fundamental-engineering-optimization-methods.pdf

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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 74 Linear Programming Methods The problem is first converted to standard LP form by changing the sign of the objective function and adding slack variables ݏ ǡ ݏ to the LE constraints. The resulting optimization variables, cost coefficients, and constraint coefficients are given below: ൌ ሾ ݔ ݔ ݏ ݏ ሿǡ ൌ ሾെ͵ െ ʹ Ͳ Ͳሿǡ ࡭ ൌ ቂ ʹ ʹ ͳ ͵ ͳ Ͳ Ͳ ͳ ቃ ǡ ࢈ ൌ ቂ ͳʹ ͳ͸ The steps of the Simplex algorithm for the problem are shown below: Step 1: ൌ ቂ ݏ ݏ ቃ ൌ ቂ ͳʹ ͳ͸ ቃ ǡ ࢞ ൌ ቂ ݔ ݔ ቃ ൌ ቂ Ͳ Ͳ ቃ ǡ ࢉ ൌ ሾͲǡͲሿǡ ࢉ ൌ ሾെ͵ǡ െʹሿǡ ࡮ ൌ ࡵǡ ࢈ ൌ ࢈ ൌ ቂ ͳʹ ͳ͸ ቃǡ ݖ Ƹ ൌ Ͳǡ ൌ ቂ ݏ ݏ ቃ ൌ ቂ ͳʹ ͳ͸ ቃ ǡ ࢞ ൌ ቂ ݔ ݔ ቃ ൌ ቂ Ͳ Ͳ ቃ ǡ ࢉ ൌ ሾͲǡͲሿǡ ࢉ ൌ ሾെ͵ǡ െʹሿǡ ࡮ ൌ ࡵǡ ࢈ ൌ ࢈ ൌ ቂ ͳʹ ͳ͸ ቃǡ ݖ Ƹ ൌ Ͳǡ Update: ݔ ՚ ͸ǡ ࢞ ՚ ቂ Ͳ Ͷ ቃ ǡ ݖ Ƹ ՚ െͳͺ Step 2: ൌ ቂ ݔ ݏ ቃ ൌ ቂ ͸ Ͷ ቃ ǡ ࢞ ൌ ቂ ݔ ݏ ቃ ൌ ቂ Ͳ Ͳ ቃ ǡ ࢉ ൌ ሾെ͵ǡ Ͳሿǡ ࢉ ൌ ሾെʹǡ Ͳሿǡ ࡮ ൌ ቂ ʹ ʹ Ͳ ͳ ቃ ǡ ࡺ ൌ ቂ ͳ ͵ ͳ Ͳ ቃ ǡ ࢈ ൌ ቂ ͸ Ͷ ቃǡ ൌ ሾെ͵Ȁʹǡ Ͳሿǡ ࢉ ൌ ሾെͳȀʹǡ ͵Ȁʹሿǡ ݔ ൌ ݔ ǡ ܣ ൌ ቂ ͳȀʹ ʹ ቃ ǡ ൜ ೔ǡభ ǣ ܣ ௜ǡଵ ൐ Ͳൠ ൌ ሼͳʹǡ ʹሽǡ ܣ ௣ǡ௤ ൌ ܣ ଶǡଶ Update: ݔ ՚ ʹǡ ࢞ ՚ ቂ ͷ Ͳ ቃ ǡ ݖ Ƹ ՚ െͳͻ Step 3: ൌ ቂ ݔ ݔ ቃ ൌ ቂ ͷ ʹ ቃ ǡ ࢞ ൌ ቂ ݏ ݏ ቃ ൌ ቂ Ͳ Ͳ ቃ ǡ ࢉ ൌ ሾെ͵ǡ െʹሿǡ ࢉ ൌ ሾͲǡ Ͳሿǡ ࡮ ൌ ቂ ʹ ʹ ͳ ͵ ቃ ǡ ࡺ ൌ ࡵǡ ൌ ሾെͷȀͶǡ െͳȀͶሿǡ ࢉ ൌ ሾͷȀͶǡ ͳȀͶሿ Since all ܿ Ƹ ൐ Ͳ ³ an optimal has been reached and ݖ ௢௣௧ ൌ െͳͻ ² 5.3.2 Tableau Implementation of the Simplex Algorithm It is customary to use tableaus to capture the essential information about the LP problem. A tableau is an augmented matrix that includes the constraints matrix, the right-hand-side (rhs), and the coefficients of the cost function (represented in the last row). Each preceding row of the tableau represents a constraint equation, and each column represents a variable. The tableau method implements the simplex algorithm by iteratively computing the inverse of the basis ( B ) matrix.
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Download free eBooks at bookboon.com Fundamental Engineering Optimization
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