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.
cpj: |
preventive maintenance costs on production line j (in dollars) |
cfj: |
corrective maintenance costs on production line j (in dollars) |
sij: Capj: |
setup costs of product i on production line j (in dollars) capacity available on production line j per period (in time units per period) |
pij: |
processing time of product i on production line j (in time units per ton) |
Rij: Dit: |
production rate of product i on production line j (in tons per period) |
RT: |
length of preventive and corrective maintenance (in periods) |
λjt: |
probability of breakdown on production line j in period t, when the previous |
cjmt: |
expected breakdown cost in period t on production line j, when the previous |
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, φijt 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 , λjt , multiplied by the corrective
maintenance costs, or : cjmt= cfj*P[_xj =t ] = cfj*λjt , with _xj equalling the life of production line j.