Introduction to Operations Research

Linear Programming
Optimization is an important and fascinating area of management science and operations research. It helps to do less work, but gain more.Linear programming (LP) is a central topic in optimization. It provides a powerful tool in modeling many applications. LP
has attracted most of its attention in optimization during the last six decades for two main reasons:
Applicability: There are many real world applications that can be modeled as linear programming;
Solvability: There are theoretically and practically efficient techniques for solving large-scale problems.

Each optimization problem consists of three elements:
? decision variables: describe our choices that are under our control;
? objective function: describes a criterion that we wish to minimize
(e.g., cost) or maximize (e.g., profit);
? constraints: describe the limitations that restrict our choices for decision variables.

Problem Statement: A company makes two products (say, P and Q) using two machines (say, A and B). Each unit of P that is produced
requires 50 minutes processing time on machine A and 30 minutes processing time on machine B. Each unit of Q that is produced requires 24 minutes processing time on machine A and 33 minutes processing time on machine B. Machine A is going to be available for 40 hours and machine B is available for 35 hours. The profit per unit of P is $25 and the profit per unit of Q is $30. Company policy is to determine the production quantity of each product in such a way as to maximize the total profit given that the
available resources should not be exceeded.
The aim is to formulate the problem of deciding how much of each product to make in the current week as an LP.