" (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...

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.

Bowersox and Closs state that revenue elasticity replication in simulation of profit enhancement "can be achieved by a combination of probabilities and dynamics." (nd) Advanced simulation methodologies are utilized in this case which is quite complex and stated as an example are the complexities that exist between: (1) price, demand, and cost of providing service for a market segment; (2) volume shipped and price to ship for a segment; and (3) service level offered and segment demand as illustrated by the desire for market presence. (Bowersox and Closs, nd)

The work of Chang and Makatsoris (nd) entitled: "Supply Chain Modeling Using Simulation" reviews the today's marketing and relates that it is "highly competitive" and an environment in which manufacturers face the challenge of reducing manufacturing cycle time, delivery lead-time and inventory reduction." (Chang and Makatsoris, 2000) a typical supply chain is shown in the figure located in Appendix a and on page 12 of this study. Figure 2 located just following the previously stated figure and also located on page 12 of this study is a listing of the Functionalities as well as the Areas of Supply Chain Management as stated in the work of Chang and Makatsoris, 2000) These include the areas of: (1) demand planning; (2) master planning; (3) procurement; (4) transportation; and (5) manufacturing. Expected benefits of supply chain management are stated to include those as follows: (1) material is coordinated better and no utilization loss occurs due to lack of needed materials. This is referred to as "throughput improvements." (Chang and Makatsoris, 2000) (2) Reduction of cycle time by examining constraints and alternate methods or processes in the supply chain ultimately reducing the time to cycle; (3) reduction of the costs of inventory through "demand and supply visibility" which effectively brings lower the requirement of holding inventory at high levels and in the environment of uncertainty; (4) Capacity to know when to purchase material upon the bases of demands of customers, logistics including capacity to produce and other materials needed to produce; (5) transportation that is optimized (optimizing logistics and vehicle loads); (6) sales increases with "real time visibility across the supply chain (alternate routings, alternate capacity) enables to increase order fill rate: (7) "Analysis of the supply chain management can help to predict propagation of disturbance to downstream"; and (8) Responsiveness of customer increases resulting in the "...understanding [of] the capability to deliver based on availability of materials, capacity and logistics. (Chang and Makatsoris, 2000)

The work of Laporte, Gendreau, Potvin, and Semet (2000) entitled: Classical and Modern Heuristics for the Vehicle Routing Problem" conducts a survey of heuristics for the 'Vehicle Routing Problem' reports that there have been "...several "families of heuristics proposed for the VRP." (Laporte, Gendreau, Potvin, and Semet, 2000) These are classified as: (1) classical heuristics (developed between 1960-1990); and (2) metaheuristics (over the…