Operations Management Inventory management models 1 What is Inventory? Stock of items kept to meet future demand Purpose of inventory management how many units to order when to order 2
Types of Inventory Raw materials Purchased parts and supplies Work-in-process (partially completed) products (WIP) Items being transported Tools and equipment 3 Inventory s Ordering cost cost of replenishing inventory Holding cost cost of holding an item in inventory Stockout or shortage cost temporary or permanent loss of sales when demand cannot be met Item cost 4
RFID: Radio Frequency Identification An automatic identification method, relying on storing and remotely retrieving data using devices called RFID tags. An RFID tag is an object that can be attached to or incorporated into a product, animal, or person for the purpose of identification using radio waves. 5 RFID in retail On a typical afternoon in a typical US supermarket 8.2% of the item are out of stock and this number is nearly doubled for items that are advertised In 34% of stockouts consumers refuse to buy and alternative product Stockouts in US supermarkets cause between $7 to $12 billion sales Andersen Consulting Since January 2005, Wal-Mart has required its top 100 suppliers to apply RFID labels to all shipments. 6
RFID for kids TOKYO Stunned by the kidnapping of a teenage girl, a rural Japanese city plans to use a satellite-linked tracking system to help parents find their children. The northern city of Murakami has asked two security companies to provide the service for the families of 2,700 elementary and junior high school students, said Kenkichi Kimura, an official on the city's Board of Education. With the new service, students will carry devices that will send out signals allowing their parents to pinpoint where they are through a Web site on the Internet, Mr. Kimura said Thursday. It will use a combination of technologies provided by mobile phone companies and the Global Positioning System, a U.S. satellite navigation service used by everyone from hikers to ship captains. The device also will be equipped with a button that can be pushed to call for help. "If you are in a big city, people will come to help if you call for help," Mr. Kimura said. "But here, students walk to school in the mountains and rice fields. We need the latest device." Associated Press Other uses: fraud identification, secure building access, computer access, storage of medical records, anti-kidnapping initiatives and a variety of lawenforcement applications. Just after the operation to insert the RFID tag was completed 7 Economic Order Quantity Model On-hand Inventory Q Q/2 Reorder Point, R Demand Rate, D Place Order Receive order Time Between Orders (Order Cycle) = Q/D Average Cycle Inventory, Q/2 Time Lead Time L The EOQ model is for independent demand and looks at the trade off between ordering cost and carrying cost of inventory. 8
Trade-off in EOQ Model: Inventory Level vs. Number of Orders Many orders, low inventory level Few orders, high inventory level On-hand Inventory On-hand Inventory Q Q Time Time 9 Economic Order Quantity - Model Assumptions Single product or item Demand rate known and constant Lead time known and constant Inventory holding cost based on average inventory Ordering, or setup costs are constant Item produced in lots, or purchased in orders Each lot or order received in single delivery No backorders are allowed No quantity discounts are allowed Price per unit is constant 10
EOQ Model (Constant Demand, No Shortages) TC = total annual inventory cost D = annual demand (units / year) Q = order quantity (units) C O = cost of placing an order or setup cost ($) C H = annual inventory carrying cost ($ / unit /year) Total Annual Inventory = Annual Ordering TC = (D / Q) C O + (Q / 2) C H + Annual Holding 11 Relationships for Basic EOQ (Constant Demand, No Shortages) Annual EOQ balances carrying costs and ordering costs in this model. Q* Order Quantity Total Carrying Ordering 12
Economic Order Quantity Results (Constant Demand, No Shortages) Economic Order Quantity= Q* = Number of Orders per year = D / Q* 2 D C O C H Length of order cycle = Q* / D Total cost = TC = (D / Q*) C O + (Q* / 2) C H = = 2 C H C 0 D 13 Determining When to Reorder Quantity to order (how much ) determined by EOQ model Reorder point (when ) determined by finding the inventory level that is adequate to protect the company from running out during delivery lead time With constant demand and constant lead time, reorder point is exactly the amount that will be sold during the lead time. Example: Milk sold at rate of 12 bottles / day ( = d ) Delivery lead time of 2 days ( = LT ) R = d x LT = (12) (2) = 24 bottles 14
EOQ Example D = 1,000 units per year C o = $20 per order C H = $8.33 per unit per month = $100 per unit per year EOQ: ( )( 20) 2 1000 Q* = = 20 100 Number of orders per year = 1000/20 = 50 orders Length of order cycle = 365 days/50 orders = 7.3 days Total cost = (2*1000*100*20) 1/2 = $2,000 15 Robustness of EOQ model Annual TC Very Flat Curve - Good!! Q*- Q Q* Q*+ Q Total Order Quantity Would have to mis-specify Q* by quite a bit before total annual inventory costs would change significantly. 16
In-Class Exercise Annual Demand = 10,000 units Days per year considered in average daily demand = 365 to place an order = $10 Holding cost per unit per year = 10% of cost per unit Lead time = 10 days per unit = $15 Determine the economic order quantity and the reorder point. When the inventory level reaches 274, order 365 units. Total (holding and setup) = $548 17 Variations of EOQ Some assumptions so far Constant price Certain and constant demand rate Instantaneous replenishment Some variations of EOQ EOQ with quantity discount EOQ with non-instantaneous replenishment EOQ with backorders 18
Example: EOQ with Quantity Discounts Order Size (Cases) 0 to 1,000 1,001 to 2,000 2,001 + Discount 0% 5% 10% Unit $24.00 $22.80 $21.60 Can you think of a situation where that happens? 19 EOQ with Quantity Discounts Order Quantity: Q < Q 1 Q 1 Q < Q 2 Q 2 Q Unit : C 1 C 2 C 3 Annual Total @ C 1 Total @ C 3 Total @ C 2 Q 1 Q 2 Order Quantity 20
EOQ with Price Discounts: Finding the Optimal Order Quantity Step Step 1 Step Step 2 Step Step 3 TC = (D / Q*) C O + (Q* / 2) C H + DC I Need to add procurement cost For each unit cost (C I ), compute the optimal order quantity using the EOQ formula (Q*). Remember that C H may also change as C I changes. If Q* is below the minimum order quantity for that unit cost, adjust Q* up to the minimum. If Q* is within the quantity range, keep it. If Q* is above the maximum order quantity, drop the unit cost from further consideration - cannot be optimal For each order quantity from step 2, calculate the total annual cost (TC, as above). The order quantity with the minimum total cost is the optimal order quantity. 21 Variations of EOQ Some assumptions so far Constant price Certain and constant demand rate Instantaneous replenishment Some variations of EOQ EOQ with quantity discounts EOQ with non-instantaneous replenishment EOQ with backorders 22
EOQ w/ Non-Instantaneous Replenishment (POQ - Production Order Quantity) Q* Production Rate P and Demand Rate D Stock level Demand Rate, D Max stock without production Max. Stock Q (1 - D/P) Production begins Production ends No production Time 23 EOQ (Uniform Replenishment Rate, Constant Demand, No Shortages) Total Annual Inventory = Annual Setup + Annual Holding TC = (D/Q) C S + (Q/2)(1 - D/P) C H where: C S D P = setup cost for each lot = usage (demand) rate = production rate 24
EOQ (Uniform Replenishment Rate, Constant Demand, No Shortages) Max stock level Holding costs Setup costs Optimal Production Lot Size: = = = Q * Q D Q = ( C H Q 2 1 C S - D 1 - ( P) C H D P ) 2 D C S D 1 - ( P) 25 Variations of EOQ Some assumptions so far Constant price Certain and constant demand rate Instantaneous replenishment Some variations of EOQ EOQ with quantity discounts EOQ with non-instantaneous replenishment EOQ with backorders 26
EOQ with backorders Max. Stock (Q* - B) (ROP) Backorder level (B*) Stock level Lead Time Average stock level (Q*-B)/2 * (Q*-B*)/Q* Tempo Average backorder (B*/2)*B*/Q* 27 EOQ with backorders TC TC D C Q S = Q OPT DC = + D Q S + 2DS H + b H b Total annual inventory cost Annual demand per unit Order quantity Setup costs or cost of placing an order Please note the different notation Q - B Q - B H + 2 Q R = d L - B R L H b B OPT = B B b 2 Q 2DS H b H + b Reorder point Lead time Annual invent. costs per unit Stockout cost per unit 28
Probabilistic models Demand is not known But follows a normal distribution Remaining EOQ assumptions are valid Consider service level and safety stock Service level = 1 Probability of stockout Higher service level requires higher level of service level The higher the safety stock, the higher the ROP 29 Probabilistic inventory models Fixed-quantity system (Q system) Same order amount each time When inventory decreases to the ROP, a new order for Q units is placed Inventory must be continuously monitored Fixed-period system (P system) A order is placed at the end of a given period The on-hand inventory is counted The amount necessary to bring total inventory up to a target level is ordered 30
Probabilistic models Optimal quantity ROP Stock Service level ROP P(Stockout) SS X Safety Stock (SS) Place order Lead Time Order is received Time 31 Probabilistic models: EOQ and ROP Q OPT _ = 2DS H R = dl + zσ SS = zσ L L D Annual demand S Set-up / ordering cost H Holding cost _ R Reorder point d Daily average demand (constante) L Lead time (constant) z # standard deviations σl Standard deviation of demand during L SS Safety stock L 2 σ L = σ d i = Lσ d i=1 32
Fixed-period system How much to order? On-hand inventory Safety Stock (SS) Period L T Period Max level of stock Time Lead Time 33 Fixed-period system How much to order? q = average demand (T+L) + SS On-hand inventory q= d(t+l)+ z _ σ T+ L -I Standard deviation for L and T σ = T + Lσ T+L d 34
Fixed-period system Example How many units should be ordered? 35 Fixed-period system Example 36
Comparing fixed-period system (P) and fixed-quantity system (Q) Q SS for period L P SS for period T+L Higher SS for P system P not very flexible P easier to implement Not necessary to continuously verify the stock level Allows Just-In-Time 37