INTRODUCTION TO OPERATION RESEARCH

According to Churchman, Aackoff, and Aruoff operations research is defined as: “the application of scientific methods, techniques and tools to operation of a system with optimum solutions to the problems,” where 'optimum' refers to the best possible alternative. It helps in the decision-making process in complicated situations in various fields, such as industrial, academic, and government organizations.

SCOPE: Defense operations, industry, planning, agriculture, hospital, transport, etc.

FEATURES: System-oriented, interdisciplinary team approach, scientific methods to solve problems, offers quantitative solutions, etc.

PHASES: Judgement, research and action.

METHODOLOGY: Definition, construction, solution, validation and implementation.

TECHNIQUES: Linear programming, inventory control method, goal programming, queuing model, transportation model.

LINEAR PROGRAMMING

Linear Programming (LP) is a mathematical technique designed to help managers in their planning and decision making. Linear programming focuses on obtaining the best possible output from a given set of limited resources. Linear programming is used to use the resources in an organization in a most efficient manner.

FUNCTIONAL AREAS: Production and operations management is to determine the quantities of each product that should be produced.

Marketing is to determine how many advertisements to place in each medium. Distribution is to determine the shipping pattern that minimizes total costs.

ADVANTAGES DISADVANTAGESOptimum utilization of productive resources. Objective functions and constraints are not linear.

Quality of decisions by LPP technique. No guarantee of integer value solutions.Provides practically applicable solutions. Effect of time and uncertainty are not taken into

consideration.In production process, high lighting of bottlenecks. Large scale problems are not solved.

A few common requirements of LPP are: Decision variables and their relationship, well-defined objective function, existence of alternative courses of action and non-negative conditions on decision variables.

GRAPHICAL ANALYSIS OF LPP

SIMPLEX METHOD

The simplex method simply selects the optimal solution amongst the set of feasible solutions of the problem. The simplex algorithm is an iterative procedure for finding the optimal solution to a linear programming problem. The simplex algorithm considers only those feasible solutions which are provided by the corner points.

CHARACTERISTICS: All constraints are equations. The right hand side element of each constraint equation is non-negative. All the variables are non-negatives. The objective function is of maximization type.

DUALITY IN LPP

Every Linear Programming Problem (LPP) is associated with another linear programming problem involving the same data and optimal solutions. The two problems are said to be duals of each other. One problem is called the primal, while the other problem is called the dual. The concept of duality is useful to obtain additional information about the variation in the optimal solution.

ECONOMIC INTERPRETATION: The linear programming problem is thought of as a resource allocation model, where the objective is to maximize revenue or profit subject to limited resources. The associated dual problem offers interesting economic interpretations of the linear programming resource allocation model.

SENSITIVITY ANALYSIS: The management of a company is interested in knowing the impact of changes in the input parameter values on the optimal solution. This process is called sensitivity analysis or post optimality analysis. The results of sensitivity analysis establishes upper and lower bounds for input parameter values within which they can vary without causing major changes in the current optimal solution. Sensitivity analysis allows us to figure out which data offers a significant impact on the results. This in turn allows us to concentrate on getting accurate data for those items, or at least running through several scenarios with various values of the crucial data, to get an idea of the range of possible outcomes.

TRANSPORTATION PROBLEM

Every Transportation model is an important class of linear programs. For a given supply at each source and a given demand at each destination, the model studies the minimization of the cost of transporting a commodity from a number of sources to several destinations. To facilitate the presentation and solution, the general transportation problem is normally portrayed in a tabular form.

INTEGER PROGRAMMING PROBLEM

The IPP is a special case of Linear Programming Problem (LPP), where all or some variables are constrained to assume non-negative integer values. Integer LP problems are those in which some or all of the variables are restricted to integer (or discrete) values. An integer LP problem has important applications. Capital budgeting, construction scheduling, plant location and size, routing and shipping schedule, batch size, capacity expansion, fixed charge, etc are a few problems which demonstrate the areas of application of integer programming.

TYPES: Pure, mixed and zero-one.

QUEUING THEORY

Queuing theory is a collection of mathematical models of various queuing systems. It is based on probability concepts. It gives an indication of the capability of a

given system and the possible changes in its performance with modification to the system. All the constraints of the process are not taken into account in the formulation of a queuing model. The application of queuing theory cannot be viewed as an optimization process as there is no maximization or minimization of an objective function.

CONSTITUETS: Arrival pattern of customers can be regular as in case of an appointment system of a doctor or flow of components on a conveyor belt. The regular pattern of arrivals is neither very common nor

very easy to deal with mathematically.

Completely random arrivals: If the number of potential customers is infinitely large, then probability of an arrival in the next interval of time will not depend upon the number of customers already in the system. When the arrivals are completely random, they follow the Poisson distribution, which equals to the average number of arrivals per unit time. Sometimes it is necessary to distinguish between groups of customers, such as male and female callers, or large and small aircrafts during the arrivals.