eem2046 - 1011 t3 - or tutorial questions

9
EEM2046 Engineering Mathematics IV Tutorial – O.R. Trimester 3, Session 2010/11 1 EEM2046 Engineering Mathematics IV Tutorial – Operations Research 1. Construct the mathematical model for the following problems. Do Not solve the problems. (a) Univesal Electric manufactures and sells two models of lamps, L 1 and L 2 , the profit being $15 and $10, respectively. The process involves two workers W 1 and W 2 who are available for this kind of work 100 and 80 hours per month, respectively. W 1 assembles L 1 in 20 minutes and L 2 in 30 minutes. W 2 paints L 1 in 20 minutes and L 2 in 10 minutes. Assuming that all lamps made can be sold without difficulty, determine production figures that maximize the profit.* (b) Hardbrick Company has two kilns. Kiln I can produce 3000 grey bricks, 2000 red bricks, and 300 glazed bricks daily. For kiln II the corresponding figures are 2000, 5000 and 1500. Daily operating costs of kilns I and II are $400 and $600, respectively. Find the number of days of operation of each kiln so that the operation cost in filling an order of 9000 grey, 1700 red and 4500 glazed bricks is minimized.* (c) Food A and B have 700 and 500 calories, contain 10g and 35g of protein, cost $1.50 per unit and $2.00 per unit, respectively. Find the minimum cost diet of at least 3 100 calories containing at least 100 g of protein.* *Kreyszig, Advanced Engineering Mathematics, John Wiley (8 th ed.). (d) The National Science Foundation (NSF) has received 4 proposals from professors to undertake new research in OR methods. Each proposal can be accepted for funding next year at the level (in thousands of dollars) shown in the following table or rejected. A total of $1 million is available for the year. Proposal 1 2 3 4 Funding 700 400 300 600 Score 85 70 62 93 Formulate a mathematical model to decide what projects to accept to maximize total score within the available budget using decision variables x j = 1 if proposal j is selected and = 0 otherwise. Ronald, Optimization in Operations Research, Prentice Hall. (e) A television manufacturing company has to decide on the number of 27- and 20-inch sets to be produced at one of its factories. Market research indicates that at most 40 of the 27-inch sets and 10 of the the 20-inch sets can be sold per month. The maximum number of work hours available is 500 per month. A 27-inch set set requires 20 work hours, and a 20-inch set requires 10 work hours. Each 27-inch set sold produces a profit of $120, and each 20-inch set produces a profit of $80. A wholesaler has agreed to purchase all the television sets produced if

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Page 1: EEM2046 - 1011 T3 - Or Tutorial Questions

EEM2046 Engineering Mathematics IV Tutorial – O.R.

Trimester 3, Session 2010/11 1

EEM2046 Engineering Mathematics IV Tutorial – Operations Research

1. Construct the mathematical model for the following problems. Do Not solve the

problems. (a) Univesal Electric manufactures and sells two models of lamps, L1 and L2,

the profit being $15 and $10, respectively. The process involves two workers W1 and W2 who are available for this kind of work 100 and 80 hours per month, respectively. W1 assembles L1 in 20 minutes and L2 in 30 minutes. W2 paints L1 in 20 minutes and L2 in 10 minutes. Assuming that all lamps made can be sold without difficulty, determine production figures that maximize the profit.*

(b) Hardbrick Company has two kilns. Kiln I can produce 3000 grey bricks,

2000 red bricks, and 300 glazed bricks daily. For kiln II the corresponding figures are 2000, 5000 and 1500. Daily operating costs of kilns I and II are $400 and $600, respectively. Find the number of days of operation of each kiln so that the operation cost in filling an order of 9000 grey, 1700 red and 4500 glazed bricks is minimized.*

(c) Food A and B have 700 and 500 calories, contain 10g and 35g of protein,

cost $1.50 per unit and $2.00 per unit, respectively. Find the minimum cost diet of at least 3 100 calories containing at least 100 g of protein.*

*Kreyszig, Advanced Engineering Mathematics, John Wiley (8th ed.).

(d) The National Science Foundation (NSF) has received 4 proposals from professors to undertake new research in OR methods. Each proposal can be accepted for funding next year at the level (in thousands of dollars) shown in the following table or rejected. A total of $1 million is available for the year.

Proposal 1 2 3 4 Funding 700 400 300 600 Score 85 70 62 93

Formulate a mathematical model to decide what projects to accept to maximize total score within the available budget using decision variables xj = 1 if proposal j is selected and = 0 otherwise.

Ronald, Optimization in Operations Research, Prentice Hall.

(e) A television manufacturing company has to decide on the number of 27- and 20-inch sets to be produced at one of its factories. Market research indicates that at most 40 of the 27-inch sets and 10 of the the 20-inch sets can be sold per month. The maximum number of work hours available is 500 per month. A 27-inch set set requires 20 work hours, and a 20-inch set requires 10 work hours. Each 27-inch set sold produces a profit of $120, and each 20-inch set produces a profit of $80. A wholesaler has agreed to purchase all the television sets produced if

Page 2: EEM2046 - 1011 T3 - Or Tutorial Questions

EEM2046 Engineering Mathematics IV Tutorial – O.R.

Trimester 3, Session 2010/11 2

the numbers do not exceed the maxima indicated by the market research.

Hillier, F.S, Introduction to Operations Research.

(f) An engineer is designing a 30meter2 (area of the base) open rectangular pool for processing the chemical waste produced from the manufacturing process. Due to the structure of the machine that is going to be placed in the pool, she wishes to design the pool to be 3 meter deep, the length should be at least twice its width, and its width is at most 4 meter. The engineer wishes to choose an optimal design that minimizes cost by minimizing the concrete area of the pool walls. Suppose that the decision variables x1 and x2 are defined as the length of the pool and the width of the pool respectively, formulate a mathematical model using the above decision variables.

2. Describe and graph the feasible region determined by the following inequalities.

(a) 0 8

84

2

21

21

≥≤+

−≤−

xxxxx

(b)

0 55 2

2

21

21

21

2121

≥≥+≥−−

≤+−

x,xxxxx

xx

(c)

04 2 50 5105 5

21

21

21

21

≥≥+≤+≤+−

x,xxxxxxx

(d)

0 5

1010 4 8

21

21

21

1

21

≥−≥−

≥+≥≥+

x,xxx

xxx

xx

3. Consider the feasible region in question 2(a), give an example of objective function

such that the problem has (a) unique optimal solution (b) multiple optimal solution (c) unbounded solution

4. Consider the region in question 2(c),

(a) Maximize 21 255 xxf += (b) Minimize 21 255 xxf +=

[(5/7, 8 4/7), 217 6/7; (2, 0), 10]

5. (a) Maximize x1 + 3x2 in the region of question 2(d) (b) Minimize x1 + x2 in the region of question 2(d).

[unbounded solution; x= 4+34/9 t, y = 4 – 34/9 t, 0 ≤ t ≤ 1, 8]

6. Consider the linear programming problem:

( )( )

.x,x,xxxxxxx:

xxxz

0 2 10102 1 63 2 subject to

Maximize

321

321

321

321

≥≤++≤−+

−−=

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EEM2046 Engineering Mathematics IV Tutorial – O.R.

Trimester 3, Session 2010/11 3

(a) By introducing the slack variables x4 and x5 to the inequalities ( )1 and ( )2 respectively, express the problem in standard form.

(b) Write down the basic solution corresponding to basic variables x1 and x3. Is this basic solution feasible? Give reasons for your answer.

(c) Given that the remaining basic solutions ( )54321 ,,,, xxxxx are as follows:

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ).0,3,0,1,0

,0,0,,,0 ,0,0,0,,,10,0,0,2,0 ,0,4,0,0,5,10,6,0,0,0 ,16,0,6,0,0 ,4,0,0,0,3,0,16,10,0,0

1330

1316

74

715 −−−

Determine the extreme points of the feasible region for the problem. (d) Hence, or otherwise, determine the optimal value and the optimal solution.

[ (3,0,0), 3] (Trimester 3, Session 2001/2002 PEM2046 Supplementary Examination)

7. Solve the following linear-programming problems by means of the simplex

method: (a) Maximize z = 2x1 + 4x2 + 3x3 (b) Maximize z =4x1 + 3x2 + 6x3

subject to: subject to: 3x1 +4x2 + 2x3 ≤ 60 3x1 + x2 +3x3 ≤ 30

2x1 + x2 + 2x3 ≤ 40 2x1+2x2 +3x3 ≤ 40 x1 + 3x2 + 2x3 ≤ 80 x1≥ 0, x2≥ 0, x3 ≥ 0

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

(c) Maximize 4y1 + 5y2 subject to:

–y1 + y2 ≤ 4 y1 – y2 ≤ 10 y1, y2 ≥ 0

(d) Minimize 2y1 –3y2 subject to: y1 + y2 ≤ 4

y1 – y2 ≤ 6 y1, y2 ≥ 0

(e) Minimize z = 4x1 – 10x2 – 20x3

subjec to: 3x1 + 4x2 + 5x3 ≤ 60

2x1 + x2 ≤ 20 2x1 +3x3 ≤ 30 x1, x2, x3 ≥ 0

(f) Maximize 30x1 – 4x2 subject to:

5x1 – x2 ≤ 30 x1 ≤ 5 x1 ≥ 0, x2 .urs

[(0, 6 2/3, 16 2/3), :z=76 2/3; (0,10,6 2/3), z =70; unbounded solution; (0,4), −12; (0, 2.5, 10), −225; (5, −5), 170]

8. By using the simplex method, solve the following linear programming problem

with multiple optimal solutions. Hence, give two optimal non-basic feasible solutions.

0 30106 8 4 :tosubject

53Maximize

21

21

21

21

≥≤+≤+

+

x,xxxxx

xx

(Trimester 3, Session 2000/2001 PEM2046 Final Examination)

[optimal solutions: ,tx1725

1= ,tx

17153

2−= .t 10 ≤≤ ]

[optimal non-basic solutions: ,tx1725

1= ,tx

17153

2−= 0<t<1]

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EEM2046 Engineering Mathematics IV Tutorial – O.R.

Trimester 3, Session 2010/11 4

9. Convert the following problem to its dual problem:

(a) Minimize: z = 15x1 + 12x2 subject to: x1 + x2 ≥ 1.5

2x1 + 3x2 ≥ 5 x1 , x2 ≥ 0

(b) Maximize: z = 5x1 + 12x2 + 4x3 subject to: x1 + 2x2 + x3 ≤ 10

2x1 − x2 + 3x3 = 8 x1 , x2 , x3 ≥ 0

(c) Maximize: z = 4x1 – x2 + 2x3 subject to: x1 + x2 ≤ 5

2x1 + x2 ≤ 7 2x2 + x3 ≥ 6 x1 + x3 = 4 x1 ≥ 0, x2 , x3 .urs

(d) Minimize: w = 4y1 + 2y2 + y3 subject to: y1 + 2y2 + y3 ≤ 6

y1 – y2 + 2y3 ≤7 y1, y2 ≥ 0, y3 .urs

10. Consider the following linear programming problem:

Maximize z = 4x1 + x2 subject to: 3x1 + 2x2 ≤ 6

6x1 + 3x2 ≤ 10 x1, x2 ≥ 0

Suppose that in solving this problem, row 0 of the optimal tableau is found to be z + 2x2 + s2 = 20/3. Use the Dual theorem to prove that the computations must be incorrect.

11. Consider the following problem: Maximize z = 3x1 + x2 + 4x3 subject to: 6x1 + 3x2 + 5x3 ≤ 25 3x1 + 4x2 + 5x3 ≤ 20

xi ≥ 0, i = 1, 2, 3. The corresponding final set of equations yielding the optimal solution is

z + 2x2 + 1/5x4 + 3/5x5 = 17 x1 − 1/3x2 + 1/3x4 − 1/3x5 = 5/3 x2 + x3 − 1/5x4 + 2/5x5 = 3

with x4 and x5 the slack variables for the first and second constraint. Identify the optimal solution and the optimal value for the dual problem from the final set of equations.

[ (1/5,3/5), 17] You are known to conduct sensitivity analysis by independently investigating each of the following changes in the original model. For each change, test for feasibility and for optimality (do not reoptimize)

(a) Change the coefficient of x2 in the objective function to 4 (b) Change the coefficient of x3 in the objective function to 3 (c) Change the coefficient of x2 in constraint 2 from 4 to 1 (d) Change the coefficient of x1 in constraint 1 from 6 to 10

[not optimal but feasible, optimal & feasible, optimal & feasible, optimal & feasible]

12. Consider the following LP allocation model: Maximize: 3x1 + 2x2

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EEM2046 Engineering Mathematics IV Tutorial – O.R.

Trimester 3, Session 2010/11 5

Subject to: 4x1 + 3x2 ≤ 12 (resource 1) 4x1 + x2 ≤ 8 (resource 2) 4x1 − x2 ≤ 8 (resource 3) x1, x2 ≥ 0

with the optimum tableau given by:

Basic z x1 x2 s1 s2 s3 Solution z 1 0 0 5/8 1/8 0 17/2 x2 0 0 1 ½ −1/2 0 2 x1 0 1 0 −1/8 3/8 0 3/2 s3 0 0 0 1 −2 1 4

(a) If there is a chance to increase the availability of resources. Which resource should

be given the priority for an increase in level?

(b) Conduct the following sensitivity analysis independently. In each case, identify the

new optimal value.

(i) The coefficients in the objective function change from [3 2] to [4 3]. (ii) The coefficients in the objective function change from [3 2] to [5 4]. (iii)The RHS of the constraints change from [12 8 8]T to [15 10 5 ]T. (iv) The RHS of the constraints change from [12 8 8]T to [8 10 10 ]T.

[ (a) 1 (b) 12, 16, 10 5/8, infeasible]

13. Consider the following problem. Maximize z = 3 x1 + x2 + 4 x3 subject to : 6x1 + 3x2 + 5x3 ≤ 25 3x1 + 4x2 + 5x3 ≤ 20 xi ≥ 0, i =1,2,3.

The corresponding final set of equations yielding the optimal solution is z + 2x2 + 1/5 x4 + 3/5 x5 = 17 x1 – 1/3 x2 + 1/3 x4 − 1/3 x5 = 5/3 x2 + x3 – 1/5 x4 + 2/5 x5 = 3

with x4 and x5 the slack variables for the first and second constraint. Find the new optimal solution if a new variable xnew has been introduced into the model as follow:

(i) Maximize z = 3 x1 + x2 + 4 x3 + 2xnew Subject to : 6x1 + 3x2 + 5x3 + 3xnew ≤ 25 3x1 + 4x2 + 5x3 + 2xnew ≤ 20 xi ≥ 0, i =1,2,3, xnew ≥ 0.

(ii) Maximize z = 3 x1 + x2 + 4 x3 + 2 1/2xnew Subject to : 6x1 + 3x2 + 5x3 + 4xnew ≤ 25 3x1 + 4x2 + 5x3 + 3xnew ≤ 20 xi ≥ 0, i =1,2,3, xnew ≥ 0.

[(i) 0, 0, 2, 5 (ii) 5/3, 0, 3, 0]

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EEM2046 Engineering Mathematics IV Tutorial – O.R.

Trimester 3, Session 2010/11 6

14. JC lives in New York City, but he plans to drive to Los Angeles to seek fame and fortune.

JC’c funds are limited , so he has decided to spend each night on his trip at a friend’s house. JC has a friend in Columbus, Nashville, Louisville, Kansas City, Omaha, Dallas, San Antonio, and Denver. JC knows that after one day’s drive he can reach Columbus, Nashville or Louisville. After two days of driving, he can reach Kansas City, Omaha or Dallas. After three days of driving, he can reach San Antonio, or Denver. Finally, after four days of driving, he can reach Los Angeles. To minimize the number of miles traveled, where should JC spend each night of the trip? The actual mileage between cities are given below.

[shortest path:2870 miles; New York – Columbus – Kansas City – Denver – LA]

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EEM2046 Engineering Mathematics IV Tutorial – O.R.

Trimester 3, Session 2010/11 7

15. By using dynamic programming, find the shortest path from node 1 to node 9 in the network shown below. Hence, find the shortest path from node 3 to node 9.

(Trimester 2, Session 1999/2000 PEM2062 Supplementary Examination) [6.75, 1-2-6-8-9;

The shortest path from node 3 is 3-4-5-9, 3-4-8-9 or 3-6-8-9.]

16. Suppose that a 10-lb knapsack is to be filled with items listed as follows. To maximize total benefit, how should the knapsack be filled?

[Max:25, one unit type 1 & two units type 2]

Weight Benefit Item 1 4 lb 11 Item 2 3 lb 7 Item 3 5 lb 12

17. You, an engineering consultant, receive offers from three different companies.

Each company is willing to employ you on a part-time basis for as many days per week as you are prepared to give. The fees offered are shown below.

Number of Days

Company 1 (RM)

Company 2 (RM)

Company 3 (RM)

0 0 0 0 1 450 445 440 2 880 890 880 3 1320 1330 1320 4 1775 1770 1770 5 2200 2200 2210

As a matter of policy, you just work for five days weekly. By using dynamic programming, determine an optimal policy on how you should devote your five working days to these three companies so as to maximize your weekly income.

(Trimester 1, Session 2001/2002 PEM2062 Final Examination) [RM2220. (1,0,4); (1,2,2); (1,3,1); (1,4,0); (4,1,0)]

18. A vending machine company currently operates a 2-year-old machine at a certain

location. The following table gives estimates of upkeep, replacement cost, and income (all in dollars) for any machine at this location, as functions of the age of the machine.

1

4

3

2

5

6

9

8 8

7

2

0.5

0.75

3

1

4

2

1

3

2

2.5 3

3

3 3

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EEM2046 Engineering Mathematics IV Tutorial – O.R.

Trimester 3, Session 2010/11 8

Age, u 0 1 2 3 4 5 Income, I(u) 10 000 9 500 9 200 8 500 7 300 6 100 Maintenance, M(u) 100 400 800 2000 2800 3300 Replacement, R(u) … 3 500 4 200 4 900 5 800 5 900

As a matter of policy, no machine is ever kept past its sixth anniversary and replacement are only with new machines. Determine a replacement policy that will maximize the total profit from this one location over the next 4 years.

[keep-replace-keep-keep]

19. Apply Dijkstra’s algorithm to the following graphs to find shortest paths from vertex 1 to all other vertices.

(a) (b)

[(a) 0,1,4,4,3,5,6; (b) 0,9,7,8,4,14] 20. Find a shortest spanning tree T by Kruskal’s algorithm. Sketch T.

[T consists of edges (1,2),(1,8),(8,7),(8,5),(5,4),(4,3),(3,6)]

6

5

4

1

2

3

3

10

1

5

15

4

2 6

3

1 4

3 2

8 5

6

14

16

12 12

15

16

15

13

14

14

7 14

13

1

2

4

3

5

6

7

1

5

4

1

4

2

3

2

2

1 2

4

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EEM2046 Engineering Mathematics IV Tutorial – O.R.

Trimester 3, Session 2010/11 9

21. Find a shortest spanning tree T by Prim’s algorithm. Sketch T.

[T consists of edges (1,2),(1,3),(1,4),(2,6),(3,5)]

22. Apply Dijkstra’s algorithm to the following graph, starting at vertex 1. Draw the

Dijkstra tree that results and indicate the distance from vertex 1 to each of the other vertices.

Fig. 1 23. Apply Prim’s algorithm to Fig. 1 starting at vertex 1. Draw the minimum spanning

tree that results and indicate the total weight

1

4

3

2

5

6

6

4 12

20

14

8

12

2

6

1

3

4

2

5

7

4

5 8

6

2

4

5

9

7

6 8 6

3

4

2

7 1