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Lecture

Lecture. 4. MGMT 650 Network Models – Shortest Path Project Scheduling Forecasting. Shortest Path Problem. Belongs to class of problems typically known as network flow models What is the “best way” to traverse a network to get from one point to another as cheaply as possible?

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Lecture

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  1. Lecture 4 MGMT 650 Network Models – Shortest Path Project Scheduling Forecasting

  2. Shortest Path Problem • Belongs to class of problems typically known as network flow models • What is the “best way” to traverse a network to get from one point to another as cheaply as possible? • Network consists of nodes and arcs • For example, consider a transportation network • Nodes represent cities • Arcs represent travel distances between cities • Criterion to be minimized in the shortest path problem not limited to distance • Other criteria include time and cost

  3. Example: Shortest Route • Find the Shortest Route From Node 1 to All Other Nodes in the Network: 5 2 5 6 4 3 2 7 7 3 1 3 1 5 2 6 4 6 8

  4. Management Scientist Input

  5. Example Solution Summary NodeMinimum DistanceShortest Route 2 4 1-2 3 6 1-4-3 4 5 1-4 5 8 1-4-3-5 6 11 1-4-3-5-6 7 13 1-4-3-5-6-7

  6. Applications • Stand alone applications • Emergency vehicle routing • Urban traffic planning • Telecommunications • Sub-problems in more complex settings • Allocating inspection effort in a production line • Scheduling operations • Optimal equipment replacement policies • Personnel planning problem

  7. Optimal Equipment Replacement Policy • The Erie County Medical Center allocates a portion of its budget to purchase newer and more advanced x-ray machines at the beginning of each year. • As machines age, they break down more frequently and maintenance costs tend to increase. • Furthermore salvage values decrease. • Determine the optimal replacement policy for ECMC • that minimizes the total cost of buying, selling and operating the machine over a planning horizon of 5 years, • such that at least one x-ray machine must be in service at all times.

  8. Lecture 4 Project Scheduling Chapter 10

  9. Project Management • How is it different? • Limited time frame • Narrow focus, specific objectives • Why is it used? • Special needs • Pressures for new or improves products or services • Definition of a project • Unique, one-time sequence of activities designed to accomplish a specific set of objectives in a limited time frame

  10. Project Scheduling: PERT/CPM • Project Scheduling with Known Activity Times • Project Scheduling with Uncertain Activity Times

  11. PERT/CPM • PERT • Program Evaluation and Review Technique • CPM • Critical Path Method • PERT and CPM have been used to plan, schedule, and control a wide variety of projects: • R&D of new products and processes • Construction of buildings and highways • Maintenance of large and complex equipment • Design and installation of new systems

  12. PERT/CPM • PERT/CPM is used to plan the scheduling of individual activities that make up a project. • Projects may have as many as several thousand activities. • A complicating factor in carrying out the activities is that some activities depend on the completion of other activities before they can be started.

  13. PERT/CPM • Project managers rely on PERT/CPM to help them answer questions such as: • What is the total time to complete the project? • What are the scheduled start and finish dates for each specific activity? • Which activities are critical? • must be completed exactly as scheduled to keep the project on schedule? • How long can non-critical activities be delayed • before they cause an increase in the project completion time?

  14. Project Network • Project network • constructed to model the precedence of the activities. • Nodes represent activities • Arcs represent precedence relationships of the activities • Critical path for the network • a path consisting of activities with zero slack

  15. 0 2 4 6 8 10 12 14 16 18 20 Locate new facilities Interview staff Hire and train staff Select and order furniture Remodel and install phones Furniture setup Move in/startup Activity Planning and Scheduling

  16. Orderfurniture Furnituresetup Locatefacilities B F A Move in Remodel E S G Hire andtrain Interview D C Project Network – An Example 6 weeks 3 weeks 8 weeks 11 weeks 1 week 9 weeks 4 weeks

  17. Critical Path Management Scientist Solution

  18. Uncertain Activity Times • Three-time estimate approach • the time to complete an activity assumed to follow a Beta distribution • An activity’s mean completion time is: t = (a + 4m + b)/6 • a = the optimistic completion time estimate • b = the pessimistic completion time estimate • m = the most likely completion time estimate • An activity’s completion time variance is 2 = ((b-a)/6)2

  19. Uncertain Activity Times • In the three-time estimate approach, the critical path is determined as if the mean times for the activities were fixed times. • The overall project completion time is assumed to have a normal distribution • with mean equal to the sum of the means along the critical path, and • variance equal to the sum of the variances along the critical path.

  20. Example

  21. Management Scientist Solution

  22. Key Terminology • Network activities • ES: early start • EF: early finish • LS: late start • LF: late finish • Used to determine • Expected project duration • Slack time • Critical path

  23. A 0 7 7 0 7 C 0 6 6 1 7 B 7 10 3 7 10 D 6 9 3 7 10 E 10 12 2 10 12 Example: Two Machine Maintenance Project Immediate Completion ActivityDescriptionPredecessorsTime (wks) A Overhaul machine I --- 7 B Adjust machine I A 3 C Overhaul machine II --- 6 D Adjust machine II C 3 E Test system B,D 2 Start

  24. Normal Costs and Crash Costs

  25. Linear Program for Minimum-Cost Crashing Let: Xi = earliest finish time for activity i Yi= the amount of time activity i is crashed 10 variables, 12 constraints Crash activity A by 2 days Crash activity D by 1 day Crash cost = 200 + 150 = $350 Crash activity A by 1 day Crash activity E by 1 day Crash cost = 100 + 250 = $350

  26. Lecture 4 Forecasting Chapter 16

  27. Forecasting - Topics • Quantitative Approaches to Forecasting • The Components of a Time Series • Measures of Forecast Accuracy • Using Smoothing Methods in Forecasting • Using Trend Projection in Forecasting

  28. Time Series Forecasts • Trend - long-term movement in data • Seasonality - short-term regular variations in data • Cycle – wavelike variations of more than one year’s duration • Irregular variations - caused by unusual circumstances

  29. Forecast Variations Irregularvariation Trend Cycles 90 89 88 Seasonal variations

  30. Smoothing/Averaging Methods • Used in cases in which the time series is fairly stable and has no significant trend, seasonal, or cyclical effects • Purpose of averaging - to smooth out the irregular components of the time series. • Four common smoothing/averaging methods are: • Moving averages • Weighted moving averages • Exponential smoothing

  31. Example of Moving Average • Sales of gasoline for the past 12 weeks at your local Chevron (in ‘000 gallons). If the dealer uses a 3-period moving average to forecast sales, what is the forecast for Week 13? • Past Sales WeekSalesWeekSales 1 17 7 20 2 21 8 18 3 19 9 22 4 23 10 20 5 18 11 15 6 16 12 22

  32. Management Scientist Solutions MA(3) for period 4 = (17+21+19)/3 = 19 Forecast error for period 3 = Actual – Forecast = 23 – 19 = 4

  33. MA(5) versus MA(3)

  34. Exponential Smoothing • Premise - The most recent observations might have the highest predictive value. • Therefore, we should give more weight to the more recent time periods when forecasting. Ft+1 = Ft + (At - Ft), Formula 16.3

  35. Linear Trend Equation Suitable for time series data that exhibit a long term linear trend • Ft = Forecast for period t • t = Specified number of time periods • a = Value of Ft at t = 0 • b = Slope of the line Ft Ft = a + bt a 0 1 2 3 4 5 t

  36. Linear Trend Example Linear trend equation F11 = 20.4 + 1.1(11) = 32.5 Sale increases every time period @ 1.1 units

  37. Actual vs Forecast Linear Trend Example 35 30 25 20 Actual Actual/Forecasted sales 15 Forecast 10 5 0 1 2 3 4 5 6 7 8 9 10 Week F(t) = 20.4 + 1.1t

  38. Measure of Forecast Accuracy • MSE = Mean Squared Error

  39. Forecasting with Trends and Seasonal Components – An Example • Business at Terry's Tie Shop can be viewed as falling into three distinct seasons: (1) Christmas (November-December); (2) Father's Day (late May - mid-June); and (3) all other times. • Average weekly sales ($) during each of the three seasons during the past four years are known and given below. • Determine a forecast for the average weekly sales in year 5 for each of the three seasons. Year Season1234 1 1856 1995 2241 2280 2 2012 2168 2306 2408 3 985 1072 1105 1120

  40. Management Scientist Solutions

  41. Interpretation of Seasonal Indices • Seasonal index for season 2 (Father’s Day) = 1.236 • Means that the sale value of ties during season 2 is 23.6% higher than the average sale value over the year • Seasonal index for season 3 (all other times) = 0.586 • Means that the sale value of ties during season 3 is 41.4% lower than the average sale value over the year

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