Mathematical programming model (a)

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First, the basic knowledge of mathematical programming models

1, generally in the form of mathematical programming models

Simple optimization models tend to be monohydric or polyhydric, unconstrained maximum value equality constraints or problems. In the engineering, economic and financial management, scientific research and many other areas of daily life, people often encounter the following problems: structural design to choose the size of the material under the conditions meet the strength requirements, making it the lightest total weight; resource allocation to be developed under the constraints of limited resources assigned amount for each user, so that the total effective resources to produce the maximum; production plans in accordance with product and process customer demand, the development of raw materials, spare parts and other orders, production schedules and quantity, to minimize the cost of the highest profits. The essence of the above series of questions is: In a series of objective or subjective constraints, one or more indicators seek the concern of the maximum (or minimum).Conduct research on such issues by the method of mathematical modeling, resulting in a series of equality and inequality constraints, so one or more target function reaches a maximum (or minimum) mathematical model, namely mathematical programming models.
Mathematical programming models generally need to consider the following three elements:
(1) decision variables: it is usually studied problems require solutions for those unknowns, usually with n-dimensional vector , where xj j-th issue of the decision variables. When the assignment of X which is usually called a solution to this problem.
(2) the objective function: it is usually the research questions required to achieve maximum (or minimum) of the mathematical expression (those) index, which is a function of the decision variables, denoted by f (X).
(3) constraints: a study on the issue of the decision variables X constraints given, X allowed range of values referred to as D, i.e. X∈D, D is called the feasible region. D a set of equations used on the decision variable X hi (X) = 0 (i = 1,2, ..., p) and inequalities gi (X) ≤ (≥) 0 (j = p + 1, p + 2, ..., m) defined, respectively known as equality constraints and inequality constraints.
Mathematical programming model can be expressed by the following general form:

where the max (min) is seeking the maximum or minimum of the objective function f (X) means, st is "bound to" means.
Since the total equality constraints can be converted to inequality constraints, greater than or equal Constrained can be converted to less constraint, then the general form of a mathematical programming model is simplified in turn:

2, feasible solutions and the optimal solution of a mathematical programming model

The general form of the mathematical programming model, the feasible region can be expressed as:

to meet the constraints of the solution to row Domain D feasible solution point is called mathematical programming model; the objective function f (X) reaches a maximum or minimum viable Solutions can row manipulation target domain D function f (X) reaches a maximum or minimum point is called the optimal solution mathematical programming model.
Mathematical programming model **Solving natureIs to select feasible domain D in the objective function optimal point for solving mathematical programming model is a lot of research and solution methods described in operations research, but overall for solving mathematical programming model is more complex and cumbersome, solving some problems very difficult. According to mathematical programming model practical problems created, its structure is often very complex and large amount of data, you can not use conventional methods to solve the current mathematical software has been developed more mature,MATLAB software can help solve or LINGO**。

3 basic types of mathematical programming models

Classification of mathematical programming models more, there will be a mathematical programming model is divided in the following manner:
(1) a linear programming model: the objective function and constraints are linear functions of mathematical programming models;
(2) integer programming model: decision variables required linear programming model takes an integer value; and
(3) non-linear programming model: objective function or constraint condition mathematical programming model nonlinear function; and
(4) multiple objective planning model: having a plurality of mathematical programming model objective function.
It focuses on the above four categories of mathematical programming models. The others are dynamic programming models can be found in Tsinghua made "operational research" teaching .

The next time will be updated linear programming model, integer programming model, nonlinear programming models, like to see the contents of the follow-up, see the next time to explain! ! ! Stay tuned. . . . .

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