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Routing strategies in unreliable production networks are an essential tool to meet given demands and to avoid high inventory levels. Therefore, we are interested in studying stateindependent and state-dependent control policies to maximize the total throughput of the production network. Different from M/M/1 queuing theory, the underlying model is based on partial and ordinary differential equations with random breakdowns capturing the time-varying behavior of the system. The key idea is to numerically compare suitable routing strategies with results computed by nonlinear optimization techniques. We comment on the efficiency of the proposed methods and their qualitative behavior as well.
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ROUTING STRATEGIES IN PRODUCTION NETWORKS WITH
RANDOM BREAKDOWNS
SIMONE G ¨
OTTLICHAND SEBASTIAN K ¨
UHN
Abstract. Routing strategies in unreliable production networks are an essential tool to meet given
demands and to avoid high inventory levels. Therefore we are interested in studying state-independent
and state-dependent control policies to maximize the total throughput of the production network.
Different to M/M/1 queuing theory the underlying model is based on partial and ordinary differential
equations with random breakdowns capturing the time varying behavior of the system. The key idea
is to numerically compare suitable routing strategies with results computed by nonlinear optimization
techniques. We comment on the efficiency of the proposed methods and their qualitative behavior as
well.
Key words. Production networks, differential equations, random breakdowns, routing strategies,
optimal control.
AMS subject classifications. 90B15, 65Mxx, 90C30
1. Introduction
Continuous models for the modeling, simulation and optimization of production
networks has become an important research field during the last decades. In contrast to
widely used models based on discrete optimization approaches [30, 33], discrete event
simulations [3, 26] or queuing theory [4, 8], continuous models allow for a detailed
time-dependent description of the production process using quantities such as the part
density or the flow of goods [5, 10, 11, 12, 13].
Time continuous network models of serial networks have been introduced in [2] for
the first time. Therein, the authors rigorously derived a differential equation, namely a
conservation law, for the part density from a discrete event simulation. In [19, 20], this
model has been reformulated by installing buffer of infinite size in front of each individ-
ual processors. So far, these models have been mostly considered from the deterministic
point of view, but it is possible to include stochastic effects in a straightforward way. For
instance, under certain assumptions for the availability of processors, averaged densities
can be either computed analytically [16] or numerically [22] using Monte-Carlo simula-
tions. In both approaches, random breakdowns of processors are modeled as capacity
drops at exponentially distributed points in time. We briefly describe the coupling of
the stochastic process to the dynamics of the production system in Section 2.
For optimization purposes, the computation of the maximal throughput or the
minimal buffer loads are of main interest. There exists a broad variety of literature
related to this topic with focus on the optimal routing of goods or cars [20, 24, 28], inflow
optimization [14] or demand tracking [25]. However, the combination of continuous
randomly perturbed production models and mathematical optimization issues has been
less investigated yet. In other words, the challenge we face here is the optimal control
of a nonlinear stochastic model relying on differential equations. That means we need
This work was financially supported by the DAAD project “Transport network modeling and
analysis” (Project-ID 57049018) and by the BMBF project KinOpt. Special thanks go to Stephan
Martin, Thorsten Sickenberger for fruitful discussions and Markus Erbrich for his help in generating
sample scenarios.
University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany,
email(goettlich@math.uni-mannheim.de)
University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany,
email(kuehn@math.uni-mannheim.de)
1
2PRODUCTION NETWORKS WITH RANDOM BREAKDOWNS
to think about suitable optimization strategies and algorithms as well. We emphasize
on different solution approaches for the optimal routing problem, where the overall
goal is to efficiently distribute goods through the system to achieve high throughputs.
Major applications for the optimal routing problem are for example packets flow on
data networks [7] or traffic flow on road networks [6, 9, 15, 23, 28].
In this work, our contribution will follow two central ideas. Due to the complexity
of our modeling approach, a detailed analytical study of the routing problem is hardly
possible. Therefore, we stick in our investigations to a numerical study and propose
routing strategies (or policies) in a heuristic manner on the one hand and optimal so-
lutions obtained by nonlinear optimization on the other hand. The control strategies
may either depend on the current state of the system or not. In this way we are able to
include the time varying behavior of the system more precisely. We still see in Section 4
how these additional information will influence the system optimum. Similar ideas can
be found for example in queuing theory, where a variety of literature related to rout-
ing decisions exists, see [1, 27, 31, 32] for an overview. However, these techniques do
not directly apply to our approach due to the fluctuations resulting from the random
breakdowns of processors. Motivated by queuing theory, we develop problem-adapted
routing strategies and approximate expectations of the system using a large number
of Monte-Carlo runs. To assess the impact of the results achieved we present an algo-
rithm to solve the stochastic control problem directly. The latter can be interpreted
as an optimization model restricted by differential equations. It is numerically solved
using a rolling time horizon approach to really include all occurring random failures,
cf. Section 3. This is non-standard, computationally very costly and often gets stuck
at local approximations. To remedy this drawback heuristic routing strategies offer an
alternative and less expensive way to approximate or even reach a system optimum, see
computational experiments in Section 4. From a numerical point of view we try to find
the most suitable strategy to reach high outputs and low buffer loads while taking also
into account the network topology and different arrival rates.
2. Modeling of production networks and routing strategies
In this section we briefly discuss a mathematical model to describe the flow of
goods in production networks with random breakdowns of processors originally intro-
duced in [22]. Here, breakdowns are modeled by a two-state process with exponentially
distributed switching times between on-and-off states. We also present several routing
strategies or policies to distribute the product flow through the system. We mainly
distinguish between two types of strategies: state-independent and state-dependent
policies.
2.1. Stochastic network model with random breakdowns
The modeling and numerical simulation of a stochastic time-dependent production
model including random breakdowns is presented in [22]. In this work, this model is
used to describe the fundamental dynamic behavior and coupled to routing strategies
or control policies, respectively.
To introduce the model, we first set a couple of notations. With (V,A) we denote
a directed graph consisting of a set of arcs Aand a set of vertices Vand define N=|V|,
M=|A|. For any fixed vertex v∈ V, the set of ingoing arcs is denoted by δ
vand the set
of outgoing arcs by δ+
v, see Figure 2.3. Each processor is represented by an arc e∈ A
with an associated queue or buffer in front of it. We assume that each processor has
a non-physical length, the so-called degree of completion described by the variable x.
The degree of completion is normed to the unit interval [0,1],where x= 0 indicates the
entering and x=1 the exiting of parts. A vertex v∈ V without any predecessor represents
S. G ¨
OTTLICH AND S. K ¨
UHN 3
an inflow point to the production network. We denote the set of all these vertices by
Vin ={v∈ V | |δ
v|= 0}. The time-varying influx is externally given and denoted by Gv
in(t)
for all v∈ Vin. Accordingly, we define Vout ={v∈ V | |δ+
v|= 0}as the set of all vertices,
where goods leave the production network. Furthermore, let s:A→V map an arc onto
its vertex of origin.
The considered time span is [0, T ]. We assume that processors may breakdown
eventually and get restarted again within the time horizon T. Following [22], we define
a two-state stochastic process
re:R0×→ {0,1}
t×ω7−re(t,ω)(2.1)
for each processor e∈ A indicating whether the processor is on, i.e. re(t, ω) = 1, or off,
i.e. re(t,ω) = 0. Intermediate states are not possible, see Figure 2.1. Furthermore, we
initialize the states of the processors by
re(0,ω) = re
0,(2.2)
where we usually choose all processors being on, i.e. re
0= 1.
t
re(t,ω)
tt+τe
0
1
Fig. 2.1: Realization of a two-state-process (2.1).
The state process redepends both on the time tand the random sample ωΩ.
Thus for a fixed time t0, re(t,·) is a binary random variable, whereas for a fixed
random sample ωΩ, re(·,ω) is a realization of the state process. We call a change
in the state re(t,ω) of a processor e∈ A at time taswitching. To model these, we
assume that the switchings are independent of the queue load, the load of the processor
and the state of other processors. This allows us to introduce the mean time between
failures (MTBF) τe
on and the mean repair time (MRT) τe
off for each processor e. The
former describes the mean time for a switching from re= 1 to re= 0, while the latter
defines the mean time for a processor being broken, i.e. re= 0, before switching back to
operating, i.e. re= 1. Then, for each processor, the time ∆τebetween two switchings
at tand t+τeis chosen randomly from the exponential distribution with density
function Exp(t;λ) and the rate parameter
λ=λ(re(t,ω)) = (1 e
on if re(t,ω)=1,
1e
off if re(t,ω)=0.(2.3)
4PRODUCTION NETWORKS WITH RANDOM BREAKDOWNS
Having the modeling of breakdowns at hand, we can introduce the stochastic pro-
duction network model as follows. We assume that each processor e∈ A works with
a constant velocity veand has a maximal processing rate µemeasured in parts per
unit time. The density of products ρe(x,t, ω) is governed by the continuity equation
x[0,1], t0, ω
tρe(x,t,ω) + xfe(ρe(x,t,ω)) = 0, ρ(x,0,ω) = ρe
0(x),(2.4a)
where the flux function feis given by
fe(ρe) = minnve·ρe(x,t,ω),µe·re(t,ω)o.(2.4b)
This means in particular, if the processor eis not broken, i.e. re= 1, the density of
goods ρe(x, t, ω) is transported with velocity veand the flux is less or equal than the
maximal processing rate µe. On the other hand, if the processor eis broken, i.e. re= 0,
no goods are processed at all and the flux is zero.
Each processor ehas the possibility to store goods that cannot be processed imme-
diately in a queue qe(t,ω), see Figure 2.2.
vtρe+xfe(ρe)=0
qe
ge
in ge
out
Fig. 2.2: A processing unit is composed of an ordinary differential equation describ-
ing the load of a queue qecoupled to the dynamics of the processor governed by a
conservation law.
The inflow to this queue is denoted by the function ge
in(t, ω) and the outflow of
the queue by the function ge
out(t, ω). The dynamics of the queue are determined by
the difference of its inflow ge
in(t, ω) and its outflow ge
out(t, ω). Thus, the load of the
queue qe(t,ω) is given by the rate equation
tqe(t,ω) = ge
in(t, ω)ge
out(t, ω), qe(0,ω) = qe
0.(2.5)
For the inflow to the queue ge
in(t, ω) we remark that if the origin s(e) of proces-
sor eis a inflow point to the network, i.e. s(e)∈ Vin , the inflow is given by the inflow
function Gv
in(t). On the other hand, if the origin of processor eis an inner vertex, i.e.
s(e)/∈ Vin, the inflow is given by the sum of all incoming flows multiplied by the distri-
bution or routing parameter As(e),e(t), which describes the percentage of flow sent to
processor e, cf. Figure 2.3. Routing parameters are a degree of freedom in simulation
models and will be the controls for optimization purposes in Section 3.
For a vertex vand any outgoing processor eδ+
vthe routing parameters Av,e are
defined as follows.
Definition 2.1 (Distribution rates). For any vertex v∈ V with |δ+
v| 6=and any
processor eδ+
vthe distribution rate Av,e(t)should fulfill the two conditions for all t0:
S. G ¨
OTTLICH AND S. K ¨
UHN 5
δ(v)
v
1
3
2
δ+(v)
Av,5
5
Av,4
4
Fig. 2.3: Illustrations of δ±
vas well as of the distribution rates Av,4(t) and Av,5(t) =
1Av,4, respectively.
(i) 0Av,e(t)1, and
(ii) Peδ+
vAv,e(t)=1.
Now, we are able to replace the inflow ge
in(t, ω) in equation (2.5) by
ge
in(t, ω) =
As(e),e(t)P
¯eδ
s(e)
f¯e(ρ¯e(1,t,ω)) if s(e)/∈ Vin ,
Gs(e)
in (t) if s(e)∈ Vin .
(2.6)
The outflow ge
out(t, ω) appearing in (2.5) can also be specified: If the queue qe(t,ω)
is empty, the outflow equals the minimum of the ingoing flow ge
in(t, ω) and the maximal
capacity µe·re(t,ω) of the processor. If the queue is filled, the queue is reduced with
maximal capacity of the processor. This yields
ge
out(t, ω) = (min{ge
in(t, ω),µe·re(t,ω)}if qe(t)=0,
µe·re(t,ω) if qe(t)>0.(2.7)
Summarizing, the stochastic simulation network model is given by the equations:
n(2.2),(2.4),(2.5),(2.6),(2.7).(2.8)
Note that the stochastic network model (2.8) also captures a deterministic pro-
duction network. Therefore we initialize all processors e∈ A with states re
0= 1 and set
τe
on =. Consequently, the processors do not switch and keep operating for the whole
time. This fact will be used in the numerical experiments in Section 4 later on.
2.2. Routing strategies
A crucial point in production network models is the distribution of incoming goods
among the outgoing processors, cf. (2.5) and (2.6). In the following, we introduce two
different types of control strategies for the simulation model (2.8) that are compared to
solutions obtained by solving an optimization model in Section 4. In our approach, we
distinguish between state-independent (shortly s-i) and state-dependent (shortly s-d)
routing strategies that take the current states of the processors into account. Certainly,
all strategies, or distribution rates respectively, need to fulfill the properties stated in
Definition 2.1. In the following we will write αs(e),e instead of As(e),e (t) whenever the
distribution rates are time-independent.
6PRODUCTION NETWORKS WITH RANDOM BREAKDOWNS
2.2.1. State-independent strategies
ottlich et al. [22] consider a very intuitive way to control the incoming flow among
outgoing processors, i.e. the flow is equally distributed according to the number of
outgoing processors. We call this strategy the s-i uniform control strategy.
Definition 2.2 (s-i uniform control). Let v∈ V be an inner vertex and eδ+
v, i.e.
v=s(e)the ongoing processors. Then, we define the s-i uniform control by
αs(e),e
uniform =1
|δ+
s(e)|.(1)
Note that the s-i uniform control (1) is time-independent since δ+
vonly depends on the
topology of the network.
Sticking to topology dependent and time-independent constant controls, we define
the s-i capacity control strategy. Here, the distribution rates are proportional to the
maximal capacities µeof outgoing processors.
Definition 2.3 (s-i capacity control). Let v∈ V be an inner vertex and eδ+
vthe
ongoing processors with capacity µe. Then, we define the s-i capacity control by
αs(e),e
capacity =µe
P
¯eδ+
s(e)
µ¯e.(2)
These rates are again time-independent, because the maximal capacities µeare constant
parameters and will not change in time.
In the presence of random breakdowns of processors, the mean availability of each
processor is another important characteristic that should be included in the routing
strategy. For a single processor eEthe mean availability can be computed by
τe
mean =τe
on
τe
on +τe
off
,
where τe
on >0 is the mean time between failures and τe
off >0 is the mean repair time, cf.
Figure 2.1. For a processor e, which is not considered to break down, we set τe
mean = 1.
Including the mean availability, we end up with a more elaborate routing strategy taking
both the capacity and the availability into account.
Definition 2.4 (s-i availability control). Let v∈ V be an inner vertex and eδ+
v
the ongoing processors with the capacity µeand mean availability τe
mean. Then, we define
the s-i availability control by
αs(e),e
availability =µe·τe
mean
P
¯eδ+
s(e)
µ¯e·τ¯e
mean
.(3)
Note that Strategy (3) is also time-indepedent, because the mean availabilities do not
change in time.
Another state-independent, but now time-dependent, strategy might additionally
depend on the queue load. Let the inverse relative queue load of a processor eEbe
defined by
qe
rel(t) = (µe/qe(t) if qe(t)> µe,
1 else ,
S. G ¨
OTTLICH AND S. K ¨
UHN 7
where qe(t) is the current queue load and µethe maximum capacity. This leads to
the property 0 < q e
rel(t)1 since all processors have positive maximal capacities µe>0.
So the relative queue load qe
rel(t) can be seen as the percentage of queue load which is
cleared from the queue in one time step. Clearly, the lower the value of qe
rel(t) gets, the
worse is the routing of goods.
This gives rise to another routing strategy extending the former state-independent
strategies by relative queue loads.
Definition 2.5 (s-i queuing control). Let v∈ V be an inner vertex and eδ+
v
the ongoing processors with the capacity µe, mean availability τe
mean and relative queue
load qe
rel(t)for t0. Then, we define the s-i queuing control by
As(e),e
queue(t) = µe·τe
mean ·qe
rel(t)
P
¯eδ+
s(e)
µ¯e·τ¯e
mean ·q¯e
rel(t).(4)
Note that the distribution strategy (4) is time-dependent but still independent from
the state of processors.
2.2.2. State-dependent strategies
State-dependent strategies compared to state-independent strategies are mainly
concerned with random breakdown scenarios. For instance, if a processor is broken
at time t, i.e., re(t) = 0, goods should be no longer fed into this processor since other-
wise they pile up and the corresponding queue starts to increase. For this reason, we are
now interested in state-dependent (or s-d) strategies that do not distribute goods into
broken processors unless all ongoing processors are broken. This is done by converting
state-independent (Section 2.2.1) into state-dependent strategies. We start with the
uniform control.
Definition 2.6 (s-d uniform control). Let v∈ V be an inner vertex and eδ+
vthe
ongoing processors with state re(t)for t0. Then, we define the s-d uniform control by
As(e),e
uniform(t) =
1/P
¯eδ+
s(e)
r¯e(t)if P
¯eδ+
s(e)
r¯e(t)>0,
1/|δ+
s(e)|if P
¯eδ+
s(e)
r¯e(t)=0.(5)
Here, in the first case the sum indicates the number of processors eδ+
s(e)which are
operating at time t0.
In the same way we design state-dependent counterparts of (2), (3) and (4). Ac-
cording to the s-d uniform control (5), we introduce a second case that resembles the
state-independent control if all ongoing processors are down.
Definition 2.7 (s-d capacity control). Let v∈ V be an inner vertex and eδ+
vthe
ongoing processors with the capacity µeand state re(t)for t0. Then, we define the
s-d capacity control by
As(e),e
capacity (t) =
µe·re(t)/P
¯eδ+
s(e)µ¯e·r¯e(t)if P
¯eδ+
s(e)
r¯e(t)>0,
µe/P
¯eδ+
s(e)
µ¯eif P
¯eδ+
s(e)
r¯e(t)=0.(6)
8PRODUCTION NETWORKS WITH RANDOM BREAKDOWNS
Definition 2.8 (s-d availability control). Let v∈ V be an inner vertex and eδ+
v
the ongoing processors with the capacity µe, mean availability τe
mean and state re(t)for
t0. Then, we define the s-d availability control by
As(e),e
availability(t) =
µe·τe
mean ·re(t)/P
¯eδ+
s(e)µ¯e·τ¯e
mean ·r¯e(t)if P
¯eδ+
s(e)
r¯e(t)>0,
µe·τe
mean/P
¯eδ+
s(e)µ¯e·τ¯e
meanif P
¯eδ+
s(e)
r¯e(t)=0.(7)
Definition 2.9 (s-d queuing control). Let v∈ V be an inner vertex and eδ+
v
the ongoing processors with the capacity µeand mean availability τe
mean. Furthermore,
let qe
rel(t)be the relative queue load and re(t)the state for each processor eδ+
vand
t0. Then, we define the s-d queuing control by
As(e),e
queuing(t) =
µe·τe
mean ·qe
rel(t)·re(t)/P
¯eδ+
s(e)
µ¯e·τ¯e
mean ·q¯e
rel(t)·r¯e(t)if P
¯eδ+
s(e)
r¯e(t)>0,
µe·τe
mean ·qe
rel(t)/P
¯eδ+
s(e)
µ¯e·τ¯e
mean ·q¯e
rel(t)if P
¯eδ+
s(e)
r¯e(t) = 0 .
(8)
Finally, additional to the state-dependent strategies (5) – (8), we consider the
following non-trivial adaption. The idea is to allocate goods also to an in-operating
processor as long as the relative queue load is smaller than a constant c. If this processor
is on again it can start directly at its maximum capacity which might be higher than its
average workload. We expect that this strategy helps to regain the lost time when the
processor was off. Therefore, we intend to distribute the flow among processors ethat
are in progress, i.e. re(t) = 1, and have a relative queue load of at least c, i.e. qe
rel > c (cf.
cases 2 and 3 in (9)). In the case where all processors are down or the relative queue
load is smaller than c(case 1 in (9)), we distribute the flow according to Definition 2.5.
Definition 2.10 (advanced s-d control). Let v∈ V be an inner vertex and eδ+
v
the ongoing processors with the capacity µeand mean availability τe
mean. Furthermore,
let qe
rel(t)be the relative queue load and re(t)the state for each processor eδ+
vand
t0. Then, we define the advanced s-d control with the user defined constant 0c1
by
As(e),e
adv (t) =
As(e),e
queue(t)if eδ+
s(e):re(t)=0 or qe
rel(t)< c ,
µe·τe
mean·qe
rel(t)
P¯eδ+
s(e),q¯e
rel(t)>c,r¯e(t)=1(µ¯e·τ¯e
mean·q¯e
rel(t))if re(t)=1 and qe
rel(t)> c ,
0else.
(9)
We remark that the the parameter 0 c1 switches between state-dependent and
independent controls. For instance, c=0 reproduces the s-d queuing control (8) and c=
1 the s-i queuing control (4). Preliminary tests have shown that for the choice c=1/2 the
results of stratgey (9) lie in-between the results of the bounding strategies (4) and (8),
with a tendency to the better performing one. For that reason, the computations in
Section 4 are performed using c= 1/2.
S. G ¨
OTTLICH AND S. K ¨
UHN 9
2.3. Numerical solution method
We remark that the stochastic network model (2.8) is piecewise deterministic, i.e.
there occurs no further stochasticity between the switching points. This property is in
particular used to solve the model (2.8) numerically while following the idea of Gille-
spie [17, 18] who first studied piecewise deterministic processes (PDPs). The proposed
Stochastic Simulation Algorithm (SSA) has been successfully adapted to production
networks with random breakdowns [22] and unreliable flow lines [21]. For our simula-
tion purposes, we use the version from [22] and modify Algorithm 4.2 therein to fit to
our routing strategies defined above (cf. Algorithm 1).
Algorithm 1 Pseudocode: adapted stochastic sampling algorithm
Input: [0, T ] real interval. Initial data for t= 0. Strategy for distribution rates
Output: Simulation of the production network for one realization ωof model (2.8) on
[0, T ].
1: while t
i< T do
2: Sample next switching point t
i+1.
3: Compute solution in the interval t
i,t
i+1using Euler steps for the updates of
the queues and an upwind scheme for the update on the arcs. The distribution
rates at the vertices are given by the predefined strategy.
4: Set i=i+1.
5: end while
3. An approximate optimization algorithm
We intend to compare the distribution strategies discussed in Section 2.2 to approx-
imate solutions to the following optimization problem:
max
Av,e /min
Av,e J(ρ)
s.t. (2.8), (3.1)
where we consider two different objective functions. On the one hand the approximate
maximization of throughput corresponds to the approximate maximization of flow at
the end of the processors eEout leading to a sink, i.e.
max
Av,e J(ρ) = Zt
i+τhor
t
iX
eEout
fe(ρe(1,t,ω)) dt. (10)
This choice of objective function only aims at high outputs at exiting arcs but disregards
the queue loads inside the network. To tackle this problem we consider as a second
objective, namely the approximate minimization of queues
min
Av,e J(ρ) = X
eE
1
2Zt
i+τhor
t
iqe(t)2dt. (11)
This type of objective minimizes not only the total amount of stored goods but also
maximizes the throughput at the same time. We choose a quadratic formulation in (11)
as in [20] to achieve better convergence.
Apparently, the problem (3.1) is constrained by differential equations, i.e. the
stochastic production model (2.8), and the controls to be determined are the distri-
bution rates Av,e.
10 PRODUCTION NETWORKS WITH RANDOM BREAKDOWNS
As we have seen in Section 2, the model (2.8) is accompanied by stochastic pro-
cesses, i.e. the random failures of processors, and therefore the optimization model (3.1)
is also stochastic. To exploit the underlying piecewise deterministic structure of the
model equations (2.8), we solve the control problem on a rolling time horizon. That
means, at each switching point t
iwe fix the states reof the processors efor the time
interval [t
i,t
i+τhor], where τhor >0 is a prediction parameter defining the length of the
considered time horizon. For this time interval we optimize the objective function lead-
ing to constant distribution rates in the interval [t
i,t
i+τhor]. The utilized numerical
algorithm to solve the optimization problem (3.1) is the MATLAB routine fmincon [29].
We repeat this procedure for each switching point t
ito obtain a solution on [0, T ] (see
Algorithms 2, 3). Consequently, this leads to an approximate optimization algorithm,
as the computational solution depends on the prediction parameter τhor.
Algorithm 2 Pseudocode: rolling time horizon
Input: [0, T ] real interval. Initial data for t= 0.
Output: Distribution rates for one realization ωof model (2.8) on [0, T ].
1: while t
i< T do
2: Sample next switching point t
i+1.
3: Compute solution in the interval [ t
i,t
i+τhor) using Algorithm 3.
4: Set i=i+1.
5: end while
Algorithm 3 Pseudocode: approximate solution for one time interval
Input: t
isampled switching times; τhor time horizon; re(t
i) states of the processors e
E;ρe(t) and qe(t) for 0 tt
iand for all processors eE.
Output: Av,e(t), ρe(t) and qe(t) for 0 tt
i+τhor and for all eE.
1: Fix states ¯re(t)= re(t
i) for tt
i.
2: Set αv,e
0=Av,e(t
i) as initial distribution rates.
3: Apply fmincon to solve (3.1) for αv,e
opt within [t
i,t
i+τhor] with re= ¯reusing αv,e
0as
initial rates.
4: Set Av,e(t) = αv,e
opt for ¯
t
itt
i+τhor.
Note that the time horizon τhor should be chosen large enough in order to provide
reliable results. If τhor is chosen too small, i.e. smaller than the longest path from a
source to a sink, no flow is distributed along this path. Furthermore τhor must be larger
than the mean time between the switchings to ensure the definition of Av,e(t) for all t
(see Algorithm 3, Step 4).
4. Numerical results
In this section we present the numerical results comparing the different routing
heuristics and the approximate optimization procedures proposed in Section 2.2 and 3,
respectively. We start with a qualitative study of different network topologies since the
routing highly depends on the underlying geometry and parameter configuration. To
cover reasonable settings we distinguish between three different types of networks: the
diamond network, the cascade network and a non-symmetric network. They are typi-
cally characterized by their size and capacity allocation. Another crucial ingredient in
our experiments is the choice of the arrival rates (or inflow functions). For each network
sample we consider the following two scenarios: The first inflow function is constant
S. G ¨
OTTLICH AND S. K ¨
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for all times and fixed to 80% of the network’s deterministic bottleneck capacity µeb,
whereas the second inflow function repeats a cycle of delivering 100% for 30 time steps
and stopping the inflow for 10 time steps for five times. Consequently we refer to the
two functions as constant inflow and stop-go inflow, respectively. The constant inflow is
used to analyze how the proposed routing strategies behave for non-time varying filling
of the system. In contrast, the stop-go inflow is highly fluctuating to see how the routing
strategies react and respond on variations in time, cf. Figures 4.1 and 4.2. Note, that
the total constant inflow RT
0Gv1
in (t) dt= 0.8µeb·Tis 5% larger than the total stop-go
inflow RT
0Gv1
in (t) dt= 0.75µeb·T.
t
Gv1
in(t)
80 ·· ·
0
0.8·µeb
Fig. 4.1: Constant inflow profile.
t
Gv1
in(t)
30 40 60 80 ·· ·
0
µeb
Fig. 4.2: Stop-go inflow profile.
We close the section with a comparison of computation times, where we particularly
analyze the influence of the network size and its parameters. To fairly compare the
running times, all simulations and optimizations have been performed on a PC equipped
with 32GB Ram, Intel(R) Xeon(R) CPU E3-1280 @ 3.60GHz.
For our investigations, we use a spatial discretization of ∆x= 1/9 to solve the
model (2.8) and (3.1) as well. Setting the velocities to ve=1 for all processors, we
end up with the step size ∆t= 1/9 due to the CFL condition ∆tx. For all test cases
we consider a total time horizon of T= 200.
4.1. Diamond network The first network to be considered is the so-called
diamond network consisting of 8 arcs, see Figure 4.3. Here, the capacities of each pro-
cessor eare illustrated together with its index. We also color-coded the availability of the
processors. The green arcs represent deterministic processors with availability τe
mean = 1.
We marked the processors (arcs) slightly failing, i.e. with an availability of τe
mean = 95%,
in yellow and the ones most prone to failure (availability τe
mean = 75%) in red (cf. also
Table 4.1). The diamond network consists of two vertices, namely v3and v4, where the
flux has to be controlled according to a routing policy or approximate optimization.
Table 4.1: Parameters for the diamond network.
processors parameters
type indices availability τe
mean mean up τe
on mean down τe
off
green 1,8 1 0
yellow 2,4,6,7 0.95 47.5 2.5
red 3,5 0.75 30 10
First, we present the results for the constant influx, i.e., we consider for all times t
Gv1
in (t) = 32 (4.1)
12 PRODUCTION NETWORKS WITH RANDOM BREAKDOWNS
v1v2v3
v4
v5
v6v7
1,40 2,40
3,30
4,20
5,20
6,10
7,30
8,40
e,µe= index, capacity
Fig. 4.3: The diamond network. The values at the arcs describe the capacities of
the processors. Processors with the same configuration share the same color.
at processor e= 1. This equals 80% of the networks deterministic bottleneck capacity
(cf. Figure 4.1).
In Figure 4.4, the total outflow of the network for one sample ω
ZT
0X
eEout
fe(ρe(1,t,ω)) dt(4.2)
at T= 200 is shown for all nine heuristic routing strategies (blue squares, strategies (1)–
(9)) and the two approximate optimization approaches (white squares, strategies (10),
(11)). To improve readability, we use this notation throughout this section. Further-
more, we provide results for the deterministic model (red dots) as a benchmark solution
for the aforementioned computations. In this way, we directly observe the influence of
the stochasticity on the single strategies.
The sum of all queues within the network over the whole time horizon and for
sample ω
M
X
e=1 ZT
0
qe(t,ω) dt(4.3)
is presented in Figure 4.5 using the same order as above. The maximal queue length
max
eEmax
0tTqe(t,ω) (4.4)
arising at one processor within the network is presented in Table 4.2.
S. G ¨
OTTLICH AND S. K ¨
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optimization
routing strategies
deterministic
12 3 4 5 6 7 8 9 10 11
5600
5700
5800
5900
6000
6100
6200
6300
Fig. 4.4: Total outflow of the diamond
network at T=200 for all control strate-
gies and constant inflow.
optimization
routing strategies
deterministic
1 2 3 4 5 6 7 8 9 10 11
0
1 x105
2 x105
3 x105
4 x105
5 x105
6 x105
7 x105
Fig. 4.5: Sum of all queues within the di-
amond network for all control strategies
and constant inflow.
Table 4.2: Maximum queues of the diamond network for a constant inflow.
objective strategy
1 2 3 4 5 6 7 8 9 10 11
maximum queue 162 287 205 98 292 181 233 92 98 188 68
The second inflow function we consider for 0tTis plotted in Figure 4.2 and
given by
Gv1
in (t) = (40 if 0 mod(t, 40) <30 ,
0 if 30 mod(t,40) <40.(4.5)
In Figures 4.6, 4.7 and Table 4.3 again the total outflow of the network at T= 200,
the sum of all queues and the maximal occurring queue lengths are presented.
optimization
routing strategies
deterministic
12 3 4 5 6 7 8 9 10 11
5500
5600
5700
5800
5900
6000
Fig. 4.6: Total outflow of the diamond
network at T=200 for all control strate-
gies and stop-go inflow.
optimization
routing strategies
deterministic
1 2 3 4 5 6 7 8 9 10 11
0
1 x105
2 x105
3 x105
4 x105
5 x105
6 x105
7 x105
Fig. 4.7: Sum of all queues within the di-
amond network for all control strategies
and stop-go inflow.
14 PRODUCTION NETWORKS WITH RANDOM BREAKDOWNS
Table 4.3: Maximum queues of the diamond network for a stop go inflow.
objective strategy
1234567891011
maximum queue 168 276 206 103 274 190 232 117 99 202 96
From Figures 4.4 and 4.5, we see that in the deterministic case all strategies work
equally well for the constant inflow (4.1). Additionally, queues remain empty. This
means the available capacity of the network is sufficient to process all incoming parts.
This changes for the stop-go inflow (4.5), where queues start to build up for various
cases. For both scenarios, the strategies in the stochastic case lead to completely dif-
ferent results. Obviously, the s-i capacity control (2) is worst in all three performance
measures and for both inflow functions. This is due to processors e= 3,5 both having
higher capacity but lower availability than processors e= 4,6. Consequently, the flow is
mainly distributed in those processors and gets stuck more often. The best performing
strategies for both inflows and for all three performance measures are s-i queuing (4),
s-d queuing (8) as well as the advanced s-d strategy (9). Note that the latter is a
mixture of the earlier ones.
We also remark, that the state-dependent strategies (8) and (9) work slightly better
than the state-independent strategy (4), but the difference is rather small. Neverthe-
less, the state-dependent strategies yield better results compared to state-independent
strategies. This is also true for the capacity controls, for instance, cf. s-d capacity con-
trol (6) with s-i capacity control (2). For all other control pairs (state-independent vs.
state-dependent) the respective state-independent control (i.e. str. (1) and (3)) perform
much better than their state-dependent analogues (str. (5) and (7)). The maximum
queue values in Table 4.3 are important for the design of inventories. However, they are
observed at single points in time and do not reflect the total utilization in the considered
time horizon.
Concerning the approximate optimization approaches, we detect that the quadratic
objective function (11) yields the best overall performance. We note that the opti-
mization problem (3.1) with objective function (10) performs even worse than the best
heuristics. This is due to the state-dependency of the approximate optimization algo-
rithm and the fact that the extrapolation from the current point in time may lead to a
bad prediction in the worst case.
4.2. A cascade network The second network to be considered is a large sym-
metric cascade network with 27 arcs shown in Figure 4.8. The main ingredient of this
network are the first and second layer of processors (e= 6,...,23) consisting of nine pro-
cessors each. In the first layer three processors lead from each of the three vertices v3,
v4and v5to each of the three vertices v6,v7and v8of the second layer. Herby, the
processors heading ”straight” down are more prone to failure than all the other proces-
sors of this layer. In the second layer the processors leaving vertices v6and v8are those
most likely to fail. As depicted in Table 4.4 the first five and last four processors are
deterministic and do not fail.
As before, we first present results for the constant inflow (cf. Figure 4.1) which is
now
Gv1
in (t) = 72 (4.6)
in Figures 4.9, 4.10 and Table 4.5. The performance measures are again the total outflow
S. G ¨
OTTLICH AND S. K ¨
UHN 15
v0v1
v2
v3v4v5
v6v7v8
v9v10 v11
v12
v13
1,90
2,90
3,30 4,30 5,30
6,10
7,10
8,10 9,10
10,10
11,10 12,10
13,10
14,10
15,10
7,10
16,10
17,10
18,10
19,10
20,10
21,10
22,10
23,10
24,30 25,30 26,30
27,90
e,µe= index, capacity
Fig. 4.8: The cascade network.
Table 4.4: Parameters for the cascade network.
processor parameters
type indices τe
mean τe
on τe
off
green 1–5, 24–27 1 1 0
yellow 6,8,9,11,12,14,18–20 0.95 47.5 2.5
red 7,10,13,15–17,21–23 0.75 30 10
of the network (4.2), the sum of queues within the network (4.3) and the maximal queue
lengths (4.4).
16 PRODUCTION NETWORKS WITH RANDOM BREAKDOWNS
1 2 3 4 5 6 7 8 9 10 11
1.2 x104
1.3 x104
1.4 x104
Fig. 4.9: Total outflow of the cascade net-
work at T=200 for all control strategies
and constant inflow.
1 2 3 4 5 6 7 8 9 10 11
0
0.5 x106
1 x106
1.5 x106
2 x106
Fig. 4.10: Sum of all queues within the
cascade network for all control strategies
and constant inflow.
Table 4.5: Maximum queues of the cascade network for a constant inflow.
objective strategy
1234567891011
maximum queue 258 258 333 332 296 296 363 325 333 145 150
Second, we consider the stop-go-inflow for 0t200 (cf. Figure 4.2)
Gv1
in (t) = (90 if 0 mod(t, 40) <30 ,
0 if 30 mod(t,40) <40 (4.7)
applied to the cascade network and show the results in Figures 4.11, 4.12 and Table 4.6.
1 2 3 4 5 6 7 8 9 10 11
1.2 x104
1.3 x104
Fig. 4.11: Total outflow of the cascade
network at T=200 for all control strate-
gies and stop-go inflow.
1 2 3 4 5 6 7 8 9 10 11
0
0.5 x106
1 x106
1.5 x106
Fig. 4.12: Sum of all queues within the
cascade network for all control strategies
and stop-go inflow.
Due to the full symmetry of the network, the uniform and capacity control strategy
yield the same results for the state-independent and -dependent case respectively. From
S. G ¨
OTTLICH AND S. K ¨
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Table 4.6: Maximum queues of the cascade network for a stop go inflow.
objective strategy
1234567891011
maximum queue 229 229 282 220 252 252 305 228 218 156 128
Figures 4.9 and 4.11, we see that despite of the availability control (Str. (7)) the state-
dependent controls perform better than their state-independent analogues when consid-
ering the total outflow of the network. While the performance of the different strategies
in terms of the total outflow (4.2) is not significant for the constant inflow (4.6), we
observe a change for the stop-go inflow (4.7). The controls also differ concerning the
sum of all queues (4.3) for both inflow functions (see Figures 4.10, 4.12). The tendency
is once more that the state-dependent controls perform better than the corresponding
state-independent ones. However, the state-independent strategies lead to acceptable
limits of the maximal queue length (4.4) (cf. Tables 4.5, 4.6). This is due to the choice of
the availability of the processors in the first and second layer. Since the state-dependent
controls try to avoid uncertain processors, more flow is led to the vertices v6and v8.
Consequently in the next processors, which are all uncertain, the queues start to in-
crease. This drawback is avoided by the state-independent controls. Additionally, we
observe that the maximal queue length of the state-independent controls arise at pro-
cessors of the first layer (those leaving v3,v4and v5) while the state-dependent controls
have maximal queues at processors of the second layer (those leaving v6,v7and v8).
For the stop-go inflow (4.7), the sum of queues as well as the maximal queue lengths are
smaller than for the constant inflow (4.6). The stop-go inflow also favors the queuing
controls and the advanced control over the uniform controls. Those controls are able to
exploit the stopping in the inflow to clear the queues.
Different to the diamond network example, the approximate optimization (10)
yields an improvement of 6% in the total outflow, of 80% for the sum of all queues
of 70% for the maximum queue length in case of constant inflow (4.6). This shows
that in contrast to the heuristics strategies, which consider only local criteria for the
distribution of flow, the approximate optimization process includes global information
of the network. Thus, the approximation algorithm is able to detect and avoid uncer-
tain processors in the second layer that the heuristics do not to see. For the stop-go
inflow (4.7) the approximate optimization still performs better than the heuristics, but
the benefit is less (2%, 25% and 40%).
Note that due to the time limit of 1 month (744 hours), only computational results
for 14 optimization runs can be presented for the objective function (11). Therefore we
do not observe an improvement compared to (10).
4.3. Non-symmetric network Finally, we consider a non-symmetric network
shown in Figure 4.13 with parameters given in Table 4.7. In this network, there is only
one (deterministic) feeding processor e= 1 and the processors leading to the sinks v10
and v11, i.e. processors e= 17 and e= 16 are also reliable. Furthermore, the processors
e= 3,4,7 and e= 13 are unreliable with an availability of τe
mean = 0.75 while all other
processors have an availability of τe
mean = 0.95. The network is non-symmetric with
respect to the number of linked processors at each vertex as well as their capacities (cf.
Figures 4.8 and 4.13). Nevertheless, the non-symmetric network is arranged in such a
way that in the deterministic case all flow entering the network could be completely
processed given appropriate distributions at vertices.
18 PRODUCTION NETWORKS WITH RANDOM BREAKDOWNS
Table 4.7: Parameters for the non-symmetric network.
processor parameters
type indices τe
mean τe
on τe
off
green 1,16,17 1 1 0
yellow 2,5,6,8–12,14,15 0.95 47.5 2.5
red 3,4,7,13 0.75 30 10
v1
v2
v3v4v5v6
v7v8
v9
v10 v11
1,100
2,40
3,10 4,30
5,20
6,20
7,20
8,20
9,10 10,15
11,15
12,20 13,20
14,45 15,10
16,30
17,70
e,µe= index, capacity
Fig. 4.13: The non-symmetric network.
As a constant inflow to the non-symmetric network we consider (cf. Figure 4.1)
Gv1
in (t) = 80 .(4.8)
The numerical results for the total outflow (4.2), the sum of queues (4.3) and the
maximum queue length for (4.8) are presented in Figures 4.14, 4.15 and Table 4.8,
respectively.
S. G ¨
OTTLICH AND S. K ¨
UHN 19
1 2 3 4 5 6 7 8 9 10 11
1.1 x104
1.2 x104
1.3 x104
1.4 x104
1.5 x104
1.6 x104
Fig. 4.14: Total outflow of the non-
symmetric network at T=200 for all con-
trol strategies and constant inflow.
12 3 4 5 6 7 8 9 10 11
0
0.5 x106
1 x106
1.5 x106
2 x106
Fig. 4.15: Sum of all queues within the
non-symmetric network for all control
strategies and constant inflow.
Table 4.8: Maximum queues of the non-symmetric network for a constant inflow.
objective strategy
1 2 3 4 5 6 7 8 9 10 11
maximum queue 2500 619 518 631 1849 584 861 635 634 277 231
Lastly, we consider the non-symmetric network with the stop-go inflow (cf. Fig-
ure 4.2)
Gv1
in (t) = (100 if 0 mod(t, 40) <30 ,
0 if 30 mod(t,40) <40 (4.9)
for 0 t200 and show the results in Figures 4.11, 4.12 and Table 4.6.
1 2 3 4 5 6 7 8 9 10 11
1.1 x104
1.2 x104
1.3 x104
1.4 x104
1.5 x104
Fig. 4.16: Total outflow of the non-
symmetric network at T=200 for all con-
trol strategies and stop-go inflow.
1 2 3 4 5 6 7 8 9 10 11
0
1 x106
2 x106
3 x106
4 x106
Fig. 4.17: Sum of all queues within the
non-symmetric network for all control
strategies and stop-go inflow.
20 PRODUCTION NETWORKS WITH RANDOM BREAKDOWNS
Table 4.9: Maximum queues of the non-symmetric network for a stop go inflow.
objective strategy
1 2 3 4 5 6 7 8 9 10 11
maximum queue 2327 535 426 452 1772 480 665 466 451 382 260
From Figures 4.14 – 4.17 we see that while the deviation between the other strate-
gies are small (10%) both the s-i uniform (1) as well as the s-d uniform control (5)
yield poor results for both inflow functions. We even recognize that already in the de-
terministic case those strategies perform worse than all other strategies in the stochastic
regime. This is due to the non-symmetric structure of the network, where the uniform
strategy naively distributes flow into capacity-restricted processors. E.g. processor e= 3
is filled with the same amount of flow as processor e= 2 despite e= 3 having a quarter
of the capacity of e=2 and a lower availability as well. The other strategies are able
to avoid such situations and yield similar results with the queuing controls (4), (8)
and, more preferable, the advanced control (9) strategy. For both inflow functions the
availability controls (3), (7) perform slightly worse than the average. But for the stop-
go inflow (4.9) we see that those controls perform best relative to their deterministic
solution as this result is already below average.
Both approximate optimization approaches are able to outperform the heuristics for
the constant inflow function (4.8). The approximate minimization of the queues (11)
even performs slightly better than the approximate maximization of the outflow (10).
As seen for the other networks before, the advantage of the approximate optimization
algorithms becomes less for the stop-go inflow (4.9).
While the approximate maximization of the outflow (10) yields worse results con-
cerning the total outflow and the queue loads e.g. compared to strategy (9), the ap-
proximate minimization of the queues (11) results in better objective values than all
other strategies. Concerning the maximum queue lengths both approximate algorithms
are able to outperform the heuristics significantly (10%).
4.4. Runtime analysis
To conclude our work we compare the computing times for different network ge-
ometries and inflow patterns. We present runtimes for all strategies summed up over
all Monte Carlo runs in Table 4.10. Note that the runtimes are given in hours. 1
Table 4.10: Total computation times in hours for according number of Monte Carlo
runs.
topology strategy
network MC-runs influx 1 2 3 4 5 6 7 8 9 10 11
diamond 100 constant 0.33 0.32 0.32 0.34 0.34 0.35 0.36 0.36 0.37 16.6 16.6
stop-go 0.33 0.33 0.32 0.35 0.35 0.35 0.36 0.37 0.37 16.6 16.6
cascade 30/14 constant 0.63 0.63 0.63 0.66 0.66 0.67 0.69 0.70 0.70 180 744
stop-go 0.63 0.63 0.63 0.66 0.66 0.67 0.69 0.70 0.70 240 744
non-symmetric 30 constant 0.94 0.94 0.94 0.98 0.99 1.01 1.03 1.05 1.06 144 360
stop-go 0.94 0.94 0.94 0.99 0.99 1.01 1.03 1.05 1.07 200 360
From Table 4.10 we see that the heuristic strategies are two–three orders of mag-
nitude faster than the approximate optimization algorithms. While both approxima-
tions are equally slow when considering the diamond network, the approximate outflow
maximization (10) provides significantly faster results than the approximate queue min-
imization (11) in case of larger networks. The time limit for all computations was 1
S. G ¨
OTTLICH AND S. K ¨
UHN 21
month, i.e. 744 hours. Therefore the approximate optimization algorithm (11) has been
stopped after 14 runs only. Qualitatively, we observe that sometimes the flow maximiz-
ing approximation (10) performs worse than the best heuristic. In contrast to that the
queue minimizing approximation (11) performs better for almost all cases. While there
is no difference in the runtime depending on the choice of the inflow function for the
diamond network, the runtime increases drastically for the approximation algorithms
using the stop-go inflow.
Comparing the computing times of the heuristic routing strategies (1)–(9) we
see that they differ according to their time-dependence. More precisely, the state-
independent controls (1), (2) and (3) (uniform, capacity and availability) are only
dependent on the network structure and can thus be computed in advance. This is time
efficient and therefore those strategies are the fastest ones. The s-i queuing control (4)
is independent of the state of processors but dependent on the relative queue load qe
rel(t)
at time t. Due to this time-dependence it cannot be computed in advance. But obvi-
ously the computation is still faster than the state-dependent controls (5), (6) and (7)
(uniform, capacity and availability), which are the fastest state-dependent strategies.
The most costly state-dependent controls are the s-d queuing (8) and advanced s-d con-
trol (9) which not only depend on the state but also on the relative queue length. The
maximal deviation between the slowest and fastest computation is at most 14%. We
point out that in symmetric networks such as the diamond (Figure 4.3, Section 4.1) or
the cascade (Figure 4.8, Section 4.2) the s-i uniform control (1) is the strategy with the
fastest runtime on the one hand and the best performance concerning all three objectives
on the other hand. Switching to a non-symmetric network (Figure 4.13, Section 4.3)
the uniform control strategy fails and the slight increase in runtime of more advanced
strategies is compensated by a large increase in the overall performance (at least 30%).
Conclusion. Summarizing, we observe that the advanced distribution strategies
are a good tool to control production networks with random breakdowns. The qual-
itative behavior and the runtimes are very promising compared to the approximate
optimization algorithms. In total, the best choice is the distribution strategy (9), cf.
Table 4.11. Additionally, we can also note that the state-independent strategies perform
slightly worse than their state-dependent counterparts.
Table 4.11: Classification of strategies, showing the best strategies for combinations of
networks and inflows.
networks diamond cascade non-symmetric
inflows
constant (4), (8), (9) (5), (6) (4), (6), (9)
stop-go (4), (8), (9) (4), (9) (4), (9)
Future work might include a study for other objective functions and different net-
work dynamics. Another open question is the application of the proposed routing strate-
gies to other randomly disturbed networks problems, e.g. the bounded buffer problem
in [21].
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