Vogel's Approximation Method (VAM) is a method for solving transportation problems that considers penalties (opportunity costs) associated with not shipping to cells with the lowest costs. It works as follows:
1. Compute penalties for each row and column based on the difference between the lowest and second lowest costs.
2. Ship to the cell in the row or column with the largest penalty, removing that row or column.
3. Recompute penalties and repeat until all shipments are allocated. VAM finds an initial solution close to optimal with few iterations.
The document describes a transportation problem and its solution. A transportation problem aims to minimize the cost of distributing goods from multiple sources to multiple destinations, given supply and demand constraints. It describes the basic components and phases of solving a transportation problem, including obtaining an initial feasible solution and then optimizing the solution using methods like the stepping stone method. The stepping stone method traces paths between cells on the transportation table to find negative cost cycles, and adjusts values to further optimize the solution.
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.
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 presents information about the transportation problem and the North West Corner Method for solving it. It includes an introduction to transportation problems, definitions of key concepts, examples of applications, and steps for solving balanced and unbalanced problems using the North West Corner Method. It also provides an example problem from a medical supply company shipping catheters from production facilities to warehouses.
The document discusses the simplex method for solving linear programming problems. It introduces some key terminology used in the simplex method like slack variables, surplus variables, and artificial variables. It then provides an overview of how the simplex method works for maximization problems, including forming the initial simplex table, testing for optimality and feasibility, pivoting to find an optimal solution. Finally, it provides an example application of the simplex method to a sample maximization problem.
Vogel's Approximation Method & Modified Distribution MethodKaushik Maitra
Vogel's Approximation Method (VAM) and Modified Distribution Method (MODI) are used to solve transportation problems. VAM computes penalties for each row and column to select the cell with the lowest cost to allocate units until constraints are satisfied, producing an initial basic feasible solution. MODI determines if the solution is optimal and identifies non-basic variables to consider, allowing it to find the true optimal solution. It is applied after VAM to a manufacturing company's transportation problem of supplying raw materials across plants and destinations.
Modified distribution method (modi method)Dinesh Suthar
The document describes the Modified Distribution Method (MODI Method) for finding the optimal transportation plan. It involves the following steps: 1) Determine an initial basic feasible solution, 2) Calculate dual variables to find opportunity costs, 3) Select the cell with most negative opportunity cost to add to the solution, 4) Draw a closed loop and update values along the loop until all opportunity costs are non-negative, indicating optimality. The example shows applying the MODI Method to find the least-cost shipment plan to meet brick orders from two plants. The optimal solution ships a total of 80 tons at a cost of Rs. 2,490.
The stepping stone method is used to find the optimal feasible solution to a transportation problem after an initial basic feasible solution is found. It involves systematically checking each unused cell to see if shipping one additional unit to that cell would lower total transportation costs. This is done by forming closed loops around the cell and calculating the net change in cost. If a cell is found that lowers costs, units are reallocated until no further improvements can be made. The document provides an example problem that is solved step-by-step using this method, ultimately arriving at a lower optimal transportation cost than the initial basic feasible solution.
The document describes a transportation problem and its solution. A transportation problem aims to minimize the cost of distributing goods from multiple sources to multiple destinations, given supply and demand constraints. It describes the basic components and phases of solving a transportation problem, including obtaining an initial feasible solution and then optimizing the solution using methods like the stepping stone method. The stepping stone method traces paths between cells on the transportation table to find negative cost cycles, and adjusts values to further optimize the solution.
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.
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 presents information about the transportation problem and the North West Corner Method for solving it. It includes an introduction to transportation problems, definitions of key concepts, examples of applications, and steps for solving balanced and unbalanced problems using the North West Corner Method. It also provides an example problem from a medical supply company shipping catheters from production facilities to warehouses.
The document discusses the simplex method for solving linear programming problems. It introduces some key terminology used in the simplex method like slack variables, surplus variables, and artificial variables. It then provides an overview of how the simplex method works for maximization problems, including forming the initial simplex table, testing for optimality and feasibility, pivoting to find an optimal solution. Finally, it provides an example application of the simplex method to a sample maximization problem.
Vogel's Approximation Method & Modified Distribution MethodKaushik Maitra
Vogel's Approximation Method (VAM) and Modified Distribution Method (MODI) are used to solve transportation problems. VAM computes penalties for each row and column to select the cell with the lowest cost to allocate units until constraints are satisfied, producing an initial basic feasible solution. MODI determines if the solution is optimal and identifies non-basic variables to consider, allowing it to find the true optimal solution. It is applied after VAM to a manufacturing company's transportation problem of supplying raw materials across plants and destinations.
Modified distribution method (modi method)Dinesh Suthar
The document describes the Modified Distribution Method (MODI Method) for finding the optimal transportation plan. It involves the following steps: 1) Determine an initial basic feasible solution, 2) Calculate dual variables to find opportunity costs, 3) Select the cell with most negative opportunity cost to add to the solution, 4) Draw a closed loop and update values along the loop until all opportunity costs are non-negative, indicating optimality. The example shows applying the MODI Method to find the least-cost shipment plan to meet brick orders from two plants. The optimal solution ships a total of 80 tons at a cost of Rs. 2,490.
The stepping stone method is used to find the optimal feasible solution to a transportation problem after an initial basic feasible solution is found. It involves systematically checking each unused cell to see if shipping one additional unit to that cell would lower total transportation costs. This is done by forming closed loops around the cell and calculating the net change in cost. If a cell is found that lowers costs, units are reallocated until no further improvements can be made. The document provides an example problem that is solved step-by-step using this method, ultimately arriving at a lower optimal transportation cost than the initial basic feasible solution.
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 duality theory in linear programming (LP). It explains that for every LP primal problem, there exists an associated dual problem. The primal problem aims to optimize resource allocation, while the dual problem aims to determine the appropriate valuation of resources. The relationship between primal and dual problems is fundamental to duality theory. The document provides examples of primal and dual problems and their formulations. It also outlines some general rules for constructing the dual problem from the primal, as well as relations between optimal solutions of primal and dual problems.
The document describes the Modi method for solving transportation problems. It involves finding the unused route with the largest negative improvement index to determine the best way to ship units. The key steps are to construct a transportation table, find the initial basic feasible solution, identify occupied and unoccupied cells, calculate opportunity costs for unoccupied cells, select the cell with the largest negative opportunity cost, and assign units until reaching the optimal solution. The method is demonstrated on two example problems.
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.
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 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.
This document provides an overview of game theory and two-person zero-sum games. It defines key concepts such as players, strategies, payoffs, and classifications of games. It also describes the assumptions and solutions for pure strategy and mixed strategy games. Pure strategy games have a saddle point solution found using minimax and maximin rules. Mixed strategy games do not have a saddle point and require determining the optimal probabilities that players select each strategy.
The Least Cost Method is another method used to obtain the initial feasible solution for the transportation problem. Here, the allocation begins with the cell which has the minimum cost. The lower cost cells are chosen over the higher-cost cell with the objective to have the least cost of transportation.
This document discusses various applications of linear programming (LP) in areas such as marketing, finance, and operations management. In marketing, LP can be used for media selection and marketing research problems. It provides examples of using LP to determine an optimal advertising plan that maximizes exposure quality within a budget. In finance, LP is used for portfolio selection and financial planning problems. Operations management applications covered include make-or-buy decisions, production scheduling, workforce assignment, and blending 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.
This presentation discusses transportation model optimization techniques. It introduces transportation models and methods for solving transportation problems to reach an optimal solution. Specifically, it covers what optimization is, transportation models and their applications, phases of solution such as obtaining initial and optimal solutions, and methods like the North-West Corner rule. It provides an example problem demonstrating solutions using different methods and showing the Vogel's Approximation Method achieves the lowest transportation cost. The presentation concludes that this technique can optimize scheduling, production, investment and other problems by minimizing transportation or distribution costs.
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 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 describes a transportation problem involving transporting a commodity from multiple sources to multiple destinations at minimum cost. It provides definitions for a balanced and unbalanced transportation problem. For the given problem, it solves it using the North-West Corner Rule and Vogel's Approximation Method to find initial basic feasible solutions. The North-West Corner Rule allocates shipments cell by cell from the top left. Vogel's Method assigns penalty costs to each cell based on the difference from lowest cost and selects the minimum penalty route.
Transportation Method
Initial Basic Feasible Solution-IBFS
North West Corner Method--NWCM ,
Least Cost Method--LCM and
Vogel’s Approximation Method--VAM
Optimality Test using Modified Distribution Method-MODI method.
Variation in transportation
Unbalance Supply and Demand
Degeneracy and its resolution
Maximization Problem
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.
1) The document discusses three methods for finding an initial basic feasible solution to transportation problems: the North-West Corner method, Least Cost/Matrix Minimum method, and Vogel's Approximation Method.
2) The North-West Corner method works by sequentially filling cells beginning from the northwest corner based on minimum supply or demand values.
3) The Least Cost method selects the cell with the minimum unit cost and allocates the maximum possible units to it, eliminating satisfied rows and columns until complete.
4) Vogel's Approximation Method identifies row and column penalties based on cost differences and sequentially fills the cell in the row or column with the largest penalty.
The document outlines the presentation topic of Modified Distribution Method (MODI Method) for solving transportation problems. It first discusses the prerequisite methods of Least Cost Method, Vogel's Approximation Method and North-West Corner Method. It then explains the steps of MODI Method which involves setting up cost matrices for unallocated cells and introducing dual variables to find the implicit cost and evaluate unoccupied cells to determine if the initial solution can be improved. The document provides an example problem and solution to demonstrate the application of MODI 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 duality theory in linear programming (LP). It explains that for every LP primal problem, there exists an associated dual problem. The primal problem aims to optimize resource allocation, while the dual problem aims to determine the appropriate valuation of resources. The relationship between primal and dual problems is fundamental to duality theory. The document provides examples of primal and dual problems and their formulations. It also outlines some general rules for constructing the dual problem from the primal, as well as relations between optimal solutions of primal and dual problems.
The document describes the Modi method for solving transportation problems. It involves finding the unused route with the largest negative improvement index to determine the best way to ship units. The key steps are to construct a transportation table, find the initial basic feasible solution, identify occupied and unoccupied cells, calculate opportunity costs for unoccupied cells, select the cell with the largest negative opportunity cost, and assign units until reaching the optimal solution. The method is demonstrated on two example problems.
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.
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 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.
This document provides an overview of game theory and two-person zero-sum games. It defines key concepts such as players, strategies, payoffs, and classifications of games. It also describes the assumptions and solutions for pure strategy and mixed strategy games. Pure strategy games have a saddle point solution found using minimax and maximin rules. Mixed strategy games do not have a saddle point and require determining the optimal probabilities that players select each strategy.
The Least Cost Method is another method used to obtain the initial feasible solution for the transportation problem. Here, the allocation begins with the cell which has the minimum cost. The lower cost cells are chosen over the higher-cost cell with the objective to have the least cost of transportation.
This document discusses various applications of linear programming (LP) in areas such as marketing, finance, and operations management. In marketing, LP can be used for media selection and marketing research problems. It provides examples of using LP to determine an optimal advertising plan that maximizes exposure quality within a budget. In finance, LP is used for portfolio selection and financial planning problems. Operations management applications covered include make-or-buy decisions, production scheduling, workforce assignment, and blending 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.
This presentation discusses transportation model optimization techniques. It introduces transportation models and methods for solving transportation problems to reach an optimal solution. Specifically, it covers what optimization is, transportation models and their applications, phases of solution such as obtaining initial and optimal solutions, and methods like the North-West Corner rule. It provides an example problem demonstrating solutions using different methods and showing the Vogel's Approximation Method achieves the lowest transportation cost. The presentation concludes that this technique can optimize scheduling, production, investment and other problems by minimizing transportation or distribution costs.
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 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 describes a transportation problem involving transporting a commodity from multiple sources to multiple destinations at minimum cost. It provides definitions for a balanced and unbalanced transportation problem. For the given problem, it solves it using the North-West Corner Rule and Vogel's Approximation Method to find initial basic feasible solutions. The North-West Corner Rule allocates shipments cell by cell from the top left. Vogel's Method assigns penalty costs to each cell based on the difference from lowest cost and selects the minimum penalty route.
Transportation Method
Initial Basic Feasible Solution-IBFS
North West Corner Method--NWCM ,
Least Cost Method--LCM and
Vogel’s Approximation Method--VAM
Optimality Test using Modified Distribution Method-MODI method.
Variation in transportation
Unbalance Supply and Demand
Degeneracy and its resolution
Maximization Problem
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.
1) The document discusses three methods for finding an initial basic feasible solution to transportation problems: the North-West Corner method, Least Cost/Matrix Minimum method, and Vogel's Approximation Method.
2) The North-West Corner method works by sequentially filling cells beginning from the northwest corner based on minimum supply or demand values.
3) The Least Cost method selects the cell with the minimum unit cost and allocates the maximum possible units to it, eliminating satisfied rows and columns until complete.
4) Vogel's Approximation Method identifies row and column penalties based on cost differences and sequentially fills the cell in the row or column with the largest penalty.
The document outlines the presentation topic of Modified Distribution Method (MODI Method) for solving transportation problems. It first discusses the prerequisite methods of Least Cost Method, Vogel's Approximation Method and North-West Corner Method. It then explains the steps of MODI Method which involves setting up cost matrices for unallocated cells and introducing dual variables to find the implicit cost and evaluate unoccupied cells to determine if the initial solution can be improved. The document provides an example problem and solution to demonstrate the application of MODI Method.
Lecture 7 transportation problem finding initial basic feasible solutionAathi Suku
This document describes three methods for finding an initial basic feasible solution to a transportation problem:
1. The North-West Corner Method which allocates values starting from the top-left cell and proceeds row-by-row, column-by-column.
2. The Least Cost Method which allocates to the cell with the lowest unit cost, exhausting that row or column before proceeding.
3. Vogel's Approximation Method which calculates penalties for each row/column and allocates to the row/column with the highest penalty.
It provides an example problem and shows that Vogel's Method results in the lowest total transportation cost compared to the other two methods.
Vogel's approximation method is an iterative method used to find an initial basic feasible solution for the transportation problem. It works by calculating penalties for each row and column based on the difference between lowest costs. It then selects the row or column with the highest penalty and allocates the minimum supply or demand to the cell in that row/column with the lowest cost. This process repeats, eliminating satisfied rows and columns, until all constraints are met. The document provides examples demonstrating how to apply Vogel's method step-by-step to solve transportation problems.
This document contains the problem set for the 4th CusContest competition in computer engineering and systems at the Universidad Nacional de San Antonio Abad del Cusco. It includes 5 problems related to counting triangles, designing reliable networks, maximizing integrals, minimizing the cost of breaking a chocolate bar into pieces, and identifying arithmetic or geometric sequences. For each problem, the input and output formats are clearly defined, and examples are provided.
The document discusses linear programming models and the transportation problem. It defines linear programming as optimally allocating limited resources under certain assumptions. The transportation problem involves shipping goods from multiple origins to destinations in a way that minimizes costs. Basic concepts are defined, including feasible and basic feasible solutions. Methods for solving transportation problems are described, such as the North-West Corner Rule, Lowest Cost Entry Method, and Vogel's Approximation Method. Degeneracy in transportation problems and moving toward optimality are also covered.
This document discusses transportation problems and their solutions. It begins by outlining the objectives of transportation problems, which is to minimize transportation costs while meeting supply and demand constraints. It then provides an introduction and mathematical formulation of transportation problems. The document explains how to represent transportation problems in a standard table and defines key terms. It describes methods to find the initial basic feasible solution, including the Northwest Corner Rule, Least Cost Method, and Vogel's Approximation Method. The document concludes by explaining how to find the optimal basic solution using the MODI or Modified Distribution Method.
This document discusses transportation problems and their solution using linear programming models. Transportation problems aim to minimize the cost of transporting goods from multiple sources to multiple destinations, given supply and demand constraints. The document introduces transportation models and Vogel's Approximation Method, an iterative procedure for finding an initial basic feasible solution to transportation problems by assigning flows and eliminating sources and destinations until all constraints are satisfied.
The document describes transportation problems and how to formulate and solve them using linear programming. It provides an example of a transportation problem involving supplying electricity from three power plants to four cities. Key steps include: (1) defining decision variables for amounts shipped, (2) writing the objective function to minimize total shipping costs, (3) specifying supply and demand constraints, and (4) using methods like the Northwest Corner method to find an initial basic feasible solution. The transportation simplex method is then used to iteratively improve the solution by entering and exiting variables from the basis.
This document presents a transportation problem faced by Epsilon Computers Co. in distributing desktop computers to universities. There are 3 distribution warehouses that can supply a total of 400 computers. Three universities have a total demand of 400 computers. The costs of shipping computers from each warehouse to each university are provided. The objective is to determine the quantities to ship from each warehouse to each university to minimize total shipping costs, using methods like Northwest Corner Rule, Least Cost Method, Stepping Stone Method and Vogel's Approximation Method. The Stepping Stone Method is explained in detail through multiple steps of allocating quantities and computing opportunity costs to arrive at an optimal solution.
This document discusses transportation problems and their solution using linear programming. It provides examples to illustrate key concepts. Specifically:
- Transportation problems involve minimizing the cost of transporting goods between multiple sources and destinations, given supply/demand amounts and per-unit shipping costs. They can be modeled as linear programs.
- The document outlines a two-phase algorithm to solve transportation problems, beginning with obtaining an initial feasible solution and then iteratively moving toward optimality. It provides examples to demonstrate these solution methods.
- Key steps include using the northwest corner or minimum cost rules to generate an initial tableau, then calculating reduced costs and identifying stepping stone paths to improve the solution in each iteration.
This document discusses the transportation problem and methods for solving it. It begins with an example of distributing goods from 3 canneries to 4 warehouses. It then defines the transportation problem as distributing goods from sources to destinations at minimum cost. It provides the formulation for the transportation problem as a linear program. It discusses properties of transportation problems and methods for constructing an initial basic feasible solution, including the Northwest Corner Rule and Vogel's Approximation Method. It also discusses using the transportation simplex method to solve transportation problems.
Transportation and Assiggnment problem pptHailemariam
The transportation problem in operational research aims to find the most economical way of transporting goods from multiple sources to multiple destinations. On the other hand, the assignment problem focuses on assigning tasks, jobs, or resources one-to-one. Both of these problems are usually solved through linear programming techniques. The transportation problem is commonly approached through simplex methods, and the assignment problem is addressed using specific algorithms like the Hungarian method. In this article, we will learn the difference between transportation problems and assignment problems with the help of examples. Transportation Problems and Assignment Problems are types of Linear Programming Problems. Transportation Problem deals with the optimal distribution of goods or resources from multiple sources to multiple destinations. While Assignment Problem deals with allocating tasks, jobs, or resources one-to-one.
These LPP methods are used for cost minimization, resource allocation, supply chain management, workforce planning, facility location, time management, and decision-making support.
This article will briefly discuss the difference between transportation problems and assignment problems based on different parameters.
The document discusses the transportation problem and methods to solve it. It begins by defining the transportation problem as finding the optimal transportation schedule to minimize transportation costs when distributing a product from multiple sources to multiple destinations. It then describes three methods to obtain an initial basic feasible solution: Northwest Corner Rule, Least Cost Method, and Vogel Approximation Method. The document concludes by explaining the Modi Method, which improves the initial solution by calculating opportunity costs and adjusting cell values along closed paths to find the optimal solution.
The document provides instructions for using Excel to analyze the relationship between two variables, the odometer reading (X) and selling price (Y) of used cars. It describes how to produce a scatter plot and regression line to model the relationship, and how to interpret the results including the slope, intercept, standard error, coefficient of determination (R2), and testing whether there is a significant linear relationship between the variables.
This document discusses transportation problems and network optimization problems. It provides examples and explanations of how to formulate transportation problems as linear programs. Specifically, it describes a transportation problem faced by a power company with 3 plants and 4 cities. It provides the supply and demand amounts for each plant and city and the shipping costs. It then formulates this problem as a linear program to minimize total shipping costs subject to supply and demand constraints. The document also describes general transportation problems and different methods for finding basic feasible solutions, including the Northwest Corner Method, Minimum Cost Method, and Vogel's Method.
Matrices and row operations 12123 and applications of matricesmayaGER
This document discusses using loop analysis to solve electrical circuits. It describes the 5 step process: 1) counting loops, 2) choosing independent loop currents, 3) writing Kirchhoff's voltage law for each loop, 4) solving the system of equations to find loop currents, and 5) reconstructing branch currents from loop currents. It then provides an example solution of a circuit with 3 loops and loop currents of 0.245 A, 0.111 A, and 0.117 A.
Vogel's Approximation Method (VAM) is a 3 step process for solving transportation problems:
1) Compute penalties for each row and column based on smallest costs.
2) Identify largest penalty and assign lowest cost variable the highest possible value, crossing out exhausted row or column.
3) Recalculate penalties and repeat until all requirements are satisfied.
The document discusses assignment problems and provides examples to illustrate how to solve them. Assignment problems involve allocating jobs to people or machines in a way that minimizes costs or maximizes profits. The key steps to solve assignment problems are: (1) construct a cost matrix, (2) perform row and column reductions to obtain zeros, (3) draw lines to cover zeros and determine optimal assignments. Traveling salesman problems, which involve finding the lowest cost route to visit all cities once, can also be formulated as assignment problems.
How to Create User Notification in Odoo 17Celine George
This slide will represent how to create user notification in Odoo 17. Odoo allows us to create and send custom notifications on some events or actions. We have different types of notification such as sticky notification, rainbow man effect, alert and raise exception warning or validation.
Get Success with the Latest UiPath UIPATH-ADPV1 Exam Dumps (V11.02) 2024yarusun
Are you worried about your preparation for the UiPath Power Platform Functional Consultant Certification Exam? You can come to DumpsBase to download the latest UiPath UIPATH-ADPV1 exam dumps (V11.02) to evaluate your preparation for the UIPATH-ADPV1 exam with the PDF format and testing engine software. The latest UiPath UIPATH-ADPV1 exam questions and answers go over every subject on the exam so you can easily understand them. You won't need to worry about passing the UIPATH-ADPV1 exam if you master all of these UiPath UIPATH-ADPV1 dumps (V11.02) of DumpsBase. #UIPATH-ADPV1 Dumps #UIPATH-ADPV1 #UIPATH-ADPV1 Exam Dumps
The Science of Learning: implications for modern teachingDerek Wenmoth
Keynote presentation to the Educational Leaders hui Kōkiritia Marautanga held in Auckland on 26 June 2024. Provides a high level overview of the history and development of the science of learning, and implications for the design of learning in our modern schools and classrooms.
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 3)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
Lesson Outcomes:
- students will be able to identify and name various types of ornamental plants commonly used in landscaping and decoration, classifying them based on their characteristics such as foliage, flowering, and growth habits. They will understand the ecological, aesthetic, and economic benefits of ornamental plants, including their roles in improving air quality, providing habitats for wildlife, and enhancing the visual appeal of environments. Additionally, students will demonstrate knowledge of the basic requirements for growing ornamental plants, ensuring they can effectively cultivate and maintain these plants in various settings.
Creativity for Innovation and SpeechmakingMattVassar1
Tapping into the creative side of your brain to come up with truly innovative approaches. These strategies are based on original research from Stanford University lecturer Matt Vassar, where he discusses how you can use them to come up with truly innovative solutions, regardless of whether you're using to come up with a creative and memorable angle for a business pitch--or if you're coming up with business or technical innovations.
Brand Guideline of Bashundhara A4 Paper - 2024khabri85
It outlines the basic identity elements such as symbol, logotype, colors, and typefaces. It provides examples of applying the identity to materials like letterhead, business cards, reports, folders, and websites.
8+8+8 Rule Of Time Management For Better ProductivityRuchiRathor2
This is a great way to be more productive but a few things to
Keep in mind:
- The 8+8+8 rule offers a general guideline. You may need to adjust the schedule depending on your individual needs and commitments.
- Some days may require more work or less sleep, demanding flexibility in your approach.
- The key is to be mindful of your time allocation and strive for a healthy balance across the three categories.
Decolonizing Universal Design for LearningFrederic Fovet
UDL has gained in popularity over the last decade both in the K-12 and the post-secondary sectors. The usefulness of UDL to create inclusive learning experiences for the full array of diverse learners has been well documented in the literature, and there is now increasing scholarship examining the process of integrating UDL strategically across organisations. One concern, however, remains under-reported and under-researched. Much of the scholarship on UDL ironically remains while and Eurocentric. Even if UDL, as a discourse, considers the decolonization of the curriculum, it is abundantly clear that the research and advocacy related to UDL originates almost exclusively from the Global North and from a Euro-Caucasian authorship. It is argued that it is high time for the way UDL has been monopolized by Global North scholars and practitioners to be challenged. Voices discussing and framing UDL, from the Global South and Indigenous communities, must be amplified and showcased in order to rectify this glaring imbalance and contradiction.
This session represents an opportunity for the author to reflect on a volume he has just finished editing entitled Decolonizing UDL and to highlight and share insights into the key innovations, promising practices, and calls for change, originating from the Global South and Indigenous Communities, that have woven the canvas of this book. The session seeks to create a space for critical dialogue, for the challenging of existing power dynamics within the UDL scholarship, and for the emergence of transformative voices from underrepresented communities. The workshop will use the UDL principles scrupulously to engage participants in diverse ways (challenging single story approaches to the narrative that surrounds UDL implementation) , as well as offer multiple means of action and expression for them to gain ownership over the key themes and concerns of the session (by encouraging a broad range of interventions, contributions, and stances).
1. Vogel’s Approximation Method (VAM)
The Vogel’s Approximate Method takes into account not only the least cost
Cij but also the costs that just exceed Cij. Various steps of the method are
given below:
Step I:
For each row of the transportation table identify the smallest and the next to
smallest cost. Determine the difference between them for each row. These
are called penalties (opportunity cost). Put them along side of the
transportation table by enclosing them in parenthesis against the respective
rows. Similarly compute these penalties for each column.
Step II:
Identify the row or column with the largest penalty among all the rows and
columns. If a tie occurs, use any arbitrary tie breaking choice. Let the
greatest penalty correspond to ith row and let Cij be the smallest cost in the ith
row. Allocate the largest possible amount Xij = min. (ai, bj) in the (i, j) of the
cell and cross off the ith row or jth column in the usual manner.
Step III:
Re-compute the column and row penalty for the reduced transportation table
and go to step II. Repeat the procedure until all the requirements are
satisfied.
Remarks:
1. A row or column ‘difference’ indicates the minimum unit penalty
incurred by failing to make an allocation to the smallest cost cell in
that row or column.
2. It will be seen that VAM determines an initial basic feasible solution
which is very close to the optimum solution that is the number of
iterations required to reach optimum solution is minimum in this case.
2. Examples:
1. Balanced Transportation Problem
SupplyP Q R S
A 18 21 12 15016
B 17 14 13 16019
C 32 11 10 9015
Demand 140 120 90 50 400
Solution:
In this example, Total Supply = Total Demand. So, this is balanced
transportation problem.
P Q R S Supply Penalities
A 18 21 12 150 416
B 17 14 13 160 119
C 32 11 10 90 115
Demand 140 120 90 50 400
Penalities 1 7 1 2
For each row and column of the transportation table determine the penalties
and put them along side of the transportation table by enclosing them in
parenthesis against the respective rows and beneath the corresponding columns.
3. Select the row or column with the largest penalty i.e. (7) (marked with an arrow)
associated with second column and allocate the maximum possible amount to the
cell (3,2) with minimum cost and allocate an amount X32=Min(120,90)=90 to it.
This exhausts the supply of C. Therefore, cross off third row. The first reduced
penalty table will be:
Supply PenalitiesP Q R S
A 18 21 12 150 416
B 17 14 13 160 119
Demand 140 30 90 50 310
Penalities 1 1 7 1
In the first reduced penalty table the maximum penalty of rows and columns
occurs in column 3, allocate the maximum possible amount to the cell (2,3) with
minimum cost and allocate an amount X23=Min(90,160)=90 to it . This exhausts
the demand of R .As such cross off third column to get second reduced penalty
table as given below.
Supply PenalitiesP Q S
A 18 12 150 416
B 17 19 13 160 4
191
919
Demand 140 30 50 220
Penalities 1 1 1
In the second reduced penalty table there is a tie in the maximum penalty
between first and second row. Choose the first row and allocate the maximum
possible amount to the cell (1,4) with minimum cost and allocate an amount
4. X14=Min(50,150)=50 to it . This exhausts the demand of S so cross off fourth
column to get third reduced penalty table as given below:
Supply PenalitiesP Q
A 18 100 216
B 17 19 70 2
191
919
Demand 140 30 170
Penalities 1 1
The largest of the penalty in the third reduced penalty table is (2) and is
associated with first row and second row. We choose the first row arbitrarily
whose min. cost is C11 = 16. The fourth allocation of X11=min. (140, 100) =100 is
made in cell (1, 1). Cross off the first row. In the fourth reduced penalty table i.e.
second row, minimum cost occurs in cell (2,1) followed by cell (2,2) hence
allocate X21=40 and X22=30. Hence the whole allocation is as under:
SupplyP Q R S
A
100
18 21
50
12
150
16
B
40 30 90
14 13
160
17 19
C
32
90
11 10
90
15
Demand 140 120 90 50 400
The transportation costaccording to the above allocation is given by
Z = 16 × 100 + 12 × 50 + 17 × 40 +19 × 30 + 14 × 90 + 11 × 90 = 5700
5. 2. Unbalanced Transportation Problem
A B C Supply
P 4 8 8 76
Q 16 24 16 82
R 8 16 24 77
Demand 72 102 41
Solution:
In this example, Total Supply is 235 and Total Demand is 225. Here
Total Supply > Total Demand. So, this is unbalanced transportation
problem. To solve this problem we need to make it balanced
transportation problem. Here we introduce dummy column with
transportation cost including the dummy. Thus new balanced
transportation table is as:
A B C Dummy Supply
P 4 8 8 0 76
Q 16 24 16 0 82
R 8 16 24 0 77
Demand 72 102 41 200
6. A B C Dummy Supply Penalities
P 4 8 8 0 76 4
Q 16 24 16 0 82 16
R 8 16 24 0 77 8
Demand 72 102 41 20
Penalities 4 8 8
For each row and column of the transportation table determine the penalties
and put them along side of the transportation table by enclosing them in
parenthesis against the respective rows and beneath the corresponding columns.
Select the row or column with the largest penalty i.e. (16) (marked with an
arrow) associated with second row and allocate the maximum possible amount to
the cell (2,4) with minimum cost and allocate an amount X24=Min(20,82)=20 to it.
This exhausts the demand of dummy. Therefore, cross off forth column. The first
reduced penalty table will be:
A B C Supply Penalities
P 4 8 8 76 4
Q 16 24 16 62 8
R 8 16 24 77 8
Demand 72 102 41
Penalities 4 8 8
7. In the first reduced penalty table there is a tie in the maximum penalty
between second row, third row, second column and third column . Choose the
second column and allocate the maximum possible amount to the cell (1,2) with
minimum cost and allocate an amount X12=Min(102,76)=76 to it . This exhausts
the supply of B so cross off first row to get second reduced penalty table as given
below:
A B C Supply Penalities
Q 16 24 16 82 8
R 8 16 24 77 8
Demand 72 26 41
Penalities 8 8 8
In the second reduced penalty table there is a tie in the maximum penalty
between all rows and columns. Choose the first column and allocate the maximum
possible amount to the cell (2,2) with minimum cost and allocate an amount
X12=Min(72,77)=72 to it . This exhausts the demand of A so cross off first column
to get third reduced penalty table as given below:
B C Supply Penalities
Q 24 16 82 8
R 16 24 5 8
Demand 102 41
Penalities 8 8
8. In the third reduced penalty table there is a tie in the maximum penalty
between all rows and columns. Choose the first column and allocate the maximum
possible amount to the cell (1,2) with minimum cost and allocate an amount
X12=Min(41,82)=41 to it . This exhausts the demand of C so cross off second
column to get fourth reduced penalty table. In the fourth reduced penalty table i.e.
second row, minimum cost occurs in cell (2,1) followed by cell (1,1) hence
allocate X21=5 and X22=21. Hence the whole allocation is as under:
A B C Dummy Supply
P
4
76
8 8 0
76
Q
16
21 41
16
20
0
82
24
R
72 5
16 24 0
77
8
Demand 72 102 41 20
The transportation costaccording to the above allocation is given by
Z = 8 × 76 + 24 × 21 + 16× 41 + 0 × 20 + 8 × 72 + 16 × 5 = 2424
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