A production model and maintenance planning model for the process industry



zjt1-mj,t-1    j=1,...,M, t=1,...,T                           (2)

zjtmjt    j=1,...,M, t=1,...,T                           (3)

zjt(1-mj,t-1)+mjt-1    j=1,...,M, t=1,...,T                    (4)

The constraint sets (2), (3) and (4) also fix zjt at either 0 or 1.

To make sure that no jobs can be planned during a preventive maintenance (which requires RT
periods of time) we need the next constraints to make m
jk one for every
k=t,...,t+RT-1:

zjtmjk    j= 1,...,M, t= 1,...,T, k = t,...,t+RT-1                  (5)

Besides constraints (1) through (5), some other key constraints are needed. A fundamental
material balance equation for each period is stated as:

Production + beginning inventory - ending inventory = demand

Or mathematically:

M Rijδijt+(I+i,t-1-Ii-,t-1)-(I+it-Ii-t)=Dit   i=1,...,N t=1,...,T(6)

j=1

where inventory is defined as Iit+ - Iit- , or positive net inventory minus backorders.

Rij is defined as the capacity of production line j divided by the processing time of product i on
production line
j, or: Rij= Capj / pij.

Maintenance and production cannot be scheduled in the same period on a certain production line.
Constraint (7) prevents this from happening:

N δijt+mjt1 j=1,...,M t = 0,...,T                      (7)

i=1

The variable yjmt must equal one when the last preventive maintenance job on production line j
occurred in period m and we are now in period t. To ensure this, we define yjmt as:

yjmt = (1-mjt)(1-mj,t-1)...(1-mj,m+1 )mjm

This is again nonlinear, so we use the same technique as applied in constraints (2) through (4):

yjmt1-mjk   j=1,..., M, t=1,...,T, k=m+1,...,t                 (8)

yjmtmjm    j=1,..., M, t=1,...,T, m=0,...,t-1                   (9)

t

yjmt(1-mjk)+mjm-(t-m) j=1,..., M, t=1,...,T, m=0,...,t-1      (10)

k=m+1

where yjmt is greater than or equal to 0.

To initialize some of the variables, we introduce a dummy period 0, where the previous
preventive maintenance was performed and where inventory was zero.

mj,0 = 1 j= 1,..., M

(11)


2

The general rule for linearizing expressions like x1...xk is as follows: replace x1...xk with y and add the next three constraints:

1) y<_ xj, 1_< j<_ k

2) y>_ Σkj=1 xj - (k-1)

3) y>_ 0



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