fundamental-engineering-optimization-methods.pdf

ଵ ² since by assumption ݎ ሺ? ³ can be

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ିଵ ࢈ ൐ ૙ ² Since, by assumption, ݎܽ݊݇ ሺ࡭ሻ ൌ ݉ ³ , can be selected from the various permutations of the columns of A . Further, since each basic solution has exactly m non-zero components, the total number of basic solutions is finite, and is given as: ݊ ݉ ቁ ൌ ௡Ǩ ௠Ǩሺ௡ି௠ሻǨ ² The number of BFS is smaller than number of basic solutions and can be determined by comparing the objective function values at the various basic solutions. The Basic Theorem of Linear Programming (e.g., Arora, p. 201) states that if there is a feasible solution to the LP problem, there is a BFS; and if there is an optimum feasible solution, there is an optimum BFS. The basic LP theorem implies that an optimal BFS must coincide with one of the vertices of the feasible region. This fact can be used to compare the objective function value at all BFSs, and find the optimum by comparison if the number of vertices is small. Finally, there can also be multiple optimums if an active constraint boundary is parallel to the level curves of the cost function.
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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 72 Linear Programming Methods 5.3 The Simplex Method The simplex method iteratively solves the standard LP problem. It does so by starting from a known BFS and successively moving to an adjacent BFS that carries a lower objective function value. Each move involves replacing a single variable in the basis with a new variable, such that the objective function value decreases. The previously nonbasic variable entering the basis is termed as entering basic variable (EBV), and the one leaving it is termed as leaving basic variable (LBV). An optimum is reached when no neighboring BFS with a lower objective function value can be found. 5.3.1 The Simplex Algorithm In order to mathematically formulate the simplex algorithm, let ൌ ሾ࢞ ǡ ࢞ represent a non-basic solution to the LP problem, and let the constraints be expressed as: ࡮࢞ ൅ ࡺ࢞ ൌ ࢈ ² Then, we can solve for DV± ൌ ࡮ ିଵ ሺ࢈ െ ࡺ࢞ ሻǡ and substitute it in the objective function to obtain: ݖ ൌ ࢉ ିଵ ࢈ ൅ ሺࢉ െ ࢉ ିଵ ࡺሻ࢞ ൌ ࢟ ࢈ ൅ ࢉ ൌ ݖ Ƹ ൅ ࢉ (5.4) where ൌ ࢉ ିଵ defines a vector of simplex (Lagrange) multipliers, ൌ ࢉ െ ࢟ represents the reduced costs for the nonbasic variables (reduced costs are zero for the basic variables), and ݖ Ƹ ൌ ࢟ represents the objective function value corresponding to a basic solution, where ݕ ൐ Ͳ represents an active constraint.
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