For the following linear programming problem, determine the optimal solution using the graphical solution method. Max −X + 2Y s.t. 6X − 2Y ≤ 3 −2X + 3Y ≤ 8 X + Y ≤ 3 X, Y ≥ 0 (X, Y) =
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| Max | −X + 2Y | ||||
| s.t. | |||||
| 6X − 2Y | ≤ | 3 | |||
| −2X + 3Y | ≤ | 8 | |||
| X + Y | ≤ | 3 | |||
| X, Y ≥ 0 |
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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.Seas Beginning sells clothing by mail order. An important question is when to strike a customer from the companys mailing list. At present, the company strikes a customer from its mailing list if a customer fails to order from six consecutive catalogs. The company wants to know whether striking a customer from its list after a customer fails to order from four consecutive catalogs results in a higher profit per customer. The following data are available: If a customer placed an order the last time she received a catalog, then there is a 20% chance she will order from the next catalog. If a customer last placed an order one catalog ago, there is a 16% chance she will order from the next catalog she receives. If a customer last placed an order two catalogs ago, there is a 12% chance she will order from the next catalog she receives. If a customer last placed an order three catalogs ago, there is an 8% chance she will order from the next catalog she receives. If a customer last placed an order four catalogs ago, there is a 4% chance she will order from the next catalog she receives. If a customer last placed an order five catalogs ago, there is a 2% chance she will order from the next catalog she receives. It costs 2 to send a catalog, and the average profit per order is 30. Assume a customer has just placed an order. To maximize expected profit per customer, would Seas Beginning make more money canceling such a customer after six nonorders or four nonorders?Find the indicated maximum or minimum value of the objective function in the linear programming problem. Minimize g = 10x + 6y subject to the following. x + 2y ≥ 10 2x + y ≥ 11 x + y ≥ 9 x ≥ 0, y ≥ 0
- Find the optimal solution for the following problem. (Round your answers to 3 decimal places.) Minimize C = 8x + 8y subject to 7x + 9y ≥ 16 10x + 10y ≥ 22 and x ≥ 0, y ≥ 0. What is the optimal value of x? What is the optimal value of y? What is the minimum value of the objective function?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.)Consider the following constraints from a two-variable Linear Program. (1) X ≥ 0 (2) Y ≥ 0 (3) 5X + 4Y ≤ 50 (4) 5X - 2Y ≤ 20 If constraints (3) and (4) are binding, what is the optimal solution (X, Y)? Answer choices (6, 5) (9, 5) (0, 20) (20, 0)
- Find the optimal solution for the following problem. (Round your answers to 3 decimal places.) Minimize C = 14x + 7y + 8z subject to 6x + 12y + 19z ≥ 64 17x + 24y + 9z ≥ 128 and x ≥ 0, y ≥ 0, z ≥ 0. What is the optimal value of x? What is the optimal value of y? What is the optimal value of z? What is the minimum value of the objective function?Solve these problems using graphical linear programming and answer the questions that follow. Usesimultaneous equations to determine the optimal values of the decision variables.a. Maximize Z = 4x1 + 3x2Subject toMaterial 6x1 + 4x2 ≤ 48 lbLabor 4x1 + 8x2 ≤ 80 hrx1, x2 ≥ 0 b. Maximize Z = 2x1 + 10x2Subject toDurability 10x1 + 4x2 ≥ 40 wkStrength 1x1 + 6x2 ≥ 24 psi Time 1x1 + 2x2 ≤ 14 hrx1, x2 ≥ 0 c. Maximize Z = 6A + 3B (revenue)Subject toMaterial 20A+ 6B ≤ 600 lbMachinery 25A+ 20B ≤ 1,000 hr Labor 20A+ 30B ≤ 1,200 hrA, B ≥ 0 (1) What are the optimal values of the decision variables and Z?Construct one example for each of the following types of two-variable linear programs. Feasible with more than one optimal solution. That is, more than one set of (x, y) that gives the same optimal objective value. Provide an algebraic formulation, graph, decision variable quantities, and optimal objective value (if any).
- Consider the following linear programming problem: MIN Z = 3x1 + 2x2 Subject to: 2x1 + 3x2 ≥ 12 5x1 + 8x2 ≥ 37 x1, x2 ≥ 0 What is minimum cost and the value of x1 and x2 at the optimal solution?Construct one example for each of the following types of two-variable linear programs. •Feasible with a unique optimal solution. That is, one set of (x, y) that gives one optimal objective value. Provide an algebraic formulation, graph, decision variable quantities, and optimal objective value (if any).Solve using the duality linear programming method of the following problem:Object Function: F = X1+X2+4X3Subjected to:X1+2X2+3X3 ≥ 1152X1+X2+8X3 ≥ 200X1+X3 ≥ 50X1,X2, X3 ≥ 0 Note: Please make sure to use the duality linear programming method.