OPERATIONS MANAGEMENT Inventory Management Stochastic Inventory Control Models W-16-01 DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR AGENDA • Stochastic inventory control models • Periodic review model (BH, Chapter 11.2) • Continuous review model (BH, Chapter 11.3) DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR CYCLE SERVICE LEVEL VS FILL RATE DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR STOCHASTIC CONTINUOUS REVIEW SYSTEM Q Replenishment cycle ROP L DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY Q Make an order of size Q Make an order of size Q INVENTORY MANAGEMENT L ATA JALILI MARAND ASSISTANT PROFESSOR Time PERFORMANCE MEASURES (Cycle) service level customer 11) fraction of replenishment cycles that end with all demand being met (used in BH, Chapter = Probability of not having a stock out Fill rate available DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY fraction of orders that are (immediately) filled from inventory. INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR CYCLE SERVICE LEVEL VS. FILL RATE (1) “Inventory cycle Cycle 1 2 3 4 5 6 7 8 9 10 Demand 18 23 24 15 26 16 14 19 24 21 Sum 200 DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY Shortage ----5 --------7 ----- service level” = 1-2/10 = 80% “Fill rate” service level = 1-(5+7)/200 = 94% 12 INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR CYCLE SERVICE LEVEL VS. FILL RATE (2) Cycle 1 2 3 4 5 6 7 8 9 10 Demand 18 23 24 15 26 16 14 19 24 21 Sum 200 DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY Shortage ----5 --------5 2 --- “Inventory cycle service level” = 1-3/10 = 70% “Fill rate” service level = 1-(5+7)/200 = 94% 12 INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR CYCLE SERVICE LEVEL VS. FILL RATE (3) “Inventory cycle Cycle 1 2 3 4 5 6 7 8 9 10 Demand 18 23 24 15 26 16 14 19 24 21 Sum 200 DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY Shortage ----5 --------9 ----- service level” = 1-2/10 = 80% “Fill rate” service level = 1-(5+9)/200 = 93% 14 INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR CONTINUOUS REVIEW MODEL (ROP, Q) DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR REORDER POINT CALCULATION (1) The Reorder point, ROP, represents the inventory available to meet all demand that arises over a period of length . We determine ROP such that the business is protected against uncertainties in demand and lead time over a period of length . Protection interval = L DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR REORDER POINT CALCULATION (2) Reorder point was previously calculated as: ROP = Average demand during lead time Lead time Inventory Reorder point time DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR REORDER POINT CALCULATION (3) Now we consider safety stock: ROP = Average demand during lead time + Safety stock Lead time Inventory Reorder point Safety Stock DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR time NOTATION Demand Lead time Deman d during lead time DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY d = stochastic variable; demand per time unit 𝑑ҧ = average demand per time unit 𝜎𝑑 = standard deviation of demand per time unit L = stochastic variable; Lead time ത 𝐿 = average lead time 𝜎𝐿 = standard deviation of lead time 𝑑𝐿 = stochastic variable; demand during lead time 𝜇𝑑𝐿 ത= average demand during lead time = 𝑑ҧ 𝐿 𝜎𝑑𝐿 2𝜎 2 = standard deviation of demand during lead time ത𝜎𝑑2 + 𝑑ҧ = ඥ𝐿 𝐿 INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR DEMAND DURING LEAD TIME 𝑁 (𝜇 𝑑 , 𝜎 𝑑 ) 𝐿 𝐿 dL Standard deviation of demand during lead time d L Average demand during lead time Now, what is the risk of a stockout? X% SS d DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY L ROP ROP d L SS INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR Demand during lead time CALCULATING ROP (1) Given the target service level (), the reorder point equals the smallest ROP for which the following equation holds: Now, assume is normally distributed with mean and standard deviation . The equation above can be rewritten as follows: is a random variabe with standard normal distribution. Let denote the -score corresponding to the service level , then we have DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT Always rounded up to hit (at least) the service level ATA JALILI MARAND ASSISTANT PROFESSOR CALCULATING ROP (2) 𝑁 (0,1) 𝑁 (𝜇 𝑑 , 𝜎 𝑑 ) 𝐿 𝜇𝑑 𝐿 𝐿 0 𝑅𝑂𝑃 Distribution of DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY Distribution of INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR 𝑧 CALCULATING ROP (3) 𝑆1 𝑧 DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR CALCULATING ROP (4) 𝜇 𝑑 =𝑑 𝐿 𝐿 𝜎 𝑑 =√ 𝐿 𝜎 + 𝑑 𝜎 2 𝑑 𝐿 2 2 𝐿 2 d 2 ROP d L z d L dL z L d DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR 2 L MODIFICATION TO CYCLE SERVICE LEVEL Let denote the cycle service level and denote the expected number of cycles with shortage per year. Then, the following equation holds: or equivalently DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR EXAMPLE 11.3 (1) Demand Lead time d 16 L 9 d 3 L 2 Cycle service level =95% Demand during lead time dL dL 16 × 9 144 σ dL L2d d2L2 9 × 32 162 × 22 33.24 Always rounded up to hit (at least) the service level ROP dL z × dL 144 1.645 × 33.24 198,677 199 . DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY Z = NORMSINV (0.95) INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR EXAMPLE 11.3 (2) 250 200 150 ROP SS 100 50 INVENTORY MANAGEMENT 0. 98 0. 96 0. 94 0. 92 0. 9 0. 88 0. 86 0. 84 DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY 0. 82 0. 8 0 ATA JALILI MARAND ASSISTANT PROFESSOR SUMMARY is given Calculate the safety factor Calculate the safety stock Calculate the reorder point DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR PERIODIC REVIEW MODEL (RP, R) DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR Order received Order placed L 𝒕𝟎 DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY 𝒕 𝟎+𝑳 INVENTORY MANAGEMENT RP L 𝒕𝟎+𝑹𝑷 ATA JALILI MARAND ASSISTANT PROFESSOR RP + RP + L Deficiency at time + RP + L depends on and the aggregate demand in the period from to + RP + L (length ) On-hand inventory STOCHASTIC PERIODIC REVIEW SYSTEM (1) R L 𝒕𝟎+𝟐𝑹𝑷 Time STOCHASTIC PERIODIC REVIEW SYSTEM (2) • When an order is placed, we aim at having the sum of quantity ordered and the inventory on hand (=inventory position) equal to the restocking level, R. • Once an order is placed, the replenishment lot arrives after the lead time L (Here we assume L to be constant). • The next reorder period is RP time units after the first order, and the next order will arrive after RP + L units of time (Here we assume RP to be given). How is the restocking level R established? DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR STOCHASTIC PERIODIC REVIEW SYSTEM (3) The Restocking level (order-up-to level), R, represents the inventory available to meet all demand that arises over period to . Protection interval = RP + L DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR CALCULATING R (1) Given the target service level (), the restocking level equals the smallest R for which the following equation holds: Now, assume is normally distributed with mean and standard deviation . The equation above can be rewritten as follows: is a random variabe with standard normal distribution. Let denote the -score corresponding to the service level , then we have Always rounded up to hit (at least) the service level DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR CALCULATING R (2) 𝑆1 𝑧 DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR EXAMPLE 11.1 Cycle service level DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY Average daily demand md 18.46 Std. Dev of daily demand sd 11.09 Review period Lead time RP L RP+L Cyc le service level S1 Average demand during RP+L Std. Dev demand during RP+L m RP+L s RP+L Safety factor z Order- up-to level R INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR 10 3 13 95% EXAM 2019 DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR EXAM 2019 (1) DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR EXAM 2019 (2) DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR NEXT? DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR IN THE NEST SESSION We will learn solve exam assignments from previous years. DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY INVENTORY MANAGEMENT ATA JALILI MARAND ASSISTANT PROFESSOR DEPARTMENT OF ECONOMIC S AND BUSINESS ECONOMIC S A ARHUS UNIVERSITY
