# Chapter 4: The Mechanics of the Simplex Method

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 4: The Mechanics of the Simplex Method The simplex method is a remarkably simple and elegant algorithmic engine for solving linear programs. In this chapter we will examine the internal mechanics of the simplex method as formalized in the simplex tableau, a table representation of the basis at any cornerpoint. The tableau contains all the information that is needed to decide on the exchange of variables that drives the movement between cornerpoints as the simplex method advances. Using the tableau, you can solve small LPs by hand, though this is tedious and error-prone. More importantly, the tableau calculations are similar to the calculations carried out by computer implementations of the simplex method. You need a good grasp of the tableau calculations to understand how the computer does its work on large LPs, how the computer calculations can go wrong, and why certain modifications to the basic method permit faster and larger solutions by computer. Let s review what we know so far about the simplex method by presenting a high level view of the algorithm, as in Figure 4.1. Phase 1: Find an initial cornerpoint feasible solution (basic feasible solution). If none is found, then the model is infeasible, so exit. Note: If dealing with a standard form LP, the origin is always a basic feasible solution, so you can always start there. If it is not a standard form LP, then you must use a special phase 1 procedure that we will study later. Phase 2: Iterate until the stopping conditions are met..2.1 Are we optimal yet? Look at the current version of the objective function to see if an entering basic variable is available. If none is available, then exit with the current basic feasible solution as the optimum solution..2.2 Select entering basic variable: choose the nonbasic variable that gives the fastest rate of increase in the objective function value. 2.3 Select the leaving basic variable by applying the Minimum Ratio Test. 2.4 Update the equations to reflect the new basic feasible solution. 2.5 Go to Step 2.1. Figure 4.1: Overview of the simplex method. The Simplex Tableau The Acme Bicycle Company problem is a standard form LP, so we know that the origin is a basic feasible solution (feasible cornerpoint). Here is the simplex tableau for the basic feasible solution for ABC at the origin: 1

2 basic eqn. Z x 1 x 2 s 1 s 2 s 3 RHS MRT Z never s s s Tableau 4.1: ABC at the origin. The tableau is just a way of writing the equation versions of the objective function and the constraints in a nice aligned manner. The second column, titled eqn. no., gives a label to each of the equations. Equation 0 is the objective function, and equations 1-3 are the three constraint equations. The RHS column is the right hand side of the equation, where the constant goes. For example, equation 3 could be written out in full as: or more simply as: 0Z + 1x 1 +1x 2 + 0s 1 + 0s 2 + 1s 3 = 4 x 1 + x 2 + s 3 = 4 which you will recognize as the equality version of the original constraint on the metal finishing machine production limit. The objective function is also rewritten to align with the constraints. The original version was Z = 15x x 2, but we have moved all of the variables to the left hand side to create this equivalent version: Z 15x 1 10x 2 = 0. The MRT column is for the minimum ratio test. We will fill this column in as we begin the simplex calculations. The basic variable column shows the single basic variable that occurs in each equation. The most important thing about Tableau 4.1 is that it is in proper form (remember, standard form is a type of LP, but proper form is the way that a tableau is written). A tableau in proper form has these characteristics: exactly 1 basic variable per equation. the coefficient of the basic variable is always exactly +1, and the coefficients above and below the basic variable in the same column are all 0. Z is treated as the basic variable for the objective function row (equation 0). A big advantage of proper form is that you can always read the current solution directly from the tableau. This is because there is exactly one basic variable per row, and the coefficient of that variable is +1. All the other variables in the row are nonbasic (set to zero, remember?), so the value of any variable is just given by the value shown in the right hand side column. For example, in Tableau 4.1, the value of s 3 is 4. What s the value of x 1? Well, x 1 does not appear in the list of basic variables, which means it must be nonbasic, and nonbasic variables are always zero, hence x 1 is zero in this basic feasible solution. What s the value of the objective function in this tableau? Because Z is the 2

3 basic variable for equation 0, we can read it s current value from the RHS column: its value is 0. Now we can use the tableau to keep track of our calculations as we go through the steps of phase 2 of the simplex method as described in Figure 4.1. Step 2.1: Are we optimal yet? We are optimal if no entering basic variable is available. Recall that the entering basic variable chosen is the one that gives the fastest rate of increase in the objective function value. In the original version of the objective function, Z = 15x x 2, the obvious entering basic variable is x 1. In the tableau, however, we have moved all of the variables to the left hand side of the equation, to obtain the form Z 15x 1 10x 2 = 0. Because of this the rule is a little bit different: choose the variable in the objective function row that has the most negative value as the entering basic variable. In this step, then, just check to see if there are any negative coefficients in the objective function row. There are two potential entering basic variables in the tableau at this point, so we are not optimal yet: the simplex iterations must continue. Of course, we are only selecting from among currently nonbasic variables at this point: these are the only ones that could enter the basis. A nice feature of proper form is that only nonbasic variables can have a nonzero value in the objective function row, so you never have to worry about accidentally selecting a basic variable (which is already in the basic set) as the entering basic variable. Proper form enforces this by requiring that each basic variable have all zeroes in its column, except for the row in which it is the basic variable. Step 2.2: Select the entering basic variable As described above, when using the simplex tableau, the entering basic variable is the nonbasic variable in the objective function row (equation 0) that has the most negative coefficient. In Tableau 4.1, this is x 1 whose objective function coefficient of 15 is more negative than the objective function coefficient of 10 for x 2. The column for the entering basic variable is called the pivot column. The pivot column is highlighted for ease of reference in Tableau 4.2. Recall that the entering basic variable, which is currently nonbasic and therefore set to zero, will now be allowed to take on a positive value. Step 2.3: Select the leaving basic variable The minimum ratio test is used to determine the leaving basic variable. Recall that this test determines which constraint most limits the increase in the value of the entering basic variable. The most limiting constraint is the one whose basic variable is driven to zero first as the entering basic variable increases in value. 3

4 For the minimum ratio test, look only at the entries in the pivot column (the column for the entering basic variable) for the constraint rows, and calculate: There are two special cases: (RHS)/(coefficient of entering basic variable) if the coefficient of the entering basic variable is zero: enter no limit in the minimum ratio test column if the coefficient of the entering basic variable is negative: enter no limit in the minimum ratio test column. The minimum ratio test is never applied to the objective function row. Why? The objective function row is not a constraint, so it can never limit the increase in the value of the entering basic variable. The objective function always goes along for the ride, just measuring the value of the objective at any point that it is given. As usual in the minimum ratio test, the leaving basic variable is associated with the row that has the minimum value of the ratio test. This row is called the pivot row and is highlighted for ease of reference in Tableau 4.2. At this point, we have determined that we are not optimal, and have selected an entering basic variable and a leaving basic variable. These can now be swapped so that we can move to the next basic feasible solution (feasible cornerpoint). The calculations so far are summarized in Tableau 4.2. basic eqn. Z x 1 x 2 s 1 s 2 s 3 RHS MRT Z never s /1 = 2 s no limit s /1 = 4 Tableau 4.2: Updated ABC at the origin. Step 2.4: Update the tableau Now that the entering and leaving basic variables are known, we can move to the new and better basic feasible solution. This is easily done by putting the tableau back into proper form. It is not in proper form now because the exchange of the entering and leaving basic variables has not yet been done, so for example, s 1 still shows as a basic variable when it should now be nonbasic, and the column for x 1, the new basic variable does not have zeros everywhere except in its own row. Here are the steps for putting the tableau back into proper form: Step 2.4.1: In the column entitled basic variable, replace the leaving basic variable listed for the pivot row by the entering basic variable. In Tableau 4.2 the basic variable listed for equation 1 is s 1, but in Tableau 4.3 it is x 1. 4

5 Step 2.4.2: The tableau element where the pivot row and the pivot column intersect is known as the pivot element. This will be the coefficient associated with the new basic variable for this row, so if it is not already +1, then we must divide all of the elements in the pivot row by the pivot element to obtain a +1 in the pivot element position. Luckily, we already have a +1 in the pivot element position in Tableau 4.2. Step 2.4.3: Now we need to eliminate all of the coefficients in the pivot column except the pivot element. This is done by simple Gaussian operations. To remove the pivot column element in some row k for example, proceed as follows: (new row k) = (row k) (pivot column coefficient in row k) (pivot row) Remember that the pivot row has been updated by now so that the pivot element is +1. For example, to clean out the -15 that appears in the objective function row, new Z coefficient= 1 - (-15) 0 = 1 new x 1 coefficient = (-15) 1 = 0 new x 2 coefficient = (-15) 0 = -10 new s 1 coefficient = 0 - (-15) 1 = 15 new s 2 coefficient = 0 - (-15) 0 = 0 new s 2 coefficient = 0 - (-15) 0 = 0 new RHS = 0 - (-15) 2 = 30 Always follow the procedure given above to make sure that there is never any change in the Z column, because Z will always be the basic variable for the objective function, so you must always maintain a +1 in the equation 0 row under the Z column. Tableau 4.3 shows the tableau after it has been put into proper form by eliminating all of the elements in the pivot row except the +1 for the pivot element. This now summarizes the second basic feasible solution (or cornerpoint, if thinking graphically). basic eqn. Z x 1 x 2 s 1 s 2 s 3 RHS MRT Z never x s s Tableau 4.3: The second basic feasible solution. As before, the values of the basic variables at the second feasible solution can be read directly from Tableau 4.3. For example, the value of x 1 is now 2. Recalling that nonbasic variables are always zero, the new basic feasible solution is (x 1,x 2,s 1,s 2,s 3 ) = (2,0,0,3,2). The value of Z can also be read directly from the tableau because it is the basic variable for equation 0: its value now is 30, which is an improvement over the value at the initial basic feasible solution. 5

6 In putting the tableau into proper form, we have reached the second basic feasible solution. This process of moving from one basic feasible solution to the next is called pivoting, and so we have just completed one pivot. Completing the ABC solution We can see from Tableau 4.3 that have not yet reached the optimal solution because there are still negative coefficients available in the objective function row, equation 0. So the solution process continues for another iteration, as follows: Step 2.2: x 2 is the entering basic variable (see Tableau 4.4). Step 2.3: the minimum ratio test shows that s 3 is the leaving basic variable (see Tableau 4.4). Step 2.4: Tableau 4.5 shows the revised tableau in proper form. basic eqn. Z x 1 x 2 s 1 s 2 s 3 RHS MRT Z never x no limit s /1 = 3 s /1 = 2 Tableau 4.4: The second iteration. basic eqn. Z x 1 x 2 s 1 s 2 s 3 RHS MRT Z never x s x Tableau 4.5: The optimal tableau. Applying Step 2.1 shows that we have reached the optimum solution in Tableau 4.5: there are no negative variable coefficients in the objective function. This means that there is no improving direction in which to move. We can now read the optimum solution from the tableau (remembering that any variables not shown in the basic variables column must be nonbasic and hence zero): (x 1,x 2,s 1,s 2,s 3 ) = (2,2,0,1,0). The optimum value of the objective function is the value of Z shown in the tableau: 50. The optimum solution shows that the Acme Bicycle Company should produce 2 mountain bikes per day and 2 racing bicycles per day for a total daily profit of \$50 per day. Special Cases in Tableau Manipulation There are several unusual cases that arise as you progress from tableau to tableau while carrying out the simplex method. Each of these special cases has an interesting interpretation. 6

7 Tie for the Entering Basic Variable Suppose that there are two or more nonbasic variables that have the same most negative value in the objective function row. For example, suppose that the objective function for the Acme Bicycle Company problem was actually Z = 15x x 2 (i.e. Z 15x 1 15x 2 = 0). What is the interpretation? This means that both x 1 and x 2, if allowed to become basic and increase in value, would both increase Z at the same rate. See Figure 4.2 for an example of a cornerpoint at which the rate of increase would be the same for two entering basic variables. So how to handle this situation? Simple: choose the entering basic variable arbitrarily from among those tied at the most negative value, because there is no good way to determine in advance which selection will reach the optimum solution in the smallest number of pivots. Figure 4.3 shows an example in which two variables are tied, but for which many more pivots are required to reach the optimum via one choice (clockwise) than via the other choice (counterclockwise). Figure 4.2: A tie for the entering basic variable. In fact, the choice of the most negative objective function coefficient is only a heuristic method (rule of thumb) for choosing the entering basic variable. It usually provides a good entering basic variable, but not always. On the other hand it is very quick. In commercial solvers for very large LPs, the solver may not even calculate the objective function coefficients for all of the variables. Instead it might only calculate them for say 1000 of the variables in a 500,000 variable problem, and take the most negative from among those Next time around it might calculate the objective function coefficients for the next 1000 variables and choose the entering basic variable only from among that The only time it might calculate the objective function coefficients for all 500,000 variables is at the optimum, when it must check that there are no negative coefficients left at all. Tie for the Leaving Basic Variable Figure 4.3: One entering basic variable may be better than another. Recall the mantra: nonbasic, variable set to zero, corresponding constraint is active. This should remind you that the choosing the leaving basic variable via the minimum ratio test is the same as finding out which constraint you first bump into as the entering 7

8 basic variable increases in value. So what does it mean when there is a tie during the minimum ratio test? As Figure 4.4 shows, it must mean that there are two constraints (with their A C corresponding basic variables) that are current basic tied as the first constraints you bump B feasible solution into. Choosing the basic variable associated with constraint A or the basic variable associated with constraint B as the leaving basic variable will define the same new cornerpoint. Technically speaking, the basic feasible solutions Figure 4.4: A tie for the leaving basic variable. will be different, because each will define a different partitioning of the variables into basic and nonbasic sets. So how do we choose between the two possibilities for the leaving basic variable? Again, simply choose arbitrarily. The variable that is not chosen as the leaving basic variable will remain basic, but will have a calculated value of zero. In contrast, the variable chosen as the leaving basic variable will of course be forced to zero by simplex. Both variables must be zero simultaneously because both constraints are active at that point; it s just that simplex only needs one of them to define the basic feasible solution. When there is a tie for the leaving basic variable as we have described, the basic feasible solution defines what is known as a degenerate solution. Degenerate solutions can lead to an infinite loop of solutions that traps the simplex solution method; this is known as cycling. This does not happen in 2-dimensional problems, so it is difficult to show a simple diagram. But here is the general idea of how cycling happens, if it could happen in two dimensions (refer to Figure 4.4): first define the basic feasible solution using constraints A and C, then pivot to a new basic feasible solution defined by constraints B and C, then pivot back to the basic solution defined by constraints A and C, then continue around this loop infinitely. Cycling is actually fairly rare in real problems, and most good-quality commercial linear programming solvers have special code to detect and exit from cycling behavior. One of the most often used anti-cycling routines for network LPs was in fact invented by former Carleton University professor William Cunningham. Now suppose that all of the minimum ratio tests are tied at no limit : what does this mean? The interpretation is straightforward: no constraint puts a limit on the increase in the value of the entering basic variable! Because of this, there is then no limit on the increase in the value of the objective function. Your result might then show that you can make an infinite amount of profit for the Acme Bicycle Company, for example. Sounds great, but it usually means that you forgot a constraint. You may have assumed, for example, that you can sell whatever quantity of widgets you manufacture, or that a key ingredient is available in infinite quantities. Problems of this type are called unbounded, and have an unbounded solution. Figure 4.5 gives an example of an unbounded problem. 8

9 An unbounded problem is recognized while pivoting when all of the entries in the pivot column (the entering basic variable column) are zero or negative, and hence place no limit on the increase in the value of the entering basic variable. At the Optimum, the Coefficients of Some Nonbasic Variables are Zero in the Objective Function Row You expect that the objective function coefficients of the basic variables will be zero because putting the tableau into proper form requires that those coefficients be zeroed. But the coefficients of the nonbasic variables are Figure 4.5: An unbounded linear program. normally nonzero, and in fact are normally positive at the optimum (if any are negative, you are not yet at the optimum). So what does it mean if a nonbasic variable has a coefficient of zero in the objective function at the optimum? It means that choosing this variable as the entering basic variable increases Z at the rate of zero, i.e. that choosing this variable as the entering basic variable has no effect on Z! But choosing such a variable as the entering basic variable means that you will pivot to a different basic feasible solution. And the resulting new basic feasible solution will have the same value of Z. Two different basic feasible solutions, same value of Z? That means that there must be multiple optimum solutions. You will have the same value of the objective function at both of the two basic feasible solutions and at any point between those two solutions. Remember that there may in fact be more than two basic feasible solutions with the same value of Z. You may wish to find the other basic feasible solutions. Sometimes one solution is preferred over another for non-quantifiable reasons, so you would like to see a selection of solutions that give the optimum value. To see the other optima, choose one of the nonbasic variables whose objective function coefficient is zero as the entering basic variable, and pivot to another basic feasible solution as you normally would. 9

### Chapter 5: Solving General Linear Programs

Chapter 5: Solving General Linear Programs So far we have concentrated on linear programs that are in standard form, which have a maximization objective, all constraints of type, all of the right hand

### Chap 4 The Simplex Method

The Essence of the Simplex Method Recall the Wyndor problem Max Z = 3x 1 + 5x 2 S.T. x 1 4 2x 2 12 3x 1 + 2x 2 18 x 1, x 2 0 Chap 4 The Simplex Method 8 corner point solutions. 5 out of them are CPF solutions.

### Special Situations in the Simplex Algorithm

Special Situations in the Simplex Algorithm Degeneracy Consider the linear program: Maximize 2x 1 +x 2 Subject to: 4x 1 +3x 2 12 (1) 4x 1 +x 2 8 (2) 4x 1 +2x 2 8 (3) x 1, x 2 0. We will first apply the

### Simplex method summary

Simplex method summary Problem: optimize a linear objective, subject to linear constraints 1. Step 1: Convert to standard form: variables on right-hand side, positive constant on left slack variables for

### Algebraic Simplex Method - Introduction Previous

Algebraic Simplex Method - Introduction To demonstrate the simplex method, consider the following linear programming model: This is the model for Leo Coco's problem presented in the demo, Graphical Method.

### Chapter 6: Sensitivity Analysis

Chapter 6: Sensitivity Analysis Suppose that you have just completed a linear programming solution which will have a major impact on your company, such as determining how much to increase the overall production

### Graphical method. plane. (for max) and down (for min) until it touches the set of feasible solutions. Graphical method

The graphical method of solving linear programming problems can be applied to models with two decision variables. This method consists of two steps (see also the first lecture): 1 Draw the set of feasible

### Section Notes 3. The Simplex Algorithm. Applied Math 121. Week of February 14, 2011

Section Notes 3 The Simplex Algorithm Applied Math 121 Week of February 14, 2011 Goals for the week understand how to get from an LP to a simplex tableau. be familiar with reduced costs, optimal solutions,

### The Graphical Simplex Method: An Example

The Graphical Simplex Method: An Example Consider the following linear program: Max 4x 1 +3x Subject to: x 1 +3x 6 (1) 3x 1 +x 3 () x 5 (3) x 1 +x 4 (4) x 1, x 0. Goal: produce a pair of x 1 and x that

### The Simplex Solution Method

Module A The Simplex Solution Method A-1 A-2 Module A The Simplex Solution Method The simplex method is a general mathematical solution technique for solving linear programming problems. In the simplex

### Linear Inequalities and Linear Programming. Systems of Linear Inequalities in Two Variables

Linear Inequalities and Linear Programming 5.1 Systems of Linear Inequalities 5.2 Linear Programming Geometric Approach 5.3 Geometric Introduction to Simplex Method 5.4 Maximization with constraints 5.5

### 7.1 Modelling the transportation problem

Chapter Transportation Problems.1 Modelling the transportation problem The transportation problem is concerned with finding the minimum cost of transporting a single commodity from a given number of sources

### Chapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints

Chapter 6 Linear Programming: The Simplex Method Introduction to the Big M Method In this section, we will present a generalized version of the simplex method that t will solve both maximization i and

### Definition of a Linear Program

Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

### Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs

Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs These questions are from previous years and should you give you some idea of what to expect on Quiz 1. 1. Consider the

### CHAPTER 17. Linear Programming: Simplex Method

CHAPTER 17 Linear Programming: Simplex Method CONTENTS 17.1 AN ALGEBRAIC OVERVIEW OF THE SIMPLEX METHOD Algebraic Properties of the Simplex Method Determining a Basic Solution Basic Feasible Solution 17.2

### Standard Form of a Linear Programming Problem

494 CHAPTER 9 LINEAR PROGRAMMING 9. THE SIMPLEX METHOD: MAXIMIZATION For linear programming problems involving two variables, the graphical solution method introduced in Section 9. is convenient. However,

### Simplex Method. Introduction:

Introduction: In the previous chapter, we discussed about the graphical method for solving linear programming problems. Although the graphical method is an invaluable aid to understand the properties of

### 7.4 Linear Programming: The Simplex Method

7.4 Linear Programming: The Simplex Method For linear programming problems with more than two variables, the graphical method is usually impossible, so the simplex method is used. Because the simplex method

### LINEAR PROGRAMMING P V Ram B. Sc., ACA, ACMA Hyderabad

LINEAR PROGRAMMING P V Ram B. Sc., ACA, ACMA 98481 85073 Hyderabad Page 1 of 19 Question: Explain LPP. Answer: Linear programming is a mathematical technique for determining the optimal allocation of resources

### 1 Solving LPs: The Simplex Algorithm of George Dantzig

Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

### The Simplex Method. What happens when we need more decision variables and more problem constraints?

The Simplex Method The geometric method of solving linear programming problems presented before. The graphical method is useful only for problems involving two decision variables and relatively few problem

### Chapter 2: Introduction to Linear Programming

Chapter 2: Introduction to Linear Programming You may recall unconstrained optimization from your high school years: the idea is to find the highest point (or perhaps the lowest point) on an objective

### LINEAR PROGRAMMING THE SIMPLEX METHOD

LINEAR PROGRAMMING THE SIMPLE METHOD () Problems involving both slack and surplus variables A linear programming model has to be extended to comply with the requirements of the simplex procedure, that

### Linear Programming Notes VII Sensitivity Analysis

Linear Programming Notes VII Sensitivity Analysis 1 Introduction When you use a mathematical model to describe reality you must make approximations. The world is more complicated than the kinds of optimization

### The Graphical Method: An Example

The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,

### OPRE 6201 : 2. Simplex Method

OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2

### Linear Programming. March 14, 2014

Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1

### 3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max

SOLUTION OF LINEAR PROGRAMMING PROBLEMS THEOREM 1 If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore,

### Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

### IEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2

IEOR 4404 Homework # Intro OR: Deterministic Models February 14, 011 Prof. Jay Sethuraman Page 1 of 5 Homework #.1 (a) What is the optimal solution of this problem? Let us consider that x 1, x and x 3

### Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood

PERFORMANCE EXCELLENCE IN THE WOOD PRODUCTS INDUSTRY EM 8720-E October 1998 \$3.00 Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood A key problem faced

### The Simplex Method. yyye

Yinyu Ye, MS&E, Stanford MS&E310 Lecture Note #05 1 The Simplex Method Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/

### 3 Does the Simplex Algorithm Work?

Does the Simplex Algorithm Work? In this section we carefully examine the simplex algorithm introduced in the previous chapter. Our goal is to either prove that it works, or to determine those circumstances

### Chapter 2 Solving Linear Programs

Chapter 2 Solving Linear Programs Companion slides of Applied Mathematical Programming by Bradley, Hax, and Magnanti (Addison-Wesley, 1977) prepared by José Fernando Oliveira Maria Antónia Carravilla A

### max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x

Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study

### Reduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:

Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in

### 4.4 The Simplex Method: Non-Standard Form

4.4 The Simplex Method: Non-Standard Form Standard form and what can be relaxed What were the conditions for standard form we have been adhering to? Standard form and what can be relaxed What were the

### Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

### Section Notes 4. Duality, Sensitivity, Dual Simplex, Complementary Slackness. Applied Math 121. Week of February 28, 2011

Section Notes 4 Duality, Sensitivity, Dual Simplex, Complementary Slackness Applied Math 121 Week of February 28, 2011 Goals for the week understand the relationship between primal and dual programs. know

### Degeneracy in Linear Programming

Degeneracy in Linear Programming I heard that today s tutorial is all about Ellen DeGeneres Sorry, Stan. But the topic is just as interesting. It s about degeneracy in Linear Programming. Degeneracy? Students

### Solving Linear Programs

Solving Linear Programs 2 In this chapter, we present a systematic procedure for solving linear programs. This procedure, called the simplex method, proceeds by moving from one feasible solution to another,

### Min-cost flow problems and network simplex algorithm

Min-cost flow problems and network simplex algorithm The particular structure of some LP problems can be sometimes used for the design of solution techniques more efficient than the simplex algorithm.

### Solving Systems of Linear Equations

LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

### Operation Research. Module 1. Module 2. Unit 1. Unit 2. Unit 3. Unit 1

Operation Research Module 1 Unit 1 1.1 Origin of Operations Research 1.2 Concept and Definition of OR 1.3 Characteristics of OR 1.4 Applications of OR 1.5 Phases of OR Unit 2 2.1 Introduction to Linear

### Chapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS 3.2 TERMINOLOGY

Chapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS Once the problem is formulated by setting appropriate objective function and constraints, the next step is to solve it. Solving LPP

### Chapter 5.1 Systems of linear inequalities in two variables.

Chapter 5. Systems of linear inequalities in two variables. In this section, we will learn how to graph linear inequalities in two variables and then apply this procedure to practical application problems.

### 3.5 Reduction to Standard Form

3.5 Reduction to Standard Form 77 Exercise 3-4-4. Read the following statements carefully to see whether it is true or false. Justify your answer by constructing a simple illustrative example (for false),

### Linear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.

1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that

### 4.3 The Simplex Method and the Standard Maximization Problem

. The Simplex Method and the Standard Maximiation Problem Question : What is a standard maximiation problem? Question : What are slack variables? Question : How do you find a basic feasible solution? Question

### Linear Programming Problems

Linear Programming Problems Linear programming problems come up in many applications. In a linear programming problem, we have a function, called the objective function, which depends linearly on a number

### The Transportation Problem: LP Formulations

The Transportation Problem: LP Formulations An LP Formulation Suppose a company has m warehouses and n retail outlets A single product is to be shipped from the warehouses to the outlets Each warehouse

### I. Solving Linear Programs by the Simplex Method

Optimization Methods Draft of August 26, 2005 I. Solving Linear Programs by the Simplex Method Robert Fourer Department of Industrial Engineering and Management Sciences Northwestern University Evanston,

### Linear Programming. April 12, 2005

Linear Programming April 1, 005 Parts of this were adapted from Chapter 9 of i Introduction to Algorithms (Second Edition) /i by Cormen, Leiserson, Rivest and Stein. 1 What is linear programming? The first

### SUPPLEMENT TO CHAPTER

SUPPLEMENT TO CHAPTER 6 Linear Programming SUPPLEMENT OUTLINE Introduction and Linear Programming Model, 2 Graphical Solution Method, 5 Computer Solutions, 14 Sensitivity Analysis, 17 Key Terms, 22 Solved

### Linear Programming: Theory and Applications

Linear Programming: Theory and Applications Catherine Lewis May 11, 2008 1 Contents 1 Introduction to Linear Programming 3 1.1 What is a linear program?...................... 3 1.2 Assumptions.............................

### 1 Gaussian Elimination

Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 Gauss-Jordan reduction and the Reduced

### 10 Linear Programming

10 Linear Programming Learning Objectives: After studying this unit you will be able to: Formulate the Linear Programming Problem Solve the Linear Programming Problem with two variables using graphical

### Solving Linear Programs in Excel

Notes for AGEC 622 Bruce McCarl Regents Professor of Agricultural Economics Texas A&M University Thanks to Michael Lau for his efforts to prepare the earlier copies of this. 1 http://ageco.tamu.edu/faculty/mccarl/622class/

### Using the Simplex Method in Mixed Integer Linear Programming

Integer Using the Simplex Method in Mixed Integer UTFSM Nancy, 17 december 2015 Using the Simplex Method in Mixed Integer Outline Mathematical Programming Integer 1 Mathematical Programming Optimisation

### Linear Programming. Solving LP Models Using MS Excel, 18

SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting

### Outline. Linear Programming (LP): Simplex Search. Simplex: An Extreme-Point Search Algorithm. Basic Solutions

Outline Linear Programming (LP): Simplex Search Benoît Chachuat McMaster University Department of Chemical Engineering ChE 4G03: Optimization in Chemical Engineering 1 Basic Solutions

### 0.1 Linear Programming

0.1 Linear Programming 0.1.1 Objectives By the end of this unit you will be able to: formulate simple linear programming problems in terms of an objective function to be maximized or minimized subject

### value is determined is called constraints. It can be either a linear equation or inequality.

Contents 1. Index 2. Introduction 3. Syntax 4. Explanation of Simplex Algorithm Index Objective function the function to be maximized or minimized is called objective function. E.g. max z = 2x + 3y Constraints

### Linear Programming in Matrix Form

Linear Programming in Matrix Form Appendix B We first introduce matrix concepts in linear programming by developing a variation of the simplex method called the revised simplex method. This algorithm,

### Sensitivity Analysis 3.1 AN EXAMPLE FOR ANALYSIS

Sensitivity Analysis 3 We have already been introduced to sensitivity analysis in Chapter via the geometry of a simple example. We saw that the values of the decision variables and those of the slack and

### Duality in Linear Programming

Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow

### Numerical Methods Lecture 2 Simultaneous Equations

Numerical Methods Lecture 2 Simultaneous Equations Topics: matrix operations solving systems of equations Matrix operations: Mathcad is designed to be a tool for quick and easy manipulation of matrix forms

### Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

### 6 Gaussian Elimination

G1BINM Introduction to Numerical Methods 6 1 6 Gaussian Elimination 61 Simultaneous linear equations Consider the system of linear equations a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 +

### Linear Programming: Chapter 5 Duality

Linear Programming: Chapter 5 Duality Robert J. Vanderbei October 17, 2007 Operations Research and Financial Engineering Princeton University Princeton, NJ 08544 http://www.princeton.edu/ rvdb Resource

### The Canonical and Dictionary Forms of Linear Programs

The Canonical and Dictionary Forms of Linear Programs Recall the general optimization problem: Minimize f(x) subect to x D (1) The real valued function f is called the obective function, and D the constraint

### Lecture 4 Linear Programming Models: Standard Form. August 31, 2009

Linear Programming Models: Standard Form August 31, 2009 Outline: Lecture 4 Standard form LP Transforming the LP problem to standard form Basic solutions of standard LP problem Operations Research Methods

### LINEAR PROGRAMMING MODULE Part 2 - Solution Algorithm

LINEAR PROGRAMMING MODULE Part 2 - Solution Algorithm Name: THE SIMPLEX SOLUTION METHOD It should be obvious to the reader by now that the graphical method for solving linear programs is limited to models

### LINEAR PROGRAMMING PROBLEM: A GEOMETRIC APPROACH

59 LINEAR PRGRAMMING PRBLEM: A GEMETRIC APPRACH 59.1 INTRDUCTIN Let us consider a simple problem in two variables x and y. Find x and y which satisfy the following equations x + y = 4 3x + 4y = 14 Solving

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Linear Programming IS 601

Linear Programming IS 01 e-optimization.com A Resource Page on Optimization Distribution System at Proctor and Gamble Proctor and Gamble needed to consolidate and re-design their North American distribution

### 56:171 Operations Research Midterm Exam Solutions Fall 2001

56:171 Operations Research Midterm Exam Solutions Fall 2001 True/False: Indicate by "+" or "o" whether each statement is "true" or "false", respectively: o_ 1. If a primal LP constraint is slack at the

### Theory of Linear Programming

Theory of Linear Programming Debasis Mishra April 6, 2011 1 Introduction Optimization of a function f over a set S involves finding the maximum (minimum) value of f (objective function) in the set S (feasible

### 1 Determinants and the Solvability of Linear Systems

1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely side-stepped

### 4.6 Linear Programming duality

4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal

### 56:171. Operations Research -- Sample Homework Assignments Fall 1992 Dennis Bricker Dept. of Industrial Engineering University of Iowa.

56:171 Operations Research -- Sample Homework Assignments Fall 1992 Dennis Bricker Dept. of Industrial Engineering University of Iowa Homework #1 (1.) Linear Programming Model Formulation. SunCo processes

### 5 Systems of Equations

Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate

### Basic solutions in standard form polyhedra

Recap, and outline of Lecture Previously min s.t. c x Ax = b x Converting any LP into standard form () Interpretation and visualization () Full row rank assumption on A () Basic solutions and bases in

### (a) Let x and y be the number of pounds of seed and corn that the chicken rancher must buy. Give the inequalities that x and y must satisfy.

MA 4-4 Practice Exam Justify your answers and show all relevant work. The exam paper will not be graded, put all your work in the blue book provided. Problem A chicken rancher concludes that his flock

### 1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.

Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S

### 18. Linear optimization

18. Linear optimization EE103 (Fall 2011-12) linear program examples geometrical interpretation extreme points simplex method 18-1 Linear program minimize c 1 x 1 +c 2 x 2 + +c n x n subject to a 11 x

### Mathematical Programming

1 The Add-in constructs models that can be solved using the Solver Add-in or one of the solution add-ins provided in the collection. When the Math Programming add-in is installed, several new command lines

### Sensitivity Report in Excel

The Answer Report contains the original guess for the solution and the final value of the solution as well as the objective function values for the original guess and final value. The report also indicates

### 2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system

1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables

### Lecture notes 1: Introduction to linear and (mixed) integer programs

Lecture notes 1: Introduction to linear and (mixed) integer programs Vincent Conitzer 1 An example We will start with a simple example. Suppose we are in the business of selling reproductions of two different

### Geometry of Linear Programming

Chapter 2 Geometry of Linear Programming The intent of this chapter is to provide a geometric interpretation of linear programming problems. To conceive fundamental concepts and validity of different algorithms

### Introduction to Linear Programming.

Chapter 1 Introduction to Linear Programming. This chapter introduces notations, terminologies and formulations of linear programming. Examples will be given to show how real-life problems can be modeled

### Chapter 1, Operations Research (OR)

Chapter 1, Operations Research (OR) Kent Andersen February 7, 2007 The term Operations Research refers to research on operations. In other words, the study of how to operate something in the best possible

### 6. Mixed Integer Linear Programming

6. Mixed Integer Linear Programming Javier Larrosa Albert Oliveras Enric Rodríguez-Carbonell Problem Solving and Constraint Programming (RPAR) Session 6 p.1/40 Mixed Integer Linear Programming A mixed

### Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.

Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required

### MTS OPERATIONS RESEARCH...

MTS 363 - OPERATIONS RESEARCH... Dr.I.A.Osinuga Recommended Texts 1. Operations Research (an introduction) Handy A. Taha (3 rd Edition) 2. Optimization (Theory and applications) S.S. Rao 3. Principles