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Managing Uncertainty in the Supply Chain: Safety Inventory Susan Cholette DS855 – Fall 2006 11-1 Outline • The role of safety inventory in a supply chain • Determining the appropriate level of safety inventory • Impact of supply uncertainty on safety inventory • Impact of aggregation on safety inventory • Impact of replenishment policies on safety inventory • Estimating and managing safety inventory in practice • Managing safety inventory in a multi-echelon supply chain will implicitly be covered in Chapter 16: SC-coordination 11-2 The Role of Safety Inventory in a Supply Chain • Forecasts are never completely accurate • If average demand is 1000 units per week, every once in a while actual demand is 1000. But about half the time actual demand will be greater than 1000, and about half the time actual demand will be less than 1000 • If you kept only enough inventory in stock to satisfy average demand, half the time you would run out • Safety inventory: Inventory carried for the purpose of satisfying demand that exceeds the amount forecasted in a given period 11-3 Role of Safety Inventory • Average inventory is cycle inventory + safety inventory • The fundamental tradeoff: • Raising the level of safety inventory provides higher levels of product availability and customer service • Raising the level of safety inventory also raises the level of average inventory and therefore increases holding costs • Very important in high-tech or other industries where obsolescence is a significant risk (where the value of inventory, such as PCs, can drop in value). i.e. Compaq vs. Dell in PCs • As cycle inventory had a cost of hCQ/2, a 100 lot order of $10 wine (at 20% cost of capital ) had an annual holding cost of ___? What would the safety stock cost be to hold safety stock of 100 bottles? 11-4 Determining the Appropriate Level of Safety Inventory • Two questions that we need to ask 1. What is the appropriate level of safety inventory to carry? 2. What actions can be taken to improve product availability while reducing safety inventory? • We will discuss the following • Demand uncertainty • Product availability • Replenishment policies • Cycle service level and fill rate • Determining safety level given desired cycle service level or fill rate • Determining the impact of required product availability and uncertainty on safety inventory 11-5 Measuring Demand Uncertainty • Appropriate level of safety inventory determined by: • supply or demand uncertainty • desired level of product availability • Demand has a systematic component and a random component • The estimate of the random part is the measure of demand uncertainty and is usually measured by the standard deviation of demand • Notation: D or m = Average demand per period (day or week most common) sD = standard deviation of demand per period L = lead time = time between when an order is placed and received Coefficient of variation is the size of uncertainty relative to the demand: cv = sD / m = std_dev-of_demand/ mean_demand • You can ignore the covariance equation, r, in the textbook, as for all lectures, homeworks and quizzes/final we will assume demands are independent between regions/stores/days and thus will have no measurable correlation effects 11-6 Measuring Product Availability: Terms • Product availability: a firm’s ability to fill a customer’s order out of the available inventory • Not Rainchecks or “We’ll Fed-Ex it to you free of S/H” • Out-of-stock: (OOS) the product is no longer available, we “run out” • not a problem per se if no customer demand the product before our next order comes in • Fill Rate (fr): fraction of demand that is satisfied from inventory • Can relate to product or orders (multiple products) • We will focus on customer demand for a single item in 855 • Cycle service level: (CSL or just SL) the fraction of replenishment cycles that end with all customer demand met 11-7 Replenishment Policies Replenishment policy: decisions regarding when to reorder and how much to reorder • Continuous review: inventory is continuously monitored and an order of size Q is placed when the inventory level reaches the reorder point ROP • Periodic review: inventory is checked at regular (periodic) intervals and an order is placed to raise the inventory to a specified threshold (the “order-up-to” level) (a.k.a. Fixed Order Intervals) • We will first discuss Continuous Review, and then briefly cover Periodic Review towards the end 11-8 Continuous Review Policy: Safety Inventory and Cycle Service Level L: Lead time for replenishment- if it remains invariant D: Average demand per unit D L DL time (sometimes m) sD: Standard deviation of demand per period sL LsD DL: Mean demand during lead ss F 1 (CSL) s L S time sL: Standard deviation of ROP D L ss demand during lead time CSL: Cycle service level (also CSL F ( ROP, D L , s L ) denoted %SL or SL) ss:Safety stock Average Inventory = Q/2 + ss ROP: Reorder point 11-9 Review: Using Standard Normal Distributions • Recall from BUS786 (and statistics- DS512) z = (D-m)/s Once you know z, you can determine SL (and vice versa) How? • Option 1: The Standard Normal can be referenced in Excel, F(z)=NORMSDIST(z) gives %SL i.e. NORMSDIST(1.65) = .95 F-1(SL) = NORMSINV(SL) gives the z value corresponding to the %SL, i.e. NORMSINV(.99) = 2.33 You can use the “regular” normal distribution shown in the book, but it is easier to calculate the z value and just use the Standard Normal. • See next slide for Option 2: Table-Lookups On any test or quiz you will be provided sample values or a table 11-10 Option 2: Table Look-ups for Standard Normal • If we discover z = 1.32, our SL = 90.66% • What z does an 80% SL correspond to? 11-11 Examples 11.1+11.2: Estimating Safety Inventory (Continuous Review Policy) Example: Weekly demand for PalmPCs averages 2,500 with a standard deviation of 500 units. We place an order of 10,000 units when we drop to 6000 units, and the order takes 2 weeks to arrive. 1. What is our average inventory? 2. What is the average time a unit spends on the shelf? 3. What is our chance of running out of stock before the order arrives? • 1. DL = DL = (2500)*(2) = 5000 sL = sqrt(LT)* sL = 1.41*500 = 707 ss = ROP - DL = 6000 - 5000 = 1000 Cycle inventory = Q/2 = 10000/2 = 5000 Average Inventory = cycle inventory + ss = 5000 + 1000 = 6000 • 2. Average Flow Time = Avg inventory / throughput = 6000/2500 = 2.4 weeks • 3. %SL = NORMSDIST (ss/sL) = NORMSDIST(1000/707) = 92% (This value can also be determined from a probability distribution table) • So we have an 8% chance of running out 11-12 Estimating Unmet Demand: Fill Rate • Fill Rate: Proportion of customer demand satisfied from stock ESC • Stock-out occur when demand during fr 1 lead time exceeds the reorder point Q • ESC is the expected shortage per cycle ss (average demand in excess of reorder ESC ss * (1 F S ) s L point in each replenishment cycle) • ss is the safety inventory ss • Q is the order quantity, which is the s L f s L S average demand, D, and so can be used interchangeably ESC = -ss{1-NORMDIST(ss/sL, 0, 1, 1)} + sL NORMDIST(ss/sL, 0, 1, 0) 11-13 Example 11.3: Evaluating Fill Rate 1. This example can also be performed in Excel • Examples on sheets 1 and 2 in Ch11_ss_inv.xls Given ss = 1,000, Q = 10,000, sL = 707, Fill Rate (fr) = ? ESC = -ss{1-NORMDIST(ss/sL, 0, 1, 1)} + sL NORMDIST(ss/sL, 0, 1, 0) = -1,000{1-NORMDIST(1,000/707, 0, 1, 1)} + 707* NORMDIST(1,000/707, 0, 1, 0) = 25.13 For every order cycle, we expect to be short about 25 units fr = 1- ESC/Q = 1- (25.13)/10,000 = 0.9975 So only .25% of demand is unmet (yet have a mere 92% CSL!) 2. Second (easier!!) option for calculation • Look up E(z), given z or SL on Unit Normal Loss Table • I will provide you a copy of this Table for quizzes and tests • ESC = E(z) sL • Overall Fill Rate = 1- ESC/Q 11-14 Service Level and Fill Rate • Fill Rate and Service Level are not the same! • The Fill Rate increases as Service Level increases, but is affected by other factors such as… • Standard Deviation of Demand • Lead Time • Order Size • Stock-outs themselves (hence CSL) are not the problem- if we run out of inventory, but have no customers until the next order comes in, we have no lost sales- so no problem! • For most real-life situations, Fill Rates usually turn out to be much higher than Service Levels 11-15 Example 11.4: Evaluating Safety Inventory Given CSL Demand for LegosTM: D = 2,500/week; sD = 500/week L = 2 weeks; Q = 10,000; CSL = 0.90 Calculations show: DL = 5000, sL = 707 (from earlier example) ss = FS-1(CSL)sL = [NORMSINV(0.90)](707) = 906 • this value can also be determined from a Normal probability distribution table ROP = DL + ss = 5000 + 906 = 5906 11-16 Evaluating Safety Inventory Given Desired Fill Rate D = 2500/wk, sD = 500/wk, Q = 10000, LT = 2wks If desired fill rate is 97.5%, what safety inventory should be held? • ESC = (1 - fr)Q = 250 • We aren’t going to attempt to take the inverse of the ESC function(!), so we have two options: See sheet 2 of Ch11_ss_inv.xls Option 1) Using Excel, plug different values of SS in- the larger the SS, the lower the ESC. Option 2) Solve for E(z), given ESC = E(z) sL Then look up closest z on the lookup table. E(z) = 250/707 = .35 -> z = .1 (or a CSL of 54%) • Discussion: how can CSL be so low for a high Fill Rate? • BTW, it is possible to have negative values for z. This is when you order less than you expect to be able to sell. 2. Get SS = 67 units • What happens when we increase our desired fill rate? 11-17 Determine Safety Inventory for a Desired Fill Rate (try different values of ss) Desired Fill Rate Necessary Safety Inventory 97.5% 67 98.0% 183 98.5% 321 99.0% 499 99.5% 767 11-18 Impact of Supply Uncertainty • Everything we’ve done so far assumes that our suppliers will deliver the product within the specified LT. But what if that is not the case and LT is variable? (Assume normal distribution) • D: Average demand per period • sD: Standard deviation of demand per period • L: Average lead time • sL: Standard deviation of lead time D L DL s L Ls D s 2 D 2 2 L 11-19 Example: Impact of Supply Uncertainty Daily Demand for Computers: D = 2,500/day; sD = 500/day But now Lead time is variable: L = 7 days; sL = 7 days Our order and SL policies: Q = 10,000; CSL = 0.90; DL = DL = (2500)(7) = 17500 sL L s 2 D 2 sL D 2 (7) 5002 (2500)2 (7)2 17550 So ss = F-1s(CSL)sL = NORMSINV(0.90) x 17550 = 22,491 computers Open example on sheet 3 of ch11-ss-inv.xls 11-20 Impact of Supply Uncertainty • Given demand averages 2500/day with sD = 500/day and that average LT = 7 days Safety inventory when sL = 0 days is 1,695 Safety inventory when sL = 1 is 3,625 Safety inventory when sL = 2 is 6,628 Safety inventory when sL = 3 is 9,760 Safety inventory when sL = 4 is 12,927 Safety inventory when sL = 5 is 16,109 Also, compare to LT = 14 days, with sL = 0 is 2398 11-21 Impact of Required Product Availability and Uncertainty on Safety Inventory • As desired product availability (as measured by service level or fill rate) increases, required safety inventory increases • Demand uncertainty (sL) increases, required safety inventory increases • Managerial levers to reduce safety inventory without reducing product availability include: • reducing supplier lead time, L or reduce variability in lead time (better relationships with suppliers) • reducing uncertainty in demand, sL (better forecasts, better information collection and use) • 9/2005 CSCMP Forum: Market conditions • Ghiradelli’s clients’ #1 concern: On-time delivery, neither late or early 11-22 Impact of Using Periodic Review Instead of Continuous Review Policies • To date we’ve assumed that we can re-order when stock drops to a ROP. But what if we can order only at fixed, pre-determined intervals? • Instead of setting Q, now use an Order-up-to-level (OUL) that we place every T periods, where OUL = : D(L+T) +ss – A • A = on-hand inventory, where, generally, we’d expect: A = ss + D*L • We can determine safety stock, ss = z* sT+L where: • D: Average demand per period • sD: Standard deviation of demand per period • L: Average lead time • T: Review Interval DT L D(T L) s T L s D T L 11-23 Example: Periodic Review Policy • Take the demand distribution from the Legos™ example and assume that Lead time is constant at 1 week, but that we are only allowed to place an order every 4 weeks. How does our Safety stock differ from using ROP policy? See Sheet 5 in ch11_ss_inv.xls D = 2,500/wk; sD = 500/wk L = 2 weeks T = 4 weeks, CSL = 0.90; DL+T = D(L+T) = (2500)(2+4) =15,000 s T L s D T L 500 * 4 2 1225 SS s D T L * normsinv(CSL) *1.28 1570 • Every 4 weeks we order up to the level of 16750 units (order size adjusted downward by existing inventory) • Our safety stock is 1570 • If we could order with ROP, our Safety stock would be 906 boxes, or 58% of what is required now. If annual H is only $.1/box, the difference in costs is $66. 11-24 Cycle and Safety Stock Inventory: Periodic Review Policy • What is our average cycle inventory? Not in book Cycle stock = .5* D*T, same as with ROP • Given SS needs are higher, What are reasons we might use Periodic Review? 11-25 Impact of Aggregation on Safety Inventory • Aggregation is a potentially powerful way to reduce safety inventory and, thus, costs, without impacting Service Level • It is also called consolidation or risk-pooling • Some of the possible methods to achieve it: 1. Aggregation through consolidation 2. Information centralization 3. Specialization 4. Product substitution 5. Component commonality 6. Postponement 11-26 Formulae for Impact of Aggregation n D Di C i 1 n s C D s i 1 2 i s Ls C L C D ss F s (CSL) s L 1 C Will not use covariance formulae 11-27 Impact of Aggregation (Example 11.7) Car Dealer : 4 dealership locations (disaggregated) D = 25 cars; sD = 5 cars; L = 2 weeks; desired CSL=0.90 • What would the effect be on safety stock if the 4 outlets are consolidated into 1 large (aggregated) location? At each disaggregated outlet: For L = 2 weeks, sL = 7.07 cars ss = Fs-1(CSL) x sL = (z=1.28) x 7.07 = 9.06 • Each outlet must carry 9 cars as safety stock, so safety inventory for the 4 outlets in total is 4*9 = 36 cars 11-28 Impact of Aggregation, cont. One outlet (aggregated option): DC = D1 + D2 + D3 + D4 = 25+25+25+25 = 100 cars/wk sRC = Sqrt(52 + 52 + 52 + 52) = 10 sLC = sDC Sqrt(L) = (10)Sqrt(2) = (10)(1.414) = 14.14 ss = Fs-1(CSL=.9) x sLC = (z=1.28) x 14.14 =18.12 or about 18 cars What is the factor of improvement in Safety Stock with aggregation? • Caveat: If covariance, r does not equal 0 (demand is not completely independent), the impact of aggregation is not as great • What are some situations where covariance is very likely to be present and cannot be ignored? • In this class, we will assume covariance is negligible 11-29 Generalization: Consolidating n Identical Facilities • The optimal order quantity (EOQ) increases by a factor of n • The average inventory decreases by a factor of 1/ n • True of both cycle and safety stock inventory • The total number of setups decreases by a factor of 1/ n • This translates to a proportional decrease in setup/order costs • The total cost decreases by a factor of 1/ n - Where total costs = carrying costs of cycle stock, + carrying costs of safty stock + order costs Note that the cycle stock at the combined facility is larger by a factor of n than the cycle stock at a single pre-consolidation facility. But, because there would were n of these pre-consolidation cycle stocks, the total inventory is smaller after consolidation. 11-30 Impact of Aggregation • If number of independent stocking locations decreases by n, the expected level of safety inventory will be reduced by square root of n (square root law) • E-commerce retailers can attempt to take advantage of aggregation (Amazon) more easily compared to bricks and mortar retailers (Borders) • Aggregation has two major disadvantages: • Increase in response time to customer order • Increase in transportation cost to customer • Some e-commerce firms (such as Amazon) have reduced aggregation to mitigate these disadvantages • Open Question: How might we get some of the same benefits of aggregation without the disadvantages? 11-31 Information Centralization • Virtual aggregation • Information system that allows access to current inventory records in all warehouses from each warehouse • Most orders are filled from closest warehouse • In case of a stock-out, another warehouse can fill the order • Better responsiveness, lower transportation cost, higher product availability, but reduced safety inventory 11-32 Specialization • Stock all items in each location or stock different items at different locations? • Different products may have different demands in different locations (e.g., snow shovels) • There can be benefits from aggregation • E.g. Barnes and Noble- use of kiosks for low-volume items • Benefits of aggregation can be affected by: • coefficient of variation of demand (higher cv yields greater reduction in safety inventory from centralization) • value of item (high value items provide more benefits from centralization) 11-33 Value of Aggregation at Grainger (Table 11.4) Motors Cleaner Mean demand 20 1,000 SD of demand 40 100 Disaggregate cv 2 0.1 Value/Unit $500 $30 Disaggregate ss $105,600,000 $15,792,000 Aggregate cv 0.05 0.0025 Aggregate ss $2,632,000 $394,770 Holding Cost $25,742,000 $3,849,308 Saving Saving / Unit $7.74 $0.046 11-34 Product Substitution • Use of one product to satisfy another product’s demand • Manufacturer-driven: one-way substitution • Ship a 120Gig HD instead of 100Gig HD • Customer-driven: two-way substitution • Buy 180 tablet bottle of Advil instead of 90 tablet bottle, or buy store brand • Analysis and proper product placement are necessary for substitution to be fully effective • Clothing retailers: Design collection so several tops match several pants (Zara) • Caveats (not in text) • “Substitution is not as prevalent as grocers would like” (H.Dunn, Inventory Management Expert and 855 guest lecturer, 9/30/2003) • ”There are certain items which a grocery store simply must have on its shelf. We've seen someone push a nearly-full cart down the detergent aisle, see the empty slot where Tide was, and walk out of the store leaving the cart by the empty Tide slot. [The moral is] people expect certain things when it comes to service, and one of those is a standard item or brand… no one wants to be the one responsible for letting the store run out of Tide.” (Robert Knedlik, 855 Student who worked in Alberson’s IT Dept.) 11-35 Component Commonality • Using common components in a variety of different products • Can be an effective approach to exploit aggregation and reduce component inventories • Can be an effective approach to reduce component inventories • Used extensively in electronics (Dell) and automotive (Toyota) • Clothing manufacturers: Sports Obermeyers’ zippers (remove unnecessary differentiation) • The cost savings from expanding usage from 2 to 3 products is much higher than expanding from 4 to 5 products • See example on sheet 4 of Ch11-ss-inv.xls 11-36 Postponement • The ability of a supply chain to delay product differentiation or customization until closer to the time the product is sold • Goal is to have common components in the supply chain for most of the push phase and move product differentiation as close to the pull phase as possible • An analysis of the potential cost savings from postponement is errr… postponed until Chapter 12 • Examples: • Dell in electronics • Benetton and Mango both use gray fabric for garment dyeing 11-37 Estimating and Managing Safety Inventory in Practice 1. Account for the fact that supply chain demand is lumpy 2. Adjust inventory policies if demand is seasonal 3. Use simulation to test inventory policies first…. • Simulation is essential to evaluate complex policies and is useful to examine implications of simple ones (will see examples in Ch12_ • Why use Simulation? see Dr. Savage’s* “Flaw of Averages” http://www.stanford.edu/~savage/flaw/Article.htm 4. Then start with a limited pilot before rolling out company- wide! 5. Monitor service levels 6. Focus on reducing safety inventories (but don’t forget #5!) • Dr. Savage is the Dave Barry of Decision Science. If you are studying accounting, or want to read a humorous but disturbingly relevant article on FASB: http://www.stanford.edu/dept/MSandE/faculty/savage/AccountingRemarks.pdf 11-38 Summary of Learning Objectives • What is the role of safety inventory in a supply chain? • What are the factors that influence the required level of safety inventory? • What are the different measures of product availability? • What managerial levers are available to lower safety inventory and improve product availability? 11-39