Advertisements
Advertisements
प्रश्न
Give mathematical form of Assignment problem
Advertisements
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
APPEARS IN
संबंधित प्रश्न
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 |
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.
The assignment problem is said to be unbalance if ______
Fill in the blank :
When an assignment problem has more than one solution, then it is _______ optimal solution.
In an assignment problem, if number of column is greater than number of rows, then a dummy column is added.
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?
State whether the following statement is True or False:
The objective of an assignment problem is to assign number of jobs to equal number of persons at maximum cost
Three jobs A, B and C one to be assigned to three machines U, V and W. The processing cost for each job machine combination is shown in the matrix given below. Determine the allocation that minimizes the overall processing cost.
| Machine | ||||
| U | V | W | ||
| Jobs | A | 17 | 25 | 31 |
| B | 10 | 25 | 16 | |
| C | 12 | 14 | 11 | |
(cost is in ₹ per unit)
Find the optimal solution for the assignment problem with the following cost matrix.
| Area | |||||
| 1 | 2 | 3 | 4 | ||
| P | 11 | 17 | 8 | 16 | |
| Salesman | Q | 9 | 7 | 12 | 6 |
| R | 13 | 16 | 15 | 12 | |
| S | 14 | 10 | 12 | 11 | |
Choose the correct alternative:
North – West Corner refers to ______
