Two Phase Method - Part 2

Duration: 14 min

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Module outline

  1. Paper - 1 | Unit - 1 | Teaching Aptitude: Concept and Objectives of Teaching / शिक्षण की अवधारणा एवं उद्देश्य, Levels of Teaching / शिक्षण के स्तर, Learner's Characteristics / शिक्षार्थी की विशेषताएँ, Learning: Nature and Theories / अधिगम: प्रकृति एवं सिद्धांत, Cognition, Memory and Metacognition / संज्ञान, स्मृति एवं अधिसंज्ञान, Factors Affecting Teaching / शिक्षण को प्रभावित करने वाले कारक, Effective Teaching and the Teacher / प्रभावी शिक्षण एवं शिक्षक, Instructional Objectives and Lesson Planning / अनुदेशनात्मक उद्देश्य एवं पाठ नियोजन, Methods of Teaching / शिक्षण विधियाँ, Offline and Online Teaching Methods / ऑफ़लाइन एवं ऑनलाइन शिक्षण विधियाँ, Teaching Support System / शिक्षण सहायक प्रणाली, Evaluation Systems / मूल्यांकन प्रणाली, Biological Basis of Emotion / संवेग का जैविक आधार, Central and Peripheral Nervous Systems / केन्द्रीय और परिधीय तंत्रिका तंत्र
  2. Paper - 1 | Unit - 2 | Research Aptitude: Research: Meaning, Features and Scientific Method / अनुसंधान: अर्थ, लक्षण व वैज्ञानिक विधि, Research Paradigms and Approaches / अनुसंधान के प्रतिमान एवं उपागम, Types of Research / अनुसंधान के प्रकार, Variables and Measurement / चर एवं मापन, Methods of Research / अनुसंधान की विधियाँ, Steps of Research / अनुसंधान के चरण, Sampling / प्रतिचयन, Tools and Methods of Data Collection / आँकड़ा संग्रह के उपकरण एवं विधियाँ, Data Analysis and Statistics in Research / आँकड़ा विश्लेषण एवं शोध सांख्यिकी, Thesis and Article Writing / शोध-प्रबंध एवं लेख लेखन, Application of ICT in Research / अनुसंधान में ICT का अनुप्रयोग, Research Ethics / शोध नैतिकता
  3. Paper - 1 | Unit - 3 | Comprehension: Literal Comprehension, Vocabulary and Language in Context, Inferential Comprehension, Critical Comprehension, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions)
  4. Paper - 1 | Unit - 4 | Communication: Communication: Meaning, Nature and Process / संचार: अर्थ, प्रकृति एवं प्रक्रिया, Types and Levels of Communication / संचार के प्रकार एवं स्तर, Verbal and Non-verbal Communication / मौखिक एवं अमौखिक संचार, Semiotics and Meaning / संकेत विज्ञान एवं अर्थ, Models and Theories of Communication / संचार के मॉडल एवं सिद्धांत, Effective Communication / प्रभावी संचार, Barriers to Communication / संचार की बाधाएँ, Classroom Communication / कक्षा संचार, Mass Media: Types and Evolution / जनसंचार माध्यम: प्रकार एवं विकास, Mass Media and Society / जनसंचार एवं समाज, Digital and New Media / डिजिटल एवं नव माध्यम, General Awareness (Legacy Paper 1) / सामान्य जागरूकता (पुराना पेपर 1)
  5. Paper - 1 | Unit - 5 | Mathematical Reasoning and Aptitude: Number System and Simplification / संख्या पद्धति एवं सरलीकरण, Algebra and Ages / बीजगणित एवं आयु, Percentage / प्रतिशतता, Ratio, Proportion, Partnership and Mixtures / अनुपात, समानुपात, साझेदारी एवं मिश्रण, Averages / औसत, Profit, Loss and Discount / लाभ, हानि एवं बट्टा, Simple and Compound Interest / साधारण एवं चक्रवृद्धि ब्याज, Time and Work / समय एवं कार्य, Speed, Time and Distance / चाल, समय एवं दूरी, Mensuration and Geometry / क्षेत्रमिति एवं ज्यामिति, Series / श्रेणी, Coding-Decoding / कूटलेखन-कूटवाचन, Relationships, Directions and Ranking / संबंध, दिशा एवं क्रम, Arrangements, Puzzles, Clock and Calendar / व्यवस्था, पहेलियाँ, घड़ी एवं कैलेंडर, Mathematical Foundations, जीवनी, Philosophy of the Constitution / संविधान का दर्शन, Analysis, Indian Polity and Governance (Legacy Unit 10) / भारतीय राजव्यवस्था और शासन (पुराना यूनिट 10)
  6. Paper - 1 | Unit - 6 | Logical Reasoning: Structure of Arguments / युक्ति की संरचना, Terms, Definitions and Uses of Language / पद, परिभाषा एवं भाषा के उपयोग, Categorical Propositions / निरुपाधिक प्रतिज्ञप्तियाँ, Categorical Syllogism / निरुपाधिक न्यायवाक्य, Venn Diagrams / वेन आरेख, Deduction, Induction and Analogy / निगमन, आगमन एवं सादृश्यता, Fallacies / तर्कदोष, Indian Logic: Pramanas / भारतीय तर्कशास्त्र: प्रमाण, Anumana and Vyapti / अनुमान एवं व्याप्ति, Hetvabhasa / हेत्वाभास, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Aristotle / अरस्तू, Western Schools of Philosophy / पाश्चात्य दर्शन की विचारधाराएँ, Kant / कांट, Sadhana Paada, Vaiśeṣika / वैशेषिक
  7. Paper - 1 | Unit - 7 | Data Interpretation: Data and Graphical Representation / आँकड़े एवं आलेखीय निरूपण, Data Interpretation: Tables / आँकड़ा निर्वचन: तालिका
  8. Paper - 1 | Unit - 8 | Information and Communication Technology: Computer Fundamentals and Hardware, Memory, Storage and Data Representation, Computer Software, Office Applications, Multimedia and File Formats, Computer Networks, Internet and World Wide Web, Email and Online Communication, Cyber Security, Digital Ethics, IPR and Society, Cloud and Distributed Computing, Emerging Technologies, ICT in Education and Governance, Cyber Threats and Malware Attacks, Role of Internet and Web Services, Electronic Data Interchange and E-Commerce, Internet Fundamentals, Web Design & Development, Web Publishing & Hosting, Emerging Technologies, Terms & Abbreviations, Society, Law & Ethics, Keyboard Shortcuts, Website, Browser & Services, Protection of Geographical Indications / भौगोलिक संकेतों का संरक्षण
  9. Paper - 1 | Unit - 9 | People Development and Environment: Ecosystems and Biodiversity / पारितंत्र और जैव विविधता, Human and Environment Interaction / मानव और पर्यावरण की अंतःक्रिया, Development and Environment / विकास और पर्यावरण, Natural Resources – Forests, Water and Soil / प्राकृतिक संसाधन: वन, जल और मृदा, Energy Resources / ऊर्जा संसाधन, Air Pollution / वायु प्रदूषण, Water Pollution and Treatment / जल प्रदूषण और उपचार, Noise Pollution / ध्वनि प्रदूषण, Waste Management / अपशिष्ट प्रबंधन, Environmental Health and Toxicology / पर्यावरणीय स्वास्थ्य और विषविज्ञान, Climate Change and Global Environmental Issues / जलवायु परिवर्तन और वैश्विक पर्यावरणीय समस्याएँ, Natural Hazards and Disaster Management / प्राकृतिक प्रकोप और आपदा प्रबंधन, Indian Environmental Laws and Policies / भारतीय पर्यावरण कानून और नीतियाँ, International Environmental Agreements / अंतर्राष्ट्रीय पर्यावरण समझौते, Environment / पर्यावरण, Geography / भूगोल, Factory and Industry Systems / कारखाना और उद्योग व्यवस्थाएँ, Earth Materials, Surface Features and Processes, Communicable Diseases and Immunization / संक्रामक रोग और प्रतिरक्षण
  10. Paper - 1 | Unit - 10 | Higher Education System: Education in Ancient India / प्राचीन भारत में शिक्षा, Ancient Centres of Higher Learning / प्राचीन भारत के उच्च शिक्षा केंद्र, Colonial Education Policies and Commissions / औपनिवेशिक काल की शिक्षा नीतियाँ और आयोग, Colonial Colleges, Universities and Reformers / औपनिवेशिक कॉलेज, विश्वविद्यालय व सुधारक, Education Commissions and Policies after Independence / स्वतंत्र भारत में शिक्षा आयोग और नीतियाँ, National Education Policy 2020 / राष्ट्रीय शिक्षा नीति 2020, Types of Higher Education Institutions / उच्च शिक्षा संस्थानों के प्रकार, Learning Programmes and Academic Credits / अधिगम कार्यक्रम और अकादमिक क्रेडिट, Professional, Technical and Skill-Based Education / व्यावसायिक, तकनीकी और कौशल आधारित शिक्षा, Value Education and Environmental Education / मूल्य शिक्षा और पर्यावरण शिक्षा, Governance, Administration and Financing / शासन, प्रशासन और वित्तपोषण, Regulation, Accreditation and Ranking / उच्च शिक्षा का नियमन, प्रत्यायन और रैंकिंग
  11. Paper 2 | Unit 1 | Discrete Structures and Optimization: Propositional and Predicate Logic, Set Theory, Relations, Functions, Permutation, Combination and Probability / क्रमचय, संचय एवं प्रायिकता, Graph Theory, Group Theory, Digital Systems & Boolean Basics, Boolean Expression, Boolean Minimization, Optimization, Markov Chains
  12. Paper 2 | Unit 2 | Computer System Architecture: Logic Gates & Hardware, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep, Basics of COA, Register Transfer and Microoperations, Programming the Basic Computer, Instr Formats & Modes, Control Unit Design, Pipelining, Input Output Organisation, Cache Memory Organization, Multiprocessors
  13. Paper 2 | Unit 3 | Programming Languages and Computer Graphics: Language Design, C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, HTML Basics, XML, JavaScript-Basics, Java, Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, 3-D Object Representation, Geometric Transformations and Viewing, OOPS with C++, Java Fundamentals, Array, String, IO & References, OOPS with Java
  14. Paper 2 | Unit 4 | Database Management Systems: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control, Database Recovery, ORDBMS, Database Security & Authorization, Query Processing & Optimization, Enhanced Data Models, Data Warehousing & Mining, Big Data Systems, NoSQL
  15. Paper 2 | Unit 5 | System Software and Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Windows OS, Linux OS, Security, Distributed Systems, Virtual Machines
  16. Paper 2 | Unit 6 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  17. Paper 2 | Unit 7 | Data Structures and Algorithms: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  18. Paper 2 | Unit 8 | Theory of Computation and Compilers: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Complexity Theory, Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
  19. Paper 2 | Unit 9 | Data Communication and Computer Networks: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware basics, Network Security, Cloud Computing, Mobile Computing
  20. Paper 2 | Unit 10 | Artificial Intelligence: Approaches to AI, Search Algorithms, Game Playing, Knowledge Representation, Planning, Multi Agent Systems, Fuzzy Sets, Natural Language Processing, Artificial Neural Networks, Genetic Algorithms
  21. Full Notes: Paper 1 Full Notes
  22. Paper 1 | Full Mock Tests:
  23. Paper 2 | Full Mock Tests:
  24. Paper 1 | Previous Year Papers:
  25. Paper 2 | Previous Year Papers:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

The video is a lecture on solving a linear programming problem using the Two Phase Method, a technique for handling constraints that do not have an initial basic feasible solution. The instructor begins by presenting the problem, which includes an objective function to maximize and a set of constraints. He explains that the standard Simplex Method cannot be applied directly because the constraints are of the 'greater than or equal to' type, which prevents the immediate identification of a basic feasible solution. To overcome this, he introduces the Two Phase Method. In Phase I, an auxiliary objective function is created to minimize the sum of artificial variables (A1), with the goal of driving this sum to zero. The instructor sets up the initial simplex tableau for Phase I, showing the coefficients of the variables, including the artificial variable A1, and the right-hand side values. He then demonstrates the first iteration of the Simplex Method, identifying the entering variable (the one with the most negative coefficient in the objective row) and the leaving variable (determined by the minimum ratio test). The pivot element is identified, and the instructor proceeds to perform row operations to create a new tableau. The process continues, with the instructor updating the tableau, calculating new values, and checking for optimality. The video shows the step-by-step progression of the algorithm, including the calculation of ratios and the updating of the solution vector. The instructor's goal is to find a feasible solution where the artificial variable A1 is zero, indicating that a basic feasible solution has been found for the original problem. The video concludes with the final tableau of Phase I, setting the stage for Phase II, where the original objective function is optimized using the feasible basis found in Phase I.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a whiteboard displaying a linear programming problem. The instructor, standing to the right, begins to explain the problem. The objective is to maximize Z = 5x1 - x2 + 3x3, subject to the constraints: 2x1 + x2 - x3 = 20, 6x1 + 5x2 + 10x3 = 76, and 8x1 - 3x2 + 6x3 = 50, with all variables being non-negative. The instructor notes that the constraints are of the 'greater than or equal to' type, which means the standard Simplex Method cannot be applied directly because there is no initial basic feasible solution. He introduces the Two Phase Method as a solution. The first step is to create an auxiliary problem to find a basic feasible solution. The objective for Phase I is to minimize Z' = A1, where A1 is an artificial variable. The instructor writes the initial simplex tableau for Phase I, with the objective row showing coefficients for the variables x1, x2, x3, A1, S1, S2, S3, and the solution column. The tableau shows the coefficients of the constraints and the right-hand side values. The instructor points to the tableau, explaining that the goal is to make the artificial variable A1 zero, which would mean a feasible solution has been found for the original problem.

  2. 2:00 – 5:00 02:00-05:00

    The instructor continues to work on the Phase I simplex tableau. He identifies the entering variable by looking for the most negative coefficient in the objective row (Z' - A1). The coefficient for A1 is -1, which is the most negative, so A1 is the entering variable. He then performs the minimum ratio test to determine the leaving variable. The ratios are calculated by dividing the right-hand side (Soln) by the corresponding coefficient in the A1 column. The ratios are 20/2 = 10, 76/5 = 15.2, and 50/(-3) = -16.67. The minimum positive ratio is 10, which corresponds to the first row. Therefore, S1 is the leaving variable. The pivot element is the intersection of the A1 column and the S1 row, which is 2. The instructor circles this pivot element. He then begins to perform row operations to make the pivot element 1 and to eliminate the other entries in the A1 column. He divides the first row by 2 to make the pivot element 1. The new first row becomes: 1, 0.5, -0.5, 1, 0, 0, 0, 10. He then uses this new row to eliminate the A1 coefficient in the other rows. For the second row, he subtracts 5 times the new first row from the original second row. For the third row, he adds 3 times the new first row to the original third row. The instructor writes down the new values for the second and third rows, updating the tableau.

  3. 5:00 – 10:00 05:00-10:00

    The instructor has updated the tableau after the first iteration of Phase I. He now checks for optimality. The objective row (Z' - A1) has coefficients: 0, 0.5, -0.5, 0, 0, 0, 0, 10. Since there is a negative coefficient (-0.5) in the x3 column, the solution is not optimal, and another iteration is needed. The entering variable is x3. He performs the minimum ratio test for x3. The ratios are 10 / (-0.5) = -20, 76 / 10 = 7.6, and 50 / 6 = 8.33. The minimum positive ratio is 7.6, which corresponds to the second row. Therefore, S2 is the leaving variable. The pivot element is the intersection of the x3 column and the S2 row, which is 10. The instructor circles this pivot element. He then performs row operations. He divides the second row by 10 to make the pivot element 1. The new second row becomes: 0, 0.5, 1, 0, 0.1, 0, 0, 7.6. He then uses this new row to eliminate the x3 coefficient in the other rows. For the first row, he adds 0.5 times the new second row to the original first row. For the objective row, he adds 0.5 times the new second row to the original objective row. The instructor writes down the new values for the first row and the objective row, updating the tableau. He then checks for optimality again. The objective row now has coefficients: 0, 0.75, 0, 0, 0.05, 0, 0, 13.8. Since all coefficients are non-negative, the solution is optimal for Phase I. The value of the artificial variable A1 is 0, which means a basic feasible solution has been found for the original problem.

  4. 10:00 – 14:02 10:00-14:02

    The instructor now transitions to Phase II of the Two Phase Method. He removes the artificial variable A1 and its column from the tableau, as it is no longer needed. He also removes the Phase I objective row (Z' - A1) and replaces it with the original objective function, Z = 5x1 - x2 + 3x3. The new objective row is: 0, 0, 0, 0, 0, 0, 0, 0, but with the coefficients of the original objective function. The instructor writes the new objective row as: 5, -1, 3, 0, 0, 0, 0, 0. He then checks for optimality. The coefficients in the objective row are: 5, -1, 3, 0, 0, 0, 0. Since there are negative coefficients (-1 and 3), the solution is not optimal. The entering variable is x2 (the one with the most negative coefficient). He performs the minimum ratio test for x2. The ratios are 10 / 0.5 = 20, 7.6 / 0.5 = 15.2, and 50 / 1 = 50. The minimum positive ratio is 15.2, which corresponds to the second row. Therefore, S2 is the leaving variable. The pivot element is the intersection of the x2 column and the S2 row, which is 0.5. The instructor circles this pivot element. He then performs row operations. He divides the second row by 0.5 to make the pivot element 1. The new second row becomes: 0, 1, 2, 0, 0.2, 0, 0, 15.2. He then uses this new row to eliminate the x2 coefficient in the other rows. For the first row, he subtracts 0.5 times the new second row from the original first row. For the objective row, he adds 1 times the new second row to the original objective row. The instructor writes down the new values for the first row and the objective row, updating the tableau. He then checks for optimality again. The objective row now has coefficients: 5, 0, 5, 0, 0.2, 0, 0, 15.2. Since all coefficients are non-negative, the solution is optimal. The optimal solution is x1 = 0, x2 = 15.2, x3 = 0, and the maximum value of Z is 15.2.

The video provides a comprehensive, step-by-step demonstration of the Two Phase Method for solving a linear programming problem. It begins by identifying the challenge of an initial infeasible solution due to 'greater than or equal to' constraints. The method is broken down into two distinct phases. Phase I is dedicated to finding a basic feasible solution by minimizing the sum of artificial variables. The instructor meticulously walks through the first iteration of the Simplex Method, identifying the pivot element, performing row operations, and checking for optimality. The process is repeated until the artificial variable is driven to zero, confirming a feasible solution. The video then transitions to Phase II, where the original objective function is optimized using the feasible basis found in Phase I. The instructor demonstrates the final iteration, showing how the optimal solution is reached. The entire process is illustrated on a whiteboard, with clear labeling of the tableau, variables, and calculations, making it an excellent educational resource for understanding this advanced optimization technique.

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