- Matoušek J
- Gärtner B

Springer Berlin Heidelberg, (2007), 29-40

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In Section 2.7 we encountered a situation in which among all feasible solutions of a linear program, only those with all components integral are of interest in the practical application. A similar situation occurs quite often in attempts to apply linear programming, because objects that can be split into arbitrary fractions are more an exception than the rule. When hiring workers, scheduling buses, or cutting paper rolls one somehow has to deal with the fact that workers, buses, and paper rolls occur only in integral quantities. Sometimes an optimal or almost-optimal integral solution can be obtained by simply rounding the components of an optimal solution of the linear program to integers, either up, or down, or to the nearest integer. In our paper-cutting example from Section 2.7 it is natural to round up, since we have to fulfill the order. Starting from the optimal solution x 1 = 48.5, x 5 = 206.25, x 6 = 197.5 of the linear program, we thus arrive at the integral solution x 1 = 49, x 5 = 207, and x 6 = 198, which means cutting 454 rolls. Since we have found an optimum of the linear program, we know that no solution whatsoever, even one with fractional amounts of rolls allowed, can do better than cutting 452.5 rolls. If we insist on cutting an integral number of rolls, we can thus be sure that at least 453 rolls must be cut. So the solution obtained by rounding is quite good. However, it turns out that we can do slightly better. The integral solution x 1 = 49, x 5 = 207, x 6 = 196, and x 9 = 1 (with all other components 0) requires cutting only 453 rolls. By the above considerations, no integral solution can do better. In general, the gap between a rounded solution and an optimal integral solution can be much larger. If the linear program specifies that for most of 197 bus lines connecting villages it is best to schedule something between 0.1 and 0.3 buses, then, clearly, rounding to integers exerts a truly radical influence. The problem of cutting paper rolls actually leads to a problem with a linear objective function and linear constraints (equations and inequalities), but the variables are allowed to attain only integer values. Such an optimization

CITATION STYLE

APA

Matoušek, J., & Gärtner, B. (2007). Integer Programming and LP Relaxation. In *Understanding and Using Linear Programming* (pp. 29–40). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-30717-4_3

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