Project Management PERT: Time Analysis and Critical Path Determination

1. Project Management PERT ___________________________________________________________________________ Operations Research  Jan Fábry

2. Expected completion time Project Management PERT • Probabilistic time – 3 estimates Optimistic Pessimistic Most likely ___________________________________________________________________________ Operations Research  Jan Fábry

3. Most Likely Expected Optimistic Pessimistic Time 0 aij mij μij bij Project Management PERT • Activity time β-distribution ___________________________________________________________________________ Operations Research  Jan Fábry

4. Project Management PERT • Expected completion time • Standard deviation ___________________________________________________________________________ Operations Research  Jan Fábry

5. Critical path Project Management PERT 1) Using CPM ( tij= μij) 2) Expected duration of the project Variance of the project duration ___________________________________________________________________________ Operations Research  Jan Fábry

6. T M -σ M M + σ Project Management PERT Duration of the project - normal distribution M – expectedduration T – realduration σ– standarddeviation ___________________________________________________________________________ Operations Research  Jan Fábry

7. probabilty of completion time within TD p Project Management PERT 3) Probability analysis desired time TD desired time TD ___________________________________________________________________________ Operations Research  Jan Fábry

8. completiontime zp Project Management PERT 3) Probability analysis desired probability p desired probability p ___________________________________________________________________________ Operations Research  Jan Fábry

9. Project Management PERT Example - Music DM, Inc. ___________________________________________________________________________ Operations Research  Jan Fábry

10. Project Management PERT Example - Music DM, Inc. ___________________________________________________________________________ Operations Research  Jan Fábry

11. H K B D2 J A D G I D1 C F E 8 5 6 4 7 3 9 10 2 1 Project Management PERT Example - Music DM, Inc. Critical path: ___________________________________________________________________________ Operations Research  Jan Fábry

12. Project Management PERT Example - Music DM, Inc. Critical activities ___________________________________________________________________________ Operations Research  Jan Fábry

13. Expected duration of the project M = 48 Variance of the project duration  2 = 9 Std. deviation of the project duration  = 3 Project Management PERT Example - Music DM, Inc. ___________________________________________________________________________ Operations Research  Jan Fábry

14. Time (days) 45 48 51 Project Management PERT Example - Music DM, Inc. Duration of the project ___________________________________________________________________________ Operations Research  Jan Fábry

15. Project Management PERT Example - Music DM, Inc. Probability analysis Desired time TD = 45 ___________________________________________________________________________ Operations Research  Jan Fábry

16. zp = 1.645 Project Management PERT Example - Music DM, Inc. Probability analysis Desired probability = 0.95 ___________________________________________________________________________ Operations Research  Jan Fábry

17. Inventory Models ___________________________________________________________________________ Operations Research  Jan Fábry

18. Inventory Models Inventory • Stored for use in future • Examples of inventories - raw material - finished goods - semi-finished products - spare parts ___________________________________________________________________________ Operations Research  Jan Fábry

19. Inventory Models Reasons for Maintaining Inventories • Fluctuating demand • Delayed supply • Savings on ordering cost • Benefits on large quantities • Inflation ___________________________________________________________________________ Operations Research  Jan Fábry

20. Inventory Models Management • How much to order? • When to order? • Objective – minimize total cost ___________________________________________________________________________ Operations Research  Jan Fábry

21. frequency of ordering cost Inventory Models Partial Inventory Cost • Ordering & setup cost – fixed charge • Unit purchasing cost – variable cost quantity discounts ___________________________________________________________________________ Operations Research  Jan Fábry

22. Inventory Models Partial Inventory Cost • Holding & carrying cost – variable cost - storage cost (space) - store keeping operations (movement) - insurance & taxes - interest (invested capital ) - oportunity cost - spoilage or obscolescence ___________________________________________________________________________ Operations Research  Jan Fábry

23. Inventory Models Partial Inventory Cost • Shortage (stockout) cost – variable cost - idled production (idled machine) - placing special expensive expediting order - loss of customers due to late deliveries ___________________________________________________________________________ Operations Research  Jan Fábry

24. Inventory Models Inventory Terminology • Inventory level Available size of the inventory • Demand & depletion Rate of demand  Depletion rate ___________________________________________________________________________ Operations Research  Jan Fábry

25. Lead time Time Placing order Receiving shipment&Replenishment Inventory Models Inventory Terminology • Reordering Reorder point = signal inventory level ___________________________________________________________________________ Operations Research  Jan Fábry

26. Receiving shipment Shortage !! Lead time Time Optimal reorder date Inventory level = 0 Placing order Inventory Models Inventory Terminology • Shortage ___________________________________________________________________________ Operations Research  Jan Fábry

27. Inventory Models Inventory Terminology • Safety stock Buffer being built to prevent shortage In case of probabilistic demand ___________________________________________________________________________ Operations Research  Jan Fábry

28. Inventory Models Inventory Models • Deterministic models Rate of demand is known with certainty • Probabilistic models Probabilistic demand (random variable) ___________________________________________________________________________ Operations Research  Jan Fábry

29. Probability density function – varies in time Increasing Level of Mathematical Difficulty Deterministic Probabilistic Static Dynamic Stationary Nonstationary Probability density function – unchanhged over time Rate of demand - known with certainty - not constant Rate of demand - known with certainty - constant Inventory Models Demand Classification Demand ___________________________________________________________________________ Operations Research  Jan Fábry

30. Inventory Models Deterministic Models • Basic economic order quantity model (EOQ model) • EOQ model with back orders allowed (with planned shortages) • Economic production lot size model • EOQ model with quantity discounts ___________________________________________________________________________ Operations Research  Jan Fábry

31. Inventory Models Probabilistic Models • Probabilistic model with continuous demand • Single-period decision model ___________________________________________________________________________ Operations Research  Jan Fábry