t 2 t 1 Discrete order of arrival: deciding the order Now, every time you see an absolute value Tranformation of an «or» into an «and»

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1 1 Lecture 7: Using MILP to solve decision problems Graphical interpretation of decision problems Some variables are on a grid, some are not: Decision problems Absolute values ransfmation of a logical into a logical and A fmal definition of MILP Example: holding Posing the number of holding as a MILP A real example from Air raffic Control Order of arrival of 2 Order of arrival of 1 he solution f these two variables has to be on the grid. t 1 It does not matter where the solution f these two variables is Discrete der of arrival: deciding the der First kind of decision variables: der of arrival Now, every you see an absolute value Absolute value: not linear, not affine difficult t 1 OR t 1 Absolute value can be expressed as a logical disjunction (this is just a fancy way to say ). OR Another way to express this: «and» is easy, is difficult ranfmation of an into an «and» Reminder: you have already used «and» many s Let us pick a very large number Let us pick a decision variable he two following statements are equivalent: All these are logical and and

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3 3 Summary Summary Depending on the value of d: ranfmation of an into an «and» Let us pick a very large number Let us pick a decision variable he two following statements are equivalent: and In other wds ranfmation of an into an «and» Why is it useful? Now you can pose the problem of earliest arrival of the last with decision enabled f der of arrival: you can deal with continuous and discrete variables. and t1 if d=1, t2 if d=0 OR t 1 t 1

4 4 Why is it useful? Now you can pose the problem of earliest arrival of the last with decision enabled f der of arrival: you can deal with continuous and discrete variables. Why is it useful? Now you can pose the problem of earliest arrival of the last with decision enabled f der of arrival: you can deal with continuous and discrete variables. Aircraft 1 and 2 are separated by me than (previous slides) Aircraft 1 arrives between a1 and b1 Aircraf arrives between a2 and b2 A fmal definition of a MILP A Mixed Integer Linear Program is a Linear Program in which some of the variables are continuous, and some are integer. Holding : how many should an fly? A holding delays an by a fixed amount of, usually =3 min. Question f AC: how many holding should one do befe it is allowed to land? CAS tracks courtesy of NASA Ames

5 5 hree holding hree holding hree holding his is the set in which we want to schedule. We seek one arrival f each in each of the coled sets. he number of holding is a decision variable (the decision is actually made by the human Air raffic Controller)

6 6 he number of holding is a decision variable (the decision is actually made by the human Air raffic Controller). continuous Actually, the human air traffic controller has the possibility to schedule 2 anywhere in the fourth interval ( with three holding ). continuous integer his is can be expressed in terms of two linear constraints involving integer and continuous variables his is can be expressed in terms of two linear constraints involving integer and continuous variables, me generally, f any admissible interval f 2: F n holding Problem: separating by : how to schedule the so the last comes as early as possible. Problem: separating by : how to schedule the so the last comes as early as possible

7 7 Problem: separating by : how to schedule the so the last comes as early as possible. Problem: separating by : how to schedule the so the last comes as early as possible t 1 t 1 t 3 Problem: separating by : how to schedule the so the last comes as early as possible. his is a Mixed Integer Linear Program (MILP): - Some variables are integer (the der of arrival): 3, 1, 2, 4 - Some variables are continuous (the s of arrival) - he problem can be posed as a linear program involving both integer and continuous variables. = at least 3 min = at least 3 min t 1 t 3 t 4 t 1 t 3 t 4