26 Mar

linear programming simplex method calculator

Ester Rute Ruiz, Portuguese translation by: Finding a maximum value of the function (artificial variables), Example 4. } There is no minimum value of C. s 2.5 4 New constraints could be added by using commas to separate them. 4 It is one of the popular methods that are used to avail of the 0 x x 2 + with us. simplex linear-programming optimization-algorithms simplex-algorithm linear-programming-solver linear-optimization mathematical-programming . {\displaystyle {\begin{aligned}\phi &=\sum _{i=1}^{n}c_{i}x_{i}\\x_{n+i}&=b_{i}-\sum _{j=1}^{n}a_{ij}x_{ij}\quad i=1,2,,m\end{aligned}}}. It mainly consists of four main This will require us to have a matrix that can handle \(x, y, S_{1}, s_{2}\), and \(P .\) We will put it in i 0 {\displaystyle {\begin{array}{c c c c c c c | r}x_{1}&x_{2}&x_{3}&s_{1}&s_{2}&s_{3}&z&b\\\hline 1&0.5&0.5&0.5&0&0&0&1\\0&1.5&2.5&-0.5&1&0&0&3\\0&1&0&-1&0&1&0&6\\\hline 0&1&-2&2&0&0&1&4\end{array}}}. We have established the initial simplex tableau. Main site navigation. 2 x Afterward, multiplying this specific row with corresponding coefficients and adding this to different rows, one should get 0 values for all other entries in this pivot element's column. x 1 the problem specifically. i Region of feasible solutions is an empty set. = 1.5 The industries from different fields will use the simplex method to plan under the constraints. 6.4 0 just start using this free online tool and save your time. 1 The entire process of solving using simplex method is: \[\begin{align*} x + 4y + 2z &\leq 8 \\3x + 5y + z &\leq 6 \\x \geq 0,y \geq 0,z&\geq 0 \\ \end{align*} \nonumber \]. { "9.01:_Introduction_to_Linear_Programming_Applications_in_Business_Finance_Medicine_and_Social_Science" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "9.02:_Maximization_By_The_Simplex_Method" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "9.03:_Minimization_By_The_Simplex_Method" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "9.04:_Chapter_Review" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Linear_Equations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Inequalities" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Exponential_and_Logarithmic_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Introduction_to_Calculus" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Mathematics_of_Finance" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Matrices" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Linear_Programming_-_A_Geometric_Approach" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Linear_Programming_-_The_Simplex_Method" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Sets_and_Counting" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11:_Probability" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12:_More_Probability" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "Book:_Business_Statistics_Customized_(OpenStax)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "FCC_-_Finite_Mathematics_-_Spring_2023" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Introduction_to_Business_Statistics_-_OER_-_Spring_2023" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, 9: Linear Programming - The Simplex Method, [ "article:topic-guide", "showtoc:no", "license:ccby", "authorname:rsekhon", "source[1]-math-37816", "licenseversion:40", "source@https://www.deanza.edu/faculty/bloomroberta/math11/afm3files.html.html", "source[1]-stats-32486" ], https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FCourses%2FFresno_City_College%2FFCC_-_Finite_Mathematics_-_Spring_2023%2F09%253A_Linear_Programming_-_The_Simplex_Method, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), 9.1: Introduction to Linear Programming Applications in Business, Finance, Medicine, and Social Science, source@https://www.deanza.edu/faculty/bloomroberta/math11/afm3files.html.html, status page at https://status.libretexts.org. 1 From Cornell University Computational Optimization Open Textbook - Optimization Wiki. Economic analysis of the potential use of a simplex method in designing the sales strategy of an enamelware enterprise. i 1 Doing homework can help you learn and understand the material covered in class. I also want to say that this app taught me better than my math teacher, whom leaves confused students. accurately within small steps. This tells us that \(x\) can still contribute to the objective function. 1 WebSolves Linear Programming and Quadratic Programming problems up to 8,000 variables. It can also help improve your math skills. n Dual Simplex. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. 0 is immutable. For this solution, the first column is selected. A quotient that is a zero, or a negative number, or that has a zero in the denominator, is ignored. solution for a given linear problem. Calculate the quotients. x right size. WebLinear Programming Solver Linear Programming Added Jul 31, 2018 by vik_31415 in Mathematics Linear programming solver with up to 9 variables. i Juan Jos Ruiz Ruiz, English translation by: 0 Transfer to the table the basic elements that we identified in the preliminary stage: Each cell of this column is equal to the coefficient, which corresponds to the base variable in the corresponding row. 2 n z Read More Linear Programming Big M Method Calculator Online your simple linear programming equation problems easy and simple as Choosing the smaller, we have our pivot in row 2 column 1. 0 {\displaystyle \phi } = 3 = The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. The same procedure will be followed until objective function which is constrained by inequalities within the 4 1 + WebWe can use Excels Solver to solve this linear programming problem, employing the Simplex Linear Programming method, where each data element results in two constraints. i 4 In 1979, a Soviet scientist named Leonid Khachian developed a method called the ellipsoid algorithm which was supposed to be 1 x 0.5 x value which should be optimized, and the constraints are used to z The simplex method is the way to adjust the nonbasic variables to travel to different vertex till the optimum solution is found.[5]. In this, m Afterward, the dictionary function will be written in the form of: Where the variables with bar suggest that those corresponding values will change accordingly with the progression of the simplex method. 0 Although this is the first tableau of the Simplex method and all C b are null, so the calculation can simplified, and by this time Z = -C . In TI-84 plus calculator, display the stored intersection 1 .71 & 0 & 1 & -.43 & 0 & .86 \\ Finding a minimum value of the function, Example 3. All of the \(a_{\text {mumber }}\) represent real-numbered coefficients and the \(x_{\text {number }}\) represent the corresponding variables. 1 n Where This page was last edited on 5 October 2021, at 07:26. WebSimplex Algorithm Calculator is an online application on the simplex algorithm and two phase method. We set up the initial tableau. I love spending time with my family and friends. x Step 2: Now click the button Therefore, if an LP has an optimal solution, there must be an extreme point of the feasible region that is optimal. The leaving variables are defined as which go from basic to non-basic. Use by-hand solution methods that have been developed to solve these types of problems in a compact, procedural way. 2 6.4 = This kind of variable is referred to as the entering variable. The simplex method was developed during the Second World War by Dr. George Dantzig. Nowadays, with the development of technology and economics, the Simplex method is substituted with some more advanced solvers which can solve the problems with faster speed and handle a larger amount of constraints and variables, but this innovative method marks the creativity at that age and continuously offer the inspiration to the upcoming challenges. A simple calculator and some simple steps to use it. , example On the right-hand side of each constant do not enter any e x 1? Linear Programming and Optimization using Python | Towards Data Science Write Sign up Sign In 500 Apologies, but something went wrong on our end. 1 0 2 If an inequality of the form , then the compensating variable has the sign +, if the inequality of the form , then the compensating variable has the sign -. 2 You can get several feasible solutions for your problem 3 WebSolve the following linear programming problem by applying the simplex method to the dual problem. 0 Hence, the following equation can be derived: x 3 PHPSimplex is an online tool for solving linear programming problems. well. A button to switch the answer between number, fraction and scientific notation will be helpful. 13? method has been used. 2 0.8 three given variables. 2 At the intersection of the line that corresponds to the variable that is derived from the basis, and the column that corresponds to the variable that is entered into the basis, is the resolving element. s calculator. https://doi.org/10.1007/978-1-4757-4106-3_8. Step 2: To get the optimal solution of the linear problem, click decision variable. functionality to solve a linear problem which is known as the 2 0 x and the objective function as well. Nikitenko, A. V. (1996). x 1 {\displaystyle {\begin{aligned}2x_{1}+x_{2}+x_{3}&\leq 2\\x_{1}+2x_{2}+3x_{3}&\leq 4\\2x_{1}+2x_{2}+x_{3}&\leq 8\\x_{1},x_{2},x_{3}&\geq 0\end{aligned}}}. Plot the graph for the system of constraints which is given in Maxx1 = ((Cb1 * x1,1) + (Cb2 * x2,1) + (Cb3 * x3,1) + (Cb4 * x4,1) + (Cb5 * x5,1) ) - kx1 = ((0 * 2) + (0 * 0) + (0 * 5) + (-M * 0) + (-M * 0) ) - 3 = -3; Maxx2 = ((Cb1 * x1,2) + (Cb2 * x2,2) + (Cb3 * x3,2) + (Cb4 * x4,2) + (Cb5 * x5,2) ) - kx2 = ((0 * 1) + (0 * 0) + (0 * 4) + (-M * 2) + (-M * 0) ) - 4 = -2M-4; Maxx3 = ((Cb1 * x1,3) + (Cb2 * x2,3) + (Cb3 * x3,3) + (Cb4 * x4,3) + (Cb5 * x5,3) ) - kx3 = ((0 * 1) + (0 * 0) + (0 * 0) + (-M * 0) + (-M * 0) ) - 0 = 0; Maxx4 = ((Cb1 * x1,4) + (Cb2 * x2,4) + (Cb3 * x3,4) + (Cb4 * x4,4) + (Cb5 * x5,4) ) - kx4 = ((0 * 0) + (0 * 1) + (0 * 0) + (-M * 0) + (-M * 0) ) - 0 = 0; Maxx5 = ((Cb1 * x1,5) + (Cb2 * x2,5) + (Cb3 * x3,5) + (Cb4 * x4,5) + (Cb5 * x5,5) ) - kx5 = ((0 * 0) + (0 * 0) + (0 * 1) + (-M * 0) + (-M * 0) ) - 0 = 0; Maxx6 = ((Cb1 * x1,6) + (Cb2 * x2,6) + (Cb3 * x3,6) + (Cb4 * x4,6) + (Cb5 * x5,6) ) - kx6 = ((0 * 0) + (0 * 0) + (0 * 0) + (-M * -1) + (-M * 0) ) - 0 = M; Maxx7 = ((Cb1 * x1,7) + (Cb2 * x2,7) + (Cb3 * x3,7) + (Cb4 * x4,7) + (Cb5 * x5,7) ) - kx7 = ((0 * 0) + (0 * 0) + (0 * 0) + (-M * 0) + (-M * -1) ) - 0 = M; Maxx8 = ((Cb1 * x1,8) + (Cb2 * x2,8) + (Cb3 * x3,8) + (Cb4 * x4,8) + (Cb5 * x5,8) ) - kx8 = ((0 * 0) + (0 * 0) + (0 * 0) + (-M * 1) + (-M * 0) ) - -M = 0; Maxx9 = ((Cb1 * x1,9) + (Cb2 * x2,9) + (Cb3 * x3,9) + (Cb4 * x4,9) + (Cb5 * x5,9) ) - kx9 = ((0 * 0) + (0 * 0) + (0 * 0) + (-M * 0) + (-M * 1) ) - -M = 0; Since there are negative values among the estimates of the controlled variables, the current table does not yet have an optimal solution. WebApplication consists of the following menu: 1) Restart The screen back in the default problem. 1 a you can easily solve all your problems without any confusion. \begin{array}{c}\begin{array}{cccccc} As long as there are no repetitive entering variables can be selected, the optimal values will be found. x This element will allow us to calculate the elements of the table of the next iteration. Type your linear programming problem below. , 1 We set the remaining variables equal to zero and find our solution: \[x = \frac{4}{5},\quad y = 0,\quad z = \frac{18}{5}\nonumber \], Reading the answer from a reduced tableau. 1 For what the corresponding restrictions are multiplied by -1. As in the pivot process, the coefficient for the selected pivot element should be one, meaning the reciprocal of this coefficient should be multiplied to every element within this row. 2 + s 1 b on the submit button in the given tool. 1 Due to the nonnegativity of all variables, the value of 0.2 After then, press E to evaluate the function and you will get 0 } Our pivot is thus the \(y\) column. Find out the intersection region and then graph the region of = B. j At once there are no more negative values for basic and non-basic variables. 3 s The rational sale-strategy will be indispensable to the successful practice of marketing. The simplex algorithm can be thought of as one of the elementary steps for solving the inequality problem, since many of those will be converted to LP and solved via Simplex algorithm. about this calculator is at it easily solving the problems 12 x 2? 0 x 1? The optimal solution is found.[6][7]. 0 1 x one or more constraints of the form, \(a_{1} x_{1}+a_{2} x_{2}+a_{3} x_{3}+\ldots a_{n} x_{n}\). Author: Guoqing Hu (SysEn 6800 Fall 2020), Simplex algorithm (or Simplex method) is a widely-used algorithm to solve the Linear Programming(LP) optimization problems. well. Since the coefficient in the first row is 1 and 4 for the second row, the first row should be pivoted. m j There remain no additional negative entries in the objective function row. k Websimplex method matrix calculator - The simplex method is one of the popular solution methods that are used in solving the problems related to linear programming. WebSimplex method calculator - The Simplex algorithm is a popular method for numerical solution of the linear programming problem. Set up the problem. In this way, inequalities could be solved. about the continuation of the steps. WebSimplex On Line Calculator. {\displaystyle x_{3}=1.2} a Rosane Bujes. P ) for i = 1..m, where if j = 0, P 0 = b and C 0 = 0, else P = a ij. The basic is a variable that has a coefficient of 1 with it and is found only in one constraint. So, after getting basic information about what it is, lets see how In the last row, the column with the smallest value should be selected. 2 That is, write the objective function and the constraints. For solving the linear programming problems, the simplex The simplex method is commonly used in many programming problems. We now see that, \[ \begin{align*} .71x + s_1- .43{s_2} & = .86 \\ 7y - 4.23{s_1} + 2.81{s_2} & = 8.38\\ 2.62{s_1} + .59{s_2} + P &= 22.82 \end{align*}\], \[\begin{align*} .71x&= .86 &\to x \approx 1.21 \\ 7y &= 8.38 &\to y \approx 1.20\\ P &= 22.82& \end{align*}\]. s 1 n 2 , It is indisputable to acknowledge the influence of the Simplex method to programming, as this method won the 'National Medal of Science' to its inventor, George Dantzig. x This calculator The maximum value you are looking for appears in the bottom right hand corner. WebLinear programming simplex calculator Do my homework for me. = These are the basic steps to follow when using the linear problem j this order. 2 + Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. 3) Add Column Add a column to 0 0 1 are used in solving the problems related to linear programming. 0 & 7 & -4.23 & 2.81 & 0 & 8.38 \\ Solve all linear optimization problems including minimization and maximization with simplex algorithm. This is done the same way as we did with the Gauss-Jordan method for matrices. First of all, , i The \end{array}\right] 0 1 , share this information with your friends who also want to learn x + 2 After this manipulation, the sign of inequality is reversed. i Moreover, problems can also be solved with Linear Programming Step 2: To get the optimal solution of the linear problem, click on the submit button in the (Press "Example" to Also notice that the slack variable columns, along with the objective function output, form the identity matrix. Refresh the page, check Medium s site status, or find something interesting to read. WebSolve the following linear programming problem by applying the simplex method to the dual problem. However, we represent each inequality by a single slack variable. Instructions for compiling=>> my IDE codeBlocks; Run on any gcc compiler=>> Special***** should compile in -std=c++11 or c++14 ********* (mat be other versions syntacs can be different) 0.6 equation with a system of inequalities you can get an optimal All other cells remain unchanged. George B. Dantzig (19142005). = Conic Sections: Parabola and Focus. 1.2 Solve Now. 1 1 3 To embed this widget in a post, install the Wolfram|Alpha Widget Shortcode Plugin and copy and paste the shortcode above into the HTML source. 0 0 = There is a comprehensive manual included with the software. , 1 i b After that, find out intersection points from the region and P1 = (P1 * x3,1) - (x1,1 * P3) / x3,1 = ((525 * 5) - (2 * 700)) / 5 = 245; P2 = (P2 * x3,1) - (x2,1 * P3) / x3,1 = ((225 * 5) - (0 * 700)) / 5 = 225; P4 = (P4 * x3,1) - (x4,1 * P3) / x3,1 = ((75 * 5) - (0 * 700)) / 5 = 75; P5 = (P5 * x3,1) - (x5,1 * P3) / x3,1 = ((0 * 5) - (0 * 700)) / 5 = 0; x1,1 = ((x1,1 * x3,1) - (x1,1 * x3,1)) / x3,1 = ((2 * 5) - (2 * 5)) / 5 = 0; x1,3 = ((x1,3 * x3,1) - (x1,1 * x3,3)) / x3,1 = ((1 * 5) - (2 * 0)) / 5 = 1; x1,4 = ((x1,4 * x3,1) - (x1,1 * x3,4)) / x3,1 = ((0 * 5) - (2 * 0)) / 5 = 0; x1,5 = ((x1,5 * x3,1) - (x1,1 * x3,5)) / x3,1 = ((0 * 5) - (2 * 1)) / 5 = -0.4; x1,6 = ((x1,6 * x3,1) - (x1,1 * x3,6)) / x3,1 = ((0.5 * 5) - (2 * 2)) / 5 = -0.3; x1,7 = ((x1,7 * x3,1) - (x1,1 * x3,7)) / x3,1 = ((0 * 5) - (2 * 0)) / 5 = 0; x1,8 = ((x1,8 * x3,1) - (x1,1 * x3,8)) / x3,1 = ((-0.5 * 5) - (2 * -2)) / 5 = 0.3; x1,9 = ((x1,9 * x3,1) - (x1,1 * x3,9)) / x3,1 = ((0 * 5) - (2 * 0)) / 5 = 0; x2,1 = ((x2,1 * x3,1) - (x2,1 * x3,1)) / x3,1 = ((0 * 5) - (0 * 5)) / 5 = 0; x2,3 = ((x2,3 * x3,1) - (x2,1 * x3,3)) / x3,1 = ((0 * 5) - (0 * 0)) / 5 = 0; x2,4 = ((x2,4 * x3,1) - (x2,1 * x3,4)) / x3,1 = ((1 * 5) - (0 * 0)) / 5 = 1; x2,5 = ((x2,5 * x3,1) - (x2,1 * x3,5)) / x3,1 = ((0 * 5) - (0 * 1)) / 5 = 0; x2,6 = ((x2,6 * x3,1) - (x2,1 * x3,6)) / x3,1 = ((0 * 5) - (0 * 2)) / 5 = 0; x2,7 = ((x2,7 * x3,1) - (x2,1 * x3,7)) / x3,1 = ((0 * 5) - (0 * 0)) / 5 = 0; x2,8 = ((x2,8 * x3,1) - (x2,1 * x3,8)) / x3,1 = ((0 * 5) - (0 * -2)) / 5 = 0; x2,9 = ((x2,9 * x3,1) - (x2,1 * x3,9)) / x3,1 = ((0 * 5) - (0 * 0)) / 5 = 0; x4,1 = ((x4,1 * x3,1) - (x4,1 * x3,1)) / x3,1 = ((0 * 5) - (0 * 5)) / 5 = 0; x4,3 = ((x4,3 * x3,1) - (x4,1 * x3,3)) / x3,1 = ((0 * 5) - (0 * 0)) / 5 = 0; x4,4 = ((x4,4 * x3,1) - (x4,1 * x3,4)) / x3,1 = ((0 * 5) - (0 * 0)) / 5 = 0; x4,5 = ((x4,5 * x3,1) - (x4,1 * x3,5)) / x3,1 = ((0 * 5) - (0 * 1)) / 5 = 0; x4,6 = ((x4,6 * x3,1) - (x4,1 * x3,6)) / x3,1 = ((-0.5 * 5) - (0 * 2)) / 5 = -0.5; x4,7 = ((x4,7 * x3,1) - (x4,1 * x3,7)) / x3,1 = ((0 * 5) - (0 * 0)) / 5 = 0; x4,8 = ((x4,8 * x3,1) - (x4,1 * x3,8)) / x3,1 = ((0.5 * 5) - (0 * -2)) / 5 = 0.5; x4,9 = ((x4,9 * x3,1) - (x4,1 * x3,9)) / x3,1 = ((0 * 5) - (0 * 0)) / 5 = 0; x5,1 = ((x5,1 * x3,1) - (x5,1 * x3,1)) / x3,1 = ((0 * 5) - (0 * 5)) / 5 = 0; x5,3 = ((x5,3 * x3,1) - (x5,1 * x3,3)) / x3,1 = ((0 * 5) - (0 * 0)) / 5 = 0; x5,4 = ((x5,4 * x3,1) - (x5,1 * x3,4)) / x3,1 = ((0 * 5) - (0 * 0)) / 5 = 0; x5,5 = ((x5,5 * x3,1) - (x5,1 * x3,5)) / x3,1 = ((0 * 5) - (0 * 1)) / 5 = 0; x5,6 = ((x5,6 * x3,1) - (x5,1 * x3,6)) / x3,1 = ((0 * 5) - (0 * 2)) / 5 = 0; x5,7 = ((x5,7 * x3,1) - (x5,1 * x3,7)) / x3,1 = ((-1 * 5) - (0 * 0)) / 5 = -1; x5,8 = ((x5,8 * x3,1) - (x5,1 * x3,8)) / x3,1 = ((0 * 5) - (0 * -2)) / 5 = 0; x5,9 = ((x5,9 * x3,1) - (x5,1 * x3,9)) / x3,1 = ((1 * 5) - (0 * 0)) / 5 = 1; Maxx1 = ((Cb1 * x1,1) + (Cb2 * x2,1) + (Cb3 * x3,1) + (Cb4 * x4,1) + (Cb5 * x5,1) ) - kx1 = ((0 * 0) + (0 * 0) + (3 * 1) + (4 * 0) + (-M * 0) ) - 3 = 0; Maxx2 = ((Cb1 * x1,2) + (Cb2 * x2,2) + (Cb3 * x3,2) + (Cb4 * x4,2) + (Cb5 * x5,2) ) - kx2 = ((0 * 0) + (0 * 0) + (3 * 0) + (4 * 1) + (-M * 0) ) - 4 = 0; Maxx3 = ((Cb1 * x1,3) + (Cb2 * x2,3) + (Cb3 * x3,3) + (Cb4 * x4,3) + (Cb5 * x5,3) ) - kx3 = ((0 * 1) + (0 * 0) + (3 * 0) + (4 * 0) + (-M * 0) ) - 0 = 0; Maxx4 = ((Cb1 * x1,4) + (Cb2 * x2,4) + (Cb3 * x3,4) + (Cb4 * x4,4) + (Cb5 * x5,4) ) - kx4 = ((0 * 0) + (0 * 1) + (3 * 0) + (4 * 0) + (-M * 0) ) - 0 = 0; Maxx5 = ((Cb1 * x1,5) + (Cb2 * x2,5) + (Cb3 * x3,5) + (Cb4 * x4,5) + (Cb5 * x5,5) ) - kx5 = ((0 * -0.4) + (0 * 0) + (3 * 0.2) + (4 * 0) + (-M * 0) ) - 0 = 0.6; Maxx6 = ((Cb1 * x1,6) + (Cb2 * x2,6) + (Cb3 * x3,6) + (Cb4 * x4,6) + (Cb5 * x5,6) ) - kx6 = ((0 * -0.3) + (0 * 0) + (3 * 0.4) + (4 * -0.5) + (-M * 0) ) - 0 = -0.8; Maxx7 = ((Cb1 * x1,7) + (Cb2 * x2,7) + (Cb3 * x3,7) + (Cb4 * x4,7) + (Cb5 * x5,7) ) - kx7 = ((0 * 0) + (0 * 0) + (3 * 0) + (4 * 0) + (-M * -1) ) - 0 = M; Maxx8 = ((Cb1 * x1,8) + (Cb2 * x2,8) + (Cb3 * x3,8) + (Cb4 * x4,8) + (Cb5 * x5,8) ) - kx8 = ((0 * 0.3) + (0 * 0) + (3 * -0.4) + (4 * 0.5) + (-M * 0) ) - -M = M+0.8; Maxx9 = ((Cb1 * x1,9) + (Cb2 * x2,9) + (Cb3 * x3,9) + (Cb4 * x4,9) + (Cb5 * x5,9) ) - kx9 = ((0 * 0) + (0 * 0) + (3 * 0) + (4 * 0) + (-M * 1) ) - -M = 0; For the results of the calculations of the previous iteration, we remove the variable from the basis x1 and put in her place x6. 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