Strategies of manufacture 1 Strategies of manufacture 2 Regular programming Crash programming Regular programming 20 Crash mmI According to the minimax criterion, the saddle point is 8. that is manufacture I and 2 both hoice Regular programming 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 he 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 3 Start-up cost, S 50.000 40.000 70,000 60.000 Marginal revenue. S 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 (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 points) Solution: the MiP model maxz=70x1+60x2+90x3+80x4-50000y1-40000y2-70000y3-60000y4 y1+y2+y3+y4≤2 y3≤y+y2 ≤1,+ s15x+3x2+6x3+4x≤6000+M 4x1+6x2+3x3+5x4≤6000+M(1-y) My x2≥0,y,y2=0Or 5. Use the BIP branch-and-bound algorithm to solve the following problem interactively(153 Strategies of manufacture 1 Strategies of manufacture 2 Regular programming Crash programming Regular programming 8 20 Crash programming 6 -8 According to the minimax criterion, the saddle point is 8. that is manufacture 1 and 2 both choice Regular programming. 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 5 3 6 4 x1 + x2 + x3 + x4 ≤ Or 6000 4x1 + 6x2 + 3x3 + 5x4 ≤ Introduce auxiliary binary variables to formulate an MIP model for this problem. (10 points) Solution: the MIP model is ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ ≥ = ≤ + + + ≤ + − + + + ≤ + ≤ + ≤ + + + + ≤ = + + + − − − − 0, , 0 1 4 6 3 5 6000 (1 ) 5 3 6 4 6000 2 . . max 70 60 90 80 50000 40000 70000 60000 1 2 3 4 1 2 3 4 4 1 2 3 1 2 1 2 3 4 1 2 3 4 1 2 3 4 x y y or x My x x x x M y x x x x My y y y y y y y y y y st Z x x x x y y y y i i i i 5. Use the BIP branch-and-bound algorithm to solve the following problem interactively (15 points)