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ROUTING STRATEGIES IN PRODUCTION NETWORKS WITH

RANDOM BREAKDOWNS∗

SIMONE G ¨

OTTLICH†AND 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:R≥0×Ω−→ {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)

t∗t∗+∆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 ω∈Ω.

Thus for a ﬁxed time t≥0, re(t,·) is a binary random variable, whereas for a ﬁxed

random sample ω∈Ω, 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 t∗aswitching. 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 t∗and t∗+∆τeis chosen randomly from the exponential distribution with density

function Exp(t;λ) and the rate parameter

λ=λ(re(t∗,ω)) = (1/τ e

on if re(t∗,ω)=1,

1/τe

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], t≥0, ω∈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.

v∂tρe+∂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 e∈δ+

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 e∈δ+

vthe distribution rate Av,e(t)should fulﬁll the two conditions for all t≥0:

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) =

1−Av,4, respectively.

(i) 0≤Av,e(t)≤1, and

(ii) Pe∈δ+

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

¯e∈δ−

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

G¨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 e∈δ+

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 e∈δ+

vthe

ongoing processors with capacity µe. Then, we deﬁne the s-i capacity control by

αs(e),e

capacity =µe

P

¯e∈δ+

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 e∈Ethe 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 e∈δ+

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

¯e∈δ+

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 e∈Ebe

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 e∈δ+

v

the ongoing processors with the capacity µe, mean availability τe

mean and relative queue

load qe

rel(t)for t≥0. Then, we deﬁne the s-i queuing control by

As(e),e

queue(t) = µe·τe

mean ·qe

rel(t)

P

¯e∈δ+

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 e∈δ+

vthe

ongoing processors with state re(t)for t≥0. Then, we deﬁne the s-d uniform control by

As(e),e

uniform(t) =

1/P

¯e∈δ+

s(e)

r¯e(t)if P

¯e∈δ+

s(e)

r¯e(t)>0,

1/|δ+

s(e)|if P

¯e∈δ+

s(e)

r¯e(t)=0.(5)

Here, in the ﬁrst case the sum indicates the number of processors e∈δ+

s(e)which are

operating at time t≥0.

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 e∈δ+

vthe

ongoing processors with the capacity µeand state re(t)for t≥0. Then, we deﬁne the

s-d capacity control by

As(e),e

capacity (t) =

µe·re(t)/P

¯e∈δ+

s(e)µ¯e·r¯e(t)if P

¯e∈δ+

s(e)

r¯e(t)>0,

µe/P

¯e∈δ+

s(e)

µ¯eif P

¯e∈δ+

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 e∈δ+

v

the ongoing processors with the capacity µe, mean availability τe

mean and state re(t)for

t≥0. Then, we deﬁne the s-d availability control by

As(e),e

availability(t) =

µe·τe

mean ·re(t)/P

¯e∈δ+

s(e)µ¯e·τ¯e

mean ·r¯e(t)if P

¯e∈δ+

s(e)

r¯e(t)>0,

µe·τe

mean/P

¯e∈δ+

s(e)µ¯e·τ¯e

meanif P

¯e∈δ+

s(e)

r¯e(t)=0.(7)

Definition 2.9 (s-d queuing control). Let v∈ V be an inner vertex and e∈δ+

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 e∈δ+

vand

t≥0. Then, we deﬁne the s-d queuing control by

As(e),e

queuing(t) =

µe·τe

mean ·qe

rel(t)·re(t)/P

¯e∈δ+

s(e)

µ¯e·τ¯e

mean ·q¯e

rel(t)·r¯e(t)if P

¯e∈δ+

s(e)

r¯e(t)>0,

µe·τe

mean ·qe

rel(t)/P

¯e∈δ+

s(e)

µ¯e·τ¯e

mean ·q¯e

rel(t)if P

¯e∈δ+

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 e∈δ+

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 e∈δ+

vand

t≥0. Then, we deﬁne the advanced s-d control with the user deﬁned constant 0≤c≤1

by

As(e),e

adv (t) =

As(e),e

queue(t)if ∀e∈δ+

s(e):re(t)=0 or qe

rel(t)< c ,

µe·τe

mean·qe

rel(t)

P¯e∈δ+

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 ≤c≤1 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 e∈Eout leading to a sink, i.e.

max

Av,e J(ρ) = Zt∗

i+τhor

t∗

iX

e∈Eout

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

e∈E

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 ≤t≤t∗

iand for all processors e∈E.

Output: Av,e(t), ρe(t) and qe(t) for 0 ≤t≤t∗

i+τhor and for all e∈E.

1: Fix states ¯re(t)= re(t∗

i) for t≥t∗

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∗

i≤t≤t∗

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 ¨

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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 ∆t≤∆x. 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∈Ω

ZT

0X

e∈Eout

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∈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

e∈Emax

0≤t≤Tqe(t,ω) (4.4)

arising at one processor within the network is presented in Table 4.2.

S. G ¨

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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 0≤t≤Tis 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 ¨

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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 0≤t≤200 (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 ¨

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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 ¨

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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 ≤t≤200 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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