The document discusses assignment problems and the Hungarian method for solving them. It begins by introducing the concept of assignment problems where the goal is to assign n jobs to n workers in a way that maximizes profit or efficiency. It then provides the mathematical formulation of an assignment problem as minimizing a cost function subject to constraints. The bulk of the document describes the Hungarian method, a multi-step algorithm for finding optimal assignments. It involves row/column reductions, finding a complete assignment of zeros, drawing lines to cover remaining zeros, and modifying the cost matrix to increase the number of zeros. An example is provided to illustrate the method.
GAME THEORY
Terminology
Example : Game with Saddle point
Dominance Rules: (Theory-Example)
Arithmetic method – Example
Algebraic method - Example
Matrix method - Example
Graphical method - Example
Operations research (OR) began during World War II when scientists applied analytical methods to solve complex military problems. Since then, OR has expanded to help organizations with strategic decision-making. OR uses interdisciplinary teams and quantitative techniques like linear programming to build mathematical models of systems. These models help optimize resource allocation and identify optimal solutions. OR aims to improve systems through objective, data-driven analysis and continues providing value as new problems emerge over time.
The document discusses the travelling salesman problem (TSP) and assignment problems in operations research. It provides an introduction to assignment problems, describing how they involve allocating resources like people, jobs, or teachers to minimize costs. It then gives an example of an assignment problem involving assigning four men to four jobs. The document also provides an overview of the TSP, explaining that it involves finding the shortest route to visit all cities on a list once. Finally, it discusses some applications of the TSP in areas like planning, logistics, and manufacturing.
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.
Operation research history and overview application limitationBalaji P
This document provides an overview of operation research (OR). It discusses OR topics like quantitative approaches to decision making, the history and definition of OR, common OR models like linear programming and network flow programming, and applications of OR. It also explains problem solving, decision making, and quantitative analysis approaches. OR aims to apply analytical methods to help make optimal decisions for complex systems and problems.
This is a special type of LPP in which the objective function is to find the optimum allocation of a number of tasks (jobs) to an equal number of facilities (persons). Here we make the assumption that each person can perform each job but with varying degree of efficiency. For example, a departmental head may have 4 persons available for assignment and 4 jobs to fill. Then his interest is to find the best assignment which will be in the best interest of the department.
Sequencing problems in Operations ResearchAbu Bashar
The document discusses sequencing problems and various sequencing rules used to optimize outputs when assigning jobs to machines. It describes Johnson's rule for minimizing completion time when scheduling jobs on two workstations. Johnson's rule involves scheduling the job with the shortest processing time first at the workstation where it finishes earliest. It provides an example of applying Johnson's rule to schedule five motor repair jobs at the Morris Machine Company across two workstations. Finally, it discusses Johnson's three machine rule for sequencing jobs across three machines.
GAME THEORY
Terminology
Example : Game with Saddle point
Dominance Rules: (Theory-Example)
Arithmetic method – Example
Algebraic method - Example
Matrix method - Example
Graphical method - Example
Operations research (OR) began during World War II when scientists applied analytical methods to solve complex military problems. Since then, OR has expanded to help organizations with strategic decision-making. OR uses interdisciplinary teams and quantitative techniques like linear programming to build mathematical models of systems. These models help optimize resource allocation and identify optimal solutions. OR aims to improve systems through objective, data-driven analysis and continues providing value as new problems emerge over time.
The document discusses the travelling salesman problem (TSP) and assignment problems in operations research. It provides an introduction to assignment problems, describing how they involve allocating resources like people, jobs, or teachers to minimize costs. It then gives an example of an assignment problem involving assigning four men to four jobs. The document also provides an overview of the TSP, explaining that it involves finding the shortest route to visit all cities on a list once. Finally, it discusses some applications of the TSP in areas like planning, logistics, and manufacturing.
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.
Operation research history and overview application limitationBalaji P
This document provides an overview of operation research (OR). It discusses OR topics like quantitative approaches to decision making, the history and definition of OR, common OR models like linear programming and network flow programming, and applications of OR. It also explains problem solving, decision making, and quantitative analysis approaches. OR aims to apply analytical methods to help make optimal decisions for complex systems and problems.
This is a special type of LPP in which the objective function is to find the optimum allocation of a number of tasks (jobs) to an equal number of facilities (persons). Here we make the assumption that each person can perform each job but with varying degree of efficiency. For example, a departmental head may have 4 persons available for assignment and 4 jobs to fill. Then his interest is to find the best assignment which will be in the best interest of the department.
Sequencing problems in Operations ResearchAbu Bashar
The document discusses sequencing problems and various sequencing rules used to optimize outputs when assigning jobs to machines. It describes Johnson's rule for minimizing completion time when scheduling jobs on two workstations. Johnson's rule involves scheduling the job with the shortest processing time first at the workstation where it finishes earliest. It provides an example of applying Johnson's rule to schedule five motor repair jobs at the Morris Machine Company across two workstations. Finally, it discusses Johnson's three machine rule for sequencing jobs across three machines.
Models of Operations Research is addressedSundar B N
Introduction, Meaning and Characteristics of Operations Research is addressed.
MODELS IN OPERATIONS RESEARCH, Classification of Models, degree of abstraction, Purpose Models, Predictive models, Descriptive models, Prescriptive models, Mathematic / Symbolic models, Models by nature of an environment, Models by the extent of generality, Models by Behaviour, Models by Method of Solution, Models by Method of Solution, Static and dynamic models, Iconic models Iconic models, Analogue models.
Subscribe to Vision Academy for Video Assistance
https://www.youtube.com/channel/UCjzpit_cXjdnzER_165mIiw
The document provides an overview of linear programming, including its applications, assumptions, and mathematical formulation. Some key points:
- Linear programming is a tool for maximizing or minimizing quantities like profit or cost, subject to constraints. 50-90% of business decisions and computations involve linear programming.
- Applications in business include production, personnel, inventory, marketing, financial, and blending problems. The objective is to optimize variables like costs, profits, or resources while meeting constraints.
- Assumptions of linear programming include certainty, linearity/proportionality, additivity, divisibility, non-negativity, finiteness, and optimality at corner points.
- A linear programming problem is modeled mathemat
The document discusses the Simplex method for solving linear programming problems involving profit maximization and cost minimization. It provides an overview of the concept and steps of the Simplex method, and gives an example of formulating and solving a farm linear programming model to maximize profits from two products. The document also discusses some complications that can arise in applying the Simplex method.
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.
1) The document discusses the Hungarian method for solving assignment problems by finding the optimal assignment of jobs to machines that minimizes costs.
2) It provides examples of using the Hungarian method to solve assignment problems by finding minimum costs in the cost matrix and obtaining a feasible assignment with zero costs.
3) The optimal solution is determined by selecting the assignments indicated by the cells with zero costs in the final cost matrix after applying the Hungarian method steps.
The document discusses 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 provides an introduction to operations research. It discusses that operations research is a systematic approach to decision-making and problem-solving that uses techniques like statistics, mathematics, and modeling to arrive at optimal solutions. It also briefly outlines some primary tools used in operations research like statistics, game theory, and probability theory. The document then gives a short history of operations research, noting that it originated in the UK during World War II to analyze problems like radar systems. It concludes with discussing the scope and applications of operations research in fields like management, regulation, and economics.
The document discusses transportation problems and their optimization using linear programming. It begins by explaining that transportation problems aim to optimally transport goods from supply origins to demand destinations at minimum cost while satisfying supply and demand constraints. The document then discusses how balanced transportation problems have equal total supply and demand, while unbalanced problems introduce dummy variables to balance totals. It provides examples of unbalanced problems where supply exceeds demand and vice versa, and how dummy columns/rows are added to balance the problems and find optimal solutions.
The document discusses rules for constructing network diagrams that represent project operations. Some key rules include: arrows represent activities and circles represent events; networks flow from left to right with activities only starting once all preceding activities are complete; each activity must start and end at a circle; event numbers increase from tail to head; networks can only have one initial and one terminal node; dummy and parallel activities are not permitted; and looping or burst events are also not allowed.
The least cost method is used to obtain an initial feasible solution for the transportation problem. It works by allocating units to the cell with the lowest cost first without exceeding supply or demand, then crosses out the exhausted row or column. This process repeats until all units are allocated. The method provides an accurate solution while considering transportation costs, but it does not guarantee finding the true optimal solution and may not be systematic in cases of tied costs.
This is a presentation from video on 'Introduction to Operations Research' available at the end of this presentations and directly at https://youtu.be/PSOW3_gX2OU
Topics like Organisations of Operations Research, History of Operations Research Role of Operations Research(OR), Scope of Operations Research(OR), Characteristics of Operations Research(OR), Attributes of Operations Research(OR).
This video also talks about Models of Operations Research
• Degree of abstraction
o Mathematical models
o Language models
o Concrete models
• Function
o Descriptive models
o Predictive models
o Normative models
• Time Horizon
o Static models
o Dynamic models
• Structure
o Iconic or physical models
o Analog or schematic models
o Symbolic or mathematical models
• Nature of environment
o Deterministic models
o Probabilistic models
• Extent of generality
o General model
o Specific models
This document presents 15 quantitative techniques and tools: Linear Programming, Queuing Theory, Inventory Control Method, Net Work Analysis, Replacement Problems, Sequencing, Integer Programming, Assignment Problems, Transportation Problems, Decision Theory and Game Theory, Markov Analysis, Simulation, Dynamic Programming, Goal Programming, and Symbolic Logic. It provides a brief overview of each technique, describing its purpose and typical applications.
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 is a project report on an inventory model submitted by three students. It includes:
1. An introduction to inventory, defining it as stock held for future production or sales in raw material, semi-finished, and finished forms. The objective is to minimize total costs or maximize profits.
2. A section on economic order quantity (EOQ) that describes costs of holding too much or too little inventory and assumptions like uniform demand. It includes a table calculating EOQ for different order quantities.
3. The EOQ formula and an example calculation of EOQ for Frooti beverages based on given demand and costs.
4. A re-order level calculation for Pad
This document discusses various multi-criteria decision making (MCDM) methods. It describes the objectives and steps in MCDM methodology. Three MCDM methods are explained in detail: Compromise Programming (CP), Preference Ranking Organisation METHod of Enrichment Evaluation (PROMETHEE), and the Weighted Average Method. An example is provided to illustrate the application of each method.
The document provides an overview of operations research techniques. It discusses:
- Operations research aims to improve decision-making through methods like simulation, optimization, and data analysis.
- Major applications include production scheduling, inventory control, transportation planning, and more.
- The techniques were developed in World War II and are now used widely in business for problems like resource allocation, forecasting, and process improvement.
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 transportation problems and their formulation as linear programming problems. It describes the basic transportation problem as involving transporting goods from sources to destinations while minimizing costs, subject to supply and demand constraints. Various methods for constructing an initial basic feasible solution are presented, including the northwest corner method and Vogel's approximation method. The stepping stone method for finding the optimal solution through iterative improvements is also covered.
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.
Models of Operations Research is addressedSundar B N
Introduction, Meaning and Characteristics of Operations Research is addressed.
MODELS IN OPERATIONS RESEARCH, Classification of Models, degree of abstraction, Purpose Models, Predictive models, Descriptive models, Prescriptive models, Mathematic / Symbolic models, Models by nature of an environment, Models by the extent of generality, Models by Behaviour, Models by Method of Solution, Models by Method of Solution, Static and dynamic models, Iconic models Iconic models, Analogue models.
Subscribe to Vision Academy for Video Assistance
https://www.youtube.com/channel/UCjzpit_cXjdnzER_165mIiw
The document provides an overview of linear programming, including its applications, assumptions, and mathematical formulation. Some key points:
- Linear programming is a tool for maximizing or minimizing quantities like profit or cost, subject to constraints. 50-90% of business decisions and computations involve linear programming.
- Applications in business include production, personnel, inventory, marketing, financial, and blending problems. The objective is to optimize variables like costs, profits, or resources while meeting constraints.
- Assumptions of linear programming include certainty, linearity/proportionality, additivity, divisibility, non-negativity, finiteness, and optimality at corner points.
- A linear programming problem is modeled mathemat
The document discusses the Simplex method for solving linear programming problems involving profit maximization and cost minimization. It provides an overview of the concept and steps of the Simplex method, and gives an example of formulating and solving a farm linear programming model to maximize profits from two products. The document also discusses some complications that can arise in applying the Simplex method.
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.
1) The document discusses the Hungarian method for solving assignment problems by finding the optimal assignment of jobs to machines that minimizes costs.
2) It provides examples of using the Hungarian method to solve assignment problems by finding minimum costs in the cost matrix and obtaining a feasible assignment with zero costs.
3) The optimal solution is determined by selecting the assignments indicated by the cells with zero costs in the final cost matrix after applying the Hungarian method steps.
The document discusses 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 provides an introduction to operations research. It discusses that operations research is a systematic approach to decision-making and problem-solving that uses techniques like statistics, mathematics, and modeling to arrive at optimal solutions. It also briefly outlines some primary tools used in operations research like statistics, game theory, and probability theory. The document then gives a short history of operations research, noting that it originated in the UK during World War II to analyze problems like radar systems. It concludes with discussing the scope and applications of operations research in fields like management, regulation, and economics.
The document discusses transportation problems and their optimization using linear programming. It begins by explaining that transportation problems aim to optimally transport goods from supply origins to demand destinations at minimum cost while satisfying supply and demand constraints. The document then discusses how balanced transportation problems have equal total supply and demand, while unbalanced problems introduce dummy variables to balance totals. It provides examples of unbalanced problems where supply exceeds demand and vice versa, and how dummy columns/rows are added to balance the problems and find optimal solutions.
The document discusses rules for constructing network diagrams that represent project operations. Some key rules include: arrows represent activities and circles represent events; networks flow from left to right with activities only starting once all preceding activities are complete; each activity must start and end at a circle; event numbers increase from tail to head; networks can only have one initial and one terminal node; dummy and parallel activities are not permitted; and looping or burst events are also not allowed.
The least cost method is used to obtain an initial feasible solution for the transportation problem. It works by allocating units to the cell with the lowest cost first without exceeding supply or demand, then crosses out the exhausted row or column. This process repeats until all units are allocated. The method provides an accurate solution while considering transportation costs, but it does not guarantee finding the true optimal solution and may not be systematic in cases of tied costs.
This is a presentation from video on 'Introduction to Operations Research' available at the end of this presentations and directly at https://youtu.be/PSOW3_gX2OU
Topics like Organisations of Operations Research, History of Operations Research Role of Operations Research(OR), Scope of Operations Research(OR), Characteristics of Operations Research(OR), Attributes of Operations Research(OR).
This video also talks about Models of Operations Research
• Degree of abstraction
o Mathematical models
o Language models
o Concrete models
• Function
o Descriptive models
o Predictive models
o Normative models
• Time Horizon
o Static models
o Dynamic models
• Structure
o Iconic or physical models
o Analog or schematic models
o Symbolic or mathematical models
• Nature of environment
o Deterministic models
o Probabilistic models
• Extent of generality
o General model
o Specific models
This document presents 15 quantitative techniques and tools: Linear Programming, Queuing Theory, Inventory Control Method, Net Work Analysis, Replacement Problems, Sequencing, Integer Programming, Assignment Problems, Transportation Problems, Decision Theory and Game Theory, Markov Analysis, Simulation, Dynamic Programming, Goal Programming, and Symbolic Logic. It provides a brief overview of each technique, describing its purpose and typical applications.
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 is a project report on an inventory model submitted by three students. It includes:
1. An introduction to inventory, defining it as stock held for future production or sales in raw material, semi-finished, and finished forms. The objective is to minimize total costs or maximize profits.
2. A section on economic order quantity (EOQ) that describes costs of holding too much or too little inventory and assumptions like uniform demand. It includes a table calculating EOQ for different order quantities.
3. The EOQ formula and an example calculation of EOQ for Frooti beverages based on given demand and costs.
4. A re-order level calculation for Pad
This document discusses various multi-criteria decision making (MCDM) methods. It describes the objectives and steps in MCDM methodology. Three MCDM methods are explained in detail: Compromise Programming (CP), Preference Ranking Organisation METHod of Enrichment Evaluation (PROMETHEE), and the Weighted Average Method. An example is provided to illustrate the application of each method.
The document provides an overview of operations research techniques. It discusses:
- Operations research aims to improve decision-making through methods like simulation, optimization, and data analysis.
- Major applications include production scheduling, inventory control, transportation planning, and more.
- The techniques were developed in World War II and are now used widely in business for problems like resource allocation, forecasting, and process improvement.
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 transportation problems and their formulation as linear programming problems. It describes the basic transportation problem as involving transporting goods from sources to destinations while minimizing costs, subject to supply and demand constraints. Various methods for constructing an initial basic feasible solution are presented, including the northwest corner method and Vogel's approximation method. The stepping stone method for finding the optimal solution through iterative improvements is also covered.
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.
The document discusses the assignment problem and various methods to solve it. The assignment problem involves assigning jobs to workers or other resources in an optimal way according to certain criteria like minimizing time or cost. The Hungarian assignment method is described as a multi-step algorithm to find the optimal assignment between jobs and workers/resources. It involves creating a cost matrix and performing row and column reductions to arrive at a matrix with zeros that indicates the optimal assignment. The document also briefly discusses handling unbalanced and constrained assignment problems.
The document discusses the assignment problem, which aims to assign jobs to persons so that time, cost, or profit is optimized. It describes the model of an assignment problem using a 3x3 matrix. It outlines the steps to solve an assignment problem, including checking if it is balanced, subtracting row/column minimums, and using the Hungarian method if the initial assignment is not complete. The Hungarian method involves tick marks and strikethroughs to identify the minimum cost assignment. Finally, it notes there can be maximization problems, unbalanced matrices, alternative solutions, and restricted assignments.
RESOURCE ALLOCATION AND STORAGE IN MOBILE USING CLOUD COMPUTINGSathmica K
This document discusses resource allocation and storage in mobile computing using cloud computing. It proposes using a reservation plan and the Hungarian method for virtual machine deployment to efficiently allocate resources. An optical code resources provisioning approach is also implemented using stochastic integer programming to optimize resource scheduling in the cloud. The goal is to reduce reservation and expenditure costs while improving resource utilization for mobile cloud applications.
The document discusses solving an assignment problem to minimize crew waiting time for an airline with flights between Delhi and Kolkata. It involves:
1) Calculating waiting times for all flight pairings assuming crews based in Delhi or Kolkata;
2) Developing an opportunity cost matrix from the minimum waiting times;
3) Applying row and column reduction to the matrix;
4) Finding an optimal assignment of 4 flight pairings that minimizes total waiting time of 40.5 hours.
Solving Transportation Problem in Operations ResearchChandan Pahelwani
This document presents a transportation problem involving 3 production facilities and 4 warehouses. The facilities have weekly production capacities of 7, 10, and 18 units. The warehouses have weekly demands of 5, 8, 7, and 15 units. The transportation costs between each facility-warehouse pair are given.
Using the Vogel's Approximation Method, an initial basic feasible solution is found allocating specific facilities to meet warehouse demands. The MODI method is then used to test for optimality. Some reallocations are made to improve the solution.
The optimal solution allocates production from the facilities to warehouses to meet demands at a total transportation cost of Rs. 900.
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.
The Kuhn-Munkres algorithm is used to find the maximum weighted matching in a weighted bipartite graph. It works by finding an initial feasible labeling and matching. It then searches for augmenting paths in the equality graph to improve the matching, or improves the labeling if no augmenting path exists to obtain a larger equality graph. The algorithm repeats this process until finding a perfect matching. An example applies the algorithm to find the optimal matching for a weighted bipartite graph.
The document discusses applying linear programming to solve employee assignment problems. It describes the assignment problem as assigning personnel to tasks to optimize resources like time and cost. It provides an example of assigning 5 candidates to 4 work jobs to minimize the total time. It defines decision variables and constraints, formulates the objective function to minimize total time, and explains how to solve it using the Hungarian method algorithm. The algorithm involves subtracting row/column minimums, drawing lines to cover all zeros, checking for optimality, and adjusting uneovered elements iteratively until optimal. It also discusses converting a maximization assignment problem to minimization to use the Hungarian method.
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 different methods to solve assignment problems including enumeration, integer programming, transportation, and Hungarian methods. It provides examples of balanced and unbalanced minimization and maximization problems. The Hungarian method is described as having steps like row and column deduction, assigning zeros, and tick marking to find the optimal assignment with the minimum cost or maximum profit. A sample problem demonstrates converting a profit matrix to a relative cost matrix and using the Hungarian method to find the optimal solution.
Operations research : Assignment problem (One's method) presentationPankaj Kumar
The document describes the process of assigning jobs to machines to maximize total return or minimize total cost. It involves creating an assignment matrix with the return/cost of assigning each job to each machine. The matrix is transformed through operations like finding minimum/maximum elements in rows and columns to create a feasible assignment with all ones. If this is not possible after the transformations, additional steps like drawing lines through ones are taken to find the optimal assignment. An example optimal assignment that yields a total return of 50 is provided.
The document discusses transportation and assignment models in operations research. The transportation model aims to minimize the cost of distributing a product from multiple sources to multiple destinations, while satisfying supply and demand constraints. The assignment model finds optimal one-to-one matching between sources and destinations to minimize costs. Some solution methods for transportation problems include the northwest corner method, row minima method, column minima method, and least cost method. The Hungarian method is commonly used to solve assignment problems by finding the minimum cost matching.
The document discusses different types of layout strategies used in operations management. It describes layouts for offices, retail stores, warehouses, and manufacturing facilities. For manufacturing facilities, it outlines process-oriented layouts where similar machines are grouped, work cell layouts that focus on single products, and product-oriented layouts that optimize personnel and machine utilization for repetitive production. Good layouts consider factors like material handling, space usage, and employee and customer flows.
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.
This document discusses operational research and assignment techniques. It begins by defining operational research as a scientific approach to problem solving aimed at reducing costs. It then provides examples of how operational research is used in various sectors like transportation, healthcare, and banking. The document outlines the characteristics and limitations of operational research. It proceeds to define assignment techniques as a way to allocate jobs and resources in the lowest cost manner. The remainder of the document details the specific steps involved in solving an assignment problem using a matrix-based approach to find an optimal solution.
This document outlines the job description for a Proposal Manager. The Proposal Manager is responsible for preparing proposals from initial concept through final presentation to customers. Key responsibilities include developing proposal plans, obtaining input from other departments, defining the scope of work, and leading proposal turnover meetings. The ideal candidate has a engineering degree, 5 years of industry experience including 3 years in project engineering or management, and skills in organization, communication, working in teams, and computer applications like Microsoft Office.
Linear Programming Problems {Operation Research}FellowBuddy.com
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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.
This document presents a statistical approach for solving a two-objective fuzzy assignment problem where costs and times are represented as triangular fuzzy numbers. The methodology proposes using Pascal's triangle to determine the coefficients of the fuzzy numbers and then applying a simple probability approach to obtain solutions. A numerical example is provided to illustrate the approach. Statistical tests like F-test and t-test are suggested to analyze and compare the proposed Pascal triangle method to existing graded mean integration representation techniques for solving fuzzy assignment problems.
This document discusses greedy algorithms and divide and conquer algorithms. It provides examples of problems that can be solved using each approach and outlines the general solutions. For greedy algorithms, it explains that they make locally optimal choices at each step without considering future possibilities. For divide and conquer, it explains the three main steps of dividing the problem into subproblems, solving the subproblems recursively, and merging the solutions. It also provides an example problem of finding the number of inversions in an array by modifying the merge sort algorithm.
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.
[This sheet must be completed and attached to the last page of.docxhanneloremccaffery
[This sheet must be completed and attached to the last page of your homework]
ISE 421
Operations Research II
Term 161
Homework #1
Student Name ID# Signature
Homework Guidelines
To receive full credit, you should make sure you follow the following guidelines.
Homework Presentation:
• Every main problem should be answered on a different page.
• You should submit the solutions for the first two problems only.
• All pages of your homework should be in chronological order.
• Your name, and the homework number should be clearly indicated.
Modeling Questions:
• Clearly define all the variables in one group. Then clearly define all the parameters in another group. Then display
the final model in the standard style (Objective, Constraints, Restriction on Domain). You can use ABCD, and
EVER OLD CARD mnemonic if desired.
ISE-421 HW-1
Problem #1
Suppose that the decision variables of a mathematical programming model are defined as:
xi,j,t := acers of land plot i allocated to crop j in year t
Ct := the funds in SAR donated by the governament at the begining of year t
Rj,t := the revenue generated from crop j in $ at the end of year t
where i = 1, . . . , 47; j = 1, . . . , 9; t = 1, . . . , 10.
Use summation (
∑
) and enumeration (∀) indexed notation to write expressions for each of
the following systems of constraints in terms of these decision variables, and determine how many
constraints belong to each system. You need to define additional variables to model the following
constraints. Assume $1 = 3.75SAR. In addition assume appropriate information wherever neces-
sary.
(a) The acres allocated in each plot i cannot exceed the available acreage (call it Ai) in any year.
(b) At least 1000 total acres must be devoted to corn (corp j = 4) in each year.
(c) At least one-third of the total acreage planted over 10 years must be in soybeans (corp j = 2).
(d) Either rice (corp j = 9) or wheat (corp j = 8) should be planted in a given year.
(e) Grapes (corp j = 7) should be planted in a year, when the current funds from the government
and the total revenue from the previous year is greater than or equal to 38000 SAR.
(f) In the odd years (t = 1, 3, . . . , 9) land plot 32 is unusable.
(g) On the same land plot, there should be at least a two years of difference between corn and rice
crops plantation.
(h) If soybeans are planted in a land plot, then no other crops should be planted on the same land
plot.
(i) Every plot must be used for planting in a given year.
(j) In every year, there should be at least 7 different crops.
1
ISE-421 HW-1
Problem #2
Consider the following IP problem.
maximize :
14 ∗ x1 + 22 ∗ x2 + 12 ∗ x3 + 10 ∗ x4
subject to :
50 ∗ x1 + 70 ∗ x2 + 40 ∗ x3 + 30 ∗ x4 ≤ 100
10 ∗ x1 + 60 ∗ x2 + 50 ∗ x3 + 60 ∗ x4 ≤ 80
6 ∗ x1 + 1 ∗ x2 + 3 ∗ x3 + 7 ∗ x4 ≤ 9
xi ∈ {0, 1} ∀ i = 1, . . . , 4
(a) Write the LP relaxation of the above model.
(b) Get the optimal objective function value of the LP relaxation from Tabl.
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1. 62
ASSIGNMENT PROBLEMS
62.1 INTRODUCTION
Imagine, if in a printing press there is one machine and one
operator is there to operate. How would you employ the worker?
Your immediate answer will be, the available operator will
operate the machine.
Again suppose there are two machines in the press and two
operators are engaged at different rates to operate them. Which
operator should operate which machine for maximising profit?
Similarly, if there are n machines available and n persons are
engaged at different rates to operate them. Which operator should
be assigned to which machine to ensure maximum efficiency?
While answering the above questions we have to think about
the interest of the press, so we have to find such an assignment
by which the press gets maximum profit on minimum investment.
Such problems are known as "assignment problems".
In this lesson we will study such problems.
62.2 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
2. 98::Mathematics
62.3 FORMULATION OF THE PROBLEM
Let there are n jobs and n persons are available with different
skills. If the cost of doing jth work by ith person is c ij. Then the
cost matrix is given in the table 1 below:
Jobs
1 2 3 ........ j ........ n Table
Persons 1
1 c 11 c 12 c 13 ........ c 1j ........ c 1n
2 c 21 c 22 c 23 ........ c 2j ........ c 2n
. . . . ............ .............
. . . . ............ .............
. . . . ............ .............
. . . . ............ .............
i c i1 c i2 c i3 ........ c ij ........ c in
. . . .
. ............. ............
. . .
. ............. ............
. . .
. ............. .............
. . .
n c n1 c n2 c n3 ........ c nj ........ c nn
Now the problem is which work is to be assigned to whom so
that the cost of completion of work will be minimum.
Mathematically, we can express the problem as follows:
n n
To minimize z (cost) = Σ Σ c ij x ij; [ i=1,2,...n; j=1,2,...n] ...(1)
i=1 j=1
where x ij =
{ 1; if ith person is assigned jth work
0; if ith person is not assigned the jth work
with the restrictions
n
(i) Σ xij =1; j=1,2,...n., i.e., ith person will do only one work.
i=1
n
(ii) Σ xij=1; i=1,2,...n.,i.e., jth work will be done only by one person.
j=1
3. Assignment Problems :: 99
62.4 ASSIGNMENT ALGORITHM
(The Hungarian Method)
In order to find the proper assignment it is essential for us
to know the Hungarian method. This method is dependent upon
two vital theorems, stated as below.
Theorem 1: If a constant is added (or subtracted) to every element
of any row (or column) of the cost matrix [cij] in an assingment
problem then an assingment which minimises the total cost for the
new matrix will also minimize the total cost matrix.
Theorem 2: If all cij ≥ 0 and there exists a solution
xij = X ij such that
∑ c ij xij = 0.
i
then this solution is an optimal solution, i.e., minimizes z.
The computational proecdure is given as under:
Step I (A) Row reduction:
Subtract the minimum entry of each row from all the
entires of the respective row in the cost matrix.
∑ (B) Column reduction:
j
After completion of row reduction, subtract the minimum
entry of each column from all the entires of the
respective column.
Step II Zero assignment:
(A) Starting with first row of the matrix received in first
step, examine the rows one by one until a row containing
exactly one zero is found. Then an experimental
assignment indicated by ‘ ’ is marked to that zero.
Now cross all the zeros in the column in which the
assignment is made. This procedure should be adopted
for each row assignment.
(B) When the set of rows has been completely examined, an
identical procedure is applied successively to columns.
Starting with column 1, examine all columns until a
column containing exactly one zero is found. Then make
an experimental assignment in that position and cross
other zeros in the row in which the assignment was made.
Continue these successive operations on rows and columns
until all zero’s have either been assigned or corssed-out.
4. 100 :: Mathematics
Now there are two possibilities:
(a) Either all the zeros are assigned or crossed out, i.e., we get
the maximal assignment.
or
(b) At least two zeros are remained by assignment or by crossing
out in each row or column. In this situation we try to exclude
some of the zeros by trial and error method.
This completes the second step. After this step we can get
two situations.
(i) Total assigned zero’s = n
The assignment is optimal.
(ii) Total assigned zero’s < n
Use step III and onwards.
Step III: Draw of minimum lines to cover zero’s
In order to cover all the zero’s at least once you may
adopt the following procedure.
(i) Marks (√) to all rows in which the assignment has
not been done.
(ii) See the position of zero in marked (√) row and then
mark (√) to the corresponding column.
(iii) See the marked (√) column and find the position of
a s s i g n e d z e r o ’ s a n d t h e n m a r k ( √) t o t h e
corresponding rows which are not marked till now.
(iv) Repeat the procedure (ii) and (iii) till the completion
of marking.
(v) Draw the lines through unmarked rows and marked
columns.
Note: If the above method does not work then make an
arbitrary assignment and then follow step IV.
Step IV: Select the smallest element from the uncovered elements.
(i) Subtract this smallest element from all those
elements which are not covered.
(ii) Add this smallest element to all those elements
which are at the intersection of two lines.
Step V: Thus we have increased the number of zero’s. Now,
5. Assignment Problems :: 101
modify the matrix with the help of step II and find the
required assignment.
This procedure will be more clear by the following examples.
Example A:
Four persons A,B,C and D are to be assigned four jobs I, II, III
and IV. The cost matrix is given as under, find the proper
assignment.
Man A B C D
Jobs
I 8 10 17 9
II 3 8 5 6
III 10 12 11 9
IV 6 13 9 7
Solution :
In order to find the proper assignment we apply the Hungarian
algorithm as follows:
I (A) Row reduction
Man A B C D
Jobs
I 0 2 9 1
II 0 5 2 3
III 1 3 2 0
IV 0 7 3 1
I (B) Column reduction
Man A B C D
Jobs
I 0 0 7 1
II 0 3 0 3
III 1 1 0 0
IV 0 5 1 1
6. 102 :: Mathematics
II(A) and (B) Zero assignment
Man A B C D
Jobs
I 0 0 7 1
II 0 3 0 3
III 1 1 0 0
IV 0 5 1 1
In this way all the zero’s are either crossed out or assigned.
Also total assigned zero’s = 4 (i.e., number of rows or columns).
Thus, the assignment is optimal.
From the table we get I → B; II → C: III → D and IV → A.
Example B:
There are five machines and five jobs are to be assigned and the
associated cost matrix is as follows. Find the proper assignment.
Machines
I II III IV V
A 6 12 3 11 15
B 4 2 7 1 10
Jobs C 8 11 10 7 11
D 16 19 12 23 21
E 9 5 7 6 10
Solution:
In order to find the proper assignment, we apply the Hungarian
method as follows:
IA (Row reduction)
Machines
I II III IV V
A 3 9 0 8 12
B 3 1 6 0 9
Jobs C 1 4 3 0 4
D 4 7 0 11 9
E 4 0 2 1 5
7. Assignment Problems :: 103
IB (Column reduction)
Machines
I II III IV V
A 2 9 0 8 8
B 2 1 6 0 5
Jobs C 0 4 3 0 0
D 3 7 0 11 5
E 3 0 2 1 1
II (Zero assignment)
Machines
I II III IV V
A 2 9 0 8 8
B 2 1 6 0 5
Jobs C 0 4 3 0 0
D 3 7 0 11 5
E 3 0 2 1 1
From the last table we see that all the zeros are either
assigned or crossed out, but the total number of assignment,
i.e., 4<5 (number of jobs to be assigned to machines). Therefore,
we have to follow step III and onwards as follows:
Step III
Machines
I II III IV V
A 2 9 0 8 8 √
B 2 1 6 0 5
Jobs C 0 4 3 0 0
D 3 7 0 11 5 √
E 3 0 2 1 1
Step IV: Here, the smallest element among the uncovered elements
is 2.
8. 104 :: Mathematics
(i) Subtract 2 from all those elements which are not
covered.
(ii) Add 2 to those entries which are at the junction of two
lines.
Complete the table as under:
Machines
I II III IV V
A 0 7 0 6 6
B 2 1 8 0 5
Jobs C 0 4 5 0 0
D 1 5 0 9 3
E 3 0 4 1 1
Step V. using step II again
Machines
I II III IV V
A 0 7 0 6 6
B 2 1 8 0 5
Jobs C 0 4 5 0 0
D 1 5 0 9 3
E 3 0 4 1 1
Thus, we have got five assignments as required by the problem.
The assignment is as follows:
A → I, B → IV, C → V, D → III and E → II.
This assignment holds for table given in step IV but from
theorem 1 it also holds for the original cost matrix. Thus from
the cost matrix the minimum cost = 6+1+11+12+5=Rs.35.
Note:
If we are given a maximization problem then convert it into
minimization problem, simply, multiplying by -1 to each entry in
the effectiveness matrix and then solve it in the usual manner.
9. Assignment Problems :: 105
Example C:
Solve the minimal assignment problem whose effectiveness matrix
is given by
A B C D
I 2 3 4 5
II 4 5 6 7
III 7 8 9 8
IV 3 5 8 4
Solution:
In order to find the proper assignment, we apply the Hungarian
method as follows:
Step I (A) A B C D
I 0 1 2 3
II 0 1 2 3
III 0 1 2 1
IV 0 2 5 1
I(B) A B C D
I 0 0 0 2
II 0 0 0 2
III 0 0 0 0
IV 0 1 3 0
Step II:
Since single zero neither exist in columns nor in rows, it is
usually easy to make zero assignments.
While examining rows successively, it is observed that row 4
has two zeros in both cells (4,1) and (4,4).
Now, arbitrarily make an experiemental assignment to one of
these two cells, say (4,1) and cross other zeros in row 4 and
column 1.
10. 106 :: Mathematics
The tables given below show the necessary steps for reaching
the optimal assignment I → B, II → C, III → D, IV → A.
A B C D
I 0 0 0 2
II 0 0 0 2
III 0 0 0 0
IV 0 1 3 0
A B C D
I 0 0 0 2
II 0 0 0 2
III 0 0 0 0
IV 0 1 3 0
A B C D
I 0 0 0 2
II 0 0 0 2
III 0 0 0 0
IV 0 1 3 0
Other optimal assignments are also possible in this example.
I → A, II → B, III → C, IV → D
I → C, II → B, III → A, IV → D
I → C, II → B, III → D, IV → A
I → B, II → C, III → A, IV → D
} (each has cost 20)
11. Assignment Problems :: 107
INTEXT QUESTIONS 62.1
1. An office has four workers, and four tasks have to be
performed. Workers differ in efficiency and tasks differ in
their intrinsic difficulty. Time each worker would take to
complete each task is given in the effectiveness matrix.
How the tasks should be allocated to each worker so as
to minimise the total man-hour?
Wrokers
I II III IV
A 5 23 14 8
B 10 25 1 23
Tasks C 35 16 15 12
D 16 23 21 7
2. A taxi hire company has one taxi at each of five depots
a,b,c,d and e. A customer requires a taxi in each town,
namely A,B,C,D and E. Distances (in kms) between depots
(origins) and towns (Destinations) are given in the following
distance matrix:
a b c d e
A 140 110 155 170 180
B 115 100 110 140 155
C 120 90 135 150 165
D 30 30 60 60 90
E 35 15 50 60 85
How should taxis be assigned to customers so as to
minimize the distance travelled?
12. 108 :: Mathematics
WHAT YOU HAVE LEARNT
Flow chart for Assignment problem
START
Construct the effectivness
matrix if not already given
Row reduction
Column reduction
No Is Yes
zero assignment
possible
(i) Draw minimum number
of lines to cover all
the zeros
(ii) Choose the least uncovered
element
(iii)Subtract this from the
uncovered elements and
add it to the elements ASSIGNMENT
at intersection of the lines. Put square over the
zero and cross out
all zeros (if any) of
No Is Yes the corresponding
Zero assignment column.
Possible ↓
SOLUTION
Add the elements
of the given matrix
correspond to
each square
↓
STOP
13. Assignment Problems :: 109
TERMINAL QUESTIONS
1. Solve the following assignment problems:
(a) Workers
I II III IV
A 1 19 10 4
Jobs B 6 21 7 19
C 31 12 11 8
D 12 19 17 3
(b) Persons
I II III IV
A 15 17 24 16
Jobs B 10 15 12 13
C 17 19 18 16
D 13 20 16 14
(c) Persons
A B C D
I 8 26 17 11
Jobs II 14 29 5 27
III 40 21 20 17
IV 19 26 24 10
14. 110 :: Mathematics
2. Find the proper assignment of the assignment problem whose
cost matrix is given as under.
I II III IV V
A 10 6 4 8 3
B 2 11 7 7 6
C 5 10 11 4 8
D 6 5 3 2 5
E 11 7 10 11 7
3. Solve the following assignment problem.
I II III IV
1 8 9 10 11
2 10 11 12 13
3 13 14 15 13
4 9 11 14 10
ANSWERS TO INTEXT QUESTIONS
62.1
1. I → A, II → C, III → B, IV → D
2. A → e, B → c, C → b, D → a, E → d
min distance (km) = 180+110+90+30+60=470 km.
ANSWERS TO TERMINAL QUESTIONS
1. (a) A → I, B → III, C → II, D → IV
(b) A → II, B → III, C → IV, D → I
min cost = 17+12+16+13=58
(c) I → A, II → C, III → B, IV → D
2. A → V, B → I, C → IV, D → III, E → II
3. 1 → II, 2 → III, 3 → IV, 4 → I
or
1 → III, 2 → II, 3 → IV, 4 → I