Linear programming examples and solutions ppt

Introduction linear programming is constrained optimization, where the constraints and the objective function are all linear. Solve linear programming problems using the simplex method. The above stated optimisation problem is an example of linear programming problem. Chapter 4 duality given any linear program, there is another related linear program called the dual. How to solve a linear programming problem using the graphical. Or when you have a project delivery you make strategies to make your team work efficiently for ontime delivery. Hale company manufactures products a and b, each of which requires two processes, grinding and polishing. Each intersection point is the the solution to a 3. To make a dress requires 2 1 hour of cutting and 20 minutes of stitching. Solution decision variables completely describe the decisions to be made in this case, by manager. Their goal is to minimize cost, subject to meeting the minimum nutritional requirements imposed by law. It turns out that the solutions to linear programming problems provide interesting economic information.

March 31, 2007 mathematics department brown university 151 thayer street providence, ri 02912 abstract we describe linear programming, an important generalization of linear algebra. In order to illustrate some applicationsof linear programming,we will explain simpli ed \realworld examples in section 2. This gure also illustrates the fact that a ball in r2 is just a disk and its boundary. Linear programming problems carpenter problem teaching. Linear programming 507 given sum by the dealer in purchasing chairs and tables is an example of an optimisation problem as well as of a linear programming problem. Some worked examples and exercises for grades 11 and 12 learners.

Any linear program can in fact be transformed into an equivalent linear program in standard form. Any finite optimization algorithm should terminate in one. For example, an increase in manpower resource will result in an increase in work output. Examplesoflinear programmingproblems formulate each of the following problems as a linear programming problem by writing down the objective function and the constraints. Linear programming linear programming mathematical. Michel goemans 1 basics linear programming deals with the problem of optimizing a linear objective function subject to linear equality and inequality constraints on the decision variables. Linear programming deals with this type of problems using inequalities and graphical solution method. Introduction lpp, in which constraints may also have and signs, we introduce a new type of variable, called the artificial variable. This should provide insight into the scope of integer programming applications and give some indication of why many practitioners feel that the integer programming model is one of the most important models in management science. A general linear programming problem need not have a feasible solution. The simplex method, for example, is an algorithm for solving the class of linear programming problems. Model formulation and graphical solution joseph george konnully prepared by slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. A number of algorithms for other types of optimization problems work by solving lp problems as subproblems. Basic linear programming concepts linear programming is a mathematical technique for finding optimal solutions to problems that can be expressed using linear equations and inequalities.

Linear programming is an optimization technique for a system of linear constraints and a linear objective function. Applications of linear programming are everywhere around you. The feasible region of the linear programming problem is empty. Optimization models make it easier to solve complex organizationwide problems. This procedure, called the simplex method, proceeds by moving from one feasible solution to another, at each step improving the value of the objective function. If it does have a feasible solution, it need not have an optimal. Linear programming is a special case of mathematical programming also known as mathematical optimization. Methods of solving inequalities with two variables, system of linear inequalities with two variables along with linear programming and optimization are used to solve word and application problems where. However, there are constraints like the budget, number of workers, production capacity, space, etc. Solving problems in practice or using solvers glpk or. Linear programming, graphically weve seen examples of problems that lead to linear constraints on some unknown quantities.

All the variables in the current dictionary solution are nonnegative. It involves an objective function, linear inequalities with subject to constraints. Linear programming is used to successfully model numerous real world situations, ranging. A maximization model example graphical solutions of linear programming models a minimization model example irregular types of linear. This understanding translates to important insights about many optimization problems and algorithms. Objective understand the basic concepts of linear programming models and learn. If the quantity to be maximizedminimized can be written. Using excel to solve linear programming problems technology can be used to solve a system of equations once the constraints and objective function have been defined. Linear programming used to obtain optimal solutions to problems that. What is meant by the unit cost in linear programming problems. Solving linear programs 2 in this chapter, we present a systematic procedure for solving linear programs. Ppt linear programming powerpoint presentation free to. Linear programming solution examples linear programming example 1997 ug exam. Several word problems and applications related to linear programming are presented along with their solutions and detailed explanations.

These variables are fictitious and cannot have any physical meaning. In this chapter, we will develop an understanding of the dual linear program. A maximization model example graphical solutions of linear. First, we will discuss integer programming formulations. Define and discuss the linear programming technique, including assumptions of linear programming and accounting data used therein. Apr 10, 2014 in this lesson we learn how to solve a linear programming problem using the graphical method with an example. Linear programming problem ppt download slideplayer. You use linear programming at personal and professional fronts. It is concerned with one production process at a time. You are using linear programming when you are driving from home to work and want to take the shortest route. A feasible solution satisfies all the problems constraints.

Linear programming solution to get some insight into solving lps consider the two mines problem that we had before the lp formulation of the problem was. All lp problems have constraints that limit the degree to which the objective can be pursued. Linear programming is a mathematical technique for solving constrained maximization and minimization problems, when there are many constraints and the objective function to be optimized, as well as the constraints faced, are linear i. Make problems amenable to mathematical and computer solution. Linear programming with two variables 191 the fundamental theorem of linear programming if the feasible region for a linear programming problem is nonempty and convex, and if the objective function has a maximum or minimum value within that set, then that maximum or minimum will always correspond to at least one corner point of. Fundamental theorem of linear programming to solve a linear programming problem, we first need to know the fundamental theorem of linear programming. The mathematical technique for solving lp problems was developed by. Vanderbei october 17, 2007 operations research and financial engineering princeton university. Burtonville burns 3000 tons of trash per day in three elderly incinerators. The below mentioned article provides an overview on the application of linear programming to the theory of firm.

This paper will cover the main concepts in linear programming, including examples when appropriate. They would like to offer some combination of milk, beans, and oranges. An objective function defines the quantity to be optimized, and the goal of linear programming is to find the values of the variables that maximize or minimize the objective function. It is called programming because the goal of the calculations help you choose a program of action. Each unit of x that is produced requires 50 minutes processing time on machine a and 30 minutes processing time on machine b.

Linear programming applications of linear programming. The neoclassical theory of the firm analyses the problem of decisionmaking with one or two variables at a time. We will now discuss how to find solutions to a linear programming problem. In most of the examples in this section, both the maximum and minimum will be found. Linear programming applications in construction sites. If a linear programming problem represents a companys profits, then a maximum amount of profit is desired. Jul 04, 20 22 topics linear programming an overview model formulation characteristics of linear programming problems assumptions of a linear programming model advantages and limitations of a linear programming. A small business enterprise makes dresses and trousers. The simplex method, for example, is an algorithm for solving the class of linearprogramming problems. In linear programming, we formulate our reallife problem into a mathematical model. Integer programming 9 mit massachusetts institute of. To make a trousers requires 15 minutes of cutting and 2 1 hour of stitching. Excel has an addin called the solver which can be used to solve systems of equations or inequalities.

In matrix form, a linear program in standard form can be written as. Linear programming lp, also called linear optimization is a method to achieve the best outcome such as maximum profit or lowest cost in a mathematical model whose requirements are represented by linear relationships. A nice way to introduce pupils to exam style questions as question broken down into parts. Linear programming has many practical applications in transportation, production planning. Introduction linear programming is a widely used mathematical.