Introduction to linear programming

Introduction to linear programming

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  1 Presented by: AkshatMishra BBA (3rd Sem) Submitted to: Mr. Durgesh Batra
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   Linear Programming. Linear Programming Problem.  Problem formulation.  Guidelines for model formulations.  Solved Example. 2
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   The useof the word “programming” hereThe use of the word “programming” here means “choosing a course of action.”means “choosing a course of action.”  Linear programming involves choosing aLinear programming involves choosing a course of action when the mathematical modelcourse of action when the mathematical model of the problem contains only linear functions.of the problem contains only linear functions. 3
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   The maximizationor minimization of some quantity is the objective in all linear programming problems.  All LP problems have constraints that limit the degree to which the objective can be pursued.  A feasible solution satisfies all the problem's constraints.  An optimal solution is a feasible solution that results in the largest possible objective function value when maximizing (or smallest when minimizing).  A graphical solution method can be used to solve a linear program with two variables. 4
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   If boththe objective function and the constraints are linear, the problem is referred to as a linear programming problem.  Linear functions are functions in which each variable appears in a separate term raised to the first power and is multiplied by a constant (which could be 0).  Linear constraints are linear functions that are restricted to be "less than or equal to", "equal to", or "greater than or equal to" a constant. 5
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   Problem formulationor modeling is the process of translating a verbal statement of a problem into a mathematical statement.  Formulating models is an art that can only be mastered with practice and experience.  Every LP problems has some unique features, but most problems also have common features. 6
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   Understand theproblem thoroughly.  Describe the objective.  Describe each constraint.  Define the decision variables.  Write the objective in terms of the decision variables.  Write the constraints in terms of the decision variables. 7
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   XYZ ltd.can invest Rs40,000 in production and use 85 hours of labor. To manufacture one unit of product “A” requires 15 minutes of labor, and to manufacture one unit of product “B” requires 9 minutes of labor. The company wants to maximize its profit. How many units of product “A” and product “B” should it manufacture? What is the maximized profit? 8
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   Since theprofit to be maximized depend on the number of product “A” and “B”, our decision variables are: x1 = number of product “A” produced; x2 = number of product “B” produced;  We want to maximize profit: i.e. 30x1 + 20x2 Subject to the constraints: Money: 40x1 + 60x2 ≤ 40,000 labor: 15x1 + 9x2 ≤ 5,100 Non-negativity: x1,x2 ≥ 0 9
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   Note thelast constraint: x1,x2 ≥ 0 of product “B” produced:.The unknowns x1 and x2 are called decision variables.The function 30x1+20x2 to be maximized is called the objective function.  What we have now is a Linear Program. maximum z = 30x1 + 20x2 40x1 + 60x2 ≤ 40;000 15x1 + 9x2 ≤ 5;100 x1; x2 ≥ 0 10
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