The Doctor Rostering Problem - Asser Fahrenholz

Chapter 4. Solving the DRP 26
decreases over time. Given, a temperature T , an incumbent value z(S) and a neighbor
value z( S), the probability of accepting a neighborhood, in a minimisation problem, is
commonly chosen to be the Metropolis criterion:
pmin(z(S), z( S), T ) =
1 if z( S) ≤ z(S)
exp( z( S)−z(S)
T ) otherwise
When the algorithm starts and the temperature is still high, the criterion accepts arbi-
trarily large deteriorations in solution value, but as the temperature drops only smaller
deteriorations are accepted and finally, when the temperature reaches zero, no deteri-
orations are accepted. At this point, the SA is essentially a basic, hill climbing (or
descending), LS. The temperature is controlled by some scheme, also implemented in
this project, that can be as simple as T = α · T , where α ∈ [0, 1[. This scheme is a static
scheme, but one can also choose a dynamic scheme, such as one based on the ratio of
accepted solutions.
Algorithm 4.4 outlines the basic SA steps.
Algorithm 4.4 Simulated Annealing - require: Schedule S, start temperature T
S ← S
while ¬Stop do
S ← choose S from N(S)
if z(S) > z( S) then
S ← S
else if exp( z( S)−z(S)
T
S ← S
end if
T ← Update(T )
end while
) > random[0,1) then
The initial value and update of the temperature in each iteration is one of the parameters
that need to be tuned to each problem. In this project I set T = 1000 and α = 0.999.
These values was accepted after due tuning and allows the temperature to drop towards
a defined end-temperature in a steady manner. An example is given in figure C.2 in
appendix C.2.5.
4.3.5 Heuristic bounds
It can be useful to describe the bounds of the search algorithm which is chosen. I do
however feel, that proving the bound of the approximations described above is out of
the scope of this project and I thus leave it for future considerations. Chapter 7 includes
(M)