Optimization Theory MS Mathematics/MSc Mathematics 4 Semester/Term BZU — Bahauddin Zakariya University

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Bahauddin Zakariya university, Multan

(Final-Exams)

MC-278-20-22

M.Sc. Mathematics (4th semesten)

Session (202g-23)

Time Allowed: 2- hours

Paper Code: Math 404

Paper: Optimization Theory

Maximum Marks: 60

SECTION I

(SHORT QUESTIONS)

Q.1: Write short answers/ solve the following questions (5x4)

i: Detine the following.

• Point of inflexion

• Hyper plane in »-dimensional space

i: Find the maxima and minima, if any, of the function: /(x) = 4x7 - 18x7 + 27x -7

iii: Find Euler-Langrange Equation for the functional

iy Write the dual for the following primal LPP.

Minimize Zp = 22; + 972 + 2g subject 10zy + 4x2 + 2x, ≥ 5, 3x; + *2 + 2xy ≥ 4

With x1,

22. Xy 20

vr Write the Khun-Tucker conditions for the following problem.

Minimize (x), subject to g(x) ≥ 0, with x, ≥

0.

SECTION II

(LONG QUESTIONS)

(5x8)

0.2, The profit on a contract of road laying is estimated as 26x; + 20xg + 4xyxg - 3x,2 -4x,2, where xi

and xy denote the labour cost and the material cost respectively. Find the values of labour cost and

material cost that can maximize the profit on the road.

Of Minimize Z = 3x + 4x} subjectto 2xy - 3xy = 10, with xuX. ≥ 0 using Lagrangian multiplier

method.

Q.4: Solve the following LPP problem using simplex method.

Maximize Z- 10x, + 8x=

Subject to

* + 2x, ≤ 1000, 3, ≤300 X ≤ 500 and 221x2 ≥ 0

Q.s. Discuss the degeneracy in simplex method and how you can break the tie in the following cases.

(i) Tie occurs between leaving variables (il) Tie occurs between incoming variables

0.6: Ai-Khair Flour Mill has four branches A, B, C and D and four warehouses 1, 2, 3 and

4. Production

, demand and transportation costs are given below

MC- 278-20-22-200..

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43

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50.

Raees Khan

DEC 12, 2023

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