Continuous Review Inventory System

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Continuous Review Inventory System

  1. 1. Lot size reorder point systems (Q, R) system
  2. 2. Introduction <ul><li>Generalize EOQ model with reorder point R for the case where demand is stochastic </li></ul><ul><li>Multi-period newsboy problem was not realistic for 2 reasons: </li></ul><ul><ul><li>No ordering cost </li></ul></ul><ul><ul><li>No lead time </li></ul></ul><ul><li>(Q,R) system with stochastic demand are common in practice </li></ul><ul><li>Form the basis of many commercial inventory systems </li></ul>
  3. 3. Changes in Inventory Over Time for Continuous-Review (Q, R) System Fig. 5-5
  4. 4. (Q,R) inventory system <ul><li>The systems is continuous review </li></ul><ul><li>Demand is random and stationary </li></ul><ul><li>Fixed lead time </li></ul><ul><li>Cost involved </li></ul><ul><ul><li>K : ordering cost </li></ul></ul><ul><ul><li>h : holding cost per unit per unit time </li></ul></ul><ul><ul><li>c : cost per item </li></ul></ul><ul><ul><li>p : shortage cost per unit of unsatisfied demand </li></ul></ul>
  5. 5. Inventory Model <ul><li>Decision variables: Q and R </li></ul><ul><li>Costs </li></ul><ul><ul><li>Holding cost </li></ul></ul><ul><ul><li>Set up (ordering cost) </li></ul></ul><ul><ul><li>Penalty (shortage) cost </li></ul></ul><ul><ul><li>Proportional ordering cost (cost of items ordered) </li></ul></ul>
  6. 6. Holding cost λ τ R- λτ Q + R- λτ Q + R - λτ R- λτ Q/2 + R - λτ
  7. 7. Penalty cost x
  8. 8. Expected number of shortages
  9. 9. Total cost function Holding cost Ordering cost Shortage cost
  10. 10. Necessary conditions for optimality
  11. 11. Optimal solution
  12. 12. Service Level in (Q,R) systems <ul><li>Difficult to determine an exact value of p </li></ul><ul><li>A substitute for penalty cost is a service level </li></ul><ul><li>Two types of service level are considered </li></ul><ul><ul><li>Type 1 service level </li></ul></ul><ul><ul><li>Type 2 service level </li></ul></ul>
  13. 13. Type 1 service level <ul><li>In this case we specify the probability of no shortage in the lead time </li></ul><ul><li>Symbol is used to represent this probability </li></ul><ul><li>In this case </li></ul><ul><ul><li>Determine R to satisfy the equation F(R) = </li></ul></ul><ul><ul><li>Set Q = EOQ </li></ul></ul>
  14. 14. Interpretation of <ul><li>The proportion of cycles in which no shortage occurs </li></ul><ul><li>Appropriate when a shortage occurrence has the same consequence regardless of its time or amount </li></ul><ul><li>Not how service level is interpreted in most applications </li></ul><ul><li>Different items have different cycle lengths  this measure will not be consistent among different products making the choice of alpha difficult </li></ul>
  15. 15. Type 2 service level <ul><li>Measures the proportion of demands that are met from stock </li></ul><ul><li>Symbol β is used to represent this proportion </li></ul><ul><li>n(R)/Q is the average fraction of demands that stock out each cycle </li></ul><ul><li>n(R)/Q = 1 - β </li></ul>
  16. 16. Approximate solution with Type 2 service level constraint <ul><li>Set Q= EOQ </li></ul><ul><li>Find R to solve n(R)=EOQ(1 – β ) </li></ul>

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