To solve a linear programming problem using the Simplex Method, follow these steps:### (A) Introduce Slack, Surplus, and Artificial Variables and Formulate the Modified Problem**Objective:**\[ \text{Maximize} \quad P = 9x_1 + 7x_2 \]**Subject to:**\[ x_1 + x_2 = 2 \]\[ -x_1 + 3x_2 \leq 20 \]\[ x_1, x_2 \geq 0 \]**Introducing Variables:**1. For the equality constraint $ x_1 + x_2 = 2 $: - Introduce an artificial variable $ A_1 $.2. For the inequality $ -x_1 + 3x_2 \leq 20 $: - Introduce a slack variable $ s_2 $.**Modified Problem:**\[ \text{Maximize} \quad P = 9x_1 + 7x_2 - M a_1 \]**Subject to:**\[ x_1 + x_2 - s_1 + a_1 = 2 \]\[ -x_1 + 3x_2 + s_2 = 20 \]\[ x_1, x_2, s_1, s_2, a_1 \geq 0 \]### (B) Write the Preliminary Simplex Tableau for the Modified Problem and Find the Initial Simplex Tableau**Initial Simplex Tableau:**| Constraints | $ x_1 $ | $ x_2 $ | $ s_1 $ | $ s_2 $ | $ a_1 $ | P | RHS ||-------------|-----------|-----------|-----------|-----------|-----------|----|-----|| $ s_1 $ | 1 | 1 | 1 | 0 | 1 | 0 | 2 || $ s_2 $ | -1 | 3 | 0 | 1 | 0 | 0 | 20 || PL | -9 | -7 | 0 | 0 | M | 1 | 0 |### Detailed Explanation of the Tableau:- **Columns

Given : A problem Maximize P=9x1+7x2 subject to the constraintsx1+x2≥2-x1+3x2≤20 andx1,x2≥0 To find…

Answered: (A) Introduce slack, surplus, and…

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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Determine whether the following statement is true or false, and explain why. A surplus variable of a linear programming problem can never have a negative value during the intermediate steps of the simplex method. Choose the correct answer below. O A. The statement is true. A surplus variable always has a positive value during every step of the simplex method, including intermediate steps. B. The statement is false. A surplus variable cannot have a negative value at the beginning or end of the simplex method. It almost always has a negative value during at least some of the intermediate steps. C. The statement is true. A surplus variable can never have a negative value during any step of the simplex method, including intermediate steps. D. The statement is false. A surplus variable cannot have a negative value at the beginning or end of the simplex method. It may have a negative value during an intermediate step.
Solve the following differential equation: yde+xdy 1-x²y² +ædr=0. = constant.
Use the Simplex Method to solve the following problem. Show all work. 2. A contractor builds three types of houses; the Aries, the Belfair and the Wexford. Each house requires one lot and the following table gives the number of labor-hours and the amount of capital needed for each type of house, as well as the profit on the sale of each house. There are 12 lots, 47,500 labor-hours and $3,413,000 available for the contractor's use. Belfair 3,700 $279,600 $25,000 Wexford 5,000 Aries 3,000 $205,000 $20,000 Labor-hours Capital Profit $350,000 $30,000 a. Building how many of each type of house will maximize her profit b. What is the maximum possible profit?
Formulate but do not solve the following exercise as a linear programming problem. A financier plans to invest up to $400,000 in two projects. Project A yields a return of 10% on the investment of x dollars, whereas Project B yields a return of 13% on the investment of y dollars. Because the investment in Project B is riskier than the investment in Project A, the financier has decided that the investment in Project B should not exceed 40% of the total investment. How much should she invest in each project to maximize the return on her investment P in dollars? MaximizeP= subject to the constraints ? amount available for investment? allocation of funds?
2. Solve by Lagrange Multiplier Method: Clearly identify your constraint function and your system. You may use either approach you learned in class and on the video. Mr. Eppolito has S360 to spend on fencing to fence a rectangular garden. The fencing for the north and south sides of the garden costs $10/ft. The fencing for the east and west sides costs $15/ft. Find the dimensions that will allow Mr. Eppolito to fence in the largest-sized garden. Draw a diagram of this scenario. Write the method you will use to solve it, then do so, showing all the necessary equations.
The problem below involves three variables. Solve it with the simplex method, Excel, or some other technology.A medical clinic performs three types of medical tests that use the same machines. Tests A, B, and C take 15 minutes, 30 minutes, and 1 hour, respectively, with respective profits of $120, $260, and $530. The clinic has four machines and four workers available. One person is qualified to do test A, two to do test B, and one to do test C. If the clinic has a rush of customers for these tests, how many of each type should it schedule in a 8-hour day to maximize its profit? ? A tests ? B tests ? C tests
Simplex method and elementary row of operations if necessary. Please show steps.
Use the non-standard simplex method to solve. Show work by setting up the initial simplex tableau (with the columns labeled), circling the first pivot element, and giving the final simplex tableau. If there is no solution, then state which type: Not Feasible or Unbounded. If the solution is unique, then provide the solution with context. If there are infinite solutions, then describe the line segment containing the solutions by providing the two end points. Maximize: P=0.15x+0.17y subject to: 1.10x+1.30y<=5400 x+y>=5000 x>=0, y>=0
Solve the linear programming problem using the simplex method. Maximize P = -x₁ + 2x₂ subject to x₁ + x₂ ≤2 -X₁ + 3x₂ ≤18 X₁-4x2 ≤8 X₁, X₂ 20 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. OA. The maximum value of P is P= (Simplify your answers.) OB. There is no optimal solution. when x₁ = *** and X₂=

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