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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
Solve the following minimal assignment problem and hence find minimum time where '- ' indicates that job cannot be assigned to the machine :
| Machines | Processing time in hours | ||||
| 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 |
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.
Choose the correct alternative:
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State whether the following statement is True or False:
In assignment problem each worker or machine is assigned only one job
Choose the correct alternative:
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| a | b | c | d | e | |
| A | 160 | 130 | 175 | 190 | 200 |
| B | 135 | 120 | 130 | 160 | 175 |
| C | 140 | 110 | 155 | 170 | 185 |
| D | 50 | 50 | 80 | 80 | 110 |
| E | 55 | 35 | 70 | 80 | 105 |
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| 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 |
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| 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 |
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| Workers | Packing of | |||
| Books | Toys | Crockery | Cutlery | |
| A | 3 | 11 | 10 | 8 |
| B | 13 | 2 | 12 | 12 |
| C | 3 | 4 | 6 | 1 |
| D | 4 | 15 | 4 | 9 |
