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Branch &amp; Bound

Branch &amp; Bound - Contents 8 The Branch and Bound...

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Contents 8 The Branch and Bound Approach 375 8.1 The Di ff erence Between Linear and Integer Programming Models . . . . . . . . . . . . 375 8.2 The Three Main Tools in the Branch and Bound Approach377 8.3 The Strategies Needed to Apply the Branch and Bound Approach . . . . . . . . . . . . . . . . . . . . . . . . . 380 8.3.1 The Lower Bounding Strategy . . . . . . . . . . 381 8.3.2 The Branching Strategy . . . . . . . . . . . . . 382 8.3.3 The Search Strategy . . . . . . . . . . . . . . . 385 8.4 The 0 1 Knapsack Problem . . . . . . . . . . . . . . . 393 8.5 The General MIP . . . . . . . . . . . . . . . . . . . . . 405 8.6 B&B Approach for Pure 0 1 IPs . . . . . . . . . . . . 409 8.7 Advantages and Limitations of the B&B Approach, Re- cent Developments . . . . . . . . . . . . . . . . . . . . 417 8.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 420 8.9 References . . . . . . . . . . . . . . . . . . . . . . . . . 423 i

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Chapter 8 The Branch and Bound Approach This is Chapter 8 of “Junior Level Web-Book for Optimization Models for decision Making” by Katta G. Murty. 8.1 The Di ff erence Between Linear and Integer Programming Models The algorithms that we discussed in earlier chapters for linear pro- grams, and some recently developed algorithms such as interior point methods not discussed in this book, are able to solve very large scale LP models arising in real world applications within reasonable times (i.e., within a few hours of time on modern supercomputers for truly large models). This has made linear programming a highly viable prac- tical tool. If a problem can be modeled as an LP with all the data in it available, then we can expect to solve it and use the solution for decision making; given adequate resources such as computer facilities and a good software package, which are becoming very widely available everywhere these days. Unfortunately, the situation is not that rosy for integer and com- 375
376 Ch.8. Branch And Bound binatorial optimization models. The research e ff ort devoted to these areas is substantial, and it has produced very fundamental and elegant theory, but has not delivered algorithms on which practitioners can place faith that exact optimum solutions for large scale models can be obtained within reasonable times. Certain types of problems, like the knapsack problem, and the traveling salesman problem (TSP), seem easier to handle than others. Knapsack problems involving 10,000 or more 0 1 variables and TSPs involving a few thousands of cities, have been solved very successfully in at most a few hours of computer time on modern parallel process- ing supercomputers by implementations of branch and bound methods discussed in this chapter custom-made to solve them using their special structure. But for many other types of problems discussed in Chapter 7, only moderate sized problems may be solvable to optimality within these times by existing techniques. Real world applications sometimes lead to large scale problems. When faced with such problems, prac- titioners usually resort to heuristic methods which may obtain good solutions in general, but cannot guarantee that they will be optimal.

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