In equation 1, we see that the production in station
1 is the same as its capacity. Equation 2 determines the
WIP available at time t as a function of variables
at previous time. Specifically, the WIP available is equal
to the WIP available at the previous time plus any new
production from station 1 less any withdrawals for production
at station 2.
In equation 3, we limit the production for stations 2
through n to the stations capacity or the amount
of material available from WIP or new production. Note
that this equation has the effect that the WIP will never
become negative. Equation 4 is like equation 2, but computes
the WIP for all but the last station of production.
Equation 5 computes the production at the last station.
We include no WIP equation for the last station, because
all units produced leave the system. The production at
the last station is the production for the entire system.
For the equations to have meaning for t = 1, values
for all variables must be specified for t = 0.
There is no upper limit for the value of t, but
normally some maximum T is specified. We call this
the time horizon.
If the capacities of the stations were known for all
times within the time horizon, and we assume initial values
of the variables for time 0, the system of equations defined
above could be easily solved for any finite time horizon.
The values of the WIP variables and production variables
are completely determined by the capacities. Since each
variable at time t depends only on variables at
time t - 1 and selected additional variables at
time t, we can solve the equations for all the
variables for t =1, then solve for the variables
at t = 2, and so on until t = T.
For this solution to be easily obtained the variables
for any given t must be ordered so the value of
one variable depends only on the variables earlier in
the order. For this example the order is:
For any given t, the variables are
easily determined if considered from left to right.