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October 21, 2014. This is the author’s manuscript of a paper that has been submitted for

publication. The contents of the paper may change as the paper goes through the review process.

Allocating resources to enhance resilience

Cameron A. MacKenzie, Defense Resources Management Institute, Naval Postgraduate School,

camacken@nps.edu

Christopher W. Zobel, Dept. of Business Information Technology, Virginia Tech, czobel@vt.edu

ABSTRACT

This paper constructs a framework to help a decision maker allocate resources to increase his or

her organization’s resilience to a system disruption, where resilience is measured as a function of

the average loss per unit time and the time needed to recover full functionality. Enhancing

resilience prior to a disruption involves allocating resources from a fixed budget to reduce the

value of one or both of these characteristics. We first look at characterizing the optimal resource

allocations associated with several standard allocation functions. Because the resources are being

allocated before the disruption, however, the initial loss and recovery time may not be known

with certainty. We thus also apply the optimal resource allocation model for resilience to three

models of uncertain disruptions: (1) independent probabilities, (2) dependent probabilities, and

(3) unknown probabilities.

KEYWORDS: resilience, resource allocation, nonlinear programming

mailto:camacken@nps.edumailto:czobel@vt.edu

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1 INTRODUCTION

The capacity for resilience is an important characteristic of any real-world system that is subject

to the possibility of disruptions. Given the increasing number of natural and man-made disasters,

and their growing impact on our globally interconnected infrastructure systems, improving

system resilience and assessing the level and type of resources that must be committed in order

to do so effectively is of growing importance. This requires quantifying a system’s resilience and

the relative cost effectiveness of different approaches for increasing that resilience, subject to

constraints on resource availability.

Although the term resilience has a variety of different definitions in the literature, depending on

the discipline, a resilient system is frequently considered to be one which is able to withstand,

absorb, and successfully recover from the impact of some sort of disruption to normal operations

(NRC, 2012; Obama, 2013). Quantifying system or organizational resilience thus requires

measuring both the amount of loss and the length of recovery time (Bruneau et al., 2003; Zobel,

2011). From the standpoint of making investments to improve resilience, some activities will

have a larger impact on reducing loss while others will more directly and significantly impact the

recovery time. This paper considers the tradeoffs between these two types of investments from a

resource allocation perspective: given a fixed budget, which investment should a decision maker

choose to make to improve the overall resilience of a given system.

This paper makes the following contributions to the growing literature on resilience.

Mathematical models determine the optimal allocation of resources toward reducing loss and

improving recovery time in order to maximize system resilience prior to a disruption. We prove

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the conditions under which a decision maker should allocate the entire budget either to reducing

loss or to reducing the time to recover for four different allocation functions. Since the

optimization model assumes that resources are being allocated prior to a disruption, we also

explore the impact of uncertainty on model parameters for (1) independent probabilities, (2)

dependent probabilities, and (3) unknown probabilities. Theoretical results from the models and

a simulation that compares optimal allocations given the different resource allocation models

demonstrate how the assumptions inherent in the models should influence the decision about

allocating resources.

We begin our discussion with a brief literature review to establish the context for the resilience

enhancing, resource allocation framework. Following a discussion of the modeling formulation,

we examine four characteristic allocation functions under certainty and under uncertainty, and

we discuss the results of a simulation analysis focused on understanding the implications of

different resource allocation schemes. The paper concludes by expanding on the mathematical

results to provide general guidelines for decision makers while recognizing the importance and

limitations of modeling assumptions.

2 LITERATURE REVIEW

The capacity for a system to withstand, absorb, and recover from a disruption can depend on a

number of different physical, social, and economic factors. This multi-dimensional nature of the

concept of resilience lends itself to different types of analysis by researchers in different

disciplines. In the social sciences, for example, indicator variables are typically used to capture

the breadth of societal or economic characteristics that support resilient behavior (Chan & Wong,

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2007; Cutter et al., 2008), whereas in engineering it is more easily quantifiable physical

measures of changing system performance that are typically analyzed (Ayyub, 2014; Pant et al.,

2014b).

In this paper, we adopt the engineering-based view that resilience is a process that can be

measured and monitored with respect to changes in system performance over time. Much of the

this research is related to measuring resilience as a function of the calculated area beneath the

performance curve (see Figure 1), with the amount of loss suffered and the length of time until

recovery serving as important related parameters (Cimellaro et al., 2010; Zobel, 2010). Under

this approach, resilience, R, may be represented as follows, where T represents the length of time

that the system is in a state of loss and �̅� represents the average loss per unit time during the

disruption (Zobel et al., 2012):

𝑅∗(�̅�, 𝑇) = 1 −�̅�𝑇

𝑇∗ . (1)

The parameter 𝑇∗ in eq. (1) represents the maximum possible recovery time (if �̅� or 𝑇 equal 0,

then there are no losses and the organization’s resilience is 1). This particular formulation will

apply in general even if the system’s recovery trajectory is non-monotonic because the average

loss per unit time depends on the recovery trajectory. As the average loss in system performance,

�̅� is the fraction of performance lost from the non-disrupted system, and 0 ≤ �̅� ≤ 1. From an

operational standpoint, a decision maker can improve resilience by allocating resources to lessen

the average impact and/or to decrease the time to recovery.

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Figure 1: Measuring disaster resilience

This basic idea has been extended in a number of ways, including explicitly modeling the

tradeoffs between loss and recovery time (Zobel, 2011a; Zobel et al., 2012), modeling

interconnected infrastructure resilience (MacKenzie & Barker 2013; Baroud et al. 2014a),

capturing multi-dimensionality (Bruneau & Reinhorn, 2007; Zobel, 2011b), and tracking the

disaster recovery process (Zobel, 2014). Most of these research efforts ultimately tend to

concentrate on the characterization of resilience so as to better understand system behavior.

Simulation, for example, is often used to compare a system’s response to a disruptive event

under alternative control policies (MacKenzie et al., 2012; Zobel et al., 2012), and techniques

like regression are used to try to uncover the theoretical relationship between decisions and

outcomes (MacKenzie & Barker, 2013).

A related viewpoint depicts resilience as a time-dependent metric defined as the ratio of recovery

at a given time to the loss in performance (Henry & Ramirez-Marquez, 2012). Such a metric can

be used to identify the most important components in a network, such as a transportation or

power network, with the goal of prioritizing investments to protect the most influential

System performance

(fraction of nominal value)

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components (Barker et al., 2013). Incorporating uncertainty with this metric produces a

stochastic measure of resilience, and the optimal recovery activity can be identified via

simulation (Pant et al., 2014a) or by comparing the percentiles of different probability

distributions of resilience generated by different recovery activities (Baroud et al., 2014b).

Although the motivation behind measuring resilience typically focuses on helping decision

makers prepare and respond to disruptions, less research has concentrated on how resources

should be allocated to improve an organization or system’s resilience. The literature studying

how to optimize system resilience usually defines activities to pursue or components to protect

and then uses a numerical method such as simulation to select the best alternative (e.g., Barker et

al., 2013; Pant et al., 2014a; Baroud et al., 2014b). In a real-world context, however, it is

important to keep in mind that there are limits not only on how quickly a system can recover

from a disruption, but also on the physical and financial resources available to help offset the

disaster’s impacts. A process manager will thus need to focus on determining the most effective

way to allocate his or her resources in order to improve system resilience. This paper seeks to

help a decision maker determine the most effective alternative by mapping resources to the

resilience metric in eq. (1) via mathematical functions. Non-linear programming is used to

analytically determine the optimal allocation of resources in order to provide a broader

application and lead to more general results than a numerical method tied to a specific

application.

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3 ALLOCATIION MODEL TO ENHANCE RESILIENCE

Given a system that will be directly impacted by a disruption, let us assume that functions are

known which describe the effect of allocating resources to reducing that system’s average loss

per unit time: �̅�(𝑧�̅�), where 𝑧�̅� is the amount allocated to lessen the initial impact, and to

reducing its recovery time: 𝑇(𝑧𝑇), where 𝑧𝑇 is the amount allocated to reduce the recovery time.

The total budget available is Z. The decision maker may then maximize system resilience by

solving the following problem:

maximize 𝑅∗(�̅�(𝑧�̅�), 𝑇(𝑧𝑇))

subject to 𝑧�̅� + 𝑧𝑇 ≤ 𝑍

𝑧�̅� , 𝑧𝑇 ≥ 0

(2)

where 𝑅∗ is defined as in eq. (1). In general, the functional forms for �̅�(𝑧�̅�) and 𝑇(𝑧𝑇) should

obey certain properties. First of all, the first derivatives, 𝑑�̅�

𝑑𝑧�̅� and

𝑑𝑇

𝑑𝑧𝑇, should be less than 0,

which means that the functions are decreasing functions and as more resources are allocated to �̅�

or 𝑇 the average loss and recovery time should continue to decrease. Furthermore, the second

derivatives 𝑑2�̅�

𝑑𝑧�̅�2 and

𝑑2𝑇

𝑑𝑧𝑇2 should be greater than or equal to 0, which signifies constant returns or

marginal decreasing improvements as more resources are allocated. The first dollar allocated to

reduce �̅� or 𝑇 thus should be at least as effective as the next dollar allocated.

Given these constraints, several functional forms could describe the effectiveness of allocating

resources to �̅� or to 𝑇. Although one may not invest directly in reducing the average system loss

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or the system recovery time, the projects or activities to which one allocates those resources

would typically impact one or both of these characteristics. The model assumes that a resource

only reduces one of the factors, and future research can examine the allocation if a resource can

simultaneously benefit both factors. Despite this limitation, if the optimization model in (2)

reveals that 70% of the resources should be allocated to reduce �̅� and 30% to reduce 𝑇, a good

heuristic may be to select a project in which 70% of the benefits reduce the average losses and

30% reduce the recovery time.

The following discussion explores four such forms for �̅�(𝑧�̅�) and 𝑇(𝑧𝑇) and studies the

conditions under which (a) all of the resources should be allocated to reduce either �̅� or 𝑇, or (b)

the budget should be divided between the two factors. Rules for allocating resources are

developed first for models with certainty and then for models that incorporate uncertainty in one

or more of the parameters.

4 MODEL UNDER CERTAINTY

Let us first assume that all parameters are known with certainty. The functional forms that we

will consider are linear, exponential, quadratic, and logarithmic. The parameters �̂� and �̂�

represent the baseline average loss and time to recovery if no resources are allocated.

4.1 Linear allocation function

A linear allocation function rests on the assumption that the decrease in one of the resilience

factors is constant, no matter how small that factor becomes. It thus assumes constant returns to

scale. Mathematically, this gives �̅�(𝑧�̅�) = �̂� − 𝑎�̅�𝑧�̅� and 𝑇(𝑧𝑇) = �̂� − 𝑎𝑇𝑧𝑇, where 𝑎�̅� > 0 and

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𝑎𝑇 > 0 are parameters describing the effectiveness of allocating resources. After substituting the

linear allocation functions into eq. (1), the objective function in (2) becomes

maximize 1 −(�̂� − 𝑎�̅�𝑧�̅�)(�̂� − 𝑎𝑇𝑧𝑇)

𝑇∗. (3)

In order to simplify the following proposition, we assume that �̂� ≥ 𝑎�̅�𝑍 and �̂� ≥ 𝑎𝑇𝑍. If that

assumption proves to be incorrect, then it is trivial to see that perfect resilience, 𝑅∗ = 1, can be

achieved by reducing one of the factors to zero.

Proposition 1. The decision maker should allocate resources to reduce only one factor. He or she

should choose 𝑧�̅� = 𝑍 if 𝑎�̅� �̂�⁄ ≥ 𝑎𝑇 �̂�⁄ and choose 𝑧𝑇 = 𝑍 otherwise.

Proof. Because of the symmetry in the optimization problem, it suffices to just show for one

parameter. After expanding the product, optimizing eq. (3) is equivalent to:

minimize �̂��̂� − 𝑎�̅��̂�𝑧�̅� − 𝑎𝑇�̂�𝑧𝑇 + 𝑎�̅�𝑎𝑇𝑧�̅�𝑧𝑇 . (4)

Assume that 𝑎�̅� �̂�⁄ ≥ 𝑎𝑇 �̂�⁄ , which implies 𝑎�̅��̂� ≥ 𝑎𝑇�̂�. Because 𝑎�̅��̂� and 𝑎𝑇�̂� are the two

negative components, it is clear that that 𝑧�̅� = 𝑍 and 𝑧𝑇 = 0 is the optimal allocation. If 𝑎𝑇 �̂�⁄ >

𝑎�̅� �̂�⁄ , the opposite is true, and all the resources should be used to reduce 𝑇. ∎

If a linear function is an appropriate model for representing the effectiveness of allocating

resources, then the decision maker should focus his or her resources to reduce the factor whose

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initial parameter is the smallest, assuming the effectiveness parameters are relatively equal. For

example, if the average loss �̂� is very small but the time to recovery �̂� is large, then the decision

maker should allocate resources to reducing the average loss.

4.2 Exponential allocation function

In contrast to linear allocation, an exponential allocation function assumes that each resilience

factor is always reduced by same fractional amount for a given investment. In other words, if a

given investment reduces an average loss of �̂�1 by one half, then the same investment would also

reduce an average loss of �̂�2 by one half, regardless of the actual values for �̂�1 and �̂�2. Unlike

the linear allocation function, an exponential allocation function assumes marginally decreasing

improvements and that the first dollar allocated is more effective than the second dollar.

The exponential allocation functions for the two factors may be represented by: �̅�(𝑧�̅�) =

�̂� exp(−𝑎�̅�𝑧�̅�), and 𝑇(𝑧𝑇) = �̂� exp(−𝑎𝑇𝑧𝑇), where 𝑎�̅� > 0 and 𝑎𝑇 > 0 describe the

effectiveness of allocating resources.

Proposition 2. The decision maker should choose 𝑧�̅� = 𝑍 if 𝑎�̅� ≥ 𝑎𝑇 and 𝑧𝑇 = 𝑍 otherwise.

Proof. Without loss of generality, assume 𝑎�̅� ≥ 𝑎𝑇. After substituting the exponential functions

into eq. (1), the objective function in (2) becomes

maximize 1 −�̂��̂�exp(−𝑎�̅�𝑧�̅� − 𝑎𝑇𝑧𝑇)

𝑇∗. (5)

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Eq. (5) is equivalent to minimizing exp(−𝑎�̅�𝑧�̅� − 𝑎𝑇𝑧𝑇), which leads to setting 𝑧�̅� = 𝑍. The

same argument applies for the other case. ∎

A decision maker who relies on either the exponential and linear functions to guide allocation

decisions should therefore allocate resources to reduce only one of the two factors. Whereas the

linear allocation function focuses attention on reducing the smallest initial factor �̂� or �̂�, a

decision maker using the exponential allocation function should focus resources on the factor

with the largest effectiveness parameter, regardless of the initial values.

4.3 Quadratic allocation function

Similar to an exponential function, a quadratic allocation function assumes marginal decreasing

returns. The marginal return on allocating resource decreases at a rate linear to the amount of

resources allocated. The quadratic allocation functions for the two factors are: �̅�(𝑧�̅�) = �̂� −

𝑏�̅�𝑧�̅� + 𝑎�̅�𝑧�̅�2 and 𝑇(𝑧𝑇) = �̂� − 𝑏𝑇𝑧𝑇 + 𝑎𝑇𝑧𝑇

2 where 𝑎�̅� , 𝑎𝑇 > 0 and 𝑏�̅� , 𝑏𝑇 ≥ 0. In order to

ensure decreasing functions, it must be true that 𝑧�̅� ≤ 𝑏�̅� (2𝑎�̅�)⁄ and 𝑧𝑇 ≤ 𝑏𝑇 (2𝑎𝑇)⁄ .

After taking the product of �̅�(𝑧�̅�) and 𝑇(𝑧𝑇) and replacing 𝑧𝑇 with 𝑍 − 𝑧�̅�, we minimize

�̅�(𝑧�̅�) = 𝐴𝑧�̅� + 𝐵𝑧�̅�2 + 𝐶𝑧�̅�

3 + 𝐷𝑧�̅�4 (6)

where

𝐴 = �̂�𝑏𝑇 − �̂�𝑏�̅� − 2�̂�𝑎𝑇𝑍+𝑏�̅�𝑏𝑇𝑍 − 𝑏�̅�𝑎𝑇𝑍2

𝐵 = �̂�𝑎𝑇 − 𝑏�̅�𝑏𝑇 + 2𝑏�̅�𝑎𝑇𝑍 + �̂�𝑎�̅� − 𝑎�̅�𝑏𝑇𝑍 + 𝑎�̅�𝑎𝑇𝑍2

𝐶 = −𝑏�̅�𝑎𝑇 + 𝑎�̅�𝑏𝑇 − 2𝑎�̅�𝑎𝑇𝑍

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𝐷 = 𝑎�̅�𝑎𝑇 .

Minimizing �̅�(𝑧�̅�) is equivalent to maximizing 𝑅∗(�̅�, 𝑇) in eq. (2). Eq. (6) is a quartic equation.

As 𝑧�̅� approaches negative and positive infinity, eq. (6) approaches positive infinity because

𝐷 > 0. Thus, �̅�(𝑧�̅�) has at most two local minima and at most one local maximum. We label the

first local minimum 𝑧�̅� = 𝑧1 and the second local minimum 𝑧�̅� = 𝑧2.

Unlike the linear and exponential allocation functions, the quadratic allocation function may

result in an optimal allocation of resources to reduce both factors, which occurs at either 𝑧1 or 𝑧2.

We describe conditions when it is optimal to allocate to reduce either one factor (𝑧�̅� = 0 or 𝑍)

and when it is optimal to allocate to reduce both factors (𝑧�̅� = 𝑧1 or 𝑧2).

Proposition 3. The follow conditions should guide the decision in determining how to optimally

allocate resources:

1. If 𝐴 > 0 and �̅�′(𝑍) > 0

𝑧�̅� = {𝑧2, 0 < 𝑧2 < 𝑍 and �̅�(𝑧2) ≤ 0 0, otherwise

2. If 𝐴 > 0 and �̅�′(𝑍) < 0

𝑧�̅� = {0, 0 ≤ �̅�(𝑍)𝑍, otherwise

3. If 𝐴 < 0 and �̅�′(𝑍) > 0

𝑧�̅� = {𝑧1, 𝑧1 > 0, 𝑧2 < 𝑍, and �̅�(𝑧1) ≤ �̅�(𝑧2)𝑧1, 𝑧2 > 𝑍𝑧2, otherwise

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4. If 𝐴 < 0 and �̅�′(𝑍) < 0

𝑧�̅� = {𝑧1, 0 < 𝑧1 < 𝑍 and �̅�(𝑧1) ≤ �̅�(𝑍)

𝑍, otherwise

where �̅�′(𝑧�̅�) = 𝐴 + 2𝐵𝑧�̅� + 3𝐶𝑧�̅�2 + 4𝐷𝑧�̅�

3.

Proof. Condition 1. If 𝐴 > 0, �̅�(0) is increasing because �̅�′(0) = 𝐴. Since �̅�(𝑧�̅�) approaches

positive infinity as 𝑧�̅� approaches negative infinity, �̅�′(𝑧�̅�) < 0 for a very negative 𝑧�̅�. �̅�′(0) >

0, which means that the first local minimum 𝑧1 occurs when 𝑧�̅� < 0. If 𝑧2 < 𝑍, then 𝑧�̅� = 𝑧2 is a

global minimum if �̅�(𝑧2) ≤ 0 because �̅�(0) = 0. Otherwise, 𝑧�̅� = 0 is the global minimum

because �̅�(𝑧�̅�) is increasing when 𝑧�̅� = 𝑍 if �̅�′(𝑍) > 0.

Condition 2. As with the first condition, if 𝐴 > 0, the first local minimum 𝑧1 occurs when

𝑧�̅� < 0. Since �̅�(𝑧�̅�) approaches positive infinity as 𝑧�̅� approaches positive infinity, �̅�′(𝑧�̅�) > 0

for a very positive 𝑧�̅�. If �̅�′(𝑍) < 0, the second local minimum 𝑧2 occurs when 𝑧�̅� > 𝑍. Thus,

only two possible minima occur when 𝑧�̅� = 0 or 𝑍. If �̅�(𝑍) ≥ 0, a global minimum occurs when

𝑧�̅� = 0. Otherwise, the global minimum occurs when 𝑧�̅� = 𝑍.

Condition 3. If 𝐴 < 0, then �̅�(0) is decreasing and 𝑧�̅� = 0 cannot be a minimum. If �̅�′(𝑍) > 0,

then �̅�(𝑍) is increasing and 𝑧�̅� = 𝑍 cannot be a minimum. Thus, at least one local minimum

exists between 0 and 𝑍. If 0 < 𝑧1, 𝑧2 < 𝑍 and �̅�(𝑧1) ≤ �̅�(𝑧2), 𝑧�̅� = 𝑧1 is a global minimum. If

𝑧2 > 𝑍, then 0 < 𝑧1 < 𝑍 and the global minimum occurs at 𝑧�̅� = 𝑧1. If neither of those

conditions are met, then 𝑧1 < 0, which implies that 𝑧2 < 𝑍, and 𝑧�̅� = 𝑧2 is the global minimum.

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Condition 4. As with the third condition, if 𝐴 < 0 then 𝑧�̅� = 0 cannot be a minimum. As with

the second condition, if �̅�′(𝑍) < 0, the 𝑧2 > 𝑍. If 𝑧1 > 0 and �̅�(𝑧1) ≤ �̅�(𝑍), then 𝑧�̅� = 𝑧1 is a

global minimum. If 𝑧1 < 0 or �̅�(𝑧1) > �̅�(𝑍), then 𝑧�̅� = 𝑍 is the global minimum because �̅�(𝑍)

is decreasing. ∎

Calculating 𝑧1 and 𝑧2 involves finding the roots of a cubic equation �̅�′(𝑧�̅�). Although a closed

formula exists for solving a cubic equation, expressing the solution in terms of 𝐴, 𝐵, 𝐶, and 𝐷

provides little insight into 𝑧1 and 𝑧2.

4.4 Logarithmic allocation function

The logarithmic allocation functions are: �̅� = �̂� − 𝑎�̅� log(1 + 𝑏�̅�𝑧�̅�) and 𝑇 = �̂� − 𝑎𝑇 log(1 +

𝑏𝑇𝑧𝑇) where 𝑎�̅� , 𝑏�̅� , 𝑎𝑇 , 𝑏𝑇 > 0. The function is strictly decreasing if the parameters are greater

than 0. The logarithmic allocation function also assumes marginal decreasing returns, but the

marginal reduction in a factor decreases at a quicker rate than that of the quadratic allocation

function.

The logarithmic allocation function may also result in situations where it is optimal to allocate

resources to reduce both factors. Maximizing the resilience function is equivalent to minimizing

�̃�(𝑧�̅�) = log(�̂� − 𝑎�̅� log(1 + 𝑏�̅�𝑧�̅�)) + log(�̂� − 𝑎𝑇 log(1 + 𝑏𝑇[𝑍 − 𝑧�̅�])) (7)

where 𝑧𝑇 is replaced by 𝑍 − 𝑧�̅�.

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As with the quadratic allocation function, the logarithmic allocation function may result in an

optimal allocation of resources to reduce both factors. Calculating a solution that reduces both

factors involves finding the roots to the numerator of the first derivative of �̃�(𝑧�̅�):

�̃�′(𝑧�̅�)

=𝑎𝑇𝑏𝑇(�̂� − 𝑎�̅� log(1 + 𝑏�̅�𝑧�̅�))(1 + 𝑏�̅�𝑧�̅�) − 𝑎�̅�𝑏�̅�(�̂� − 𝑎𝑇 log(1 + 𝑏𝑇[𝑍 − 𝑧�̅�]))(1 + 𝑏𝑇[𝑍 − 𝑧�̅�])

(�̂� − 𝑎�̅� log(1 + 𝑏�̅�𝑧�̅�))(�̂� − 𝑎𝑇 log(1 + 𝑏𝑇[𝑍 − 𝑧�̅�]))(1 + 𝑏�̅�𝑧�̅�)(1 + 𝑏𝑇[𝑍 − 𝑧�̅�]).

(8)

Before describing the conditions under which it is optimal to allocate resources to reduce both

factors, we determine the maximum number of local minima and maxima for �̃�(𝑧�̅�). We assume

that �̂� > 𝑎�̅� log(1 + 𝑏�̅�𝑍) and �̂� > log(1 + 𝑏𝑇𝑍) to ensure that �̃�(𝑧�̅�) is continuous and

differentiable over the domain of 𝑧�̅�. If this assumption is violated, the decision maker should

allocate resources until either �̅�(𝑧�̅�) or 𝑇(𝑧𝑇) equals 0.

Lemma 1. �̃�(𝑧�̅�) has at most two local interior minima.

Proof. The number of local extrema for �̃�(𝑧�̅�) equals to the number of solutions to 𝑧�̅� when the

numerator of �̃�′(𝑧�̅�) in eq. (8) equals 0. The first derivative of the numerator of �̃�′(𝑧�̅�) is

�̃�′′(𝑧�̅�) = 𝑏�̅�𝑏𝑇(�̂�𝑎𝑇 + �̂�𝑎�̅� − 𝑎�̅�𝑎𝑇 log([1 + 𝑏�̅�𝑧�̅�][1 + 𝑏𝑇(𝑍 − 𝑧�̅�)]) − 2𝑎�̅�𝑎𝑇). (9)

After setting �̃�′′(𝑧�̅�) = 0, we rearrange eq. (9) so that the equation is quadratic in 𝑧�̅�.

Thus, �̃�′′(𝑧�̅�) = 0 has at most two solutions for 𝑧�̅�, which implies that �̃�′(𝑧�̅�) has at most one

local maximum and one local minimum. This implies that �̃�′(𝑧�̅�) = 0 has at most three solutions

16

for 𝑧�̅�, and �̃�(𝑧�̅�) has at most three local extrema. At least one of these local extrema must be a

local maximum. Consequently, at most two local interior minima exist for �̃�(𝑧�̅�). ∎

Lemma 2. �̃�(𝑧�̅�) has at most one local interior maximum.

Proof. We can conclude that �̃�(𝑧�̅�) has at most two local interior maxima by the same reasoning

as in Lemma 1. To show that �̃�(𝑧�̅�) has at most one local interior maximum, we multiply terms

in eq. (9):

�̃�′′(𝑧�̅�) = 𝑏�̅�𝑏𝑇(�̂�𝑎𝑇 + �̂�𝑎�̅� − 𝑎�̅�𝑎𝑇 log(1 + 𝑏𝑇𝑍 + [𝑏�̅� + 𝑏𝑇𝑍 − 𝑏𝑇]𝑧�̅� − 𝑏�̅�𝑏𝑇𝑧�̅�2)

− 2𝑎�̅�𝑎𝑇).

(10)

Since − log(∙) is convex and nonincreasing and 1 + 𝑏𝑇𝑍 + [𝑏�̅� + 𝑏𝑇𝑍 − 𝑏𝑇]𝑧�̅� − 𝑏�̅�𝑏𝑇𝑧�̅�2 is

concave, the expression −𝑎�̅�𝑎𝑇 log(1 + 𝑏𝑇𝑍 + [𝑏�̅� + 𝑏𝑇𝑍 − 𝑏𝑇]𝑧�̅� − 𝑏�̅�𝑏𝑇𝑧�̅�2) is convex (Boyd

& Vandeberghe, 2006). Thus �̃�′′(𝑧�̅�) is convex in 𝑧�̅�.

If �̃�(𝑧�̅�) has three local extrema, then �̃�′′(𝑧�̅�) = 0 must have two solutions for 𝑧𝑋 or two roots.

In this case, the convexity of �̃�′′(𝑧�̅�) implies that �̃�′′(𝑧�̅�) > 0 when 𝑧�̅� is less than the smaller

root, �̃�′′(𝑧�̅�) < 0 when 𝑧�̅� is greater than the smaller root but less than the larger root, and

�̃�′′(𝑧�̅�) > 0 when 𝑧�̅� is greater than the larger root. Thus, �̃�(𝑧�̅�) is initially convex, then

concave, and then convex. Consequently, the first extreme point must be a local minimum, the

second extreme point must be a local maximum, and the third extreme point must be a local

minimum.

17

If �̃�(𝑧�̅�) only has two local extrema, one of them must be a local minimum because of the

continuity of �̃�(𝑧�̅�) over the domain of 𝑧�̅�. Therefore, �̃�(𝑧�̅�) has at most one local interior

maximum. ∎

Since �̃�(𝑧�̅�) has at most two local minima and one local maximum, we label the local minimum

that occurs before the local maximum as 𝑧1 and the local minimum that occurs after the local

maximum as 𝑧2. The conditions with the logarithmic allocation function that should guide the

decision about the optimal allocation are identical to the conditions with the quadratic allocation

function, where �̃�(𝑧�̅�) replaces �̅�(𝑧�̅�) and �̃�′(𝑧�̅�) replaces �̅�′(𝑧�̅�).

Proposition 4. The following conditions should guide the decision in determining how to

optimally allocate resources:

1. If �̃�′(0) > 0 and �̃�′(𝑍) > 0

𝑧�̅� = {𝑧2, 𝑧2 < 𝑍 and �̃�(𝑧2) ≤ log(�̂�) + log(�̂� − 𝑎𝑇 log(1 + 𝑏𝑇𝑍))

0, otherwise

2. If �̃�′(0) > 0 and �̃�′(𝑍) < 0

𝑧�̅� = {0, �̂�(�̂� − 𝑎𝑇 log(1 + 𝑏𝑇𝑍)) ≤ �̂�(�̂� − 𝑎�̅� log(1 + 𝑏�̅�𝑍))

𝑍, otherwise

3. If �̃�′(0) < 0 and �̃�′(𝑍) > 0

𝑧�̅� = {𝑧1, 𝑧1 > 0, 𝑧2 < 𝑍, and �̃�(𝑧1) ≤ �̃�(𝑧2)𝑧1, 𝑧2 > 𝑍𝑧2, otherwise

4. If �̃�′(0) < 0 and �̃�′(𝑍) < 0

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𝑧�̅� = {𝑧1, 𝑧1 > 0 and �̃�(𝑧1) ≤ log(�̂� − 𝑎�̅� log(1 + 𝑏�̅�𝑍)) + log(�̂�)

𝑍, otherwise

Proof. The proof is similar to that for Proposition 3. Condition 1. If �̃�′(0) > 0, �̃�(𝑧�̅�) is

increasing when 𝑧�̅� = 0. Because �̃�(𝑧�̅�) is increasing, we know the first extreme point (if one

exists) is a local maximum. If two local minima exist, 𝑧1 < 0. If 𝑧2 < 𝑍, then 𝑧�̅� = 𝑧2 is a global

minimum if �̃�(𝑧2) ≤ log(�̂�) + log(�̂� − 𝑎𝑇 log(1 + 𝑏𝑇𝑍)) = �̃�(0). Otherwise, 𝑧�̅� = 0 is the

global minimum because �̃�(𝑧�̅�) is increasing when 𝑧�̅� = 𝑍 if �̃�′(𝑍) > 0.

Condition 2. As with the first condition, if �̃�′(0) > 0, the first local minimum 𝑧1 is less than 0 if

it exists. �̃�′(𝑍) < 0 implies that �̃�(𝑍) is increasing. If 𝑧2 were less than 𝑍, this would mean that

two local maxima exist because the first local maximum is less than 𝑧2. From Lemma 2, we

know that at most one local maximum exists. Thus, 𝑧2 > 𝑍 and the only two possible minima

occur when 𝑧�̅� = 0 or 𝑍. If �̂�(�̂� − 𝑎𝑇 log(1 + 𝑏𝑇𝑍)) ≤ �̂�(�̂� − 𝑎�̅� log(1 + 𝑏�̅�𝑍)), �̃�(0) ≤ �̃�(𝑍),

and a global minimum occurs when 𝑧𝑋 = 0. Otherwise, the global minimum occurs when

𝑧�̅� = 𝑍.

Condition 3. If �̃�′(0) < 0, then �̃�(0) is decreasing and 𝑧�̅� = 0 cannot be a minimum. If �̃�′(𝑍) >

0, then �̅�(𝑍) is increasing and 𝑧�̅� = 𝑍 cannot be a minimum. Thus, at least one local minimum

exists between 0 and 𝑍. If 0 < 𝑧1, 𝑧2 < 𝑍 and �̃�(𝑧1) ≤ �̃�(𝑧2), 𝑧�̅� = 𝑧1 is a global minimum. If

𝑧2 > 𝑍, then 0 < 𝑧1 < 𝑍 and the global minimum occurs at 𝑧�̅� = 𝑧1. If neither of those

conditions are met, then 𝑧1 < 0, which implies that 𝑧2 < 𝑍, and 𝑧�̅� = 𝑧2 is the global minimum.

19

Condition 4. As with the third condition, if �̃�′(0) < 0 then 𝑧�̅� = 0 cannot be a minimum. As

with the second condition, if �̃�′(𝑍) < 0, then 𝑧2 > 𝑍. If 𝑧1 > 0 and �̃�(𝑧1) ≤ log(�̂� −

𝑎�̅� log(1 + 𝑏�̅�𝑍)) + log(�̂�) = �̃�(𝑍), then 𝑧�̅� = 𝑧1 is a global minimum. If 𝑧1 < 0 or �̃�(𝑧1) >

log(�̂� − 𝑎𝑋 log(1 + 𝑏�̅�𝑍)) + log(�̂�) = �̃�(𝑍), then 𝑧�̅� = 𝑍 is the global minimum since �̃�(𝑍) is

decreasing. ∎

Calculating the values for 𝑧1 and 𝑧2, if they both exist, is a little more complicated with the

logarithmic allocation function than with the quadratic allocation function. The conditions in

Proposition 4 can be used to determine whether 𝑧1, 𝑧2, neither, or both should be calculated.

Many algorithms exist to find the zeroes of an equation, and these algorithms can be applied to

eq. (8) to find the solutions when �̃�′(𝑧�̅�) = 0.

To summarize Section 4, both linear and exponential allocation functions should result in

allocating resources to reduce only one factor. Which factor is determined by the initial values

and effectiveness parameters for the linear allocation function but is governed by only the

effectiveness parameter for the exponential allocation function. Quadratic or logarithmic

allocation functions may result in an optimal allocation to reduce both factors, and the conditions

for how to allocate resources under those allocation schemes are outlined in the propositions.

The simulation in Section 6 numerically compares the optimal allocation using a logarithmic

versus a quadratic allocation function.

Preferences of the decision maker could also change the resource allocation decisions. As

described in Zobel (2011), the decision maker may have preferences that alter the shape of the

20

resilience curves and change the objective function beyond minimizing the product of average

loss and recovery time. The resource allocation decisions could change depending on those

preferences.

5 MODEL UNDER UNCERTAINTY

Because the decision maker needs to allocate resources prior to a disruption, the nature and

possible consequences of the disruption will not be known for certain. The average loss and time

to recovery if no resources are allocated and the parameters describing the effectiveness of

allocating resources may all be uncertain. This section assumes that the parameters �̂�, �̂�,

𝑎�̅� , 𝑏�̅�, 𝑎𝑇 , and 𝑏𝑇 are uncertain. We explore three situations with uncertainty: (1) the uncertain

variables have known probability distributions and are independent of each other; (2) the random

variables are probabilistically dependent; and (3) a robust optimization method if probability

distributions are unknown.

5.1 Allocation assuming independence

If the decision maker can assign probability distributions to all of the uncertain parameters, the

decision maker’s objective could be to maximize expected resilience, 𝐸[𝑅∗(�̅�(𝑧�̅�), 𝑇(𝑧𝑇))]. This

subsection assumes that the random variables are independent of each other and explores how

the allocation decisions may change compared to the previous cases with certainty.

If the allocation functions are linear, quadratic, or logarithmic, it is mathematically possible that

�̅�(𝑧�̅�) < 0 or 𝑇(𝑧𝑇) < 0 for a given realization of the random parameters even if 𝐸[�̅�(𝑧�̅�)] ≥ 0

and 𝐸[𝑇(𝑧𝑇)] ≥ 0 for all 0 ≤ 𝑧�̅� , 𝑧𝑇 ≤ 𝑍. Because it is physically impossible to have a negative

21

impact or a negative time to recovery, the expected resilience with independence equals 1 −

𝐸[�̅�(𝑧�̅�)+]𝐸[𝑇(𝑧𝑇)

+] 𝑇∗⁄ where �̅�(𝑧�̅�)+ = max{�̅�(𝑧�̅�), 0} and 𝑇(𝑧𝑇)

+ = max{𝑇(𝑧𝑇), 0}.

Consequently, it may be optimal to allocate resources to reduce both factors even if the

allocation functions are linear. As will be discussed in Section 6, if the allocation functions are

quadratic or logarithmic, it is more likely the decision maker should allocate to reduce both

factors when accounting for uncertainty than when assuming the parameters are certain.

If the allocation functions are exponential, �̅�(𝑧�̅�) and 𝑇(𝑧𝑇) will always be positive so the max

function is unnecessary. Although the decision maker should allocate to reduce only one factor

under certainty, he or she may allocate to reduce both factors under uncertainty.

Proposition 5. If the allocation functions are exponential, the decision maker should allocate to

reduce both factors if and only if

max{𝐸[−𝑎�̅�], 𝐸[−𝑎𝑇]} < min {𝐸[−𝑎�̅� exp(−𝑎𝑋𝑍)]

𝐸[exp(−𝑎�̅�𝑍)],𝐸[−𝑎𝑇 exp(−𝑎𝑇𝑍)]

𝐸[exp(−𝑎𝑇𝑍)]}

(11)

Proof. Because the random variables are independent, maximizing expected resilience is

equivalent to

minimize log 𝐸[exp(−𝑎�̅�𝑧�̅�)] + log 𝐸[exp(−𝑎𝑇𝑧𝑇)]. (12)

22

Each component of eq. (12) is identical to a cumulant-generating function, which is a convex

function (Boyd & Vandeberghe, 2006). A solution that satisfies the Karush-Kuhn-Tucker

conditions is the unique minimum to eq. (12).

The optimal solution must satisfy the following equations:

𝐸[−𝑎�̅�exp(−𝑎�̅�𝑧�̅�)]

𝐸[exp(−𝑎�̅�𝑧�̅�)]− 𝜆𝑋 =

𝐸[−𝑎𝑇exp(−𝑎𝑇𝑧𝑇)]

𝐸[exp(−𝑎𝑇𝑧𝑇)]− 𝜆𝑇

(13)

𝑧𝑋 + 𝑧𝑇 = 𝑍

where 𝜆�̅� , 𝜆𝑇 ≥ 0 are the Lagrange multipliers for the non-negativity constraints for 𝑧�̅� and 𝑧𝑇,

respectively. The Lagrange multiplier 𝜆�̅� > 0 if and only if 𝑧�̅� = 0 and 𝜆𝑇 > 0 if and only if

𝑧𝑇 = 0.

We first prove that if eq. (11) is true, it is optimal to allocate resources to reduce both factors.

Without loss of generality, assume that 𝐸[−𝑎�̅�] = max{𝐸[−𝑎�̅�], 𝐸[−𝑎𝑇]} and

𝐸[−𝑎𝑇 exp(−𝑎𝑇𝑍)] 𝐸[exp(−𝑎𝑇𝑍)]⁄ = min {𝐸[−𝑎�̅� exp(−𝑎�̅�𝑍)]

𝐸[exp(−𝑎�̅�𝑍)],

𝐸[−𝑎𝑇 exp(−𝑎𝑇𝑍)]

𝐸[exp(−𝑎𝑇𝑍)]} and that

𝐸[−𝑎�̅�] < 𝐸[−𝑎𝑇 exp(−𝑎𝑇𝑍)] 𝐸[exp(−𝑎𝑇𝑍)]⁄ . If 𝑧�̅� = 0 and 𝑧𝑇 = 𝑍, 𝜆�̅� > 0 and 𝜆𝑇 = 0.

But then eq. (13) would not be true because 𝐸[−𝑎�̅�] < 𝐸[−𝑎𝑇 exp(−𝑎𝑇𝑍)] 𝐸[exp(−𝑎𝑇𝑍)]⁄ .

Thus, 𝑧�̅� = 0 and 𝑧𝑇 = 𝑍 is not an optimal allocation.

If 𝑧�̅� = 𝑍 and 𝑧𝑇 = 0, then 𝜆�̅� = 0 and 𝜆𝑇 > 0. In order for eq. (13) to be true, it must be true

that 𝐸[−𝑎�̅� exp(−𝑎�̅�𝑍)] 𝐸[exp(−𝑎�̅�𝑍)]⁄ < 𝐸[−𝑎𝑇]. Due to the convexity of the optimization

23

problem, the gradient is strictly increasing in each variable and

𝐸[−𝑎𝑋] < 𝐸[−𝑎�̅� exp(−𝑎�̅�𝑍)] 𝐸[exp(−𝑎�̅�𝑍)]⁄ . Since 𝐸[−𝑎�̅�] ≥ 𝐸[−𝑎𝑇], it must be true that

𝐸[−𝑎�̅� exp(−𝑎�̅�𝑍)] 𝐸[exp(−𝑎�̅�𝑍)]⁄ > 𝐸[−𝑎𝑇]. Thus 𝑧�̅� = 𝑍 and 𝑧𝑇 = 0 is not an optimal

allocation.

If 𝐸[−𝑎�̅�] = max{𝐸[−𝑎�̅�], 𝐸[−𝑎𝑇]} and

𝐸[−𝑎�̅� exp(−𝑎�̅�𝑍)] 𝐸[exp(−𝑎�̅�𝑍)]⁄ = min {𝐸[−𝑎�̅� exp(−𝑎�̅�𝑍)]

𝐸[exp(−�̅�𝑍)],

𝐸[−𝑎𝑇 exp(−𝑎𝑇𝑍)]

𝐸[exp(−𝑎𝑇𝑍)]}, the same

argument applies. If 𝑧�̅� = 0 and 𝑧𝑇 = 𝑍, eq. (13) would not be true because 𝐸[−𝑎�̅�] <

𝐸[−𝑎�̅� exp(−𝑎�̅�𝑍)] 𝐸[exp(−𝑎𝑋𝑍)]⁄ ≤ 𝐸[−𝑎𝑇 exp(−𝑎𝑇𝑍)] 𝐸[exp(−𝑎𝑇𝑍)]⁄ . If 𝑧�̅� = 𝑍 and

𝑧𝑇 = 0, eq. (13) would not be true because

𝐸[−𝑎𝑇] ≤ 𝐸[−𝑎�̅�] < 𝐸[−𝑎�̅� exp(−𝑎�̅�𝑍)] 𝐸[exp(−𝑎�̅�𝑍)]⁄ .

Therefore, if eq. (11) is true, the only solution that satisfies eq. (13) is when 𝑧�̅� , 𝑧𝑇 > 0, and it is

optimal to allocate resources to reduce both factors.

We next prove that if eq. (11) is not true, then either 𝑧�̅� or 𝑧𝑇 equals 0. Assume without loss of

generality that 𝐸[−𝑎�̅�] = max{𝐸[−𝑎�̅�], 𝐸[−𝑎𝑇]} and 𝐸[−𝑎𝑇 exp(−𝑎𝑇𝑍)] 𝐸[exp(−𝑎𝑇𝑍)]⁄ =

min {𝐸[−𝑎�̅� exp(−𝑎�̅�𝑍)]

𝐸[exp(−𝑎�̅�𝑍)],

𝐸[−𝑎𝑇 exp(−𝑎𝑇𝑍)]

𝐸[exp(−𝑎𝑇𝑍)]} but that 𝐸[−𝑎�̅�] ≥ 𝐸[−𝑎𝑇 exp(−𝑎𝑇𝑍)] 𝐸[exp(−𝑎𝑇𝑍)]⁄ .

If 𝑧�̅� > 0 and 𝑧𝑇 > 0, then 𝜆�̅� = 𝜆𝑇 = 0. Such a solution does not satisfy eq. (13) because each

gradient is strictly increasing and the gradient when 𝑧�̅� = 0, 𝐸[−𝑎�̅�], is greater than or equal to

the gradient when 𝑧𝑇 = 𝑍, 𝐸[−𝑎𝑇 exp(−𝑎𝑇𝑍)] 𝐸[exp(−𝑎𝑇𝑍)]⁄ .

24

Because each gradient is strictly increasing, it is impossible that

𝐸[−𝑎�̅�] ≥ 𝐸[−𝑎�̅� exp(−𝑎�̅�𝑍)] 𝐸[exp(−𝑎�̅�𝑍)]⁄ . If eq. (11) is not true, it cannot be true that

𝐸[−𝑎�̅�] = max{𝐸[−𝑎�̅�], 𝐸[−𝑎𝑇]} and

𝐸[−𝑎�̅� exp(−𝑎�̅�𝑍)] 𝐸[exp(−𝑎�̅�𝑍)]⁄ = min {𝐸[−𝑎�̅� exp(−𝑎�̅�𝑍)]

𝐸[exp(−𝑎�̅�𝑍)],

𝐸[−𝑎𝑇 exp(−𝑎𝑇𝑍)]

𝐸[exp(−𝑎𝑇𝑍)]}. Thus, if eq. (11)

is not true, 𝑧�̅� > 0 and 𝑧𝑇 > 0 is not an optimal allocation. ∎

If the conditions of Proposition 5 are satisfied, the decision maker should choose 𝑧�̅� such that

𝐸[(𝑎𝑇 − 𝑎�̅�)exp([𝑎𝑇 − 𝑎�̅�]𝑧�̅� − 𝑎𝑇𝑍)] = 0, (14)

which is derived from the conditions for optimality in eq. (13) and by replacing 𝑧𝑇 with 𝑍 − 𝑧�̅�.

5.2 Allocation assuming dependence

The allocation decisions may change if the random variables are probabilistically dependent.

This subsection assumes that the decision maker still desires to maximize expected resilience,

but the uncertain parameters �̂�, �̂�, 𝑎�̅� , 𝑏�̅� , 𝑎𝑇 , and 𝑏𝑇 are dependent. As with independence,

calculating expected resilience must account for the mathematical possibility that �̅�(𝑧�̅�) < 0 or

𝑇(𝑧𝑇) < 0 when the allocation functions are linear, quadratic, or logarithmic. Expected

resilience equals 1 − 𝐸[�̅�(𝑧�̅�)+𝑇(𝑧𝑇)

+] 𝑇∗⁄ , and it may be optimal to allocate to reduce both

factors for these allocation functions.

25

If 𝑃(�̂� < 𝑎�̅�𝑍) = 0 and 𝑃(�̂� < 𝑎𝑇𝑍) = 0 when the allocation functions are linear, it is always

optimal to allocate resources to only one factor. This can be seen by multiplying the objective

function in eq. (3) where the decision maker should minimize

𝐸[�̂��̂�] − 𝐸[𝑎�̅��̂�]𝑧�̅� − 𝐸[𝑎𝑇�̂�]𝑧𝑇 + 𝐸[𝑎�̅�𝑎𝑇]𝑧�̅�𝑧𝑇 (15)

Because 𝐸[𝑎�̅��̂�]𝑧�̅� and 𝐸[𝑎𝑇�̂�]𝑧𝑇 are the two negative components, it is clear that 𝑧�̅� = 𝑍

minimizes the above equation if 𝐸[𝑎�̅��̂�] ≥ 𝐸[𝑎𝑇�̂�] and 𝑧𝑇 = 𝑍 minimizes eq. (15) if 𝐸[𝑎𝑇�̂�] ≥

𝐸[𝑎�̅��̂�].

If the allocation functions are exponential, the optimization problem is convex in the two

decision variables, but the allocation rules described in Proposition 5 do not necessarily hold

because of the dependent random variables. Unlike the situation when assuming independence,

the random variables �̂� and �̂� may influence the optimal allocation if these variables are

probabilistically dependent with 𝑎�̅� or 𝑎𝑇.

5.3 Robust allocation

If probability distributions cannot be determined for the uncertain parameters, the decision maker

may choose a “robust” solution where he or she maximizes the worst-case resilience, i.e.,

maximize min 𝑅∗(�̅�(𝑧�̅�), 𝑇(𝑧𝑇)). This subsection assumes that bounds on each parameter or

known such that

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�̂� ≤ �̂� ≤ �̅̂�

(16)

�̂� ≤ �̂� ≤ �̅̂�

𝑎�̅� ≤ 𝑎�̅� ≤ 𝑎�̅�

𝑎𝑇 ≤ 𝑎𝑇 ≤ 𝑎𝑇

𝑏�̅� ≤ 𝑏�̅� ≤ 𝑏�̅�

𝑏𝑇 ≤ 𝑏𝑇 ≤ 𝑏𝑇

The robust allocation follows the same principles as the allocation under certainty but replaces

the certain parameters with the worst-case values of the parameters. If the allocation functions

are linear, the decision maker should follow Proposition 1 and allocate the budget 𝑍 to the factor

that corresponds to max{𝑎�̅� �̅̂�⁄ , 𝑎𝑇 �̅̂�⁄ }. If the allocation functions are exponential, allocate 𝑍 to

the factor that corresponds to max{𝑎�̅�, 𝑎𝑇}. If the allocation functions are quadratic, Proposition

3 applies after assigning �̂� = �̅̂�, 𝑎�̅� = 𝑎�̅�, 𝑏�̅� = 𝑏�̅�, �̂� = �̅̂�, 𝑎𝑇 = 𝑎𝑇, and 𝑏𝑇 = 𝑏𝑇. If the

allocation functions are logarithmic, the decision maker should follow the rules in Proposition 4

after assigning �̂� = �̅̂�, 𝑎�̅� = 𝑎�̅�, 𝑏�̅� = 𝑏�̅�, �̂� = �̅̂�, 𝑎𝑇 = 𝑎𝑇, and 𝑏𝑇 = 𝑏𝑇.

6 SIMULATION

Simulation can be used to explore the implications of using one type of allocation versus another

and how accounting for uncertainty impacts the allocation of resources to enhance resilience.

The simulation is used to solve for the parameters �̂�, �̂�, 𝑎�̅� , 𝑏�̅� , 𝑎𝑇 , and 𝑏𝑇 for each allocation

function. Each simulation trial randomly selects three points for �̅� and 𝑇 where �̅�~𝑈(0,1),

𝑇~𝑈(0, 𝑇∗), and 𝑇∗ = 30. Each trial sets �̂� equal to the maximum �̅� and �̂� equal to the

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maximum 𝑇, and the other two points correspond to different allocation amounts for 𝑧�̅� and 𝑧𝑇,

each of which can vary between 0 and 𝑍 = 10. We calculate parameters for each of the four

allocation functions that has the best fit to the three points for �̅� and the three points for 𝑇. Since

each allocation function’s parameters rely on the same underlying data, we can compare results

from the different allocation functions. Each simulation trial also explores allocation with

uncertainty by randomly selecting multiple parameters for the same allocation function.

We calculate the optimal allocation for 𝑧�̅� and 𝑧𝑇 that maximizes resilience or expected

resilience for the four different allocation functions with certainty, under uncertainty with

independence, under uncertainty with dependence, and with a robust allocation. The simulation

is run 5000 times.

Table I depicts the results of these simulations. We allow for the possibility that allocating

resources can completely eliminate the risk by reducing either �̅� or 𝑇 to 0, and the simulation

results show that perfect resilience (i.e., 𝑅∗(�̅�, 𝑇) = 1) is achieved most often when using a

linear allocation function with certainty. The linear function often achieves perfect resilience

because it assumes constant marginal benefit. The quadratic allocation function with certainty

achieves perfect resilience in 27% of the simulations whereas the logarithmic allocation function

only achieves perfect resilience in 6% of the simulations. Although the exponential allocation

function should never achieve perfect resilience, almost 6% of the simulations with an

exponential allocation function result in resilience where the calculation is so close to 1 that it

exceeds the precision of the computer. Perfect resilience is rarely achieved if the optimization

accounts for uncertainty, regardless of the allocation function deployed.

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Table I Simulation results

Certainty

Uncertainty with

independence Uncertainty with dependence Robust allocation

Lin Exp Quad Log Lin Exp Quad Log Lin Exp Quad Log Lin Exp Quad Log

Percentage of simulations

where 𝑅∗(�̅�, 𝑇) = 1 36.7 5.6 27.1 6.2 0.4 1.9 0.04 0.04 2.0 1.9 3.6 2.3 0.4 1.8 0.3 0.04

Average resilience 0.979 0.980 0.978 0.977 0.959 0.972 0.963 0.960 0.960 0.972 0.969 0.960 0.819 0.892 0.839 0.803

Percentage of simulations

where 𝑧�̅� > 0 and 𝑧𝑇 > 0 given 𝑅∗(�̅�, 𝑇) < 1

0.0 0.0 64.6 56.3 8.8 26.1 84.7 85.2 17.6 23.3 54.8 81.7 0.0 0.0 46.3 54.8

Average absolute

difference between 𝑧�̅� and 𝑧𝑇

10.0 10.0 9.0 8.4 9.7 8.6 4.1 5.9 9.2 8.7 7.2 6.3 10.0 10.0 7.9 7.7

Note: Lin = linear allocation function; Exp = exponential allocation function; Quad = quadratic allocation function; Log = logarithmic allocation function

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The average optimal resilience is very similar for all four allocation functions under certainty

with 𝑅∗(�̅�, 𝑇) = 0.98. When uncertainty is introduced, the average expected resilience ranges

between 0.96 and 0.97 for the four allocation functions, with very little difference between

expected resilience if the probabilities are independent or dependent. However, if probabilities

cannot be determined and the robust allocation method is used, the average resilience drops

significantly and ranges between 0.8 and 0.9 for the four allocation functions. This is expected

because robust allocation uses the worst-case parameters. If the decision maker is following a

robust allocation, he or she may be incentivized to request a larger budget in order to achieve

resilience closer to 1 even if the worst case has a very low chance of occurrence. Using an

exponential allocation function partially alleviates this problem because the resilience for a

robust allocation with an exponential function is larger than resilience with the other allocation

functions.

The simulation results provide insight into how frequently a decision maker should allocate

resources to reduce one factor (�̅� or 𝑇) versus both factors. If perfect resilience is not achieved,

the decision maker should reduce both factors in 65% of the simulations with a quadratic

allocation function with certainty and in 56% of the simulations with a logarithmic allocation

function. Thus, assuming that the benefit from allocating resources follows a quadratic or

logarithmic allocation function as opposed to a linear or exponential function should result in

allocating to reduce both factors a little more than half the time.

If the optimization problem incorporates uncertainty, Proposition 5 is satisfied in 26% of the

simulations with an exponential allocation function and probabilistic independence. In these

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cases, it is optimal to allocate to reduce both factors. The decision maker should also allocate to

reduce both factors with a quadratic or logarithmic allocation function more often when

accounting for uncertainty with independence than if the parameters are certain. If the allocation

functions are exponential or logarithmic, the decision maker should allocate resources to reduce

both �̅� and 𝑇 at about the same frequency whether or not the random parameters are dependent.

However, the quadratic allocation function with dependent random variables results in an

optimal allocation to reduce both factors less often than when assuming independence (from 85

to 55% of the simulations). The reason for this change is likely due to a requirement in the

simulation that each 𝑏�̅� (2𝑎�̅�)⁄ ≥ 𝑍 and each 𝑏𝑇 (2𝑎𝑇)⁄ ≥ 𝑍 in order to ensure marginally

decreasing or constant returns for the quadratic allocation function.

It is also useful to compare among the different situations about how much should be allocated to

𝑧�̅� versus 𝑧𝑇. For the linear allocation function, the average absolute difference between 𝑧�̅� and

𝑧𝑇 becomes slightly smaller when the parameters are uncertain, but the average difference

between 𝑧�̅� and 𝑧𝑇 still exceeds 9 (when the budget is 10). Similarly, the average absolute

difference between 𝑧�̅� and 𝑧𝑇 is 8.6 when maximizing expected resilience with exponential

allocation functions. Although it may be optimal to allocate resources to reduce both factors

when incorporating uncertainty with either the linear or exponential allocation function, the

decision maker should still allocate the vast majority of the resources to either 𝑧�̅� or 𝑧𝑇 most of

the time.

Incorporating uncertainty with the quadratic or logarithmic allocation functions results in a big

change to whether one factor should receive the majority of the resources. The average absolute

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difference between 𝑧�̅� and 𝑧𝑇 with certainty is 9.0 for the quadratic function and 8.4 with the

logarithmic function. If the model assumes uncertainty with independence, the average absolute

difference becomes 4.1 with the quadratic function and 5.9 with the logarithmic function. A

difference of 4.0 indicates the factor that receives the most resources should receive 70% of the

resources and the other factor should receive 30%. The average absolute difference increases for

both allocation functions when the random variables are dependent to 7.2 and 6.3, respectively,

and for the robust allocation to 7.9 and 7.7, respectively. If the decision maker believes the

parameters that determine resilience are uncertain, he or she should distribute resources more

evenly between impact and recovery time than if the parameters are certain.

7 CONCLUSIONS

Based on a widespread definition and mathematical formula for resilience, allocating resources

to enhance resilience can either lessen the impacts from a disruption or improve recovery time.

How much a decision maker should allocate to one of those two factors depends to a large extent

on how he or she believes resources affects those factors and on the level and type of uncertainty.

If resources reduce a factor in either a linear or exponential manner with no uncertainty, the

decision maker should allocate all the resources to reduce only one factor. The marginal benefits

for quadratic and logarithmic allocation functions with certainty decrease more rapidly than

linear or exponential functions, and it is often optimal to allocate resources to lessen the impacts

and improve recovery with quadratic and logarithmic functions.

Practically all situations that call for enhancing resilience will involve some uncertainty, and

maximizing expected resilience means that a decision maker should allocate resources to reduce

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both factors more often than if he or she is maximizing resilience with certainty. If probabilities

cannot be determined and the decision maker wishes to maximize the worst-case resilience, he or

she should follow the same rules as for allocation under certainty but the optimal resilience in the

robust case will be less than the optimal resilience in the certain case.

Trying to apply the resilience model in preparation for a potential disruption to a business or

region will likely suffer from a lack of quality data. Making assumptions is necessary, and the

results from the simulation shed light on how these assumptions impact the optimal allocation. If

the decision maker requires a certain level of resilience (such as 0.95), it may not matter which

allocation function is used because they all result in similar levels of resilience given a fixed

budget. A robust allocation plan that optimizes for the worst case requires a larger budget to

meet the same target.

Because of the assumption that resilience is calculated as the product of impact and recovery

time, a good heuristic is to focus resources on the factor with a small initial value and high

effectiveness. That heuristic is complicated when uncertainty is considered. Especially in an

uncertain environment, the allocation functions play a large role in determining how much

should be allocated to reduce the impacts versus improving the recovery time. Assuming that the

effectiveness and initial value parameters are uncertain but independent should mean that

resources frequently are allocated to reduce both factors, especially when it is believed that the

marginal benefits decrease rapidly as in the case of the quadratic or logarithmic functions.

Dividing the resources in an approximately equal manner is often a good strategy under this

assumption. If dependence among parameters is assumed, allocating resources to reduce both

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factors often remains a good strategy, but the decision maker should invest a higher proportion of

the budget in one of the factors.

Allocating resources to lessen the impacts may include system hardening, building redundancy,

and better emergency response systems that can reduce the impacts. Allocating resources to

improve recovery time may focus on activities such as improving capabilities for repairing and

rebuilding and prepositioning supplies that will be needed for recovery. Determining which

activities to fund prior to a disruption poses challenges for decision makers at both a national and

local level and within individual businesses. The resilience model, the analytical solutions

derived from the model, and the simulation results can provide guidance to a decision maker

seeking to determine which of these activities contributes most toward enhancing resilience.

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