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 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.
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.
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.
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 discusses transportation problems and their solution. It begins by stating the aim is to find an optimal transportation schedule that minimizes transportation costs. It then provides an example transportation problem table and defines key terms. The remainder of the document explains assumptions of transportation models, their applications, and steps to solve them. It covers obtaining an initial basic feasible solution using methods like the Northwest Corner Rule, Least Cost Method, and Vogel's Approximation Method. It also discusses obtaining the optimal basic solution using the Stepping Stone Method.
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.
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 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.
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.
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.
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 discusses transportation problems and their solution. It begins by stating the aim is to find an optimal transportation schedule that minimizes transportation costs. It then provides an example transportation problem table and defines key terms. The remainder of the document explains assumptions of transportation models, their applications, and steps to solve them. It covers obtaining an initial basic feasible solution using methods like the Northwest Corner Rule, Least Cost Method, and Vogel's Approximation Method. It also discusses obtaining the optimal basic solution using the Stepping Stone Method.
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.
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.
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.
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.
This document discusses decision theory and decision-making under conditions of certainty, uncertainty, and risk. It defines key decision-making concepts like available courses of action, states of nature or outcomes, payoffs, and expected monetary value. Methods for decision-making under uncertainty include the maximin, maximax, minimax regret, Hurwitz, and Bayes criteria. Decision-making under risk involves assigning probabilities to outcomes and selecting the action with the largest expected payoff value or smallest expected opportunity loss.
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 describes how to solve an assignment problem to maximize overall performance using the Hungarian method. It involves converting the maximization problem into a minimization problem by subtracting all values from the highest value or changing the signs of all values. The Hungarian method is then applied which involves reducing the matrix to find the optimal assignment that minimizes the total value by selecting the smallest values in rows and columns and drawing lines to cover zeros.
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 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.
The document discusses linear programming, which is a method for optimizing a linear objective function subject to linear equality and inequality constraints. It describes how to formulate a linear programming problem by defining the objective function and constraints in terms of decision variables. It also discusses graphical and algebraic solution methods, including identifying an optimal solution at an extreme point of the feasible region. Applications of linear programming are mentioned in areas like business, industry, and marketing.
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 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.
1) The document discusses the Hungarian method for solving assignment problems. It involves minimizing the total cost or maximizing the total profit of assigning resources like employees or machines to activities like jobs.
2) The method includes steps like developing a cost matrix, finding the opportunity cost table, making assignments to zeros in the table, and revising the table until an optimal solution is reached.
3) There are examples showing the application of these steps to problems with unique and multiple optimal solutions, as well as an unbalanced problem with more resources than activities.
GAME THEORY
Terminology
Example : Game with Saddle point
Dominance Rules: (Theory-Example)
Arithmetic method – Example
Algebraic method - Example
Matrix method - Example
Graphical method - Example
The document discusses various models and methods used for project selection. It begins by describing non-numeric models such as sacred cow, operating necessity, competitive necessity, and product line extension. It then discusses numeric scoring models including unweighted 0-1 factor model, unweighted factor scoring model, and weighted factor scoring model. Finally, it discusses financial models used for project selection, focusing on models that evaluate profitability. The document provides an overview of different approaches organizations can take when selecting projects.
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 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 presentation is made to represent the basic transportation model. The aim of this presentation is to implement the transportation model in solving transportation problem.
The document discusses using goal programming to solve a scheduling problem of allocating study time for two exams. The goal is to spend no more than 4 hours studying and achieve at least a B grade in one exam and an A in the other. Initially, the model allocates 1 hour to the first exam and 3 hours to the second. When priorities are changed to emphasize grades over study time, the optimal solution changes to allocating 3 hours to the first exam and 1 hour to the second.
This document discusses key concepts in decision theory and decision making under uncertainty. It begins by defining decision theory and describing the degree of certainty in decision making problems. It then outlines elements of decision analysis like states of nature, chance occurrences governed by probabilities, and payoff matrices. An example involving production decisions for a dairy product is provided. The document also discusses criteria for decision making under uncertainty like Laplace, maximin, maximax, Hurwicz, and regret. It concludes by covering expected monetary value, expected opportunity loss, expected value of perfect information, and decision trees as approaches to decision making under risk.
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.
This document discusses the Hungarian method for solving assignment problems. It begins by explaining the assignment problem and providing an example with 2 machines and 2 operators. It then describes the Hungarian method, which finds the optimal assignment in polynomial time. The method is explained through a 4x4 example, with the key steps being: 1) row and column reductions, 2) finding a complete assignment or using adjustments to enable it. The document also notes the method works for balanced or minimization problems, and how maximization problems can be converted. It concludes by providing a professor-subject example and walking through applying the Hungarian method to find the optimal assignment.
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.
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.
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.
This document discusses decision theory and decision-making under conditions of certainty, uncertainty, and risk. It defines key decision-making concepts like available courses of action, states of nature or outcomes, payoffs, and expected monetary value. Methods for decision-making under uncertainty include the maximin, maximax, minimax regret, Hurwitz, and Bayes criteria. Decision-making under risk involves assigning probabilities to outcomes and selecting the action with the largest expected payoff value or smallest expected opportunity loss.
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 describes how to solve an assignment problem to maximize overall performance using the Hungarian method. It involves converting the maximization problem into a minimization problem by subtracting all values from the highest value or changing the signs of all values. The Hungarian method is then applied which involves reducing the matrix to find the optimal assignment that minimizes the total value by selecting the smallest values in rows and columns and drawing lines to cover zeros.
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 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.
The document discusses linear programming, which is a method for optimizing a linear objective function subject to linear equality and inequality constraints. It describes how to formulate a linear programming problem by defining the objective function and constraints in terms of decision variables. It also discusses graphical and algebraic solution methods, including identifying an optimal solution at an extreme point of the feasible region. Applications of linear programming are mentioned in areas like business, industry, and marketing.
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 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.
1) The document discusses the Hungarian method for solving assignment problems. It involves minimizing the total cost or maximizing the total profit of assigning resources like employees or machines to activities like jobs.
2) The method includes steps like developing a cost matrix, finding the opportunity cost table, making assignments to zeros in the table, and revising the table until an optimal solution is reached.
3) There are examples showing the application of these steps to problems with unique and multiple optimal solutions, as well as an unbalanced problem with more resources than activities.
GAME THEORY
Terminology
Example : Game with Saddle point
Dominance Rules: (Theory-Example)
Arithmetic method – Example
Algebraic method - Example
Matrix method - Example
Graphical method - Example
The document discusses various models and methods used for project selection. It begins by describing non-numeric models such as sacred cow, operating necessity, competitive necessity, and product line extension. It then discusses numeric scoring models including unweighted 0-1 factor model, unweighted factor scoring model, and weighted factor scoring model. Finally, it discusses financial models used for project selection, focusing on models that evaluate profitability. The document provides an overview of different approaches organizations can take when selecting projects.
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 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 presentation is made to represent the basic transportation model. The aim of this presentation is to implement the transportation model in solving transportation problem.
The document discusses using goal programming to solve a scheduling problem of allocating study time for two exams. The goal is to spend no more than 4 hours studying and achieve at least a B grade in one exam and an A in the other. Initially, the model allocates 1 hour to the first exam and 3 hours to the second. When priorities are changed to emphasize grades over study time, the optimal solution changes to allocating 3 hours to the first exam and 1 hour to the second.
This document discusses key concepts in decision theory and decision making under uncertainty. It begins by defining decision theory and describing the degree of certainty in decision making problems. It then outlines elements of decision analysis like states of nature, chance occurrences governed by probabilities, and payoff matrices. An example involving production decisions for a dairy product is provided. The document also discusses criteria for decision making under uncertainty like Laplace, maximin, maximax, Hurwicz, and regret. It concludes by covering expected monetary value, expected opportunity loss, expected value of perfect information, and decision trees as approaches to decision making under risk.
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.
This document discusses the Hungarian method for solving assignment problems. It begins by explaining the assignment problem and providing an example with 2 machines and 2 operators. It then describes the Hungarian method, which finds the optimal assignment in polynomial time. The method is explained through a 4x4 example, with the key steps being: 1) row and column reductions, 2) finding a complete assignment or using adjustments to enable it. The document also notes the method works for balanced or minimization problems, and how maximization problems can be converted. It concludes by providing a professor-subject example and walking through applying the Hungarian method to find the optimal assignment.
This document discusses solving assignment problems using the Hungarian method. It provides an 8-step process for solving both balanced and unbalanced assignment problems to minimize or maximize the objective. For balanced problems, the steps include reducing the matrix, finding possible assignments based on zeros, and covering and updating the matrix if no optimal solution is found. For unbalanced problems, dummy rows or columns are added to create a balanced matrix before applying the same steps. Examples demonstrate solving both minimization and maximization problems.
This is one of the topic covered here to give a flavour of the Operations Research(OR) topics covered in the CD ROM.This ebook will be available by the end of September 2014 on snapdeal website.The OR topics covered are simplified through a number of solved illustrations and will be useful to BMS,MMS.MBA and CA students.
The Hungarian method is an algorithm that solves assignment problems in polynomial time. It was developed by Harold Kuhn in 1955. The method finds the optimal assignment between two equally sized sets that maximizes the total value of assignments. It works by constructing and updating a cost matrix through row and column reductions until the optimal assignment is revealed.
The document discusses the stable marriage problem and its solution using the Gale-Shapley algorithm. The stable marriage problem aims to match pairs of men and women for marriage such that there are no two people who would both rather be matched with each other over their assigned partners. The Gale-Shapley algorithm solves this problem by having the men and women iteratively propose to their preferred partners until all pairs are in stable marriages. An example with 5 men and 5 women ranked by their preferences is provided to illustrate the algorithm's steps.
This document discusses assignment problems and the Hungarian method for solving them. Assignment problems involve assigning n jobs to n workers or machines in a way that minimizes costs or maximizes effectiveness. The Hungarian method is an algorithm that can be used to find the optimal assignment. It involves row and column reductions on the cost or effectiveness matrix, followed by finding a complete assignment with zeros or modifying the matrix to create more zeros until a complete assignment is possible. Examples demonstrate applying the Hungarian method to solve different assignment problems step-by-step.
Assignment Chapter - Q & A Compilation by Niraj ThapaCA Niraj Thapa
My name is Niraj Thapa. I have compiled Assignment Chapter including SM, PM & Exam Questions of AMA.
You feedback on this will be valuable inputs for me to proceed further.
The document summarizes the Hungarian algorithm, an optimization method for solving assignment problems. It describes an example of an unbalanced assignment problem with 4 jobs and 3 workers, where a dummy worker is added. The algorithm proceeds in steps: (1) subtracting the minimum element from each row, (2) looking for a circled zero, (3) drawing lines around zeros if none are circled, (4) subtracting the minimum uncovered element and adding to intersecting lines, then (5) repeating step 3 until an optimal assignment is found. Applying these steps to the example results in an optimal assignment with total cost of 16.
This document discusses the matching problem and provides an overview of algorithms and methods for solving it. It begins with an agenda that covers illustrating the problem, relevant definitions, modeling it mathematically, selecting algorithms like the Hungarian method, use cases, exercises, solving it using Excel, and reviewing literature. It then goes into more detail on each section, providing examples, visualizations, the formal problem model, and a step-by-step example of applying the Hungarian method to solve a matching problem.
The topic of assignment is a critical problem in mathematics and is further explored in the real
physical world. We try to implement a replacement method during this paper to solve assignment problems with
algorithm and solution steps. By using new method and computing by existing two methods, we analyse a
numerical example, also we compare the optimal solutions between this new method and two current methods. A
standardized technique, simple to use to solve assignment problems, may be the proposed method
This document discusses assignment problems and how to solve them using the Hungarian method. Assignment problems involve efficiently allocating people or resources to tasks when only one task can be assigned to each person. 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. The document provides an example using this method to assign 5 workers to 5 jobs with the goal of minimizing total work hours.
Chapter 3.Simplex Method hand out last.pdfTsegay Berhe
This document provides material on solving linear programming problems using the simplex method. It begins with an introduction to the simplex method and how it can be used to solve linear programming problems analytically. It then presents the steps for solving a problem using the simplex method, including determining a starting basic feasible solution, selecting entering and leaving variables, and performing elementary row operations to arrive at the optimal solution. An example problem is also presented to illustrate how to set up and solve a linear programming problem using the simplex method.
The document provides an overview of the simplex algorithm for solving linear programming problems. It begins with an introduction and defines the standard format for representing linear programs. It then describes the key steps of the simplex algorithm, including setting up the initial simplex tableau, choosing the pivot column and pivot row, and pivoting to move to the next basic feasible solution. It notes that the algorithm terminates when an optimal solution is reached where all entries in the objective row are non-negative. The document also briefly discusses variants like the ellipsoid method and cycling issues addressed by Bland's rule.
The document discusses the Hungarian method for solving assignment problems. It begins by defining an assignment problem as minimizing the cost of completing jobs by assigning workers to tasks, where each job is assigned to exactly one worker. It then outlines the steps of the Hungarian method, which involves constructing a cost matrix, subtracting rows and columns to find zeros, and using the zeros to determine the optimal assignment. Finally, it provides an example and lists some applications of the Hungarian method like assigning machines, salespeople, contracts, teachers, and accountants.
This document discusses assignment problems in operations research. It begins by defining assignment problems as linear programming problems that involve assigning resources like jobs, machines or tasks to workers or projects in the most efficient way, typically to minimize costs or time. It provides examples of assignment problems and explains how they can be modeled as zero-one programming problems or transportation problems. The document then describes the Hungarian method for solving assignment problems, which involves setting up a cost table and finding the optimal assignments by covering zeros. It also mentions some special cases that can occur in assignment problems and provides an example solved using Excel Solver.
This document discusses the assignment problem and provides an example of how to solve it using the Hungarian method. It begins by defining the assignment problem and providing examples of problems that can be modeled as assignment problems. It then explains how to use the Hungarian method, which involves two phases - first reducing rows and columns to get zeros, then finding the minimum number of lines to cover all zeros. An example of using the Hungarian method on a sales assignment problem is worked through step-by-step to find the optimal assignment.
The document discusses the simplex algorithm for solving linear programming problems. It begins with an introduction and overview of the simplex algorithm. It then describes the key steps of the algorithm, which are: 1) converting the problem into slack format, 2) constructing the initial simplex tableau, 3) selecting the pivot column and calculating the theta ratio to determine the departing variable, 4) pivoting to create the next tableau. The document provides examples to illustrate these steps. It also briefly discusses cycling issues, software implementations, efficiency considerations and variants of the simplex algorithm.
This document describes an approach for solving Sudoku puzzles through a step-by-step process of improving an initial brute force solver. It begins with a simple representation of Sudoku grids and validity checking, then introduces techniques to prune invalid choices, collapse partial grids, and expand single choices to iteratively reduce the search space. The final solver uses these techniques to efficiently solve any newspaper Sudoku puzzle instantly by avoiding blocked states that cannot lead to a solution.
The document provides information on operations research and the assignment problem. It discusses the steps to solve an assignment problem, which include: (1) writing the problem in a matrix form, (2) obtaining a reduced cost matrix through row and column operations, and (3) making assignments on a one-to-one basis by considering zeros in rows and columns. It also addresses issues like unbalanced matrices, maximization problems, and infeasible assignments.
Similar to Chapter 1 Assignment Problems (DS) (1).pptx (20)
This document outlines the product life cycle (PLC) model, including its key stages of introduction, growth, maturity, and decline. It provides details on how products, styles, fashions, and fads may progress differently through the PLC. Marketing strategies are discussed for each stage, with an emphasis on adapting strategies based on where a product is in its life cycle. Examples are given of products that have successfully navigated the PLC stages as well as those that have declined.
The document summarizes the top 5 truck companies in India:
1. Tata Motors, founded in 1945, is India's largest truck manufacturer with a revenue of ₹319,247 crore and uses advertisements across print, TV, radio and online with the tagline "Connecting Aspirations".
2. Ashok Leyland, founded in 1948, is endorsed by cricketer Mahendra Singh Dhoni and uses campaigns like "Driver's Anthem" with the tagline "Engineering your Tomorrows".
3. Mahindra & Mahindra was founded in 2005 and advertises on TV, print and exhibitions with the tagline "Spark the New".
4. E
Economic globalization is defined as the integration and interlinking of economies across borders through movements of capital, technology, goods, and services. Key agents driving globalization are multinational corporations and market forces. Globalization leads to increased specialization through global commodity chains and production networks but also greater financial vulnerability, income inequality, and environmental issues for developing nations. International organizations need to safeguard developing country interests and reforms are required to manage the consequences of globalization.
The document provides information about the induction program for the MBA batch of 2022-23 at S.S Dhamankar Institute of Management located at Pune Vidyarathi Griha's College of Engineering. It includes details about the vision, mission and facilities available at the institute like the library and laboratories. It also mentions the faculty members, past industrial visits, cultural and technical events conducted at the institute. Furthermore, it lists the top rank holders and placement details of past batches.
Dabur produces health care, hair care, oral care, skin care, home care, and food products. Its vision is dedicated to household health and well-being, and its mission is to provide global quality services while maintaining an India advantage. Objectives include sustainable practices, natural solutions from ayurveda and herbs, and shareholder returns. Goals are to be the preferred brand for target consumers' needs and become a global ayurvedic leader. A SWOT analysis identifies strengths in R&D, product development, rural penetration, and overseas presence, while weaknesses include uneven profitability and low rural category penetration. Opportunities exist in brand initiatives, urban products, and shifting from unbranded to branded
Sahara India Pariwar is an Indian conglomerate established in 1978 with interests in finance, infrastructure, housing, media and entertainment. In 2010, two Sahara companies raised thousands of crores from investors through optionally fully convertible debentures without approval from SEBI as required. SEBI ordered Sahara to repay investors within 3 months, but Sahara failed to comply. The Supreme Court later ruled in favor of SEBI, ordering Sahara to provide details of over 2.5 crore investors to SEBI, though the documents were found to be incomplete and unrealistic, raising suspicions of money laundering.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
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Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...Diana Rendina
Librarians are leading the way in creating future-ready citizens – now we need to update our spaces to match. In this session, attendees will get inspiration for transforming their library spaces. You’ll learn how to survey students and patrons, create a focus group, and use design thinking to brainstorm ideas for your space. We’ll discuss budget friendly ways to change your space as well as how to find funding. No matter where you’re at, you’ll find ideas for reimagining your space in this session.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
2. 2
Objectives
After completion of this lesson you will be able to:
• formulate the assignment problem
• know Hungarian method to find proper assignment
• employ Hungarian method to find proper assignment
3. 3
The Assignment Problem
The Assignment Problem: Suppose we have n resources to which we want
to assign to n tasks on a one-to-one basis. Suppose also that we know the cost
of assigning a given resource to a given task. We wish to find an optimal
assignment–one which minimizes total cost.
4. 4
Mathematical Model of Assignment Problem
The Mathematical Model: Let 𝐶𝑖,𝑗 be the cost of assigning the 𝑖𝑡ℎresource to
the 𝑗𝑡ℎ task. We define the cost matrix to be the n × n matrix
C=
𝐶1,1 𝐶1,2 … … . 𝐶1,𝑛
𝐶2,1 𝐶2,2 … … 𝐶2,𝑛
… . .
𝐶𝑛,1
… . .
𝐶𝑛,1
… . 𝐶1,1
… … 𝐶𝑛,𝑛
An assignment is a set of n entry positions in the cost matrix, no two of which
lie in the same row or column. The sum of the n entries of an assignment is its cost. An
assignment with the smallest possible cost is called an optimal assignment.
5. 5
Mathematical Model of Assignment Problem
The Mathematical Model:
Mathematically, we can express the problem as follows:
To Minimize z(cost)= 𝒊=𝟏
𝒏
𝒋=𝟏
𝒏
𝒄𝒊𝒋𝒙𝒊𝒋 ……… {i=1,2,3…….n , j=1,2,3…. n}
Where 𝒙𝒊𝒋 =
1 ; 𝑖𝑓 𝑖𝑡ℎ 𝑝𝑒𝑟𝑠𝑜𝑛 𝑖𝑠 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑡𝑜 𝑗𝑡ℎ 𝑤𝑜𝑟𝑘
0 ; 𝑖𝑓 𝑖𝑡ℎ 𝑝𝑒𝑟𝑠𝑜𝑛 𝑖𝑠 𝑛𝑜𝑡 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑡𝑜 𝑗𝑡ℎ 𝑤𝑜𝑟𝑘
With the restrictions
i) 𝒊=𝟏
𝒏
𝒙𝒊𝒋=1; j=1,2,….n, i.e. 𝑖𝑡ℎ
𝑝𝑒𝑟𝑠𝑜𝑛 will do only one work
ii) 𝒋=𝟏
𝒏
𝒙𝒊𝒋=1; i=1,2,….n, i.e. 𝑗𝑡ℎ
𝑤𝑜𝑟𝑘 will be done only by one person.
6. 6
The Hungarian Method: The following algorithm applies the above theorem to a given n × n cost matrix to find
an optimal assignment.
Step 1. Subtract the smallest entry in each row from all the entries of its row.
Step 2. Subtract the smallest entry in each column from all the entries of its column.
Step 3. Draw lines through appropriate rows and columns so that all the zero entries of the cost
matrix are covered and the minimum number of such lines is used.
Step 4. Test for Optimality: (i) If the minimum number of covering lines is n, an optimal assignment
of zeros is possible and we are finished. (ii) If the minimum number of covering lines is less than n,
an optimal assignment of zeros is not yet possible. In that case, proceed to Step 5.
Step 5. Determine the smallest entry not covered by any line. Subtract this entry from each uncovered
row, and then add it to each covered column. Return to Step 3.
The Hungarian Method
7. 7
Flowchart to solve Assignment Problem
Start
Prepare the Assignment Table
Is it a
balance
Problem ?
Add Dummy rows or columns
It is a
Maximization
Problem ?
Convert it into a minimization
problem by subtracting all the
elements from the largest element
No
No
Yes
Yes
8. 8
Flowchart to solve Assignment Problem
Yes
Obtain the reduced cost table. For this:
1) Subtract the Minimum element in each row from all the
elements of that row and then
2) Subtract the Minimum element in each column from all the
elements of that row and then
Does the number of
lines draw equal the
order of the matrix ?
Convert it Subtract the
smallest uncovered element
from all the uncovered
elements. Add it to the
elements that lies at the
intersection of two lines.
Keep the remaining
elements unchanged in the
revised cost table.
No
Draw minimum number of lines to cover all the zeroes in the table
Optimum Solution is obtained
Stop
9. 9
We must determine how jobs should be assigned to machines to minimize setup
times, which are given below:
Example 1
Job 1 Job 2 Job 3 Job 4
Machine 1 14 12 15 15
Machine 2 21 18 18 22
Machine 3 14 17 12 14
Machine 4 6 5 3 6
10. 10
Step 1: Find the minimum element in each row of the cost matrix. Form a new matrix by subtracting this
cost from each row i.e. subtract 12 from 1st row, 18 from 2nd row, 12 from 3rd row and 3 from 4th row
respectively
Example 1
Job 1 Job 2 Job 3 Job 4
Machine 1 14 12 15 15
Machine 2 21 18 18 22
Machine 3 14 17 12 14
Machine 4 6 5 3 6
Job 1 Job 2 Job 3 Job 4
Machine 1 2 0 3 3
Machine 2 3 0 0 4
Machine 3 2 5 0 2
Machine 4 3 2 0 3
Row Reduction
11. 11
Step 2: Find the minimum element in each column of the cost matrix. Form a new matrix by subtracting
this cost from each column i.e. subtract 2 from 1st col, 0 from 2nd col, 0 from 3rd col and 2 from 4th col
respectively
Example 1
Job 1 Job 2 Job 3 Job 4
Machine 1 2 0 3 3
Machine 2 3 0 0 4
Machine 3 2 5 0 2
Machine 4 3 2 0 3
Job 1 Job 2 Job 3 Job 4
Machine 1 0 0 3 1
Machine 2 1 0 0 2
Machine 3 0 5 0 0
Machine 4 1 2 0 1
Column
Reduction
12. 12
Step 3: Cover all the zeroes of the
matrix with the minimum number
of horizontal or vertical lines.
Example 1
Job 1 Job 2 Job 3 Job 4
Machine 1 0 0 3 1
Machine 2 1 0 0 2
Machine 3 0 5 0 0
Machine 4 1 2 0 1
1
2
3
4
Step 4: Since the minimal number of lines is 4, an optimal assignment of zeroes is
possible.
No. of Rows / Columns = No of Lines
4 = 4
13. 13
Step 5 : Once the No. of Rows / Columns = No of Lines conditions are satisfied Find out the optimum
solution
Process to Find out Optimum Solution: Select the row or column which contain exactly one zero
If you found single zero in row then cut all the zeroes in their column and If you found single zero in
Column then cut all the zeroes in row.
In the given example it can be seen that there is single 0 in the row 4th and col 4th . If you select the 0
from row 4th then cancel all zeroes in the 3rd col.
Example 1
Job 1 Job 2 Job 3 Job 4
Machine 1 0 0 3 1
Machine 2 1 0 0 2
Machine 3 0 5 0 0
Machine 4 1 2 1
0
Job 1 Job 2 Job 3 Job 4
Machine 1 0 0 3 1
Machine 2 1 0 0 2
Machine 3 0 5 0 0
Machine 4 1 2 1
0
14. 14
Similarly you can perform the operation for Remaining
Matrix
Example 1
Job 1 Job 2 Job 3 Job 4
Machine 1 0 3 1
Machine 2 1 0 2
Machine 3 0 5 0
Machine 4 1 2 1
0
0
0
0
Step 6: Once we perform the allocation final assignment
is as follows
Note: For Time Kindly check Original Matrix in given
question
Machine Jobs Time
Machine 1 1 14
Machine 2 2 18
Machine 3 4 14
Machine 4 3 03
Total Time 49 Hrs
15. 15
Solve the following assignment problem for minimization
Example 2
Job 1 Job 2 Job 3 Job 4 Job 5
Workers A 8 8 8 11 12
B 3 9 18 13 6
C 10 7 2 2 2
D 7 11 9 7 12
E 7 9 10 4 12
16. 16
Step 1: Find the minimum element in each row of the cost matrix. Form a new matrix by subtracting this cost from each
row i.e. subtract 8 from 1st row, 3 from 2nd row, 2 from 3rd row, 7 from 4th row and 4 from 5th row respectively
Example 2
Row Reduction
Job 1 Job 2 Job 3 Job 4 Job 5
A 8 8 8 11 12
B 3 9 18 13 6
C 10 7 2 2 2
D 7 11 9 7 12
E 7 9 10 4 12
Job 1 Job 2 Job 3 Job 4 Job 5
A 0 0 0 3 4
B 0 6 15 10 3
C 8 5 0 0 0
D 0 4 2 0 5
E 3 5 6 0 8
17. 17
Step 2: Find the minimum element in each column of the cost matrix. Form a new matrix by subtracting this cost from
each column i.e. subtract 0 from 1st col, 0 from 2nd col, 0 from 3rd col, 0 from 4th col and 0 from 5th respectively
Example 2
Col Reduction
Job 1 Job 2 Job 3 Job 4 Job 5
A 0 0 0 3 4
B 0 6 15 10 3
C 8 5 0 0 0
D 0 4 2 0 5
E 3 5 6 0 8
Job 1 Job 2 Job 3 Job 4 Job 5
A 0 0 0 3 4
B 0 6 15 10 3
C 8 5 0 0 0
D 0 4 2 0 5
E 3 5 6 0 8
18. 18
Step 3: Cover all the zeroes of the
matrix with the minimum number
of horizontal or vertical lines.
Example 2
1
2
3
4
Step 4: Since the minimal number of lines is 4, and total number of rows/ columns are
5 hence we need to perform improvement.
No. of Rows / Columns ≠ No of Lines
5 ≠ 4
( Note: Kindly check the difference in Ex. 1 and Ex.2)
Job 1 Job 2 Job 3 Job 4 Job 5
A 0 0 0 3 4
B 0 6 15 10 3
C 8 5 0 0 0
D 0 4 2 0 5
E 3 5 6 0 8
19. 19
Note: The numbers which are covered with
the red lines are called as covered element
and remaining are called as uncovered
elements
Step 5: Process for Improvement: Select
the smallest no from all uncovered element.
Subtract this smallest no from all uncovered
elements and add only at intersection of two
line are happened and keep all other covered
element as it is.
Example 2
1
2
3
4
Job 1 Job 2 Job 3 Job 4 Job 5
A 0 0 0 3 4
B 0 6 15 10 3
C 8 5 0 0 0
D 0 4 2 0 5
E 3 5 6 0 8
In the above example 2 can be subtracted
from all uncovered elements and add only at
cell A-Job1, A-job4, C-Job1 and C-Job 4.
20. 20
Improvement 1:
Example 2
1
2
3
4
Job 1 Job 2 Job 3 Job 4 Job 5
A 0 0 0 3 4
B 0 6 15 10 3
C 8 5 0 0 0
D 0 4 2 0 5
E 3 5 6 0 8
In the above example 2 can be subtracted
from all uncovered elements and add only at
cell A-Job1, A-job4, C-Job1 and C-Job 4.
Job 1 Job 2 Job 3 Job 4 Job 5
A 2 0 0 5 4
B 0 4 13 10 1
C 10 5 0 2 0
D 0 2 0 0 3
E 3 3 4 0 6
After
Improvement
Note: After Improvement start the repeat process with cover zeroes
21. 21
Improvement 1:
Example 2
1
2 3
4
Since the minimal number of lines is 5, an optimal
assignment of zeroes is possible.
No. of Rows / Columns = No of Lines
5 = 5
Job 1 Job 2 Job 3 Job 4 Job 5
A 2 0 0 5 4
B 0 4 13 10 1
C 10 5 0 2 0
D 0 2 0 0 3
E 3 3 4 0 6
5
22. 22
Step 6 : Once the No. of Rows / Columns = No of Lines conditions are satisfied Find out the optimum
solution
Example 2
Job 1 Job 2 Job 3 Job 4 Job 5
A 2 0 5 4
B 4 13 10 1
C 10 5 0 2
D 0 2 0 3
E 3 3 4 6
0
0
0
0
0
Step 7: Once we perform the allocation final
assignment is as follows
Note: For Time Kindly check Original Matrix in
given question
Worker Jobs Time
A 2 05
B 1 03
C 5 02
D 3 09
E 4 04
Total Time 23 Hrs
23. 23
Examples For Practice
Solve the following assignment problem for minimization
Machines
1 2 3 4 5
Jobs A 8 8 8 11 12
B 4 5 6 3 4
C 12 11 10 9 8
D 18 21 18 17 15
E 10 11 10 8 12
Job Machines
A B C D E
1
2
3
4
5
30 37 40 28 40
40 24 27 21 36
40 32 33 30 35
25 38 40 36 36
29 62 41 34 39
Ans: 45 Hrs
Note: You have to perform Improvement till the following condition has to be satisfied
No of Rows/ Columns = No. of Lines
Ans: 149 Hrs
24. 24
1) Unbalanced Problem
2) Multiple Optimum Solution
3) Maximization Problem
4) Prohibited Assignment Problem ( Restricted Problem)
Special Cases in Assignment Problem
25. 25
1) Unbalanced Problem:
When the number of rows is not equal to the number of columns it is called as an
unbalanced assignment problem. Here we add required numbers of dummy rows or
columns with all its element as 0, to the matrix so as to make it square matrix (i.e.
balanced).
Eg:
Special Cases in Assignment Problem
1 2 3 4 5 6
A 12 10 15 22 18 8
B 10 18 25 15 16 12
C 11 10 3 8 5 9
D 6 14 10 13 13 12
E 8 12 11 7 13 10
1 2 3 4 5 6
A 12 10 15 22 18 8
B 10 18 25 15 16 12
C 11 10 3 8 5 9
D 6 14 10 13 13 12
E 8 12 11 7 13 10
F 0 0 0 0 0 0
As you can seen in the above example total number of rows are 5 and columns are 6 hence its is unbalanced
problem. We can balance the problem after adding dummy row i.e. F with all its elements as ‘0’
After Balance
26. 26
Let consider the previous example after balancing the problem matrix look like this
Example on Unbalanced Problem
1 2 3 4 5 6
A 12 10 15 22 18 8
B 10 18 25 15 16 12
C 11 10 3 8 5 9
D 6 14 10 13 13 12
E 8 12 11 7 13 10
1 2 3 4 5 6
A 12 10 15 22 18 8
B 10 18 25 15 16 12
C 11 10 3 8 5 9
D 6 14 10 13 13 12
E 8 12 11 7 13 10
F 0 0 0 0 0 0
Note: Before Solving any problem in assignment make sure that it has to be balance i.e.
No of Rows = No of Columns if it is not then balance it.
After Balance
27. 27
Example on Unbalanced Problem
1 2 3 4 5 6
A 12 10 15 22 18 8
B 10 18 25 15 16 12
C 11 10 3 8 5 9
D 6 14 10 13 13 12
E 8 12 11 7 13 10
F 0 0 0 0 0 0
Row Subtraction
Step 1: Find the minimum element in each row of the cost matrix. Form a new matrix by subtracting this cost from
each row i.e. subtract 8 from 1st row, 10 from 2nd row, 3 from 3rd row, 6 from 4th row ,7 from 5th row and 0 from th
6th row respectively
1 2 3 4 5 6
A 4 2 7 14 10 0
B 0 8 15 5 6 2
C 8 7 0 5 2 6
D 0 8 4 7 7 5
E 1 5 4 0 6 3
F 0 0 0 0 0 0
28. 28
Example on Unbalanced Problem
Column Subtraction
Step 2: Find the minimum element in each column of the cost matrix. Form a new matrix by subtracting this cost
from each column. As you can seen in the matrix each column has smallest element is ‘0’
1 2 3 4 5 6
A 4 2 7 14 10 0
B 0 8 15 5 6 2
C 8 7 0 5 2 6
D 0 8 4 7 7 5
E 1 5 4 0 6 3
F 0 0 0 0 0 0
1 2 3 4 5 6
A 4 2 7 14 10 0
B 0 8 15 5 6 2
C 8 7 0 5 2 6
D 0 8 4 7 7 5
E 1 5 4 0 6 3
F 0 0 0 0 0 0
29. 29
Example on Unbalanced Problem
Step 3: Cover all the zeroes of the matrix with the minimum number of horizontal or vertical lines.
1 2 3 4 5 6
A 4 2 7 14 10 0
B 0 8 15 5 6 2
C 8 7 0 5 2 6
D 0 8 4 7 7 5
E 1 5 4 0 6 3
F 0 0 0 0 0 0 1
2
3
4
Step 4: Since the minimal number of
lines is 5, and total number of rows/
columns are 6 hence we need to
perform improvement.
No. of Rows / Columns ≠ No of Lines
6 ≠ 5
5
30. 30
Example on Unbalanced Problem
1 2 3 4 5 6
A 4 2 7 14 10 0
B 0 8 15 5 6 2
C 8 7 0 5 2 6
D 0 8 4 7 7 5
E 1 5 4 0 6 3
F 0 0 0 0 0 0 1
2
3
4
Step 5: Process for Improvement: Select
the smallest no from all uncovered element.
Subtract this smallest no from all uncovered
elements and add only at intersection of two
line are happened and keep all other covered
element as it is.
In the given example 2 can be subtracted
from all uncovered elements and add only at
cell E1,E3,E6 and F1,F3,F6. Keep remaining
covered element as it is
5
31. 31
Example on Unbalanced Problem
1 2 3 4 5 6
A 4 2 7 14 10 0
B 0 8 15 5 6 2
C 8 7 0 5 2 6
D 0 8 4 7 7 5
E 1 5 4 0 6 3
F 0 0 0 0 0 0 1
2
3
4 5
Improvement 1:
1 2 3 4 5 6
A 4 0 7 12 8 0
B 0 6 15 3 4 2
C 8 5 0 3 0 6
D 0 6 4 5 5 5
E 3 5 6 0 6 5
F 2 0 2 0 0 2
After Improvement
Note: After Improvement start the repeat process with cover zeroes
32. 32
Example on Unbalanced Problem
1
2
3
4
5
Improvement 1:
1 2 3 4 5 6
A 4 0 7 12 8 0
B 0 6 15 3 4 2
C 8 5 0 3 0 6
D 0 6 4 5 5 5
E 3 5 6 0 6 5
F 2 0 2 0 0 2
Step 6: Since the minimal number of
lines is 5, and total number of rows/
columns are 6 hence we need to
perform improvement number 2.
No. of Rows / Columns ≠ No of Lines
6 ≠ 5
33. 33
Example on Unbalanced Problem
1
2
3
4
5
Improvement 2:
1 2 3 4 5 6
A 4 0 7 12 8 0
B 0 6 15 3 4 2
C 8 5 0 3 0 6
D 0 6 4 5 5 5
E 3 5 6 0 6 5
F 2 0 2 0 0 2
Step 7: Process for Improvement: Select
the smallest no from all uncovered element.
Subtract this smallest no from all uncovered
elements and add only at intersection of two
line are happened and keep all other covered
element as it is.
In the given example 2 can be subtracted
from all uncovered elements and add only at
cell A1,A4,C1,C4,F2 and F4. Keep
remaining covered element as it is
34. 34
Example on Unbalanced Problem
1
2
3
4
5
Improvement 2:
1 2 3 4 5 6
A 4 0 7 12 8 0
B 0 6 15 3 4 2
C 8 5 0 3 0 6
D 0 6 4 5 5 5
E 3 5 6 0 6 5
F 2 0 2 0 0 2
1 2 3 4 5 6
A 6 0 7 14 8 0
B 0 4 13 3 2 0
C 10 5 0 5 0 6
D 0 4 2 5 3 3
E 3 3 4 0 4 3
F 4 0 2 0 0 2
After Improvement
Note: After Improvement start the repeat process with cover zeroes
35. 35
Example on Unbalanced Problem
1
2
3
4
5
Improvement 2:
1 2 3 4 5 6
A 6 0 7 14 8 0
B 0 4 13 3 2 0
C 10 5 0 5 0 6
D 0 4 2 5 3 3
E 3 3 4 0 4 3
F 4 0 2 0 0 2
6
Since the minimal number of lines is 6, an optimal
assignment of zeroes is possible.
No. of Rows / Columns = No of Lines
6 = 6
36. 36
Example on Unbalanced Problem
Step 8 : Once the No. of Rows / Columns = No of Lines conditions are satisfied Find out the
optimum solution
1 2 3 4 5 6
A 6 0 7 14 8 0
B 0 4 13 3 2 0
C 10 5 0 5 0 6
D 0 4 2 5 3 3
E 3 3 4 0 4 3
F 4 0 2 0 0 2
0
0
0
0
0
0
Step 9: Once we perform the allocation final assignment is
as follows
Note: For Time Kindly check Original Matrix in given
question
Worker Jobs Time
A 2 10
B 6 12
C 3 03
D 1 06
E 4 13
F 5 00
Total Time 38 Hrs
37. 37
Step 8 : Once the No. of Rows / Columns = No of Lines conditions are satisfied Find out the optimum
solution
Example 2
Job 1 Job 2 Job 3 Job 4 Job 5
A 2 0 5 4
B 4 13 10 1
C 10 5 0 2
D 0 2 0 3
E 3 3 4 6
0
0
0
0
0
Step 7: Once we perform the allocation final
assignment is as follows
Note: For Time Kindly check Original Matrix in
given question
Worker Jobs Time
A 2 05
B 1 03
C 5 02
D 3 09
E 4 04
Total Time 23 Hrs
38. 38
1) Unbalanced Problem
Solve the following assignment problem ?
Special Cases in Assignment Problem
J
O
B
S
Workers
M1 M2 M3 M4
A 17 23 27 31
B 7 12 16 18
C 9 14 18 21
J
O
B
S
Workers
M1 M2 M3 M4
A 17 23 27 31
B 7 12 16 18
C 9 14 18 21
D 0 0 0 0
After Balance
39. 39
1) Unbalanced Problem
Step 1: Row Subtraction
Special Cases in Assignment Problem
J
O
B
S
Workers
M1 M2 M3 M4
A 17 23 27 31
B 7 12 16 18
C 9 14 18 21
D 0 0 0 0
After Row Subtraction
J
O
B
S
Workers
M1 M2 M3 M4
A 0 0 10 14
B 0 5 9 11
C 0 5 9 12
D 0 0 0 0
40. 40
1) Unbalanced Problem
Step 1: Column Subtraction
Special Cases in Assignment Problem
After Column Subtraction
J
O
B
S
Workers
M1 M2 M3 M4
A 0 0 10 14
B 0 5 9 11
C 0 5 9 12
D 0 0 0 0
J
O
B
S
Workers
M1 M2 M3 M4
A 0 0 10 14
B 0 5 9 11
C 0 5 9 12
D 0 0 0 0
41. 41
1) Unbalanced Problem
Step 3 : Cover zeroes
Special Cases in Assignment Problem
No of Rows ≠ 𝑵𝒐 𝒐𝒇 𝑳𝒊𝒏𝒆𝒔
4 ≠ 3
J
O
B
S
Workers
M1 M2 M3 M4
A 0 0 10 14
B 0 5 9 11
C 0 5 9 12
D 0 0 0 0 1
2
3
42. 42
1) Unbalanced Problem
Improvement No. 1
Special Cases in Assignment Problem
After Improvement
J
O
B
S
Workers
M1 M2 M3 M4
A 0 0 10 14
B 0 5 9 11
C 0 5 9 12
D 0 0 0 0 1
2
3
J
O
B
S
Workers
M1 M2 M3 M4
A 5 0 10 14
B 0 0 4 6
C 0 0 4 7
D 0 0 0 0
43. 43
1) Unbalanced Problem
Improvement No. 1
Special Cases in Assignment Problem
No of Rows ≠ 𝑵𝒐 𝒐𝒇 𝑳𝒊𝒏𝒆𝒔
4 ≠ 3
1
2
3
J
O
B
S
Workers
M1 M2 M3 M4
A 5 0 10 14
B 0 0 4 6
C 0 0 4 7
D 0 0 0 0
44. 44
1) Unbalanced Problem
Improvement No. 2
Special Cases in Assignment Problem
After Improvement
1
2
3
J
O
B
S
Workers
M1 M2 M3 M4
A 5 0 10 14
B 0 0 4 6
C 0 0 4 7
D 0 0 0 0
J
O
B
S
Workers
M1 M2 M3 M4
A 5 0 6 10
B 0 0 0 2
C 0 0 0 3
D 4 4 0 0
45. 45
1) Unbalanced Problem
Improvement No. 2
Special Cases in Assignment Problem
No. of Rows= No of line
4 = 4
2
1
3
J
O
B
S
Workers
M1 M2 M3 M4
A 5 0 6 10
B 0 0 0 2
C 0 0 0 3
D 4 4 0 0 4
46. 46
1) Unbalanced Problem
Optimum Solution
Special Cases in Assignment Problem
J
O
B
S
Workers
M1 M2 M3 M4
A 5 0 6 10
B 0 0 0 2
C 0 0 0 3
D 4 4 0 0
0
0
0
0
0
0
Worker Jobs Time
A M2 23
B M1 07
C M3 18
D M4 0
Total Time 48 Hrs
Solution 1
Worker Jobs Time
A M2 23
B M3 16
C M1 09
D M4 00
Total Time 48 Hrs
Solution 2
47. 47
2) Multiple Optimum Solution
After making the assignment (ie. after making the zeroes with square ) to the single unmarked in
all possible rows and columns, it is found that the two or more rows or columns still contain more than one
unmarked zeroes then the problem has multiple optimal solution. To get alternate solutions:
i) Select the row or column containing maximum number of unmarked zeroes (after making assignments
to the single unmarked zeroes.)
ii) Select one of the zeros and mark it with a square ( ).
iii) Cancel all other zeros in its row as well as column.
iv) Proceed further in usual manner to make other assignments.
v) Repeat the procedure by making assignment to each of the zeroes in the row or column selected in (i)
above separately to get the alternate solutions.
vi) All these alternate solutions gives the same optimal value.
Special Cases in Assignment Problem
48. 48
As we can find that the given problem is un balanced ( No of Rows ≠ 𝑁𝑜 𝑜𝑓 𝐶𝑜𝑙𝑢𝑚𝑛𝑠) hence here we add
dummy row i.e. D with all its element as ‘0’.
Example On Multiple Optimum Solution
Machine
M1 M2 M3 M4
Jobs
A 17 23 27 31
B 7 12 16 18
C 9 14 18 21
Machine
M1 M2 M3 M4
Jobs
A 17 23 27 31
B 7 12 16 18
C 9 14 18 21
D 0 0 0 0
49. 49
Example On Multiple Optimum Solution
Step 1: Find the minimum element in each row of the cost matrix. Form a new matrix by subtracting this cost from
each row i.e. subtract 17 from 1st row, 7 from 2nd row, 9 from 3rd row and 0 from 4th row respectively
Machine
M1 M2 M3 M4
Jobs
A 17 23 27 31
B 7 12 16 18
C 9 14 18 21
D 0 0 0 0 Machine
M1 M2 M3 M4
Jobs
A 0 6 10 14
B 0 5 9 11
C 0 5 9 12
D 0 0 0 0
Row Reduction
50. 50
Example On Multiple Optimum Solution
Step 2: Find the minimum element in each column of the cost matrix. Form a new matrix by subtracting this cost from
each column i.e. subtract 0 from 1st col, 0 from 2nd col, 0 from 3rd col, and 0 from 4th respectively
Machine
M1 M2 M3 M4
Jobs
A 0 6 10 14
B 0 5 9 11
C 0 5 9 12
D 0 0 0 0 Machine
M1 M2 M3 M4
Jobs
A 0 6 10 14
B 0 5 9 11
C 0 5 9 12
D 0 0 0 0
Column Reduction
51. 51
Example On Multiple Optimum Solution
Machine
M1 M2 M3 M4
Jobs
A 0 6 10 14
B 0 5 9 11
C 0 5 9 12
D 0 0 0 0
Step 3: Cover all the zeroes of the matrix with the minimum number of horizontal or vertical lines.
1
2
Step 4: Since the minimal number of lines is 5,
and total number of rows/ columns are 6 hence
we need to perform improvement number 1.
No. of Rows / Columns ≠ No of Lines
4 ≠ 𝟐
52. 52
Example On Multiple Optimum Solution
Machine
M1 M2 M3 M4
Jobs
A 0 6 10 14
B 0 5 9 11
C 0 5 9 12
D 0 0 0 0 1
2
Step 5: Process for Improvement: Select the
smallest no from all uncovered element. Subtract
this smallest no from all uncovered elements and
add only at intersection of two line are happened
and keep all other covered element as it is.
In the above example 5 can be subtracted
from all uncovered elements and add only at
cell D-M1
53. 53
Example On Multiple Optimum Solution
Machine
M1 M2 M3 M4
Jobs
A 0 6 10 14
B 0 5 9 11
C 0 5 9 12
D 0 0 0 0 1
2
Step 5: Process for Improvement: Select the
smallest no from all uncovered element. Subtract
this smallest no from all uncovered elements and
add only at intersection of two line are happened
and keep all other covered element as it is.
In the above example 5 can be subtracted
from all uncovered elements and add only at
cell D-M1
54. 54
Example On Multiple Optimum Solution
Machine
M1 M2 M3 M4
Jobs
A 0 6 10 14
B 0 5 9 11
C 0 5 9 12
D 0 0 0 0 1
2
Improvement No 1
Machine
M1 M2 M3 M4
Jobs
A 0 1 5 9
B 0 0 4 6
C 0 0 4 7
D 5 0 0 0
Improvement No 1
Note: After Improvement start the repeat process with cover zeroes
55. 55
Example On Multiple Optimum Solution
1
2
Improvement No 1
Machine
M1 M2 M3 M4
Jobs
A 0 1 5 9
B 0 0 4 6
C 0 0 4 7
D 5 0 0 0
3
Step 6: Since the minimal number of
lines is 3, and total number of rows/
columns are 4 hence we need to
perform improvement number 2.
No. of Rows / Columns ≠ No of Lines
4 ≠ 𝟑
56. 56
Example On Multiple Optimum Solution
1
2
Improvement No 2
Machine
M1 M2 M3 M4
Jobs
A 0 1 5 9
B 0 0 4 6
C 0 0 4 7
D 5 0 0 0
3
Step 7: Process for Improvement: Select
the smallest no from all uncovered element.
Subtract this smallest no from all uncovered
elements and add only at intersection of two
line are happened and keep all other covered
element as it is.
In the given example 4 can be subtracted
from all uncovered elements and add only at
cell D-M1 and D-M2. Keep remaining
covered element as it is
57. 57
Example On Multiple Optimum Solution
1
2
Improvement No 2
Machine
M1 M2 M3 M4
Jobs
A 0 1 5 9
B 0 0 4 6
C 0 0 4 7
D 5 0 0 0
3
Machine
M1 M2 M3 M4
Jobs
A 0 1 1 5
B 0 0 0 2
C 0 0 0 3
D 9 4 0 0
Improvement No 2
Note: After Improvement start the repeat process with cover zeroes
58. 58
Example On Multiple Optimum Solution
1
2
Improvement No 2
3
Machine
M1 M2 M3 M4
Jobs
A 0 1 1 5
B 0 0 0 2
C 0 0 0 3
D 9 4 0 0 4
Since the minimal number of lines is 4, an optimal
assignment of zeroes is possible.
No. of Rows / Columns = No of Lines
4 = 4
59. 59
Example On Multiple Optimum Solution
Machine
M1 M2 M3 M4
Jobs
A 0 1 1 5
B 0 0 0 2
C 0 0 0 3
D 9 4 0 0
Step 6 : Once the No. of Rows / Columns = No of Lines conditions are satisfied Find out the optimum
solution
0
0
As you can seen in above matrix we did not find the single ‘0’ either in row or column.
Such type of problem has multiple solution (i.e. More than one solution)
Note: Multiple Optimum Solution problem can be identified only at the time of final allocation
60. 60
Example On Multiple Optimum Solution
Machine
M1 M2 M3 M4
Jobs
A 0 1 1 5
B 0 0 0 2
C 0 0 0 3
D 9 4 0 0
Here Job B can be assigned to either Machine M2 or M3. If you assign Job B to Machine M2 then
Job C can be assigned to M3 or you can assign Job B to Machine M3 and Job C can be assigned to M2.
This can be represented in and
0
0
0
0
0
0
61. 61
Example On Multiple Optimum Solution
Machine
M1 M2 M3 M4
Jobs
A 0 1 1 5
B 0 0 0 2
C 0 0 0 3
D 9 4 0 0
0
0
0
0
0
0
Step 7: Once we perform the allocation final assignment is as follows
Jobs Machine Time
A M1 17
B M2 12
C M3 18
D M4 00
Total Time 47 Hrs
Solution No-1
Jobs Machine Time
A M1 17
B M3 16
C M2 14
D M4 00
Total Time 47 Hrs
Solution No-2
62. 62
Examples For Practice
Q. Solve the following assignment problem
for minimization
Ans: 55 Hrs
Note: You have to perform Improvement till the following condition has to be satisfied
No of Rows/ Columns = No. of Lines
Ans: 32 Hrs
Worker
A B C D E
Jobs
I 16 13 17 19 20
II 14 12 13 16 17
III 14 11 12 17 18
IV 5 5 8 8 11
V 5 3 8 8 10
Q. In a particular plant there are 4 machines to be
installed. There are 5 vacant places available. The
costs of installation of machines at different vacant
places are given in the following table. Find the
optimum assignment.
Places
Machines A B C D E
M1 9 11 15 10 11
M2 12 9 10 15 9
M3 10 13 14 11 7
M4 14 8 12 7 8
63. 63
3) Maximization Problem
Hungarian method can be used for maximization problem as follows:
converting it into equivalent minimization problem as follows:
i) Convert the given profit matrix into relative loss matrix. By subtracting all its element from the largest
element including it.
ii) Add a dummy row or column to it ( with 0 unit profit for its cells) if necessary.
iii) Locate the largest per unit profit figure in the table and subtract all profit figure ( including itself ) from it
to get an equivalent relative loss matrix.
iv) Solve it further as a normal Hungarian method to get optimum assignment
v) To find the total maximum profit consider the original profit elements for the respective assignment.
Maximization Keyword: Profit, Sales, Production, Revenue
Minimization Keyword:- Loss, Time, Space, Expenditure, Cost, Defects
Special Cases in Assignment Problem
64. 64
Ques. The data given in the table refer to production in certain units: Solve the following
assignment problem
Example on Maximization Problem
OPE
RAT
OR
MACHINES
A B C D
1 10 5 7 8
2 11 4 9 10
3 8 4 9 7
4 7 5 6 4
5 8 9 7 5
As we can find out that the given problem is not balanced. Hence here we add dummy column i.e E with all its
element as ‘0’
Note: Before Solving any problem
in assignment make sure that it has
to be balance i.e.
No of Rows = No of Columns if it
is not then balance it.
65. 65
Step 1: Balance the given problem
Example on Maximization Problem
Note: Before Solving any problem in assignment make sure that it has to be balance i.e.
No of Rows = No of Columns if it is not then balance it.
After Balance
A B C D
1 10 5 7 8
2 11 4 9 10
3 8 4 9 7
4 7 5 6 4
5 8 9 7 5
A B C D E
1 10 5 7 8 0
2 11 4 9 10 0
3 8 4 9 7 0
4 7 5 6 4 0
5 8 9 7 5 0
66. 66
Step 2: To solve the maximization problem first it has to be converted in to minimization by subtracting all its
element from the largest element.
In given problem the largest element in matrix is 11 hence we can subtract all the element from 11
Example on Maximization Problem
Once the problem is convert into minimization remaining process is same as we have done in earlier examples
Maximization to
Minimization
A B C D E
1 1 6 4 3 11
2 0 7 2 1 11
3 3 7 2 4 11
4 4 6 5 7 11
5 3 2 4 6 11
A B C D E
1 10 5 7 8 0
2 11 4 9 10 0
3 8 4 9 7 0
4 7 5 6 4 0
5 8 9 7 5 0
67. 67
Step 3: Find the minimum element in each row of the cost matrix. Form a new matrix by subtracting this cost from each row
i.e. subtract 1 from 1st row, 0 from 2nd row, 2 from 3rd row, 4 from 4th row and 2 from 5th row respectively.
Example on Maximization Problem
A B C D E
1 1 6 4 3 11
2 0 7 2 1 11
3 3 7 2 4 11
4 4 6 5 7 11
5 3 2 4 6 11
A B C D E
1 0 5 3 2 10
2 0 7 2 1 11
3 1 5 0 2 9
4 0 2 1 3 7
5 1 0 2 4 9
Row Subtraction
68. 68
Step 4: Find the minimum element in each column of the cost matrix. Form a new matrix by subtracting this cost from each
column. i.e. subtract 0 from 1st col, 0 from 2nd col, 0 from 3rd col, 1 from 4th col and 7 from 5th row respectively.
Example on Maximization Problem
A B C D E
1 0 5 3 2 10
2 0 7 2 1 11
3 1 5 0 2 9
4 0 2 1 3 7
5 1 0 2 4 9
Column Subtraction
A B C D E
1 0 5 3 1 3
2 0 7 2 0 4
3 1 5 0 1 2
4 0 2 1 2 0
5 1 0 2 3 2
69. 69
Step 5: Cover all the zeroes of the matrix with the minimum number of horizontal or vertical lines
Example on Maximization Problem
1 3
2
A B C D E
1 0 5 3 1 3
2 0 7 2 0 4
3 1 5 0 1 2
4 0 2 1 2 0
5 1 0 2 3 2
4
5
Step 6: Since the minimal number of lines is 4,
and total number of rows/ columns are 5 hence
an optimal assignment of zeroes is possible.
No. of Rows / Columns = No of Lines
5 = 5
70. 70
Example on Maximization Problem
Step 7 : Once the No. of Rows / Columns = No of Lines conditions are satisfied Find out the optimum
solution
0
0
A B C D E
1 0 5 3 1 3
2 0 7 2 4
3 1 5 0 1 2
4 0 2 1 2 0
5 1 2 3 2
0
0
0
Step 8: Once we perform the allocation final
assignment is as follows
Note: For Time Kindly check Original Matrix in
given question
Operator Machine Production
1 A 10
2 D 10
3 C 09
4 E 00
5 B 09
Total Time 38
71. 71
Q.1 The following table gives the profit of
assignment in Rupees. Also give the optimal
profit.
Examples For Practice on Maximization
Ans: 280
Jobs
J1 J2 J3 J4 J5
M1 50 60 40 30 45
M2 35 55 45 55 40
M3 40 45 50 35 35
M4 60 40 55 40 30
M5 45 35 45 50 55
Q.2 The marketing director of a multi unit company
is faced with a problem of assigning 5 senior
managers to 6 zones. From past experience he
knows that the efficiency percentage judged by
sales, operating cost etc.. depends on manager zone
combination. The efficiency of different managers is
given below
M
A
N
A
G
E
R
ZONES
1 2 3 4 5 6
A 73 91 87 82 78 80
B 81 85 69 76 74 85
C 75 72 83 84 78 91
D 93 96 86 91 83 82
E 90 91 79 89 69 76
Find out which zone will be managed by a junior
manager due to non-availability of a senior manager
72. 72
4) Prohibited Assignment Problem:
• A prohibited assignment problem is a problem which contains one or more constraints. It is also know as
restricted assignment problem. Suppose in case 𝑖𝑡ℎ
person is restricted to perform 𝑗𝑡ℎ
then constrained
assignment occur in the cell (i,j) of the cost matrix.
This are indicated by putting a dash (-) or cross (x) at the positions. To solve the problem:
i) For minimization problem we assume ( +M or + ∞ ) for the prohibited positions and solve further as usual.
ii) For maximization problem we assume ( -M or - ∞ ) for the prohibited positions and solve further as usual.
This are indicated by : -, X, M, ∞
Eg.
Special Cases in Assignment Problem
Clerk
s
Jobs
A B C D
1 4 7 8 6
2 * 8 7 4
3 3 * 8 3
4 6 6 4 2
As it can be seen in the given example clerk 2 is
not able to perform job A and clerk 3 is not able to
perform job B hence there is no point to assign Job
A and Job to Clerk 1 and 2 respectively that’s why
it can be denoted as *
73. 73
Step 1: Find the minimum element in each row of the cost matrix. Form a new matrix by subtracting this cost from each
row i.e. subtract 4 from 1st row, 4 from 2nd row, 3 from 3rd row and 2 from 4th row respectively.
Example on Prohibited Roots
Clerks Jobs
A B C D
1 4 7 8 6
2 * 8 7 4
3 3 * 8 3
4 6 6 4 2
Row Reduction
Clerks Jobs
A B C D
1 0 3 4 2
2 * 4 3 0
3 0 * 5 0
4 4 4 2 0
74. 74
Step 2: Find the minimum element in each column of the cost matrix. Form a new matrix by subtracting this cost from
each column i.e. subtract 0 from 1st col, 3 from 2nd col, 2 from 3rd col and 0 from 4th col respectively
Example on Prohibited Roots
Column Reduction
Clerks Jobs
A B C D
1 0 3 4 2
2 * 4 3 0
3 0 * 5 0
4 4 4 2 0 Clerks Jobs
A B C D
1 0 0 2 2
2 * 1 1 0
3 0 * 3 0
4 4 1 0 0
75. 75
Step 3: Cover all the zeroes of the matrix with the minimum number of horizontal or vertical lines.
Example on Prohibited Roots
Clerks Jobs
A B C D
1 0 0 2 2
2 * 1 1 0
3 0 * 3 0
4 4 1 0 0
1
2
3
4
Step 4: Since the minimal number of
lines is 4, and total number of rows/
columns are 4 hence we need to
perform improvement.
No. of Rows / Columns = No of Lines
4 = 4
76. 76
Step 8 : Once the No. of Rows / Columns = No of Lines conditions are satisfied Find out the optimum solution
Example on Prohibited Roots
Clerks Jobs
A B C D
1 0 0 2 2
2 * 1 1 0
3 0 * 3 0
4 4 1 0 0
0
Step 9: Once we perform the allocation final assignment is
as follows
Note: For Time Kindly check Original Matrix in given
question
Clerks Jobs Time
1 B 07
2 D 04
3 A 03
4 C 04
Total Time 18 Hrs
0
0
0
77. 77
Q.1 Solve the following assignment problem
Examples For Practice on Prohibited Roots
Ans: 66
PER
SON
JOBS
J1 J2 J3 J4 J5
P1 27 18 * 20 21
P2 31 24 21 12 17
P3 20 17 20 * 16
P4 20 28 20 16 27
Q.2 Solve the following assignment problem
Task
Machines
X Y X W P
A 19 21 25 20 21
B 27 24 * 25 24
C * 24 27 24 20
D 22 16 20 15 16
78. 78
Extra Problems for Practice
Q.1 Solve the following assignment problem:
1 2 3 4 5
A 4 6 10 5 6
B 7 4 8 5 4
C 12 6 9 6 2
D 9 3 7 2 3
E 6 5 5 3 8
Q.2 Five jobs are to be assigned to five persons
A,B,C,D, E, The time taken (in Minutes) by each of
them on each job is given below. Workout the
optimal assignment as the minimum total time
taken.
Jobs
1 2 3 4 5
Perso
n
A 16 13 17 19 20
B 14 12 13 16 17
C 14 11 12 17 18
D 5 5 8 8 11
E 3 3 8 8 10
Q.3 Solve the following assignment problem and
obtain minimum cost that job can be performed
1 2 3 4 5
A 25 18 32 20 21
B 34 25 21 12 17
C 20 17 20 32 16
D 20 28 20 16 27
79. 79
• Assigning teaching fellows to time slots
• Assigning airplanes to flights
• Assigning project members to tasks
• Determining positions on a team
• Assigning brides to grooms (once called the marriage problem)
• Machine allocation for optimum space utilization
Applications of AP
80. 80
1. A job assignment problem is unbalanced when
A) Each worker can perform only one job B) A worker can not perform all the jobs but can do only some of the jobs
C) The number of jobs and the number of workers are the same D) The number of jobs is not same as the number of workers
Multiple Choice Questions
2. In case multiple zeroes are obtained in all rows and columns
A) No solution is possible for the problem B) A unique solution is exist for the problem
C) The problem has infeasible solutions D) The problem has multiple solutions
3. Balancing of an unbalanced assignment problem involve
A) The introduction of dummy column B) The introduction of dummy row
C) The introduction of dummy row or a dummy column D) All the above
4. In assignment problem at what condition optimum solution is occurred
A) No of rows / columns = no of lines B) No of rows = no of columns
C) No of rows / columns ≠ no of lines D) No of the above