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can be used to evaluate summary measures under
various scenarios, such as the probability of an
excessive wait as is relevant for our research, to
construct minimum coverage requirements.
However, the non-stationary nature of
demand for WAST assistance renders the queueing
model analytically intractable, i.e. there are no
closed-form expressions by which one can evaluate
various performance metrics over time, so both
quick approximation techniques and more
computationally expensive numerical methods have
been developed to adequately deal with time-
dependent and priority demand, and ultimately
produce minimum hourly coverage requirements
that satisfy the response time targets. The most basic
type of analysis can be achieved using a SIPP
(Stationary Independent Period by Period)
approximation (see for example Green et al, 2001).
SIPP estimates the time-dependent behaviour by
first segmenting the operation period into distinct
shifts, and finding the average arrival rate in each
shift. Then treating each shift independently and
assuming the system settles down to steady-state
(operates at a consistent level) within each period,
closed-form formulae can be used to calculate the
number of staff required for each shift and match the
coverage requirements to the demand levels.
Recognising that the approximation method
requires many assumptions and fails to account for
the dependency that exists between periods, we have
also investigated a numerical method which
produces accurate estimates at the expense of
computation speed by accurately tracking the
movement of customers through the service system
using a set of differential-difference equations to
predict the number of patients awaiting and
receiving assistance at all time points. Balancing the
ability of the approximation method to provide
rough solutions rapidly, and the advantage of the
numerical method to produce accurate predictions at
the expense of computation speed, the ultimate
methodology we have proposed to WAST is a novel
hybrid method which employs both methods to
produce coverage requirements (Vile et al, 2013).
3.3 Scheduling and Rostering
Finally, with the minimum hourly coverage
requirements produced, we have investigated shift
scheduling and rostering techniques that can be used
to optimise the shift pattern and assign staff to shifts.
Both problems have been formulated in terms of
integer linear programs, which may be incorporated
as part of the capacity planning tool, and solved
using exact methods and heuristic search techniques
(using random descent and simulated annealing).
The heuristic method is helpful for inclusion in the
developed spreadsheet tool in case no commercial IP
solvers are available, such as in use by WAST.
Various objectives can be chosen to construct
optimised functions, such as minimising the total
labour hours used or crew size, and any number of
constraints can be added to the model to develop
potential schedules. Whilst the shift schedule may be
optimised prior to the application of a rostering
model, our research has acknowledged the benefit in
simultaneously constructing the shift schedule and
roster, due to complex working time directives
which can prevent crews from working certain shift
patterns of the optimised shift schedule.
The workforce capacity planning and
scheduling tool which amalgamates all of the above
techniques into a single integrated model, has been
designed with a user-friendly interface with
parameters that may be flexibly adjusted by the user
to provide staffing recommendations for various
scenarios that satisfy the response time targets.
While taking into account the importance of
accurately estimating future demand, the need to
develop OR methodology to evaluate service quality
in time-dependent priority multi-server systems, and
generate efficient shift schedules, the tool:
a) Incorporates time-series methods that
adequately account for the stochastic nature of
demand to produce accurate forecasts of future
demand;
b) Provides both accurate and approximate
evaluations of system performance over time;
c) Permits a certain service quality to be met as
inexpensively as possible by generating an
efficient staffing function that accurately
matches resources to fluctuating demand levels;
d) Assigns staff to shifts in an efficient manner,
whilst adhering to governmental regulations and
working time directives;
e) Is user-friendly and practical; so it could be
used to inform WAST staffing decisions and
readily adopted by planners to optimise
resources independently.
4 AMBULANCE LOCATION
AND DEPLOYMENT
We now turn our attention to location and
deployment of the vehicles. An EMS system can be
thought of as a priority queueing system with arrival
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