Logistics simulation models and applications

¶ … 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 as a function of the travelled distance or of the travel time; fixed costs associated with vehicles and drivers can be considered, and therefore the number of vehicles can also be minimized." (Rizzoli, Oliverio, Montemanni and Gambardella, 2004)

According to Rizzoli, Oliverio, Montemanni and Gambardella objectives are that which "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 as a function of the traveled distance or of the travel time; fixed costs associated with vehicles and drivers can be considered, and therefore the number of vehicles can also be minimized." (Rizzoli, Oliverio, Montemanni and Gambardella, 2004) Vehicle efficiency is another objective to consider and this is stated to be expressed as "the percentage of load capacity" and it is held that the higher the load capacity the better. The objective function is also used in representation of 'soft' constraints described as constraints "...which can be violated paying a penalty." (Rizzoli, Oliverio, Montemanni and Gambardella, 2004) Both independent variables and dependent variables are contained within the objective function and under the planner's control the independent variables are stated to be decision variables and the dependent variables are stated to be the "consequence of the assumed decisions." (Rizzoli, Oliverio, Montemanni and Gambardella, 2004)

The problem's solution is stated to be given "by the decision variables returning to the best evaluation of the objective function." (Rizzoli, Oliverio, Montemanni and Gambardella, 2004) in the case of VRP the decisions is that which define how the visits to the customers will be sequenced and specifically through defining a set of routes. In order to discover the values which are to be assigned as decision variables needed is a model of the vehicle routing system which is a model 'defined by the constraints that establish the relationships among independent and dependent variable and set limits of variable's values.

Stated as inclusive in the elements that serve to "define and constrain the model" are the elements relating to: (1) the road network (which describes the connectivity among customers and depots; (2) the vehicles (transporting goods between customers and depots on the road network; (3) the customers (placing orders and receiving goods). (Rizzoli, Oliverio, Montemanni and Gambardella, 2004) the road network is stated by Rizzoli, Oliverio, Montemanni and Gambardella to be presented as a graph in which "depots and customers are placed on nodes and the edges represent the distance, in space and/or time between two nodes." (Rizzoli, Oliverio, Montemanni and Gambardella, 2004)

Rizzoli, Oliverio, Montemanni and Gambardella state that the road network graph may be obtained from a map that details the distribution area with the depots and customers geo-referenced on it. Shortest routes can be discovered through use of standard algorithms in regards to time and distance between all node couples enabling the distance matrix to be constructed. Depending on the metric that is adopted various VRP instances may arise and that stated example is in relation to travel time and it depending on the time of day which means that the Time Dependent VRP is encountered.

When the various elements of the problem are combined it is possible to define "a whole family of different VRPs." (Rizzoli, Oliverio, Montemanni and Gambardella, 2004) Some of these are those as follows: (1) Capacitated Vehicle Routing Problem (CVRP); (2) VRP with Time Windows (VRPTW); (3) Time Dependent Variable of VRP with Time Windows (TDVRPTX); (4) the VRP with Pickup and Delivery (VRPPD); and (5) the Dynamic VRP (DVRP). (Rizzoli, Oliverio, Montemanni and Gambardella, 2004)

The work of Bowersox and Closs entitled: "Simulation in Logistics: A Review of Present Practice and a Look to the Future" states that there are generally three model categories used in logistics planning which are the following: (1) Analytic; (2) Heuristic; and (3) Simulation. (nd) Analytic models are stated to use mathematic models to make identification of the best solution to the problem being analyzed however the models that use heuristic or simulation procedures are stated to use "numerical techniques to quantify specific problem solutions." (Bowersox and Closs, nd)

The distinctive feature of simulation is stated to be the capacity of simulation to "include stochastic situations. In most logistical planning situations, uncertainty and resulting variance are significant considerations." (Bowersox and Closs, nd) the capability of simulation technologies is the incorporation of variance "across either a dynamic or static planning horizon." (Bowersox and Closs, nd) Stated differently "probability can be introduced into analyses dealing with a specific point in time problem (warehouse location) or across time (inventory/customer service relationships). " (Bowersox and Closs, nd) Uncertainty is effectively dealt with by simulations therefore these are used frequently in solving problems in which there is a requirement of space and time integration. Stated as an example is that of network inventory. Logistic planning situations are either: (1) structural; or (2) operational. (Bowersox and Closs, nd)

Structural problems are typically characterized as location of the facility and design of the distribution channel. Stated as typical operational analyses are: (1) integration of customer service; (2) inventory; (3) transportation and (4) production. (Bowersox and Closs, nd) Structural analysis involves "the number of facilities and the channel design relationships facilities and/or channel participants." (Bowersox and Closs, nd) Facility analysis has as its focus the "geographical location and arrangement of the production, warehouse and to a lesser extent retail stores." (Bowersox and Closs, nd) Stated to be an issue that is "closely related to the facility structure" is that of the channel design used in supporting marketing.

Operational analysis is stated to be the second in the planning and evaluation categories used in simulation and it is stated that operational analysis "considers spatial product positioning and alternative timing. Operational analysis is typically focused on the "integration of raw material and finished goods inventory, service levels, and production planning." (Bowersox and Closs, nd) Bowersox and Closs additionally related that simulation tools are "becoming increasingly dynamic to capture the interaction between level of service and potential revenue generation." (nd) Due to the fact that there are economies of scale in logistics which are substantial "increased demand offers the potential to reduce costs, which in turn, offers potential incentives to increase demand." (nd) it is necessary to evaluate this cycle in terms of finance to make sufficient strategic marketing plans.