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प्रश्न
Give mathematical form of Assignment problem
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उत्तर
Consider the problem of assigning n jobs to n machines (one job to one machine).
Let Cij be the cost of assigning ith job to the jth machine and xij represents the assignment of ith job to the jth machine.
Then, xij = `{{:(1",", "if" "i"^"th" "job is assigned to" "j"^"th" "machine"),(0",", "if" "i"^"th" "job is assigned to" "j"^"th" "machine"):}`
| Machines | ||||||
| 1 | 2 | … | n | Supply | ||
| 1 | `""^((x_11))"C"_11` | `""^((x_12))"C"_12` | … | `""^((x_(1n)))("C"_(1n))` | 1 | |
| 2 | `""^((x_21))"C"_21` | `""^((x_22))"C"_22` | … | `""^((x_(2n)))("C"_(2n))` | 1 | |
| Jobs | : | : | : | : | 1 | |
| m | `""^((x_"ij"))"C"_("n"1)` | `""^((x_(m2)))"C"_("n"1)` | … | `""^((x_"ij"))("C"_"nn")` | 1 | |
| Demand | b1 | b2 | … | bn | ||
xij is missing in any cell means that no assignment is made between the pair of job and machine.
i.e xij = 0.
xij is presents in any cell means that an assignment is made their.
In such cases xij = 1
The assignment model can written in LPP as follows:
Minimize Z = `sum_("i" = 1)^"m", sum_("j" = 1)^"n" "C"_"ij" "X"_"ij"`
Subject to the constrains
`sum_("i" = 1)^"n" "X"_"ij"` = 1, j = 1, 2, …. n
`sum_("i" = 1)^"n" "X"_"ij"` = 1, i = 1, 2, …. n and xij =0 or 1 for all i, j
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संबंधित प्रश्न
Suggest optimum solution to the following assignment. Problem, also find the total minimum service time.
Service Time ( in hrs.)
| Counters | Salesmen | |||
| A | B | C | D | |
| W | 41 | 72 | 39 | 52 |
| X | 22 | 29 | 49 | 65 |
| Y | 27 | 39 | 60 | 51 |
| Z | 45 | 50 | 48 | 52 |
Solve the following minimal assignment problem :
| Machines | A | B | C | D | E |
| M1 | 27 | 18 | ∞ | 20 | 21 |
| M2 | 31 | 24 | 21 | 12 | 17 |
| M3 | 20 | 17 | 20 | ∞ | 16 |
| M4 | 21 | 28 | 20 | 16 | 27 |
The assignment problem is said to be unbalance if ______
The assignment problem is said to be balanced if ______.
The objective of an assignment problem is to assign ______.
Fill in the blank :
When an assignment problem has more than one solution, then it is _______ optimal solution.
State whether the following statement is True or False:
In assignment problem each worker or machine is assigned only one job
What is the Assignment problem?
A departmental head has four subordinates and four tasks to be performed. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. His estimates of the time each man would take to perform each task is given below:
| Tasks | |||||
| 1 | 2 | 3 | 4 | ||
| Subordinates | P | 8 | 26 | 17 | 11 |
| Q | 13 | 28 | 4 | 26 | |
| R | 38 | 19 | 18 | 15 | |
| S | 9 | 26 | 24 | 10 | |
How should the tasks be allocated to subordinates so as to minimize the total manhours?
A job production unit has four jobs P, Q, R, S which can be manufactured on each of the four machines I, II, III and IV. The processing cost of each job for each machine is given in the following table :
| Job | Machines (Processing cost in ₹) |
|||
| I | II | III | IV | |
| P | 31 | 25 | 33 | 29 |
| Q | 25 | 24 | 23 | 21 |
| R | 19 | 21 | 23 | 24 |
| S | 38 | 36 | 34 | 40 |
Complete the following activity to find the optimal assignment to minimize the total processing cost.
Solution:
Step 1: Subtract the smallest element in each row from every element of it. New assignment matrix is obtained as follows :
| Job | Machines (Processing cost in ₹) |
|||
| I | II | III | IV | |
| P | 6 | 0 | 8 | 4 |
| Q | 4 | 3 | 2 | 0 |
| R | 0 | 2 | 4 | 5 |
| S | 4 | 2 | 0 | 6 |
Step 2: Subtract the smallest element in each column from every element of it. New assignment matrix is obtained as above, because each column in it contains one zero.
Step 3: Draw minimum number of vertical and horizontal lines to cover all zeros:
| Job | Machines (Processing cost in ₹) |
|||
| I | II | III | IV | |
| P | 6 | 0 | 8 | 4 |
| Q | 4 | 3 | 2 | 0 |
| R | 0 | 2 | 4 | 5 |
| S | 4 | 2 | 0 | 6 |
Step 4: From step 3, as the minimum number of straight lines required to cover all zeros in the assignment matrix equals the number of rows/columns. Optimal solution has reached.
Examine the rows one by one starting with the first row with exactly one zero is found. Mark the zero by enclosing it in (`square`), indicating assignment of the job. Cross all the zeros in the same column. This step is shown in the following table :
| Job | Machines (Processing cost in ₹) |
|||
| I | II | III | IV | |
| P | 6 | 0 | 8 | 4 |
| Q | 4 | 3 | 2 | 0 |
| R | 0 | 2 | 4 | 5 |
| S | 4 | 2 | 0 | 6 |
Step 5: It is observed that all the zeros are assigned and each row and each column contains exactly one assignment. Hence, the optimal (minimum) assignment schedule is :
| Job | Machine | Min.cost |
| P | II | `square` |
| Q | `square` | 21 |
| R | I | `square` |
| S | III | 34 |
Hence, total (minimum) processing cost = 25 + 21 + 19 + 34 = ₹`square`
