# Efficient algorithms for some combinatorial problems

#### Abstract

This dissertation considers three closely related combinational optimization problems and proposes efficient algorithms for their solutions. These are: Bulk Pickup/Delivery problem: The objective of this problem is to minimize the total transportation cost given a uniform fleet of vehicles at a depot and a collection of customer orders. Each order consists of a certain amount of a bulk product to be picked up and delivered at designated points. Each piece of an order must be assigned to a specific shift, vehicle and driver for transportation. We present an efficient heuristic algorithm based on the solution to a transportation problem, to obtain a feasible solution to the problem. The heuristic has been applied to a real instance of this problem arising in the operations of the Shanghai Truck Transportation Corporation. The test results showed that the gap between the feasible solution ratio ((total loaded distance)/(total distance)) and an upper bound on the ratio is about 1.3%. Generalized Traveling Salesman Problem: It is a particular Traveling Salesman Problem in which the requirement that all cities be visited is relaxed to allow missing cities at a penalty cost. We present an optimization algorithm comprised of a Lagrangean Relaxation algorithm in which tight subtour elimination constraints are dualized into the objective function, various upper bounding procedures and a Branch-and-Bound strategy. The test results show this algorithm to be very efficient in solving this problem. Node-Weighted Steiner Tree Problem (NWSTP): It is a natural extension to the Steiner Tree Problem by the addition of node-associated weights. In this dissertation, we concentrate on a special case of NWSTP called Single Point Weighted Steiner Tree Problem (WSTP) suggested by Segev (1987), where the set of nodes, which must be included in the solution tree, consists of a single node, and all node weights are nonpositive. We introduce a T-Set-based algorithm for solving the WSTP. The test results show that our algorithm can not only be used for improving the worst case of Beasley's SST (Shortest Spanning Tree problem with additional constraints)-based algorithm (1987), but is also more efficient.

#### Subject Area

Transportation|Mathematics education|Computer science|Operations research

#### Recommended Citation

Tang, Baoxing, "Efficient algorithms for some combinatorial problems" (1990). *Dissertations available from ProQuest*. AAI9026658.

https://repository.upenn.edu/dissertations/AAI9026658