Elad H. Kivelevitch

An ant colony optimization technique for solving min–max Multi-Depot Vehicle Routing Problem

Abstract The Multi-Depot Vehicle Routing Problem (MDVRP) involves minimizing the total distance traveled by vehicles originating from multiple depots so that the vehicles together visit the specified customer locations (or cities) exactly once. This problem belongs to a class of Nondeterministic Polynomial Hard (NP Hard) problems and has been used in literature as a benchmark for development of optimization schemes. This article deals with a variant of MDVRP, called min–max MDVRP, where the objective is to minimize the tour-length of the vehicle traveling the longest distance in MDVRP. Markedly different from the traditional MDVRP, min–max MDVRP is of specific significance for time-critical applications such as emergency response, where one wants to minimize the time taken to attend any customer. This article presents an extension of an existing ant-colony technique for solving the Single Depot Vehicle Routing Problem (SDVRP) to solve the multiple depots and min–max variants of the problem. First, the article presents the algorithm that solves the min–max version of SDVRP. Then, the article extends the algorithm for min–max MDVRP using an equitable region partitioning approach aimed at assigning customer locations to depots so that MDVRP is reduced to multiple SDVRPs. The proposed method has been implemented in MATLAB for obtaining the solution for the min–max MDVRP with any number of vehicles and customer locations. A comparative study is carried out to evaluate the proposed algorithms performance with respect to a currently available Linear Programming (LP) based algorithm in literature in terms of the optimality of solution. Based on simulation studies and statistical evaluations, it has been demonstrated that the ant colony optimization technique proposed in this article leads to more optimal results as compared to the existing LP based method.

A Market-based Solution to the Multiple Traveling Salesmen Problem

This paper describes a market-based solution to the problem of assigning mobile agents to tasks. The problem is formulated as the multiple depots, multiple traveling salesmen problem (MTSP), where agents and tasks operate in a market to achieve near-optimal solutions. We consider both the classical MTSP, in which the sum of all tour lengths is minimized, and the Min-Max MTSP, in which the longest tour is minimized. We compare the market-based solution with direct enumeration in small scenarios, and show that the results are nearly optimal. For the classical MTSP, we compare our results to linear programming, and show that the results are within 1 % of the best cost found by linear programming in more than 90 % of the runs, with a significant reduction in runtime. For the Min-Max case, we compare our method with Carlsson’s algorithm and show an improvement of 5 % to 40 % in cost, albeit at an increase in runtime. Finally, we demonstrate the ability of the market-based solution to deal with changes in the scenario, e.g., agents leaving and entering the market. We show that the market paradigm is ideal for dealing with these changes during runtime, without the need to restart the algorithm, and that the solution reacts to the new scenarios in a quick and near-optimal way.

Ant colony optimization technique to solve the min-max Single Depot Vehicle Routing Problem

This paper implements a swarm intelligence based algorithm called ant colony optimization to solve the min-max Single Depot Vehicle Routing Problem (SDVRP). A traditional SDVRP tries to minimize the total distance travelled by all the vehicles to all customer locations. The min max SDVRP, on the other hand, tries to minimize the maximum distance travelled by any vehicle. This problem is of specific significance for time-critical applications where one wants to minimize the time taken to attend any customer. The algorithm developed is an extension of SDVRP algorithm developed by Bullnheimer et al. in 1997 based upon ant colony optimization. A computer simulation model using the MATLAB is developed. A comparative study is carried out to evaluate the proposed algorithms performance with respect to the algorithm developed by Carlsson et al. in terms of the optimality of solution and time taken to reach the solution.

Market-Based Solution to the Allocation of Tasks to Agents

Tasks allocation is a fundamental problem in multiagent systems. We formulate the problem as a multiple traveling salesmen problem (MTSP), which is an extension to the well known traveling salesman problem (TSP), both considered to be NP-hard combinatorial optimization problems. We propose a solution in which agents interact in an economic market to win tasks situated in an environment. The agents strive to minimize required costs, defined as either the total distance traveled by all agents or the maximum distance traveled by any agent. Using a set of simple market operations, the agents come up with a solution for task allocation. In this work we examine the processing speed of the market-based solution (MBS), as well as the quality vs. optimal solutions achieved using enumeration for a 3 agents by 8 tasks scenario. We show that the MBS is both quick and close to optimal. We then show that the MBS can be scaled to more complicated problems, by comparing its results with results from genetic algorithm (GA) and clustering. We also show the robustness of the MBS to changes in the scenario, e.g. the addition and removal of tasks or agents.

A Hierarchical Market Solution to the Min-Max Multiple Depots Vehicle Routing Problem

The problem of assigning a group of Unmanned Aerial Vehicles (UAVs) to perform spatially distributed tasks often requires that the tasks will be performed as quickly as possible. This problem can be defined as the Min–Max Multiple Depots Vehicle Routing Problem (MMMDVRP), which is a benchmark combinatorial optimization problem. In this problem, UAVs are assigned to service tasks so that each task is serviced once and the goal is to minimize the longest tour performed by any UAV in its motion from its initial location (depot) to the tasks and back to the depot. This problem arises in many time-critical applications, e.g. mobile targets assigned to UAVs in a military context, wildfire fighting, and disaster relief efforts in civilian applications. In this work, we formulate the problem using Mixed Integer Linear Programming (MILP) and Binary Programming and show the scalability limitation of these formulations. To improve scalability, we propose a hierarchical market-based solution (MBS). Simulation results demonstrate the ability of the MBS to solve large scale problems and obtain better costs compared with other known heuristic solution.

Genetic Fuzzy Trees and their Application Towards Autonomous Training and Control of a Squadron of Unmanned Combat Aerial Vehicles

This study introduces the technique of Genetic Fuzzy Trees (GFTs) through novel application to an air combat control problem of an autonomous squadron of Unmanned Combat Aerial Vehicles (UCAVs) equ...

Multi-Agent Maze Exploration

Mazes have intrigued the human mind for thousands of years, and have been used to measure mental abilities of laboratory animals. In recent years mazes have been used to measure the artificial intelligence of robots by examining their ability to traverse mazes using maze exploration and solution algorithms. We use a simulation of a multi-agent system and show that it is beneficial to use a group of several robots in maze exploration. Based on Tarry’s behavioral algorithm we demonstrate that the group performance improves and becomes more robust as the number of robots in the group increases. In addition, simulation results yield that the amount of data transfer required for group coordination can be minimized to a small set of data items, which is independent of either the number of robots in the group or the maze size. Thus, our solution can be scaled up to mazes and/or groups of any size.

A Binary Programming Solution to the Multiple-Depot, Multiple Traveling Salesman Problem with Constant Profits.

We present a solution to the Multiple Depots, Multiple Traveling Salesmen Problem (MTSP) with constant prots, which is a generalization of the single traveling salesman problem (TSP) in the following ways: there are several salesmen, the salesmen originate at various initial locations called depots, and the cities are visited only if it is protable to visit them, i.e., the benet of visit is higher than the cost of travel to that city. The solution is based on a binary programming formulation of the problem and solving this formulation using a generic branch-and-bound algorithm. We present results that show the ability of the proposed solution to solve problems with various levels of city benets and compare these results to other MTSP variants.

Comparing the Robustness of Market-Based Task Assignment to Genetic Algorithm

In previous works we developed a market-based solution to the problem of assigning mobile agents to tasks. We showed that this solution is near-optimal, quick, and can naturally incorporate fuzzy logic to deal with uncertainty in task locations to reduce sensitivity. So far we have only hypothesized that the market-based can naturally handle changes in the scenario and do it more eciently than other near-optimal algorithms. In this work we show that the market-based solution is very robust, and handles changes in the scenario better than a commonly used genetic algorithm. This is true both for cases when the changes occur at relatively low rates and even more so at higher rates. Thus, we show that the market-based solution is a robust task assignment algorithm.

On The Scalability of the Market-Based Solution to the Multiple Traveling Salesmen Problem

In previous works we developed a market-based solution to the problem of assigning mobile agents to tasks. We showed that this solution is near-optimal, quick, and can naturally incorporate fuzzy logic to deal with uncertainty in task locations to reduce sensitivity. In this work we develop a hierarchical market and show that it provides results that are more optimal than the results of the Carlsson algorithm for the Min-Max Multiple Depots Multiple Traveling Salesmen Problem. We also demonstrate the ability of this algorithm to solve problems with thousands of cities and a hundred salesmen.