Vehicle routing optimization software has been implemented on a personal computer for Quality Stores, a chain of retail stores. Customized route generation procedures are combined with an efficient branch-and-bound procedure to obtain optimal routings and assignment of routes to trucks. Both the data base maintenance routines and optimization runs are completely menu-driven, enabling personnel with no prior exposure to computers or optimization to operate the software unaided.
We introduce an effective branch-and-bound algorithm for solving the set partitioning problem. The new algorithm employs a new multiplier-adjustment-based bounding procedure, and a complementary branching strategy which results in relatively small search trees. Computational results based on 20 moderately sized crew scheduling problems indicate that our new algorithm is on average 16.6 times faster than the popular code, SETPAR. The improvements are mainly due to the bounding procedure, which is fast, easy to use, and provides tight lower bounds. On average, the bounds are 97.6% of the optimal objective value of the linear programming relaxation after only five iterations, and 98.5% after ten iterations. Moreover, the lower bounds are observed to be monotonically nondecreasing. We also apply the technique of variable elimination, which is very effective in reducing the size of the problems. On average, 89% of the variables are eliminated from the problem at the root node.
SYNOPTIC ABSTRACTThe minimum cost perfect 2-matching problem is defined on a weighted undirected graph Q = (V,t',c) and is the problem of finding a minimum weight subset of edges £ ~ £ such that each vertex is incident to exactly two edges in £. The problem appears as a relaxation subproblem in routing environments where the distance matrices are symmetric. This paper presents a multiplier adjustment approach for obtaining a quick lower bound on the linear programming relaxation of the minimum cost perfect 2-matching problem. The lower bound is constructed by generating a feasible solution to the dual of the linear programming (LP) relaxation. On a collection of 100 random Euclidean and non-Euclidean problems, the value of the lower bound as a percent of the optimal LP objective value averaged over 97 percent with a standard deviation of only 1.15. The approach is also shown to compare favorably with alternative methods for generating fast lower bounds for the symmetric Traveling Salesman Problem.
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