Basic introduction Chinese name: 0- 1 planning mbth: zero-one? Programming essence: A special integer programming with only 0 or 1. Application scope: solving mutually exclusive programming problems, etc. Also known as: introduction of binary variables, application, mutually exclusive programming problem, constraint, fixed cost problem, allocation problem, solution method, zero-one integer programming, introduction of 0- 1 programming is a special form of integer programming. This programming decision variable only takes the value of 0 or 1, so it is called 0- 1 variable or binary variable, because a non-negative integer can be expressed as several 0- 1 variables by binary notation. 0- 1 variable can quantitatively describe the logical relationship, sequence relationship and mutual exclusion constraints between discrete variables reflected by on-off phenomenon, take-off phenomenon, existence phenomenon and non-existence phenomenon, so 0- 1 programming is very suitable for describing and solving many problems that people care about, such as line design, factory site selection, production planning, travel shopping, knapsack problem, personnel arrangement, code selection and reliability. In fact, all integer programming with bounded variables can be transformed into 0- 1 programming. 0- 1 planning has been concerned by people for decades because of its profound background and wide application. 0- 1 programming is mainly used to solve mutually exclusive programming problems, constrained mutually exclusive problems, fixed cost problems and assignment problems. The application of mutually exclusive planning problems 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 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 0- 1 programming is Figure 2 and Figure 3, where max represents the maximum value; S.T. stands for "bound"; Z is the objective function; Aj is the investment amount of various products. There are m mutually exclusive constraints (≤ type) ai1x1+ai2x2+…+a in x n ≤ B I (I = 1, 2, …, m). In order to ensure that only one of the m constraints works, m variables y i from 0 to 1 and a large enough constant m are introduced. Construct m+1 constraint: ai1x1+ai2x2+…+a in x n ≤ b i+Y I m y1+y 2+…+y m = m-1because only one Y I can take a value. Transport two kinds of goods, the quantities are x 1 and x 2, respectively. The volume of goods transported by car shall not exceed b 1, and the weight of goods transported by ship shall not exceed b 2. That is, a11x1+a12x2 ≤ b1(vehicle transportation), a 2 1 x 1+a 22 x 2 ≤ b 2 (sea transportation). If only one mode of transportation can be used, these two constraints are mutually exclusive. In order to unify in a problem, the variable y i of 0- 1 is quoted. Suppose that the above constraints are transformed into the following constraint sets in Figures 4 and 5: a11x12x2 ≤ b1+y1m 21x/kloc-0. 2 ≤ b 2+y 2m y 1+y 2 = 2-1,where m is a large enough number, the constraint conditions of vehicle transportation can be obtained from formula1when vehicle transportation is adopted, and from formula 2 when vehicle transportation is adopted. Therefore, the above mutually exclusive constraints are replaced by a set of simultaneous constraints. The problem of fixed cost cannot be solved by general linear programming, so 0- 1 programming is needed. There are n production modes to choose from, x i is the output when I mode is adopted, c i is the variable cost of each product when I mode is adopted, and k i is the fixed cost when I mode is adopted. The total cost of each mode of production is (I = 1, 2, ..., n). In order to discuss the objective function in Figure 6, the variable y i of 0- 1 is introduced, that is, the mathematical model of 0- 1 programming is min in Figures 7 and 8, and m is a large enough constant. The assignment problem involves several people completing several tasks, but because the nature of the task and everyone's expertise are different, which person should be assigned to complete which task in order 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 cij (>; 0)( i, j = 1, 2, ..., n) indicates efficiency (or time, cost, etc.). ) When sending the i-th person to finish the j-th task. Citing the variable x ij of 0- 1, let the mathematical model of assignment problem in Figure 9 be Figure 10 Figure 1 1. The 1 constraint means that J task can only be completed by 1 person, and the second constraint means that I person can only complete 1 task. The solution of assignment problem can be written in matrix form (x ij), and the sum of elements in each row and column is 1. Solution The main method to solve 0- 1 programming is implicit enumeration method (such as branch and bound method). There are some more effective methods for some special problems, such as the Hungarian law invented by D. Koenig, which is more convenient and effective to solve the assignment problem. There are generally three solutions to the 0- 1 programming problem, namely transformation method, exhaustive method and implicit enumeration method. Using transformation method to solve special 0- 1 programming problem. The exhaustive method is to check every combination of variable values of 0 or 1, compare the objective function values, and find the optimal solution. It is necessary to check 2 n variable combinations. For n> 10, which 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. This method is called implicit enumeration. When solving the 0- 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. In general, the order of x i in objective function and constraint conditions should be rearranged according to the increasing order of x i coefficients in objective function to simplify calculation. Zero-One Integer Programming [Zero-One Integer Programming] 0- 1 Integer Programming is the simplest kind of integer programming, that is, the vector knapsack problem with variables of only 0 or 1 and x as (1, 0) is a typical zero-one integer programming. Zero-one integer programming can be solved by branch-and-bound method, which is briefly described as follows. Let n integer variables be 0- 1, the original problem is that its relaxed linear programming is to remember its optimal solution value as. 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, then 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 this integer solution is the optimal solution of the original problem. When the above conditions are not met, the second variable is decomposed for the subprogram whose optimal value is less than or less than the existing integer solution value and whose solution is non-integer. The principle of the following steps is the same, and the specific algorithms often use different skills.