Branch and Bound technique to solve Integer Linear Programming
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The document discusses integer linear programming (ILP) and its classification as a NP-hard problem, focusing on the branch and bound technique as a solution method. This involves dividing the problem into smaller sub-problems and estimating their solutions to find optimal outcomes. Examples of problems that can be formulated as ILPs are also provided, showcasing the practical application of these concepts.
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Critical Thinking Project - Knowing the world
Critical Thinking Project - Knowing the worldKaivalya Shah The document discusses a study where the authors interviewed 17 individuals to gather their opinions on Narendra Modi, focusing on various logical fallacies present in their responses. Key concepts include composition fallacy, appeal to emotion, and argument from analogy, demonstrating how people's perceptions of Modi's policies and leadership are shaped by different arguments. The study aims to analyze public sentiment towards Modi's governance and the effectiveness of his policies.
Paper Review: ENERGY-EFFICIENT WIRELESS COMMUNICATIONS TUTORIAL, SURVEY, AND ...
Paper Review: ENERGY-EFFICIENT WIRELESS COMMUNICATIONS TUTORIAL, SURVEY, AND ...Kaivalya Shah This paper reviews energy-efficient wireless communications, providing an introduction to basic concepts, summarizing existing fundamental works and advanced techniques, and discussing open issues. Some techniques discussed include OFDMA networks, MIMO techniques, relay transmission, and resource allocation between signaling and data symbols. The paper concludes that while techniques like OFDMA, MIMO, and relaying can improve energy efficiency, joint optimization across techniques and consideration of practical factors is still needed, particularly for multi-user and multi-cell scenarios.
SkyViewer: An in-browser solution to fast video calling
SkyViewer: An in-browser solution to fast video callingKaivalya Shah This document summarizes the SkyViewer computer networks project created by four students. SkyViewer allows for in-browser video chatting using WebRTC without plugins. It supports major browsers and platforms. The motivation was to create a seamless video chatting website without downloads. WebRTC enables this through standards for connectivity, encryption, and adapting to network conditions. Users can chat by entering a temporary username on the website to connect in their browser in a peer-to-peer fashion. The project implements WebRTC, PubNub for signaling, and other modules to handle connections, media streams, and integration with TURN servers. Future improvements include calling features and remote access abilities.
Import Trends of India
Import Trends of IndiaKaivalya Shah The document discusses the concept of imports in India, highlighting their significance in international trade as they reflect domestic demand and economic growth. It explains the impact of exchange rates on import competitiveness, the consequences of trade deficits, and India's reliance on imports, particularly in petroleum and precious stones. Additionally, it outlines factors affecting the balance of trade and India's position as a leading importer across various commodity categories.
An Ethical Analysis of Caste based Reservation in India
An Ethical Analysis of Caste based Reservation in IndiaKaivalya Shah The document discusses caste-based reservation in India, highlighting the historical context, ethical implications, and arguments for and against the policy. It addresses concerns about meritocracy, generational benefits, and the need for a balanced change in the system to ensure equal opportunities while still assisting backward classes. The conclusion suggests that individual equality should be prioritized alongside caste-based reservations, advocating for reforms such as the introduction of economic backward classes and stricter regulations against misuse.
About the Montreal Protocol
About the Montreal ProtocolKaivalya Shah The Montreal Protocol aimed to phase out ozone depleting substances to protect the ozone layer. It was agreed in 1987 and entered into force in 1989. It has undergone several revisions to strengthen commitments. Key ozone depleting substances like CFCs and HCFCs are being phased out according to set timelines. As a result of compliance, the ozone layer is projected to recover by 2050-2070, demonstrating the Protocol's effectiveness through international cooperation.
Xerography
XerographyKaivalya Shah This document provides an overview of xerography, the process of dry photocopying. It describes how xerography works, beginning with uniformly charging a photoconductor drum coated with selenium. When a document is imaged onto the drum with a laser, the illuminated areas discharge, leaving a latent electric image. Toner particles are then deposited onto the drum, adhering to the charged image areas. The toner image is transferred to paper using an electric field and then fused using heat and pressure. Xerography was invented by Chester Carlson in 1938 and relies on the unique property of photoconductors to conduct electricity when exposed to light.
Amazon Web Services (AWS) Elastic Compute Cloud (EC2)
Amazon Web Services (AWS) Elastic Compute Cloud (EC2)Kaivalya Shah This document provides an overview of Amazon Elastic Compute Cloud (EC2) including:
- EC2 allows users to launch virtual servers in Amazon's datacenters and pay for compute capacity on a short-term basis.
- There are different types of EC2 instances optimized for various use cases like compute, storage, memory, and GPU.
- Users can launch instances using Amazon Machine Images (AMIs) or templates with pre-installed software.
- Additional services like Elastic Block Storage (EBS) provide persistent storage volumes that can be attached to EC2 instances.
Introduction to Storage technologies
Introduction to Storage technologiesKaivalya Shah The document discusses various storage technologies used for data storage, including magnetic disk drives, solid state drives (SSDs), direct attached storage (DAS), network attached storage (NAS), and storage area networks (SAN). Each technology is explained in terms of its structure, performance, and typical use cases, highlighting differences like speed and cost. The document also describes the protocols and interfaces associated with these storage solutions.
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【复刻约翰霍普金斯大学成绩单信封,Buy The Johns Hopkins University Transcripts】
购买日韩成绩单、英国大学成绩单、美国大学成绩单、澳洲大学成绩单、加拿大大学成绩单（q微1954292140）新加坡大学成绩单、新西兰大学成绩单、爱尔兰成绩单、西班牙成绩单、德国成绩单。成绩单的意义主要体现在证明学习能力、评估学术背景、展示综合素质、提高录取率，以及是作为留信认证申请材料的一部分。
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我们承诺采用的是学校原版纸张（原版纸质、底色、纹路）我们工厂拥有全套进口原装设备，特殊工艺都是采用不同机器制作，仿真度基本可以达到100%，所有成品以及工艺效果都可提前给客户展示，不满意可以根据客户要求进行调整，直到满意为止！
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Branch and Bound technique to solve Integer Linear Programming
- 1. BRANCH AND BOUND TECHNIQUE TO SOLVE INTEGER LINEAR PROGRAMMING GROUP MEMBERS AYAM AJMERA – 1401096 KAIVALYA SHAH – 1401108 RATNESH SHAH – 1401110
- 2. INTEGER PROGRAMMING AND INTEGER LINEAR PROGRAMMING Integer programming (IP) is a mathematical optimization program in which some of the variables or sometimes all the variables are restricted to be integers. Integer Linear programming (ILP) is a special case of IP where the objective function and constraints are linear. Integer programming is considered as a NP-Hard problem. A special case of 0-1 integer linear programming, is included in one the Karp’s 21 NP-complete problems. Standard form of ILP is as follows maximize 𝑐 𝑇 𝑥 subject to 𝐴𝑥 + 𝑠 = 𝑏 𝑠 ≥ 0 and 𝑥 𝜖 𝑍 𝑛
- 3. INTEGER LINEAR PROGRAMMING There are 2 variants of integer linear programming Mixed Integer Linear Programming – some variables are constrained to be integers, while others can be non-integers. Zero-One Linear Programming – variables are bounded to be either 0 or 1 i.e., binary. Some problems that can be formulated as ILP’s are as follows Travelling salesman problem Vertex cover and other covering problems Set packing and other packing problems Boolean Satisfiability
- 4. BRANCH AND BOUND ALGORITHM Branch and bound is an algorithm design paradigm for discrete and combinatorial optimization problems, as well as general real valued problems. A branch-and-bound algorithm consists of forming a rooted tree where the root is the full set of candidate solution and the leaves are the set of candidate solutions (or subset of solution set). Before enumerating the candidate solutions of a branch, the branch is checked against upper and lower bounds on the optimal solution, and is discarded if it cannot produce a better solution than the best one found so far by the algorithm. The method was firstly proposed by A. H. Land and A. G. Doig in 1960 for discrete programming. It is the most commonly used tool for solving NP-hard optimization problems.
- 5. BRANCH AND BOUND TECHNIQUE TO SOLVE ILP This is a type of divide and conquer method. The original problem is divided into multiple smaller problems (branching of a problem). This is the dividing part of the method. The conquering part is done by estimating how good a solution, we can get from these smaller problems. For this task, we may have to divide the small problem further, until we get a problem which we can handle efficiently. So, the part where the original problem is divided into smaller parts is called branching and the part where we try to estimate the goodness of the solution is called bounding.
- 6. LINEAR PROGRAMMING RELAXATION For an integer programming model P, the linear programming model we get by dropping the requirements that all the variables must be integers is called Linear Programming Relaxation. The steps are: Divide a problem into different sub-problems. Calculate the LP relaxation of a sub-problem The LP problem has no feasible solution. The LP problem has an optimal integer solution. Compare this optimal solution with the best solution we know. The LP problem has an optimal solution that is worse than the incumbent. In all the cases above, we know all we need to know about that sub-problem. We say that sub-problem is fathomed. The LP problem has an optimal solution that are not all integers, better than incumbent. In this case, we would have to divide this sub-problem further and repeat.
- 7. EXAMPLE Maximize 3x1 + 5x2 Subject to 2x1 + 4x2 < 25 x1 < 8 2x2 < 10 xi > 0, xi’s are integers
- 8. ITERATION 1
- 9. ITERATION 2
- 10. ITERATION 3
- 11. ITERATION 4
- 12. We can see that the solution of the objective function obtained from node 3F is 33.5 which is less than the solution obtained from the objective function from node 1A which is 34. So, we cannot further branch from node 3F. The final answer is obtained from node 1A which is, x1 = 8, x2 = 2, and z = 34
- 13. THANK YOU