310-2slides8 - 8.1 Duality in LP, Optimality Conditions...

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8.1 Duality in LP, Optimality Conditions Katta G. Murty Lecture slides Every LP has another LP called its dual ,whichsharesthesame data, and is derived through rational economic arguments .I n this context the original LP called the primal LP . Variables in the dual problem are di f erent from those in the primal; each dual variable is associated with a primal constraint, it is the marginal value or Lagrange multiplier corresponding to that constraint. Example: Primal Fertilizer Maker’s Problem. FERTILIZER MAKER has daily supply of: 1500 tons of RM 1 1200 tons of RM2 500 tons of RM 3 She wants to use these supplies to maximize net pro f t. DETERGENT MAKER wants to buy all of fertilizer maker’s supplies at cheapest price for his detergent process. Suppose he o f ers: 102
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$ π 1 /ton for RM1 $ π 2 /ton for RM2 $ π 3 /ton for RM3 Thesepr icesarethevar iab lesinh isprob lem . Tota lpaymen t comes to 1500 π 1 + 1200 π 2 +500 π 3 which he wants to minimize. FERTILIZER MAKER: won’t sell supplies unless detergent maker’s prices are competitive with each of hi-ph, lo-ph processes. Hi-ph process converts a packet of { 2 tons RM1, 1 ton RM2, and 1 ton RM3 } into $15 pro f t. In terms of detergent maker’s prices, the same packet yields $(2 π 1 + π 2 + π 3 ). So, she demands 2 π 1 + π 2 + π 3 15 for signing deal. Similarly, by analyzing lo-ph process, she demands π 1 + π 2 10 to sign deal. From these economic arguments, we see that detergent maker’s problem is: 103
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Dual Associated primal activity Minimize 1500 π 1 +1200 π 2 +500 π 3 subject to 2 π 1 + π 2 + π 3 15 Hi-ph π 1 + π 2 10 lo-ph π 1 , π 2 , π 3 , 0 From arguments, we see that if dual problem has unique opt.
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This note was uploaded on 11/06/2011 for the course ISE 421 taught by Professor Km during the Spring '11 term at King Fahd University of Petroleum & Minerals.

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310-2slides8 - 8.1 Duality in LP, Optimality Conditions...

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