高等作業研究 ( 一 )

1. 高等作業研究(一) Chapter 3 Linear Programming Methods

2. Whereas the simplex method searches for the optimal solution at the boundary of the feasible region, the latest developments are based on the idea of following a path through the interior of the feasible region until the optimum is reached. These so-called interior point methods are of great theoretical importance because they provide a bound on the computational effort required to solve a problem that is a polynomial function of its size. No polynomial bound is available for the simplex algorithm. Nevertheless, simplex codes have proven to be highly efficient in practice and remain at the center of virtually all commercial optimization packages. Introduction

3. The standard form of the model, sometimes referred to as the canonical form, is written as Standard Form of LP (1) (2) (3) Max Z= s.t. Where all bi are nonnegative.

4. Theory：If a finite optimal solution exists, there is an optimal solution at one of the extreme points or vertices of the feasible region. Definition 1: Let P be a polyhedron in n-dimensional space, written as . A vector x is an extreme point of P if we cannot find two vectors y, , both different from x, and a scalar such that x =λy + (1 -λ)z.

6. In other words,x is an extreme point if it doesn't lie on the line between any two points in P. Definition 2: Let a be a nonzero row vector in n-dimensional space, written as , and let b be a scalar. a. The set {x : ax = b} is called a hyperplane. b. The set {x : ax≧b} is called a half space.

7. Note that a hyperplane is the boundary of a corresponding halfspace. For a polyhedron in n-dimensional space, an extreme point is the intersection of n non-coplanar hyperplanes. (If more than n hyperplanes are intersected, the extreme point is degenerated.) For the linear programming problem defined by Equations (1) to (3), some of these hyperplanes will be of the form = 0.

8. The objective must be to maximize. The objective function must be linear in the variables and must not contain any constant terms. All variables must be restricted to be nonnegative. Each constraint must be written as a linear equation with the variables on the left of the equal sign and a positive constant on the right. PREPARING THE MODEL

9. Minimize { } <=> Maximize {- } <=> <=> For unrestricted variable , where 0 and

10. When more than one variable is unrestricted, where and

11. Example (Transformation)

12. Letand

14. Maximize Subject to

15. When solving two-dimensional problems graphically, we saw that solutions were always found at extreme points of the feasible region. These solutions are called basic solutions. In a problem with n variables and m constraints, a basic solution is determined by identifying m variables as basic, setting the remaining n - m (nonbasic) variables equal to zero, and solving the resultant set of simultaneous equations. GEOMETRIC PROPERTIES OF LPs

16. In order for these equations to have a unique solution and hence correspond to an extreme point, care must be taken in choosing which variables to make basic.

17. Let us consider a system of m linear equations in m unknowns, written as Ax=b. Linear Independence

18. For this system to have a unique solution, the matrix A must be invertible or nonsingular— that is, there must exist another m × m matrix B such that AB = BA = I, where I is the m x m identity matrix. Such a matrix B is called the inverse of A and is unique. It is denoted by .

19. Definition 3: Let be a collection of k column vectors, each of dimension m. We say that these vectors are linearly independent (LI)if it is not possible to find k real numbers not all zero such that , where 0 is the m-dimensional null vector; otherwise, they are called linearly dependent (LD).

20. Theorem 1: Let A be a square matrix. The following statements are equivalent. The determinant of A is nonzero. The matrix A is invertible, as is its transpose . The rows and columns of A are linearly independent. For every vector b, the linear system Ax = b has a unique solution.

21. Consider a system of equations Ax = b, where x is an n-dimensional vector, b is an m-dimensional vector, and A is an m × n matrix. Suppose that from the n columns of A we select a set of m linearly independent columns (such a set exists if the rank of A is m). Basic Solution

22. Definition 4: Given a set of m simultaneous linear equations in n unknowns, let B be any nonsingularm×m matrix made up of columns of A. If all the n - m components of x not associated with columns of B are set equal to zero, the solution to the resulting set of equations is said to be a basic solution with respect to the basis B. The components of x associated with columns of B are called basic variables.

23. Given the assumption that A has full row rank, the system Ax = b always has a solution and, in fact, will always have at least one basic solution; however, the basic variables in a solution are not necessarily all nonzero. Definition 5: A degenerate basic solution is said to occur if one or more of the basic variables in a basic solution have values of zero.

24. Definition 6: A vector x S= {x : Ax = b, x≧0} is said to befeasible to the linear programming problem in standard form; a feasible solution that is also basic is said to be a basic feasible solution (BFS).If this solution is degenerate, it is called a degenerate basic feasible solution.

25. Theorem 2: Given a linear program in standard form, where A is an m × n matrix of rank m, a. if there is a feasible solution, there is a BFS b. if there is an optimal feasible solution, there is an optimal BFS. In general, if there are m constraints and n variables, the number of basic solutions will be bounded by the number of ways to choose m basic variables (or, equivalently, the n -m nonbasic variables) from the n variables—i.e., Foundations

26. Maximize s.t. Example

28. The upper bound on the number of basic solutions is computed to be

29. Theorem 3: For any linear program, there is a unique extreme point in the feasible region S= {x : Ax = b, x≧0} corresponding to a BFS. In addition, there is at least one BFS corresponding to each extreme point in S.

30. When more than one BFS maps into a specific extreme point, at least one of the binding constraints at that extreme point is redundant. It could be either one of the structural constraints or one of the nonnegativity restrictions. In either case, the BFSs would all be degenerate.

31. Definition 7: For a linear program in standard form with m constraints, two basic feasible solutions are said to be adjacent if they have m - 1 basic variables in common. For example, referring to Figure 3.3 we see that points #2 and #8 are adjacent. Given that m = 3, these points should have 3-1=2 basic variables in common to be adjacent according to Definition 7. (which ones?)

32. => If x is better than all its adjacent BFS’s, then x is an optimal BFS. (why?) => Convexity of Linear programming.

33. Step 1: Find an initial BFS to the linear program. As the algorithm progresses, the most recent BFS will be called the incumbent. Step 2: Determine if the incumbent is optimal. If not, move to an adjacent BFS that has a larger z value. Step 3: Return to Step 2, using the new BFS as the incumbent. Simplex Algorithm

34. Although the number of BFSs could be huge, the simplex algorithm is seen in practice to be highly efficient. Vanderbei [1996] has empirically demonstrated that in the absence of degeneracy, approximately 0.5(m + n) BFSs have to be examined before optimality is confirmed (when degeneracy is present, this number doubles)

35. Maximize s.t. Simplex Tableau

36. Move all variables to the left hand side of “ = “, and all constants to the right hand side of “ = “.

40. The marginal effects of increasing can be described by the following partial derivatives,

41. In general terms, the equations are written as where Q is the set of nonbasic variables.

42. From the general expression of the simplex form we read the objective function value and the values of the basic variables as What is the Basic Solution?

43. In general, a basic solution is feasible if Is the basic solution feasible?

44. Optimality condition: Why? Increasing the value of any nonbasic variable (from its current value 0) can not improve the current objective function value. Is the basic feasible solution optimal?

45. The current solution can be improved by increasing any nonbasic variable with a negative coefficient in Equation E0. In the general case, all nonbasic variables with < 0 are candidates for conversion to basic variables. How Can a Nonoptimal Feasible Solution Be Improved?

46. When > 0, a positive value of will drive the corresponding basic variable to zero. When more than one row has this characteristic, the nonbasic variable can increase only to the value of the minimum positive ratio if feasibility is to be guaranteed. Thus the nonbasic variable may increase by How Much Should the Nonbasic Variable Be Changed? (min. ratio test)

48. The process of obtaining the new tableau is called pivotingorpricing out the entering column. It is simply a series of linear operations on the equations that results in the simplex form of the new set of basic variables. How Is a New Simplex Form Obtained?

49. Pivoting Suppose the rth row is selected by the min. ratio test. Then,

50. Before Pivoting