The document discusses transportation problems and assignment problems in operations research. It provides:
1) An overview of transportation problems, including the mathematical formulation to minimize transportation costs while meeting supply and demand constraints.
2) Methods for obtaining initial basic feasible solutions to transportation problems, such as the North-West Corner Rule and Vogel's Approximation Method.
3) Techniques for moving towards an optimal solution, including determining net evaluations and selecting entering variables.
4) The formulation and algorithm for solving assignment problems to minimize assignment costs while ensuring each job is assigned to exactly one machine.
The document discusses linear programming, including:
1. It describes the basic concepts of linear programming, such as decision variables, constraints, and the objective function needing to be linear.
2. It explains the steps to formulate a linear programming problem, such as identifying decision variables and constraints, and writing the objective function and constraints as linear combinations of the variables.
3. It provides examples of how to write linear programming problems in standard form to maximize or minimize objectives subject to constraints.
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.
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.
This document summarizes the Stepping Stone Method used in Operations Research to determine an optimal initial basic feasible solution for transportation problems. The method treats the transportation table as a pond that can only be crossed using occupied cells as stepping stones. It involves forming closed loops through unused and occupied cells, calculating improvement indices, and reallocating units to find an optimal solution. The example problem shows applying the method to find the optimal allocation of products from warehouses to destinations.
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.
this ppt is helpful for BBA/B.tech//MBA/M.tech students.
the ppt is on simulation topic...its covers -
Meaning
Advantages & Disadvantages
Uses
Process
Monte Carlo SImulation
Advantages & Disadvantages
Its example
The transportation problem is a special type of linear programming problem where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations.
Because of its special structure, the usual simplex method is not suitable for solving transportation problems. These problems require a special method of solution.
The document 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 North West Corner method for solving transportation problems. It describes the key steps of the method which include assigning quantities to cells beginning from the top-left corner and moving either right or down until all supply and demand quantities are satisfied. An example is provided to illustrate how the North West Corner method is applied to find an initial basic feasible solution for a given transportation problem. The maximum profit for the example problem is calculated to be 27 units.
This document discusses different aspects of line and staff authority in human resource management. It defines line authority as the right of line managers to direct the work of subordinates, while staff managers are authorized to assist and advise line managers. The document also discusses different types of human resource professionals, including executives, generalists, and specialists. It outlines several key human resource management functions performed by HR managers, such as staffing, training, compensation, and employee relations.
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.
This document provides an overview of transportation, transshipment, and assignment models. It describes the structure and objectives of transportation models, which optimize the distribution of goods from supply sources to demand destinations at minimum cost. Transshipment models allow goods to be consolidated and redistributed at intermediate points. Assignment models optimize the allocation of resources to tasks. Several examples of transportation model formulations, solutions, and analyses are provided to illustrate the modeling approach.
The document discusses goal programming, which is used to solve linear programs with multiple objectives viewed as goals. It describes goal programming as attempting to reach a satisfactory level of multiple objectives by minimizing deviations between goals and what can actually be achieved given constraints. An example problem involves a hardware company with goals of achieving a $30 profit, fully utilizing wiring hours, avoiding assembly overtime, and producing at least 7 ceiling fans. The goal programming model for this problem is formulated and graphically solved to satisfy the higher priority goals as closely as possible before lower goals.
The document discusses linear programming problems and how to formulate them. It provides definitions of key terms like linear, programming, objective function, decision variables, and constraints. It then explains the steps to formulate a linear programming problem, including defining the objective, decision variables, mathematical objective function, and constraints. Several examples of formulated linear programming problems are provided to maximize profit or minimize costs subject to various constraints.
The document discusses transportation problems (TPs), which involve determining the optimal way to route products from multiple supply locations to multiple demand destinations to minimize total transportation costs. It provides the mathematical formulation of a TP as a linear programming problem (LPP) with decision variables representing the quantity transported between each origin-destination pair. Methods for solving TPs include the simplex method by formulating it as an LPP or specialized transportation methods like the northwest corner rule to find an initial feasible solution and stepping stone/modified distribution methods to check for optimality. An example TP is presented to illustrate these concepts.
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.
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.
This document discusses three methods for finding a basic feasible solution to transportation problems:
1. The North-West Corner Method which starts assigning supply from the upper left corner, comparing supply and demand and moving horizontally or vertically as needed.
2. The Least-Cost Method which assigns supply based on the lowest cost cells, striking out columns and rows as demand is met.
3. Vogel's Approximation Method which iteratively assigns supply in a round-robin fashion.
The document provides an example applying the North-West Corner Method, explaining the step-by-step process of assigning supply and tracking remaining amounts.
Applied statistics and probability for engineers solution montgomery && rungerAnkit Katiyar
This document is the copyright page and preface for the book "Applied Statistics and Probability for Engineers" by Douglas C. Montgomery and George C. Runger. The copyright is held by John Wiley & Sons, Inc. in 2003. This book was edited, designed, and produced by various teams at John Wiley & Sons and printed by Donnelley/Willard. The preface states that the purpose of the included Student Solutions Manual is to provide additional help for students in understanding the problem-solving processes presented in the main text.
The International Labour Organisation aims to promote universal and lasting peace through cooperation between governments, employers, and workers on issues of social justice. It convenes the annual International Labour Conference in Geneva, Switzerland, where the 185 member states adopt labor standards and discuss critical issues. The Governing Body acts as the ILO's executive arm by setting policy, agendas, and budgets, and elects the Director-General to head the International Labour Office secretariat.
This document discusses transportation and assignment models, specifically the transportation problem. It provides an example of a transportation problem involving Roxas Gravel Company scheduling shipments from plants to projects to minimize costs. The steps to solve this problem are outlined, which include setting up a transportation tableau, developing an initial solution, testing for improvements, and developing improved solutions iteratively until no further improvements can be made. The MODI method for finding alternative optimal solutions is also introduced.
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.
The document contains 45 multiple choice questions about linear programming problems (LPP), transportation problems, and assignment problems. Some key points covered are:
- The feasible region in a graphical LPP solution satisfies all constraints simultaneously.
- An LPP deals with problems involving a single objective.
- The optimal solution in an LPP maximizes or minimizes the objective function subject to the constraints.
- Transportation problems aim to minimize total cost and are a special case of LPPs.
- Assignment problems assign origins to destinations at minimum cost when the number of each is equal.
The document discusses various concepts related to assignment and transportation problems including:
1) The steps to solve an assignment problem using the Hungarian method.
2) Examples of assignment problems involving personnel assignment and swimmer selection.
3) The formulation of a transportation problem to minimize shipping costs involving plants, cities, supply, and demand.
4) Examples of transportation problems involving flight assignment and power plant shipping costs.
5) How to solve transshipment problems by converting them into transportation problems.
The document provides an introduction to Light Detection and Ranging (LiDAR) technology. It describes LiDAR as an active remote sensing technique that uses laser light pulses to densely sample surface areas and measure distances. Key points include:
- LiDAR systems consist of a laser scanner, GPS, and inertial navigation to measure 3D coordinates of reflective surfaces.
- LiDAR pulses can have multiple returns as they pass through vegetation like trees, recording returns from the canopy and ground.
- Data can be used to generate digital elevation models, canopy height models, and classify point clouds by elevation and surface type.
CV Template to a Journal Submission_2016Atiqa khan
This document provides guidance on key features for a CV intended for submission to a journal. It recommends keeping the CV to one page, sending it as a PDF, focusing on quantifiable accomplishments, using language from relevant job descriptions, clearly defining the objective, eliminating unnecessary details, and staying focused on results. The document then provides an example CV format that includes sections for objective, experience, skills, employment history, education, publications, projects, and references.
This document contains an assignment with three questions for a student named Atiqa Ijaz Khan. Question 1 has 10 parts asking the student to write the output of various MATLAB statements. Question 2 has 10 parts asking the student to compare the output of related MATLAB functions and explain any differences. Question 3 asks the student to evaluate 5 mathematical expressions. The assignment is providing practice with MATLAB syntax and functions.
The document discusses the transportation problem in linear programming. It defines the transportation problem as minimizing the cost of distributing products from sources to destinations to meet supply and demand constraints. It provides an example problem with 3 factories and 4 warehouses. It also defines key concepts like feasible and optimal solutions and describes common methods to find initial basic feasible solutions like the Northwest Corner Rule, Matrix Minimum Method, and Vogel Approximation Method.
This document contains an assignment submitted by Atiqa Ijaz Khan to Sir. Imran Ali at the Institute of Geology, University of the Punjab on March 15, 2014. The assignment contains 7 questions related to a Matlab assignment, but the questions themselves are not included.
This document contains questions for an assignment on vector operations in MATLAB. It includes questions about creating vectors using the colon operator with specified start and end points and step sizes, creating a column vector with values ranging from -1 to 1, accessing odd or even elements in a vector, using the clock function to get the current date and time and storing components in separate variables.
The SEEP/W modeling workflow involves 8 steps: 1) creating the problem workspace and defining analysis properties, 2) drawing domain regions or importing a CAD file, 3) defining material properties and initial pore-water pressures, 4) defining hydraulic boundary conditions, 5) drawing the mesh, 6) solving the analysis, 7) displaying computed pore-water pressure conditions, and 8) viewing results, creating plots, and generating reports. The workflow allows users to set up saturated/unsaturated seepage analyses, define time-varying conditions, and view contour plots and node results.
Instructions for Authors of IJRS (International Journal of Remote Sensing) fo...Atiqa khan
This document provides instructions for authors submitting papers to International Journal of Remote Sensing, including guidelines on manuscript preparation, submission process, copyright and authors' rights. Key points covered include formatting manuscripts with sections, references, figures and equations according to journal style; submitting manuscripts online through ScholarOne; copyright being assigned to the journal; and authors retaining rights to freely access and share their published articles.
The document describes an assignment problem involving assigning tasks to employees at Big Bazaar to maximize total effectiveness. It provides an example of assigning 5 tasks (Research, Selling, Advertising, Customer Service, Managing Budget) to 5 employees (Rohit, Raj, Nikhat, Amar, Kumal) based on their effectiveness levels for each task. The optimal assignment achieved a total effectiveness of 300. It also examines whether assigning the Advertising task to Amar was correct, which it determined was correct as it maximized total effectiveness. A second example involves scheduling seminars at TCS to minimize the number of employees unable to attend. The optimal schedule assigns seminars on specific days with a total of 70 employees unable to attend at least one
The document summarizes the dividend discount model valuation of several companies. For Con Edison, the model estimates a value of $42.37 per share based on an expected growth rate of 3%. For ABN Amro, a two-stage DDM is used with a high growth phase of 5 years at 9.73% followed by stable growth of 5%. This estimates a value of 30.87 Euros per share. For the S&P 500, a two-stage DDM estimates an intrinsic value of $526.35, significantly below the current level of 1320, indicating potential overvaluation.
This document discusses several theories of dividend decision-making:
- Walter's model states that share price is the sum of dividends and retained earnings discounted by the cost of equity. It suggests retaining earnings if return on investment exceeds the cost of equity.
- Gordon's model similarly values shares based on dividends but also incorporates the growth rate of earnings from retained profits. It argues investors prefer dividends over capital gains.
- The Miller-Modigliani hypothesis asserts that under perfect capital markets, dividend policy does not affect share price, as investors will value future cash flows regardless of payout method.
This document provides an overview of transportation and assignment problems in operations research. It discusses the key characteristics and formulations of transportation models, including how to obtain initial basic feasible solutions using different methods like the Northwest Corner Rule and Vogel's Approximation Method. It also covers testing for optimality using the Modified Distribution method and how to handle unbalanced transportation problems. For assignment problems, the document outlines the Hungarian method for obtaining optimal solutions to assignment problems and how to deal with constrained variants like unbalanced or prohibitive assignment problems.
The document discusses linear programming models and the transportation problem. It defines linear programming as optimally allocating limited resources under certain assumptions. The transportation problem involves shipping goods from multiple origins to destinations in a way that minimizes costs. Basic concepts are defined, including feasible and basic feasible solutions. Methods for solving transportation problems are described, such as the North-West Corner Rule, Lowest Cost Entry Method, and Vogel's Approximation Method. Degeneracy in transportation problems and moving toward optimality are also covered.
This document presents a modification to the traditional method for solving fixed charge transportation problems. The traditional method introduces a dummy column to balance unbalanced problems, setting the dummy column cost to zero. The modified method instead uses the maximum cost in each row as the cost for the respective dummy column position. A numerical example is provided from a previous study and the authors state their modification gives better results. The document outlines the traditional and modified mathematical formulations and provides an algorithm to solve the modified fixed charge transportation problem.
The document discusses transportation and transshipment problems, describing transportation problems as involving the optimal distribution of goods from multiple sources to multiple destinations subject to supply and demand constraints. It presents the formulation of transportation problems as linear programming problems and provides examples of different types of transportation problems including balanced vs unbalanced and minimization vs maximization problems. The document also briefly mentions transshipment problems which involve sources, destinations, and transient nodes through which goods can pass.
The Modified Distribution Method or MODI is an efficient method of checking the optimality of the initial feasible solution. MODI provides a new means of finding the unused route with the largest negative improvement index. Once the largest index is identified, we are required to trace only one closed path. This path helps determine the maximum number of units that can be shipped via the best unused route.
This document summarizes two methods for testing the optimality of solutions to transportation problems: the stepping stone method and the modified distribution (MODI) method. The stepping stone method evaluates changing allocations to unoccupied cells one by one via closed paths, checking if costs decrease. The MODI method computes opportunity costs for unoccupied cells and draws closed loops to reallocate quantities if negative costs exist, iterating until no negative costs remain. Examples demonstrate applying both methods to test initial solutions and iteratively improve them until optimality is reached.
This document describes a project report submitted by a group of MBA students at Bundelkhand Institute of Engineering and Technology Jhansi, Uttar Pradesh, India. The report analyzes a transportation problem and provides the mathematical formulation, methods to obtain an initial basic feasible solution, and the Modi method to find the optimal basic solution. It includes declarations by the students and their guide, as well as sample transportation problems solved using different methods.
The document provides information on operations research techniques, including:
1) It describes the steps of the simplex method for solving linear programming problems, including setting up an initial basic feasible solution and using pivoting to iteratively improve the solution.
2) It discusses using artificial variables and the Big M method to handle problems that lack an obvious starting basic feasible solution.
3) It presents the mathematical formulation and solution methods for the transportation problem, including the North-West Corner Rule 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
This document discusses transportation problems and their solutions. It begins by outlining the objectives of transportation problems, which is to minimize transportation costs while meeting supply and demand constraints. It then provides an introduction and mathematical formulation of transportation problems. The document explains how to represent transportation problems in a standard table and defines key terms. It describes methods to find the initial basic feasible solution, including the Northwest Corner Rule, Least Cost Method, and Vogel's Approximation Method. The document concludes by explaining how to find the optimal basic solution using the MODI or Modified Distribution Method.
OR CH 3 Transportation and assignment problem.pptxAbdiMuceeTube
This document discusses transportation and assignment problems in linear programming. It provides an overview of transportation problems, including the objective of minimizing transportation costs while meeting supply and demand constraints. It also describes three common methods for obtaining an initial solution to a transportation problem: the North-West Corner method, Least-Cost method, and Vogel's Approximation Method. Examples are provided to illustrate how each method works.
Mba i qt unit-1.2_transportation, assignment and transshipment problemsRai University
This document discusses transportation problems and their formulation as linear programs that can be solved using the simplex method. It provides examples of how to find an initial basic feasible solution using different methods like the Northwest Corner method, Minimum Cost method, and Vogel's method. It also explains how to perform pivots on the transportation tableau when applying the simplex method to solve transportation problems. Key steps include determining an entering variable, finding the associated pivot row, labeling cells as even/odd, and choosing a leaving variable based on the smallest odd cell value.
The document describes transportation problems and how to formulate and solve them using linear programming. It provides an example of a transportation problem involving supplying electricity from three power plants to four cities. Key steps include: (1) defining decision variables for amounts shipped, (2) writing the objective function to minimize total shipping costs, (3) specifying supply and demand constraints, and (4) using methods like the Northwest Corner method to find an initial basic feasible solution. The transportation simplex method is then used to iteratively improve the solution by entering and exiting variables from the basis.
This document discusses transportation problems and network optimization problems. It provides examples and explanations of how to formulate transportation problems as linear programs. Specifically, it describes a transportation problem faced by a power company with 3 plants and 4 cities. It provides the supply and demand amounts for each plant and city and the shipping costs. It then formulates this problem as a linear program to minimize total shipping costs subject to supply and demand constraints. The document also describes general transportation problems and different methods for finding basic feasible solutions, including the Northwest Corner Method, Minimum Cost Method, and Vogel's Method.
The document summarizes different methods for solving transportation problems in linear programming, which involve distributing goods from multiple sources to multiple destinations at minimum cost. It describes three common methods - the North-West Corner method, Least-Cost method, and Vogel's Approximation Method. Each method involves iteratively allocating quantities to routes based on costs until all supply is distributed and demand is met. Examples are provided to illustrate how each method solves a transportation problem step-by-step.
The document summarizes different methods for solving transportation problems in linear programming, which involve distributing goods from multiple sources to multiple destinations at minimum cost. It describes three common methods - the North-West Corner method, Least-Cost method, and Vogel's Approximation Method. Each method involves iteratively allocating quantities to routes based on costs until all supply and demand constraints are satisfied. Examples are provided to illustrate how each method solves a sample transportation problem step-by-step.
unit-5 Transportation problem in operation research ppt.pdfbizuayehuadmasu1
This document summarizes two methods for solving transportation problems in linear programming: the North-West Corner method and the Least-Cost method. The North-West Corner method begins by allocating the maximum amount possible to the northwest cell of the transportation table and proceeds cell-by-cell to the southeast. The Least-Cost method selects the cell with the lowest cost in each iteration and allocates to that cell until all supplies and demands are satisfied. Examples are provided to illustrate the application of each 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 greedy method constructs an optimal solution in stages by making locally optimal choices at each stage without reconsidering past decisions. It selects the choice that appears best at the current time without regard for its long-term consequences. The general greedy algorithm procedure selects the best choice from available inputs at each stage until a complete solution is reached. Examples demonstrate both when the greedy method succeeds in finding an optimal solution and when it fails to do so compared to alternative methods like dynamic programming.
Similar to Transportation and assignment_problem (20)
The document discusses time and space complexity analysis of algorithms. Time complexity measures the number of steps to solve a problem based on input size, with common orders being O(log n), O(n), O(n log n), O(n^2). Space complexity measures memory usage, which can be reused unlike time. Big O notation describes asymptotic growth rates to compare algorithm efficiencies, with constant O(1) being best and exponential O(c^n) being worst.
The oc curve_of_attribute_acceptance_plansAnkit Katiyar
The document discusses operating characteristic (OC) curves, which describe the probability of accepting a lot based on the lot's quality level. The typical OC curve has an S-shape, with the probability of acceptance decreasing as the percent of nonconforming items increases. Sampling plans can approach the ideal step-function OC curve as the sample size and acceptance number increase. Specific points on the OC curve correspond to acceptance quality limits and rejection quality limits.
This document discusses conceptual problems in statistics, testing, and experimentation in cognitive psychology. It identifies three main sources of variability in psychological data: (1) participant interest and motivation, (2) individual differences, and (3) potentially stochastic cognitive mechanisms. Addressing this variability poses challenges for developing normative and descriptive models of cognition and for making inferences from group-level data to individuals. The document also discusses approaches like individual differences research and modeling heterogeneous groups to help address these challenges.
The document summarizes key concepts about queuing systems and simple queuing models. It discusses:
1) Components of a queuing system including the arrival process, service mechanism, and queue discipline.
2) Performance measures for queuing systems such as average delay, waiting time, and number of customers.
3) The M/M/1 queuing model where arrivals and service times follow exponential distributions with a single server. Expressions are given for performance measures in this model.
4) How limiting the queue length to a finite number affects performance measures compared to an infinite queue system.
Scatter diagrams and correlation and simple linear regresssionAnkit Katiyar
The document discusses scatter diagrams, correlation, and linear regression. It defines key terms like predictor and response variables, positively and negatively associated variables, and the correlation coefficient. It also describes how to calculate the linear correlation coefficient and interpret it. The document shows an example of using least squares regression to fit a line to productivity and experience data. It provides formulas to calculate the slope and intercept of the regression line and how to make predictions with the line. However, predictions should stay within the scope of the observed data used to fit the model.
This document provides an introduction to queueing theory, covering basic concepts from probability theory used in queueing models like random variables, generating functions, and common probability distributions. It then discusses fundamental queueing models and relations, including Kendall's notation for describing queueing systems and Little's Law relating average queue length and waiting time. Specific queueing models are analyzed like the M/M/1, M/M/c, M/Er/1, M/G/1, and G/M/1 queues.
This document provides an introduction to queueing theory. It discusses key concepts such as random variables, probability distributions, performance measures, Little's law and the PASTA property. It then examines several common queueing models including the M/M/1, M/M/c, M/Er/1, M/G/1 and G/M/1 queues. For each model it derives the equilibrium distribution and discusses measures like mean queue length and waiting time. The goal is to provide the fundamental mathematical techniques for analyzing queueing systems.
This document provides an introduction to queueing theory. It discusses key concepts such as random variables, probability distributions, performance measures, Little's law and the PASTA property. It then examines several common queueing models including the M/M/1, M/M/c, M/Er/1, M/G/1 and G/M/1 queues. For each model it derives the equilibrium distribution and discusses measures like mean queue length and waiting time. The goal is to give an overview of basic queueing theory concepts and common single-server and multi-server queues.
Probability mass functions and probability density functionsAnkit Katiyar
This document discusses probability mass functions (pmf) and probability density functions (pdf) for discrete and continuous random variables. A pmf fX(x) gives the probability of a discrete random variable X taking on the value x. A pdf fX(x) defines the probability that a continuous random variable X falls within an interval via its cumulative distribution function FX(x). The pdf must be non-negative and have an area/sum of 1 under the curve/over all x values.
The document outlines a lesson on basic statistical concepts for comparative studies. It covers terminology used in comparative studies including factors, levels, treatments, response variables and experimental units. It discusses topics like randomization to avoid confounding, Simpson's paradox, and the difference between experiments and observational studies. Factorial experiments involving multiple factors are also introduced.
This document discusses histograms and stem-and-leaf plots for analyzing and visualizing the distribution of a single set of numerical data. It provides examples using yearly precipitation data from New York City to demonstrate how to create histograms and stem-and-leaf plots in R. Histograms partition data into bins to show the frequency or relative frequency of observations in each bin, while stem-and-leaf plots list the "stems" and "leaves" of values to show their distribution.
This document discusses inventory management for multiple items and locations. It introduces the concepts of:
1) Setting aggregate inventory policies to meet system-wide objectives when dealing with multiple items and locations.
2) Using exchange curves to analyze the tradeoffs between total inventory levels and other factors like number of replenishments and service levels. These curves allow setting parameters like order costs and carrying costs.
3) Determining optimal reorder quantities, cycle stock, and safety stock levels across an inventory system using techniques like exchange curves. This helps allocate limited inventory budgets across items to maximize performance.
The document summarizes the economic production quantity (EPQ) model and its extensions. It discusses:
1) The EPQ model balances fixed ordering costs and inventory holding costs to determine optimal production/order quantities and intervals.
2) The economic order quantity (EOQ) model is a special case where production rate is infinite and demand is met through ordering.
3) Sensitivity analysis shows how the optimal solutions change with different parameters like production rate and setup costs.
The Kano Model classifies customer needs into three categories - threshold, performance, and excitement - based on their effect on customer satisfaction. Threshold attributes are basic needs whose absence causes dissatisfaction. Performance attributes directly improve satisfaction as implementation increases. Excitement attributes unexpectedly delight customers when implemented. The model is useful for identifying needs, setting requirements, concept development, and analyzing competitors to maximize performance attributes while including excitement attributes.
This document provides an overview of basic probability and statistics concepts. It covers variables, descriptive statistics like mean and standard deviation, frequency distributions through histograms, the normal distribution, linear regression, and includes a practice test in the appendices. Key topics are qualitative and quantitative data, parameters versus statistics, measures of central tendency and dispersion, and generating frequency tables and histograms from data sets.
Conceptual foundations statistics and probabilityAnkit Katiyar
This document provides guidance for a 6th grade statistics and probability unit of study. It outlines key concepts students should understand, including developing questions that anticipate variability, understanding data distributions in terms of center, spread and shape, and summarizing and describing distributions using various graphs such as dot plots, histograms and box plots. Students learn to analyze subgroups within data sets and how to match statistical questions to the appropriate graph. The document emphasizes interpreting and constructing dot plots, histograms and box plots to display and analyze numerical data.
This document provides an overview of basic statistical concepts including populations, samples, parameters, statistics, and sampling methods. It defines key terms like population, sample, parameter, statistic, and discusses sampling methods like simple random sampling and stratified sampling. It also covers sampling variability, estimation, hypothesis testing, prediction, and issues around representative vs non-representative samples.
The document outlines 5 axioms of probability:
1) Probabilities are non-negative
2) Probabilities of mutually exclusive events add
3) The probability of the sample space is 1
It then proves 5 theorems about probability:
1) The probability of an event equals 1 minus the probability of its complement
2) The probability of the impossible event (the empty set) is 0
3) The probability of a subset is less than or equal to the probability of the larger set it is contained within
4) A probability is between 0 and 1
5) The addition law - for two events the probability of their union equals the sum of their probabilities minus the probability of their intersection
A hand kano-model-boston_upa_may-12-2004Ankit Katiyar
This document introduces the Kano Model, a framework used to classify product features based on their impact on customer satisfaction. It explains that some features are "basic" and expected, while others provide linear satisfaction proportional to quality or performance. Some "excitement" features unexpectedly delight customers. The document outlines a process to apply the Kano Model to user experience design including researching customer needs, analyzing data, plotting features on the Kano diagram, and strategizing priorities with clients. It provides an example workshop applying the model to a fictional business and discusses extending the model with personas and use cases.
This document is Leigh Slauson's dissertation on students' conceptual understanding of variability. It investigates how students understand two measures of variability - standard deviation and standard error. Two introductory statistics classes were taught, one with traditional lecture and one with hands-on active learning labs. Both classes took a pre-test and post-test to assess understanding. The analysis found that students in the active class improved their understanding of standard deviation concepts, but not standard error concepts. Interviews suggested understanding connections between data distributions and measures of variability is important for standard error. Further research is needed on students' prior knowledge of sampling distributions and the role of probability concepts.
1. Chapter 6.
1.0 TRANSPORTATION PROBLEM
The transportation problem is a special class of the linear programming problem. It
deals with the situation in which a commodity is transported from Sources to
Destinations. The objective is to determine the amount of commodity to be transported
from each source to each destination so that the total transportation cost is minimum.
EXAMPLE 1.1
A soft drink manufacturing firm has m plants located in m different cities. The total
production is absorbed by n retail shops in n different cities. We want to determine the
transportation schedule that minimizes the total cost of transporting soft drinks from
various plants to various retail shops. First we will formulate this as a linear
programming problem.
MATHEMATICAL FORMULATION
Let us consider the m-plant locations (origins) as O1 , O2 , …., Om and the n-retail shops
(destination) as D1 , D2 , ….., Dn respectively. Let ai ≥ 0, i= 1,2, ….m , be the amount
available at the ith plant Oi . Let the amount required at the jth shop Dj be bj ≥ 0, j=
1,2,….n.
Let the cost of transporting one unit of soft drink form ith origin to jth destination be Cij ,
i= 1,2, ….m, j=1,2,….n. If xij ≥ 0 be the amount of soft drink to be transported
from ith origin to jth destination , then the problem is to determine xij so as to
Minimize
m n
z = ∑∑ xij cij
i =1 j =1
Subject to the constraint
and xij ≥ 0 , for all i and j.
n
∑x
j =1
ij = ai , i = 1,2,...m
m
∑ xij = b j , j =1,2,...n.
i =1
This lPP is called a Transportation Problem.
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2. THEOREM 1.1
A necessary and sufficient condition for the existence of a feasible solution to the
transportation problem is that
m n
∑a = ∑b
i =1
i
j =1
j
Remark. The set of constraints
m n
∑ x = b and ∑ x = a
ij j ij i
i =1 j =1
Represents m+n equations in mn non-negative variables. Each variable xij appears in
exactly two constraints, one is associated with the origin and the other is associated with
the destination.
Note. If we are putting in the matrix from, the elements of A are either 0 or 1.
THE TRANSPORTATYION TABLE:
D1 D2 …… Dn supply
O1 c11 c12 ….. c1n a1
O2 c21 c22 ….. …. c2n a2
… …… ….. ….. ….. ….. :
Om cm1 cm2 …. … cmn am
Requirement b1 b2 … …. bn
Definition. (Loop). In a transportation table, an ordered set of four or more cells is said
to form a loop if :
(I) Any two adjacent cells in the ordered set lie in the same row or in the same
column.
(II) Any three or more adjacent cells in the ordered set do not lie in the same row or in
the same column.
RESULT:
A feasible solution to a transportation problem is basic if and only if the corresponding
cells in the transportation table do not contain a loop. To find an initial basic feasible
solution we apply:
(1) The North-West corner rule
(2) Vogel`s Approximation method.
2
3. 1.1 THE NORTH-WEST CORNER RULE
Step (1). The first assignment is made in the cell occupying the upper left-hand (North
West) corner of the transportation table. The maximum feasible amount is allocated there,
i.e; x11 = min( a1, b1 ) .
Step (2). If b1 > a1, the capacity of origin O 1 is exhausted but the requirement at D 1
is not satisfied. So move downs to the second row, and make the second allocation:
x21 = min ( a2 , b1 – x11 ) in the cell ( 2,1 ).
If a1 > b1 , allocate x12 = min ( a1 - x11 , b2 ) in the cell ( 1,2) .
Continue this until all the requirements and supplies are satisfied.
EXAMPL 1.1.1
Determine an initial basic feasible solution to the following transportation problem using
the North-West corner rule:
D1 D2 D3 D4 Availability
O1 6 4 1 5 14
O2 8 9 2 7 16
O3 4 3 6 2 5
Requirement 6 10 15 4
Solution to the above problem is: 6 8
6 4 1 5
2 14
8 9 2 7
1 4
4 3 6 2
Now all requirements have been satisfied and hence an initial basic feasible solution to
the transportation problem has been obtained. Since the allocated cells do not form a
loop, the feasible solution is non-degenerate. Total transportation cost with this allocation
is:
3
4. Z = 6*6 + 4*8 + 2*9 +14*2 + 1*6 +4* = 128.
VOGEL’S APPROXIMATION METHOD (VAM ).
Step 1. For each row of the transportation table, identify the smallest and the next to-
smallest costs. Determine the difference between them for each row. Display them
alongside the transportation table by enclosing them in parenthesis against the respective
rows. Similarly compute the differences for each column.
Step 2. Identify the row or column with the largest difference among all the rows and
columns. If a tie occurs, use any arbitrary tie breaking choice. Let the greatest difference
correspond to ith row and the minimum cost be Cij . Allocate a maximum feasible amount
xij = min ( ai , bj ) in the ( i, j )th cell, and cross off the ith row or jth column.
Step 3. Re compute the column and row differences for the reduced transportation table
and go to step 2. Repeat the procedure until all the rim requirements are satisfied.
Remark. VAM determines an initial basic feasible solution, which is very close to the
optimum solution.
PROBLEM 1.1.2
Obtain an initial basic feasible solution to the following transportation problem using
Vogels approximation method.
I II III IV
A 5 1 3 3 34
B 3 3 5 4 15
C 6 4 4 3 12
D 4 -1 4 2 19
21 25 17 17
1.2 MOVING TOWARDS OPTIMALITY
(1) DETERMINE THE NET-EVALUATIONS ( U-V METHOD)
4
5. Since the net evaluation is zero for all basic cells, it follows that
zij - cij = ui +vj - cij , for all basic cells (i, j). So we can make use of this relation to find
the values of ui and vj . Using the relation ui +vj = cij , for all i and j which (i,j) is a
basic cell, we can obtain the values of ui `s and vj `s. After getting the values of ui `s and
vj `s, we can compute the net-evaluation for each non-basic cell and display them in
parenthesis in the respective cells.
(2) SELECTION OF THE ELECTING VARIABLES
Choose the variable xrs to enter the basis for which the net evaluation
zrs -crs = max { zij - cij > 0} .
After identifying the entering variable xrs , form a loop which starts at the non-basic cell
(r,s) connecting only basic cells . Such a closed path exists and is unique for any non-
degenerate solution. Allocate a quantity θ alternately to the cells of the loop starting +θ
to the entering cell. The value of θ is the minimum value of allocations in the cells
having -θ.
Now compute the net-evaluation for new transportation table and continue the above
process till all the net-evaluations are positive for non-basic cells.
1.3 DEGENCY IN TRANSPORTATION PROBLEM
Transportation with m-origins and n-destinations can have m+n-1 positive basic
variables, otherwise the basic solution degenerates. So whenever the number of basic
cells is less than m + n-1, the transportation problem is degenerate.
To resolve the degeneracy, the positive variables are augmented by as many zero-valued
variables as is necessary to complete m +n –1 basic variables.
UNBALANCED TRANSPORTATION PROBLEM
If
m n
,
∑ a ≠ ∑b
i =1
i
j =1
j
The transportation problem is known as an unbalanced transportation problem. There
are two cases
Case(1).
5
6. m n
∑ a > ∑b
i =1
i
j =1
j
Introduce a dummy destination in the transportation table. The cost of transporting to
this destination is all set equal to zero. The requirement at this destination is assumed to
be equal to
m n
∑ a − ∑b
i =1
i
j =1
j.
Case (2) .
m n
∑ a < ∑b
i =1
i
j =1
j
Introduce a dummy origin in the transportation table, the costs associated with are set
equal to zero. The availability is
n m
∑b − ∑a
j =1
j
i =1
i
1.4 THE ASSIGNMENT PROBLEM
Suppose there are n-jobs for a factory and has n-machines to process the jobs. A job i
(i=1,2,…n ) when processed by machine j ( j=1,2,…n) is assumed to incur a cost c ij .The
assignment is to be made in such a way that each job can associate with one and only one
machine. Determine an assignment of jobs to machines so as to minimize the overall cost.
1.4.1 MATHEMATCAL FORMULATION
We can define xij = 0, if the ith job not assigned to jth machine.
= 1 , if the ith job is assigned to jth machine.
We can assign one job to each machine,
n n
∑x
i =1
ij = 1, and ∑ xij = 1
j =1
The total assignment cost is given by
n n
z = ∑∑ cij xij
j =1 i =1
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7. 1.4.2 THE ASSIGNMENT ALGORITHM
Step (1). Determine the effectiveness matrix. Subtract the minimum element of each row
of the given cost matrix from all of the elements of the row. Examine if there is at least
one zero in each row and in each column. If it is so, stop here, otherwise subtract the
minimum element of each column from all the elements of the column. The resulting
matrix is the starting effectiveness matrix.
Step (2). Assign the zeroes:
(a). Examine the rows of the current effective matrix successively until a row with
exactly one unmarked zero is found. Mark this zero, indicating that an assignment will be
made there. Mark all other zeroes lying in the column of above encircled zero. The cells
marked will not be considered for any future assignment. Continue in this manner until
all the rows have taken care of.
(b). Similarly for columns.
Step (3). Check for Optimality. Repeat step 2 successively till one of the following
occurs.
(a). There is no row and no column without assignment. In such a case, the current
assignment is optimal.
(b). There may be some row or column without an assignment. In this case the current
solution is not optimal. Proceed to next step.
Step (4). Draw minimum number of lines crossing all zeroes as follows. If the number of
lines is equal to the order of the matrix, then the current solution is optimal, otherwise it
is not optimal. Go to the next step>
Step (5). Examine the elements that do not have a line through them. Select the smallest
of these elements and subtract the same from all the elements that do not have a line
through them, and add this element to every element that lies in the intersection of the
two lines.
Step (6). Repeat this until an optimal assignment is reached.
PROBLEM 1.4.1
Consider the problem of assigning five jobs to five persons. The assignment costs are
given as follows:
7
8. Jobs
Person 1 2 3 4 5
A 8 4 2 6 1
B 0 9 5 5 4
C 3 8 9 2 6
D 4 3 1 0 3
E 9 5 8 9 5
1.4.4.UNBALANCEED ASSIGNMENT PROBLEM
When the cost matrix of an assignment problem is not a square matrix, i.e; number
of sources is not equal to the number of destinations, the assignment problem is called an
unbalanced assignment problem. In such problems, dummy rows or columns are added
in the matrix so as to complete it to form a square matrix.
8