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)
i. Highlight the three phases of operation research. (6 Marks) ii. Formulate the dual problem from the following primal problem. Maximize ?? = 3??1 + 2??2 + ??3 Subject to ??1 + 2??2 - ??3 = 4 2??1 - ??2 + ??3 = 8 ??1 - ??2 = 6 ??1 = 0,??2 = 0,??3 = 0 (6 Marks) iii. Given the linear programming problem Maximize ?? = 2?? + 3?? Subject to ?? + 3?? = 9 2?? + 3?? = 12 ?? = 0,?? = 0 i. Sketch the convex set of all the feasible solution. (5 Marks) ii. Determine the extreme points and the optimal solutions. (4 Marks) iii. Show that the line segment joining the optimal points will yield the maximum value of the objective function. (5 Marks) iv. Highlight the significance of duality in linear programming models. (4 Marks)
QUESTION TWO (20 MARKS)
A company can produce products A, B and C. The products yield a contribution of £8,£5 and £10 respectively. The production uses a machine which has 400 hours capacity in the next
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period. Each unit uses 2, 3 and 1 hour respectively of the machine capacity. There are only 150 units available in the period of a special component which is used singly in products A and C. 200kg only of a special alloy is available in the period. Product A uses 2kgs per unit and product C uses 4kgs per unit of the special alloy. There is agreement with a trade association to produce not more than 50 units of product B in the period. The company wishes to fund out the production plan which maximizes contribution.
Required:
a) Formulate the above as a linear programming problem. (6 Marks) b) Solve the problem formulated in (a) by simple algorithm. (14 Marks)
QUESTION THREE (20 MARKS)
a) Explain the following terms as used in linear programming models i. Objective function ii. Feasible region iii. Optimal solution iv. Constraints set. (8 Marks) b) A company has three warehouses A, B, C and four stores W, X, Y and Z. the warehouses have altogether a surplus of 150 units of a given commodity as follows A 50 B 60 C 40
The four stores together needs also 150 units of the commodity as follows; W 20 X 70 Y 50 Z 10
The cost of shipping one unit of commodity in terms of shillings from warehouse (i) to store (i) are as follows;
????????????
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 the; i. North West corner method ii. Least cost method (12 Marks)
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QUESTION FOUR (20 MARKS) a) Let ??1 = 2 3 ?????? ??2 = 1 5 be two distinct points in En. Show that the line determined by ??1 ?????? ??2 is the set of points ???????? ?? =? ??1 + 1 -? ??2;? ???????? . (8 Marks) b) Highlight two assumptions of a transportation problem. (4 Marks) c) Set up the initial simplex tableau for the linear programming problem below using the two phase method
Maximize ?? = ??1 - 2??2 - 3??3 - ??4 - ??5 + 2??6 Subject to ??1 + 2??2 + 2??3 + ??4 + ??5 = 12 ??1 + 2??2 + ??3 + ??4 + 2??5 + ??6 = 18 3??1 + 6??2 + 2??3 + ??4 + 3??6 = 24 ??1 = 0,?? = 1,2…..6 (8 Marks)
QUESTION FIVE (20 MARKS)
a) Using graphical methods, maximize ?? = 6?? + 20?? Subject to ?? + ?? = 16 ?? = 10 5?? + 15?? = 180 ?? = 0,?? = 0 (6 Marks)
b) An office manager needs to purchase new filing cabinets. All cabinets cost $40 each require 6 square feet of floor space and 8 cubic feet of files. On the other hand, each Excello cabinet cost $80 requires 8 square feet of floor space and holds 12 cubic feet. His budget permits him to spend no more than$560 on files, while the office room for no more than 72 square feet of cabinets. The manager desires the greatest storage capacity within the limitations imposed by funds and space. i. How many of each type of cabinet should he buy? (6 Marks) ii. Determine the imputed amounts of storage for each unit cost and floor space. (6 Marks) iii. Determine the shadow value of the cost and the floor space. (2 Marks)
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