So the mathematical model of this 0- 1 programming is
Where max stands for seeking 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+…+ainxn ≤ bi (I =1,2, …, m). In order to ensure that only one of the m constraints works, m 0- 1 variable yi and a large enough variable are introduced.
ai 1x 1+ai2x 2+…+ain xn≤bi+yiM
y 1+y2+…+ym=m- 1
Because only one of m yi can take the value of 0, only one constraint can work.
If two kinds of goods are transported, the quantities are x 1 and x2 respectively, the volume of goods transported by car shall not exceed b 1, and the weight of goods transported by ship shall not exceed b2, that is,
A11x1+a12x2 ≤ b1(traffic),
A2 1x 1+a22x2≤b2 (delivery).
If only one mode of transportation can be used, these two constraints are mutually exclusive. In order to unify on one issue, refer to 0- 1 variable Yi, let
Convert the above constraints into the following constraint set:
a 1 1x 1+a 12 x2≤b 1+y 1M
a2 1x 1+a22x2≤b2+y2M
y 1+y2=2- 1
Where m is a large enough number, the constraint condition of vehicle transportation can be obtained by formula y 1=0 when vehicle transportation is adopted, and it can be obtained by formula 2 when sea transportation is adopted. Therefore, the above mutually exclusive constraints are replaced by a set of simultaneous constraints. General linear programming can't solve the problem of fixed cost, so 0- 1 programming is needed. There are n production modes to choose from, xi is the output when I mode is adopted, ci is the variable cost of each product when I mode is adopted, and ki is the fixed cost when I mode is adopted. The total costs of adopting various production modes are as follows
(i= 1，2，…，n)
When constructing the objective function, the variable yi of 0- 1 is introduced for unified discussion, namely
So the mathematical model of this 0- 1 programming is
Where min means finding the minimum value and m is a constant large enough. How many people complete several tasks, but because of the different nature and specialty of tasks, 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 a coefficient matrix (also called efficiency matrix), and the elements of the matrix 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. With reference to 0- 1 variable xij, set
The mathematical model of assignment problem is
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 (xij), and the sum of elements in each row and column is 1.