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Inventory Management Operations Management Dr. Mark P. Van Oyen “You’ve got to WIP it, WIP it good!” Devo file: inven-lec.ppt 1 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). – Partially completed goods. – Goldratt warns about using “Economic Batch Quantity” for shop floor lot-sizing… which is our EOQ as you’ll see. Types of Inventory Demand 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 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.) Independent demand (Inventory). Dependent demand (MRP) Independent Demand is usually for end products (forecasted, uncertain) X Dependent Demand (MRP can back-figure release times given due dates for independent demands) B(2) C(1) Independent D(3) E(1) E(2) F(2) AA Dependent Figure 14-6 of Stevenson d e f1 f2 g1 g2 4 Functions of Inventory To meet anticipated demand. To protect against stockouts. (To meet unanticipated demand) To hedge against price increases, or to take advantage of quantity discounts. 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) 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. Despite “Zero Inventory” zealots, an appropriate amount of inventory is a good thing in almost all systems! 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) 6 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 one here: Average Backlog, Mean time to fill order (Cycle Time) Profit maximization by achieving a balance in stocking, avoiding both over-stocking and under-stocking. Controls: – Decide how much to order/produce (Q) – Decide when to order/produce (ROP) 7 The Inventory Cycle Average Profile of Inventory Level Over Time Q, Usage Actual Quan rate Usage tity Inventory on hand Reorder Point, ROP Time Receive Place Receive Place Receive order order order order order Lead time 8 Costs 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) 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 . 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!) 9 – Loss of good will, or, may need to substitute higher-cost item! 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 10 How Much to Order: Economic Order Quantity (EOQ) Models 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) We will consider 3 models: – I. EOQ: all items delivered as a batch – II. EOQ with gradual deliveries – III. EOQ with quantity discounts (uses Model I assumptions) 11 Model I: (Basic) EOQ “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 are demanded by customers. Periodically run out of parts and have to replenish inventory by ordering from suppliers. 12 EOQ Model I - Assumptions Demand, D, is known and constant. Purchase Price, P, is known and constant. Fixed setup cost per order, S, independent of Q Holding cost C=H 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. 13 Basic EOQ Model Assumptions D = Demand rate is known and constant. (units/yr) [e.g. Sell brake pads with D = 60 pads/wk * 52] P = Unit production/purchase cost - not counting setup or inventory costs ($/unit) [e.g. 2 $/pad] S = Order set-up cost, constant per order ($/order) [e.g. 28.85 $/order] C = H = Average annual carrying cost per unit ($/unit/yr.) [25% of purchase price/year = 0.50 $/yr] Q = order quantity = decision variable [e.g. ?? Brake pads] 14 Inventory Cycle for Basic EOQ Q = 600 D = 60*52 pads/yr Inventory Level Time Q/D = Length 10 wk. of Order Cycle 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 is (# orders/year) * (order setup cost) + (unit purch. cost) * (annual demand) = (D/Q)S + PD Average inventory per order cycle: (Q+0)/2 = Q/2. why? use geometry to get average inv. Level (vs. calc.) Annual carrying cost is (Q/2) C. Total annual Stocking Cost (TSC) = [annual carrying cost + annual ordering cost]: TSC = (D/Q)S + PD + (Q/2) C 16 Cost Minimization Goal The Total-Cost Curve is U-Shaped Annual Cost Ordering Costs PD QO (optimal order quantity) Order Quantity (Q) 17 Economic Order Quantity (EOQ) There is a tradeoff between carrying costs and ordering costs! 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. 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!] 18 EOQ Model I- Development It turns out that Costs are minimized where ordering and carrying costs are equal. Thus, D Q 2 DS S C Q Q 2 C 19 EOQ Model I- Example Demand = 60 pads/wk 2 DS Ordering Cost, S = $28.85/order Q C Unit Carrying Cost = 25% of purchase 2(28.85)(60)(52) price/yr. .25($2) Unit purchase price, P = $2.00/pad 600 pads 20 Model II: EOQ for Gradual Deliveries Inbasic EOQ model I, orders are assumed to be delivered instantaneously & all at once. Sometimes inventories build up gradually: Gradual Deliveries. – when the firm is both a producer and user. – Pipelines (think of water slowly filling a bathtub when the faucet is open all the way but the drain is open) – Many small shipments made as the products become available (multiple Amazon shipments for one order consisting of 12 books; The Goal illustration – getting the customer to take multiple smaller lots) EOQ Model II - Assumptions Orders are delivered (or produced) gradually at a rate of p units per day, where we choose “day” = convenient time unit as an example Demand (annual) is known and has a constant rate. d = usage rate (units/day). Units on d and p must match! Simplify Life: Convert time unit on p to “years”, then d = D = units/year demand rate. Lead Time is known and constant. Purchase Price is known and constant. 22 Model II: EOQ for Gradual Deliveries Let “TIME” be a convenient time period (day, week, month, year) p = production delivery rate (units/TIME). E.g. = 150 pads/wk. d = usage rate (units/ TIME). – d is just a conversion of D in units/year to units/ TIME. Illustration of tip: TIME = year, so p = 150*52 pads/Year., d= D. 23 Model II: EOQ for Gradual Deliveries Delivery period = Q/p = (TIME to deliver all Q products ordered) = 600/(150*52) = 4/52 years Maximum Inventory level reached = Imax = (flow rate in - flow rate out) * (time Q) = (p-d)(Q/p) in units of product = (150 - 60) (52 wks/yr) ( 4 /52 years) p d p-d Q/p 24 Inventory Cycle for EOQ for Gradual Deliveries p Cycle Time = Q/d (resupply rate) Imax Q p-d Inventory (buildup d (usage rate) Level rate) Time (p-d)Q/(pd) Buildup Period = Q/p EOQ for Gradual Deliveries TSC = (Imax/2)C + (D/Q)S + PD = (Q/2) [(p-d)/p] C + (D/Q)S + PD. 2DS p EOQ = . C p-d 26 EOQ Model II- Example Demand = 60 pads/wk Ordering Cost = 2 DS p $28.85/order Q Unit Carrying Cost = C pd 25% of purchase 2(60 * 52)(28.85) 150 price/yr. Unit purchase price = .25($2) 150 60 $2.00/pad 775 pads production rate = 150 pads/wk 27 EOQ Model II- Example Picture Imax = 465 Q=775 Inventory Level 90/wk demand of 60/wk Time 5.17 wks = Q/p 7.75 wks 28 The Concept of Reorder Point (ROP) What inventory level should trigger/ control the placement of an order? ROP = ReOrder Point = DDLT + SS Demand-During-Lead-Time + Safety-Stock If Lead Time is 3 wks., ROP = (60)(3) = 180 and inven. Falling If Lead Time is 10 wks., ROP = (90)(2.92) = 263 and inven. increasing! (this is a tricky case because of the very long lead time which requires a reorder while the inventory buildup is still in progress – it should be a rare case) 29 III. EOQ with All Units at Quantity Discount Purchase Price is known but 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. 30 Model III: Basic EOQ with 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)C + (D/Q)S + (D)(ac). – Often: C = some percent of (ac) . Find Q = EOQ that minimizes total cost. Example from G&F 10.3 (p. 387-389) D = 10,000, S = 5.50, C = 0.2 (ac). Price Schedule: Range of Q (ac) C 1-399 2.20 0.44 400-699 2.00 0.40 700+ 1.80 0.36 Assume orders are received all at once (Model I). Note: “all-units” quantity discount. – E.g. at Q = 524, ac=2.00 applies to all Q units (not just 400- 524!) 32 Computing EOQ for Ex.10.3 G&F EOQ = 2DS = 2(5.5)10,000 = 500.0 (not feasible - higher!). 2.20 C 0.44 EOQ = 2DS = 2(5.5)10,000 = 524.4 (feasible). 2.00 C 0.40 EOQ = 2DS = 2(5.5)10,000 = 552.8 (not feasible - too low!). 1.80 C 0.36 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! 33 Computing EOQ for Example 10.3 TC(Q) = (Q/2)C + (D/Q)S + (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!) 34 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!) 35 The Typical ROP System Average Demand ROP set as demand that accumulates during lead time ROP = ReOrder Point Lead Time 36 The Self-Correcting Effect- A Benign Demand Rate after ROP Hypothetical Demand Average Demand ROP Lead Time Lead Time 37 What if Demand is “brisk” after hitting the ROP? Hypothetical Demand Average Demand ROP = EDDLT + SS ROP > EDDLT Safety Stock Lead Time 38 When to Order 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. When to Order 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). DDLT, – Length of the lead time. EDDLT & – Demand and lead time variability. Std. Dev. – Degree of stockout risk acceptable to management (fill rate, order cycle Service Level) 40 The Order Cycle Service Level,(SL) The percent of the demand during the lead time (% of DDLT) the firm wishes to satisfy. This is a probability. 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. SL should not be 100% for most firms. (90%? 95%? 98%?) SL rises with the Safety Stock to a point. 41 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) LT Time 42 Variability in DDLT and SS Variabilityin 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. 43 Service Level and Stockout Risk Targetservice 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%. 44 Computing SS from SL for Normal DDLT Example 10.5 on p. 374 of Gaither & Frazier. 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) / 45 Reorder Point (ROP) Service level Risk of a stockout Probability of no stockout ROP Quantity Expected demand Safety stock 0 z z-scale 46 Area under standard Normal pdf from - to +z Z = standard Normal with mean=0, s.d. = 1 Z = (X - ) / z P(Z z) See G&F Appendix A 0 .5 See Stevenson, second from last page . 67 .75 .84 .80 P(Z <z) 1.28 .90 Standard 1.645 .95 Normal(0,1) 2.0 .98 2.33 .99 3.5 .9998 0 z z-scale 47 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 48 inventory and $ (perhaps more than lead time itself!) Summary View Holding Cost = C[ Q/2 + SS] (1) Order trigger by crossing ROP (2) Order quantity up to (SS + Q) Q+SS = Not full due to brisk Target Demand after trigger ROP = EDDLT + SS ROP > EDDLT Safety Stock Lead Time 49 Single-Period Model: Newsvendor 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, … Unsold or unused goods are not typically carried over from one period to the next; rather they are salvaged or disposed of. Model can be used to allocate time-perishable service capacity. Two costs: shortage (short) and excess (long). 50 Single-Period Model Shortage or stockout cost may be a charge for loss of customer goodwill, or the opportunity cost of lost sales (or customer!): Cs = Revenue per unit - Cost per unit. Excess (Long) cost applies to the items left over at end of the period, which need salvaging Ce = Original cost per unit - Salvage value per unit. (insert smoke, mirrors, and the magic of 51 Leibnitz’s Rule here…) The Single-Period Model: Newsvendor How do I know what service level is the best one, based upon my costs? 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) If Cs is shortage cost/unit, and Ce is excess cost/unit, then Cs SL Cs Ce 52 Single-Period Model for Normally Distributed Demand Computing the optimal stocking level differs slightly depending on whether demand is continuous (e.g. normal) or discrete. We begin with continuous case. 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. 53 Single-Period Model for Normally Distributed Demand Cs = 60 cents per liter Ce = 20 cents per liter. SL = Cs/(Cs + Ce) = 60/(60 + 20) = 0.75 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. The folks in marketing could get worried! If this is a business where stockouts lose long-term customers, then we must increase Cs to reflect the actual cost of lost customer due to stockout. 54 Single-Period Model for Continuous Demand demand is Normal(200 liters per week, variance = 10,000 liters2/wk) … so = 100 liters per week 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 () = 200 + (0.67)(100) = 267. liters. 55 Single-Period & Discrete Demand: Lively Lobsters Lively Lobsters (L.L.) receives a supply of fresh, live lobsters from Maine every day. Lively earns a profit of $7.50 for every lobster sold, but a day-old lobster is worth only $8.50. Each lobster costs L.L. $14.50. (a) what is the unit cost of a L.L. stockout? Cs = 7.50 = lost profit (b) unit cost of having a left-over lobster? Ce = 14.50 - 8.50 = cost – salvage value = 6. (c) What should the L.L. service level be? SL = Cs/(Cs + Ce) = 7.5 / (7.5 + 6) = .56 (larger Cs leads to SL > .50) Demand follows a discrete (relative frequency) distribution as given on next page. 56 Lively Lobsters: SL = Cs/(Cs + Ce) =.56 Probability that demand Cumulative will be less than or equal to x Demand follows a Relative Relative discrete Frequency Frequency (relative Demand (pmf) (cdf) frequency) distribution: 19 0.05 0.05 P(D < 19 ) 20 0.05 0.10 P(D < 20 ) 21 0.08 0.18 P(D < 21 ) Result: order 25 Lobsters, 22 0.08 0.26 P(D < 22 ) because that is 23 0.13 0.39 P(D < 23 ) the smallest 24 0.14 0.53 P(D < 24 ) amount that 25 0.10 0.63 P(D < 25 ) will serve at least 56% of 26 0.12 0.75 P(D < 26 ) the demand on 27 0.10 you do P(D < 27 ) a given night. 28 0.10 you do P(D < 28 ) 29 0.05 1.00 P(D < 29 ) 57 * pmf = prob. mass function

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