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UNIT15 INDEPENDENT DEMAND INVENTORY SYSTEMS Objectives After going through this unit, you will be able to learn • • • • • • • • • • • • • • • • • • • • • • • • • • • What is an independent demand system What is a production-distribution system The need for inventory To develop a deterministic model What are holding and ordering costs (or setup costs) Meaning of economic order quantity, reorder point, lead-time, average inventory, stock cycle. How the model behaves around the optimum point To perform sensitivity analysis What is finite production rate and how it affects the basic model The concept of planned shortages What is shortage cost and how it affects the basic model Effect of quantity discount on order quantity What is material cost What is carrying charge Effect of constraints on working capital and warehouse space What is an Optimal Policy curve Single Period Model Multiple Period Models What is uncertainty in demand and what is a stochastic model What are overstocking and under-stocking costs What are shortage costs and what is backlogging What are expected profits, shortages and demand What is demand during LT and what is variability What is a service level. Where to stock and how to determine ROP What are safety stocks Why selective control • 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15.10 15.11 15.12 What is ABC and VED analysis Introduction Basic Inventory Model Model Sensitivity Gradual Replacement Model Basic Model with Backlogging Bulk Discount Model Independent Demand System for Multiple Products Models with Uncertain. Demand Selective Control of Inventory Summary Self-Assessment Exercises Further Readings Structure 15.1 INTRODUCTION Inventory items can be broadly classified into two types. (a) Independent demand inventory items and (b) Dependent demand inventory items. The former is based on the items own usage history and statistical variations in demand, while the later is based on the production schedules of end items of which the inventoried item is a part. Hence the former is based on a replenishment philosophy while the later is based on a requirements philosophy. In this chapter we will be discussing the independent demand inventory Fig: 15.1: A Production Distributions System Figure 15.1 shows a production-distribution system (where the output is obtained by processing certain inputs). As we move along the arrows from input to output we come across the various stock points: a) Raw materials & supplies: Raw materials are needed to start the production process. Hence they are the inputs to the system. 1. There may be instances when there is a temporary increase in production requirements. If raw material is not available, production stops. Hence a raw material inventory is essential. 2. Raw materials are usually transported to the factory, site, Quite often there are delays in supply time which may disrupt operations if we don't have raw material inventory on hand. Sometimes suppliers of raw materials give bulk discounts which attracts the companies into stocking raw material even if it is not needed. 3. b) In-process inventories: During the production process raw materials are transformed into semi-finished products which are next converted into finished goods. For example, in Figure 15.1, during transformation the raw material is partly processed on A and then it moves into B. 1. It the processed material from A is not available, unit B operations get disrupted This may happen if A breaks down. Hence we have a stock point between A and B. We call this work-in-process (WIP) inventory If B breaks down, we may notice a WIP build up in between A and B, This is: unwanted inventory. This situation may however affect, other downstream operations if we don't have a stock point immediately after B. This would be our finished goods inventory. As we see here, inventories are needed for decoupling the sequential stages in the production system to maintain smooth flow. 2. c) Finished goods inventories: As we saw above, in-process inventories are converted to finished goods which go into the factory warehouse. 1. If the goods are not available in the warehouse the activities of the distributors would suffer unless they have an inventory to depend on. Similarly the retailer functioning suffers if the distributor fails to meet the requirements. Consumers have a wide range of preferences. Both distributors and retailers carry a wide range of items to meet their needs. Because of these wide ranging needs their stocking policies also differ. 2. 3. Even though the problems faced at each stage are different the basic questions asked all through while determining inventory policies are: • • How much to order, and When to order We will try to address some of the complexities mentioned above with help of several inventory models. First we will make several assumptions (see Figure 15.4) to define a idealized situation with the help of a basic inventory model. Then we will relax the assumptions and move towards more practical situations. 15.2 BASIC INVENTORY MODEL Let us take a situation from figure 15.1. We said raw materials are procured from suppliers, The company (which receives the supply) has a certain annual requirement. To satisfy this need the company resources (which cost money) are used to send orders to the supplier. The supplier processes the order (again consuming time, resource and money) and ships (either by road, rail or sea) the ordered quantity back to the company. To satisfy the annual requirement, if we order one item each time then the cost of ordering goes up. On the other extreme, if we decide to order the entire annual requirement all at once, the cost of carrying inventory goes up because we are faced with the problem of stocking the whole lot. The ordering cost may include clerical costs, follow-up costs....etc. The carrying cost is due to the huge capital (money) that is tied up. Each unit of item held in inventory for each unit of time costs us money. Thus, if we want to reduce the annual ordering cost, we would like to order in large bulks which would then increase the holding cost. Figure 15.2 and Table 15.1 show an example taking these two extreme situations. Fig. 15.2: Two Extrenue Situation Table 15.1: Two extreme Situations (Note: Situation B is much better than A. But then is this the best?) To find out if there is a still better solution, let us depend on a graphical model showing the various costs as Q varies from 0 to 12000. By substituting different values for Q we plot the results to give the holding cost curve, the ordering cost curve and the total cost curve (Figure 15.3). The minimum point on the total cost curve is given by Qo. Figure 15.3: The Economic Order Quantity This Qo is the optimal value of Q and is known as the economic order quantity (EOQ). This is the quantity for which the cost of inventory is minimum. This EOQ formula is also known as Wilson s Lot Size formula. Substituting the expression for Qo in the total cost expression and simplifying it we get: The same expression for Qo can be obtained with the help of differential calculus where we minimize the total cost expression w.r.t the variable Q. Figure 15.4 shows the basic inventory mode! along with its assumptions. Fig 15.4: Basic Inventory Model The economic order .quantity tells us how much to order in each cycle. But we would also like to know when to place the order. As we know, there is a certain time needed to process an order by both the customer and the supplier. The time between placing an order and its arrival is known as the lead time (Figure 15.4), This time (shown in the xaxis) tells us when we should be placing she order during each stock cycle. In other words, as we can see from the y-axis, there is just enough stock to last the lead time. This level of stock is known as the reorder level or reorder point ROP. Figure 15.5 elaborates the concept of ROP and it's relationship with lead time. Fig. 15.5b: ddlt > QO or LT > t (Constant LT) Fig. 15.5a & 15.5b The assumptions in Fig 15.4 make the model somewhat trivial and remove it far from reality. Real life situations are quite complex. For example we might be faced with the cases of non-instantaneous replenishment, shortages, uncertain demand patterns, resource constraints and so on. These cases can be handled by modifying the assumptions in the basic model. Before discussing these other models let us first study the sensitivity of this model. Activity A A manufacturer carries stock of an item with an annual demand of 30,000 units. Although the inventory manager cannot estimate setup cost (s) or holding cost (h) precisely. She feels that the ratio of the two is somewhere between 100 to and 150 to 1; that is 5/h=100 to s/h=150. Calculate EOQ on both conditions. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………… Activity B With annual demand of 30,000 units, s/h ratio of 100 to 1, and a lead time of ten days, what recorder point should a Macro company use? Macro is open for business 250 days per year, and sales are assumed to occur at a constant rate. What would happen if the lead time sometimes went up to 15 days? ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………… 15.3 MODEL SENSITIVITY The optimal quantity Qo = 490 seems to be a rather odd lot. Suppose we are ordering these units in a truck which can carry only 250 units at a time. In that case we would be interested to know how much extra it would cost us to order 250 units each time. Or for that matter we have a truck which can carry 1000 units say. We would like to know how this would reflect in our costs. Activity C How sensitive is the optimal Q to the s/h ratio? If s/h doubles or triples, what happens to Q*? s is the setup cost and h is the holding cost. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………… Activity D How sensitive is Q to annual demand? If annual demand doubles or triples, what happens to Q*? ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………… Fig: 15.6: Model Sensitivity 15.4 GRADUAL REPLACEMENT MODEL We will study this model where receipt of order U non-instantaneous. In the first model replenishment rate p→ ∞ Here it is finite. Under most practical situations the ordered quantity (which is either produced or supplied) is delivered over a period of time (and not instantaneously). In other words there is a gradual supply (see Fig 15.7). Let Tp be the time required to receive an order. During this time while the ordered quantity (Q) is being supplied to the warehouse, there is a simultaneous consumption due to the constant demand rate (d). Hence the rate at which the order accumulates in the warehouse is (p-d). Therefore there is a maximum inventory build-up of only Qmax and not Q Qmax will be less than Q. Let is verify this; Fig 15.7: Gradual replacement model • Qoptimal for gradual replacement is greater than EOQ for the same values of Cp Ch And R, Total cost is lower than that for the basic model. If d → p then Q→ ∞. That means we need a fully dedicated supply system to meet the demand. There is only one initial setup (a dependable system with high Cp) and the cost minimization process forces the lot size Q to be extremely high approaching ∞. There is no inventory build-up. • • Activity E Explain how the situation of the finite production (gradual replenishment) note inventory differs from the simple lot size situation. What impact does the cost of the item have on each situation? Explain. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………… 15.5 BASIC INVENTORY MODEL WITH BACKLOGGING ALLOWED We relax another assumption here by allowing shortages. We plan, to accept shortages or backorders (orders to be satisfied at a later date forcing customers to wait). Let us assume that we can make customers wait by giving them a price discount. This becomes a cost to the company. Let Cb be the cost to have one unit backordered for one year (Rs/unit/year). That is if we are asking a customer demanding one unit to wait for a year then the cost (to the company) of his waiting for one year is Cb. Figure 15.8 shows stock positions ranging from -B (demand deliberately put on backorder list) to (Q-B), the maximum level of positive inventory immediately after a lot size of Q is delivered. When amount Q is delivered B units go towards satisfying the backlogs. Intuitively, a low shortage cost would bring in a tendency to accumulate backorders (because it reduces the total holding costs). Fig 15.8: Model with backlogging • • If Cb is low, there is a tendency to accumulate backlogs If Cb = 0, it implies with no costs for backlogs there is a tendency to accumulate a huge (infinitely large) backorder which is to be satisfied by a huge single order. Note that B optimal = Q optimal = ∞ If Cb = ∞ no backorders can be tolerated. Reverts to the EOQ formula. Typically EOQ model with shortages will always have lower total costs than the basic EOQ models. • • Activity F A company orders is lot size of 2000 units. The holding cost per unit per year is $8, and the back order penalty per unit per year is $15. What should be the optimal inventory held, and what should be the maximum backorder position? ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………… Activity G With an annual demand of 2000 units, setup costs of $250, holding costs of $8 per unit per year and back order penalty costs of $24per unit per year, what is the optimal time between orders? Use 250 day working year and specify the time in days. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………… 15.6 BULK DISCOUNT MODEL In the basic model we assumed there are no quantity discounts and the material price was considered to be fixed (say p per unit). Quite often suppliers give us price breaks. For example the supplier in our example says that he would give us a discount on the unit price if we ordered in the following quantity ranges as per the following price discount schedule: If we want to take advantage of these price discounts then our order quantities will increase. Note: Because of the increased Q inventory costs of holding go up. But then what is the net effect of all these changes? Is there an overall advantage or disadvantage? To answer these questions, let us look at our changed cost expression. The total cost expression is as follows: Because of the price breaks (p;) we have a step function here. We are interested in finding out the quantity Q that is going to minimize the total cost expression. Figure 15.9 shows the total cost curve for the valid quantity ranges and the minimum cost in this curve gives the optimal quantity. When drawn to scale the minimum cost will be Rs 48520/- for a order quantity of 1000 units. We have a general procedure for calculating the optimal quantity for the bulk discount model: 1. Calculate the EOQ for each price break (in our example for P1, P2, P3,) 2. Determine if these EOQs are feasible. A feasible EOQ must fall within the 3. 4. quantity range for the corresponding price break. If the EOQ falls outside this range, it is not feasible and hence is eliminated. Calculate total cost for feasible EOQs and at price breaks. Select the quantity yielding the lowest total cost. Fig: 15.9: Total Cost Curves for model with price breaks Table 15.2 and 15.3 below summarize our example: Table 15.2: Calculation of valid EOQs P1 = Rs 5.00 Valid range EOQ Feasible? 0-499 489.90 Yes P2= Rs4.50 500-999 516.40 yes P3 = Rs 4.00 ≥1000 547.72 no Table 15.3: Total cost summary for the example problem Valid lot sizes Q = 490 (first feasible EOQ) 5.00 Cost Rs 60,000.00 cost 244.90 Q = 516 (second Q = 500 (first Q= 1000 (second feasible EOQ) price break price break quantity) quantity) 4.50 Rs 54,000.00 232.60 232.20 RS 54,464. 80 4.50 Rs 54,000.00 240.00 225.00 4.00 Rs 48,000.00 120.00 400.00 Unit price Material (price*R) Ordering (R/Q)*Cp Inventory holding 245.00 cost (Q/2)*price*Fh Total cost Rs 60,489.90 Note: As p (↓) Rs 54,465. 00 Rs 4&520.0Q order quantity Q (↑) material cost (↓) ordering cost (↓) inventory cost (either ↑ or ↓) We find there is a net advantage in ordering 1000 units for a minimum total cost of Rs 48520. Also note: As observed earlier, for EOQs ordering cost = holding cost (within rounding off errors in our example) Activity H A company annually orders one million pounds of a certain raw material for use in its own business process with annual holding costs estimated at 35% of the purchase Price of $50 per 100-Pound bag, the purchasing managers wants to decide an order size. Marginal paper work costs are $10 per order. For order of 500 bags or more the purchase price falls to $45 per bag; for orders of 1000 bags or more, the price is $40 per bag. What is the optional order size? ……………………………………………………………………………………………… ………………………………………………………………................................................ ................................................................................................................................................ .................................... 15.7 INDEPENDENT DEMAND SYSTEMS FOR MULTIPLE PRODUCTS Suppose you walk into a company and find that there are five independent demand items which need to he stocked in the warehouse. The annual requirements for each item and the price per unit is also known. The carrying charge Fh = 0.2 and the cost of ordering C = Rs 10 per order. The unit space occupied by different item types is also given. The company is currently adopting an inventory policy of ordering each item in lots thrice a year. The table below summarizes the existing data: Table 15.4: Existing inventory policy of a certain company Item (i) Annual Req, Ri. Cost perunit (pi) 12000 21000 18000 9000 21000 Rs5 8 6 9 4 Size in m3/unit 0.5 1.0 1.0 3.0 2.0 Existing Policy Existing (# orders) Policy (order qty) 3 3 3 3 3 4000 7000 6000 3000 7000 A B C D E There are several questions that come to mind now: a) • Questions regarding the existing policy (refer Table 15.5): What is the total volume requirement for each type of item in this inventory? (see Table 15.5; column 4) When all lots arrive at the same time, the total space is given by: order qty * space required per unit • What is the average space requirement (in m3) for each type? What is the total average space requirement? (see Table 15.5, column 5) Note: Average inventory Hence, Average space = (Maximum+ Minimum)/2 = (qty ordered + 0) / 2 = Average inv *.space per unit (see col 5) Total average space = 19000m 3 • What is the total "rupee volume" (or rupee value) for each type? (see Table 15.5, col 6) The rupee value becomes an aggregate measure of the inventory size. Thus different size and types of inventory item can be aggregated under one measure. The rupee value of each type is given by: Order quantity * price per unit • What is the average inventory size (in rupees) for each type? What is the total average inventory (TI) in this case? (see Table 15.5, col 7) Average inventory (in rupees) Total average inventory (TI) • Each item is ordered three times a year. Total orders (TO) • =15 What is the total inventory cost for this existing policy? (in other words, we would like to know the cost of adopting this policy) Total cost = total holding cost + total ordering cost = = = Item A orders 3 Qty 4000 [Avg inventory (in Rs)*Fh + [(# orders)*Cp] TI * Fh + TO * Cp 83500 * 0.2 +15 * 10 = Rs 16850 Total Space (in m3) 2000 Avg Space (in m3) 1000 Total Value Avg Size (in (in Rs) Rs) 20000 10000 = Average inv * price per unit = Rs 83500 What are the total orders for the current policy? (see Table 15.5, col 2) Table 15.5: Total volume and value of inventory items B C D E 3 3 3 3 15 7000 6000 3000 7000 7000 6000 9000 14000 3500 3000 4500 7000 19000 56000 36000 27000 28000 28000 18000 13500 14000 83500 b) Item A B C D E Is there a better solution? (refer Table 15.6): Table 15.6: An optimal solution EOQ (in units) 489.8979 512.3475 547.7226 316.2278 724.5688 24.4949 40.9878 32.86335 28.4605 28.98275 155.7893 Orders Avg space (in m3) 122.4745 256.1738 273.8613 474.3416 724.5688 1851.42 Avg size (in Rs) 1224.745 2049.39 1643.168 1423.025 1449.138 7789.465 • We know that EOQ is the quantity for which the inventory cost is minimum. If we take the EOQs for each type then is there a chance of minimizing the total cost? Calculate the optimal quantities for each type using the EOQ formula. Thus EOQ for each item is given by: The EOQ values are shown in Table 15.6, column 2. Columns 4 and 5 give the corresponding average space (in m3) and average size (in Rupees). Column 3 gives the number of orders (annual requirement, quantity ordered) and total orders. Total orders (TO) policy) • What is the average size of the inventory (in rupees)? Average size = (EOQ/2) * price per unit (see col 5) (compare with Rs 83500 of existing policy) • What is the average space requirement (in m3)? Average space = (EOQ/2)*space per unit (see col 4) 1851.5 m3 (compare with 19000 m3 of existing policy) • What is the cost of this optimal policy? Total cost = total holding cost + total ordering cost = orders)*Cp] = [Avg inventory (in Rs) * Fh ] + [(# TI * Fh + TO * Cp Total avg space requirement = Total average size (TI) = Rs 7789 = 155 (compare with 15 of the existing = = 7789 * 0.2 + 155 *10 1558 + 1555 = Rs3108 When compared with Rs 16850 of existing policy we find a lot of improvement. Please note that the total ordering cost is almost equal to the total holding cost (within rounding off errors). c) Now what if there are resource constraints? In real life there could be several resource constraints. For example, constraints on working capital, on storage space or on number of orders per annum. To understand such situations, let us extend our example by imposing a constraint on the amount of investment in inventory. We are told that the maximum we can invest on an average in inventory is equal to Rs 6000. The current policy has an average investment in inventory of Rs 83500 which is very high. Even the optimal policy with an average investment of 7789 is higher than this amount. So what to do now? We can handle this situation by modifying the EOQ formula as follows: EOQ formula for each item is given by: Here we are attaching a price (λ) for using the limited resource (money in this case). In other words we pay a price for each unit of the limited resource in use. Thus, for costly items the denominator is inflated more than for less costly items (e.g. λ* pD > λ*pb). For λ = 0 the formula is same as the EOQ formula. As λ increases the lot size decreases as compared to the economic order quantity. Now λ = 0.05 means we are paying 5 paise for investing each rupee in the inventory. We try to find the value of λ in an iterative manner by increasing the value of λ in steps with the objective of reducing the lot sizes, thereby reducing the amount invested in inventory. We do this until the upper limit on inventory is just met. See Table 15.7 below where we calculate optimal inventory sizes for different values of λ. Table 15.7: Order quantities and corresponding average inventory size in rupees for different values of λ EOQ (1=.05) 2190.89 3666.06 2939.39 Average size(Rs) 1095.45 1833.03 1469.69 EOQ 1 = .10 2000.00 3346.64 2683.28 Average size(Rs) 1000.00 1673.32 1341.64 EOQ (1=.137) 1887.02 3157.58 2531.70 Average size(Rs) 943.51 1578.79 1265.85 2545.58 2592.30 1272.79 1296.15 6967.11 2323.79 2366.43 1161.90 1183.22 6360.07 2192.51 2232.75 1096.26 1116.37 6000.78 Thus at λ=0.137 we are just able to satisfy the constraint on our working capital. Activity I Compute the corresponding space requirements for each of the different values of λ. Suppose there is a limitation on the average space available (being equal to 1750 m3 say). What is the modified formula now? Try to find out how the order quantities change if there is this constraint on space instead of the working capital ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………… d) What if we do not know the values of the carrying charge (F) and cost of ordering (Cp)? In most companies it is difficult to know the above mentioned values. Under such circumstances we depend on a tool known as the optimal policy curve which helps us fix inventories at an aggregate level Each point in this curve is an optimal policy. If we decided to keep the number of orders constant the1n we eah move towards a better solution by decreasing the average investment in inventory or vice-versa. The curve exchanges total average inventory size (Tl) with total orders (TO) in such a manner that (TI) * (TO) = a constant (K) where, 1 K = ∑ ( Ri * Pi ) 2 Thus, in our example (TI)*(TO) = (l/2)*(1.557.893)2 = 1213515. It is the equation to a rectangular hyperbola which is shown in Figure 15.10. Let us take the first optimal policy calculated in (b) above. Our (TI)*(TO) = 7789.465*155.7893 = 1213515 same as the constant (K) above. Aspart of your practice problem try to verify this result again by taking the optimal policy in the resource constrained problem discussed in (c) above. Fig: 15.10: Optimal Policy Curve The optimal policy curve indicates that the existing policy having a TO=15 and Tl equal to Rs 83500 is very close to the optimal policy point on the curve having a Tl equal to Rs 80901 for a TO =15. Likewise we can verify that for a TO=155 the Tl is Rs 7829. Thus it is a handy tool that facilitates quick decision making. 15.8 MODELS WITH UNCERTAIN DEMAND So far we have been discussing only the deterministic models where the demand patterns arc predicted with certainty. We also do not see any variations in the lead time. But under real life situations there are a lot of variations (both in demand as well as lead time). Here we depend on stochastic models. Stochastic models depend on historical data. By looking at past demand data we can assign weights to different levels of demand data. Higher weights indicate a greater frequency of occurrence or a greater likelihood of that event occurring (note: each level of demand is an event). Because of such uncertainty in the demand pattern (similar uncertainties are there with the lead time) we look for expected outcomes, such as, expected demand, expected revenue, expected profit, expected inventory, expected shortages and so on. We will consider the following cases where we are faced with uncertainties because of the variations: Case 1: A vendor stocks perishable items which, if not consumed, have to be either thrown or salvaged at the end of the stock period. Thus if demand falls short of the quantity stocked we incur a cost of overstocking. Some examples: • A newsboy at the news-stand stocking newspapers for the day. These papers become useless the following day. At best they can be sold as waste paper. The local bakery stocking fresh bread for the day. Monthly magazines stocked for a month. • • There could be instances where demand remains unfulfilled. These become lost opportunities which are considered as the costs of understocking. These problems are solved with the help of SINGLE PERIOD MODELS. Note that a period could be a day, week, month or even a certain lead-time. For each of these periods we must have the historical demand data. Table 15.8 below shows a couple of sample historical data: Table 8.1 (a): Daily demand data daily demand for bread (d) 30 40 50 60 70 probability of demand prob (d) .10 .20 .40 .10 .20 Table 8.1(b): Lead time demand data demand probability during lead of demand time during LT (ddlt) prob (ddlt) 30 40 50 60 70 .10 .20 .40 .10 .20 Case 2:Items stocked during a stock cycle, if not consumed, can be used in the subsequent cycle. If demand is unfulfilled then we end up having stockouts which result in either lost sales or backlogs. When customer demands are filled at a later date, they are known as backlogs (i.e. customers are willing to wait if they are given a discount on the price). These are items having longer shelf life. For these problems we depend on MULTIPLE PERIOD MODELS where we have to determine the quantity to be stocked (Q) and the reorder point (ROP). See Figure 15.11. Fig 15.11: Multiple period model Single Period Model Popularly known as the newsboy model. A newspaper vendor has to decide how much to stock so that, he can maximize his expected profits at the end of the day. The vendor depends on the following historical data (Table 15.9): Table 15.9: Data for the Newsboy model daily demand for newspaper (d) 75 probability of demand prob (d) .05 Note: 100 125 150 175 200 225 .10 .20 30 .20 .10 .05 prob(d) = probability demand will be equal to d. (eg: there is a 5% probability demand will be equal to 75) Each newspaper costs him Rs 1.50 and he sells it at Rs 2.00. Unsold papers are useless and are to be thrown. Hence the salvage value for unsold papers is zero, The vendor has to decide how much to stock so that he can maximize his profits. Because of the uncertainty in demand, the quantity stocked could be either less than, equal to or greater than the demand. If d > Q he will face a penalty for understocking. In our example, Cu = sale price This is his cost of understocking (Cu). - cost price = S-C = 2 -1.50 = Which is the cost of profits foregone. 50p He faces a penalty for overstocking. This is In our example, Co = C = his cost of overstocking (Co). Which is the Rsl.50 purchase cost of the unsold units. ifd<Q The problem is where should he stock? He has the benefit of past experience (see the historical data available above). Suppose he stocks a quantity Q = 100, Table 15.10 shows the purchase cost, revenue and profits for the various demand levels (each demand level is an event). For example, if demand = 75 men number sold = 75, revenue – 75 * 2.00 = Rs 150 and since purchase cost = 100 * 1.50 = Rs 150, profit = revenue - cost = 0 Table 15.10: The Newsboy problem (d) prob (d) Cost C*Q Number sold Revenue (#sold*S) profit (revcost) 75 100 125 150 175. 200 225 .05 .10 .20 ..30 .20 .10 .05 150 150 150 150 150 150 150 75 100 100 100 100 100 100 150 200 200 200 200 200 200 0 50 50 50 50 50 50 Because of the uncertainty in demand he is interested in the expected values of revenue and profits. His aim is to maximize expected profits. The objective is to maximize this expression. An incremental analysis approach is used to determine the optimal quantity Q. According to this analysis, for maximum profit at Q. Without getting into the details of this analysis, we state here the final result. We want the smallest value of Q which ensures that the following inequality remains valid: Cu Prob (d ≤ Q) = Co + Cu The left hand side of this inequality shows the cumulative probability distribution of demand being less than or equal to Q. Thus prob (d ≤ 100) = 0.15, prob (d ≤ 75) - 0.05, prob (d ≤ 225) = 1.00 and so on. The right hand side of this inequality introduces the setvice level concept taking into consideration the costs of overstocking raid understocking. For example, if Cu is very high as compared to Co, then the right hand side fraction also becomes very high (close to 1). In that case we would like to maintain a high service level for satisfying the demand. That means we cannot, afford to lose customers by understocking. Similarly, if Co is high compared to Cu, we cannot afford to dump large quantities in the inventory. The service level is reduced and we can afford to lose customers. In our example Co = Rs 1.50 and Cu 2 - 1.50 = 0.50 Thus Cu 0.50 = = 0.25 Co + Cu 1.50 + 0.50 Here we see a rather low service level, because the cost of understocking is quite small The smallest value of Q for which the inequality is 125 Note that Prob (d ≤ 125) = 0.35 > 0.25 * (see Tablet 5.11) Hence Qo = 125 is the optimal quantity for which we can maximize the expected profits: Expected profit (Q=125) = ∑(S*d-C*Q)*prob(d) + ∑(S-C)*Q*prob(d). = 11.875+40.625 = Rs 52.50 Table 15.11: Newsboy problem (contd) D prab(d) Prob (d ≤ Q) sd-cQO. (sd-cQ)p(d) (s-c)Qp(d) 75 100 125 150 175 200 225 0.05 0.1 0.2 0.3 0.2 0.1 0.05 0.05 0.15 0.35 0.65 0.85 0.95 1 -37.5 12.5 62.5 -1.875 1.25 12.5 18.75 12.5 6.25 3.125 11.875 40.625 the inequality is satisfied here, hence Q = 125 Then depending on historical data for demand during lead-time we try to fix the ROP using the newsboy model concept. We find the largest ROP for which the following inequality holds: Note: 1. We can arrive at the same result by selecting the largest value of Q for which the following inequality holds. 2. Instead of throwing the unsold papers, if the newspaper vendor is able to dispose them as waste papers for a certain salvage value (say L). In that case he gets a relief on his overstocking costs and the understocking costs remain the same. Thus Co = C - L and Cu = S - C. The optimal Q is obtained as before. The expression for expected profits is now modified as follows: Activity J A shop owner has four different retail locations, each featuring magazines as a major product line. The demand for a popular monthly magazine varies continuously from 500 to 1200 copies at all stores combined. Ordering is centralised and magazines can he moved easily from store to store. The magazines costs $125 hundred and sell for $2.25 each when purchased in lots at this prices the publisher accepts no returns. What should be the ordering quantity for the next period? ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………… Multiple Period Models This is similar to the basic inventory model with shortages. Only now we are faced with uncertainties in the form of either variations in demand or variations in lead time. The objectives are to decide how much to order and when to order. That is, the quantity to be ordered (Q) and the level at which the order is placed (ROP) are to be determined while minimizing the total cost of ordering, holding and shortages (or backorders). Under real life situations, companies are more vulnerable when stocks are low. Hence, as a hedge against stockouts, a buffer stock (or safety stock) is frequently added to the expected demand during lead time t; absorb variations in demand and lead-time. Note: If safety stock is high, carrying costs will go up and if safety stock is low, stockout costs may go up. Because of this vulnerability, determining the ROP is more important. In following situations (A, B and C) we will try to find the ROP: Situation A: The cost of shortages (Cs) is known Let us assume we can quantify the cost of not meeting the customer demand. Also we know the average annual requirement, holding cost and ordering (or preparation) cost. For such problems we determine the optimal order quantity (Q0) using the EOQ formula. Even though it is based on a deterministic model it helps us in approximately evaluating the cost of overstocking and understocking. Then depending on historical data for demand during lead-time we try to fix the ROP using the newsboy model concept. We find the largest ROP for which the following inequality holds: /-. We also have the lead time historical demand data: Table. 15.12: Demand data Which means the chances of a stockout should be very low. So we find ROP by choosing the largest value for which the inequality is satisfied. prob (d ≥ 50) = 0.10 > 0.0041 (see Table 15.12 above) So ROP=50. Hence (Q,ROP) is equal to (490,50). According to this policy we place an order for 490 units when the stock level during any stock cycle comes down to 50 units. Situation B: The cost of shortages (Cs) is unknown In real life situations it is difficult to quantify the cost of shortages. There are some intangible negative aspects of shortages, backlogs and backorders. In these situations we find Q as usual using the EOQ formula. To find ROP we use a service level concept. Service level specification is a policy decision and gives a certain probability of meeting the demand from inventory. For example we may have the following policies specified by management: Let us take one example and understand both the above service level concepts. The data is same as in the example discussed in Situation A Except that Cs, is unknown For this data our Qo = 490 units. Table 15.13 shows the calculations. Table 15.13: Service level concept (Examples) if x = 10% (policy) find ROP when ROP = 0, stockout 90% of time > 10% when ROP = 10, stockout 80% of time >10% when ROP = 20, stockout 50% of time > 10% when ROP = 30; stockout 30% of time > 10% when ROP = 40; stockout 10% of time = 10% Just satisfied at ROP = 40 Thus, ROP = 40 and Q = 490 if y = 99% find ROP Permissible shortages per cycle = [(100-99)/100] * 490 = 0.49* Now, When ROP=50; E(s) = Σd>ROP(d-ROP) * p(d) = (50 – 50) * 0 = 0 < 49 when ROP = 40; E(s) = (50-40)*0.1=1 >0.49 (here it, exceeds the permissible shortage) Thus, ROP = 50 and Q = 490 Situation C: Finding ROP with known demand data distributions If the demand data follows some well defined distribution (e.g. a normal or a Poisson distribution) then we can find ROP and the safety stockes as follows. All we will be doing here is to add safety stock to the expected demand during lead time to set the ROP level. The service level defined by the managerial policies help us in determining the safety stock. Now service level is the probability that the demand during lead time will be available from inventory. Figure 15.12 shows the relationship between expected demand during lead-time, reorder point and safety stock. If service level is x then the risk of a stockout is given as (1-x) as shown in the figure. For deterministic situations we said, ROP = demand rate * lead-time, Since both demand and lead-time are uncertain in nature. Demand varies because of the uncertainties in customer preferences. Lead-time could be affected by strikes, transportation difficulties did other supplier related uncertainties. The company faces stockouts if there is either a sudden surge in demand or there is an unnecessary delay in the delivery of the order quantity. Therefore when stocks " are low the company would prefer to have a safety stock as a hedge against stochastic. Hence, for stochastic models. Reorder point = Expected demand during lead-time + Safety stock ROP = E(ddlt) + SS Figure 15.12: Relationship between ROP and safety stock • Reorder point when demand is varying and lead time is constant: We assume that daily demand is uncertain, independent and normally distributed. We sum the daily variances then take the square root to get the standard deviation. ROP = d * LT + Z (σ d ) 2 * LT number of standard deviations corresponding to the service level probability • Reorder point with variable lead-time and constant demand: Since lead-time is varying we take the average lead cad-time and multiply with daily demand rate to get the average demand during lead-time. If the standard deviation of the leadtime is αLT then, ROP = d * LT + z * d * αLT • Reorder point with variable lead-time and demand: Here the total variance in lead-time demand is the sum of variance due to demand fluctuations (αd2 * LT) plus the variance in the lead-time (α LT2 * d2 ) thus, ROP = d * LT + z (σ d ) 2 * LT + (σ LT ) 2 (d ) 2 15.9 SELECTIVE CONTROL OF INVENTORY ITEMS In real life situations we have a large number of items in the inventory. It may not be possible to review (which includes counting, placing orders, receiving stock) such a large number of items. Review activities take time and cost money. Hence items are usually classified into important and less important groups. The important ones get more attention than the others. The goal is to keep stocks at low level while giving good service. One of the selective control methods is known as ABC analysis. It is governed by the Pareto principle which segregates the vital few from the trivial many. According to that principle it can be shown that around 10% of the inventory items would be accounting for around 70% to.80% of the inventory's total annual rupee value. Next 20% of the items account for around 20% of the annual rupee value and the remaining 70% around 10% of the value. For A class items close control is required. For C class items routine checks at long intervals is adequate. Similarly we have a VED classification which stands for vital, essential and desirable. Non availability of vital items would bring production to a halt. Hence they need to be stocked adequately. Stockout of essential items may result in expensive procurement and stockout of desirable items may cause minor inconvenience. More details of each inventory items are discussed in unit 14 15.10 SUMMARY We have discussed inventory control systems for independent demand items. "We have tried to understand a production-distribution system and why there is a need for the inventory function.Various inventory related costs; have been mentioned. These costs are traded-off while deciding how much to order and when to order. Models under both deterministic and probabilistic situations have been presented. 15.11 SELF-ASSESSMENT EXERCISES 1. Consider the following types of items carried in "a retail store: fight bulbs, phonograph records, refrigerated drugs. Discuss the probable cost structure for each of these items including items cost, carrying cost, ordering cost and stockout cost. Why stockout cost difficult to determine? Suggest an approach which might be used to estimate it. A student was overheard saying. "The EOQ model assumptions are so restrictive that the model would be: hard, to use in practice. Is it necessary to have a difference model for each variation in assumptions why and not? Suppose you were managing a chain of retail department stores. The inventory in, each store is computerized, but these are a large number of different items. As it top manager how would you measure, the overall inventory management performance of each, store. How would you use this information in your relationship with the individual store managers? The owner of a hotel has 32 rooms is trying to determine whether to build an addition incurr stockouts reffering customers to competitor when demand exceeds supply. The cost of maintaining a hotel room averages $ 15/day. A typical room rents for $45/ right. During the last six months, demand has averaged as follows: Range of actual Daily demand (in Rooms) 0-20 Days Demand fits Range 2. 3. 4. 5 90 50 21-30 31-40 41 on more Total 182 days 6. Sales Ordering cost Carrying Change Item cost a) b) c) 32 10 The speedy Grocery Store carries a particular brand of coffee which has the following characteristics: = = = = 10 cases per seek $10 per order 30 per cent per year $60 per case How many cases should be ordered at a time? How offer will coffee be ordered? What is the annual cost of ordering and carrying coffee? d) What factors might cause the company to order a large or smaller amount than the EOQ? 7. An appliance store carries a certain brand of TV which has, the following characteristics: = = = = = 100 units $ 25 per order 25 per cent per year $ 400 per unit 4 days Average annual sales Ordering cost Carrying cost Item cost Lead time a) b) c) 8) Determine the EOQ Standard deviation of the daily demand = 0.1 unit working days per year = 250 Calculate the reorder point for a 95% service levels assessing normal demand. How far apart would orders be placed on average? For the following total cost TC, find the optimal order quantity Q*. A is a constant. Is this a minimum on a maximum cost point? Why? TC = (27 + A)Q + 9) 10) +274 Inventory control is a national process in which decisions are offer made irrationally. Explain. Given a probability distribution of demand and a distribution of lead time, what alternatives exists for finding the probability distribution of demand during lead time? Select one attentives and explain how it works. Why is the distribution of demand during lead time important? An electrical motor housing has an annual usage rate of 75000 units/year, an 11) ordering cost of $20, annual carrying change of 15.4 per cent of the unit price. For lot sizes of fewer than 10,000 the unit price is 0.50, for 10,000 or more the unit price is $45. Delivery lead time is known with certainty to be two weeks. Determine the optional operating doctrine. 12) Demand for the local daily newspaper at a news stand is normally distributed with a daily news of 210 copies and standard deviation of 70. A newspaper sells for $25 cents and costs $20 to purchase. Day old newspapers are very seldom requested, and therefore they are destoyed. What should be the news stand's daily order be to maximise profits? A bank purchases promotional ball point pens for $3 each. The company that supplies the pens suggests that if the imprinted pens were ordered is twice the quantity, a 35 per cent discount could be arranged. At present the bank orders 100 pens every two months. Ordering costs are $12, and bank's cost of money is 18 per cent. What ordering policies shold be followed? You have a product whose average weekly sales are 600 units. By looking at the records of past demand, you find that the demand pattern has followed the distribution below: Weekly Demand Above (in units) 400 450 500 550 600 650 700 750 800 Per cent of the time Demand Above j 100 90 79 64 50 22 8 3 0 13. 14. The cost of carrying an item on inventory for one year is $1.30. Ordering cost is fixed at $72. Lead time is constant at one week. The stockout policy has been set to allow two stock outs/year on average. Determine the order quantity and the safety stock that minimised the variable cost. 15. 1 335 The monthly demand for an item was recorded as follows: 2 295 3 75 4 305 5 304 6 338 7 290 8 305 9 285 10 275 11 295 12 311 Calculate the mean absolute deviation and the standard deviation of forecast error if the forecast is 300 per month. What safety stock would be required if orders are received monthly and if two stockout occurancies is twelve months are acceptable. 15.12 FURTHER READINGS Adam, E.E. and Ebert, R.J. (1995) Production and Operation Management, fifth edition, Prentice-Hall of India Private Ltd, New Delhi. ButTa, E. S., and Sarin, R. K. (1994) Modern Production/Operations Management John Wiley & Sons, Inc. Chary, S. N., (1988) Production and Operation:; Management,T-dt'dMcGrdVJ-R\\\, New Delhi Chase, R.B., and Aquilano, N. J. (1995) Production and Operations Management: Manufacturing and Services, Richard D Irwin Inc. Hax, A C., andCandea, D. (1984) Production and Inventory Management. Prentice-Hall Inc., Englewood-Cliffs. Narasimham, S.L., Mcleavey, D.W. and Billington, P.J.(1997) Production Planning and Inventory Control, 2nd edition, Prentice Hall ')f India, N.Delhi.