The vehicle routing problem (VRP) with solution shape constraints is defined, and an interactive heuristic is proposed for its solution. A set of tours with desirable shapes—that is, that are visually attractive— possesses two characteristics. Each tour of a visually attractive set of tours is compact, and tours within the set do not cross one another. The visual attractiveness of tours is of great importance in practical routing applications and plays a central role in whether tours are adopted in actual freight transportation operations in many industries, including courier operations of large package delivery companies. Numerical exper- iments were conducted on real-world data to assess the proposed heuris- tic. Results of the experiments show that the heuristic, coupled with effective shape measures, is able to provide solutions with significantly improved layout while maintaining satisfactory results in terms of conventional VRP measures (e.g., required fleet size and total cost). The vehicle routing problem (VRP) with solution shape constraints is defined and an interactive heuristic is proposed for its solution. The VRP is defined on a complete graph G = (V, E), where V = {0, 1, . . . , n} is a vertex set representing customer locations, and E = {(i, j) ⎟ i, j ∈ V} is an edge set connecting customer locations. A fleet of identical vehicles, each with a limited capacity C, is stationed at a single depot (vertex 0). The objective of the VRP is to determine the set of tours with the overall minimum cost or, alternatively, to obtain the solution requiring the minimum number of tours (each vehicle carrying out one tour). In some formulations, each vehicle must return to the depot within maximum duration limit T, as is the case in this work. A review of solution techniques for the VRP and its variants, including, for example, the VRP with time windows, multiple depots, or a heterogeneous fleet mix, can be found in work by Crainic and Laporte (1) or by Fisher and Jaikumar (2), among others. Although numerous works have addressed a myriad of VRP variants, it appears that only one (3) has explicitly considered visual attractiveness, that is, solution shape. A set of tours with a desirable shape (that are visually attractive) as defined here possesses two characteristics: Each tour of a visually attractive set of tours is com- pact and tours within the set do not cross one another. The visual attractiveness of tours is of great importance in practical routing applications and plays a central role in whether tours are adopted in actual freight transportation operations in many industries, includ- ing courier operations of large package delivery companies (the primary focus of this study). The properties of compactness and crossing are in the same class with other operational constraints, including workload balance. Although such characteristics may not be required for implementation, tours that possess these characteristics are highly desirable. Thus, they are often treated as soft constraints, as is the case here. A tour is considered to be compact if all customer locations in the tour are within relatively small distances of one another. One could also define compactness by the maximum distance between customers in a tour. Assignment of drivers to relatively small areas aids in improving driver familiarity with the road network. Familiarity with the delivery region thereby aids in improving the efficiency of operations, reduc- ing the likelihood of missing customer appointments and improving relationships between customers and drivers. In many businesses and other enterprises, couriers, service tech- nicians, or other personnel (collectively referred to here as drivers) must serve multiple geographically dispersed customers within a given day. The service region is, and historically has been, divided into districts and the drivers are responsible for their own districts. These districts typically do not overlap. As a result, drivers and their managers who notice that routes developed by algorithms designed to solve the VRP (where districting is completed in conjunction with routing) may cross one another are reluctant to employ the resulting routes. The drivers believe that the routes must be inefficient if they cross. Since existing techniques do not explicitly consider solution shape, resulting tours are difficult to implement in practice. Figure 1 shows the optimal solution to a 45-customer benchmark VRP (4) that was formulated without consideration for solution shape. In this solution, the convex hull of Tour 1 covers a huge service area that contains Tours 2 and 3. In practice, field engineers will likely reject routing plans that have such an undesirable layout and will choose a solution with greater visual attractiveness in its place, even if the alternative solution is likely to perform significantly more poorly with respect to conventional VRP measures. In the next section, mathematical expressions for evaluating solu- tion shape are presented and implications for the design of a solution technique for the VRP with solution shape constraints are discussed. An interactive heuristic for constructing delivery tours with accept- able shape is then proposed. The performance of the heuristic is tested on a real-world package delivery problem that exists in companies like the FedEx Express Corporation (FedEx) and United Parcel Service. Results of the experiments show that the proposed heuristic, coupled with effective shape measures, is able to provide solutions with excellent solution shape, while simultaneously possessing accept- able characteristics in terms of conventional VRP measures. These Interactive Heuristic for Practical Vehicle Routing Problem with Solution Shape Constraints Hao Tang and Elise Miller-Hooks H. Tang, Operations Research Group, FedEx Express Corporation, Memphis, TN 38125. E. Miller-Hooks, Department of Civil and Environmental Engineering, University of Maryland, College Park, MD 20742. 9 Transportation Research Record: Journal of the Transportation Research Board, No. 1964, Transportation Research Board of the National Academies, Washington, D.C., 2006, pp. 9–18. 2006 FRED BURGGRAF AWARD, Planning and Environment