Operations Research Models

1
Operations Research Models

• Johannes Lüthi
• Universität der Bundeswehr München
• luethi_at_informatik.unibw-muenchen.de
• http//www.informatik.unibw-muenchen.de/inst4/luet
  hi
• 7.4.1999
• Visit of Slovenian Delegation at AStudÜbBw / IABG

2
Personal Background

• -1995 Mathematics at Univ. of Vienna,
  Austriagame theory, optimization Masters
  thesis distributed simulation
• 1995-1997 Research assistant at Dept. of Applied
  Information Systems, Univ. Vienna, Austria
  performance modeling of parallel systems(task
  graph models, queueing models)
• 1997 Dissertation analysis of queueing networks
  with parameter uncertainties and variabilities
• 1998- Univ. Fed. Armed Forces, Munich
  performance modeling of computer systems, model
  building and computer simulation

3
Overview

• Introduction
• Key aspects of OR
• Historical background
• Model types
• Application of OR Models
• Model Building
• OR Models

4
Key Aspects of OR

• Solve complex real-world problems using a
  scientific / mathematical approach
• Problems in systems consisting of
• Humans
• Machines
• Material
• Capital
• Represent system in a mathematical model
• Support people who plan and manage such systems
  in making decisions

5
Historical Background

• 1938, Great Britain Operational
  ResearchIncrease effectiveness of military
  operations (radar, submarines) using mathematical
  methods
• 1941 Linear programming for distribution
  problems (e.g. refinery problems)
• 1957 Dynamic programming
• 1959 Network models
• Also older models are used in OR

6
Model Types
7
Application of OR Models
Real System
Mathematical Model
Model Solution
Real World Solution
8
Example Continuous Model

• Observed statevolume in tank
• State changeswater flowing in and out of the
  tank

• V(t) ... Volume of water
• z(t) ... Flow in tank
• a(t) ... Flow out of tank

9
Example Discrete Model

• Observed statenumber of cars
• State changescars entering and leaving

Conceptual modelPetri net
10
Model Building

• Introduction
• Model Building
• Phases of model building
• Verification validation
• Summary
• OR Models

11
Process of Building a Model

• Problem definition
• System analysis
• Model formalization
• Implementation integration
• Analysis / experimentation
• During all steps verification validation

12
Problem Definition

• Transformation of an unsharp problem into an
  exact problem description
• Which questions have to be answered?
• E.g. what is the optimal inspection interval for
  a certain type of car?
• With what accuracy?
• E.g. optimal value ? 10

13
System Analysis

• System decomposition
• Interaction analysis
• System boundaries
• Input parameters
• Objects to be analyzed
• Output measures
• Communicative conceptual model

14
Model Formalization

• Choose modeling paradigm or language, e.g.
• Linear programming model
• Queueing model
• Petri nets
• Discrete event simulation model
• Translate communicative model into chosen paradigm

15
Model Implementation

• Implementation
• Write computer program
• Use tool to implement formal model
• Integration
• Integrate pre-built components
• Integrate model into a federation of models (e.g.
  via HLA)

16
Analysis / Experimentation

• Use the implemented model to produce results,
  e.g.
• Obtain input values that produce optimal output
  measures
• Predict system behavior
• Perform what if studies
• Build meta-models

17
Verification Validation
18
Verification

• Did we build the model right?
• Top-down modular design
• Structured walk-through
• Antibugging
• Try simplified cases
• Continuity tests
• Degeneracy tests
• Consistency tests
• Test random number generators

19
Validation

• Did we build the right model?
• Validate key aspects
• Assumptions
• Input parameter values and distributions
• Output values and conclusions
• Use comparison sources
• Expert intuition
• Real system measurements
• Theoretical results

20
Verification Validation Framework
Dirk Brade (UniBwM, brade_at_informatik.unibw-muenche
n.de)
21
Model Building - Summary
Phases
Results
Input
Problem Definition
Precise Questions
Unsharp Problem
System Analysis
Communicative Model
Observations
Model Formalization
Formal Model
Verification Validation
Modeling Paradigm/method
Implementation
Executable Model
Solution Technique
Analysis / Experimentation
Numbers, Graphs, Tables, ...
Input Parameters / Distributions
22
OR Models

• Introduction
• Model Building
• OR Models
• Optimization models
• Prediction models
• Experimentation models

23
OR Model Classification

• Optimization models
• Derive optimal parameter values directly from
  mathematical representation of the model
• Prediction models
• Derive predicted output (not necessarily optimal)
  from math. Representation
• Experimentation models
• Produce output by imitating the real system

24
Optimization Models

• Optimization Models
• Differential Calculus
• Linear Programming
• Decision Trees (not covered)
• Game Theory
• Prediction Models
• Experimentation Models

25
Differential Calculus

• Formulate output measure of interest as
  differentiable equation

• Differentiate equation

• Find extreme (optimal) values

26
Linear Programming - Concept

• Find extreme value of linear function

• Linear boundary conditions

27
Linear Programming - Solution

• Simplex Algorithm
• Transform problem into standard form
• Linear boundary conditions describe a special
  convex set
• Finite number of corner points
• Optimal solution must be at a corner
• Simplex algorithm finds optimal point in a finite
  number of steps

28
Linear Programming - Example

• Refinery Production
• Various products
• Multistage production line
• Capacities for production stages
• Different qualities of products
• Market conditions (minimum and maximum production
  for certain products)
• Linear relations
• Find optimal product mix!

29
Linear Programming - Discussion

• Very large systems can be solved(thousands of
  equations)
• Standard method for many problems in logistics
• Production optimization
• Transport problems

30
Linear Programming - Problems

• Nonlinear relations may exist
• Approximation using linear equations
• Nonlinear programming more complex, often only
  numerical solutions exist
• Problems with integer variables
• Integer optimization

31
Game Theory

• Modeling reaction in conflict situationse.g. in
  economy, politics
• Conflict partners have different strategies to
  choose from
• For every combination of chosen strategies a
  certain payoff is known
• Assume rational behavior of conflict partners

32
Game Theory - The Model

• Player A has strategies A1,...,AN
• Player B has strategies B1,...,BM
• If profit of A is loss of B Zero-sum-game
• Define payoff-matrices

33
Game Theory - Solution

• Consider strategy vectors x, y
• Optimal strategies / strategy mixes can be found
• Equilibrium points
• Evolutionary stable states

34
Game Theory - Example

• Problem U.S. oil reserves vs. possible OAPEC
  embargo
• Model with various embargo strategies for OAPEC
• no embargo
• 180/270/360 days embargo with 25/50 decreased
  oil exports
• Various U.S. strategies for using the oil reserves

35
Game Theory - Problems

• Values in payoff matrices very difficult to
  obtain or estimate
• Difficult to find a set of strategies which is
  sufficiently complete

36
Game Theory - Example, continued

• Given a fixed quantity for U.S. oil reserves,
  optimal strategies for both conflict partners can
  be computed
• This was done for oil reserves from 0 to 1550
  Mio. barrels and more than 100 scenario
  variations
• Reasonable quantities for oil reserves depending
  on U.S. oil imports from OAPEC countries have
  been derived

37
Prediction Models

• Optimization Models
• Prediction Models
• Task Graph Models
• Markov Models
• Forrester Models
• Experimentation Models

38
Prediction vs. Optimization Models

• Optimization models ? direct computation of
  optimal parameter values
• Prediction models? direct computation of output
  measures (not necessary optimal)
• Prediction models can be used for numeric
  optimization (simulation in the broader sense)

39
Task Graph Models

• Graphical representation of the tasks of a
  project
• Interdependencies of tasks
• Timing of tasks

40
Task Graphs - Analysis

• Given a scheduled project delivery date
• Compute
• earliest possible starting time
• latest possible starting time of tasks
• Compute critical path tasks for which earliest
  equals latest possible time? delay of a task in
  critical path delays the whole project!

41
Task Graphs - Example

• Planing the production of an opera play
• Tasks e.g. preparation, concept, solo
  rehearsals, ensemble rehearsals, choir
  rehearsals, stage design, costumes
• 22 tasks were defined
• Preparation, concept, solo rehearsals at the
  beginning 8 more tasks at the end were
  identified as critical

42
Task Graph Models - Discussion

• Well-known and popular for planing large projects
  such as e.g.
• Development of the Alpha jet (Dornier)
• Apollo moon missions
• Mining projects
• Building of the new railway track
  Hannover-Würzburg
• Difficult appropriate degree of detail
• Possible extension associated cost

43
Markov Models

• Define possible states S1,...,SN of a system
• Observe changes of the system in given discrete
  time steps t
• Consider transition probabilities
  pijprobability that within the next time step
  the system will change to state j, given that it
  is currently in state i
• Build transition probability matrix

44
Markov Model - Example

• 2 machines A, B working or out of order
• Repair policy if A and B out of order, A has
  repair priority

1
A and Bworking
2
3
A OK,B out of o.
B OK,A out of o.
A and Bout of o.
4
45
Markov Model - Example, continued

• Transition probability matrix

P
P2 ... Matrix for the transition probabilities
between time t0 and time t2 P3 ...
Matrix for the transition probabilities between
time t0 and time t3, etc.
46
Markov Model - Steady State

• Compute steady state probabilities

• From that, mean time in states, etc. can be
  computed

47
Markov Models - Continuous Time

• Discrete time steps Dt considered
• Let Dt approach zero
• Continous time Markov chains (CTMC)
• E.g. Queueing models

48
Queueing Models

• Network of
• Service / Work Centers
• Queues
• Customer / jobs / tasks
• Mean service time of a job at a center
• Open network arrival rate of jobsClosed
  network number of jobs in system

49
Queueing Models - Examples

• Open Model

• Closed Model

50
Queueing Models - Analysis

• Steady state measures for e.g.
• Mean queue lengths
• Mean response times for centers
• Mean system response time
• Mean system throughput
• Depending on restrictions
• Direct analytical solution
• Numerical solution
• Solution via simulation

51
Forrester Models

• Based on Control loops
• Symbols for

52
Forrester Models - Example

• Extremely simplified population model
• Assumptions
• Only married couples get children
• Constant fertility and mortality
• Constant willingness to marry
• Quantities
• Population
• Number of married couples

53
Forrester Models - Example
Willingness to marry
Flow number of marriages in t, t1
Flow number of births in t, t1
Population(t0 1000)
Number of marriedcouples (t0 250)
Fertility of married women
Flow number of deaths in t, t1
...
Mortality
54
Experimentation Models

• Optimization Models
• Prediction Models
• Experimentation Models
• Simulation in a broad sense
• Continuous Simulation
• Discrete Event Simulation

55
Simulation in a Broad Sense

• Use of optimization or prediction models for
  experimentation, e.g.
• Coverage of whole parameter spaces
• How-to-achieve models
• What-if-models
• Use of prediction models as objective functions
  for numerical optimization

56
Actual Simulation

• Why?
• Optimization and prediction models are subject to
  certain restrictions(e.g. Markov property,
  linearity)
• If such restrictions have to be violated (for
  modeling reasons), the behavior of the model can
  be simulated

57
Continuous Simulation

• Continuous models described as e.g.
• Systems of equations
• Ordinary or partial differential equations
• If such systems cannot be solved analytically,
  use numerical methods
• ? contiuous simulation

58
Discrete Simulation

• State variables of the model are changed at
  discrete points of time
• Two major possibilities
• Time driven simulatione.g. simulation of a
  discrete time Markov model
• Event driven simulatione.g. simulation of a
  continuous time Markov model (e.g. queueing model)

59
Time Driven Discrete Simulation

• Simulation procedure
• Increase virtual (simulation) time tt ? t?t
• Compute changes in interval t, t?t
• Discussion
• Problem appropriate granularity of time steps
• Well-suited for time-driven models

60
Discrete Event Simulation - Data Structure

• Virtual (or simulation) time VT
• Time stamp ordered event list
• State variables

Virtual time
Event list
State variables
VT
E1
t1
S1
SN
...
...
En
tn
61
Discrete Event Simulation - Algorithm

• Simulation procedure
• Choose scheduled event e with lowest time stamp t
• Set simulation time to VT t
• Update state variables S1,..., SN according to
  event e
• Insert new events in event list
• Cancel events from event list
• Remove event e from event list

62
Deterministic vs. Stochastic Simulation

• Deterministic simulation
• No uncertainty in parameters
• No random behavior
• Stochastic simulation
• Parameters may be characterized as random
  distributions
• Behavior may include random choices

63
Stochastic Simulation - Distributions

• Characterize model parameters (e.g. service time
  in a queueing model) as a random variable
• At each instant when this parameter is used in
  simulation, a sample value is drawn according to
  its distribution

64
Stochastic Simulation - Random Numbers

• Random variables are mathematical objects, which
  do not exist in computers
• Use random number generators (deterministic
  series of numbers) which show sufficiently
  random behavior
• Uniformity
• Independence
• Using bad random number generators can make
  simulation results invalid!

65
Stochastic Simulation - Statistics

• Every simulation run is different !
• Use multiple runs !
• Results have to be analyzed using statistical
  methods
• Mean values and variances
• Result distributions(e.g. histograms for output
  measures)
• Confidence intervals(e.g. system throughput is
  in the interval 10,15 with 95 certainty)

66
Simulation Demonstration Simul8
67
Summary 1/3 Operations Research

• Operations Research mathematical models of
  complex systems consisting of
• humans
• machines
• material
• capital

68
Summary 2/3 Model Building

• Model building process, phases
• Problem definition
• System analysis
• Model formalization
• Implementation and integration
• Analysis and/or experiments
• During all phases
• Verification
• Validation

69
Summary 3/3 OR Models

• Optimization models, e.g.
• Differential calculus
• Linear Programming
• Game theory
• Prediction models, e.g.
• Task graphs (net-plan models)
• Markov models
• Forrester models
• Experimentation models, i.e. simulation