Shri Ram College Muzaffarnagar

Question Bank of Optimization techniques

BCA 4th Sem
Fill in the Blanks:-

Q.1- The Problem where the objective is to assign a number of origins to the equal number of destinations at a number of
origins to the equal number of destinations at a minimum cost are called ‘Assignment Problem’.
n n
Q.2- In an assignment problem if xij denotes that the ith person is to be assigned the jth job then  xij = 1 and  xij = 1
i=1 i=1

Q.3-In an assignment problem with cost (cij ), if all cij  0 then a feasible solution (xij ) which satisfies
n n
  cij xij = 0 , is optimal for the problem.
i=1 j=1
Q .4- If the cost matrix of an assignment problem is not a square matrix ( number of sources is not equal to the number of
destinations), the assignment problem is called an unbalanced assignment problem.

Multiple Choice Question:-

Q.5- The complete optimal assignment problem is obtained if in the reduced cost matrix of order n the number of marked
‘ ’ zeroes is
(i) less than n (ii) greater than n (iii) exactly n (iv) none of these.

Q.6- An optimal assignment exists if the total reduced cost of the assignment is
(i) zero (ii) one (iii) two (iv) none

Q.7- In an unbalanced assignment problem to form a square matrix fictitious rows or columns are added in the matrix with
costs (i) 1 (ii) 0 (iii)  (iv) none .

Q.8- In the process of drawing minimum number of lines to cover all the zeroes of the reduced matrix we draw lines
through
(i) marked columns (ii) unmarked rows (iii) unmarked rows and marked columns (iv) none

Q.9- If there were n workers and n jobs , there would be

(i) n! solutions (ii) (n+1) ! solutions (iii) (n-1)! solutions (iv) n solutions

Q.10- If a traveling salesman wants to visit n cities then the number of possible routes is
(i) n! (ii) (n+1) ! (iii) (n-1)! (iv) n

Q.11- For a salesman who has to visit n cities . following are the ways of his tour plan
(i) n! (ii) (n+1) ! (iii) (n-1)! (iv) n

Q.12- An assignment problem can be solved by

 (i) simplex method
(ii) transportation method
(iii) both (i) and (ii)
(iv) none

Q.13- An assignment problem is considered as a particular case of a transportation problem, because

(i) the number of rows equals the number of columns.
(ii) all xij = 0 or 1.
(iii) all rim conditions are 1.
(iv) all of the above.

Q.14- An optimal assignment requires that the maximum number of lines which can be drawn through squares with zero
opportunity cost be equal to the number of
(i) rows or columns
(ii) rows and columns
(iii) rows + columns -1
(iv) none

Q.15- While solving assignment problem . an activity is assigned to a resource through a square with zero opportunity cost
because the objective is to
(i) minimize total cost of assignment
(ii) reduce the cost of assignment to zero
(iii) reduce the cost of that particular assignment to zero
(iv) all of the above.

Q.16-The method used for solving an assignment problem is called

(i) reduced matrix method
(ii) MODI method
(iii) Hungerian Method
(iv) None

Q.17- The purpose of a dummy row or column in an assignment problem is to

(i) obtain balance between total activities and total resources.
(ii) prevent a solution from becoming degenerate .
(iii) provide the means of representing a dummy problem
(iv) None.

Q.18- Maximization assignment problem is transformed into a minimization problem by

(i) adding each entry in a column from the maximum value in that column .
(ii) subtracting each entry in a column from the maximum value in that column .
(iii) subtracting entry in the table from the maximum value in that table.
(iv) any one of the above.

Q.19- The assignment problem

(i) requires that only one activity be assigned to each resource.
(ii) is a special case of transportation problem.
(iii) can be used to maximize resources.
(iv) all of the above.

Q.20- Assignment problem is a special case of transportation problem in which number of origins
(i) Equals the number of destinations .
(ii) Is greater than the number of destinations.
(iii) Is less than the number of destinations.
(iv) Is less than or equal to the number of destination.

Q.21- In marketing assignments , which of the following should be preferred .

(i) only that row have a single zero.
(ii) only that column have a single zero.
(iii) only that row/column have a single zero.
(iv) only columns having more than one zero.

Q.22- Which of the following is not correct ?

(i) It is not necessary for the aggregate demand to be equal to the aggregate supply in a transportation problem.
(ii) An unbalanced transportation problem must be converted into a balanced problem before solving it.
(iii) The cost elements in a dummy row / column shall always be taken equal to zero.
(iv) It is impossible that in some cases both, the dummy source and dummy destination,may be required to convert
an unbalanced transportation problem into a balanced one.

Q.23-An assignment problem is a special case of m x n transportation problem in which

(i) m = n
(ii) m = 2n
(iii) 2m = n
(iv) none

Q.24- In a basic feasible solution of an m by n transportation problem the number of positive allocations atmost
(i) m + n
(ii) m + n – 1
(iii) m – n
(iv) none

Q.25- The necessary and sufficient condition for the existence of a feasible solution of a transportation problem is
(i)  ai =  bj
(ii)  ai ≠ bj
(iii)  ai = 0
(iv)  bj = 0

Q.26- In a transportation problem a loop may be defined as an ordered set of atleast

(i) 3 cells
(ii) 4 cells
(iii) 5 cells
(iv) 6 cells

Q.27- If we have a feasible solution consisting of m + n -1 independent allocations , and if numbers ui and vj satisfying
cij =ui + vj for each occupied cell (i , j) then the evaluation dij corresponding to each empty cell (i , j) then the evaluation dij
corresponding to each empty cell ( i , j) is given by
(i) dij = cij – (ui + vj)
(ii) dij = cij + (ui + vj)
(iii) dij = cij – (ui - vj)
(iv) dij = cij + (ui - vj)

Q.28- To improve the current B.F.S if it is not optimal we allocate to the cell for which d ij is
(i) minimum and negative
(ii) maximum and positive
(iii) 0
(iv) none

Q.29- In a transportation problem the solution under test will be optimal if all the cell evaluations are
(i)  0 (ii) > 0 (iii) < 0 (iv)  0

Q.30- In Vogel’s approximation method we select the row or column for which the penality is the :

(i) largest (ii) smallest (iii ) zero (iv) none of these

Q.31- To find an initial B.F.S. we start with the cell (1 , 1) in :

(i) North West Corner Rule
(ii) Lowest Cost Entry Method
(iii) Vogel’s Approximation Method
(iv) None of these

Q.32- To find an initial B.F.S. by Matrix minima method, we first choose the cell with:

(i) zero cost

(ii) highest cost
(iii) lowest cost
(iv) None of these

Q.33- The transportation problem is to transport various amounts of a single homogeneous commodity , that are initially
stored at various origins , to different destinations in such a way that the total transportation cost is :
(i) minimum
(ii) maximum
(iii) may or may not be minimum
(iv) none

Q.34- A feasible solution of m by n transportation problem is said to be non degenerate basic solution if number of positive
allocations is exactly equal to :
(i) (m – n )
(ii) (m + n)
(iii) ( m+ n – 1)
(iv) None of these.

Q.35- By North West corner rule we always get a :

(i) non degenerate basic feasible solution
(ii) degenerate basic feasible solution
(iii) basic feasible solution
(iv) none

Q.36- The optimality test is applicable to a F.S. consisting of ……….. allocations in independent positions.
(i) (m – n )
(ii) (m + n)
(iii) ( m+ n – 1)
(iv) None of these.

Q.37- In a transportation problem the solution under test will be optimal and unique if all the cell evaluations are
(i)  0 (ii) > 0 (iii) < 0 (iv)  0

Q.38- The transportation problem can be regarded as a generalization of the :

(i) special type of transportation problem
(ii) assignment problem
(iii) any L.P.P
(iv) None of these

Q.39- In Vogel’s approximation method the differences of the smallest and second smallest costs in each row and column
are called :
(i) ui
(ii) vj
(iii) penalities
(iv) None of these

Q.40- In Computational procedure of optimality test we choose that ui or vj = 0 for which the corresponding row or column
has the ………………. number of individual allocations.
(i) minimum
(ii) maximum
(iii) both (i) and (ii)
(iv) none of these
Q.41- The iterative procedure of determining an optimum solution of a minimization transportation problem is known as :
(i) North West Corner Rule
(ii) Lowest Cost Entry Method
(iii) Vogel’s Approximation Method
(iv) MODI Method

Q.42- The initial basic feasible solution of the following transportation problem by North West Corner Rule is :
O1 O2 O3 Demand
D1 2 7 4 5

D2 3 3 1 8

D3 5 4 7 7

D4 1 6 2 14

Supply 7 9 18 34
(i) x11=5 , x21=2, x22=6, x32=3, x33=4, x43=14 , Rs 102

(ii) x11=5 , x21=2, x22=6, x33=3, x32=4, x41=14 , Rs 99

(iii) x11=5 , x22=2, x21=6, x33=3, x32=4, x41=14 , Rs 99

(iv) none of these

Q.43- The initial basic feasible solution of the following transportation problem by MMM is :
O1 O2 O3 Demand
D1 2 7 4 5

D2 3 3 1 8

D3 5 4 7 7

D4 1 6 2 14

Supply 7 9 18 34
(i) Rs 102
(ii) Rs 83
(iii) Rs 80
(iv) Rs 76

Q.44- The initial basic feasible solution of the following transportation problem by VAM is :
O1 O2 O3 Demand
D1 2 7 4 5

D2 3 3 1 8

D3 5 4 7 7

D4 1 6 2 14

Supply 7 9 18 34
(i) Rs 102
(ii) Rs 83
(iii) Rs 80
(iv) Rs 76
Q.45- The Optimum solution of the following transportation problem is :
O1 O2 O3 Demand
D1 2 7 4 5

D2 3 3 1 8

D3 5 4 7 7

D4 1 6 2 14

Supply 7 9 18 34
(i) Rs 102
(ii) Rs 83
(iii) Rs 80
(iv) Rs 76

Q.46- A slack variable is introduced if the given constraints has a sign :

(i) 
(ii) 
(iii) =
(iv) none of these

Q.47- Simplex Method to solve linear programming problems was doveloped by

(i) Newton
(ii) Lagranges
(iii) George Dantzig
(iv) None of these

Q.48- If a constraints has  sign , we introduce :

(i) Surplus Variable
(ii) Slack variable
(iii) Artificial variable
(iv) None of these

Q.49- In Big M Method , -M stands for :

(i) very large negative price
(ii) very large positive price
(iii) very small negative price
(iv) None of these

Q.50- The General method for solving an L.P.P. is :

(i) Simplex Method
(ii) Big M Method
(iii) Two Phase Method
(iv) None of these

Q.51- In simplex method all variables must be :

(i) negative
(ii) non negative
(iii) may or may not be negative
(iv) None of these

Q.52- In L.P.P if in the final simplex table all j < 0, then the optimal solution:
(i) is not unique
(ii) is unique
(iii) does not exist
(iv) None of these

Q.53- Big –M Method of L.P.P. was doveloped by

(i) George Dantzig
(ii) Lagrange
(iii) A . Carners
(iv)None of these

Q.54- How many artificial variables are required to solve the following L.P.P. by Big M Method ?
Max Z = - x1 – x2 s.to. 3 x1 + 2x2  30, - 2 x1 + 3 x2  - 30 , x1 + x2  5 and x1 ,x2  0

(i) one
(ii) two
(iii) three
(iv)None of these four

Q.55- In L.P.P. involves  constraints , then initial B.F.S. is obtained by introducing :

(i) decision variables
(ii) two variables
(iii) artificial variables
(iv) None of the above.

Q.56- In Big M Method of the L.P.P . , we take cost of slack , surplus and artificial variables in the objective function
respectively :
(i) 0 , 0 and M
(ii) 0 , 0 and –M
(iii) -1 , -1, -M
(iv) None of the above.

Q.57- If the primal problem is a maximization problem, its dual will be a problem of :
(i) Maximization
(ii) Minimization
(iii) Maximization or Minimization
(iv) None of these

Q.58- The dual of the dual is the :

(i) dual
(ii)primal
(iii) Both of the above
(iv) none of these

Q.59- If the dual has unbounded solution , primal has :

(i) No feasible solution
(ii)Unbounde solution
(iii) Feasible solution
(iv) None of these.

Q.60- Write the dual of the following primal problem

Max Z = x1 + 3 x2 s.to. 3 x1 + 2 x2  6, 3 x1 + 3 x2 = 4 and x1 , x2  0
(i) Min ZD = 6 y1 + y2 s.to. 3 y1 + 3 y2  1, 2 y1 + 3 y2 = 3 and y1 , y2  0
(ii) Min ZD = 6 y1 + 4y2 s.to. 3 y1 + 3 y2  1, 2 y1 + 3y2  3 and y1  0, y2 is unrestricted in sign.
(iii) Min ZD = 6 y1 + y2 s.to. 3 y1 + 3 y2  1, 2 y1 + 3 y2  3 and y1 , y2  0
( iv) None of these.

Q.61- The Word “Operation Research” is first coined by

(i) G.B.Dantzig
(ii) Mc Closky and Trefthen
(iii) Richard Bellman
( iv) Von Neuman
Q.62- Choose Availability constraint in LPP uses the mathematical symbol of
(i) <
(ii) >
(iii) 
(iv) 
Q.63 The maximum number of possible basic solutions for 'm' equations in n unknowns is
(i) m
(ii) n – m
(iii) nCm
(iv) none of these

Q.64- The solution of the LPP Max Z = 5 x1 + 7x2 s.t. 2x1 + 3 x2 ≤ 12 , 2x1 + 3 x2 ≤ 13 ,x1,x2 ≥ 0
(i) x1 = 2 , x2 = 3 , Max Z = 31
(ii) x1 = 4 , x2 = 0 , Max Z = 20
(iii) x1 = 0, x2 = 13 /3, Max Z = 91/3
(iv) none of these

Q.65- During……………. scientists and engineers models to find the solution of the problems about increasing production military
equipments , this branch of study was called as Operation Research.
(i) World War I
(ii) World War II
(iii) World War III
(iv) World War IV

Q.66- Optimization Techniques are generally studied as a part of

(i) Physics
(ii) Statistics
(iii) Probability
(iv) Operation Research

Q.67- If there is no feasible region in a LPP then we say that the problem has
(i) infinite solutions
(ii) no solution
(iii) unbounded solution
(iv) none of these

Q.68- Given the following set of equations :

x1 + 4 x2 – x3 = 3, 5x1 + 2x2 + 3x3 = 4
The B.F.S involving x1 and x2 is
(i) (5/9, 11/18, 0) (ii) (5/9 , 0, 0) (iii) (0, 11/18, 0) (iv) none of these

Q.69- The extreme points of the convex set of feasible solutions of the LPP
Max Z = 10 x1 + 15x2 s.t. x1 + x2 = 2 , 3x1 + 2 x2 ≤ 6 ,x1,x2 ≥ 0 are
(i) (2 , 0) ,(0 , 2)
 (ii) (2 , 0), ( 0,3)
(iii) (0 , 2) , (0,3)
(iv) (0,0), (0,3)
Q.70- A surplus variable is introduced if the given constraints has a sign :

(i) 
(ii) 
(iii) =
(iv) none of these
Q.71- In Standard primal form if the problem is of maximization , all the constraints involve the sign
(i) 
(ii) 
(iii) =
(iv) unrestricted
Q.72- If the standard primal problem is minimization, all the constraints involve the sign
(i) 
(ii) 
(iii) =
(iv) unrestricted
Q.73- If a finite optimal B.F.S. exists for the primal then the dual has
(i) unbounded solution
(ii) no solution
(iii) a finite feasible optimal solution
(iv) none of these
Q.74- If the primal has an unbounded solution , the dual problem has
(i) a finite optimal feasible solution
(ii) no solution
(iii) either no solution or an unbounded solution
(iv) none of these
Q.75- If the ith slack variable of the primal is positive, then the ith variable of the dual is
(i) +ive
(ii) –ive
(iii) zero
(iv) unrestricted
Q.76- If both the primal and dual problem have finite optimal solutions and Zp , ZD are the optimal values of the objective
functions of the primal and dual respectively then we have
(i) Zp > ZD
(ii) Zp < ZD
(iii) Zp = ZD
(iv) none of these
Q.77- In a single-server queuing system, which of the following represents the arrival rate of customers?
(i) λ
(ii) μ
(iii) ρ
(iv) L
Q.78- In a queuing system, if the arrival rate (λ) is 5 customers per hour and the service rate (μ) is 10 customers per hour,
what is the traffic intensity (ρ)?
(i) 0.25
(ii) 0.5
(iii) 0.75
(iv) 1
Q.79- Which of the following is Little's Law in a queuing system?
(i) L=λW
(ii) L=λ+W
(iii) L=W/λ
(iv) L=λ/W
Q.80- In an M/M/1 queue, if the arrival rate is greater than the service rate, what happens to the queue?
(i) The queue remains stable.
(ii) The queue length fluctuates but does not grow indefinitely.
(iii) The queue grows indefinitely.
(iv) The queue length reduces over time.
Q.81- In queuing theory, what does the symbol W represent?
(i) The number of customers in the system.
(ii) The arrival rate of customers.
(iii) The average time a customer spends in the system.
(iv) The service rate of the system.
Q.82- Which of the following queuing models represents a single server with Poisson arrivals and exponential service
times?
(i) M/M/1
(ii) M/G/1
(iii) M/M/C
(iv) G/M/1
Q.83- In a multi-server queuing system (M/M/c), what does the 'c' represent?
(i) The number of customers
(ii) The number of servers
(iii) The cost of service
(iv) The capacity of the system

Q.84- If the average service time (1/μ) is 8 minutes, what is the service rate (μ\muμ) per hour?
(i) 7.5 customers per hour
(ii) 8 customers per hour
(iii) 10 customers per hour
(iv) 7 customers per hour

Q.85- What is the primary objective of the replacement problem in operations research?
(i) Maximizing profit
(ii) Minimizing downtime
(iii) Minimizing total cost
(iv) Maximizing asset utilization
Q.86- Which of the following costs is generally considered when analyzing replacement decisions?
(i) Initial purchase cost
(ii) Operating and maintenance cost
(iii) Salvage value
(iv) All of the above

Q.87- In the context of replacement problems, what does "economic life" refer to?
(i) The total lifespan of an asset
(ii) The period during which an asset is most cost-effective to operate
(iii) The warranty period of an asset
(iv) The period after which an asset has no salvage value.

Q.88- When the operating cost of an asset increases with time and the salvage value decreases with time, the optimal
replacement policy is to replace the asset:
(i) At the end of its useful life
(ii) When the total cost is minimized
(iii) When the salvage value is zero
(iv) When the operating cost exceeds the initial cost
Q.89- n a replacement problem, if the cost of maintaining an old machine is higher than the annual cost of a new machine,
what should be the decision?
(i) Keep the old machine
(ii) Replace the old machine with a new one
(iii) Wait for further deterioration
(iv) Repair the old machine

Q.90- The "Present Worth Method" in replacement analysis involves:

(i) Comparing the initial costs of different assets
(ii) Calculating the present value of all future costs and revenues associated with the assets
(iii) Comparing the future salvage values of different assets
(iv) Determining the future costs without discounting

Q.91- In replacement analysis, the "Average Cost Method" is used to determine:

(i) The average cost per unit time over the asset's life
(ii) The average annual operating cost
(iii) The average cost of replacement
(iv) The average salvage value

Q.92- What is the primary objective of the job sequencing problem in operations research?
(i) Minimizing total cost
(ii) Minimizing total idle time
(iii) Maximizing total cost
(iv) Minimizing the total time required to complete a set of jobs from the start of the first job to the completion of
the last job

Q.93- What is the makespan in a job sequencing problem?

(i) The total time taken to complete all jobs
(ii) The total idle time on both machines
(iii) The total idle time on both machines
(iv) The time at which the last job is completed

Q.94- Which of the following is true when using Johnson's rule for 2 machines and n jobs?
(i) The sequence will always minimize the makespan
(ii) The sequence will always minimize idle time
(iii) The sequence will always maximize throughput
(iv) The sequence will always be in ascending order of job numbers

Q.95- In Johnson's rule, what is the first step in determining the job sequence?
(i) Calculate the total processing time for each job
(ii) Find the job with the shortest processing time
(iii) Calculate the difference in processing times for each job
(iv) Arrange the jobs in alphabetical order

Q.96- If two jobs have the same shortest processing time, how should they be sequenced according to Johnson's rule?
(i) Arbitrarily, as it does not affect the result
(ii) By their arrival time
(iii) By the machine with the shortest idle time
(iv) By their total processing time

Q.97- What is the main limitation of Johnson's rule?

(i) It can only be applied to jobs with equal processing times
(ii) It does not minimize makespan
(iii) It is only applicable to 2-machine problems
(iv) It cannot handle more than 5 jobs

Q.98-The Economic Order Quantity (EOQ) model aims to:

(i) Minimize total inventory cost
(ii) Maximize total inventory cost
(iii) Minimize carrying costs
(iv) Maximize ordering costs

Q.99- In the EOQ formula, the variable D stands for:

(i) Demand rate
(ii) Delivery time
(iii) Discount rate
(iv) Depreciation rate
Q.100- Which inventory valuation method results in the highest ending inventory value during periods of rising prices?
(i) First-In, First-Out (FIFO)
(ii) Last-In, First-Out (LIFO)
(iii) Weighted Average Cost
(iv) Specific Identification

Q.101-In the ABC analysis of inventory, which category typically contains the most critical items that require close
monitoring?
(i) A items
(ii) B items
(iii) C items
(iv) D items

Q.102-Safety stock is primarily used to:

(i) Reduce holding costs
(ii) Prevent stockouts due to demand variability and lead time uncertainty
(iii) Reduce ordering costs
(iv) Increase the EOQ
Q.103- In the Economic Order Quantity (EOQ) model, the total cost is minimized when:
(i) Holding costs are minimized
(ii) Ordering costs are minimized
(iii) The sum of holding and ordering costs is minimized
(iv) Stockout costs are minimized

Q.103- The EOQ formula is derived based on several assumptions. Which of the following is NOT one of those
assumptions?
(i) Constant demand
(ii) Instantaneous replenishment
(iii) Variable lead time
(iv) No quantity discounts

Q.104-What does the reorder point (ROP) signify in inventory management?

(i) The maximum inventory level
(ii) The minimum inventory level
(iii) The inventory level at which a new order should be placed
(iv) The average inventory level