3205-Chapter-2 - Chapter 2 The Simplex Method Also see...

Unformatted text preview: Chapter 2 The Simplex Method Also see Chapter 4 of the text book. 1 Also see Chapter 4 of the text book. 1 Basic Concept Terminology 1 1.1Basic Concept illustrate basic concept considering following problem: WeWe illustrate thethe basic concept byby considering thethe following problem: Figure 2.1:1:Feasible Figure Feasibleset setofofananLP LPproblem. problem. Let n be the number of decision variables; in this case, n = 2. To start, we introduce Let n be the number of decision variables; in this case, n = 2. To start, we introduce terminology that we shall use often: terminology that we shall use often: • Each line in the above figure is a constraint boundary. For examples, 3x1 + 2x2 = 18, = 4, line and in x1the = 0above are constraint •x1 Each figure is a boundaries. constraint boundary. For examples, 3x1 +2x2 = 18, x1 intersection = 4, and x1 point = 0 are boundaries. • The of constraint n constraint boundaries is a corner-point solution or CP solution. 1 • If a corner-point solution is feasible, it is called a corner-point feasible (CPF) solution. Otherwise, it is a corner-point infeasible solution. • Two CPF solutions are adjacent to each other if they share n 1 common constraint boundaries. In this example with n = 2, each CPF solution shares one constraint boundary with its adjacent CPF. • The line joining two CPF solutions is an edge of the feasible region. • Consider any linear programming problem that possesses at least one optimal solution. If a CPF solution has no adjacent CPF solutions that are better, then it must be an optimal solution. We call this an optimality test. 2 2 The Key Solution Concepts • The simplex method focuses solely on CPF solutions. For any problem with at least one optimal solution, finding one requires only finding a best CPF solution. The only restriction is that the problem must possess CPF solution. This is ensured if the feasible region is bounded. Since the number of feasible solutions generally is infinite, reducing the number of solutions that need to be examined to a small finite number is a tremendous simplification. • The simplex method is an iterative algorithm. • Whenever possible, the initialization of the simplex method chooses the origin (all decision variables equal to zero) to be the initial CPF solution. When there are too many decision variables to find an initial CPF solution graphically, this choice eliminates the need to use algebraic procedures to find and solve for an initial CPF solution. • Given a CPF solution, it is much quicker computationally to gather information about its adjacent CPF solutions than about other CPF solutions. Therefore, each time the simplex method performs an iteration to move from the current CPF solution to a better one., it always chooses a CPF solution that is adjacent to the current one. No other CPF solutions are considered. Consequently, the entire path followed to eventually reach an optimal solution is along the edges of the feasible region. • After the current CPF solution is identified, the simplex method examines each of the edges of the feasible region that emanate from this CPF solution. Each of these edges leads to an adjacent CPF solution at the other end, but the simplex method does not even take the time to solve for the adjacent CPF solution. Instead, it simply identifies the rate of improvement in Z that would be obtained by moving along the edge. Among the edges with a positive rate of improvement in Z, it then chooses to move along the one with the largest rate of improvement in Z. The iteration is completed by first solving for the adjacent CPF solution at the other end of this one edge and then relabeling this adjacent CPF solution as the current CPF solution for the optimality test and (if needed) the next iteration. • Last step describes how the simplex method examines each of the edges of the feasible region that emanate from the current CPF solution. This examination of an edge leads to quickly identifying the rate of improvement in Z that would be obtained by moving along the edge toward the adjacent CPF solution at the other end. A positive rate of improvement in Z implies that the adjacent CPF solution is better than the current CPF solution, whereas a negative rate of improvement in Z implies that the adjacent CPF solution is worse. Therefore, the optimality test consists simply of checking whether any of the edges give a positive rate of improvement in Z. If none do, then the current CPF solution is optimal. 3 • Consider any linear programming problem that possesses at least one optimal solution. If a CPF solution has no adjacent CPF solutions that are better, then it must be an optimal solution. We call this an optimality test. 1.2 Illustrative Example 3 Illustrative Example We 1.1using usingthe thesimplex simplex method. method. We shall only Wesolve solvethe theproblem problemininSection Fig. 2.1 only highlight highlight some some key points points here; here; the the details details will will be be discussed discussed later later in in this this chapter. chapter. key Figure Anillustration illustrationofofthe thesimplex simplexmethod. method. Figure 2.2: 2: An a. The procedure starts with the initial CFP (0, 0). For problems where (0, 0) is not feasible, we need a special procedure to look for an initial CFP; this will be discussed later in this chapter. We shall move to one of the adjacent CPF solutions 0), Last updated: October 7, 2014 Pageof2 (0, of 18 namely, (0, 6) or (4, 0). We prefer (0, 6) since it has a better objective function value. In fact, the objective function, 3x1 + 5x2 has a greater rate of improvement along x2 since x2 has a greater coefficient. b. The point (0, 6) has two adjacent CFPs, namely, (0, 0) and (2, 6). We have considered (0, 0) already in Step 0 and it has a worse objective function value than (0, 6). Hence, we move to (2, 6). c. Since (2, 6) has no better adjacent CPF solutions, we know that it is the optimal solution, according to the optimality test described previously. 4 Example 3.1 (Class work). Solve the problem using the technique introduced above: max x1 ,x2 2R 3x1 + 2x2 s.t. x1 + 2x2 6 2x1 + x2 8 x1 + x2 1 x2 2 x1 , x2 , · · · , xn 0. As an exercise, you should sketch the constraints, feasible set, and all CFP solutions by yourself before you proceed. 0. We start with the CPF solution (0, 0). The adjacent CPF solutions are (0, 1) and (4, 0). Since the objective function has a greater rate of increase for x1 , we increase x1 to 4, i.e., moving to (4, 0). 1. The point (4, 0) is not optimal since it has a better adjacent CPF solution (10/3, 4/3). Therefore, we move to that point. 2. The point (10/3, 4/3) has no better CPF solution and therefore it is the optimal solution for this problem. 5 04_089-160.qxd 4 Setting up the Simplex Method The simplex method normally is run on a computer, which can follow only algebraic instructions. Therefore, it is necessary to translate the conceptually geometric procedure just described into a usable algebraic procedure. In this section, we introduce the algebraic language of the simplex method and relate it to the concepts of the preceding sections. The algebraic procedure is based on solving systems of equations. Therefore, the first step in setting up the simplex method is to convert the functional inequality constraints to equivalent equality constraints. (The nonnegativity constraints are left as inequalities because they are treated separately.) This conversion is accomplished by introducing the slack variables. For example, given the inequality constraint 11/19/08 08:28 AM Page 95 x1 4, (1) we introduce the slack variable x3 and convert (1) into the following constraints: x1 + x3 = 4, x3 0. 4.2 SETTING UP THE SIMPLEX METHOD 95 Equation (1) is equivalent to the above two constraints. the introduction of slack variables otherfunctional functional constraints, the the original Upon the Upon introduction of slack variables for for thethe other constraints, original linear programming model for the example (shown below on the left) can now linear programming model for the example (shown below on the left) can now be replaced by be replaced by (called the equivalent model (calledform the augmented form ofshown the model) shown the equivalent model the augmented of the model) below on the right: below on the right: Augmented Form of the Model4 Original Form of the Model Maximize Z ! 3x1 # 5x2, subject to Maximize Z ! 3x1 # 5x2, subject to x1 # 2x2 $ 4 (1) 3x1 # 2x2 $ 12 (2) 2x2 3x1 # 2x2 $ 18 (3) 3x1 # 2x2 and x1 # x3 ! # x4 4 ! 12 # x5 ! 18 and x1 % 0, x2 % 0. xj % 0, for j ! 1, 2, 3, 4, 5. Although formsofofthe the model model represent exactly the same the new form Although bothboth forms represent exactly theproblem, same problem, theis new form is much more convenient for algebraic manipulation and for identification of CPF solutions. much more convenient for algebraic manipulation and for identification of CPF solutions. We We call this the augmented form of the problem because the original form has been augcall this the augmented form of the problem because form has been augmented mented by some supplementary variables needed to applythe the original simplex method. by some supplementary variables to apply thethen simplex method If a slack variable equals 0needed in the current solution, this solution lies on the constraint boundary for the corresponding functional constraint. A value greater If a slack variable equals 0 in the current solution, then this solutionthan lies0 means on the constraint that the solution lies on the feasible side of this constraint boundary, whereas a value boundary for the corresponding functional constraint. A value greater than 0 less means that the than 0 means that the solution lies on the infeasible side of this constraint boundary. A solution lies on the feasible side of this constraint boundary, or an (interior point), whereas demonstration of these properties is provided by the demonstration example in your OR a value less 0 Interpretation means thatofthe solution lies on the infeasible side of this constraint Tutorthan entitled the Slack Variables. boundary. The terminology used in Section 4.1 (corner-point solutions, etc.) applies to the original form of the problem. We now(corner-point introduce the corresponding terminology forto thethe aug-original form The terminology used previously solutions, etc.) applies mented form. of the problem. We now introduce the corresponding terminology for the augmented form. An augmented solution is a solution for the original variables (the decision variables) that has been augmented by the corresponding values of the slack variables. 6 For example, augmenting the solution (3, 2) in the example yields the augmented solution (3, 2, 1, 8, 5) because the corresponding values of the slack variables are x3 ! 1, x4 ! 8, and x5 ! 5. • An augmented solution is a solution for the original variables (the decision variables) that has been augmented by the corresponding values of the slack variables. For example, augmenting the solution (3, 2) in the example yields the augmented solution (3, 2, 1, 8, 5) because the corresponding values of the slack variables are x3 = 1, x4 = 8 and x5 = 5. • A basic solution is an augmented corner-point solution. To illustrate, consider the corner-point infeasible solution (4, 6) in Fig. 2.1. Augmenting it with the resulting values of the slack variables x3 = 0, x4 = 0 and x5 = 6 yields the corresponding basic solution (4, 6, 0, 0, 6). • The fact that corner-point solutions (and so basic solutions) can be either feasible or infeasible implies the following definition: A basic feasible (BF) solution is an augmented CPF solution. Thus, the CPF solution (0, 6) in the example is equivalent to the BF solution (0, 6, 4, 0, 6) for the problem in augmented form. Overall, the only di↵erence between basic solutions and corner-point solutions (or between BF solutions and CPF solutions) is whether the values of the slack variables are included. For any basic solution, the corresponding corner-point solution is obtained simply by deleting the slack variables. Therefore, the geometric and algebraic relationships between these two solutions are very close. Because the terms basic solution and basic feasible solution are very important parts of the standard vocabulary of linear programming, we now need to clarify their algebraic properties. For the augmented form of the example, notice that the system of functional constraints has 5 variables and 3 equations, so we have that the number of variables - number of equations = 5 - 3 = 2. This fact gives us 2 degrees of freedom in solving the system, since any two variables can be chosen to be set equal to any arbitrary value in order to solve the three equations in terms of the remaining three variables. The simplex method uses zero for this arbitrary value. Thus, two of the variables (called the nonbasic variables) are set equal to zero, and then the simultaneous solution of the three equations for the other three variables (called the basic variables) is a basic solution. These properties are described in the following general definitions. A basic solution has the following properties: • Each variable is designated as either a nonbasic variable or a basic variable. • The number of basic variables equals the number of functional constraints (now equations). Therefore, the number of nonbasic variables equals the total number of variables minus the number of functional constraints. • The nonbasic variables are set equal to zero. • The values of the basic variables are obtained as the simultaneous solution of the system of equations (functional constraints in augmented form). (The set of basic variables is often referred to as the basis.) 7 • If the basic variables satisfy the nonnegativity constraints, the basic solution is a BF solution. hil76299_ch04_089-160.qxd 11/19/08 08:28 AM Page 97 Just as certain pairs of CPF solutions are adjacent, the corresponding pairs of BF solutions also are said to be adjacent. Here is an easy way to tell when two BF solutions are adjacent: “Two BF solutions are adjacent if all but one of their nonbasic variables are the same. This implies that all but one of their basic variables also are the same, although perhaps with di↵erent numerical values”. Consequently, moving from the current BF solution to an 4.3 THE one ALGEBRA OF THE SIMPLEX METHOD adjacent one involves switching variable from nonbasic to basic and vice versa for one 97 other variable (and then adjusting the values of the basic variables to continue satisfying the Consequently, moving from the current BF solution to an adjacent one involves switchsystem of equations). variable from nonbasic basicofand vice versa for one other variable To illustrate adjacenting BFone solutions, consider onetopair adjacent CPF solutions in Fig.(and then adjusting the values of the basic variables to continue satisfying the system of 2.1: (0, 0) and (0, 6). Their augmented solutions, (0, 0, 4, 12, 18) and (0, 6, 4, 0, 6), automatiequations). cally are adjacent BF solutions. However, youBFdosolutions, not need to look 2.1 toCPF draw this in To illustrate adjacent consider one at pairFig. of adjacent solutions conclusion. Another signpost is(0, that their nonbasic variables, (x1 , x(0, (x18) are6, the Fig. 4.1: 0) and (0, 6). Their augmented solutions, 4, 12, 4, 0, 6), 2 )0,and 1 , xand 4 ), (0, automatically are adjacent BF solutions. However, you do not need to look at Fig. 4.1 to same with just the one exception — x2 has been replaced by x4 . Consequently, moving from draw this conclusion. Another signpost is that their nonbasic variables, (x , x 1 for 2) and (0, 0, 4, 12, 18) to (0, 6, 4, 0, 6) involves switching x2 from nonbasic to basic and vice versa (x1, x4), are the same with just the one exception—x2 has been replaced by x4. Consex4 . quently, moving from (0, 0, 4, 12, 18) to (0, 6, 4, 0, 6) involves switching x2 from nonWhen we deal with the invice augmented basicproblem to basic and versa for x4.form, it is convenient to consider and manipulate the objective function equation sameintime as theform, newitconstraint When we deal withatthethe problem augmented is convenientequations. to consider and manipulate the objective function equation at the same time as the new constraint Therefore, before we start the simplex method, the problem needs to be rewritten once againequations. Therefore, before we start the simplex method, the problem needs to be rewritten in an equivalent way: once again in an equivalent way: Maximize Z, subject to (0) (1) (2) (3) Z ! 3x1 ! 5x2 " 0 x1 # x3 " 4 2x2 # x4 " 12 3x1 # 2x2 # x5 " 18 and xj $ 0, for j " 1, 2, . . . , 5. It is just as if Eq. (0) actually were one of the original constraints; but because it already in equality form,one no slack variable is needed. While adding more equation, we also It is just as if Eq. (0) isactually were of the original constraints; but one because it already have added one more unknown (Z ) to the system of equations. Therefore, when is in equality form, no slack variable is needed. While adding one more equation, we alsousing Eqs. (1) to (3) to obtain a basic solution as described above, we use Eq. (0) to solve for have added one more unknown (Z) to the system of equations. Therefore, when using Eqs. Z at the same time. (1) to (3) to obtain a basic solution describedtheabove, wethe useWyndor Eq. (0) to Co. solve for Zfits atour thestanSomewhatas fortuitously, model for Glass problem same time. dard form, and all its functional constraints have nonnegative right-hand sides bi. If this had not been the case, then additional adjustments would have been needed at this point before the simplex method was applied. These details are deferred to Sec. 4.6, and we now focus on the simplex method itself. ■ 4.3 THE ALGEBRA OF THE SIMPLEX METHOD 8 We continue to use the prototype example of Sec. 3.1, as rewritten at the end of Sec. 4.2, for illustrative purposes. To start connecting the geometric and algebraic concepts of the simplex method, we begin by outlining side by side in Table 4.2 how the simplex method solves this example from both a geometric and an algebraic viewpoint. The geometric viewpoint (first presented in Sec. 4.1) is based on the original form of the model (no slack vari- 5 _ch04_089-160.qxd 8 The Algebra of the Simplex Method 11/19/08 08:28 AM Page 98 To start connecting the geometric and algebraic concepts of the simplex method, we begin by outlining side by side in the following table how the simplex method solves this example from both a geometric and an algebraic viewpoint. The geometric viewpoint is based on the original form of the model (no slack variables), so again refer to Fig. 2.1 for a visualization when you examine the second column of the table. Refer to the augmented form of the model CHAPTER 4 SOLVING LINEAR PROGRAMMING PROBLEMS: THE SIMPLEX METHOD presented at the end of last section when you examine the third column of the table. We now fill in ■ theTABLE details for each step of the third column of the following table. 4.2 Geometric and algebraic interpretations of how the simplex method solves the Wyndor Glass Co. problem Method Sequence Geometric Interpretation Initialization Choose (0, 0) to be the initial CPF solution. Optimality test Iteration 1 Step 1 Step 2 Step 3 Optimality test Iteration 2 Step 1 Step 2 Step 3 Optimality test Not optimal, because moving along either edge from (0, 0) increases Z. Move up the edge lying on the x2 axis. Stop when the first new constraint boundary (2x2 ! 12) is reached. Find the intersection of the new pair of constraint boundaries: (0, 6) is the new CPF solution. Not optimal, because moving along the edge from (0, 6) to the right increases Z. Algebraic Interpretation Choose x1 and x2 to be the nonbasic variables (! 0) for the initial BF solution: (0, 0, 4, 12, 18). Not optimal, be...
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