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Operations Management (MD021) Inventory Management Agenda Inventory Definitions Processes for Inventory Management Economic Order Quantity Inventory Models Reorder Point/Reorder Quantity Inventory Models Single Period Inventory Models Inventory Definitions Inventory is a stock or store of goods Demands determine the type of inventory Independent Demand an organization will carry. (e.g., customer demand for finished products) A Dependent Demand (e.g., necessary parts to assemble a finished good) B(4) C(2) D(2) E(1) D(3) F(2) Independent demand is uncertain Dependent demand is certain Each firm carries types of inventories relevant to its production demands Raw materials & purchased parts Partially completed goods – called work in process (WIP) Finished-goods inventories manufacturing firms Merchandise retail stores Each firm carries types of inventories relevant to its production demands Replacement parts, tools, & supplies Goods-in-transit to warehouses or customers Inventory is carried for many different reasons across different industries To meet anticipated demand Meeting demand in a timely manner enhances customer satisfaction Smooth production requirements across seasons Produce one season, sell in next (or throughout year) To decouple successive operations and maintain continuity of production Protects against machine breakdowns To protect against stock-outs Vendors do not always deliver on time Inventory is carried for many different reasons across different industries To take advantage of order cycles to minimize purchasing and inventory costs Minimum order size requirements, full truck loads To help hedge against price increases Buy now at low price, store goods for future use To permit operations to operate Operations require a certain amount of WIP inventory To take advantage of quantity discounts Vendors often give discounts when ordering large quantities Operations Strategy Having too much inventory is not good Tends to hide problems Makes it easier to live with (i.e. ignore) problems than to eliminate them Costly to maintain large stocks of inventories Opportunity costs of potentially doing something else with the money tied up in inventory Wise objectives Reduce lot sizes Reduce safety stock Reduce ordering costs and holding costs These are difficult to calculate, and often underestimated, leading to higher order sizes Objective of Inventory Control To achieve satisfactory levels of customer service while keeping inventory costs (costs of ordering and carrying inventory) within reasonable bounds Processes for Inventory Management An effective inventory management approach will have certain information A system to keep track of inventory on hand and on order A reliable forecast of demand Knowledge of order lead times and lead time variability Reasonable estimates of Holding costs Ordering costs Shortage costs A classification system for inventory items Inventory Counting Systems Periodic System Physical count of items made at periodic intervals Walgreens (1987) – manager walked around weekly, ordered everything needed across whole store Perpetual Inventory System System that keeps track of removals from inventory continuously, thus monitoring current levels of each item 2005 – grocery store scanners (bar codes) 2005 – RFID-based systems Perpetual inventory counting systems range from low-tech to high-tech Two-Bin System - Two containers of inventory; reorder when the first bin is empty Universal Product Code (UPC) – Bar code printed on a label that 0 has information about the item to which it is attached 214800 232087768 Radio Frequency Identification (RFID) – Computer chip embedded in a label on side of package, cases, or pallets Radio Frequency Identification (RFID) used in tags, chips, implants, and wristbands RFID tags are activated by RFID reader devices Example of RFID Use: Metro Future Store RFID Loyalty Card RFID Portal for Receiving Inventory RFID Enabled Tools for Counting Inventory Information Kiosks and Terminals to Find/Advertise Labels for Dynamic Pricing RFID Tagged Items Portal for Automatic Checkout Example of RFID Use: Metro Future Store RFID-enabled Advertisements RFID-based Inventory Smart Checkout Picking Store Manager’s Work Bench Lead time must be matched against expected demand Lead time: time interval between ordering and receiving the order If we expect that demand will occur on a certain day in the future, we will need to place an order several days earlier, and account for: Lead time Lead time variability Managers must estimate several types of inventory-related costs Holding (carrying) costs: cost to carry an item in inventory for a length of time, usually a year Annual cost of 20%-40% of value (unit price) of an item Ordering costs: costs of ordering and receiving inventory fixed dollar amount per order, regardless of order size Shortage costs: costs when demand exceeds supply difficult to calculate – often assumed ABC Classification System Classifying inventory according to some measure of importance and allocating control efforts accordingly. A - very important High B - mod. important A Annual C - least important $ value of items B Low C Few Many Number of Items Cycle Counting A physical count of items in inventory. Counts are conducted periodically. A items counted frequently B items counted less frequently C items counted least frequently Cycle counting management trades off inventory accuracy against costs of counting How much accuracy is needed? When should cycle counting be performed? Who should do it? Economic Order Quantity Inventory Models Economic Order Quantity (EOQ) Models Economic order quantity (EOQ) model Economic production quantity (EPQ) model Quantity discount model Assumptions of EOQ Model Only one product is involved Annual demand requirements are known Demand is even throughout the year Lead time does not vary Each order is received in a single delivery There are no quantity discounts The Inventory Cycle Profile of Inventory Level Over Time Q Usage Quantity rate on hand Reorder point Time Receive Place Receive Place Receive order order order order order Lead time Total Cost Under the Economic Order Quantity Assumptions Annual Annual Total cost = carrying + ordering cost cost Q + DS TC = H 2 Q Cost Minimization Goal The Total-Cost Curve is U-Shaped Q D TC H S Annual Cost 2 Q Ordering Costs Order Quantity QO (optimal order quantity) (Q) Deriving the EOQ Using calculus, we take the derivative of the total cost function and set the derivative (slope) equal to zero and solve for Q. 2DS 2( Annual Demand )(Order or Setup Cost ) Q OPT = = H Annual Holding Cost Minimum Total Cost The total cost curve reaches its minimum where the carrying and ordering costs are equal. 2DS 2( Annual Demand )(Order or Setup Cost ) Q OPT = = H Annual Holding Cost Economic Production Quantity (EPQ) Relevant when production is done in batches or lots Capacity to produce a part exceeds the part’s usage or demand rate Assumptions of EPQ are similar to EOQ except orders are received incrementally during production Economic Production Quantity (EPQ) Assumptions Only one item is involved Annual demand is known Usage rate is constant Usage occurs continually Production rate is constant Lead time does not vary No quantity discounts Economic Production Quantity (EPQ) Production Production & Usage & Usage Usage Usage Economic Run Size Solving a similar equation as for EOQ, we get the following equation for the optimal run size for EPQ: 2DS p Q0 H p u p = production or delivery rate u = usage rate Quantity Discounts Volume (Per Unit) Discounts 1 to 49 = $10/unit 50 to 100 = $9/unit 100 and up = $8/unit Case Discounts Single units = $10/unit Case of 10 = $90 = $9/unit Quantity Discounting Total Costs with Purchasing Cost Quantity discounts are price reductions offered to customers to induce them to buy in large quantities. Annual Annual + Purchasing TC = carrying + ordering cost cost cost Q + DS + PD TC = H 2 Q The buyer’s goal is to select the order quantity that will minimize total cost. Taking the derivative with respect to Q doesn’t change the EOQ formula Cost Adding Purchasing cost TC with PD doesn’t change EOQ TC without PD PD 0 EOQ Quantity Total Cost with Constant Carrying Costs TCa Total Cost TCb Decreasing TCc Price CC a,b,c OC EOQ Quantity Reorder Point, Reorder Quantity Inventory Models Reorder point models Goal is to place an order when the amount of inventory on hand is still sufficient to satisfy demand during the time it takes to receive that order (i.e., the lead time) When to Reorder with EOQ Ordering Reorder Point - When the quantity on hand of an item drops to this amount, the item is reordered Safety Stock - Stock that is held in excess of expected demand due to variable demand rate and/or lead time. Service Level - Probability that demand will not exceed supply during lead time. The reorder point leaves you with lead time inventory plus safety stock Quantity Maximum probable demand during lead time Expected demand during lead time ROP Safety stock reduces risk of stockout during lead time Safety stock LT Time How do we determine the reorder point quantity? Calculation accounts for … The rate of demand The lead time Demand and/or lead time variability Stock-out risk (safety stock) Assuming demand and lead time are constant (as in EOQ) Reorder Point Quantity ROP = d X LT d = demand rate (units per time) LT = lead time (in same units of time) Example Usage = 12 order forms/day Lead time = 7 days ROP = (12 forms/day)(7 days) = 84 order forms Reorder when 84 order forms are left When lead times are random, reorder point (ROP) must be adjusted The ROP based on a normal Distribution of lead time demand Service level Risk of a stockout Probability of no stockout ROP Quantity Expected demand Safety stock 0 z z-scale ROP = expected demand during lead time + safety stock When we have an estimate of standard deviation of demand during lead time ROP expected demand during lead time z dLT Example Demand during lead time = 84 forms σdLT = 2 ROP = 84 forms + zσdLT = 84 + 1.96(2) = 88 forms Reorder when 88 order forms are left When only demand is variable ROP d LT z LT d Example Usage = 12 order forms/day; σd = 3 Lead time = 7 days ROP = (12 forms/day)(7 days) + 1.96(7)0.5(3) = 84 + 15.5 = 90 Reorder when 90 order forms are left When only lead time is variable ROP d L T zd LT Example Usage = 12 order forms/day Average Lead time = 7 days; σLT = 1 ROP = (12)(7) + 1.96(12)(1) = 84 + 24 = 108 Reorder when 108 order forms are left When both demand and lead time are variable ROP d L T z L T d d 2 LT 2 2 Example Average Usage = 12 order forms/day; σd = 3 Average Lead time = 7 days; σLT = 1 ROP = (12)(7) + 1.96[(7)(9) + (144)(1)] = 84 + 1.96(14.4) = 84 + 27.7 = 112 Reorder when 112 order forms are left Single Period Inventory Model (“The Newsboy Problem”) Single Period Inventory Model Single period model: model for ordering of perishables and other items with limited useful lives how many newspapers should a newsboy on a street corner stock for a specific day? magazines fresh fruits fresh vegetables seafood cut flowers commemorative t-shirts and souvenirs spare parts Single period model balances costs of a shortage against excess costs Shortage cost: generally the unrealized profits per unit (plus loss of customer goodwill) Cshortage = Cs = revenue per unit – cost per unit Excess cost: difference between purchase cost and salvage value of items left over at the end of a period Cexcess = Ce = original cost/unit – salvage value/unit Single Period Model Continuous stocking levels Demand can be approximated using a continuous distribution Identifies optimal stocking levels Optimal stocking level balances unit shortage and excess cost Discrete stocking levels Demand can be approximated using a discrete distribution Service levels are discrete rather than continuous Desired service level is equaled or exceeded Continuous stocking level assuming a uniform demand distribution Cs Service Level C s Ce Service level represents the probability that demand will not exceed the stocking level Continuous stocking level assuming a uniform demand distribution Example: The movie “Gigli 2” will be released soon. A (somewhat crazy) retailer wants to determine the number of commemorative t- shirts to stock. Based on the rousingly successful “Gigli”, we have: Demand = uniform(1, 10) Cost/unit = $5 per t-shirt Revenue = $10 per t-shirt Salvage value = $1 per t-shirt Cs = revenue/unit – cost/unit = $10 - $5 = $5 Ce = cost/unit – salvage value/unit = $5 - $1 = $4 Service Level = Cs/(Cs+Ce) = ($5)/($5 + $4) = 0.555 The optimal stocking level must satisfy demand 55% of the time Soptimal = 1 + 0.55(10-1) = 1 + 5 = 6 t-shirts Discrete stocking levels involve inverse transform from service level to order units Here, we have a discrete uniform distribution, probability of each demand equals 0.10 Cumulative Probability 1.0 = Service Level Cs/(Cs+Ce) = 0.55 0.5 “Gigli 2” 0 t-shirts 1 2 3 4 5 6 7 8 9 10 Again, the optimal decision is to stock 6 “Gigli 2” t-shirts.