{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

MIT15_053S13_lec10

MIT15_053S13_lec10 - 15.053/8 Introduction to Integer...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
1 15.053/8 March 14, 2013 Introduction to Integer Programming Integer programming models
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2 Quotes of the Day “Somebody who thinks logically is a nice contrast to the real world.” -- The Law of Thumb Take some more tea, the March Hare said to Alice, very earnestly. I’ve had nothing yet, Alice replied in an offended tone, so I can’t take more. You mean you can’t take less, said the Hatter. It’s very easy to take more than nothing. -- Lewis Carroll in Alice in Wonderland
Background image of page 2
Combinatorial optimization problems INPUT : A description of the data for an instance of the problem FEASIBLE SOLUTIONS : there is a way of determining from the input whether a given solution x’ (assignment of values to decision variables) is feasible. Typically in combinatorial optimization problems there is a finite number of possible solutions. OBJECTIVE FUNCTION : For each feasible solution x’ there is an associated objective f(x’) . Minimization problem. Find a feasible solution x* that minimizes f( ) among all feasible solutions. 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Example 1: Traveling Salesman Problem INPUT : a set N of n points in the plane FEASIBLE SOLUTION: a tour that passes through each point exactly once. OBJECTIVE : minimize the length of the tour. 4
Background image of page 4
Example 2: Balanced Partition INPUT : A set of positive integers a 1 , …, a n FEASIBLE SOLUTION : a partition of {1, 2, … n} into two disjoint sets S and T. S ∩ T = , S T = {1, … , n} OBJECTIVE : minimize | ∑ i S a i - i T a i | 5 Example: 7, 10, 13, 17, 20, 22 These numbers sum to 89 The best split is {10, 13, 22} and {7, 17, 20}.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Example 3. Exam Scheduling INPUT : a list of subjects with a final exam; class lists for each of these subjects, and a list of times that final exams can be scheduled. Let a ij denote the number of students that are taking subjects i and j. FEASIBLE SOLUTION : An assignment of subjects to exam periods OBJECTIVE : minimize {a ij : i and j are scheduled at the same time} 6
Background image of page 6
Example 4: Maximum Clique Problem INPUT : a friendship network G = (N, A). If persons i and j are friends, then (i, j) A. FEASIBLE SOLUTION : a set S of people such that every pair of persons in S are friends. OBJECTIVE : maximize |S| 7
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Example 5: Integer programming INPUT : a set of variables x 1 , …, x n and a set of linear inequalities and equalities, and a subset of variables that is required to be integer. FEASIBLE SOLUTION : a solution x’ that satisfies all of the inequalities and equalities as well as the integrality requirements. OBJECTIVE : maximize ∑ i c i x i 8 Example: maximize 3x + 4y subject to 5x + 8y ≤ 24 x, y ≥ 0 and integer
Background image of page 8
Which of the following is false? 9 1. The Traveling Salesman Problem is a combinatorial optimization problem.
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 48

MIT15_053S13_lec10 - 15.053/8 Introduction to Integer...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online