University of Delhi 2010-2nd Year B.Sc PMCS (Physics, Mathematics, Computer Science) Prog OPTIMIZATION UNIVERSITY - Question Paper
This question paper contains 4 printed pages
B. Sc. Frog. Ill
OPERATIONAL RESEARCH OR-201 : - Optimization
(Admissions of 2005 and onwards)
Time 3 hours Maximum Marks 112
(Wnte your Roll No on the top immediately on receipt of this question paper)
Attempt any five questions
1(a) Discuss applications of Operations Research and advantages of Operations Research approach m decibion making
(b) A company has two grade of inspectors, / and II, who aie to be assigned fm a quality control inspection It is required (hat at least, 2000 items be inspected per 8 hour day Grade / inspector can check items at the rate of r'0 per horn with the accuracy of 97% while Grade II inspector can check items at the rate of 40 per hour with the actmacy of 95% The wage rate of Grade / inspector ib Rs 4 50 per hou~ and that of Grade II inspector is Rb 2 50 per hour Each time an error is made bv anv inspector, the cost to the company is one rupee The company has 10 Grade I and 5 Grade II inspectors available with it Formulate a linear programming problem to minimize the total cost of inspe( tion rind solve it graphically
2(a) Define a convex set Is the intersection of any finite number of convex ets necessarily a convex set 7 Justify
(b) If 5 and T are any tno convex sets m IP then show that for all scalars a, &, the set <\S I $T ife also a convex set
3(a) Find all basic feasible solutions for the svstem
Xi + = 8
4~ 2xj j- X4 ~ 4 Xl,X2, J3,X4 > 0
(b) Are the following constiaints consistent7 Use Simplex method to check the same
2xx 3r2 > 2
Xj + J"2
X|,X2 > 0
4(a) Solve the linear programming problem Minimize 2 2xx - x2 1 2xj subject to -[1U2I ri-4 - ri 4 x2 X3 < 6 ri < 0, X2 > 0 r-* unrestricted m sign
(b) Prove that if all - c} > 0 then thf current basic feasible solution of a linear programming problem is an optimal solution
(11 {}, 11}
5(a) Consider the linear programming problem Maximize z cr r subjer* *0 \i < b, r "> 0
Write the dual problem State and prove the complementary slackness theorem
(b) Consider the linear programming problem Maximize c 5jti + 4x2 subject to r, - 2x2 < 1 .
i j { 2x 3,
{\. *2 >0
Use dualitv relationships to show that the above problem has an unbounded
1
solution (11 r , 11)
6(a) Explain any one method to obtain a basic feasible solution of tlx; transportation problem Solve the following transportation problem for minimi/in
the total cost............................... ........
Origins Destinations Avajlabihiv
~ '
61 0 2 0 70
02 I 4 0 30
0_3__ 0 2 4 50 ___
Requirements 70 50 30
(b) Considci the pioblcm of assigning five operators to fi\e machines witli the following assignment tost matrix
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Operate* s Machines | ||||||||||||||||||||||||||||||||||||
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Assign the operators to the machines bo that the total cost is minimized
7(a) Solve the integer hneai programming problem Maxum/e r - lx] + 9x2 subject to T] +3x2 < b
7 r i + X2 < 35 U<7 ?2 < 7
<'i T2 > 0 and are integers
9 7
The non-integer optimal solution of I he problem xi x2 = ~ z 03
(b) The optimal simplex table oi the linear programming problem Maximize ; - 3xj + 4x2 f- J3 f 7xj subject to 8j?i 1 3x2 4 t? j 4- x4 < 7
2x 1 f 6 r>2 + r H 5x4 3 Xj * IX2 * 5r t 2x4 < 8 ri t2,xs,r4 > 0
is given ;ls follows
2__ 3 _ i __7 0 0 J)
c-
b x, x> r3 cj 5j v
<B Vn
3 r 1 10/19 1 9/38 1/2 0 5/38 -1 / 38 0
7 rt 5/19 0 21/19 0 1 -1/19 i/\i> 0 J) -Si 126/19 0 _59_/38 9/2 0 -1/38 -15/38 1_
Discuss the effet.t c>n the above optimal solution by changing the ies>ouree vectoi b (7,3,8) to b - (13,3,8) if tbe new solution is not feasible then obtain <1 feasible solution u&uig dual simplex method
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| Earning: Approval pending. |
