Research Proposal on "Logistics Simulation"

Research Proposal 8 pages (3243 words) Sources: 1+ Style: Harvard

[EXCERPT] . . . .

Distribution Planning Systems Based on the Traveling Salesman Problem]

The work of Applegate, Bixby, Chvatal and Cook (2007) entitled: "The Traveling Salesman Problem" states the following: "Given a set of cities along with the cost of travel between each pair of them, the traveling salesman problem, or TSP for short, is to find the cheapest way of visiting all the cities and returning to the starting point. The "way of visiting all the cities" is simply the order in which the cities are visited; the ordering is called a tour or circuit through the cities." (2007) This exercise which sounds modest according to Applegate, Bixby, Chvatal and Cook (2007) however it is

"...in fact one of the most intensely investigated problems in computational mathematics. It has inspired studies by mathematicians, computer scientists, chemists, physicists, psychologists, and a host of nonprofessional researchers. Educators use the TSP to introduce discrete mathematics in elementary, middle, and high schools, as well as in universities and professional schools. The TSP has seen applications in the areas of logistics, genetics, manufacturing, telecommunications, and neuroscience, to name just a few." (Applegate, Bixby, Chvatal, and Cook, 2007)

The origination of the assigned name "traveling salesman problem" is unknown Stated to be one of the earliest and most influential of TSP researchers was Merrill Flood of Princeton University and the RAND Corporation. It is reported that Flood revealed that he does not know "who coined the peppier name 'Traveling Salesman Problem' but that developments of this 'problem' began in the 1930s at Princeton University and that it was originally called the '48 States Problem' of Hassler Whitney.

The first written reference is stated to be in the work of Julia Robinson (1949) entitled: "On the Hamiltonian Game (a Traveling Salesman Problem)" and that it was "clear from the writing that she was not introducing the name. All we can conclude is that sometime between the 1930s or 1940s, most likely at Princeton, the TSP took on its name, and mathematicians began to study the problem in earnest." (Applegate, Bixby, Chvatal and Cook, 2007)

It is related that the Commis-Voyaguer

"explicitly described the need for good tours" in the translated version as follows: "Business leads the traveling salesman here and there, and there is not a good tour for all occurring cases; but through an expedient choice and division of the tour so much time can be won that we feel compelled to give guidelines about this. Everyone should use as much of the advice as he thinks useful for his application. We believe we can ensure as much that it will not be possible to plan the tours through Germany in consideration of the distances and the traveling back and fourth, which deserves the traveler's special attention, with more economy. The main thing to remember is always to visit as many localities as possible without having to touch them twice." (Applegate, Bixby, Chvatal and Cook, 2007)

It is related that the mode of travel that the traveling salesmen utilized varied in nature and included "horseback and stagecoach to trains and automobiles." (Applegate, Bixby, Chvatal and Cook, 2007) in each case of traveling choice "the planning of routes would often take into consideration factors other than simply the distance between the cities, but devising good TSP tours was a regular practice for the salesman on the road." Applegate, Bixby, Chvatal and Cook, 2007) it is stated that when a mathematicians is investigating a problem and seeking ideas or solutions that generally most mathematicians will take a pencil and sketch a few ideas and specifically stated is "...Geometric instances of the TSP, where cities are locations and the travel costs are distances between pairs, are tailor-made for such probing." Applegate, Bixby, Chvatal and Cook, 2007) the ability for visualization of tours and for easy manipulation of them by hand is stated to have most "certainly contributed to the widespread appeal of the problem, making the study of the TSP accessible to anyone with a pencil and clean sheet of paper." (Applegate, Bixby, Chvatal and Cook, 2007)

The work of Golden, Raghavan, and Wasil entitled: "The Vehicle Routing Problem" states that in the well-known "Vehicle Routing Problem (VRP) a set of identical vehicles, based at a central depot is to be optimally routed to supply customers with know demands subject to vehicle capacity constraints...and an important variant of the VRP arises when a fleet of vehicles characterized by different capacities and costs is available for distribution activities. The problem is known as the Mixed Fleet VRP or as the Heterogeneous Fleet VRP." (2008) the Vehicle Routing Problem (VRP) is stated as one of the "most studied combinatorial optimization problems and is concerned with the optimal design of routes to be used by a fleet of vehicles to serve a set of customers." (Golden, Raghavana and Wasil, 2008)

First proposed by Dantzig and Ramser this subject has been the focus of hundreds of papers seeking the "exact and approximate solution of the many variants of this problem including the Capacitated VRP (CVRP), in which a homogenous fleet of vehicles is available and the only constraint is the vehicle capacity or the VRP with Time Windows (VRPTW) where customers may be served within a specified time interval and the schedule of the vehicle trips needs to be determined." (Golden, Raghavana and Wasil, 2008) Golden, Raghavan and Wasil report that greater attention more recently has been given to "more complex variants of the VRP, sometimes named 'rich' VRPs, that are close to the practical distribution problems than the VRP models." (2008) These variants are stated to be characterized "...by multiple depots, multiple trips to be performed by the vehicles, multiple vehicle types or other operational issues such as loading constraints." (Golden, Raghavana and Wasil, 2008)

The work of Madsen, Larsen and Solomon entitled: "Dynamic Vehicle Routing Systems: Survey and Classification" states that Psaraftis uses the classification as follows of the static routing problem: "if the output of a certain formulation is a set of preplanned routes that are not re-optimized and are computed from inputs that do not evolve in real time." (nd) Psaraftis refers to a problem as being dynamic as follows: "if the output is not a set of routes, but rather a policy that prescribes how the routes should evolve as a function of those inputs that evolve in real-time." (Madsen, Larsen and Solomon, nd)

The Static Vehicle Routing Problem is stated to be defined by the following characteristics: (1) all information relevant to the planning of the routes is assumed to be known by the planner before the routing process begins; and (2) information relevant to the routing does not change after the routes have been constructed." (Madsen, Larsen and Solomon, nd) Madsen, Larsen and Solomon states that the dynamic counter part of the static vehicle problem as just defined can be formulated as follows: "(1) Not all information relevant to the planning of routes is known by the planner when the routing process beings; and (2) information can change after the initial routes have been constructed. (nd) Two types of requests are stated to be involved in many DVRP and those are stated as: (1) advance requests -- also referred to as static customers since these requests are received prior to the process of routing had begun; and (2) immediate requests -- referred to as dynamic customers as these appear in real-time during the route extension. (Madsen, Larsen and Solomon, nd)

Madsen, Larsen and Solomon states the "...more restricted and complex the routing problem is, the more complicated the insertion of new dynamic customers will be." (Madsen, Larsen and Solomon, nd) the example given is that inserting the new customers in a "time window constrained routing problem will usually be more difficult than in a non-time constrained problem." (Madsen, Larsen and Solomon, nd)

The work of Rizzoli, Oliverio, Montemanni and Gambardella (2004) entitled: "Any Colony Optimization for Vehicle Routing Problems: From Theory to Applications" states that the Vehicle Routing Problem is concerned with the transport of objects between the original location of manufacture and customers via a fleet of vehicles. The VRP can be translated to many domains in the world including delivery of mail, routing of school buses, collection of solid waste, distribution of heating oil, pick-up and delivery of parcel and many other such systems. Solutions to VRP are those which identify the optimal route of delivery to all customers via a fleet of vehicles and involves ensuring service to all customers within the stated constraints of operation and minimization of the cost associated with transportation. ((Rizzoli, Oliverio, Montemanni and Gambardella, 2004, paraphrased)

Rizzoli, Oliverio, Montemanni and Gambardella relate that the formulation of the VRP is as a "mathematical programming problem, defined by objective function, and a set of constraints." (Rizzoli, Oliverio, Montemanni and Gambardella, 2004) Objectives are stated to "measure the fitness of a solution. They can be multiple and often they are also conflicting. The most common objective is the minimization of transportation costs… READ MORE

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