Suppose a primal minimization problem and its dual maximization problem were solved by using the simplex method on the dualproblem, and the final simplex matrix is given.1000+in45=//250 4 -0 -34401-2205-11514540 0 17ܗ102700 2857 0 1440(a) Find the solution of the minimization problem. Use y₁, Y2, Y3 as the variables and g as the function.Y1 =Y2 =Y3 =g =(b) Find the solution of the maximization problem. Use X₁, X2, X3 as the variables and f as the function.X1 =x2 =X3 =f =

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Answered: Suppose a primal minimization problem…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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8. An upper-triangular linear system has the form a12 a13 a14 022 023 a24 0 033 034 0 0 044 a11 0 0 0 21 X2 23 by b₂ b3 ba for example, in the case of a 4 x 4 system. Such systems are easily solved by back substitution (solve the last equation for 24, then the previous equation for 23, etc.) How many multiplication operations does this require? (Count a division the same as a multipli- cation, so computing 74=b4/a44 takes 1 "multiply", for example.) Exactly how many "multiplies" does it take to solve a general n x n upper-triangular system? In particular, how many to solve an upper-triangular system of 1000 equations?
Complete the next pivot operation for the given minimization problem and then determine the values for x . Х1 X2 X3 yi C 3 1 78 4 1 1 36 -5 -1 1 (a)x1 =9, x2 = 51 (b)x1 = 9, x2 =0 (c)x1 = 0, x2 = 1/4 (d)x1 = 51, x2 = 9 N
A B (U) (O) D The augmented matrix ear equations which has 1-2 1-3 0 15 2 00 0 0 represents a system of lin- no solution. finitely many solutions. An infinitely many solutions. the unique solution □=□ 2
Suppose that Ax=b has no solution and the columns of A are linearly independent. Let t(A) denote the transpose of matrix A. Select all true statements. O If the A matrix and b vector represent data values, then the least-squares solution gives coefficients for a function that make this function the line that best fits the data values. O The system has a least-squares solution. The solution to t(A)Ax=t(A)b is the least-squares solution. O In the system Ax-b=e, it is possible for e to be the zero vector. O The system has a least squares solution but it may not be unique. O The inverse of t(A)A exists. O The least-squares solution (if it exists) yields a vector in the column space of A that is the "closest" to the vector b.
What is the corresponding system of equations? Restate the problem in terms of equations by adding slack variables or subtracting surplus variables as necessary. Then write the initial simplex tableau. X1 + X2 + X3 S1 = 69 Маximize Z-2x1 + 3x2 — 4xз X1 + X2 + X3 S2 = 48 X1 + X2 X1 20, x2 2 0, X3 0, s, 20, s2 2 0, s3 20 subject to X1 +X2 + X3 <69 + S3 = 60 X1 +X2 + X3 248 X1 +X2 < 60 Write the initial simplex tableau, omitting the z column. Fill in the missing values. X1 20, x2 2 0, X3 20 X1 X2 X3 S1 S2 $3 1 1 1 O 69 1 1 1 0 48 1 1 0| 60 4
3. Assume that you have performed Gauss-Jordan elimination on an augmented matrix and have come up with the following augmented matrix. What would be the solution to the system according to this output? If infinitely many solutions, make sure to classify the solutions in terms of a single variable. 10 1 0 1 1 0 0 0 4 -7
Solve the given linear programming problem using the Duality Method. Show work by creating the primal and dual augmented matrices, setting up the initial simplex tableau, circling the first pivot element, and giving the final simplex tableau. State the optimal solution and the optimal value for the final answer. Minimize: C= 24x+18y+24z Subject to: 3x+2y+z>=4 x+y+3z>=6 x>=0, y>=0, z>=0
4. The final simplex matrix for a problem is shown below. Give the solutions to both the maximization problem and the dual minimization problem based on this final matrix. Carefully label each solution. x2 x3 S1 S2 S3 P 1 -0.5 3 -0.7 29 1 -0.8 -2 -0.5 47 -0.6 -1 -2.4 1 53 15 7 13 1 597 O O o-
We are to minimize P = 3a+ 26 subject to the following constraints. a+b 24 3a+b 26 a, b 20 What is the dual problem in matrix form? 11 4 а. 3 1 6 3 20 1 1 4 b. 3 16 1 1 0 133 C. 1 1 2 4 6 0. d. 1 4 3 1 -3 -2 0 е. 13 -3 1 1 -2 4 6 0 1.

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