In this project, Vehicle Routing Problem (VRP) is studied. VRP is an NP-hard combinatorial optimization problem. It appears in a large number of real world situations, such as transportation of people and products, delivery services, garbage collection etc. It can be applied everywhere, for vehicles, trains, plains; that is why Vehicle Routing Problem is of great practical importance in real life. The Vehicle Routing Problems (VRPs) are the ones concerning the distribution of goods between depots and customers. In particular, the solution of VRP calls for the determination of a set of routes, each performed by a single vehicle that starts and ends at its own depot, such that all requirements of the customers are fulfilled, all operational constraints…
Traveling Salesman Problem (TSP) is one of combinatorial optimization problems. X TSP is NP-hard problem which defined as a set of cities and each city should be visited once with minimum tour length. This paper solved this problem using Firefly Algorithm (FA) and k-means clustering by three steps: cluster the nodes, finding optimal path in each cluster and connect the clusters. The first step is to divide all nodes into sub-problems using k-means clustering, the second step is to use FA to find…
The artificial bee colony (ABC), an optimization technique is based upon the intelligent moving behavior of honey bee swarm was proposed by Karaboga in 2005. This kind of new Meta heuristic is inspired by the clever foraging behavior of honey bee swarm. The criteria presented in the work is for numerical function optimization. The advantage of ABC is that the global search ability in the algorithm is implemented by introducing neighborhood source production mechanism. Rao et al.…
all the tours.
* Temperature is cooled by the predetermined cooling factor in each iteration.
* Once the process ends we have the best possible .
C. Pseudo Code
1. Choose an initial tour S
2. Choose a temperature = > 0
3. Repeat :
a.Choose a new tour S’
b. Let = , where is energy (length) of tour
c. If , accept new tour i.e.,
d. Else if , accept new tour
e. Else reject new tour
f. Reduce temperature according to cooling factor
4. Until termination conditions are met
Ant and Grasshopper
Should the ant give food to the grasshopper?
Yes and no. Now I say both for a variety of different reasons. I have a feeling that the ant in this situation is not mean enough to just let the grasshopper go hungry. But is the grasshopper allowed to just get off without any punishment for not working hard? This is a classic argument of mercy verses justice. The ant can't show the grasshopper the mercy that he desires and at the same time have…
What is the maximum value the queues attain? Can we control the distribution such that we achieve a maximal outflow? What is the maximal load of the factory? How long does it take to process a certain type of snack or cookie? The question we ask is how to control the flow through the network so that a maximum number of goods can be produced and storage costs are minimized. In a simulation only the product flow propagates according to user-defined distribution rates. The simplest choice is…
Divide and Conquer Strategies:
Divide and conquer is an algorithm which design paradigm based on multi-branched recursion. This designed paradigm consists of following phases:
1) Break the problem (divide): Breaking the problem into several sub-problems that are smaller in size.
2) Solve the sub problem(conquer) : Solve the sub-problem recursively .
3) Combine the solutions (Merge): Combine solutions to subproblems to create a solution to the original problem.
This technique is the basis of…
A. Mumford-Shah segmentation functional
Mumford and Shah proposed this segmentation method based on a variational platform. Let Ω certainly be a bounded open amount of region R in addition to u_0 is preliminary image data. Segmentation of this kind of image into homogeneous items is completed via the search for a pair of components (u, K), where K is a number of contours, and u is usually a piecewise smooth approximation of u_0.The minimization of an energy functional (u, K) in a way that u…
4 The Simplex Method
As we have seen, a linear programming problem forms a convex polygon in the best possible scenario. It is imperative to obtain a process that would assist in determining the optimal solution without the need to examine the graphical representation. The need for an algorithm that would perform such process was essential in the early days of the formulations of linear programming problems. Although challenging, the task was accomplished by a mathematician of the twentieth…
provide details of the four modules of this invention. However, any one module of this invention can’t solve our problem alone. It is the interaction of all parts that allow the user to understand and improve the optimization models.
We begin by defining the optimization problem. Our objective is to maximize or minimize the sum of objective terms by selecting optimal feasible levels of the independent variables. The set of process limits or constraints define the feasible region. For…