江西财经大学 05-06学年第二学期期末考试试卷 试卷代码:03883B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)In transportation problem, there are three methods to construct an initial BF solution, they are ①②③ (b)The four assumptions of linear programming are ②③④ (c)Write the following special linear programming model OThe general model of transportation problem is 2The general model of assignee problem is 3The general model of min-cost flow problem is 2. An investor has money-making activities A and B available at the beginning of each of the next 5 years(call them years 1 to 5). Each dollar invested in A at the beginning of a year returns $1. 40(a profit of $0. 40)two years later(in time for immediate reinvestment). Each dollar invested in B at the beginning of a year returns $1.70 three years later In addition, money-making activities C and d will each be available at one time in the future. Each dollar invested in C at the beginning of year 2 returns $1.90 at the end of year 5. Each dollar invested in d at the beginning of year 5 returns 1. 30 at the nd of year 5 The investor begins with $60000 and wishes to know which investment plan maximizes the amount of money that can be accumulated by the beginning of year 6 Formulate a linear programming model for this problem. (10 points 3. The BUILD-FAST COMPANY has agreed to supply is best customer with three widgets during each of the next 3 weekS, even though producing them will require some overtime work. The relevant production data are as follows week Maximum production, Maximum production, Production cost per Regular time Overtime unit, Regular time(s)
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：03883B 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) In transportation problem, there are three methods to construct an initial BF solution, they are ① ; ② ③ . (b) The four assumptions of linear programming are: ① ; ② ; ③ ④ 。 (c)Write the following special linear programming model: ①The general model of transportation problem is ②The general model of assignee problem is . ③The general model of min-cost flow problem is . 2. An investor has money-making activities A and B available at the beginning of each of the next 5 years (call them years 1 to 5). Each dollar invested in A at the beginning of a year returns $1.40 (a profit of $0.40) two years later (in time for immediate reinvestment). Each dollar invested in B at the beginning of a year returns $1.70 three years later. In addition, money-making activities C and D will each be available at one time in the future. Each dollar invested in C at the beginning of year 2 returns $1.90 at the end of year 5. Each dollar invested in D at the beginning of year 5 returns 1.30 at the end of year 5. The investor begins with $60000 and wishes to know which investment plan maximizes the amount of money that can be accumulated by the beginning of year 6. Formulate a linear programming model for this problem. (10 points) 3. The BUILD-FAST COMPANY has agreed to supply is best customer with three widgets during each of the next 3 weeks, even though producing them will require some overtime work. The relevant production data are as follows: week Maximum production, Regular time Maximum production, Overtime Production cost per unit, Regular time($)
2 300 The cost per unit produced with overtime for each week is $100 more than that for regular time. The cost of storage is $50 per unit for each week. There is already an nventory of two widgets on hand currently, but the company does not want to retain any widgets in inventory after 3 weeks Management wants to know how many units should be produced in each week in rder to minimize costs Formulate this problem as a transportation problem by constructing the appropriate cost and requirements table. (10 points) 4. Consider the following problem (10 points) Maximize Z=6x,+x+2x subject to 4x1-2x2-x≤3 2x,+-x3≤1 ≥0,x2≥0,x3≥0 Let x4, 5, and x6 denote the slack variables for the respective constraints. After you apply the simplex method, a portion of the final simplex tableau is as follow Basic variable Coefficient of XI X2 X3 X4 X5 Right side X3 2 0 Use the fundamental insight to identify the missing numbers in the final simplex tableau 5. Consider the following problem(20 points) Maximize Z=2x,+7x,-3x x1+3x2+4x3≤30 subject to x,+4x2-x,<10 x1≥0,x2≥0,x3≥0 The corresponding final set of equations yielding the optimal solution is
2 1 2 3 2 2 1 2 1 2 300 500 400 The cost per unit produced with overtime for each week is $100 more than that for regular time. The cost of storage is $50 per unit for each week. There is already an inventory of two widgets on hand currently, but the company does not want to retain any widgets in inventory after 3 weeks. Management wants to know how many units should be produced in each week in order to minimize costs. Formulate this problem as a transportation problem by constructing the appropriate cost and requirements table. (10 points) 4.Consider the following problem (10 points) ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ ≥ ≥ ≥ + + ≤ − − − ≤ + + ≤ = + + 0, 0, 0 1 2 1 2 3 2 3 4 2 2 2 1 2 2 6 2 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 x x x x x x x x x x x x subject to Maximize Z x x x Let x4,x5, and x6 denote the slack variables for the respective constraints. After you apply the simplex method, a portion of the final simplex tableau is as follows: Coefficient of : Basic variable Eq. Z X1 X2 X3 X4 X5 X6 Right side Z (0) 1 2 0 2 X5 (1) 0 1 1 2 X3 (2) 0 -2 0 4 X1 (3) 0 1 0 -1 Use the fundamental insight to identify the missing numbers in the final simplex tableau. 5. Consider the following problem (20 points) ⎪ ⎩ ⎪ ⎨ ⎧ ≥ ≥ ≥ + − ≤ + + ≤ = + − 0, 0, 0 4 10 3 4 30 2 7 3 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 The corresponding final set of equations yielding the optimal solution is
x2+x3 +2x。=20 x2+5x3+x4-x5=20 (2) x1+4x2-x3+x5=10 Now you are to conduct sensitivity analysis by independently investigating each of the following changes in the original model. For each change, use the sensitivity analysis procedure to determine whether the previous optimal solution is till optimal (a) Change the right-hand sides to/6 2 (b)Change the coefficients of x to a,3= 3 2 (c) Introduce a new variable X6 with coefficients a,I (d) Introduce a new constraint 3x,+2x,+3x3 <25 Change constraint 2 to x,+2x2+2x, <35 6. Solve the following parametric linear programming problem (15 points) Maxz()=(10-46)x1+(4-6)x2+(7+0) 3x1+x,+2x3≤7 s2x1+x2+3x1≤5 0 0,x2≥0 when 0=0, the final simplex tableau is Coefficient of Ba asic variabie E X2 X3 X4 5 Right side (0) 0 3 24 7. The BETTER PRODUCTS COMPANY has decided to initiate the production of four new products, using three plants that currently have excess production capacity The products require a comparable production effort per unit, so the availabl production capacity of the plants is measured by the number of units of any product that can be produced per day, as given in the following table. The bottom row gives the required production rate per day to meet projected sales. Each plant can produce any of these products, except that plant 2 cannot produce product 3. However, the variable costs per unit of each product differ from plant to plant. Management now
3 (2) 4 10 (1) 5 20 (0) 2 20 1 2 3 5 2 3 4 5 2 3 5 + − + = − + + − = + + + = x x x x x x x x Z x x x Now you are to conduct sensitivity analysis by independently investigating each of the following changes in the original model. For each change, use the sensitivity analysis procedure to determine whether the previous optimal solution is till optimal. (a) Change the right-hand sides to ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ =⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 30 20 2 1 b b (b) Change the coefficients of x3 to ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ 2 3 2 23 13 3 a a c (c) Introduce a new variable x6 with coefficients ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡− = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ 2 1 3 21 11 6 a a c (d) Introduce a new constraint 3x1 + 2x2 + 3x3 ≤ 25 Change constraint 2 to x1 + 2x2 + 2x3 ≤ 35 6. Solve the following parametric linear programming problem (15 points) ⎪ ⎩ ⎪ ⎨ ⎧ ≥ ≥ ≥ + + ≤ + + ≤ = − + − + + 0, 0, 0 2 3 5 3 2 7 . . ( ) (10 4 ) (4 ) (7 ) 1 2 3 1 2 3 1 2 3 1 2 3 x x x x x x x x x st MaxZ θ θ x θ x θ x when θ=0, the final simplex tableau is Coefficient of : Basic variable Eq. Z X1 X2 X3 X4 X5 Right side Z (0) 1 0 0 3 2 2 24 X1 (1) 0 1 0 -1 1 -1 2 X2 (2) 0 0 1 5 -2 3 1 7. The BETTER PRODUCTS COMPANY has decided to initiate the production of four new products, using three plants that currently have excess production capacity. The products require a comparable production effort per unit, so the available production capacity of the plants is measured by the number of units of any product that can be produced per day, as given in the following table. The bottom row gives the required production rate per day to meet projected sales. Each plant can produce any of these products, except that plant 2 cannot produce product 3. However, the variable costs per unit of each product differ from plant to plant. Management now
needs to make a decision on how to split up the production of the products among plants (15 points) Unit cost for product Capacity available 2 tI Plant 4 28 75 Plant 2 Plant3 27 45 Production rate 8. Consider a project whose activities and required times are as given: (10 points) activity Required Time(Minutes) (1,2) 5 (4,5) 6 (6,7) 2 (78 (8,9) 6 (4,7) a. Construct the project network for this project b. Find the earliest time. latest time. and slack for each event as well as the slack for each activity. Also identify the critical path If the precedence relationship implied by dummy activity(4, 7)could be removed, by how much would the project completion time be reduced?
4 needs to make a decision on how to split up the production of the products among plants. (15 points) Unit cost for product 1 2 3 4 Capacity available Plant 1 41 27 28 24 75 Plant 2 40 29 — 23 75 Plant 3 37 30 27 21 45 Production rate 20 30 30 40 8.Consider a project whose activities and required times are as given: (10 points) Activity Required Time (Minutes) (1,2) (2,3) (3,4) (4,5) (5,9) (2,6) (6,7) (7,8) (8,9) (4,7) 5 5 5 5 5 6 2 5 6 0 a. Construct the project network for this project. b. Find the earliest time, latest time , and slack for each event as well as the slack for each activity. Also identify the critical path. c. If the precedence relationship implied by dummy activity (4,7) could be removed, by how much would the project completion time be reduced?