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### iastate-HW1-07F

Course: IE 361, Spring 2006
School: Iowa State
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Word Count: 413

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Hw1 IE312 Solution Section 3.1 1. Define: x1: number of acres of corn planted x2: number of acres of wheat planted max z = 30x1 + 100x2 s.t. x1 + x2 7 (Land Constraint) 4x1 + 10x2 40 (Labor Constraint) 10x1 30 (Govt. Constraint) x1 0, x2 0 (Sign Constraint) 2a. at (2, 3) check all the constraints until some constraint is violated. x1 + x2 = 2 + 3 =5 7 OK 4x1 + 10x2 = 30 40 OK 10x1 = 10* 2 = 20 30 Violated x1...

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Hw1 IE312 Solution Section 3.1 1. Define: x1: number of acres of corn planted x2: number of acres of wheat planted max z = 30x1 + 100x2 s.t. x1 + x2 7 (Land Constraint) 4x1 + 10x2 40 (Labor Constraint) 10x1 30 (Govt. Constraint) x1 0, x2 0 (Sign Constraint) 2a. at (2, 3) check all the constraints until some constraint is violated. x1 + x2 = 2 + 3 =5 7 OK 4x1 + 10x2 = 30 40 OK 10x1 = 10* 2 = 20 30 Violated x1 0, x2 0 No, government constraint is violated. 2b. 4x1 + 10x2 = 46 40 No; Labor constraint is not satisfied. 2c. No, x2 0 is not satisfied. 2d. Yes, all constraints and sign restrictions are satisfied. 3. Define: x1: number of bushel of corn produced x2: number of bushel of wheat produced get the number of acres of corn planted: x1/10 get the number of acres of wheat planted: x1/25 max z = 3x1 + 4x2 s.t. x1/10 + x2/25 7 4x1/10 + 10x2/25 40 x1 30 x1 0, x2 0 (Land Constraint) (Labor Const.) (Govt. Const.) (Sign Constraint) 4. Define : x1 = Number of Type 1 Trucks produced daily x2 = Number of Type 2 Trucks produced daily max z = 300x1 + 500x2 s.t. x1/800 + x2/700 1 x1/1500 + x2/1200 1 (Paint Shop Const.) (Assembly Shop Const.) 1 x1 0, x2 0 Painting shop finishes 800 truck 1 in one day. So it takes 1/800 day to paint a truck 1. Likewise, one truck 2 need 1/700 day to finish it. One day, the total time spend in two truck types will be x 1/800 + x2/700. Section 3.2 1. EF is 4x1 + 10x2 = 40, CD is x1 = 3, and AB is x1 + x2 = 7. The feasible region is bounded by ACGH. Pick a feasible point, e.g. (4, 0), get Z = 120. We have 120 = 30x 1 + 100x2, pick x1 = 3, then x2 = 0.3, draw isoprofit line 30x1 + 100x2 = 120 as shown in dotted line. Point G is optimal. At G the constraints 10x1 30 and 4x1 + 10x2 40 are binding. Thus optimal solution has x1 = 3, x2 = 2.8 and z = 30(3) + 100(2.8) = 370. 2 2. AB is x1/800 + x2/700 = 1. CD is x1/1500 + x2/1200 = 1. Feasible region is bounded by ABE. Dotted line is z = 300x 1 + 500x2 = 150,000. Moving isoprofit line up and to right we find optimal solution to be where x1 0 and Paint Shop constraint are binding. Thus x1 = 0, x2 = 700, z = 350,000 is optimal solution. 5. Define: xD = number of desks produced xC = number of chairs produced max z = 40xD + 25xC s.t. 4xD + 3xC 20 2xD - xC 0 xD 0, xC 0 (Material Constraint) (Marketing Constraint) (Sign Constraint) Graphical Solution: AB is 4xD + 3xC = 20, CD is 2xD - xC = 0. The feasible region is bounded by AGC. Dotted line is z = 40xD + 25xC = 120. Moving isoprofit line up and to right we find optimal solution at G (2, 4) where Material Constraint and Marketing Constraint are binding. The optimal solution is at xD = 2, xC = 4, z = 40*2+25*4 = \$180. 3 4
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