Inventory Management

Agenda Independent Demand Inventory Dependent vs. independent demand Basic Economic Order Quantity (EOQ) model. Also known as Economic Lot Size Model Models with Demand and Supply Uncertainty Fixed ordering costs: the base-stock model (s,S) No fixed ordering costs: the base-stock model (S) Risk pooling

Why do companies hold inventory? Why might they avoid doing so? To meet anticipated customer demand To account for differences in production timing (smoothing) To protect against uncertainty (demand surge, price increase, lead time slippage) To maintain independence of operations (buffering) To take advantage of economic purchase order size WHY NOT? Requires additional space Opportunity cost of capital Spoilage / obsolescence

Independent vs. Dependent Demand Independent Demand (Demand not related to other items or the final end-product) Ford Taurus Dependent Demand (Derived demand items for component parts, subassemblies, raw materials, etc.) Body Assy. Wheel Assy. (4) E(1) Wheel (1) Tire (1) 6

Two Decisions in Inventory Management When is it time to reorder? If it is time to reorder, how much?

Economic Order Quantity Model: Where it all started…. Time Between Orders (Cycle Time) T = Q/D Demand Rate, D Q On-hand Inventory Average Cycle Inventory, Q/2 Q/2 Time

Basic EOQ Assumptions Constant Demand Rate Instantaneous replenishment Orders received in full after lead-time. Constant Unit Price (no discounts)

Economic Order Quantity Cost Model: Constant Demand, No Shortages TC = total annual inventory cost D = annual demand (units / year) Q = order quantity (units) K = cost of placing an order or setup cost ($) h = annual inventory carrying cost ($ / unit /year) Total Annual Inventory Cost = Annual Ordering Cost Annual Holding Cost + TC = (D / Q) K + (Q / 2) h

Trade-off in EOQ Model: Inventory Level vs. Number of Orders Many orders, low inventory level On-hand Inventory Q Time Q Few orders, high inventory level On-hand Inventory Time

Cost Relationships for Basic EOQ (Constant Demand, No Shortages) Total Cost Ordering Cost TC – Annual Cost Carrying Cost Q* EOQ balances carrying costs and ordering costs in this model. Order Quantity (how much)

EOQ Results (How Much to Order) (Constant Demand, No Shortages) 2 D K h Economic Order Quantity Number of Orders per year Length of order cycle Total cost = D / Q* T = Q* / D = TC = (D / Q*) K + (Q* / 2) h

EOQ Example (How Much) EOQ: D = 1,000 units per year K = $20 per order h = $8.33 per unit per month BE CAREFUL! = $100 per unit per year EOQ: Number of orders per year = 1000/20 = 50 orders Length of order cycle = T = 365 days/50 orders = 7.3 days Total cost = 20(1000/20)+100*20/2 = $2,000

EOQ Example (cont.) (D = 1,000, K = $20, h = $100) Question: What if the company can only order in multiples of 12? (That is, order either 0 or 12 or 24 or 36, etc…)? Answer: Q* = 20. Closest matches (above and below) are 12 and 24. Need to compute TC for both, and decide: TC(Q) = (D / Q) K + (Q / 2) h Q = 12  TC(12) = 20(1000/12)+100*12/2 = $2,266.67 Q = 24  TC(24) = 20(1000/24)+100*24/2 = $2,033.33 So, the company should order in lots of Q = 24

Robustness of EOQ model Annual Cost Very Flat Curve - Good!! Total Cost DTC Q*-DQ Q*+DQ Q* Order Quantity Would have to mis-specify Q* by quite a bit before total annual inventory costs would change significantly. Example

Example: EOQ Robustness Suppose that in the last problem, you have mis-specified the order costs by 100% and the holding costs by 50%. That is, K used in the computation = $40/order (actual cost = $20 / order) h used in computation = $150 / unit / year (actual = $ 100) Then, using these wrong costs, you would have gotten Your actual TC (computed substituting Q’ into TC, using correct costs of K = $20, and h = $100: Only 1% above minimum TC!

Variations of EOQ Some assumptions so far Some variations of EOQ Instantaneous replenishment (zero lead time) Certain and constant demand rate Constant price Some variations of EOQ Positive lead times and uncertain lead times Uncertain demand EOQ with quantity discounts

EOQ with Positive Lead Time On-hand Inventory Time Q Demand Rate, D Time Between Orders (Cycle Time) T = Q/D Average Cycle Inventory, Q/2 Reorder Point, s Place Order Receive order Lead Time, L Q/2

Determining When to Reorder Quantity to order (how much…) determined by EOQ 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, (EOQ assumptions), the reorder point is exactly the amount that will be sold during the lead time. Example: Daily demand (d) = 1,000/365 = 2.74 /day Delivery lead time (L) of 2 days s = d*L = (2.74) (2) = 5.5  6 units

Effects of Demand / Lead Time Variability on Reorder Point (When) Variable demand Expected demand at average demand rate d QUESTION: How much inventory is needed during lead time L? s Safety Stock level KEY POINT: s is larger when there is uncertainty about demand or L Place order Receive order L

Calculation of Appropriate Safety Stock Level Safety stock: stock carried to provide a level of protection against stockouts due to uncertainty of demand during lead time Stockout Criterion: Find s such that the probability of stockout (during the lead time) is  Demand during lead time is a random variable Estimate distribution from historical data (build histogram of demand + frequencies) Normal is frequently used if distribution is unknown

Computing s … Assumption: Demand during lead-time is normally distributed 1- Probability {Demand during lead-time < s} = Probability distribution of demand during lead time Service Level 1-  s

Computing s: Taking Advantage of the Normal Distribution  1- s Probability distribution of demand during lead time: Mean = ; Std Dev =    1- z 1 -  .90 .95 .98 .99 .999 z 1.28 1.64 2.05 2.33 3.09 From normal table or, in Excel, use: =normsinv (0.90)

Issue… The parameters  and  refer to mean and standard deviation of demand during lead time Normally, companies have statistics on demand and lead time per unit of time (say, days, weeks, months) AVG = average demand per unit of time STD = standard deviation of demand per unit of time AVGL = average lead time STDL = standard deviation of lead time Just be consistent: if demand is given on a certain time unit, say, days, then use lead time in the same time unit (in this case, days) How to we compute  and  from AVG, STD, AVGL, and STDL?

More specifically…. If lead time is constant, If demand is constant, Standard deviation of demand during lead time () Safety factor (std normal table) Mean demand during lead time () Safety stock SS If lead time is constant, If demand is constant, Note: This is a very good approximation even when demand is not normally distributed.

The (s,S) Policy: When There Are Fixed Ordering Costs placed arrives Average demand during lead time Safety Stock s S s should be set to cover the lead time demand and together with a safety stock that insures the stock out probability is  (When) S depends on the fixed order cost – EOQ (How much)

The (s,S) Policy: Fixed Ordering Costs Need to define inventory position (IP) IP = On-Hand + On-Order– Backorder Order when: inventory position (IP) drops below s Order how much: bring IP to S (“big S”) Compute Q using the EOQ formula, using mean demand D = AVG (be careful about units…): Set S = s + Q

Example: (s,S) Model Consider inventory management for a certain SKU at Home Depot. Supply lead time is variable (since it depends on order consolidation with other stores) and has a mean of 5 days and standard deviation of 2 days. Daily demand for the item is variable with a mean of 30 units and a coefficient of variation of 0.20. Assume a 95% service level. There are fixed ordering costs that are estimated at $50. Assume that holding costs are 15% of the product cost ($80) per year. Also, assume that the store is open 360 days a year. Propose an inventory policy for this SKU.

Solution Variable definitions and preliminary calculations: Assume 95% service level  z = 1.64 Compute s Compute Q

Example: (s,S) Model Consider inventory management for a certain SKU at WalMart. Supply lead time is variable and has a mean of 1 week and standard deviation of 2 weeks. Weekly demand for the item is variable with a mean of 125 units and a standard deviation of 50. Assume a 90% service level. There are fixed ordering costs that are estimated at $30. Assume that holding costs are 20% of the product cost ($40) per year. Also, assume that the store is open 52 weeks a year. Propose an inventory policy for this SKU.

Solution Variable definitions and preliminary calculations: Assume 90% service level  z = 1.28 Compute s Compute Q

The Base-Stock Policy s: No Fixed Ordering Costs Inventory policy: keep IP constant at s units (s is called the base stock level). When: IP drops below s So, s is also the reorder point for this model How much: order to bring IP back to s Example: suppose inventory level on-hand is 10, s = 20, and there are 2 units already in order. Then, IP = 10 + 2 = 12 units. The firm should order 20 – 12 = 8 units.

Example: Base-Stock Model s Consider inventory management for a certain SKU at King Soopers Supply lead time is variable and has a mean of 2 days and standard deviation of 4 days. Daily demand for the item is variable with a mean of 24 units and a coefficient of variation of 0.30. Propose an inventory policy for this SKU. Assume a 98% service level. Assume 98% service level  z = 2.05

Summary of Inventory Models Use EOQ How much: Q (EOQ formula) When: d*L (reorder point) yes Use (s, S) policy How much: necessary to bring IP back to S, where S = s + Q (Q is from EOQ formula) When: IP drops below s (base-stock policy formula) Is demand rate and lead time constant? yes Are there fixed ordering costs? no no Use base stock (s) policy When: IP drops below s How much: necessary to bring IP back to s

Risk Pooling Means and variances are additive Stock is based on std. Deviations Square root law: stock for combined demands is less than the combined stocks

HP Example: Benefits of a Universal Product Because of a different power supplies, HP had two laser printers, one for Europe and one for N. America. A universal product (with a universal power supply) has been proposed, but costs $30 extra. Is it worthwhile? Below is monthly demand for HP for the two markets (in thousands). Assume a one-month constant lead-time (STDL = 0) for both markets. N. America N(200,60) Europe N(150,50) Consider z = 2 (~ 98% of service level)

HP Example (cont.): Benefits of a Universal Product Demand seen by HP (NA and Europe)