A production model and maintenance planning model for the process industry

0 otherwise

I⁺it:    Net inventory of product i at the end of period t (in tons of product)

I^-it:     Backorders of product i at the end of period t (in tons of product)

The variable yjmt is needed, because when preventive maintenance is performed, the production
line is assumed to be as good as new, so that the possibility of a breakdown solely depends on the
last period in which maintenance was performed.

The parameters of the model are listed in table 2.

Table 2: Parameters Used in the Production and Maintenance Planning Model

The constraints:

Although a large number of binary variables are defined, only δijt and mjt need to be constrained
to 0 or 1. The remaining variables may be uniquely defined 0 or 1 through the use of other
constraints. For example, φ_ij_t is uniquely determined by the following constraint:

φijt≥δijt-δij,t-1 i=1,..., N, j=1,..., M, t=1,...,T                       (1)

Note that, if δijt=1 and δij,t-1=0 (product i is produced in period t but not in period t-1), φijt=1, as it
will never be greater than 1 because of its positive objective coefficient. Similar arguments apply
when δijt=0 and δij,t-1=1. When both δijt and δij,t-1 are zero or one, the positive objective coefficient
of φijt will make sure that φijt is zero.

The variable zjt denotes the period in which a preventive maintenance job starts, and defined as:
zjt = (1 - mj,t-1 )mjt . This is a nonlinear expression, but can be solved by introducing the following
three constraints²:

Expected breakdown cost in period t on production line j, where the previous preventive maintenance job
ended in period m, is calculated as the probability of a breakdown in period t , λ_jt , multiplied by the corrective
maintenance costs, or : c_jmt= cf_j*P[_x_j =t ] = cf_j*λ_jt , with _x_j equalling the life of production line j.

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