either manufacturer, the second product could then be ready for sale in an additional 9 months For either product line, if both manufacturers market their improved models simultaneously, it is estimated that manufacturer 1 would increase its share of the total future sales of this product by 8 percent of the total(from 25 to 33 percent). Similarly, manufacturer I would increase its share by 20, 30, and 40 percent of the total if it marketed the product sooner than manufacturer 2 by 2, 6, and 8 months, respectively. On the other hand manufacturer I would lose 4, 10, 12, and 14 percent of the total if manufacturer 2 marketed it sooner by 1, 3, 7, and 10 months, respectively Formulate this problem as a two-person, zero-sum game, and then determine which strategy the respective manufacturers should use according to the minimax criterion. (10 4. The research and development division of a company has been developing four possible new product lines. Management must now make a decision as to which of these four products actually will be produced and at what levels. Therefore, they have asked the or department to formulate a mathematical programming model to find the most profitable product mix A substantial cost is associated with beginning the production of any product, as given in the first row of the following table. The marginal net revenue from each unit produced is given in the second row of the table Product Parallel Units Start-up cost, 50,000 40,000 70.000 60.000 Marginal revenue,S70 Let the continuous decision variables x1, 2, 3, and x4 be the production levels of products 1, 2, 3 and 4, respectively. Management has imposed the following policy constraints on these varlables (1)No more than two of the products can be produced (2) Either product 3 or 4 can be produced only if either product 1 or 2 is produced (3) Either5x1+3x2+6x3+4x4≤6000 4x1+6x,+3x3+5x4≤6000 Introduce auxiliary binary variables to formulate an MIP model for this problem. (10 5. Use the BlP branch-and-bound algorithm to solve the following problem interactively(15 points) Max z= 2x,-x+5x,-3x,+4x 7x3-5x4+4x≤6 s.t.x 2x3-4x4+2x5≤0 x is binary2 either manufacturer, the second product could then be ready for sale in an additional 9 months. For either product line, if both manufacturers market their improved models simultaneously, it is estimated that manufacturer 1 would increase its share of the total future sales of this product by 8 percent of the total (from 25 to 33 percent). Similarly, manufacturer 1 would increase its share by 20,30,and 40 percent of the total if it marketed the product sooner than manufacturer 2 by 2, 6, and 8 months, respectively. On the other hand, manufacturer 1 would lose 4, 10,12, and 14 percent of the total if manufacturer 2 marketed it sooner by 1,3,7, and 10 months, respectively. Formulate this problem as a two-person, zero-sum game, and then determine which strategy the respective manufacturers should use according to the minimax criterion. (10 points) 4. The research and development division of a company has been developing four possible new product lines. Management must now make a decision as to which of these four products actually will be produced and at what levels. Therefore, they have asked the OR department to formulate a mathematical programming model to find the most profitable product mix. A substantial cost is associated with beginning the production of any product, as given in the first row of the following table. The marginal net revenue from each unit produced is given in the second row of the table. Product Parallel Units 1 2 3 4 Start-up cost,$ Marginal revenue,$ 50,000 70 40,000 60 70,000 90 60,000 80 Let the continuous decision variables x1,x2,x3, and x4 be the production levels of products 1,2,3 and 4, respectively. Management has imposed the following policy constraints on these variables: (1) No more than two of the products can be produced. (2) Either product 3 or 4 can be produced only if either product 1 or 2 is produced (3) Either 6000 5x1 + 3x2 + 6x3 + 4x4 ≤ Or 6000 4x1 + 6x2 + 3x3 + 5x4 ≤ Introduce auxiliary binary variables to formulate an MIP model for this problem. (10 points) 5. Use the BIP branch-and-bound algorithm to solve the following problem interactively (15 points) ⎪ ⎩ ⎪ ⎨ ⎧ − + − + ≤ − + − + ≤ = − + − + x is binary x x x x x x x x x x st Max Z x x x x x j 2 4 2 0 3 2 7 5 4 6 . . 2 5 3 4 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5