[PDF] Chapter 6Linear Programming: The Simplex Method

Chapter 6Linear Programming: The Simplex Method

Simplex Tableau The simplex method utilizes matrix representation of the initial system while performing search for the optimal solution This matrix repre-sentation is called simplex tableau and it is actually the augmented matrix of the initial systems with some additional information

Asymptotics for the distributions of subtableaux in Young and

the up-down tableau is actually a standard tableau, and approaches 0 if has fewer than k cells 1 Introduction Let be a partition of k,andT a standard Young tableau of shape WesaythatT is a subtableau of a larger Young tableau Y if the entries 1 through k of Y form the tableau T LetP(N;T) be the probability that a randomly chosen standard

A high performance dual revised simplex solver

Update tableau N^ := N^ (1=a^ pq)a^ qa^T p Revised simplex method (RSM) Operations Form ˇT p = e T p B 1 Form a^ T p = ˇ p N Form a^ q= B 1a q Inversion of B Distinctive features Vectors e, a qarealways sparse Bmay behighly reducible B may be sparse Vectors ˇ p, a^ pand a^ qmay be sparse E cient implementations must exploit these features H

Maths GS - Tizofun Education

Maths GS Avec Tizo, j‛apprends en m‛amusant Nom : Date : tizofun com Tableau à deux entrées Title: 39-logique-tableau-a-deux-entree ai Author: Nno Created Date:

Chapter 4: Linear Programming The Simplex Method

Day 1: Learn to set up a linear programming problem with many variables and create a “simplex tableau ” Day 2: Learn to identify basic variables, read feasible solutions from a tableau, and “pivot” to manipulate your data Today – Learn to identify which variable to use as the pivot so your feasible solution gives the maximum value of

fichier exercice maths CM1 - La classe de Mallory

Complète le tableau suivant Cent-vingt-mille-quatre-cent-douze Neuf-cent-mille-quatre-vingt-dix-sept Sept-cent-neuf-mille-deux 5 –Placer, encadrer, comparer, ranger les nombres jusqu’à 999 999 Recopie le plus petit nombre de chaque série •148 612 -48 612 84 140000 → _____ • 76 201 - 7 201 - 72 601- 56 201 - 5 601 →

UNIT 4 LINEAR PROGRAMMING - SIMPLEX METHOD

Linear Program ming – 31 Simplex Method 4 2 PRINCIPLE OF SIMPLEX METHOD We explain the principle of the Simplex method with the help of the two variable linear programming problem introduced in Unit 3, Section 2

fichier exercice maths CM2 - La classe de Mallory

Complète le tableau suivant –Reconnaître la symétrie axiale Trace les axes de symétrie de ces figures (quand cela est possible) Géom 13 – Tracer une figure par symétrie Trace le symétrique de cette figure par rapport à l’axe, en utilisant le quadrillage

Nom Date LES NOMBRES JUSQUÀ 59 En te servant de lexemple

Nom Date LES NOMBRES JUSQU'À 59 En te servant de lexemple, dénombre les collections de chocolats puis écris le résultat dans la pastille et le tableau

[PDF] maths tableur troisième

[PDF] Maths tarif

[PDF] maths taux de variation

[PDF] maths terminale es exercices corrigés

[PDF] maths terminale es fonction exponentielle

[PDF] Maths Terminale S

[PDF] maths terminale s exercices corrigés livre

[PDF] maths terminale s exercices corrigés livre pdf

[PDF] maths terminale st2s statistiques

[PDF] maths terminale sti2d exercices

[PDF] maths terminale stmg exercices

[PDF] maths terminale stmg programme

[PDF] Maths TES : Suites géométriques

[PDF] Maths theoreme al-kashy

[PDF] Maths Théoreme de pythagore

Chapter 6Linear Programming: The

Simplex Method

We will now consider LP (Linear Programming) problems that involve more than 2 decision variables. We will learn an algorithm called the simplex method which will allow us to solve these kind of problems.

Maximization Problem in Standard Form

We start with dening the standard form of a linear programming problem which will make further discussion easier.

Denition.

A linear programming problem is said to be astandard max- imization problem in standard formif its mathematical model is of the following form:

MaximizeP=c1x1+c2x2+:::+cnxn

subject toa11x1+a12x2+:::+a1nxnb1 a m1x1+am2x2+:::+amnxnbm x

1;x2;:::;xn0

wherex1;x2;:::;xnare decision variables,c1;:::;cn, a

11;:::;amnare any real numbers, andb1;:::;bm0 are

nonnegative real numbers. Note:Any linear programming problem (in the form we dened earlier) can be converted intothe standard maximization problem in standard form.

Ch 6. Linear Programming: The Simplex Method

Initial System and Slack Variables

Roughly speaking, the idea of the simplex method is to represent an LP problem as a system of linear equations, and then a certain solu- tion (possessing some properties we will dene later) of the obtained system would be an optimal solution of the initial LP problem (if any exists). The simplex method denes an ecient algorithm of nding this specic solution of the system of linear equations. Therefore, we need to start with converting given LP problem into a system of linear equations. First, we convert problem constraints into equations with the help ofslack variables. Consider the following maximization problem in the standard form:

MaximizeP= 5x1+ 4x2(1)

subject to 4x1+ 2x232

2x1+ 3x224

x 1;x20 The variabless1ands2are calledslack variablesbecause each makes up the dierence (takes up the slack) between the left and right sides of an inequality.For each problem constraint of the original problem we introduce a single slack variable. 2

Ch 6. Linear Programming: The Simplex Method

Therefore, we get

4x1+ 2x2+s1=32 (2)

2x1+ 3x2+s2=24

x

1;x2;s1;s20

Note that each solution of (2) corresponds to a point in the feasible region of (1). Also note that theslack variables should be non- negativeas well. If slack variable is negative, then the right-hand side of corresponding problem constrain should be larger than the left-hand, i.e., this constraint would be violated. Now, we also add the objective function to (2) treatingPas yet another variable.

P= 5x1+ 4x2=)

Therefore, we get

4x1+ 2x2+s1= 32

2x1+ 3x2+s2= 24

5x14x2+P= 0

x

1;x2;s1;s20

The above system is called theinitial system. Again, every solution of the initial system (taking into account nonnegative constrains) cor- responds to some point in the feasible region of the original LP (and vice versa!), and in addition the initial system incorporates informa- tion about the objective function of the original LP. Therefore,one of the solution of the initial system should be an optimal solution of the original LP(if any exists). Clearly, theinitial system has innitely many solutions, so the key question is which one of these solutions is an optimal solution of the LP? 3

Ch 6. Linear Programming: The Simplex Method

Basic Solutions and Basic Feasible Solutions

We now dene two important types of solutions of the initial systems that we should focus our attention on in order to identify the optimal solution of the LP.

Denition (Basic Solution)

Given an LP withndecision variables andmconstraints, abasic solutionof the corresponding initial system is a solution of the initial systems (not taking into account nonnegative constraints) in whichnof the variablesx1;:::;xn;s1;:::;smare equal to zero. Note:the list of variablesx1;:::;xn;s1;:::;sm,nof which should be zero, does not containP.Denition (Basic Feasible Solution) If a basic solution of the initial system corresponds to a certain point in the feasible region of the original LP, then it is called a basic feasible solution.The feasible region of (1) looks like

1135791113151711357911131517

x 1x

2(8;0)(6;4)(0;8)(0;0)(12;0)(0;16)4

Ch 6. Linear Programming: The Simplex Method

In 2-dimensional case (2 decision variables), the set of basic solutions is the of pairwise intersections of boundary lines of all problem con- straints. In turn, the set of basic feasible solutions is the set of the corner points. Indeed, 5

Ch 6. Linear Programming: The Simplex Method

Theorem 1 (Fundamental Theorem of Linear Pro-

gramming: Another Version) If the optimal value of the objective function in a linear program- ming problem exists, then that value must occur at one or more

of the basic feasible solutions of the initial system.So, by checking all basic solutions for feasibility and optimality we can solve any

LP. In our example, this is quite easy because there are 6 basic solutions and just

4 of them are feasible. However, in a lot of real-world LP problems the number

of variables and the number of constraints are much higher. For example, if a problem hasn= 30 decision variables andm= 35 problem constraints, the number of possible basic solution becomes approximately31018. It will take about15 yearsfor an average modern personal computer to check all these solutions for feasibility and optimality. The simplex method describes a "smart" way to nd much smaller subset of basic solutions which would be sucient to check in order to identify the optimal solution. Staring from some basic feasible solution calledinitial basic feasible solution, the simplex method moves along the edges of the polyhedron (vertices of which are basic feasible solutions) inthe direction of increase of the objective functionuntil it reaches the optimal solution.6

Ch 6. Linear Programming: The Simplex Method

Simplex Tableau

The simplex method utilizes matrix representation of the initial system while performing search for the optimal solution. This matrix repre- sentation is calledsimplex tableauand it is actually the augmented matrix of the initial systems with some additional information. Let's write down the augmented matrix of the initial system corre- sponding to the LP (1). 2

44 2 1 0 032

2 3 0 1 024

54 0 0 10

3 5 At each step of the simplex method a particular basic feasible solution is considered. Information about this solutions is also represented in the simplex tableau. Recall that we dened a basic feasible solution as a solution withnvariables being zero. In this context, we have

Denition (Basic and Nonbasic Variables)

The variables of a basic solution that are assumed to be zero are callednonbasicvariables. All the remaining variables are called basicvariables.So at each step, we need dene the list of basic and nonbasic variables. Note that ifnnonbasic variables are assigned the value 0, the corre- sponding values of the nonbasicm+1 variables can be determined by solving the corresponding system of linear equations. Question:How do we decide which variables are basic and which are not? In particular, how can we decide what variables would be basic and what would be nonbasic on the very rst step of the simplex method (how do we choose the initial basic feasible solution)? 7

Ch 6. Linear Programming: The Simplex Method

Therefore, for our example we have

2

44 2 1 0 032

2 3 0 1 024

54 0 0 10

3 5

Pivot Operation

So far, we set up asimplex tableauand identied theinitial basic feasible solutionby determining basic and nonbasic variables. This is the rst step of the simplex method. At each further step the simplex methodsswaps one of the non- basic variables for one of the basic variables(so it moves to another vertex of the polyhedron) in the way such that the value of the objective function is improved(becomes higher). If improve- ment of the objective function is not possible, then we got an optimal solution. 8

Ch 6. Linear Programming: The Simplex Method

Since we do not choose ourselves which variables are basic but rather determine them by reading the simplex tableau, in order for such swap to happen the simplex tableau should be changed. This is done with the help ofpivot operations. However, before doing this transformation we need to decide ourselves which nonbasic variable should become basic and vice versa.

Denition (Entering and Exiting Variables)

Anonbasicvariable that is chosen to become abasicvariable at a particular step of the simplex method is calledentering variable. Abasicvariable that is chosen to become anonbasicvariable at a particular step of the simplex method is calledexiting variable.Getting back to our example 2

44 2 1 0 032

2 3 0 1 024

54 0 0 10

3 5

Let's rst select theenteringvariable:

9

Ch 6. Linear Programming: The Simplex Method

Denition (Pivot Column)

The column corresponding to theenteringvariable is called the pivot column.Now, let's select theexitingvariable: 10

Ch 6. Linear Programming: The Simplex Method

Denition (Pivot Row and Pivot Element)

The row corresponding to theexitingvariable is called thepivot row. The element at the intersection of thepivot columnand thepivot rowis called thepivot element.So, we have 2

44 2 1 0 032

2 3 0 1 024

54 0 0 10

3 511

Ch 6. Linear Programming: The Simplex Method

Now, when we know which variable is entering and which is exiting we need to perform row operations on the tableau so that thepivot ele- ment is transformed into 1 and all other elements in the column into 0's. This procedure for transforming a nonbasic vari- able into a basic variable is called apivot operation, orpivoting, and is summarized below.Performing a pivot operation has the following eects: 1. The ( entering)non basicv ariableb ecomesa b asicv ariable. 2. The ( exiting)bas icv ariableb ecomesa non basicv ariable. 3. The v alueof the ob jectivefunction is increased, or, in some cases, remains the same.

Also, note that

Do not interchange rows.

A pivot operation uses some of the same row operations as those used in GaussJordan elimination, but there is one essential dier- ence. In a pivot operation,you can never interchange two rows.12

Ch 6. Linear Programming: The Simplex Method

Getting back to our example

2

44 2 1 0 032

2 3 0 1 024

54 0 0 10

3 5 13

Ch 6. Linear Programming: The Simplex Method

14

Ch 6. Linear Programming: The Simplex Method

Interpreting the Simplex Process Geometrically

1135791113151711357911131517

x 1x

2(8;0)(6;4)(0;8)(0;0)15

Ch 6. Linear Programming: The Simplex Method

Summary

16

Ch 6. Linear Programming: The Simplex Method

More Examples

Example 1

MaximizeP= 30x1+ 40x2

subject to 2x1+x210 x 1+x27 x

1+ 2x212

x 1;x20 17

Ch 6. Linear Programming: The Simplex Method

18

Ch 6. Linear Programming: The Simplex Method

Example 2

MaximizeP= 6x1+ 3x2

subject to2x1+ 3x29 x1+ 3x212 x 1;x20 19

Ch 6. Linear Programming: The Simplex Method

Example 3

MaximizeP= 4x1+ 3x2+ 2x3

subject to 3x1+ 2x2+ 5x323

2x1+x2+x38

x

1+x2+ 2x37

x

1;x2;x30

20

Ch 6. Linear Programming: The Simplex Method

21

Ch 6. Linear Programming: The Simplex Method

22
quotesdbs_dbs5.pdfusesText_10