Jump how many programming languages are there in total navigation Jump to search “Linear problem” redirects here. It is not to be confused with Linear function. A pictorial representation of a simple linear program with two variables and six inequalities. The set of feasible solutions is depicted in yellow and forms a polygon, a 2-dimensional polytope.

A closed feasible region of a problem with three variables is a convex polyhedron. The linear programming problem is to find a point on the polyhedron that is on the plane with the highest possible value. Linear programming can be applied to various fields of study. It is widely used in mathematics, and to a lesser extent in business, economics, and for some engineering problems. Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing. In 1939 a linear programming formulation of a problem that is equivalent to the general linear programming problem was given by the Soviet economist Leonid Kantorovich, who also proposed a method for solving it.

Dantzig independently developed general linear programming formulation to use for planning problems in US Air Force. Dantzig’s original example was to find the best assignment of 70 people to 70 jobs. Linear programming is a widely used field of optimization for several reasons. Many practical problems in operations research can be expressed as linear programming problems. Standard form is the usual and most intuitive form of describing a linear programming problem. Other forms, such as minimization problems, problems with constraints on alternative forms, as well as problems involving negative variables can always be rewritten into an equivalent problem in standard form. Suppose that a farmer has a piece of farm land, say L km2, to be planted with either wheat or barley or some combination of the two.

The farmer has a limited amount of fertilizer, F kilograms, and pesticide, P kilograms. Linear programming problems can be converted into an augmented form in order to apply the common form of the simplex algorithm. This form introduces non-negative slack variables to replace inequalities with equalities in the constraints. This section does not cite any sources. Every linear programming problem, referred to as a primal problem, can be converted into a dual problem, which provides an upper bound to the optimal value of the primal problem. There are two ideas fundamental to duality theory. A linear program can also be unbounded or infeasible.

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Duality theory tells us that if the primal is unbounded then the dual is infeasible by the weak duality theorem. Likewise, if the dual is unbounded, then the primal must be infeasible. However, it is possible for both the dual and the primal to be infeasible. Revisit the above example of the farmer who may grow wheat and barley with the set provision of some L land, F fertilizer and P pesticide. The primal problem deals with physical quantities. With all inputs available in limited quantities, and assuming the unit prices of all outputs is known, what quantities of outputs to produce so as to maximize total revenue? The dual problem deals with economic values.

With floor guarantees on all output unit prices, and assuming the available quantity of all inputs is known, what input unit pricing scheme to set so as to minimize total expenditure? To each variable in the primal space corresponds an inequality to satisfy in the dual space, both indexed by output type. To each inequality to satisfy in the primal space corresponds a variable in the dual space, both indexed by input type. The coefficients that bound the inequalities in the primal space are used to compute the objective in the dual space, input quantities in this example.

The coefficients used to compute the objective in the primal space bound the inequalities in the dual space, output unit prices in this example. Both the primal and the dual problems make use of the same matrix. In the primal space, this matrix expresses the consumption of physical quantities of inputs necessary to produce set quantities of outputs. In the dual space, it expresses the creation of the economic values associated with the outputs from set input unit prices. Since each inequality can be replaced by an equality and a slack variable, this means each primal variable corresponds to a dual slack variable, and each dual variable corresponds to a primal slack variable. This relation allows us to speak about complementary slackness.

Sometimes, one may find it more intuitive to obtain the dual program without looking at the program matrix. Since this is a minimization problem, we would like to obtain a dual program that is a lower bound of the primal. In other words, we would like the sum of all right hand side of the constraints to be the maximal under the condition that for each primal variable the sum of its coefficients do not exceed its coefficient in the linear function. Note that we assume in our calculations steps that the program is in standard form. However, any linear program may be transformed to standard form and it is therefore not a limiting factor.

0, such that the matrix A and the vectors b and c are non-negative. Covering and packing LPs commonly arise as a linear programming relaxation of a combinatorial problem and are important in the study of approximation algorithms. Finding a fractional coloring of a graph is another example of a covering LP. In this case, there is one constraint for each vertex of the graph and one variable for each independent set of the graph. It is possible to obtain an optimal solution to the dual when only an optimal solution to the primal is known using the complementary slackness theorem. So if the i-th slack variable of the primal is not zero, then the i-th variable of the dual is equal to zero. Likewise, if the j-th slack variable of the dual is not zero, then the j-th variable of the primal is equal to zero.