fundamental-engineering-optimization-methods.pdf

Application of fonc gives if and if ݔ כ ? ఓ li

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Application of FONC gives: if and if ݔ כ ఓ௖ LI ܿ ൐ Ͳǡ DQG ݔ כ ൌ ݔ LI ܿ ൌ Ͳ ² The resulting dual maximization problem is defined as: ߮ሺߤሻ ൌ ʹ σ ඥܿ ܮ ξ ߤ െ ߤ ଵ଺ ௜ୀଵ ൅ ܿ where c is a constant. Application of FONC then gives: ߤ ൌ ൫σ ඥܿ ܮ ² For the given data, a closed-form solution is obtained as: ߤ כ ൌ ͳ͵ͷͺǤʹǡ ݂ כ ൌ ͳ͵ͷͺǤʹ ǡ ࢞ ൌ ሾͷǤ͸͹ ͶǤʹͷ ͶǤʹͷ ʹǤͺͶ ʹǤͺͶ ͳǤͶʹ ͳǤͶʹ ͳͲ ି଺ ͳǤͲ͸ ͳǤͲ͸ ͳǤͲ͸ ͳͲ ି଺ ͳǤ͹͹ ͳǤ͹͹ ͳǤ͹͹ ͳǤ͹͹ሿ in.
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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 68 Linear Programming Methods 5 Linear Programming Methods Linear programming (LP) problems form an important subclass of the optimization problems. The distinguishing feature of the LP problems is that the objective function and the constraints are linear functions of the optimization variables. LP problems occur in many real-life economic situations where profits are to be maximized or costs minimized with constraints on resources. Specialized procedures, such as the Simplex method, were developed to solve the LPP. The simplex method divides the variables into basic and nonbasic, the latter being zero, in order to develop a basic feasible solution (BFS). It then iteratively updates basic variables thus generating a series of BFS, each of which carries a lower objective function value than the previous. Each time, the reduced costs associated with nonbasic variables are inspected to check for optimality. An optimum is reached when all the reduced costs are non-negative. Learning Objectives: The learning objectives in this chapter are: 1. Understand the general formulation of a linear programming (LP) problem 2. Learn the Simplex method to solve LP problems and its matrix/tableau implementation 3. Understand the fundamental duality properties associated with LP problems 4. Learn sensitivity analysis applied to the LP problems 5. Grasp the formulation of KKT conditions applied to linear and quadratic programming problems 6. Learn to formulate and solve the linear complementarity problem (LCP) 5.1 The Standard LP Problem The general LP problem is described in terms of minimization (or maximization) of a scalar objective function of n variables, that are subject to m constraints. These constraints may be specified as EQ (equality constraints), GE (greater than or equal to inequalities), or LE (less than or equal to inequalities). The variables themselves may be unrestricted in range, specified to be non-negative, or upper and/or lower bounded. Mathematically, the LP problem is expressed as: RU ሻ ݖ ൌ σ ܿ ݔ ௝ୀଵ 6XEMHFW WR± σ ܽ ௜௝ ݔ ሺ൑ǡ ൌǡ ൒ሻܾ ௝ୀଵ ǡ ݅ ൌ ͳǡʹǡ ǥ ǡ ݉ ݔ ൒ ݔ ௝௅ ´IRU VRPH ݆ µ³ ݔ ൑ ݔ ௝௎ ´IRU VRPH ݆ µ³ ݔ IUHH ´IRU VRPH ݆ µ
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