Inventory Management

1. Inventory Management A Basic Introduction for Operations & Supply Chain Management Dr. Mark P. Van Oyen file: inven-lec.ppt

2. Types of Inventory • 1. Raw materials and purchased parts. • 3. Finished goods. • Supplier • Distributor • Retailer • 4. Replacement parts, tools, and supplies. We are NOT talking about:Work in process (WIP). • Shop-floor control of partially completed goods. • Often dealt with using PUSH (MRP II, ERP) or PULL (Kanban, CONWIP, Bucket Brigades, etc.) • Goldratt warns about using “Economic Batch Quantity” for shop floor lot-sizing… which is our EOQ as you’ll see.

3. Customers, demand centers sinks Field Warehouses: stocking points Sources: plants vendors ports Regional Warehouses: stocking points Supply Inventory & warehousing costs Production/ purchase costs Transportation costs Transportation costs Inventory & warehousing costs

4. Perspectives on Inventory • Inventory becomes an increasingly large part of total assets and managerial focus as we move down the supply chain. • Suppliers • Manufacturers • Wholesalers & • Distributors • 4. Retailers

5. Goals:Reduce Cost, Improve Service • By effectively managing inventory: • Xerox eliminated $700 million inventory from its supply chain • Wal-Mart became the largest retail company utilizing efficient inventory management (many new business processes and distribution, logistics, IS, & warehousing innovations all for the net impact of excellent inventory management) • GM has reduced parts inventory and transportation costs by 26% annually

6. Functions of Inventory • To meet anticipated demand. • Display, provide customer hands-on experience (dealer inventory) • To protect against stockouts. (variation, unanticipated demand) • To smooth production requirements. • Level aggregate production plan is an example. • To take advantage of order cycles. • Efficiency in fixed costs of ordering or producing (e.g. share delivery truck or economy of scale in batch production) • To de-couple operations, providing smooth flow between operations • WIP inventory between successive manufacturing steps or supply chain echelons (buffer stock) permits operations to continue during periods of breakdowns/strikes/storms. • To hedge against price increases, or to take advantage of quantity discounts. [caution – hazardous to corporate health] • Despite “Zero Inventory” zealots, an appropriate amount of inventory is a necessary evil in almost all systems!

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

8. Objectives/Controls of Inventory Control • 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 upon in this class: Average Backlog, Mean time to fill order (Cycle Time) • Profit maximization by achieving a balance in stocking, avoiding both over-stocking and under-stocking. • It boils down to 2 control decisions: • Decide how much to order/produce (Q) • Decide when to order/produce (ROP)

9. Costs • h = Carrying (or holding) cost (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) • S= 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) • Independent of order quantity (Q) from supplier. • OR view this as production costs if we are the manufacturer

10. Costs • Purchase cost per unit of inventory (P) • We model only a variable cost from supplier • If we make the parts ourselves, S captures production setup cost, while P captures the variable cost of production • Shortage costs. (used in Newsvendor Model with demand uncertainty when 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! • Do not underestimate shortage costs, or you will settle for less market share than you really want

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

12. Part I: How Much to Order: Economic Order Quantity (EOQ) • Objective: Identify the optimal order quantity that minimizes the sum of certain annual costs that vary with order size. • Assumption: Fixed order quantity systems (don’t change order size dynamically over time) • CERTAIN demand(no random market behavior) • We will consider 2 models: • EOQ: all items delivered as a batch • EOQ-QD: EOQ with quantity discounts (uses EOQ assumptions)

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

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

15. EOQ Model - Assumptions • Demand, D, is known and constant. • Purchase Price, P, is known and constant. • Fixed setup cost per order, A, independent of Q • Holding cost h (or C) per unit per year. • 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). • There will be no stockouts, no backorders, no uncertainties! • Lead Time is known and constant.

16. Basic EOQ Model Assumptions • D = 3120 (units/yr) Annual demand rate [e.g. Sell brake pads with D = 60 pads/wk * 52 wks/yr.] • P = 2 $/pad Unit production/purchase cost - not counting setup or inventory costs ($/unit) • A = 28.85 $/order Order set-up cost, constant per order, for any order size • h (or C) = Average annual carrying cost per unit ($/unit/yr.) [25% of purchase price = 0.50 $/yr] • Q = order quantity = decision variable [e.g. How many brake pads to order]

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

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

19. Inventory Cycle for Basic EOQ Q = 600 Inventory Level D = 60*52 pads/yr Q/2 = Ave Inventory = 300. Time Q/D = Length of Order Cycle = 10 wk.

20. Cost Minimization Goal The Total Stocking Cost (TSC) Curve is U-Shaped Annual Cost h * q/2 Ordering Costs PD PD, cost to purchase stock 0 Q (optimal) Order Quantity Used (q)

21. Economic Order Quantity (EOQ) • There is a tradeoff between carrying costs and ordering costs! • EOQ is the value, Q, that minimizes TSC. In this sense, the Q determined by the EOQ formula is OPTIMAL given a simplistic model. • 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) h = (D/Q)A. [this is not a general method!]

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

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

24. EOQ-QD: EOQ with “All Units” at Quantity Discount • Purchase Price is knownbut varies with the amount purchased. The price is the same for all units. Also, holding cost may be fraction of price! • Demand and Lead time both known and constant. • Lots are delivered in FULL as in Model I. The problem is solved as a series of problems - one for each price break. (1) solve for EOQ for each possible price and modify the Q value to be feasible at that price, then (2) compute resulting TSC and finally (3) search for the lowest cost.

25. EOQ-QD: Quantity Discounts (All Units) • Quantity discounts offered by the supplier to the buyer occur when the unit purchase price of the product (actual cost = ac) decreases as the quantity purchased (Q) increases. • Assume orders are received all at once (Model I). • Total Cost = (Q/2)h + (D/Q)A + (D)(ac). • Careful: h may be some percent of (ac) . • Find Q = EOQ that minimizes total cost.

26. Example • D = 10,000, A = 5.50, h = 0.2(ac). • Price Schedule: • Note: “all-units” quantity discount. • E.g. at Q = 524, ac=2.00 applies to all Q units (not just 400-524!)

27. Computing EOQ for each range • ac = 2.20 yields optimal EOQ at a level that deserves a better price/volume, so this is the one situation in which we can disqualify the possibility of this range containing our answer! • ac = 2.00 yields EOQ = 524.4 which is feasible. • ac = 1.80 yields EOQ = 552.8 which is NOT feasible. We then take the closest quantity that will give us that price range: • Q = 700 for ac = 1.80. Next step: cost these out - answer has to be one of them!

28. Computing EOQ for Example TC(Q) = (Q/2)h + (D/Q)A + (D)ac. • TC(Q = 524.4) = (524.4/2)(0.40) + (10,000/524.4)(5.5) + 10,000(2.00) =20,209.76. • TC(Q = 700) = (700/2)(0.36) + (10,000/700)(5.5) + 10,000(1.8) = 18,204.57 (min). • Conclusion: EOQ = 700. (even though the EOQ solution was not feasible at that price!)

29. Part II: When to Order? Inventory Management Under Uncertainty • Demand or Lead Time or both uncertain • Even “good” managers are likely to run out once in a while (a firm must start by choosing a service level/fill rate) • When can you run out? • Only during the Lead Time if you monitor the system. • 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!)

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

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

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

33. When to Order: A (Q,r) Policy • The basic EOQ models address how much to order: Q • Now, we address when to order. • 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. ROP is referred to as “r” in the (Q,r) policy

34. When to Order: r = ROP • 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) • 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.

35. The Order Cycle Service Level, (SL) • SL = % of demand during the lead time (% of DDLT) the firm wishes to satisfy. SL = probability that random customer during lead time is served • SL must be set according to Shortage Cost • This is NOT annual service level (or fill rate), since that averages over all time periods and will be a larger number than SL. • SL should not be 100% for most firms. (90%? 95%? 98%?) SL increases as Safety Stock increases, but with diminishing returns (due to shape of demand distribution)

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

37. Variability in DDLT and SS • 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. • We will emphasize Normal (continuous) distributions to model variable DDLT, but discrete distributions are common as well. • SS buffers against stockout during lead time.

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

39. Reminder:The Normal Distribution Standard Deviation = 5 Standard Deviation = 10 Average = 30

40. Computing SS from SL for Normal DDLT • Example: 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) / 

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

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

43. Computing SS from SL for Normal DDLT to provide SL = 95%. • ROP = EDDLT + SS = EDDLT + z (). z is the number of standard deviations SS is set above EDDLT, which is the mean of DDLT. • 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 • If we double the s.d. to about 278, SS would double! • Lead time variability reduction can same a lot of inventory and $ (perhaps more than lead time itself!)

44. Analysis of Std Dev (DDLT) • The reorder point (s) must account for deviations from average (call this safety stock - SS). There are two standard ways: • View 1: Summarize the std. deviation of “Demand During Lead Time (DDLT) as . • View 2: (more detailed) STD = std. dev. of demand for 1 day or 1 unit of time, so std. deviation of “Demand During Lead Time (DDLT)=STD  LT • View 2 has SS= z (STD  LT)where z is chosen from statistical tables to ensure that the probability of stockouts during leadtime is100%-SL. ROP = EDDLT + SS =LTAVG + zSTDLT

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

46. The (s,S) Policy: Continuous Review • (s, S) Policy: Whenever the inventory position drops below a certain ReOrder Point, s (aka ROP), we order to raise the inventory position to level S = s + Q. • The reorder point is a function of: • The Lead Time • Average demand • Demand variability • Service level

47. A View of (s, S) Policys= ROP, S= ROP + Q S Inventory Position Lead Time Lead Time Inventory Level s 0 Time

48. Inventory & Supply Chain • So far, we’ve approached the analysis from the perspective of Purchasing, ordering from a supplier • This is “local optimization” • Concept: The battle between supply chains has more impact on your success than your battles with your direct competitors!?!?

49. Risk Pooling • Consider these two systems: Market One Warehouse One LT W1 Ship Times Supplier Warehouse Two Market Two LT W2 Ship Times Market One Centralized Warehouse Ship Times Supplier Market Two

50. Risk Pooling • For the same service level, which system will require more inventory? Why? • For the same total inventory level, which system will have better service? Why? • What are the factors that affect these answers?