirical demand distribution. As an illustration, we will use item PN102-504 (we have made an arbitrary part number, different from the actual number used by the company). The historical data shows that 71% of months has no demand for this item. About 3.5% has a demand of 1 or 2, 10.7% has a demand of 3 or 4 and the rest is 5 or 6. To generate the demand, a uniform (0,1) random number is use. If random number falls between 0 and 0.710 then demand is set to be zero. If random number is between 0.710 and 0.745 then demand is either 1 or 2 with an equal probability, and so on. Let D(t) = demand in period t, obtained through the above procedure. We then do the following steps: Update inventory: I(t) = I(t-1) + R(t) D(t) B(t) Update on-hand inventory: O(t) = max (0, I(t)) Update backlog: B(t) = max (0, -I(t)) Calculate inventory position: IP(t) = I(t) + O(t) Determine ordering decision: If IP(t) < s, Q = S IP(t), otherwise Q = 0 Update on order quantity: O(t) = O(t-1)+Q(t)-R(t) Determine order receipt schedule: R(t+L) = Q Where I(t)is inventory at the end of period t, O(t) is on hand inventory, B(t) is backlog in period t, R(t) is schedule receive in period t, Q(t) is order quantity in period t, and L lead time. As an illustration, we will use one part called PN30-802 (we have made an arbitrary part number, different from the actual number used by the company). The baseline parameters obtained from the power approximation is s = 5 and S = 6, with an intention to achieve a service level of 95%. We then do a parameter search by lowering and increasing both s and S values around their baselines. The result is shown in table 1. Assuming that service level 95% is required, then we have to exclude those alternatives with actual service level below 95%. Figure 3 shows that the remaining candidates are those at the right side of the vertical dotted line. Among the remaining candidates, we then have to again exclude those that are not at the efficient frontier. Efficient frontier can simply be defined as a curve connecting the most efficient candidates. The most efficient candidates can be visually seen as those at the outer side of group. By looking at figure 3, there appears to be only 4 candidates are remaining. Now, the decision maker has to make a choice among the 4 remaining candidates. Since the service level for each part should be dependent on part criticality, choosing a solution with a service level much higher than the initial target would be inappropriate. Thus, as long as the service level target has been set carefully with consideration of item criticality then the simple rule of thumb is to choose a candidate with lowest cost in the efficient frontier. Efficient frontier Figure 3 - Service level (horizontal) and cost (vertical) for each candidate

3. DEVELOPMENT OF DECISION SUPPORT SYSTEM A spreadsheet-based decision support system (DSS) has been developed to assist decision maker in performing the above procedure until the best candidate is obtained. We have tested the DSS with the data of 20 seat components. The tests show that the DSS has worked well and practical enough to use. 4. CONCLUDING REMARKS In this study, a decision support system has been developed to obtain the best replenishment policy for spare parts. As the demand pattern for most spare parts typically is lumpy, it is hardly ever true that it follows any theoretical distribution such as Normal or Poisson. The analytical models for inventory policy is normally assumed that the demand follows Normal, Poisson, or any other standard distribution. This has lead this study to use simulation to obtain the best inventory parameters. We use a periodic review (R, s, S) inventory model and the parameter s (the reorder point) and S (the maximum stock) is obtained through simulation. The results suggest that the values of s and S obtained by simulation are always better than those from analytical models in terms of costs and service level. REFERENCES Archibald, B. C., dan Silver, E. A. (1978). (s, S) policies under continuous review and discrete compound Poisson demand. Management Science 24, pp. 899 904. Babai, M. Z., Syntetos, A. A., dan Teunter, R. H. (2010). On the empirical performance of (T, s, S) heuristics. European Journal of Operational Research 202, pp. 466 472. Croston, J. D. (1972). Forecasting and stock control for intermittent demand. Operational Research Quarterly 23, pp. 289 303. Huiskonen, J. (2001). Maintenance spare parts logistics: Special characteristics and strategic choices. International Journal of Production Economics 71, pp. 125 133. Nenes, G., Panagiotidou, S., dan Tagaras, G. (2010). Inventory Management of multiple items with irregular demand: A case study. European Journal of Operational Research 205, pp. 313 324. Pujawan, I N. dan Kingsman, B. G. (2003). Properties of lot sizing rules under lumpy demand. International Journal of Production Economics 88, pp. 295-307. Regattieri, A., Gamberi, M., Gamberini, R., dan Manzini, R. (2005). Managing lumpy demand for aircraft spare parts. Journal of Air Transport Management 11, pp. 426 431. Rustenburg, W. D., van Houtum, G. J., dan Zijm, W. H. M. (2001). Spare parts management at complex technology-based organizations: An agenda for research. International Journal of Production Economics 71, pp. 177 193. Syntetos, A. A., dan Boylan, J. E. (2005). The accuracy of intermitten demand estimates. International Journal of Forecasting 21, pp. 303 314. Teunter, R. H., Syntetos, A. A., dan Babai, M. Z. (2010). Determining order-up-to levels under periodic review for compound binomial (intermittent) demand. European Journal of Operational Research 203, pp. 619 624. AUTHOR BIOGRAPHIES Nyoman Pujawan is professor of supply chain engineering at the Department of Industrial Engineering, Sepuluh Nopember Institute of Technology (ITS), Surabaya - Indonesia. He received his PhD from Lancaster University, UK. He has published papers in such journals as European Journal of Operational Research, International Journal of Production Economics, Production Planning and Control, Business Process Management Journal, and many others. He is the Editor-in-Chief of Operations and Supply Chain Management: An International Journal. He can be reached at< pujawan@ie.its.ac.id>. Niniet Indah Arvitrida is a Lecturer at the Department of Industrial Engineering, Sepuluh Nopember Institute of Technology, Surabaya - Indonesia. She received her bachelor and master degrees, both in Industrial Engineering, from ITS. Bathamas P. Asihanto graduated from the Department of Industrial Engineering, Sepuluh Nopember Institute of Technology, Surabaya - Indonesia in 2010.

Table 1 Simulation results for different parameters s and S for item PN30-802. INPUT RESULTS Experiment Theoretical Actual Holding Order Shortage Total No. S s SL SL Cost Cost Cost Cost 1 8 92,50% 6,00 88,50% $138.342 $6.668 $68.376 $213.386 2 9 92,50% 6,00 91,74% $162.522 $6.337 $45.562 $214.421 3 10 92,50% 6,00 93,20% $178.029 $5.368 $37.686 $221.083 4 11 92,50% 6,00 94,47% $193.524 $4.634 $32.450 $230.608 5 12 92,50% 6,00 94,64% $204.378 $3.892 $32.780 $241.050 6 9 95,00% 7,00 91,97% $164.961 $6.668 $43.010 $214.639 7 10 95,00% 7,00 94,50% $190.116 $6.337 $27.346 $223.799 8 11 95,00% 7,00 95,68% $206.061 $5.368 $22.682 $234.111 9 12 95,00% 7,00 96,42% $221.937 $4.634 $20.240 $246.811 10 13 95,00% 7,00 96,02% $232.845 $3.892 $20.966 $257.703 11 10 97,50% 8,00 94,80% $192.624 $6.668 $25.300 $224.592 12 11 97,50% 8,00 96,82% $218.538 $6.337 $15.202 $240.077 13 12 97,50% 8,00 97,58% $234.840 $5.368 $13.156 $253.364 14 13 97,50% 8,00 97,53% $250.938 $4.634 $12.342 $267.914 15 14 97,50% 8,00 97,49% $261.726 $3.892 $12.188 $277.806 16 11 99,00% 9,00 97,07% $221.139 $6.668 $13.838 $241.645 17 12 99,00% 9,00 98,51% $247.659 $6.337 $8.184 $262.180 18 13 99,00% 9,00 98,35% $264.189 $5.368 $7.810 $277.367 19 14 99,00% 9,00 98,49% $280.272 $4.634 $6.886 $291.792 20 15 99,00% 9,00 98,53% $291.048 $3.892 $6.644 $301.584