Model Formulation

Before, explaining the underlying model, given set of notations are required to be described;

• n = number of parts;
• m = no. of machines;
• = no. of operations;
• due_i = due date of part i;
• busy_i = busy time of machine i;
• t_j = machining time of operation j.
• M = A huge positive number;
• C_ij = completion time of operation j of part i.
• A_ij is a decision variable equals 1, if machine I is chosen to perform the operation j, otherwise 0.
• B_ij is a decision variable equals 1, whether operation j belongs to part i, otherwise 0.
• P_ij is a decision variable equals 1, whether operation i precedes the operation j of same part, otherwise 0.
• M_ij is a decision variable equals 1, whether operation 1 precedes operation j on similar machine.

The objective function minimizes total of completion time of each operation

In order to optimize the afore-mentioned objective, several constraints desired to be satisfied, as follows:

                                                C_ij  - t_j  > busy_i ..... Eq(14)

 All the operations can be started after the busy time of the machine to that it is assigned.

                                                 x_ij  - x_{i(j - 1)}  > t_j..... Eq(15)

 For the operations of the similar parts, the completion time of recent operation should be larger than its precedence operation by at least the processing time of the recent operation.

                                              x_ij  > due_i  ∀ j ∈ o  ..... Eq(16)

All the operations of the job should be processed prior to the due date allocated to the job.