Optimization of Design Lecturer:Dung-An Wang Lecture 8
Lecture outline Reading: Ch8 of text Today s lecture 2
8.1 LINEAR FUNCTIONS Cost Function Constraints 3
8.2 The standard LP problem Only equality constraints are treated in standard linear programming 4
Summation Form of the Standard LP Problem 5
Matrix Form of the Standard LP Problem 6
8.2.2 Transcription to Standard LP Non-Negative Constraint Limits Treatment of <= Type Constraints Treatment of >= Type Constraints 7
Unrestricted Variables Require that all design variables to be nonnegative in the standard LP problem If a design variable x j is unrestricted in sign, it can always be written as the difference of two non-negative variables 8
EXAMPLE 8.1 CONVERSION TO STANDARD LP FORM 9
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BASIC CONCEPTS the optimum solution for the LP problem always lies on the boundary of the feasible set. The solution is at least at one of the vertices of the convex feasible set (called the convex polyhedral set). 11
Convexity of LP Since all functions are linear in an LP problem, the feasible set defined by linear equalities or inequalities is convex (Section 4.8). Also, the cost function is linear, so it is convex. Therefore, the LP problem is convex, and if an optimum solution exists, it is a global optimum, as stated in Theorem 4.10. 12
Write the linear system in a tableau 13
EXAMPLE 8.5 PIVOT STEP Obtain a new canonical form by interchanging the roles of the variables x1 and x4 (i.e., make x1 a basic variable and x4 a nonbasic variable). 14
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Simplex method Searches through only the basic feasible solutions and stops once an optimum solution is obtained 16
Determine three basic solutions using the Gauss-Jordan elimination method 17
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8.5 THE SIMPLEX METHOD EXAMPLE 8.7 STEPS OF THE SIMPLEX METHOD 21
The graphical solution to the problem Optimum solution along line CD. z*=4. 22
1. Convert the problem to the standard form the cost function expression as the last row. the cost function is in terms of only the nonbasic variables x1 and x2. This is one of the basic requirements of the Simplex method that the cost function always be in terms of the nonbasic variables When the cost function is only in terms of the nonbasic variables, then the cost coefficients in the last row are the reduced cost coefficients 23
2. Initial basic feasible solution 24
3. Optimality check If all of the nonzero entries of the cost row are nonnegative, then we have an optimum solution because the cost function cannot be reduced any further and the Simplex method is terminated. Since there are negative entries in the nonbasic columns of the cost row, the current basic feasible solution is not optimum 25
4. Selection of a nonbasic variable to become basic Select a variable associated with the smallest value in the cost row 26
5. Selection of a basic variable to become nonbasic Selection of the row with the smallest ratio as the pivot row maintains the feasibility of the new basic solution (all x i >= 0). 27
6. Pivot step 28
7. Optimum solution This solution is identified as point D in the figure. We see that the cost function has been reduced from 0 to -4. The coefficients in the nonbasic columns of the last row are non-negative so no further reduction of the cost function is possible. Thus, the foregoing solution is the optimum point. Note that for this example, only one iteration of the Simplex method gave the optimum solution. In general, more iterations are needed until all coefficients in the cost row become non-negative 29
EXAMPLE 8.8 SOLUTION BY THE SIMPLEX METHOD 30
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8.6.1 Artificial Variables When there are >= type constraints in the linear programming problem, surplus variables are subtracted from them to transform the problem into the standard form. The equality constraints, if present, are not changed because they are already in the standard form. For such problems, an initial basic feasible solution cannot be obtained by selecting the original design variables as nonbasic (setting them to zero), as is the case when there are only <= type constraints. To obtain an initial basic feasible solution, the Gauss- Jordan elimination procedure can be used to convert Ax=b to the canonical form. 34
However, an easier way is to introduce non-negative auxiliary variables for the >= type and equality constraints, define an auxiliary LP problem, and solve it using the Simplex method. The auxiliary variables are called artificial variables. These are variables in addition to the surplus variables for the >= type constraints. They have no physical meaning, but with their addition we obtain an initial basic feasible solution for the auxiliary LP problem by treating the artificial variables as basic along with any slack variables for the <= type constraints. All other variables are treated as nonbasic (i.e., set to zero). 35
EXAMPLE 8.12 INTRODUCTION OF ARTIFICIAL VARIABLES 36
standard form 37
canonical form introduce non-negative artificial variables 38
8.6.2 Artificial Cost Function The artificial variable for each equality and >= type constraint is introduced to obtain an initial basic feasible solution for the auxiliary problem. These variables have no physical meaning and need to be eliminated from the problem. To eliminate the artificial variables from the problem, we define an auxiliary cost function called the artificial cost function and minimize it subject to the constraints of the problem and the non-negativity of all of the variables. 39
The artificial cost function is simply a sum of all the artificial variables and will be designated as w 40
8.6.3 Definition of the Phase I Problem Since the artificial variables are introduced simply to obtain an initial basic feasible solution for the original problem, they need to be eliminated eventually. This elimination is done by defining and solving an LP problem called the Phase I problem. The objective of this problem is to make all the artificial variables nonbasic so they have zero value. 41
EXAMPLE 8.13 PHASE I OF THE SIMPLEX METHOD 42
EXAMPLE 8.13 PHASE I OF THE SIMPLEX METHOD 43
Now the Simplex iterations are carried out using the artificial cost row to determine the pivot element. In the initial tableau, the element a 31 =1 is identified as the pivot element 44
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Since both of the artificial variables have become nonbasic, the artificial cost function has attained a zero value. This indicates the end of Phase I of the Simplex method. Now we can discard the artificial cost row and the artificial variable columns x 5 and x 6 in Table 8.17 and continue with the Simplex iterations using the real cost row to determine the pivot column. 47
EXAMPLE 8.14 USE OF ARTIFICIAL VARIABLES FOR >= TYPE CONSTRAINTS 48
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