DEPARTMENT/SEMESTER ME VII Sem COURSE NAME Operation Research Manav Rachna College of Engg.

I NTRODUCTION TO O PERATION R ESEARCH Manav Rachna College of Engg.

` What is Operations Reseach? Operations Research (OR) started just before World War II in Britain with the establishment of teams of scientists to study the strategic and tactical problems involved in military operations. The objective was to find the most effective utilization of limited military resources by the use of quantitative techniques. Following the war, numerous peacetime applications emerged, leading to the use of OR and management science in many industries and Occupations. Definitions Operations Research (OR) is the study of mathematical models for complex organizational systems. Optimization is a branch of OR which uses mathematical techniques such as linear and nonlinear programming to derive values for system variables that will optimize performance.

D EFINITIONS Operations Research (OR) is the study of mathematical models for complex organizational systems. Optimization is a branch of OR which uses mathematical techniques such as linear and nonlinear programming to derive values for system variables that will optimize performance.

Manav Rachna College of Engg. Evolution The first formal activities of Operations Research (OR) were initiated in England during World War II, when a team of British scientists set out to make scientifically based decisions regarding the best utilization of war materiel. After the war, the ideas advanced in military operations were adapted to improve efficiency and productivity in the civilian sector. This chapter will familiarize you with the basic terminology of operations research, including mathematical modeling, feasible solutions, optimization, and iterative computations

M ODELS Linear Programming Typically, a single objective function, representing either a profit to be maximized or a cost to be minimized, and a set of constraints that circumscribe the decision variables. The objective function and constraints all are linear functions of the decision variables. Software has been developed that is capable of solving problems containing millions of variables and tens of thousands of constraints. Network Flow Programming A special case of the more general linear program. Includes such problems as the transportation problem, the assignment problem, the shortest path problem, the maximum flow problem, and the minimum cost flow problem. Very efficient algorithms exist which are many times more efficient than linear programming in the utilization of computer time and space resources. Manav Rachna College of Engg.

M ODELS Integer Programming Some of the variables are required to take on discrete values. NP-hard: Most problems of practical size are very difficult or impossible to solve. Nonlinear Programming The objective and/or any constraint is nonlinear. In general, much more difficult to solve than linear. Most (if not all) real world applications require a nonlinear model. In order to be make the problems tractable, we often approximate using linear functions Manav Rachna College of Engg.

M ODELS Dynamic Programming A DP model describes a process in terms of states, decisions, transitions and returns. The process begins in some initial state where a decision is made. The decision causes a transition to a new state. Based on the starting state, ending state and decision a return is realized. The process continues through a sequence of states until finally a final state is reached. The problem is to find the sequence that maximizes the total return. Objectives with very general functional forms may be handled and a global optimal solution is always obtained. "Curse of dimensionality" - the number of states grows exponentially with the number of dimensions of the problem. Manav Rachna College of Engg.

M ODELS Simulation It is often difficult to obtain a closed form expression for the behavior of a stochastic system. Simulation is a very general technique for estimating statistical measures of complex systems. A system is modeled as if the random variables were known. Then values for the variables are drawn randomly from their known probability distributions. Each replication gives one observation of the system response. By simulating a system in this fashion for many replications and recording the responses, one can compute statistics concerning the results. The statistics are used for evaluation and design. Manav Rachna College of Engg.

M ODELS Inventory Theory Inventories are materials stored, waiting for processing, or experiencing processing. When and how much raw material should be ordered? When should a production order should be released to the plant? What level of safety stock should be maintained at a retail outlet? How is in-process inventory maintained in a production process? Reliability Theory Attempts to assign numbers to the propensity of systems to fail. Estimating reliability is essentially a problem in probability modeling. Extremely important in the telecommunications and networking industry. Manav Rachna College of Engg.

ART OF MODELING Manav Rachna College of Engg.

PHASES OF AN OR STUDY The principal phases for implementing OR in practice include: 1. Definition of the problem 2. Construction of the model 3. Solution of the model 4. Validation of the model 5. Implementation of the solution. Manav Rachna College of Engg.

M ATHEMATICAL P ROGRAMMING A mathematical model consists of: Decision Variables, Constraints, Objective Function, Parameters and Data The general form of a math programming model is: Min or Max f(x 1 ; …… ; x n ) such that g i (x 1 ;…… ; x n ) ≤ bi ≥ bi = b i x є X Linear program (LP): all functions f and gi are linear and X is continuous. Integer program (IP): X is discrete. Manav Rachna College of Engg.

M ATHEMATICAL P ROGRAMMING A solution is an assignment of values to variables. A feasible solution is an assignment of values to variables such that all the constraints are satisfied. The objective function value of a solution is obtained by evaluating the objective function at the given solution. An optimal solution (assuming minimization) is one whose objective function value is less than or equal to that of all other feasible solutions. Manav Rachna College of Engg.

O UTLINE What is Operations Research? Optimization Problems and Applications Personal Examples Manav Rachna College of Engg.