Lecture 5 Updated

1. Lecture 5 Updated MaysoonIsleem

2. A simple inventory system • As a policy, the inventory level is reviewed periodically and new items are ordered from a supplier, if necessary. • ** When items are ordered, the facility incurs an ordering cost that is the sum of a fixed setup cost independent of the amount ordered plus an item cost proportional to the number of items ordered. • This periodic inventory review policy is defined by two parameters, conventionally denoted s and S.

3. A simple inventory system • s is the minimum inventory level | if at the time of review the current inventory level is below the threshold s then an order will be placed with the supplier to replenish the inventory. If the current inventory level is at or above s then no order will be placed. • S is the maximum inventory level | when an order is placed, the amount ordered is the number of items required to bring the inventory back up to the level S. • The (s; S) parameters are constant in time with 0 ≤ s < S.

4. A simple inventory system • Conceptual Model: • Periodic Inventory Review • Inventory review is periodic • Items are ordered, if necessary, only at review times • (s, S) are the min,max inventory levels, 0 ≤ s < S • We assume periodic inventory review • Search for (s, S) that minimize cost • Transaction Reporting • Inventory review after each transaction • Significant labor may be required • Less likely to experience shortage

5. A simple inventory system Conceptual Model: • Inventory System Costs • Holding cost: for items in inventory • Shortage cost: for unmet demand • Setup cost: fixed cost when order is placed • Item cost: per-item order cost • Ordering cost: sum of setup and item costs • Additional Assumptions • Back ordering is possible • No delivery lag • Initial inventory level is S • Terminal inventory level is S

6. A simple inventory system (Because we have assumed that back ordering is possible, if the demand during the ith time interval is greater than the inventory level at the beginning of the interval (plus the amount ordered, if any)

7. A simple inventory system

8. Time evolution of inventory level

9. SIS with sample demands • Let (s, S) = (20, 60) and consider n = 12 time intervals

10. Output statistics • we must address the issue of what statistics should be computed to measure the performance of a simple inventory system. • our objective is to analyze these statistics and, by so doing, better understand how the system operates. • Average demand and average order *For the example ￣d = ￣o = 305/12 ≃ 25.42 items per time interval.

11. Flow Balance

12. Flow Balance • Average demand and order must be equal • • Ending inventory level is S • • Over the simulated period, all demand is satisfied • • Average “flow” of items in equals average “flow” of items out • The inventory system is flow balanced

13. Constant Demand Rate • The holding cost and shortage cost are proportional to time- averaged inventory levels . • To compute these averages it is necessary to know the inventory level for all t, not just at the inventory review times. • Therefore, we assume that the demand rate is constant between review times so that the continuous time evolution of the inventory level is piecewise linear

14. Constant Demand Rate

15. Inventory Level As A Function Of Time

16. Inventory Decrease Is Not Linear • Inventory level at any time t is an integer • l(t) should be rounded to an integer value • l(t) is a stair-step, rather than linear, function

17. Time-Averaged Inventory Level Case 2: If l (t) becomes negative at some time • ￣l+i is the time-averaged holding level • ￣l−i is the time-averaged shortage level

18. Case 1: No Back Ordering

19. Case 1: No Back Ordering

20. Case2: Back Ordering

21. Case2: Back Ordering

22. Time -Averaged Inventory Level • Time-averaged holding level and time-averaged shortage level The time-averaged inventory level is

23. Computational Model • sis1 is a trace-driven computational model of the SIS • Computes the statistics • ￣d, ￣o, ￣l +, ￣l − • and the order frequency ￣u Consistency check: compute ￣o and ￣d separately, then compare

24. Example 1.3.4: Executing SIS1

25. Operating Costs

26. Case Study • Automobile dealership that uses weekly periodic inventory review • The facility is the showroom and surrounding areas • The items are new cars • The supplier is the car manufacturer “...customers are people convinced by clever advertising that their lives will be improved significantly if they purchase a new car from this dealer.” (S. Park) • Simple (one type of car) inventory system

27. Case Study • Limited to a maximum of S = 80 cars • Inventory reviewed every Monday • If inventory falls below s = 20, order cars sufficient to restore to S • For now, ignore delivery lag • Costs:

28. Per-Interval Average Operating Costs • The average operating costs per time interval are

29. Per-Interval Average Operating Costs • The average total operating cost per time interval is their sum • For the stats and costs of the hypothetical dealership:

30. Cost Minimization • By varying s (and possibly S), an optimal policy can be determined • Optimal ⇐⇒ minimum average cost • Note that ￣o = ￣d, and ￣d depends only on the demands • Hence, item cost is independent of (s, S) • Average dependent cost is avg setup cost + avg holding cost + avg shortage cost • Let S be fixed, and let the demand sequence be fixed • If s is systematically increased, we expect: • average setup cost and holding cost will increase as s increases • average shortage cost will decrease as s increases • average dependent cost will have ‘U’ shape, yielding an optimum

31. Simulation Results