Probabilistic OR Models and their Formulations

Prepared by Prof.Dr.Fetih YILDIRIM Certainty and Uncertainty Measurements Randomness Experiments- Random and nonrandom experiments What is probability? What is a model? What are the possible advantages of having

models? Probabilistic and unprobabilistic Models Probabilistic models in OR Formulations of Some Probabilistic OR models

Introduction to experimentation

An experiment may be required to assist in clarifying a number of questions,some of a general nature and others more specific examples:

(i)How widely applicable is a particular theory?(ii)At what temperature does a newly prepared alloy mely?(iii)Can a technique for measuring a particular quantity be

improved kupon?(iv)What happens to the magnetic properties of a material when

it is cooled to very low temperatures?

The importance of experiments in science and engineering Experimental work The reasons for doing experiments Possibhle difficulties

Stages of a typical Experiment The aim The plan Preparation Preliminary experiment Collecting data Repeatability Analysis of data What do the data tell you? Reporting the experiment

Dealing with Uncertainties

What are the uncertainties? Uncertainty in a single measurement: It is important to be

aware of resolution, reading and calibration uncertainties when attempting to quote the uncertianty in a single measurement.Such uncertainties exist whenever a measurement is made in an experiment. However, to be able to get a real feel for the variability in measurement, more than one measurement should be made of each quantity. Where this is possible we can use some results from statistical analysis to allow us to quantify experimental uncertainties.

True value, accuracy and precision:If a value is obtained from an experiment is called accurate, then

it is close to the true value but,unless given, the uncertainty could be of any magnitude.

If a value is obtained from an experiment is called precise, then it has a small uncertainty, but this does not mean that it is close to the true value.

Cont.’n of Dealing with Uncertainties

If a value is obtained from anexperiment is called both accurate and precise, then it is close to true value and with a small unceratinty. We would like our experimental data to fall into this category.

Systematic and random uncertainties

Probabilistic models

A model means a system that simulates an object under consideration.

A probabilistic model is a model that produces different outcomes with different probabilities – it can simulate a whole class of objects, assigning each an associated probability.

In bioinformatics, the objects usually are DNA or protein sequences and a model might describe a family of related sequences

Probabilistic models

Probabilistic statements are fundamental to many communities: Science Engineering Medicine

Probabilities are meaningful only within the context of a stochastic model, which itself has a context (not necessarily probabilistic).

Bayesian networks are an example of a stochastic modeling technique for specifying dependencies among random variables.

Probability is the language for expressing the experimental results.

Decision Support

A decision tree can be used for specifying a logical decision.

Decisions may involve uncertain observations and dependent observations so a simple decision tree will not be accurate.

Influence diagrams Bayesian network extended with utility

functions and with variables representing decisions

The objective is to maximize the expected utility.

Stochastic modeling techniques

Logic programming Data modeling Statistics Programming languages World Wide Web

Logic Programming: ICL

Independent Choice Logic Expansion of Probabilistic Horn abduction to

include a richer logic (including negation as failure), and choices by multiple agents.

Extends logic programs, Bayesian networks, influence diagrams, Markov decision processes, and game theory representations.

Did not address ease of use

Operations Research

OR is a scientific approach to decision making that involves the operations of organizational systems.

"Research on Operations" OR involves creative scientific research into the

fundamental properties of operations. Operations research is also concerned with the practical management of the organization.

It attempts to resolve the conflicts of interest among the components of the organization in a way that is best for the organization as a whole.

OR attempts to find the best or optimal solution to the problem under consideration.

Team Approach - OR team to undertake a full-fledged study

OR-team needs to include specialists in mathematics, statistics and probability theory, economics, business administration, electronic computing, engineering and the physical sciences, the behavioral sciences, and the special techniques of OR.

 OR is concerned with optimal decision making

in, and modeling of, deterministic and probabilistic systems that originate from real life.

Typical OR Approach is:

1) Formulating the problem. - From vague description, determine objectives, goals, and constraints

2) Constructing a mathematical model to represent the system under study. - decision variables, objective function, (mathematical) constraints

3) Deriving a solution from the model. - optimizing or satisfying, post-optimality analysis

4) Testing the model and the solution derived from it. - validity

5) Establishing controls over the solution. - sensitivity analysis

6) Putting the solution to work: implementation.

OR Methods Used in

• Mathematical Programming(LP, IP, NLP, Multi-Objective)

• Transportation Problem• Game Theory• Dynamic Programming• PERT-CPM• Queuing Theory• Forecasting• Inventory• Decision Analysis• Reliability• Simulation

Game Theory

Prisoner 2

Prisoner 1

Confess

Don'tConfess

Confess (-5, -5) (0, -20)

Don'tConfess (-20, 0) (-1,-1)

QUEUEING MODELS

A queuing problem arises whenever the demand for customer service cannot perfectly be matched by a set of well-defined service facilities.

Purpose of Queueing Model To help design a system that will minimize a

stated measure of performance such as the sum of the costs of customer waiting and costs of idle facilities.

Service Facility

Customers Arriving

Serviced Customers Leaving

Discouraged Customers Leaving

QUEUEING THEORY

Simple Queueing Theory

Queueing Models Kendall notation Steady state analysis Performance measures Different queue models

Queues and components

Queues are frequently used in simulations. Population: The entity (“customers”) that

requires service Server: The entity that provides the service Queue: The entity that temporarily holds the

waiting “customers” before they are served. Events: arrival, service, and leaving.

Purpose of Queueing Models Most models are to determine the

level of service Two major factors:

Cost of providing service: cannot afford many idle servers.

Cost of customer dissatisfaction: customers will leave if queue is too long.

Tradeoff between these 2 factors

Characteristics of Queue models Calling population

infinite population: leads to simpler model, useful when number of potential “customers” >> number of “customers” in system.

Finite population: arrival rate is affected by the number of customers already in the system.

System capacity The number of customers that can be in the queue or under

service. An infinite capacity means no customer will exit prematurely.

Arrival process For infinite population, arrival process is defined by the

interarrival times of successive customers Arrivals can be scheduled or at random times, Poisson dist’n

is used frequently for random arrivals, and scheduled arrivals usually use a constant interarrival rate.

Characteristics of Queue models Queue behaviour

describes how the customer behaves while in the queue waiting

• balking - leave when they see the line is too long• renege - leave after being in the queue for too long• jockey - move from one queue to another

Queue discipline FIFO - first in first out (most common) FILO - first in last out (stack) SIRO - service in random order SPT - shortest processing time first PR - service based on priority

Characteristics of Queue models Service Times

random: mainly modeled by using exponential distribution or truncated normal distribution (truncate at 0).

Constant Service mechanism describes how the servers are

configured. Parallel - multiple servers are operating and take

customer in from the same queue. Serial - customers have to go through a series of

servers before completion of service combinations of parallel and serial.

Kendall Notations

Kendall defined the notations for parallel server systems

A / B / c / N / K

A: interarrival distribution type B: service time distribution type Common symbols for A, B are M for exponential, D for

constant, Ek for Erlang, G for general or arbitrary. c: for number of parallel servers N: for system capacity K: for size of calling population.