-1 programming is a special kind of integer programming whose decision variables only take the value of or 1. When dealing with some planning problems in economic management, if the decision variables are -1 variables, that is, logical variables, the problems that need to be discussed separately in various situations can be discussed in one question. Basic Introduction Chinese Name: -1 Planning mbth: zero-one? Programming essence: the application scope of a special kind of integer programming that only takes the value of or 1: solving mutually exclusive planning problems, etc. Also known as: introduction of binary variables, application, mutually exclusive planning problems, constraints, fixed cost problems, assignment problems, solving methods, zero-one integer programming, introduction -1 programming is a special form of integer programming. The decision variables of this kind of programming only take the value of or 1, so it is called -1 variable or binary variable, because a non-negative integer can be represented by several -1 variables in binary notation. -1 variables can quantitatively describe the logical relationship, sequence relationship and mutually exclusive constraints between discrete variables reflected by such phenomena as on and off, take and discard, existence and none, so -1 planning is very suitable for describing and solving many problems that people care about, such as circuit design, factory location, production planning, travel shopping, knapsack problem, personnel arrangement, code selection, reliability and so on. In fact, all integer programming with bounded variables can be transformed into -1 programming. Because of its profound background and wide application, -1 planning has been paid attention to by people for decades. -1 programming is mainly used to solve mutually exclusive planning problems, constraint mutually exclusive problems, fixed cost problems and assignment problems. The application of mutually exclusive planning issues such as determining investment projects, selecting investment sites, and deciding to put into production products. There are several products, the profit of each product after put into production is c j, the investment limit is B, and the value of decision variable xj is specified as Figure 1, then the mathematical model of this -1 programming is Figure 2, in which max represents the maximum value; S.t. stands for "bound by"; Z is the objective function; Aj is the investment amount of various products. There are m mutually exclusive constraints (≤ type) AII1x1+AII2x2+…+AIinXN ≤ BI (I = 1,2, …, m) in the constraint problem. In order to ensure that only one of these m constraints works, M -1 variables y i and a large enough constant m are introduced, Construct m +1 constraints AI1x1+AI2x2+…+AINXN ≤ BI+YIMYY1+Y2+…+YM = M-1 Because only one of M YIs can take the value of , only one constraint can work. If two kinds of goods are transported, the quantities are x 1 and x 2, respectively, the volume of the goods shall not exceed b 1 when transported by car, and the weight of the goods shall not exceed b 2 when transported by ship, that is, a 11 x 1+a 12 x 2 ≤ b 1 (transported by car) and a 21 x 1+a 22 x 2 ≤ b 2 (transported by ship). If only one mode of transportation can be used, these two constraints are mutually exclusive. In order to unify in a problem, reference -1 variable y i, Suppose Figure 4 and Figure 5 transform the above constraints into the following set of constraints: A11x1+A12x2 ≤ B1+Y1mA21x1+A22x2 ≤ B2+Y2MY1+Y2 = 2-1 where m is For a large enough number, y 1 = is used for vehicle transportation, and the constraint condition of vehicle transportation is obtained from the first formula, y 2 = is used for shipping, and the constraint condition of shipping is obtained from the second formula. Therefore, the above mutually exclusive constraints are replaced by a set of simultaneous constraints. The fixed cost problem cannot be solved by using general linear programming, so -1 programming is needed. There are n modes of production to choose from, x i is the output when the I mode is adopted, c i is the variable cost of each product when the I mode is adopted, and k i is the fixed cost when the I mode is adopted. The total costs of adopting various modes of production are (I = 1, 2, ..., n) Figure 6 When the objective function is formed, In order to unify the discussion in a problem, the -1 variable y i is introduced, that is, the mathematical model of this -1 programming is shown in Figure 7 and Figure 8, where min represents the minimum value and m is a sufficiently large constant. The assignment problem involves several people completing several tasks. However, due to the different nature and expertise of each person, which person should be assigned to complete which task, so as to maximize the overall efficiency or minimize the total time spent. This kind of problem is called assignment problem, also known as assignment problem. The assignment problem must give the coefficient matrix (also called the efficiency matrix), and the element of the matrix, Cij (>; )( i, j =1, 2, …, n) indicates the efficiency (or time, cost, etc.) when sending the i-th person to complete the j-th task. Citing -1 variable x ij, let the mathematical model of the assignment problem in Figure 9 be Figure 1 and Figure 11. The first constraint indicates that the J task can only be completed by one person, and the second constraint indicates that the I person can only complete one task. The solution of the assignment problem can be written in matrix form (x ij), and the sum of the elements in each row and column is 1. The method of solving -1 programming is mainly implicit enumeration method (such as branch and bound method). There are some more effective methods for some special problems, for example, the Hungarian method invented by D. Koenig is more convenient and effective to solve the assignment problem. There are generally three solutions to -1 programming problems, namely, transformation method, exhaustive method and implicit enumeration method. Transformation method is used to solve special -1 programming problems. Exhaustion is to check every combination of variable values of or 1 and compare the values of objective functions to find the optimal solution, which requires checking 2 n combinations of variable values. For n> 1, this is almost impossible. Therefore, some methods are often designed, and only a part of the combination of variable values can be checked to get the optimal solution of the problem. Such a method is called implicit enumeration. When solving the -1 programming problem by implicit enumeration method, a corresponding inequality should be added as an additional constraint condition according to the properties of the objective function, which is called a filter condition to reduce the number of operations. Generally, the order of x i in the objective function and constraint conditions should be rearranged according to the increasing order of the coefficient of x i in the objective function to simplify the calculation. Zero-one integer programming [Zero-one integer programming] -1 integer programming is the simplest kind of integer programming, that is, the vector knapsack problem with variables only taking or 1 and x being (1,) is a typical zero-one integer programming. Zero-one integer programming can be solved by branch and bound method, which can be briefly described as follows. Let n integer variables be -1, and remember that the original problem is that its relaxed linear programming is to remember that its optimal solution value is . The first step is to solve the relaxed linear programming of two subproblems. If the optimal solution value is, and both solutions are integer solutions, one of them is the optimal solution. If one of them is an integer solution and the corresponding optimal value is less than or equal to the optimal value of the other subproblem, then the integer solution is the optimal solution of the original problem. When none of the above conditions are satisfied, the second variable is decomposed for the subprogram whose optimal value is smaller or smaller than the existing integer solution value and whose solution is non-integer. The principle of the following steps is the same, and the specific algorithm often adopts different skills.