# probabilistic demand copy

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- 1. Individual Items with Probabilistic Demand

2. TERMINOLOGY On-hand stock: Physically present on shelf. Cannot be -ve number. Net stock: (on-hand) - (Backorders) Inventory position: (on-hand) +(on order) -(Backorders) - (Committed) Safety stock: Average level of net stock just before a replenishment arrives 3. Backorder vs. Lost sales Complete backordering: Government organizations, wholesale-retail link of few distribution systems (exclusive dealership) Complete lost sales: Retail-consumer link mostly for FMCGs Combination of the two Numerical value of SS depends on the degree to which backorders or lost sales occur 4. Three Key Issues Howoftentheinventorystatusshoulddetermined? When a replenishment order should be placed? How large the replenishment order should be?be 5. 4 Questions for Inventory Policies How important is the item? Can, or should, the stockstatus be reviewedcontinuously or periodically? What form should the inventory policy take? What specific cost or service objectives should be set? 6. IMPORTANCE OF THE ITEM A items comprise roughly 20% of the total number of items, but represent 80% of the dollar sales volume B items comprise roughly 30% of the items and 15% of the dollar volume C items comprise of balance 50% of items and only 5% of the total dollar volume 7. CONTINUOUS vs. PERIODIC Howoftenshouldtheinventorystatusbedetermined? - Periodic ReviewTransactions Reporting - Continuous review COMPARISON Items may be produced on the same pieceofequipment, purchased from same supplier or shipped in the same transportation mode - Periodic Review 8. CONTINUOUS Vs. PERIODIC COMPARISON Periodic reviewallows a reasonable prediction ofthe level of the workload on the staff. While the replenishment can be done at any moment in continuous review making it less predictable Continuousreview is expensive. Especially in fastmoving goods (FMCG). So reviewing errors and costs are high 9. CONTINUOUS Vs. PERIODIC COMPARISON For slow movinggoods, periodic review is stillbetter as it can trace any pilferage or spoilage during the review period but if it continuous review it doesnt detect unless a transaction happens Continuousreview, to provide the same level ofcustomer service, requires less safety stock than periodic review 10. Four Types of Control Systems Order Point, Order Quantity (s, Q) System Order Point, Order-up-to-level (s, S) System Periodic-Review, Order-up-to-level (R, S) System (R, s, S) System 11. (s, Q) System Continuous review (R = 0) Inventory position and notNet stock is used totrigger an order.Two-binsystem: Amountinthesecondbincorresponds to order point.(s, Q) system is easy to understand even for the stock clerk 12. (s, Q) System One disadvantage is that it cannot accommodate a large order. For ex., if Q=10 units, D=15 units occurring whenInventory Levelthe position is 1 unit just above s. Order in multiples. s+Qs ALBTime 13. (s, S) System Orderquantity is variable enough to raise theposition to order-up-to level SFor unit-sized demand, (s,S) is same as (s,Q) system. Inventory LevelSs ALBTime 14. (s, S) System Also called as min-max system as inventory position, except for momentary drop is always between s & S.Optimal value of (s, S) is difficult to find and is most of the times arbitrarily fixed. 15. Used(R, S) System by those companies not using computercontrolsControlprocedure is that every R units of time,enough is ordered to raise the inventory position to SCarrying costs are high Provides opportunity to adjust the position to S if demand pattern is changing with time 16. S0R LL 17. (R, s, S) System Combination of (R, S) and (s, S) systems. Every R units of time, check the inventory position, if it is below s, then order to raise the level to S. If the level is above s, nothing is done until the next review.(s, S) is special case where R=0 (R, s, S) is a periodic version of (s, S) system 18. (R, s, S) System (R, s, S)provides the lower total cost comprisingreplenishment, holding and shortage costs than any other method but is difficult to get all 2 parameters 19. Specific Cost & Service Objectives Two types of risks to be balanced: If demand is large, stockout may occur If demand is lower, then large inventory is carried 20. Specific Cost & Service Objectives Four methods of modeling these are: Safety stocks established throughuse of simpleapproach Safety stocks based on minimizing cost Safety stocks based on customer service levels Safety stocks based on aggregate consideration 21. SINGLE PERIOD MODEL Christmas trees Cost (C) = $4 per tree and SPrice (S) = $9, Profit = $10000, so demand was around 2000 units Xbar = 2000 and = 100 trees Let salvage value be V = $1 Marginal Profit (MP) on a tree = S - C = 9 - 4 = $5 Marginal Loss (ML) on a tree = C - V = 4 - 1 = $3 To order Q rather than Q - 1, the expected profit on marginal tree must be expected loss 22. SINGLE PERIOD MODEL Let p be the probability of selling the marginal tree Profit criterion is expected profit expected loss, i.e. p(MP) (1 - p)ML p(MP) ML - p(ML) p(MP+ML) ML Minimum acceptable probability of selling the Qth tree is given by p ML / (MP + ML) or SOR ML / (MP + ML) Q = xbar + zSOR 23. MULTI PERIOD MODEL Service Level and Safety Stocks: In reality, sometimes it is difficult to calculate ML (Stockout costs which has intangibles) Whenstockout costs are not available, surrogate is thecustomer service level Service level can be classified in two ways: i) Order Service Level (OSL) and ii) Unit Service Level (USL) 24. ORDER SERVICE LEVEL (OSL) Proportion of cycles that customer demand was satisfied. Order Service Level represents the probability of not having a stockout during the placement of order. Suppose OSL = 0.90 and number of orders/cycles = D/Q = 20, then customer demand will be satisfied in OSL * No. Of orders = 0.90 * 20 = 18 orders. It also means that during 2 orders, stockout can be expected. (0.10 * 20 or 20 - 18 = 2) OSOR doesnt tell how many units were short or not filled during any cycle 25. UNIT SERVICE LEVEL (USL) USL indicates the percentage of units of demand filled during any period of time, whereas the USOR specifies the quantities of units unfilled or short during that period. Suppose USL = 0.95 and D = 5000 units for an item. Then 0.95 * 5000 = 4750 units of customer demand will be filled on an average during the year. It also means that 250 units of stockout can be expected. i.e. If there are 10 orders then approx 25 units per order will be short of. 26. PERCENT ORDER SERVICE LEVEL Land MADL are obtained from forecasting systems,then L must be calculated i.e. L = 1.25MAD. Suppose the following details are available,L Suppose the= 200, MADL = 32, L= 5 days, D = 10000, S = 1500 and h = 30, then using EOQ we get, Sqrt(2DS / h) = Sqrt ((2*10000*1500)/30) = 1000, so D/Q = 10 orders or cycles Suppose the manager is willing to accept a 0.375 probability of stockout on any cycle. Then OSL = 1 - 0.375 = 0.625 + Safety Stock = L + Z0.375 * L L And the ROP = 27. PERCENT UNIT SERVICE LEVEL Percent Unit Service Level tells us what percentage of units demanded can be supplied from stock. It is usually called as fill rate.LLx0s x - R = (z - k)kz 28. PERCENT UNIT SERVICE LEVEL USOR = E(X - R) / Q USOR = g(k) / Q g(k) = (Q * USOR) / Specify USL, then find g(k), then k, then Safety Stock, then R Suppose USL = 0.99, then USOR = 0.01, then g(k) = Q*0.01/ , LLLthen go to the table, pick k value corresponding to calculated g(k), then calculate ROP = xL + kL 29. BACKORDER COST When we are provided with stockout costs and not the service levels, then how to calculate safety stock Increasing the safety stock by one more unit has the same effect as adding one more unit to reorder level s Raisingthe reorder point by 1 unit will cost (holding cost)(Q*h) / D = (1000*30) / 10000 = $3 per cycle If we do not add one more unit and suffer the stockout, then backorder penalty is given by $b / unit with SOR probability. 30. PerBACKORDER COST cycle marginal cost of adding 1 unit to R = per cyclemarginal cost of not adding 1 unit to R Qh / D = OSOR*(b) So, b = Qh/ (D * OSOR) i.e. Backorder cost b At 99% USL, 14 units were safety stock, so, z = 14 where = LL40. z * 40 = 14 leading to z = 0.35 leading to OSOR = 0.363 or 36.3% b = (30 * 1000) / (0.363 * 10000) = $8.26

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