Solving the Vehicle Routing Problem with Genetic ALgorithm and Simulated Annealing

This Thesis Work will concentrate on a very interesting problem, the Vehicle Routing Problem (VRP). In this problem, customers or cities have to be visited and packages have to be transported to each of them, starting from a basis point on the map. The goal is to solve the transportation problem, to be able to deliver the packages-on time for the customers,-enough package for each Customer,-using the available resources- and – of course - to be so effective as it is possible.Although this problem seems to be very easy to solve with a small number of cities or customers, it is not. In this problem the algorithm have to face with several constraints, for example opening hours, package delivery times, truck capacities, etc. This makes this problem a so called Multi Constraint Optimization Problem (MCOP). What’s more, this problem is intractable with current amount of computational power which is available for most of us. As the number of customers grow, the calculations to be done grows exponential fast, because all constraints have to be solved for each customers and it should not be forgotten that the goal is to find a solution, what is best enough, before the time for the calculation is up. This problem is introduced in the first chapter: form its basics, the Traveling Salesman Problem, using some theoretical and mathematical background it is shown, why is it so hard to optimize this problem, and although it is so hard, and there is no best algorithm known for huge number of customers, why is it a worth to deal with it. Just think about a huge transportation company with ten thousands of trucks, millions of customers: how much money could be saved if we would know the optimal path for all our packages.Although there is no best algorithm is known for this kind of optimization problems, we are trying to give an acceptable solution for it in the second and third chapter, where two algorithms are described: the Genetic Algorithm and the Simulated Annealing. Both of them are based on obtaining the processes of nature and material science. These algorithms will hardly ever be able to find the best solution for the problem, but they are able to give a very good solution in special cases within acceptable calculation time.In these chapters (2nd and 3rd) the Genetic Algorithm and Simulated Annealing is described in details, from their basis in the “real world” through their terminology and finally the basic implementation of them. The work will put a stress on the limits of these algorithms, their advantages and disadvantages, and also the comparison of them to each other.Finally, after all of these theories are shown, a simulation will be executed on an artificial environment of the VRP, with both Simulated Annealing and Genetic Algorithm. They will both solve the same problem in the same environment and are going to be compared to each other. The environment and the implementation are also described here, so as the test results obtained.Finally the possible improvements of these algorithms are discussed, and the work will try to answer the “big” question, “Which algorithm is better?”, if this question even exists.

University Scheduling using Genetic Algorithm

The automated timetabling and scheduling is one of the hardest problem areas. This isbecause of constraints and satisfying those constraints to get the feasible and optimizedschedule, and it is already proved as an NP Complete (1) [1]. The basic idea behind this studyis to investigate the performance of Genetic Algorithm on general scheduling problem underpredefined constraints and check the validity of results, and then having comparative analysiswith other available approaches like Tabu search, simulated annealing, direct and indirectheuristics [2] and expert system. It is observed that Genetic Algorithm is good solutiontechnique for solving such problems and later analysis will prove this argument. The programis written in C++ and analysis is done by using variation in various parameters.

University Timetabling using Genetic Algorithm

The field of automated timetabling and scheduling meeting all the requirementsthat we call constraints is always difficult task and already proved as NPComplete. The idea behind my research is to implement Genetic Algorithm ongeneral scheduling problem under predefined constraints and check the validityof results, and then I will explain the possible usage of other approaches likeexpert systems, direct heuristics, network flows, simulated annealing and someother approaches. It is observed that Genetic Algorithm is good solutiontechnique for solving such problems. The program written in C++ and analysisis done with using various tools explained in details later.

Optimal Path Searching through Specified Routes using different Algorithms

To connect different electrical, network and data devices with the minimum cost and shortest path, is a complex job. In huge buildings, where the devices are placed at different locations on different floors and only some specific routes are available to pass the cables and buses, the shortest path search becomes more complex. The aim of this thesis project is, to develop an application which indentifies the best path to connect all objects or devices by following the specific routes.To address the above issue we adopted three algorithms Greedy Algorithm, Simulated Annealing and Exhaustive search and analyzed their results. The given problem is similar to Travelling Salesman Problem. Exhaustive search is a best algorithm to solve this problem as it checks each and every possibility and give the accurate result but it is an impractical solution because of huge time consumption. If no. of objects increased from 12 it takes hours to search the shortest path. Simulated annealing is emerged with some promising results with lower time cost. As of probabilistic nature, Simulated annealing could be non optimal but it gives a near optimal solution in a reasonable duration. Greedy algorithm is not a good choice for this problem. So, simulated annealing is proved best algorithm for this problem. The project has been implemented in C-language which takes input and store output in an Excel Workbook

On statistical bounds of heuristic solutions to location problems

Solutions to combinatorial optimization problems, such as problems of locating facilities, frequently rely on heuristics to minimize the objective function. The optimum is sought iteratively and a criterion is needed to decide when the procedure (almost) attains it. Pre-setting the number of iterations dominates in OR applications, which implies that the quality of the solution cannot be ascertained. A small, almost dormant, branch of the literature suggests using statistical principles to estimate the minimum and its bounds as a tool to decide upon stopping and evaluating the quality of the solution. In this paper we examine the functioning of statistical bounds obtained from four different estimators by using simulated annealing on p-median test problems taken from Beasley’s OR-library. We find the Weibull estimator and the 2nd order Jackknife estimator preferable and the requirement of sample size to be about 10 being much less than the current recommendation. However, reliable statistical bounds are found to depend critically on a sample of heuristic solutions of high quality and we give a simple statistic useful for checking the quality. We end the paper with an illustration on using statistical bounds in a problem of locating some 70 distribution centers of the Swedish Post in one Swedish region. 

A stopping rule while searching for optimal solution of facility-location

Solutions to combinatorial optimization, such as p-median problems of locating facilities, frequently rely on heuristics to minimize the objective function. The minimum is sought iteratively and a criterion is needed to decide when the procedure (almost) attains it. However, pre-setting the number of iterations dominates in OR applications, which implies that the quality of the solution cannot be ascertained. A small branch of the literature suggests using statistical principles to estimate the minimum and use the estimate for either stopping or evaluating the quality of the solution. In this paper we use test-problems taken from Baesley's OR-library and apply Simulated Annealing on these p-median problems. We do this for the purpose of comparing suggested methods of minimum estimation and, eventually, provide a recommendation for practioners. An illustration ends the paper being a problem of locating some 70 distribution centers of the Swedish Post in a region.

Confidence in heuristic solutions?

Solutions to combinatorial optimization problems frequently rely on heuristics to minimize an objective function. The optimum is sought iteratively and pre-setting the number of iterations dominates in operations research applications, which implies that the quality of the solution cannot be ascertained. Deterministic bounds offer a mean of ascertaining the quality, but such bounds are available for only a limited number of heuristics and the length of the interval may be difficult to control in an application. A small, almost dormant, branch of the literature suggests using statistical principles to derive statistical bounds for the optimum. We discuss alternative approaches to derive statistical bounds. We also assess their performance by testing them on 40 test p-median problems on facility location, taken from Beasley’s OR-library, for which the optimum is known. We consider three popular heuristics for solving such location problems; simulated annealing, vertex substitution, and Lagrangian relaxation where only the last offers deterministic bounds. Moreover, we illustrate statistical bounds in the location of 71 regional delivery points of the Swedish Post. We find statistical bounds reliable and much more efficient than deterministic bounds provided that the heuristic solutions are sampled close to the optimum. Statistical bounds are also found computationally affordable.

How do different densities in a network affect the optimal location of service centers?

The p-median problem is often used to locate p service centers by minimizing their distances to a geographically distributed demand (n). The optimal locations are sensitive to geographical context such as road network and demand points especially when they are asymmetrically distributed in the plane. Most studies focus on evaluating performances of the p-median model when p and n vary. To our knowledge this is not a very well-studied problem when the road network is alternated especially when it is applied in a real world context. The aim in this study is to analyze how the optimal location solutions vary, using the p-median model, when the density in the road network is alternated. The investigation is conducted by the means of a case study in a region in Sweden with an asymmetrically distributed population (15,000 weighted demand points), Dalecarlia. To locate 5 to 50 service centers we use the national transport administrations official road network (NVDB). The road network consists of 1.5 million nodes. To find the optimal location we start with 500 candidate nodes in the network and increase the number of candidate nodes in steps up to 67,000. To find the optimal solution we use a simulated annealing algorithm with adaptive tuning of the temperature. The results show that there is a limited improvement in the optimal solutions when nodes in the road network increase and p is low. When p is high the improvements are larger. The results also show that choice of the best network depends on p. The larger p the larger density of the network is needed. 

An empirical test of the gravity p-median model

A customer is presumed to gravitate to a facility by the distance to it and the attractiveness of it. However regarding the location of the facility, the presumption is that the customer opts for the shortest route to the nearest facility.This paradox was recently solved by the introduction of the gravity p-median model. The model is yet to be implemented and tested empirically. We implemented the model in an empirical problem of locating locksmiths, vehicle inspections, and retail stores ofv ehicle spare-parts, and we compared the solutions with those of the p-median model. We found the gravity p-median model to be of limited use for the problem of locating facilities as it either gives solutions similar to the p-median model, or it gives unstable solutions due to a non-concave objective function.

Distance measure and the p-median problem in rural areas

The p-median model is used to locate P facilities to serve a geographically distributed population. Conventionally, it is assumed that the population patronize the nearest facility and that the distance between the resident and the facility may be measured by the Euclidean distance. Carling, Han, and Håkansson (2012) compared two network distances with the Euclidean in a rural region witha sparse, heterogeneous network and a non-symmetric distribution of thepopulation. For a coarse network and P small, they found, in contrast to the literature, the Euclidean distance to be problematic. In this paper we extend their work by use of a refined network and study systematically the case when P is of varying size (2-100 facilities). We find that the network distance give as gooda solution as the travel-time network. The Euclidean distance gives solutions some 2-7 per cent worse than the network distances, and the solutions deteriorate with increasing P. Our conclusions extend to intra-urban location problems.

Network density and the p-median solution

The p-medianmodel is commonly used to find optimal locations of facilities for geographically distributed demands. So far, there are few studies that have considered the importance of the road network in the model. However, Han, Håkansson, and Rebreyend (2013) examined the solutions of the p-median model with densities of the road network varying from 500 to 70,000 nodes. They found as the density went beyond some 10,000 nodes, solutions have no further improvements but gradually worsen. The aim of this study is to check their findings by using an alternative heuristic being vertex substitution, as a complement to their using simulated annealing. We reject the findings in Han et al (2013). The solutions do not further improve as the nodes exceed 10,000, but neither do the solutions deteriorate.

How does the use of different road networks effect the optimal location of facilities in rural areas?

The p-median problem is often used to locate P service facilities in a geographically distributed population. Important for the performance of such a model is the distance measure. Distance measure can vary if the accuracy of the road network varies. The rst aim in this study is to analyze how the optimal location solutions vary, using the p-median model, when the road network is alternated. It is hard to nd an exact optimal solution for p-median problems. Therefore, in this study two heuristic solutions are applied, simulating annealing and a classic heuristic. The secondary aim is to compare the optimal location solutions using dierent algorithms for large p-median problem. The investigation is conducted by the means of a case study in a rural region with an asymmetrically distributed population, Dalecarlia. The study shows that the use of more accurate road networks gives better solutions for optimal location, regardless what algorithm that is used and regardless how many service facilities that is optimized for. It is also shown that the simulated annealing algorithm not just is much faster than the classic heuristic used here, but also in most cases gives better location solutions.

How does data quality in a network affect heuristic solutions?

To have good data quality with high complexity is often seen to be important. Intuition says that the higher accuracy and complexity the data have the better the analytic solutions becomes if it is possible to handle the increasing computing time. However, for most of the practical computational problems, high complexity data means that computational times become too long or that heuristics used to solve the problem have difficulties to reach good solutions. This is even further stressed when the size of the combinatorial problem increases. Consequently, we often need a simplified data to deal with complex combinatorial problems. In this study we stress the question of how the complexity and accuracy in a network affect the quality of the heuristic solutions for different sizes of the combinatorial problem. We evaluate this question by applying the commonly used p-median model, which is used to find optimal locations in a network of p supply points that serve n demand points. To evaluate this, we vary both the accuracy (the number of nodes) of the network and the size of the combinatorial problem (p). The investigation is conducted by the means of a case study in a region in Sweden with an asymmetrically distributed population (15,000 weighted demand points), Dalecarlia. To locate 5 to 50 supply points we use the national transport administrations official road network (NVDB). The road network consists of 1.5 million nodes. To find the optimal location we start with 500 candidate nodes in the network and increase the number of candidate nodes in steps up to 67,000 (which is aggregated from the 1.5 million nodes). To find the optimal solution we use a simulated annealing algorithm with adaptive tuning of the temperature. The results show that there is a limited improvement in the optimal solutions when the accuracy in the road network increase and the combinatorial problem (low p) is simple. When the combinatorial problem is complex (large p) the improvements of increasing the accuracy in the road network are much larger. The results also show that choice of the best accuracy of the network depends on the complexity of the combinatorial (varying p) problem.

Testing the gravity p-median model empirically

Regarding the location of a facility, the presumption in the widely used p-median model is that the customer opts for the shortest route to the nearest facility. However, this assumption is problematic on free markets since the customer is presumed to gravitate to a facility by the distance to and the attractiveness of it. The recently introduced gravity p-median model offers an extension to the p-median model that account for this. The model is therefore potentially interesting, although it has not yet been implemented and tested empirically. In this paper, we have implemented the model in an empirical problem of locating vehicle inspections, locksmiths, and retail stores of vehicle spare-parts for the purpose of investigating its superiority to the p-median model. We found, however, the gravity p-median model to be of limited use for the problem of locating facilities as it either gives solutions similar to the p-median model, or it gives unstable solutions due to a non-concave objective function.

Simulation model using meta heuristic algorithms for achieving optimal arrangement of storage bins in a sawmill yard

Bin planning (arrangements) is a key factor in the timber industry. Improper planning of the storage bins may lead to inefficient transportation of resources, which threaten the overall efficiency and thereby limit the profit margins of sawmills. To address this challenge, a simulation model has been developed. However, as numerous alternatives are available for arranging bins, simulating all possibilities will take an enormous amount of time and it is computationally infeasible. A discrete-event simulation model incorporating meta-heuristic algorithms has therefore been investigated in this study. Preliminary investigations indicate that the results achieved by GA based simulation model are promising and better than the other meta-heuristic algorithm. Further, a sensitivity analysis has been done on the GA based optimal arrangement which contributes to gaining insights and knowledge about the real system that ultimately leads to improved and enhanced efficiency in sawmill yards. It is expected that the results achieved in the work will support timber industries in making optimal decisions with respect to arrangement of storage bins in a sawmill yard.