# Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board chapter 7 - Assignment Problem and Sequencing [Latest edition]

## Chapter 7: Assignment Problem and Sequencing

Exercise 7.1Exercise 7.2Miscellaneous Exercise 7Part IPart II
Exercise 7.1 [Pages 118 - 119]

### 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]

Exercise 7.1 | Q 1 | Page 118

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.

Exercise 7.1 | Q 2 | Page 118

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
Exercise 7.1 | Q 3 | Page 118

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.

Exercise 7.1 | Q 4 | Page 119

Four new machines M1, M2, M3 and M4 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 M2 cannot be placed at C and M3 cannot be placed at A. The cost matrix is given below.

 Machines Places A B C D E M1 4 6 10 5 6 M2 7 4 – 5 4 M3 – 6 9 6 2 M4 9 3 7 2 3

Find the optimal assignment schedule

Exercise 7.1 | Q 5 | Page 119

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.

Exercise 7.1 | Q 6 | Page 119

In the modification of a plant layout of a factory four new machines M1, M2, M3 and M4 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 M2 cannot be placed at C and M3 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 M1 9 11 15 10 11 M2 12 9 – 10 9 M3 – 11 14 11 7 M4 14 8 12 7 8

Find the optimal assignment schedule.

Exercise 7.2 [Pages 125 - 126]

### 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]

Exercise 7.2 | Q 1 | Page 125

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
Exercise 7.2 | Q 2 | Page 125

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.

Exercise 7.2 | Q 3 | Page 125

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.

Exercise 7.2 | Q 4 | Page 125

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
Exercise 7.2 | Q 5 | Page 125

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
Exercise 7.2 | Q 6.1 | Page 125

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
Exercise 7.2 | Q 6.2 | Page 126

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
Exercise 7.2 | Q 7 | Page 126

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.

Miscellaneous Exercise 7 [Pages 126 - 128]

### 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]

Miscellaneous Exercise 7 | Q 1.01 | Page 126

Choose the correct alternative :

In sequencing, an optimal path is one that minimizes _______

• Elapsed time

• Idle time

• Both (a) and (b)

Miscellaneous Exercise 7 | Q 1.02 | Page 126

Choose the correct alternative :

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

Miscellaneous Exercise 7 | Q 1.03 | Page 126

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

Miscellaneous Exercise 7 | Q 1.04 | Page 126

Choose the correct alternative :

If there are n jobs and m machines, then there will be_______ sequences of doing the jobs.

• mn

• m(n!)

• n

• (n!)m

Miscellaneous Exercise 7 | Q 1.05 | Page 126

Choose the correct alternative :

The Assignment Problem is solved by

• Simplex method,

• Hungarian method

• Vector method,

• Graphical method,

Miscellaneous Exercise 7 | Q 1.06 | Page 126

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

Miscellaneous Exercise 7 | Q 1.07 | Page 126

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

Miscellaneous Exercise 7 | Q 1.08 | Page 126

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.

Miscellaneous Exercise 7 | Q 1.09 | Page 127

Choose the correct alternative :

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)

Miscellaneous Exercise 7 | Q 1.1 | Page 127

Choose the correct alternative :

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

Miscellaneous Exercise 7 | Q 1.11 | Page 127

Choose the correct alternative :

The assignment problem is said to be balanced if it is a ______.

• Square matrix

• Rectangular matrix

• Unit matrix

• Triangular matrix

Miscellaneous Exercise 7 | Q 1.12 | Page 127

Choose the correct alternative :

In an assignment problem if number of rows is greater than number of columns then

• Row with cost 1 is added

• Column with cost 1 is added

Miscellaneous Exercise 7 | Q 1.13 | Page 127

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)

Miscellaneous Exercise 7 | Q 1.14 | Page 127

Choose the correct alternative :

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

Miscellaneous Exercise 7 | Q 2.01 | Page 127

Fill in the blank :

An assignment problem is said to be unbalanced when _______.

Miscellaneous Exercise 7 | Q 2.02 | Page 127

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.

Miscellaneous Exercise 7 | Q 2.03 | Page 127

Fill in the blank :

For solving an assignment problem the matrix should be a _______ matrix.

Miscellaneous Exercise 7 | Q 2.04 | Page 127

Fill in the blank :

If the given matrix is not a _______ matrix, the assignment problem is called an unbalanced problem.

Miscellaneous Exercise 7 | Q 2.05 | Page 127

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.

Miscellaneous Exercise 7 | Q 2.06 | Page 127

Fill in the blank :

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 _______.

Miscellaneous Exercise 7 | Q 2.07 | Page 127

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 _______.

Miscellaneous Exercise 7 | Q 2.08 | Page 127

Fill in the blank :

Maximization assignment problem is transformed to minimization problem by subtracting each entry in the table from the _______ value in the table.

Miscellaneous Exercise 7 | Q 2.09 | Page 127

Fill in the blank :

When an assignment problem has more than one solution, then it is _______ optimal solution.

Miscellaneous Exercise 7 | Q 2.1 | Page 127

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 _______

Miscellaneous Exercise 7 | Q 2.11 | Page 127

Fill in the blank :

In Hungarian Method, only _______ task can be assigned to each facility.

Miscellaneous Exercise 7 | Q 2.12 | Page 127

Fill in the blank :

In an assignment problem, a solution having _______ total cost is an optimum solution.

Miscellaneous Exercise 7 | Q 2.13 | Page 127

Fill in the blank :

In maximization type, all the elements in the matrix are subtracted from the _______ element in the matrix.

Miscellaneous Exercise 7 | Q 2.14 | Page 127

Fill in the blank :

In a sequencing problem, all machines are of _______ types.

Miscellaneous Exercise 7 | Q 2.15 | Page 127

Fill in the blank :

An _______ is a special type of linear programming problem.

Miscellaneous Exercise 7 | Q 3.01 | Page 127

State whether the following is True or False :

One machine - one job is not an assumption in solving sequencing problems.

• True

• False

Miscellaneous Exercise 7 | Q 3.02 | Page 128

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

Miscellaneous Exercise 7 | Q 3.03 | Page 128

State whether the following is True or False :

To convert the assignment problem into a maximization problem, the smallest element in the matrix is deducted from all other elements.

• True

• False

Miscellaneous Exercise 7 | Q 3.04 | Page 128

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

Miscellaneous Exercise 7 | Q 3.05 | Page 128

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

Miscellaneous Exercise 7 | Q 3.06 | Page 128

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

Miscellaneous Exercise 7 | Q 3.07 | Page 128

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

Miscellaneous Exercise 7 | Q 3.08 | Page 128

State whether the following is True or False :

In an assignment problem, if number of column is greater than number of rows, then a dummy column is added.

• True

• False

Miscellaneous Exercise 7 | Q 3.09 | Page 128

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

Miscellaneous Exercise 7 | Q 3.1 | Page 128

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

Miscellaneous Exercise 7 | Q 3.11 | Page 128

State whether the following is True or False :

In assignment problem, each facility is capable of performing each task.

• True

• False

Miscellaneous Exercise 7 | Q 3.12 | Page 128

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

Miscellaneous Exercise 7 | Q 3.13 | Page 128

State whether the following is True or False :

It is not necessary to express an assignment problem into n x n matrix.

• True

• False

Miscellaneous Exercise 7 | Q 3.14 | Page 128

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

Miscellaneous Exercise 7 | Q 3.15 | Page 128

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

Part I [Pages 128 - 129]

### 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]

Part I | Q 1 | Page 128

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?

Part I | Q 2 | Page 128

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?

Part I | Q 3 | Page 128

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
Part I | Q 4 | Page 128

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.

Part I | Q 5 | Page 129

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.

Part I | Q 6 | Page 129

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 E1 6 4 5 7 8 E2 7 – 8 6 9 E3 8 6 7 9 10 E4 5 7 – 4 6 E5 9 5 3 10 –
Part II [Pages 129 - 130]

### 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]

Part II | Q 1 | Page 129

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.

Part II | Q 2 | Page 129

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
Part II | Q 3 | Page 129

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.

Part II | Q 4 | Page 129

Solve the following problem :

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.

Part II | Q 5 | Page 130

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

## Chapter 7: Assignment Problem and Sequencing

Exercise 7.1Exercise 7.2Miscellaneous Exercise 7Part IPart II

## 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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