Inventory Models (II) under SC environment By Dr. Debadyuti Das

The Multi-Period Continuous Review [ROP or Q or (s, S)] Model There are five determinants of reorder point quantity. The rate of demand The lead time The extent of demand variability The extent of lead time variability The degree of stockout risk acceptable to the management (safety stock) If demand and lead time are both constant, the reorder point is simply ROP = d x LT

A View of (s, S) Policy S Inventory Level s Time Inventory Position Lead Time Lead Time Inventory Level s Time

Safety Stock Quantity Maximum probable demand Expected demand LT Time Expected demand during lead time Maximum probable demand ROP Quantity Safety stock Safety stock reduces risk of stockout during lead time

Reorder Point The ROP based on a normal Risk of a stockout Service level Probability of no stockout Expected demand Safety stock z Quantity z-scale The ROP based on a normal Distribution of lead time demand

Notations D = average demand per unit time d = standard deviation of daily demand dLT = standard deviation of demand during lead time LT = replenishment lead time in days LT = standard deviation of lead time h = holding cost of one unit for one day K = fixed cost SL = Service level (for example, 95%). This implies that the probability of stocking out is 5% SS = Safety inventory

Notations ROP = Reorder point The reorder point (s) has two components: To account for average demand during lead time: LT D To account for deviations from average (we call this safety stock) z  STD   d Since there is a fixed cost, we order more than up to the reorder point: Q=(2 K D)/h The total order-up-to level is: S=Q+s

Three possible situations Case 1: Only demand is variable Case 2: Only lead time is variable Case 3: Both demand and lead time are variable

Case1: Example The distributor has historically observed weekly demand of: D = 44.6 d = 32.1 Replenishment lead time is 2 weeks, and desired service level SL = 97% Average demand during lead time is: 44.6  2 = 89.2 Safety Stock is: 1.88  32.1  2 = 85.3 Reorder point is thus 175, or about 3.9 weeks of supply at warehouse and in the pipeline

Case1: Example (Contd.) Q=679 Order-up-to level (S) thus equals: Reorder Point + Q = 175+679 = 854 Cycle inventory = Q/2 = 340 Average Inventory = cycle inventory + SS = 340 + 85 = 425 Average Flow Time = Avg inventory / throughput = 425 /44.6 = 9.53

Case 2: Example Only lead time is variable D = 600, Average lead time (LT) = 6 S.D. of LT (LT) = 2 Desired service level = 90 % Demand during average LT = (D X LT)= (600 X 6) = 3600 Safety stock = z D (LT) = 1.28 X 600 X 2 = 1536 ROP = 5136

Case 3: Example Both demand and lead time are variable Average demand (D) = 25, Average lead time (LT) = 10 d = 3, LT = 2 Desired service level = 95 % ROP = D X LT + Z (LT d2 + D2 LT2) = 25 X 10 + 1.65 10(3)2 + (25)2 (2)2 =334

Impact of demand and supply uncertainty on safety stock Average demand S.D. of demand Average lead time S.D. of L.T. Safety stock Safety stock in days Remarks 100 30 15 5 1026 10.3 Base case 232 2.3 No supply uncertainty 1000 10 No demand uncertainty 1006 Reduce demand uncertainty 2.5 526 5.3 Reduce supply uncertainty 7.5 1003 Reduction in lead time

Periodic Review Policy (P Model or Fixed order Model or Base stock policy) Orders are placed at fixed time intervals Order quantity for next interval? Suppliers might encourage fixed intervals May require only periodic checks of inventory levels Risk of stockout

Fixed order Model At each review, inventory position is raised to the base-stock level. The base-stock level includes two components: Average demand during r+L days (the time until the next order arrives): (r+L)*D Safety stock during that time: z* d * r+L

Base-Stock Policy Inventory Level Time r L Inventory Position Base-stock Level Inventory Position r L

Issues involved in FOI Model Items from the same supplier may yield savings in: Ordering Packing Shipping costs May be practical when inventories cannot be closely monitored However, Requires a larger safety stock Increases carrying cost Costs of periodic reviews

Risk pooling (Centralization vs Decentralization) Ordering cost: Rs. 257 per order Purchase price: Rs. 30 Inventory carrying cost per unit per year: 20% of Rs. 30 i.e. Rs 6 Annual demand: 30000 Q*: 1600 Cycle stock: Q*/2: 800 Service level: 97.7 % d: 100; d : 30; L: 15 Transportation cost in centralization case: Rs 1/unit Transportation cost in decentralization case: Rs 1.10/unit

Risk Pooling (Centralization vs Decentralization) For the same service level, which system will require more inventory? Why? For the same total inventory level, which system will have better service? Why? What are the factors that affect these answers?

Risk pooling (Centralization vs Decentralization) Analysis of decentralization vs centralization Decentralized system Centralized system Remarks No. of stock points 16 1 Cycle stock 800 3200 Increases cycle stock by 4 times Safety stock 232 928 Increases by 4 times Total inventory (232 + 800) x 16 = 16,512 928 + 3200 = 4128 Total inventory carrying cost 16512 x 6 = 99,072 4128 x 6 = 24,768 Incremental transportation cost 300 x 100 x 16 x 0.1 = 48,000