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3. Obtain initial basic feasible solution of the following transportation problem using:
i) North-west corner method and
ii) Matrix-minima method.
To
From
A B C Supply
X 7 3 4 2
Y 2 1 3 3
Z 3 4 6 5
Demand 4 1 5 10
Also find the optimum solution for both initial solutions
i) North-west corner method and
ii) Matrix-minima method.
To
From
A B C Supply
X 7 3 4 2
Y 2 1 3 3
Z 3 4 6 5
Demand 4 1 5 10
Also find the optimum solution for both initial solutions
a) A firm manufacturers three products A, and B C . The profit on these products are `
3, ` 2, ` 4 respectively. The firm has two machines and the required processing time
in minutes for each machine on each product is given below.
Product
A B C
Machine X 4 3 5
Y 2 2 4
Machine X and Y have 2000 and 1500 minutes respectively. The firm must
manufacture 100 A’s, 200 B’s and 50 C’s but no more than 150 A’s. Set up an L.P.
model to maximize the profit. Solve the problem graphically. (6)
b) Write the dual of the following LPP:
1 2 3 max Z = 2x + 3x −5x
Subject to
2 2 3 x1 + x2 − x3 ≤
3 3 4 x1 + x2 − x3 ≤
4 5 7 6 x1 + x2 − x3 =
and , 0 x1
x2 ≥ , and 3
x unrestricted.
3, ` 2, ` 4 respectively. The firm has two machines and the required processing time
in minutes for each machine on each product is given below.
Product
A B C
Machine X 4 3 5
Y 2 2 4
Machine X and Y have 2000 and 1500 minutes respectively. The firm must
manufacture 100 A’s, 200 B’s and 50 C’s but no more than 150 A’s. Set up an L.P.
model to maximize the profit. Solve the problem graphically. (6)
b) Write the dual of the following LPP:
1 2 3 max Z = 2x + 3x −5x
Subject to
2 2 3 x1 + x2 − x3 ≤
3 3 4 x1 + x2 − x3 ≤
4 5 7 6 x1 + x2 − x3 =
and , 0 x1
x2 ≥ , and 3
x unrestricted.
1. Which of the following statements are true? Give reasons for your answers.
i) A balanced transportation problem has a feasible solution.
ii) In LLP, an unrestricted primal variable changes into an equality constraint in its dual.
iii) When maximin value of the game is less than or equal to minimax values of the
game, then saddle point exists.
iv) S = {(x, y 2:) x + y ≤ 4 or }4 x + 2y ≤ is a convex set.
v) If a dual LPP is unbounded, then the primal LPP is bounded.
vi) The solution to a transportation problem with m rows (supplies) and n-columns
(destinations) is feasible if number of positive allocation are m + n .
vii) If there are n workers and n jobs there would be n! solutions of assigning jobs to
the workers.
viii) Game theory models are classified by the number of strategies.
ix) Every LPP has an optimal solution.
x) If two constraints do not intersect in the positive quadrant of the graph, then the
solution is bounded.
i) A balanced transportation problem has a feasible solution.
ii) In LLP, an unrestricted primal variable changes into an equality constraint in its dual.
iii) When maximin value of the game is less than or equal to minimax values of the
game, then saddle point exists.
iv) S = {(x, y 2:) x + y ≤ 4 or }4 x + 2y ≤ is a convex set.
v) If a dual LPP is unbounded, then the primal LPP is bounded.
vi) The solution to a transportation problem with m rows (supplies) and n-columns
(destinations) is feasible if number of positive allocation are m + n .
vii) If there are n workers and n jobs there would be n! solutions of assigning jobs to
the workers.
viii) Game theory models are classified by the number of strategies.
ix) Every LPP has an optimal solution.
x) If two constraints do not intersect in the positive quadrant of the graph, then the
solution is bounded.
1. Which of the following statements are true? Give reasons for your answers.
i) A balanced transportation problem has a feasible solution.
ii) In LLP, an unrestricted primal variable changes into an equality constraint in its dual.
iii) When maximin value of the game is less than or equal to minimax values of the
game, then saddle point exists.
iv) S = {(x, y 2:) x + y ≤ 4 or }4 x + 2y ≤ is a convex set.
v) If a dual LPP is unbounded, then the primal LPP is bounded.
vi) The solution to a transportation problem with m rows (supplies) and n-columns
(destinations) is feasible if number of positive allocation are m + n .
vii) If there are n workers and n jobs there would be n! solutions of assigning jobs to
the workers.
viii) Game theory models are classified by the number of strategies.
ix) Every LPP has an optimal solution.
x) If two constraints do not intersect in the positive quadrant of the graph, then the
solution is bounded.
i) A balanced transportation problem has a feasible solution.
ii) In LLP, an unrestricted primal variable changes into an equality constraint in its dual.
iii) When maximin value of the game is less than or equal to minimax values of the
game, then saddle point exists.
iv) S = {(x, y 2:) x + y ≤ 4 or }4 x + 2y ≤ is a convex set.
v) If a dual LPP is unbounded, then the primal LPP is bounded.
vi) The solution to a transportation problem with m rows (supplies) and n-columns
(destinations) is feasible if number of positive allocation are m + n .
vii) If there are n workers and n jobs there would be n! solutions of assigning jobs to
the workers.
viii) Game theory models are classified by the number of strategies.
ix) Every LPP has an optimal solution.
x) If two constraints do not intersect in the positive quadrant of the graph, then the
solution is bounded.
9. a) The following table of values of )x(f is given
Find )2.0( f′ using an h(0 )
2
method (using all the three values.) (2)
b) Evaluate by Simpson’s one-third rule an approximate value of ∫
+
1
0
sin x cos x dx using 7
ordinates. (4)
c) Determine the value of the integral
∫
= +
1
0
2 2/1
I 1(x x ) dx
by composite trapezoidal rule with 3 and 5 ordinates. Improve the result by using
extrapolation technique. (4)
10. a) Find the solution of the difference equation yk+2 − 4yk+1 + 4yk = ;0 k = ,1,0 K. Also find
the particular solution when 1 y0 = and 6 y1 = . (2)
b) Solve the IVP, ; )4(y 4
x 4y
1
y
2
=
−
′ = using Euler’s method. Find )2.4(y with h = 2.0
and 1.0 and extrapolate the value )2.4(y . (2)
c) Solve the IVP
y 1 y , )0(y 0
2
′ = + =
using classical R-K method of h(0 )
4
. Find )4.0(y taking h = 2.0 . Compare the solution
obtained with the exact solution and find the error. (6)
x 0.2 0.3 0.4
f(x) 1.2214 1.3499 1.4918
Find )2.0( f′ using an h(0 )
2
method (using all the three values.) (2)
b) Evaluate by Simpson’s one-third rule an approximate value of ∫
+
1
0
sin x cos x dx using 7
ordinates. (4)
c) Determine the value of the integral
∫
= +
1
0
2 2/1
I 1(x x ) dx
by composite trapezoidal rule with 3 and 5 ordinates. Improve the result by using
extrapolation technique. (4)
10. a) Find the solution of the difference equation yk+2 − 4yk+1 + 4yk = ;0 k = ,1,0 K. Also find
the particular solution when 1 y0 = and 6 y1 = . (2)
b) Solve the IVP, ; )4(y 4
x 4y
1
y
2
=
−
′ = using Euler’s method. Find )2.4(y with h = 2.0
and 1.0 and extrapolate the value )2.4(y . (2)
c) Solve the IVP
y 1 y , )0(y 0
2
′ = + =
using classical R-K method of h(0 )
4
. Find )4.0(y taking h = 2.0 . Compare the solution
obtained with the exact solution and find the error. (6)
x 0.2 0.3 0.4
f(x) 1.2214 1.3499 1.4918
8. a) Find the value of )5( f′ from the following table
(3)
x 0 0.5 1 1.5 2
f(x) 1.0 1.75 3 4.75 7
x 4 7 10 12
y –1 1 2 4
x 0 2 3 4 7 9
f(x) 4 26 58 112 466 922
5
b) Find the value of the constants c ,a in the numerical differentiation formula ,b
y x( ) ay x( )h by x( ) cy x( )h ′
i = i − + i + i +
such that the method is of highest possible order. Derive the corresponding Richardson
extrapolation scheme. (3)
c) Using Stirling’s formula find the number of persons at age 35 years, given
y 512, y 439, y 346, y 243 20 = 30 = 40 = 50 =
where, x
y represents the number of persons at age x years in a life table.
(3)
x 0 0.5 1 1.5 2
f(x) 1.0 1.75 3 4.75 7
x 4 7 10 12
y –1 1 2 4
x 0 2 3 4 7 9
f(x) 4 26 58 112 466 922
5
b) Find the value of the constants c ,a in the numerical differentiation formula ,b
y x( ) ay x( )h by x( ) cy x( )h ′
i = i − + i + i +
such that the method is of highest possible order. Derive the corresponding Richardson
extrapolation scheme. (3)
c) Using Stirling’s formula find the number of persons at age 35 years, given
y 512, y 439, y 346, y 243 20 = 30 = 40 = 50 =
where, x
y represents the number of persons at age x years in a life table.
6. a) Find the interpolating polynomial that fits the following data:
(3)
b) Using the Lagrange’s form of an interpolating polynomial find the value of x when y = 3
from the following table of values:
(3)
c) Using Lagrange’s interpolation formula, prove that ) y y y(3.0 y ) y(2.0 y 1 = 3 − 5 − −3 + −3 − −5
approximately. (4)
7. a) Given log 654 .2 8156, log 658 .2 8182, log 659 .2 8189, log 661 .2 8202 10 = 10 = 10 = 10 = , find
656 log10 . (3)
b) Prove that the third divided differences with arguments d ,a of the function ,b ,c
x
1
is equal
to
abcd
1
− . (3)
c) Determine the spacing h in a table of equally spaced values of the function 3
)x(f = x
between 0 and 1, so that quadratic interpolation in this table yields accuracy of 6
1 10−
× . (4)
8. a) Find the value of )5( f′ from the following table
(3)
x 0 0.5 1 1.5 2
f(x) 1.0 1.75 3 4.75 7
x 4 7 10 12
y –1 1 2 4
(3)
b) Using the Lagrange’s form of an interpolating polynomial find the value of x when y = 3
from the following table of values:
(3)
c) Using Lagrange’s interpolation formula, prove that ) y y y(3.0 y ) y(2.0 y 1 = 3 − 5 − −3 + −3 − −5
approximately. (4)
7. a) Given log 654 .2 8156, log 658 .2 8182, log 659 .2 8189, log 661 .2 8202 10 = 10 = 10 = 10 = , find
656 log10 . (3)
b) Prove that the third divided differences with arguments d ,a of the function ,b ,c
x
1
is equal
to
abcd
1
− . (3)
c) Determine the spacing h in a table of equally spaced values of the function 3
)x(f = x
between 0 and 1, so that quadratic interpolation in this table yields accuracy of 6
1 10−
× . (4)
8. a) Find the value of )5( f′ from the following table
(3)
x 0 0.5 1 1.5 2
f(x) 1.0 1.75 3 4.75 7
x 4 7 10 12
y –1 1 2 4
b) Find the eigenvalue of the matrix A, nearest to 2 and also the corresponding eigenvector
using four iterations of the inverse power method where
−
− −
−
=
0 1 4
1 4 1
4 1 0
A (6)
5. a) i) Set up the Gauss-Seidel iteration scheme in matrix form for solving the system of
equations
7 2x x 1 − 2 =
1 x 2x x − 1 + 2 − 3 =
1 x 2x − 2 − 3 =
ii) Show that this iteration scheme converges and find the rate of convergence.
iii) Perform two iterations of this method taking the zero vector as the initial
approximation. (6)
b) Find the inverse of the matrix
= − −
1 2 4
2 3 5
3 1 2
A
using LU decomposition method
using four iterations of the inverse power method where
−
− −
−
=
0 1 4
1 4 1
4 1 0
A (6)
5. a) i) Set up the Gauss-Seidel iteration scheme in matrix form for solving the system of
equations
7 2x x 1 − 2 =
1 x 2x x − 1 + 2 − 3 =
1 x 2x − 2 − 3 =
ii) Show that this iteration scheme converges and find the rate of convergence.
iii) Perform two iterations of this method taking the zero vector as the initial
approximation. (6)
b) Find the inverse of the matrix
= − −
1 2 4
2 3 5
3 1 2
A
using LU decomposition method
3. a) Find all the roots of the polynomial 0 x 6x 11x 6
3 2
− + − = by the Graeffe’s root squaring
method using three squarings. (7)
b) Find the constants α, β and γ in the method
[ ] ( ) ( ) ( 2 )
1
( ) f x f x h f x h
h
f x ′
k = α k + β k + + γ k +
so that it is of maximum order. Find the values of h s.t.
max|round off error| = max |TE | .
3 2
− + − = by the Graeffe’s root squaring
method using three squarings. (7)
b) Find the constants α, β and γ in the method
[ ] ( ) ( ) ( 2 )
1
( ) f x f x h f x h
h
f x ′
k = α k + β k + + γ k +
so that it is of maximum order. Find the values of h s.t.
max|round off error| = max |TE | .
Solve dy/dx = y^3+3x^2y ÷ x^3+3xy^2
