In several practical cases the cost matrix satisfies the triangle inequality, such that cik + ckj ? cij for any i, j, k ? V." (Toth and Vigo, 1998, cited in Vural (2003). Vural (2003) states that the primary attributes within the configuration of the majority of VRP problems are those as follows:

(1) Number of vehicles;

(2) Vehicles homogeneity/heterogeneity;

(3) Time windows;

(4) Backhauls;

(5) Splitting/Unsplitting of Load;

(6) Single Depot/Multi Depot;

(7) Static/Dynamic Service Needs; (8) Precedence/Coupling Constraints. (Vural, 2003)

Heuristic and Meta-heuristic Models

The work of Badr (nd) Solving Dynamic Vehicle Routing: An Alternative Metaheuristic Approach" states that Dynamic Vehicle Routing Problem (DVRP) can be considered a good example of a distribution context, because of the fact that intelligent manipulation of real-time information can distinguish between one company and another by superior on-time service. Problems of both generic and vehicle routing (VRP) and dynamic vehicle routing (DVRP) are identical. But in VRP all routing and demand information are certain known prior to the day of operation, whereas in DVRP part of or all of the necessary information is available only at the day of operation." (Badr, nd) the DVRP significance is stated to be "crystallized by the variety of environments it can model." (Badr, nd)

The work of Gambardella, Taillard and Agazzi (1999) entitled: "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows" states that one of the most successful and exact methods for the CVRP is the method known as the: "…k-tree method of (Fisher, 1994) that succeeded in solving a problem with 71 customers. However, there are smaller instances that have not been exactly solved yet. To treat larger instances, or to compute solutions faster, heuristic methods must be used." (Gambardella, Taillard and Agazzi, 1999)

Tabu and Genetic Search

Among the best heuristic methods are tabu searches (Taillard, 1993, Rochat and Taillard, 1995, Rego and Roucairol, 1996, Xu and Kelly, 1996, Toth and Vigo, 1998) and large neighborhood search (Shaw, 1998). The CVRP can be extended in many ways." (Gambardella, Taillard and Agazzi, 1999) the service time si for each customer (with so = 0) and a time limit over the duration of each tour can be considered. The goal is again to search for a set of tours that minimizes the sum of the travel times." (Gambardella, Taillard and Agazzi, 1999)

Gambardella, Taillard and Agazzi (1999) states that in addition to the CVRP features, included in this problem is

"…for the depot and for each customer ci (i = 0,..., n) a time window [bi, ei] during which the customer has to be served (with b0 the earliest start time and e0 the latest return time of each vehicle to the depot) the tours are performed by a fleet of v identical vehicles. The additional constraints are that the service beginning time at each node ci (i = 1,..., n) must be greater than or equal to bi, the beginning of the time window, and the arrival time at each node ci must be lower than or equal to ei, the end of the time window. In case the arrival time is less than bi, the vehicle has to wait till the beginning of the time window before starting servicing the customer." (Gambardella, Taillard and Agazzi, 1999)

Planning under Certainty and Uncertainty

The work of Kelywegt and Shapiro (2000) entitled: "Stochastic Optimization" states that decisions are often made by decision makers "in the presence of uncertainty. Decision problems are often formulated "as optimization problems and thus in many situations decision makers wish to solve optimization problems which depend on parameters which are unknown." (Kleywegt and Shapiro, 2000) Formulation and solution of such type problems are generally very difficult "both conceptually and numerically.' (Kleywegt and Shapiro, 2000)

The conceptual stage of modeling contains difficulty since there are various ways that formalization of the uncertainty can be modeled formally. The attempt in formulating optimization problems is to identify a suitable trade-off between "the realism of the optimization model, which usually affects the usefulness and quality of the obtained decisions, and the tractability of the problem, so that it could be solved analytically or numerically." (Kleywegt and Shapiro, 2000) Kleywegt and Shapiro state a static optimization problem as follows in relation to operation under uncertainty:

"Suppose we want to maximize an objective function G (x, ?), where x denotes the decision to be made, X denotes the set of all feasible decisions, ? denotes an outcome that is unknown at the time the decision has to be made, and ? denotes the set of all possible outcomes." (Kleywegt and Shapiro, 2000)

Kelywegt and Shapiro state that there are

"…several approaches for dealing with optimization under uncertainty" for example in the case of the company...

Upon the selling period beginning there is not enough remaining time to change the purchase or manufacture decision so the product quantity is a 'given' and the decision made prior to the selling period remains regardless of whether much more of the product could have been sold. Therefore, the situation is such that the decision has to have been made prior to the knowledge of the eventual outcome is known to the decision maker. (Kleywegt and Shapiro, 2000)
Stochastic & Dynamic Simulation-based Planning in DPS

The work of Ganesh, Dhanlakshmi, Thangavelu, Parthiban (2009) entitled: 'Hybrid Artificial Intelligence Heuristics and Clustering Algorithm for Combinatorial asymmetric Traveling Salesman Problem" states that stochastic and/or dynamic information in most real life applications "occurs parallel to the routes being carried out." (Ganesh, Dhanlakshmi, Thangavelu, and Parthiban 2009)

Real-life examples of stochastic and/or dynamic routing problems include the distribution of oil to private households, the pick-up of courier mail/packages and the dispatching of buses for the transportation of elderly and handicapped people." (Ganesh, Dhanlakshmi, Thangavelu, and Parthiban, 2009)

It is related by Ganesh, Dhanlakshmi, Thangavelu, and Parthiban, 2009) that in these specific examples unknown may include:

(1) customer profiles;

(2) time to begin service;

(3) geographic location; and

4) actual demand and these factors may not be known when planning begins or even at the time service has begun for the customers with advance requests.

Stated as two distinct features result in the planning of routes that are of high quality in this environment more complex are those of:

(1) constant change; and (2) time horizon. (Ganesh, Dhanlakshmi, Thangavelu, and Parthiban, 2009)

Simulated Annealing

Simulated annealing (SA) is stated in the work of Ganesh, Dhanlakshmi, Thangavelu, and Parthiban (2009) to be a "generalization of a Monte Carlo method for examining the equation of state and frozen states of n-body systems. The concept is based on the manner in which liquids freeze or metal recrystalize in the process of annealing." (Ganesh, Dhanlakshmi, Thangavelu, and Parthiban, 2009)

The annealing process involves a melt generally at a high temperature "and disordered, is slowly cooled so that the system at any time is approximately in thermodynamic equilibrium." (Ganesh, Dhanlakshmi, Thangavelu, and Parthiban,, 2009) the procession of cooling results in the system becoming more ordered and approaching a ground state that is termed as 'frozen'…at T=0. Therefore this process can be viewed as "an adiabatic approach to the lowest energy state." (Ganesh, Dhanlakshmi, Thangavelu, and Parthiban,, 2009)

Ganesh, Dhanlakshmi, Thangavelu, and Parthiban (2009) state that in order to use this specific analogy when referring to thermo dynamic systems in attempting mathematical optimization solutions the following elements are required:

(1) a description of possible system configurations, i.e. some way of representing a solution to the minimization (maximization) problem, usually this involves some configuration of parameters that represent a solution;

(2) a generator of random changes in a configuration; these changes are typically solutions in the neighborhood of the current configuration, for example, a change in one of the parameters."

Capacitated Vehicle Routing

The work of Ormerod and Slavin (nd) entitled: "Human Solutions to the Capacitated Vehicle Routing Problem" states that the Vehicle Routing Problem (VRP) "arises naturally in transportation, distribution and logistics." The focus of the Vehicle

Routing Problem (VRH) is a given set of customers with needs (loads) and vehicles (capacity limited) and finding the shortest set of tours from the depot(s) that collect/deliver specified loads. Other concerns are stated as the Intersection of TSP and bin packing problems. The work of Matson, Miller and Matson (1999) entitled: "A Capacitated Vehicle Routing Problem for Just-in-time delivery" reports a work that has its focus on the "formulation and solution of the problem of planning vehicle routes for material delivery within the premises of a plant working under a just-in-time production system. The unique characteristic of this problem is that the quantity to be delivered at each of the demand nodes is a function of the route taken by the vehicle assigned to serve that node." ( ) Matson, Miller and Matson (1999) state as follows concerning the just-in-time system:

"The JITCVRP can be defined as follows. There…