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Sma2109:Applied Mathematics Question Paper
Sma2109:Applied Mathematics
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2013
QUESTION ONE – (30 MARKS)
(a) Discuss the three phases of operation research. (6 Marks) (b) Using graphical method, solve the following Linear Programming problem. ???????????? ?? = 6?? + 20?? ?????????????? ???? ?? + ?? = 16 ?? = 10 5?? + 15?? = 180 ?? = 10, ?? = 0 (6 Marks) (c) Given the linear programming problem. max ?? = 2?? + 3?? ??.?? ?? + 3?? = 9 2?? + 3?? = 12 ?? = 0, = 0 (i) Sketch the convex set of all the feasible solutions. (3 Marks) (ii) Determine the extreme points and the optimal solutions. (3 Marks) (iii) Show that the line segment joining the optional points will yield the maximum value of the objective function. (4 Marks)
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(d) Formulate the dual problem from the following primal problem and solve the dual using the simplex algorithm ???????????????? ?? = 3??1 + 2??2 ?????????????? ???? ??1 + 3??2 = 6 2??1 + ??2 = 3 ??1 = 0, ??1 = 0 (8 Marks)
QUESTION TWO – (20 MARKS)
(a) Using Simplex method, solve the following ???????????????? ?? = 3?? + 4?? + 5?? ?????????????? ???? 2?? + 4?? + 3?? = 80 4?? + 2?? + ?? = 48 ?? + ?? + 2?? = 40 ??,??,?? = 0 (12 Marks)
(b) Determine and show the region enclosed by the hyper planes in R2 defined by the linear equations. ?? - 3?? = -2 2?? - ?? = 1 ?? + 2?? = 8 (8 Marks)
QUESTION THREE – (20 MARKS)
(a) Discuss four assumptions of linear programming models. (8 Marks) (b) A company how three warehouses ??,??,?? and four stores ??,??,?? & ??. The warehourse have altogether a surplus of 150 units of a given commodity as follows A 50 B 60 C 40
The four stores together needs 150 units of the commodity as follows. W 20 X 70 Y 50 Z 10
The cost of shipping of one unit of commodity in tens of shillings from warehouses i to store j are as follows.
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Stores/Warehouse W X Y Z A 50 150 70 60 B 80 70 90 10 C 15 87 79 81
Determine the initial feasible solution using;
(i) Least cost method (ii) Vogel’s approximation method. (12 Marks)
QUESTION FOUR – (20 MARKS)
(a) Highlight the significance of: (i) Duality in linear programming models. (4 Marks) (ii) Sensitivity analysis (4 Marks)
(b) Suppose an animal breeder needs at least 6 units per day of nutrients A and at least 3 units of nutrients B. The breeder can choose between 2 different feeds, feed 1 and feed 2. Determine the minimum cost for the breeder if each bag of feed 1 cost $3 and provides 1 unit of nutrient A and 2 unit of B, while each bag of feed 2 costs $2 and provides 3 units of nutrient and of 1 of B.
(Hint: Use simplex method for optimal solution and carry out sensitivity analysis on the solutions) (12 Marks)
QUESTION FIVE – (20 MARKS)
(a) Set up the initial simplex tablean for the linear programming problem below using the two phase method. Max ?? = ??1 - 2??2 - 3??3 - ??4 - ??5 - 2??6
??.?? ??1 + 2??2 + 2??3 + ??4 + ??5 = 12 ??1 + 2??2 + ??3 + ??4 + 2??5 + ??6 = 18
???? = 0 , ?? = 1,2….6 (10 Marks)
(a) A company employs service engineers based at various locations throughout the country to service and repair their equipment installed in customer’s premises. Four requests for service have been received and the company finds that four engineers are available. The distance each of the engineers is from the various customers is given in the following
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table and the company wishes to assign engineers to customers to minimize the total distance to be travelled Customers
Engineers
Advise the company on the optimal assignment of engineers to customers (10 Marks)
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