2I. Introduction What is inventory? Types of Inventories: stored resource used to satisfy current or future demandTypes of Inventories:Raw Materials/ComponentsIn-Process Goods (WIP)Finished GoodsSupplies
3Introduction Inventory Related Costs: Holding Cost -- cost to carry a unit in inventory for a length of time (annual), includes interest opportunity cost, insurance, taxes, depreciation, obsolescence, deterioration, May be expressed as a percentage of unit price or as a dollar amount per unit
4Introduction Inventory Related Costs (continued): Order Cost -- Cost of ordering and receiving inventory, Include determining how much is needed, preparing invoices, shipping costs, inspecting goods upon receipt for quantity and quality, Generally expressed as a fixed dollar amount, regardless of order sizeInventory may also influence purchasing costInventory is costly
5Introduction Inventory Related Costs (continued): Shortage Cost-- result when demand exceeds the inventory on hand, Include the opportunity cost of not making a sales, loss of customer goodwill, late charges, and in the case of internal customers, the cost of lost production or downtime, difficult to measure, thus may have be subjectively estimated
6Introduction Why Hold Inventories? Meet anticipated demand Lead time – the time period between place an order until receive the orderAverage lead time demand is considered as anticipate demandProtect against stock-outSafety stock – more than average lead time demand inventory
7Introduction Why Hold Inventories (continued)? De-couple successive operations - separate production from distributionWine production and inventorySmooth production processSnowmobile production and inventoryBuy/Produce in economic lot sizes - take advantage of quantity discountsHedge against price increases
8IntroductionJIT Inventory – minimum inventory needed to keep a system running, small lot sizesAdvantageslower inventory costseasy to identify problems and potential problemsDisadvantagesrequires accurate timing and cooperationbreakdowns stop everything
9Introduction Inventory Classification A Identify important Annual$ volumeof itemsABCHighLowFewManyNumber of ItemsInventory ClassificationIdentify importantItems and more inventorycontrol on important itemsMeasure of importance:ABC analysis:A = 70-80% of total inventory value, but only 15% of itemsB = 15-25% of total inventory value, but 30% of itemsC = 5% of total inventory value, but 55% of items
10Introduction Monitor Inventory As important as demand forecast for decision makingUniversal Product Code - Bar code printed on a label that has information about the item to which it is attachedCycle counting: taking physical counts of items and reconciling with records on a continual rotating basis, regular inventory audits, ABC approach
11Introduction Inventory Systems Objective: minimize annual total inventory cost and maintain satisfied service level.service level: probability of no shortageTotal Inventory Cost is not Inventory CostAnnual total inventory cost (TC) = annual product cost + annual inventory costAnnual product cost = annual demand * unit priceAnnual inventory cost = annual holding cost + annual setup (order) cost + annual shortage cost
12Introduction Possible performance measures customer satisfaction number of backorders/lost salesnumber of customer complaintsinventory turnoverratio of annual cost of goods sold to average inventory investmentdays of inventoryexpected number of days of sales that can be supplied from existing inventory
13Introduction Requirements for Effective Inventory Management : A system to keep track of the inventory on hand and on orderA classification system for inventory itemsA reliable forecast of demand that includes an measure of forecast errorReasonable estimates of inventory holding costs, ordering costs, and shortage costsKnowledge of lead times and lead time variability
14Introduction1. Continuous (Perpetual) Review System: (event-triggered)Monitor the inventory level all the time, order a fixed quantity (Q) when the inventory level drops to the reorder point (ROP)Calculate: Q and ROPRe-Order Point (ROP) – an inventory level when actual inventory drops to it will trigger an activity of re-order.
15Introduction 2. Periodic Review System: (time-triggered) Place an order every fixed period T. Each time bring the current inventory to a target level MCalculate: T and M3. Advantages and Disadvantages?
16Introduction Dependent and Independent Demand: Dependent demand: derived demand, lumpy (subassemblies and components)carsIndependent demand: from customer side, smooth (end items and finished goods)tires
17II. Inventory Models On Order Quantity Model Basics (consider as annual)Total Cost (TC) = Product Cost + Inventory CostInventory Cost = Holding Cost+ Setup (Order) Cost + Shortage CostTC = Product Cost + Holding Cost
18Inventory Models On Order Quantity Product Cost = Annual Demand * Unit PriceHolding Cost = average inventory level * Holding Cost per unit per yearOrdering Cost = # of orders * Setup Cost per order# of orders = annual demand / order quantityShortage Cost = Shortage Cost per unit* average # of shortage per yearBest Order Quantity = a quantity that minimizes TC
19Inventory Models On Order Quantity EOQ Model (Economic Order Quantity), Fixed-Order-Quantity ModelAssumptionsThere is one product typeDemand is known and constantLead time is known and constantReceipt of inventory is instantaneous (one batch, same time)Shortage is not allowed
20EOQ Model (continued) Q Lead time Reorder point Place order Receive
21EOQ Model (continued) Notation and Terminology Q = order quantity(# of pieces per order)Q0 = Economic Order Quantity (EOQ)D = demand for the time period considered (units per year)S = setup/order cost ($ per order)H = holding cost per unit per year ($ per unit per year)in general proportional to the price, H = I*P
22EOQ Model (continued) Notation and Terminology (continued) I = Interest rate (expanded) (% per year)P = unit price ($ per unit)IC = inventory cost = setup cost + holding costTC = IC + product costFind Out EOQ
23EOQ Model (continued) Average Inventory Level = Holding Cost = Number of orders per year =Setup (Order) Cost =Shortage Cost = 0, why?
24EOQ Model (continued) Product Cost = IC = Total Cost (TC) = Minimize TC Minimize IC, why?
25EOQ Model (continued)Observation: at the best order quantity EOQ (Q0),holding cost = setup costSolve Q0, we have
26EOQ Model (continued) The Inventory Cost Curve is U-Shaped Annual Cost Carrying CostsAnnualOrdering CostsQO(EOQ)Order Quantity (Q)
27EOQ Model (continued) Example: Annual demand = 10,000 unit/year, ordering cost = $50/order, unit cost (price) = $4/unit, expanded interest rate = 25%/year. EOQ? TC at EOQ?
28EOQ Model (continued) Sensitivity of IC with related to Q -- Example (continued)Avg.InventoryHolding Cost# of ordersper yearOrderCostICQ(Q/2)(Q/2)*H(D/Q)(D/Q)*S+(D/Q)*S500250$25020$1,000$1,2501000$500101500750$7506.667$333$1,083
29EOQ Model (continued) Conclusion: Thinking Challenge: 1. Inventory cost curve is flat around EOQ2. Flatter when Q increases than when Q decreases from EOQThinking Challenge:If the order quantity Q = 2*EOQ, by how much IC will increase?
30EOQ Model (continued) Sensitivity of EOQ with related to D, H, S, P, I 1. Insensitive to parameter change2. Directions?
31EPQ ModelEPQ (Economic Production Quantity) Model: Fixed Order Quantity Model with Incremental ReplenishmentProblem description:AssumptionsThere is one product typeDemand is known and constantReceipt of inventory is gradual and at a constant replenishment (production) rateShortage is not allowed
32EPQ Model (continued) Q Production rate - usage rate Usage rate Quantityon handUsagerateReorderpointTimeStart toproduceFinishproductionStart toproduceProduction run length
33EPQ Model (continued) Notation and Terminology Qp = production quantity(# of pieces/production run)Qp0 = Best production quantity (EPQ)p = daily production rate (units per day)d = daily demand rate (units per day)D = demand rate (units per year)S = production setup (order) cost($ per setup)H = holding cost per unit per year (again H = I*P in general)T = production run length = Q/p
34EPQ Model (continued) Maximum Inventory Level = Average Inventory Level =Annual Holding Cost =
35EPQ Model (continued) Number of production runs per year = Order Cost =IC =TC =Minimize TC Minimize IC, why?
36EPQ Model (continued) Observation: at EPO, holding cost = setup cost Best Production Quantity (EPQ) formula:
37EPQ Model (continued) Remarks: EPQ > EOQ (why?) Example: D=2000 unit/year, S=$5/setup, H=$0.4/unit/year, p=100 unit/day, 200 working days/year. Find the best production batch size and the # of production runs/year.
38EOQ with discount EOQ with Discount Model: Assumptions: same as with EOQ, plus discount on all unitsTerminologyPrice breaks: the smallest order quantity to receive a discount priceFeasibility: the order quantity matching the claimed price is feasible, otherwise infeasible.
39EOQ with discount (continued) Example:Order Price$2.1/unit$2.0Great equal 700 $1.9Idea is to compare TC curves under different prices - why TC?
40EOQ with discount (continued) Order QuantityTotal Cost Curvefor Price 1Total Cost Curvefor Price 2$ costTotal Cost Curvefor Price 3400700
41EOQ with discount (continued) Order QuantityTotal Cost Curvefor Price 1Total Cost Curvefor Price 2$ costTotal Cost Curvefor Price 3400700
42EOQ with discount (continued) Observations:EOQ with a lower price, if feasible, is better than any order quantity with the same or higher price.Potential best order quantity: cheapest feasible EOQ, price breaks associated with lower prices.
43EOQ with discount (continued) Solution Procedure:1. Find the feasible EOQ with cheapest possible price.2. Calculate TCs of the EOQ (from Step 1) and price breaks above EOQ.3. Pick the order quantity with lowest TC
44EOQ with discount (continued) Example (continued) Annual demand = 10,000 unit/year, order cost = $5.5/order. Assuming holding costs are proportional to unit prices and annual interest rate = 20%. Find the best order quantity.
45III. Models on Reorder Points - When to Order? Find ROP (Re-Order Point)ROP depends on:Lead Time: time between placing and receiving an orderDemand Distribution: how uncertainDesired Service Level: probability of no shortage = 1-P(s), where P(s) = probability of shortage
46Models on Reorder Points - When to Order ? (continued) Constant Demand Rate:Constant daily demand rate = d, Lead time = L daysROP = d * L = Lead time demandRemark:no uncertainty in demandservice level = 100%safety stock = 0
47Models on Reorder Points - When to Order ? (continued) Variable Demand with Stable Average RateHow continuous review system works?Lead time demand: demand during the lead timeROP Lead time demand ==>ROP < Lead time demand ==>ROP = Average lead time demand + Safety Stock = m + SS
48Models on Reorder Points - When to Order ? (continued) Remarks:Higher the desired service level --->More uncertain the demand --->Two methods to determine the SS
49Models on Reorder Points - When to Order ? (continued) 1. Determine SS and ROP based on shortage cost inf. (if available)SS increases Holding cost ? Shortage cost ?Best SS minimizes total inventory cost
50Models on Reorder Points - When to Order ? (continued) 1. Determine SS and ROP based on shortage cost inf. (continued)-- Example: Consider a light switch carried by Litely. Litely sells 1,350 of these switches per year, and places order for 300 of these switches at a time. The carrying cost per unit per year is calculated as $5 while the stock out cost is estimated at $6 ($3 lost profit per switch and another $3 lost in goodwill, or future sales loss). Find the best SS level and ROP for Litely.
51Models on Reorder Points - When to Order ? (continued) Determine SS and ROP based on shortage cost inf. (continued)1. Determine SS and ROP based on demand inf. during each lead time period:Lead Time Demand51015202530Probability0.10.150.2
52Models on Reorder Points - When to Order ? (continued) Determine SS and ROP based on shortage cost inf. (continued)If SS = 0, ROP = m = 15 switches
53Models on Reorder Points - When to Order ? (continued) Determine SS and ROP based on shortage cost inf. (continued)# of orders per year =For no safety stock, Litely has the following shortage table. Why?Shortage Levelno shortage51015Probability0.5188.8.131.52
54Models on Reorder Points - When to Order ? (continued) Determine SS and ROP based on shortage cost inf. (continued)Determine the best SS in following tableSafety stockAdd. HoldingcostAvg.shortage(per order)Annualshortage costTotal cost51015
55Models on Reorder Points - When to Order ? (continued) 2. Determine ROP and SS based on lead time demand distribution and desired service level:
56Models on Reorder Points - When to Order ? (continued) Case 1. Empirical Lead time demand distribution-- Example:Lead Time DemandFrequencyProbabilityROPService Level3245678
57Models on Reorder Points - When to Order ? (continued) Find R and SS to achieve the service level of 85% and 95%, respectively.
58Models on Reorder Points - When to Order ? (continued) Case 2. Lead time demand is Normally distributed with (m, )SS = , ROP = m + SS, z = single tail normal score of desired service level.( is the standard deviation)Example:Lead time demand is Normally distributed with mean = 4 and standard deviation = 3. Find ROP and SS to achieve the service level of 85% and 95%, respectively.
59IV. Single Period Model and Marginal Analysis (Newsvendor Problem)
60Homework (Additional problems) Problem 1: A toy manufacturer uses approximately 36,000 silicon chips annually. The chips are used at a steady rate during the 240 days the plant operates. Annual holding cost is 50 cents per chip, and ordering cost (per order) is $25/order. Assume that each of their orders comes in one batch. Determine:a. .the best order quantityb. demonstrate that your order quantity is optimal by showing that annual ordering costs = annual holding costsc. the average inventory leveld. the number of orders per yeare. the number of working days between orders (Hint: days between orders = # days in a year / # of orders per year. Why?)
61Homework (Additional problems) Problem 2. The Dine Corporation is both a producer and a user of brass couplings. The firm operates 200 days a year and uses the couplings at a steady rate of 50 per day. Couplings can be produced at a rate of 150 per day. Inventory holding cost is estimated at $5 per unit per year. Machine setup costs are $40 per production run. Determine:a. the best production run sizeb. demonstrate that your production run size is optimal by showing that annual set up costs = annual holding costs (Hint: find the formula of holding and setup cost for EPQ model in my lecture note.)c. the maximum inventory level (Hint: find the formula in the derivation of EPQ)d. the number of production runs per yeare. the cycle time and the production time within each cycle (Hint: cycle time is given by Q/d and production time is given by Q/p. Why? Think before using the formula)
62Homework (Additional problems) A small manufacturing firm used roughly 3,400 pounds of chemical dye each year. Currently the firm purchases 300 pounds per order and pays $3 per pound. The supplier has just announced that orders of 1,000 pounds or more will be filled at a price of $2.5 per pound. The manufacturing firm incurs a cost of $100 each time it submits an order and assigns an annual holding cost of 20% of the purchase price per pound.a. determine the best order size that minimizes the total costb. if the supplier offered the discount at 2,500 pounds instead of at 1,000 pounds, what order size would minimize total cost?
63Homework (Additional problems) Problem 4: A product is ordered four times every year. Inventory carrying cost is $20 per unit per year, and the cost of shortage for each unit is $40. Given the following demand probabilities during the reorder periodLead Time Demand4080120160Probability0.10.250.3
64Homework (Additional problems) Problem 4 (continued)a) What is the average lead time demand?b) What would be the reorder point without safety stock?c) What would be the probabilities of the following shortage levels if the company uses the reorder point without safety stock?
65Homework (Additional problems) Problem 4 (continued)d) Follow the Litely example in my lecture to find out the best safety stock level to minimize the total cost.e) What is the reorder point to achieve the 95% service level? What is the associated safety stock? (Hint: you need to follow the example in my lecture note under Case 1)Shortage Level4080Probability