Lecture23

# Lecture23 - Advanced Operations Research Techniques IE316...

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Unformatted text preview: Advanced Operations Research Techniques IE316 Lecture 23 Dr. Ted Ralphs IE316 Lecture 23 1 Reading for This Lecture Bertsimas Sections 10.2, 10.3, 11.1, 11.2 IE316 Lecture 23 2 The Importance of Formulation The most vital aspect of branch and bound is obtaining good lower bounds . In this respect, not all formulations are created equal . Choosing the right one is critical. A typical MILP can have many alternative formulations. Each formulation corresponds to a different polyhedron enclosing the integer points that are feasible for the problem. The more closely the polyhedron approximates the convex hull of the integer solutions, the better the bound will be . IE316 Lecture 23 3 Example: Facility Location Problem We are given n potential facility locations and m customers . There is a fixed cost c j of opening facility j . There is a cost d ij associated with serving customer i from facility j . We have two sets of binary variables. y j is 1 if facility j is opened, 0 otherwise. x ij is 1 if customer i is served by facility j , 0 otherwise. Here is one formulation: min n X j =1 c j y j + m X i =1 n X j =1 d ij x ij s.t. n X j =1 x ij = K i m X i =1 x ij my j j x ij ,y j { , 1 } i,j IE316 Lecture 23 4 Example: Facility Location Problem Here is another formulation for the same problem: min n X j =1 c j y j + m X i =1 n X j =1 d ij x ij s.t. n X j =1 x ij = K i x ij y j i,j x ij ,y j { , 1 } i,j Notice that the set of integer solutions contained in each of the polyhedra is the same ( why ?). However, the second polyhedra strictly includes the first one. Therefore, the second polyhedra will yield better lower bounds and be better for branch and bound. Notice that the second formulation includes more constraints, but will likely solve more quickly . IE316 Lecture 23 5 Formulation Strength and Ideal Formulations Consider two formulations A and B for the same ILP....
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## This note was uploaded on 08/06/2008 for the course IE 316 taught by Professor Ralphs during the Fall '08 term at Lehigh University .

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Lecture23 - Advanced Operations Research Techniques IE316...

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