Lecture 6 in which we introduce the theory of duality in linear programming. Formally, you can formulate a dual of any integer program, e. I use great textbook linear programming and network flows by bazaraa ii ed on the page 240 the author states that for every primal problem, regardless of its type canonical or standard, dual problem can be found by using the following converting table. Using duality and sensitivity analysis to interpret linear programming solutions j. Chapter 4 duality given any linear program, there is another related linear program called the dual. In discussing duality, it is common to depart from the standard equality form of the lp given in section 4. Sometimes applied to theorems, as for example desargues theorem and pappus theorem.
Blair ce, jeroslow r 1977 the value function of a mixed integer program 1. The dual model is derived by construction from the standard inequality form of linear programming model as shown in tables 1 and 2. Inputs simply enter your linear programming problem as follows 1 select if the problem is maximization or minimization 2 enter the cost vector in the space provided, ie in boxes labeled with the ci. Lips is based on the efficient implementation of the modified simplex method that solves large scale problems.
The work of solving an lp by the dual simplex method is about the same as of by the revised primal simplex method. Simplex algorithm calculator is an online application on the simplex algorithm and two phase method. Linear program solver lips is an optimization package oriented on solving linear, integer and goal programming problems. Lecture 6 1 the dual of linear program stanford cs theory. Primal dual relationships in linear programming duality. Duality in linear programming linear programming duality duality theorem. Standard form means, all the variables in the problem should be nonnegative and. Linear programming lp, involves minimizing or maximizing a linear objective function subject to bounds, linear equality, and inequality constraints. This understanding translates to important insights about many optimization problems and algorithms. A typical problem of linear programming is to maximize the linear function.
For this purpose there are computational tools that assist in applying the graphical model, like tora, iortutorial and geogebra within this context we will present a series of linear programming. Note that you can add dimensions to this vector with the menu add column or. Linear programming problems arise pervasively in science and engineering. Sometimes even applied to a method like simplex and dual simplex methods in linear programming. If any of the two problems has an infeasible solution, then the value of the objective. Linear programming is the mathematical problem of finding a. Duality in linear programming standard form for linear. Duality in linear programming is essentially a unifying theory that develops the relationships between a given linear program and another related linear program stated in terms of variables with this shadowprice interpretation. The two programmes are very closely related and optimal solution of.
Department of management engineering technical university of denmark. As already mentioned in chapter 1, many books cover this topic, including dantzig dan63, lasdon las70, and vanderbei van08. Duality in linear programming is essentially a unifying theory that develops the relationships between a given linear program and another related linear program stated in terms of variables with this shadowprice. Linear programming simplex algorithm, duality and dual. By linear programming webmaster on january 15, 2015 in linear programming lp the dual model of a linear programming problem consists of an alternative modeling instance that allows us to recover the information of the. Optimum for lp defines natural upper bound on the optimum for ilp. Linear programming, lagrange multipliers, and duality. Because it is often possible to solve the related linear program with the shadow prices as the variables in place of, or in conjunction with, the original linear program, thereby taking advantage of some computational efficiencies. How to solve a linear programming problem using the dual simplex method duration.
Example problems include blending in process industries, profit maximization in manufacturing, portfolio optimization in finance, and scheduling in energy and transportation. The tora optimization software for linear programming are explained below. Then the claim follows directly from duality theorem in linear programming. Knowledge of duality allows one to develop increased insight into lp solution interpretation. Consider the linear programming problem in standard form.
Given any linear program, there is another related linear program called the dual. Using the simplex method to solve linear programming maximization problems j. The original problem is called primal programme and the corresponding unique problem is called dual programme. Maximize ctx subject to ax b primal x 0 its dual linear program is. Characteristics of dual problem, advantages of duality. The graphical method graphic solving is an excellent alternative for the representation and solving of linear programming models that have two decision variables.
If either the primal of dual problem has a solution, then the other also has a solution and their optimum values are equal. In this method, we keep transforming the value of basic variables to get maximum value for the objective function. If any of the two problems has only an infeasible solution then the value of the objective function of the other. Using duality and sensitivity analysis to interpret linear. If either the primal or dual problem has a solution then the other also has a solution and their optimum values are equal. Linear programming is used as a mathematical method for determining and planning for the best outcomes and was developed during world war ii by leonid kantorovich in 1937. In order to distinguish it from the dual, the original linear program of interest in this case, the one involving decisions on quantities of cars and trucks to build in order to maximize pro. From the main menu, select linear programming option, and enter the input values of the previously discussed problem as shown in the following figure.
The simplex method is an iterative procedure for getting the most feasible solution. It was a method used to plan expenditures and returns in a way that reduced costs for the military and possibly caused the opposite for the enemy. The idea of duality for cones is almost the same as the standard idea of geometric duality. This lectures notes focus on the duality in linear programming, and give an example of the dual problem for maximum ow problem. Duality 7 duality of an lpp with equality, mixed constraints and a variable unrestricted in sign duration. Linear programming is a branch of mathematical programming. Linear programming course engineering courses purdue. At each iteration of the dual simplex method, we require. In such a situation, it is useful to think of profit as a function of a vector. In this chapter, we will develop an understanding of the dual linear program. Linearprogramming problems are mathematical models of numerous problems that have technical and economic.
Music let us now look at linear programming duality from a geometry perspective. So the problem can be solved by solving the dual of lp. When a solution is obtained for a linear program with the revised simplex method, the solution to a second model. Linear programming, or lp, is a method of allocating resources in an optimal way. Given a primal lp, the following algorithm can be used to construct its dual lp. Strong duality of linear programming emory university. Before solving for the duality, the original linear programming problem is to be formulated in its standard form. It is a difference whether one can dualize or not or that a duality theory holds or not. Actually, the linear programming duality theorem says much more than that. Simplex method jg convert inequalities by adding slack variables. Strong duality of linear programming hao huang for an m nmatrix a, a vector c2rn and another vector b2rm. Dual simplex using lingo software and sas is as follows. Slaters conditions holds if the primal is feasible, i. Since the 1980s, new interior point methods have been used as well.
Duality in linear programming wiley online library. Since the 1940s, linear programming problems have been and still are solved by the simplex method and its variants. They will also obtain a broad exposure to the theoretical underpinnings of linear optimization. Now since strong duality holds in lp, it depends on a case by case basis which one would you solve for. Linear programming applications of linear programming. Duality in mixed integer linear programs operations. Since the problem d is a linear program, it too has a dual. For lp solvers lagrangian duality is somewhat analogous to lp duality. The problem p has an optimal solution if and only if the dual problem d has an optimal solution.
Duality in linear programming problems your article library. Duality in linear programming in quantitative techniques. Duality in linear programming has the following major characteristics. Do any linear programming optimization software kits use. In solving any linear program by the simplex method, we also determine the shadow prices associated with the. The linear programming dual of the last problem is the problem 11, 12. Geometry of lp duality linear programming duality coursera. Sometimes applied to spaces, for example the dual space of a vector space. Also, when solving the dual of any problem, one simultaneously solves the primal. Lp duality and the primaldual method in the previous few lectures we have seen examples of lprounding, a method for obtaining approximation algorithms that involves solving a linear programming relaxation of the problem at hand and rounding the solution.