© Hajime Mizuyama Production & Operations Management #1 @AGU Lec.5: Inventory Control and Management (2) • Inventory control problem • Fixed size ordering system • Fixed interval ordering system 

© Hajime Mizuyama Course Schedule #1 Date Contents Outline of production management (1): How to conceptualize production systems Outline of production management (2): Framework of production planning and control Inventory control and management (1): Economic order quantity (EOQ) and the role of safety stock Inventory control and management (2): Conventional inventory control models Production planning and linear programming MRP (1): Bill of materials (BOM), priority planning, capacity planning, etc. MRP (2): Lot sizing and dynamic order quantity (DOQ) 

© Hajime Mizuyama Inventory Control Problem Order placement Replenishment Order receipt Delivery Demand side Supply side Inventory level Time When & how many? At the beginning of time period t + LT +1. (= At the end of time period t + LT.) At the end of time period t. 

© Hajime Mizuyama Available Stock Stock on hand Stock on order Backorder Available stock = Stock on hand + Stock on order - Backorder 

© Hajime Mizuyama Fixed Size Ordering System (Ordering Point System) Order Replenishment Time Procurement lead-time Order size (Fixed) Order interval (Varied) Inventory level Ordering point Order Order Replenishment Replenish- ment Procurement lead-time Procurement lead-time 

© Hajime Mizuyama Safety stock Fixed Size Ordering System (Ordering Point System) Order Replenishment Time Procurement lead-time Order size (Fixed) Order interval (Varied) Inventory level Ordering point Order Order Replenishment Replenish- ment Procurement lead-time Procurement lead-time 

© Hajime Mizuyama Safety Stock Level Probability distribution of the inventory level Demand uncertainty in this period should be covered by the safety stock (= Procurement lead-time: LT ) Time Inventory level Ordering point Order Replenishment 

© Hajime Mizuyama Safety Stock Level Probability distribution of the inventory level Time Inventory level Ordering point Replenishment Order In discrete time setting, order delay may occur. The delay is less than a day, and hence the period to be covered is less than LT +1. LT Demand uncertainty in this period should be covered by the safety stock 

© Hajime Mizuyama Safety stock Safety Stock Level Stockout probability Demand uncertainty in this period should be covered by the safety stock (< LT +1) Inventory level Ordering point Order Replenishment Time Probability distribution of the inventory level LT 

© Hajime Mizuyama Stockout prob. Safety stock (S! = 𝑘 $ 𝐿𝑇 + 1 $ 𝜎𝐷 ) Ordering point (𝑠 = (𝐿𝑇 + 1) $ 𝐷 + 𝑆! ) 5% k=1.65 1% k=2.33 Service level: • The probability that all the orders received from the demand side in a specified period can be fulfilled immediately (= 1 - Stockout prob.). • It can be calculated from the inventory level at the beginning of and the probability distribution of the demand quantity in the period. Safety Stock and Service Level Probability density Demand Average demand = (𝐿𝑇 + 1) $ 𝐷 

© Hajime Mizuyama Probability distribution of uncertain demand per day 𝑁(𝐷 = 200, 𝜎! " = 100) Procurement lead time (LT) 3 days Safety stock level & ordering point for keeping service level at 95% Numerical Example 

© Hajime Mizuyama Inventory Control Operations How to determine ordering timing • Monitor the level of available stock continuously, and if it becomes lower than the ordering point s, trigger an order. How to determine ordering quantity • Calculate EOQ based on the average demand, setup cost and holding cost, and order (approximately) this quantity (Q = EOQ) every time. • When underlying conditions, such as average demand or costs, change, update EOQ accordingly. 

© Hajime Mizuyama Double bin system • Prepare two bins (containers) of equal storage capacity, and, when one of them becomes empty, trigger an order of the size of the bin (container). • For its simplicity, it is widely used in practice. • This system can be regarded as a fixed size ordering system with an additional constraint: 𝑄 = 𝑠 A Variant of Fixed Size Ordering System 

© Hajime Mizuyama Fixed Interval Ordering System (Periodic Ordering System) 1/10 1/25 2/10 2/25 3/10 3/25 Lead-time Inventory level Order interval (Fixed) Order Order Order Repleni- shment Repleni- shment Repleni- shment Order size (Varied) Time Order cycle 

© Hajime Mizuyama Safety stock Fixed Interval Ordering System (Periodic Ordering System) 1/10 1/25 2/10 2/25 3/10 3/25 Lead-time Order cycle Inventory level Order interval (Fixed) Order Order Order Repleni- shment Repleni- shment Repleni- shment Order size (Varied) Time 

© Hajime Mizuyama Safety Stock Level Probability distribution of the inventory level Inventory level Order Replenish- ment Order Replenish- ment Time Demand uncertainty in this period should be covered by the safety stock (= The period until the next order is fulfilled) 

© Hajime Mizuyama Safety Stock Level Probability distribution of the inventory level Time Demand uncertainty in this period should be covered by the safety stock (= The period until the next order is fulfilled) Inventory level 

© Hajime Mizuyama Safety stock Safety Stock Level Stockout probability Probability distribution of the inventory level Inventory level Time Order-up-to level Demand uncertainty in this period should be covered by the safety stock (= The period until the next order is fulfilled) 

© Hajime Mizuyama Stock-out prob. Safety stock 𝑆! = 𝑘 $ 𝐿𝑇 + 𝐶𝑇 $ 𝜎𝐷 Order-up-to level 𝑆 = (𝐿𝑇 + 𝐶𝑇) $ 𝐷 + 𝑆! 5% k=1.65 1% k=2.33 Service level • The probability that all the orders received from the demand side in a specified period can be fulfilled immediately (= 1 - Stockout prob.). • It can be calculated from the inventory level at the beginning of and the probability distribution of the demand quantity in the period. Safety Stock and Service Level Probability density Demand Average demand = (𝐿𝑇 + 𝐶𝑇) $ 𝐷 

© Hajime Mizuyama Probability distribution of uncertain demand per day 𝑁(𝐷 = 200, 𝜎! " = 100) Procurement lead time (LT) 3 days Order cycle time (CT) 6 days Safety stock level & order-up-to level for keeping service level at 95% Numerical Example 

© Hajime Mizuyama Inventory Control Operations How to determine ordering timing • Order periodically, e.g., every Monday, on nth day in every month, etc. How to determine ordering quantity • Q = Expected demand until the next order is fulfilled + Safety stock - Available stock at the time of ordering When this is treated as a constant value, it is called the order-up-to level S. 

© Hajime Mizuyama (R, s, S) system • Check the level of available stock periodically (with cycle time R), and, if and only if the level is lower than s, trigger an order so as to bring the level back up to S. • The level s is usually set as: 𝑠 = (𝐿𝑇 + 𝐶𝑇/2) 3 𝐷 + 𝑆# • This system can be regarded as a fixed interval ordering system with an additional constraint on the ordering lot size: 𝑄 > (𝑆 − 𝑠) = 𝐷×𝐶𝑇/2 A Variant of Fixed Interval Ordering System 

© Hajime Mizuyama Fixed size ordering system Fixed interval ordering system Parameters Order quantity: Q Ordering point: s Order cycle: R Order-up-to level: S Order interval Varied Fixed Order quantity Fixed Varied Merit Suitable for lot production/delivery Easy to coordinate with production plan Comparison between Fixed-Size and Fixed-Interval Systems