This document provides an overview of linear programming (LP), including its key characteristics and applications. LP aims to optimally allocate limited resources to achieve objectives. It involves defining decision variables, an objective function to maximize/minimize, and constraints on the resources. Common applications include production planning, finance, marketing, and more. The document also discusses various LP solving techniques like the graphical method, algebraic method, simplex method, and their use of concepts like the feasible region, basic feasible solutions, and optimality conditions.
Assignment Chapter - Q & A Compilation by Niraj ThapaCA Niraj Thapa
My name is Niraj Thapa. I have compiled Assignment Chapter including SM, PM & Exam Questions of AMA.
You feedback on this will be valuable inputs for me to proceed further.
This presentation is made to represent the basic transportation model. The aim of this presentation is to implement the transportation model in solving transportation problem.
- Just-in-time (JIT) manufacturing was first developed by Toyota to reduce waste by supplying the right materials to production lines only when needed and in the minimum necessary amounts. It aims to eliminate overproduction and excess inventory that do not add value.
- Key aspects of JIT include a pull-based production system controlled through kanban signals, small lot sizes, low setup times, continuous flow, and close partnerships with suppliers. Implementing JIT exposes problems in production so they can be addressed, improving quality over time. It requires stable production schedules and reliable equipment and processes.
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.
This document presents 15 quantitative techniques and tools: Linear Programming, Queuing Theory, Inventory Control Method, Net Work Analysis, Replacement Problems, Sequencing, Integer Programming, Assignment Problems, Transportation Problems, Decision Theory and Game Theory, Markov Analysis, Simulation, Dynamic Programming, Goal Programming, and Symbolic Logic. It provides a brief overview of each technique, describing its purpose and typical applications.
GAME THEORY
Terminology
Example : Game with Saddle point
Dominance Rules: (Theory-Example)
Arithmetic method – Example
Algebraic method - Example
Matrix method - Example
Graphical method - Example
Capacity Requirements Planning (CRP) is a technique to project resource needs for workstations. It takes inputs like planned orders and outputs a load profile for each work center. The load profile compares released orders to work center capacity to identify underloads and overloads. CRP helps determine timing of capacity expansion using strategies like capacity lead, lag, or average. It also provides information to adjust capacity through methods like adding shifts or outsourcing work. The goal is to balance load and capacity to prevent bottlenecks.
Business Application of Operation ResearchAshim Roy
This document discusses a project on the business applications of operations research. It begins with an acknowledgment section thanking teachers and parents for their support. The main body provides an abstract, introduction and overview of operations research. It discusses the early history and development of OR, and provides examples of its applications in business such as optimizing supply chain management and power grid operations. The document outlines the various techniques, methods, and areas where OR is applied to improve decision making and efficiency.
Assignment Chapter - Q & A Compilation by Niraj ThapaCA Niraj Thapa
My name is Niraj Thapa. I have compiled Assignment Chapter including SM, PM & Exam Questions of AMA.
You feedback on this will be valuable inputs for me to proceed further.
This presentation is made to represent the basic transportation model. The aim of this presentation is to implement the transportation model in solving transportation problem.
- Just-in-time (JIT) manufacturing was first developed by Toyota to reduce waste by supplying the right materials to production lines only when needed and in the minimum necessary amounts. It aims to eliminate overproduction and excess inventory that do not add value.
- Key aspects of JIT include a pull-based production system controlled through kanban signals, small lot sizes, low setup times, continuous flow, and close partnerships with suppliers. Implementing JIT exposes problems in production so they can be addressed, improving quality over time. It requires stable production schedules and reliable equipment and processes.
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.
This document presents 15 quantitative techniques and tools: Linear Programming, Queuing Theory, Inventory Control Method, Net Work Analysis, Replacement Problems, Sequencing, Integer Programming, Assignment Problems, Transportation Problems, Decision Theory and Game Theory, Markov Analysis, Simulation, Dynamic Programming, Goal Programming, and Symbolic Logic. It provides a brief overview of each technique, describing its purpose and typical applications.
GAME THEORY
Terminology
Example : Game with Saddle point
Dominance Rules: (Theory-Example)
Arithmetic method – Example
Algebraic method - Example
Matrix method - Example
Graphical method - Example
Capacity Requirements Planning (CRP) is a technique to project resource needs for workstations. It takes inputs like planned orders and outputs a load profile for each work center. The load profile compares released orders to work center capacity to identify underloads and overloads. CRP helps determine timing of capacity expansion using strategies like capacity lead, lag, or average. It also provides information to adjust capacity through methods like adding shifts or outsourcing work. The goal is to balance load and capacity to prevent bottlenecks.
Business Application of Operation ResearchAshim Roy
This document discusses a project on the business applications of operations research. It begins with an acknowledgment section thanking teachers and parents for their support. The main body provides an abstract, introduction and overview of operations research. It discusses the early history and development of OR, and provides examples of its applications in business such as optimizing supply chain management and power grid operations. The document outlines the various techniques, methods, and areas where OR is applied to improve decision making and efficiency.
Product, process, fixed and group layoutsAjith Antony
This document discusses and compares different types of manufacturing facility layouts: process layout, product layout, group layout, and fixed position layout. It provides details on when each layout type is typically used and their advantages and disadvantages. A process layout groups machines by function, while a product layout arranges machines in sequential order of operations. A group layout combines aspects of process and product layouts. A fixed position layout is used when large, identical items are manufactured and materials remain in a fixed position.
The document discusses various aspects of maintenance management including definitions, objectives, types of maintenance, reliability concepts, modern maintenance methods, and total productive maintenance pillars. It defines maintenance as work to keep equipment in proper working order and prevent failures. The main types of maintenance discussed are breakdown, preventive, and predictive maintenance. Modern maintenance methods include reliability centered maintenance, six sigma maintenance, and total productive maintenance.
Inventory management involves determining optimal inventory levels to balance costs and meet demand. There are three main types of inventory - raw materials, work in progress, and finished goods. The economic order quantity model helps determine how much to order by balancing ordering costs, carrying costs, and shortage costs. Maintaining appropriate inventory levels decouples processes, provides product variety for customers, and allows for quantity discounts.
This document provides an overview of line balancing. It defines line balancing as assigning tasks to workstations to equalize workloads and cycle times. The objectives of line balancing include equalizing workloads, identifying bottlenecks, establishing production speeds, determining staffing needs, and reducing costs. Common methods are described along with concepts such as precedence diagrams, work content, and cycle times. A simple example is provided to demonstrate how to balance a line to increase efficiency from 79% to 84% by redistributing tasks among five workstations. The document emphasizes eliminating waste to further improve efficiency.
The document discusses two main types of production systems: intermittent and continuous. Intermittent production involves producing goods in small batches based on customer orders, with irregular start/stop cycles. Continuous production aims to produce goods constantly to meet forecasted demand at large scale using standardized processes. Specific intermittent systems include project production (complex one-time orders), job production (custom single units), and batch production (producing in lots based on orders or forecasts). Continuous systems emphasize mass production of standardized goods and process production of a single product.
The document defines and describes the components of a production system. A production system combines various inputs like materials, labor, machines, and information and transforms them through a process to produce finished goods and services. The key components are facilities, which include the factory, equipment, and layout, and manufacturing support systems, which encompass how work and machines are organized. The aim of a production system is to provide the right products, in the proper quantities, at the needed time and location, and at a reasonable cost.
This document discusses operations scheduling. It begins by introducing operations scheduling and explaining that it involves assigning jobs, resources, and sequencing operations while accounting for deviations. It then discusses key performance measures for schedules such as job flow time, makespan, past due jobs, work-in-process inventory, total inventory, and utilization. The document proceeds to list objectives and functions of operations scheduling such as efficient resource use, on-time delivery, and minimizing costs and inventory. Finally, it briefly outlines types of scheduling like forward and backward, and methods like Johnson's algorithm and the index method.
Queueing Theory is the mathematical study of waiting lines in systems where demand for service exceeds the available resources. A pioneer in the field was Agner Krarup Erlang who applied its principles to telecommunications. The document discusses key concepts in queueing theory including arrival and service processes, queue configurations, performance measures and examples of real-world applications. It also covers limitations of classical queueing models in fully representing complex real systems.
The document discusses different types of plant layouts, including process layout, product layout, combination layout, fixed position layout, and group layout. It provides details on the key characteristics and advantages and disadvantages of each layout type. It also includes an example of a company that is evaluating its layout and proposes which type of layout may be most suitable.
Production System and Production Facilitiessanket394
The ppt takes you through some of the production facilities and functions that are part of manufacturing process. And helps in carrying out the manufacturing process and functions more efficeiantly.
The document discusses different types of production systems and factors that influence process selection. It describes four main types of production systems: project, job, batch, and mass production. It also discusses intermittent and continuous manufacturing systems. Key factors that influence process selection include variety, volume, flexibility, and expected output. Process selection impacts capacity planning, facility layout, equipment design, and work design.
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.
Operations Management : Line BalancingRohan Bharaj
This presentation gives us details about the different methods of Line Balancing.
It also gives an example of Ford Motors and how Line Balancing helped Ford become a powerhouse in the early 1900s
The Least Cost Method is another method used to obtain the initial feasible solution for the transportation problem. Here, the allocation begins with the cell which has the minimum cost. The lower cost cells are chosen over the higher-cost cell with the objective to have the least cost of transportation.
Here are the steps to solve this EPQ problem:
1) Demand per year = 48,000 wheels
2) Production rate per day = 800 wheels
3) Setup cost = $45
4) Carrying cost per wheel = $1
5) Number of working days per year = 240
6) Using the EPQ formula:
EPQ = √(2 * Demand * Setup cost / Carrying cost per unit)
= √(2 * 48,000 * $45 / $1)
= √432,000 = 208 wheels
7) Cycle time = EPQ / Production rate per day
= 208 / 800 = 0.26 days = 6.
The document provides guidance on keeping moving forward, even when facing challenges. It advises that if one cannot fly (move quickly), they should run, if they cannot run they should walk, and if unable to walk then crawl. But the key message is to keep moving in whatever way possible. The quote is attributed to civil rights leader Martin Luther King Jr.
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.
This document outlines the key topics in operations management layout strategies. It begins with an overview of McDonald's innovations in layout design over the years. The strategic importance of layout decisions is discussed as well as considerations for good layout design such as material handling, capacity, and flows. Different types of layout strategies are also summarized, including office, retail, warehouse, project/fixed position, process-oriented, work cell, and repetitive/product-oriented layouts. Specific examples and key issues are provided for each type.
The document provides an overview of linear programming models (LPM), including:
- Defining the key components of an LPM, such as the objective function, decision variables, constraints, and parameters.
- Explaining the characteristics and assumptions of an LPM, such as linearity, divisibility, and non-negativity.
- Describing methods for solving LPMs, including graphical and algebraic (simplex) methods.
- Providing examples of formulating LPMs to maximize profit based on constraints like available resources.
UNIT-2 Quantitaitive Anlaysis for Mgt Decisions.pptxMinilikDerseh1
This document provides an overview of linear programming problems (LPP). It discusses the key components of linear programming models including objectives, decision variables, constraints, and parameters. It also covers formulation of LPP, graphical and simplex solution methods, duality, and post-optimality analysis. Various applications of linear programming in areas like production, marketing, finance, and personnel management are also highlighted. An example problem on determining optimal product mix given resource constraints is presented to illustrate linear programming formulation.
Product, process, fixed and group layoutsAjith Antony
This document discusses and compares different types of manufacturing facility layouts: process layout, product layout, group layout, and fixed position layout. It provides details on when each layout type is typically used and their advantages and disadvantages. A process layout groups machines by function, while a product layout arranges machines in sequential order of operations. A group layout combines aspects of process and product layouts. A fixed position layout is used when large, identical items are manufactured and materials remain in a fixed position.
The document discusses various aspects of maintenance management including definitions, objectives, types of maintenance, reliability concepts, modern maintenance methods, and total productive maintenance pillars. It defines maintenance as work to keep equipment in proper working order and prevent failures. The main types of maintenance discussed are breakdown, preventive, and predictive maintenance. Modern maintenance methods include reliability centered maintenance, six sigma maintenance, and total productive maintenance.
Inventory management involves determining optimal inventory levels to balance costs and meet demand. There are three main types of inventory - raw materials, work in progress, and finished goods. The economic order quantity model helps determine how much to order by balancing ordering costs, carrying costs, and shortage costs. Maintaining appropriate inventory levels decouples processes, provides product variety for customers, and allows for quantity discounts.
This document provides an overview of line balancing. It defines line balancing as assigning tasks to workstations to equalize workloads and cycle times. The objectives of line balancing include equalizing workloads, identifying bottlenecks, establishing production speeds, determining staffing needs, and reducing costs. Common methods are described along with concepts such as precedence diagrams, work content, and cycle times. A simple example is provided to demonstrate how to balance a line to increase efficiency from 79% to 84% by redistributing tasks among five workstations. The document emphasizes eliminating waste to further improve efficiency.
The document discusses two main types of production systems: intermittent and continuous. Intermittent production involves producing goods in small batches based on customer orders, with irregular start/stop cycles. Continuous production aims to produce goods constantly to meet forecasted demand at large scale using standardized processes. Specific intermittent systems include project production (complex one-time orders), job production (custom single units), and batch production (producing in lots based on orders or forecasts). Continuous systems emphasize mass production of standardized goods and process production of a single product.
The document defines and describes the components of a production system. A production system combines various inputs like materials, labor, machines, and information and transforms them through a process to produce finished goods and services. The key components are facilities, which include the factory, equipment, and layout, and manufacturing support systems, which encompass how work and machines are organized. The aim of a production system is to provide the right products, in the proper quantities, at the needed time and location, and at a reasonable cost.
This document discusses operations scheduling. It begins by introducing operations scheduling and explaining that it involves assigning jobs, resources, and sequencing operations while accounting for deviations. It then discusses key performance measures for schedules such as job flow time, makespan, past due jobs, work-in-process inventory, total inventory, and utilization. The document proceeds to list objectives and functions of operations scheduling such as efficient resource use, on-time delivery, and minimizing costs and inventory. Finally, it briefly outlines types of scheduling like forward and backward, and methods like Johnson's algorithm and the index method.
Queueing Theory is the mathematical study of waiting lines in systems where demand for service exceeds the available resources. A pioneer in the field was Agner Krarup Erlang who applied its principles to telecommunications. The document discusses key concepts in queueing theory including arrival and service processes, queue configurations, performance measures and examples of real-world applications. It also covers limitations of classical queueing models in fully representing complex real systems.
The document discusses different types of plant layouts, including process layout, product layout, combination layout, fixed position layout, and group layout. It provides details on the key characteristics and advantages and disadvantages of each layout type. It also includes an example of a company that is evaluating its layout and proposes which type of layout may be most suitable.
Production System and Production Facilitiessanket394
The ppt takes you through some of the production facilities and functions that are part of manufacturing process. And helps in carrying out the manufacturing process and functions more efficeiantly.
The document discusses different types of production systems and factors that influence process selection. It describes four main types of production systems: project, job, batch, and mass production. It also discusses intermittent and continuous manufacturing systems. Key factors that influence process selection include variety, volume, flexibility, and expected output. Process selection impacts capacity planning, facility layout, equipment design, and work design.
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.
Operations Management : Line BalancingRohan Bharaj
This presentation gives us details about the different methods of Line Balancing.
It also gives an example of Ford Motors and how Line Balancing helped Ford become a powerhouse in the early 1900s
The Least Cost Method is another method used to obtain the initial feasible solution for the transportation problem. Here, the allocation begins with the cell which has the minimum cost. The lower cost cells are chosen over the higher-cost cell with the objective to have the least cost of transportation.
Here are the steps to solve this EPQ problem:
1) Demand per year = 48,000 wheels
2) Production rate per day = 800 wheels
3) Setup cost = $45
4) Carrying cost per wheel = $1
5) Number of working days per year = 240
6) Using the EPQ formula:
EPQ = √(2 * Demand * Setup cost / Carrying cost per unit)
= √(2 * 48,000 * $45 / $1)
= √432,000 = 208 wheels
7) Cycle time = EPQ / Production rate per day
= 208 / 800 = 0.26 days = 6.
The document provides guidance on keeping moving forward, even when facing challenges. It advises that if one cannot fly (move quickly), they should run, if they cannot run they should walk, and if unable to walk then crawl. But the key message is to keep moving in whatever way possible. The quote is attributed to civil rights leader Martin Luther King Jr.
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.
This document outlines the key topics in operations management layout strategies. It begins with an overview of McDonald's innovations in layout design over the years. The strategic importance of layout decisions is discussed as well as considerations for good layout design such as material handling, capacity, and flows. Different types of layout strategies are also summarized, including office, retail, warehouse, project/fixed position, process-oriented, work cell, and repetitive/product-oriented layouts. Specific examples and key issues are provided for each type.
The document provides an overview of linear programming models (LPM), including:
- Defining the key components of an LPM, such as the objective function, decision variables, constraints, and parameters.
- Explaining the characteristics and assumptions of an LPM, such as linearity, divisibility, and non-negativity.
- Describing methods for solving LPMs, including graphical and algebraic (simplex) methods.
- Providing examples of formulating LPMs to maximize profit based on constraints like available resources.
UNIT-2 Quantitaitive Anlaysis for Mgt Decisions.pptxMinilikDerseh1
This document provides an overview of linear programming problems (LPP). It discusses the key components of linear programming models including objectives, decision variables, constraints, and parameters. It also covers formulation of LPP, graphical and simplex solution methods, duality, and post-optimality analysis. Various applications of linear programming in areas like production, marketing, finance, and personnel management are also highlighted. An example problem on determining optimal product mix given resource constraints is presented to illustrate linear programming formulation.
This document provides an introduction to operations research. It defines operations research as a scientific approach to decision making that seeks to determine how best to operate a system under conditions of allocating scarce resources. The document discusses the origin and applications of operations research. It also outlines some common operations research techniques like linear programming, transportation problems, assignment problems, and PERT-CPM. Finally, it provides definitions of operations research from different authors and discusses the scope and methodology of operations research.
The document summarizes key concepts regarding linear programming problems. It discusses:
1. Linear programming problems aim to optimize an objective function subject to constraints. They can model many practical operations research problems.
2. The document provides an example problem of determining production levels to maximize profit. It demonstrates formulating the problem as a mathematical model and solving it graphically and with the simplex method.
3. The simplex method solves linear programming problems by examining vertex points of the feasible solution space. It involves setting up the problem in standard form and using minimum ratio and pivot element calculations to systematically search for an optimal solution.
This document contains answers to assignment questions on operations research. It defines operations research and describes types of operations research models including physical and mathematical models. It also outlines the phases of operations research including the judgment, research, and action phases. Additionally, it provides explanations and examples of linear programming problems and their graphical solution method, as well as addressing how to solve degeneracies in transportation problems and explaining the MODI optimality test procedure.
For a good business plan creative thinking is important. A business plan is very important and strategic tool for entrepreneurs. A good business plan not only helps entrepreneurs focus on specific steps necessary for them to make business ideas succeed, but it also helps them to achieve short-term and long-term objectives. As an inspiring entrepreneur who is looking towards starting a business, one of the businesses you can successfully start without much stress is book servicing café.
Importance:
Nowadays, network plays an important role in people’s life. In the process of the improvement of the people’s living standard, people’s demand of the life’s quality and efficiency is more higher, the traditional bookstore’s inconvenience gradually emerge, and the online book store has gradually be used in public. The online book store system based on the principle of providing convenience and service to people.
With the online book servicing café, college student do not need to blindly go to various places to find their own books, but only in a computer connected to the internet log on online book servicing café in the search box, type u want to find of the book information retrieval, you can efficiently know whether a site has its own books, if you can online direct purchase, if not u can change the home book store to continue to search or provide advice to the seller in order to supply. This greatly facilitates every college student saving time.
The online book servicing café’s main users are divided into two categories, one is the front user, and one is the background user. The main business model for Book Servicing Café relies on college students providing textbooks, auctions, classifieds teacher evaluations available on website. Therefore, our focus will be on the marketing strategy to increase student traffic and usage. In turn, visitor volume and transactions will maintain the inventory of products and services offered.
Online bookstore system i.e. Book Servicing Café not only can easily find the information and purchase books, and the operating conditions are simple, user-friendly, to a large extent to solve real-life problems in the purchase of the books.
When you shop in online book servicing cafe, you have the chance of accessing and going through customers who have shopped at book servicing café and review about the book you intend to buy. This will give you beforehand information about that book.
While purchasing or selling books at the book servicing café, you save money, energy and time for your favorite book online. The book servicing café will offer discount coupons which help college students save money or make money on their purchases or selling. Shopping for books online is economical too because of the low shipping price.
Book servicing café tend to work with multiple suppliers, which allows them to offer a wider variety of books than a traditional retail store without accruing a large, costly inventory which will help colle
This document provides an overview of a faculty development program on operations management focusing on supply chain analytics. It discusses the origin and evolution of supply chain management. It also defines different types of analytics including data analytics and big data analytics. The document outlines various decision science models such as quantitative, qualitative, simulation-based, and advanced models. It provides examples of applications of decision science in various organizations. Finally, it introduces linear programming techniques for prescriptive analytics and provides an example linear programming model to maximize profit from production.
The document discusses linear programming and its key concepts. It begins by defining linear programming as using a mathematical model to allocate scarce resources to maximize profit or minimize cost. It then provides the steps to solve linear programming problems: [1] identify the problem as solvable by LP, [2] formulate a mathematical model, [3] solve the model, and [4] implement the solution. The document also discusses modeling techniques like defining decision variables, objective functions, and constraints. It provides examples of LP formulations and solutions using both graphical and algebraic methods. Finally, it discusses special issues that can arise like infeasible, unbounded, and redundant solutions or the existence of multiple optimal solutions.
The document discusses linear programming and its key concepts. It begins by defining linear programming as using a mathematical model to allocate scarce resources to maximize profit or minimize cost. It then provides the steps to solve linear programming problems: [1] identify the problem as solvable by LP, [2] formulate a mathematical model, [3] solve the model, and [4] implement the solution. The document also discusses modeling techniques like defining decision variables, objective functions, and constraints. It provides examples of LP formulations and solutions using both graphical and algebraic methods. Finally, it discusses special issues that can arise like infeasible, unbounded, and redundant solutions or the existence of multiple optimal solutions.
This document provides an overview of linear programming (LP). It begins with a brief introduction defining LP as a technique for determining optimal resource allocation to achieve objectives. The history of LP is then summarized, noting its development in 1947 to solve military planning problems. Key aspects of LP are outlined, including decision variables, constraints, and the objective function. Common applications are listed such as manufacturing, finance, and agriculture. An example diet problem is illustrated to demonstrate solving an LP formulation. The assignment problem as a type of LP is also described. The assumptions, methods, and limitations of LP are discussed. Finally, duality in LP is defined as analyzing a problem and its equivalent dual problem from different perspectives.
Linear programming class 12 investigatory projectDivyans890
This document provides an introduction to linear programming, including its definition, characteristics, formulation, and uses. Linear programming is a technique for determining an optimal plan that maximizes or minimizes an objective function subject to constraints. It involves expressing a problem mathematically and using linear algebra to determine the optimal values for the decision variables. Common applications of linear programming include production planning, portfolio optimization, and transportation scheduling.
A brief study on linear programming solving methodsMayurjyotiNeog
This document summarizes linear programming and two methods for solving linear programming problems: the graphical method and the simplex method. It outlines the key components of linear programming problems including decision variables, objective functions, and constraints. It then describes the steps of the graphical method and simplex method in solving linear programming problems. The graphical method involves plotting the feasible region and objective function on a graph to find the optimal point. The simplex method uses an algebraic table approach to iteratively find the optimal solution.
This document discusses optimization of objects and systems. It defines optimization as making something as fully perfect by finding alternative solutions to problems. It discusses how optimization is used everyday consciously or subconsciously to reach the best possible outcome with available resources. The document outlines different types of optimization problems including linear and nonlinear, single and multi-objective, constrained and unconstrained. It also discusses analytical and numerical optimization methods, including those with and without derivatives like the complex method, flexible tolerance method, and hillclimb method. Numerical methods are useful for problems that cannot be solved analytically. The document concludes with discussing methods that use first and second derivatives like Newton's method and sequential quadratic programming methods.
The document defines linear programming and its key components. It explains that linear programming is a mathematical optimization technique used to allocate limited resources to achieve the best outcome, such as maximizing profit or minimizing costs. The document outlines the basic steps of the simplex method for solving linear programming problems and provides an example to illustrate determining the maximum value of a linear function given a set of constraints. It also discusses other applications of linear programming in fields like engineering, manufacturing, energy, and transportation for optimization.
STUDY ON PROJECT MANAGEMENT THROUGH GENETIC ALGORITHMAvay Minni
This document describes using a genetic algorithm to solve resource constrained project scheduling problems. It begins with an introduction explaining that planning and scheduling projects involves managing many possible solutions and resource allocations. It then provides sections on genetic algorithms, the basic genetic algorithm process, and why genetic algorithms are suitable for this type of optimization problem. The document outlines the general formulation of resource constrained project scheduling as a linear programming problem and provides an example problem scenario. It includes flowcharts and discusses implementing the proposed genetic algorithm solution methodology.
The document describes linear programming and its applications. It defines linear programming as using mathematical models to allocate limited resources to maximize profit or minimize cost. Key aspects covered include:
- The components of a linear programming model including decision variables, constraints, objective function and parameters.
- The steps to formulate a linear programming problem including identifying variables and constraints, writing the objective function and ensuring models follow linear programming assumptions.
- Examples of linear programming applications like product mix problems, portfolio selection, blending problems.
- The format of a general linear programming model.
- Assumptions required for linear programming like proportionality, additivity, divisibility and certainty of parameters.
Let x1 = Number of units of M1 produced
x2 = Number of units of M2 produced
ii) Write the constraints:
4x1 + 2x2 ≤ 80 (Grinding constraint)
2x1 + 5x2 ≤ 180 (Polishing constraint)
x1, x2 ≥ 0
iii) Write the objective function:
Maximize Z = 3x1 + 4x2
iv) Solve the LP problem graphically or by simplex method to find the optimal solution.
Operational research models can help organizations in various sectors. Some key examples include:
1) British Telecom used an OR model to schedule over 40,000 field engineers, saving $150 million annually from 1997-2000.
2) Continental Airlines developed a crew scheduling model to help resume normal operations just days after 9/11.
3) Ford Motor Company reduced annual prototype costs by $250 million using an optimization model to share prototype vehicles between testing needs.
Linear programming is a process used to optimize a linear objective function subject to linear constraints. It can be applied to problems in manufacturing, diets, transportation, allocation and more. Key components include decision variables, constraints, and an objective function. The process involves formulating the problem, identifying variables and constraints, solving using graphical or simplex methods, and interpreting the optimal solution. Linear programming provides a tool for modeling real-world problems mathematically and determining the best outcome.
The document summarizes the simplex method for solving linear programming problems involving maximization. It involves 12 steps: 1) Formulating the LPP, 2) Introducing slack, surplus and artificial variables, 3) Formulating the initial basic solution, 4) Constructing the initial simplex table, 5) Checking for positive elements in the Cj-Zj row, 6) Identifying the incoming basic variable, 7) Choosing the incoming basic variable if multiple positives exist, 8) Identifying the outgoing basic variable, 9) Constructing the next simplex table using row operations, 10) Completing the new simplex table, 11) Repeating steps 5-10, and 12) Terminating when the
A publisher has contracted an author to produce a textbook. The production process involves the author submitting a manuscript and files, editing, sample page and cover design, artwork, formatting, and printing. The critical path through the network is the author submitting the manuscript, editing, formatting, artwork approval, plate production, and binding, taking 17 weeks total to complete the project.
This flowchart outlines an optimization process to find an optimal solution. It starts with finding an initial basic solution, then checks if that solution is optimal. If it is optimal, that solution is the final answer. If not, the process seeks a better solution to try and find the optimal one.
The document outlines a strategic management model that includes four main stages: strategic intent, formulation, implementation, and evaluation. It involves analyzing internal and external environments to determine a vision, mission, goals and objectives. Strategies are then formulated, implemented through resource allocation and structure, and evaluated for effectiveness with feedback into reformulation.
The document presents a linear programming problem to determine the optimal production mix for two products (P1 and P2) that maximizes profit. The products have different processing times and resource requirements on milling and drilling machines, which have limited weekly hours. The problem is formulated as a linear program to maximize total profit subject to the machine hour constraints. Slack variables are introduced and the problem is solved using the simplex method to find the optimal production levels of 50 units of P1 and 20 units of P2, yielding maximum profit of Rs. 20,500.
This document presents a linear programming problem involving assigning quality inspectors to minimize total inspection costs. There are two types of inspectors (Grade I and Grade II) with different inspection rates and accuracy. The objective is to minimize total costs based on wages, inspection pieces, and error costs with constraints on minimum inspection pieces and available inspectors.
This document formulates a linear programming problem to determine the optimal production quantities of Products P1 and P2 given machine time and contribution margin constraints. Product P1 takes 4 hours on machine M1 and 2 hours on M2, while Product P2 takes 2 hours on M1 and 4 hours on M2. The objective is to maximize total contribution by choosing the quantities x1 and x2 subject to the 60 hours available on M1, 48 hours on M2, and non-negativity constraints.
The document describes a production problem involving two products (P1 and P2) that are manufactured using two machines (M1 and M2). P1 requires 4 hours on M1 and 2 hours on M2, while P2 requires 2 hours on M1 and 4 hours on M2. The goal is to determine the optimal quantities of P1 and P2 to maximize total contribution, given 60 hours available on M1 and 48 hours on M2. This problem is modeled as a linear programming problem and graphically solved by plotting the constraint lines and finding their intersection point.
The document contains information about game theory including pure strategies, mixed strategies, and solving games. It provides examples of games represented as payoff matrices and discusses applying the principles of dominance, algebraic methods for 2x2 games, graphical methods for 2xn and mx2 games, and linear programming for mxn games. It also includes an example analyzing a 6x6 game modeling the Allied invasion of Normandy in WWII.
Four teams will participate in a game involving selecting strategies of A or B. The aim is to score the maximum dividends. Scoring is based on the number of As and Bs selected. The document then explains the Prisoner's Dilemma game theory concept where two prisoners can either cooperate or betray each other, and discusses why rational individuals may not cooperate even if it is in their best interest to do so.
The document discusses how to formulate the dual of a primal linear programming problem. It provides 10 steps for converting a primal maximization problem into a dual minimization problem. As an example, it formulates the dual of the primal problem: Maximize z = -5x1 + 2x2 subject to x1 - x2 ≥ 2 and 2x1 + 3x2 ≤ 5, with non-negativity constraints. The dual is formulated as: Minimize z = -2y1 + 5y2 subject to -y1 + 2y2 ≥ -5 and y1 + 3y2 ≥ 2, with non-negativity constraints on the dual variables y1 and y2.
A finance manager is considering drilling a well on their property. Based on past data, there is a 70% chance of finding water at 20 meters depth, and a 20% chance of finding water between 20-25 meters if no water is found at 20 meters. The costs to drill are Rs.500 per meter plus Rs.15,000 to buy water externally if the well is not drilled. The optimal decision tree strategy is to first drill to 20 meters, and if no water, then drill further to 25 meters, resulting in an expected cost of Rs. 11,350.
The grocer must decide how many cases of milk to stock for tomorrow's demand. Each case sold yields a profit of Rs.3, but unsold cases at the end of the day lose Rs.5. Historical demand data shows the number of cases demanded and the probability of each quantity. The optimal decision can be determined by calculating the expected monetary value (EMV) of stocking different quantities of milk based on the probabilities and outcomes. The expected profit for the grocer if they stock the quantity with the highest EMV is Rs.47.7.
A fast food chain wants to build four new stores and received bids from six construction companies. The document shows the bid amounts in a table and describes using the Hungarian method to determine the optimal assignment of companies to stores that minimizes the total cost. The method involves reducing the table through successive steps to reveal a unique solution with no remaining zeros. The result assigns each store to a single construction company to minimize the total cost for building all four stores.
Operations research (OR) is a tool used to increase the effectiveness of managerial decisions. It can help with profit maximization, production management like determining optimal product mix and scheduling, financial management, marketing management, and personnel management. Some common OR models include linear programming, transportation, assignment, and sequencing problems. OR uses mathematical techniques like linear programming, decision theory, game theory, queuing theory, simulation, network analysis, and inventory models.
From Concept to reality : Implementing Lean Managements DMAIC Methodology for...Rokibul Hasan
The Ready-Made Garments (RMG) industry in Bangladesh is a cornerstone of the economy, but increasing costs and stagnant productivity pose significant challenges to profitability. This study explores the implementation of Lean Management in the Sampling Section of RMG factories to enhance productivity. Drawing from a comprehensive literature review, theoretical framework, and action research methodology, the study identifies key areas for improvement and proposes solutions.
Through the DMAIC approach (Define, Measure, Analyze, Improve, Control), the research identifies low productivity as the primary problem in the Sampling Section, with a PPH (Productivity per head) of only 4.0. Using Lean Management techniques such as 5S, Standardized work, PDCA/Kaizen, KANBAN, and Quick Changeover, the study addresses issues such as pre and post Quick Changeover (QCO) time, improper line balancing, and sudden plan changes.
The research employs regression analysis to test hypotheses, revealing a significant correlation between reducing QCO time and increasing productivity. With a regression equation of Y = -0.000501X + 6.72 and an R-squared value of 0.98, the study demonstrates a strong relationship between the independent variables (QCO downtime and improper line balancing downtime) and the dependent variable (productivity per head).
The findings suggest that by implementing Lean Management practices and addressing key productivity inhibitors, RMG factories can achieve substantial improvements in efficiency and profitability. The study provides valuable insights for practitioners, policymakers, and researchers seeking to enhance productivity in the RMG industry and similar manufacturing sectors.
Maximize Your Efficiency with This Comprehensive Project Management Platform ...SOFTTECHHUB
In today's work environment, staying organized and productive can be a daunting challenge. With multiple tasks, projects, and tools to juggle, it's easy to feel overwhelmed and lose focus. Fortunately, liftOS offers a comprehensive solution to streamline your workflow and boost your productivity. This innovative platform brings together all your essential tools, files, and tasks into a single, centralized workspace, allowing you to work smarter and more efficiently.
Designing and Sustaining Large-Scale Value-Centered Agile Ecosystems (powered...Alexey Krivitsky
Is Agile dead? It depends on what you mean by 'Agile'. If you mean that the organizations are not getting the promised benefits because they were focusing too much on the team-level agile "ways of working" instead of systemic global improvements -- then we are in agreement. It is a misunderstanding of Agility that led us down a dead-end. At Org Topologies, we see bright sparks -- the signs of the 'second wave of Agile' as we call it. The emphasis is shifting towards both in-team and inter-team collaboration. Away from false dichotomies. Both: team autonomy and shared broad product ownership are required to sustain true result-oriented organizational agility. Org Topologies is a package offering a visual language plus thinking tools required to communicate org development direction and can be used to help design and then sustain org change aiming at higher organizational archetypes.
Many companies have perceived CRM that accompanied by numerous
uncoordinated initiatives as a technological solution for problems in
individual areas. However, CRM should be considered as a strategy when
a company decides to implement it due to its humanitarian, technological
and process-related effects (Mendoza et al., 2007, p. 913). CRM is
evolving today as it should be seen as a strategy for maintaining a longterm relationship with customers.
A CRM business strategy includes the internet with the marketing,
sales, operations, customer services, human resources, R&D, finance, and
information technology departments to achieve the company’s purpose and
maximize the profitability of customer interactions (Chen and Popovich,
2003, p. 673).
After Corona Virus Disease-2019/Covid-19 (Coronavirus) first
appeared in Wuhan, China towards the end of 2019, its effects began to
be felt clearly all over the world. If the Coronavirus crisis is not managed
properly in business-to-business (B2B) and business-to-consumer
(B2C) sectors, it can have serious negative consequences. In this crisis,
companies can typically face significant losses in their sales performance,
existing customers and customer satisfaction, interruptions in operations
and accordingly bankruptcy
This presentation, "The Morale Killers: 9 Ways Managers Unintentionally Demotivate Employees (and How to Fix It)," is a deep dive into the critical factors that can negatively impact employee morale and engagement. Based on extensive research and real-world experiences, this presentation reveals the nine most common mistakes managers make, often without even realizing it.
The presentation begins by highlighting the alarming statistic that 70% of employees report feeling disengaged at work, underscoring the urgency of addressing this issue. It then delves into each of the nine "morale killers," providing clear explanations and illustrative examples.
1. Ignoring Achievements: The presentation emphasizes the importance of recognizing and rewarding employees' efforts, tailored to their individual preferences.
2. Bad Hiring/Promotions & Broken Promises: It reveals the detrimental effects of poor hiring and promotion decisions, along with the erosion of trust that results from broken promises.
3. Treating Everyone Equally & Tolerating Poor Performance: This section stresses the need for fair treatment while acknowledging that employees have different needs. It also emphasizes the importance of addressing poor performance promptly.
4. Stifling Growth & Lack of Interest: The presentation highlights the importance of providing opportunities for learning and growth, as well as showing genuine care for employees' well-being.
5. Unclear Communication & Micromanaging: It exposes the frustration and resentment caused by vague expectations and excessive control, advocating for clear communication and employee empowerment.
The presentation then shifts its focus to the power of recognition and empowerment, highlighting how a culture of appreciation can fuel engagement and motivation. It provides actionable takeaways for managers, emphasizing the need to stop demotivating behaviors and start actively fostering a positive workplace culture.
The presentation concludes with a strong call to action, encouraging viewers to explore the accompanying blog post, "9 Proven Ways to Crush Employee Morale (and How to Avoid Them)," for a more in-depth analysis and practical solutions.
A comprehensive-study-of-biparjoy-cyclone-disaster-management-in-gujarat-a-ca...Samirsinh Parmar
Disaster management;
Cyclone Disaster Management;;
Biparjoy Cyclone Case Study;
Meteorological Observations;
Best practices in Disaster Management;
Synchronization of Agencies;
GSDMA in Cyclone disaster Management;
History of Cyclone in Arabian ocean;
Intensity of Cyclone in Gujarat;
Cyclone preparedness;
Miscellaneous observations - Biparjoy cyclone;
Role of social Media in Disaster Management;
Unique features of Biparjoy cyclone;
Role of IMD in Biparjoy Prediction;
Lessons Learned; Disaster Preparedness; published paper;
Case study; for disaster management agencies; for guideline to manage cyclone disaster; cyclone management; cyclone risks; rescue and rehabilitation for cyclone; timely evacuation during cyclone; port closure; tourism closure etc.
m249-saw PMI To familiarize the soldier with the M249 Squad Automatic Weapon ...LinghuaKong2
M249 Saw marksman PMIThe Squad Automatic Weapon (SAW), or 5.56mm M249 is an individually portable, gas operated, magazine or disintegrating metallic link-belt fed, light machine gun with fixed headspace and quick change barrel feature. The M249 engages point targets out to 800 meters, firing the improved NATO standard 5.56mm cartridge.The SAW forms the basis of firepower for the fire team. The gunner has the option of using 30-round M16 magazines or linked ammunition from pre-loaded 200-round plastic magazines. The gunner's basic load is 600 rounds of linked ammunition.The SAW was developed through an initially Army-led research and development effort and eventually a Joint NDO program in the late 1970s/early 1980s to restore sustained and accurate automatic weapons fire to the fire team and squad. When actually fielded in the mid-1980s, the SAW was issued as a one-for-one replacement for the designated "automatic rifle" (M16A1) in the Fire Team. In this regard, the SAW filled the void created by the retirement of the Browning Automatic Rifle (BAR) during the 1950s because interim automatic weapons (e.g. M-14E2/M16A1) had failed as viable "base of fire" weapons.
Early in the SAW's fielding, the Army identified the need for a Product Improvement Program (PIP) to enhance the weapon. This effort resulted in a "PIP kit" which modifies the barrel, handguard, stock, pistol grip, buffer, and sights.
The M249 machine gun is an ideal complementary weapon system for the infantry squad platoon. It is light enough to be carried and operated by one man, and can be fired from the hip in an assault, even when loaded with a 200-round ammunition box. The barrel change facility ensures that it can continue to fire for long periods. The US Army has conducted strenuous trials on the M249 MG, showing that this weapon has a reliability factor that is well above that of most other small arms weapon systems. Today, the US Army and Marine Corps utilize the license-produced M249 SAW.
Small Business Management An Entrepreneur’s Guidebook 8th edition by Byrd tes...ssuserf63bd7
Small Business Management An Entrepreneur’s Guidebook 8th edition by Byrd test bank.docx
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Neal Elbaum Shares Top 5 Trends Shaping the Logistics Industry in 2024Neal Elbaum
In the ever-evolving world of logistics, staying ahead of the curve is crucial. Industry expert Neal Elbaum highlights the top five trends shaping the logistics industry in 2024, offering valuable insights into the future of supply chain management.
3. Linear Programming
• Programming is another term for planning.
• In LP we study about planning and allocation
of resources.
• The allocation of resources (men, machine,
materials) is necessary because they are
limited in supply.
4. • In LP we are concerned with the following
definition of Economics given by Lionel
Robbins:
• Economics is the science which studies
human behaviour as a relationship
between ends and scarce means which
have alternative uses.
• ‘Ends’ are the objectives to be achieved and
resources are to be allocated so as to achieve
the objectives, that is, the resources have
alternative applications.
5. • LP aims to answer the question – ‘How to
allocate resources?’
• There must be a definite criterion, a measure
of effectiveness for optimum allocation of
resources.
• Resource generates a separate constraint.
• Constraints on the utilisation of resources can
also be expressed as linear equations or linear
inequalities.
6. • This gives us a Linear Programming Problem.
• The term ‘linear’ means a relationship of
proportionality.
7. LP: Origin
• The origin of LP can be traced to the early
1920’s when the celebrated economist
Leontiff discovered the Input/Output Model.
• The more recent advances can be traced to
1947 when the discovery of the Simplex
method of LP (by George Dantzig)
revolutionised the science of OR.
8. LPP: Essential Characteristics
• The measure of effectiveness (spells out the
objective that we are trying to achieve) which
is generally profit maximisation.
• Resources are limited in supply.
• Linearity, both of the objective function as
well as of the constraints.
9. • Alternative courses of action. An LPP
generates the alternatives. Only when there
are more than one solutions, will we decide
on the optimum one.
• Variables are interrelated and not
independent of each other.
• Variables are non-negative in nature, that is
they are either zero or positive.
10. • Divisibility or continuity of the variables: when
solving an LPP, the variables can take non-
integral values also.
• As far as uncontrollable variables are
concerned, they are supposed to be
deterministic, that is, they are known for
certain.
• In the short run, they are supposed to be
constant.
11. • Examples of uncontrollable variables:
• Magnitude or quantum of each resource
available.
• Technological coefficients.
• Input coefficients.
• Output coefficients.
12. Application of LPP
• LP is considered a very versatile technique
capable of handling a large variety of
management problems:
– Production.
– Finance.
– Marketing.
– Human Resource.
13. • Production.
• Optimal product mix.
• Fluid blending.
• General blending problems (blending
liquor, proteins, calories, etc., to prepare
baby food).
• Production scheduling.
• Production planning.
14. • Finance.
• Portfolio selection.
• Profit planning.
• Marketing.
• Media selection.
• Physical distribution of products.
• Market share.
• Deciding the channels of distribution.
15. • Personnel. This field has very few applications
of LP, the main of them being Staffing.
• OR also has a number of applications in the
field of agriculture – both at macro and micro
levels.
16. Mathematical Formulation of LP
Model for Product-Mix Problems
• Two products, namely P1 and P2 are being
manufactured. Each product has to be processed
through two machines M1 and M2. One unit of
product P1 consumes 4 hours of time on M1 and 2
hours of time on M2. Similarly, one unit of P2
consumes 2 hours of time on M1 and 4 hours of
time on M2. 60 hours of time is available on M1
and 48 hours on M2. The per unit contribution
margin of P1 is 8 and of P2 is 6. Determine the
number of units of P1 and P2 to be manufactured
so as to maximise total contribution.
17. A company has two grades of inspectors I and II who are
to be assigned for a quality control inspection. It is
required that at least 2000 pieces be inspected per 8
hour day. Grade I inspectors can check pieces @ 40 per
hour with an accuracy of 97%. Grade II inspectors check
@ 30 pieces per hour with an accuracy of 95%. The wage
rate of Grade I inspectors is Rs.5 per hour while that of a
Grade II inspector is Rs.4 per hour. An error made by an
inspector costs Rs.3 to the company. There are only 9
Grade I inspectors and 11 Grade II inspectors available in
the company. The company wishes to assign work to the
available inspectors so as to minimise the total cost of
inspection. Formulate the LP model for the problem and
solve it by the graphical method.
18. A company, makes two types of leather belts.
Belt A is a high quality belt and belt B is of lower
quality. The respective profits are Rs.40 and
Rs.30. Each belt of type A requires twice as
much time as a belt of type B and if all the belts
were of type B, the company could make 1000
belts per day. The supply of leather is sufficient
for only 800 belts per day, both A and B
combined. Belt A requires a fancy buckle of
which only 400 per day are available. There are
700 buckles per day available for belt B. Set up
the LPP and solve it by the graphical method.
28. Iso-Profit Line
• The line representing the objective function is
called the iso-profit line.
• We give a random value to z (objective
function) and draw an iso-profit line.
• We keep increasing this value and successive
iso-profit lines will be parallel to the first one,
and above it.
29. • There will ultimately be one such iso-profit
line that will just touch one of the extremities
or boundaries of the feasible polygon.
• Any other iso-profit line above this line will
not pass through or touch the feasible
polygon.
• Therefore the maximum value of z will be
reached at one of the extremity or boundary
of the feasible polygon.
• The reverse process may be adopted for a
minimisation problem.
30. Extreme Point Theorem
• The optimum solution to an LPP lies at one of
the extremities of the feasible polygon,
provided there exists a solution to the LPP
which is UNIQUE, FINITE and OPTIMAL.
36. Trial and Error Method
• This and the subsequent methods are based
on another theorem – The Basis Theorem.
37. Basis Theorem of LP
• If in a system of equations we have ‘m’
variables and ‘n’ constraints, m being greater
than n, then the solution obtained by putting
m–n of the variables as zero results in a corner
point. The solution at the corner point is
known as a Basic Solution.
38. • With extreme point theorem, we examined all the
extreme points but not all the basic solutions.
• Basic solutions can be feasible or infeasible.
• Extreme point theorem gave only the feasible basic
solutions.
• x1, x2, etc., are structural variables as they constitute
the structure of the LPP.
• Any other variables used (S1, S2, etc.:
Slack/Surplus Variables; A1, A2, etc.: Artificial
Variables) are non-structural variables.
• Those variables given the value zero are known as non-
basic variables.
• The non-zero variables are known as basic variables.
39. Algebraic Method
• The T & E and the algebraic methods
constitute the basis of the Simplex Method.
41. Simplex Method of Solving LP
Problems
• Discovered by George Dantzig.
• The Initial Simplex Table serves two purposes
– gives a tabular representation of the LPP
and provides an initial basic feasible solution.
• First three columns of the simplex table are
common to all problems.
• The number of remaining columns depends
upon the number of variables involved.
42. • Simplex Criterion I: Used to identify the
incoming variable (Optimum Column).
• Simplex Criterion II: Used to identify the
outgoing variable (Replaced Row).
• Pivot Element: The element lying at the
intersection of the optimum column and the
replaced row.
• Criterion for optimality in a maximisation
problem: All elements in the Cj-Zj row should
be either 0 or negative.
43. • Augmented Matrix: The coefficient matrix is
augmented by the column vector of the RHS
constants.
• We perform ‘row operations’ in such a
manner so as to obtain an identity matrix in
place of the coefficient matrix.
45. The Big ‘M’ Method
• Structural and slack/surplus variables can
reenter the basis.
• An artificial variable driven out of the basis
cannot reenter it.
• Therefore, if an artificial variable has been
driven out of the basis, the terms in its column
need not be computed.
46. • In minimisation problem we reach the optimal
solution if all the elements of the Cj-Zj row are
either positive or zero.
• In maximisation problem, the objective
function coefficient of an artificial variable is
‘– M.’
• In minimisation problem, the objective
function coefficient of an artificial variable is
‘+M.’
47. The Three Exceptions
• Multiple Optimal Solutions: Cj-Zj element of a
non-basic variable is zero.
• Unbounded Solution: All elements of optimum
column are negative.
• Infeasible Solution: If the criterion for
optimality is being satisfied and yet there is an
artificial variable in the basis.
48. A manufacturer of 3 products tries to follow a
policy of producing those products which
contribute most to profit. However, there is also a
policy of recognising certain minimum sales
requirements. Currently these are:
Product X: 20 units/week.
Product Y: 30 units/week.
Product z: 60 units/week.
There are 3 producing departments. The
production times in hours/unit in each department
and the total time available for each week in each
department are:
49. Deptt. Time Reqd. (hrs.) Total Hrs.
AvailableX Y Z
1 0.25 0.20 0.15 420
2 0.30 0.40 0.50 1048
3 0.25 0.30 0.25 529
50. The contributions per unit of products X, Y and Z
are Rs.10, 9 and 8, respectively. The company
has scheduled 1558 units of X, 30 units of Y and
60 units of Z for production in the following
week.
You are required to state whether the present
schedule is optimum and if it is not, then what it
should be? What recommendation should be
made to the firm regarding its production
facilities?
51. Degeneracy
• Degeneracy is revealed when a basic variable
acquires a zero value (rather than a negative
or a positive value).
53. Sensitivity Analysis
• Also known as Post-Optimality Analysis.
• Undertaken to explore the effect of changes in
the LP parameters on the optimal solution.
• For example, to study the effect of costs and
price change, diminishing resources becoming
readily available or technological advances, on
the optimal LP solution, sensitivity analysis
suggests the changes in the data used to build
the model...
54. Sensitivity Analysis
• Concerned with the extent of sensitivity of the
optimum solution to an LP for change in one
or more of:
– The profit or cost coefficients of the objective
function,
– The LHS coefficients of the variables in the
constraints, and
– The RHS quantities of the constraints.
55. A firm produces three products A, B, C. Unit contributions
of the products are Rs.5, 10, 8 respectively. Each unit of
product A requires 3kg of material, 5 machine hours and 2
labour hours; each unit of product B requires 4kg of
material, 4 machine hours and 4 labour hours and each unit
of product C requires 2kg of material, 4 machine hours and
5 labour hours.
Everyday 60kg of material, 72 machine hours and 100
labour hours are available. Find out the best production
strategy. Investigate the effect on the solution of the
following:
i.An increase of 12 machine hours.
ii.A decrease of 6kg of material.
iii.3 units of product A are to be produced.
64. Cj 5 10 8 0 0 0
Cb BV SV x1 x2 x3 S1 S2 S3
10x2 8 1/3 1 0 1/3 - 1/6 0
8x3 10 2/3 0 1 - 1/3 5/12 0
0S3 18 -8/3 0 0 1/3 -17/12 1
Zj 160 26/3 10 8 2/3 5/3 0
Cj-Zj -11/3 0 0 - 2/3 -5/3 0
Sensitivity Analysis
3 units of product A
are to be produced
(variable x1)
65. Duality
• LPPs always exist in pairs.
• Associated to every maximisation problem
there is a complementary minimisation
problem and vice versa.
• The original problem is known as primal.
• Its complementary problem is known as dual.
• The names primal and dual are
interchangeable.
66. Reasons for Studying Duality
• Dual may provide a computational shortcut to
the primal.
• Solution to primal also gives the solution to its
dual.
• Economic interpretations can be drawn from
dual’s solution.
67. Rules for Formulating Dual
1. If primal is a maximisation problem, its dual
will be a minimisation problem, and vice
versa.
2. No. of dual variables = no. of primal
constraints.
3. No. of dual constraints = no. of primal
variables.
68. Rules of Formulating Dual
4. The transpose of the coefficient matrix of the
primal is the coefficient matrix of the dual.
5. The direction of constraints of the dual is the
reverse of the direction of constraints in the
primal.
6. If the kth
primal constraint is a strict equality,
then the kth
dual variable will be unrestricted
in sign.
69. Rules of Formulating Dual
7. If the ith
primal variable is unrestricted in sign,
then the ith
dual constraint will be a strict
equality.
8. If the primal is a maximisation problem, then
before formulating the dual all the
constraints should be converted into ‘≤ type’.
If the primal is a minimisation problem, then
before formulating the dual all the
constraints should be converted into ‘≥ type’.
70. Rules for Formulating Dual
9. The objective function coefficients of the
primal become the RHS constants of the
constraints of the dual and the RHS constants
of the primal constraints become the
objective function coefficients of the
variables in the dual.
10.If a variable is unrestricted in sign, it can be
written as the difference between two non-
negative variables.
77. • If primal is from sellers point of view
(maximise profits), dual is from the buyers
point of view (minimise expenses).
• Slack variables in the primal (or the structural
variables of the dual) would give the imputed
values of the resources or shadow
prices/artificial accounting prices/opportunity
cost.
• Slack variables of the primal correspond to the
structural variables of the dual and vice versa.
78. • If the primal has m structural and n slack
variables, then in the dual there will be n
structural and m surplus variables.
• Cj-Zj row of the primal gives the solution
values of the dual variables.
82. A housewife asks a butcher to grind up several
cuts of beef to form a blend of not less than
17.6% protein and 14.8% fat. The butcher may
use several available cuts. What cuts and in
what quantities should he use if he wants to
minimise the cost of the cuts he buys and satisfy
the housewife’s requirements.
The protein and fat contents in percentage in
each pound of meat are given below:
83. A B C
Protein (%) 20 12 14
Fat (%) 10 18 22
Cost/lb (cents) 110 150 122
84. A foundry is faced with the problem of scheduling production and subcontracting of
three products. Each of the products required casting, machining and assembly
operations. Casting operations for products 1 and 2 could be subcontracted but the
casting for product 3 required special equipment which precludes the use of
subcontractors. If cast in the foundry, each of product 1 requires 6 minutes of casting
time and product 3 requires 8 minutes. Machining times required per unit of products
1, 2 and 3 are 6, 3 and 8 minutes respectively and the assembly times are 3, 2 and 2
minutes respectively.
The foundry has available 8000 minutes of casting time; 12000 minutes of machining
time and 10000 minutes of assembly time per week. The overall profit obtained from
unit of products 1, 2 and 3 sold are Rs.7, Rs.10 and Rs.11 respectively with castings
produced in the foundry. With castings obtained from subcontractors, the profits per
unit of products 1 and 2 are Rs.5 and Rs.9 respectively. There is a restriction that the
foundry must supply at least 3000 units of products 1, 2 and 3 taken together, per
week. How should the foundry schedule its production and subcontracting so as to
maximise its profits?
85. XYZ Chemicals Ltd. must produce 10000 kg of a
special mixture for a customer. The mix consists
of ingredients X1, X2 and X3. X1 costs Rs.8 per kg, X2
costs Rs.10 per kg and X3 costs Rs.11 per kg. No
more than 3500 kg of X1 can be used and at least
1500 kg of X2 must be used. Also, at least 2000 kg
of X3 is required.
1.Calculate the number of kg for each ingredient
to use in order to minimise total cost for 10000
kg.
2.Calculate the lowest possible total cost.