# lec21 - 15.053 z Tuesday May 8 Decision Trees The example...

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15.053 Tuesday, May 8 z Decision Trees The example in this lecture is a modification of the example in Bertsimas and Freund: Data, Models, and Decisions. 1

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Quote of the Day Truly successful decision making relies on a balance between deliberate and instinctive thinking. Malcolm Gladwell Blink: The Power of Thinking Without Thinking Every decision has a cost. Do I have to make this decision at all or can I move on to the next thing? What we decided to leave out is almost as important as what we put in. Joshua Schachter 2
Newsvendor Problem z News vendor Phyllis orders newspapers for sale ± ordered newspapers cost 20¢ each ± she sells newspapers for 25¢ each ± unsold newspapers are worthless ± Prob(demand = j) = 1/5 for j = 6 to 10. r ij is the “net revenue of ordering i papers if the demand is j. r ij = 25 i – 20 i = 5 i if i j r ij = 25 j – 20 i if i > j The newsvendor problem is a classic model in operations management. It models a situation in which one purchases goods in advance of sales, and unsold items are valueless. But it could also be used for times in which unsold items are disposed of for a small revenue. 3

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4 30 30 30 30 30 10 35 35 35 35 -10 15 40 40 40 -30 -5 20 45 45 -50 -25 0 25 50 6 7 8 9 10 Papers Demanded Papers Ordered 6 7 8 9 10 r ij is the value of ordering i papers if the demand is j. We consider a simple case in which we order 6 to 10 newspapers and we sell 6 to 10 newspapers. Of course, one could order 5 or fewer or 11 or greater, but that would clearly not be optimal.
5 30 30 30 30 30 10 35 35 35 35 -10 15 40 40 40 -30 -5 20 45 45 -50 -25 0 25 50 6 7 8 9 10 Papers Demanded Papers Ordered 6 7 8 9 10 Dominated Actions 25 25 25 25 25 5 “Ordering 5” is dominated by “ordering 6” “Ordering 11” is dominated by “ordering 10” -70 -45 -20 5 30 11 “Ordering i” dominates “ordering k” if r ij r kj for all j. Ordering 5 is not as profitable as ordering 6. So, “ordering 5” is dominated by “ordering 6”, regardless of what the demand is. Similarly, ordering “ordering 11” is not as profitable as ordering 10.

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6 30 30 30 30 30 10 35 35 35 35 -10 15 40 40 40 -30 -5 20 45 45 -50 -25 0 25 50 6 7 8 9 10 Papers Demanded Papers Ordered 6 7 8 9 10 The Maximin Criterion maximin criterion : choose the action with the largest guaranteed return maximize i min {r ij : j = 6 to 10} = 30. 30 10 -10 -30 -50 min (r ij ) We now consider various metrics for measuring the quality of a solution. The maximin criteria is very conservative, and is arguably very pessimistic. For each number that you order, we consider the worst possible profit. This is stored in the column min (r ij ). We then maximize over all possible orders. In the newsvendor problem, this will always lead to ordering the minimum possible demand for newspapers.
7 30 30 30 30 30 10 35 35 35 35 -10 15 40 40 40 -30 -5 20 45 45 -50 -25 0 25 6 7 8 9 10 Papers Demanded Papers Ordered 6 7 8 9 10 The Maximax Criterion maximax criterion : choose the action with the largest possible return maximize i max {r ij : j = 6 to 10} = 50.

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lec21 - 15.053 z Tuesday May 8 Decision Trees The example...

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