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Formulation as a linear programming problem
Because the assignment problem is a special case of the transportation problem, a linear programming formulation can be developed.
We need a decision variable for each arc. For the arc from node I to node j we define:
We obtain the constraints by considering each node in turn:
(i) Since each worker is assigned to one and only one job we have:
for A xA1 â€¡ xA2 â€¡ xA3 Ë† 1 (1a)
for B xB1 â€¡ xB2 â€¡ xB3 Ë† 1 (1b)
for C xC1 â€¡ xC2 â€¡ xC3 Ë† 1 (1c)
(ii) Since each job is assigned to one and only one worker we have:
for 1 xA1 â€¡ xB1 â€¡ xC1 Ë† 1 (2a)
for 2 xA2 â€¡ xB2 â€¡ xC2 Ë† 1 (2b)
for 3 xA3 â€¡ xB3 â€¡ xC3 Ë† 1 (2c)
To obtain the objective function we consider completion times:
for A 17xA1 â€¡ 10xA2 â€¡ 12xA3 (3a)
for B 9xB1 â€¡ 8xB2 â€¡ 10xB3 (3b)
for C14xC1 â€¡ 4xC2 â€¡ 7xC3 (3c)
The sum of the three completion times gives us the total number of days needed to complete the three jobs. The objective is therefore to minimise:
Z Ë† 17xA1 â€¡ 10xA2 â€¡ 12xA3
â€¡ 9xB1 â€¡ 8xB2 â€¡ 10xB3 (4)
â€¡ 14xC1 â€¡ 4xC2 â€¡ 7xC3
The linear programming problem is then: minimise Z, given by equation (4), subject to the set of constraints given by equations (1) and (2). The conditions (1) and (2) ensure that the variables are 0 or 1 and so no non-negativity condition is required.
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