Linear Programming

In mathematics, the problems of linear programming (PL) are problems of optimization where the function objective and the Contrainte S are all linear. Nevertheless, the majority of the results presented here are also true if the objective is a monotonous Fonction increasing of each variable considered. The linear programming also indicates the manner of solving the problems of PL.

The linear programming is a central field of optimization, because the problems of PL are the easiest problems of optimization - all the constraints being linear there. Many real problems of Operations research can be expressed as a problem of PL. For this reason a great number of algorithms for the resolution of other problems of optimization are founded on the solution to problem linear.

The linear programming term supposes that the solutions to be found must be represented in real variables. If it is necessary to use discrete variables in the modeling of the problem, one then speaks about linear programming of integers (PLNE). It is important to know that the latter are definitely more difficult to be solved that the PL with continuous variables.

Example

Let us consider a farmer who has grounds, of surface equal to H hectares, in which it can plant corn and corn. The farmer has a quantity E of manure and I of insecticide. The corn requires a quantity E ₁ of manure per hectare and I ₁ of insecticide per hectare. The corresponding quantities for corn are noted E ₂ and I ₂.

That is to say P ₁ the selling price of corn and P ₂ that of corn. If one notes by X ₁ and X ₂ the number of hectares to be planted out of corn and corn, then the optimal number of hectares to be planted out of corn and corn can be expressed like a linear program:

Theory

From a geometrical point of view, the linear constraints form a convex Polyèdre . If the function objective is it also linear, all the local optima are also total optima; that remains true if it is monotonous increasing on each variable considered, the linear case representing only one particular case whose property is not used besides.

There are two cases where there does not exist optimal solution. First is when the constraints are contradicted mutually (for example $((X \ Ge 2) \ wedge (X \ the 1))$ ). In such a case, the polytope is empty and there is no optimal solution since there is no solution of the whole. The PL is then unfeasible .

The polyhedron can also be not-limited in the direction defined by the function objective (for example $x_1 + 3 x_2$ such as $x_1 \ Ge 0$ , $x_2 \ Ge 0$ , $x_1 + x_2 \ Ge 10$ ). In this case, there is no optimal solution since it is possible to build solutions satisfying the constraints with values arbitrarily high (or low) of the function objective.

Apart from these two cases (which is finally rare in the practical problems), the optimum is always reached at a top of the polytope. However, the optimum is not necessarily single: it is possible to have a whole of optimal solutions corresponding to an edge or a face of the polytope, even with the polytope in entirety.

Duality

All the linear programs can be written in the following form:

$\begin{array}{rrll}
\ max Z = & c^tx & & \ \
s.c & Ax & \ leq& B \ \
& X & \ geq&0
\end{array}$

Where C and X are vectors of size N, B a vector of size m, and has a matrix of size mXn. If one indicates this representation under the term of forme primale, one then indicates under the term of forme duale the following problem:

$\begin{array}{rrll}
\ min W = & b^ty & & \ \
s.c & A^ty & \ geq& C \ \
& there & \ geq&0
\end{array}$

Where has, B and C are the same ones and there a vector of size Mr. (Note: The exact details of the two representations vary much from one work to another)

The two problems are very strongly dependant. If one of them has an optimal solution, then the other too. Moreover, the two solutions have then the same value (w*=z*). If one of them not-is limited, the other does not have a solution.

In addition to its theoretical interest, the dual problem has very interesting economic applications. With each primal constraint corresponds a dual variable. The value of this variable in the optimal solution represents the marginal Coût associated with the primal constraint.

Algorithms of the linear programming

The Algorithme of the simplex makes it possible to solve the problems of PL by first of all building a realizable solution which is a top of the polytope then while moving according to the edges of the polytope to reach tops for which the ideal value is increasingly large, until reaching the optimum. Although this algorithm is effective in practice and that it is ensured to find the optimum, its behavior in the worst case can be bad. It is thus possible to build a PL for which the method of the simplex requires an exponential number of stages in the size of the problem. Thus, during several years, knowledge if the PL were a Np-complete problem or polynomial remained an open-ended question.

The first polynomial algorithm for the PL was proposed by Leonid Khachiyan in 1979. It is based on the Méthode of the ellipsoid in nonlinear optimization previously suggested by Naum Shor. This method is itself a generalization of the method of the ellipsoid in convex Optimization due to Arkadi Nemirovski (Prix John von Neumann 2003), and to D. Yudin.

However, the practical effectiveness of the algorithm of Khachiyan is disappointing: the algorithm of the simplex is practically increasingly more powerful. On the other hand, this result encouraged research in the methods of interior point. In opposition to the algorithm of the simplex which considers only the border of the polytope defined by the constraints, the methods of interior point evolve/move inside the polytope.

In 1984, NR. Karmarkar proposes the projective method. It is the first effective algorithm at the same time in theory and in practice. Its complexity in the worst case is polynomial and the experiments on the practical problems show that the method can reasonably be compared with the algorithm of the simplex. Since then, several methods of interior point were proposed and studied. One of the most famous methods is the predictive/corrective Méthode which functions very well in practice even if its theoretical study is still imperfect.

For the practical resolution of problems of ordinary PL, it is currently common to regard as equivalent them (good) codes based on the methods derived from the simplex or the interior point. Moreover, for the solution to problem of big size, a technique as the Génération of columns can appear extremely effective.

The solveurs based on the PL are used more and more for the optimization of various industrial problems such as the optimization of flows of transport or the production planning. However, the models of PL prove insufficient to represent many problems, the linear programming of integers (PLNE) then makes it possible to model a great number of additional problems, in particular the problems Np-complete S.

Linear programming of integers

A problème of linear programming entiers (PLNE) numbers is not a linear program in the direction where its field of realisability is not a polyhedron but a discrete whole of points. However, one can describe it as a PL to which one adds the additional constraint that certain variables can take only whole values. One distinguishes the mixed linear program with integer and continuous variables from the whole program with all his integer variables.

The PLNE is a Np-complete problem because of many Np-complete problems can be expressed like PLNE (for example to find stable in a graph $G= (V, E)$ amounts finding a vector $X \ in \ {0,1 \} ^E$ satisfactory $x_u +x_v \ the 1$ for any edge $UV \ in E$ ). Of course, the algorithms described above for the PL do not solve the problems of PLNE. Algorithmiquement thus, the resolution of a PLNE is differently more difficult that of a PL which however plays a crucial role as for their resolution, mainly for two reasons. Firstly, relieving continues of a PLNE (it is the PLNE without the constraints of integrity) is a PL which can be solved effectively and to provide a thus limits dual (in the not-realizable direction). The algorithms of resolution of PLNE, such as the algorithms by Separation and evaluation are based on this relieving continues to decrease to the maximum the enumeration of the solutions. Secondly, the Theorem Optimization/Separation of Grötschel, Lovasz and Schrijver makes it possible to solve in practice by the PL the whole problems which one knows a good description polyèdrale (i.e. which one can separate the time constraints polynomial). It is the principle of operation of the secant algorithms of Plans,

Applications

The linear programming is primarily applied to solve problems of optimization in the medium and long term (strategic and tactical problems, in the vocabulary of operations research). The scopes of application of these problems are very numerous as well in the nature of the tackled problems (Planification and controls production, Distribution in Réseau X) as in the sectors of industry: manufacturing industry, energy (Oil, Gas, electricity, Nuclear), Transport S (air, road and railway), Telecommunication S, forest industry, Finance.

Applications in oil

See also: Oil, Refining of oil, procurement, production and distribution Plan of oil

The technique of the linear programming is usually applied in oil industry. It is one of industries, if it is not the principal one which uses daily the PL (linear programming). It is the tool which makes it possible the refiner to make the optimal determination of production of a Raffinerie. With this intention, the program must take account of a certain number of constraints such as:

rough available, their outputs and qualities of the cuts,
specifications of the products to be manufactured,
limitations of outlets for certain products,
capacities of the units,
modes of adjustments of the installations,
storage capacities available.

The PL can also be used in other fields of the refining, for example:

calculations of the optimal composition of the mixtures of products (containing hydrocarbon, gas oils, fuels) by taking account of the specifications.
optimization in the use of the installations,
calculations of obtaining the best pre-heating of the crudes and the loads,
determination of best balance “steamer-electricity” of a refinery.

Apart from the refineries, one can use the PL in operations research for:

to build plans with average and short length/terms of an oil company,
to optimize the operation of a fleet of tankers and the installation of the products.

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