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Solutions for Chapter 7: Assignment Problem and Sequencing
Below listed, you can find solutions for Chapter 7 of Maharashtra State Board Balbharati for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board.
Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 7 Assignment Problem and Sequencing Exercise 7.1 [Pages 118 - 119]
A job production unit has four jobs A, B, C, D which can be manufactured on each of the four machines P, Q, R and S. The processing cost of each job for each machine is given in the following table:
Jobs | Machines (Processing Cost in ₹) |
|||
P | Q | R | S | |
A | 31 | 25 | 33 | 29 |
B | 25 | 24 | 23 | 21 |
C | 19 | 21 | 23 | 24 |
D | 38 | 36 | 34 | 40 |
Find the optimal assignment to minimize the total processing cost.
Five wagons are available at stations 1, 2, 3, 4, and 5. These are required at 5 stations I, II, III, IV, and V. The mileage between various stations are given in the table below. How should the wagons be transported so as to minimize the mileage covered?
I | II | III | IV | V | |
1 | 10 | 5 | 9 | 18 | 11 |
2 | 13 | 9 | 6 | 12 | 14 |
3 | 3 | 2 | 4 | 4 | 5 |
4 | 18 | 9 | 12 | 17 | 15 |
5 | 11 | 6 | 14 | 19 | 10 |
Five different machines can do any of the five required jobs, with different profits resulting from each assignment as shown below:
Job | Machines (Profit in ₹) | ||||
A | B | C | D | E | |
1 | 30 | 37 | 40 | 28 | 40 |
2 | 40 | 24 | 27 | 21 | 36 |
3 | 40 | 32 | 33 | 30 | 35 |
4 | 25 | 38 | 40 | 36 | 36 |
5 | 29 | 62 | 41 | 34 | 39 |
Find the optimal assignment schedule.
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
A company has a team of four salesmen and there are four districts where the company wants to start its business. After taking into account the capabilities of salesmen and the nature of districts, the company estimates that the profit per day in rupees for each salesman in each district is as below:
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 |
Find the assignment of salesman to various districts which will yield maximum profit.
In the modification of a plant layout of a factory 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 of locating a machine at a place (in hundred rupees) is as follows.
Machines | Location | ||||
A | B | C | D | E | |
M_{1} | 9 | 11 | 15 | 10 | 11 |
M_{2} | 12 | 9 | – | 10 | 9 |
M_{3} | – | 11 | 14 | 11 | 7 |
M_{4} | 14 | 8 | 12 | 7 | 8 |
Find the optimal assignment schedule.
Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 7 Assignment Problem and Sequencing Exercise 7.2 [Pages 125 - 126]
A machine operator has to perform two operations, turning and threading on 6 different jobs. The time required to perform these operations (in minutes) for each job is known. Determine the order in which the jobs should be processed in order to minimize the total time required to complete all the jobs. Also find the total processing time and idle times for turning and threading operations.
Job | 1 | 2 | 3 | 4 | 5 | 6 |
Time of turning | 3 | 12 | 5 | 2 | 9 | 11 |
Time for threading | 8 | 10 | 9 | 6 | 3 | 1 |
A company has three jobs on hand. Each of these must be processed through two departments, in the order AB where
Department A: Press shop and
Department B: Finishing
The table below gives the number of days required by each job in each department
Job | I | II | III |
Department A | 8 | 6 | 5 |
Department B | 8 | 3 | 4 |
Find the sequence in which the three jobs should be processed so as to take minimum time to finish all the three jobs. Also find idle time for both the departments.
An insurance company receives three types of policy application bundles daily from its head office for data entry and tiling. The time (in minutes) required for each type for these two operations is given in the following table:
Policy | 1 | 2 | 3 |
Data Entry | 90 | 120 | 180 |
Filing | 140 | 110 | 100 |
Find the sequence that minimizes the total time required to complete the entire task.
There are five jobs, each of which must go through two machines in the order XY. Processing times (in hours) are given below. Determine the sequence for the jobs that will minimize the total elapsed time. Also find the total elapsed time and idle time for each machine.
Job | A | B | C | D | E |
Machine X | 10 | 2 | 18 | 6 | 20 |
Machine Y | 4 | 12 | 14 | 16 | 8 |
Find the sequence that minimizes the total elapsed time to complete the following jobs in the order AB. Find the total elapsed time and idle times for both the machines.
Job | I | II | IIII | IV | V | VI | VII |
Machine A | 7 | 16 | 19 | 10 | 14 | 15 | 5 |
Machine B | 12 | 14 | 14 | 10 | 16 | 5 | 7 |
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 |
Find the optimal sequence that minimizes total time required to complete the following jobs in the order ABC. The processing times are given in hrs.
Job | 1 | 2 | 3 | 4 | 5 |
Machine A | 5 | 7 | 6 | 9 | 5 |
Machine B | 2 | 1 | 4 | 5 | 3 |
Machine C | 3 | 7 | 5 | 6 | 7 |
A publisher produces 5 books on Mathematics. The books have to go through composing, printing and binding done by 3 machines P, Q, R. The time schedule for the entire task in proper unit is as follows.
Book | A | B | C | D | E |
Machine P | 4 | 9 | 8 | 6 | 5 |
Machine Q | 5 | 6 | 2 | 3 | 4 |
Machine R | 8 | 10 | 6 | 7 | 11 |
Determine the optimum time required to finish the entire task.
Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 7 Assignment Problem and Sequencing Miscellaneous Exercise 7 [Pages 126 - 128]
In sequencing, an optimal path is one that minimizes _______.
Elapsed time
Idle time
Both Elapsed time and Idle time
Ready time
If job A to D have processing times as 5, 6, 8, 4 on first machine and 4, 7, 9, 10 on second machine then the optimal sequence is : ______.
CDAB
DBCA
BCDA
ABCD
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 maximize the cost
If there are n jobs and m machines, then there will be_______ sequences of doing the jobs.
mn
m(n!)
n^{m }
(n!)^{m}
The Assignment Problem is solved by ______.
Simplex method
Hungarian method
Vector method
Graphical method
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
Choose the correct alternative :
To use the Hungarian method, a profit maximization assignment problem requires ______.
Converting all profits to opportunity losses
A dummy person or job
Matrix expansion
Finding the maximum number of lines to cover all the zeros in the reduced matrix
Choose the correct alternative :
Using Hungarian method the optimal assignment obtained for the following assignment problem to minimize the total cost is:
Agent | Job | |||
A | B | C | D | |
1 | 10 | 12 | 15 | 25 |
2 | 14 | 11 | 19 | 32 |
3 | 18 | 21 | 23 | 29 |
4 | 15 | 20 | 26 | 28 |
1 – C, 2 – B, 3 – D, 4 – A
1 – B, 2 – C, 3 – A, 4 – D
1 – A, 2 – B, 3 – C, 4 – D
Using Hungarian method the optimal assignment obtained for the following assignment problem to minimize the total cost is: 1 – D, 2 – A, 3 – B, 4 – C.
The assignment problem is said to be unbalance if ______
Number of rows is greater than number of columns
Number of rows is lesser than number of columns
Number of rows is equal to number of columns
Both (a) and (b)
The assignment problem is said to be balanced if ______.
Number of rows is greater than number of columns
Number of rows is lesser than number of columns
Number of rows is equal to number of columns
If the entry of row is zero
Choose the correct alternative :
The assignment problem is said to be balanced if it is a ______.
Square matrix
Rectangular matrix
Unit matrix
Triangular matrix
Choose the correct alternative :
In an assignment problem if number of rows is greater than number of columns then
Dummy column is added
Dummy row is added
Row with cost 1 is added
Column with cost 1 is added
Choose the correct alternative :
In a 3 machine and 5 jobs problem, the least of processing times on machine A, B and C are 5, 1 and 3 hours and the highest processing times are 9, 5 and 7 respectively, then it can be converted to a 2 machine problem if order of the machines is:
B-A-C,
A-B-C
C - B - A
Both (B) and (C)
The objective of an assignment problem is to assign ______.
Number of jobs to equal number of persons at maximum cost.
Number of jobs to equal number of persons at minimum cost
Only the maximize cost
Only to minimize cost
Fill in the blank :
An assignment problem is said to be unbalanced when _______.
Fill in the blank :
When the number of rows is equal to the number of columns then the problem is said to be _______ assignment problem.
Fill in the blank :
For solving an assignment problem the matrix should be a _______ matrix.
Fill in the blank :
If the given matrix is not a _______ matrix, the assignment problem is called an unbalanced problem.
Fill in the blank :
A dummy row(s) or column(s) with the cost elements as _______ is added to the matrix of an unbalanced assignment problem to convert into a square matrix.
The time interval between starting the first job and completing the last job including the idle time (if any) in a particular order by the given set of machines is called _______.
Fill in the blank :
The time for which a machine j does not have a job to process to the start of job is called _______.
Maximization assignment problem is transformed to minimization problem by subtracting each entry in the table from the _______ value in the table.
Fill in the blank :
When an assignment problem has more than one solution, then it is _______ optimal solution.
Fill in the blank :
The time required for printing of four books A, B, C and D is 5, 8, 10 and 7 hours while its data entry requires 7, 4, 3 and 6 hrs respectively. The sequence that minimizes total elapsed time is _______
Fill in the blank :
In Hungarian Method, only _______ task can be assigned to each facility.
Fill in the blank :
In an assignment problem, a solution having _______ total cost is an optimum solution.
Fill in the blank :
In maximization type, all the elements in the matrix are subtracted from the _______ element in the matrix.
Fill in the blank :
In a sequencing problem, all machines are of _______ types.
Fill in the blank :
An _______ is a special type of linear programming problem.
One machine - one job is not an assumption in solving sequencing problems.
True
False
State whether the following is True or False :
If there are two least processing times for machine A and machine B, priority is given for the processing time which has lowest time of the adjacent machine.
True
False
To convert the assignment problem into a maximization problem, the smallest element in the matrix is deducted from all other elements.
True
False
State whether the following is True or False :
The Hungarian method operates on the principle of matrix reduction, whereby the cost table is reduced to a set of opportunity costs.
True
False
State whether the following is True or False :
In a sequencing problem, the processing times are dependent of order of processing the jobs on machines.
True
False
State whether the following 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 is True or False :
Using the Hungarian method, the optimal solution to an assignment problem is found when the minimum number of lines required to cover the zero cells in the reduced matrix equals the no of persons.
True
False
In an assignment problem, if number of column is greater than number of rows, then a dummy column is added.
True
False
State whether the following is True or False :
The purpose of dummy row or column in an assignment problem is to obtain balance between total number of activities and total number of resources.
True
False
State whether the following is True or False :
One of the assumptions 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 is True or False :
In assignment problem, each facility is capable of performing each task.
True
False
State whether the following is True or False
In number of lines (horizontal on vertical) > order of matrix then we get optimal solution.
True
False
State whether the following is True or False :
It is not necessary to express an assignment problem into n x n matrix.
True
False
State whether the following is True or False :
In a sequencing problem, a machine can process more than one job at a time.
True
False
State whether the following is True or False :
The time involved in moving a job from one machine to another is negligibly small.
True
False
Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 7 Assignment Problem and Sequencing Part I [Pages 128 - 129]
Solve the following problem :
A plant manager has four subordinates, and four tasks to be performed. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. This estimate of the time each man would take to perform each task is given in the effectiveness matrix below.
I | II | III | IV | |
A | 7 | 25 | 26 | 10 |
B | 12 | 27 | 3 | 25 |
C | 37 | 18 | 17 | 14 |
D | 18 | 25 | 23 | 9 |
How should the tasks be allocated, one to a man, as to minimize the total man hours?
Solve the following problem :
A dairy plant has five milk tankers, I, II, III, IV and V. These 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 | 175 |
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 :
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 :
The estimated sales (tons) per month in four different cities by five different managers are given below:
Manager | Cities | |||
P | Q | R | S | |
I | 34 | 36 | 33 | 35 |
II | 33 | 35 | 31 | 33 |
III | 37 | 39 | 35 | 35 |
IV | 36 | 36 | 34 | 34 |
V | 35 | 36 | 35 | 33 |
Find out the assignment of managers to cities in order to maximize sales.
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.
Solve the following problem :
A chartered accountant’s firm has accepted five new cases. The estimated number of days required by each of their five employees for each case are given below, where - means that the particular employee cannot be assigned the particular case. Determine the optimal assignment of cases of the employees so that the total number of days required to complete these five cases will be minimum. Also find the minimum number of days.
Employee | Cases | ||||
I | II | III | IV | V | |
E_{1} | 6 | 4 | 5 | 7 | 8 |
E_{2} | 7 | – | 8 | 6 | 9 |
E_{3} | 8 | 6 | 7 | 9 | 10 |
E_{4} | 5 | 7 | – | 4 | 6 |
E_{5} | 9 | 5 | 3 | 10 | – |
Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 7 Assignment Problem and Sequencing Part II [Pages 129 - 130]
Solve the following problem :
A readymade garments manufacturer has to process 7 items through two stages of production, namely cutting and sewing. The time taken in hours for each of these items in different stages are given below:
Items | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
Time for Cutting | 5 | 7 | 3 | 4 | 6 | 7 | 12 |
Time for Sewing | 2 | 6 | 7 | 5 | 9 | 5 | 8 |
Find the sequence in which these items are to be processed through these stages so as to minimize the total processing time. Also find the idle time of each machine.
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 |
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.
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.
Solve the following problem :
A foreman wants to process 4 different jobs on three machines: a shaping machine, a drilling machine and a tapping machine, the sequence of operations being shaping-drilling-tapping. Decide the optimal sequence for the four jobs to minimize the total elapsed time. Also find the total elapsed time and the idle time for every machine.
Job | Shaping (Minutes) | Drilling (Minutes) | Trapping (Minutes) |
1 | 13 | 3 | 18 |
2 | 18 | 8 | 4 |
3 | 8 | 6 | 13 |
4 | 23 | 6 | 8 |
Solutions for Chapter 7: Assignment Problem and Sequencing
Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board chapter 7 - Assignment Problem and Sequencing
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Concepts covered in Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board chapter 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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