This document summarizes a research article that applies linear programming to maximize profit for Johnsons Nigeria Limited's bakery division. The article establishes a linear programming model to determine the optimal production mix of four bread sizes to maximize total profit given production capacity constraints. An initial simplex tableau is constructed and the optimal solution is obtained through iterative pivoting. The optimal solution shows producing three units of large bread yields the maximum profit of 150 naira. Sensitivity analysis confirms the optimal solution remains valid if the large bread's profit coefficient remains between 50-30 naira.
Linear programming is a mathematical technique used to optimize a linear objective function subject to linear equality and inequality constraints. It has broad applications in business and economics for allocating scarce resources optimally. Some key points:
- George Dantzig developed the simplex method in 1947, making linear programming problems tractable.
- Linear programming is used widely in industries for problems like transportation, production planning, blending, and portfolio selection to maximize profits or minimize costs.
- It provides an objective way to identify bottlenecks and ensure the best use of limited resources like time, labor, and machines.
Application of linear programming technique for staff training of register se...Enamul Islam
This study aims to minimize training costs for staff at Patuakhali Science and Technology University using linear programming. It identifies two decision variables (permanent and non-permanent staff to be trained) and develops constraints based on time available and staff in different departments. The linear programming model is solved to find the optimal solution: 1 permanent staff should be sent for 5 days of training among departments to minimize costs. The research suggests this approach can help determine optimal staffing levels for future training programs.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
1) The document discusses definitions and characteristics of operations research (OR). It provides definitions of OR from several leaders and pioneers in the field that describe OR as applying scientific methods to optimize complex systems.
2) Key characteristics of OR mentioned are that it takes a team approach using quantitative techniques, aims to help executives make optimal decisions, relies on mathematical models, and uses computers to analyze models.
3) Limitations of OR discussed include that it is time-consuming, practitioners may lack industrial experience, and solutions can be difficult to communicate to non-technical executives. Linear programming is introduced as a prominent OR technique.
The document defines operations research and discusses its history and applications. It originated during World War II to optimize limited military resources. Operations research uses mathematical modeling to aid decision making. It defines problems, formulates models, derives solutions, and implements recommendations. Some key applications include allocating scarce resources optimally and minimizing costs and wait times.
An Optimal Solution to the Linear Programming Problem using Lingo Solver: A Case Study of an
Apparel Production Plant of Sri Lanka.................................................................................................1
Z. A. M. S. Juman and W. B. Daundasekara
Analysis of BT and SMS based Mobile Malware Propagation ................................................................. 16
Prof. R. S. Sonar and Sonal Mohite
Behavioral Pattern of Internet Use among University Students of Pakistan........................................... 25
Amir Manzoor
BER Analysis of BPSK and QAM Modulation Schemes using RS Encoding over Rayleigh Fading Channel
.................................................................................................................................................................... 37
Faisal Rasheed Lone and Sanjay Sharma
Harnessing Mobile Technology (MT) to Enhancy the Sustainable Livelihood of Rural Women in
Zimbabwe: Case of Mobile Money Transfer (MMT) ................................................................................ 46
Samuel Musungwini, Tinashe Gwendolyn Zhou, Munyaradzi Zhou, Caroline Ruvinga and Raviro Gumbo
Design and Evaluation of a Comprehensive e-Learning System using the Tools on Web 2.0 ................ 58
Maria Dominic, Anthony Philomenraj and Sagayaraj Francis
Critical Success Factors for the Adoption of School Administration and Management System in South
African Schools ...............................................................................................................................74
Mokwena Nicolas Sello
Efficient and Trust Based Black Hole Attack Detection and Prevention in WSN ................................... 93
Research on Lexicographic Linear Goal Programming Problem Based on LINGO and ...paperpublications3
Abstract: Lexicographic Linear Goal programming within a pre-emptive priority structure including Column-dropping Rule has been one of the useful techniques considered in solving multiple objective problems. The basic ideas to solve goal programming are transforming goal programming into single-objective linear programming. An optimal solution is attained when all the goals are reached as close as possible to their aspiration level, while satisfying a set of constraints. One of the Goal Programming algorithm – the Lexicographic method including Column-dropping Rule and the method of LINGO software are discussed in this paper. Finally goal programming model are applied to the actual management decisions, multi-objective programming model are established and used LINGO software and Column-dropping Rule to achieve satisfied solution.Keywords: Goal programming, Lexicographic Goal programming, multi-objective, LINGO software, Column-dropping Rule.
Title: Research on Lexicographic Linear Goal Programming Problem Based on LINGO and Column-Dropping Rule
Author: N. R. Neelavathi
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
Linear programming is a mathematical technique used to optimize a linear objective function subject to linear equality and inequality constraints. It has broad applications in business and economics for allocating scarce resources optimally. Some key points:
- George Dantzig developed the simplex method in 1947, making linear programming problems tractable.
- Linear programming is used widely in industries for problems like transportation, production planning, blending, and portfolio selection to maximize profits or minimize costs.
- It provides an objective way to identify bottlenecks and ensure the best use of limited resources like time, labor, and machines.
Application of linear programming technique for staff training of register se...Enamul Islam
This study aims to minimize training costs for staff at Patuakhali Science and Technology University using linear programming. It identifies two decision variables (permanent and non-permanent staff to be trained) and develops constraints based on time available and staff in different departments. The linear programming model is solved to find the optimal solution: 1 permanent staff should be sent for 5 days of training among departments to minimize costs. The research suggests this approach can help determine optimal staffing levels for future training programs.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
1) The document discusses definitions and characteristics of operations research (OR). It provides definitions of OR from several leaders and pioneers in the field that describe OR as applying scientific methods to optimize complex systems.
2) Key characteristics of OR mentioned are that it takes a team approach using quantitative techniques, aims to help executives make optimal decisions, relies on mathematical models, and uses computers to analyze models.
3) Limitations of OR discussed include that it is time-consuming, practitioners may lack industrial experience, and solutions can be difficult to communicate to non-technical executives. Linear programming is introduced as a prominent OR technique.
The document defines operations research and discusses its history and applications. It originated during World War II to optimize limited military resources. Operations research uses mathematical modeling to aid decision making. It defines problems, formulates models, derives solutions, and implements recommendations. Some key applications include allocating scarce resources optimally and minimizing costs and wait times.
An Optimal Solution to the Linear Programming Problem using Lingo Solver: A Case Study of an
Apparel Production Plant of Sri Lanka.................................................................................................1
Z. A. M. S. Juman and W. B. Daundasekara
Analysis of BT and SMS based Mobile Malware Propagation ................................................................. 16
Prof. R. S. Sonar and Sonal Mohite
Behavioral Pattern of Internet Use among University Students of Pakistan........................................... 25
Amir Manzoor
BER Analysis of BPSK and QAM Modulation Schemes using RS Encoding over Rayleigh Fading Channel
.................................................................................................................................................................... 37
Faisal Rasheed Lone and Sanjay Sharma
Harnessing Mobile Technology (MT) to Enhancy the Sustainable Livelihood of Rural Women in
Zimbabwe: Case of Mobile Money Transfer (MMT) ................................................................................ 46
Samuel Musungwini, Tinashe Gwendolyn Zhou, Munyaradzi Zhou, Caroline Ruvinga and Raviro Gumbo
Design and Evaluation of a Comprehensive e-Learning System using the Tools on Web 2.0 ................ 58
Maria Dominic, Anthony Philomenraj and Sagayaraj Francis
Critical Success Factors for the Adoption of School Administration and Management System in South
African Schools ...............................................................................................................................74
Mokwena Nicolas Sello
Efficient and Trust Based Black Hole Attack Detection and Prevention in WSN ................................... 93
Research on Lexicographic Linear Goal Programming Problem Based on LINGO and ...paperpublications3
Abstract: Lexicographic Linear Goal programming within a pre-emptive priority structure including Column-dropping Rule has been one of the useful techniques considered in solving multiple objective problems. The basic ideas to solve goal programming are transforming goal programming into single-objective linear programming. An optimal solution is attained when all the goals are reached as close as possible to their aspiration level, while satisfying a set of constraints. One of the Goal Programming algorithm – the Lexicographic method including Column-dropping Rule and the method of LINGO software are discussed in this paper. Finally goal programming model are applied to the actual management decisions, multi-objective programming model are established and used LINGO software and Column-dropping Rule to achieve satisfied solution.Keywords: Goal programming, Lexicographic Goal programming, multi-objective, LINGO software, Column-dropping Rule.
Title: Research on Lexicographic Linear Goal Programming Problem Based on LINGO and Column-Dropping Rule
Author: N. R. Neelavathi
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
The document provides an introduction to operation research techniques, specifically linear programming. It discusses the key components of a linear programming model, including decision variables, the objective function, and constraints. It also outlines the basic assumptions required for linear programming, such as certainty of parameters, proportionality of relationships, divisibility of resources, and optimization. The document serves to introduce readers to the structure and assumptions of linear programming as an operations research technique.
An Optimal Solution To The Linear Programming Problem Using Lingo Solver A C...Jessica Thompson
This document summarizes a research paper that presents a mathematical model to minimize production costs for a t-shirt manufacturing plant in Sri Lanka. The model seeks to determine the optimal number of machine operators, workers, and raw materials needed while satisfying operational constraints. Data was collected through questionnaires and interviews and a linear programming model was formulated with decision variables, objective function, and constraints. The optimal solution to the model was found using LINGO Solver software.
Nonlinear Programming: Theories and Algorithms of Some Unconstrained Optimiza...Dr. Amarjeet Singh
Nonlinear programming problem (NPP) had become an important branch of operations research, and it was the mathematical programming with the objective function or constraints being nonlinear functions. There were a variety of traditional methods to solve nonlinear programming problems such as bisection method, gradient projection method, the penalty function method, feasible direction method, the multiplier method. But these methods had their specific scope and limitations, the objective function and constraint conditions generally had continuous and differentiable request. The traditional optimization methods were difficult to adopt as the optimized object being more complicated. However, in this paper, mathematical programming techniques that are commonly used to extremize nonlinear functions of single and multiple (n) design variables subject to no constraints are been used to overcome the above challenge. Although most structural optimization problems involve constraints that bound the design space, study of the methods of unconstrained optimization is important for several reasons. Steepest Descent and Newton’s methods are employed in this paper to solve an optimization problem.
OR is defined as a scientific approach to optimal decision making through modelling of
deterministic and probabilistic systems that originate from real life.
Scientific approach: LPP, PERT/CPM, Queueing model, NLP, DP,MILP, Game
theory, heuristic programming.
Deterministic system: - a system which gives the same result for a particular set of
input, no matter how many times we recalculate it
Operational research (OR) is the application of advanced analytical techniques to improve decision making. It involves using tools from mathematics like algorithms, statistics, and modeling techniques to find optimal solutions to complex problems. Some common OR techniques include linear programming, network flow programming, integer programming, nonlinear programming, dynamic programming, and stochastic programming. OR has many applications in business for issues like inventory planning, production scheduling, financial management, and risk management. It helps organizations make better decisions around areas like sequencing jobs, production scheduling, and introducing new products/facilities. OR allows for more systematic and analytical decision making with less risk of errors.
The document provides an overview of the history and applications of operations research (OR). It discusses:
- OR originated in the UK during World War II when scientists were called upon to apply a scientific approach to military operations and allocate scarce resources effectively.
- The success of OR in the military spread its use to other government departments and industries.
- Today, OR uses quantitative techniques like mathematical modeling, computer analysis and simulation to help organizations like the military, businesses, transportation and more make optimal decisions. It breaks problems down and finds the best solutions.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
LITERATURE REVIEW OF OPTIMIZATION TECHNIQUE BUSINESS: BASED ON CASEIAEME Publication
In today’s complex business world, decision making plays a vital role in the
success of any business. The simplex method, an operation research technique is
widely used to finding solutions in many real world problems. This paper is an attempt
to get an insight about the various application of optimization techniques in business.
It is a conceptual research based on various literatures available. This study is based
on different cases applied on selected sectors, viz., industrial, financial, resource
allocation, agriculture, marketing and personnel management area.
The linear programming problem seeks to minimize the objective function z = x1 + x2 + x3 subject to three constraints. The two-phase simplex method is used to solve the problem. In phase I, an artificial variable is introduced to convert inequality constraints to equations in order to find a feasible solution. Phase I results in x1 entering the basis. In phase II, the original objective function is optimized subject to the constraints, resulting in a maximum value of Z = 6 attained when x1 = 2, x2 = 0.
The document discusses using the linear programming technique to aid in decision making for marketing and finance problems. It provides an example of using linear programming to determine the optimal allocation of advertising budgets across multiple media (television, radio, newspaper) to maximize total audience reach given budget constraints. Linear programming can be applied to problems in marketing mix determination, financial decision making, production scheduling, and more. It also briefly describes the simplex method for solving linear programming problems.
The operation research book that involves all units including the lpp problems, integer programming problem, queuing theory, simulation Monte Carlo and more is covered in this digital material.
The document discusses problem solving approaches and techniques in operations research. It defines operations research as using quantitative methods to assist decision-makers in designing, analyzing, and improving systems to make better decisions. The scientific approach involves studying differences between past and present cases while considering new environmental factors. Some quantitative techniques mentioned include break-even point analysis, financial analysis, and decision theory. The document also provides examples of linear programming models and their components.
An Integrated Solver For Optimization ProblemsMonica Waters
This document presents an integrated solver called SIMPL that combines mixed integer linear programming, global optimization, and constraint programming techniques. SIMPL uses an algorithmic framework called search-infer-and-relax that encompasses various optimization methods. It also uses constraint-based modeling where constraints define how techniques are combined to solve the problem. The paper demonstrates that SIMPL can match or exceed the computational advantages of customized integrated solvers by solving production planning, product configuration, and machine scheduling problems.
Recent DEA Applications to Industry: A Literature Review From 2010 To 2014inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Developing of decision support system for budget allocation of an r&d organiz...eSAT Publishing House
1) The document describes developing a decision support system for budget allocation of an R&D organization using a performance-based goal programming model.
2) It analyzes nine years of budget data from the organization and finds a wide gap between allocated funds and funds utilized.
3) The proposed model assesses R&D programs based on priority and risk factors using fuzzy set theory, and aims to allocate budgets in a more realistic and accurate manner than the previous approach.
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.
The document provides an overview of operations research (OR), including its history, methodology, tools and techniques, and applications. It discusses how OR began during World War II to analyze military operations and optimize resource allocation. The seven main steps of the OR methodology are described. Common OR tools include linear programming, game theory, decision theory, queuing theory, inventory models, simulation, and dynamic programming. Finally, the document outlines some example applications of OR in fields like accounting, construction, and facilities planning.
Unit I (8 Hrs)
Introduction to Linear Programming – Various definitions, Statements of basic
theorems and properties, Advantages Limitations and Application areas of Linear
Programming, Linear Programming -Graphical method, - graphical solution
methods of Linear Programming problems, The Simplex Method: -the Simplex
Algorithm, Phase II in simplex method, Primal and Dual Simplex Method, Big-M
Method
Unit II (8 Hrs)
Transportation Model and its variants: Definition of the Transportation Model
-Nontraditional Transportation Models-the Transportation Algorithm-the Assignment
Model– The Transshipment Model
Unit III (8 Hrs)
Network Models: Basic differences between CPM and PERT, Arrow Networks,
Time estimates, earliest completion time, Latest allowable occurrences time,
Forward Press Computation, Backward Press Computation, Representation in
tabular form, Critical Path, Probability of meeting the scheduled date of completion,
Various floats for activities, Critical Path updating projects, Operation time cost trade
off Curve project,
Selection of schedule based on :- Cost analysis, Crashing the network
Sequential model & related problems, processing n jobs through – 1 machine & 2
machines
Unit IV (8 Hrs)
Network Models: Scope of Network Applications – Network definitions, Goal
Programming Algorithms, Minimum Spanning Tree Algorithm, Shortest Route
Problem, Maximal flow model, Minimum cost capacitated flow problem
Unit V (8 Hrs)
Decision Analysis: Decision - Making under certainty - Decision - Making under
Risk, Decision
under uncertainty.
Unit VI (8 Hrs)
Simulation Modeling: Monte Carlo Simulation, Generation of Random Numbers,
Method for
Gathering Statistical observations
This document discusses optimization problems in engineering applications. It begins by defining optimization and describing how it can be applied to engineering problems to minimize costs or maximize benefits. Some examples of engineering applications that can be optimized are described, such as designing structures for minimum cost or maximum efficiency. The document then discusses procedures for solving optimization problems, including recognizing and defining the problem, constructing a model, and implementing solutions. It also describes different types of optimization problems and methods for solving linear programming problems, including the graphical and simplex methods.
Article Paragraph Example. How To Write A 5 ParagrapBrittany Allen
The documents present differing views of America's history with race relations. President Reagan argues that Americans should have a positive view of their country's history and values of morality. Senator Obama says modern Americans must acknowledge the struggles and inequalities in the country's past that have guided how Americans relate to each other and the world today. The documents disagree on how Americans should think about their country's history regarding race.
Exploring Writing Paragraphs And Essays By John LanganBrittany Allen
The document provides instructions for requesting writing assistance from HelpWriting.net in 5 steps:
1. Create an account with a password and email.
2. Complete a 10-minute order form with instructions, sources, deadline, and attach a sample for style imitation.
3. Review bids from writers and choose one based on qualifications, history, and feedback, then pay a deposit.
4. Review the completed paper and authorize full payment or request revisions if needed.
5. Choose HelpWriting.net for original, high-quality content with the option of revisions and a refund if plagiarized.
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The document provides an introduction to operation research techniques, specifically linear programming. It discusses the key components of a linear programming model, including decision variables, the objective function, and constraints. It also outlines the basic assumptions required for linear programming, such as certainty of parameters, proportionality of relationships, divisibility of resources, and optimization. The document serves to introduce readers to the structure and assumptions of linear programming as an operations research technique.
An Optimal Solution To The Linear Programming Problem Using Lingo Solver A C...Jessica Thompson
This document summarizes a research paper that presents a mathematical model to minimize production costs for a t-shirt manufacturing plant in Sri Lanka. The model seeks to determine the optimal number of machine operators, workers, and raw materials needed while satisfying operational constraints. Data was collected through questionnaires and interviews and a linear programming model was formulated with decision variables, objective function, and constraints. The optimal solution to the model was found using LINGO Solver software.
Nonlinear Programming: Theories and Algorithms of Some Unconstrained Optimiza...Dr. Amarjeet Singh
Nonlinear programming problem (NPP) had become an important branch of operations research, and it was the mathematical programming with the objective function or constraints being nonlinear functions. There were a variety of traditional methods to solve nonlinear programming problems such as bisection method, gradient projection method, the penalty function method, feasible direction method, the multiplier method. But these methods had their specific scope and limitations, the objective function and constraint conditions generally had continuous and differentiable request. The traditional optimization methods were difficult to adopt as the optimized object being more complicated. However, in this paper, mathematical programming techniques that are commonly used to extremize nonlinear functions of single and multiple (n) design variables subject to no constraints are been used to overcome the above challenge. Although most structural optimization problems involve constraints that bound the design space, study of the methods of unconstrained optimization is important for several reasons. Steepest Descent and Newton’s methods are employed in this paper to solve an optimization problem.
OR is defined as a scientific approach to optimal decision making through modelling of
deterministic and probabilistic systems that originate from real life.
Scientific approach: LPP, PERT/CPM, Queueing model, NLP, DP,MILP, Game
theory, heuristic programming.
Deterministic system: - a system which gives the same result for a particular set of
input, no matter how many times we recalculate it
Operational research (OR) is the application of advanced analytical techniques to improve decision making. It involves using tools from mathematics like algorithms, statistics, and modeling techniques to find optimal solutions to complex problems. Some common OR techniques include linear programming, network flow programming, integer programming, nonlinear programming, dynamic programming, and stochastic programming. OR has many applications in business for issues like inventory planning, production scheduling, financial management, and risk management. It helps organizations make better decisions around areas like sequencing jobs, production scheduling, and introducing new products/facilities. OR allows for more systematic and analytical decision making with less risk of errors.
The document provides an overview of the history and applications of operations research (OR). It discusses:
- OR originated in the UK during World War II when scientists were called upon to apply a scientific approach to military operations and allocate scarce resources effectively.
- The success of OR in the military spread its use to other government departments and industries.
- Today, OR uses quantitative techniques like mathematical modeling, computer analysis and simulation to help organizations like the military, businesses, transportation and more make optimal decisions. It breaks problems down and finds the best solutions.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
LITERATURE REVIEW OF OPTIMIZATION TECHNIQUE BUSINESS: BASED ON CASEIAEME Publication
In today’s complex business world, decision making plays a vital role in the
success of any business. The simplex method, an operation research technique is
widely used to finding solutions in many real world problems. This paper is an attempt
to get an insight about the various application of optimization techniques in business.
It is a conceptual research based on various literatures available. This study is based
on different cases applied on selected sectors, viz., industrial, financial, resource
allocation, agriculture, marketing and personnel management area.
The linear programming problem seeks to minimize the objective function z = x1 + x2 + x3 subject to three constraints. The two-phase simplex method is used to solve the problem. In phase I, an artificial variable is introduced to convert inequality constraints to equations in order to find a feasible solution. Phase I results in x1 entering the basis. In phase II, the original objective function is optimized subject to the constraints, resulting in a maximum value of Z = 6 attained when x1 = 2, x2 = 0.
The document discusses using the linear programming technique to aid in decision making for marketing and finance problems. It provides an example of using linear programming to determine the optimal allocation of advertising budgets across multiple media (television, radio, newspaper) to maximize total audience reach given budget constraints. Linear programming can be applied to problems in marketing mix determination, financial decision making, production scheduling, and more. It also briefly describes the simplex method for solving linear programming problems.
The operation research book that involves all units including the lpp problems, integer programming problem, queuing theory, simulation Monte Carlo and more is covered in this digital material.
The document discusses problem solving approaches and techniques in operations research. It defines operations research as using quantitative methods to assist decision-makers in designing, analyzing, and improving systems to make better decisions. The scientific approach involves studying differences between past and present cases while considering new environmental factors. Some quantitative techniques mentioned include break-even point analysis, financial analysis, and decision theory. The document also provides examples of linear programming models and their components.
An Integrated Solver For Optimization ProblemsMonica Waters
This document presents an integrated solver called SIMPL that combines mixed integer linear programming, global optimization, and constraint programming techniques. SIMPL uses an algorithmic framework called search-infer-and-relax that encompasses various optimization methods. It also uses constraint-based modeling where constraints define how techniques are combined to solve the problem. The paper demonstrates that SIMPL can match or exceed the computational advantages of customized integrated solvers by solving production planning, product configuration, and machine scheduling problems.
Recent DEA Applications to Industry: A Literature Review From 2010 To 2014inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Developing of decision support system for budget allocation of an r&d organiz...eSAT Publishing House
1) The document describes developing a decision support system for budget allocation of an R&D organization using a performance-based goal programming model.
2) It analyzes nine years of budget data from the organization and finds a wide gap between allocated funds and funds utilized.
3) The proposed model assesses R&D programs based on priority and risk factors using fuzzy set theory, and aims to allocate budgets in a more realistic and accurate manner than the previous approach.
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.
The document provides an overview of operations research (OR), including its history, methodology, tools and techniques, and applications. It discusses how OR began during World War II to analyze military operations and optimize resource allocation. The seven main steps of the OR methodology are described. Common OR tools include linear programming, game theory, decision theory, queuing theory, inventory models, simulation, and dynamic programming. Finally, the document outlines some example applications of OR in fields like accounting, construction, and facilities planning.
Unit I (8 Hrs)
Introduction to Linear Programming – Various definitions, Statements of basic
theorems and properties, Advantages Limitations and Application areas of Linear
Programming, Linear Programming -Graphical method, - graphical solution
methods of Linear Programming problems, The Simplex Method: -the Simplex
Algorithm, Phase II in simplex method, Primal and Dual Simplex Method, Big-M
Method
Unit II (8 Hrs)
Transportation Model and its variants: Definition of the Transportation Model
-Nontraditional Transportation Models-the Transportation Algorithm-the Assignment
Model– The Transshipment Model
Unit III (8 Hrs)
Network Models: Basic differences between CPM and PERT, Arrow Networks,
Time estimates, earliest completion time, Latest allowable occurrences time,
Forward Press Computation, Backward Press Computation, Representation in
tabular form, Critical Path, Probability of meeting the scheduled date of completion,
Various floats for activities, Critical Path updating projects, Operation time cost trade
off Curve project,
Selection of schedule based on :- Cost analysis, Crashing the network
Sequential model & related problems, processing n jobs through – 1 machine & 2
machines
Unit IV (8 Hrs)
Network Models: Scope of Network Applications – Network definitions, Goal
Programming Algorithms, Minimum Spanning Tree Algorithm, Shortest Route
Problem, Maximal flow model, Minimum cost capacitated flow problem
Unit V (8 Hrs)
Decision Analysis: Decision - Making under certainty - Decision - Making under
Risk, Decision
under uncertainty.
Unit VI (8 Hrs)
Simulation Modeling: Monte Carlo Simulation, Generation of Random Numbers,
Method for
Gathering Statistical observations
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3. Review bids from writers and choose one based on qualifications, history, and feedback, then pay a deposit.
4. Review the completed paper and authorize full payment or request revisions if needed.
5. Choose HelpWriting.net for original, high-quality content with the option of revisions and a refund if plagiarized.
The document provides instructions for how to request and complete an assignment writing request on the HelpWriting.net website. It outlines a 5-step process: 1) Create an account; 2) Complete an order form with instructions and deadline; 3) Review bids from writers and select one; 4) Review the completed paper and authorize payment; 5) Request revisions to ensure satisfaction and receive a refund if plagiarized.
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Application of Linear Programming to Profit Maximization (A Case Study of.pdf
1. Advances in Mathematical & Computational Sciences Journal,
Vol.7 No.1,March 2019
11
Article Citation Format
Oyekan, E.A. & Temisan, G.O. (2019):
Application of Linear Programming to Profit Maximization
(A Case Study of Johnsons Nig. Ltd).
Journal of Advances in Mathematical & Computational Sciences
Vol. 7, No. 1. Pp 11-20
Application of Linear Programming to Profit Maximization
(A Case Study of Johnsons Nig. Ltd)
*
Oyekan Ezekiel A. & Temisan Gabriel O.
Department of Mathematical Sciences,
Ondo State University of Science and Technology (OSUSTECH),
Okitipupa, Ondo State, Nigeria
E-mail: ea.oyekan@osustech.edu.ng, temisangabriel@gmail.com
ABSTRACT
Companies exist not just for the provision of goods and services but to maximize profits and drive wealth to
their owners. As that is what guarantees its continuous existence, productivity and expansion. The lack of
effective and sustainable profit maximization results in the liquidation of organizations, especially the small
scale production companies. This work demonstrates the pragmatic use of linear programming methods in
maximization of profit at Johnsons Nig. Ltd, Bakery division wherein the four various Kings Size bread were
subjected to statistical analysis via the Simplex method in a bid to determine which bread size assures more
objective value contribution and gives maximum profit at a given level of production capacity. The optimal
solution obtained showed that the production of three units of large Kings bread size with the objective
coefficient of fifty naira will yield an objective value contribution of One hundred and fifty naira. To test the
feasibility of the solution, Sensitivity analysis was carried out, the result of which revealed that for the optimal
solution to remain comparatively unchanged the coefficient of 𝑥 must remain between 50 to 30. Otherwise,
its contribution to profit will be zero, which will in turn necessitate its discontinuity.
Keyword: Linear programming, Objective functions, Constraints, Slack variables, Basic variables, Optimal solution.
1. INTRODUCTION
It is Impossible to properly define Linear Programming without first exploring the field of Operations
research. This is because linear programming is one of the prominent techniques adopted in this field of
study. Operation Research, abbreviated ‘OR’ for short is a scientific approach to decision making, which
seeks to determine how best to design and operate a system, under conditions requiring the allocation of
scarce resources (Oyekan, 2015).
Article Progress Time Stamps
Article Type: Research Article
Manuscript Received 9th
March, 2019
Review Type: Blind
Final Acceptance: 27th March, 2019
Article DOI: dx.doi.org/10.22624/AIMS/MATHS/V7N1P2
2. Advancesin Mathematical & Computational Sciences Journal,
Vol.7 No.1,March 2019
12
Operation research as a field of study provides a set of algorithms that acts as tools for effective problem
solving and decision making in chosen application areas. OR has extensive applications in engineering,
business and public systems and is also used extensively by manufacturing and service industries in decision
making. The history of OR as a field of study dates back to World War II when the British military
asked scientists to analyze military problems. In fact, the Second World War was perhaps the first
time when people realized that scarce resources can be used effectively and allocated efficiently. [2]
The application of mathematics and scientific method to military applications was called “Operations
Research”; but today, it is also called “Management Science”. In general, the term Management
Science also includes Operations Research; in fact, these two terms are used interchangeably.
Companies exist to maximize profits and drive wealth to their owners. As that is what guarantees its continuous
existence, productivity and expansion. Businesses can only grow and expand when profit is made as this is
the major reason people go into business. According to George Dantzig, one of the mathematically proven
ways to ensure profit maximization is the Linear Programming Method [4]. Major management decisions
involve trying to make the most effective use of Organization resources [1]. These resources include
Machinery, Labour, Money, Time, Warehouse space or Raw materials to produce goods (machinery,
furniture, food or cooking) or service (schedules for machinery and production, advertising, policies or
investment decision). Linear Programming (LP) is a widely used mathematical technique designed to help
managers in planning and decision making relative to resource allocations [3]. And also provide relatively
simple and realistic solutions to these problems. A wide variety of production, finance, marketing, and
distribution problems have been formulated in the linear programming framework.
Linear Programming (LP) also called Linear Optimization is a technique which is used to solve mathematical
problems in which the relationships are linear in nature [9]. And as its name implies “Linear programming”,
it is designed for models with linear objective and constraint functions. The term “programming” indicates
the solution method which can be carried out by an iterative process in which a researcher advances from
one solution to a better solution, until a final solution is reached which cannot be improved upon. This final
solution is termed the optimal solution of the LP problem, which is a solution that fulfills both the constraints
of the problem and the set objective. A LP model can be designed and solved to determine the best course
of action as in a product mix, production schedule, blending problems, etc. subject to the available constraints
[7]. Generally, the objective function may be of maximization of profit (which is the focus of this research
work) or minimization of costs or labor hours. Moreover, the model also consists of certain structural
constraints which are set of conditions that the optimal solution should justify. Examples of the structural
constraints include the raw material constraints, production time constraint, and skilled labour constraints to
mention a few.
2. LITERATURE REVIEW
Linear Programming was developed as a discipline in 1940’s, motivated initially by the need to solve complex
planning problems in war time operations. This was possible by the recognition that most practical planning
problems could be reformulated mathematically as finding a solution to a system of linear inequalities. The
problem of solving a system of linear inequalities dates back at least as far as Fourier Joseph (1768 –1830),
after whom the method of Fourier-Motzkin elimination is named.
3. Advancesin Mathematical & Computational Sciences Journal,
Vol.7 No.1,March 2019
13
The three founders of the subject are considered to be Leonid Kantorovich, the Russian Mathematician who
developed the earliest linear programming problems in 1939, George B. Dantzig, who published the simplex
method in 1947, and John von Neumann, who developed the theory of the duality in the same year. The
earliest linear programming was first developed by Leonid Kantorovich, a Russian mathematician, in 1939.
It was used during World War II to plan expenditures and returns in order to reduce costs to the army and
increase losses to the enemy. The method was kept secret until 1947 when George B. Dantzig published the
simplex method and John von Neumann developed the theory of duality as a linear optimization solution,
and applied it in the field of game theory. Postwar, its development accelerated rapidly as many industries
found its use in their daily planning.
The Simplex method, which is used to solve linear programming, was developed by George B. Dantzig in
1947 as a product of his research work during World War II when he was working in the Pentagon with the
Mil. Most linear programming problems are solved with this method. He extended his research work to
solving problems of planning or scheduling dynamically overtime, particularly planning dynamically under
uncertainty. This method has been the standard technique for solving a linear program since the 1940's. In
brief, the simplex method passes from vertex to vertex on the boundary of the feasible polyhedron, repeatedly
increasing the objective function until either an optimal solution is found, or it is established that no solution
exists. In principle, the time required might be an exponential function of the number of variables, and this
can happen in some contrived cases. In practice, however, the method is highly efficient, typically requiring a
number of steps which is just a small multiple of the number of variables. Linear programs in thousands or
even millions of variables are routinely solved using the simplex method on modern computers.
Efficient, highly sophisticated implementations are available in the form of computer software packages. In
1979, Leonid Khaciyan presented the ellipsoid method, guaranteed to solve any linear program in a number
of steps which is a polynomial function of the amount of data defining the linear program. Consequently, the
ellipsoid method is faster than the simplex method in contrived cases where the simplex method performs
poorly. In practice, however, the simplex method is far superior to the ellipsoid method. In 1984, Narendra
Karmarkar introduced an interior-point method for linear programming, combining the desirable theoretical
properties of the ellipsoid method and practical advantages of the simplex method. Its success initiated an
explosion in the development of interior-point methods.
These do not pass from vertex to vertex, but pass only through the interior of the feasible region. Though this
property is easy to state, the analysis of interior-point methods is a subtle subject which is much less easily
understood than the behavior of the simplex method. Interior-point methods are now generally considered
competitive with the simplex method in most, though not all, applications, and sophisticated software
packages implementing them are now available. Whether they will ultimately replace the simplex method in
industrial applications is not clear.
Conclusively, the development of linear programming has been ranked among the most important scientific
advances of the mid-20th
century, and its assessment is generally accepted. Its impact since 1950 has been
extra ordinary and has saved thousands or millions of dollars of many production companies.
(Wikipedia, the free encyclopedia).
4. Advances in Mathematical & Computational Sciences Journal,
Vol.7 No.1,March 2019
14
3. METHODOLOGY
The method adopted is the Simplex (Tableau) Algorithm. Simplex algorithm is an iterative procedure that
examines the vertices of the feasible region to determine the optimal value of the objective function [6]. This
method is the principal algorithm used in solving LPP consisting of two or more decision variables. It involves
a sequence of exchange so that the trial solution proceeds systematically from one vertex to another in k, each
step produces a feasible solution [5]. This procedure is stopped when the volume of cT
x is no longer increased
as a result of the exchange. Listed below are the procedures required in the afore mentioned exchange:
Step 1: Setting up the Initial Simplex Tableau
In developing the initial simplex tableau, convert the constraints into equations by introducing slack
variables to the inequalities. So that the problem can be re-written in standard form as a maximization
problem. Such that, we can find the initial basic feasible solution by setting the decision variables
n
x
x
x ...,
,
, 2
1 to zero in the constraints we get the basic feasible solution and the objective function
becomes 0
Z .
Step 2: Optimality Process
Having setup the initial simplex tableau, determine the entering variable (key column) and the
departing variable (key row). From the Cj – Zj row we locate the column that contains the largest
positive number and this becomes the Pivot Column. In each row divide the value in the R.H.S by
the positive entry in the pivot column (ignoring all zero or negative entries) and the smallest one of
these ratios gives the pivot row. The number at the intersection of the pivot column and the pivot
row is called the pivot. Now, divide the entries of that row in the matrix by the pivot and use row
operation to reduce all other entries in the pivot column, apart from the pivot, to zero.
Step 3: The Stopping Criterion
The simplex method will always terminate in a finite number of steps when the necessary condition
for optimality is reached. The optimal solution to a maximum linear program problem is reached
when all the entries in the net evaluation row, that is Cj – Zj, are all negative or zero.
4. DATA PRESENTATION AND ANALYSIS
Johnsons Nigeria Limited, Bakery division, produces four sizes of Kings Bread namely: small, medium, short-
long and large. The dough for each bread size requires the following ingredients: flour, sugar, salt, yeast,
butter, water, oxidizing agent and improver. The dough is baked in an electric oven at 200℃ and lowered to
150℃ for a fine bread crust. According to the manager, the bakery’s production of each unit bread size
fluctuates and presumptuously based upon the previous day market. Now, the product mix will be determined
based on the data obtained in an interview with the Bakery’s manager and a member of its production staff
in the early month of August, 2017.
5. Advancesin Mathematical & Computational Sciences Journal,
Vol.7 No.1,March 2019
15
Table 1: Bakery’s (estimated) daily production capacity
Bread size Dough input
(kg)
Chamber capacity
(in layers i.e. l1,l2,l3)
Production output
(Cups)
Small 150 200 600
Medium 240 130 390
Short-Long 340 110 330
Large 600 100 300
In an attempt to determine the quantity of dough required for producing a unit of each bread size, the dough
input was divided by the total number of baked bread cups. The result of which is tabulated alongside the
baking time (per cup), average available resources and the unit profit per cup size.
Table 2: Available resources per unit production
Bread size Dough
(kg)
Baking time
(Min)
Profit
(₦)
Small 0.25 5 10
Medium 0.62 8 20
Short-Long 1.03 8 30
Long 2 10 50
Available resources 175 30
The linear programming model can now be stated as follows:
Maximize: Z = 4
3
2
1 50
30
20
10 x
x
x
x
Subject to: 175
2
03
.
1
62
.
0
25
.
0 4
3
2
1
x
x
x
x
(1)
30
10
8
8
5 4
3
2
1
x
x
x
x
with 0
1
x , 0
2
x , 0
3
x and 0
4
x
for all non-negative conditions
By introducing the slack variables in the objective function above the inequalities becomes equality and can
thus be written as:
Maximize: Z = 2
1
4
3
2
1 0
0
50
30
20
10 s
s
x
x
x
x
Subject to: 175
2
03
.
1
62
.
0
25
.
0 1
4
3
2
1
s
x
x
x
x
(2)
30
10
8
8
5 2
4
3
2
1
s
x
x
x
x
0
1
x , 0
2
x , 0
3
x and 0
4
x
0
1
s and 0
2
s
For which an initial simplex tableau is setup below
6. Advances in Mathematical & Computational Sciences Journal,
Vol.7 No.1,March 2019
16
Table 3: Initial Solution
B 1
x 2
x 3
x 4
x 1
s 2
s XB
1
s 0.25 0.62 1.03 2 1 0 175
2
s 5 8 2 10 0 1 30
Cj - Zj -10 -20 -30 -50 0 0
The bottom (Cj - Zj) row of table 4.3 contains the net profit per unit obtained by introducing one unit of a
given variable into the solution.
Pivoting
The key column is 4
x being the least (most) negative of all entries in (Cj - Zj) row of table 3. To obtain
the key row, a ratio test is carried out as follows:
𝜃 = = 87.5
𝜃 = = 3
Therefore, the key row is 2
s being the least positive of the above values.
Now, to construct a new table, we find the pivotal entry at the intersection of the entry variable column and
the departing variable-row. And then use the pivotal element for elimination (to get zero). Thus, we obtain a
new simplex table by entering 4
x into the solution and removing 2
s variable, now (old) R2. Performing the
row operation below yields table 4
(New)𝑅 : 𝑅 − 2𝑁𝑅
(New)𝑅 : 𝑅 ×
(New)𝑅 : 𝑅 + 50𝑁𝑅
Table 4: First Iteration
B 1
x 2
x 3
x 4
x 1
s 2
s XB
1
s -0.75 -0.98 0.57 0 1 0.2 169
4
x 0.5 0.8 0.8 1 0 0.1 3
Cj - Zj 15 20 10 0 0 5 150
Since (Cj - Zj)row of table 4 has no negative entry in the column of variables. Therefore, this is the case of
optimal solution. From the (XB) column of the table, we have 3
4
x , 0
3
2
1
x
x
x , 169
1
s 0
2
s and
the maximum value of Z=150.
4.1 Results
The linear programming model was solved using the simplex algorithm earlier discussed and the optimal
solution is given as shown in Table 4. By which we have that, the production of three units of large Kings
7. Advances in Mathematical & Computational Sciences Journal,
Vol.7 No.1,March 2019
17
Bread size with the objective coefficient of fifty naira will give an objective value contribution of One hundred
and fifty naira. This result is not substantive enough to reach a valid conclusion, thereby necessitating the
need to carry out Sensitivity Analysis.
4.2 Application of Sensitivity Analysis
When a mathematical model is used to describe reality, approximations are made. The world as we know it
is more complicated than the kinds of optimization problems that we are able to solve. Linearity assumptions
usually are significant approximations. Another important approximation comes because one cannot be sure
of the data being put into the model. Moreover, information may change. Sensitivity analysis is a systematic
study of how sensitive solutions are to (small) changes in the data. In other words, sensitivity analysis deals
with finding out the amount by which we can change the input data for the output of our linear programming
model to remain comparatively unchanged [10].
The basic idea is to be able to give answers to questions of the form:
1. If the objective function coefficient changes, how does the solution change?
2. If the available resources changes, how does the solution change?
3. If a constraint is added to the problem, how does the solution change?
For the purpose of this study, we’ll answer question (1) by evaluating what happens when one parameter of
the problem changes.
4.2.1 Changing the Objective function coefficient
What we do here is to change the coefficient of the objective function and hold the constraints fixed, but only
one coefficient in the objective function. We consider the linear programming model earlier formulated
whose solution is, 3
0 4
3
2
1
x
and
x
x
x and 150
Z . From this solution, the relevant non-basic
variables are .
,
, 3
2
1 x
and
x
x
Now, we consider what happens to the solution if the coefficient of a non-basic variable decreases. For
example, if the coefficient of 3
x in the objective function was reduced from 30 to 25. So that the objective
function is:
Maximize: Z = 4
3
2
1 50
25
20
10 x
x
x
x
(3)
the solution of which does not changes. Below, we consider the following cases:
Case 1: What happens to the solution if the coefficient of the non-basic variable is raised?
Intuitively, raising it a little bit should not matter, but raising the coefficient a lot might induce a change in the
value of x in a way that makes 𝑥 > 0. So, for a non-basic variable, it is expected that the solution will continue
to be valid for a range (say 5 to 15) of values for coefficients of non-basic variables. If the coefficient increases
enough (say ≥ 20), then the solution may change.
8. Advances in Mathematical & Computational Sciences Journal,
Vol.7 No.1,March 2019
18
Figure 1: Changes in solution via increase in non-basic variable
Case 2: What happens to the solution if the coefficient of the basic variable decreases?
This situation differs entirely from the previous. The change makes the variable contribute less profit. It is
expected that a sufficiently large reduction brings about a change in the solution. For example, if the coefficient
of 𝑥 in the objective function of the model formulated was 30 instead of 50. So that the objective function
is:
Maximize: Z = 4
3
2
1 30
30
20
10 x
x
x
x
(4)
for which we might (possibly) have to set 𝑥 = 0 instead of 𝑥 = 3. On the other hand, a small reduction
(say 5 to 15) in 𝑥 ′𝑠 objective function coefficient would typically not cause a change in the solution. In
contrast to the case of the non-basic variable, such a change will have a negative impact on the value of the
objective function. The value is computed by plugging in x into the objective function, if 𝑥 = 3, then the
coefficient of 𝑥 goes down from 150 to 90 (assuming that the solution remains the same).
From the above discussions, it is evident that if the coefficient of a basic variable goes up, then the value goes
up and we can still use the variable. Since the value of the problem always changes whenever there is a change
in the coefficient of the basic variable, intuitively, there should be a range of values of the coefficient of the
objective function (a range that includes the original value) in which the solution of the problem does not
change. Outside of this range, the solution will change.
9. Advances in Mathematical & Computational Sciences Journal,
Vol.7 No.1,March 2019
19
Figure 2: Changes in solution via decrease in basic variable
Based on the sensitivity analysis carried out, we now know to what extent a change in the input data induces
a change in the optimal solution. And for the optimal solution to remain comparatively unchanged the
coefficient of x4 must remain between the ranges of 50 to 30. Otherwise, its contribution to profit will be zero.
Thereby changing the optimal solution, such that .
5
.
112
75
.
3
,
0 3
4
2
1
Z
and
x
x
x
x
5. CONCLUSION
This study has succeeded in shedding more light on the profitability of using linear programming techniques
in production over any known theory of profit maximization. Based on the data obtained from the company,
it was discovered that if the company is to maximize their profit, the production of large size Kings bread has
to stand at three units. Since it assures an objective value contribution of one hundred and fifty naira,
compared to the initial (total) contribution of one hundred and ten naira. Furthermore, the sensitivity analysis
provided the range at which the optimal solution changes due to a change in the input parameter. Wherein,
we considered the cases of changes in the objective function coefficient, available resources and the addition
of a new constraint and how these changes affects the optimal solution. By means of this analysis, an alternative
optimal solution was obtained with an objective value contribution of one hundred and twelve naira fifteen
kobo, whose marginal value as compared to the original solution is thirty seven naira fifty kobo.
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Vol.7 No.1,March 2019
20
This therefore depicts that the optimal solution earlier obtained is substantive enough to reach a valid
conclusion. Hence, the production of three large Kings Bread size with an objective coefficient of fifty naira
assures a profit margin of one hundred and fifty naira. If only the unit production stood within the specified
time range.
Thus, if there happens to be a sufficiently large reduction in the objective coefficient of the large Kings Bread
size such that a unit of the short-long bread size gives a profit higher than the former, discontinuing its
production is advised. Since its contribution to profit would be zero.
Researchers with interest in this subject area can consider the blending problem associated with selection of
raw materials to produce the best combination at minimum cost.
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