The Simplex Method

11 Key excerpts on "The Simplex Method"

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  Introduction to Linear Optimization and Extensions with MATLAB

  • Roy H. Kwon(Author)
  • 2013(Publication Date)
  • CRC Press

    (Publisher)

  3 The Simplex Method 3.1 Introduction In this chapter The Simplex Method, which is an important and well-known method to solve linear programming problems, is developed. The Simplex Method was conceived by Dantzig (1948), still remains a powerful class of methods, and is often the main strategy for solving linear programs in commer-cial software. We know by the Fundamental Theorem of Linear Programming in Chapter 2 that if an LP has a finite optimal solution, then it can be attained at an extreme point and therefore at some basic feasible solution. The basic strategy of The Simplex Method is to explore the extreme points of the feasible region of an LP to find the optimal extreme point. However, in practice, The Simplex Method will in most cases not need to explore all possible extreme points before finding an optimal one. The strategy of The Simplex Method is as follows: given an initial basic feasible solution, The Simplex Method deter-mines whether the basic feasible solution is optimal. If it is optimal, then the method terminates, else, another basic feasible solution is generated whose objective function value is better or no worse than the previous one, optimal-ity is checked and so on, until an optimal basic feasible solution is obtained. If the LP is unbounded, then The Simplex Method will be able to detect this and return with the recession direction along which the objective function is unbounded. The Simplex Method requires that a linear program is in standard form. A high-level description of the method is summarized below (the finer details of each step will be subsequently developed). Simplex Method Step 0: Generate an initial basic feasible solution x (0) . Let k = 0 and go to Step 1. Step 1: Check optimality of x ( k ) . If x ( k ) is optimal, then STOP and return x ( k ) as the optimal solution, else go to Step 2. Step 2: Check whether the LP is unbounded; if so STOP, else go to Step 3.

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  Invitation to Linear Programming and Game Theory

  • David C. Vella(Author)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  5 The Simplex Algorithm In Chapter 3, we considered two distinct geometric approaches to solving linear programming problems. Since geometry is easiest to visualize in two dimensions, these approaches were limited in scope to m×2 or 2×n problems. However, while these techniques do not apply directly to larger problems, during our discussion of the corner point theorem, we did glean enough from these techniques to say that the following points hold true for a general linear programming problem: • The set of feasible points is always a simple, convex polytope in the decision space, also known as a simplex X. • An optimal point, if one exists, can always be found at one of the corners of the simplex X. • The corners of X correspond to the basic, feasible solutions to Eq. (3.6), or, rather, to a version of Eq. (3.6) that has m unit vectors U j instead of two, for the general m × n problem: x 1 A 1 + x 2 A 2 + · · · + x n A n + S 1 U 1 + S 2 U 2 + · · · + S m U m = R. (5.1) In order to be able to solve larger linear programming problems, we need a technique that is more algebraic than the techniques of Chapter 3. We might imagine the problem as a two-step process. First, we somehow find a feasible point P (that is any point in X) , and from there, we seek to improve our choice until it becomes optimal. If our initial choice of a feasible point P happens to be a basic solution to Eq. (5.1), there is a beautiful technique developed by George Dantzig in 1948, which today is known as the simplex algorithm. It is the most widely applied method for solving linear programming problems, and it is very efficient. If the initial choice P is feasible but not a basic solution to Eq. (5.1), there are several techniques for moving from this choice toward the optimal point. Such techniques are termed interior point methods. Most interior point methods use some sophisticated mathematics, such as multivariable calculus or projective geometry.

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  Numerical Linear Approximation in C

  • Nabih Abdelmalek, William A. Malek(Authors)
  • 2008(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  39 Chapter 3 Linear Programming and the Simplex Algorithm 3.1 Introduction This chapter is a tutorial one. Its purpose is to introduce the reader to the subject of Linear Programming, which is used as the main tool for solving all the approximation problems in this book, with the exception of some least squares approximation problems. In general, linear programs deal with optimization problems in real life, such as those found in industry, transportation, economics, numerical analysis and other fields [6, 8, 10, 11]. A linear programming problem is an optimization problem in which a linear function of a set of variables is to be maximized (or minimized) subject to a set of linear constraints. It may be formulated as follows. Find the n variables x 1 , x 2 , …, x n that maximize the linear function (3.1.1) z = c 1 x 1 + c 2 x 2 + … + c n x n subject to the m linear constraints (conditions) a 11 x 1 + a 12 x 2 + … + a 1n x n η 1 b 1 a 21 x 1 + a 22 x 2 + … + a 2n x n η 2 b 2 (3.1.2) … … … a m1 x 1 + a m2 x 2 + … + a mn x n η m b m where η i , i = 1, 2 …, m, is a ≤ , ≥ , or = sign. The following n constraints are also specified (3.1.3) x 1 > 0, x 2 > 0, …, x n > 0 The following is a common linear programming problem that 40 Numerical Linear Approximation in C occurs in industry. The so-called simplex method solves the problem via Gauss-Jordan elimination processes . Example 3.1 A firm has 3 workshops, one for parts, one for wiring and one for testing. The firm produces 2 different products A and B. Each product has to undergo an operation in each workshop consecutively. One unit of product A requires 1, 1, and 2 hours in the 3 workshops respectively. One unit of product B requires 1, 3, and 1 hours in the 3 workshops respectively. The workshops are available 20, 50 and 30 hours per week respectively. The profit in selling one unit of product A is $20 and in selling one unit of product B is $30.

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  Optimization techniques I

  • Max Cerf(Author)
  • 2023(Publication Date)
  • EDP Sciences

    (Publisher)

  384 Optimization techniques 5.1 Simplex The Simplex Method is specific to linear programming problems. This method developed by G. Dantzig in the late 1940s exploits the properties of the solution of a linear problem to reduce the search domain. It has proven to be extremely efficient on most applications. Improvements made since 1950 have extended its field of application to large problems and to mixed linear programming. 5.1.1 Standard form The standard form of a linear programming problem is as follows:  n T m n m n x Ax b min z c x s.t. with A , b , c x 0   = =     (5.1) The cost function and the constraints are linear. The variables x are positive. The number m of constraints is less than the number n of variables and the matrix A is assumed to have full rank: rang(A) m n =  . These assumptions ensure the existence of feasible solutions. The problem (5.1) is denoted by (LP). Any linear programming problem can be put into the standard form (5.1) using the following transformations: • an inequality constraint is transformed into an equality by introducing a positive slack variable: ( ) T T T x a 1 b a x x' b a x b x' x' 0 x' 0    =   + =               (5.2) • a lower bound l x is treated by changing the variable, an upper bound u x is treated by introducing a positive slack variable: l l u l u l u l l u l x x x , x 0 x x x 0 x x x x x x x x x x , x 0 x x '' x x , x '' 0   = −       −  −     −    = −      + = −   (5.3) • a free variable is expressed as the difference of two positive variables: x x x '' x' with x' 0 , x'' 0   = −   (5.4) Linear programming 385 Example 5-1: Putting a linear problem into standard form Consider the linear problem (LP) 1 2 3 1 2 1 2 3 1 2 3 x ,x ,x 1 2 3 x 3x 5 min x 2x 3x s.t.

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  Mathematics

  An Applied Approach

  • Michael Sullivan, Abe Mizrahi(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  One of these methods is The Simplex Method, the subject of this chapter. The Simplex Method is a way to solve linear program- ming problems involving many inequalities and variables. This method was developed by George Dantzig in 1946 and is particularly well suited for computerization. In 1984, Narendra Karmarkar of Bell Laboratories discovered a way of solving large linear programming problems that improves on The Simplex Method. A discussion of LINDO, a software package that closely mimics The Simplex Method, may be found in Appendix B. PREPARING FOR THIS SECTION Before getting started, review the following: 4.1 > Row Operations (Section 2.3, pp. 67 – 68) > Linear Programming (Section 3.2, pp. 171– 178) OBJECTIVES 1 Determine a maximum problem is in standard form 2 Set up the initial simplex tableau 3 Perform pivot operations 4 Analyze a tableau The Simplex Tableau; Pivoting 196 Chapter 4 Linear Programming: Simplex Method Standard Form of a Maximum Problem A linear programming problem in which the objective function is to be maximized is referred to as a maximum problem. Such problems are said to be in standard form provided two conditions are met: Standard Form of a Maximum Problem Condition 1 All the variables are nonnegative. Condition 2 Every other constraint is written as a linear expression that is less than or equal to a positive constant.

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  Linear and Nonlinear Programming with Maple

  An Interactive, Applications-Based Approach

  • Paul E. Fishback(Author)
  • 2009(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 2 The Simplex Algorithm 2.1 The Simplex Algorithm Considered to be the classical approach for solving LPs, the simplex algorithm was developed in 1947 by George Dantzig, who used it to solve program-ming problems for the Air Force. Such logistics problems addressed a variety of issues including, for example, the scheduling of troop deployments and the delivery of supplies. 2.1.1 An Overview of the Algorithm We begin by revisiting the standard form of the FuelPro LP: maximize z = 4 x 1 + 3 x 2 (2.1) subject to x 1 + s 1 = 8 2 x 1 + 2 x 2 + s 2 = 28 3 x 1 + 2 x 2 + s 3 = 32 x 1 , x 2 , s 1 , s 2 , s 3 ≥ 0 . We will rewrite the LP as an “augmented matrix,” so to speak. That is, we will write the system of equations given by the constraints as an augmented matrix, and we will add a top row that records the value of our objective function. The result is shown in Table 2.1. TABLE 2.1: Initial tableau for the FuelPro LP z x 1 x 2 s 1 s 2 s 3 RHS 1 -4 -3 0 0 0 0 0 1 0 1 0 0 8 0 2 2 0 1 0 28 0 3 2 0 0 1 32 29 30 Chapter 2. The Simplex Algorithm Sometimes Table 2.1 is referred to as the “simplex tableau.” In a nutshell, the simplex algorithm works as follows: 1. We begin by finding an initial basic feasible solution (BFS). For many of the initial problems we shall encounter, this task is readily accomplished by setting all decision variables equal to zero. 2. We use the top row of the tableau to determine which nonbasic variable should then become positive in order to increase the objective function most rapidly. Because this variable will switch from nonbasic to basic, we call it the entering variable . 3. We then find the basic variable that swaps places with the entering variable and becomes nonbasic. 4. Once we determine which variables trade roles, we perform elementary row operations, whose purpose is to yield a new tableau containing updated values for all variables, as well as the objective function.

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  Linear Mathematics

  A Practical Approach

  • Patricia Clark Kenschaft(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  Chapter 6 THE SIMPLEX ALGORITHM

  6.1 Solving Standard Linear Programming Problems Using the Simplex Algorithm

  The linear programming problems in Chapter 5 included both nonnegativity constraints and other constraints. We shall refer to the constraints in any linear programming problem that are not nonnegativity constraints as significant constraints . These will be the constraints of primary interest, since whenever The Simplex Method is used, all the independent variables are assumed to be nonnegative . Thus the nonnegativity constraints are often not written, but simply understood.

  A linear programming problem is said to be standard if the objective function is to be maximized and if all the significant constraints are of the form a1 x1 + a2 x2 + … + an xn ≤ b where the ai and b are constants (b ≥ 0) and the x

  i

  are variables. Section 5.2 and 5.3 discussed only standard linear programming problems, as will this section and the next. In Section 7.1 we shall see that every standard linear programming problem can be expressed in the form “Maximize P = DX when AX ≤ B,” where D is a row vector, B is a column vector, A is a matrix, and all the elements in B are nonnegative.

  An algorithm is a method for solving a particular type of routine problem. The simplex algorithm (or method) is a mathematical technique that was developed in the middle of the twentieth century for the purpose of solving linear programming problems. This section shows how to use the simplex algorithm to solve standard linear programming problems. The next section explains more of the reasoning behind the algorithm, and Section 6.3 shows how it can be adapted to linear programming problems that are not in standard form.

  The simplex algorithm uses a basic idea from the tabular method-that of examining vertices of the feasible region. But only certain vertices are examined by the simplex algorithm. Each such vertex is described by a simplex tableau; a series of tableaux are written, each one describing a vertex adjacent to the one described by the previous tableau. In each tableau the value of the objective function is increased over the value in the previous tableau. When the objective function can increase no more, we have reached the maximum, and the location of the corresponding vertex can be read off the tableau. A standard linear programming problem always includes the origin as one of its feasible points; thus the first tableau can and will always describe the origin.

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  Finite Mathematics

  An Applied Approach

  • Michael Sullivan(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  OUTLINE 217 5 In Chapter 4 we described a geometric method (using graphs) for solving linear programming problems. Unfortunately, this method is useful only when there are no more than two variables and the number of constraints is small. Since most practical linear programming problems involve systems of several thousand linear inequalities containing several thousand variables, more sophisticated techniques need to be used. One of these methods is The Simplex Method, the subject of this chapter. The Simplex Method is a way to solve linear programming problems involving many inequalities and variables. Developed by George Dantzig in 1946, it is particularly well suited for computerization. In 1984, Narendra Karmarkar of Bell Laboratories discovered a way of solving large linear programming problems that led to methods that compete with and often outperform The Simplex Method. A discussion of LINDO, a software package that performs The Simplex Method, is provided in Appendix B. That hiking trip to Yosemite is still going to happen. But when you told the others who are going about your trail mix of unsalted peanuts and raisins, they all said “borrring.” It was decided to use the unsalted peanuts and raisins mixed with M&Ms and salted mini-pretzels. Now it’s more complicated to determine in what proportions these four ingredients should be mixed, and there remain the issues of carbohydrates, protein, calories, and fat. Fortunately, you also finished The Simplex Method before midterm, so you can handle the problem just fine. The Chapter Project at the end of the chapter will guide you. 5.1 The Simplex Tableau; Pivoting 5.2 The Simplex Method: Solving Maximum Problems in Standard Form 5.3 Solving Minimum Problems Using the Duality Principle 5.4 The Simplex Method for Problems Not in Standard Form • Chapter Review • Chapter Project • Mathematical Questions from Professional Exams A Look Back, A Look Forward Linear Programming: Simplex Method

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  Introductory Mathematical Economics

  • Adil H. Mouhammed(Author)
  • 2020(Publication Date)
  • Routledge

    (Publisher)

  Chapter Seven

  Linear Programming I: The Simplex Method

  Economics was defined by Lionel Robbins (1932) as the science of studying the allocation of scarce economic resources among competing ends, that is, economics is concerened with making choices and decisions. Similarly, linear programming is a mathematical theory by which the available economic resources can be allocated efficiently for achieving a certain goal.

  Linear programming was developed independently by Leonid Kantorovich and T.C. Koopmans; both shared the Nobel prize in economics in 1975 for developing this mathematical model (Kantorovich 1960,1965; Koopmans 1951). Kantorovich was able to find a mathematical model by which the available economic resources in the Soviet Union could be allocated efficiently. Kantorovich, however, was not able to develop an algorithm by which a linear mathematical program can be solved. This task was left to George Dantzig to develop in 1947 (Dantzig 1963). Since then economists such as Dorfman (1953), Dorfman et al. (1958), and Baumol (1977) have applied the model of linear programming to a variety of real world problems in transportation, accountancy, finance, diet, the military, agriculture, and human resources, to mention a few.

  A Problem’s Formulation and Assumptions

  The most difficult part of linear programming is the formulation of a particular segment of the real world in a linear mathematical programming context (Taha 1987; Thompson 1976; Lapin 1991). In fact, the formulation of a linear programming problem is an art that must be learned from experience rather than a textbook. Mathematically, a mathematical linear program is written as

  Maximize Z = c1 X1 + c2 X2 + c3 X3 + ... + cn Xn

  Subject to

  a 11

  X 1

  +

  a

  1 2

  X 2

  + ... +

  a ln

  X n

  ≤

  b 1

  a 21

  X 1

  +

  a 22

  X 2

  + ... +

  a

  2 n

  X n

  ≤

  b 3

  a

  3 1

  X 1

  +

  a

  3 2

  X 2

  + ... +

  a

  3 n

  X n

  ≤

  b 3

  ...................................

  a

  m 1

  X 1

  +

  a

  m 2

  X 2

  + ... +

  a

  m n

  X n

  ≤

  b m

  X 1

  ≥ 0 ,  

  X 2

  ≥ 0

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  Critical Path Analysis and Linear Programming

  • Mik Wisniewski, Jonathan H Klein(Authors)
  • 2017(Publication Date)
  • Red Globe Press

    (Publisher)

  However, we need to be cautious about removing redundant constraints. Sometimes sensitivity analysis on redundant constraints can highlight useful management information. 28 Linear Programming Summary In this chapter we have introduced the basic LP model, its formulation, its solution and interpretation and sensitivity analysis. In the next chapter we shall extend our analysis to allow us to examine larger and more realistic LP problems. Linear Programming: an Introduction 29 We have seen how the graphical solution method can be used to find the optimal solution to a two variable LP problem and how we can carry out sensitivity analysis. Clearly, in a business environment, we will never be faced with such a simple problem so in this chapter we introduce a general-purpose solution method, known as The Simplex Method. An understanding of the principles of The Simplex Method is important since most computer based LP packages use this approach. Without this understanding, it will be difficult to understand how solutions are determined and how the output from such packages can properly be interpreted. The Simplex formulation Recollect that for our basic profit maximization problem that we used in Chapter 2 the formulation is: Max. 5B + 6.5D subject to: 1B + 1D ≤ 1200 3B + 6D ≤ 4830 8B + 4D ≤ 5200 B,D ≥ 0 Although we already know the optimal solution, we shall use The Simplex Method to confirm this. The Simplex Method is a solution process that iteratively solves sets of equations until an optimal solution is found. We must first transform constraint inequalities into equations. Introducing slack variables Consider the first constraint: 1B + 1D ≤ 1200 30 The Simplex Method 3 We can rewrite this as: 1B + 1D + 1S 1 = 1200 where the new variable, S 1 , is known as a slack variable. A slack variable is introduced into the Simplex formulation to transform an inequality of the form ≤ into an equation.

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  Mathematical Programming Methods for Geographers and Planners

  • James Killen(Author)
  • 2021(Publication Date)
  • Routledge

    (Publisher)

  via the graphical method in the previous section is retained as the illustrative example.

  4.4.1 Standard Form Representation

  The Simplex Method comprises an algebraic method for finding the optimal solution to a linear programming problem. Given that most algebraic methods deal with equalities rather than inequalities, it is not surprising that The Simplex Method commences by restating the problem to be solved in terms of equations.

  Reconsider the first constraint of the problem of the previous section i.e.

  2

  X 1

  + 3

  X 2

  ≤ 600

  ---- (4.43)

  Because the left hand side of this equation is generally less than the right hand side, it is necessary to add a quantity to the left hand side to convert it to an equality. Denoting this quantity as SI gives the revised condition:

  2

  X 1

  + 3

  X 2

  +

  S 1

  = 600

  ---- (4.44)

  A value for S1 is fixed automatically if values for X1 and X2 are given; if X1 = X2 = 100, S1 = 100; if X1 = 250 and X2 = 10, S1 = 70. As can be seen, S1 makes up the difference or ‘slack’ between the left and right hand sides of (4.43) . For this reason, it is usually referred to as a slack variable. Converting the three inequalities (4.37) –(4.39) to equations by inserting slack variables denoted S1 , S2 , S3 respectively yields:

  2

  X 1

  + 3

  X 2

  +

  S 1

  =

  600

  )

  X 1

  +

  X 2

  +

  S 2

  =

  225

  )

  5

  X 1

  + 4

  X 2

  +

  S 3

  =

  1000

  )

  ---- (4.45)

  Consider now the objective (4.36) and the expression obtained by subtracting it from a quantity Xo and setting the resultant expression to zero i.e.

  X o

  − 3

  X 1

  − 4

  X 2

  = 0

  ---- (4.46)

  Suppose that X1 = 30 and X2 = 40. Then for (4.46) to hold true, Xo = 250 which is in fact the value the objective would have for those values of X1 and X2 i.e. the return the housebuilder would receive if those numbers of each housetype were built. Again, suppose that X1 =75 and X2 =150 which is in fact the optimal solution as determined in the last section. Then, automatically, Xo =825 which is in fact the optimal value of the objective. (4.46) is in fact defined in such a way that if X1 and X2 are known, Xo takes the value of the objective corresponding to those vagues of X1 and X2 . Combining (4.45) and (4.46) , the complete problem (4.36) -(4.40)

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