Simulation approach in stock control of products with sporadic demand

Transcription

1 Simulation approach in stock control of products with sporadic demand Jakub Dyntar, Eva Kemrová, Ivan Gros The stock management of products with sporadic demand is one of the main problems for many entrepreneurs. It is possible to find cases of the sporadic demand in the area of car, aircraft and unique assembly lines spare parts manufacturing and distribution. The problems with assessing the sporadic demand are caused not only by significant variability and a relatively small demand but, above all, by prolonged periods with no demand at all. Classical forecasting methods (for example exponential smoothing, moving average methods, regression analysis, etc.) used in common supply management systems are ineffective when applied to sporadic demand for mainly the following reasons: Classical methods do not take into account the importance of zero demand periods. Classical methods are not focused on the distribution function forecast of demand during the order lead time period, which is very important for the effective flow of such materials. Application of inappropriate methods in sporadic demand product stock management leads to insufficient stock and the inability to fulfill orders consequently causing significant economic loss. Situations leading to high stocks of these products have a similarly negative effect on the management efficiency. The importance of forecasting influence is obvious from Figure 1, showing the typical structure of spare parts distribution system. The distributor has to maintain high stock levels in order to be able to meet orders of repair shops etc., because lead times for spare parts delivery reorders form manufacturers are relative long. Low increase in accuracy of forecasting leads in the above mentioned system to significant decrease of stock levels. Fig.1: Supply system Spare parts, medicaments or capital goods manufacturers Lead time 1-3 months Distributors Stock level??? Order level??? Lead time 1-3 days Service Part of the research plan MSM Jakub Dyntar, MSc.,PhD., Institute of Chemical Technology Prague, jakub.dyntar@vscht.cz Eva Kemrová, MSc., Institute of Chemical Technology Prague, eva.kemrova@vscht.cz Prof. Ivan Gros, MSc.,CSc., Institute of Chemical Technology Prague, ivan.gros@vscht.cz 1

2 Introduction Croston s method and its modifications are the most commonly used methods in sporadic demand of product stock management systems. This method eliminates the drawbacks of classical exponential smoothing and secures sufficient stock levels during order lead time period. The advantage of this method is its reliability and robustness, but also the relative simplicity in computer processing. Croston s method solves only the question of the reorder point, i.e. when to demand restocking in order to remove the possibility of stock-out. The method does not solve the problem of restocking delivery volume and the mechanics of ordering. The questions are how to refill stocks and what level of restocking deliveries to implement in order to secure economic efficiency while still maintaining demanded service levels. One of the promising ways of solving stated problems is to apply the dynamic simulation method. The authors workplace has had experience in implementing this method even in other areas of management. The aim of this article is to introduce sporadic demand product stock management method based on dynamic simulation, which would offer simple and easily interpretable answers on basic questions connected to effective stock management, which are: reorder stock level assessment, replenishment orders volume assessment, choice of appropriate ordering manner, optimal stock level assessment. 1 Theoretical background of research The majority of common stock management systems utilize restocking level as the leading variable. This variable represents the level of stock capable, with certain probability, of meeting the demand during the time period required for order fulfillment. The calculation of reorder point is often based on the average demand and its variable assessment using forecasting methods. Further on we will sum up forecasting methods used for reorder stock level determination. 1.1 Exponential smoothing Simple exponential smoothing (Brown, 1959), alternatively exponential smoothing for time series with trend (Holt, 1957) or seasonal fluctuations (Winters, 1960) belong to the most commonly used demand forecasting methods in common stock management systems. These methods provide unsatisfactory results when applied to sporadic demand, the main reason being the inability to take note of the zero demand periods importance. Exponential smoothing model can be described by the following equation system: e t = y t y t-1, (1) y t = y t-1 e t, (2) m t = (1-) m t-1 + e t, (3) y t = demand in time t 2

3 y t = average demand forecast in time t e t = forecast error m t = average deviation of forecast deficiency Demand forecast, equations (1) and (2) is therefore weighted average of past demand volume, is forecast coefficient, originally inverted value of time series length used for demand forecast. Equation (4) than determines the calculation of reorder point level R t : R t = y t + k m t, (4) k = safety factor dependent on demand distribution type 1.2 Croston s method Croston (1972) suggested modification of exponential smoothing for sporadic demand product time series. The core of this method is not only the estimation of average demand volume, but also estimation of time interval length between two non-zero demands. Following set of equations describes Croston s method: e t = y t z t-1, (5) z t = z t-1 e t, (6) m t = (1-) m t-1 + e t, (7) p t = p t-1 (1 )+ q, (8) y t = z t / p t, (9) R t = y t + k m t, (10) q = 1, (11) q = q + 1, (12) z t = the average demand forecast in time t p t = the estimation of time interval length between two non-zero demands q = number of periods between two non-zero demand periods The difference between Croston s approach and exponential smoothing is that the estimation of average demand volume takes place only in the non-zero demand periods. If the demand equals zero, average demand volume is the same as in the previous period. Rao (1973) pointed out the mistake in deriving some of Croston s method attributes, without any effect on the model described by equations (5)-(12). Many authors have proven better results of Croston s method when compared to exponential smoothing. Willemain et al. (1994) compared the efficiency of Croston s method and exponential 3

4 smoothing and found that Croston s method achieves better results, even though the benefit was insignificant in some cases. Similar results can be found in the work of Johnston and Boylan (1996), who also pointed out that Croston s method leads to better results if the average interval between non-zero demand periods is higher than Sani and Kingsman (1997) have tested various methods of demand forecasting on real data from spare parts warehouse in Great Britain and found that best result are achieved by using the method of moving average followed by Croston s method. Their study is extremely valuable because it is one of the few works focused on the economic efficiency of the supplying process. 1.3 Croston s method modifications Even with its positive attributes, Croston s method suffers a major drawback. Syntetos and Boylan (2001) have noted that the demand volume estimation is positively deviated and have suggested a modification of Croston s method. This modification consists in adjusting the equation (9): y t = (1-/2) z t / p t (13) Improvement of the methods efficiency has been proven by Syntetos and Boylan (2005, 2006) or Syntetos, Boylan and Croston (2005). Levén and Segersted (2004) have tried to create an universal approach applicable to common and sporadic demand by modifying equation (9) of original Croston s method and designed equation for estimating average demand volume: y t = z t / p t + (1-) y t-1 (14) However, this modification leads to even higher positive deviations, proven in the work of Teunter and Sani (2009). These authors have pointed out that Syntetos and Boylan s modification removes the positive deviation of the original Croston s method, but in some cases their approach can lead to negative deviation, which they called dumping effect. Teunter and Sani point out that Croston s methods give best results when the portion of zero demand periods is relatively low, while Synteto s and Boylan s modification gives good results in cases a high portion of zero demand periods occurs. Based on Synteto s and Boylan s work, Teunter and Sani designed their own modification of Croston s method (9): y t = (1-/2) z t / (p t - /2) (15) Testing their modification on randomly generated data and comparing it with Croston s method, Teunter and Sani arrived at several interesting conclusions. Firstly they managed to prove that the average demand volume estimation described by equation (15) is not deviated. Furthermore, the deviations of original and modified Croston s method were proved to be caused by the probability of demand and forecast coefficient choice, while distribution type and variability of demand volume is insignificant. Low deviation of the method also suppresses the necessity of time series classification into groups, as suggested by Syntetos, Boylan and Croston (2005), significantly simplifying the method for computer processing 4

5 Pravděpodobnost[%] Probability 1.4 Simulation method bootstrapping In 2004 Smart, Willemain and Schwarz have introduced a brand new approach in sporadic demand product stock management. Their simulation method, known also as bootstrapping is not aimed at the average demand volume forecast like Croston s method and its modifications, but it estimates the distribution function of these volumes. The principle of this method is simple. From timeline obtained in the past, random k-sets of demands are chosen, k is the order lead time length. Sums of k generated values are created by theoretically possible demanded volumes during the order lead time term. If a sufficient number of generated k-sets is available, it is possible to create demand frequency distribution during order lead time term and its distribution function. Reorder point level for required level is then easily identified on the x axis. The example of demand distribution function during order lead time term and reorder point identification for 2 periods and required service level 98% is presented in Figure 2: Fig. 2: Demand distribution function during order fulfillment term 100% Distribuční Distribution funkce function poptávek (LeadTime (TVO = 2) = 2) 80% 60% 40% Reorder point 20% 0% Demand Poptávka [Pieces] [ks] Smart s method is simple and fitting for computer processing. Unfortunately its contribution has not been sufficiently proved, pointed out by Gardner and Koehler (2005) in their commentary. 1.5 Dynamic simulation Dynamic simulation presents an approach capable of taking into account various random factors and complex logic connections, which mathematic models are able to describe only to a limited measure. Simulation can be described as the creation of a logic-mathematical model of the real object, its aim being the description of object, determination of its function and estimation of its future behavior. The object simulation model, created by computer, enables the user to estimate system behavior during internal and external condition changes, optimize process regarding set criteria (profit, costs, reliability, etc.), to compare various alternatives of the process arrangement and choose an arrangement with appropriate efficiency. Computer model removes the risk of negative impact on real systems and provides required values, matching the aims of the simulation study. The main advantages of simulations can be summarized by the following points: 5

6 Simulation allows user to test suggested variants in advance without the necessity of allocating resources for their implementation. Simulation allows user to slow down or accelerate time. Simulation helps to find the reasons for the phenomena taking place and enables its detailed study. Simulation model offers the possibility of creating scenarios and provides answers to questions what happens if. Simulation helps to verify efficiency of planned investments before their actual realization. There are many specialized software applications based on dynamic simulation. We have good working experiences with products such as Witness or SIMUL8, providing an environment for simulation of even complex models. The core of these software products is a set of predefined elements, connected by logical parameters generated in a simple programming language. The advantage of these applications is the possibility of visualization of simulated systems. However, less complex problems can be successfully solved using Visual Basic for Applications, a part of MS Excel. 2 Dynamic simulation of stock management 2.1 Basic model Dynamic simulation of stock management is based on the recap of past warehouse stock movement under conditions of the chosen stock management system. The simulation model input is past demand time series of product and order lead time term. Stock management system is described by stock replenishment system, i.e. when to generate replenishment order and how to determine its size. Warehouse stock movements are the fulfilled needs of customers (stock decrease) and replenishment order arrivals (stock increase). The entire system of stock movement for automobile spare part demand time series is shown in Table 1. Let us consider the order lead time term of the length of 2 periods and stock management system with constant replenishment order of 5 pieces and reorder point of 2 pieces. Initial stock of given stock product in period 1 is 2 pieces. Tab. 1: Spare part movement in Q-System of stock management Period t Starting stock P t [Pieces] Demand S t [Pieces] Generate order Q [Pieces] 5 5 Order arrival O t [Pieces] 5 Missing amount C t [Pices] Final stock level K t [Pices] Starting state of system in each period is, with the exception of period 1, described by final stock level of preceding period. Order generated in the period corresponding to order lead time term is added to that of the stock and then appropriate demand is subtracted. The following step is to find out if it is necessary to create a new order. From the logic of the chosen system it is 6

7 obvious that the order will be generated in case the difference between starting stock level and demand increased by eventual replenishment order arrival is below the reorder point. With the exception of cases when replenishment order was generated in preceding periods and are being realized. Final stock level is used in the next step as the starting state of the system and the entire calculation is repeated. It is obvious that for set parameters (reorder point= 2 pieces, replenishment order = 5 pieces) 2 replenishment orders would be generated in period t=1 and t=8 in this stock management system. The key point of dynamic simulation is the choice of the stock management system and determination of optimal level of parameters, representing the system. If we are to decide if the chosen solution meets required efficiency, we need to determine the set of solutions (i.e. group of stock management systems) and criteria of evaluation. This problem will be examined more closely. 2.2 Stock management system choice Three basic stock management systems are described in literature. Their main characteristics are (Winston, 1994): Q-system, reorder point model with constant order quantity Operating parameter of this system is reorder point, while replenishment order size is constant. Order is generated if stock level, represented by final stock level is below reorder point. P- system, periodic review inventory system with upper replenish level and constant reorder period In P-system orders are placed at intervals with constant length, while replenishment order size is calculated by: Q = x h K t, (16) x h = upper reorder level K t = warehouse stock level in time t PQ-system PQ-system combines both mentioned systems. Term of order generation is based on reorder point level, replenishment order size is calculated by equation (16). Operating parameters of mentioned stock management systems are shown in Table 2: Tab. 2: Operating parameters of stock management systems System Q PQ P Operating parameters Reorder point, constant replenishment order Reorder point, dynamic replenishment order size Ordering interval length, dynamic replenishment order size 7

8 2.3 Simulation efficiency evaluation Let us find an answer to the question of how to determine whether the chosen solution represented by chosen stock management system and combination of decision parameters will secure required efficiency of the whole system. The first step will be the survey weather the customer demands were fulfilled completely or not in each period. If we assume the situation described it Table 1, missing amount C t in period t will be defined as: and total missing amount C: C t = S t O t P t if S t > O t + P t, (17) C T t1 To assess fulfillment of customers demands we use the simple indicator service level SL: Ct (18) SL = (1 - C/S)100%, (19) S T S t t1 = total demanded quantity in period 1,2,... T Therefore service level indicator means what percentage of total demanded quantity can be immediately released from stock. To calculate economic efficiency of chosen system we can use stock purchasing and maintenance costs (Winston, 1994) and taking into account a possible penalization for inability to fulfill required service level. Stock purchasing costs are expressed as a function of viable orders number: N o = o n o, (20) o = number of realized orders n o = costs for one order Maintenance stock level costs can be determined as a function of average stock level x p : T = period length c = product unite price N s = x p T c n s, (21) n s = maintenance stock costs as % of average stock in monetary units in period of length T Total purchasing and maintenance cost can be easily formulated as: N tot = N o + N s + P, (22) P = the effect of inability to fulfill demanded service level 8

9 Average stock level can be calculated as arithmetic average of final stock levels in periods 1, 2...T: T Kt t x p 1 (23) T If the organization uses continuous stock level monitoring, it is possible to use more accurate methods of average stock level calculation. 2.4 Decision parameters assessment of chosen stock management system and their optimization By the choice of the stock management system from the possible set we determine the manner of replenishment order generation, time of their generation and their size will be assessed by systems decision parameters calculations. With the exception of P-system, the order generation assessment is different, the moment determined by term of stock level decrease on reorder point can be assessed by some of the forecasting methods. Replenishment order size can be obtained by repeating the simulation in an appropriate parameters value range by choosing decision parameters combinations securing required service level. The other possibility of operating parameters assessing is to repeat the simulation in fittingly defined values ranges of all decision parameters. Stock management system choice and operating parameters calculation can be schematically described as follows: Fig. 3: Stock management system choice and combination of decision operating parameters generation Choice of stock management system Is chosen system a P-system? NO YES Reorder point calculation method Forecasting method Generate decision parameters combinations Combine reorder point with other decision parameters Simulate warehouse stock movement Calculate SL by formula (19) Note operating parameters combinations with corresponding SL 9

10 It is obvious that by repeating the simulation for various stock management systems and various combinations of decision parameters we obtain multiple possible solutions securing required service level. If we add viable order number monitoring and average stock level calculation to the simulation model, we can optimize it by using the formula (22). The aim of optimization is to find a stock management system with minimal purchasing and maintenance costs while retaining required service level described by the formula (19). 3 Application of dynamic simulation on sporadic demand products The table below shows demanded quantities of a specific automobile spare part in previous periods. Tab 3: Automobile spare part demand timeline Period t S t Period t S t Using dynamic simulation in the setting of MS Excel, the authors have tried to find a stock management system characterized by decision parameters combinations with minimal purchasing and maintenance costs while retaining required service level. Values of input parameters used in calculations are shown in Table 4. Tab. 4: Experiment input parameters c 185 Euro/piece n s 25% % from average stock level in Euro yearly n o 37 Euro/1 order P S 98 Pieces T 50 Number of time periods SL 98% % Lead time 3 Time periods Starting stock level 9 Pieces The choice of system was limited to Q-system, PQ-system and P-system, characterized by decision parameters shown in Table 2. These parameters were calculated using total simulation or the reorder point calculation method. Implemented experiment variants including the operating parameters calculation are shown in Table 5. Tab. 5: Implemented experiment variants Variant Description Signal level Replenishment order size 1 Q&B Bootstrapping Smart-Willemain (B) Dynamic simulation 2 PQ&B Bootstrapping Smart-Willemain Dynamic simulation 3 Q&ES Exponential soothing (ES) Dynamic simulation 4 PQ&ES Exponential soothing Dynamic simulation 5 Q&CR Crostons method (CR) Dynamic simulation 6 PQ&CR Crostons method Dynamic simulation 7 Q&SB Syntetos-Boylan method (SB) Dynamic simulation 8 PQ&SB Syntetos-Boylan method Dynamic simulation 10

11 9 Q&LS Levén-Segersteds method (LS) Dynamic simulation 10 PQ&LS Levén-Segersteds method Dynamic simulation 11 Q&TS Teunter-Sanihs method (TS) Dynamic simulation 12 PQ&TS Teunter-Sanish method Dynamic simulation 13 Q Dynamic simulation Dynamic simulation 14 PQ Dynamic simulation Dynamic simulation 15 P Dynamic simulation Dynamic simulation Reorder point calculation realized with the aid of forecasting methods was carried out using forecasting coefficient = 0.1; k = 3;.The total stock management system simulation (i.e. determination of both parameters using simulation) was carried out by using the total enumeration method in the value range of 1-98 for decision parameters of the Q and PQ-system and ordering interval range 1-50 for the P-system. Results of the experiment are shown in Table 6. Variant Signal level [pieces/priod] Tab. 6: Experiment outputs Q [pieces] x p [pieces] N s[euro] N o[euro] N tot[euro] C [pieces] Description US 14 PQsystem , % 4 PQstm+SES , % 6 PQstm+Croston , % 8 PQstm+S&B , % 10 PQstm+L&S , % 12 PQstm+T&S , % 13 Qsystem , % 2 PQstm+Bting , % 15 Psystem , % 5 Qstm+Croston , % 7 Qstm+S&B , % 11 Qstm+T&S , % 3 Qstm+SES , % 9 Qstm+L&S , % 1 Qstm+Bting , % Simulated variants were arranged by ascending values of total costs. The best variant would be PQ-system with reorder point 15 pieces and replenishment order size 30 pieces. The outputs have verified the assumption that decisive parameter values obtained by total simulations provide better, or at least the same efficiency of a given system compared to values obtained by the combination of a forecasting method for calculating the reorder point and dynamic simulation for assessing restocking order level. It is obvious that when using "Total Enumeration" method in total simulation of stock movement, a combination containing reorder point level obtained by the forecasting method will be involved when sufficient parameters value range is used. This fact evokes not only the question of how to determine decision parameter value range in order to find optimal value of criterial function, but also in order to search the range of possible solutions. Total simulation of stock movement will become rational only when system efficiency is increased compared to using forecasting methods to calculate restocking point and time required for calculation. This problem is largely insignificant in relation to products with sporadic demand because of the relatively low demand numbers in the individual period. Total demand is therefore not great, leading to a relatively narrow range of operating parameter values, and therefore 11

12 minimal time is required for calculation. The following objective of the authors is to modify the dynamic simulation method to make it universally applicable even for a large portfolio of existing items, not showing the attributes of sporadic demand. Conclusion The aim of this article was to introduce dynamic simulation as an effective method of sporadic demand product stock management system. The authors created 15 algorithms working on the basis of stock movement simulation in the setting of the given stock management system and additional decision parameter optimizations using cost function. Issuing from experiments carried out on real auto part demand time series, the hypothesis has proven that operating parameter values obtained using total simulations provide better, or at least the same efficiency of the given system compared to values obtained by using the combination of a number of predicting method for the calculation the restocking point. The ability of dynamic simulation to find an optimal stock management system characterized by operating parameters combinations predetermines this method universally not only for sporadic demand products stock management but also for stock management. The problem of simulation approach in the current form lies in the manner of searching for possible solutions, the excessive time requirement is for that reason unsuitable for managing large portfolio of stock with varying demand. It is obvious that further research activities of the authors will be aimed at targeting this drawback. Even when considering its drawbacks, dynamic simulation is a promising method, capable of efficiently managing the process relevant to restocking and maintaining the inventory and, therefore, contributes to lowering the cost associated with stocks. In the case of many organizations the costs are enormous and, therefore, even a small decrease of stock levels represents considerable savings. It is therefore necessary to pay special attention to the development of appropriate stock management methods. 12

13 Appendix Q -system simulation model created in language VBA: Sub Qsystem() EndSub 'Initialization of variables Q, Signal, StartStock, LeadTime, T For t = 1 to T 'Set P(t) If t = 1 then P(t) = StartStock Else P(t) =K(t-1) End If 'Delivery arrival If oo = t Then P(t) = P(t) + Q oo = 0 End If 'Dispatch of demanded quantity If P(t) >= S(t) Then P(t) = P(t) - S(t) 'Sufficient stock ElseIf P(t) < S(t) Then 'Stock-out C = C + S(t) - P(t) P(t) = 0 End If 'Order generation If P(t) <= Signal And oo = 0 Then oo = t + LeadTime + 1 End If 'Set K(t) K(t) = P(t) Next 13

14 References [1] Brown, R. G.: Statistical forecasting for inventory control, Mc Graw-Hill, New York, 1959 [2] Croston, J. D.: Forecasting and stock control for intermittent demands, Operational research quarterly 23, , 1972 [3] Gardner, Koehler: Comments on a patented bootstrapping method for forecasting intermittent demand, International Journal of Forecasting 21, , 2005 [4] Holt, C. C.: Forecasting seasonal and trends by exponentially weighted averages, Carnegie Institute of technology, Pittsburg, Pennsylvania, 1957 [5] Johnston, Boylan: Forecasting for items with intermittent demand, Journal of the operational research society 47, , 1996 [6] Levén, Segersted: Inventory control with a modified Croston procedure and Erlang distribution, International journal of production economics 90, , 2004 [7] Rao, A. V.: A comment on: Forecasting and stock control for intermittent demands, Operational research quarterly 24, , 1973 [8] Sani, Kingsman: Selecting the best periodic inventory control and demand forecasting methods for low demand items, Journal of the operational research society 48, , 1997 [9] Syntetos, Boylan: On the bias of intermittent demand estimates, International journal of production economics 71, , 2001 [10] Syntetos, Boylan: The accuracy of intermittent demand estimates, International journal of forecasting 21, , 2005 [11] Syntetos, Boylan, Croston: On the categorisation of demand patterns, Journal of the operational research society 56, , 2005 [12] Teunter, R.H., Sani B.: On the bias of Croston s forecasting method, European Journal of Operational Research, Volume 194, Issue 1, 1 April 2009, Pages , ISSN [13] Willemain, Smart, Shockor, DeSautels: Forecasting intermittent demand in manufacturing: a comparative evaluation of Croston s method, International journal of forecasting 10, , 1994 [14] Willemain, Smart, Schwarz: A new approach to forecasting intermittent demand for service parts inventories, International journal of forecasting 20, , 2004 [15] Winston, W. L.: Operations research: applications and algorithms, ITP, 1994 [16] Winters, P. R.: Forecasting sales by exponentially weighted moving averages, Mgmt Sci. 6, 324,

15 Simulation approach in stock control of products with sporadic demand Jakub Dyntar, Eva Kemrová, Ivan Gros ABSTRACT Croston s method and its modifications are the most commonly used methods in sporadic demand of product stock management systems. This method eliminates the drawbacks of classical exponential smoothing and secures sufficient stock levels during order lead time period. The disadvantage of Croston s method is the fact that it solves only the question of the reorder point but does not solve the problem of restocking delivery volume and the mechanism of ordering. The questions are how to refill stocks and what level of restocking deliveries to implement in order to secure economic efficiency while still maintaining demanded service levels. One of the promising ways of solving stated problems is to apply the dynamic simulation method. The aim of this article is to introduce sporadic demand product stock management method based on dynamic simulation, which would offer simple and easily interpretable answers on basic questions connected to effective stock management. Keywords: Forecasting, Simulation, Inventory Management, Sporadic Demand Jel classification: C53 15