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Chapters
Chapter 1.2: Matrices
Chapter 1.3: Differentiation
Chapter 1.4: Applications of Derivatives
Chapter 1.5: Integration
Chapter 1.6: Definite Integration
Chapter 1.7: Application of Definite Integration
Chapter 1.8: Differential Equation and Applications
Chapter 2.1: Commission, Brokerage and Discount
Chapter 2.2: Insurance and Annuity
Chapter 2.3: Linear Regression
Chapter 2.4: Time Series
Chapter 2.5: Index Numbers
Chapter 2.6: Linear Programming
▶ Chapter 2.7: Assignment Problem and Sequencing
Chapter 2.8: Probability Distributions
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Solutions for Chapter 2.7: Assignment Problem and Sequencing
Below listed, you can find solutions for Chapter 2.7 of Maharashtra State Board SCERT Maharashtra Question Bank for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022.
SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2.7 Assignment Problem and Sequencing Q.1
MCQ [1 Mark]
Choose the correct alternative:
The assignment problem is generally defined as a problem of ______
maximization
minimization
allocation
restriction
Choose the correct alternative:
Assignment Problem is special case of ______
Linear Programming Problem
Unbalanced Problem
Restricted Problem
Sequencing Problem
Choose the correct alternative:
The assignment problem is solved by ______
Simplex method
Hungarian method
Vector method
Graphical method
Choose the correct alternative:
The cost matrix of an unbalanced assignment problem is not a ______
unit matrix
triangular matrix
rectangular matrix
square matrix
Choose the correct alternative:
When an assignment problem has more than one solution, then it is ______
double optimal solution
infinite optimal solution
multiple optimal solution
dual optimal solution
Choose the correct alternative:
The assignment problem is said to be balanced if ______
no. of rows = no. of columns
no. of rows ≠ no. of columns
no. of rows < no. of columns
no. of rows > no. of columns
Choose the correct alternative:
Machine | Job | |||
A | B | C | D | |
I | 5 | 6 | 8 | 4 |
II | 4 | 7 | 9 | 10 |
The optimal sequence for above data is
CDBA
DBCA
BCDA
ABCD
Choose the correct alternative:
In sequencing, an optimal path that minimizes ______
Elapsed time
Idle time
Both elapsed time and idle time
Ready time
Choose the correct alternative:
If there are 3 machines A, B and C, conditions for reducing a 3 machine problem to a 2 machine problem with respect to minimum processing time is ______
Min A_{i} ≥ Max B_{i} OR Min C_{i} ≥ Max B_{i}, i = 1, 2, 3…n
Min A_{i} ≤ Max B_{i} OR Min C ≤ Max B_{i}, i = 1, 2, 3…n
Max A_{i} ≥ Min B_{i} OR Max B ≥ Min A_{i}, i = 1, 2, 3…n
Max A_{i} ≤ Min B_{i} OR Max B ≤ Min A_{i}, i = 1, 2, 3…n
Choose the correct alternative:
The objective of sequencing problem is ______
to find the order in which jobs are to be made
to find the time required for the completing all the job on hand
to find the sequence in which jobs on hand are to be processed to minimize the total time required for processing the jobs
to maximization the cost
If there are n jobs and m machines, then there will be _______ sequence of doing jobs.
mn
m(n!)
n^{m}
(n!)^{m}
Choose the correct alternative:
In solving 2 machine and n jobs sequencing problem, the following assumption is wrong
No passing is allowed
Processing times are known
Handling time is negligible
The time of passing depends on the order of machining
SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2.7 Assignment Problem and Sequencing Q.2
Fill in the blanks [1 Mark]
If the given matrix is ______ matrix, the assignment problem is called balanced problem
An unbalanced assignment problems can be balanced by adding dummy rows or columns with ______ cost
A ______ assignment problem does not allow some worker(s) to be assign to some job(s)
In an assignment problem if number of rows is greater than number of columns, then dummy ______ is added
The Hungarian method is an ______ algorithm that solves an assignment problem
In assignment problem each worker or machine is ______ capable of handling any job
In sequencing problems one has to ______ the total processing time or cost
In sequencing problem the time which required to complete all the jobs i.e. entire task is called ______
______ time is the time when the machine is available but is waiting for a job to be processed
In sequencing problem the time required to transfer a job from one machine to another is ______
Book | A | B | C | D |
Printing | 5 | 8 | 10 | 7 |
Data Entry | 7 | 4 | 3 | 6 |
The optimum sequence for the above data is ______
In sequencing problem one has to determine the ______ in which jobs has to processed through the machines
SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2.7 Assignment Problem and Sequencing Q.3
[1 Mark]
State whether the following statement is True or False:
The objective of an assignment problem is to assign number of jobs to equal number of persons at maximum cost
True
False
State whether the following statement is True or False:
To convert the assignment problem into maximization problem, the smallest element in the matrix is to deducted from all other elements
True
False
State whether the following statement is True or False:
Optimal assignments are made in the Hungarian method to cells in the reduced matrix that contain a zero
True
False
State whether the following statement is True or False:
In assignment problem, if number of columns is greater than number of rows, then a dummy row is added
True
False
State whether the following statement is True or False:
The Hungarian method is used to assign n jobs on 2 machines to get the optimal sequence
True
False
State whether the following statement is True or False:
In assignment problem each worker or machine is assigned only one job
True
False
State whether the following statement is True or False:
One of the assumptions is made while sequencing n jobs on 2 machines is: two jobs must be loaded at a time on any machine
True
False
State whether the following statement is True or False:
The Total Elapsed is the time required to complete all the jobs i.e. entire task
True
False
State whether the following statement is True or False:
The idle time for a machine is the when machine is available but is not waiting for any job to be processed
True
False
State whether the following statement is True or False:
In sequencing problem the processing times are dependent of order of processing the jobs on machine
True
False
State whether the following statement is True or False:
In sequencing problem each job once started on any machine must be processed still its completion
True
False
State whether the following statement is True or False:
In sequencing problem each machine is of different type
True
False
SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2.7 Assignment Problem and Sequencing Q.4
Attempt the following questions [3 or 4 Marks]
Four new machines M_{1}, M_{2}, M_{3} and M_{4} are to be installed in a machine shop. There are five vacant places A, B, C, D and E available. Because of limited space, machine M_{2} cannot be placed at C and M_{3} cannot be placed at A. The cost matrix is given below.
Machines | Places | ||||
A | B | C | D | E | |
M_{1} | 4 | 6 | 10 | 5 | 6 |
M_{2} | 7 | 4 | – | 5 | 4 |
M_{3} | – | 6 | 9 | 6 | 2 |
M_{4} | 9 | 3 | 7 | 2 | 3 |
Find the optimal assignment schedule
Solve the following problem :
Solve the following assignment problem to maximize sales:
Salesman | Territories | ||||
I | II | III | IV | V | |
A | 11 | 16 | 18 | 15 | 15 |
B | 7 | 19 | 11 | 13 | 17 |
C | 9 | 6 | 14 | 14 | 7 |
D | 13 | 12 | 17 | 11 | 13 |
Solve the following problem :
Consider the problem of assigning five operators to five machines. The assignment costs are given in following table.
Operator | Machine | ||||
1 | 2 | 3 | 4 | 5 | |
A | 6 | 6 | – | 3 | 7 |
B | 8 | 5 | 3 | 4 | 5 |
C | 10 | 4 | 6 | – | 4 |
D | 8 | 3 | 7 | 8 | 3 |
E | 7 | 6 | 8 | 10 | 2 |
Operator A cannot be assigned to machine 3 and operator C cannot be assigned to machine 4. Find the optimal assignment schedule.
Find the assignments of salesman to various district which will yield maximum profit
Salesman | District | |||
1 | 2 | 3 | 4 | |
A | 16 | 10 | 12 | 11 |
B | 12 | 13 | 15 | 15 |
C | 15 | 15 | 11 | 14 |
D | 13 | 14 | 14 | 15 |
A dairy plant has five milk tankers, I, II, III, IV and V. Three milk tankers are to be used on five delivery routes A, B, C, D and E. The distances (in kms) between the dairy plant and the delivery routes are given in the following distance matrix.
I | II | III | IV | V | |
A | 150 | 120 | 175 | 180 | 200 |
B | 125 | 110 | 120 | 150 | 165 |
C | 130 | 100 | 145 | 160 | 170 |
D | 40 | 40 | 70 | 70 | 100 |
E | 45 | 25 | 60 | 70 | 95 |
How should the milk tankers be assigned to the chilling center so as to minimize the distance travelled?
Solve the following problem :
Find the sequence that minimizes the total elapsed time to complete the following jobs. Each job is processed in order AB.
Machines | Jobs (Processing times in minutes) | ||||||
I | II | III | IV | V | VI | VII | |
Machine A | 12 | 6 | 5 | 11 | 5 | 7 | 6 |
Machine B | 7 | 8 | 9 | 4 | 7 | 8 | 3 |
Determine the sequence for the jobs so as to minimize the processing time. Find the total elapsed time and the idle times for both the machines.
Find the optimal sequence that minimizes total time required to complete the following jobs in the order ABC. The processing times are given in hours.
Jobs | I | II | III | IV | V | VI | VII |
Machine A | 6 | 7 | 5 | 11 | 6 | 7 | 12 |
Machine B | 4 | 3 | 2 | 5 | 1 | 5 | 3 |
Machine C | 3 | 8 | 7 | 4 | 9 | 8 | 7 |
Solve the following problem :
Five jobs must pass through a lathe and a surface grinder, in that order. The processing times in hours are shown below. Determine the optimal sequence of the jobs. Also find the idle time of each machine.
Job | I | II | III | IV | V |
Lathe | 4 | 1 | 5 | 2 | 5 |
Surface grinder | 3 | 2 | 4 | 3 | 6 |
A toy manufacturing company produces five types of toys. Each toy has to go through three machines A, B, C in the order ABC. The time required in hours for each process is given in the following table.
Type | 1 | 2 | 3 | 4 | 5 |
Machine A | 16 | 20 | 12 | 14 | 22 |
Machine B | 10 | 12 | 4 | 6 | 8 |
Machine C | 8 | 18 | 16 | 12 | 10 |
Solve the problem for minimizing the total elapsed time.
Find the sequence that minimizes total elapsed time to complete the following jobs in the order XY. Find the total elasped time and idle times for each machine.
Jobs | A | B | C | D | E |
Machine X | 10 | 2 | 18 | 6 | 20 |
Machine Y | 4 | 12 | 14 | 16 | 8 |
SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2.7 Assignment Problem and Sequencing Q.5
Activities [4 marks]
For the following assignment problem minimize total man hours:
Subordinates | Required hours for task | |||
I | II | III | IV | |
A | 7 | 25 | 26 | 10 |
B | 12 | 27 | 3 | 25 |
C | 37 | 18 | 17 | 14 |
D | 18 | 25 | 23 | 9 |
Subtract the `square` element of each `square` from every element of that `square`
Subordinates | Required hours for task | |||
I | II | III | IV | |
A | 0 | 18 | 19 | 3 |
B | 9 | 24 | 0 | 22 |
C | 23 | 4 | 3 | 0 |
D | 9 | 16 | 14 | 0 |
Subtract the smallest element in each column from `square` of that column.
Subordinates | Required hours for task | |||
I | II | III | IV | |
A | `square` | `square` | 19 | `square` |
B | `square` | `square` | 0 | `square` |
C | `square` | `square` | 3 | `square` |
D | `square` | `square` | 14 | `square` |
The lines covering all zeros is `square` to the order of matrix `square`
The assignment is made as follows:
Subordinates | Required hours for task | |||
I | II | III | IV | |
A | 0 | 14 | 19 | 3 |
B | 9 | 20 | 0 | 22 |
C | 23 | 0 | 3 | 0 |
D | 9 | 12 | 14 | 0 |
Optimum solution is shown as follows:
A → `square, square` → III, C → `square, square` → IV
Minimum hours required is `square` hours
Solve the following problem of sequencing for minimizing the total elapsed time and idle time for both the machines.
Job | P | Q | R | S | T | U |
M_{1} | 1 | 4 | 6 | 3 | 5 | 2 |
M_{2} | 3 | 6 | 8 | 8 | 1 | 5 |
The optimal sequence of the jobs as follows:
Total elasped time is obtained as follows:
Job sequence |
Machine A | Machine B | Idle time for Machine B |
||
Time In |
Time Out |
Time In |
Time Out |
||
P | `square` | 1 | 1 | `square` | `square` |
U | `square` | 3 | 4 | `square` | `square` |
S | `square` | 6 | 9 | `square` | `square` |
Q | `square` | 10 | 17 | `square` | `square` |
R | `square` | 16 | 23 | `square` | `square` |
T | `square` | 21 | 31 | `square` | `square` |
Total elapsed time T = `square` minutes
Idle time for Machine A = T – `square` = `square` minutes
Idle time for Machine B = `square`
Solutions for Chapter 2.7: Assignment Problem and Sequencing
SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 2.7 - Assignment Problem and Sequencing
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Concepts covered in 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 2.7 Assignment Problem and Sequencing are Assignment Problem, Hungarian Method of Solving Assignment Problem, Special Cases of Assignment Problem, Sequencing Problem, Types of Sequencing Problem, Finding an Optimal Sequence.
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