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.

cpj:

preventive maintenance costs on production line j (in dollars)

cfj:
h
i:
l
i:

corrective maintenance costs on production line j (in dollars)
inventory costs of product
i per ton per period (in dollars)
backorder costs of product
i per ton per period (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)
demand of product
i in period t (in tons)

RT:

length of preventive and corrective maintenance (in periods)

λjt:

probability of breakdown on production line j in period t, when the previous
maintenance was done in period 0

cjmt:

expected breakdown cost in period t on production line j, when the previous
preventive maintenance job ended in period
m 1

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:
z
jt = (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.



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