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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 , which shares the same data, and is derived through rational economic arguments . In this context the original LP called the primal LP . Variables in the dual problem are di ff 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 fi t. DETERGENT MAKER wants to buy all of fertilizer maker’s supplies at cheapest price for his detergent process. Suppose he o ff ers: 102

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\$ π 1 /ton for RM1 \$ π 2 /ton for RM2 \$ π 3 /ton for RM3 These prices are the variables in his problem. Total payment 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 fi 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
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. sol., it is the marginal value vector for primal.

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