Integer programming model
Basic knowledge of integer programming model
Linear programming model of decision variables value can be any non-negative real number, but modeling many practical problems, only when the value of decision variables is only meaningful integer . For example, the number of number of pieces, machine products, the number of loading vehicles, such as the number of work done, fractions or decimals solution is clearly unreasonable. Require all or part of the decision variable values for the integer linear programming model, called integer programming model .
The value of all decision variables are integer integer programming model, called pure integer programming model ; the value is only required part of the decision variables as integer integer programming model, known as mixed integer programming model ; requires decision variable can take only 0 or 1 integer programming model values, is called 0-1 programming model .
**Pure integer programming model** The general form is:
0-1 Programming ModelThe general form:
integer programming model is much more difficult to solve than solving linear programming model, integer programming model to solve the difficulties is that when the number of dimensions (the number of decision variables and constraints) model increases, the calculation of the amount of the explosive (ie, exponentially) increases .
Here are some **About integer programming model commonly used method for solving**：
(1) enumeration method and the implicit enumeration : commonly used in the 0-1 programming model, but it is not feasible when the high dimensionality of the model.
(2) the branch and bound method : pure integer programming model for solving mixed integer programming model and apply more feasible.
(3) Cutting Plane Method : pure integer programming model for solving mixed integer programming model and apply more feasible.
More detailed algorithm for solving integer programming model, see Tsinghua made "operational research" materials. In the actual modeling, for integer programming model can be solved by means of LINGO, MATLAB and other mathematical software.
Assignment Problem
Question: has a task to send someone to complete, but because of the nature and mission of personal expertise, the efficiency of each person to complete different tasks (or elapsed time) is different. What should I ask people to send to accomplish which tasks, the overall efficiency of the highest (or least total time spent)? Such problem is called assignment problem.
ANSWER: The set represents the people of the time required to complete the first task, representing the total time used. The decision variables are set as follows:
Due to the requirements of the problem, some people have to go to each task is completed, that there
has to be dispatched per capita task, namely
to the total time used 
as the objective function, assignment problem boils down to the following planning 0-1 model:
for solving the assignment problem, Hungarian algorithm has been more mature (see Tsinghua made "operational research" materials). In the actual modeling, the available LINGO, MATLAB and other mathematical software to solve it.
Integer Programming Modeling Examples
Two loaded flatbed problem
has seven specifications of the box to be installed up to two railroad flat cars. The width and height of the box is the same, but the thickness (t, in cm) and weight (w, in kg) are different. Table 6-5 shows the thickness of each box, and the number of weight. Each section has a 10.2m long flatbed place can be used to install the package (like bread that), load of 40 tons. Because of local cargo, for C5, C6, C7 total number of classes is a package particularly limited: the space occupied by this kind of boxes (thickness) should not exceed 302.7cm. Try to design a loading scheme, such that a minimum wasted space.
1. The model assumes that
(1) all kinds of packaging box is the same width and height, and width to meet loading, high demands, the number of crates flatbed can hold only on the thickness and weight of the packing box;
(2 ) loading each package can not be split;
(3) each package may be distributed in different flatbed;
(4) different containers can be placed on the same flatbed truck, and can ensure that no space left.
2. model
(1) decision variables
is provided xij（i=1,2,...,7; j=1,2）showing the first jsection of the vehicle load iquantity kind of the box to xijdecision variables, xijonly take a nonnegative integer value.
(2) constraints
to nirepresent the first inumber of pieces of the kind of box available loading, packing number two vehicles can not exceed the number of available loading member, so are:
to ljrepresent the first jsection of the vehicle available packing length, tirepresents the ikind of packaging the thickness of the box, each section of the vehicle can not be longer than the length of the loaded vehicle can provide, so it is:
in wirepresents the ikind of package weight, Wjrepresents the jsection of the vehicle load, each section can be loaded vehicle weight vehicle can not exceed bear the load, so it is:
for the C5, C6, C7 particularly limited, the total number of classes is denoted by box S (S = 302.7cm), it has:
(3) the objective function
minimum wasted space, i.e. loaded crates the maximum total thickness. The total thickness of loaded crates as the objective function, the total thickness of loaded crates expression:
The above analysis, mathematical model of two flatbed truck loading problems integer programming model is as follows:
3. model solution
using LINGO software to get the optimal solution:
the optimal target value 
, namely Click optimal loading plan two flatbed total free space 0.6 cm.
Further, the model is fully applicable to the different load and two flatbed case available loading space.
Connect two articles:
mathematical programming model (a)
mathematical programming model (B): linear programming model
After the update later a planning model of mathematical programming models. . . Stay tuned. . . .
