Operations Research I Lecture 1-3 Chapter 1 Dr. Ayham Jaaron First semester 2013/2014

Outline What is OR? Mathematical modeling Linear programming (LP) Formulating linear programming

Background To understand what operations research (OR) is today, one must know something of its history and evolution World War II : British military leaders asked scientists and engineers to analyze several military problems Management of materials, convoy, bombing, antisubmarine, and mining operations. As these teams were generally assigned to the commanders in charge of military operations, they were called operations research (OR) teams.

Background .. Cont’d At the end of the war, many of the scientists who worked in the military operations research units returned to civilian life in universities and industries. They started applying the OR methodology to solve complex management problems in industries. Petroleum companies were the first to make use of OR models for solving large-scale production and distribution problems In the universities, advancements in OR techniques were made that led to the further development and applications of OR. Much of the postwar development of OR took place in the United States

What is OR? “Operational research is the application of the methods of science to complex problems arising in the direction and management of large systems of men, machines, materials and money in industry, business, government, and defense. The distinctive approach is to develop a scientific model of the system, incorporating measurement of factors such as chance and risk, with which to predict and compare the outcomes of alternative decisions, strategies or controls. The purpose is to help management determine its policy and actions scientifically”. The Operational Research Society of Great Britain “Operations research is concerned with scientifically deciding how to best design and operate man–machine systems, usually under conditions requiring the allocation of scarce resources”. The Operations Research Society of America

Branches of OR Deterministic – input data are known (e.g. mathematical programming models). Non-deterministic – input data have uncertainties (probabilistic or stochastic models)

Phases of OR (deterministic) Formulation of the problem Identify and construct an appropriate mathematical model Finding a solution to the model Analyze the solution Validate Implement

Mathematical modeling and programming This is concerned with the optimum allocation of limited resources among competitive activities under a set of constraints To maximize/minimize an objective function Types of mathematical programming Linear programming Non-linear programming Integer programming Dynamic programming

Modeling

Mathematical modeling and programming Model: an abstraction (simplification) of an actual problem that captures the major characteristics of the problem . Types of models Quantitative model – mathematical, LP , etc Qualitative model – narrative , graphs, drawings , etc Physical model

Class Task: 5 minutes Given a collection of numbers, partition them into two groups such that the difference in the sums is as small as possible. Example: 7, 10, 13, 17, 20, 22 These numbers sum to 89

An optimization problem Given a collection of numbers, partition them into two groups such that the difference in the sums is as small as possible. Example: 7, 10, 13, 17, 20, 22 These numbers sum to 89 I can split them into {7, 10, 13, 17} sum is 47 {20, 22} sum is 42 Difference = 5 Can we do better?

Some Skills for Operation Researchers Modeling Skills Take a real world situation, model it using mathematics Methodological Toolkit Optimization Decide whether the problem is maximization or minimization.

Optimization is Everywhere It is embedded in language, and part of the way we think. firms want to maximize value to shareholders people want to make the best choices We want the highest quality at the lowest price When playing games, we want the best strategy When we have too much to do, we want to optimize the use of our time etc. TASK 2: Take 3 minutes with your partner to brainstorm on where optimization might be used. (business, or sports, or personal uses, or politics, or …)

Linear programming (LP) Mathematical programming models whose objective functions are linear, whose constraints are linear inequalities or equalities, and whose variables take continuous values. minimize or maximize a linear objective subject to linear equalities and inequalities

Formulating an LP model Read the problem carefully Define the decision variables (The decision variables are the variables whose specification describes the solution for the problem. It typically comprises the set of decisions to be made). 3. Determine the objective function 4. Determine the constraints

Terminologies Decision variables: In general, these are quantities you can control to improve your objective which should completely describe the set of decisions to be made. Constraints: Limitations on the values of the decision variables. Objective Function: Value measure used to rank alternatives Seek to maximize or minimize this objective examples: maximize Productivity, minimize cost