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Question
Give mathematical form of Assignment problem
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Solution
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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