Inventory models Nur Aini Masruroh

Outline  Introduction  Deterministic model  Probabilistic model

Introduction  What is inventory?  It is usable but idle resource  Keeping of physical goods or commodities for the purpose of satisfying demand over a specified time horizon.  Over stocking  Higher invested capital per unit time  Less frequent occurrence of placement of orders  Less frequent occurrence of shortages  Under stocking  Lower invested capital per unit time  Increase the frequency of ordering  Higher risk of running out of stock

The Functions of Inventory  To ”decouple” or separate various parts of the production process  To provide a stock of goods that will provide a “selection” for customers  To take advantage of quantity discounts  To hedge against inflation and upward price changes

 Higher costs  Item cost (if purchased)  Ordering (or setup) cost  Costs of forms, clerks’ wages etc.  Holding (or carrying) cost  Building lease, insurance, taxes etc.  Difficult to control  Hides production problems Disadvantages of Inventory

Inventory Process stage Demand Type Number & Value Other Raw Material WIP Finished Goods Independent Dependent A Items B Items C Items Maintenance Operating © T/Maker Co. Inventory Classifications

 Divides on-hand inventory into 3 classes  A class, B class, C class  Basis is usually annual $ volume  $ volume = Annual demand x Unit cost  Policies based on ABC analysis  Develop class A suppliers more  Give tighter physical control of A items  Forecast A items more carefully ABC Analysis

% of Inventory Items Classifying Items as ABC % Annual $ UsageA B C Class% $ Vol% Items A8015 B 30 C 555

Independent versus Dependent Demand  Independent demand - demand for item is independent of demand for any other item  Dependent demand - demand for item is dependent upon the demand for some other item

Inventory decision  How should it be ordered for stocking?  When should it be ordered?

Basic characteristics of inventory systems  Economic parameters  Holding cost  Ordering cost  Setup cost  Shortage cost  Purchase price  Selling price  Demand  Ordering cycle  Delivery lags or lead times

Economic parameters  Holding costs - associated with holding or “carrying” inventory over time  Ordering (setup) costs - Involve the fixed charge associated with the placement of an order or with the initial preparation of a production system.  Shortage costs - The penalty costs incurred as a result of running out of stock  Loss in customers goodwill  Loss in sales, etc.  Purchase price - This parameter is of interest when quantity discounts or price breaks can be secured for orders above a certain quantity  Selling price- This parameter is of interest when the demand on the commodity may be affected by the quantity stocked

Holding (Carrying) Costs  Obsolescence  Insurance  Extra staffing  Interest  Pilferage  Damage  Warehousing  Etc.

Ordering Costs  Supplies  Forms  Order processing  Clerical support  Etc.

Symbols of inventory model  Q = Order quantity per order  K = Setup cost per order  d = Demand rate  h = Holding cost per unit per unit time  c = Purchase price or cost per unit  s = Shortage cost per unit per unit time  t = Inventory cycle

EOQ assumptions:  Known and constant demand  Known and constant lead time  Instantaneous receipt of material  No quantity discounts  Only order (setup) cost and holding cost  No stockouts Basic EOQ model

Inventory Usage Over Time Time Inventory Level Average Inventory (Q*/2) 0 Minimum inventory Order quantity = Q (maximum inventory level) Usage Rate

EOQ Model How Much to Order? Order quantity Annual Cost Holding Cost Curve Total Cost Curve Order (Setup) Cost Curve Optimal Order Quantity (Q*) Minimum total cost

 More units must be stored if more are ordered Purchase Order DescriptionQty. Microwave1 Order quantity Purchase Order DescriptionQty. Microwave1000 Order quantity Why Holding Costs Increase

Cost is spread over more units Example: You need 1000 microwave ovens DescriptionQty. Microwave1 Purchase Order DescriptionQty. Microwave1 Purchase Order DescriptionQty. Microwave1 Purchase Order Description Qty. Microwave 1 1 Order (Postage $ 0.33)1000 Orders (Postage $330) Order quantity Purchase Order Description Qty. Microwave1000 Why Order Costs Decrease

EOQ Model When To Order Reorder Point (ROP) Time Inventory Level Average Inventory (Q*/2) Lead Time Optimal Order Quantity (Q*) Inventory holding cost per cycle: hQ 2 /(2d)

 Total cost per cycle TC(Q) = Acquisition costs + Holding cost = K + cQ + hQ 2 /(2d)  Total cost per unit time TCU(Q) = TC(Q)/t = Kd/Q + cd + hQ/2  where t = Q/d.  To minimize the total cost per unit time, we differentiate TCU(Q) with respect to Q and set it equal to zero  This gives  The order cycle length, Deriving EOQ Model

The Reorder Point (ROP) Curve Q* ROP (Units) Slope = units/day = d Lead time = L Time (days) Inventory level (units)

EOQ Model with Shortages Allowed Assumptions:  A continuous, constant and known rate of demand d  Constant lead time L  Constant unit price or cost c  No interaction between items  Infinite planning horizon  Unfilled demands are backlogged at the cost of s per unit per unit time  Split delivery not allowed  No limit on capital availability

EOQ Model with Shortages Allowed

Single item economic production quantity (EPQ) Assumptions:  Continuous, constant and known rate of demand d  Continuous, constant and known rate of production p  P > d  Constant lead time L  Constant unit price or cost c  No interaction between items  Infinite planning horizon  Shortages are not permitted  Production and consumption can occur simultaneously  No limit on production capacity

Single item economic production quantity (EPQ)

 Answers how much to order & when to order  Allows quantity discounts  Reduced price when item is purchased in larger quantities  Other EOQ assumptions apply  Trade-off is between lower price & increased holding cost Quantity Discount Model

Quantity Discount Schedule Discount Number Discount QuantityDiscount (%) Discount Price (P) 10 to 999No discount$ ,000 to 1,9994$ ,000 and over5$4.75

Quantity Discount – How Much to Order

 Answer how much & when to order  Allow demand to vary  Follows normal distribution  Other EOQ assumptions apply  Consider service level & safety stock  Service level = 1 - Probability of stockout  Higher service level means more safety stock  More safety stock means higher ROP Probabilistic Models

Probabilistic Models When to Order? Reorder Point (ROP) Optimal Order Quantity X Safety Stock (SS) Time Inventory Level Lead Time SS ROP Service Level P(Stockout) Place order Receive order Frequency

 Answers how much to order  Orders placed at fixed intervals  Inventory brought up to target amount  Amount ordered varies  No continuous inventory count  Possibility of stockout between intervals  Useful when vendors visit routinely  Example: P&G representative calls every 2 weeks Fixed Period Model

Inventory Level in a Fixed Period System Various amounts (Q i ) are ordered at regular time intervals (p) based on the quantity necessary to bring inventory up to target maximum ppp Q1Q1Q1Q1 Q2Q2Q2Q2 Q3Q3Q3Q3 Q4Q4Q4Q4 Target maximum Time On-Hand Inventory

Time Inventory Level Target maximum Period Fixed Period Model When to Order?