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Received: 16 May 2019 Accepted: 12 June 2019

DOI: 10.1002/pamm.201900252

Optimal inﬂow control in transport systems with uncertain demands -

A comparison of undersupply penalties

Kerstin Lux1,∗and Simone Göttlich1,∗∗

1University of Mannheim, 68131 Mannheim, Germany

We address the challenging task of setting up an optimal production plan taking into account uncertain demand. The transport

process is represented by the linear advection equation and the uncertain demand stream is captured by an Ornstein-Uhlenbeck

process (OUP). With a model predictive control approach, we determine the optimal inﬂow. We use two types of undersupply

penalties and compare the average undersupply as well as the number of undersupply cases in a numerical simulation.

© 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim

1 Stochastic optimal control problem

Accounting for uncertainty is crucial in the modeling of many real-world processes. One example is the increasing interest in

solutions accounting for large ﬂuctuations in renewable energy generation (see e.g. [1]). Another source of uncertainty in the

production context arises in terms of the unknown demand for goods (see e.g. [2]). A reliable supply is crucial to avoid high

costs of short-term external purchase and for the company’s reputation. We focus on the optimal inﬂow control in transport

systems avoiding undersupply. We use the stochastic optimal control framework set up in [3].

1.1 Supply system and demand dynamics

We consider a supply system where goods are transported from left (x= 0) to right (x= 1) with a constant velocity λ, i.e.

goods feeded into the system need 1

/λtime units to pass from left to right. Mathematically, this transportation process is

described by the linear advection equation with initial and boundary condition given by

zt+λzx= 0, x ∈(0,1), t ∈[0, T ], z(x, 0) = 0, z(0, t) = u(t), t ∈[0, T −1

/λ]., (1)

where u∈L2([0, T −1

/λ]) is the inﬂow control. We set u(0) = 0. The output of the system is y(t) = z(1, t). We aim at

determing the inﬂow control u(t)such that the resulting supply y(t)optimally matches the uncertain demand (Yt)t∈[0,T ].

The demand dynamics are described by an Ornstein-Uhlenbeck process (OUP) being the unique strong solution of

dYt=κ(µ(t)−Yt)dt +σdWt, Y0=y0.(2)

Wtis a one-dimensional Brownian motion, σ > 0, κ > 0are constants, and y0is the initial demand. The OUP is a popular

stochastic process in demand modeling (see e.g. [4] for electricty demand modeling). Its mean reverting property allows the

interpretation as random ﬂuctuations around a given mean demand level µ(t)possibly obtained from historical demand data.

For further details and an extension of it including the possibility of jumps, please see [3].

The resulting stochastic optimal control (SOC) problem can be stated as

min

u∈L2([0,T −1

/λ]) ZT

1

/λ

OF (Ys, t0, yt0, y(s))ds subject to (1) and (2).(3)

Here, we assume u(t)to be Fˆ

ti-measurable for t∈[ˆ

ti,ˆ

ti+1], where ˆ

ti=i·∆tup,i∈ {0,1,··· ,T−1

/λ

/∆tup}, and

∆tup ∈[0, T −1

/λ]is the update frequency. Then, 0 = ˆ

t0<ˆ

t1<···<ˆ

tn≤T−1

/λspeciﬁes the grid of update times, where

the current demand at the market is observed. Accordingly, we subdivide our optimization horizon [0, T ]into subintervals

[ˆ

ti,ˆ

ti+1]and solve the SOC problem of type (3) thereon with updated state of the supply system and updated initial demand

(see control method CM2 in [3]).

1.2 Choice of cost function

Our focus is the choice of the cost function for the optimization. We will compare different types of undersupply penalties in

terms of different cost functions OF (Ys, t0, yt0, y(s)). The ﬁrst one is taken from [5]:

OFpenC(Ys, t0, yt0, y(s)) = E(Ys−y(s))2|Yt0=yt0

|{z }

I

+α·E(Ys−y(s))2|Ys> y(s)∧Yt0=yt0

|{z }

II

.(4)

∗Corresponding author: e-mail: klux@mail.uni- mannheim.de, phone: +49 621 181 2472

∗∗ e-mail: goettlich@uni-mannheim.de

This is an open access article under the terms of the Creative Commons Attribution License 4.0, which permits use,

distribution and reproduction in any medium, provided the original work is properly cited.

PAMM ·Proc. Appl. Math. Mech. 2019;19:e201900252. www.gamm-proceedings.com 1 of 2

https://doi.org/10.1002/pamm.201900252 © 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim

2 of 2 Section 15: Uncertainty quantiﬁcation

Iis of tracking type and II represents a multiple αof the expected quadratic undersupply given the information of being in

an undersupply case. The multiple αgives the possibilty to control the intensity of penalization. For the effect of different

intensities on the optimal supply, we refer the reader to [5]. We formulate an alternative penalization of undersupply as

follows:

OFpenI(Ys, t0, yt0, y(s)) = E(Ys−y(s))2|Yt0=yt0

|{z }

I

+α·E(Ys−y(s))2Ys>y(s)|Yt0=yt0

|{z }

III

.(5)

In contrast to (4), it might be of greater practical relevance for a producer to also account for the probability that an undersupply

occurs. Note that the difference between the two objective functions is determined by the latter probability P(Ys> y(s)):

(OFpenC −OFpenI )(Ys, t0, yt0, y(s)) =α·E(Ys−y(s))2|Ys> y(s)∧Yt0=yt0·(1 −P(Ys> y(s))) ≥0.

We always have OFpenC(Ys, t0, yt0, y(s)) ≥OFpenI(Ys, t0, yt0, y (s)) with equality if P(Ys> y(s)) = 1, i.e. undersupply is

penalized less for OFpenI. That goes along with two intuitive hypothesis numerically analyzed in the Section 2:

(H1) The average undersupply E(y∗(s)−Ys)Ys>y(s)|Yt0=yt0for OFpenI is higher than that for OFpenC .

(H2) The number of undersupply cases is expected to be higher for OFpenI.

2 Numerical results

With only a slight modiﬁcation of the reformulation of equation (5) in [5], we again have a deterministic reformulation of

the SOC problem (3). We use the numerical procedure set up in [5] adapted to the alternative cost function OFpenI to test

our hypothesis (H1) and (H2) numerically. We take 103Monte Carlo repetitions for the following parameter setting: T= 1,

λ= 4,µ(t) = 2 + 3 ·sin(2πt),κ= 3,σ= 2,y0= 1.

0.25 0.375 0.5 0.625 0.75 0.875 1

Time

0

50

100

150

200

250

300

350

(a) Number of undersupply cases

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time

-0.2

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

(b) Average undersupply

Fig. 1: Comparison of OFpenI with O FpenC

In Figure 1, we see that the number of undersupply cases is higher for OFpenI . The average undersupply (negative value)

resulting from OFpenI lies clearly below the one based on the optimization with respect to OFpenC . Both the binary measure

of undersupply as well as the quantitative height of the average undersupply show the higher undersupply penalization by

OFpenC, and numerically conﬁrm our hypotheses (H1) and (H2). However, the increasing or decreasing tendency is the same

for both measures indicating that the use of either one is reasonable.

Acknowledgements The authors are grateful for the support of the German Research Foundation (DFG) within the project “Novel models

and control for networked problems: from discrete event to continuous dynamics” (GO1920/4-1).

References

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SIAM Rev. 56(3), 461–495 (2014).

[2] J. L. Higle and K.G. Kempf, Production planning under supply and demand uncertainty: a stochastic programming approach, in:

Stochastic programming , Internat. Ser. Oper. Res. Management Sci., Vol. 150 (Springer, New York, 2011), pp. 297–315.

[3] S. Göttlich, R. Korn, and K. Lux, Optimal control of electricity input given an uncertain demand, arXiv preprint: 1810.05480, 2018.

[4] M. T. Barlow A diffusion model for electricity prices, Math. Finance 12(4), 287–298 (2002).

[5] S. Göttlich, R. Korn, and K. Lux, Optimal inﬂow control penalizing undersupply in transport systems with uncertain demands, arXiv

preprint: 1901.09653, 2019.

© 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim www.gamm-proceedings.com