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Chapter 7

Chapter 7. Individual Items with Probabilistic Demand. Individual Items with Probabilistic Demand. Average demand remains approximately constant with time.

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Chapter 7

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  1. Chapter 7 Individual Items with Probabilistic Demand

  2. Individual Items with Probabilistic Demand • Average demand remains approximately constant with time. • An appropriate reallocation of buffer (or safety) stocks which are kept to meet unexpected fluctuations in demand can lead to a significant improvement in the service. • Single-stage problem

  3. Terminology Stock level • On-hand stock: Stock on the shelf • Can never be negative • Net stock: (On-hand) – (Backorders) • Can be negative • Inventory position: (On-hand) + (On-order) – (Backorders) – (Committed) • On-order: requisitoned but not yet received by the stocking point • Committed: cannot be used for other purposes in the short run • Safety stock (buffer stock): Average level of the net stock just before a replenishment arrives

  4. Terminology Backorders vs. Lost sales • Complete backordering: Wholesale-retailer link of some distribution systems • Complete lost sales: Retailer-consumer link In most practical situations, combination of these two extremes occur.

  5. Key Issues to be Resolved by a Control System • How often the inventory status should be determined. • frequently  costly • otherwise  may be backorders • When a replenishment order should be placed. • early  carry inventory • late  inadequate customer service • How large the replenishment order should be. • too much  low fixed cost, high inventory level • Similar to EOQ

  6. Key Issues to be Resolved by a Control System To answer these questions, several things must be determined: • How important is the item? (ABC Classification) • Can, or should, the stock status be reviewed continuously or periodically? • What form should the inventory policy take? • What specific cost or service objectives should be set?

  7. Continuous vs. Periodic Review • Continuous review: Inventory status is always known. • In reality, not required. • Each transaction (shipment, receipt, demand, etc.) updates the system (called transactions reporting). • Point-of-sales (POS) data collection systems (involving electronic scanners) permit transactions reporting. • Periodic review: Stock status is determined every R time units. • R: review interval (time that elapses between two consecutive moments at which we know the stock level) • Ex: coffee machine refilled once every 2 days • Physical limitations determine R.

  8. Continuous vs. Periodic Review Advantages and Disadvantages • Periodic review is more appealing when items in a coordinated group can be given the same R, e.g. all items from the same supplier scheduled for review every Thursday. • Periodic review allows a prediction of the level of workload on the staff involved. The load is less predictable in continuous review. • Continuous review is more expensive in terms of reviewing costs and reviewing errors. • To provide the same level of customer service, continuous review requires less safety stock, hence lower carrying costs.

  9. Example Demand for an item over the past 6 months has been 10, 80, 240, 130, 100 and 40 units, respectively. the reorder cost is $50 and holding cost is $1 a unit a month, and any orders placed in one month become available in the following month. How good is an ordering policy based on average values? (Backorders are not allowed.) D = 100 units/month A = $50/order Holding cost = $1/unit/month L = 1 month

  10. Example

  11. The policy is to order 100 units whenever inventory position (on hand + on order) falls to or below 100 units. Orders are placed every month when the closing stock is below 100 units and the delivery is available to meet demand in the following month.

  12. Form of the Inventory Policy • 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

  13. Order-Point, Order-Quantity (s,Q) System Continuous review system, R=0. A fixed quantity Q is ordered whenever the inventory position drops to the reorder point s or lower. Inventory position, not the net stock, is used to trigger an order. Two bin system.

  14. Order-Point, Order-Quantity (s,Q) System Advantages • Simple • Errors are less likely to occur • Production requirements for the supplier are predictable Disadvantages • Not effective in coping with the situation in which transactions are large • Q may not raise inventory position above s

  15. Order-Point, Order-Quantity (s,Q) System Assumptions • Demand is probabilistic. But average demand rate changes very little with time. • A replenishment order of size Q is placed when inventory position is exactly at s, the order point. • All demand transactions are of unit size OR • Undershoots of order point are negligible compared with DL (lead time demand) • Crossing of orders not permitted (constant L satisfies this assumption)

  16. Order-Point, Order-Quantity (s,Q) System Assumptions • Forecast errors  N(0,L) • When we need Q, we will assume that Q is predetermined. Effects of s and Q are not independent. The best value of Q depends on s and vice versa. • The cost of control system does not depend on the value of s.

  17. Order-Point, Order-Quantity (s,Q) System How to reduce SS by changing the givens? • Ways to reduce L: faster transportation, closer suppliers • Ways to reduce variability of demand: improve forecast accuracy, provide customer incentives for specific purchase times and quantities. • Reduce required service level.

  18. Order-Point, Order-Up-to-Level (s,S) System Continuous review. An order is placed when inventory position drops to s or lower. A variable replenishment quantity is used, ordering enough to raise the inventory position up to S. If all demand amount is 1, (s,Q) and (s,S) are identical, S=s+Q. Frequently called min-max system. Best (s,S) system is better than best (s,Q) system but more difficult. (s,S) is frequently used but parameters are set arbitrarily. Variable order quantity is a disadvantage.

  19. Periodic Review, Order-Up-to-Level (R,S) System Every R units of time, enough is ordered to raise the inventory position to S. Used in companies not using computer control. Preferred to order point systems in terms of coordinating the replenishment of related items. Carrying costs are higher than continuous review systems. (R,S) is equivalent to (s,Q) if following transformations are made. (s,Q)(R,S) s S Q DR L R+L

  20. Periodic Review, Order-Up-to-Level (R,S) System The Choice of R When computing S, R is known. Determination of R is equivalent to determination of EOQ as a time supply. • The cost of reviewing inventory must be included in A. • R is restricted to a reasonably small number of feasible discrete values. R is often dictated by external factors, like frequency of deliveries.

  21. Periodic Review, Order-Up-to-Level (R,S) System Assumptions Assumptions for (s,Q) system are valid. Additionally, • Negligible chance of no demand between reviewsan order is placed at every review. • R is predetermined. Unlike in (s,Q) system, the assumption of unit sized transactions is not needed.

  22. (R,s,S) System Check the system every R units of time. If it is at or below s, then order enough to raise to S. Otherwise, do nothing. R=0  (s,S) s=S-1  (R,S) The best (R,s,S) system produces a better TC than any other system.

  23. Form of the Inventory Policy

  24. Specific Cost and Service Objectives Trade-off: Risk of stockout and risk of high inventory Four methods to determine safety stocks.

  25. Methods to Determine Safety Stocks 1. SS established through the use of a simple-minded approach: • Common safety factor for all items SS = k . L k: safety factor L: standard deviation of errors of total demand forecast over L • Common time supply Safety stocks of a broad range of items are set equal to the same time supply.

  26. Methods to Determine Safety Stocks 2. SS based on cost minimization: • Fixed cost (B1) per stockout occasion B1 is independent of the magnitude or duration of stockout. • Fractional charge (B2) per unit short When units short are produced during overtime. • Fractional charge (B3) per unit short per unit time Spare part where shortage causes a machine to be idle. • Charge (B4) per customer line item short Ex: customer orders 10 items but 9 items are in stock; the supplier is short 1 customer line item.

  27. Methods to Determine Safety Stocks 3. SS based on customer service: • Probability (P1) of no stockout per replenishment cycle - cycle service level Using a common P1 Using a common k P1 : fraction of cycles where no stockout occurs • Fraction (P2) of demand to be satisfied routinely from shelf - fill rate  Using P2 and B3 are equivalent

  28. Methods to Determine Safety Stocks 3. SS based on customer service: • Fraction (P3) of time during which net stock is positive – ready rate Ex: equipment for emergency purposes (military hardware) Under Poison demand, P3 P2 • Average time between stockout occasions (TBS) = average number of stockout occasions per year

  29. Methods to Determine Safety Stocks 4. SS based on aggregate considerations: • Allocation of a given total safety stock among items to minimize the expected total stockout occasions per year (ETSOPY) • Allocation of a given total safety stock among items to minimize the expected total value of shortages per year (ETVSPY)

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