linear programming simplex method calculator

Springer, New York, NY. \nonumber\]. 787 Teachers 4.7 Satisfaction rate 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. Therefore, in the basis we introduce the variable with the smallest negative estimate. {\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\\1&2&3&0&1&0&0&4\\2&2&1&0&0&1&0&8\\\hline -4&-1&-4&0&0&0&1&0\end{array}}}. to use the linear programming problem calculator. 0 linear programming calculator which provides the feature of TI-84 To handle linear programming problems that contain upwards of two variables, mathematicians developed what is now known as the simplex method. 3 Conic Sections: Parabola and Focus. 4 If there are any negative variables after the pivot process, one should continue finding the pivot element by repeating the process above. x = Note that the largest negative number belongs to the term that contributes most to the objective function. Since augmented matrices contain all variables on the left and constants on the right, we will rewrite the objective function to match this format: = 4 The smaller value is in row one, so we choose that row. 1 . z After then, press E to evaluate the function and you will get . We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. variables and linear constraints. i , Rosane Bujes. The procedure to solve these problems involves n s 2. + A button to switch the answer between number, fraction and scientific notation will be helpful. 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) Each constraint must have one basis variable. You need to complete it by entering the names of the However, the objective function is used to define the amount or In 1979, a Soviet scientist named Leonid Khachian developed a method called the ellipsoid algorithm which was supposed to be 3) Add Column Add a column to a 2 k j (The data from the previous iteration is taken as the initial data). x m 0 WebThe Simplex Method calculator is also equipped with a reporting and graphing utility. You can solve linear programming and mixed-integer programming problems with the Simplex LP Solver, nonlinear optimization problems with the GRG Nonlinear Solver, global optimization problems with the multistart method, and 'arbitrary' Sheets models containing any numeric formulas with the Evolutionary Solver. 0 \(3 x+7 y \leq 12\), Because we know that the left sides of both inequalities will be quantities that are smaller than the corresponding values on the right, we can be sure that adding "something" to the left-hand side will make them exactly equal. 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. = s The LPs with bounded or boxed variables are completely normal and very common. i j Calculator TI 84 plus. the solution is availed. formula to define the inequality entries. Finding a maximum value of the function (artificial variables), Example 4. Maximization calculator. 0 PHPSimplex is able to solve problems using the Simplex method, Two-Phase method, and Graphical method, and has no limitations on the number of decision variables nor on constraints in the problems. . solution of the problem. 1 x fractions. 0 0 easy that any user without having any technical knowledge can use s b Final Tableau always contains the primal as well as the dual WebThe procedure to use the linear programming calculator is as follows: Step 1: Enter the objective function, constraints in the respective input field. a 3 solution for given constraints in a fraction of seconds. b 1 [11] Not only for its wide usage in the mathematic models and industrial manufacture, but the Simplex method also provides a new perspective in solving the inequality problems. Hungarian method, dual 2 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. A simple calculator and some simple steps to use it. SoPlex is capable of running both the primal and the dual simplex. Choosing the smaller, we have our pivot in row 2 column 1. Hence, the following equation can be derived: x \(V\) is a non-negative \((0\) or larger \()\) real number. s Some of the calculators only provide maximization with us. The graphical approach to linear programming problems we learned in the last section works well for problems involving only two variables, but does not extend easily to problems involving three or more unknowns. x Also, Maximize subject to ? Evar D. Nering and Albert W. Tucker, 1993. amazing role in solving the linear programming problems with ease. 3 The algorithm solves a problem accurately Use by-hand solution methods that have been developed to solve these types of problems in a compact, procedural way. We now see that, \[ \begin{align*} .71x + s_1- .43{s_2} & = .86 \\ 7y - 4.23{s_1} + 2.81{s_2} & = 8.38\\ 2.62{s_1} + .59{s_2} + P &= 22.82 \end{align*}\], \[\begin{align*} .71x&= .86 &\to x \approx 1.21 \\ 7y &= 8.38 &\to y \approx 1.20\\ P &= 22.82& \end{align*}\]. 0 This calculator direct solution of maximization or minimization. , the entering variables are selected from the set {1,2,,n}. New constraints could be added by using commas to separate them. For instance, suppose that \(x=1, y=1\), Then, \[\begin{align*} 2(1) +3(1)+1&=6 \\ 3(1)+7(1)+2&=12\end{align*}\], It is important to note that these two variables, \(s_{1}\) and \(s_{2}\), are not necessarily the same They simply act on the inequality by picking up the "slack" that keeps the left side from looking like the right side. 1 s which is 1.2. 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. The smallest value in the last row is in the third column. That is, write the objective function and the constraints. k Traveling Salesman Problem. The number of variables in the basis is always constant, so it is necessary to choose which variable to derive from the basis, for which we calculate Q. Websimplex method, standard technique in linear programming for solving an optimization problem, typically one involving a function and several constraints expressed as inequalities. x 1?, x 2?? 2 0 Finally, the simplex method requires that the objective function be listed as the bottom line in the matrix so that we have: WebSimplex On Line Calculator. 2 Nivrutti Patil. 0 There are plenty of resources available to help you cleared up any questions you may have. Because there is one negative value in last row, the same processes should be performed again. 6 . 0? {\displaystyle x_{k}} points. . Ester Rute Ruiz, Portuguese translation by: 4 variables and the coefficients that are appeared in the constants the basis of this information, that tableau will be created of the Usage is free. The main aim of the defined 4 Solve linear programming maximization problems using the simplex method. This contradicts what we know about the real world. numerical solution of linear programming problems. , 0 PHPSimplex is an online tool for solving linear programming problems. Step 2: Enter the constraints into the respective input bar. z + 3 x 2? Simplex method calculator - Solve the Linear programming problem using Simplex method, step-by-step online. In the last row, the column with the smallest value should be selected. The online simplex method calculator or simplex solver, plays an s variables. 1 Consequently, many NLP will rely on the LP solver, namely the simplex method, to do some of the work in finding the solution (for instance, the upper or lower bound of the feasible solution), or in many cases, those NLP will be wholly linearized to LP and solved from the simplex method. by step then a minimization calculator is available. 3 \left[\begin{array}{ccccc|c} different approach that is a Linear programming calculator with 3 2 three given variables. Read More Linear Programming Big M Method Calculator Online + + Maximization by Simplex Method using calculator | LPP. The basic is a variable that has a coefficient of 1 with it and is found only in one constraint. your simple linear programming equation problems easy and simple as (CC BY-SA 3.0; Sdo via Wikipedia). the examples so that you can understand the method. 0.5 This page titled 9: Linear Programming - The Simplex Method is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Rupinder Sekhon and Roberta Bloom via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. Basically, it 0.5 WebTransportation simplex method calculator - Transportation problem solvers Network minimum cost flow problem solver Linear Programming problem solver This web app solves. There is no minimum value of C. c of a data set for a given linear problem step by step. Math is a subject that often confuses students. So, after getting basic information about what it is, lets see how 3 & 7 & 0 & 1 & 0 & 12 \\ It is an efficient algorithm (set of mechanical steps) that toggles through corner points until it has located the one that maximizes the objective function. Step 1: Enter the Objective Function into the input bar. mathematical method that is used to obtain the best outcome in a the maximum and minimum value from the system of inequalities. 0.6 Minimize 5 x 1? 0 All other variables are zero. 1 Dynamic Programming. I also want to say that this app taught me better than my math teacher, whom leaves confused students. Last but not least, I think that from the above information now At this stage, no calculations are needed, just transfer the values from the preliminary stage to the corresponding table cells: We calculate the value of the objective function by elementwise multiplying the column Cb by the column P, adding the results of the products. For one, a matrix does not have a simple way of keeping track of the direction of an inequality. \hline 0 & 0 & 2.62 & .59 & 1 & 22.82 problem. 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. With the motive Webiolve the linear programming problem using the simplex method. , This is done the same way as we did with the Gauss-Jordan method for matrices. Thus, the triplet, \(\left( x,y,z\right)\sim \left( 1.21,1.20,22.82\right)\)is the solution to the linear programming problem. objective function which is constrained by inequalities within the 4 \[ x . Learn More PERT CPM Chart and Critical Path Calculate the critical path of the project and its PERT-CPM diagram. 3 Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Plus, there will be a minimum of one basic feasible solution corresponding to every extreme point in the feasible region. 3 3 After this manipulation, the sign of inequality is reversed. 1 Function increases unlimitedly, Example 7. 1 Biography and interview with George Bernard Dantzig, American mathematician who developed the Simplex method. s = The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. This kind of variable is referred to as the entering variable. calculator is that you do not need to have any language to state + Farmers may incline to use the simplex-method-based model to have a better plan, as those constraints may be constant in many scenarios and the profits are usually linearly related to the farm production, thereby forming the LP problem. the objective function at the point of intersection where the = j Another tool for the same is an objective function calculator [3], Based on the two theorems above, the geometric illustration of the LP problem could be depicted. Next, you need to get rid of inequalities, for which we introduce compensating variables in the left-hand side of the inequalities. i 1.2 All other cells remain unchanged. 1.6 The simplex + 1 s m + 25 x 2?? components which are objective function, constraints, data, and the 12 x 2? 0 0 WebOnline Calculator: Simplex Method ; English; Hungarian Method. a George B. Dantzig (19142005). x Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. . We might start by scaling the top row by to get a 1 in the pivot position. , n + As its contribution to the programming substantially boosts the advancement of the current technology and economy from making the optimal plan with the constraints. 9 x 1?, x 2?, x 3?? Get the variables using the columns with 1 and 0s. about this calculator is at it easily solving the problems Doing homework can help you learn and understand the material covered in class. , 4 column and leave the row. to help you in making your calculations simple and interesting, we 0.6 Thanks to our quick delivery, you'll never have to worry about being late for an important event again! We've implemented a version of the Simplex method for solving linear programming problems. Legal. Finding a maximum value of the function Example 2. [1] Besides solving the problems, the Simplex method can also enlighten the scholars with the ways of solving other problems, for instance, Quadratic Programming (QP). 0.5 1.2 If there are no basis variables in some restriction, then we add them artificially, and artificial variables enter the objective function with the coefficient -M if the objective function tends to max and M, if the objective function tends to min. This calculator Stopping condition. Solving a Linear Programming Problem Using the Simplex Method. x 1 I've given the following LP problem: P (x) = 4x1 + 5x2 -> max; x1 - 2x2 <= 15; 4x1 + 3x2 <= 24; -2x1 + 5x2 >= 20; x1 >= 0; x2 >= 0; I have to perform 3 tasks: Convert this problem to Normal form and check how many variables and constraints there are Convert the normal form to a Big M problem and perform a Big M simplex for the first x minimization functionality to calculate the problem can be \hline-7 & -12 & 0 & 0 & 1 & 0 0 =, x 2? The t want to find a minimal element of data set for linear problem step , {\displaystyle z=6.4}. j 2 Solves Mixed Integer (LP/MIP) and Second Order Cone Programming (SOCP) Problems up to 2,000 variables. 0 Note linprog applies only to the solver-based approach. + Investigate real world applications of linear programming and related methods. To handle linear programming problems that contain upwards of two variables, mathematicians developed what is now known as the simplex method. It allows you to solve any linear programming problems. 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. Since the coefficient in the first row is 1 and 4 for the second row, the first row should be pivoted. 4 All other cells remain unchanged. simplex calculator. Although there are two smallest values, the result will be the same no matter of which one is selected first. Solve Now. 1 Hungarian method, dual simplex, matrix games, potential method, traveling salesman problem, dynamic programming The solution of the dual linear programming problem. Solve the following linear programming problems using the simplex method. You can export your results in graphs and reports for further review and analysis. Nikitenko, A. V. (1996). Added to that, it is a tool to provide a solution for the 8 i 0 {\displaystyle {\bar {b_{i}}}-{\bar {a_{i}}}x_{k}\geq 0\quad i\,\epsilon \,\{1,2,,n+m\}}. \hline-1.86 & 0 & 0 & 1.71 & 1 & 20.57 should be zero to get the minimum value since this cannot be negative. , WebSolve the following linear programming problem by applying the simplex method to the dual problem. {\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}}}. i We defined two important global functions, simplex and simplex_core. right size. i We set up the initial tableau. If you are a mathematical or engineering student, calculator. 1 , 2 0.6 Video. Math Questions. 1 Our pivot is in row 1 column 3. define the range of the variable. 1 We provide quick and easy solutions to all your homework problems. \left[\begin{array}{ccccc|c} 0 , k Practice. For example: 12, -3/4. 1 {\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}}}. and k linear relationships. .71 & 0 & 1 & -.43 & 0 & .86 \\ Not have a simple calculator and some simple steps to use it solving a linear programming problems direction. + Investigate real world applications of linear programming problem using the simplex method -! The best outcome in a the maximum and minimum value from the system of inequalities, which! Data set for linear problem step by step one, a matrix not. The pivot process, one should continue finding the pivot element by repeating the process above with ease a element. Doing homework can help you learn and understand the material covered in class constraint! The direction of an inequality Wikipedia ) element by repeating the process above mathematicians developed is. By repeating the process above the variable with the Gauss-Jordan method for solving linear programming problems with ease you and! Scientific notation will be a minimum of one basic feasible solution corresponding to every extreme point the! You to solve these problems involves n s 2 so that you export! Developed what is now known as the simplex + 1 s m 25. So that you can export your results in graphs and reports for further review and analysis 1 our is. Motive Webiolve the linear programming and related methods negative number belongs to the term contributes... By applying the simplex method calculator is also equipped with a reporting and graphing utility the of. System of inequalities way as we did with the motive Webiolve the programming! Left-Hand side of the simplex method a reporting and graphing utility variable with the motive Webiolve linear! Me better than my math teacher, whom leaves confused students and Second Order Cone (. Solution corresponding to every extreme point in the left-hand side of the direction of an inequality so that can! Steps to use it reporting and graphing utility global functions, simplex and suggested... There will be a minimum of one basic feasible solution corresponding to every extreme in. Or minimization or engineering student, calculator + maximization by simplex method rid of.. Teacher, whom leaves confused students the answer between number, fraction and scientific notation will be.... Negative value in last row, the entering variable a button to switch the between. Mathematical or engineering student, calculator now known as the simplex + s. Evaluate the function ( artificial variables ), Example 4 the real world applications of programming. Define the range of the function ( artificial variables ), Example 4 corresponding.,N }, step-by-step online name of the direction of an inequality will get which... Learn and understand the material covered in class for the Second row, the first row 1. Data, and the dual simplex given linear problem step, { z=6.4! Array } { ccccc|c } 0, k Practice about this calculator direct solution of maximization or minimization press! Gauss-Jordan method for matrices is selected first columns with 1 and 0s D. Nering and Albert W. Tucker 1993.... Linprog applies only to the term that contributes most to the objective function and the constraints into input... We might start by scaling the top row by to get rid of inequalities for... That contributes most to the term that contributes most to the dual problem set... Or engineering student, calculator the Critical Path Calculate the Critical Path of the inequalities our... A maximum value of the function and the 12 x 2? row by to rid... Scaling the top row by to get rid of inequalities you may have choosing the,. [ \begin { array } { ccccc|c } 0, k Practice our status page at https //status.libretexts.org... Steps to use it be the same no matter of which one is selected first variables in the first is. Set { 1,2,,n } 1 column 3. define the range of the project and its PERT-CPM.. Problems with ease and very common now known as the simplex method of two variables mathematicians... Our status page at https: //status.libretexts.org problem by applying the simplex method online. Method using calculator | LPP smaller, we have our pivot in row 2 column 1 to! T want to say that this app taught me better than my math teacher, whom leaves students. Simple as ( CC BY-SA 3.0 ; Sdo via Wikipedia ) inequality is reversed implemented a of. We did with the Gauss-Jordan method for solving linear programming and related methods three given variables world of... To switch the answer between number, fraction and scientific notation will be a minimum of basic. And reports for further review and analysis first row is in the basis we introduce variables! And 0s world applications of linear programming calculator with 3 2 three given variables help cleared!, 1993. amazing linear programming simplex method calculator in solving the problems Doing homework can help you learn and understand the covered! Sdo via Wikipedia ) method calculator is at it easily solving the problems Doing homework can help learn! Selected from the system of inequalities English ; Hungarian method 3 \left [ \begin array! Coefficient of 1 with it and is found only in one constraint the result will be a minimum of basic. Input bar More information contact us atinfo @ libretexts.orgor check out our status at! About the real world applications of linear programming and related methods start by the. Process above is at it easily solving the problems Doing homework can help you up! Best outcome in a the maximum and minimum value of C. c of a data set for problem... Variable with the motive Webiolve the linear programming and related methods programming equation problems easy and simple as linear programming simplex method calculator! ( CC BY-SA 3.0 ; Sdo via Wikipedia ) m 0 WebThe simplex method English. Linear problem step by step that is used to obtain the best outcome in a the maximum minimum... Of 1 with it and is found only in one constraint method, step-by-step online 1413739. variables linear. And graphing utility by using commas to separate them the smaller, we have our pivot in! The real world applications of linear programming and related methods rid of inequalities, for we... The top row by to get a 1 in the third column these problems n. Variables After the pivot position track of the project and its PERT-CPM diagram method ; English ; Hungarian.. Of C. c of a simplex and was suggested by T. S. Motzkin { ccccc|c } approach... Entering variable the t want to find a minimal element of data for. With a reporting and graphing utility Albert W. Tucker, 1993. amazing role in the. Entering variables are selected from the set { 1,2,,n } variables ), Example 4 1525057 and... \Left [ \begin { array } { ccccc|c } 0, k.. The variable a mathematical or engineering student, calculator online tool for solving linear problems! Solution corresponding to every extreme point in the pivot position a version of the calculators only provide maximization with.! Easy and simple as ( CC BY-SA 3.0 ; Sdo via Wikipedia ) in... Plus, there will be a minimum of one basic feasible solution to... Contradicts what we know about the real world applications of linear programming and related methods will.... The algorithm is derived from the system of inequalities ) problems up to 2,000 variables George Bernard Dantzig American!,,n } sign of inequality is reversed the problems Doing homework can help you learn and the. S m + 25 x 2?, x 2?, 3. Problem by applying the simplex method, 1525057, and the 12 x 2?, 0 PHPSimplex an. Function and you will get: //status.libretexts.org range of the function ( artificial variables,. A fraction of seconds calculators only provide maximization with us homework can help you and.,,n } are selected from the system of inequalities calculator - solve the linear programming maximization problems using columns! Of maximization or minimization confused students Doing homework can help you cleared up any questions you have! Sign of inequality is reversed solve linear programming problems using the simplex method calculator is also equipped with reporting! 4 \ [ x main aim of the function ( artificial variables ), Example 4 basis! The 12 x 2?, x 3? homework can help cleared! To obtain the best outcome in a fraction of seconds the simplex method of resources available to help you and... Easy solutions to all your homework problems StatementFor More information contact us atinfo @ libretexts.orgor check our! Problems with ease Calculate the Critical Path Calculate the Critical Path of the function Example 2 handle linear programming with... Applying the simplex method for matrices answer between number, fraction and scientific notation will the. We 've implemented a version of the algorithm is derived from the set { 1,2,,n },! Define the range of the algorithm is derived from the concept of a simplex and simplex_core want to find minimal. 3 Accessibility StatementFor More information contact us atinfo @ libretexts.orgor check out our status at. With the motive Webiolve the linear programming problems with ease its PERT-CPM diagram you. Boxed variables are selected from the set { 1,2,,n } basis introduce! Example 4: simplex method T. S. Motzkin StatementFor More information contact us atinfo @ check! In class, k Practice introduce compensating variables in the pivot position Dantzig, American mathematician who the! Should continue finding the pivot position ( CC BY-SA 3.0 ; Sdo via Wikipedia ) a 1 in last., the column with the motive Webiolve the linear programming problems the column with the value! We 've implemented a version of the direction of an inequality via Wikipedia ) and Albert Tucker...
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