INVENTORY MODELS COMPARISON IN VARIABLE DEMAND ENVIRONMENTS

INVENTORY MODELS COMPARISON IN VARIABLE DEMAND ENVIRONMENTS Miguel Cezar Santoro, Gilberto Freire, Álvaro Eusébio Hernandez Depto. de Engenharia de Produção da Escola Politécnica da Universidade de São Paulo Cidade Universitária, São Paulo, SP, Brasil Abstract This paper presents a comparative cost performance study of four inventory models. One of them, referred as Requirement Planning, operates using demand forecast to quantify the acquisition decisions. The other three inventory models are the Order Up to Maximum Inventory Level, Base Stock and Fixed Lot Size. The Requirement Planning and Base Stock models operate with minimal purchase order quantity referred as Minimum Net Requirements. Firstly, the forecast for the active model is obtained by adjusting the Single Exponential Smoothing or the Holt s linear method to the historical data. After that, operations of the four inventory models are simulated, using search methods to minimize the cost performance of each one. The models are ranked using the mean total cost criterion. An analysis is carried out to verify the fixed purchase cost effect on the performances. Conclusions show the best overall model can change with the Purchase cost. Kanban - Base Stock model presents poor results. Keywords: inventory, simulation, search methods 1 INTRODUCTION Competition has led operations managers to look for cost efficiency in all systems. Inventories are no exception. The success of the Japanese methods and ideas in the last 20 years showed the value of concepts and systems like Just in Time (JIT) and Kanban for this efficiency. Unless JIT is usually understood as a production system, it implies the concept of production or acquisition of an item just at the moment of use, neither before nor after this moment. This ideal condition is the most efficient when inventory quantities and demand satisfaction are the goals of the system. The system has no inventory and no shortage. But this condition is not sufficient to assure cost efficiency. JIT concept often implies a large number of acquisition/production orders, which can result in large operation costs when the purchase/production fixed costs are high. The most common inventory models decide the moment and the quantity to replenish inventory using some current values of the system parameters. They react to a system state, justifying the adjective reactive that is used here to identify them. Reactive models need safety stocks to face the total demand variability during the time they need to react, and only when demand variability and reaction time are close to zero do they attain the JIT condition. Another class of inventory models, using forecasts to decide when and how much to buy or to produce, can be seen in Material Requirements Planning systems (MRP). Named here Requirements Planning (RP), this active model can anticipate the demand, adjusting the quantities and the times close to the moment of use, the JIT condition. It needs safety stocks to face the forecast errors, instead of all demand variability, which the reactive models need to face. It thus seems more efficient in quantities than the reactive models. But it is not enough to assure cost efficiency. This observation raises the following question: Under what conditions do demand variability and forecast errors equally impact the cost performance of the inventory models? A comparative cost performance study of 3 reactive models and the RP was conducted to answer this question. Sixty different demand environments were created to test the models performances. Simulation and search methods were used to optimize models parameters to ensure, for each model in each demand environment, the use of the best cost efficiency condition. 2 LITERATURE REVISION The first reference to forecasting applied to inventory is Brown [1]. This author recommends the use of forecasting to calculate reactive inventory models parameters to achieve better efficiency. Eilon et al. [2] continued this approach. Studies about inventory models during the 60s and 70s adopted demand behavior based on static Normal and Poisson distributions, which allowed convenient mathematical treatment. Browne and Zipkin [3] studied the limits of this traditional approach. Agarwall [4] is an exception in this period, showing that the environment changes are important and alter the best model for inventory management. The use of forecasting in inventory returned in Lee and Adam [5]. The impact of the forecast errors in MRP systems was studied, comparing several reactive inventory models performances applied to the dependent items. The main conclusions were: the forecast errors of the independent item hardly impact the inventory performance of the dependent items. Period Order Quantity (POQ) was the model with the best overall performance. It is important to note in this study that: The forecast errors were generated by static Normal distributions. The impact of the errors was studied on the dependent items and not on the independent item. Simulation is used as an experimental method to evaluate the models performance. Armstrong [6] and Goddard [7] presented another point of view. The improvements in performance by forecasting systems implementation would not pay for the costs and problems this implementation could produce. Goddard, in his Let s scrap forecasting paper, showed the importance of reducing (items) lead times to achieve better systems performances. But the demands complexity was still out of

the discussion and only reactive inventory models were considered. Gardner [8], using real demand historical data of independent items, studied the performance of the inventory system based on the classical Economic Lot Size model. Defining the lots acquisitions based on forecast demands, generated by four different forecasting models, he calculated trade off curves between inventory investment and customer lead time. Simulation of the system operation during the period of historical demand data was used in the calculations. His main conclusion was the importance of forecasting errors to the amount of inventory investment needed to achieve a specific customer lead time. The smaller the error, the smaller the investment needed. The study is important, too, due to the adoption of the joint performance measurement of the forecasting and inventory models. Fildes and Beard [9] studied the forecasting use in the production and inventory control. They analyzed the typical characteristics of inventory data and several forecasting models, and concluded that researchers neglected the production and inventory control area, the commercial systems had inadequate forecasting models and users experimented unnecessary large errors, inventories and poor demand satisfaction. The contributions of the studies discussed above were critical to the experimental modeling used in this study. 3 FORECASTING MODELS Both forecasting models selected use the smoothing logic. Table 1 presents the projection equations for forecast calculation. These models are analyzed in details in Makridakis [10] and Hanke and Reitsch [11]. Based on them, smoothing methods are used to quantify the Q, I and F parameters, balancing the importance of old and new demand data in the forecasts. Table 1: Forecasting models selected. Smoothing Model Curve Projection Method Constant Simple with α d t+k = q t Simple with α and Trend d t+k = q t + k.i t β (Holt) At each period, Q, and I are recalculated, using the real demand occurred (V) and Table 2 formulas. The models adjusted to the item demand series become the selection of the corresponding α, β and γ for each of them, using the search and simulation routine. Table 2: Models formula for period to period calculations. Model Formula Curve Constant Trend q t = α v t + (1- α)q t-1 = q t-1 + α(v t - q t-1) q t = α v t + (1- α) (q t-1 + i t-1) i t = β (q t - q t-1) + (1- β) i t-1 Again, Mariachis [10] and Hanker and Retch [11] provide the optimization method, using 24 initial periods to calculate the first Q and I values, and the last 36 periods to adjust the parameters to the minimum forecast error, measured by Mean Absolute Deviation (MAD). 4 INVENTORY MODELS Three reactive inventory models, detailed below, were selected for the comparative test. 4.1 Periodic Up to Order Level model In this model, the lot size is calculated to ensure that the total stock quantity never exceeds a defined quantity, the Order Level (old) parameter. The decision rule is: poq = ( ol st) if st rl (1) poq = 0 (Zero) if st > rl (2) Where poq ol st rl is the Purchase Order Quantity, the lot size to replenish the inventory is the Order Level parameter is the current stock at the decision moment, including the purchased lots still arriving is the Reorder Level, the quantity parameter that starts the purchase activity. The periodic review occurs at the end of each period. 4.2 Periodic Base Stock model The decision rule for this model is poq ( ol st) = (3) This simple rule implies stock replenishment every time the system reviews the inventory. It is a particular case of the precedent model, the Reorder Level is equal to the Order Level parameter. Like the other, it is a model with a 1-period review. 4.3 Periodic Economic Order Quantity model This traditional model has the following decision rule poq = n. ls if st rl (4) poq = 0 (zero) if st > rl (5) ls n is the fixed lot size, usually calculated using some economic criteria is a number of lots (LS sized) that assure the st>=rl after this purchase Again, the replenish decision occurs at every 1 period. 4.4 Requirements Planning Model This active model uses the following decision rule: poq = nr if > 0 poq = 0 (zero) if 0 nr nr (zero) (6) nr (zero) (7) is the Net Requirements needed to satisfy the forecast demand of the time defined by the Lead Time plus Reviewing Period. The general NR formula is: lt+ rp lt 1 nrt t+ lt = dt t i poqt + i lt t i st + ss nr t lt rp, (8) i= 1, + j= 1, + is the Net Requirements need to satisfy the demand during the next LT + RP periods is the system date when the decision is taken is Lead Time of the item is the Reviewing Period

d is the Demand forecast poq is the Purchase Order Quantity s is the stock of items ss is the Safety Stock The first key in the parameters is the date when the system takes the decision, which generates the value parameter. The second key refers to the deadline to that decision. The following example helps to understand the model operation. tries to end the last period with only the Safety Stock in the inventory. Safety Stock is the single parameter for this model in its pure form. 4.5 Minimum Net Requirements (MNR) The Base Stock models and the Requirements Planning Model tend to order with low quantities due to their decision rules. This is a great difficulty in environments the purchase cost is high. To improve the competitiveness of these 3 models, a second parameter is introduced, the Minimum Net Requirements. It is the smallest quantity the system must order when a quantity different from zero is needed. If a quantity greater than MNR is needed, the model orders this quantity instead of MNR. The new decision rules for the models are Periodic Base Stock models ( (10) poq = ( ol st) if ol st) mnr ( (11) poq = mnr if ol st) < mnr Figure 1: Example of Requirements Planning model operation The Net Requirements value in this example is = ( 30+ 40+ 30+ 20) (20+ 30) 20+ 10= 60 nr (9) The Order decided in the current t = 0 needs to satisfy the next 4 periods, because the lot will only arrive at the end of period 3, to be used in period 4. To order only the necessary quantity, the POQ decided in the past (and not received) and the current stock is subtracted from the initial sum. Finally, a safety stock is added to prevent shortages due to forecasting errors. Therefore, this model always Historical series data set Demand of 36 periods of 400 items Unit price of each item Shortage cost of each item Lead time of each item Start Requirements Planning Model poq = nr if nr mnr (12) poq = mnr if < nr < mnr poq = 0 (zero) if 0 0 (13) nr (zero) (14) These modified rules were used in all simulations of this study. Simulation Parameters Data set Forecasting models Search and simulation routine Best parameters search for each forecasting model Forecasting errors calculations Forecasting models ranking Best forecasting model End Inventory models Search and Simulation routine Best parameters search for each inventory model Operation cost calculation Inventory models ranking Results report Next item Figure 2: Experimental design for the comparative study

5 MODELING Figure 2 presents the experimental design for models comparison. Visual Basic Applications software and Excel worksheets were the basic tools used for all routines. Actual historical data series of 400 items, each one with 36 periods, are used to compare the inventory models. This demand data set was used to select the best between the 2 forecasting models, which provided the demand forecast in the active inventory model. The decision variable was the forecasting error, measured by the Mean Absolute Deviation (MAD), calculated in the last 24 periods of each item. The initial 12 periods were used to calculate the parameters seeds required in the search routine. Then, the 4 inventory models were simulated using the same demand data set. To eliminate the cost inefficiency due to incorrect parameters values, a searchand- simulation sequence was repeated until the lowest operation cost was achieved for each model. Consequently, the comparison of the models was made in the best models conditions. Since the decision variable in this routine was the total operation cost of 36 periods, a set of costs was supplied to the software: Purchase cost cost incurred each time a purchase order is decided upon - 5 values were used in the experiment: 1.0 (low level), 3.0, 5.0, 7.0 and 9.0 (high level) for all items; Holding cost cost incurred in holding one unit of the item during 1 period this cost was equal to 1% of the item acquisition price, informed to the system in the simulation parameters data set. Shortage cost - cost incurred per period for each unit not supplied due to unavailable inventory the values adopted for each item were the actual values used in the company data were taken. The operation cost was the sum of these costs averages, considering only the periods the model was running on phase. The actual item lead time was informed in the simulation parameters data set. 6 RESULTS Table 3 presents the simulation results for the 5 purchase cost values. The 4 inventory models were ranked (ascending order) in each item using the total operation cost. Table 3 - % of the 400 items in which the model had the best (lower) cost performance PURCHASE COST UP TO ORDER LEVEL BASE STOCK ECONOMIC QUANTITY REQUIRE- MENTS PLANNING 1,0 31,5% 6,8% 36,8% 25,0% 3,0 29,0% 11,8% 32,8% 26,5% 5,0 28,3% 11,5% 35,3% 25,0% 7,0 31,0% 11,8% 30,3% 27,0% 9,0 28,5% 12,0% 29,0% 30,5% The Figure 3 presents the same data in a graphical form. The model showed the best performance for lower purchase cost values. Its initial advantage is reduced with the growing, with both Up to Order Level and Requirements Planning showing better performance in higher values. Base Stock presents the worst performance among the 4 models. % of Inventory Models.. 1st Places Inventory Model 1st Places % x 40% 35% 30% 25% 20% 15% 10% 5% 0% Up to Order Level Base Stock Requirements Planning Figure 3 Inventory Models performance expressed in % of the 400 items in which the model had the lower Operation Cost Figure 4 shows the sum of the 400 items Operation Costs for each model, plotted against the used in each simulation run. The Operation Cost growing is expected, since the is part of it. The Up to Order Level sum values were the lowest for all Purchase Cost values calculated for the models, but this fact isn t enough to assure the best performance at items level, showed in Figure 3. The Operation Costs sum of the best model (1st. place) for each item is included in the chart to evaluate the total performance improvement due to the selection. Operation Cost Sum.. ($ 000) 18000 17000 16000 15000 14000 Operation Cost Sum x Up to Order 1st Place Sum Base Stock Resource Planning Figure 4 Inventory Models Operation Cost performance comparison, using the 400 items Operation Cost sum. Finally, Figure 5 shows the models performances measured by Inventory Average, valued by items prices and calculated for each model and item. Again, these averages sum are plotted against the and the sum of the best model for each item is included. The best performance of Requirements Planning model was expected, since the use of forecasts leads to lower inventory levels. Like the Up to Order model and the Operation Cost analysis, this fact isn t enough to assure good cost performance at items level. The improvement is significant when the best model is considered at items level.

Inventory Average Sum.. ($ 000).. 1480 1460 1440 1420 1400 1380 1360 1340 1320 1300 1280 Inventory Average Sum x Up to Order Level Stock Base Resource Planning 1st Place Figure 6 - Inventory Average Sum of the models 7 CONCLUSIONS The use of actual items data in inventory systems simulation limits the general inferences one could make. Nevertheless, jointly with simulation, is a good tool for systems performance in specific environments. The model had the best performance in s low values when the number of items the model get 1st. place is the criterion used. But only 32 to 36% of the items have this model as the best. Similar results were got in high s values for the Up to Order Level and Requirements Planning models. It is clear that any model is the best for the majority of the items. If this strategy is used, Up to Order Level is the recommended model. Analyzing the Inventory Average, the Requirements Planning model had the best performance. The use of forecasts can explain this fact. When this characteristic is important for the inventory system, that model should be used. But, again, significant improvements can be achieved by the selection of the best model for each item. Base Stock model had poor overall performance. Better results were expected, since this model is the base of Kanban system, largely recommended currently to reduce process inventory. Its Inventory average results were practically the same of the Up to Order Level, and higher than the Requirements Planning. And the quantity of the 400 items in which it had the lowest Operation Cost was 6 to 12%, poor results for this popular model. In few words, the results of the simulations show the importance of the models correct selection for each item in inventory systems. The improvements in overall performance are significant and probably pay the selection cost implementation. [5} Lee T. S., Adam Jr. E.E., Sept. 1986, Forecasting error evaluation in material requirements planning (MRP) production-inventory systems, Management Science, 1186 1205. [6] Armstrong J. S., Jan. Feb. 1986, Research on forecasting: a quarter-century review, 1960-1984, Interfaces, v. 16, 52 66. [7] Goddard W. E., Sept. 1989, Let s scrap forecasting, Modern Materials Handling, v.39, 39. [8] Gardner Jr. E. S., April 1990, Evaluating Forecast Performance in an Inventory Control System, Management Science, v. 36, #. 4. [9] Fildes R., Beard C., 1992, Forecasting Systems for Production and Inventory Control, International Journal of Operation & Production Management, v.12, #.4, 4-27. [10] Makridakis S., Wheelright S.C., 1998, Forecasting- Methods and Applications, 3rd. ed., New York, Wiley. [11] Hanke J. E., Reitsch A. G., Business Forecasting, 1998, New Jersey, Prentice Hall. 8 REFERENCES [1] Brown R.G., 1959, Statistical Forecasting for Inventory Control, New York, McGraw-Hill. [2] Eilon et al., 1970, Adaptive Limits in Inventory Control, Management Science, v.16, n.8,. 533-548. [3] Browne S., Zipkin P., 1991, Inventory models with continuous, stochastic demands, The Annals of Applied Probability, v.1, 419-435. [4] Agarwall S.C., 1974, A review of Current Inventory Theory and its Applications, International Journal of Operations Research, v.12, 443-482.