江西财经大学 05-06学年第二学期期末考试试卷 试卷代码:03883A卷 课时:64 课程名称:运筹学I(英) 适用对象:04管理科学专业 1. Fill in the blanks (10 points) (a)the four assumptions of linear programming are ①②③④ (b)the difference between simplex method and dual simplex method are: (c)the duality theorem stated that the only possible relationship between the primal and the dual problems are ①②③ 2. We want to select an advertising strategy to reach two types of customers homemakers in families with over $25,000 annual income and homemakers in families with under $25,000 income. We feel that people in the first group will purchase twice as much of our product as people in the second, and our goal is to maximize purchases. We may advertise either on TV or in a magazine; one unit ofT advertising costs $40,000 and reaches approximately 20,000 people in the first group and 80,000 in the second. One unit of advertising in the magazine costs $24,000 and reaches 60,000 people in the first group and 30,000 in the second We require that at least 6 units of Tv advertising be used and that no more than 12 units of magazine advertising be used for policy reasons. The advertising budget is 360,000 Formulate this problem as a linear programming problem, defining all variables used.(10 points) 3.A manufacturing firm has discontinued the production of a certain unprofitable product line. This act created considerable excess production capacity. Management is considering devoting this excess capacity to one or more of three products; call them ducts 1, 2, and 3. The available capacity on the machines that might limit output is summarized in the following table
1 江西财经大学 05-06 学年第二学期期末考试试卷 试卷代码：03883A 卷 课时：64 课程名称：运筹学 I（英） 适用对象：04 管理科学专业 1.Fill in the blanks. (10 points) (a) the four assumptions of linear programming are: ① ; ② ; ③ ④ 。 (b) the difference between simplex method and dual simplex method are: ① ; ② ③ . (c) the duality theorem stated that the only possible relationship between the primal and the dual problems are: ① . ② . ③ . 2.We want to select an advertising strategy to reach two types of customers: homemakers in families with over $25,000 annual income and homemakers in families with under $25,000 income. We feel that people in the first group will purchase twice as much of our product as people in the second, and our goal is to maximize purchases. We may advertise either on TV or in a magazine; one unit of TV advertising costs $40,000 and reaches approximately 20,000 people in the first group and 80,000 in the second. One unit of advertising in the magazine costs $24,000 and reaches 60,000 people in the first group and 30,000 in the second. We require that at least 6 units of TV advertising be used and that no more than 12 units of magazine advertising be used for policy reasons. The advertising budget is 360,000. Formulate this problem as a linear programming problem, defining all variables used. (10 points) 3.A manufacturing firm has discontinued the production of a certain unprofitable product line. This act created considerable excess production capacity. Management is considering devoting this excess capacity to one or more of three products; call them products 1,2, and 3. The available capacity on the machines that might limit output is summarized in the following table:
Machine Type Available Time(in Machine Hours per Week) Milling machine 500 350 Grinder The number of machine hours required for each unit of the respective products is as Machine type Product Product 2 Product Milling machi Lathe 4 0 The sales department indicates that the sales potential for products I and 2 exceed the maximum production rate and that the sales potential for product 3 is 20 units per week. The unit profit would be $50, $20, and $25, respectively, on products 1, 2 and 3 the objective is to determine how much of each product the firm should produce to maximize profit Formulate a li model for this probler 4. For a linear programming x1+5x2+2x3≤b ≤b2 x3≥0 In this model, bi, b2 are unknown parameters, and the final simplex table is Basic variable E Coefficient of X3 X4 s Right side (1)0 abc Please determine the parameters in b1, b2 LP model and a, b, c, d, e in final simplex table.(15 5. Consider the transportation problem having the following cost and requirements table Destination supply 6 Source 4 44252 Demand Determine the optimal solution (15 points) 6. For a linear programming model (15 points)
2 Machine Type Available Time (in Machine Hours per Week) Milling machine Lathe Grinder 500 350 150 The number of machine hours required for each unit of the respective products is as follows: Machine Type Product 1 Product 2 Product 3 Milling machine Lathe Grinder 9 5 3 3 4 0 5 0 2 The sales department indicates that the sales potential for products 1 and 2 exceeds the maximum production rate and that the sales potential for product 3 is 20 units per week. The unit profit would be $50, $20, and $25, respectively, on products 1,2 and 3. the objective is to determine how much of each product the firm should produce to maximize profit. Formulate a linear programming model for this problem. (10 points) 4. For a linear programming In this model, b1,b2 are unknown parameters，and the final simplex table is： Coefficient of : Basic variable Eq. Z X1 X2 X3 X4 X5 Right side Z (0) 1 0 a 7 d e 150 X1 (1) 0 1 b 2 1 0 30 X5 (2) 0 0 c -8 -1 1 10 Please determine the parameters in b1,b2 LP model and a, b, c, d, e in final simplex table。（15 points） 5. Consider the transportation problem having the following cost and requirements table: Destination 1 2 3 4 Supply 1 3 7 6 4 5 Source 2 2 4 3 2 2 3 4 3 8 5 3 Demand 3 3 2 2 Determine the optimal solution. (15 points) 6. For a linear programming model (15 points) ⎪ ⎩ ⎪ ⎨ ⎧ ≥ − − ≤ + + ≤ = + + 0 5 6 5 2 5 2 3 1 2 3 1 2 3 2 1 2 3 1 1 2 3 x x x x x x b x x x b x x x , , s.t. max z
3x1+2x,+5 XI x2+x3≤430 s.t. x1+4x2≤420 ≥0 Let x4, Xs and x denote the slack variables for the respective constraints. The final simplex tableau is as follows Basic variable E Coefficient of Right side (1)0 X3 03/20 0 1/2 0 230 2 0-2 (a) Identify the optimal solution from this table (b)Identify the optimal solution for the dual problem (c)Introduce a new constraint x,+2x2+3x3<480, Determine whether the previous optimal solution is till optimal (d) If we want the previous optimal solution to be always optimal, how much does the right-side of constraint 1 b, increase? (e) If a new variable Xnew has been introduced into the model, Xnew coefficient is 27 324 Determine whether the previous optimal solution is till optimal 7. Solve the parameter programming as follow: (15 points) Max==2x,+x2 x1s10+20 x1+x2≤25-0 x,<10+26 x1≥0,x2≥0 8. At a small but growing airport, the local airline company is purchasing a new tractor for a tractor-trailer train to bring luggage to and from the airplanes. A new mechanized luggage system will be installed in 3 years, so the tractor will not be needed after that. However, because it will receive heavy use, so that the running and maintenance costs will increase rapidly as the tractor ages, it may still be economical to replace the tractor after 1 or 2 years. The following table gives the total net discounted cost associated with purchasing a tractor(purchase price minus trade-in allowance, plus running and maintenance costsO at the end of year I and
3 ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ≥ + ≤ + ≤ + + ≤ = + + 0 4 420 3 2 460 2 430 3 2 5 1 2 3 1 2 1 3 1 2 3 1 2 3 x x x x x x x x x x x x x , , s.t max z Let x4,x5, and x6 denote the slack variables for the respective constraints. The final simplex tableau is as follows: Coefficient of : Basic variable Eq. Z X1 X2 X3 X4 X5 X6 Right side Z (0) 1 4 0 0 1 2 0 X2 (1) 0 -1/4 1 0 1/2 -1/4 0 100 X3 (2) 0 3/2 0 1 0 1/2 0 230 X6 (3) 0 2 0 0 -2 1 1 20 (a) Identify the optimal solution from this table. (b) Identify the optimal solution for the dual problem. (c) Introduce a new constraint 480 x1 + 2x2 + 3x3 ≤ , Determine whether the previous optimal solution is till optimal. (d) If we want the previous optimal solution to be always optimal, how much does the right-side of constraint 1 b1 increase? . (e) If a new variable Xnew has been introduced into the model, Xnew coefficient is ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ 4 2 3 9 37 27 17 7 a a a c . Determine whether the previous optimal solution is till optimal. 7.Solve the parameter programming as follow: (15 points) ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ≥ ≥ ≤ + + ≤ − ≤ + = + 0, 0 10 2 25 10 2 . . 2 1 2 2 1 2 1 1 2 x x x x x x st Maxz x x θ θ θ 8. At a small but growing airport, the local airline company is purchasing a new tractor for a tractor-trailer train to bring luggage to and from the airplanes. A new mechanized luggage system will be installed in 3 years, so the tractor will not be needed after that. However, because it will receive heavy use, so that the running and maintenance costs will increase rapidly as the tractor ages, it may still be more economical to replace the tractor after 1 or 2 years. The following table gives the total net discounted cost associated with purchasing a tractor (purchase price minus trade-in allowance, plus running and maintenance costs0 at the end of year I and
trading it in at the end of year j (where year 0 is now) 0 10 21 12 minimize the total cost for the tractors over 3 years. (10 points) oulo The problem is to determine at what times (if any) the tractor should be replaced to
4 trading it in at the end of year j (where year 0 is now). j 1 2 3 0 8 18 31 i 1 10 21 2 12 The problem is to determine at what times (if any) the tractor should be replaced to minimize the total cost for the tractors over 3 years. (10 points)