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

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Bahandlin Zakariya universin Mutton: MC- 2.28-19-21

(Final-Exams)

M. Se Mathematics (4ih

-Semester)

Time Allowed: 2f hours

Paper Code: Math 104

Session (2019.21)

Puper: Optimization Theory

Maximum Marks: 60

SECTION I

(SHORT QUESTIONS)

2.1: Write short answers/solve the following questions: (5x4)

i Define the Titlowing:

• Convex Polytope

• Extreme Points

#: Determine the muxima and minmo I'any, of the function /(x) = 4x* - 4x4 11

iti: Find Euler-Lagrange Equation for the timctional

1 - V-T

iv: Write the dual for the following primal LPP.

Maximize 7, = 2x, + 3xg + 5x_ subject to xy + 12 + 13 ≤7. xg + 2x4:2%9 S13.

3x1 - 12 + 1; ≤5

With

*1.X2. 6, 200

y: Write the Khun-Tucker conditions for the following probiem.

Minimize (x), subject 10 g (x) 2 0, with x, 2 0.

SECTION!!

(LONG QUESTIONS) (5x8)

02; Minimie 7 = 302 + 4x5

subject to 2x,-3x_ 3x_ = with with xx 20 using Lagronge multiplier

wetlind.

03: The profit on a contract of road laying is estimated as 26x, + 2012 + 4x2 - 3x,* -4x:, where n

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

muterial cost that cun masimize the profit on the mood.

Oa: Salve the following LPP problem using simplex method.

Miximize x = x, +x, + 3xg + 2x4

Subject to x, + 2x2 - 3xg + 5x, ≤

4. 5x, - 2x0 +6x, S8,

0.5: Discuss the falluwing special cases of Simplex method

1 Enbounded Solution (il) Degeneracy Gill Multiple Optimum Solution

0,6: Panju Tour Mill has four branches A, B, Cund D and four warchouses 1,

2. 3 and

4. Production,

demand and transportation costs are given below:

Pintactico (Toms)

Demand (Tens)

170-

-30

3-15

D- 41

Find the optoe coinion of the shove trm pottation moblem.

Mr-228-19-21-20.

O