This document provides an overview of Chapter 4 on the Simplex Method for solving linear programming problems. It begins with an introduction to the simplex method and its development by George Dantzig. The chapter outline lists topics like linear programs in standard form, basic feasible solutions, and the simplex algorithm. It then discusses concepts such as slack and surplus variables, and conditions for applying the simplex method. Examples are provided to demonstrate converting problems to standard form and finding basic feasible solutions. The summary provides a high-level view of the key topics and concepts covered in the chapter on using the simplex method to solve linear programs.
The Big-M method is a variation of the simplex method for solving linear programming problems with "greater-than" constraints. It works by introducing artificial variables with a large coefficient M to transform inequality constraints into equality constraints, creating an initial feasible solution. The transformed problem is then solved via simplex elimination to arrive at an optimal solution while eliminating artificial variables. The document provides an example problem demonstrating the step-by-step Big-M method process of setting up and solving a linear program with inequalities.
The document discusses the Simplex method for solving linear programming problems involving profit maximization and cost minimization. It provides an overview of the concept and steps of the Simplex method, and gives an example of formulating and solving a farm linear programming model to maximize profits from two products. The document also discusses some complications that can arise in applying the Simplex method.
This document discusses several types of complications that can occur when solving linear programming problems (LPP), including degeneracy, unbounded problems, multiple optimal solutions, infeasible problems, and redundant or unrestricted variables. It provides examples and explanations of how to identify each type of complication and the appropriate steps to resolve it such as introducing slack or artificial variables, breaking ties, or setting unrestricted variables equal to the difference of two non-negative variables.
Linear Programming Problems : Dr. Purnima PanditPurnima Pandit
Linear programming problems involve optimizing an objective function subject to constraints on variables. They can be modeled and solved using techniques like the simplex method. The simplex method works by moving from one basic feasible solution to an adjacent extreme point through an exchange of variables in and out of the basis. It begins with an initial basic feasible solution and proceeds iteratively until an optimal solution is reached.
The Big M Method is used to solve linear programming problems with inequality constraints. It involves multiplying inequality constraints to make the right hand side positive, introducing surplus and slack variables, and adding a large penalty term M to the objective for any artificial variables. The example problem is solved using this method in multiple iterations of the simplex algorithm to find the optimal solution.
- The document discusses duality theory and sensitivity analysis in linear programming.
- Duality theory states that for every linear programming problem (LPP), there is a corresponding dual LPP. The dual problem can be constructed from the primal problem using specific rules. Solving one problem provides the solution to the other.
- Sensitivity analysis determines how changes in the coefficients or right-hand side values of the LPP affect the optimal solution. It identifies the ranges that parameters can vary without impacting the optimal values of variables. This provides insight into the robustness of the optimal solution.
This document provides examples of constructing the dual problem of a linear programming primal problem and solving it using the two-phase simplex method. It first presents the rules for constructing the dual problem and then works through two examples. The first example derives the dual problem from the primal and solves it using the two-phase method. The second example shows how to find the optimal dual solution given the optimal primal solution using two methods - using the objective coefficients of the primal variables or using the inverse of the primal basic variable matrix.
The Big-M method is a variation of the simplex method for solving linear programming problems with "greater-than" constraints. It works by introducing artificial variables with a large coefficient M to transform inequality constraints into equality constraints, creating an initial feasible solution. The transformed problem is then solved via simplex elimination to arrive at an optimal solution while eliminating artificial variables. The document provides an example problem demonstrating the step-by-step Big-M method process of setting up and solving a linear program with inequalities.
The document discusses the Simplex method for solving linear programming problems involving profit maximization and cost minimization. It provides an overview of the concept and steps of the Simplex method, and gives an example of formulating and solving a farm linear programming model to maximize profits from two products. The document also discusses some complications that can arise in applying the Simplex method.
This document discusses several types of complications that can occur when solving linear programming problems (LPP), including degeneracy, unbounded problems, multiple optimal solutions, infeasible problems, and redundant or unrestricted variables. It provides examples and explanations of how to identify each type of complication and the appropriate steps to resolve it such as introducing slack or artificial variables, breaking ties, or setting unrestricted variables equal to the difference of two non-negative variables.
Linear Programming Problems : Dr. Purnima PanditPurnima Pandit
Linear programming problems involve optimizing an objective function subject to constraints on variables. They can be modeled and solved using techniques like the simplex method. The simplex method works by moving from one basic feasible solution to an adjacent extreme point through an exchange of variables in and out of the basis. It begins with an initial basic feasible solution and proceeds iteratively until an optimal solution is reached.
The Big M Method is used to solve linear programming problems with inequality constraints. It involves multiplying inequality constraints to make the right hand side positive, introducing surplus and slack variables, and adding a large penalty term M to the objective for any artificial variables. The example problem is solved using this method in multiple iterations of the simplex algorithm to find the optimal solution.
- The document discusses duality theory and sensitivity analysis in linear programming.
- Duality theory states that for every linear programming problem (LPP), there is a corresponding dual LPP. The dual problem can be constructed from the primal problem using specific rules. Solving one problem provides the solution to the other.
- Sensitivity analysis determines how changes in the coefficients or right-hand side values of the LPP affect the optimal solution. It identifies the ranges that parameters can vary without impacting the optimal values of variables. This provides insight into the robustness of the optimal solution.
This document provides examples of constructing the dual problem of a linear programming primal problem and solving it using the two-phase simplex method. It first presents the rules for constructing the dual problem and then works through two examples. The first example derives the dual problem from the primal and solves it using the two-phase method. The second example shows how to find the optimal dual solution given the optimal primal solution using two methods - using the objective coefficients of the primal variables or using the inverse of the primal basic variable matrix.
The Simplex Method is an algorithm for solving linear programming problems. It involves setting up the problem in standard form, constructing an initial simplex tableau, and then iteratively selecting pivot columns and performing row operations until an optimal solution is found. The method terminates when all indicators in the tableau are positive or zero, at which point the basic and non-basic variables can be identified to read the optimal solution.
The document discusses duality theory in linear programming (LP). It explains that for every LP primal problem, there exists an associated dual problem. The primal problem aims to optimize resource allocation, while the dual problem aims to determine the appropriate valuation of resources. The relationship between primal and dual problems is fundamental to duality theory. The document provides examples of primal and dual problems and their formulations. It also outlines some general rules for constructing the dual problem from the primal, as well as relations between optimal solutions of primal and dual problems.
The document describes the graphical method for solving linear programming problems with two decision variables. It provides the step-by-step procedure which involves plotting the constraints on a graph to identify the feasible region, determining the corner points of this region which represent the feasible solutions, substituting these points into the objective function to find the optimal value, and identifying the optimal solution. It then provides examples demonstrating this process and different types of solutions that can arise such as unbounded, infeasible, and optimal.
The document provides an overview of the simplex method for solving linear programming problems. It discusses:
- The simplex method is an iterative algorithm that generates a series of solutions in tabular form called tableaus to find an optimal solution.
- It involves writing the problem in standard form, introducing slack variables, and constructing an initial tableau.
- The method then performs iterations involving selecting a pivot column and row, and applying row operations to generate new tableaus until an optimal solution is found.
- It also discusses how artificial variables are introduced for problems with non-strict inequalities and provides an example solved using the simplex method.
The document provides an outline of topics related to linear programming, including:
1) An introduction to linear programming models and examples of problems that can be solved using linear programming.
2) Developing linear programming models by determining objectives, constraints, and decision variables.
3) Graphical and simplex methods for solving linear programming problems.
4) Using a simplex tableau to iteratively solve a sample product mix problem to find the optimal solution.
Linear programming - Model formulation, Graphical MethodJoseph Konnully
The document discusses linear programming, including an overview of the topic, model formulation, graphical solutions, and irregular problem types. It provides examples to demonstrate how to set up linear programming models for maximization and minimization problems, interpret feasible and optimal solution regions graphically, and address multiple optimal solutions, infeasible solutions, and unbounded solutions. The examples aid in understanding the key steps and components of linear programming models.
The document provides an overview of the simplex method for solving linear programming problems with more than two decision variables. It describes key concepts like slack variables, surplus variables, basic feasible solutions, degenerate and non-degenerate solutions, and using tableau steps to arrive at an optimal solution. Examples are provided to illustrate setting up and solving problems using the simplex method.
The document discusses the simplex method for solving linear programming problems. It introduces some key terminology used in the simplex method like slack variables, surplus variables, and artificial variables. It then provides an overview of how the simplex method works for maximization problems, including forming the initial simplex table, testing for optimality and feasibility, pivoting to find an optimal solution. Finally, it provides an example application of the simplex method to a sample maximization problem.
The document summarizes the simplex method for solving linear programming problems. It provides examples to demonstrate how to set up the simplex tableau, choose entering and departing variables at each iteration, and arrive at the optimal solution. The key steps are to rewrite the objective function, convert inequalities to equalities using slack variables, choose pivots to make coefficients zero, and iterate until an optimal basic feasible solution is found.
The simplex method is a linear programming algorithm that can solve problems with more than two decision variables. It works by generating a series of solutions, called tableaus, where each tableau corresponds to a corner point of the feasible solution space. The algorithm starts at the initial tableau, which corresponds to the origin. It then shifts to adjacent corner points, moving in the direction that optimizes the objective function. This process of generating new tableaus continues until an optimal solution is found.
This document summarizes the two phase simplex method for solving linear programming problems. In phase I, artificial variables are introduced to convert infeasible problems into feasible problems. The objective is to minimize the artificial variables. If the minimum is zero, the original problem is feasible and phase II begins. Phase II uses the original objective function and simplex method to find an optimal solution. An example problem is provided to illustrate the two phase method.
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 transportation problem is a special type of linear programming problem where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations.
Because of its special structure, the usual simplex method is not suitable for solving transportation problems. These problems require a special method of solution.
This presentation is trying to explain the Linear Programming in operations research. There is a software called "Gipels" available on the internet which easily solves the LPP Problems along with the transportation problems. This presentation is co-developed with Sankeerth P & Aakansha Bajpai.
By:-
Aniruddh Tiwari
Linkedin :- http://in.linkedin.com/in/aniruddhtiwari
The Big M Method is a variant of the simplex method for solving linear programming problems. It introduces artificial variables and a large number M to convert inequalities into equalities. The transformed problem is then solved using the simplex method, eliminating artificial variables until an optimal solution is found. However, the method has drawbacks in determining a sufficiently large M value and not knowing feasibility until optimality is reached. It is inferior to the two-phase method and not used in commercial solvers.
This document contains lecture slides on nonlinear programming from lectures given at MIT. It discusses two main issues in nonlinear programming: 1) characterizing solutions through necessary and sufficient conditions using concepts like Lagrange multipliers and sensitivity analysis, and 2) computational methods for finding solutions through iterative algorithms. It provides examples of application areas for nonlinear programming like data networks, production planning, and engineering design. It outlines topics covered in the first lecture, including duality theory and the relationship between linear and nonlinear programming.
Linear programming deals with optimizing a linear objective function subject to linear constraints. It involves determining the values of decision variables to maximize or minimize the objective function. The general linear programming model involves maximizing or minimizing a linear combination of n decision variables subject to m linear constraints, along with non-negativity restrictions on the decision variables. Formulating a linear programming problem involves identifying decision variables, expressing constraints and the objective function linearly in terms of the variables, and adding non-negativity restrictions.
This document discusses transportation problems and three methods to solve them: the North West Corner Method, Least Cost Method, and Vogel Approximation Method. The objective of transportation problems is to minimize the cost of distributing products from sources to destinations while satisfying supply and demand constraints. The document provides examples to illustrate how each method works step-by-step to arrive at a basic feasible solution.
The document discusses linear programming, which is a mathematical modeling technique used to allocate limited resources optimally. It provides examples of linear programming problems and their formulation. Key aspects covered include defining decision variables and constraints, developing the objective function, and interpreting feasible and optimal solutions. Graphical and algebraic solution methods like the simplex method are also introduced.
The document discusses convex functions and related concepts. It defines convex functions and provides examples of convex and concave functions on R and Rn, including norms, logarithms, and powers. It describes properties that preserve convexity, such as positive weighted sums and composition with affine functions. The conjugate function and quasiconvex functions are also introduced. Key concepts are illustrated with examples throughout.
This document provides information about operation research and the graphical method for solving linear programming problems. It defines operation research and describes the general structure of a linear programming model, which includes decision variables, the objective function, and constraints. The document then formulates a sample linear programming problem and provides the step-by-step graphical solution, identifying the feasible region and finding the optimal solution that maximizes the objective function.
The document provides information about linear programming problems (LPP), including:
- LPPs involve optimization of a linear objective function subject to linear constraints.
- Graphical and algebraic methods can be used to find the optimal solution, which must occur at a corner point of the feasible region.
- The simplex method is an algorithm that moves from one corner point to another to optimize the objective function.
- Examples are provided to illustrate LPP formulation, graphical solution, and use of the simplex method to iteratively find an optimal solution.
The Simplex Method is an algorithm for solving linear programming problems. It involves setting up the problem in standard form, constructing an initial simplex tableau, and then iteratively selecting pivot columns and performing row operations until an optimal solution is found. The method terminates when all indicators in the tableau are positive or zero, at which point the basic and non-basic variables can be identified to read the optimal solution.
The document discusses duality theory in linear programming (LP). It explains that for every LP primal problem, there exists an associated dual problem. The primal problem aims to optimize resource allocation, while the dual problem aims to determine the appropriate valuation of resources. The relationship between primal and dual problems is fundamental to duality theory. The document provides examples of primal and dual problems and their formulations. It also outlines some general rules for constructing the dual problem from the primal, as well as relations between optimal solutions of primal and dual problems.
The document describes the graphical method for solving linear programming problems with two decision variables. It provides the step-by-step procedure which involves plotting the constraints on a graph to identify the feasible region, determining the corner points of this region which represent the feasible solutions, substituting these points into the objective function to find the optimal value, and identifying the optimal solution. It then provides examples demonstrating this process and different types of solutions that can arise such as unbounded, infeasible, and optimal.
The document provides an overview of the simplex method for solving linear programming problems. It discusses:
- The simplex method is an iterative algorithm that generates a series of solutions in tabular form called tableaus to find an optimal solution.
- It involves writing the problem in standard form, introducing slack variables, and constructing an initial tableau.
- The method then performs iterations involving selecting a pivot column and row, and applying row operations to generate new tableaus until an optimal solution is found.
- It also discusses how artificial variables are introduced for problems with non-strict inequalities and provides an example solved using the simplex method.
The document provides an outline of topics related to linear programming, including:
1) An introduction to linear programming models and examples of problems that can be solved using linear programming.
2) Developing linear programming models by determining objectives, constraints, and decision variables.
3) Graphical and simplex methods for solving linear programming problems.
4) Using a simplex tableau to iteratively solve a sample product mix problem to find the optimal solution.
Linear programming - Model formulation, Graphical MethodJoseph Konnully
The document discusses linear programming, including an overview of the topic, model formulation, graphical solutions, and irregular problem types. It provides examples to demonstrate how to set up linear programming models for maximization and minimization problems, interpret feasible and optimal solution regions graphically, and address multiple optimal solutions, infeasible solutions, and unbounded solutions. The examples aid in understanding the key steps and components of linear programming models.
The document provides an overview of the simplex method for solving linear programming problems with more than two decision variables. It describes key concepts like slack variables, surplus variables, basic feasible solutions, degenerate and non-degenerate solutions, and using tableau steps to arrive at an optimal solution. Examples are provided to illustrate setting up and solving problems using the simplex method.
The document discusses the simplex method for solving linear programming problems. It introduces some key terminology used in the simplex method like slack variables, surplus variables, and artificial variables. It then provides an overview of how the simplex method works for maximization problems, including forming the initial simplex table, testing for optimality and feasibility, pivoting to find an optimal solution. Finally, it provides an example application of the simplex method to a sample maximization problem.
The document summarizes the simplex method for solving linear programming problems. It provides examples to demonstrate how to set up the simplex tableau, choose entering and departing variables at each iteration, and arrive at the optimal solution. The key steps are to rewrite the objective function, convert inequalities to equalities using slack variables, choose pivots to make coefficients zero, and iterate until an optimal basic feasible solution is found.
The simplex method is a linear programming algorithm that can solve problems with more than two decision variables. It works by generating a series of solutions, called tableaus, where each tableau corresponds to a corner point of the feasible solution space. The algorithm starts at the initial tableau, which corresponds to the origin. It then shifts to adjacent corner points, moving in the direction that optimizes the objective function. This process of generating new tableaus continues until an optimal solution is found.
This document summarizes the two phase simplex method for solving linear programming problems. In phase I, artificial variables are introduced to convert infeasible problems into feasible problems. The objective is to minimize the artificial variables. If the minimum is zero, the original problem is feasible and phase II begins. Phase II uses the original objective function and simplex method to find an optimal solution. An example problem is provided to illustrate the two phase method.
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 transportation problem is a special type of linear programming problem where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations.
Because of its special structure, the usual simplex method is not suitable for solving transportation problems. These problems require a special method of solution.
This presentation is trying to explain the Linear Programming in operations research. There is a software called "Gipels" available on the internet which easily solves the LPP Problems along with the transportation problems. This presentation is co-developed with Sankeerth P & Aakansha Bajpai.
By:-
Aniruddh Tiwari
Linkedin :- http://in.linkedin.com/in/aniruddhtiwari
The Big M Method is a variant of the simplex method for solving linear programming problems. It introduces artificial variables and a large number M to convert inequalities into equalities. The transformed problem is then solved using the simplex method, eliminating artificial variables until an optimal solution is found. However, the method has drawbacks in determining a sufficiently large M value and not knowing feasibility until optimality is reached. It is inferior to the two-phase method and not used in commercial solvers.
This document contains lecture slides on nonlinear programming from lectures given at MIT. It discusses two main issues in nonlinear programming: 1) characterizing solutions through necessary and sufficient conditions using concepts like Lagrange multipliers and sensitivity analysis, and 2) computational methods for finding solutions through iterative algorithms. It provides examples of application areas for nonlinear programming like data networks, production planning, and engineering design. It outlines topics covered in the first lecture, including duality theory and the relationship between linear and nonlinear programming.
Linear programming deals with optimizing a linear objective function subject to linear constraints. It involves determining the values of decision variables to maximize or minimize the objective function. The general linear programming model involves maximizing or minimizing a linear combination of n decision variables subject to m linear constraints, along with non-negativity restrictions on the decision variables. Formulating a linear programming problem involves identifying decision variables, expressing constraints and the objective function linearly in terms of the variables, and adding non-negativity restrictions.
This document discusses transportation problems and three methods to solve them: the North West Corner Method, Least Cost Method, and Vogel Approximation Method. The objective of transportation problems is to minimize the cost of distributing products from sources to destinations while satisfying supply and demand constraints. The document provides examples to illustrate how each method works step-by-step to arrive at a basic feasible solution.
The document discusses linear programming, which is a mathematical modeling technique used to allocate limited resources optimally. It provides examples of linear programming problems and their formulation. Key aspects covered include defining decision variables and constraints, developing the objective function, and interpreting feasible and optimal solutions. Graphical and algebraic solution methods like the simplex method are also introduced.
The document discusses convex functions and related concepts. It defines convex functions and provides examples of convex and concave functions on R and Rn, including norms, logarithms, and powers. It describes properties that preserve convexity, such as positive weighted sums and composition with affine functions. The conjugate function and quasiconvex functions are also introduced. Key concepts are illustrated with examples throughout.
This document provides information about operation research and the graphical method for solving linear programming problems. It defines operation research and describes the general structure of a linear programming model, which includes decision variables, the objective function, and constraints. The document then formulates a sample linear programming problem and provides the step-by-step graphical solution, identifying the feasible region and finding the optimal solution that maximizes the objective function.
The document provides information about linear programming problems (LPP), including:
- LPPs involve optimization of a linear objective function subject to linear constraints.
- Graphical and algebraic methods can be used to find the optimal solution, which must occur at a corner point of the feasible region.
- The simplex method is an algorithm that moves from one corner point to another to optimize the objective function.
- Examples are provided to illustrate LPP formulation, graphical solution, and use of the simplex method to iteratively find an optimal solution.
This document provides an introduction to optimization problems. It discusses key concepts like convex and non-convex functions, global vs local optima, and solving convex programs graphically and analytically. Specifically, it shows how to represent an optimization problem with an objective function and constraints, defines convexity, and explains that convex programs can be solved to find the global optimum while non-convex programs may only find a local optimum. It also demonstrates solving a simple convex program graphically by finding the point where the objective function contour is tangent to the constraint, and analytically by setting the gradients of the objective and constraints equal using Lagrange multipliers.
Determination of Optimal Product Mix for Profit Maximization using Linear Pro...IJERA Editor
This paper demonstrates the use of liner programming methods in order to determine the optimal product mix for
profit maximization. There had been several papers written to demonstrate the use of linear programming in
finding the optimal product mix in various organization. This paper is aimed to show the generic approach to be
taken to find the optimal product mix.
Determination of Optimal Product Mix for Profit Maximization using Linear Pro...IJERA Editor
This document demonstrates using linear programming to determine the optimal product mix for a manufacturing firm to maximize profit. The firm produces n products using m raw materials. The problem is formulated as a linear program to maximize total profit subject to raw material constraints. The optimal solution is found using the simplex method and provides the quantities of each product (v1, v2, etc.) that maximize total profit (z0). The solution may show some product quantities as zero, indicating those products should not be produced to maximize profit under the given constraints.
The document describes the simplex method for solving linear programming problems. It begins by explaining that the graphical solution can only be used for problems with 2-3 variables, so an algebraic procedure is needed for more variables. It then introduces the simplex method, which requires the problem to be in standard form. The rest of the document discusses how to transform any linear program into standard form and defines key terms used in the simplex method like basic and non-basic variables, basic feasible solutions, degenerate solutions, and optimal solutions.
This document introduces basic concepts in optimization, including:
- Local and global optima are defined, with local optima being points where no nearby points have lower objective values, and global optima having no other feasible points with lower values.
- Numerical methods are used to find optima by iteratively improving search along feasible directions from a starting point.
- Convex and concave functions and sets are defined, with convex functions/sets having important implications for optimization.
This document provides an overview of linear programming problems and methods for solving them. It defines a linear programming problem and describes how to write it in standard form with decision variables and constraints. It then explains the simplex method, including how to form an initial tableau and iterate to reach an optimal solution. Finally, it introduces the Big-M method for handling problems with inequality constraints by adding artificial variables with large penalty coefficients. An example demonstrates both simplex and Big-M methods.
Linearprog, Reading Materials for Operational Research Derbew Tesfa
The document discusses linear programming (LP), which involves optimizing a linear objective function subject to linear constraints. It provides examples of LP problems, such as production planning and transportation problems. It defines key LP concepts like the feasible region, basic solutions, basic variables, and degenerate basic feasible solutions. It also describes how to transform any LP problem into standard form and discusses properties of optimal solutions.
This document discusses solving a linear programming problem to maximize profit given two decision variables (number of newspaper and social media ads) and two constraints (budget and work hours). It presents the decision variables, objective function to maximize profit, constraints, graphing the feasible region, identifying the extreme points, and using simultaneous equations and the multiplication method to find the optimal solution where the constraints intersect at (76.92, 9.23). This maximizes profit of $326 given the budget of $240 and 100 work hours available.
This document discusses various techniques for applying the simplex method to optimize linear programs with different formulations, including:
1. Using a tabular format to systematically perform calculations in each iteration of the simplex method.
2. Adapting the simplex method to handle problems with equality constraints or minimization objectives using an artificial variable approach.
3. Using Big-M or two-phase methods to convert inequality constraints into proper form for the simplex method.
4. Conducting sensitivity analysis to determine how changes to parameters like resources would impact the optimal objective value.
The document provides information about linear programming, including:
- Linear programming is a technique to optimize allocation of scarce resources among competing demands. It involves determining variables, constraints, and an objective function.
- The linear programming model consists of linear objectives and constraints, where variables have a proportionate relationship (e.g. increasing labor increases output proportionately).
- Essential elements of a linear programming model include limited resources, an objective to maximize or minimize, linear relationships between variables, identical resources/products, and divisible resources.
- Linear programming problems can be solved graphically by plotting constraints and objective function to find the optimal point, or algebraically using the simplex method through iterative tables.
This document discusses linear programming techniques for managerial decision making. Linear programming can determine the optimal allocation of scarce resources among competing demands. It consists of linear objectives and constraints where variables have a proportionate relationship. Essential elements of a linear programming model include limited resources, objectives to maximize or minimize, linear relationships between variables, homogeneity of products/resources, and divisibility of resources/products. The linear programming problem is formulated by defining variables and constraints, with the objective of optimizing a linear function subject to the constraints. It is then solved using graphical or simplex methods through an iterative process to find the optimal solution.
The document discusses integer programming and various methods to solve integer linear programming problems. It provides:
1) An overview of integer programming, defining it as an optimization problem where some or all variables must take integer values.
2) Three main types of integer programming problems - pure, mixed, and 0-1 integer problems.
3) Four methods for solving integer linear programming problems: rounding, cutting-plane, branch-and-bound, and additive algorithms.
4) A detailed example applying the cutting-plane and branch-and-bound methods to solve a sample integer programming problem.
This document presents new certified optimal solutions found by the Charibde algorithm for six difficult benchmark optimization problems. Charibde combines an evolutionary algorithm and interval-based methods in a cooperative framework. It has achieved optimality proofs for five bound-constrained problems and one nonlinearly constrained problem. These problems are highly multimodal and some had not been solved before even with approximate methods. The document also compares Charibde's performance to other state-of-the-art solvers, showing it is highly competitive while providing reliable optimality proofs.
CHAPTER 6 System Techniques in water resuorce ppt yadesa.pptxGodisgoodtube
This document discusses linear programming techniques for solving optimization problems in water resource systems. It begins with definitions of optimization and its uses. Linear programming is introduced as a popular optimization technique. The key aspects of linear programming covered include: the formulation of linear programming problems by defining variables, objectives and constraints; common methods for solving problems, including graphical and simplex methods; and an example problem demonstrating how to set up and solve a linear programming optimization problem to maximize total net benefits from allocating land between two crops.
This document discusses and compares different methods for solving assignment problems. It begins with an abstract that defines assignment problems as optimally assigning n objects to m other objects in an injective (one-to-one) fashion. It then provides an introduction to the Hungarian method and a new proposed Matrix Ones Assignment (MOA) method. The body of the document provides details on modeling assignment problems with cost matrices, formulations as linear programs, and step-by-step explanations of the Hungarian and MOA methods. It includes an example solved using the Hungarian method.
A Comparative Analysis Of Assignment ProblemJim Webb
This document provides a comparative analysis of different methods for solving assignment problems, including the Hungarian method and a new proposed Matrix Ones Assignment (MOA) method. It first introduces assignment problems and describes their applications. It then explains the Hungarian method in detail through examples. Finally, it outlines the steps of the new MOA method, which aims to create ones in the assignment matrix to find optimal assignments. The document compares the two approaches and provides an example solved using the MOA method.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
Phenomics assisted breeding in crop improvementIshaGoswami9
As the population is increasing and will reach about 9 billion upto 2050. Also due to climate change, it is difficult to meet the food requirement of such a large population. Facing the challenges presented by resource shortages, climate
change, and increasing global population, crop yield and quality need to be improved in a sustainable way over the coming decades. Genetic improvement by breeding is the best way to increase crop productivity. With the rapid progression of functional
genomics, an increasing number of crop genomes have been sequenced and dozens of genes influencing key agronomic traits have been identified. However, current genome sequence information has not been adequately exploited for understanding
the complex characteristics of multiple gene, owing to a lack of crop phenotypic data. Efficient, automatic, and accurate technologies and platforms that can capture phenotypic data that can
be linked to genomics information for crop improvement at all growth stages have become as important as genotyping. Thus,
high-throughput phenotyping has become the major bottleneck restricting crop breeding. Plant phenomics has been defined as the high-throughput, accurate acquisition and analysis of multi-dimensional phenotypes
during crop growing stages at the organism level, including the cell, tissue, organ, individual plant, plot, and field levels. With the rapid development of novel sensors, imaging technology,
and analysis methods, numerous infrastructure platforms have been developed for phenotyping.
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Leonel Morgado
Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
1. Linear Optimization(MATH 2062)
Dereje Tigabu(MSc.)
Department of Mathematics
Debark University
October 7, 2020
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 1 / 107
2. Chapter 4
Simplex Method1
1
Checking the results of a decision against its expectations shows executives
what their strengths are, where they need to improve, and where they lack
knowledge or information. Peter Drucker
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 2 / 107
3. Outline of the Chapter
1 Introduction
2 Linear Programs in Standard Form
3 Basic Feasible Solutions
4 Fundamental Theorem of Linear Programming
5 Algebra of the Simplex Method
6 The simplex Algorithm
7 Degeneracy and Finiteness of Simplex Algorithm
8 Finding a Starting Basic Feasible Solution
Two -phase method
Big - M method
9 Some Complications and Their Resolution
Unrestricted Variables
Tie for Entering Basic Variable (Key Column)
Tie for Leaving Basic Variable (Key Row) Degeneracy
10 Types of Linear Programming Solutions
Alternative (Multiple) Optimal Solutions
Unbounded Solution
Infeasible Solution
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 3 / 107
4. 1. Introduction
We shall discuss a procedure called the simplex method for solving
an LP model of such problems.
This method was developed by G B Dantzig in 1947.
For LP problems with several variables, we may not be able to graph
the feasible region, but the optimal solution will still lie at an extreme
point of the many-sided, multidimensional figure (called an
n-dimensional polyhedron) that represents the feasible solution space.
The simplex method examines these extreme points in a systematic
manner, repeating the same set of steps of the algorithm until an
optimal solution is found.
It is also called the iterative method.
Since the number of extreme points of the feasible solution space are
finite, the method assures an improvement in the value of objective
function as we move from one iteration (extreme point) to another
and achieve the optimal solution in a finite number of steps.
The method also indicates when an unbounded solution is reached.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 4 / 107
5. Slack variables : If the constraints are in the form of ”≤”, then we
add variable to the left hand side of inequality to make equality. This
variables are called slack variables.
Remark: A slack variable represents unused resource, either in the
form of time on a machine, labour hours, money, warehouse space
or any number of such resources in various business problems. Since
these variables yield no profit, therefore such variables are added to
the original objective function with zero coefficients.
Surplus variables : If the constraints are in the form of ” ≥ ” ,
then we subtract variables to the left hand side of inequality to make
equality. This variables are called the surplus variables.
Remark: A surplus variable represents amount by which solution values
exceed a resource. These variables are also called negative slack variables.
Surplus variables like slack variables carry a zero coefficients in the objective
function.(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 5 / 107
6. Conditions for Application of Simplex Method
1 The right hand side of each of the constraint bi should be
non-negative. If LPP has a negative constraint, we should
convert it to positive by multiplying both sides by -1.
2 Each of the decision variables of the problem should be
non-negative. If one of the choice variables is not feasible we
can’t apply the Simplex method. Therefore feasibility is the
necessity condition for application of the Simplex method.
3 The inequality constraint of resources or any other activities
must be converted in to equality equations by adding slack
variables (≤ type inequality) or by subtracting surplus variables
(≥ type inequality) to the left of the inequality.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 6 / 107
7. 2. Linear Programs in Standard Form
The standard form have the following characteristics.
(i) All the constraints should be expressed as equations by adding
slack or surplus and/ or artificial variables.
(ii) The right hand side of each constraint should be made of
non-negative, if it is not, this should be done by multiplying
both sides of the resulting constraint by -1.
Minimization and Maximization Problems
Max(Z) = −Min(−Z).
After the optimization of the new problem is completed, the
objective value of the old problem is -1 times the optimal
objective value of the new problem.
Let Z = −Z. After the Z value is found, replace Z = −Z .
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 7 / 107
8. Consider an LP model,
Max/Min Z = cT
X
S.t : Ax(≤, ≥)b
xi ≥ 0
The standard form of the Linear programming problem is expressed as
Optimize(Max or Min)Z = CT X + 0S
Subject to AX ± S = b.
and X, S ≥ 0.
where CT = (c1, c2, ..., cn) is the row vector, X = (x1, x2, ..., xn)T
b = (b1, b2, ..., bm)T and S = (s1, s2, ..., sm)T are column vectors.
and A =
a11 a12 . . . a1n
a21 a22 . . . a2n
. . .
. . . . . .
. . .
am1 am2 . . . amn
is the m × n matrix of coefficients of variables of x1, x2, ..., xn in the
constraints.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 8 / 107
9. Example 1
Transform the following LPP into standard form of LPP.
Maximize Z = x1 − x2 + x3
Subject to x1 + x2 − 3x3 ≥ 4
2x1 − 4x2 + x3 ≥ −5
x1 + 2x2 − 2x3 ≤ 3
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
Solution:
Since the second constraint have a negative value on the right we can
multiply both side by (-1) so,
Maximize Z = x1 − x2 + x3 + 0x4 + 0x5 + 0x6
subject to x1 + x2 − 3x3 − x4 = 4
−2x1 + 4x2 − x3 + x5 = 5
x1 + 2x2 − 2x3 + x6 = 3
x1, x2, x3, x4, x5, x6 ≥ 0
Therefore, the above problem is standard LPP.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 9 / 107
10. Variables Unrestricted In Sign
The difference of two non-negative variables is a variable unrestricted
in sign.
Let x1 and x2 be two non-negative variables. The difference of these
two variables is a variable x3 i.e x3 = x1 − x2 which is unrestricted in
sign.
If x1 > x2, then x3 > 0.
If x1 < x2, then x3 < 0.
If x1 = x2 , then x3 = 0.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 10 / 107
11. Example 2.
Write down the following LPP, where the variables are non-negative in
standard LPP form
Maximize Z = 2x1 + 3x2 − x3
Subject to 4x1 + x2 + x3 ≥ 4
7x1 + 4x2 − x3 ≤ 25,
x1, x3 ≥ 0, x2 unrestricted sign.
Solution: Since x2 is unrestricted sign, we will convert x2 by x2 = x2 − x2 ,
where x2, x2 ≥ 0.
Maximize Z = 2x1 + 3x2 − 3x2 − x3 + 0x4 + 0x5
Subject to 4x1 + x2 − x2 + x3 − x4 = 4
7x1 + 4x2 − 4x2 − x3 + x5 = 25,
x1, x3, x4, x5, x2, x2 ≥ 0,
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 11 / 107
12. 3. Basic Feasible Solutions
Consider the system Ax = b and x ≥ 0, where A is an m × n matrix
and b is any m vector.
Suppose that the rank(A, b) = rank(A) = m. After possibly
rearrangement of the columns of A, let A = [B, N] where B is an
m × m invertible matrix and N is m × (n − m) matrix.
The point X =
XB
XN
, where XB = B−1b and XN = 0 is called a
basic solution of the system.
If XB ≥ 0, then X is called a basic feasible solution of the system.
Here B is called the basic matrix (or simply the basis ) and N is
called the non basic matrix.
The components of XB are called basic variables, and the
components XN are called non basic variables.
If XB > 0, then X is called a non degenerate basic feasible
solution and if at least one component of XB is zero, then X is called
a degenerate basic feasible solution.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 12 / 107
13. Example 1
Consider the polyhedral set defined by the following inequalities. Find
the basic solution.
(a)
x1 + x2 ≤ 6
x2 ≤ 3
x1, x2 ≥ 0
(b)
x + y ≤ 3
2x − y ≤ 4
x, y ≥ 0
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 13 / 107
14. Solution
(a) By introducing the slack variables x3 and x4 , the problem is put in the
following standard format:
x1 + x2 + x3 = 6
x2 + x4 = 3
x1, x2, x3, x4 ≥ 0
Note that, the constraint matrix
A = [a1 a2 a3 a4] =
1 1 1 0
0 1 0 1
From forgoing definition, basic feasible solutions corresponding to finding
a 2 × 2 basic matrix B with non-negative B−1b. The following are possible
ways of extracting B out of A.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 14 / 107
17. These basic feasible solutions, projected in R2 that is in the (x1, x2) space
give rise to the following four points.
3
3
,
6
0
,
0
3
,
0
0
these points are precisely the extreme points of the feasible region.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 17 / 107
18. In this example, the possible number of basic feasible solutions is bounded
by the number of ways of extracting two columns out of four columns to
form the basis. therefore, the number of basic feasible solutions is less
than or equal to
4
2
= 4!
2!(4−2)! = 6.
Out of these six possibilities, one point violates the non-negativity of B−1b.
furthermore, a1 and a3 could not have been used to form a basic since
a1 = a3 =
1
0
are linearly dependent, and hence the matrix
1 1
0 0
does not qualify as a basis. This leaves four basic feasible solutions.
(b)
x + y + z = 3
2x − y + w = 4
x, y, z, w ≥ 0
1 1 1 0
2 −1 0 1
x
y
z
w
=
3
4
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 18 / 107
19. Here, 4
2 = 6 possible square matrix obtained from the given system.
If we select column 3 and 4 and assign zero value to the variable
associated with column 1and 2, z = 3, w = 4. So, (0, 0, 3, 4) is basic
solution and x, y is non-basic variable and z and w is basic variable.
B =
1 0
0 1
. XB =
z
w
=
3
4
and XN =
x
y
=
0
0
Again take column 1 and 2 and assign zero value to the variable associated
with column 3 and 4, B =
1 1
2 −1
and XB = B−1b and follow the same
fashion as above.
In general, the number of basic feasible solutions is less than or
equal to
n
m
= n!
m!(n−m)!
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 19 / 107
20. Example 2: Degenerate basic feasible solutions
Consider the following system of inequalities
x1 + x2 ≤ 6
x2 ≤ 3
x1 + 2x2 ≤ 9
x1, x2 ≥ 0
Solution: After adding the slack variables x3, x4 and x5, we get
x1 + x2 + x3 = 6
x2 + x4 = 3
x1 + 2x2 + x5 = 9
x1, x2, x3, x4, x5 ≥ 0
A = [a1, a2, a3, a4, a5] =
1 1 1 0 0
0 1 0 1 0
1 2 0 0 1
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 20 / 107
22. Note that these basic feasible solution give rise to the same point obtained
by B = [a1, a2, a3]. It can be also checked the other basic feasible solution
with basis B = [a1, a2, a5] is give by
XB =
x1
x2
x5
=
3
3
0
, XN =
x3
x4
=
0
0
Note that all the three foregoing bases represent the single extreme point
or basic feasible solution (x1, x2, x3, x4, x5) = (3, 3, 0, 0, 0). This basic
feasible solution is degenerate since each associated basis involves a basic
variable at level zero.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 22 / 107
23. 4. Fundamental Theorem of Linear Programming
Theorem: For an arbitrary linear programming in standard form of
the following are true.
i . If there is no optimal solution, then the problem is either
infeasible or unbounded
ii . If a feasible solution exists, then a basic feasible solution exists
iii . If an optimal solution exists, then a basic optimal solution
exists
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 23 / 107
24. Key to the Simplex Method
The key to the simplex method lies in recognizing the optimality of a given
extreme point solution based on local considerations without having to
(globally) enumerate all extreme points or basic feasible solutions.
Consider the following linear programming problem :
LP: Minimize CT X
Subject to AX = b
X ≥ 0
where A is an m × n matrix with rank m, suppose that we have a basic
feasible solution
B−1b
0
whose objective value Z0 is given by
Z0 = CT B−1b
0
= (CT
B , CT
N )
B−1b
0
= CT
B B−1
b (3.1)
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 24 / 107
25. Now, let XB and XN denote the set of basic and non basic variables for
the given basis and X =
XB
XN
be an arbitrary feasible solution. Then
feasibility requires that XB ≥ 0, XN ≥ 0 and that b = AX = BXB + NXN.
Multiplying by B−1 and rearranging the terms, we get
B−1
(b = BXB + NXN)
B−1
b = XB + B−1
NXN
XB = B−1
b − B−1
NXN
XB = B−1
b −
j∈J
B−1
aj xj
XB = B−1
b −
j∈J
yj xj
XB = b −
j∈J
yj xj (3.2)
Where b = B−1
b, yj = B−1
aj and J is the current set of the indices of the non
basic variables.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 25 / 107
26. Letting Z denote the objective function value. we get
Z = CT X
= (CT
B , CT
N )
XB
XN
= CT
B XB + CT
N XN
= CT
B (B−1b − j∈J B−1aj xj ) + j∈J cj xj
= CT
B (B−1b) − j∈J CT
B B−1aj xj + j∈J cj xj
= Z0 − j∈J zj xj + j∈J cj xj
= Z0 − j∈J(zj − cj )xj (3.3)
Where zj = CT
B B−1aj for each non basic variable
Using the foregoing transformation, the linear programming problem may
be written as
Minimize Z = Z0 −
j∈J
(zj − cj )xj
subject to
j∈J
yj xj + XB = B−1
b
xj ≥ 0, j ∈ J, and XB ≥ 0 (3.4)
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 26 / 107
27. 5. Algebra of the Simplex Method
Consider the representation of the linear programming LP in the non
basic variable space written in equality form as in equation (3.4).
If (zj − cj ) ≤ 0 for j ∈ J, then xj = 0 for j ∈ J and XB = B−1b is
optimal for the LP. otherwise, while holding (p-1) non basic variable
fixed at zero, the simplex method considers increasing the remaining
variables, say xk.
Naturally we would like zk − ck to be positive and perhaps the most
positive of all the zj − cj , j ∈ J.
Now fixing xj = 0 for j ∈ J − {k} we obtain from equation (3.4)that
z = z0 − (zk − ck)xk (3.5)
and
xB1
xB2
.
.
.
xBr
.
.
.
xBm
=
b1
b2
.
.
.
br
.
.
.
bm
−
y1k
y2k
.
.
.
yrk
.
.
.
ymk
xk (3.6)
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 27 / 107
28. If yik ≤ 0, then xBi
increases as xk increase and so xBi
continues to be
non-negative.
If yik ≥ 0 then xBi
will decrease as xk increases.
In order to satisfy non-negativity, xk is increase until the first point at
which some basic variable xBr drops to zero.
Examining equation (3.6), it is then clear that the first basic variable
dropping to zero corresponds to the minimum of bi
yik
for positive yik.
more precisely, we can increase xk until
xk = br
yrk
= minimum1≤i≤m{ bi
yik
: yik > 0} (3.7)
In the absence of degeneracy, br > 0 and hence xk = br
yrk
> 0, from
equation (3.5) and the fact that zk − ck > 0, it then follows z < z0
and the objective function strictly improves.
As xk increases from level 0 to br
yrk
, a new feasible solution obtained.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 28 / 107
29. Substituting xk = br
yrk
in equation (3.6) gives the following point:
xBi
= bi − yik
yrk
br , i = 1, 2, ..., m
xk = br
yrk
(3.8)
and all other xj - variables are zero.
From equation (3.8), xBr = 0 and hence, at most m variables are
positive.
The corresponding columns in A are
aB1 , aB2 , ..., aBr−1 , ak, aBr+1 , ..., aBm .
Note that these columns are linearly independent. since yrk = 0.
(Recall that if aB1 , aB1 , ..., aBm are linearly independent, and if ak
replaces aBr , then the new columns are linearly independent if and
only if yrk
= 0.
Therefore, the point given by equation (3.8) is a basic feasible
solution.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 29 / 107
30. Recall that a basic variable xB that first drops to zero is called a
blocking variable because it blocks the further increase of xk. Thus,
xk enters the basis and xB leaves the basis.
To summarize, we have algebraically described an iteration, that is,
the process of transforming from one basis to an adjacent basis.
This is done by increasing the value of a nonbasic variable xk with
positive zk − ck and adjusting the current basic variables.
In the process, the variable xB drops to zero.
The variable xk hence enters the basis and xB leaves the basis.
In the absence of degeneracy the objective function value strictly
decreases, and hence the basic feasible solutions generated are
distinct.
Because there exists only a finite number of basic feasible solutions,
the procedure would terminate in a finite number of steps.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 30 / 107
31. Example 1.
Minimize x1 + x2
subject to x1 + 2x2 ≤ 4
x2 ≤ 1
x1, x2 ≥ 0
Solution: Introducing the slack variables x3 and x4 to put the problem in
a standard form.
This leads to the following constraint matrix A :
A = [a1, a2, a3, a4] =
1 2 1 0
0 1 0 1
Consider the basic feasible solution corresponding to B = [a1, a2].
In other words, x1 and x2 are the basic variables, while x3 and x4 are
the nonbasic variables.
The representation of the problem in this nonbasic variable space as
in Equation (3.4) with J= 3, 4 may be obtained as follows.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 31 / 107
33. Hence, the required representation of the problem is
Minimize 3 − x3 + x4
subject to x3 − 2x4 + x1 = 2
x4 + x2 = 1
x1, x2, x3, x4 ≥ 0
Since z3 − c3 > 0, then the objective function improves by increasing
x3, the modified solution is given by
XB = B−1b − B−1a3x3
x1
x2
=
2
1
−
1
0
x3
The maximum value of x3 is 2 (any larger value of x3 will force x1 to
be negative).
Therefore the new basic feasible solution is (x1, x2, x3, x4) = (0, 1, 2, 0)
Here, x3 enters the basis and x1 leaves the basis.
Note that the new point has an objective value equal to 1, which is
an improvement over the previous objective value of 3. the
improvement is precisely (z3 − c3)x3 = 2.
Remark: CB is the coefficient of the basic variable in the objective of the
LP.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 33 / 107
34. Termination with an optimal solution
Consider the following problem, where A is an m × n matrix with rank m.
Minimize CT X
Subject to AX = b
X ≥ 0
Suppose that X∗ is a basic feasible solution with basis B; that is
X∗ =
B−1b
0
Let Z∗ denote the objective value at X∗, that is Z∗ = CT
B B−1b. Suppose
further that Zj − Cj ≤ 0, for all non basic variables, and hence there are no
non basic variables that are eligible to enter the basis. let x be any feasible
solution with objective function value z, then from equation (3.3) we have
Z∗ − Z = j∈J(Zj − Cj )xj (3.9)
Because Zj − Cj ≤ 0 and Xj ≥ 0 for all variables, then Z∗ ≤ Z, and so X∗
is an optimal basic feasible solution, since if (Zj − Cj ) ≤ 0 for all j ∈ J,
then the current basic feasible solution is optimal.
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35. Example 2.
Minimize 2x1 − x2
subject to −x1 + x2 ≤ 2
2x1 + x2 ≤ 6
x1, x2 ≥ 0
Introducing the slack variables x3 and x4. This leads to the following
constraints
−x1 + x2 + x3 = 2
2x1 + x2 + x4 = 6
x1, x2, x3, x4 ≥ 0
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37. Hence, the objective improves by holding x3 non- basic and
introducing x4 in the basis.
Then x3 is kept at zero level, x4 is increased and x1 and x2 are
modified according to equation (3.10). we see that x4 can be
increased to 4, at which x1 drops to zero.
Any further increase of x4 results in violating the non-negativity of x1
and so x1 is the blocking variable.
With x4 = 4 and x3 = 0, the modified value of x1 and x2 are 0 and 2
respectively.
The new basic feasible solution is
(x1, x2, x3, x4) = (0, 2, 0, 4)
Note that a4 replaces a1, that is x1 drop from the basic and x4 enters
the basis.
The new set of basic and non basic variables and their values are
given as :
XB =
xB4
xB2
=
x4
x1
=
4
2
, XN =
x3
x1
=
0
0
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 37 / 107
38. Moving from old to the new basic feasible solution is illustrated in
figure given below, note that as x4 increases by one unit x1 decrease
by 1/3 unit and x2 decrease by 1/3 unit, that is we move in the
direction (-1/3, -1/3) in the (x1, x2) space.
This continues until we are blocked by the non negativity restriction
x1 ≥ 0.
At this point x1 drops to zero and leaves the basis.
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39. The simplex Algorithm
The key solution concepts
The simplex method focuses on corner point feasible(CPF) solutions.
The simplex method is an iterative algorithm(a systematic solution
procedure that keeps repeating a fixed series of steps, called an
iteration, until a desired result has been obtained ) with the following
structure.
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40. Whenever possible, the initialization of the simplex method chooses
the origin point(all decision variables equal to zero) to be the initial
CPF solution
Given a CPF solution, it is much quicker computationally to gather
information about its adjacent CPF solutions than about other CPF
solutions. Therefore, each time the simplex method performs an
iteration to move from the current CPF solution to a better one, it
always chooses a CPF solution that is adjacent to the current one.
After the current CPF solution is identified, the simplex method
examines each of the edges of the feasible region that emanates from
this CPF solution. Each of these edges leads to an adjacent solution
at the other end, but the simplex method doesn’t even take the time
to solve for adjacent CPF solutions, instead, it simply identifies the
rate of improvement in Z that would be obtained by moving along the
edge. and then chooses to move along the one with largest positive
rate of improvement.
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41. A positive rate of improvement in Z implies that the adjacent CPF
solution is better than the current one, whereas a negative rate of
improvement in Z implies that the adjacent CPF solution is worse.
Therefore, the optimality test consists simply checking whether any of
the edges give a positive rate of improvement in Z. if none do, then
the current CPF solution is optimal.
The Simplex Method in Tabular Form
Steps
Initialization:
1 Convert (Transform) all the constraints to equality by introducing
slack, surplus, and artificial variables as follows
Constraint type Variable to be added
≤ +slack(s)
≥ -surplus(s)+artificial(A)
= + artificial(A)
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42. The artificial variable refers to the kind of variable which is introduced in
the linear program model to obtain the initial basic feasible solution. It is
utilized for the equality constraints and for the greater than or equal
inequality constraints.
1 Construct the initial simplex tableau
Cj c1 . . . cn 0 . . . 0 0 . . . 0
CB BV X1 . . . Xn S1 . . . Sn A1 . . . An RHS
0 S b1
. . .
. . .
. . .
0 A bm
Z Z val
Zj − Cj
2 Test for optimality :
Case 1: In maximization problem the current bfs is optimal if every
element in the last row of the simplex tableau is nonnegative
Case 2 : In minimization problem the current bfs is optimal if every
element in the last row of the simplex tableau is non positive
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 42 / 107
43. Iteration :
Step 1 : Determine the entering basic variable by selecting the
variable (automatically a non basic variable ) with the most negative
value (in case of maximization)or the most positive(in case of
minimization) in the last row (Z-row). Put a box around the column
below this variable, and call it the ” pivot column”.
Step 2 : Determine the leaving basic variable by applying the
minimum ratio test as following:
1 Pick out each coefficients in the pivot column that is strictly positive
2 Divide each of these coefficients into the right hand side entry for the
same row
3 Identify the row that has the smallest of these ratios
4 The basic variable for that row is the leaving variable, so replace that
variable by the entering variable in the basic variable column of the next
simplex tableau. Put a box around this row and call it the ”pivot row”.
Step 3: Solve for the new basic feasible solution by using elementary
row operation(multiply or divide a row by a nonzero constant, add or
subtract a multiple of one row another row)to construct a new
simplex tableau, and then return the optimality test.
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44. Example 1.
Solve the following problem using the simplex method
Maximize z = 3x1 + 5x2
Subject to x1 ≤ 4
2x2 ≤ 12
3x1 + 2x2 ≤ 18
x1, x2 ≥ 0
Solution: 1. Write the standard form of the above linear programming
problem.
Maximize z = 3x1 + 5x2
Subject to x1 + s1 = 4
2x2 + s2 = 12
3x1 + 2x2 + s3 = 18
x1, x2, s1, s2, s3 ≥ 0
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45. 2. Initial tableau
C 3 5 0 0 0
CB XB x1 x2 s1 s2 s3 b(RHS)
0 s1 1 0 1 0 0 4
0 s2 0 2 0 1 0 12
0 s3 3 2 0 0 1 18
Zj − Cj -3 -5 0 0 0
The basic feasible solution at the initial tableau is (0,0,4,12,18) where:
x1 = 0, x2 = 0, s1 = 4, s2 = 12, s3 = 18 and z = 0. Where s1, s2, s3 are
basic variable and x1 and x2 are non-basic variables. The solution at the
initial tableau is associated to the origin point at which all the decision
variables are zero.
Optimality test: By investigating the last row of the initial tableau, we
find that there are some negative numbers. Therefore, the current solution
is not optimal.
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46. Iteration
Step 1: Determine the entering variable by selecting the variable with the
most negative in the last row. From the initial tableau, in the last row
((Zj − Cj ) row), the coefficient of X1 is -3 and the coefficient of X2 is -5;
therefore, the most negative is -5. consequently, X2 is the entering
variable. X2 is surrounded by a box and it is called the pivot column.
Step 2: Determining the leaving variable by using the minimum ratio test
as follows:
C 3 5 0 0 0
CB XB x1 x2(Entering) s1 s2 s3 b(RHS)
0 s1 1 0 1 0 0 4
0 s2(Leaving) 0 2 0 1 0 12
0 s3 3 2 0 0 1 18
Zj − Cj -3 -5 0 0 0
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47. Step 3: solving for the new BF solution by using the eliminatory row
operations as follows:
C 3 5 0 0 0
CB XB x1 x2 s1 s2 s3 b(RHS)
0 s1 1 0 1 0 0 4
5 x2 0 1 0 1/2 0 6
0 s3 3 0 0 -1 1 6
Zj − Cj -3 0 0 5/2 0
This solution is not optimal, since there is a negative numbers in the last
row. Apply the same rules we will obtain this solution:
C 3 5 0 0 0
CB XB x1 x2 s1 s2 s3 b(RHS)
0 s1 0 0 1 1/3 -1/3 2
5 x2 0 1 0 1/2 0 6
3 x1 1 0 0 -1/3 1/3 2
Zj − Cj 0 0 0 3/2 1 36
This solution is optimal; since there is no negative solution in the last row:
basic variables are x1 = 2, x2 = 6 and s1 = 2; the nonbasic variables are
s2 = s3 = 0, and the optimal value Z = 36.
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48. Example 2.
Minimize x1 + x2 − 4x3
subject to x1 + x2 + 2x3 ≤ 9
x1 + x2 − x3 ≤ 2
−x1 + x2 + x3 ≤ 4
x1, x2, x3 ≥ 0
Solution : Introducing the non- negative slack variables s1, s2 and s3. The
problem becomes the following :
Minimize x1 + x2 − 4x3
subject to x1 + x2 + 2x3 + s1 = 9
x1 + x2 − x3 + s2 = 2
−x1 + x2 + x3 + s3 = 4
x1, x2, x3, s1, s2, s3 ≥ 0
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49. Since b ≥ 0, then we can choose our initial basis as B = [a4, a5, a6] = I3,
and we indeed have B−1b = b ≥ 0. This gives the following initial tableau:
Iteration 1
C 1 1 -4 0 0 0 0
CB XB x1 x2 x3 s1 s2 s3 b(RHS)
0 s1 1 1 2 1 0 0 9
0 s2 1 1 -1 0 1 0 2
0 s3 -1 1 1 0 0 1 4
Zj − Cj -1 -1 4 0 0 0
The initial basic feasible solution in the above table is
x1 = 0, x2 = 0, x3 = 0, but it is not optimal, since there is positive element
the last row of the simplex table (Zj − Cj ). Identify the entering and
leaving variable
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50. Iteration 2
C 1 1 -4 0 0 0 0
CB XB x1 x2 x3 s1 s2 s3 b(RHS)
0 s1 1 1 2 1 0 0 9
0 s2 1 1 -1 0 1 0 2
0 s3 -1 1 1 0 0 1 4
Zj − Cj -1 -1 4 0 0 0
From the above simplex table s3 is the leaving variable and x3 is the
entering variable.
C 1 1 -4 0 0 0 0
CB XB x1 x2 x3 s1 s2 s3 b(RHS)
0 s1 3 -1 0 1 0 -2 1
0 s2 0 2 0 0 1 1 6
-4 x3 -1 1 1 0 0 1 4
Zj − Cj 3 -5 0 0 0 -4
The initial basic feasible solution in the above table is x1 = 0, x2 = 0, x3 = 4, but
it is not optimal, since there is positive element in the last row of the simplex
table (Zj − Cj ).(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 50 / 107
51. Iteration 3
C 1 1 -4 0 0 0 0
CB XB x1 x2 x3 s1 s2 s3 b(RHS)
0 s1 3 -1 0 1 0 -2 1
0 s2 0 2 0 0 1 1 6
-4 x3 -1 1 1 0 0 1 4
Zj − Cj 3 -5 0 0 0 -4
From the above simplex table s1 is the leaving variable and x1 is the
entering variable.
C 1 1 -4 0 0 0
CB XB x1 x2 x3 s1 s2 s3 b(RHS)
1 x1 1 -1/3 0 1/3 0 -2/3 1/3
0 s2 0 2 0 0 1 1 6
-4 x3 0 2/3 1 1/3 0 1/3 13/3
Zj − Cj 0 -4 0 -1 0 -2
This is an optimal tableau since Zj − Cj ≤ 0 in the last row. The optimal solution
is given by x1 = 1/3, x2 = 0, x3 = 13/3, with z = -17.
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52. Degeneracy and Finiteness of Simplex Algorithm
Degeneracy: 1. only one component of basic feasible solution
corresponding to the basic variable is zero.
2. at least two component of basic feasible solution corresponding to the
basic variables are zero.
Example 1:
Maximize 5x1 + 3x2
Subject to x1 + x2 ≤ 2
5x1 + 2x2 ≤ 10
x1 + 8x2 ≤ 12
x1, x2 ≥ 0
Solution :The standard linear programming problem for the above problem is
Maximize 5x1 + 3x2
Subject to x1 + x2 + s1 = 2
5x1 + 2x2 + s2 = 10
x1 + 8x2 + s3 = 12
x1, x2, s1, s2, s3 ≥ 0
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53. Now we can construct the simplex tableau
C 5 3 0 0 0
CB XB x1 x2 s1 s2 s3 b(RHS)
0 s1 1 1 1 0 0 2 ←
0 s2 5 2 0 1 0 10
0 s3 3 8 0 0 1 12
Zj − Cj ↑-5 -3 0 0 0
The initial basic feasible solution of the LP problem is x1 = 0, x2 = 0, but
it can not be an optimal solution, since there is a negative element in the
last row of the simplex table. from the above table we can identify the
entering and leaving variables, thus x1 is the entering variable and s1 is the
leaving variable.
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54. By applying elementary row operation we can get the following simplex
tableau
C 5 3 0 0 0
CB XB x1 x2 s1 s2 s3 b(RHS)
5 x1 1 1 1 0 0 2
0 s2 0 -3 -5 1 0 0
0 s3 0 5 -3 0 1 6
Zj − Cj 0 2 5 0 0
Thus the optimal solution is (x1, x2, s1, s2, s3) = (2, 0, 0, 0, 6) and the
optimal value is 10, since there exist a zero value of basic variable ,it is
degenerate case of finiteness of simplex algorithm.
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55. Infinite number of solutions
The alternative optimal solution can be obtained by considering the zj − cj
row of the simplex table. That is, zj − cj = 0 for some non-basic variable
columns in the optimal simplex table.
Example
Maximize 10x1 + 20x2
subject to x1 ≤ 10
x2 ≤ 6
2x1 + 4x2 ≤ 36
x1, x2 ≥ 0
Solution: The standard linear programming problem of the above problem
is
Maximize 10x1 + 20x2
subject to x1 + s1 = 10
x2 + s2 = 6
2x1 + 4x2 + s3 = 36
x1, x2, s1, s2, s3 ≥ 0
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 55 / 107
57. C 10 20 0 0 0
CB XB x1 x2 s1 s2 s3 b(RHS)
0 s1 0 0 1 2 -1/2 4
20 x2 0 1 0 1 0 6
10 x1 1 0 0 -2 1/2 6
Zj − Cj 0 0 0 0 5 180
We have basic feasible solution (x1, x2) = (6, 6) with optimal value z =
180. We apply a new simplex step with Zj − Cj = 0 for a non- basic
variable.
C 10 20 0 0 0
CB XB x1 x2 s1 s2 s3 b(RHS)
0 s1 0 0 1 2 -1/2 4
20 x2 0 1 0 1 0 6
10 x1 1 0 0 -2 1/2 6
Zj − Cj 0 0 0 0 5 180
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 57 / 107
58. C 10 20 0 0 0
CB XB x1 x2 s1 s2 s3 b(RHS)
0 s2 0 0 1/2 1 -1/4 2
20 x2 0 1 -1/2 0 1/4 4
10 x1 1 0 1 0 0 10
Zj − Cj 0 0 0 0 5 180
Now we have another basic solution (x1, x2) = (10, 4), but the optimal
values remains 180.
In our case, since (x1, x2) = (6, 6) and (x1, x2) = (10, 4) are solutions all
points of the segment (x1, x2)(x1, x2) = λ(6, 6) + (1 − λ)(10, 4), λ ∈ [0, 1]
are solutions.
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59. Finding a Starting Basic Feasible Solution
In certain cases, it is difficult to obtain an initial basic feasible solution of
the given LP problem. Such cases arise
1 when the constraints are of the ≤ type,
n
j=1
aij xj ≤ bi , xj ≥ 0
and value of few right-hand side constants is negative [i.e. bi < 0].
After adding the non-negative slack variable si (i = 1, 2, . . ., m), the
initial solution so obtained will be si = −bi for a particular resource, i.
This solution is not feasible because it does not satisfy non-negativity
conditions of slack variables (i.e. si ≥ 0).
2 when the constraints are of the ≥ type,
n
j=1
aij xj ≥ bi , xj ≥ 0
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60. After adding surplus (negative slack) variable si , the initial solution so
obtained will be −si = bi or si = −bi
n
j=1
aij xj − si = bi , xj ≥ 0, si ≥ 0
This solution is not feasible because it does not satisfy non-negativity
conditions of surplus variables (i.e. si ≥ 0).
In such a case, artificial variables, Ai (i = 1, 2, . . ., m) are added to
get an initial basic feasible solution.
The resulting system of equations then becomes:
n
j=1
aij xj − si + Ai = bi
xj , si , Ai ≥ 0, i = 1, 2, ..., m
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61. These are m simultaneous equations with (n + m + m) variables (n
decision variables, m artificial variables and m surplus variables).
An initial basic feasible solution of LP problem with such constraints
can be obtained by equating (n + 2m - m) = (n + m) variables equal
to zero.
Thus the new solution to the given LP problem is: Ai = bi (i = 1, 2,
. . . , m), which is not the solution to the original system of
equations because the two systems of equations are not equivalent.
Thus, to get back to the original problem, artificial variables must be
removed from the optimal solution.
There are two methods for removing artificial variables from the solution.
Two-Phase Method
Big-M Method or Method of Penalties
Remark Before the optimal solution is reached, all artificial variables must
be dropped out from the solution mix. This is done by assigning
appropriate coefficients to these variables in the objective function. These
variables are added to those constraints with equality (=) and greater than
or equal to (≥) sign.
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62. Two -phase method
In the first phase of this method, the sum of the artificial variables is
minimized subject to the given constraints in order to get a basic
feasible solution of the LP problem.
The second phase minimizes the original objective function starting
with the basic feasible solution obtained at the end of the first phase.
Since the solution of the LP problem is completed in two phases, this
method is called the two-phase method.
Advantages of the method
No assumptions on the original system of constraints are made, i.e.
the system may be redundant, inconsistent or not solvable in
non-negative numbers.
It is easy to obtain an initial basic feasible solution for Phase I.
The basic feasible solution (if it exists) obtained at the end of phase I
is used as initial solution for Phase II
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63. Steps of the Algorithm: Phase I
Step 1 If all the constraints in the given LP problem are less than or equal
to (≤) type, then Phase II can be directly used to solve the problem.
Otherwise, the necessary number of surplus and artificial variables are
added to convert constraints into equality constraints.
Step 2: Assign zero coefficient to each of the decision variables (xj ) and
to the surplus variables; and assign - 1 coefficient to each of the artificial
variables. This yields the following auxiliary LP problem.
Maximize Z∗ = m
i=1(−1)Ai
subject to the constraints
m
i=1
aij xj + Ai = bi , i = 1, 2, ..., m
and xj , Aj ≥ 0
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64. Step 3: Apply the simplex algorithm to solve this auxiliary LP problem.
The following three cases may arise at optimality.
(i) Max Z∗ = 0 and at least one artificial variable is present in the basis
with positive value. This means that no feasible solution exists for
the original LP problem.
(ii) Max Z∗ = 0 and no artificial variable is present in the basis. This
means that only decision variables(xj ’s) are present in the basis and
hence proceed to Phase II to obtain an optimal basic feasible solution
on the original LP problem.
(iii) Max Z∗ = 0 and at least one artificial variable is present in the basis
at zero value. This means that a feasible solution to the auxiliary LP
problem is also a feasible solution to the original LP problem. In
order to arrive at the basic feasible solution, proceed directly to
Phase II or else eliminate the artificial basic variable and then proceed
to Phase II.
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65. Remark Once an artificial variable has left the basis, it has served
its purpose and can, therefore, be removed from the simplex table.
An artificial variable is never considered for re-entry into the basis.
Phase II
Assign actual coefficients to the variables in the objective function
and zero coefficient to the artificial variables which appear at zero
value in the basis at the end of Phase I.
The last simplex table of Phase I can be used as the initial simplex
table for Phase II.
Then apply the usual simplex algorithm to the modified simplex table
in order to get the optimal solution to the original problem.
Artificial variables that do not appear in the basis may be removed.
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66. Remark
1 All artificial vectors are not present in the bases which
indicates that all artificial vectors at zero level at the optimal
stage, thus the solution obtained is a basic feasible solution.
2 Some artificial vectors are present in the basis and some
artificial vectors are at positive level at the optimal stage. In
that case there are no feasible solution to the problem.
3 All artificial are at zero level but at least one artificial vector is
present in the basis at optimal stage. Here the solution under
test is an optimal solution. Here the converted equations are
consistent but some of the constraints may be redundant. By
redundancy means the system has more than enough
constraints.
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67. Example 1.
Solve the following Linear programming problems by using two phase
method
Maximize Z = 3x1 − x2
subject to 2x1 + x2 ≥ 2
x1 + 3x2 ≤ 2
x2 ≤ 4
x1, x2 ≥ 0
Solution:Phase I Convert the above problem to Standard form of LPP
Maximize Z = 3x1 − x2/Z∗ = −A1
subject to 2x1 + x2 − s1 + A1 = 2
x1 + 3x2 + s2 = 2
x2 + s3 = 4
x1, x2, s1, s2, s3, A1 ≥ 0
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68. Now, we will Max Z∗ = −A1 till zero in order to remove the variable A1,
where A1 is an artificial variable.
Maximize Z∗ = 0x1 − 0x2 + 0s1 + 0s2 + 0s3 − 1A1
subject to 2x1 + x2 − s1 + A1 = 2
x1 + 3x2 + s2 = 2
x2 + s3 = 4
x1, x2, s1, s2, s3, A1 ≥ 0
Cj 0 0 0 0 0 -1
CB XB x1 x2 s1 s2 s3 A1 b(RHS) Min ratio bi /yik
-1 A1 2 1 -1 0 0 1 2 1 ←
0 s2 1 3 0 1 0 0 2 2
0 s3 0 1 0 0 1 0 4 -
Zj − Cj ↑-2 -1 1 0 0 0 Z∗ = −2
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69. Cj 0 0 0 0 0 -1
CB XB x1 x2 s1 s2 s3 A1 b(RHS)
0 x1 1 1/2 -1/2 0 0 × 1
0 s2 0 5/2 1/2 1 0 × 1
0 s3 0 1 0 0 1 × 4
Zj − Cj 0 0 0 0 0 × Z∗ = 0
Since all Zj − Cj = 0 and no artificial vector appears in the the basis, we
proceed to phase II.
Phase II
Cj 3 -1 0 0 0
CB XB x1 x2 s1 s2 s3 b(RHS)
3 x1 1 1/2 -1/2 0 0 1
0 s2 0 5/2 1/2 1 0 1 ←
0 s3 0 1 0 0 1 4
Zj − Cj 0 5/2 ↑ -3/2 0 0 Z = 3
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 69 / 107
70. Cj 3 -1 0 0 0
CB XB x1 x2 s1 s2 s3 b(RHS)
3 x1 1 3 0 1 0 2
0 s1 0 5 1 2 0 2
0 s3 0 1 0 0 1 4
Zj − Cj 0 10 0 3 0 Z = 6
Since all Zj − Cj ≥ 0, optimal basic feasible solution is obtained, therefore
the solution is Max Z = 6, x1 = 2, x2 = 0.
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71. Example 2.
Maximize Z = 5x1 + 8x2
subject to 3x1 + 2x2 ≥ 3
x1 + 4x2 ≥ 4
x1 + x2 ≤ 5
x1, x2 ≥ 0
Solution: The standard linear programming problem
Maximize Z = 5x1 + 8x2
subject to 3x1 + 2x2 − s1 + A1 = 3
x1 + 4x2 − s2 + A2 = 4
x1 + x2 + s3 = 5
x1, x2, s1, s2, s3, A1, A2 ≥ 0
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73. Cj 0 0 0 0 0 -1 -1
CB XB x1 x2 s1 s2 s3 A1 A2 b(RHS) min rat
-1 A1 5/2 0 -1 1/2 0 1 × 1 2/
0 x2 1/4 1 0 -1/4 0 0 × 1
0 s3 3/4 0 0 1/4 1 0 × 4 16
Zj − Cj ↑ −5/2 0 1 -1/2 0 0 × Z∗ = −1
Cj 0 0 0 0 0 -1 -1
CB XB x1 x2 s1 s2 s3 A1 A2 b(RHS) min ratio
0 x1 1 0 -2/5 1/5 0 × × 2/5
0 x2 0 1 1/10 -3/10 0 × × 9/10
0 s3 0 0 3/10 1/10 1 × × 37/10
Zj − Cj 0 0 0 0 0 × × Z∗ = 0
Since all Zj − Cj ≥ 0, Maz zx = 0, and no artificial vector appears in the
basis, we proceed to phase II.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 73 / 107
74. Phase II
Cj 5 8 0 0 0
CB XB x1 x2 s1 s2 s3 b(RHS) min ratio bi /yik
5 x1 1 0 -2/5 1/5 0 2/5 2 →
8 x2 0 1 1/10 -3/10 0 9/10 1
0 s3 0 0 3/10 1/10 1 37/10 37
Zj − Cj 0 0 -6/5 ↑ −7/5 0 Z = 46/5
Cj 5 8 0 0 0
CB XB x1 x2 s1 s2 s3 b(RHS) min ratio bi /yik
0 s2 5 0 -2 1 0 2 -
8 x2 3/2 1 -1/2 0 0 3/2 -
0 s3 -1/2 0 1/2 0 1 7/2 7→
Zj − Cj 7 0 ↑ −4 0 0 Z = 12
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 74 / 107
75. Cj 5 8 0 0 0
CB XB x1 x2 s1 s2 s3 b(RHS) min ratio bi /yik
0 s2 3 0 0 1 2 16
8 x2 1 1 0 0 1/2 5
0 s1 -1 0 1 0 2 7
Zj − Cj 3 0 0 0 4 Z = 40
Since all Zj − Cj ≥ 0, optimal basic feasible solution is obtained. Therefore
the solution is Max Z = 40, x1 = 0, x2 = 5.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 75 / 107
76. Big - M method
Big-M method is another method of removing artificial variables from
the basis.
In this method, large undesirable (unacceptable penalty) coefficients
to artificial variables are assigned from the point of view of the
objective function.
If the objective function Z is to be minimized, then a very large
positive price (called penalty) is assigned to each artificial variable.
Similarly, if Z is to be maximized, then a very large negative price
(also called penalty) is assigned to each of these variables.
The penalty is supposed to be designated by - M, for a maximization
problem, and + M, for a minimization problem, where M > 0.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 76 / 107
77. The Big-M method for solving an LPP can be summarized in
the following steps:
Step 1: Express the LP problem in the standard form by adding slack
variables, surplus variables and/or artificial variables. Assign a zero
coefficient to both slack and surplus variables. Then assign a very large
coefficient + M (minimization case) and - M (maximization case) to
artificial variable in the objective function.
Step 2: The initial basic feasible solution is obtained by assigning zero
value to decision variables, x1, x2, ..., etc.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 77 / 107
78. Step 3: Calculate the values of zj − cj in last row of the simplex table and
examine these values.
1 If all zj − cj ≤ 0, then the current basic feasible solution is optimal.
2 If for a column, k, zk − ck is most positive and all entries in this
column are negative, then the problem has an unbounded optimal
solution.
3 If one or more zj − cj > 0 (minimization case), then select the
variable to enter into the basis (solution mix) with the largest positive
zj − cj value (largest per unit increase in the objective function
value). This value also represents the opportunity cost of not having
one unit of the variable in the solution. That is,
zk − ck = Max{zj − cj : zj − cj > 0}
Step 4: Determine the key row and key element in the same manner as
discussed in the simplex algorithm.
Step 5: Continue with the procedure to update solution at each iteration
till optimal solution is obtained.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 78 / 107
79. Remarks At any iteration of the simplex algorithm one of the fol-
lowing cases may arise:
1 If at least one artificial variable is a basic variable (i.e., variable
that is present in the basis) with zero value and the coefficient
it M in each zj − cj ( j = 1, 2, . . ., n) values is non-positive,
then the given LP problem has no solution. That is, the
current basic feasible solution is degenerate.
2 If at least one artificial variable is present in the basis with a
positive value and the coefficients M in each zj − cj ( j = 1, 2,
. . ., n) values is non-positive, then the given LP problem has
no optimum basic feasible solution. In this case, the given LP
problem has a pseudo optimum basic feasible solution.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 79 / 107
80. Example 1.
Solve the following linear programming problem by using Big-M method.
Maximize Z = −2x1 − x2
subject to 3x1 + x2 = 3
4x1 + 3x2 ≥ 6
x1 + 2x2 ≤ 4
x1, x2 ≥ 0
Solution : The standard linear programming problem of the above LP
Maximize Z = −2x1 − x2 + 0s1 + 0s2 − MA1 − MA2
subject to 3x1 + x2 + A1 = 3
4x1 + 3x2 − s1 + A2 = 6
x1 + 2x2 + s2 = 4
x1, x2, s1, s2, A1, A2 ≥ 0
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 80 / 107
88. Cj → 3 -3 2 1 -M -M
CB ↓ XB ↓ x1 x1 x2 x3 A1 A2 b(RHS) min ratio
1 x3 0 0 -5/3 1 × × 14/3
3 x1 1 -1 4/3 0 × × 11/3 -
Zj − Cj → 0 0 1/3 0 × × Z = 47/3
Since all Zj − Cj ≥ 0, optimal basic feasible solution is obtained
x1 = 11/3, x1 = 0
x1 = x1 − x1 = 11/3 − 0 = 11/3
Therefore the solution is Max Z = 47/3, x1 = 11/3, x2 = 0, x3 = 14/3.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 88 / 107
89. Some Complications and Their Resolution
Unrestricted Variables
Let variable xr be unrestricted in sign. We define two new variables say xr
and xr such that
xr = xr − xr ; &xr , xr ≥ 0
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90. Tie for Entering Basic Variable (Key Column)
While solving an LP problem using simplex method two or more
columns of simplex table may have same zj − cj value (positive or
negative depending upon the type of LP problem).
In order to break this tie, the selection for key column (entering
variable) can be made arbitrary.
However, the number of iterations required to arrive at the optimal
solution can be minimized by adopting the following rules.
(i) If there is a tie between two decision variables, then the selection can
be made arbitrarily.
(ii) If there is a tie between a decision variable and a slack (or surplus)
variable, then select the decision variable to enter into basis.
(iii) If there is a tie between two slack (or surplus) variables, then the
selection can be made arbitrarily
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 90 / 107
91. Tie for Leaving Basic Variable (Key Row) Degeneracy
While solving an LP problem a situation may arise where either the
minimum ratio to identify the basic variable to leave the basis is not
unique or value of one or more basic variables in the xB becomes
zero.This causes the problem of degeneracy.
In order to break the tie in the minimum ratios, the selection can be
made arbitrarily.
However, the number of iterations required to arrive at the optimal
solution can be minimized by adopting the following rules.
(i) Divide the coefficients of slack variables in the simplex table where
degeneracy is seen by the corresponding positive numbers of the key
column in the row, starting from left to right.
(ii) Compare the ratios in step (i) from left to right column wise, select the
row that contains the smallest ratio.
Remark When there is a tie between a slack and artificial variable to
leave the basis, preference should be given to the artificial variable for
leaving the basis.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 91 / 107
92. Example
Solve the following LP problem
Maximize Z = 3x1 + 9x2
subject to the constraints
x1 + 4x2 ≤ 8,
x1 + 2x2 ≤ 4
and x1, x2 ≥ 0
Solution Adding slack variables s1 and s2 to the constraints, the problem
can be expressed as
Maximize Z = 3x1 + 9x2 + 0s1 + 0s2
subject to the constraints
x1 + 4x2 + s1 = 8,
x1 + 2x2 + s2 = 4
and x1, x2, s1, a2 ≥ 0
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 92 / 107
93. The initial basic feasible solution is given in Table . Both variables s1 and
s2 are eligible to leave the basis as the minimum ratio is same, i.e. 2, so
there is a tie among the ratio in rows s1 and s2.This causes the problem of
degeneracy. To obtain the key row for resolving degeneracy, apply the
following procedure:
cj 3 9 0 0
CB XB x1 x2 s1 s2 b(RHS) min. ratio
0 s1 1 4 1 0 8 8/4=2
0 s2 1 2 0 1 4 4/2=2
Zj − Cj -3 ↑-9 0 0 0
Dividing the coefficients of slack variables s1 and s2 by the corresponding
elements in the key column as shown below in the table.
x2 column
Row (Key Column) s1 s2
s1 4 1/4=1/4 0/4=0
s2 2 0/2=0 1/2=1/2
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 93 / 107
94. Comparing the ratios of Step (ii) from left to right column wise, the
minimum ratio (i.e., 0/2 = 0) occurs in the s2-row. Thus, variable s2 is
selected to leave the basis. The new solution is shown in Table
cj 3 9 0 0
CB XB x1 x2 s1 s2 b(RHS)
0 s1 -1 0 1 -2 0
9 x2 1/2 1 0 1/2 2
Zj − Cj 3/2 0 0 9/2 18
In Table above , all Zj − Cj ≤ 0. Hence, an optimal solution is arrived at.
The optimal basic feasible solution is: x1 = 0, x2 = 2 and Max Z = 18.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 94 / 107
95. Types of Linear Programming Solutions
While solving any LP problem using simplex method, at the stage of
optimal solution, the following three types of solutions may exist:
Alternative (Multiple) Optimal Solutions
Unbounded Solution
Infeasible Solutions
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 95 / 107
96. Alternative (Multiple) Optimal Solutions
The zj − cj values in the simplex table indicates the contribution in
the objective function value by each unit of a variable chosen to enter
into the basis.
Also, Also, an optimal solution to a maximization(minimization) LP
problem is reached when all zj − cj ≥ 0(zj − cj ≤ 0).
But, if zj − cj = 0 for a non-basic variable column in the optimal
simplex table and such nonbasic variable is chosen to enter into the
basis, then another optimal solution so obtained will show no
improvement in the value of objective function.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 96 / 107
97. Example
Solve the following LP problem.
Maximize Z = 6x1 + 4x2
subject to the constraints
2x1 + 3x2 ≤ 30,
3x1 + 2x2 ≤ 24
,
x1 + x2 ≥ 3
and
x1, x2 ≥ 0
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 97 / 107
98. Solution Adding slack variables s1, s2, surplus variable s3 and artificial
variable A1 in the constraint set,the LP problem becomes
Maximize Z = 6x1 + 4x2 + 0s1 + 0s2 + 0s3 − MA1
subject to the constraints
2x1 + 3x2 + s1 = 30,
3x1 + 2x2 + s2 = 24
,
x1 + x2 − s3 + A1 = 3
and
x1, x2, s1, s2, A1 ≥ 0
The optimal solution: x1 = 8, x2 = 0 and Max Z = 48 for this LP problem
is shown in Table below.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 98 / 107
99. Cj 6 4 0 0 0
CB XB x1 x2 s1 s2 s3 b(RHS) Ratio
0 s1 0 5/3 1 -2/3 0 14 42/5 ←
0 s3 0 -1/3 0 1/3 1 5
6 x1 1 2/3 0 1/3 0 8 12
zj − cj 0 ↑ 0 0 2 0 48
In Table above , z2 − c2 = 0 corresponds to a non-basic variable, x2. Thus, an
alternative optimal solution can also be obtained by entering variable x2 into the
basis and removing basic variable, s1 from the basis. The new solution is shown in
Table below.
Cj 6 4 0 0 0
CB XB x1 x2 s1 s2 s3 b(RHS)
4 x2 0 1 3/5 -2/5 0 42/5
0 s3 0 0 1/5 1/5 1 39/5
6 x1 1 0 -2/5 3/5 0 12/5
zj − cj 0 0 0 2 0 48
The optimal solution shown in Table above is: x1 = 12/5, x2 = 42/5 and Max Z
= 48. Since this optimal solution shows no change in the value of objective
function, it is an alternative solution. Once again, z3 − c3 = 0 corresponds to
nonbasic variable, s1. This indicates that an alternative optimal solution exists.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 99 / 107
100. Unbounded Solution
In a maximization LP problem, if zj − zj < 0(zj − cj > 0 for a
minimization case) corresponds to a non-basic variable column in
simplex table, and all aij values in this column are negative, then
minimum ratio to decide basic variable to leave the basis can not be
calculated.
It is because negative value in denominator would indicate the entry
of a non-basic variable in the basis with a negative value (an
infeasible solution).
Also,a zero value in the denominator would result in a ratio having an
infinite value and would indicate that the value of non-basic variable
could be increased infinitely with any of the current basic variables
being removed from the basis.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 100 / 107
101. Example
Solve the following LP problem.
Maximize Z = 3x1 + 5x2
subject to the constraints
x1 − 2x2 ≤ 6,
3x1 ≤ 10
,
x2 ≥ 1
and
x1, x2 ≥ 0
Solution:Adding slack variables s1, s2 , surplus variable s3 and artificial
variable A1 in the constraint set.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 101 / 107
102. Then the standard form of LP problem becomes
Maximize Z = 3x1 + 5x2 + 0s1 + 0s2 + 0s3 − MA1
subject to the constraints
x1 − 2x2 + s1 = 6,
3x1 + s2 = 10
,
x2 − s3 + A1 = 1
and
x1, x2, s1, s2, s3, A1 ≥ 0
The initial solution to this LP problem is shown in Table below.
Cj 3 5 0 0 0 -M
CB xB x1 x2 s1 s2 s3 A1 b(RHS) min. ratio
0 s1 1 -2 1 0 0 0 6 -
0 s2 1 0 0 1 0 0 10 -
-M A1 0 1 0 0 -1 1 1 1 ←
zj − cj -3 ↑-5-M 0 0 M 0 M
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 102 / 107
103. Iteration 1: Since z2 − c2 ≥ 0, non-basic variable x2 is chosen to enter into
the basis in place of basic variable A1. The new solution is shown in Table
below.
Cj 3 5 0 0 0 -M
CB xB x1 x2 s1 s2 s3 A1 b(RHS)
0 s1 1 0 1 0 -2 2 8
0 s2 1 0 0 1 0 0 10
5 x2 0 1 0 0 -1 1 1
zj − cj -3 0 0 0 -5 M+5 5
In Table above, z5 − c5 = −5 is largest negative, so variable s3 should enter
into the basis. But, coefficients in the s3 column are all negative or zero.
This indicates that s3 cannot be entered into the basis. However,the value
of s3 can be increased infinitely without removing any one of the basic
variables. Further, since s3 is associated with x2 in the third constraint, x2
will also be increased infinitely because it can be expressed as
x2 = 1 + s3 − A1. Hence, the solution to the given problem is unbounded.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 103 / 107
104. Infeasible Solution
Lp models with inconsistent constraints have no feasible solution.
This situation can never occur if all the constraints are of the type ≤
with nonnegative right hand sides because the slacks provide a
feasible solution.
For other types of constraints we use artificial variables.
Although the artificial variables are penalized in the objective function
to force them to zero at the optimum, this can occur only if the
model has a feasible space.
Otherwise, at least one artificial variables will be positive in the
optimum iteration.
If LP problem solution does not satisfy all of the constraints, then
such a solution is called infeasible solution.
Also, infeasible solution occurs when all zj − cj values satisfy optimal
solution condition but at least one of artificial variables appears in the
basis with a positive value.
This situation may occur when an LP model is either improperly
formulated or more than two of the constraints are incompatible.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 104 / 107
105. Example
Solve the following LP problem.
Maximize Z = 6x1 + 4x2
subject to the constraints
x1 + x2 ≤ 5,
x2 ≥ 8
and
x1, x2 ≥ 0
Solution:Adding slack, surplus and artificial variables, the standard form
of LP problem becomes
Maximize Z = 6x1 + 4x2 + 0s1 + 0s2 − MA1
subject to the constraints
x1 + x2 + s1 = 5,
x2 − s2 + A1 = 8
and x1, x2, s1, s2, A1 ≥ 0
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 105 / 107
106. The initial solution to this LP problem is shown in Table below.
Cj 6 4 0 0 -M
CB XB x1 x2 s1 s2 A1 b(RHS) min. ratio
6 x1 1 1 1 0 0 5 5/1←
-M A1 0 1 0 -1 1 8 8/1
zj − cj 0 ↑2-M 6 M 0 30-8M
Iteration 1: Since z2 − c2 = 2 − M(≤ 0), non-basic variable x2 is chosen to enter
into the basis to replace basic variable x1. The new solution is shown in Table
below.
Cj 6 4 0 0 -M
CB XB x1 x2 s1 s2 A1 b(RHS)
4 x2 1 1 1 0 0 5
-M A1 -1 0 -1 -1 1 3
zj − cj M-2 0 4+M M 0 20-3M
In table above, since all zj − cj ≥ 0, the current solution is optimal. But this
solution is not feasible for the given LP problem because values of decision
variables are: x1 = 0 and x2 = 5 violates second constraint,x2 ≥ 8. The presence
of artificial variable A1 = 3 in the solution also indicates that the optimal solution
violates the second constraint (x2 ≥ 8) by 3 units.
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 106 / 107
107. I THANK YOU!!
(Department of Mathematics Debark University )Linear Optimization(MATH 2062) October 7, 2020 107 / 107