APME_6 - Applied Probability Methods for Engineers Slide...

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Click to edit Master subtitle style Applied Probability Methods for Engineers Slide Set 6
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Click to edit Master subtitle style Chapter 13 of Winston’s Operations Research Book Decision Making under Uncertainty
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Newsvendor Example n A newsvendor sells newspapers each day n Cost of a paper to vendor is 20¢ n Revenue from selling a paper is 25¢ n Unsold papers are worthless n Sales are uniform between 6 and 10 (S = {6, 7, 8, 9, 10}) n If i papers are purchased by vendor, and j papers are demanded, profit is rij, where n rij = 25i – 20i = 5i, if i ≤ j n rij = 25j – 20 i, if i ≥ j n What should the vendor do?
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Table of rewards n Why did we not consider ordering 0 – 5 papers? Papers Demanded Papers ordered 6 7 8 9 10 6 30 ¢ 30 ¢ 30 ¢ 30 ¢ 30 ¢ 7 10 ¢ 35 ¢ 35 ¢ 35 ¢ 35 ¢ 8 - 10 ¢ 15 ¢ 40 ¢ 40 ¢ 40 ¢ 9 - 30 ¢ - 5 ¢ 20 ¢ 45 ¢ 45 ¢ 10 - 50 ¢ - 25 ¢ 0 ¢ 25 ¢ 50 ¢
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Maximin Criterion n Chooses action ai with largest value of Papers Demanded Papers ordered 6 7 8 9 10 6 30 ¢ 30 ¢ 30 ¢ 30 ¢ 30 ¢ 7 10 ¢ 35 ¢ 35 ¢ 35 ¢ 35 ¢ 8 - 10 ¢ 15 ¢ 40 ¢ 40 ¢ 40 ¢ 9 - 30 ¢ - 5 ¢ 20 ¢ 45 ¢ 45 ¢ 10 - 50 ¢ - 25 ¢ 0 ¢ 25 ¢ 50 ¢ Maximin
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Maximax Criterion n Chooses action ai with largest value of Papers Demanded Papers ordered 6 7 8 9 10 6 30 ¢ 30 ¢ 30 ¢ 30 ¢ 30 ¢ 7 10 ¢ 35 ¢ 35 ¢ 35 ¢ 35 ¢ 8 - 10 ¢ 15 ¢ 40 ¢ 40 ¢ 40 ¢ 9 - 30 ¢ - 5 ¢ 20 ¢ 45 ¢ 45 ¢ 10 - 50 ¢ - 25 ¢ 0 ¢ 25 ¢ 50 ¢ Maximax
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Minimax Regret n For each possible state sj, find action i*(j) that maximizes rij n For any action ai and state sj, opportunity loss (or regret) is ri*(j),j – rij n Subtract from max column value Papers Demanded Papers ordered 6 7 8 9 10 6 30 ¢ (0) 30 ¢ (5) 30 ¢ (10) 30 ¢ (15) 30 ¢ (20) 7 10 ¢ (20) 35 ¢ (0) 35 ¢ (5) 35 ¢ (10) 35 ¢ (15) 8 - 10 ¢ (40) 15 ¢ (20) 40 ¢ (0) 40 ¢ (5) 40 ¢ (10) 9 - 30 ¢ (60) - 5 ¢ (40) 20 ¢ (20) 45 ¢ (0) 45 ¢ (5) 10 - 50 ¢ (80) - 25 ¢ (60) 0 ¢ (40) 25 ¢ (20) 50 ¢ (0) Minimax Regret
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Maximize Expected Value Papers Demanded Papers ordered 6 7 8 9 10 Expected Value 6 30 ¢ 30 ¢ 30 ¢ 30 ¢ 30 ¢ 30 ¢ 7 10 ¢ 35 ¢ 35 ¢ 35 ¢ 35 ¢ 30 ¢ 8 - 10 ¢ 15 ¢ 40 ¢ 40 ¢ 40 ¢ 25 ¢ 9 - 30 ¢ - 5 ¢ 20 ¢ 45 ¢ 45 ¢ 15 ¢ 10 - 50 ¢ - 25 ¢ 0 ¢ 25 ¢ 50 ¢ 0 ¢ Max Expected Value
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Utility Theory n Consider a lottery where with probability pi you win ri n Tree representation of lottery (1/4, $500; ¾, $0): 1/4 3/4 $500 $0
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n Suppose you had to choose between two lotteries L1 and L2: n L1 n Would you prefer L1, with expected value $10K, or L2, with expected value $15K? $10,000
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APME_6 - Applied Probability Methods for Engineers Slide...

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