Graduate Program in Business Information Systems Inventory Decisions with Certain Factors Aslı Sencer

A Retailer’s Plea If I order too little, I make no profit. If I order too much, I may go broke. Every product is different. Help me!

Aslı Sencer Why do we control inventory?  Inventories represent a vast segment of total economic activity.  Even minor improvements can create large savings. How do we control inventory?  Application of optimization techniques  Information processing and retrieval techniques

Aslı Sencer Decisions of an inventory policy  If there is no production, i.e., pure inventory system How much to order? Order quantity When to order? Reorder quantity Ex:Order Q=100 units when the inventory level drops to r=15 units.  If there is also production When to start/stop production?

Aslı Sencer An inventory system

Aslı Sencer Elements of Inventory Decisions  Costs:  Ordering and Procurement costs  Inventory holding or carrying costs  Inventory shortage costs  Demand structure  How does it vary? Certain, uncertain?  Supply structure  Any capacity limitations, defectives, number of suppliers?  Lead times:  Certain, uncertain?

Aslı Sencer Ordering and Procurement Costs  Represent all expenses incurred in ordering or manufacturing items related to Acquisition Transportation Collecting, sorting, placing the items in the storage Managerial and clerical costs associated with order placement.  Ordering costs are fixed, independent of the order size.  Procurement costs depend on the order size.

Aslı Sencer Holding or Carrying Costs  Expenses incurred during the storage of items. Physical Costs: Warehouse operation costs, insurence, property taxes. Pilferage, spoilage, obsolescence Opportunity cost of investing in inventory rather than investing somewhere else, ex. in a bank.  Inventory costs are variable costs that depend on the order size.

Aslı Sencer Shortage Costs  Occur whenever the demand is not satisfied. Order is either “backordered” or “lost”.  Backordering Costs: Fixed cost of extra managerial work. Loss of customer goodwill: Variable cost that depends on duration of backorder.  Lost Sales Costs: Marginal profit that the item would have earned. Loss of customer goodwill.

Aslı Sencer Demand Structure  Continuous versus discrete demand Ex: Natural gas consumption in houses Detergent consumption in houses  Deterministic (certain) versus stochastic (uncertain) demand Ex: Order quantities for the next months are 20,30,10,50. Order quantities in a month are normally distributed with mean 25 and variance 4.  Constant versus dynamic demand Ex: Demand quantities for the next months are 20, 21, 20, 19 Demand quantities for the next months are 20, 50, 10, 2

Aslı Sencer Supply Structure  Any defectives? If the received lot includes defective items this brings uncertainty  Any capacity limitations? Do we fully receive what we order?  Number of suppliers, fixed or variable?

Aslı Sencer Lead time  Time elapsed between the order delivery and order receipt.  Can be constant or stochastic. Ex: Lead time is 10 days. Lead time is between 8-12 days.

Aslı Sencer The Economic Order Quantity EOQ-Model  Decision variable:Q = Order Quantity  Parameters: k = Fixed cost per order ($/order) A = Annual number of items demanded (unit/year) c = Unit cost of procuring an item ($/unit) h = Annual cost of holding a dollar in inventory ($/$/year)  Objective is to “minimize total annual cost”. Total = Ordering + Holding + Procurement Annual costCost

Aslı Sencer EOQ Inventory Policy Average Inv. Level

Aslı Sencer Assumptions of Classical EOQ Model  Demand rate is constant or stable.  There is infinite supply availability.  Lead time is constant or zero.  No quantity discounts are made.  All demand is met on time, no backordering, no stockout.

Aslı Sencer Costs of EOQ Model  Total ordering cost is the number of orders times the cost per order:  Total holding cost is the cost per item held 1 year times the average inventory:  The annual procurement cost is the product of annual demand and unit cost: Procurement cost = Ac

Aslı Sencer Annual Cost of EOQ-Model  Here Ac is not a relevant cost and thus dropped.  Minimize Total Annual Inventory Cost:

Aslı Sencer Optimal Solution of EOQ  Optimal solution is the economic order quantity  Optimal Total Cost

Aslı Sencer Example:The House of Wines and Liquors Allex Mullen decides that the first task in utilizing inventory models is to determine the value of model parameters:  Annual demand 5200 cases of beer  $10 telephone charge for ordering  Purchase cost is $1.5/case beer +shipping cost $0.5/case  10%bank interest, 5%state franchise tax, 5% theft insurance rate How many should he order, how often, and at what annual relevant inventory cost?

Aslı Sencer Solution: The economic order quantity is  The inventory cycle duration is T = Q/A = 510/5200 = year or 36 days  The total annual relevant inventory cost is:

Aslı Sencer Robustness of EOQ Model  EOQ is a robust model with respect to the estimation errors in A, k, c or h.  Let A actual =4 A estimated Then EOQ actual =2 A estimated Since

Aslı Sencer Ex: The House of Wines and Liquors  Alex Mullen applies EOQ to another product, a particular variety of Chilean wine that sells 1000 cases annually. The cost is $20 per case. A telephone call to Chile to place an order costs $100. The holding costs are the same as for Tres Equis Beer.

Aslı Sencer Ex: T = Q/A = 24/1000 =.224 year or 82 days

Aslı Sencer Optimal Inventory Policy with Backordering Orders placed during shortages are backordered.

Aslı Sencer Optimal Inventory Policy with Backordering S: Quantity on hand when a shipment arrives. P: Cost of being one item short for a year Optimal order quantity and order level:

Aslı Sencer Example:The House of Wines and Liquors-Backorders The marketing department tells Alex that beer is a convenience product that can not be backordered, so sale is lost! However some wine customers are connoisseurs who are willing to order out-of-stock items. Nevertheless, the store owner will incur some penalty cost if there is a shortage of wine. Suppose that retailer suffers lost profit on future business equal to $0.01/unit each day that a wine is on backorder. What should be the optimal ordering policy if backordering is allowed? Solution: The order quantity is computed: p = $.01×365 = $3.65/unit/year.

Aslı Sencer  The order level S is  The relevant cost is smaller than before, why? Example: Solution

Aslı Sencer Is backordering better?  Fewer orders are placed when there is backordering.  Average inventory level is smaller. Backorders/cycle= Q* – S*=324 – 154 = 170 units/cycle. Proportion of demand not satisfied on time =(Q*-S*)/Q*=170/324= 52.5%  The results suggest that: Retailers will run short in each cycle. But can they get away with it?  So backordering must make sense!

Aslı Sencer Imputed Shortage Penalty An alternative approach for establishing an inventory policy is based on achieving a desired service level. Service Level, L is the proportion of demand met on time Imputed shortage penalty p= hcL 1  L

Aslı Sencer As p increases EOQ is more robust $36 imputed shortage penalty L=90% EOQ with no backordering P L=47.5% Q* S* $3.65 A=1000 units/yrk=$100/order c=$20/unit h= $0.20/$/year

Aslı Sencer Economic Production-Quantity Model The inventory model may be extended to finding the optimal production quantity.

Aslı Sencer  B: Annual production rate  K: Production setup cost.  c: Variable production cost per unit.  Total Annual Cost:  Economic Production Quantity: Economic Production-Quantity Model

Aslı Sencer Example: Water Wheelies have annual demand of A =100,000 units and are made at the rate of B = 500,000 units. Production costs are k = $2,000/setup and c = $5/unit variable. It costs h = $.40/year to tie up a dollar.  Economic production quantity is  Total relevant cost is TC(8,944)

Aslı Sencer More Elaborate Models  Incorporate a second one-time shortage penalty.  Add additional products.  Incorporate uncertainty regarding:  Demand  Lead-time for delivery of order  Incorporate lost sales  Extend to single period products

Aslı Sencer Economic Order Quantity Model (Figure 15-3)

Aslı Sencer Sensitivity Analysis (Figure 15-6)

Aslı Sencer Graphing the Sensitivity Analysis (Figure 15-7)

Aslı Sencer Backordering Model (Figure 15-9)

Aslı Sencer Production Model (Figure 15-13)