DOI: 10.3303/CET2188110 
 

 
 
 

 

 

 

 
 
 

 
 
 

 
 
 

 

 

 
 
 

 
 
 

 
 
 

 
 
 

 
 
 

 
 

 
 
 

 
 

Paper Received: 4 May 2021; Revised: 21 September 2021; Accepted: 6 October 2021 
Please cite this article as: Jia X., Zhang Y., Aviso K.B., Jia X., Tan R.R., Klemeš J.J., Wang F., 2021, Vulnerability Analysis for Park-Wide Water 
Management Using Dynamic Inoperability Input-Output Model, Chemical Engineering Transactions, 88, 661-666  DOI:10.3303/CET2188110 

CHEMICAL ENGINEERING TRANSACTIONS

VOL. 88, 2021

A publication of 

The Italian Association 
of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Petar S. Varbanov, Yee Van Fan, Jiří J. Klemeš

Copyright © 2021, AIDIC Servizi S.r.l.

ISBN 978-88-95608-86-0; ISSN 2283-9216

Vulnerability Analysis for Park-Wide Water Management 
Using Dynamic Inoperability Input-Output Model 

Xiaoping Jiaa, Yanmei Zhanga, Kathleen Avisob, Xuexiu Jiac, Raymond Tanb,
Jiří Jaromír Klemešc, Fang Wangd,* 
a School of Environment and Safety Engineering, Qingdao University of Science and Technology, Qingdao, China 
b Chemical Engineering Department, De La Salle University, Manila, Philippines 
c Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic 
d Sino-German Engineering College, Qingdao University of Science and Technology, Qingdao, China 

wangf@qust.edu.cn 

Park-wide water management is an important way to conserve water resources. However, the enhancement of 
inter-plant connection will weaken the system stability and strengthen the cascade effect of disaster losses. In
this work, a dynamic inoperability input-output model is applied to park-wide water network to predict the impact 
of disturbances. The inoperability is expressed as the degree of water loss. The dependence of different water 
networks on freshwater is presented. This work analyzes the impact of freshwater supply disturbance on the 
water volume. By simulating the time-varying trajectory of inoperability, the indicators (i.e., resistance to
disturbance, adaptability, and recoverability) are determined to evaluate the vulnerability of the water network 
in each scenario. The results show that under the same freshwater disturbance, water networks that are less
dependent on freshwater are more robust but weaker in recoverability. In the case of freshwater disturbance, 
the possibility of disruptions in the participating plants of the without integrated optimization water network is 
about 1.5 times that of the integrated optimization water network. 

1. Introduction

Interdependent Infrastructure System is a system formed by several interrelated and interacting infrastructures, 
which provide the most basic and important safeguards for human society (Barker and Haimes, 2009). In recent 
years, various types of disasters which result in emergencies (i.e., earthquakes, fires, floods, tornadoes, terrorist
attacks) have been occurring more frequently. This has resulted in huge impacts on the infrastructure systems.
Supply-driven disturbances can affect the services provided by critical infrastructure. Snowstorms for example
can disrupt the function of power supply systems, transportation systems, and prevent these systems from 
continuing to function properly (Ghosh, 1958).
Vulnerability studies include the assessment of system disturbances, sensitivity analysis of system response to 
disturbances, the potential impact of disturbances on the system, and system adaptation analysis. Its main 
purpose is to identify the vulnerable points in the system and then to defend or protect them so that economic
losses due to emergency disturbances are minimized. Based on the Leontief Input-output Model, Haimes and 
Jiang (2001) introduced the concept of ‘inoperability’, and developed the first generation of the Inoperability
Input-output Model (IIM) and introduced resilience coefficients to explain the Dynamic Inoperability Input-output 
Model (DIIM). DIIM can be utilized in examining the capability of systems to recover after a disruption. 
The water network of an industrial park is one of the key infrastructures. If the freshwater supply is reduced due
to the impact of emergencies, and no other emergency water management measures are taken, the park-wide 
water supply will be reduced and the desired level of production may no longer be guaranteed. Therefore, in the 
context of network complexity caused by park-wide water integration optimization, there is an urgent demand 
for system vulnerability studies. This paper proposes an improved DIIM to account for the vulnerability of the
designed park-wide water network to emergency event disturbance as an effective tool for assessing system
stability. The work proposes relevant time indicators to represent the vulnerability performance: robustness, 
adaptability, recoverability, in order to evaluate the vulnerability of the park-wide water network. Then, the

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vulnerable points of each scenario are identified to provide guidance to the park-wide management in taking 
emergency measures to minimize the inoperability caused by deepening connections between plants. 

2. Method

IIM was originally developed to study economic equilibrium behavior. In recent years, with the deepening of 
research, it has been extended to the study of population, energy, environment and other fields. Because the 
material flow exchange of each plant satisfies the conservation of energy and material, this work applies the 
DIIM to specific water networks example, analyzes the inoperability change trajectory of water plants under 
particular initial disturbance, and provides data support for the vulnerability analysis of water network. 

2.1 Dynamic Inoperability Input-output Model 

Haimes (2005) discusses the theory and methodology supporting the development of the Inoperability Input-
output Model (IIM). Dynamic Inoperability Input-output Model (DIIM) as an extension that supplements and 
complements the static IIM. The normalized DIIM equation proposed in that paper is as follows. 

dp(t)

dt
=-K

(S)
[p(t)-A

(S)*
p(t)-z*(t)] (1) 

Where, p(t) is a vector that denotes the level of system inoperability at time t; K(S)  denotes the resilience
coefficient matrix that express the recoverability after disturbance; z* denotes supply disturbance vector caused 
by emergency events; A(S)*denotes deriving interdependence matrix, converted from Eq (2) – Eq (5):

A
(S)*

=diag(x̂)-1A
(s)

diag(x̂) (2) 

A
(S)

=(A
*
)
T (3) 

A
*
=[(diag(x̂)-1)A(diag(x̂))] (4) 

A=aij=
xij

xi
(5) 

Where, x̂  denotes planned production degree; A denotes technical coefficient matrix (direct consumption 
coefficient matrix), xj denotes the output of sector j, xij denotes the amount of input from sector i to sector j. 
Eq (1) is the general expression for the DIIM, which is a standard first order linear differential equation. When 
the initial value is p (0), its continuous solution can be obtained: 

p(t)=e
-K
(S)
(I-A

(S)*
)t

p(0)+ ∫ K
(S)

t

0

e
-K
(S)
(I-A

(S)*
)(t-T)

z*(T)dT] (6) 

When z* is fixed, then: 

p(t)= (I-A
(S)*

) z*+e-K
(S)

(I-A
(S)*

)t[p(0)- (I-A
(S)*

) z*] (7) 

The discrete solution is as follows: 

p(t+1)-p(t)=-K
(S)

[p(t)-A
(S)*

p(t)-z*(t)] (8) 

2.2 Resilience 

DIIM can also be used to describe the dynamic recovery process of the correlation infrastructure sector, 
assuming that: 

p
i
(0)>0 p

j
(0)=0     i≠j, z*=0 (9) 

then the recovery trajectory of sector i is obtained by solving the continuous equation： 

p
i
(t)=e

-Ki
(s)
[1-a

ii
(S)*

t]
p

i
(0) (10) 

Assuming that the time Ti of sector i is estimated by experts or obtained from field investigations, from the initial 
disturbance pi (0)>0 was back to pi (Ti), the resilience coefficient of sector i can be obtained using Eq (11): 

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ki
(S)

=
-ln[p

i
(Ti/pi(0))]

Ti(1-aii
(S)*

)
(11) 

The larger the resilience coefficient is, the faster the recovery will be. The resilience coefficient of the sector will 
be evaluated by restoring to the ideal level through maintenance (machine failure). The resilience of the actual 
infrastructure sector i is composed of two parts: its own recovery and associated recovery with other sectors. 
The correlation recovery rate is mainly determined by the dependence coefficient. Resilience emphasizes the 
ability to recover quickly, with the system begin both adaptable and resistant to disturbance (Geng, 2014). The 
resilience model is a process curve that describes the recovery of the whole system from the beginning of the 
system disturbance (Zhang et al., 2018). 

3. Case study

Industrial park has developed many integrated optimization water networks to save water，while potentially 
increased the disaster cascading effect.  To explore and compare the disaster propagation of water networks, 
assuming no external outflow of water from the park, only internal consumption, according to the water volume 
balance to establish the water input and output table for water network. Then calculate the degree of 
interdependence between the water plant, expressed by A. Simulating the disaster propagation process with 
the help of DIIM. From there, the vulnerability of each water network can be obtained, including resistance to 
disturbances, adaptability and recoverability. The ultimate intention is to provide park managers with water 
network vulnerability point materials. 

3.1 Case description 

A water network case study (Aviso et al., 2010) is revisited here. The industrial park consists of a maintenance 
plant (M), an organic chemical plant (O) and a plastics manufacturing plant (P), an external fresh water supplier 
(F), a wastewater treatment facility for biological treatment (A) and a storage tank (H). 
Scenario 1. Initially, the wastewater stream from M and O is sent for biological treatment (A). After treatment, it 
is sent to a water storage tank (H). The wastewater from P goes directly to H due to its higher quality. It is then 
discharged to the environment or elsewhere by H. The technical coefficient (i.e., direct consumption coefficient) 
matrix A1, each element represents the degree of dependence of one sector on another sector, and the deriving 
interdependence matrix A(S)*1, as the basis for subsequent calculations, are obtained from the water exchange 
data in the case. 

Table 1: technical coefficient and deriving interdependence matrix A1,  A(S)*1

Plant M O P A H F 
M 0, 0 0, 0 0, 0 0.0115, 0 0, 0 0, 1 
O 0, 0 0, 0 0, 0 0.9885, 0 0, 0 0, 1 
P 0, 0 0, 0 0, 0 0, 0 0.5757, 0 0, 1 
A 0, 0.0115 0, 0.9885 0, 0 0, 0 0.4243, 0 0, 0 
H 0, 0 0, 0 0, 0 0, 0.5757 0, 0.4243 0, 0 
F 1, 0 1, 0 1, 0 0, 0 0, 0 0, 0 

Scenario 2. The objective of the developed model is to minimize the total system cost, including purchase and 
treatment water cost. Aviso (2010) applied mathematical programming model to solve for the results. It is show 
that a total cost saving of 11.42 % compared to the scenario 1, P savings are 35.22 %, M and O are 5.64 % and 
0.6 %. Based on optimized water network obtained, the relevant coefficients were calculated. Table 2 displays 
the technical coefficient matrix A2 and the deriving interdependence matrix A(S)*2. 

Table 2: technical coefficient and deriving interdependence matrix A2,  A(S)*2 

Plant M O P A H B F 
M 0, 0 0, 0 0, 1 0.0115, 0 0, 0 0, 0 0, 0 
O 0, 0 0, 0 0, 0.4403 0.9885, 0 0, 0 0, 0 0, 0.5597 
P 1, 0 0.4403, 0 0, 0 0, 0 0.4765, 0 0, 1 0, 0 
A 0, 0.0115 0, 0.9885 0, 0 0, 0 0.5235, 0 0, 0 0, 0 
H 0, 0 0, 0 0, 0.4765 0, 0.5235 0, 0 1, 0 0, 0 
B 0, 0 0, 0 1, 0 0, 0 0, 1 0, 0 0, 0 
F 0.5597, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 

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3.2 Scenario setting 

In order to compare the vulnerability with and without integrated optimization water networks, two scenarios 
were set up to obtain relevant comparative indicators. 
Scenario 1: Assuming that freshwater is affected by an emergency event that generates an initial disturbance, 
Within the time τ: 

∫ zF
* (t)dt=20% ,p

i
(0)=0

τ

0

, 𝑖 = 𝑀,𝑂,𝑃,𝐴,𝐻,𝐹 (12) 

Scenario 2: Based on the actual investigation, it is obtained that when the inoperability level of P is larger than 
10 %, it will cause the system to be unable to guarantee the minimum production level, considering the following 
emergency measures before T(p(T)=10 %): restoring F or finding a new alternative water source. Simulations 
are performed to obtain the inoperability recovery trajectory. 
MATLAB simulation is applied to iteratively obtain the trajectory of inoperability over time for each plant in each 
scenario. In this study, inoperability indicates the ratio of water loss and planned water supply. It is a 
dimensionless scalar in the range of 0 to 1. 

3.3 supply-driven DIIM 

Assuming that each plant has a resilience coefficient of 0.1, the DIIM equation describing each plant can be 
derived as follows (in the scenario 1): 

{
 
 
 

 
 
 

p
M1
(t+1)=p

M1
(t)-0.1[p

M1
(t)-p

F1
(t)-zM1

* (t)]

p
O1
(t+1)=p

O1
(t)-0.1[p

O1
(t)-p

F1
(t)-zO1

* (t)]

p
P1
(t+1)=p

P1
(t)-0.1[p

P1
(t)-p

F1
(t)-zP1

* (t)]

p
A1
(t+1)=p

A1
(t)-0.1[p

A1
(t)-0.0115p

M1
(t)-0.9885p

O1
(t)-zA1

* (t)]

p
H1
(t+1)=p

H1
(t)-0.1[p

H1
(t)-0.5757p

P1
(t)-0.4243p

A1
(t)-zH1

* (t)]

p
F
(t+1)=p

F
(t)-0.1[p

F
(t)-zF

* (t)]

(13) 

3.4 Simulation results 

From Figure 1 and Figure 3, it shows the propagation curve of the failure of each plant when a disturbance 
occurs in freshwater，it can be seen that reaching the new equilibrium, the level of inoperability of each plant is 
0.2, same as freshwater initial disturbance. Due to the different order of water transfer and dependence on F, 
the time to reach the new equilibrium varies among plants. According to Figure 1, the time used to reach 
equilibrium for each plant: F<M=O=P<A<H, the longer the time, the stronger the resistance to damage, so the 
resistance to damage: H>A>P=O=M>F; The first-class water plant for F in scenario 2 is only O, by comparing 
the data in Table 1 and Table 2, the dependence of the water plants on F is drastically reduced in scenario 2 
compared to the scenario1. From Figure 3, the time used to reach the new equilibrium: F<O<A<H<B<P<M, so 
the resistance to damage: M>P>B>H>A>O>F. The difference between the two scenarios is due to the increase 
in the regeneration plant of scenario 2 and the internal water use of the park resulting in a change in the 
dependence on F and the sequence of water transfer. 
System robustness measured by average time to new equilibrium, so the robustness of scenario 2 is greater 
than that of scenario 1. It shows that the integrated optimization scenario increases the resistance of the water 
network to freshwater disturbances. Although the final inoperability is the same for each scenario, the time 
experienced varies. 
From Figure 1, when t=17 h, the inoperability of P reaches 10.36 %. Therefore, emergency management 
measures are required to restore freshwater or activate the backup water source of P before t=17 h. From Figure 
2, the time used to return the ideal state: H>A>M=O=P, so the recoverability: F>M=O=P>A>H. Similarly, from 
Figure 4, the time used to return the ideal state: M>P>B=H>A>O>F, so the recoverability: F>O>A>H=B>P>M; 
System recoverability is measured by the average time it takes for each plant to recover from maximum 
inoperability to the ideal state. So, the recoverability of scenario 1 is greater than that of scenario 2. The 
adaptability is measured by the maximum inoperability that the system can achieve throughout the disturbance 
and recovery process. So, the adaptability of scenario 2 is greater than that of scenario 1. It shows that the 
fluctuations of the optimization scenario are smaller than the without optimization scenario for the same 
disturbance conditions. 

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Figure 1: scenario 1 disturbance propagation Figure 2:  scenario 1 restored curve 

Figure 3: scenario 2 disturbance propagation Figure 4: scenario 2 restored curve 

3.5 Emergency management for scenario 2 

Vulnerability point analysis. Scenario 2 (with integrated optimization), which satisfies the maximum economic 
benefits of park, has the stronger resistance to freshwater disturbance, but the weaker recoverability; Scenario 
1(without integrated optimization) has the stronger recoverability, but the weaker resistance to freshwater 
disturbance. Therefore, it is necessary to perform management measures for the scenario 2 to increase the 
stability of the water network structure. 

Figure 5: Inoperability change for the resilience coefficient of 0.6 for scenario 2 

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Scenario 2 is based on the maximum water recycling rate obtained between water plants, which has little 
dependence on freshwater. When the freshwater suddenly decreases, the spread speed of disasters between 
plants is slow, the resistance is the strongest, but the recoverability is the weakest. In order to increase the 
recoverability, one of the measures is to increase the resilience coefficient of the plant itself are taken to shorten 
the recovery time. According to the simulation, the resistance to damage and recoverability of the scenario were 
improved, as shown in Figure 5. 

4. Conclusions
When the freshwater supply is reduced due to an emergency, the water balance of plants is broken due to the 
ripple effect. This paper uses the Dynamic Inoperability Input-output Model to investigate the vulnerability of 
different water networks. Concepts related to vulnerability including robustness, adaptability and recoverability 
are introduced to characterize the corresponding performance from the perspective of time. Integrated 
optimization scenarios enhanced inter-plant recycling and weakened the dependence on freshwater. When the 
freshwater is reduced, some extension of the time to reach the new equilibrium or restore the ideal state. The 
stronger the resistance to disturbance is, the more time the water management department in the park can take 
emergency measures. The model is generic and provides vulnerability analysis support for integrated 
programming. 

Acknowledgements 

The authors gratefully acknowledge the financial support provided by National Key R&D Program of China 
(2020YFE0201400) and the National Natural Science Foundation of China (41771575). 
This research was also supported by a project “Sustainable Process Integration Laboratory - SPIL”, project 
No. CZ.02.1.01/0.0/0.0/15_003/0000456 funded by EU as “CZ Operational Programme Research, 
Development and Education,” Priority 1: Strengthening capacity for quality research

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