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nma_matlab.m source code

Contents

List of Figures

List of Tables

1 Introduction

This is a description of a Matlab function called nma_simplex.m that implements the matrix based simplex algorithm for solving standard form linear programming problem. It supports phase one and phase two.

The function solves (returns the optimal solution \(x^{\ast }\) of the standard linear programming problem given by\[ \min _x J(x) = c^T x \] Subject to \begin{align*} Ax &= b\\ x & \geq 0 \end{align*}

The constraints have to be in standard form (equality), which results after adding any needed surplus and/or slack variables. The above is equivalent to Matlab’s \(A_{eq},b_{eq}\) used with the standard command linprog.

This function returns the final tableau, which contains the final solution. It can print all of the intermediate tableau generated and the basic feasible solutions generated during the process by passing an extra flag argument. These are generated as it runs through the simplex algorithm.

The final tableau contains the optimal solution \(x^{\ast }\) which can be read directly from the tableau. Examples below illustrate how to call this function and how to read the solution from the final tableau.

The tableau printed on the screen have this format

The optimal \(x^{\ast }\) is read directly by looking at the columns in \(A\) that make up the identity matrix. With the debug flag set, the optimal \(x^{\ast }\) is also displayed on the screen.

For example, if the final tableau was the following

Then \(x_1=1.4\) and \(x_2=3.8\). This means the optimal solution is \[ x^{\ast }= \begin{pmatrix} 1.4\\ 3.8\\ 0\\ 0 \end{pmatrix} \]

The function accepts \(A,b,c\) and a fourth parameter which is a flag ( true or false). If the flag is true, then each tableau is printed as the algorithm searches for the optimal \(x\) solution, and it also prints each \(x\) found at each step.

The following are few example showing how to use this function to solve linear programming problems, and comparing the answer to Matlab’s linprog to verify they are the same.

2 Examples

2.1 Example 1

A vendor selling rings and bracelets. A ring has 3 oz. gold., 1 oz. silver. Bracelet has 1 oz. gold, 2 oz. silver. Profit on a ring is $4 and the profit on bracelet is $5. Initially we have 8 oz. gold and 9 0z silver. How many rings and bracelets to produce to maximize profit? Note: we need integer LP to solve this. But for now we can ignore this to illustrate the use of this function. \begin{align*} x_{1} & =\text{number of rings}\\ x_{2} & =\text{number of bracelets}\\ J\left (x\right ) & =4x_{1}+5x_{2} \end{align*}

Since we want to maximize \(J(x)\), then we change the sign \[ J(x) = -4x_1 - 5x_2 \] With \(x_{i}\geq 0\). The constraints are \(3x_1+x_2 \leq 8\) and \(x_1+2 x_2\leq 9\). We start by converting to standard linear programming model by adding slack variables. This results in standard model\[ \min _{x} c^T x =\min _x \begin{pmatrix} -4 & -5 & 0 & 0 \end{pmatrix}\begin{pmatrix} x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\end{pmatrix} \] subject to \begin{align*} Ax & =b \\\begin{pmatrix} 3 & 1 & 1 & 0\\ 1 & 2 & 0 & 1 \end{pmatrix}\begin{pmatrix} x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\end{pmatrix} & =\begin{pmatrix} 8\\ 9 \end{pmatrix} \end{align*}

Here is the call and result returned which is the final tableau

Which returns

We see from above that the optimal \(x\) is \begin{align*} \begin{pmatrix} x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\end{pmatrix} = & \begin{pmatrix} 1.4000\\ 3.8000\\ 0\\ 0\end{pmatrix} \end{align*}

Since \( J(x) = 4 x_1 + 5x_2 \) then we calculate \(J(x)=4(1.4)+5(3.8)\) which gives the optimal objective function of \(24.4\) dollars. This mean the vendor should make \(1.4\) rings and \(3.8\) bracelets for maximum profit.

The same solution using Matlab’s linprog is

which gives

Which is the same.

2.2 Example 2

Given \begin{align*} A_{eq} &= \begin{pmatrix} 2 & 2 & 1 & 0 & 0\\ 2 & 2 & 0 & -1& 0\\ -1.5 & 1 & 0 & 0 & -1 \end{pmatrix} \end{align*}

And \begin{align*} b_{eq} &= \begin{pmatrix} 10&4&0 \end{pmatrix} \end{align*}

And \begin{align*} c^T &= \begin{pmatrix} \frac{1}{30}&\frac{1}{15}&0&0&0 \end{pmatrix} \end{align*}

The following gives the solution

With the output printed on the console as

From the above, we see that the solution is \begin{align*} \begin{pmatrix} 0.8\\ 1.2\\ 6\\ 0\\ 0 \end{pmatrix} \end{align*}

Given the optimal solution, the optimal objective function is now known.

To see each step and each \(x\) solution found, set the last argument to true.

Here is the result

Using Matlab linprog

Which gives

Which is the same solution.

2.3 Example 3

minimize \(2 x_1 + 3 x_2\) subject to \begin{align*} 4 x_1+2 x_2 &\geq 12\\ x_1+4 x_2 &\geq 6 \\ x_i \geq 0 \end{align*}

We convert the problem to standard form, which results in minimize \(2 x_1 + 3 x_2\) subject to \begin{align*} 4 x_1+2 x_2 -x_3 &= 12\\ x_1+4 x_2 -x_4 &=6 \end{align*}

with \(x_i \geq 0\). Therefore \begin{align*} A =& \begin{pmatrix} 4&2&-1&0\\ 1&4&0&-1 \end{pmatrix} \end{align*}

And \begin{align*} b =& \begin{pmatrix} 12\\ 6 \end{pmatrix} \end{align*}

And \(c^T = \begin{pmatrix} 2&3&0&0 \end{pmatrix}\). To solve this using nma_simplex the commands are

And the solution is

We see from the last tableau that \(x_1=2.5714\) and \(x_2=0.8671\). The optimal \(x\) is also printed in the display since the the debug flag is true.

The optimal objective function is \begin{align*} J(x)&=2(2.5714)+3(0.8671) \\ &=7.7441 \end{align*}

2.4 Example 4

maximize \(3 x_1 + 5 x_2\) subject to

\begin{align*} x_1 + 5 x_2 & \leq 40 \\ 2 x_1 + x_2 & \leq 20 \\ x_1 + x_2 & \leq 12 \\ x_i \geq 0 \end{align*}

Introducing slack and surplus variables and converting to standard form gives

minimize \(- 3 x_1 - 5 x_2\) subject to \begin{align*} x_1 + 5 x_2 + x_3 &= 40 \\ 2 x_1 + x_2 + x_4 &= 20 \\ x_1 + x_2 + x_5 &= 12 \\ x_i \geq 0 \end{align*}

Therefore \begin{align*} A =& \begin{pmatrix} 1&5&1&0&0\\ 2&1&0&1&0\\ 1&1&0&0&1 \end{pmatrix} \end{align*}

And \begin{align*} b =& \begin{pmatrix} 40\\ 20\\ 12 \end{pmatrix} \end{align*}

And \(c^T = \begin{pmatrix} -2&-5&0&0&0 \end{pmatrix}\). To solve this using nma_simplex

And the solution is

We see that \(x_1=5\) and \(x_2=7\) and \(x_4=3\) with all others zero. We always read the solution from the identity matrix inside the final tableau. All other \(x_i\) are zero. Hence the optimal solution is \[ \begin{pmatrix} 5\\ 7\\ 0\\ 3\\ 0 \end{pmatrix} \] And the corresponding optimal objective function is \(3 x_1+5 x_2=3(5)+5(7)=50\)

Result using Matlab linprog is

2.5 Example 5

maximize \(2 x_1 + 5 x_2\) subject to

\begin{align*} x_1 & \leq 4 \\ x_2 & \leq 6 \\ x_1 + x_2 & \leq 8 \\ x_i \geq 0 \end{align*}

Introducing slack and surplus variables and converting to standard form we now have the following problem

minimize \(- 2 x_1 - 5 x_2\) subject to \begin{align*} x_1 + x_3 &= 4 \\ x_2 + x_4 &= 6 \\ x_1 + x_2 + x_5 &= 8 \\ x_i \geq 0 \end{align*}

Therefore \begin{align*} A =& \begin{pmatrix} 1&0&1&0&0\\ 0&1&0&1&0\\ 1&1&0&0&1 \end{pmatrix} \end{align*}

And \begin{align*} b =& \begin{pmatrix} 4\\ 6\\ 8 \end{pmatrix} \end{align*}

And \(c^T = \begin{pmatrix} -2&-5&0&0&0 \end{pmatrix}\). To solve this using nma_simplex

And the solution is

We see that \(x_1=2\) and \(x_2=6\) and \(x_3=2\) with all others zero. We always read the solution from the identity matrix inside the final tableau. All other \(x_i\) are zero. Hence the optimal solution is \[ \begin{pmatrix} 2\\ 6\\ 2\\ 0\\ 0 \end{pmatrix} \] And the corresponding optimal objective function is \(2 x_1+5 x_2=2(2)+5(6)=34\)

Result using Matlab’s linprog

3 Appendix

3.1 References

1. Lecture notes, ECE 719 optimal systems, Univ. Wisconsin, Madison spring 2016 by Professor B. Ross Barmish
2. An introduction to Optimization. Edwin Chong, Stanislaw Zak. Wiley publication, NY 1996.