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.
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 transportation problem and its solution methodology. It states that the transportation problem seeks to minimize the total shipping costs of transporting goods from multiple origins to destinations, given the unit shipping costs. It is solved in two phases - obtaining an initial feasible solution and then moving toward the optimal solution. Several methods are described for obtaining the initial feasible solution, including the Northwest Corner method, Least Cost method, and Vogel's Approximation Method. The document also discusses testing the initial solution for optimality using methods like the Stepping Stone method and Modified Distribution method.
The document discusses transportation problems and their solutions. It defines transportation problems as dealing with assigning origins to destinations to maximize effectiveness. It outlines the history of transportation models and some common applications. It then describes the standard process of formulating a transportation problem and several algorithms for solving transportation problems, including the North West Corner Rule, Row Minima Method, Column Minima Method, Least Cost Method, and Vogel's Approximation Method.
The MODI method is used to find the optimal solution to a transportation problem in 3 steps:
1) Obtain an initial basic feasible solution using the Matrix Minimum method
2) Evaluate unoccupied cells to find their opportunity costs by calculating implicit costs as the sum of dual variables for each row and column
3) Find the most negative opportunity cost and draw a closed path, then adjust quantities along the path to make an unoccupied cell occupied and recalculate, repeating until all costs are non-negative
Vogel's Approximation Method (VAM) is a 3 step process for solving transportation problems:
1) Compute penalties for each row and column based on smallest costs.
2) Identify largest penalty and assign lowest cost variable the highest possible value, crossing out exhausted row or column.
3) Recalculate penalties and repeat until all requirements are satisfied.
The document discusses 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.
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.
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 transportation problem and its solution methodology. It states that the transportation problem seeks to minimize the total shipping costs of transporting goods from multiple origins to destinations, given the unit shipping costs. It is solved in two phases - obtaining an initial feasible solution and then moving toward the optimal solution. Several methods are described for obtaining the initial feasible solution, including the Northwest Corner method, Least Cost method, and Vogel's Approximation Method. The document also discusses testing the initial solution for optimality using methods like the Stepping Stone method and Modified Distribution method.
The document discusses transportation problems and their solutions. It defines transportation problems as dealing with assigning origins to destinations to maximize effectiveness. It outlines the history of transportation models and some common applications. It then describes the standard process of formulating a transportation problem and several algorithms for solving transportation problems, including the North West Corner Rule, Row Minima Method, Column Minima Method, Least Cost Method, and Vogel's Approximation Method.
The MODI method is used to find the optimal solution to a transportation problem in 3 steps:
1) Obtain an initial basic feasible solution using the Matrix Minimum method
2) Evaluate unoccupied cells to find their opportunity costs by calculating implicit costs as the sum of dual variables for each row and column
3) Find the most negative opportunity cost and draw a closed path, then adjust quantities along the path to make an unoccupied cell occupied and recalculate, repeating until all costs are non-negative
Vogel's Approximation Method (VAM) is a 3 step process for solving transportation problems:
1) Compute penalties for each row and column based on smallest costs.
2) Identify largest penalty and assign lowest cost variable the highest possible value, crossing out exhausted row or column.
3) Recalculate penalties and repeat until all requirements are satisfied.
The document discusses 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.
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 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.
The document discusses assignment problems and the Hungarian method for solving them. It begins by introducing the concept of assignment problems where the goal is to assign n jobs to n workers in a way that maximizes profit or efficiency. It then provides the mathematical formulation of an assignment problem as minimizing a cost function subject to constraints. The bulk of the document describes the Hungarian method, a multi-step algorithm for finding optimal assignments. It involves row/column reductions, finding a complete assignment of zeros, drawing lines to cover remaining zeros, and modifying the cost matrix to increase the number of zeros. An example is provided to illustrate the method.
The document summarizes transportation problems and their formulation as linear programming problems. It discusses how transportation problems involve optimizing the shipment of goods from sources to destinations given supply and demand constraints. Different methods for finding an initial basic feasible solution are presented, including the northwest corner method, least cost method, and Vogel's approximation method. Transportation problems can be balanced, where total supply equals demand, or unbalanced. Key concepts like feasible and optimal solutions are also defined.
Transportation Problem in Operational ResearchNeha Sharma
The document discusses the transportation problem and methods for finding its optimal solution. It begins by defining key terminology used in transportation models like feasible solution, basic feasible solution, and optimal solution. It then outlines the basic steps to obtain an initial basic feasible solution and subsequently improve it to reach the optimal solution. Three common methods for obtaining the initial solution are described: the Northwest Corner Method, Least Cost Entry Method, and Vogel's Approximation Method. The document also addresses how to solve unbalanced transportation problems and provides examples applying the methods.
Vogel's Approximation Method & Modified Distribution MethodKaushik Maitra
Vogel's Approximation Method (VAM) and Modified Distribution Method (MODI) are used to solve transportation problems. VAM computes penalties for each row and column to select the cell with the lowest cost to allocate units until constraints are satisfied, producing an initial basic feasible solution. MODI determines if the solution is optimal and identifies non-basic variables to consider, allowing it to find the true optimal solution. It is applied after VAM to a manufacturing company's transportation problem of supplying raw materials across plants and destinations.
The document summarizes the transportation problem and various methods to solve it. It discusses the transportation problem aims to find the optimal transportation schedule to minimize transportation costs. It describes the North West Corner Method, Least Cost Method, and Vogel's Approximation Method to solve transportation problems. It provides steps for the Vogel's Approximation Method, which includes checking for a basic feasible solution and revising solutions using a loop method if positive check numbers exist.
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.
The document discusses linear programming, which is a mathematical modeling technique used to allocate limited resources optimally. It provides examples of linear programming problems and their formulation. Key aspects covered include defining decision variables and constraints, developing the objective function, and interpreting feasible and optimal solutions. Graphical and algebraic solution methods like the simplex method are also introduced.
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.
This presentation is trying to explain the Linear Programming in operations research. There is a software called "Gipels" available on the internet which easily solves the LPP Problems along with the transportation problems. This presentation is co-developed with Sankeerth P & Aakansha Bajpai.
By:-
Aniruddh Tiwari
Linkedin :- http://in.linkedin.com/in/aniruddhtiwari
The document provides an outline of topics related to linear programming, including:
1) An introduction to linear programming models and examples of problems that can be solved using linear programming.
2) Developing linear programming models by determining objectives, constraints, and decision variables.
3) Graphical and simplex methods for solving linear programming problems.
4) Using a simplex tableau to iteratively solve a sample product mix problem to find the optimal solution.
The document discusses the Hungarian method for solving assignment problems. It begins by defining an assignment problem and providing examples. It then explains the steps of the Hungarian method, which involves reducing the cost matrix to find the optimal assignment that minimizes total cost. Three example problems are provided and solved using the Hungarian method. The key steps are row reduction, column reduction, and eliminating zeros with lines to reach the optimal solution.
8. normal distribution qt pgdm 1st semesterKaran Kukreja
The document discusses the normal distribution and its key properties. It explains that the normal distribution is a limiting case of the binomial distribution when the number of trials is large. It has a bell-shaped symmetrical curve centered around the mean. The normal distribution is uniquely defined by its mean and standard deviation. The document also covers how to convert between a normal distribution and the standard normal distribution and how to find probabilities using the standard normal distribution table.
The document outlines the presentation topic of Modified Distribution Method (MODI Method) for solving transportation problems. It first discusses the prerequisite methods of Least Cost Method, Vogel's Approximation Method and North-West Corner Method. It then explains the steps of MODI Method which involves setting up cost matrices for unallocated cells and introducing dual variables to find the implicit cost and evaluate unoccupied cells to determine if the initial solution can be improved. The document provides an example problem and solution to demonstrate the application of MODI Method.
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.
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.
This document discusses transportation problems and three methods to solve them: the North West Corner Method, Least Cost Method, and Vogel Approximation Method. The objective of transportation problems is to minimize the cost of distributing products from sources to destinations while satisfying supply and demand constraints. The document provides examples to illustrate how each method works step-by-step to arrive at a basic feasible solution.
The 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 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 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 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.
The document discusses assignment problems and the Hungarian method for solving them. It begins by introducing the concept of assignment problems where the goal is to assign n jobs to n workers in a way that maximizes profit or efficiency. It then provides the mathematical formulation of an assignment problem as minimizing a cost function subject to constraints. The bulk of the document describes the Hungarian method, a multi-step algorithm for finding optimal assignments. It involves row/column reductions, finding a complete assignment of zeros, drawing lines to cover remaining zeros, and modifying the cost matrix to increase the number of zeros. An example is provided to illustrate the method.
The document summarizes transportation problems and their formulation as linear programming problems. It discusses how transportation problems involve optimizing the shipment of goods from sources to destinations given supply and demand constraints. Different methods for finding an initial basic feasible solution are presented, including the northwest corner method, least cost method, and Vogel's approximation method. Transportation problems can be balanced, where total supply equals demand, or unbalanced. Key concepts like feasible and optimal solutions are also defined.
Transportation Problem in Operational ResearchNeha Sharma
The document discusses the transportation problem and methods for finding its optimal solution. It begins by defining key terminology used in transportation models like feasible solution, basic feasible solution, and optimal solution. It then outlines the basic steps to obtain an initial basic feasible solution and subsequently improve it to reach the optimal solution. Three common methods for obtaining the initial solution are described: the Northwest Corner Method, Least Cost Entry Method, and Vogel's Approximation Method. The document also addresses how to solve unbalanced transportation problems and provides examples applying the methods.
Vogel's Approximation Method & Modified Distribution MethodKaushik Maitra
Vogel's Approximation Method (VAM) and Modified Distribution Method (MODI) are used to solve transportation problems. VAM computes penalties for each row and column to select the cell with the lowest cost to allocate units until constraints are satisfied, producing an initial basic feasible solution. MODI determines if the solution is optimal and identifies non-basic variables to consider, allowing it to find the true optimal solution. It is applied after VAM to a manufacturing company's transportation problem of supplying raw materials across plants and destinations.
The document summarizes the transportation problem and various methods to solve it. It discusses the transportation problem aims to find the optimal transportation schedule to minimize transportation costs. It describes the North West Corner Method, Least Cost Method, and Vogel's Approximation Method to solve transportation problems. It provides steps for the Vogel's Approximation Method, which includes checking for a basic feasible solution and revising solutions using a loop method if positive check numbers exist.
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.
The document discusses linear programming, which is a mathematical modeling technique used to allocate limited resources optimally. It provides examples of linear programming problems and their formulation. Key aspects covered include defining decision variables and constraints, developing the objective function, and interpreting feasible and optimal solutions. Graphical and algebraic solution methods like the simplex method are also introduced.
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.
This presentation is trying to explain the Linear Programming in operations research. There is a software called "Gipels" available on the internet which easily solves the LPP Problems along with the transportation problems. This presentation is co-developed with Sankeerth P & Aakansha Bajpai.
By:-
Aniruddh Tiwari
Linkedin :- http://in.linkedin.com/in/aniruddhtiwari
The document provides an outline of topics related to linear programming, including:
1) An introduction to linear programming models and examples of problems that can be solved using linear programming.
2) Developing linear programming models by determining objectives, constraints, and decision variables.
3) Graphical and simplex methods for solving linear programming problems.
4) Using a simplex tableau to iteratively solve a sample product mix problem to find the optimal solution.
The document discusses the Hungarian method for solving assignment problems. It begins by defining an assignment problem and providing examples. It then explains the steps of the Hungarian method, which involves reducing the cost matrix to find the optimal assignment that minimizes total cost. Three example problems are provided and solved using the Hungarian method. The key steps are row reduction, column reduction, and eliminating zeros with lines to reach the optimal solution.
8. normal distribution qt pgdm 1st semesterKaran Kukreja
The document discusses the normal distribution and its key properties. It explains that the normal distribution is a limiting case of the binomial distribution when the number of trials is large. It has a bell-shaped symmetrical curve centered around the mean. The normal distribution is uniquely defined by its mean and standard deviation. The document also covers how to convert between a normal distribution and the standard normal distribution and how to find probabilities using the standard normal distribution table.
The document outlines the presentation topic of Modified Distribution Method (MODI Method) for solving transportation problems. It first discusses the prerequisite methods of Least Cost Method, Vogel's Approximation Method and North-West Corner Method. It then explains the steps of MODI Method which involves setting up cost matrices for unallocated cells and introducing dual variables to find the implicit cost and evaluate unoccupied cells to determine if the initial solution can be improved. The document provides an example problem and solution to demonstrate the application of MODI Method.
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.
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.
This document discusses transportation problems and three methods to solve them: the North West Corner Method, Least Cost Method, and Vogel Approximation Method. The objective of transportation problems is to minimize the cost of distributing products from sources to destinations while satisfying supply and demand constraints. The document provides examples to illustrate how each method works step-by-step to arrive at a basic feasible solution.
The 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 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 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 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.
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 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 various methods for improving the performance of multiplication operations, including using shifts and adds instead of actual multiplication, and Booth's algorithm. It examines these methods through examples of multiplying pairs of hexadecimal numbers. Booth's algorithm works by repeatedly adding or subtracting the multiplicand based on examining pairs of bits in the multiplier, allowing multiplication to be performed with only shifts. The document also covers non-restoring and non-performing division algorithms.
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.
Gaussian elimination is a method for solving systems of linear equations. It involves two main steps: forward elimination and back substitution. Forward elimination reduces the equations to an upper triangular system with one unknown in each equation. Back substitution then solves for the unknowns, starting from the last equation. Partial pivoting may be used to avoid division by zero and reduce rounding errors. It involves row swapping based on the largest element in each column during forward elimination.
The document discusses numerical methods for finding roots of equations and integrating functions. It covers root-finding algorithms like the bisection method, Regula Falsi method, modified Regula Falsi, and secant method. These algorithms iteratively find roots by narrowing the interval that contains the root. The document also discusses numerical integration techniques like the trapezoidal rule to approximate the area under a curve without having a closed-form solution. It notes the tradeoffs between different root-finding algorithms in terms of speed, accuracy, and ability to guarantee convergence.
The document describes the linear programming model and the simplex method for solving linear programming problems.
The linear programming model involves defining decision variables, an objective function to maximize, and constraints on the variables. The simplex method is then used to iteratively find the optimal solution. It involves setting up an initial tableau, identifying entering and leaving variables, performing row operations to create new tableaus, and checking for optimality until reached.
As an example, the document formulates a linear programming model to maximize profits for a company producing two products subject to resource constraints. It then applies the simplex method through multiple iterations to arrive at the optimal solution of 270 units of product 1 and 75 units of product 2 for maximum profits
The document describes the linear programming model and the simplex method for solving linear programming problems.
The linear programming model involves maximizing a linear objective function subject to linear inequality constraints involving decision variables. The simplex method is then used to solve linear programming problems by iteratively arriving at optimal feasible solutions.
The method involves setting up an initial tableau with slack variables, then selecting entering and leaving variables at each iteration to improve the objective function value, arriving at a final optimal solution where all coefficients in the objective function are positive. An example problem demonstrates applying the simplex method graphically and through tableau iterations to find the optimal product mix for a company.
The document describes the linear programming model and the simplex method for solving linear programming problems.
The linear programming model involves maximizing a linear objective function subject to linear inequality constraints. Decision variables, constraints, and the objective function are defined.
The simplex method is then described as the process of iteratively finding feasible solutions and improving the objective function value until an optimal solution is reached. The method involves setting up a simplex tableau, identifying entering and leaving variables, and performing row operations to derive new tableaus until an optimal solution is identified where all coefficients in the objective function are positive.
An example problem of determining a product mix is presented to demonstrate applying the linear programming model and solving it step-by
The document describes the linear programming problem and the simplex method for solving it. It provides an example problem of determining the optimal product mix for two products to maximize total income. The summary is:
(1) The example problem involves determining the optimal levels of two products given constraints on raw materials, storage space, and production time to maximize total income.
(2) The simplex method is applied by setting up the linear programming model, identifying entering and leaving variables, and performing row operations to iteratively find a better solution until reaching an optimal solution.
(3) For the example, the optimal solution found through three iterations of the simplex method is to produce 270 units of the first product and 75
MATLAB sessions: Laboratory 2
MAT 275 Laboratory 2
Matrix Computations and Programming in MATLAB
In this laboratory session we will learn how to
1. Create and manipulate matrices and vectors.
2. Write simple programs in MATLAB
NOTE: For your lab write-up, follow the instructions of LAB1.
Matrices and Linear Algebra
⋆ Matrices can be constructed in MATLAB in different ways. For example the 3 × 3 matrix
A =
8 1 63 5 7
4 9 2
can be entered as
>> A=[8,1,6;3,5,7;4,9,2]
A =
8 1 6
3 5 7
4 9 2
or
>> A=[8,1,6;
3,5,7;
4,9,2]
A =
8 1 6
3 5 7
4 9 2
or defined as the concatenation of 3 rows
>> row1=[8,1,6]; row2=[3,5,7]; row3=[4,9,2]; A=[row1;row2;row3]
A =
8 1 6
3 5 7
4 9 2
or 3 columns
>> col1=[8;3;4]; col2=[1;5;9]; col3=[6;7;2]; A=[col1,col2,col3]
A =
8 1 6
3 5 7
4 9 2
Note the use of , and ;. Concatenated rows/columns must have the same length. Larger matrices can
be created from smaller ones in the same way:
c⃝2011 Stefania Tracogna, SoMSS, ASU
MATLAB sessions: Laboratory 2
>> C=[A,A] % Same as C=[A A]
C =
8 1 6 8 1 6
3 5 7 3 5 7
4 9 2 4 9 2
The matrix C has dimension 3 × 6 (“3 by 6”). On the other hand smaller matrices (submatrices) can
be extracted from any given matrix:
>> A(2,3) % coefficient of A in 2nd row, 3rd column
ans =
7
>> A(1,:) % 1st row of A
ans =
8 1 6
>> A(:,3) % 3rd column of A
ans =
6
7
2
>> A([1,3],[2,3]) % keep coefficients in rows 1 & 3 and columns 2 & 3
ans =
1 6
9 2
⋆ Some matrices are already predefined in MATLAB:
>> I=eye(3) % the Identity matrix
I =
1 0 0
0 1 0
0 0 1
>> magic(3)
ans =
8 1 6
3 5 7
4 9 2
(what is magic about this matrix?)
⋆ Matrices can be manipulated very easily in MATLAB (unlike Maple). Here are sample commands
to exercise with:
>> A=magic(3);
>> B=A’ % transpose of A, i.e, rows of B are columns of A
B =
8 3 4
1 5 9
6 7 2
>> A+B % sum of A and B
ans =
16 4 10
4 10 16
10 16 4
>> A*B % standard linear algebra matrix multiplication
ans =
101 71 53
c⃝2011 Stefania Tracogna, SoMSS, ASU
MATLAB sessions: Laboratory 2
71 83 71
53 71 101
>> A.*B % coefficient-wise multiplication
ans =
64 3 24
3 25 63
24 63 4
⋆ One MATLAB command is especially relevant when studying the solution of linear systems of dif-
ferentials equations: x=A\b determines the solution x = A−1b of the linear system Ax = b. Here is an
example:
>> A=magic(3);
>> z=[1,2,3]’ % same as z=[1;2;3]
z =
1
2
3
>> b=A*z
b =
28
34
28
>> x = A\b % solve the system Ax = b. Compare with the exact solution, z, defined above.
x =
1
2
3
>> y =inv(A)*b % solve the system using the inverse: less efficient and accurate
ans =
1.0000
2.0000
3.0000
Now let’s check for accuracy by evaluating the difference z − x and z − y. In exact arithmetic they
should both be zero since x, y and z all represent the solution to the system.
>> z - x % error for backslash command
ans =
0
0
0
>> z - y % error for inverse
ans =
1.0e-015 *
-0.4441
0
-0.88 ...
The document discusses linear programming and provides examples to illustrate the process. It explains that linear programming involves optimizing a linear objective function subject to linear constraints. There are three basic components: decision variables, an objective to optimize, and constraints. Examples show how to formulate the objective function and constraints as linear equations or inequalities. The optimal solution is found by analyzing the feasible region defined by the constraints and determining which corner point gives the best value for the objective function.
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.
The document discusses the greedy method algorithmic approach. It provides an overview of greedy algorithms including that they make locally optimal choices at each step to find a global optimal solution. The document also provides examples of problems that can be solved using greedy methods like job sequencing, the knapsack problem, finding minimum spanning trees, and single source shortest paths. It summarizes control flow and applications of greedy algorithms.
- The document discusses matrices, including definitions, operations, and examples of matrix addition, subtraction, transposition, and multiplication. It also covers linear programming, defining it as a method to optimize a mathematical model to achieve the best outcome.
- Key concepts covered include the definitions of a matrix and its elements, how to perform basic operations like addition and subtraction on matrices, and how matrices are multiplied using the dot product of rows and columns. Linear programming is introduced as a method using linear relationships to find the maximum or minimum value of an objective function.
Similar to Quantitative Analysis For Decision Making (20)
2. The Hungarian method is a
combinatorial optimization algorithm which
solves the assignment problem in
polynomial time and which anticipated later
primal-dual methods. It was developed and
published by Harold Kuhn in 1955, who gave
the name "Hungarian method" because the
algorithm was largely based on the earlier
works of two Hungarian mathematicians:
Dénes Kőnig and Jenő Egerváry.
3. 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
maximizing 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"
4. Phase 1: Row and column
reductions
Step 0: Consider the given cost matrix
Step 1: Subtract the minimum value of each
row from the entries of that row, to obtain
the next matrix.
Step 2: Subtract the minimum value of each
column from the entries of that column , to
obtain the next matrix.
Treat the resulting matrix as the input for
phase 2.
7. Step 1: From each entry of a row, we subtract
the minimum value in that row and get the
following reduced cost matrix:
3 1 0 2
3 2 0 1
3 4 0 2
1 2 0 1
Column q1=1 q2=1 q3=0 q4=1
Minimum
8. Step 2: From each entry of a column, we
subtract the minimum value in that column
and get the following reduced cost matrix:
2 0 0 1
2 1 0 0
2 3 0 1
0 1 0 0
9. Step 3: Now we test whether an assignment
can be made as follows. If such an assignment
is possible, it is the optimal assignment.
•Examine the first row. If there is only one zero
in that row, then make an ( ) and cross ( )
all the other zeros in the column passing
through the surrounded zero and draw a
vertical line on that column
•Then starting with the first column if there is
one zero then make an ( ) cross all the zero
in that row & draw horizontal line on that row
cont till all zero are crossed or even assignment
10. Step 3(a) gives the following table.
2 0 0 1
2 1 0 0
2 3 0 1
0 1 0 0
Step 3(b): Now repeat the above procedure for
columns. (Remember to interchange row and
column in that step.)
12. If there is now a surrounded zero in each row
and each column, the optimal assignment is
obtained.
In our example, there is a surrounded zero in
each row and each column and so the optimal
assignment is: Hrs
Worker 1 is assigned to Job 2 = 6
Worker 2 is assigned to Job 4 = 4
Worker 3 is assigned to Job 3 = 4
Worker 4 is assigned to Job 1 = 6
Minimum total time = 20 hrs
The optimal solution is unique
13. If the final stage is reached (that is all the
zeros are either surrounded or crossed) and
if there is no surrounded zero in each row
and column, it is not possible to get the
optimal solution at this stage. We have to do
some more work. Again we illustrate with a
numerical example.
Solve the following unbalanced assignment
problem (Only one job to one man and only
one man to one job): 7 5 8 4
5 6 7 4
8 7 9 8
14. Since the problem is unbalanced, we add a
dummy worker 4 with cost 0 and get the
following starting cost matrix:
Job Row Min
7 5 8 4 p1=4
Worker 5 6 7 4 p2=4
8 7 9 8 p3=7
0 0 0 0 p4=0 Dummy
Applying Step 1, we get the reduced cost
matrix
15. 3 1 4 0
1 2 3 0
1 0 2 1
0 0 0 0
Now Step 2 is Not needed. We now apply
Step 3(a) and get the following table.
16. 3 1 4 0
1 2 3 0
1 0 2 1
0 0 0 0
Now all the zeros are either surrounded or
crossed but there is no surrounded zero in
Row 2. Hence assignment is NOT possible.
We go to Step 4.
17. Step 4(b) Select the smallest element, say, u, from
among all elements uncovered by all the lines.
In our example, u = 1
Step 4(c) Now subtract this u from all uncovered
elements but add this to all elements that lie at the
intersection of two lines
3 1 4 0
1 2 3 0
1 0 2 1
0 0 0 0
18. Doing this, we get the table:
2 1 3 0
0 2 2 0
0 0 1 1
0 1 0 1
19. Step 5: Reapply Step 3.
We thus get the table
2 1 3 0
0 2 2 0
0 0 1 1
0 1 0 1
Thus the optimum allocation is:
W1 → J4 W2 → J1 W3 → J2 W4 → J3
Hence Job 3 is not done by any (real) worker.
And the optimal cost = 4+5+7+0 = 16
The optimal assignment is unique
20. MAXIMIZATION TYPE
•Hungarian method is valid for balanced &
minimization type
•The assignment problem can be converted
to minimization by finding the opportunity
loss
•The opportunity loss matrix is found by
subtracting all the element of the matrix from
the largest element
21. Consider the assignment problem
•Efficiencyof each professor to teach each
subject as follow :
SUBJECT
1 2 3 4
10 5 9 15
A
Professor 6 - 3 12
B
C 16 8 5 9
Find which professor to be assigned to which subject so
that total efficiency can be maximize . (-) indicates that
professor b cannot be assigned to sub 2 also find sub for
which we do not have professor
22. Since the problem is unbalanced, we add a
dummy professor D with cost 0 and get the
following starting cost matrix:
SUBJECT
1 2 3
4
10 5 9 15
A
Professor B 6 - 3 12
C 16 8 5 9
0 0 0 0 Dummy
D
23. OPPORTUNITY LOSS MATRIX
•Subtracting all the element of the matrix
from the largest element that is 16
•We get this table SUBJECT
1 2 3 4
6 11 7 1
A
Professor 10 - 13 4
B
C 0 8 11 7
D 16 16 16 16
24. APPLY HUNGARIAN METHOD
Apply Step 1 ,step 2 is not needed
Doing this, we get the table:
SUBJECT
1 2 3 4
A 5 10 6 0
Professor B 6 - 6 0
C 0 8 11 7
0 0 0 0
D
Complete assignment is not formed
25. NOW SUBTRACT MINIMUM ELEMENT FROM ALL
UNCOVERED ELEMENTS BUT ADD THIS TO ALL
ELEMENTS THAT LIE AT THE INTERSECTION OF
TWO LINES
SUBJECT
1 2 3 4
A 5 10 6 0
Professor B 6 - 6 0
C 0 8 11 7
0 0 0 0
D
26. Doing this, we get the table:
SUBJECT
1 2 3 4
5 4 0 0
A
6 - 3 0
Professor B
0 2 5 7
C
6 0 0 6
D
Complete assignment is formed
27. •The Optimal assignment is
• Professor Subject Efficiency
• A 3 9
• B 4 12
• C 1 16
• D 2 0
•Maximum total efficiency 37
•The optimal assignment is unique
•Subject 2 is not assigned to any professor