problem solving using simplex method

Chapter 5: Linear Programming with the Simplex Method

One of the most significant advancements in linear programming is the simplex method, developed by George Dantzig. This algorithm provides a systematic approach to finding the optimal solution to linear programming problems. In this article, we will explore the simplex method, its key concepts, and how it is applied to solve linear programming problems.

Overview of the Linear Programming with the Simplex Method

The simplex method is a systematic approach to traverse the vertices of the polyhedron containing feasible solutions in a linear programming problem. It aims to find the optimal solution by iteratively improving the objective function value. This method is considered one of the greatest inventions of modern times due to its broad applicability in solving business-related problems.

Formulating the Original Linear Programming Problem

To illustrate the simplex method, let’s consider a furniture production problem. We want to maximize the revenue generated by producing chairs and tables, subject to constraints on the availability of mahogany and labor. The original problem can be formulated as follows:

• Maximize Revenue = 45×1 + 80×2
• 5×1 + 20×2 ≤ 400 (Mahogany constraint)
• 10×1 + 15×2 ≤ 450 (Labor constraint)
• x1, x2 ≥ 0 (Non-negativity constraint)

In this formulation, x1 represents the number of chairs produced, x2 represents the number of tables produced, and the objective function maximizes the total revenue. The constraints limit the consumption of mahogany and labor within the available capacities.

Transforming into Standard Form

To apply the simplex method, we transform the original problem into the standard form, which involves converting the inequalities into equalities. We introduce slack variables to represent the surplus capacity of each constraint. In this case, we add h1 and h2 as slack variables for the mahogany and labor constraints, respectively. The problem in standard form becomes:

• Maximize Revenue = 45×1 + 80×2 + 0h1 + 0h2
• 5×1 + 20×2 + h1 = 400 (Mahogany constraint)
• 10×1 + 15×2 + h2 = 450 (Labor constraint)
• x1, x2, h1, h2 ≥ 0 (Non-negativity constraint)

The slack variables h1 and h2 represent the unused capacities of mahogany and labor, respectively. We still aim to maximize the revenue while satisfying these transformed equalities.

Basic Feasible Solutions and Canonical Form

A basic feasible solution is an initial production plan that satisfies all the constraints, with some variables set to zero. In our case, the initial solution where no chairs or tables are produced (x1=0, x2=0) represents a basic feasible solution. This solution generates zero revenue, as expected.

The non-basic variables in a basic feasible solution are set to zero, while the basic variables take positive values. In our initial solution, h1 and h2 are the basic variables, representing the unused capacities of mahogany and labor. The non-basic variables are x1 and x2, which are set to zero. This configuration is called a basic feasible solution and is an important concept in linear programming.

To express the problem in canonical form, we represent the basic variables (h1 and h2) in terms of the non-basic variables (x1 and x2). Similarly, we substitute the non-basic variables in the objective function. This process is known as pivoting. The canonical form of the problem becomes:

• Maximize z = 45×1 + 80×2 + 0h1 + 0h2
• x1 = 0 + (1/5)h1 – (4/5)h2 (Transformed mahogany constraint)
• x2 = 0 – (2/5)h1 + (3/5)h2 (Transformed labor constraint)
• x1, x2, h1, h2 ≥ 0

In the canonical form, the objective function and constraints are expressed with respect to the basic variables x1 and x2. The reduced costs associated with the non-basic variables x1 and x2 (coefficients in the objective function) indicate the potential improvement in the objective function value if these variables enter the basis.

Iteration and Optimal Solution

In each iteration of the simplex method, we analyze the non-basic variables and their coefficients. If all the non-basic variables have coefficients ≤ 0, the current solution is optimal. Negative reduced costs associated with the non-basic variables indicate that increasing these variables would decrease the objective function value.

If there is a non-basic variable with a positive reduced cost, we choose the one with the largest coefficient to enter the basis. To determine the maximum value for this variable, we perform the minimum ratio test using the transformed equations. The minimum ratio identifies the constraint that limits the increase of the non-basic variable while staying feasible.

After selecting the entering variable, we perform pivoting to express the problem in canonical form with respect to the new basic variables. This process continues iteratively until we reach an optimal solution or determine that the problem is unbounded.

The simplex method provides a systematic approach to solving linear programming problems by iteratively improving the objective function value. By transforming the problem into the standard form and expressing it in canonical form, we can identify basic feasible solutions and optimize the objective function. The simplex method is a fundamental tool in linear programming, enabling efficient optimization in various industries and applications.

Download the complete  Linear Programming Tutorial Series slide deck .

View the entire series:

• Welcome: Linear Programming Tutorial
• Chapter 1: Mathematical Programming
• Chapter 2: Introduction to Linear Programming
• Chapter 3: Mixed Integer Linear Programming Problems
• Chapter 4: Furniture Factory Problem
• Chapter 5: Simplex Method
• Chapter 6: Modeling and Solving Linear Programming Problems
• Chapter 7: Sensitivity Analysis of Linear Programming Problems
• Chapter 8: Multiple Optimal Solutions
• Chapter 9: Unbounded Linear Programming Problems
• Chapter 10: Infeasible Linear Programming Problems
• Chapter 11: Linear Programming Overview – Further Considerations
• Chapter 12: Duality in Linear Programming
• Chapter 13: Optimality Conditions
• Chapter 14: Dual Simplex Method

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simplex method

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• Mathematics LibreTexts - Simplex Method
• The University of Texas at Dallas - Atlas Service Portal - The Simplex Method in Tabular Form
• Iowa State University - The Simplex Method

simplex method , standard technique in linear programming for solving an optimization problem, typically one involving a function and several constraints expressed as inequalities. The inequalities define a polygonal region, and the solution is typically at one of the vertices. 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. In practice, problems often involve hundreds of equations with thousands of variables, which can result in an astronomical number of extreme points. In 1947 George Dantzig , a mathematical adviser for the U.S. Air Force, devised the simplex method to restrict the number of extreme points that have to be examined. The simplex method is one of the most useful and efficient algorithms ever invented, and it is still the standard method employed on computers to solve optimization problems.

First, the method assumes that an extreme point is known. (If no extreme point is given, a variant of the simplex method, called Phase I, is used to find one or to determine that there are no feasible solutions.) Next, using an algebraic specification of the problem, a test determines whether that extreme point is optimal. If the test for optimality is not passed, an adjacent extreme point is sought along an edge in the direction for which the value of the objective function increases at the fastest rate. Sometimes one can move along an edge and make the objective function value increase without bound. If this occurs, the procedure terminates with a prescription of the edge along which the objective goes to positive infinity . If not, a new extreme point is reached having at least as high an objective function value as its predecessor. The sequence described is then repeated. Termination occurs when an optimal extreme point is found or the unbounded case occurs. Although in principle the necessary steps may grow exponentially with the number of extreme points, in practice the method typically converges on the optimal solution in a number of steps that is only a small multiple of the number of extreme points.

To illustrate the simplex method, consider the example of a factory producing two products, x 1 and x 2 . If the profit on the second type is twice that on the first, then x 1 + 2 x 2 represents the total profit. The function x 1 + 2 x 2 is known as the objective function.

Clearly, the profit will be highest if the factory devotes its entire production capacity to making the second type of commodity. In a practical situation, however, this may not be possible; a set of constraints is introduced by such factors as availability of machine time, labour, and raw materials. For example, if the second type of commodity requires a raw material that is limited so that no more than five can be made in any batch, then x 2 must be less than or equal to five; i.e., x 2 ≤ 5. If the first commodity requires another type of material limiting it to eight per batch, then x 1 ≤ 8. If x 1 and x 2 take equal time to make and the machine time available allows a maximum of 10 to be made in a batch, then x 1 + x 2 must be less than or equal to 10; i.e., x 1 + x 2 ≤ 10.

Two other constraints are that x 1 and x 2 must each be greater than or equal to zero, because it is impossible to make a negative number of either; i.e., x 1 ≥ 0 and x 2 ≥ 0. The problem is to find the values of x 1 and x 2 for which the profit is a maximum. Any solution can be denoted by a pair of numbers ( x 1 , x 2 ); for example, if x 1 = 3 and x 2 = 6, the solution is (3, 6). These numbers can be represented by points plotted on two axes, as shown in the figure . On this graph the distance along the horizontal axis represents x 1 and that along the vertical represents x 2 . Because of the constraints given above, the feasible solutions must lie within a certain well-defined region of the graph. For example, the constraint x 1 ≥ 0 means that points representing feasible solutions lie on or to the right of the x 2 axis. Similarly, the constraint x 2 ≥ 0 means that they also lie on or above the x 1 axis . Application of the entire set of constraints gives the feasible solution set, which is bounded by a polygon formed by the intersection of the lines x 1 = 0, x 2 = 0, x 1 = 8, x 2 = 5, and x 1 + x 2 = 10. For example, production of three items of commodity x 1 and four of x 2 is a feasible solution since the point (3, 4) lies in this region. To find the best solution, however, the objective function x 1 + 2 x 2 = k is plotted on the graph for some value of k , say k = 4. This value is indicated by the broken line in the figure. As k is increased, a family of parallel lines are produced, and the line for k = 15 just touches the constraint set at the point (5, 5). If k is increased further, the values of x 1 and x 2 will lie outside the set of feasible solutions. Thus, the best solution is that in which equal quantities of each commodity are made. It is no coincidence that an optimal solution occurs at a vertex, or “extreme point,” of the region. This will always be true for linear problems, although an optimal solution may not be unique. Thus, the solution of such problems reduces to finding which extreme point (or points) yields the largest value for the objective function.

In the simplex method, the problem is first put into canonical form by converting the linear inequalities into equalities by introducing “slack variables” x 3 ≥ 0 (so that x 1 + x 3 = 8), x 4 ≥ 0 (so that x 2 + x 4 = 5), x 5 ≥ 0 (so that x 1 + x 2 + x 5 = 10), and the variable x 0 for the value of the objective function (so that x 1 + 2 x 2 − x 0 = 0). The problem may then be restated as that of finding nonnegative quantities x 1 , …, x 5 and the largest possible x 0 satisfying the resulting equations. One obvious solution is to set the objective variables x 1 = x 2 = 0, which corresponds to the extreme point at the origin. If one of the objective variables is increased from zero while the other one is fixed at zero, the objective value x 0 will increase as desired (subject to the slack variables satisfying the equality constraints). The variable x 2 produces the largest increase of x 0 per unit change; so it is used first. Its increase is limited by the nonnegativity requirement on the variables. In particular, if x 2 is increased beyond 5, x 4 becomes negative.

At x 2 = 5, this situation produces a new solution—( x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ) = (10, 0, 5, 8, 0, 5)—that corresponds to the extreme point (0, 5) in the figure. The system of equations is put into an equivalent form by solving for the nonzero variables x 0 , x 2 , x 3 , x 5 in terms of those variables now at zero; i.e., x 1 and x 4 . Thus, the new objective function is x 1 − 2 x 4 = −10, while the constraints are x 1 + x 3 = 8, x 2 + x 4 = 5, and x 1 − x 4 + x 5 = 5. It is now apparent that an increase of x 1 while holding x 4 equal to zero will produce a further increase in x 0 . The nonnegativity restriction on x 3 prevents x 1 from going beyond 5. The new solution—( x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ) = (15, 5, 5, 3, 0, 0)—corresponds to the extreme point (5, 5) in the figure. Finally, since solving for x 0 in terms of the variables x 4 and x 5 (which are currently at zero value) yields x 0 = 15 − x 4 − x 5 , it can be seen that any further change in these slack variables will decrease the objective value. Hence, an optimal solution exists at the extreme point (5, 5).

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Linear Programming Optimization: The Simplex Method

Part 3: the algorithm under the hood.

Jarom Hulet

Towards Data Science

Up until now, this series has covered the basics of linear programming. In this article, we are going to move from basic concepts into the details under the hood! This article will cover the simplex method, which is the algorithm that is often used to solve linear programming problems. While we will solve a simple linear programming example by hand with the simplex method, our focus will be on the intuition of the algorithm rather than memorizing the algorithmic steps (we have computers for that kind of stuff!).

Here is what we will cover:

Why the simplex method is needed

• Moving from graphical solutions to algebraic
• Demonstrating how the simplex method works with a simple example

Here is the link to the list that contains all of the articles I’ve written so far for this series:

Linear Programming

Written by Jarom Hulet

Data Scientist | Self-Proclaimed Data Philosopher

Text to speech

Simplex Method for Solution of L.P.P (With Examples) | Operation Research

After reading this article you will learn about:- 1. Introduction to the Simplex Method 2. Principle of Simplex Method 3. Computational Procedure 4. Flow Chart.

Introduction to the Simplex Method :

Simplex method also called simplex technique or simplex algorithm was developed by G.B. Dantzeg, An American mathematician. Simplex method is suitable for solving linear programming problems with a large number of variable. The method through an iterative process progressively approaches and ultimately reaches to the maximum or minimum values of the objective function.

Principle of Simplex Method :

It has not been possible to obtain the graphical solution to the LP problem of more than two variables. For these reasons mathematical iterative procedure known as ‘Simplex Method’ was developed. The simplex method is applicable to any problem that can be formulated in-terms of linear objective function subject to a set of linear constraints.

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The simplex method provides an algorithm which is based on the fundamental theorem of linear programming. This states that “the optimal solution to a linear programming problem if it exists, always occurs at one of the corner points of the feasible solution space.”

The simplex method provides a systematic algorithm which consist of moving from one basic feasible solution to another in a prescribed manner such that the value of the objective function is improved. The procedure of jumping from vertex to the vertex is repeated. The simplex algorithm is an iterative procedure for solving LP problems.

It consists of:

(i) Having a trial basic feasible solution to constraints equation,

(ii) Testing whether it is an optimal solution,

(iii) Improving the first trial solution by repeating the process till an optimal solution is obtained.

Computational Procedure of Simplex Method :

The computational aspect of the simplex procedure is best explained by a simple example.

Consider the linear programming problem:

Maximize z = 3x 1 + 2x 2

Subject to x 1 + x 2 , ≤ 4

x 1 – x 2 , ≤ 2

x 1 , x 2 , ≥ 4

< 2 x v x 2 > 0

The steps in simplex algorithm are as follows:

Formulation of the mathematical model:

(i) Formulate the mathematical model of given LPP.

(ii) If objective function is of minimisation type then convert it into one of maximisation by following relationship

Minimise Z = – Maximise Z*

When Z* = -Z

(iii) Ensure all b i values [all the right side constants of constraints] are positive. If not, it can be changed into positive value on multiplying both side of the constraints by-1.

In this example, all the b i (height side constants) are already positive.

(iv) Next convert the inequality constraints to equation by introducing the non-negative slack or surplus variable. The coefficients of slack or surplus variables are zero in the objective function.

In this example, the inequality constraints being ‘≤’ only slack variables s 1 and s 2 are needed.

Therefore given problem now becomes:

The first row in table indicates the coefficient c j of variables in objective function, which remain same in successive tables. These values represent cost or profit per unit of objective function of each of the variables.

The second row gives major column headings for the simple table. Column C B gives the coefficients of the current basic variables in the objective function. Column x B gives the current values of the corresponding variables in the basic.

Number a ij represent the rate at which resource (i- 1, 2- m) is consumed by each unit of an activity j (j = 1,2 … n).

The values z j represents the amount by which the value of objective function Z would be decreased or increased if one unit of given variable is added to the new solution.

It should be remembered that values of non-basic variables are always zero at each iteration.

So x 1 = x 2 = 0 here, column x B gives the values of basic variables in the first column.

So 5, = 4, s 2 = 2, here; The complete starting feasible solution can be immediately read from table 2 as s 1 = 4, s 2 , x, = 0, x 2 = 0 and the value of the objective function is zero.

Flow Chart of Simplex Method :

Linear Programming: Simplex Method

• Post last modified: 22 July 2022
• Reading time: 6 mins read
• Post category: Operations Research

What is Simplex Method Linear Programming?

The simplex method is an algorithm used to calculate the optimal solution to an LP problem. It is a systematically performed iterative procedure to identify the optimal solution from the set of feasible solutions. You might remember that in the graphical solution, the unique optimal solution to the LP problem occurred at a corner point or vertex of the feasible region.

The simplex algorithm also starts at one corner point of the feasible region and at each iteration moves to an adjacent vertex in sequence, until the corner point corresponding to the optimal solution is reached.

Table of Content

• 1 What is Simplex Method Linear Programming?
• 2 Introduction LP Simplex Method
• 3 Important Terms of Linear Programming for Simplex Method
• 4 Steps for Solving Linear Programming using Simplex Method

Introduction LP Simplex Method

In the previous article, you studied how to solve linear programming problems graphically . You also studied some special cases in the previous chapter.

The graphical approach is not applicable to problems with more than two variables are involved. The simplex method is more suitable for solving LP problems in three or more variables, or problems involving many constraints. The simplex method is a mathematical solution technique where the model is formulated as a tableau on which a series of repetitive mathematical steps are performed to reach the optimal solution.

The simplex method was developed in 1947 by George B. Dantzig. He put forward the simplex method for obtaining an optimal solution to a linear programming problem, i.e., for obtaining a non-negative solution of a system of m linear equations in n variables which maximises a given linear functional of the vector of variables.

It is one of the most universally applied mathematical techniques, the popularity of the simplex method comes from the fact that it can indicate at each phase if the solution is optimal and if the solution can be improved and what that improved solution would be.

All LP problems can be solved using the simplex method. It is much more adaptable to computers than the graphical method, therefore, it is more suited for complex problems despite being mathematically more complex. Using the simplex method, a decision maker can also identify degeneracy, unbounded solutions, alternate solutions, and infeasible solutions along with redundant constraints.

Important Terms of Linear Programming for Simplex Method

• Pivot column : In a row-echelon matrix, the first non-zero entry of each row is called a pivot, and the columns where pivots occur are called pivot columns or key columns. This is the column with the most negative index number, and it shows the entering variable in the basis.
• Pivot row : It is the row which contains the smallest non-negative ratio is called the pivot row or key row. This row has the smallest quotient obtained after dividing the values of quantity column by key column for each row. It shows the exiting variable from the basis.
• Pivot element/ number : The pivot element of a matrix is selected first by an algorithm to do certain computations. The pivot element is at the intersection of the pivot column with pivot row.
• Simplex tableau : The simplex tableau organises the model into a form that simplifies the application of the mathematical steps. An LP problem in standard form can be represented as a tableau of the form given below:
• Basis : It is the set of variables not constrained to equal zero in the current basic solution. Basic variables are those variables which make up the basis.
• Non-basic variables : These are all variables other than basic variables.
• Iteration: This refers to the steps performed to progress from one feasible solution to another in simplex method.
• Cj Row : The coefficients of the variables in the objective function occur in this row.
• Zj Row : The Zj row element shows the increase or decrease in objective function value when one unit of that variable is brought into the solution.
• Zj – Cj Row : It is also called the index row; the elements of this row depict net contribution/ loss per unit when one unit of that vari- able is brought into the solution.

Steps for Solving Linear Programming using Simplex Method

• To apply the simplex method to solve an LP problem, the problem first needs to be put into the standard form. For this, the inequalities in constraints must be replaced by equalities by adding slack variables.
• Now, organise a simplex tableau using slack variables.
• Select a pivot column i.e., the column that has the smallest number in the last row.
• Divide each element in the right most columns with the corresponding element in the pivot column. The row with the smallest non-negative quotient is the pivot row.
• Locate the pivot element/number at the intersection of the pivot row and pivot column.
• Calculate new values for the pivot row by dividing every number in the pivot row by the pivot element.
• Calculate new values for each remaining row using the formula: (New row number) = (no in old row) – (no in old row above or below pivot number) x (corresponding no in the new row)
• The goal is to have no negative indicators in the first row. The simplex method is iterative, i.e., we repeat steps 5, 6 and 7 until all numbers on the first row are positive.

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Simplex Method Calculator – Two Phase Online 🥇

Simplex method calculator - free version, members-only content, do you already have a membership, get membership.

The free version of the calculator shows you each of the intermediate tableaus that are generated in each iteration of the simplex method, so you can check the results you obtained when solving the problem manually.

Advanced Functions of the simplex method online calculator – Two-Phase

Let's face it, the simplex method is characterized by being a meticulous and impractical procedure, because if you fail in an intermediate calculation you can compromise the final solution of the problem. In that sense, it is important for the student to know the step by step procedure to obtain each of the values in the iterations. Thus, in PM Calculators we have improved our application to include a complete step-by-step explanation of the calculations of the method. You can access this tool and others (such as the big m calculator and the graphical linear programming calculator ) by becoming a member of our membership .

Within the functionality that this application counts we have:

• Ability to solve exercises with up to 20 variables and 50 constraints.
• Explanation of how to determine the optimality condition.
• Explanation of the criteria to establish the feasibility condition.
• Detail of the calculations performed to obtain the vector of reduced costs, the pivot row and the other rows of the table.
• For exercises with artificial variables it becomes a two-phase method calculator .
• Explanation of the special cases such as unbounded and infeasible solutions.

You can find complete examples of how the application works in this link .

How to use the simplex method online calculator

To use our tool you must perform the following steps:

• Enter the number of variables and constraints of the problem.
• Select the type of problem: maximize or minimize .
• Enter the coefficients in the objective function and the constraints. You can enter negative numbers, fractions, and decimals (with point).
• Click on “Solve”.
• The online software will adapt the entered values ​​to the standard form of the simplex algorithm and create the first tableau .
• Depending on the sign of the constraints, the normal simplex algorithm or the two phase method is used.
• We can see step by step the iterations and tableaus of the simplex method calculator.
• In the last part will show the results of the problem.

We have considered for our application to solve problems with a maximum of 20 variables and 50 restrictions; this is because exercises with a greater number of variables would make it difficult to follow the steps using the simplex method. For problems with more variables, we recommend using other method.

Below we show some reference images of the step by step and the result of the following example:

The following problem is required to be maximized:

Objective Function Z = 3X 1 + 2X 2

Subject to the following restrictions

2X 1 + X 2 ≤ 18 2X 1 + 3X 2 ≤ 42 X 1 , X 2 ≥ 0

We enter the number of variables and constraints:

Enter the coefficients of the equations / inequalities of the problem and click on Solve:

Next you will see the step by step in obtaining the solution as well as the calculation of the vector of reduced costs:

The calculation of the values of the pivot row:

Until the final result:

Final reflection

Our free simplex minimizing and maximizing calculator is being used by thousands of students every month and has become one of the most popular online Simplex method calculators available. In addition, our full version has been helping hundreds of students study and do their homework faster and giving them more time to devote to their personal activities.

If you have questions about it or find an error in our application, we will appreciate if you can write to us on our contact page .

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4.3.1: Minimization By The Simplex Method (Exercises)

• Last updated
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• Page ID 37872

• Rupinder Sekhon and Roberta Bloom
• De Anza College

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SECTION 4.3 PROBLEM SET: MINIMIZATION BY THE SIMPLEX METHOD

In problems 1-2, convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method.

1) \[\begin{aligned} \text { Minimize } & \mathrm{z}=6 \mathrm{x}_{1}+8 \mathrm{x}_{2} \\ \text { subject to } & 2 \mathrm{x}_{1}+3 \mathrm{x}_{2} \geq 7 \\ & 4 \mathrm{x}_{1}+5 \mathrm{x}_{2} \geq 9 \\ & \mathrm{x}_{1}, \mathrm{x}_{2} \geq 0 \end{aligned} \nonumber \]

2) \[\begin{array}{ll} \text { Minimize } & \mathrm{z}=5 \mathrm{x}_{1}+6 \mathrm{x}_{2}+7 \mathrm{x}_{3} \\ \text { subject to } & 3 \mathrm{x}_{1}+2 \mathrm{x}_{2}+3 \mathrm{x}_{3} \quad \geq 10 \\ & 4 \mathrm{x}_{1}+3 \mathrm{x}_{2}+5 \mathrm{x}_{3} \geq 12 \\ &\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3} \geq & 0 \end{array} \nonumber \]

In problems 3-4, convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method.

3) \[\begin{array}{lr} \text { Minimize } & \mathrm{z}=4 \mathrm{x}_1+3 \mathrm{x}_2 \\ \text { subject to } & \mathrm{x}_{1}+\mathrm{x}_{2} \geq 10 \\ & 3 \mathrm{x}_{1}+2 \mathrm{x}_{2} \geq 24 \\ & \mathrm{x}_{1}, \mathrm{x}_{2} \geq 0 \end{array} \nonumber \]

4) A diet is to contain at least 8 units of vitamins, 9 units of minerals, and 10 calories. Three foods, Food A, Food B, and Food C are to be purchased. Each unit of Food A provides 1 unit of vitamins, 1 unit of minerals, and 2 calories. Each unit of Food B provides 2 units of vitamins, 1 unit of minerals, and 1 calorie. Each unit of Food C provides 2 units of vitamins, 1 unit of minerals, and 2 calories. If Food A costs $3 per unit, Food B costs $2 per unit and Food C costs $3 per unit, how many units of each food should be purchased to keep costs at a minimum?

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Example (part 1): Simplex method

Solve using the Simplex method the following problem:

Consider the following steps:

A change is made to the variable naming, establishing the following correspondences:

• x becomes X 1
• y becomes X 2

As the independent terms of all restrictions are positive no further action is required. Otherwise there would be multiplied by "-1" on both sides of the inequality (noting that this operation also affects the type of restriction).

The inequalities become equations by adding slack , surplus and artificial variables as the following table:

In this case, a slack variable (X 3 , X 4 and X 5 ) is introduced in each of the restrictions of ≤ type, to convert them into equalities, resulting the system of linear equations:

• Match the objective function to zero. Z - 3·X 1 - 2·X 2 - 0·X 3 - 0·X 4 - 0·X 5 = 0

The initial tableau of Simplex method consists of all the coefficients of the decision variables of the original problem and the slack, surplus and artificial variables added in second step (in columns, with P 0 as the constant term and P i as the coefficients of the rest of X i variables), and constraints (in rows). The C b column contains the coefficients of the variables that are in the base.

The first row consists of the objective function coefficients, while the last row contains the objective function value and reduced costs Z j - C j .

The last row is calculated as follows: Z j = Σ(C bi ·P j ) for i = 1..m, where if j = 0, P 0 = b i and C 0 = 0, else P j = a ij . 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 j = -C j .

If the objective is to maximize, when in the last row (indicator row) there is no negative value between discounted costs (P 1 columns below) the stop condition is reached.

In that case, the algorithm reaches the end as there is no improvement possibility. The Z value (P 0 column) is the optimal solution of the problem.

Another possible scenario is all values are negative or zero in the input variable column of the base. This indicates that the problem is not limited and the solution will always be improved.

Otherwise, the following steps are executed iteratively.

First, input base variable is determined. For this, column whose value in Z row is the lesser of all the negatives is chosen. In this example it would be the variable X 1 (P 1 ) with -3 as coefficient.

If there are two or more equal coefficients satisfying the above condition (case of tie), then choice the basic variable.

The column of the input base variable is called pivot column (in green color).

Once obtained the input base variable, the output base variable is determined. The decision is based on a simple calculation: divide each independent term (P 0 column) between the corresponding value in the pivot column, if both values are strictly positive (greater than zero). The row whose result is minimum score is chosen.

If there is any value less than or equal to zero, this quotient will not be performed. If all values of the pivot column satisfy this condition, the stop condition will be reached and the problem has an unbounded solution (see Simplex method theory ).

In this example: 18/2 [=9] , 42/2 [=21] and 24/3 [=8]

The term of the pivot column which led to the lesser positive quotient in the previous division indicates the row of the slack variable leaving the base. In this example, it is X 5 (P 5 ), with 3 as coefficient. This row is called pivot row (in green ).

If two or more quotients meet the choosing condition (case of tie), other than that basic variable is chosen (wherever possible).

The intersection of pivot column and pivot row marks the pivot value , in this example, 3.

The new coefficients of the tableau are calculated as follows:

• In the pivot row each new value is calculated as: New value = Previous value / Pivot
• In the other rows each new value is calculated as: New value = Previous value - (Previous value in pivot column * New value in pivot row)

So the pivot is normalized (its value becomes 1), while the other values of the pivot column are canceled (analogous to the Gauss-Jordan method).

Calculations for P 4 row are shown below:

The tableau corresponding to this second iteration is:

• 6.1. The input base variable is X 2 (P 2 ), since it is the variable that corresponds to the column where the coefficient is -1.
• 6.2. To calculate the output base variable, the constant terms P 0 column) are divided by the terms of the new pivot column: 2 / 1/3 [=6] , 26 / 7/3 [=78/7] and 8 / 1/3 [=24]. As the lesser positive quotient is 6, the output base variable is X 3 (P 3 ).
• 6.3. The new pivot is 1/3.

• 6.1. The input base variable is X 5 (P 5 ), since it is the variable that corresponds to the column where the coefficient is -1.
• 6.2. To calculate the output base variable, the constant terms (P 0 ) are divided by the terms of the new pivot column: 6/(-2) [=-3] , 12/4 [=3] , and 6/1 [=6]. In this iteration, the output base variable is X 4 (P 4 ).
• 6.3. The new pivot is 4.

It is noted that in the last row, all the coefficients are positive, so the stop condition is fulfilled.

The optimal solution is given by the val-ue of Z in the constant terms column (P 0 column), in the example: 33. In the same column, the point where it reaches is shown, watching the corresponding rows of input decision variables: X 1 = 3 and X 2 = 12.

Undoing the name change gives x = 3 and y = 12.

Solve with PHPSimplex .

PHPSimplex Version 0.81

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Developed by: Daniel Izquierdo Granja Juan José Ruiz Ruiz

English translation by: Luciano Miguel Tobaria

French translation by: Ester Rute Ruiz

Portuguese translation by: Rosane Bujes