decision analysis

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
3
1
1
5
2
6
6
4
8
4
Management Scientist Input
5
Example Solution Summary

• Node Minimum Distance Shortest
  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.

Year Purchase Cost (000)
1 170
2 190
3 210
4 250
5 300
Age Maintenance cost (000) Salvage value (000)
1 50 20
2 97 15
3 182 10
4 380 0
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
• RD 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
Planning and Scheduling
16
Project Network An Example
6 weeks
3 weeks
8 weeks
11 weeks
1 week
9 weeks
4 weeks
17
Management Scientist Solution
18
Uncertain Activity Times

• Three-time estimate approach
• the time to complete an activity assumed to
  follow a Beta distribution
• An activitys 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 activitys 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
Activity Immediate Predecessor Optimistic Time (a) Most Likely Time (m) Pessimistic Time (b)
A -- 4 6 8
B -- 1 4.5 5
C A 3 3 3
D A 4 5 6
E A 0.5 1 1.5
F B,C 3 4 5
G B,C 1 1.5 5
H E,F 5 6 7
I E,F 2 5 8
J D,H 2.5 2.75 4.5
K G,I 3 5 7
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
Example Two Machine Maintenance Project
Immediate Completion Activity
Description Predecessors
Time (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
Activity Normal Time Normal Cost () Crash Time Crash Cost () Maximum Reduction in Time Crash Cost per day ()
A Overhaul Machine I 7 500 4 800 3 (800-500)/3 100
B Adjust machine I 3 200 2 350 1 150
C Overhaul Machine II 6 500 4 900 2 200
D Adjust machine II 3 200 1 500 2 150
E Test System 2 300 1 550 1 250
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
  years 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
• Week Sales Week
  Sales
• 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 (172119)/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.

Ft1 Ft ?(At - Ft), Formula 16.3
35
Linear Trend Equation
Suitable for time series data that exhibit a long
term linear trend
Ft
Ft a bt
a

• Ft Forecast for period t
• t Specified number of time periods
• a Value of Ft at t 0
• b Slope of the line

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 _at_ 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
• Season 1 2
  3 4
• 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 (Fathers 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