Consider the following problem Minimize Z = 4x₁ + 12x₂ + 12x3 subject to X1 + 3x3 ≥ 3 2x2 + 2x322 and x₁ ≥ 0, x₂ ≥ 0, x3 ≥ 0. (a) Convert this problem into an augmented form to get ready for applying the dual simplex method. (b) Use the dual simplex method manually to solve this problem.
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- Another way to derive a demand function is to break the market into segments and identify a low price, a medium price, and a high price. For each of these prices and market segments, we ask company experts to estimate product demand. Then we use Excels trend curve fitting capabilities to fit a quadratic function that represents that segments demand function. Finally, we add the segment demand curves to derive an aggregate demand curve. Try this procedure for pricing a candy bar. Assume the candy bar costs 0.55 to produce. The company plans to charge between 1.10 and 1.50 for this candy bar. Its marketing department estimates the demands shown in the file P07_47.xlsx (in thousands) in the three regions of the country where the candy bar will be sold. What is the profit-maximizing price, assuming that the same price will be charged in all three regions?This problem is based on Motorolas online method for choosing suppliers. Suppose Motorola solicits bids from five suppliers for eight products. The list price for each product and the quantity of each product that Motorola needs to purchase during the next year are listed in the file P06_93.xlsx. Each supplier has submitted the percentage discount it will offer on each product. These percentages are also listed in the file. For example, supplier 1 offers a 7% discount on product 1 and a 30% discount on product 2. The following considerations also apply: There is an administrative cost of 5000 associated with setting up a suppliers account. For example, if Motorola uses three suppliers, it incurs an administrative cost of 15,000. To ensure reliability, no supplier can supply more than 80% of Motorolas demand for any product. A supplier must supply an integer amount of each product it supplies. Develop a linear integer model to help Motorola minimize the sum of its purchase and administrative costs.Given the following LP: Min 4x1 + 6x2 s.t 2x1 + 2x2 ≥ 3, x1 + 3x2 ≥ 2, x1 + 3x2 ≥ 0, Find its dual LP. Use graphical method or the Simplex algorithm, solve the primal LP in (1). Use graphical method or the Simplex algorithm, solve the dual LP (that you find in Part A). Find its dual LP.
- Analyze algebraically what special case in simplex application is present in each of the LP model below. Give an explanation to support your answer. a) Maximize z = 4x1 + 2x2 Subject to: 2x1 - x2 ≤ 2 3x1 - 4x2 ≤ 8 x1, x2 ≥ 0b) Maximize z = 3x1 + 2x2 Subject to: 4x1 - x2 ≤ 8 4x1 + 3x2 ≤ 12 4x1 + x2 ≤ 8 x1, x2 ≥ 0Consider the following LP model in standard form, with a row for the objective function Z. a) Put it into Canonical form ( or Simplex Tableau form) with basic variables X1, X2 , and X3. b) Determine the association BFS (Basic Feasible Solution) and the new formula for the objective function Z Minimize 10X1 + 4X2 Sujbject to 3X1 + 2X2 - X3 = 60 7X1 + 2X2 - X4 = 84 3X1 + 6X2 -X5 = 72 X1, X2, X3 , X4 , X5 >= 0We have 60 meters of fence and want to fence a triangular shaped area. Please formulate an NLP (do not try to solve) that will enable us to maximize the fenced area (Hint: The area of a triangle with sides of length a, b, and c is ( s (s – a) (s – b) (s – c))1/2, where s is half the parameter of the triangle).
- Solve the following problem after finding its dual. Min z = x1 - 3x2 + 3x3 s.t. 3x1 - x2 + 2 x3 ≤ 7 2x1 + 4x2 ≥ -12 -4x1 + 3x2 + 8x3 ≤ 10 x1, x2, x3 ≥ 0Consider the following all-integer linear program. Max 5x1 + 8x2 s.t. 6x1 + 5x2 ≤ 25 10x1 + 4x2 ≤ 40 1x1 + 2x2 ≤ 8 x1, x2 ≥ 0 and integer (a) Graph the constraints for this problem. Use points to indicate all feasible integer solutions. (b) Find the optimal solution to the LP Relaxation. (Round your answers to three decimal places.) at (x1, x2) = Using this solution, round down to find a feasible integer solution. at (x1, x2) = (c) Find the optimal integer solution. at (x1, x2) = Is it the same as the solution obtained in part (b) by rounding down? YesNoConsider the following linear programming model with 4 regular constraints:Maximize 3X + 5Y (a) Draw your graph in the space below:subject to: 4X + 4Y ≤ 48 (constraint #1) 4X + 3Y ≤ 50 (constraint #2) 2X + 1Y ≤ 20 (constraint #3) X ≥ 2 (constraint #4) X, Y ≥ 0 (non-negativity constraints)(a) Which of the constraints is redundant? Constraint #______.Justify by drawing a graph similar to Figure 7.14 on p.263.(b) Is point (9,3) a feasible solution? _____. Explain your answer (by analyzing each of the constraints).Constraint #1: _______________________________________________________________Constraint #2: _______________________________________________________________Constraint #3: _______________________________________________________________Constraint #4: ______________________________________________________________
- Consider the following all-integer linear program: Max 5x1 + 8x2 s.t. 6x1 + 5x2 ≤ 28 11x1 + 5x2 ≤ 46 x1 + 2x2 ≤ 8 x1, x2 ≥ 0 and integer Find the optimal solution to the LP Relaxation. If required, round your answers to two decimal places. x1= fill in the blank 2 x2= fill in the blank 3 Optimal Solution to the LP Relaxation fill in the blank 4 Round down to find a feasible integer solution. If your answer is zero enter “0”. x1= fill in the blank 5 x2= fill in the blank 6 Feasible integer solution fill in the blank 7 Find the optimal integer solution. If your answer is zero enter “0”. x1= fill in the blank 8 x2= fill in the blank 9 Optimal Integer Solution fill in the blank 10Consider the following all-integer linear program: Max 5x1 + 8x2 s.t. 6x1 + 5x2 ≤ 28 11x1 + 5x2 ≤ 46 x1 + 2x2 ≤ 8 x1, x2 ≥ 0 and integer Find the optimal solution to the LP Relaxation. If required, round your answers to two decimal places. x1= fill in the blank 2 x2= fill in the blank 3 Optimal Solution to the LP Relaxation fill in the blank 4 Round down to find a feasible integer solution. If your answer is zero enter “0”. x1= fill in the blank 5 x2= fill in the blank 6 Feasible integer solution fill in the blank 7 Find the optimal integer solution. If your answer is zero enter “0”. x1= fill in the blank 8 x2= fill in the blank 9 Optimal Integer Solution fill in the blank 10 Is it the same as the solution obtained in part (b) by rounding down?Find the optimal solution for the following problem. Maximize C = 4x + 12y subject to 3x + 5y ≤ 12 6x + 2y ≤ 10 and x ≥ 0, y ≥ 0. What is the optimal value of x? What is the optimal value of y? (Round your answer to 3 decimal places.) What is the maximum value of the objective function? (Round your answer to 3 decimal places.)