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Introduction to Decision Theory and Operations Research

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Rona Pfeffer

on 4 August 2014

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Transcript of Introduction to Decision Theory and Operations Research

Introduction to Decision Theory and Operations Research
Rona Pfeffer
Fall 2014-2015
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Linear Programming
Formulating LP Problems
Solving LP Problems
The Simplex Algorithm
Sensitivity Analysis
integer Programming
Network theory
Dynamic Programming
Decision theory
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2
3
The Transportation Problem
Linear Programming Model
Assumptions
Let's begin with the following "Diet Problem":
The Standard LP Model:
Class Exercise
Formulate the following problems using an LP model
The Assignment Problem
assumptions
formulation
Transportation Problem - Example
class exercise
find the solution for the above problem, for different values of C1
General
Shortest Path
Maximum Flow
Minimum Spanning Tree
shortest path example
The recursive formula
Class exercise
Probabilistic DP
The IP Model
OR definition
example
product
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2
3
4
volume
22
30
50
70
profit
10
15
20
30
per box
$ per box
Maximize Z = 10X1 + 15X2 + 20X3 + 30X4

subject to:

22X1 + 30X2 + 50X3 + 70X4 <=100
Optimization Problems
Maximize Z = 10X1 + 15X2 + 20X3 + 30X4
Maximize Z = 10X1 + 15X2 + 20X3 + 30X4

subject to:

22X1 + 30X2 + 50X3 + 70X4 <=100
(a non-linear function)
22X1 + 30X2 + 50X3 + 70X4 <=100
history
psychology
economics
computer science
engineering
mathematical model
components of a mathematical model:
decision variables
objective function
constraints
parameters
related disciplines
mathematics
statistics
management
course site
contact information
where and when
assignments
reading materials
prerequisites
General Information
introduction
1. graphic solution method
4. using excel's solver
2. solution for LP problems
3. the case of a single constraint
Types of problems on networks
Course Outline
Formulating IP Problems
Set-Covering
Bin-Packing
Solving IP Problems
Rounding?
Branch and Bound
Branch and Bound
Routing Problems
Locating problems
Planning Problems
1 function Dijkstra(Graph, source):
2 dist[source] := 0 // Initializations
3 for each vertex v in Graph:
4 if v ≠ source
5 dist[v] := infinity // Unknown distance from source to v
6 previous[v] := undefined // Predecessor of v
7 end if
8 PQ.add_with_priority(v,dist[v])
9 end for
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12 while PQ is not empty: // The main loop
13 u := PQ.extract_min() // Remove and return best vertex
14 for each neighbor v of u: // where v has not yet been removed from PQ.
15 alt = dist[u] + length(u, v)
16 if alt < dist[v]
17 dist[v] := alt
18 previous[v] := u
19 PQ.decrease_priority(v,alt)
20 end if
21 end for
22 end while
23 return previous[]
DIjkstra's Algorithm
Ford-Fulkerson's Algorithm
Example
Max-Flow - Min Cut
LP Formulation
Example
IP Formulation
Pseudo-Code
Complexity
Full transcript