Post on 13-Dec-2015
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Overview Introduction
Example problem
Results for Traveling Salesman Problem
Future Work
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Introduction Linear search strategy called cut-and-solve At each step:
A chunk of the solution space is cut away and solved, providing incumbent solutions
A relaxed solution is found for remaining solution space
Iterate until relaxed solution is greater than or equal to incumbent
Cut-and-solve may not be useful for simple problem instances
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Introduction
Use cut-and-solve to solve Linear Programs (LPs)
LPs are useful for modeling: Traveling Salesman Problem Constraint Satisfaction Problem Minimum cost flow problem
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Introduction
Asymmetric Traveling Salesman Problem (ATSP) can be used to model: No-wait flowshop Stacker crane Tilted drilling machine Computer disk read head Robotic motion Pay phone coin collection
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Introduction
Minimize Z = cij xij
s.t.: xij = 1 for j = 1,…,n
xij = 1 for i = 1,…,n
xij <= |W| - 1, for all proper non-empty subsets W of V
xij = {0,1}
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Introduction NP-hard Frequently solved using search trees
Branch-and-bound Branch-and-cut
Search strategy Best-first search Depth-first search
Cut-and-solve has minimal memory requirements and no “wrong” subtrees
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Introduction
ATSP solution space is a high-dimensional convex polyhedron
See paper for algorithm
Simple 2D problem for example
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Example
Minimize Z = y – 4/5 x
s.t.: x >= 0y <= 3y + 13/6 x <= 9y – 5/13 x >= 1/14y + 3/5 x >= 6/5x,y integers
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Example
x >= 0y <= 3y + 13/6 x <= 9y – 5/13 x >= 1/14y + 3/5 x >= 6/5x,y integers
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Example
Minimize Z = y – 4/5 x
x = 2y = 1Z = -0.6
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Example
Minimize Z = y – 4/5 x
x = 3.5y = 1.4Z = -1.4
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Example
Minimize Z = y – 4/5 x
x = 2.6y = 1.0Z = -1.1
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Example
New incumbent solution:
x = 2y = 1Z = -0.6
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Example
Minimize Z = y – 4/5 x
x = 1.0y = 0.6Z = -0.2
Incumbent solution:Z = -0.6
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Example
Keep new constraints and incumbent solution from one iteration to the next
Incumbent yields pruning opportunities
No children to choose between Need to decide size of cut
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ATSP results
Implemented using cplex with default settings
Compared with Concorde (Applegate, Bixby, Chvatal, & Cook)
Concorde designed for STSP Used 2-node transformation
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ATSP results
Testbed: All 27 TSPLIB instances 6 each: rtilt, stilt, crane, disk, coin, shop,
and super (Fischetti, Lodi, & Toth, 2002)
Cut-and-solve required 29.7% as much time as Concorde