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प्रश्न
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 |
उत्तर
Step 1:
Since it is a maximization problem, subtract each of the elements in the table from the largest element, i.e., 16
| Salesman | District | |||
| 1 | 2 | 3 | 4 | |
| A | 0 | 6 | 4 | 5 |
| B | 4 | 3 | 1 | 1 |
| C | 1 | 1 | 5 | 2 |
| D | 3 | 2 | 2 | 1 |
Step 2: Row minimum
Subtract the smallest element in each row from every element in its row.
The matrix obtained is given below:
| Salesman | District | |||
| 1 | 2 | 3 | 4 | |
| A | 0 | 6 | 4 | 5 |
| B | 3 | 2 | 0 | 0 |
| C | 0 | 0 | 4 | 1 |
| D | 2 | 1 | 1 | 0 |
Step 3: Column minimum
Here, each column contains element zero.
∴ Matrix obtained by column minimum is same as above matrix.
Step 4:
Draw minimum number of vertical and horizontal lines to cover all zeros.
First cover all rows and columns which have maximum number of zeros.
| Salesman | District | |||
| 1 | 2 | 3 | 4 | |
| A | 0 | 6 | 4 | 5 |
| B | 3 | 2 | 0 | 0 |
| C | 0 | 0 | 4 | 1 |
| D | 2 | 1 | 1 | 0 |
Step 5:
From step 4, minimum number of lines covering all the zeros are 4, which is equal to order of the matrix, i.e., 4.
∴ Select a row with exactly one zero, enclose that zero in () and cross out all zeros in its respective column.
Similarly, examine each row and column and mark the assignment ().
∴ The matrix obtained is as follows:
| Salesman | District | |||
| 1 | 2 | 3 | 4 | |
| A | 0 | 6 | 4 | 5 |
| B | 3 | 2 | 0 | 0 |
| C | 0 | 0 | 4 | 1 |
| D | 2 | 1 | 1 | 0 |
Step 6:
The matrix obtained in step 5 contains exactly one assignment for each row and column.
∴ Optimal assignment schedule is as follows:
| Salesman | District | Profit (in ₹) |
| A | 1 | 16 |
| B | 3 | 15 |
| C | 2 | 15 |
| D | 4 | 15 |
∴ The maximum profit
= 16 + 15 + 15 + 15
= ₹ 61
संबंधित प्रश्न
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
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 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.
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 :
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.
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 :
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.
To convert the assignment problem into a maximization problem, the smallest element in the matrix is deducted from all other elements.
State whether the following is True or False
In number of lines (horizontal on vertical) > order of matrix then we get optimal solution.
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)
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
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
State whether the following statement is true or false:
To convert a maximization-type assignment problem into a minimization problem, the smallest element in the matrix is deducted from all elements of the matrix.
A marketing manager has list of salesmen and territories. Considering the travelling cost of the salesmen and the nature of territory, the marketing manager estimates the total of cost per month (in thousand rupees) for each salesman in each territory. Suppose these amounts are as follows:
| 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 |
Find the assignment of salesman to territories that will result in minimum cost.
To solve the problem of maximization objective, all the elements in the matrix are subtracted from the largest element in the matrix.
Three new machines M1, M2, M3 are to be installed in a machine shop. There are four vacant places A, B, C, D. Due to limited space, machine M2 can not be placed at B. The cost matrix (in hundred rupees) is as follows:
| Machines | Places | |||
| A | B | C | D | |
| M1 | 13 | 10 | 12 | 11 |
| M2 | 15 | - | 13 | 20 |
| M3 | 5 | 7 | 10 | 6 |
Determine the optimum assignment schedule and find the minimum cost.
