1. 1 Inventory Management Must-know Fundamentals Operations Management Dr. Mark P. Van Oyen “Just the facts, Ma’am” Sgt. Friday “You’ve got to WIP it, WIP it good!” Devo file: inven-lec.ppt

2. Types of Inventory u 1. Raw materials and purchased parts. u 3. Finished goods. –Supplier –Distributor –Retailer u 4. Replacement parts, tools, and supplies. We are NOT talking about: Work in process (WIP). –Partially completed goods. –Goldratt warns about using “Economic Batch Quantity” for shop floor lot-sizing… which is our EOQ as you’ll see.

3. Types of Inventory Demand u Independent demand items have demands which are not linked to the demand for other products. (Inventory focus) –Finished goods or end items (Saturns, laser pointers, –These demands are typically forecasted. We are NOT talking about u Dependent demand items = Subassemblies or component parts. –Derived from the demand of the independent demand items (finished goods) –Often dealt with using PUSH (MRP II, ERP) or PULL (Kanban, CONWIP, Bucket Brigades, etc.)

4. 4 Independent AA d e f1f2 g1 g2 Dependent Independent demand (Inventory). Dependent demand (MRP) Independent Demand is usually for end products (forecasted, uncertain) X B(2) C(1) D(3)E(1) E(2) F(2) Dependent Demand (MRP can back-figure release times given due dates for independent demands) Figure 14-6 of Stevenson

5. Functions of Inventory u To meet anticipated demand. u To protect against stockouts. (To meet unanticipated demand) u To take advantage of quantity discounts, or to hedge against price increases. u To smooth production requirements. –Level aggregate production plan is an example. u To take advantage of order cycles. –Efficiency in fixed costs of ordering or producing (e.g. share delivery truck) u To de-couple operations. –WIP inventory between successive manufacturing steps or supply chain echelons (buffer stock) permits operations to continue during periods of breakdowns/strikes/storms. u Despite “Zero Inventory” zealots, an appropriate amount of inventory is a good thing in almost all systems!

6. 6 Tradeoffs in the Size of Inventories u Inventories that are too high are expensive to carry, and they tie up capital. u Warehousing, transportation u Greater risk of defects, even when carefully inventoried u Market shifts leave seller with many unwanted parts u Inventories that are too low can result in u reduced operational efficiency (machine starvation) u poor service (delayed service, product substitution) u lost sales (customer refuses raincheck, finds a more reliable supplier)

7. 7 Objectives/Controls of Inventory Control u Achieve satisfactory levels of customer service while keeping inventory costs under control. –Fill rate (probability of meeting a demand from inventory) –Inventory turnover ratio = Annual Sales / Average Inventory Level = D / WIP –Others not focused one here: Average Backlog, Mean time to fill order (Cycle Time) u Profit maximization by achieving a balance in stocking, avoiding both over-stocking and under-stocking. u Controls: –Decide how much to order/produce (Q) –Decide when to order/produce (ROP)

8. 8 Profile of Inventory Level Over Time Inventory on hand Q, Quan tity Receive order Place order Receive order Place order Receive order Lead time Reorder Point, ROP Average Usage rate Time Actual Usage The Inventory Cycle

9. 9 Costs u Holding costs or carrying costs (applied per unit of inventory) –Associated with keeping inventory for a period of time. –Capital tied up –Warehousing and transportation –Perishable products -OR- expected risk of damage to product –Note that easiest calculation is (Holding cost rate/unit/yr) * (Average annual inventory level) u Ordering OR production costs (Fixed order set up + variable unit costs) –For ordering and receiving inventory if ordering from a supplier, (Delivery charges, postage & handling, cost of purchasing dept. labor) – OR production costs if we are the manufacturer. u Shortage costs. (used in Newsvendor Model with demand uncertainty) –Demand exceeds supply of inventory and a stockout occurs. –Opportunity costs (lost sales and LOST CUSTOMERS!) –Loss of good will, or, may need to substitute higher-cost item!

10. 10 The Inventory Management Problem u Determine Inventory Policy: How much to order or make? (Q) When to order or make? (Reorder point) How much to store in safety stock? u To: Minimize the cost of the inventory system

11. 11 How Much to Order: Economic Order Quantity (EOQ) Models u Objective: Identify the optimal order quantity that minimizes the sum of certain annual costs that vary with order size. u Assumption: Fixed order quantity systems (don’t change order size dynamically over time) u We will consider here only the FIRST of 3 models: –I. EOQ: all items delivered as a batch –II. EOQ with gradual deliveries –III. EOQ with quantity discounts (uses Model I assumptions)

12. 12 Model I: (Basic) EOQ u “How Many Parts to Make at Once” by Ford Harris –Original 1913 version of this model. u Very simple - makes a lot of restrictive assumptions. It’s not realistic at first glance, but it is! Its usefulness keeps it alive! u Nicely illustrates basic tradeoffs that exist in any inventory management problem. u Basic Model Scenario : u Own a warehouse from which parts (brake pads) are demanded by customers. u Periodically run out of parts and have to replenish inventory by ordering from reliable suppliers.

13. 13 EOQ Model I - Assumptions u Demand, D, is known and constant. u Purchase Price, P, is known and constant. u Fixed setup cost per order, S, independent of Q u Holding cost C=H per unit per year. u Lots of size Q are delivered in full (Production Perspective: produce items and hold them in FGI until production is completed, at which time we ship them out). u There will be no stockouts, no backorders, no uncertainties! u Lead Time is known and constant.

14. 14 Basic EOQ Model Assumptions u D = Demand rate is known and constant. (units/yr) [e.g. Sell brake pads with D = 60 pads/wk * 52] u P = Unit production/purchase cost - not counting setup or inventory costs ($/unit) [e.g. 2 $/pad] u S = Order set-up cost, constant per order ($/order) [e.g. 28.85 $/order] u C = H = Average annual carrying cost per unit ($/unit/yr.) [25% of purchase price/year = 0.50 $/yr] u Q = order quantity = decision variable [e.g. ?? Brake pads]

15. Inventory Cycle for Basic EOQ Time Inventory Level Q = 600 Q/D = Length of Order Cycle D = 60*52 pads/yr 10 wk.

16. 16 Total Annual Cost Calculation u Order Frequency = F = D/Q = reciprocal of period e.g. (60 pads/wk)* 52 /600 pads = 52/10. (orders/yr) u Annual ordering cost is (# orders/year) * (order setup cost) + (unit purch. cost) * (annual demand) = (D/Q)S + PD u Average inventory per order cycle: (Q+0)/2 = Q/2. why? use geometry to get average inv. Level (vs. calc.) u Annual carrying cost is (Q/2) C. u Total annual Stocking Cost (TSC) = [annual carrying cost + annual ordering cost]: TSC = (D/Q)S + PD + (Q/2) C

17. 17 Cost Minimization Goal The Total-Cost Curve is U-Shaped Ordering Costs QOQO Order Quantity (Q) Annual Cost ( optimal order quantity) PD

18. 18 Economic Order Quantity (EOQ) u There is a tradeoff between carrying costs and ordering costs! u EOQ is the value of Q that minimizes TSC. In this sense, the Q determined by the EOQ formula is OPTIMAL given a simplistic model. u FYI only: EOQ is found by either: –Using calculus and solving (d/dQ)TSC = 0, the points of graph with zero slope are either local maxima & minima, or, –Observing that the EOQ occurs where carrying and ordering costs are equal, i.e., by solving (Q/2) C = (D/Q)S. [this is not a general method!]

19. 19 EOQ Model I- Development It turns out that Costs are minimized where ordering and carrying costs are equal. Thus,

20. 20 EOQ Model I- Example Demand = 60 pads/wk Ordering Cost, S = $28.85/order Unit Carrying Cost = 25% of purchase price/yr. Unit purchase price, P = $2.00/pad

21. 21 The Concept of Reorder Point (ROP) u What inventory level should trigger/ control the placement of an order? u ROP = ReOrder Point = DDLT + SS Demand-During-Lead-Time + Safety-Stock u If Lead Time is 3 wks., ROP = (60)(3) = 180

22. 22 Inventory Management Under Uncertainty u Demand or Lead Time or both uncertain u Even “good” managers are likely to run out once in a while (a firm must start by choosing a service level/fill rate) u When can you run out? –Only during the Lead Time if you monitor the system. u Solution: build a standard ROP system based on the probability distribution on demand during the lead time (DDLT), which is a r.v. (collecting statistics on lead times is a good starting point!)

23. 23 The Typical ROP System ROP set as demand that accumulates during lead time Lead Time Average Demand ROP = ReOrder Point

24. 24 The Self-Correcting Effect- A Benign Demand Rate after ROP ROP Lead Time Average Demand Lead Time Hypothetical Demand

25. 25 What if Demand is “brisk” after hitting the ROP? ROP > Lead Time Average Demand Hypothetical Demand Safety Stock EDDLT ROP = EDDLT + SS

26. When to Order u The basic EOQ models address how much to order: Q u Now, we address when to order. u Re-Order point (ROP) occurs when the inventory level drops to a predetermined amount, which includes expected demand during lead time (EDDLT) and a safety stock (SS): ROP = EDDLT + SS.

27. 27 When to Order u SS is additional inventory carried to reduce the risk of a stockout during the lead time interval (think of it as slush fund that we dip into when demand after ROP (DDLT) is more brisk than average) u ROP depends on: –Demand rate (forecast based). –Length of the lead time. –Demand and lead time variability. –Degree of stockout risk acceptable to management (fill rate, order cycle Service Level) DDLT, EDDLT & Std. Dev.

28. 28 The Order Cycle Service Level,(SL) u The percent of the demand during the lead time (% of DDLT) the firm wishes to satisfy. This is a probability. u This is not the same as the annual service level, since that averages over all time periods and will be a larger number than SL. u SL should not be 100% for most firms. (90%? 95%? 98%?) u SL rises with the Safety Stock to a point.

29. 29 Safety Stock LT Time Expected demand during lead time (EDDLT) Maximum probable demand during lead time (in excess of EDDLT) defines SS ROP Quantity Safety stock (SS)

30. 30 Variability in DDLT and SS u Variability in demand during lead time (DDLT) means that stockouts can occur. –Variations in demand rates can result in a temporary surge in demand, which can drain inventory more quickly than expected. –Variations in delivery times can lengthen the time a given supply must cover. u We will emphasize Normal (continuous) distributions to model variable DDLT, but discrete distributions are common as well. u SS buffers against stockout during lead time.

31. 31 Service Level and Stockout Risk u Target service level (SL) determines how much SS should be held. –Remember, holding stock costs money. u SL = probability that demand will not exceed supply during lead time (i.e. there is no stockout then). u Service level + stockout risk = 100%.

32. 32 Computing SS from SL for Normal DDLT u Example 10.5 on p. 374 of Gaither & Frazier. u DDLT is normally distributed a mean of 693. and a standard deviation of 139.: –EDDLT = 693. –s.d. (std dev) of DDLT =  = 139.. –As computational aid, we need to relate this to Z = standard Normal with mean=0, s.d. = 1 »Z = (DDLT - EDDLT) / 

33. 33 Reorder Point (ROP) ROP Risk of a stockout Service level Probability of no stockout Expected demand Safety stock 0z Quantity z-scale

34. 34 Area under standard Normal pdf from -  to +z Standard Normal(0,1) 0z z-scale P(Z <z) Z = standard Normal with mean=0, s.d. = 1 Z = (X -  ) /  See G&F Appendix A See Stevenson, second from last page

35. 35 Computing SS from SL for Normal DDLT to provide SL = 95%. u ROP = EDDLT + SS = EDDLT + z (  ). z is the number of standard deviations SS is set above EDDLT, which is the mean of DDLT. u z is read from Appendix B Table B2. Of Stevenson - OR- Appendix A (p. 768) of Gaither & Frazier: –Locate.95 (area to the left of ROP) inside the table (or as close as you can get), and read off the z value from the margins: z = 1.64. Example: ROP = 693 + 1.64(139) = 921 SS = ROP - EDDLT = 921 - 693. = 1.64(139) = 228 u If we double the s.d. to about 278, SS would double! u Lead time variability reduction can same a lot of inventory and $ (perhaps more than lead time itself!)

36. 36 Single-Period Model: Newsvendor u Used to order perishables or other items with limited useful lives. –Fruits and vegetables, Seafood, Cut flowers. –Blood (certain blood products in a blood bank) –Newspapers, magazines, … u Unsold or unused goods are not typically carried over from one period to the next; rather they are salvaged or disposed of. u Model can be used to allocate time-perishable service capacity. u Two costs: shortage (short) and excess (long).

37. 37 Single-Period Model u Shortage or stockout cost may be a charge for loss of customer goodwill, or the opportunity cost of lost sales (or customer!): C s = Revenue per unit - Cost per unit. u Excess (Long) cost applies to the items left over at end of the period, which need salvaging C e = Original cost per unit - Salvage value per unit. (insert smoke, mirrors, and the magic of Leibnitz’s Rule here…)

38. 38 The Single-Period Model: Newsvendor u How do I know what service level is the best one, based upon my costs? u Answer: Assuming my goal is to maximize profit (at least for the purposes of this analysis!) I should satisfy SL fraction of demand during the next period (DDLT) u If C s is shortage cost/unit, and C e is excess cost/unit, then

39. 39 Single-Period Model for Normally Distributed Demand u Computing the optimal stocking level differs slightly depending on whether demand is continuous (e.g. normal) or discrete. We begin with continuous case. u Suppose demand for apple cider at a downtown street stand varies continuously according to a normal distribution with a mean of 200 liters per week and a standard deviation of 100 liters per week: –Revenue per unit = $ 1 per liter –Cost per unit = $ 0.40 per liter –Salvage value = $ 0.20 per liter.

40. 40 Single-Period Model for Normally Distributed Demand u C s = 60 cents per liter u C e = 20 cents per liter. u SL = C s /(C s + C e ) = 60/(60 + 20) = 0.75 u To maximize profit, we should stock enough product to satisfy 75% of the demand (on average!), while we intentionally plan NOT to serve 25% of the demand. u The folks in marketing could get worried! If this is a business where stockouts lose long-term customers, then we must increase C s to reflect the actual cost of lost customer due to stockout.

41. 41 Single-Period Model for Continuous Demand u Continuous example continued: –75% of the area under the normal curve must be to the left of the stocking level. –Appendix shows a z of 0.67 corresponds to a “left area” of 0.749 –Optimal stocking level = mean + z (s.d.) = 200 + (0.67)(100) = 267. liters.

42. 42 Tiny Tot Toys: A discrete example u Ave. Dem. = 6/day; D = 6 * 365 u Demand distribution will be given (discrete) u S = $10/order u C = H= $2/unit/yr. u Cs = $3.25/unit = penalty per unit stocked out u C e = ?? $/unit = average cost of having left-over inventory at the end of period, based on C=H and variability in demand. 1. Use deterministic EOQ model to set Q 2. Use Newsvendor Model to set Safety Stock

43. 43 Tiny Tot Toys - Optimizing Q u Ave. Dem. = D = 6*365 = 2190/yr u S = $10/order u C = H= $2/unit/yr. 1. Use deterministic EOQ model to set Q

44. 44 Use Newsvendor Model to set SL 2. We need to capture holding cost and inventory levels in C e = avg. holding cost incurred over one order cycle (Why? Longer time frame. By the time we determine the excess, we have already ordered the next resupply. We’ll carry the excess for one cycle, at which point we decrease the reorder size to get things back on schedule.) u C e = C*[Q/D] = 2 $/yr/unit * [25 days / (365 days/yr)] = 0.1370 $/unit, u Cs = 3.25 $/unit was given.

45. 45 Wrap-up: SL for discrete distr. u Tiny Tot Toys desires SL = 96%. u Demand is not Normally distributed; rather, past data shows the following:

46. 46 Wrap-up: EOQ & SL for discrete distr. u Done: Already determined Q=150 u Now, set ROP based upon desired SL u SS is chosen to keep stockout probability to (1 – SL) or less. u ROP = EDDLT + SS = 60 + SS = 95

47. 47 Summary View ROP > Lead Time Holding Cost = C[ Q/2 + SS] (approx.) (1)Order trigger by crossing ROP (2)Order a quantity up to (SS + Q) Safety Stock EDDLT ROP = EDDLT + SS Q+SS = Target Not full due to brisk Demand after trigger