network flows and linear programming

... Drawing the Linear Model Standard Graph Number of Y units 2500 2000 1500 1000 500 0 500 1000 1500 Number of X units 2000 2500 Adding the Linear Constraints Standard Graph: Constraints ... constraint levels Linear Programming: Summary       Use when an optimum solution is required, from constrained resources Express the objective function and the constraints as linear equations ... manually applied  As the Simplex method is tedious, and prone to error, the solution is best found with computer software such as Excel Solver  The standard Excel spreadsheet needs to specially adapted...

Ngày tải lên: 31/10/2014, 14:56

Ngày tải lên: 25/03/2015, 14:04

... scale, multiple goals and investment risk Explanations of the ‘Extension’ Ideas I    More Activities and Constraints: this notion deals with more complex resource mixes, and more constraints, ... mix; or an overall risk measure such as ‘variance could be targeted and minimized 15 Advanced LP Applications: Summary Linear programming can be used to solve selection problems from amongst competing ... level of analysis requires imaginative definitions of both goals and constraints, and an appreciation of Linear Programming methodology 16 ...

Ngày tải lên: 23/03/2014, 04:20

... major parts: Linear Programming, Unconstrained Problems, and Constrained Problems The last two parts together comprise the subject of nonlinear programming Linear Programming Linear programming ... A linear programming problem is characterized, as the name implies, by linear functions of the unknowns; the objective is linear in the unknowns, and the constraints are linear equalities or linear ... PROGRAMS INTRODUCTION A linear program (LP) is an optimization problem in which the objective function is linear in the unknowns and the constraints consist of linear equalities and linear inequalities...

Ngày tải lên: 27/06/2014, 17:20

... summarize the analogies between linear algebra and linear programming in tabular form: Linear algebra Linear programming Basic problem system of linear equations system of linear inequalities Algorithm ... applications of linear programming from various fields, each with motivation and some background material Chapter contains remarks on software available for linear programming and on the literature Linear ... Linear Programming and Linear Algebra The basics of linear algebra can be regarded as a theory of systems of linear equations Linear algebra considers many other things as well, but systems of linear...

Ngày tải lên: 03/07/2014, 21:10

... major parts: Linear Programming, Unconstrained Problems, and Constrained Problems The last two parts together comprise the subject of nonlinear programming Linear Programming Linear programming ... A linear programming problem is characterized, as the name implies, by linear functions of the unknowns; the objective is linear in the unknowns, and the constraints are linear equalities or linear ... PROGRAMS INTRODUCTION A linear program (LP) is an optimization problem in which the objective function is linear in the unknowns and the constraints consist of linear equalities and linear inequalities...

Ngày tải lên: 06/08/2014, 15:20

... solution Fundamental theorem of linear programming Given a linear program in standard form (11) where A is an m × n matrix of rank m, 2.4 The Fundamental Theorem of Linear Programming 21 i) if there ... FUNDAMENTAL THEOREM OF LINEAR PROGRAMMING In this section, through the fundamental theorem of linear programming, we establish the primary importance of basic feasible solutions in solving linear programs ... understanding of the result The main link between the algebraic and geometric theories is the formal relation between basic feasible solutions of linear inequalities in standard form and extreme...

Ngày tải lên: 06/08/2014, 15:20

... linear programs, however, and it is necessary to develop a means for determining one so that the simplex method can be initiated Interestingly (and fortunately), an auxiliary linear program and ... initially, a last row consisting of the ci ’s and a right-hand side of zero can be appended to the standard array to represent this additional equation Using standard pivot operations, the elements in ... matrix formulation is also a natural setting for the discussion of dual linear programs and other topics related to linear programming A preliminary observation in the development is that the tableau...

Ngày tải lên: 06/08/2014, 15:20

... artificial basis technique; Dantzig, Orden and Wolfe [D8], Orchard-Hays [O1], and Dantzig [D4] for the revised simplex method; and Charnes and Lemke [C3] and Dantzig [D5] for upper bounds The synthetic ... several fixed origins and destinations so as to minimize transportation cost while satisfying demand Referring to (6) and (7) of Chapter 2, the problem is in standard form, and hence the asymmetric ... Theorem of linear programming states that the converse is also true, and that, in fact, the two regions in Fig 4.1 actually have a common point; there is no “gap.” Duality Theorem of Linear Programming...

Ngày tải lên: 06/08/2014, 15:20

... 4.7 REDUCTION OF LINEAR INEQUALITIES Linear programming is in part the study of linear inequalities, and each progressive stage of linear programming theory adds to our understanding of this important ... 12 and xi i = subject to 17 Given the linear programming problem in standard form (3) suppose a basis B and the corresponding (not necessarily feasible) primal and dual basic solutions x and ... understanding of how the modern theory of linear programming evolved, and help make clear how complexity theory impacts linear programming However, the reader may wish to consider them optional and...

Ngày tải lên: 06/08/2014, 15:20

... chapters Not only have nonlinear methods improved linear programming, but interiorpoint methods for linear programming have been extended to provide new approaches to nonlinear programming This chapter ... to show how this merger of linear and nonlinear programming produces elegant and effective methods These ideas take an especially pleasing form when applied to linear programming Study of them ... for linear programming is to use nonlinear programming techniques of analysis and methodology The analysis is often based on differentiation of the functions defining the problem Traditional linear...

Ngày tải lên: 06/08/2014, 15:20

... [T11], and others The homogeneous and self-dual embedding method can be found in Ye et al [Y2], Luo et al [L18], Andersen and Ye [A5], and many others It is also implemented in most linear programming ... interior-point linear programming algorithms They include Bazaraa, Jarvis and Sherali [B6], Bertsekas [B12], Bertsimas and Tsitsiklis [B13], Cottle [C6], Cottle, Pang and Stone [C7], Dantzig and Thapa ... mathematicians, and this method was later generalized to form the primal-dual method for linear programming 6.6 BASIC NETWORK CONCEPTS We now begin a study of an entirely different topic in linear programming: ...

Ngày tải lên: 06/08/2014, 15:20

... −20 178 Chapter Transportation and Network Flow Problems ⎡ ⎢ ⎢ C=⎢ ⎢ ⎣ 3 5 5 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ In this system points and are net suppliers, points and are net demanders, and point is neither Any of the ... are based on representing the network structure more compactly and assigning flows to arcs and simplex multipliers to nodes 6.8 MAXIMAL FLOW A different type of network problem, discussed in ... nondegenerate solution Network flow problems represent another important class of linear programming problems The transportation problem can be generalized to a minimum cost flow problem in a network This...

Ngày tải lên: 06/08/2014, 15:20

... define corresponding notions of average linear and average superlinear convergence Although the above array of definitions can be further embellished and expanded, it is quite adequate for our purposes ... 13 Let rk k=0 and ck k=0 be sequences of real numbers Suppose rk → average linearly and that there are constants c > and C such that c ck C for all k Show that ck rk → average linearly 14 Prove ... solution set and let A be an algorithm on X A continuous real-valued function Z on X is said to be a descent function for and A if it satisfies i) if x ii) if x ∈ and y ∈ A x , then Z y < Z x and y...

Ngày tải lên: 06/08/2014, 15:20

... xk − f ∗ Clearly a/ A < and hence there is linear convergence Notice if that in fact is chosen very close to and is chosen very close to 1, then the stopping condition demands that the be restricted ... point x4 is calculated from the quadratic fit in the standard way and f x4 is measured Assuming (as in the figure) that x2 < x4 < x3 , and accounting for the unimodal nature of f , there are ... be replaced by f x4 , and by construction, and the assumption that f is unimodal, it will replace a strictly larger value Of course, at x∗ = x∗ x∗ x∗ we have A x∗ = x∗ and hence Z A x∗ = Z x∗...

Ngày tải lên: 06/08/2014, 15:20

... practice Newton’s Method and Logarithms Interior point methods of linear and nonlinear programming use barrier functions, which usually are based on the logarithm For linear programming especially, ... Boyd and Vandenberghe [B23] The numerical problem considered in the example is a standard one See Faddeev and Faddeeva [F1] 8.8 For good reviews of modern Newton methods, see Fletcher [F9] and ... = xk − k gk where 1− and k k k 1+ k minimizes f xk − gk over a) Find the convergence rate of the algorithm in terms of a and A, the smallest and largest eigenvalues of Q, and the tolerance Hint:...

Ngày tải lên: 06/08/2014, 15:20

... standard steepest descent step from xk , and then xk+1 is taken to be the minimum point on the line connecting xk−1 and yk The process is continued for n steps and then restarted with a standard ... Daniel [D1] and Faddeev and Faddeeva [F1] 9.5 The partial conjugate gradient method presented here is identical to the so-called s-step gradient method See Faddeev and Faddeeva [F1] and Forsythe ... Faddeev and Faddeeva [F1], and Luenberger [L8] The result stated in Exercise is due to Hestenes and Stiefel [H10] Exercise is due to Powell [P6] For the solution to Exercise 10, see Faddeev and Faddeeva...

Ngày tải lên: 06/08/2014, 15:20

... DESCENT AND NEWTON’S METHOD In this section we digress from the study of quasi-Newton methods, and again expand our collection of basic principles We present a combination of steepest descent and ... formula, and for this reason it is now generally preferred It can be noted that both the DFP and the BFGS updates have symmetric rank two corrections that are constructed from the vectors pk and Hk ... the solution, and hence convergence will be superlinear Unfortunately, while the methods are certainly excellent, their convergence characteristics require more careful analysis, and this will...

Ngày tải lên: 06/08/2014, 15:20

... examples of nonlinear programming in industrial operations and business decision making Many of these are nonlinear versions of the kinds of examples that were discussed in the linear programming ... nonlinear programming problems of the form minimize subject to fx h1 x = h2 x = g1 x g2 x 0 hm x = gp x x ∈ ⊂ En (1) m and gj j = p where m n and the functions f , hi i = are continuous, and ... Larger, more complex, nonlinear programming problems arise frequently in modern applied analysis in a wide variety of disciplines Indeed, within the past few decades nonlinear programming has advanced...

Ngày tải lên: 06/08/2014, 15:20

... the demand for the three sizes are independent and the demand for each size is uniformly distributed between zero and three thousand pairs He wishes to allocate his shoe budget of four thousand ... convex programming problems are both pervasive and relatively well treated by theory and numerical methods The corresponding theory also motivates many approaches to general nonlinear programming ... the sets A and B in E m+1 as A= r y r≥ B= r y r ≤ f∗ y = y y∈C A is the epigraph of (see Section 7.6) and B is the vertical line extending below f ∗ and aligned with the origin Both A and B are...

Ngày tải lên: 06/08/2014, 15:20