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Learning Objectives
In this section, you will learn to solve linear programming maximization problems using the Simplex Method:
- Identify and set up a linear program in standard maximization form
- Convert inequality constraints to equations using slack variables
- Set up the initial simplex tableau using the objective function and slack equations
- Find the optimal simplex tableau by performing pivoting operations.
- Identify the optimal solution from the optimal simplex tableau.
In the last chapter, we used the geometrical method to solve linear programming problems, but the geometrical approach will not work for problems that have more than two variables. In real life situations, linear programming problems consist of literally thousands of variables and are solved by computers. We can solve these problems algebraically, but that will not be very efficient. Suppose we were given a problem with, say, 5 variables and 10 constraints. By choosing all combinations of five equations with five unknowns, we could find all the corner points, test them for feasibility, and come up with the solution, if it exists. But the trouble is that even for a problem with so few variables, we will get more than 250 corner points, and testing each point will be very tedious. So we need a method that has a systematic algorithm and can be programmed for a computer. The method has to be efficient enough so we wouldn't have to evaluate the objective function at each corner point. We have just such a method, and it is called the simplex method.
The simplex method was developed during the Second World War by Dr. George Dantzig. His linear programming models helped the Allied forces with transportation and scheduling problems. In 1979, a Soviet scientist named Leonid Khachian developed a method called the ellipsoid algorithm which was supposed to be revolutionary, but as it turned out it is not any better than the simplex method. In 1984, Narendra Karmarkar, a research scientist at AT&T Bell Laboratories developed Karmarkar's algorithm which has been proven to be four times faster than the simplex method for certain problems. But the simplex method still works the best for most problems.
The simplex method uses an approach that is very efficient. It does not compute the value of the objective function at every point; instead, it begins with a corner point of the feasibility region where all the main variables are zero and then systematically moves from corner point to corner point, while improving the value of the objective function at each stage. The process continues until the optimal solution is found.
To learn the simplex method, we try a rather unconventional approach. We first list the algorithm, and then work a problem. We justify the reasoning behind each step during the process. A thorough justification is beyond the scope of this course.
We start out with an example we solved in the last chapter by the graphical method. This will provide us with some insight into the simplex method and at the same time give us the chance to compare a few of the feasible solutions we obtained previously by the graphical method. But first, we list the algorithm for the simplex method.
THE SIMPLEX METHOD
- Set up the problem. That is, write the objective function and the inequality constraints.
- Convert the inequalities into equations. This is done by adding one slack variable for each inequality.
- Construct the initial simplex tableau. Write the objective function as the bottom row.
- The most negative entry in the bottom row identifies the pivot column.
- Calculate the quotients. The smallest quotient identifies a row. The element in the intersection of the column identified in step 4 and the row identified in this step is identified as the pivot element. The quotients are computed by dividing the far right column by the identified column in step 4. A quotient that is a zero, or a negative number, or that has a zero in the denominator, is ignored.
- Perform pivoting to make all other entries in this column zero. This is done the same way as we did with the Gauss-Jordan method.
- When there are no more negative entries in the bottom row, we are finished; otherwise, we start again from step 4.
- Read off your answers. Get the variables using the columns with 1 and 0s. All other variables are zero. The maximum value you are looking for appears in the bottom right hand corner.
Now, we use the simplex method to solve Example 3.1.1 solved geometrically in section 3.1.
Example \(\PageIndex{1}\)
Niki holds two part-time jobs, Job I and Job II. She never wants to work more than a total of 12 hours a week. She has determined that for every hour she works at Job I, she needs 2 hours of preparation time, and for every hour she works at Job II, she needs one hour of preparation time, and she cannot spend more than 16 hours for preparation. If she makes $40 an hour at Job I, and $30 an hour at Job II, how many hours should she work per week at each job to maximize her income?
Solution
In solving this problem, we will follow the algorithm listed above.
STEP 1. Set up the problem. Write the objective function and the constraints.
Since the simplex method is used for problems that consist of many variables, it is not practical to use the variables \(x\), \(y\), \(z\) etc. We use symbols \(x_1\), \(x_2\), \(x_3\), and so on.
Let
- \(x_1\) = The number of hours per week Niki will work at Job I and
- \(x_2\) = The number of hours per week Niki will work at Job II.
It is customary to choose the variable that is to be maximized as \(Z\).
The problem is formulated the same way as we did in the last chapter.
\[\begin{array}{ll}
\textbf { Maximize } & \mathrm{Z}=40 \mathrm{x}_{1}+30 \mathrm{x}_{2} \\
\textbf { Subject to: } & \mathrm{x}_{1}+\mathrm{x}_{2} \leq 12 \\
& 2 \mathrm{x}_{1}+\mathrm{x}_{2} \leq 16 \\
& \mathrm{x}_{1} \geq 0 ; \mathrm{x}_{2} \geq 0
\end{array}\nonumber \]
STEP 2. Convert the inequalities into equations. This is done by adding one slack variable for each inequality.
For example to convert the inequality \(x_1 + x_2 ≤ 12\) into an equation, we add a non-negative variable \(y_1\), and we get
\[x_1 + x_2 + y_1 = 12 \nonumber \]
Here the variable \(y_1\) picks up the slack, and it represents the amount by which \(x_1 + x_2\) falls short of 12. In this problem, if Niki works fewer than 12 hours, say 10, then \(y_1\) is 2. Later when we read off the final solution from the simplex table, the values of the slack variables will identify the unused amounts.
We rewrite the objective function \(Z = 40x_1 + 30x_2\) as \(- 40x_1 - 30x_2 + Z = 0\).
After adding the slack variables, our problem reads
\[\begin{array}{ll}
\text { Objective function } & - 40x_1 - 30x_2 + Z = 0 \\
\text { Subject to constraints: } & x_1 + x_2 + y_1 = 12 \\
& 2x_1 + x_2 + y_2 = 16 \\
& x1 ≥ 0; x2 ≥ 0
\end{array} \nonumber \]
STEP 3. Construct the initial simplex tableau. Each inequality constraint appears in its own row. (The non-negativity constraints do not appear as rows in the simplex tableau.) Write the objective function as the bottom row.
Now that the inequalities are converted into equations, we can represent the problem into an augmented matrix called the initial simplex tableau as follows.
Here the vertical line separates the left hand side of the equations from the right side. The horizontal line separates the constraints from the objective function. The right side of the equation is represented by the column C.
The reader needs to observe that the last four columns of this matrix look like the final matrix for the solution of a system of equations. If we arbitrarily choose \(x_1 = 0\) and \(x_2 = 0\), we get
\[\left[\begin{array}{ccccc}
y_{1} & y_{2} & Z & | & C \\
1 & 0 & 0 & | & 12 \\
0 & 1 & 0 & | & 16 \\
0 & 0 & 1 & | & 0
\end{array}\right]\nonumber \]
which reads
\[y_1 = 12 \quad y_2 = 16 \quad Z = 0 \nonumber \]
The solution obtained by arbitrarily assigning values to some variables and then solving for the remaining variables is called the basic solution associated with the tableau. So the above solution is the basic solution associated with the initial simplex tableau. We can label the basic solution variable in the right of the last column as shown in the table below.
STEP 4. The most negative entry in the bottom row identifies the pivot column.
The most negative entry in the bottom row is -40; therefore the column 1 is identified.
Question Why do we choose the most negative entry in the bottom row?
Answer The most negative entry in the bottom row represents the largest coefficient in the objective function - the coefficient whose entry will increase the value of the objective function the quickest.
The simplex method begins at a corner point where all the main variables, the variables that have symbols such as \(x_1\), \(x_2\), \(x_3\) etc., are zero. It then moves from a corner point to the adjacent corner point always increasing the value of the objective function. In the case of the objective function \(Z = 40x_1+ 30x_2\), it will make more sense to increase the value of \(x_1\) rather than \(x_2\). The variable \(x_1\) represents the number of hours per week Niki works at Job I. Since Job I pays $40 per hour as opposed to Job II which pays only $30, the variable \(x_1\) will increase the objective function by $40 for a unit of increase in the variable \(x_1\).
STEP 5. Calculate the quotients. The smallest quotient identifies a row. The element in the intersection of the column identified in step 4 and the row identified in this step is identified as the pivot element.
Following the algorithm, in order to calculate the quotient, we divide the entries in the far right column by the entries in column 1, excluding the entry in the bottom row.
The smallest of the two quotients, 12 and 8, is 8. Therefore row 2 is identified. The intersection of column 1 and row 2 is the entry 2, which has been highlighted. This is our pivot element.
Question Why do we find quotients, and why does the smallest quotient identify a row?
Answer When we choose the most negative entry in the bottom row, we are trying to increase the value of the objective function by bringing in the variable \(x_1\). But we cannot choose any value for \(x_1\). Can we let \(x_1 = 100\)? Definitely not! That is because Niki never wants to work for more than 12 hours at both jobs combined: \(x_1 + x_2 ≤ 12\). Can we let \(x_1 = 12\)? Again, the answer is no because the preparation time for Job I is two times the time spent on the job. Since Niki never wants to spend more than 16 hours for preparation, the maximum time she can work is 16 ÷ 2 = 8.
Now you see the purpose of computing the quotients; using the quotients to identify the pivot element guarantees that we do not violate the constraints.
Question Why do we identify the pivot element?
Answer As we have mentioned earlier, the simplex method begins with a corner point and then moves to the next corner point always improving the value of the objective function. The value of the objective function is improved by changing the number of units of the variables. We may add the number of units of one variable, while throwing away the units of another. Pivoting allows us to do just that.
The variable whose units are being added is called the entering variable, and the variable whose units are being replaced is called the departing variable. The entering variable in the above table is \(x_1\), and it was identified by the most negative entry in the bottom row. The departing variable \(y_2\) was identified by the lowest of all quotients.
STEP 6. Perform pivoting to make all other entries in this column zero.
In chapter 2, we used pivoting to obtain the row echelon form of an augmented matrix. Pivoting is a process of obtaining a 1 in the location of the pivot element, and then making all other entries zeros in that column. So now our job is to make our pivot element a 1 by dividing the entire second row by 2. The result follows.
To obtain a zero in the entry first above the pivot element, we multiply the second row by -1 and add it to row 1. We get
To obtain a zero in the element below the pivot, we multiply the second row by 40 and add it to the last row.
We now determine the basic solution associated with this tableau. By arbitrarily choosing \(x_2 = 0\) and \(y_2 = 0\), we obtain \(x_1 = 8\), \(y_1 = 4\), and \(z = 320\). If we write the augmented matrix, whose left side is a matrix with columns that have one 1 and all other entries zeros, we get the following matrix stating the same thing.
\[\left[\begin{array}{ccccc}
\mathrm{x}_{1} & \mathrm{y}_1 & \mathrm{Z} & | & \mathrm{C} \\
0 & 1 & 0 & | & 4 \\
1 & 0 & 0 & | & 8 \\
0 & 0 & 1 & | & 320
\end{array}\right] \nonumber \]
We can restate the solution associated with this matrix as \(x_1 = 8\), \(x_2 = 0\), \(y_1 = 4\), \(y_2 = 0\) and \(z = 320\). At this stage of the game, it reads that if Niki works 8 hours at Job I, and no hours at Job II, her profit Z will be $320. Recall from Example 3.1.1 in section 3.1 that (8, 0) was one of our corner points. Here \(y_1 = 4\) and \(y_2 = 0\) mean that she will be left with 4 hours of working time and no preparation time.
STEP 7. When there are no more negative entries in the bottom row, we are finished; otherwise, we start again from step 4.
Since there is still a negative entry, -10, in the bottom row, we need to begin, again, from step 4. This time we will not repeat the details of every step, instead, we will identify the column and row that give us the pivot element, and highlight the pivot element. The result is as follows.
We make the pivot element 1 by multiplying row 1 by 2, and we get
Now to make all other entries as zeros in this column, we first multiply row 1 by -1/2 and add it to row 2, and then multiply row 1 by 10 and add it to the bottom row.
We no longer have negative entries in the bottom row, therefore we are finished.
Question Why are we finished when there are no negative entries in the bottom row?
Answer The answer lies in the bottom row. The bottom row corresponds to the equation:
\[\begin{array}{l}
0 x_{1}+0 x_{2}+20 y_{1}+10 y_{2}+Z=400 \quad \text { or } \\
z=400-20 y 1-10 y 2
\end{array}\nonumber \]
Since all variables are non-negative, the highest value \(Z\) can ever achieve is 400, and that will happen only when \(y_1\) and \(y_2\) are zero.
STEP 8. Read off your answers.
We now read off our answers, that is, we determine the basic solution associated with the final simplex tableau. Again, we look at the columns that have a 1 and all other entries zeros. Since the columns labeled \(y_1\) and \(y_2\) are not such columns, we arbitrarily choose \(y_1 = 0\), and \(y_2 = 0\), and we get
\[\left[\begin{array}{ccccc}
\mathrm{x}_{1} & \mathrm{x}_{2} & \mathrm{Z} & | & \mathrm{C} \\
0 & 1 & 0 & | & 8 \\
1 & 0 & 0 & | & 4 \\
0 & 0 & 1 & | & 400
\end{array}\right] \nonumber \]
The matrix reads \(x_1 = 4\), \(x_2= 8\) and \(z = 400\).
The final solution says that if Niki works 4 hours at Job I and 8 hours at Job II, she will maximize her income to $400. Since both slack variables are zero, it means that she would have used up all the working time, as well as the preparation time, and none will be left.
FAQs
How do you solve simplex method minimization? ›
- Set up the problem.
- Write a matrix whose rows represent each constraint with the objective function as its bottom row.
- Write the transpose of this matrix by interchanging the rows and columns.
- Now write the dual problem associated with the transpose.
The first row consists of the objective function coefficients, while the last row contains the objective function value and reduced costs Zj - Cj. The last row is calculated as follows: Zj = Σ(Cbi·Pj) for i = 1.. m, where if j = 0, P0 = bi and C0 = 0, else Pj = aij.
How do you solve for maximization? ›- Write the objective function.
- Write the constraints. ...
- Graph the constraints.
- Shade the feasibility region.
- Find the corner points.
- Determine the corner point that gives the maximum value.
Any solution to the minimization problem will be a solution to the maximization problem and conversely. (Note that the value of the maximization problem will be −1 times the value of the minimization problem.) In summary: to change a max problem to a min problem, just multiply the objective function by −1.
How do you calculate minimization? ›Cost Minimization Analysis - Key takeaways
In the cost minimization formula, the marginal product of labor divided by the wage rate equals the marginal product of capital divided by the rental price of capital.
In minimization problems, the variables are the constraints and the goal is to find a solution that satisfies all of the constraints as closely as possible. In maximization problems, the variables are the constraints and the goal is to find a solution that maximizes or maximize some criterion.
What is simplex method for dummies? ›Simplex method is an approach to solving linear programming models by hand using slack variables, tableaus, and pivot variables as a means to finding the optimal solution of an optimization problem. Simplex tableau is used to perform row operations on the linear programming model as well as for checking optimality.
What is the max Z in simplex method? ›. Hence the solution is optimal. The optimal solution is Max Z=11 at x1 =3, x2 =1.
What is Phase 1 of simplex method? ›Phase I: minimize the sum of the artificial variables, starting from the BFS where the absolute value of the artificial variable for each constraint, or of the slack variable in case there is no artificial variable, is equal to that of the right-hand side.
What does maximizing an equation mean? ›maximizing: means finding the largest (or maximum) value the quantity can be. minimizing: means finding the smallest (or minimum) value the quantity can be.
What is simplex method in statistics? ›
The simplex method is a systematic procedure for testing the vertices as possible solutions. Some simple optimization problems can be solved by drawing the constraints on a graph. However, this method is useful only for systems of inequalities involving two variables.
What is the optimal condition for maximization problem in simplex method? ›Optimality condition: The entering variable in a maximization (minimization) problem is the non-basic variable having the most negative (positive) coefficient in the Z-row. The optimum is reached at the iteration where all the Z-row coefficient of the non-basic variables are non-negative (non-positive).
What is an example of minimization? ›In other words, minimizing is when we frame something to be lesser than it is by denying or dismissing its significance. Minimization can be a conscious process. For instance, a bully might deliberately downplay his rude remarks to avoid any consequences for his actions and claim that he was merely joking.
What is the maximization or minimization of the quantity? ›The maximization or minimization of some quantity is the objective in all linear programming problems. A feasible solution satisfies all the problem's constraints. An optimal solution is a feasible solution that results in the largest possible objective function value when maximizing (or smallest when minimizing).
What is the maximization problem? ›A linear programming (LP) problem is called a standard maximization problem if: We are to find the maximum (not minimum) value of the objective function. All the decision variables x1, x2, ..., xn are constrained to be non-negative. All further constraints have the form bx1 + bx2 + ..
How can we solve a maximization assignment problem? ›Maximization assignment problem is transformed into minimization problem by. Solution: The given maximization problem is converted into minimization problem by subtracting from the highest sales value (i.e., 41) with all elements of the given table.
Which is the problem of maximizing or minimizing a linear function? ›Linear programming (LP) or Linear Optimisation may be defined as the problem of maximizing or minimizing a linear function that is subjected to linear constraints.
Is maximization the same as optimization? ›First, what do these words mean? Maximize: “make as large or great as possible.” Optimize: “make the best or most effective use of (a situation, opportunity, or resource).”
Why simplex method is used? ›The simplex method is used to eradicate the issues in linear programming. It examines the feasible set's adjacent vertices in sequence to ensure that, at every new vertex, the objective function increases or is unaffected.
Why is it called the simplex method? ›The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are not actually used in the method, but one interpretation of it is that it operates on simplicial cones, and these become proper simplices with an additional constraint.
What is simplex method calculator? ›
The simplex method is universal. It allows you to solve any linear programming problems. Тhe solution by the simplex method is not as difficult as it might seem at first glance. This calculator only finds a general solution when the solution is a straight line segment.
What is the maximum value of Z 3x 4y? ›∴ Maximum value of z = 3x + 4y = 140.
What is the maximum value of Z 2x 5y? ›From the above values, we can derive that the maximum value of \[z = 2x + 5y\] is \[10\] . Hence the maximum value of \[z = 2x + 5y\] is at point \[G\left( {0,2} \right)\] . Note: To solve the above question, we have to follow certain steps which are required to graph solutions of inequality.
What are the 3 variables in simplex method minimization? ›Standard form is the baseline format for all linear programs before solving for the optimal solution and has three requirements: (1) must be a maximization problem, (2) all linear constraints must be in a less-than-or-equal-to inequality, (3) all variables are non-negative.
What are the 2 phases in simplex method? ›The process of eliminating artificial variables is performed in phase-I of the solution and phase- II is used to get an optimal solution. Since the solution of LPP is computed in two phases, it is called as Two-Phase Simplex Method.
How do you use simplex method? ›- Set up the problem. ...
- Convert the inequalities into equations. ...
- Construct the initial simplex tableau. ...
- The most negative entry in the bottom row identifies the pivot column.
- Calculate the quotients. ...
- Perform pivoting to make all other entries in this column zero.
We planned out each day in order to maximize [=make the most of] our time on vacation. I rearranged the furniture to maximize the space in my small apartment.
How do you minimize and maximize equations? ›Exclude any critical points not inside the interval [a,b]. Add to the list the endpoints a,b of the interval (and any points of discontinuity or non-differentiability!) At each point on the list, evaluate the function f: the biggest number that occurs is the maximum, and the littlest number that occurs is the minimum.
How do you tell if an equation has a maximum? ›Determine whether the function will have a minimum or a maximum depending on the coefficient of the x^2 term. If the x^2 coefficient is positive, the function has a minimum. If it is negative, the function has a maximum.
How do you find the basic variable in simplex method? ›Each variable corresponds to a column in the tableau. If the column is cleared out and has only one non-zero element in it, then that variable is a basic variable. If a column is not cleared out and has more than one non-zero element in it, that variable is non-basic and the value of that variable is zero.
What are the conditions of simplex method? ›
The objective must be maximize or minimize the function. All restrictions must be equal. All variables are not negatives. The independent terms are not negatives.
Is simplex method efficient? ›The Simplex Method for solving the Linear Programming (LP) Problem, due to George Dantzig, has been an extremely efficient computational tool for almost four decades. The method has been the subject of intense investigations for many years, but some major aspects of its behavior are not fully under- stood yet.
What is the maximum possible correlation between the two variables? ›The possible range of values for the correlation coefficient is -1.0 to 1.0. In other words, the values cannot exceed 1.0 or be less than -1.0. A correlation of -1.0 indicates a perfect negative correlation, and a correlation of 1.0 indicates a perfect positive correlation.
How do you find the maximum value of a function within an interval? ›To find relative maximums, we need to find where our first derivative changes sign. To do this, find your first derivative and then find where it is equal to zero. Because we are only concerned about the interval from -5 to 0, we only need to test points on that interval.
How do you find the maximum value of two numbers? ›- Input two numbers from user. Store it in some variable say num1 and num2 .
- Check if(num1 > num2) then print num1 is maximum.
- Check if(num2 > num1) then print num2 is maximum.
- Check if(num1 == num2) then both the numbers are equal.
Under Simplex Method, the existence of multiple optimal solutions is indicated by a situation under which a non-basic variable in the final simplex table showing optimal solution to a problem, has a net zero contribution.
How do you find the alternate solution in simplex method? ›- In Simplex algorithm, alternative solutions are detected when there are 0 valued coefficients for nonbasic variables in row-0 of the optimal tableau. - If there is no nonbasic variable with a zero coefficient in row 0 of the optimal tableau, the LP has a unique optimal solution.
When using simplex method to solve maximization problem we find that the ratios for determining the pivot row are all negative then we know that the solution is <UNK>? ›If, when we are using a Simplex table to solve a maximization problem, we find that the ratios for determining the pivot row are all negative, then we know that the solution is. unbounded.
What is simplex method explain it with an example? ›Techopedia Explains Simplex Method
The simplex method is used to eradicate the issues in linear programming. It examines the feasible set's adjacent vertices in sequence to ensure that, at every new vertex, the objective function increases or is unaffected.
Under the Simplex Method, the problem is said to have no feasible solution if at least one of the artificial variable remains in the final simplex table as basic variable with non-zero quantity.
How do you find the ratio in simplex method? ›
...
| Point | The value of z |
|---|---|
| (0, 20) | 4(0) + 12(20) = 0 + 240 = 240 |
The row that has the smallest ratio contains the basic variable that will become zero the fastest as you increase the value of the entering basic variable. Since this basic variable CANNOT become negative, it is this variable that becomes the leaving basic variable.
Does simplex method can be used for optimization of 2 or more variables? ›The simplex method is a systematic procedure for testing the vertices as possible solutions. Some simple optimization problems can be solved by drawing the constraints on a graph. However, this method is useful only for systems of inequalities involving two variables.