UNIVERSITY OF CALICUT

SCHOOL OF DISTANCE EDUCATION

MULTIPLE CHOICE QUESTION BANK
MSc Mathematics: Fourth Semester
OPTIMIZATION TECHNIQUES IN OPERATIONS RESEARCH (MTH4C16)

1. The time required by each job on each machine is called ------------------ time

(a) Elapsed (b) Idle

(c) Processing (d) Average

2. In a sequencing problem the order of completion of jobs is called

(a) Completion sequence (b) Job sequence

(c) Processing order (d) job order

3. CPM means ------

(a) Critical path method (b) Cost project minimization
(c) Cost project maximisation (d) Critical Project management

4. The points at which an activity starts or ends are called

(a) Node (b) Network
(c) Project (d) Path

5. The graphical representation of project operations

(a) Critical path (b) Activities
(c) Techniques (d) Network
6. PERT means -----------------
(a) Project evaluation review techniques (b) performance evaluation report techniques
(c) Personal enquiry retrieve technique (d) Programme evaluation review technique
7. An activity (i, j) is called critical activity if and only if
(a) 𝐸𝑆𝑖 = 𝐿𝐶𝑖 , 𝐸𝑆𝑗 = 𝐿𝐶𝑗 , 𝐸𝑆𝑗 – 𝐸𝑆𝑖 = 𝐿𝐶𝑗 – 𝐿𝐶𝑖 = 𝑡𝑖𝑗
(b) , 𝐸𝑆𝑗 = 𝐿𝐶𝑗 , 𝐸𝑆𝑗 – 𝐸𝑆𝑖 = 𝐿𝐶𝑗 – 𝐿𝐶𝑖 = 𝑡𝑖𝑗
(c) 𝐸𝑆𝑖 = 𝐿𝐶𝑖 , 𝐸𝑆𝑗 – 𝐸𝑆𝑖 = 𝐿𝐶𝑗 – 𝐿𝐶𝑖 = 𝑡𝑖𝑗
(d) 𝐸𝑆𝑖 = 𝐿𝐶𝑖 , 𝐸𝑆𝑗 = 𝐿𝐶𝑗

8. The slope of crashing an activity is given by

𝐶𝑐 +𝐶𝑛 𝐶𝑐 −𝐶𝑛
(a) 𝑇𝑛 +𝑇𝑐
(b) 𝑇𝑛

𝐶𝑐 −𝐶𝑛 𝐶𝑐
(c) 𝑇𝑛 −𝑇𝑐
(d) 𝑇𝑛 −𝑇𝑐

9. The increment cost of expediting that activity per unit period is called
 (a)Crashing activity (b) slope of crashing activity
(c) Marginal activity (d) Crash duration
10. In the network of the project for which the activities E precedes G and H means
(a) 𝐸 → 𝐺 → 𝐻 (b) 𝐻 → 𝐺 → 𝐸
(c) 𝐸 → 𝐺, 𝐸 → 𝐻 (d) 𝐺 → 𝐸, 𝐻 → 𝐸
11. The Dynamic Programming was invented by…..
(a) Euler (b) Richard E. Bellman
(c) Kuhn Tucker (d) Bernoulli
12. An optimal policy has the property that whatever the initial state and initial decision are the
remaining decisions must constitute an optimal policy with regard to the state resulting from
the preceding decision, is called
(a) Forward Recursion (b) Backward Recursion
(c) Dynamic method (d) Bellman’s optimality

13. 𝐹𝑖 (𝑥𝑖 ) = 𝑜𝑝𝑡𝑦𝑖 [𝐹𝑖−1 (𝑥𝑖−1 ) + 𝑓𝑖 (𝑦𝑖 )], 𝑖 = 2, 3, . . . , 𝑁,is called the …..
(a) Backward recursion formula (b) Forward recursion formula
(c) Optimality criteria (d)Critical equation
14. A quadratic form XTAX is said to be positive definite if
(a) 𝑋 𝑇 𝐴𝑋 > 0 ∀𝑋 ≠ 0 (b) 𝑋 𝑇 𝐴𝑋 ≥ 0 ∀𝑋 ≠ 0 𝑎𝑡𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝑋 ≠ 0, 𝑋 𝑇 𝐴𝑋 = 0

(c) 𝑋 𝑇 𝐴𝑋 < 0 ∀𝑋 ≠ 0 (d) 𝑋 𝑇 𝐴𝑋 ≤ 0 ∀𝑋 ≠ 0 𝑎𝑡𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝑋 ≠ 0, 𝑋 𝑇 𝐴𝑋 = 0

15. A quadratic form XTAX is said to be positive semi definite if
(a) 𝑋 𝑇 𝐴𝑋 > 0 ∀𝑋 ≠ 0 (b) 𝑋 𝑇 𝐴𝑋 ≥ 0 ∀𝑋 ≠ 0 𝑎𝑡𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝑋 ≠ 0, 𝑋 𝑇 𝐴𝑋 = 0

(c) 𝑋 𝑇 𝐴𝑋 < 0 ∀𝑋 ≠ 0 (d) 𝑋 𝑇 𝐴𝑋 ≤ 0 ∀𝑋 ≠ 0 𝑎𝑡𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝑋 ≠ 0, 𝑋 𝑇 𝐴𝑋 =

0

16. If 𝑓(𝑋) = 𝑥12 + 2𝑥22 + 3𝑥32 then f is

(a) Positive definite (b) Negative definite
(c) Positive semi definite (d) Indefinite
2
17. If 𝑓(𝑋) = 𝑥12 − (𝑥2 − 𝑥3 ) then f is
(a) Positive definite (b) Negative definite
(c) Positive semi definite (d) Indefinite

18. If 𝑓(𝑋) = 𝑥12 − 2𝑥22 then f is

(a) Positive definite (b) Negative definite
(c) Positive semi definite (d) Indefinite

19. If 𝑓(𝑋) = 𝑋 𝑇 𝐴𝑋 is positive definite if all the principle minor of the determinant Di,
(a) 𝐷𝑖 ≤ 0 ∀𝑖 (b) 𝐷𝑖 < 0 ∀𝑖
(c) 𝐷𝑖 > 0 ∀𝑖 (d) 𝐷𝑖 ≥ 0 ∀𝑖
20. If 𝑓(𝑋) = 𝑋 𝑇 𝐴𝑋 is negative semi definite if all the principle minor of the determinant Di,
 (a) (−1)𝑖 𝐷𝑖 < 0 (b) (−1)𝑖 𝐷𝑖 ≥ 0

(c) (−1)𝑖 𝐷𝑖 ≤ 0 (d) None of these

21. The Assembly line scheduling and longest common subsequence problems are an example of
(a) Greedy algorithm (b) Branch and bound method
(c) Geometric method (d) Dynamic algorithm

22. The If 𝑓(𝑥) = −3𝑥 2 + 2𝑥22 − 3𝑥32 − 10 𝑥1 𝑥2 + 4𝑥2 𝑥3 + 6𝑥1 𝑥3 then the Matrix A can be
written as
3 −5 −3 −3 −5 −3
(a)[−5 2 −2] (b) [−5 −2 −2]
3 −2 3 3 −2 −3

3 5 −3 −3 −5 3
(c) [ 5 −2 −2] (d) [−5 2 2]
−3 −2 3 3 2 −3
−3 −5 3
23. For the matrix [−5 2 2 ] we have D2 is
3 2 −3
(a) 32 (b) -31
(c) 16 (d) -10

24. Identify the stationary points 𝑓(𝑋) = 2 + 2𝑥1 + 3𝑥2 − 𝑥12 − 𝑥22
(a) (1,1) (b) (1,2)
(c) (1, 5/2) (d) (1, 3/2)
2
25. A necessary conditions for 𝑓(𝑋) ∈ 𝐶 to have stationary points is that
(a) ∇𝑓(𝑥) = 0 (b) ∇𝑓(𝑥) ≠ 0
(c) ∇𝑓(𝑥) > 0 (d) ∇𝑓(𝑥) < 0
26. For a quadratic 𝑋 𝑇 𝐴𝑋, 𝑋 ∗ is a point of relative minimum if A is
(a) Node (b) Positive semi definite
(c) Saddle point (d) negative definite
27. For a quadratic 𝑋 𝑇 𝐴𝑋, 𝑋 ∗ is a point of relative maximum if A is
(a) Node (b) negative definite
(c) saddle point (d) Positive definite

28. For a quadratic 𝑋 𝑇 𝐴𝑋 if A is indefinite then the point X* is called

(a) Node (b) maximum
(c) Saddle point (d) minimum
29. The function 𝐿 ≡ 𝐿(𝑋, 𝜆) = 𝑓(𝑋) + ∑𝑚
𝑖=1 𝜆𝑖 𝑔𝑖 (𝑋) is called
(a) Lagrange function (b) Optimal function
(c) Constraint function (d) none of these
30. A function 𝑓((1 − 𝛼)𝑋1 + 𝛼𝑋2 ) ≤ (1 − 𝛼)𝑓(𝑋1 ) + 𝛼𝑓(𝑋2 ) then f is said to be
(a) Indefinite (b) concave
 (c) Convex (d) None of these
31. The sum of convex function is ………………
(a) Concave (b) convex
(c) Strictly convex (d) strictly concave

32. Let𝑓(𝑋) = 𝑋 𝑇 𝐴𝑋. Then 𝑓(𝑋) is convex in 𝑅 𝑛 if 𝑋 𝑇 𝐴𝑋 is

(a) Convex (b) concave
(c) Positive semi definite (d) negative semi definite
33. For an convex function every local extrema is a …………..
(a) Node (b) global extrema
(c) saddle point (d) None of these
34. A feasible solution of convex nonlinear programming problem is a………….
(a) open set (b) convex set
(c) closed set (d) empty set
35. A mathematical programming problem in which the objective function and constraint are
separable is called
(a) Linear Programming problem (b) dynamic programming problem
(c) Integer programming problem (d) Separable programming problem
36. The A function of the form 𝑓(𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) = 𝑓1 (𝑥1 ) + 𝑓2 (𝑥2 ) + . . . + 𝑓𝑛 (𝑥𝑛 ) is called
(a) Optimal (b) convex
(c) Continuous (d) Separable
37. The dual of LPP is a function of ………..
(a) Dual variable (b) primal variable
(c) Both dual and primal variable (d) None of these
38. Let f(X) and G(X) be convex differentiable functions, and let 𝑋1 ∈ 𝑃𝐹 𝑎𝑛𝑑 (𝑋2 , 𝜆) ∈ 𝑆𝐷 .
Then
(a) 𝑓(𝑋1 ) = 0 (b) 𝜙(𝑋2 , 𝜆) = 0
(c) 𝑓(𝑋1 ) ≥ 𝜙(𝑋2 , 𝜆) (d) 𝑓(𝑋1 ) ≤ 𝜙(𝑋2 , 𝐶)
39. Let f(X) and G(X) be convex differentiable functions, and let X∗ be an optimal solution of
primal, then (X*, 𝜆∗) is
(a) An optimal solution of a dual (b) An optimal solution of a primal
(c) Solution of both dual and primal (d) none of these
40. A function which has only one peak in a given interval is called
(a) Unimodal function (b) maxima of a function
(c) Lagrange function (d) minima of a function
41. The measure of effectiveness of any search technique is defined as
𝐿 𝐿
(a) 𝑛 = 1 (b) 𝑛 > 1
𝐿𝑜 𝐿𝑜
𝐿 𝐿
(c) 𝐿𝑛 ≥1 (d) 𝐿𝑛 ≤ 1
𝑜 𝑜
42. The method to find the minimum of a function of several variables
 (a) Newton-Raphson method (b) Steepest decent method
(c) Fibonacci search method (d) Dichotomous search method
43. In this method two experiments are placed as close as possible to the centre of the interval of
uncertainty

(a) Newton-Raphson method (b) Steepest decent method

(c) Fibonacci search method (d) Dichotomous search method

44. The recursion formula 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 , n > 1 is used for

(a) Newton-Raphson method (b) Steepest decent method

(c) Fibonacci search method (d) Dichotomous search method

45. Which is an extension of the Fibonacci method

(a) Newton-Raphson method (b) Steepest decent method

(c) Golden section method (d) Dichotomous search method

46. Two vectors s1 and s2 in R n are said to be conjugate vectors with respect to H if,

(a) 𝑠1𝑇 𝐻𝑠2 > 0 (b) 𝑠1𝑇 𝐻𝑠2 = 0

(c) 𝑠1𝑇 𝐻𝑠2 < 0 (d) 𝑠1𝑇 𝐻𝑠2 ≠ 0

47. Every finite set 𝑠1 , 𝑠2 , . . . , 𝑠𝑛 of conjugate vectors with respect to a symmetric and positive
definite matrix is
(a) Zero (b) Linearly dependent
(c) Linearly independent (d) Neither Linearly independent nor dependent
48. A convex function is unimodal
(a) True (b) False
(c) depends upon the condition (d) None of these
49. In the Bolzano search technique,the nth iterations of the interval of uncertainty is reduced to
2 𝑛/2 2
(a) 𝐿𝑛 = (3) 𝐿0 (b) 𝐿𝑛 = (3) 𝐿0
2 𝑛/2 2 𝑛
(c) 𝐿𝑛 = (3) 𝐿1 (d) 𝐿𝑛 = (3) 𝐿0
50. The technique used for solving problems involving posynomials is called
(a) Dynamic programming (b) Goal programming
(c) Geometric programming (d) linear programming
51. The basic tool, the arithmetic mean-geometric mean inequality ,generally utilized, in

(a) Dynamic programming (b) Goal programming

(c) Geometric programming (d) linear programming
52. Find the lower bound for 𝑓(𝑥) = 𝑥 −4 + 4𝑥 3 + 4𝑥, 𝑥 > 0
(a) 3 (b) 2
(c) 0 (d) 6
 𝑎 𝑎 𝑎
53. A function of the form 𝑓𝑖 (𝑋) = 𝑐𝑖 𝑥1 𝑖1 𝑥2 𝑖2 . . . 𝑥𝑛 𝑖𝑛 𝑖 = 1, 2, . . . , 𝑚 𝑎𝑛𝑑 𝑐𝑖 > 0 , 𝑎𝑖𝑗 ∈
𝑅, 𝑗 = 1, 2, . . . , 𝑛 is called
(a) Polynomial (b) Posynomials
(c) Fractional (d) Cartesian products
54. The geometric programming is cannot be applied in
(a) posynomial optimization (b) linear optimization
(c) Oil tank design problem (d) Gravel box design problem
55. What is a constraint?
(a) Response (b) Parameter
(c) Limitation (d) Principle
56. One of the techniques developed for solution of nonlinear programming
(a) Single programming (b) Multilinear programming
(c) Dual programming (d) Dynamic programming
57. The direct cost of project increases and indirect cost decreases if the duration of the project is
(a) No change (b) increased
(c) Medium change (d) Reduced
58. The total time involved in completion of all jobs is
(a) Critical time (b) elapsed time
(c) Period (d) optimum time
59. If a problem can be decomposed into stages and the decisions can be taken stage-wise,then it
can be solved by
(a) Geometric programming (b) Branch and bound method
(c) Dynamic programming (d) integer programming
60. Which of the following method contains least number of calculation.
(a) Geometric programming (b) Branch and bound method
(c) Dynamic programming (d) integer programming
QN No. Ans QN No. Ans QN No. Ans
1 c 26 d 51 c
2 b 27 b 52 d
3 a 28 c 53 b
4 a 29 a 54 b
5 d 30 c 55 c
6 a 31 b 56 d
7 a 32 c 57 d
8 c 33 b 58 b
9 b 34 b 59 c
10 c 35 d 60 c
11 b 36 d
12 d 37 a
13 a 38 c
14 a 39 a
15 b 40 a
16 a 41 d
17 c 42 b
18 d 43 d
19 c 44 c
20 b 45 c
21 d 46 b
22 d 47 c
23 b 48 a
24 d 49 a
25 a 50 c