2. For the Transportation Problem given by the following tableau, find an initial basic feasible solutionby Vogel's method and then find an optimal solution.L1L2L3SUPPLYS19151210S26.1323S39.31127DEMAND211425

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Answered: For the Transportation Problem given by…

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter4: Linear Programming Models

Section: Chapter Questions

Problem 135P

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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.
Consider 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 >= 0
Consider 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 primal LP problem:Maximize X1 + 2X2 – 9X3 + 8X4 – 36X5Subject to 2X2 – X3 + X4 – 3X5 ≤ 40 X1 – X2 + 2X4 – 2X5 ≤ 10 X1 ≥ 0, X2 ≤ 0, X3 ≥ 0, X4 ≥ 0, X5 ≥ 0 (a) Write the dual of the LP above, using variables Y1, Y2, etc.(b) Sketch the feasible region of the dual LP in 2 dimensions, and use the graphical method to find the dual optimalsolution (Plot an isovalue line corresponding to the feasible solution, move the line in improving direction, findthe last one touching the feasible region, and any point(s) on the intersection of the last isovalue line andfeasible region are optimal solutions)(c) Using complementary slackness conditions, - write equations which must be satisfied by the optimal primal solution X* - which primal variables must be zero?(d) Using the information in (c), determine the optimal primal solution X* (e) Compare the optimal objective values of the primal and dual solutions
Consider 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? YesNo
Consider the following set of constraints: 48Y >= 7296; 0.25 X + 12Y >= 1824, and X + Y <= 152. Pick a suitable statement for this problem: a. Solution to this problem cannot be found without the objective function. b. The feasible region is defined by a single (unique) point. c. It is a non-linear problem - unsuitable for grphical method. d. This problem has two feasible points - one is optimal for miniization problem and other is optimal for maximization problem. e. Feasible region is represented by a line and multiple feasible points are available.
Solve the following Linear programming problem using the simplex method:Maximize Z = 10X1 + 15X2 + 20X3subject to:2X1 + 4X2 + 6X3 ≤ 243X1 + 9X2 + 6X3 ≤ 30X1, X2 and X3 ≥ 0(b) Suppose X1, X2, X3 in (a) refer to number of red, blue, and green balloons respectivelywhich are produced by a company per day. And Z is the total profit obtained afterselling these balloons. Interpret your answer obtained in (a) above(c) Write the dual of the following linear programming problem:Minimize Z = 2X1 − 3X2 + 4X3subject to:3X1 + 4X2 + 5X3 ≥ 96X1 + X2 + 3X3 ≥ 47X1 − 2X2 − X3 ≤ 105x1 − 2X2 + X3 ≥ 34X1 + 6X2 − 2X3 ≥ 3X1, X2 and X3 ≥ 0
Consider the following LP problem with two constraints: 32X + 39Y >= 1248 and 17X + 24Y >= 408. The objective function is Max 13X + 19Y . What combination of X and Y will yield the optimum solution for this problem? a. 0 , 17 b. unbounded problem c. 0 , 17 d. infeasible problem e. 24 , 0
Consider the following LP problem with two constraints: 18X + 8Y >= 144and 9X + 4Y= 36. The objective function is Min 14X + 30Y . What combination of X and Y will yield the optimum solution for this problem? a. infeasible problem b. unbounded problem c. 0 , 9 d. 4 , 0 e. 2 , 4.5
We 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).
For this linear programming problem, formulate the linear programming model. Then, find the optimal solution graphically for the LP with only 2 variables. i.e: Max Z = 500x + 300y Subject to: 4x + 2y <= 60 (1st constraint) 2x + 4y <= 48 (2nd constraint) x, y >= 0 (non-negativity) A construction company manufactures bags of concrete mix from beach sand and river sand. Each cubic meter of beach sand costs ₽60 and contains 4 units of fine sand, 3 units of coarse sand, and 5 units of gravel. Each cubic meter of river sand costs ₽100 and contains 3 units of fine sand, 6 units of coarse sand, and 2 units of gravel. Each bag of concrete must contain at least 12 units of fine sand, 12 units of coarse sand, and 10 units of gravel. Find the best combination of beach sand and river sand which will meet the minimum requirements of fine sand, coarse sand, and gravel at the least cost.
4. Consider the following linear programming problem: Maximize Z=$15x + $5y, subject to (1) 2x + y ≤ 10 and (2) 4x + 3y ≤ 24 and (3) x, y ≥ 0. Will the optimal solution change if the objective function becomes Maximize Z=$15x + $20y (constraints remain the same)? Select one: a. Can't determine given the information. b. Yes, it will change. c. No, it remains the same.

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For the Transportation Problem given by the following tableau, find an initial basic feasible solution by Vogel's method and then find an optimal solution. L1 L2 L3 SUPPLY si 9. 15 12 10 S2 8 13 23