fundamental-engineering-optimization-methods.pdf

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¸²¹ ¸²¾ ¸ º º²¹ ¸ºº െࢠ º²» ¹ º º º²¿ ¾ºº From the final tableau, the optimum crop combination is given as: ݔ כ ൌ Ͳǡ ݔ כ ൌ Ͳǡ ݔ כ ൌ ͳͲͲ ³ with ݖ כ 400. Further, the shadow prices for the slack variables are: ൌ ሾͲǡ ͲǤͺሿ ³ ZLWK ݖ כ ൌ ࢟ ࢈ ൌ ͶͲͲ ² Next, without re-solving the problem, we wish to answer the following questions: a) If an additional 50 acres are added, what is the expected change in yield? The answer is found from: ݖ כ ൌ ࢟ ሺ࢈ ൅ οሻ ZKHUH οൌ ሾͷͲǡͲሿ ³ ZLWK ݖ כ ൌ ͶͲͲ ³ i.e., there is no expected change in yield. This also means that the land area constraint is not binding in the current optimum solution. b) If an additional 50 man-days are added, what is the expected change in yield? The answer is found from: ݖ כ ൌ ࢟ ሺ࢈ ൅ οሻ ZKHUH οൌ ሾͲǡ ͷͲሿ ³ ZLWK ݖ כ ൌ ͶͶͲ i.e., the yield increases by 40 units. This also means that the man-days constraint is binding in the optimum solution. Get Help Now Go to for more info Need help with your dissertation? Get in-depth feedback & advice from experts in your topic area. Find out what you can do to improve the quality of your dissertation!
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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 89 Linear Programming Methods c) If the yield/acre for tomatoes increases by 10%, how is the optimum affected? The answer is found by re-computing the reduced costs as: ൌ ࢟ ࡭ െ ࢉ ൌ ሾെͲǤͳͷǡ ʹǡ Ͳሿ ² Since a reduced cost is now negative, additional Simplex steps are needed to regain optimality. This is done and the new optimum is: ݔ כ ൌ ͺ͵Ǥ͵͵ǡ ݔ כ ൌ Ͳǡ ݔ כ ൌ Ͳ ZLWK ݖ כ ൌ ͶͳʹǤͷ ² d) If the yield/acre for cucumbers drops by 10%, how is the optimum be affected? The answer is found by re-computing the reduced costs as: ൌ ࢟ ࡭ െ ࢉ ൌ ሾͲǤ͵ǡ ʹǡ ͲǤͶሿ ² The reduced costs are non-negative, but is no more a basic variable. Regaining the basis results in reduced cost for becoming negative. Additional Simplex steps are performed to regain optimality, and the new optimum is: ݔ כ ൌ ͺ͵Ǥ͵͵ǡ ݔ כ ൌ Ͳǡ ݔ כ ൌ Ͳ ZLWK ݖ כ ൌ ͵͹ͷ ² e) If the man-hours needed to grow green peppers increase to 5/acre, how is the optimum affected? The answer is found by re-computing the reduced cost: ܿ Ƹ ൌ ݕ ܣ െ ܿ ൌ ͲǤͶ ² Since ݔ Z was non-basic and the revised reduced cost is non-negative, there is no change in the optimum solution. 5.5 Duality Theory for the LP Problems In this section we extend the Lagrangian duality (Sec. 4.5) to the LP problems. Duality theory applies to practical LP problems in engineering and economics. In engineering, for example, the primal problem in electric circuit theory may be posed in terms of electric potential, and its dual in terms of current flow.
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