3. A corporation has decided to produce three new products. Five branch plants now have excess product capacity. The unit manufacturing cost of the first product would be $31, $29, $32, $28, and $29, in plants 1, 2,3, 4, and 5, respectively. The unit manufacturing cost of the second product would be $45, $41, $46, $42, and $43 in Plants 1, 2, 3, 4, and 5, respectively. The unit manufacturing cost of the third product would be $38, $35, and $40 in Plants 1, 2, and 3, respectively, and Plants 4 and 5 do not have the capability for producing this product. Sales forecasts indicate that 600 1000, and 800 units of products 1, 2, and 3, respectively, should be produced per da Plants 1, 2, 3, 4, and 5 have the capacity to produce 400, 600, 400, 600, and 1000 units daily, respectively, regardless of the product or combinations of products involved Assume that any plant having the capability and capacity to produce them can produce any combination of the products in any quantity Management wishes to know how to allocate the new products to the plants to minimize the total manufacturing cost Formulate this problem as a transportation problem by constructing the appropriate cost and requirements table. (10 points) 4. Use the simplex method to solve the following problem(15 points) Maximize Z=5x,+3x+4 subject to3x+x,+2x3≤30 x1≥0,x2≥0,x3≥0 5 Consider the following problem(15 points) MaxZ= 2x,+7x.+4 x1+2x2+x3≤10 s【3x1+3x,+2x2≤10 ≥0,x2≥0,x3≥0 (a)Construct the dual problem for this primal problem (b)Use the dual problem to demonstrate that the optimal value of Z for the primal problem cannot exceed 25 (c) It has conjectured that x2 and x4 should be the basic variables for the optimal solution of the primal problem Directly derive this basic solution(and Z)by using Gaussian elimination. Simultaneously derive and identify the complementary basic solution for the dual problem by using Eq (0) for the primal problem 6. Consider the following parametric linear programming problem (20 points)2 3. A corporation has decided to produce three new products. Five branch plants now have excess product capacity. The unit manufacturing cost of the first product would be $31, $29,$32,$28, and $29, in plants 1,2,3,4, and 5, respectively. The unit manufacturing cost of the second product would be $45, $41,$46,$42, and $43 in Plants 1,2,3,4, and 5, respectively. The unit manufacturing cost of the third product would be $38, $35, and $40 in Plants 1,2, and 3, respectively, and Plants 4 and 5 do not have the capability for producing this product. Sales forecasts indicate that 600, 1000, and 800 units of products 1,2,and 3, respectively, should be produced per day. Plants 1,2,3,4, and 5 have the capacity to produce 400,600,400,600, and 1000 units daily, respectively, regardless of the product or combinations of products involved. Assume that any plant having the capability and capacity to produce them can produce any combination of the products in any quantity. Management wishes to know how to allocate the new products to the plants to minimize the total manufacturing cost. Formulate this problem as a transportation problem by constructing the appropriate cost and requirements table. (10 points) 4. Use the simplex method to solve the following problem (15 points) ⎪ ⎩ ⎪ ⎨ ⎧ ≥ ≥ ≥ + + ≤ + + ≤ = + + 0, 0, 0 3 2 30 2 20 5 3 4 1 2 3 1 2 3 1 2 3 1 2 3 x x x x x x x x x subject to Maximize Z x x x .5 Consider the following problem (15 points) ⎪ ⎩ ⎪ ⎨ ⎧ ≥ ≥ ≥ + + ≤ + + ≤ = + + 0, 0, 0 3 3 2 10 2 10 . . 2 7 4 1 2 3 1 2 3 1 2 3 1 2 3 x x x x x x x x x st MaxZ x x x (a) Construct the dual problem for this primal problem. (b) Use the dual problem to demonstrate that the optimal value of Z for the primal problem cannot exceed 25. (c) It has conjectured that x2 and x4 should be the basic variables for the optimal solution of the primal problem. Directly derive this basic solution (and Z) by using Gaussian elimination. Simultaneously derive and identify the complementary basic solution for the dual problem by using Eq.(0) for the primal problem. 6. Consider the following parametric linear programming problem: (20 points)