A COMPUTER-AIDED INVENTORY MANAGEMENT SYSTEM – PART 2 11 A computer-aided inventory management system – part 2: inventory level control C. Y. D. Liu and Keith Ridgway Reviews inventory policies and lot-sizing techniques in a cutting tool manufacturer Introduction In part 1 of this article the design and development of a computer-aided inventory management system (CAIMS) was described. The CAIMS system was developed for a cutting tool manufacturer, PRESTO Tools Ltd, Sheffield, with the objectives of reducing inventory investment and improving productivity, customer-service level and plant efficiency.
The CAIMS system consists of four modules incorporating analytical techniques for ABC analysis, forecasting, economic batch quantity calculation and the statistical calculation of the re-order level respectively. In this second part of the article, various inventory policies and lot-sizing techniques are reviewed and the analytical techniques used in the economic batch quantity (EBQ) and re-order level (ROL) modules are described. review. In this case, the replenishment order quantity is variable and brings the stock to a predetermined level (S ).
In the (s,S ) policy is similar to the re-order cycle policy but a variable replenishment order is only placed when the stock falls below a predetermined level (s).
In the combined re-order level and re-order cycle policy, replenishment orders are placed periodically and when the stock-on-hand falls below the re-order level. Replenishment orders placed when the re-order level is reached are of fixed size, but those placed at the review are variable. All of these systems are closely related in that a predetermined amount of stock, or period is set to trigger a fixed, or variable replenishment order.
The key task is to determine accurately the parameters required to implement a successful inventory policy, i. e. the replenishment order quantity and the re-order level. The calculation of both the re-order level and replenishment order quantity is usually dealt with in isolation. Lewis suggested that for the optimum operation of a re-order level policy, these two quantities should be calculated jointly as one directly influences the other. Lewis recommended a method originally proposed by Tate.
Tate’s formula has the advantage of being relatively insensitive to large variations in stockout costs. Lewis illustrated Tate’s formula with an example, and demonstrated that the re-order level evaluated using the joint calculation method was less than that calculated to achieve the same service level using the independent calculation method. The joint calculation method is complicated, and it is difficult to assess whether this additional computational effort is really providing an overall cost saving to the business.
Several authors including Prichard, Brown, and Ploss and Wight, have described the use of statistical techniques to calculate the re-order level. This method has been adopted in the ROL module and will be discussed later. The authors wish to thank the staff of PRESTO Tools Ltd, Sheffield for their help during the course of the project. Inventory policies Lewis identifies two basic type of inventory policy. The first, in which decisions concerning replenishment are based on the level of inventory held, is known as re-order level policy. The second based on a cycle time, is known as reorder cycle policy.
Within these two categories there are several variants including the re-order level policy with periodic reviews, the (s,S) policy and the combined re-order level, and re-order cycle policy. In the re-order level policy, an order for replenishment is placed when the stock-on-hand equals or falls below a fixed value, known as the re-order level. When a replenishment order is placed, the quantity required is fixed. The amount of stock held must be reviewed continuously. In the re-order cycle policy, the stock-on-hand is reviewed periodically and a replenishment order is placed at each Integrated Manufacturing Systems, Vol. No. 2, 1995, pp. 11-17 © MCB University Press Limited, 0957-6061 12 INTEGRATED MANUFACTURING SYSTEMS 6,2 Lot-sizing techniques Lot-sizing when set-up and holding costs are significant has been an issue in production planning, dating as far back as 1915, when F. W. Harris developed the concept of an EBQ or economic order quantity (EOQ) for purchasing. Since then, researchers have developed several models of increasing complexity to take factors such as production rate, cost of stockout and effect of price reduction[4,6,8] into account.
Saunders indicated that the main weakness in the calculation of EBQ centred on the costs used in the formula. The basic assumptions that the costs are affected by the lot-size selected and that demand and lead time are constant, may not be realistic in many situations. The EBQ formula applies to individual items and indicates a desired optimum condition for each, based on definitive assumptions regarding costs. They do not indicate the total results in either inventory or the operating conditions that can be expected. Neither do they give any consideration to changes from the present situation.
Ptak pointed out that the EBQ formula is widely used in practice because of the relative simplicity of the model and the small number of variables considered. To overcome the need for accurate costing data, Ploss and Wight developed LIMIT (lot size inventory management interpolation technique) for handling EBQ in aggregate and dealing with the constraint problems of the calculation. It is designed to handle a family of items which pass through common manufacturing facilities. Trial economic lotsizes are calculated for each group using the standard EBQ equation.
The total set-up time required for these “economic” lots is then compared with the total set-up time required for the existing lots. Consequently, new LIMIT batch quantities can be calculated, which results in a total setup time equal to the present total. Ploss and Wight concluded that a substantial reduction in the total inventory can be achieved without changing total set-up times and operating conditions. The LIMIT concept can be applied without the need to determine the precise value of the inventory carrying cost.
The EBQ formula is the easiest to implement owing to the small number of variables involved. The insensitivity of the formula to input errors is useful for companies where costing data is not readily available. As analytical techniques were being introduced for the first time, the EBQ formula was seen as a key step prior to the introduction of more complex lot-sizing techniques in the future. The economic batch quantity formula The most economical lot-size is the one which minimizes the total sum of the inventory set-up and carrying costs. q Set-up cost (Sc ).
The set-up cost (Sc ) for a batch is independent of the size of batch quantity (Q). Thus, the annual set-up costs for a forecast annual demand (D) is given by D*Sc/Q. The relationship between the set-up cost and batch quantity is shown in Figure 1. q Holding cost (H). In the simple cost model considered it is assumed that the cost of holding stock is directly proportional to the average amount of stock held (Figure 1). In a re-order level system, the average stock is given by the sum of one-half the replenishment batch quantity (Q/2) plus the safety stock (SS).
The cost of holding an item of stock can be expressed as a percentage (I ) of the unit cost of that item (C), thus the annual holding cost can be evaluated as: Annual holding cost = (Q/2 + SS) * (C*I ). (1) Hence, the complete expression for the total annual variable cost (T ) is: T = D*Sc/Q + (Q/2 + SS) * (C*I ). (2) Table I illustrates a numerical example for the 6mm jobber metric twist drill, where the values for the variables used in equation(2) are: Unit cost (C) = ? 0. 236 Holding cost (I ) = 35 per cent (expressed as a percentage of unit cost) Set-up cost (Sc) = ? 2 Annual demand (D) = 237618 (determined from forecasting module). It is assumed that the safety stock is negligible at this stage. By varying the batch size (Q) as displayed in column 1 of Table I, a series of set-up costs (column 3) and holding costs Figure 1. Cost curve for 6mm jobber Cost (000s) 4 3 2 Total cost 1 Minimum cost EBQ module The EBQ is calculated using the final forecast, set-up cost and holding cost. The technique can be applied for all products, leading to an overall reduction in the average stock holding and operating costs. Holding cost Set-up cost 0 0 5
Economic batch quantity 10 15 20 Batch quantity (000s) 25 30 A COMPUTER-AIDED INVENTORY MANAGEMENT SYSTEM – PART 2 13 (column 5) can be constructed. Thus the total annual variable cost (T ) is the sum of these as shown in column 6. The total annual variable cost (T ) is portrayed graphically in Figure 1. It can be seen that the total annual variable cost (T ) is represented by a curve which has a minimum cost value. To find this minimum cost value, the expression for the total annual cost is differentiated with respect to the batch quantity (Q ), i. e. dT/dQ = –D*Sc/Q2 + C*I/2 (3) ffect of input errors on the total annual variable cost (T ) and EBQ. Of these input variables, the annual demand (D ) is the one most likely to differ from the actual value experienced. Effect of input errors on total annual variable cost Prichard demonstrated that underestimating the annual demand or overestimating the holding cost by 50 per cent causes a 6 per cent increase in total variable cost. Even the combined effect produces only a 25 per cent error in the total cost. This suggests that the total variable cost is not particularly sensitive to errors in the input parameters.
The error in the total variable cost is only affected by the input error ratios, which may be less than the individual error ratios. This explains why the total cost curve plotted in Figure 1 is fairly flat at the bottom of the curve. Effect of input errors on EBQ Prichard demonstrated that underestimating annual demand by 50 per cent causes a 29. 3 per cent decrease in the batch quantity, while overestimating the annual demand by 100 per cent causes a 41. 4 per cent increase in batch quantity. This suggests that the EBQ formula is relatively insensitive to errors in input parameters. dT/dQ = 0 for minimum value.
If the replenishment batch quantity (Q) is redefined as the economic batch quantity (EBQ) then: EBQ = [(2*D*Sc)/(C*I)]1/2. (4) Equation (4) shows that the EBQ is directly proportional to the square root of the forecast annual demand and setup cost, or alternatively, a higher holding cost per item will decrease the EBQ. The EBQ is independent of the safety stock. Using equation (4), the economic batch quantity for the 6mm jobber drill is calculated to be 14,000. Sensitivity analysis The collection and analysis of data leading to the forecast of demand, set-up and holding cost, are processes subject to error.
In any one of the input parameters, this error will increase the total variable cost and generate inaccuracies when computing the EBQ. It is interesting to examine the ROL module A statistical method of evaluating the re-order level has been developed as one of the modules in the CAIMS system. The module is only concerned with calculating the re-order level for independent-demand items. Table I. Relationship between cost and batch size for 6mm jobber drill Batch size 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 18,000 20,000 22,000 24,000 26,000 28,000 30,000 Set-up/year 119 59 40 30 24 20 17 15 13 12 11 10 9 8 8 Set-up cost (? 3,808 1,888 1,280 960 768 640 544 480 416 384 352 320 288 256 256 Average stock 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 11,000 12,000 13,000 14,000 15,000 Holding cost (? ) 91 182 273 364 455 546 637 728 819 910 1,001 1,092 1,183 1,274 1,365 Total cost (? ) 3,899 2,070 1,553 1,324 1,223 1,186 1,181 1,208 1,235 1,294 1,353 1,412 1,471 1,530 1,621 14 INTEGRATED MANUFACTURING SYSTEMS 6,2 Re-order level with periodic review system The company has adopted the re-order level with periodic review system as illustrated in Figure 2.
This demonstrates that the safety stock is not utilized in the first two replenishment cycles. In the third cycle the rate of use increases, as indicated by the steeper line. With this higher demand, the inventory drops into the safety stock before the new supply is received. Should the demand increase even more, or the lead time become longer, the inventory could fall to zero, resulting in a stock shortage. Therefore, the re-order level must comprise an estimate of demand during lead time, plus some safety stock to protect against exceptional demand or increased lead time.
The re-order level can be expressed mathematically as: Re-order level = demand during lead time + safety-stock ROL ROL where = LTU + SS = LTU + k*(? l ) k = safety-factor (5) Figure 3. Stages in setting the re-order level Forecast demand Establish lead time Calculate number of replenishment orders Calculate demand during lead time Calculate probability of stock shortage Calculate standard deviation Determine safety factor from N distribution table Calculate standard deviation over lead time Calculate safety stock Decide permissible number of stock shortages Calculate re-order level l = standard deviation over lead time. The statistical method of evaluating the re-order level is adapted from the work of Prichard, Brown and Ploss and Wight. The key steps are illustrated in Figure 3 and described in the following sections. Demand during lead time The first step is to calculate the expected monthly demand during the lead time using the forecasting module. This demand must then be extended over the lead-time period, which usually differs from the forecast period. The standard lead time for the 6mm jobber drill is set at 40 days, which can be approximated to two months.
In January, demand during the lead time would be the sum of the February and March values. Similarly, demand during the lead time for February would be the sum of the March and April forecast figures. This method will give a variable lead-time usage and respond more quickly to seasonal changes. Setting safety stock In most practical inventory situations, the lead time is not accurately known and the demand cannot be predicted exactly. The problem is to estimate how much safety stock will be required in the re-order level (equation (5)).
Applying estimated values, either equivalent to a certain number of months’ stock or to certain percentages of lead time usage, will usually result in excess, or insufficient inventory on most items. Ploss and Wight suggest that the safety stock is a function of the following elements: q q the ability to forecast the demand accurately; the length of the lead time; the ability to forecast or control the lead time accurately; the size of the batch quantity; the service level desired. Figure 2. The re-order level with periodic review system
Quantity (000s) 100 Net inventory 80 A B 60 C 40 D Stock on hand Economic batch quantity Safety stock Lead time 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Week Re-order level q q q From equation (5), the safety stock is taken as the product of the standard deviation over the lead time and the safety factor. The method of evaluating these two variables is described as follows. Standard deviation The standard deviation is given by equation (6), where n is the number of observations. 20 ?d = [? (actual – mean)2 / (n – 1)]1/2. 6) The standard deviation for the 6mm jobber drill is evaluated as 4,904, based on a 12-month demand history. A COMPUTER-AIDED INVENTORY MANAGEMENT SYSTEM – PART 2 15 An adjustment factor must be applied when the forecast interval is different from the lead-time interval. Thus, standard deviation over the lead-time interval is given by equation (7): ? l = ? d * (Lead-time interval)1/2 . (7) Forecast interval Since the lead time for the 6mm jobber drill is two months and the forecast interval is one month, the standard deviation over the lead-time period is calculated to be 6,935.
Safety factor The safety factor (k) can be determined from the normal distribution table (see Table II) for any desired service level. The safety stock (SS) required is given by: (8) SS = k * ? l . Lewis defines the service level as the proportion of annual demand met ex-stock. Bearing in mind the concept of ABC analysis, it is logical to give class A items the highest service level as they account for at least 75 per cent of the total turnover. After consultation with the management the acceptable number of stock outs for the various categories of inventory were defined as shown in Table III.
From this the probability of not fulfilling a delivery time demand (P ) can be calculated. This is the ratio of the number of stock outs (A) to the number of replenishment orders in a year (N ) as shown in equation (9). P=A/N P = A * EBQ/Annual demand. (9) The above concept is illustrated with the example of the 6mm jobber drill. From equation (7), the standard deviation over the lead-time interval is calculated as 6,935. With an annual forecast demand of 237,618 and an EBQ of 14,000, 17 replenishment orders will be required each year.
As the 6mm jobber drill is a class A item, the acceptable number of stock shortages in a year is 0. 2, obtained from Table III. Using equation (9), the probability of not fulfilling a delivery time demand is evaluated as 1. 2 per cent, equivalent to a service level of 98. 8 per cent. With a service level of 98. 8 per cent, the safety factor required is 2. 25 from Table II. Thus, using equation (8), the safety stock (SS) for the 6mm jobber drill is evaluated as 15,604. Setting the re-order level The final step in evaluating the re-order level is to sum the demand during lead time and the safety stock.
The safety stock is assumed to be constant throughout the year so that the re-order level can be set on a month-to-month basis depending on the monthly demand forecast. When the latest demand data are obtained, the safety stock can be re-calculated. Table IV contains numerical calculations for the 6mm jobber drill. The re-order level (column 5) is the sum of the demand during lead time (column 3) and safety stock (column 4). A plot of re-order level and safety stock against the monthly period is shown in Figure 4.
If the monthly forecast is superimposed onto the same graph, (shown dotted), both clearly illustrate the same pattern except that the re-order level plot lags the monthly forecast by the lead-time period. In the case of the 6mm jobber drill, the re-order level using the traditional practice is set equivalent to three months demand (i. e. annual demand/4 = 59,404). If this is compared with the re-order levels established using statistical methods Table II. The normal distribution table Safety factor (k) 1. 00 1. 05 1. 10 1. 15 1. 20 1. 25 1. 30 1. 35 1. 40 1. 45 1. 50 1. 55 1. 60 1. 65 1. 70 1. 75 1. 80 1. 85 1. 90 1. 95 2. 0 2. 25 2. 50 2. 75 3. 00 Service level (%) 84. 1 85. 3 86. 4 87. 5 88. 5 89. 4 90. 0 91. 2 91. 9 92. 7 93. 3 94. 0 94. 5 95. 1 95. 5 96. 0 96. 4 96. 8 97. 1 97. 4 97. 7 98. 8 99. 4 99. 7 99. 9 Probability of stock shortage (%) 15. 9 14. 7 13. 6 12. 5 11. 5 10. 6 10. 0 8. 8 8. 1 7. 3 6. 7 6. 0 5. 5 4. 9 4. 5 4. 0 3. 6 3. 2 2. 9 2. 6 2. 3 1. 2 0. 6 0. 3 0. 1 Table III. Acceptable number of stock shortages per year Code A B C D Number of stock shortages (A) 1/5 1/3 1/2 1/2 16 INTEGRATED MANUFACTURING SYSTEMS 6,2 Table IV. Re-order level for 6mm jobber drill Month (1991) January February March April May June July August September
October November December Total Forecast 23,053 21,342 16,546 17,769 20,273 20,294 20,677 17,993 16,556 27,693 22,202 13,221 23,618 Lead-time demand Safety stock 37,888 34,315 38,042 40,568 40,971 38,670 34,549 44,249 49,895 35,423 15,604 15,604 15,604 15,604 15,604 15,604 15,604 15,604 15,604 15,604 Re-order level 53,492 49,919 53,646 56,172 56,575 54,274 50,153 59,853 65,499 51,027 that it enables the re-order level to be set successfully over the complete product range, and does not rely on intuitive judgement. Evaluation The CAIMS system has been implemented, tested and verified as individual modules, followed by a full system run.
Analysis of the output from each module has produced favourable results, which are summarized below. EBQ module Applying the EBQ formula over the whole range of drills provides a comparison between the current average stock and the average stock using the EBQ module. The average stock is assumed to be half the replenishment batch quantity. The use of EBQ produces an annual reduction in the number of inventory items of 54 per cent, and a cost saving of 51 per cent. As this is a one-off exercise in inventory reduction, the cost saving would not be as marked in subsequent years.
The most significant reduction is for class A items because the replenishment batch quantities are smaller. Changes in class C items are negligible as large replenishment batch quantities are used. This coincides with the concept of ABC analysis. x x x x x x x Figure 4. Re-order level for 6mm jobber Quantity (000s) 70 60 50 40 30 + + + + + Re-order level x x x Forecast + + + + + 20 10 0 Jan Safety stock The performance of the ROL module has proved to be satisfactory Management has been cautious in implementing the EBQ concept for all products because smaller batches imply that more set-ups will be required.
Analysis shows that the number of set-ups increases by 48 per cent. This will certainly reduce machine utilization, and the set-up time will become a significant element of the production lead time. It is therefore essential to introduce a set-up reduction programme to reap the benefits of small batch production. As the EBQ is now linked by formulae (with the CAIMS system) to the machine set-up time, any reduction in machine setup times will further reduce the EBQ. Hence, the use of the EBQ module and adoption of a set-up reduction programme will initiate a longer-term inventory reduction programme.
Small batch production not only reduces inventory and lead time, but also enables the manufacturing facility to be more Feb Mar Apr May Jun Jul Aug Sep Oct Month Key : + x Forecast Safety stock Re-order level (Table IV), it can be seen that the level obtained from intuitive techniques is close to the average (55,061) for the ten-month period from January to October 1991. Much care was taken in the intuitive judgement to set the re-order level at three months demand, because the 6mm jobber drill is best selling class A item.
This supports the re-order level calculated using statistical methods and suggests that the level is appropriate for the existing environment. The advantage of the statistical method is A COMPUTER-AIDED INVENTORY MANAGEMENT SYSTEM – PART 2 17 flexible and react quickly to changes in customer demand. Reducing the average inventory increases the overall stock turn ratio for drills by 146. 63 per cent. Small batch production contributes to lower carrying costs and offsets any increase in set-up costs to produce a total cost saving of 44 per cent.
ROL module The ROL module described has provided the company with a scientific and systematic method of statistically establishing the re-order level for all finished products. When applied to the whole range of drills, there is an overall increase in the re-order level quantities of 15 per cent. This is due to the higher safety stocks which are necessary when smaller replenishment batch quantities are used. Statistical techniques distribute the safety stock where it is needed most rather than applying it uniformly across the entire inventory and improve the service level provided.
The performance of the ROL module during the trial period has proved to be very satisfactory. The advantages of applying the ROL module come from the ability to measure the variation of demand and realistically determine the level of safety stock. In addition, it provides a means of updating both the safety stock and re-order levels on a routine basis for all stock items. lot-sizing techniques to be employed, such as Ploss and Wight’s LIMIT. The successful development of the ROL module has also provided the company with a scientific and systematic means of evaluating the re-order level for all finished stock.
It is important to note that the calculation of the re-order level in the ROL module and the replenishment order quantity in the EBQ module, are dealt with in isolation. Lewis recommended that these two quantities should be calculated jointly for true optimum operation of the re-order level policy. References 1. Liu, C. Y. D. and Ridgway, K. , “A computer-aided inventory management system – part 1: forecasting”, Integrated Manufacturing Systems, Vol. 6 No. 1, 1995, pp. 12-21. 2. Lewis, C. D. , Scientific Inventory Control, Butterworths, London, 1970. 3. Tate, T. B. “In defence of the economic batch quantity”, Operation Research, quarterly, Vol. 15 No. 4, 1964, p. 329. 4. Prichard, J. W. , Modern Inventory Management, John Wiley & Sons, New York, NY, 1965. 5. Brown, R. G. , Decision Rules for Inventory Management, Holt Rhinehart and Winston, New York, NY, 1967. 6. Ploss, G. W. and Wight, O. W. , Production and Inventory Control: Principles and Techniques, Prentice-Hall, Englewood Cliffs, NJ, 1967. 7. Harris, F. W. , Operation and Cost, Factory Management Series, A. W. Shaw, Chicago, IL, 1915. 8. Lockyer, K. G. , Production and Operation Management, 5th ed. , Pitman Publishing, London, 1989. . Saunders, G. , “How to use a microcomputer simulation to determine order quantity”, Production and Inventory Management, Vol 28 No. 4, New York, NY, 1987, pp. 20-3. 10. Ptak, C. A. , “A comparison of inventory models and carrying costs”, Production and Inventory Management, Vol. 4 No. 29, New York, NY, 1988. 11. Lewis, C. D. , Scientific Inventory Control, 2nd ed. , Butterworths, London, 1980. Conclusion The successful development of the EBQ module has provided the company with a scientific and systematic means of evaluating the most economical batch quantity to manufacture across the entire product range.
The sensitivity analysis demonstrates that the EBQ formula and the total cost equation are insensitive to input errors. Adjustments can be made to the EBQ without sacrificing significant savings. This would prove to be useful in companies where costing data are not readily available. The EBQ module provides a stepping stone for more complex C. Y. D. Liu is a Lecturer at the German Singapore Institute, Jurong, Singapore, and Keith Ridgway is the Ibberson Professor of Industrial Change and Regeneration in the Department of Mechanical and Process Engineering, University of Sheffield, Sheffield, UK.
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