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  • Linear Programming Theory and Applications PDF 下载

    Linear Programming Theory and Applications PDF 下载

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    1 Introduction to Linear Programming
    Linear programming was developed during World War II, when a system with
    which to maximize the efficiency of resources was of utmost importance. New
    war-related projects demanded attention and spread resources thin. “Programming” was a military term that referred to activities such as planning schedules
    efficiently or deploying men optimally. George Dantzig, a member of the U.S.
    Air Force, developed the Simplex method of optimization in 1947 in order to
    provide an efficient algorithm for solving programming problems that had linear
    structures. Since then, experts from a variety of fields, especially mathematics
    and economics, have developed the theory behind “linear programming” and
    explored its applications [1].
    This paper will cover the main concepts in linear programming, including
    examples when appropriate. First, in Section 1 we will explore simple properties, basic definitions and theories of linear programs. In order to illustrate
    some applications of linear programming, we will explain simplified “real-world”
    examples in Section 2. Section 3 presents more definitions, concluding with the
    statement of the General Representation Theorem (GRT). In Section 4, we explore an outline of the proof of the GRT and in Section 5 we work through a
    few examples related to the GRT.
    After learning the theory behind linear programs, we will focus methods
    of solving them. Section 6 introduces concepts necessary for introducing the
    Simplex algorithm, which we explain in Section 7. In Section 8, we explore
    the Simplex further and learn how to deal with no initial basis in the Simplex
    tableau. Next, Section 9 discusses cycling in Simplex tableaux and ways to
    counter this phenomenon. We present an overview of sensitivity analysis in
    Section 10. Finally, we put all of these concepts together in an extensive case
    study in Section 11.
    1.1 What is a linear program?
    We can reduce the structure that characterizes linear programming problems
    (perhaps after several manipulations) into the following form:
    3
    Minimize c1x1 + c2x2 + · · · + cnxn = z
    Subject to a11x1 + a12x2 + · · · + a1nxn = b1 a21x1 + a22x2 + · · · + a2nxn = b2 … … … … … am1x1 + am2x2 + · · · + amnxn = bm x1, x2, . . . , xn ≥ 0.
    In linear programming z, the expression being optimized, is called the objective function. The variables x1, x2 . . . xn are called decision variables, and their
    values are subject to m + 1 constraints (every line ending with a bi, plus the
    nonnegativity constraint). A set of x1, x2 . . . xn satisfying all the constraints is
    called a feasible point and the set of all such points is called the feasible region. The solution of the linear program must be a point (x1, x2, . . . , xn) in the
    feasible region, or else not all the constraints would be satisfied.
    The following example from Chapter 3 of Winston [3] illustrates that geometrically interpreting the feasible region is a useful tool for solving linear
    programming problems with two decision variables. The linear program is:
    Minimize 4×1 + x2 = z
    Subject to 3×1 + x2 ≥ 10
    x1 + x2 ≥ 5 x1 ≥ 3 x1, x2 ≥ 0. We plotted the system of inequalities as the shaded region in Figure 1. Since
    all of the constraints are “greater than or equal to” constraints, the shaded
    region above all three lines is the feasible region. The solution to this linear
    program must lie within the shaded region

     

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