The document discusses different methods to solve assignment problems including enumeration, integer programming, transportation, and Hungarian methods. It provides examples of balanced and unbalanced minimization and maximization problems. The Hungarian method is described as having steps like row and column deduction, assigning zeros, and tick marking to find the optimal assignment with the minimum cost or maximum profit. A sample problem demonstrates converting a profit matrix to a relative cost matrix and using the Hungarian method to find the optimal solution.
This presentation is trying to explain the Linear Programming in operations research. There is a software called "Gipels" available on the internet which easily solves the LPP Problems along with the transportation problems. This presentation is co-developed with Sankeerth P & Aakansha Bajpai.
By:-
Aniruddh Tiwari
Linkedin :- http://in.linkedin.com/in/aniruddhtiwari
Linear programming - Model formulation, Graphical MethodJoseph Konnully
The document discusses linear programming, including an overview of the topic, model formulation, graphical solutions, and irregular problem types. It provides examples to demonstrate how to set up linear programming models for maximization and minimization problems, interpret feasible and optimal solution regions graphically, and address multiple optimal solutions, infeasible solutions, and unbounded solutions. The examples aid in understanding the key steps and components of linear programming models.
The document discusses the transportation problem and its solution methodology. It states that the transportation problem seeks to minimize the total shipping costs of transporting goods from multiple origins to destinations, given the unit shipping costs. It is solved in two phases - obtaining an initial feasible solution and then moving toward the optimal solution. Several methods are described for obtaining the initial feasible solution, including the Northwest Corner method, Least Cost method, and Vogel's Approximation Method. The document also discusses testing the initial solution for optimality using methods like the Stepping Stone method and Modified Distribution method.
The document provides an outline of topics related to linear programming, including:
1) An introduction to linear programming models and examples of problems that can be solved using linear programming.
2) Developing linear programming models by determining objectives, constraints, and decision variables.
3) Graphical and simplex methods for solving linear programming problems.
4) Using a simplex tableau to iteratively solve a sample product mix problem to find the optimal solution.
This document discusses linear programming techniques for managerial decision making. Linear programming can determine the optimal allocation of scarce resources among competing demands. It consists of linear objectives and constraints where variables have a proportionate relationship. Essential elements of a linear programming model include limited resources, objectives to maximize or minimize, linear relationships between variables, homogeneity of products/resources, and divisibility of resources/products. The linear programming problem is formulated by defining variables and constraints, with the objective of optimizing a linear function subject to the constraints. It is then solved using graphical or simplex methods through an iterative process to find the optimal solution.
The document discusses goal programming, which is used to solve linear programs with multiple objectives viewed as goals. It describes goal programming as attempting to reach a satisfactory level of multiple objectives by minimizing deviations between goals and what can actually be achieved given constraints. An example problem involves a hardware company with goals of achieving a $30 profit, fully utilizing wiring hours, avoiding assembly overtime, and producing at least 7 ceiling fans. The goal programming model for this problem is formulated and graphically solved to satisfy the higher priority goals as closely as possible before lower goals.
NOTE:Download this file to preview as the Slideshare preview does not display it properly.
This is an introduction to Linear Programming and a few real world applications are included.
This presentation is trying to explain the Linear Programming in operations research. There is a software called "Gipels" available on the internet which easily solves the LPP Problems along with the transportation problems. This presentation is co-developed with Sankeerth P & Aakansha Bajpai.
By:-
Aniruddh Tiwari
Linkedin :- http://in.linkedin.com/in/aniruddhtiwari
Linear programming - Model formulation, Graphical MethodJoseph Konnully
The document discusses linear programming, including an overview of the topic, model formulation, graphical solutions, and irregular problem types. It provides examples to demonstrate how to set up linear programming models for maximization and minimization problems, interpret feasible and optimal solution regions graphically, and address multiple optimal solutions, infeasible solutions, and unbounded solutions. The examples aid in understanding the key steps and components of linear programming models.
The document discusses the transportation problem and its solution methodology. It states that the transportation problem seeks to minimize the total shipping costs of transporting goods from multiple origins to destinations, given the unit shipping costs. It is solved in two phases - obtaining an initial feasible solution and then moving toward the optimal solution. Several methods are described for obtaining the initial feasible solution, including the Northwest Corner method, Least Cost method, and Vogel's Approximation Method. The document also discusses testing the initial solution for optimality using methods like the Stepping Stone method and Modified Distribution method.
The document provides an outline of topics related to linear programming, including:
1) An introduction to linear programming models and examples of problems that can be solved using linear programming.
2) Developing linear programming models by determining objectives, constraints, and decision variables.
3) Graphical and simplex methods for solving linear programming problems.
4) Using a simplex tableau to iteratively solve a sample product mix problem to find the optimal solution.
This document discusses linear programming techniques for managerial decision making. Linear programming can determine the optimal allocation of scarce resources among competing demands. It consists of linear objectives and constraints where variables have a proportionate relationship. Essential elements of a linear programming model include limited resources, objectives to maximize or minimize, linear relationships between variables, homogeneity of products/resources, and divisibility of resources/products. The linear programming problem is formulated by defining variables and constraints, with the objective of optimizing a linear function subject to the constraints. It is then solved using graphical or simplex methods through an iterative process to find the optimal solution.
The document discusses goal programming, which is used to solve linear programs with multiple objectives viewed as goals. It describes goal programming as attempting to reach a satisfactory level of multiple objectives by minimizing deviations between goals and what can actually be achieved given constraints. An example problem involves a hardware company with goals of achieving a $30 profit, fully utilizing wiring hours, avoiding assembly overtime, and producing at least 7 ceiling fans. The goal programming model for this problem is formulated and graphically solved to satisfy the higher priority goals as closely as possible before lower goals.
NOTE:Download this file to preview as the Slideshare preview does not display it properly.
This is an introduction to Linear Programming and a few real world applications are included.
Transportation Problem In Linear ProgrammingMirza Tanzida
This work is an assignment on the course of 'Mathematics for Decision Making'. I think, it will provide some basic concept about transportation problem in linear programming.
The document discusses transportation and assignment problems in operations research and their solutions. It introduces transportation models and how they can be formulated as linear programs. Solution methods for finding initial feasible and optimal solutions in transportation problems are described, including the northwest corner method, least cost method, Vogel's approximation method, stepping stone method, and modified distribution method. Special cases like unbalanced problems and degeneracy are also covered. The document then shifts to discussing assignment problems as a special case of transportation problems and describes the Hungarian method for solving assignment problems.
This document discusses transportation problems and three methods to solve them: the North West Corner Method, Least Cost Method, and Vogel Approximation Method. The objective of transportation problems is to minimize the cost of distributing products from sources to destinations while satisfying supply and demand constraints. The document provides examples to illustrate how each method works step-by-step to arrive at a basic feasible solution.
The document discusses linear programming problems and how to formulate them. It provides definitions of key terms like linear, programming, objective function, decision variables, and constraints. It then explains the steps to formulate a linear programming problem, including defining the objective, decision variables, mathematical objective function, and constraints. Several examples of formulated linear programming problems are provided to maximize profit or minimize costs subject to various constraints.
This document discusses primal and dual linear programming problems. It explains that every primal problem has a corresponding dual problem that describes the original problem. The two problems are closely related, and their optimal solutions provide information about each other. It provides guidelines for converting a primal problem to its dual, such as changing the objective from maximization to minimization. The document also describes the relationship between primal and dual solutions and constraints. An example primal and dual problem are presented.
This document discusses transportation models and methods for finding an initial basic feasible solution and testing for optimality in transportation problems. It describes three methods - northwest corner, least cost, and Vogel's approximation - for obtaining an initial solution. It then explains how to test if the initial solution is optimal using the MODI or u-v method by calculating opportunity costs for unoccupied cells and finding a closed path if any cells have negative opportunity costs to obtain an improved solution. The process repeats until all opportunity costs are non-negative, indicating an optimal solution.
This document provides an introduction and overview of integer programming problems. It discusses different types of integer programming problems including pure integer, mixed integer, and 0-1 integer problems. It provides examples to illustrate how to formulate integer programming problems as mathematical models. The document also discusses common solution methods for integer programming problems, including the cutting-plane method. An example of the cutting-plane method is provided to demonstrate how it works to find an optimal integer solution.
The transportation problem is a special type of linear programming problem where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations.
Because of its special structure, the usual simplex method is not suitable for solving transportation problems. These problems require a special method of solution.
The document discusses transportation and assignment models in operations research. The transportation model aims to minimize the cost of distributing a product from multiple sources to multiple destinations, while satisfying supply and demand constraints. The assignment model finds optimal one-to-one matching between sources and destinations to minimize costs. Some solution methods for transportation problems include the northwest corner method, row minima method, column minima method, and least cost method. The Hungarian method is commonly used to solve assignment problems by finding the minimum cost matching.
The document discusses transportation problems (TPs), which involve determining the optimal way to route products from multiple supply locations to multiple demand destinations to minimize total transportation costs. It provides the mathematical formulation of a TP as a linear programming problem (LPP) with decision variables representing the quantity transported between each origin-destination pair. Methods for solving TPs include the simplex method by formulating it as an LPP or specialized transportation methods like the northwest corner rule to find an initial feasible solution and stepping stone/modified distribution methods to check for optimality. An example TP is presented to illustrate these concepts.
This document discusses duality in linear programming. It defines the dual problem as another linear program systematically constructed from the original or primal problem, such that the optimal solutions of one provide the optimal solutions of the other. The document provides rules for constructing the dual problem based on whether the primal problem is a maximization or minimization problem. It also gives examples of writing the dual of a primal problem and solving both problems to verify the optimal objective values are equal. Finally, it discusses economic interpretations of duality and the relationship between primal and dual problems and solutions.
The document discusses linear programming, which is a mathematical modeling technique used to allocate limited resources optimally. It provides examples of linear programming problems and their formulation. Key aspects covered include defining decision variables and constraints, developing the objective function, and interpreting feasible and optimal solutions. Graphical and algebraic solution methods like the simplex method are also introduced.
This document discusses nonlinear programming (NLP) problems. NLP problems involve objective functions and/or constraints that contain nonlinear terms, making them more difficult to solve than linear programs. While exact solutions cannot always be found, algorithms can typically find approximate solutions within an acceptable error range of the optimum. However, for some NLP problems there is no reliable way to find the global maximum, as algorithms may stop at a local maximum instead. The document describes different types of NLP problems and techniques for solving them, including using Excel Solver with multiple starting values to attempt finding the global rather than just local optima.
The Modified Distribution Method or MODI is an efficient method of checking the optimality of the initial feasible solution. MODI provides a new means of finding the unused route with the largest negative improvement index. Once the largest index is identified, we are required to trace only one closed path. This path helps determine the maximum number of units that can be shipped via the best unused route.
The document provides an overview of solving transportation problems using linear programming techniques. It discusses formulating the problem, finding an initial feasible solution using methods like the Northwest Corner method, testing the solution for optimality using techniques like the Stepping Stone method, and handling special cases. The document also provides an example of using the Hungarian method to solve an assignment problem by finding a minimum cost matching between resources and activities.
1) The document discusses the Hungarian method for solving assignment problems by finding the optimal assignment of jobs to machines that minimizes costs.
2) It provides examples of using the Hungarian method to solve assignment problems by finding minimum costs in the cost matrix and obtaining a feasible assignment with zero costs.
3) The optimal solution is determined by selecting the assignments indicated by the cells with zero costs in the final cost matrix after applying the Hungarian method steps.
The document discusses decision theory and decision trees. It introduces decision making under certainty, risk, and uncertainty. It defines elements related to decisions like goals, courses of action, states of nature, and payoffs. It also discusses concepts like expected monetary value, expected profit with perfect information, expected value of perfect information, and expected opportunity loss. Examples are provided to demonstrate calculating these metrics. Finally, it provides an overview of how to construct a decision tree, including defining the different node types and how to calculate values within the tree.
Transportation Problem in Operational ResearchNeha Sharma
The document discusses the transportation problem and methods for finding its optimal solution. It begins by defining key terminology used in transportation models like feasible solution, basic feasible solution, and optimal solution. It then outlines the basic steps to obtain an initial basic feasible solution and subsequently improve it to reach the optimal solution. Three common methods for obtaining the initial solution are described: the Northwest Corner Method, Least Cost Entry Method, and Vogel's Approximation Method. The document also addresses how to solve unbalanced transportation problems and provides examples applying the methods.
The document discusses transportation and transshipment problems, describing transportation problems as involving the optimal distribution of goods from multiple sources to multiple destinations subject to supply and demand constraints. It presents the formulation of transportation problems as linear programming problems and provides examples of different types of transportation problems including balanced vs unbalanced and minimization vs maximization problems. The document also briefly mentions transshipment problems which involve sources, destinations, and transient nodes through which goods can pass.
This is a special type of LPP in which the objective function is to find the optimum allocation of a number of tasks (jobs) to an equal number of facilities (persons). Here we make the assumption that each person can perform each job but with varying degree of efficiency. For example, a departmental head may have 4 persons available for assignment and 4 jobs to fill. Then his interest is to find the best assignment which will be in the best interest of the department.
The document discusses solving an assignment problem to minimize crew waiting time for an airline with flights between Delhi and Kolkata. It involves:
1) Calculating waiting times for all flight pairings assuming crews based in Delhi or Kolkata;
2) Developing an opportunity cost matrix from the minimum waiting times;
3) Applying row and column reduction to the matrix;
4) Finding an optimal assignment of 4 flight pairings that minimizes total waiting time of 40.5 hours.
Transportation Problem In Linear ProgrammingMirza Tanzida
This work is an assignment on the course of 'Mathematics for Decision Making'. I think, it will provide some basic concept about transportation problem in linear programming.
The document discusses transportation and assignment problems in operations research and their solutions. It introduces transportation models and how they can be formulated as linear programs. Solution methods for finding initial feasible and optimal solutions in transportation problems are described, including the northwest corner method, least cost method, Vogel's approximation method, stepping stone method, and modified distribution method. Special cases like unbalanced problems and degeneracy are also covered. The document then shifts to discussing assignment problems as a special case of transportation problems and describes the Hungarian method for solving assignment problems.
This document discusses transportation problems and three methods to solve them: the North West Corner Method, Least Cost Method, and Vogel Approximation Method. The objective of transportation problems is to minimize the cost of distributing products from sources to destinations while satisfying supply and demand constraints. The document provides examples to illustrate how each method works step-by-step to arrive at a basic feasible solution.
The document discusses linear programming problems and how to formulate them. It provides definitions of key terms like linear, programming, objective function, decision variables, and constraints. It then explains the steps to formulate a linear programming problem, including defining the objective, decision variables, mathematical objective function, and constraints. Several examples of formulated linear programming problems are provided to maximize profit or minimize costs subject to various constraints.
This document discusses primal and dual linear programming problems. It explains that every primal problem has a corresponding dual problem that describes the original problem. The two problems are closely related, and their optimal solutions provide information about each other. It provides guidelines for converting a primal problem to its dual, such as changing the objective from maximization to minimization. The document also describes the relationship between primal and dual solutions and constraints. An example primal and dual problem are presented.
This document discusses transportation models and methods for finding an initial basic feasible solution and testing for optimality in transportation problems. It describes three methods - northwest corner, least cost, and Vogel's approximation - for obtaining an initial solution. It then explains how to test if the initial solution is optimal using the MODI or u-v method by calculating opportunity costs for unoccupied cells and finding a closed path if any cells have negative opportunity costs to obtain an improved solution. The process repeats until all opportunity costs are non-negative, indicating an optimal solution.
This document provides an introduction and overview of integer programming problems. It discusses different types of integer programming problems including pure integer, mixed integer, and 0-1 integer problems. It provides examples to illustrate how to formulate integer programming problems as mathematical models. The document also discusses common solution methods for integer programming problems, including the cutting-plane method. An example of the cutting-plane method is provided to demonstrate how it works to find an optimal integer solution.
The transportation problem is a special type of linear programming problem where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations.
Because of its special structure, the usual simplex method is not suitable for solving transportation problems. These problems require a special method of solution.
The document discusses transportation and assignment models in operations research. The transportation model aims to minimize the cost of distributing a product from multiple sources to multiple destinations, while satisfying supply and demand constraints. The assignment model finds optimal one-to-one matching between sources and destinations to minimize costs. Some solution methods for transportation problems include the northwest corner method, row minima method, column minima method, and least cost method. The Hungarian method is commonly used to solve assignment problems by finding the minimum cost matching.
The document discusses transportation problems (TPs), which involve determining the optimal way to route products from multiple supply locations to multiple demand destinations to minimize total transportation costs. It provides the mathematical formulation of a TP as a linear programming problem (LPP) with decision variables representing the quantity transported between each origin-destination pair. Methods for solving TPs include the simplex method by formulating it as an LPP or specialized transportation methods like the northwest corner rule to find an initial feasible solution and stepping stone/modified distribution methods to check for optimality. An example TP is presented to illustrate these concepts.
This document discusses duality in linear programming. It defines the dual problem as another linear program systematically constructed from the original or primal problem, such that the optimal solutions of one provide the optimal solutions of the other. The document provides rules for constructing the dual problem based on whether the primal problem is a maximization or minimization problem. It also gives examples of writing the dual of a primal problem and solving both problems to verify the optimal objective values are equal. Finally, it discusses economic interpretations of duality and the relationship between primal and dual problems and solutions.
The document discusses linear programming, which is a mathematical modeling technique used to allocate limited resources optimally. It provides examples of linear programming problems and their formulation. Key aspects covered include defining decision variables and constraints, developing the objective function, and interpreting feasible and optimal solutions. Graphical and algebraic solution methods like the simplex method are also introduced.
This document discusses nonlinear programming (NLP) problems. NLP problems involve objective functions and/or constraints that contain nonlinear terms, making them more difficult to solve than linear programs. While exact solutions cannot always be found, algorithms can typically find approximate solutions within an acceptable error range of the optimum. However, for some NLP problems there is no reliable way to find the global maximum, as algorithms may stop at a local maximum instead. The document describes different types of NLP problems and techniques for solving them, including using Excel Solver with multiple starting values to attempt finding the global rather than just local optima.
The Modified Distribution Method or MODI is an efficient method of checking the optimality of the initial feasible solution. MODI provides a new means of finding the unused route with the largest negative improvement index. Once the largest index is identified, we are required to trace only one closed path. This path helps determine the maximum number of units that can be shipped via the best unused route.
The document provides an overview of solving transportation problems using linear programming techniques. It discusses formulating the problem, finding an initial feasible solution using methods like the Northwest Corner method, testing the solution for optimality using techniques like the Stepping Stone method, and handling special cases. The document also provides an example of using the Hungarian method to solve an assignment problem by finding a minimum cost matching between resources and activities.
1) The document discusses the Hungarian method for solving assignment problems by finding the optimal assignment of jobs to machines that minimizes costs.
2) It provides examples of using the Hungarian method to solve assignment problems by finding minimum costs in the cost matrix and obtaining a feasible assignment with zero costs.
3) The optimal solution is determined by selecting the assignments indicated by the cells with zero costs in the final cost matrix after applying the Hungarian method steps.
The document discusses decision theory and decision trees. It introduces decision making under certainty, risk, and uncertainty. It defines elements related to decisions like goals, courses of action, states of nature, and payoffs. It also discusses concepts like expected monetary value, expected profit with perfect information, expected value of perfect information, and expected opportunity loss. Examples are provided to demonstrate calculating these metrics. Finally, it provides an overview of how to construct a decision tree, including defining the different node types and how to calculate values within the tree.
Transportation Problem in Operational ResearchNeha Sharma
The document discusses the transportation problem and methods for finding its optimal solution. It begins by defining key terminology used in transportation models like feasible solution, basic feasible solution, and optimal solution. It then outlines the basic steps to obtain an initial basic feasible solution and subsequently improve it to reach the optimal solution. Three common methods for obtaining the initial solution are described: the Northwest Corner Method, Least Cost Entry Method, and Vogel's Approximation Method. The document also addresses how to solve unbalanced transportation problems and provides examples applying the methods.
The document discusses transportation and transshipment problems, describing transportation problems as involving the optimal distribution of goods from multiple sources to multiple destinations subject to supply and demand constraints. It presents the formulation of transportation problems as linear programming problems and provides examples of different types of transportation problems including balanced vs unbalanced and minimization vs maximization problems. The document also briefly mentions transshipment problems which involve sources, destinations, and transient nodes through which goods can pass.
This is a special type of LPP in which the objective function is to find the optimum allocation of a number of tasks (jobs) to an equal number of facilities (persons). Here we make the assumption that each person can perform each job but with varying degree of efficiency. For example, a departmental head may have 4 persons available for assignment and 4 jobs to fill. Then his interest is to find the best assignment which will be in the best interest of the department.
The document discusses solving an assignment problem to minimize crew waiting time for an airline with flights between Delhi and Kolkata. It involves:
1) Calculating waiting times for all flight pairings assuming crews based in Delhi or Kolkata;
2) Developing an opportunity cost matrix from the minimum waiting times;
3) Applying row and column reduction to the matrix;
4) Finding an optimal assignment of 4 flight pairings that minimizes total waiting time of 40.5 hours.
The document discusses the assignment problem, which involves assigning people, jobs, machines, etc. to minimize costs or maximize profits. It provides an example of assigning 4 men to 4 jobs to minimize total cost, walking through the Hungarian method steps. It also discusses how to handle imbalance by adding dummy rows or columns, and how to convert a maximization problem to minimization.
This document discusses assignment problems and how to solve them using the Hungarian method. Assignment problems involve efficiently allocating people to tasks when each person has varying abilities. The Hungarian method is an algorithm that can find the optimal solution to an assignment problem in polynomial time. It involves constructing a cost matrix and then subtracting elements in rows and columns to create zeros, which indicate assignments. The method is iterated until all tasks are assigned with the minimum total cost. While typically used for minimization, the method can also solve maximization problems by converting the cost matrix.
This document describes a linear programming problem (LPP) to minimize cost. The problem involves determining the optimal amounts of two fertilizer brands, SuperGro and CropQuick, to purchase to minimize total cost while meeting nitrogen and phosphate requirements. The LPP constructs decision variables for amounts of each brand, an objective function to minimize total cost, and constraints on nitrogen and phosphate levels. The optimal solution is to purchase 8 bags of CropQuick for a minimum total cost of 24.
The document discusses assignment problems and the Hungarian method for solving them. It begins by introducing the concept of assignment problems where the goal is to assign n jobs to n workers in a way that maximizes profit or efficiency. It then provides the mathematical formulation of an assignment problem as minimizing a cost function subject to constraints. The bulk of the document describes the Hungarian method, a multi-step algorithm for finding optimal assignments. It involves row/column reductions, finding a complete assignment of zeros, drawing lines to cover remaining zeros, and modifying the cost matrix to increase the number of zeros. An example is provided to illustrate the method.
The document discusses the assignment problem and various methods to solve it. The assignment problem involves assigning jobs to workers or other resources in an optimal way according to certain criteria like minimizing time or cost. The Hungarian assignment method is described as a multi-step algorithm to find the optimal assignment between jobs and workers/resources. It involves creating a cost matrix and performing row and column reductions to arrive at a matrix with zeros that indicates the optimal assignment. The document also briefly discusses handling unbalanced and constrained assignment problems.
The document discusses the assignment problem, which aims to assign jobs to persons so that time, cost, or profit is optimized. It describes the model of an assignment problem using a 3x3 matrix. It outlines the steps to solve an assignment problem, including checking if it is balanced, subtracting row/column minimums, and using the Hungarian method if the initial assignment is not complete. The Hungarian method involves tick marks and strikethroughs to identify the minimum cost assignment. Finally, it notes there can be maximization problems, unbalanced matrices, alternative solutions, and restricted assignments.
To Get any Project for CSE, IT ECE, EEE Contact Me @ 09666155510, 09849539085 or mail us - ieeefinalsemprojects@gmail.com-Visit Our Website: www.finalyearprojects.org
The document outlines 6 steps for successful research and assignment completion: 1) Define the task and questions to be answered, 2) Locate relevant information sources, 3) Select useful information to answer questions, 4) Organize notes taken, 5) Present findings effectively, and 6) Assess the completed work and one's own skills and knowledge gained. Following these steps will help students develop strong research, writing, and self-assessment abilities needed for further education and career success.
This document discusses loading jobs and scheduling work centers. It defines loading as assigning jobs to work centers to minimize costs and completion times. Infinite loading ignores capacity constraints while finite loading only assigns as much work as can be completed with available capacity. The document also discusses characteristics of high-volume and low-volume operations, input-output processes, Gantt charts, and assignment methods.
The document discusses several assignment problems involving allocating jobs or tasks to machines, employees, clerks or cities in an optimal way to minimize time, cost or maximize returns. Specific problems include assigning 5 jobs to 5 machines to minimize total time, assigning jobs to machines to maximize total returns, assigning 4 jobs to 3 staff to minimize total time, assigning clerks to tasks where some assignments are prohibited, and finding the optimal tour for a salesman to visit 5 cities while minimizing total distance traveled. Solutions to these assignment problems are to be found using techniques like the assignment algorithm, Hungarian method, column reduction, row reduction and prohibiting certain assignments.
This document discusses the meaning, importance, merits and demerits of assignments in social science teaching. It defines assignments as exercises given by teachers for students to complete outside of class. The document outlines different types of assignments and their purposes in enhancing learning. It provides characteristics of effective assignments and discusses their role in the teaching process. While assignments can help organize knowledge and prepare for exams, the document also notes potential demerits like overemphasis on facts and exam preparation over developing skills. Overall, the document presents an overview of assignments as an educational tool in social science classes.
This document provides an overview of Northern European Baroque art from 1600 to 1700. It introduces some of the major artists of the period from countries like the Netherlands, France, and England. These include Peter Paul Rubens, Anthony van Dyck, Frans Hals, Rembrandt, and Vermeer among others. Their signature works are highlighted, such as Rubens' Elevation of the Cross and Vermeer's The Letter. The document also discusses the development of landscape painting and still life genres during this era in Northern Europe.
DAM assignment - LPP formulation, Graphical solution and Simplex MethodNeha Kumar
The document describes a linear programming problem faced by a consumer products company. The company produces two sanitary napkin products, Product A and Product B, and must decide how many of each to produce to maximize profit. The objective is to maximize total profit subject to constraints on minimum production requirements, machine hours, and packaging hours. Solving the linear programming formulation reveals the optimal solution is to produce 300,000 units of Product A and 214,285 units of Product B for a maximum profit of $5,297,143.
Automating performance-based tests is challenging as it requires identifying tasks that lead to solving complex problems. Each task must have a clear endpoint that the testing software can detect to determine if the student completed it. Similarly, every aspect of the problem-solving process needs to trigger a detectable change in the system as evidence of competency. Properly designing an automated performance test involves carefully selecting tasks that allow for robust rubrics and identifying important milestones and strategies to track student progress.
Henry Fayol proposed 14 principles of management based on his experience as a mining company director from 1888 to 1918. The principles address various aspects of managing an organization such as division of labor, authority and responsibility, discipline, unity of command, and centralization vs decentralization. Considered a foundational work, Fayol's principles provided one of the earliest and most comprehensive approaches to management theory.
dynamic programming complete by Mumtaz Ali (03154103173)Mumtaz Ali
The document discusses dynamic programming, including its meaning, definition, uses, techniques, and examples. Dynamic programming refers to breaking large problems down into smaller subproblems, solving each subproblem only once, and storing the results for future use. This avoids recomputing the same subproblems repeatedly. Examples covered include matrix chain multiplication, the Fibonacci sequence, and optimal substructure. The document provides details on formulating and solving dynamic programming problems through recursive definitions and storing results in tables.
The document discusses solving assignment problems using different methods like visual method, enumeration method, transportation method, and the Hungarian method. It provides an example problem of assigning four subassemblies to four contractors to minimize total cost. The Hungarian method is used to solve this example problem, resulting in a minimum total cost of 4,900 birr by assigning: subassembly 1 to contractor 2, subassembly 2 to contractor 1, subassembly 3 to contractor 4, and subassembly 4 to contractor 3.
This document provides an overview of linear programming concepts including:
1) The mathematical formulation of a linear programming problem with an objective function and constraints.
2) The basic assumptions and graphical solution method for linear programming problems.
3) Key terms used in linear programming like feasible solution, optimal solution, corner point solution.
4) The simplex method for solving linear programming problems through an iterative process of moving between corner point solutions.
5) Sensitivity analysis and shadow prices to understand how changes to parameters impact the optimal solution.
This document discusses the assignment problem in operational research. The assignment problem involves assigning jobs to operators such that the total processing time is minimized. It presents the general format of an assignment problem as a matrix with jobs as rows and operators as columns. Cell entries represent processing times. The document describes how to formulate the problem as a binary programming problem and solve it using the Hungarian method or branch and bound algorithm. It provides an example of applying these methods to minimize the total processing time of assigning 5 operators to 5 jobs.
The document defines linear programming and its key components. It explains that linear programming is a mathematical optimization technique used to allocate limited resources to achieve the best outcome, such as maximizing profit or minimizing costs. The document outlines the basic steps of the simplex method for solving linear programming problems and provides an example to illustrate determining the maximum value of a linear function given a set of constraints. It also discusses other applications of linear programming in fields like engineering, manufacturing, energy, and transportation for optimization.
Here are 3 methods to check if a number is divisible by 7 without using the modulo (%) operator:
1. Subtract 7 repeatedly: Subtract 7 from the given number repeatedly. If at any point the number becomes 0, it is divisible by 7.
2. Digital root property: Keep adding the digits of the number until you are left with a single digit. If the digital root is 0 or 7, the original number is divisible by 7.
3. Alternate subtraction: Subtract the last digit from the remaining number. Then subtract twice the last digit from the remaining number alternately until a single digit number is obtained. If the final number is 0, the original number is divisible by 7.
So
Dynamic programming (DP) is a powerful technique for solving optimization problems by breaking them down into overlapping subproblems and storing the results of already solved subproblems. The document provides examples of how DP can be applied to problems like rod cutting, matrix chain multiplication, and longest common subsequence. It explains the key elements of DP, including optimal substructure (subproblems can be solved independently and combined to solve the overall problem) and overlapping subproblems (subproblems are solved repeatedly).
Introduction to Matrix Chain Multiplication algorithm with an example. Matrix Chain Products algorithm comes under Dynamic Programming concept. Done for the course Advanced Data Structures and Algorithms.
The document discusses backtracking and branch and bound algorithms. Backtracking incrementally builds candidates and abandons them (backtracks) when they cannot lead to a valid solution. Branch and bound systematically enumerates solutions and discards branches that cannot produce a better solution than the best found so far based on upper bounds. Examples provided are the N-Queens problem solved with backtracking and the knapsack problem solved with branch and bound. Pseudocode is given for both algorithms.
The document provides an example to formulate a linear programming problem (LPP) and solve it graphically. It first defines the steps to formulate an LPP which includes identifying decision variables, writing the objective function, mentioning constraints, and specifying non-negativity restrictions. It then gives an example problem on maximizing profit from production of two products with machine hours and input requirements. This example problem is formulated as an LPP and represented graphically to arrive at the optimal solution.
The document describes solving an unbalanced assignment problem to minimize total time for jobs. It involves 6 jobs and 5 workers, so a dummy job is added. The Hungarian method is used. The optimal assignment minimizes total time to 14 units, with worker assignments: A to job 4, B to job 1, C to job 6, D to job 5, E to job 2, and F to job 3. The document also explains prohibitive assignment problems and provides an example of solving a balanced, prohibitive problem to maximally meet pilot preferences for flight assignments.
The assignment problem is a special case of transportation problem in which the objective is to assign ‘m’ jobs or workers to ‘n’ machines such that the cost incurred is minimized.
The document discusses integer programming and various methods to solve integer linear programming problems. It provides:
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4. C11 C12 . . C1n
C21 C22 . . C2n
. . . . .
. . . . .
Cm1 . . . Cmn
Jobs (Activities)
1 2 . . n
1
2
.
.
m
Persons(Resources)
CasesCases
• m = n
• m ≠ n
• Cij
• Pij
• Cij = Cost associated with assigning ith
resource to jth
activity
Assignment ProblemAssignment Problem
5. A. Balanced Minimization → m = n with Cij
B. Unbalanced Minimization →m ≠ n with Cij
C. Balanced Maximization → m = n with Pij
D. Unbalanced Maximization →m ≠ n with Pij
Categories of Assignment ProblemsCategories of Assignment Problems
6. C11 C12 . . C1n
C21 C22 . . C2n
. . . . .
. . . . .
Cn1 . . . Cnn
Jobs (Activities)
1 2 . . n
1
2
.
.
n
Persons(Resources)
• Xij = assignment of ith
resource to jth
activity
• Assignments are made on one to one basis
A.A. Balanced Minimization ProblemBalanced Minimization Problem
7. Formulation of Assignment Problem as LPP
1 1
.
n n
i j
Min Z Cij Xij
= =
= ∑∑
1
/ 1 ( )
n
j
s t Xij for all i I
=
= −∑
1
1 ( )
n
i
Xij for all j II
=
= −∑
0 1 & .All Xij or for all i j=
9. C11 C12
C21 C22
1 2
1
2
1.1. Enumeration MethodEnumeration Method
• For n x n Problem---- n!
2! (=2)• No. of Possible solutions =
• P1J1 , P2J2 OR P1J2 , P2J1
• For 4 x 4 Problem---- 4! = 24
• For 5 x 5 Problem----5! = 120
10. • All Xij = 0 or 1 → 0-1 Integer programming.
2.2. Integer Programming MethodInteger Programming Method
• Difficult to solve manually.
• For n x n Problem → Variables = n x n.
Constraints = n+n = 2n
• For 5 x 5 Problem → Variables = 25
Constraints = 10
11. • Formulate the problem in Transportation
Problem format.
3.3. Transportation MethodTransportation Method
C11 C12 . . C1n
C21 C22 . . C2n
. . . . .
. . . . .
Cn1 . . . Cnn
1 2 . . n
1
2
.
.
n
1
1
.
1
1 1 . . 1
12. 4.4. Hungarian MethodHungarian Method
1. Row Deduction
2. Column Deduction
3. Assign zeros-If all assignments are over,
Then STOP
Else Go To Step 4
4. Adopt Tick Marking Procedure
5. Modify Cij and Go To Step 3
(Mr. D. Konig - A Hungarian Mathematician)
Steps :Steps :
17. Procedure of Assigning ZerosProcedure of Assigning Zeros
Steps to be followed, after getting at-least one zero in each row &
each column.
1. Start with 1st row. If there is only one uncrossed, unassigned
zero, assign it & cross other zeros in respective column (of
assigned zero), if they exits, else go to next row. Repeat this for
next all other rows.
2. If still uncrossed, unassigned zeros are available, start with first
column. If there is only one uncrossed, unassigned zero, assign it
and cross other zeros in respective row of assigned zero, if they
exit, else go to next column. Repeat this for next all other columns.
3. Repeat 1 & 2 until single uncrossed, unassigned zeros are
available, while going through rows & columns.
4. If still multiple uncrossed, unassigned zeros are available while
going through rows & columns, it indicates that multiple
(alternative) optimal solutions are possible.
Assign any one remaining uncrossed, unassigned zero & cross
remaining zeros in respective row & column of assigned zero.
Then go to 1.
5. If required assignments are completed then STOP, else perform
“Tick Marking Procedure”. Modify numbers & go to 1.
18. Tick marking ProcedureTick marking Procedure
(To draw minimum number of lines through zeros.)
1. Tick marks row/rows where there is no assigned zero.
2. Tick mark column/columns w.r.t. crossed zero/zeros in
marked row/rows.
3. Tick mark row/rows w.r.t. assigned zero in marked
column/columns.
4. Go to to step 2 and repeat the procedure until no zero is
available for tick marking.
Then
5. Draw lines through unmarked rows and marked columns
(Check no. of lines = No. of assigned
zeros)
19. Tick marking ProcedureTick marking Procedure
(Continue)(Continue)
How to Modify numbers ?
Find minimum number out of uncrossed numbers.
1. Add this minimum number to crossings.
2. Deduct this number from all uncrossed numbers one by one.
3. Keep crossed numbers, on horizontal & vertical lines,
except on crossing, same
20. C11 C12 C13
C21 C22 C23
. . .
. . .
C51 C52 C53
Jobs (Activities)
1 2 3
1
2
3
4
5
Persons(Resources)
• D1 and D2 Dummy Jobs are to be
introduced to balance the problem
B. Unbalanced Minimization ProblemB. Unbalanced Minimization Problem
22. Prob. 2 The personnel manager of ABC Company wants
to assign Mr. X, Mr. Y and Mr. Z to regional offices. But the
firm also has an opening in its Chennai office and would send
one of the three to that branch, if it were more economical
than a move to Delhi, Mumbai or Kolkata. It will cost Rs.
2,000 to relocate Mr. X to Chennai, Rs. 1,600 to reallocate
Mr. Y there, and Rs. 3,000 to move Mr. Z. What is the
optimal assignment of personnel to offices ?
Office
Delhi Mumbai Kolkata
Personnel
Mr. X 1,600 2,200 2,400
Mr. Y 1,000 3,200 2,600
Mr. Z 1,000 2,000 4,600
23. UBMin (AP), after adding Chennai :
Delhi Mumbai Kolkata Chennai
Mr. X 1,600 2,200 2,400 2,000
Mr. Y 1,000 3,200 2,600 1,600
Mr. Z 1,000 2,000 4,600 3,000
24. BMin (AP), after adding Dummy Row
:
Delhi Mumbai Kolkata Chennai
Mr. X 1,600 2,200 2,400 2,000
Mr. Y 1,000 3,200 2,600 1,600
Mr. Z 1,000 2,000 4,600 3,000
Dummy 0 0 0 0
25. After Row Deduction :
D M K Ch
X 0 600 800 400
Y 0 2,200 1,600 600
Z 0 1,000 3,600 2,000
Dm 0 0 0 0
26. Modified Matrix and Assignment :
D M K Ch
X 0 200 400 0
Y 0 1,800 1,200 200
Z 0 600 3,200 1,600
Dm 0 0 0 0
27. Modified Matrix and Assignment :
D M K Ch
X 200 200 400 0
Y 0 1,600 1,000 0
Z 0 400 3,000 1,400
Dm 400 0 0 0
28. Modified Matrix and Assignment :
D M K Ch
X 200 0 200 0
Y 0 1,400 800 0
Z 0 200 2,800 1,400
Dm 6 00 0 0 200
Hence, Optimal Solution is : XM, YCh, ZD
Giving Z = 2,200 + 1,600 + 1,000 = 4,800
Hence, it is economical to move Y to Chennai
29. Jobs (Activities)
1 2 3 4 5
1
2
3
4
5
Persons(Resources)
• Convert Profit Matrix into Relative Cost
Matrix
C. Balanced Maximization ProblemC. Balanced Maximization Problem
Pij
30. • How to Convert Profit Matrix into
Relative Cost Matrix ?
3. (Pij)max - Pij = RCij
1. (Pij) (-1) = RCij
2. 1/Pij = RCij
51. 0 0 0 2
Μ 1 2 0
0 Μ 1 0
4 1 1 0
Jobs (Activities)
1 2 3 4
1
2
3
4
Persons(Resources)
After Column deduction
Now modified matrix will be :
52. 0 0 0 3
Μ 0 1 0
0 Μ 1 1
3 0 0 0
Jobs (Activities)
1 2 3 4
1
2
3
4
Persons(Resources)
• Hence, this is a case of alternative optimal solutions.
• Assign any one remaining zero and cross existing zeros
in respective row and column, then apply assigning
procedure.
• Hence, one of the optimal solutions is P1J2, P2J4, P3J1,
P4J3 giving Z = 7+4+3+4 = 18
53. 0 0 0 3
Μ 0 1 0
0 Μ 1 1
3 0 0 0
Jobs (Activities)
1 2 3 4
1
2
3
4
Persons(Resources)
• Hence, another optimal solution is P1J3, P2J2, P3J1,
P4J4 giving Z = 5+8+3+2 = 18
To get another optimal solution, assign another remaining zero.
P1J3 can not be assigned, as it is already assigned before.
54. . Restriction in Assignment.
e.g. Assignment of P3 to J4 is not possible.
Then C34 = M (Big Number)
. Alternative Optimal Solution Possibility
-Already considered.
. Particular assignment is prefixed.
e.g. If P3 & J4 prefixed
Then Row3 & Column4 are deleted.
Typical Cases in Assignment ProblemsTypical Cases in Assignment Problems
56. Que. Answer each of the following questions in brief.
[ 1 ] What are the methods to solve Assignment Problems ?
Ans. : (1) Enumeration Method
(2) Integer Programming Method
(3) Transportation Method
(4) Hungarian Method
[ 2 ] How can you identify a function as Linear ?
Ans. : Power of each variable = only 1.
No multiplication of variables.
57. [ 3 ] What is the purpose of “Tick Marking Procedure” in a
method of solving Assignment Problems ?
Ans. : The purpose of “Tick Marking” procedure is to
draw minimum number of lines covering zeros.
[ 4 ] What is significance of name “Hungarian Method” ?
Ans. : It is because of D. Konig of Hungary.
58. [ 5 ] Following is Minimization Assignment Problem.
(i) Write objective equation.
(ii) Write all possible constraints.
(iii) State “Non-Negativity” conditions for this problem.
(iv) State Optimal Solution. Is it unique optimal ?
X Y
A 3 2
B 4 5
Ans. : ( i ) Min Z = 3x11 + 2x12 + 4x21 + 5x22
( ii ) x11 + x12 = 1
x21 + x22 = 1
x11 + x21 = 1
x12 + x22 = 1
( iii ) All xij = 0 or 1 OR x11, x12, x21, x22 = 0 or 1
( iv ) Optimal solution is AY, BX giving Z = 6 (unique)
59. [ 6 ] Get Optimal Solution of following Minimization
Assignment Problem. How many Optimal Solutions are
existing for this problem ?
X Y Z
A 3 2 2
B 6 5 5
C 6 1 1
X Y Z
A 3 2 2
B 6 5 5
C 6 1 1
X Y Z
A 1 0 0
B 1 0 0
C 5 0 0
60. X Y Z
A 0 0 0
B 0 0 0
C 4 0 0
X Y Z
A 0 0 0
B 0 0 0
C 4 0 0
X Y Z
A 0 0 0
B 0 0 0
C 4 0 0
X Y Z
A 0 0 0
B 0 0 0
C 4 0 0
There are 4 Optimal Solutions.
63. [ 1 ] Answer the following in brief :
( a ) What is the purpose of “Tick Marking”
procedure in solving Assignment Problem ?
( b ) What is significance of name “Hungarian”
method ?
64. [ 2 ] For the following Minimization Assignment
Problem, answer the question given below.
J1
J2
J3
P1 2 4 3
P2 5 6 6
( a ) Formulate this problem as LPP.
( b ) Get all possible Optimal Solutions.
65. [ 3 ] ( i ) State possible methods of solving
“Assignment Problem”.
( ii ) State possible methods of converting
“Profit Matrix” into “Relative Cost Matrix”.
[ 4 ] Formulate following Assignment Problem as :
( i ) LPP
( ii ) Transportation Problem
P Q
A 2 5
B 6 4
67. [ 6 ] Average time taken by operators on 4 old
machines and a new machine are tabulated below.
Management is considering to replace one of the old
machines by a new machine. Is it advantageous to
replace new machine with an old machine ? Why ?
M1
M2
M3
M4 New
O1 10 12 8 10 11
O2 9 10 8 7 10
O3 8 7 8 8 8
O4 12 13 14 14 11
68. [ 7 ] Consider the problem of assigning four operators
to four machines. The assignment costs are given
in Rupees. Operator 1 cannot be assigned to
machine 3. Also operator 3 cannot be assigned to
machine 4. Find the optimal assignment.
M1 M2 M3 M4
1 5 5 − 2
2 7 4 2 3
3 9 3 5 −
4 7 2 6 7
If 5th Machine is made available and the
respective costs to the four operators are Rs. 2, 1,
2 and 8. Find whether it is economical to replace
any of the four existing machines. If so, which ?
69. [ 8 ] There are four batsman P, Q, R, & S. The
batsman are to be selected for first three
positions P1, P2 and P3. The expected score by
these batsman at three different positions are
given as below. Decide the optimal batsman for
these three positions.
P1
P2
P3
P 42 16 27
Q 48 40 25
R 50 18 36
S 58 38 60
72. Thank youThank you
For any Query or suggestion :
Contact :
Dr. D. B. Naik
Professor & Head, Training & Placement (T&P)
S. V. National Institute of Technology (SVNIT),
Ichchhanath, Surat – 395 007 (Gujarat) INDIA.
Email ID : dbnaik@gmail.com
dbnaik@svnit.ac.in
dbnaik_svr@yahoo.com
Phone No. : 0261-2201540 (D), 2255225 (O)