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Evolutionary Bi-objective Optimization for the Dynamic

Chance-Constrained Knapsack Problem Based on Tail

Bound Objectives

Hirad Assimi 1and Oscar Harper 2and Yue Xie 3and Aneta Neumann 4and Frank Neumann 5

Abstract. Real-world combinatorial optimization problems are

often stochastic and dynamic. Therefore, it is essential to make opti-

mal and reliable decisions with a holistic approach. In this paper, we

consider the dynamic chance-constrained knapsack problem where

the weight of each item is stochastic, the capacity constraint changes

dynamically over time, and the objective is to maximize the total

proﬁt subject to the probability that total weight exceeds the capac-

ity. We make use of prominent tail inequalities such as Chebyshev’s

inequality, and Chernoff bound to approximate the probabilistic con-

straint. Our key contribution is to introduce an additional objective

which estimates the minimal capacity bound for a given stochastic

solution that still meets the chance constraint. This objective helps

to cater for dynamic changes to the stochastic problem. We apply

single- and multi-objective evolutionary algorithms to the problem

and show how bi-objective optimization can help to deal with dy-

namic chance-constrained problems.

1 INTRODUCTION

Many real-world combinatorial optimization problems involve

stochastic as well as dynamic components which are mostly treated

in isolation. However, in order to solve complex real-world problems,

it is essential to treat stochastic and dynamic aspects in a holistic ap-

proach and understand their interactions.

Dynamic components in an optimization problem may change the

objective function, constraints or decision variables over time. The

challenge to tackle a dynamic optimization problem (DOP) is to track

the moving optima when changes occur [21].

Moreover, uncertainty is pervasive in a real-world optimization

problem. The source of uncertainty may involve the nature of data,

measurement errors or lack of knowledge. Ignoring uncertainties in

solving a problem may lead to obtaining suboptimal or infeasible so-

lutions in practice [16].

Chance-constrained programming (CCP) is a powerful tool to

model uncertainty in optimization problems. It transforms an in-

equality constraint into a probabilistic constraint to ensure that the

probability of constraint violation is smaller than a limit predeﬁned

1The University of Adelaide, Australia, email: hirad.assimi@adelaide.edu.au

2The University of Adelaide, Australia, email: os-

car.harper@student.adelaide.edu.au

3The University of Adelaide, Australia, email: yue.xie@adelaide.edu.au

4The University of Adelaide, Australia, email:

aneta.neumann@adelaide.edu.au

5The University of Adelaide, Australia, email:

frank.neumann@adelaide.edu.au

by the decision-maker [3]. CCP has been applied successfully in dif-

ferent domains such as process control, scheduling and supply man-

agement where safety requirements are concerned [11].

Evolutionary algorithms (EAs) have been applied to many combi-

natorial optimization problems and demonstrate a high capability in

solving hard problems, including a wide range of real-world appli-

cations [18, 4]. Multi-objective EAs deal with several (conﬂicting)

objectives and provide a set of solutions which are non-dominated to

each other with respect to the given objective functions [7, 24, 23].

In addition to solving problems with conﬂicting objectives, several

studies have indicated that transforming a single-objective optimiza-

tion problem to a multi-objective optimization problem may lead to

obtaining better solutions. This transformation leads to obtaining a

set of non-dominated solutions instead of a single solution. There-

fore, each individual in the Pareto front contains helpful information

which can improve the performance of the algorithm in exploring the

search space [20, 24, 23].

1.1 Related work

EAs are a natural way to deal with DOPs because they are inspired by

nature which is an ever-changing environment [21]. The behaviour

of EAs has been analyzed on a knapsack problem with dynamically

changing constraint [25]. They carried out bi-objective optimization

with respect to the proﬁt and dynamic capacity constraint. They pro-

posed an algorithm to track the moving optimum and showed that

handling the constraint by a bi-objective approach can be beneﬁcial

in obtaining better solutions when the changes occur less frequently.

Their studies have been extended to analyze the EA behaviour depen-

dent on the submodularity ratio of a broad class of problems [26].

Chance-constrained knapsack problem (CCKP) is a variant of the

classical NP-hard deterministic knapsack problem where the weights

and proﬁts can be stochastic [13]. Approximation algorithm in com-

bination with a robust approach, has been applied to CCKP to ﬁnd

feasible solutions for a simpliﬁed knapsack problem [15].

Recently, Xie et al. [27] integrated inequality tails with single and

bi-objective EAs to solve CCKP. To estimate the probability of con-

straint violation, they used popular tail inequalities. They investi-

gated the behaviour of Chebyshevs inequality, and Chernoff bound

for approximation in CCKP. They also carried out bi-objective op-

timization with respect to the proﬁt and probability of constraint

violation when the capacity is static. Doerr et al. [9] have investi-

gated adaptations of classical greedy algorithms for the optimiza-

tion of submodular functions with chance constraints of knapsack

type. They have shown that the adapted greedy algorithms maintain

arXiv:2002.06766v1 [cs.NE] 17 Feb 2020

asymptotically almost the same approximation quality as in the de-

terministic setting when considering uniform distributions with the

same dispersion for the knapsack weights.

1.2 Our Contribution

In this paper, we consider the dynamic chance-constrained knapsack

problem (DCCKP) with dynamically changing constraint. We also

assume that each item in the knapsack problem has an uncertain

weight while the proﬁts are deterministic. For the dynamic compo-

nent, we follow the settings deﬁned in [25]: the knapsack capacity

changes over time every τiterations with a predeﬁned magnitude.

Moreover, for the stochastic component, we follow the approach and

settings proposed in [27] which employs inequality tails to estimate

the violation in probabilistic constraint.

Therefore, the goal in this study is to re-compute a solution of

maximal proﬁt after a dynamic change occurs to the capacity con-

straint, while the total uncertain weight can exceed the capacity with

a small probability. To beneﬁt from the bi-objective optimization, we

cannot directly apply the second objective used in previous studies

because they only considered either dynamic or stochastic aspect of

the optimization problem in isolation for the second objective func-

tion. Therefore, we introduce an objective function which deals with

uncertainties and caters for dynamic aspects of the problems. This

objective evaluates the smallest knapsack capacity bound for which

a solution would not violate the chance constraint. This objective also

can keep a set of non-dominated solutions to be used for tracking the

moving optimum. This objective makes use of tail inequalities such

as Chebyshev’s inequality and Chernoff bounds to approximate the

probabilistic constraint violation.

To solve DCCKPs, we apply a single objective EA, a modiﬁed

version of GSEMO [12] and NSGA-II [8], where the last two com-

pute a trade-offs with respect to the proﬁt and the newly introduced

objective for dealing with chance constraints. Our experimental re-

sults show that the bi-objective EAs perform better than the single

objective approaches. Introducing the additional objective function

to the problem helps the bi-objective optimization algorithm to deal

with the constraint changes as it obtains the non dominated solutions

with respect to the objective functions.

The rest of this article is organized as follows. In the next sec-

tion, we deﬁne the DCCKP and introduce the two tail inequalities

for quantifying uncertainties. Afterwards, we introduce the objec-

tive function for dealing with DCCKP and develop the bi-objective

model. Next, we report on the behaviour of single objective and bi-

objective baseline EAs in solving DCCKP. We show that bi-objective

optimization with the introduced second objective can obtain better

solutions on a wide range of instances of the DCCKP. Finally, we

ﬁnish with some concluding remarks.

2 DYNAMIC CHANCE-CONSTRAINED

KNAPSACK PROBLEM

In this section, we introduce the problem and provide Chebyshev

and Chernoff tail inequalities to estimate the probability of chance

constraint violation in the problem.

2.1 Problem Formulation

The classical knapsack problem can be deﬁned as follows. Given n

items where each item i,1≤i≤nhas a proﬁt piand a weight wi

and a knapsack capacity C. The goal is to ﬁnd a selection of items of

maximum proﬁt whose weight does not exceed the capacity bound.

A candidate solution is an element x∈ {0,1}nwhere item iis cho-

sen iff xi= 1. In this paper, we consider the stochastic and dynamic

setting for the knapsack problem where each weight is chosen inde-

pendently according to a given probability distribution. Furthermore,

the capacity bound Cchanges dynamically over time.

The search space is {0,1}nand we denote by P(x) = Pn

i=1 pixi

the proﬁt and by W(x) = Pn

i=1 wixithe weight of a solution x. We

investigate the chance-constrained knapsack problem where the goal

is to maximize P(x)under the condition that the probability that the

weight of the solution is at least as high as the capacity is at most α.

Formally, we deﬁne this constraint as

Pr [W(x)≥C]≤α

where αis a parameter that upper bounds the probability of exceed-

ing the knapsack capacity (0 <α<1).

Furthermore, the knapsack capacity in our problem is dynamic

and changes over time every τiterations. We call τthe frequency

of changes which denotes after how many iterations a change occurs

in the knapsack capacity with the magnitude of changes raccording

to some probability distributions.

2.2 Tail Bounds

Chebyshev’s inequality tail can determine a bound for a cumulative

distribution function of a random design variable. Chebyshev’s in-

equality requires to know the standard deviation of the design vari-

ables and gives a tighter bound in comparison to the weaker tails such

as Markov’s inequality. Therefore, it can be applied to any distribu-

tion if the expected weight and standard deviation of the involved

random variables are known. The standard Chebyshev inequality is

two-sided and provides tails for upper and lower bounds [2]. As

we are only interested in the probability of exceeding the weight

bound, we use a one-sided Chebyshev inequality which is also known

as Cantelli’s inequality [6]. For brevity, we refer to the one-sided

Chebyshev as Chebyshev’s inequality in this paper.

Theorem 1 (One-sided Chebyshev inequality).Let Xbe an inde-

pendent random variable, and let E(X)denote the expected weight

of X. Further, let σ2

Xbe the variance of X. Then for any λ∈R+,

we have

Pr [(X−E(X)) ≥λ]≤σ2

X

σ2

X+λ2.

Compared to Chebyshev’s inequality, Chernoff bound provides a

sharper tail with an exponential decay behavior. In order to use Cher-

noff bound, it is essential that the random variable is a summation of

independent random variables. Chernoff bound seeks a positive real

number tin order to ﬁnd the probability where the sum of indepen-

dent random variables exceeds a particular threshold [19]. Therefore,

Chernoff bound for an independent variable Xcan be given as fol-

lows based on theorem 2.3 in [17].

Theorem 2. Let X=Pn

i=1 Xibe the sum of independent random

variables Xi∈[0,1] chosen uniformly at random, and let E(X)be

the expected weight of X. For any t > 0, we have

Pr[X≥(1 + t)E(X)] ≤exp −t2

2 + 2

3tE(X).

2

3 BI-OBJECTIVE OPTIMIZATION FOR

DCCKP

In this section, we introduce a new objective-function to transform

the single-objective optimization problem into a bi-objective opti-

mization problem. We also describe (1+1)-EA and POSDC as base-

line single and bi-objective EAs.

3.1 Bi-objective Model

We redeﬁne DCCKP by introducing a new second objective function

to transform it into a bi-objective optimization problem. Therefore,

we introduce (C∗)as the stochastic bound as our second objective

function. This objective function evaluates the smallest knapsack ca-

pacity for a given solution such that it satisﬁes the predeﬁned limit

on the chance constraint. Therefore, the ﬁtness f(x)of a solution x

is given as

f(x) = (P(x), C∗(x))

where

C∗(x) = min {C|s.t. Pr[W(x)≥C]≤α}

is the smallest weight bound Csuch that the probability that the

weight W(x)of xis at least Cis at most α. Using this objective

allows to cater for dynamic changes of the weight bound of our prob-

lem. In bi-objective optimization of DCCKP, the goal is to maximize

P(x)and minimize C∗(x). Hence, we have

f(x0)f(x)iff P(x0)≥P(x)∧C∗(x0)≤C∗(x)

for the dominance relation of bi-objective optimization for two solu-

tions namely xand x0. Evaluating the chance constraint is computa-

tionally difﬁcult [1]. It has been shown that even if random variables

are from a Bernoulli distribution, calculating the probability of vio-

lating the constraint exactly is #P-complete, see Theorem 2.1 in [14].

Because it is difﬁcult to compute C∗exactly, we make use of the tail

inequalities to calculate the second objective function. For Cheby-

shev’s inequality, the stochastic bound is given as follows.

Proposition 1 (Chebyshev Constraint Bound Calculation).Let

E(W(x)) be the expected weight, σ2

W(x)be the variance of the

weight of solution xand αbe the probability bound of the chance

constraint. Then setting C∗

1(x) = E(W(x)) + σW(x)√α(1−α)

αim-

plies Pr[W(x)≥C∗

1(x)] ≤α.

Proof. Using Chebyshev’s inequality, we have

Pr [W(x)≥E(W(x)) + λ]≤σ2

W(x)

σ2

W(x)+λ2.

We set C∗

1(x) = E(W(x)) + λwhich implies

λ=σW(x)pα(1 −α)

α.

Hence, we have

Pr[W(x)≥C∗

1(x)]

=Pr "W(x)≥E(W(x) + σW(x)pα(1 −α)

α#

≤σ2

W(x)

σ2

W(x)+σW(x)√α(1−α)

α2

=α

which completes the proof.

We consider wi∈ U[E(wi)−δ, E(wi) + δ]and Pn

i=1 xidenotes

the total number of chosen items in a solution, then the stochastic

bound based on Chebyshev’s inequality for uniform distribution is

given as follows

σW(x)=δrPn

i=1 xi

3.

We substitute λas

λ=δp3α(1 −α)Pn

i=1 xi

3α

Therefore, we have

C∗

1(x) = E(W(x)) + δp3α(1 −α)Pn

i=1 xi

3α.

Moreover, to derive the second objective function by making use of

the Chernoff bound, we have:

Proposition 2 (Chernoff Constraint Bound Calculation).Let wi∈

U[E(wi)−δ, E(wi) + δ]be independent weights chosen uniformly

at random. Let E(W(x)) be the expected weight of xand αbe the

probability bound of the chance constraint. Then setting

C∗

2(x) = E(W(x))

−0.66δ

ln(α)−v

u

u

tln2(α)−9 ln(α)

n

X

i=1

xi

implies Pr[W(x)≥C∗

2(x)) ≤α.

Proof. We consider wi∈ U[E(wi)−δ, E(wi) + δ]. Then, to satisfy

Chernoff bound summation requirement, we normalize each random

weight into [0, 1] which yidenotes the normalized weight of item.

yi=wi−(E(wi)−δ)

2δ∈[0,1]

Y(x) =

n

X

i=1

yi=

n

X

i=1

wi−(E(wi)−δ)

2δxi.

Since yiis symmetric, then the total expected weight of Yis E(Y) =

1

2Pn

i=1 xi.Then, the total weight of a solution is given as

W(x) =

n

X

i=1

wixi= 2δY (x) + E(W(x)) −δ

n

X

i=1

xi.

We set

C∗

2=E(W(x)) + b

where

b=−0.66δ

ln(α)−v

u

u

tln2(α)−9 ln(α)

n

X

i=1

xi

.

Hence, the probability of violating the chance constraint. for a so-

lution is given as

Pr [W(x)≥C∗

2(x)]

=Pr "2δY (x) + E(W(x)) −δ

n

X

i=1

xi≥E(W(x)) + b#

=Pr "Y(x)≥1

2

n

X

i=1

xi+b

2δ#

=Pr Y(x)≥E(Y(x)) + b

2δ

=Pr [Y(x)≥(1 + t)E(Y(x))]

3

where

t=b

2δE(Y(x)) .

Using Chernoff bounds, we have

Pr [(Y(x)≥(1 + t)E(Y(x)))]

≤exp −t2

2 + 2

3tE(Y(x))

We have

t=−0.66 ln(α)−qln2(α)−9 ln(α)Pn

i=1 xi

Pn

i=1 xi

.

We use

ˆ

t=−0.66 ln(α)

Pn

i=1 xi≤b

2δE(Y(x)) =t

instead of twhich results in

Pr [Y(x)≥(1 + t)E(Y(x))]

≤Pr Y(x)≥(1 + ˆ

t)E(Y(x))

≤exp

−

(−0.66)2ln2α

(2E(Y(x)))2

2 + 2

3(−0.66 ln α

2E(Y(x)) )·E(Y(x))

= exp (−0.66)2·ln α·ln α

−8E(Y(x)) + 4

3(0.66 ln α)

≤α.

The last inequality holds as

−4

n

X

i=1

xi+4

3(0.66 ln α)≤(−0.66)2ln α

which completes the proof.

Note that the introduced additional objectives as C∗

1and C∗

2cal-

culate the smallest possible bound for which a solution meets the

chance constraint according to the used tail bound (Chebyshev or

Chernoff). The terms added to the expected total weight guarantee

that a given solution meets the chance constraint.

3.2 POSDC Algorithm

We adapt the algorithm proposed in [25] for our bi-objective op-

timization. We call the adapted algorithm, Pareto Optimization for

Stochastic Dynamic Constraint (POSDC) which deals with DCCKP.

POSDC (see Algorithm 1) is a baseline multi-objecitve EA which

tracks the moving optimum by storing a population in the vicinity

of the dynamic knapsack capacity. POSDC keeps a solution (x) if

C∗(x)is in [C−η, C +η], where ηdetermines the storing range.

Therefore, POSDC has two subpopulations which include feasible

and infeasible solutions (S=S−∪S+). Keeping an infeasible sub-

population helps POSDC to be prepared for the next change in the

dynamic constraint.

S−← {x∈S|C−η≤C∗(x)≤C}

S+← {x∈S|C < C ∗(x)≤C+η}.

Algorithm 1: POSDC

1Generate x∈ {0,1}nuniformly at random

2if C−η≤C∗(x)≤C+ηthen

3S←x

4else

5while S=∅do

6repair an offspring (y) by (1+1)-EA

7x←y

8if C−η≤C∗(y)≤C+ηthen

9S←x

10 while (not max iteration) do

11 if change in the capacity occurs (after τiterations) then

12 x←best solution in S

13 Update S−and S+with respect to the shifted capacity

14 if S=∅then

15 S←x

16 choose x∈Suniformly at random

17 y←create an offspring by ﬂipping each bit of x

independently with the probability of 1

n

18 if (C−η≤C∗(y)< C)∧(@z∈S−:zPOSDC y)then

19 S−←(S−∪y)\ {z∈S−|yPOSDC z}

20 else if (C≤C∗(y)≤C+η)∧(@z∈S+:zPOSDC y)

then

21 S+←(S+∪y)\ {z∈S+|yPOSDC z}

22 return best solution

POSDC generates the initial solution uniformly at random, if the gen-

erated solution is out of the storing range, then (1+1)-EA (see Algo-

rithm 2) repairs the solution and stores it in the appropriate subpopu-

lation. (1+1)-EA is a single-objective baseline EA which is described

later.

POSDC uses a mutation operator to explore the search space and

ﬁnd trade-off solutions. POSDC maintains a set of non-dominated

solutions with respect to P(x)and C∗(x)in its subpopulations. The

best solution in POSDC at each iteration is the solution with the high-

est proﬁt in S−; If S−is empty, POSDC prefers the solution with the

smallest C∗in S+.

Note that if we can compute the solutions exactly, some solutions

in S+can be feasible. However, because computing C∗in exact is

difﬁcult, we designate the optimum as the solution with the highest

proﬁt in S−.

3.3 Single Objective Approach

We only use simple baseline algorithms to make a fair comparison

between the single-objective optimization and bi-objective optimiza-

tion. (1+1)-EA and GSEMO [12] are their equivalent counterparts if

we consider identical objective functions because they use the same

mutation operator. In this study, we adapt POSDC as a variant of

GSEMO to tackle both dynamic and chance-constrained components

of the problem. Therefore, we show the efﬁciency of bi-objective op-

timization by comparing POSDC with (1+1)-EA (see Algorithm 2).

(1+1)-EA generates one potential solution uniformly at random;

In each iteration, an offspring x0is produced by ﬂipping each bit of

xwith probability 1/n [10]. The offspring x0replaces xif it is ﬁtter

with respect to the ﬁtness of a solution which is as follows,

f(1+1)(x) = (max{0, α(x)−α}, P (x))

where α(x)denotes the probability of chance constraint violation

based on Chebyshev’s inequality or Chernoff bound derived for

4

Algorithm 2: (1+1)-EA

1generate x∈ {0,1}nuniformly at random

2while termination criterion not satisﬁed do

3y←create an offspring by ﬂipping each bit of x

independently with the probability of 1

n

4if f(1+1)(y)f(1+1) (x)then

5x←y

6return x

CCKP with uniform distribution in [27] as follows,

PrChebyshev ≡Pr [W(x)≥C]≤δ2Pn

i=1 xi

δ2Pn

i=1 xi+ 3(C−E(W(x)))2,

PrChernoff ≡Pr[W(x)≥C]

≤exp −3 (C−E(W(x)))2

4δ3δPn

i=1 xi+C−E(W(x))!.

The ﬁtness function f(1+1) is in lexicographic order which means

that ﬁrst, the algorithm searches for a feasible solution according to

the chance constraint and optimizes the proﬁt afterwards. We have,

f(1+1)(x0)f(1+1) (x)

⇐⇒ (max{0, α(x0)−α}<max{0, α(x)−α})

∨((max{0, α(x0)−α}= max{0, α(x)−α})

∧(P(x0)≥P(x)))

Table 1. Corresponding weight and proﬁt interval for knapsack problems

benchmark

type weight (wi)proﬁt

Uncorrelated [1,1000] [1,1000]

Bounded strongly correlated [1,1000] E(wi) + c¸

When a change occurs in the dynamic constraint, the individual

(x) may become infeasible, and its probabilistic constraint violates

α. Therefore, (1+1)-EA mutates xto ﬁnd a feasible solution for the

newly given constraint and optimizes the proﬁt afterwards.

4 EXPERIMENTAL INVESTIGATION

In this section, we deﬁne the setup of our experimental investigation;

we apply the bi-objective optimization with the introduced objectives

and compare it with the single-objective optimization.

4.1 Experimental Setup

For this study, we use the binary knapsack test problems introduced

in [22] and later developed for dynamic knapsack problem in [25].

We consider two types of uncorrelated and bounded strongly cor-

related test problems. The latter is more difﬁcult to solve because

the proﬁt correlates with the weight [22]. Note that in our chance-

constrained setting, for bounded strongly correlated instances, we

consider the correlation between the expected weight and the proﬁt.

Table 1 lists the corresponding weight and proﬁt for each type of

knapsack instance where c¸ denotes a constant number [22].

For the Dynamic parameters of test problems, we deﬁne rwhich

determines the magnitude of changes. we consider changes accord-

ing to the uniform distribution in [−r, r]where r∈ {500,2000}to

consider the small and large magnitude of changes in the knapsack

constraint, respectively.

Also, the parameter ηfor POSDC has been considered equal to r

to cover the interval of the uniform distribution entirely for storing

desirable solutions [25].

Another dynamic parameter is the frequency parameter of τ,

which determine how many iterations there are between dynamic

constraint changes. We set τ∈ {100,1000}to observe fast and slow

changes in the constraint, respectively.

For stochastic parameters, we set α∈ {0.01,0.001,0.0001}to

consider loose and tight probability of chance-constraint violation

probability. We also set δ∈ {25,50}to assign small and large un-

certainty interval in the weight of items with uniform distribution. To

ensure that the weights of items subject to uncertainty are positive,

we also add a value of 100 to all weights to avoid negative values.

We use dynamic programming to ﬁnd the exact optimal solution

with maximal proﬁt P(x∗)for the deterministic variant of the knap-

sack problem. Therefore, we record P(x∗)for every dynamic capac-

ity change for each knapsack instances based on rand τ.

To evaluate the performance of our algorithms for DCCKP, we

consider the ofﬂine error which represents the distance between the

algorithm best-obtained solution in each iteration with respect to

P(x∗). Let xbe the best solution obtained by the considered algo-

rithm in iteration i. The ofﬂine error for iteration iis given as

φi=

P(x∗)−P(x)if Pr [W(x)≥C]≤α

(1 + Pr [W(x)≥C]) P(x∗)otherwise.

Note, that every solution xnot meeting the chance constraint receives

a higher ofﬂine error than any solution meeting the chance constraint.

The total ofﬂine error

Φ = P106

i=1 φi

106.

is the summation of ofﬂine error at each iteration divided by the num-

ber of total iterations (106).

4.2 Experimental Results

We combine the parameters of r,τ,αand, δto produce DCCKP test

problem instances for uncorrelated and bounded strongly correlated

with different types of complexities. For instance, a test problem with

r= 2000,τ= 100,α= 0.0001 and δ= 50 represents the most dif-

ﬁcult test problem; because the magnitude of dynamic change in the

knapsack capacity is large and the capacity changes very fast every

100 iterations. Also, the allowable probability of chance-constraint

violation is very tight, and the uncertainty interval in the weight of

items is big.

We apply POSDC and (1+1)-EA integrated with Chebyshev and

Chernoff inequality tails to DCCKP instances. Speciﬁcally, we in-

vestigate the following algorithms:

•(1+1)-EA with Chebyshev’s inequality: (1)

•(1+1)-EA with Chernoff bound: (2)

•POSDC with Chebyshev’s inequality: (3)

•POSDC with Chernoff bound: (4)

Each algorithm initially runs for 104warm-up iterations before the

ﬁrst change in the capacity occurs and continues for 106iterations.

5

Table 2. Statistical results of total ofﬂine error for (1+1)-EA and POSDC with small change in the dynamic constraint

r τ δ α (1+1)-EA-Chebyshev (1) (1+1)-EA-Chernoff (2) POSDC-Chebyshev (3) POSDC-Chernoff (4)

uncorrelated

Mean Std Stat Mean Std Stat Mean Std Stat Mean Std Stat

500 100 25 0.01 4232.51 475.50 2(∗),3(−),4(−)4288.23 481.60 1(∗),3(−),4(−)1485.56 177.13 1(+),2(+) ,4(∗)1381.90 183.45 1(+),2(+),3(∗)

500 100 25 0.001 5537.00 565.70 2(−),3(−),4(−)4457.91 512.51 1(+),3(−),4(−)3162.58 392.02 1(+) ,2(+),4(−)1561.59 210.73 1(+) ,2(+),3(+)

500 100 25 0.0001 9869.75 912.78 2(−),3(−),4(−)4590.04 518.27 1(+) ,3(+),4(−)7854.36 797.79 1(+) ,2(−),4(−)1718.78 243.25 1(+),2(+),3(+)

500 100 50 0.01 4862.10 512.38 2(∗),3(−),4(−)4916.24 540.29 1(∗),3(−),4(−)2266.92 281.86 1(+),2(+) ,4(∗)2062.41 291.98 1(+),2(+),3(∗)

500 100 50 0.001 7684.09 742.16 2(−),3(−),4(−)5252.78 593.48 1(+),3(∗),4(−)5442.18 616.59 1(+) ,2(∗),4(−)2414.63 346.82 1(+),2(+) ,3(+)

500 100 50 0.0001 14435.49 1539.90 2(−),3(∗),4(−)5559.46 621.60 1(+),3(+) ,4(−)13477.00 1291.40 1(∗),2(−),4(−)2730.43 400.21 1(+),2(+),3(+)

500 1000 25 0.01 2498.45 122.34 2(∗),3(−),4(−)2425.96 102.79 1(∗),3(−),4(−)1004.26 48.61 1(+),2(+),4(∗)896.22 77.77 1(+) ,2(+),3(∗)

500 1000 25 0.001 4240.41 286.22 2(−),3(−),4(−)2655.49 108.46 1(+),3(+),4(−)2900.27 130.49 1(+),2(−),4(−)1125.94 102.26 1(+),2(+) ,3(+)

500 1000 25 0.0001 8477.51 1091.61 2(−),3(∗),4(−)2837.93 119.76 1(+),3(+) ,4(−)7526.05 817.63 1(∗),2(−),4(−)1331.74 126.31 1(+),2(+) ,3(+)

500 1000 50 0.01 3342.10 178.28 2(∗),3(−),4(−)3195.27 148.83 1(∗),3(−),4(−)1904.68 83.35 1(+),2(+),4(−)1719.30 130.07 1(+),2(+) ,3(+)

500 1000 50 0.001 6440.89 611.99 2(−),3(−),4(−)3633.98 165.28 1(+),3(+),4(−)5282.87 380.55 1(+),2(−),4(−)2162.80 167.66 1(+),2(+) ,3(+)

500 1000 50 0.0001 11975.96 2404.11 2(−),3(∗),4(−)3983.10 193.16 1(+),3(+),4(−)11652.85 2129.08 1(∗),2(−),4(−)2555.44 201.20 1(+) ,2(+),3(+)

bounded-strongly-correlated

500 100 25 0.01 3287.08 390.63 2(∗),3(−),4(−)3333.50 389.40 1(∗),3(−),4(−)1523.05 166.13 1(+),2(+) ,4(∗)1400.05 125.34 1(+),2(+),3(∗)

500 100 25 0.001 4763.94 780.09 2(−),3(−),4(−)3509.95 428.48 1(+),3(∗),4(−)3251.30 454.50 1(+) ,2(∗),4(−)1583.87 142.56 1(+),2(+) ,3(+)

500 100 25 0.0001 9387.44 2060.47 2(−),3(∗),4(−)3674.83 446.95 1(+),3(+) ,4(−)7843.42 1450.53 1(∗),2(−),4(−)1745.45 156.64 1(+),2(+) ,3(+)

500 100 50 0.01 3998.90 528.67 2(∗),3(−),4(−)4052.63 516.41 1(∗),3(−),4(−)2327.44 290.44 1(+),2(+) ,4(∗)2107.63 210.29 1(+),2(+),3(∗)

500 100 50 0.001 7092.19 1348.32 2(−),3(−),4(−)4405.88 575.40 1(+),3(+),4(−)5516.23 906.14 1(+),2(−),4(−)2465.45 254.71 1(+),2(+) ,3(+)

500 100 50 0.0001 13743.81 3964.29 2(−),3(∗),4(−)4736.65 625.25 1(+),3(+) ,4(−)12936.38 3124.07 1(∗),2(−),4(−)2790.44 288.47 1(+),2(+),3(+)

500 1000 25 0.01 1971.76 244.69 2(∗),3(−),4(−)1892.49 231.54 1(∗),3(−),4(−)823.88 116.26 1(+),2(+) ,4(∗)730.56 60.10 1(+),2(+),3(∗)

500 1000 25 0.001 3455.36 486.91 2(−),3(−),4(−)2063.78 247.58 1(+),3(∗),4(−)2265.75 288.37 1(+) ,2(∗),4(−)905.61 74.11 1(+),2(+) ,3(+)

500 1000 25 0.0001 6931.91 1328.75 2(−),3(∗),4(−)2226.64 246.62 1(+),3(+) ,4(−)5709.88 949.12 1(∗),2(−),4(−)1059.64 79.37 1(+),2(+) ,3(+)

500 1000 50 0.01 2694.66 351.22 2(∗),3(−),4(−)2539.21 302.30 1(∗),3(−),4(−)1513.94 197.83 1(+),2(+) ,4(∗)1358.32 120.78 1(+),2(+),3(∗)

500 1000 50 0.001 5297.71 876.15 2(−),3(−),4(−)2883.68 335.21 1(+),3(+),4(−)4044.39 603.89 1(+),2(−),4(−)1694.02 145.32 1(+),2(+) ,3(+)

500 1000 50 0.0001 9534.51 2279.76 2(−),3(∗),4(−)3182.34 366.82 1(+),3(+) ,4(−)8831.99 1787.84 1(∗),2(−),4(−)1990.19 163.30 1(+),2(+),3(+)

Table 3. Statistical results of total ofﬂine error for (1+1)-EA and POSDC with large change in the dynamic constraint

r τ δ α (1+1)-EA-Chebyshev (1) (1+1)-EA-Chernoff (2) POSDC-Chebyshev (3) POSDC-Chernoff (4)

uncorrelated

Mean Std Stat Mean Std Stat Mean Std Stat Mean Std Stat

2000 100 25 0.01 5948.81 569.75 2(∗),3(−),4(−)6018.58 560.15 1(∗),3(−),4(−)1931.58 366.87 1(+),2(+),4(∗)1909.30 381.82 1(+) ,2(+),3(∗)

2000 100 25 0.001 6387.66 508.07 2(∗),3(−),4(−)6074.30 577.59 1(∗),3(−),4(−)3133.97 466.62 1(+),2(+),4(−)2009.58 388.93 1(+) ,2(+),3(+)

2000 100 25 0.0001 10237.76 594.11 2(−),3(−),4(−)6170.77 579.85 1(+),3(∗),4(−)7010.05 698.68 1(+) ,2(∗),4(−)2102.31 402.86 1(+),2(+) ,3(+)

2000 100 50 0.01 6328.16 563.72 2(∗),3(−),4(−)6399.28 594.11 1(∗),3(−),4(−)2476.40 410.13 1(+),2(+),4(∗)2378.09 430.97 1(+) ,2(+),3(∗)

2000 100 50 0.001 8198.35 556.15 2(−),3(−),4(−)6592.78 582.25 1(+),3(−),4(−)4963.66 603.48 1(+),2(+) ,4(−)2601.97 454.21 1(+),2(+) ,3(+)

2000 100 50 0.0001 15154.74 668.43 2(−),3(−),4(−)6794.26 590.57 1(+),3(+) ,4(−)12102.77 742.79 1(+),2(−),4(−)2806.32 467.27 1(+) ,2(+),3(+)

2000 1000 25 0.01 3027.71 377.45 2(∗),3(−),4(−)2966.82 374.67 1(∗),3(−),4(−)974.46 188.68 1(+),2(+),4(∗)874.60 190.49 1(+) ,2(+),3(∗)

2000 1000 25 0.001 4429.57 502.95 2(−),3(−),4(−)3120.19 381.19 1(+),3(∗),4(−)2556.49 361.79 1(+),2(∗),4(−)1040.90 210.60 1(+) ,2(+),3(+)

2000 1000 25 0.0001 8650.49 843.44 2(−),3(−),4(−)3255.02 416.14 1(+),3(+) ,4(−)6959.35 732.40 1(+),2(−),4(−)1186.47 235.49 1(+) ,2(+),3(+)

2000 1000 50 0.01 3714.96 442.43 2(∗),3(−),4(−)3558.83 432.43 1(∗),3(−),4(−)1704.27 270.78 1(+),2(+),4(∗)1513.22 278.61 1(+) ,2(+),3(∗)

2000 1000 50 0.001 6464.63 664.72 2(−),3(−),4(−)3872.78 476.96 1(+),3(∗),4(−)4697.95 565.23 1(+),2(∗),4(−)1845.29 322.04 1(+) ,2(+),3(+)

2000 1000 50 0.0001 13380.75 1349.08 2(−),3(∗),4(−)4159.55 509.55 1(+),3(+),4(−)12017.35 1151.98 1(∗),2(−),4(−)2138.14 367.74 1(+) ,2(+),3(+)

bounded-strongly-correlated

2000 100 25 0.01 4560.36 185.97 2(∗),3(−),4(−)4568.83 197.34 1(∗),3(−),4(−)1840.56 84.26 1(+),2(+),4(∗)1712.51 123.38 1(+),2(+) ,3(∗)

2000 100 25 0.001 5784.27 319.36 2(−),3(−),4(−)4718.52 189.73 1(+),3(−),4(−)3795.55 168.68 1(+),2(+) ,4(−)1896.19 95.79 1(+),2(+) ,3(+)

2000 100 25 0.0001 12130.92 1256.94 2(−),3(−),4(−)4879.27 180.63 1(+),3(+) ,4(−)9177.87 928.21 1(+),2(−),4(−)2063.82 79.10 1(+) ,2(+),3(+)

2000 100 50 0.01 5337.16 166.82 2(∗),3(−),4(−)5291.89 176.82 1(∗),3(−),4(−)2745.54 52.90 1(+),2(+),4(−)2484.25 54.08 1(+),2(+) ,3(+)

2000 100 50 0.001 8834.24 746.73 2(−),3(−),4(−)5653.89 184.06 1(+),3(+),4(−)6452.92 512.65 1(+) ,2(−),4(−)2862.14 42.48 1(+),2(+) ,3(+)

2000 100 50 0.0001 19641.09 2649.89 2(−),3(−),4(−)5987.13 204.54 1(+),3(+) ,4(−)15189.24 2029.02 1(+),2(−),4(−)3205.05 58.24 1(+),2(+) ,3(+)

2000 1000 25 0.01 2508.30 264.64 2(∗),3(−),4(−)2390.56 233.31 1(∗),3(−),4(−)963.19 99.04 1(+),2(+),4(−)828.44 45.06 1(+),2(+) ,3(+)

2000 1000 25 0.001 4120.10 557.29 2(−),3(−),4(−)2581.26 264.54 1(+),3(∗),4(−)2568.42 347.75 1(+),2(∗),4(−)1010.08 58.55 1(+) ,2(+),3(+)

2000 1000 25 0.0001 8550.48 1602.26 2(−),3(∗),4(−)2745.14 281.97 1(+),3(+) ,4(−)6518.39 1139.10 1(∗),2(−),4(−)1163.33 72.29 1(+),2(+),3(+)

2000 1000 50 0.01 3302.84 395.53 2(∗),3(−),4(−)3103.47 336.52 1(∗),3(−),4(−)1732.35 211.93 1(+),2(+),4(∗)1495.03 118.31 1(+) ,2(+),3(∗)

2000 1000 50 0.001 6334.26 1019.02 2(−),3(−),4(−)3455.32 379.88 1(+),3(+),4(−)4582.94 717.91 1(+) ,2(−),4(−)1844.94 154.35 1(+),2(+),3(+)

2000 1000 50 0.0001 12880.38 2979.96 2(−),3(∗),4(−)3757.97 409.95 1(+),3(+),4(−)10047.45 2229.63 1(∗),2(−),4(−)2154.25 185.00 1(+) ,2(+),3(+)

Table 4. Statistical results of total ofﬂine error for NSGA-II with large change in the dynamic constraint (r= 2000)

τ δ α uncorrelated bounded-strongly correlated

NSGA-II-Chebyshev (5) NSGA-II-Chernoff (6) NSGA-II-Chebyshev (5) NSGA-II-Chernoff (6)

Mean Std Stat Mean Std Stat Mean Std Stat Mean Std Stat

100 25 0.01 2215.77 295.97 3(∗),4(∗),6(∗)2130.59 279.40 3(∗),4(−),5(∗)2390.79 189.51 3(−),4(−),6(∗)2234.28 194.74 3(−),4(−),5(∗)

100 25 0.001 3509.35 421.03 3(∗),4(−),6(−)2268.36 289.77 3(∗),4(∗),5(+) 4399.93 374.16 3(∗),4(−),6(−)2416.83 148.89 3(∗),4(∗),5(+)

100 25 0.0001 7401.77 621.03 3(∗),4(−),6(−)2387.85 290.26 3(∗),4(∗),5(+) 9648.37 1205.88 3(∗),4(−),6(−)2652.75 166.96 3(+) ,4(∗),5(+)

100 50 0.01 2828.78 329.66 3(−),4(−),6(∗)2637.66 327.82 3(+) ,4(∗),5(+) 3342.50 234.54 3(−),4(−),6(∗)3042.16 200.33 3(∗),4(−),5(∗)

100 50 0.001 5358.32 552.16 3(∗),4(−),6(−)2905.99 352.18 3(∗),4(∗),5(∗)6993.80 744.54 3(∗),4(−),6(−)3439.87 215.47 3(+),4(∗),5(+)

100 50 0.0001 12392.67 677.71 3(∗),4(−),6(−)3150.29 392.47 3(+),4(∗),5(+) 15160.00 2418.27 3(∗),4(−),6(−)3798.48 239.52 3(+),4(∗),5(+)

1000 25 0.01 1275.93 157.45 3(−),4(−),6(∗)1170.70 157.26 3(∗),4(−),5(∗)1123.35 150.54 3(−),4(−),6(∗)1009.96 115.48 3(∗),4(−),5(∗)

1000 25 0.001 2844.91 318.58 3(∗),4(−),6(−)1333.13 168.23 3(+),4(∗),5(+) 2726.67 392.87 3(∗),4(−),6(−)1198.56 122.26 3(+) ,4(∗),5(+)

1000 25 0.0001 7228.72 700.04 3(∗),4(−),6(−)1495.80 180.51 3(+),4(∗),5(+) 6597.74 1210.95 3(∗),4(−),6(−)1360.20 150.56 3(+),4(∗),5(+)

1000 50 0.01 2016.38 233.56 3(−),4(−),6(∗)1812.49 222.10 3(∗),4(∗),5(∗)1872.84 237.35 3(∗),4(−),6(∗)1682.10 184.08 3(∗),4(∗),5(∗)

1000 50 0.001 4967.78 537.00 3(∗),4(−),6(−)2153.47 265.98 3(+),4(∗),5(+) 4679.54 753.61 3(∗),4(−),6(−)2032.80 215.07 3(+) ,4(∗),5(+)

1000 50 0.0001 12192.94 1174.98 3(∗),4(−),6(−)2447.42 306.57 3(+),4(∗),5(+) 9885.44 2350.72 3(∗),4(−),6(−)2342.47 253.88 3(+),4(∗),5(+)

6

Tables 2 and 3 report the performance of single-objective and bi-

objective optimization by the average and standard deviation of to-

tal ofﬂine error for 30 independent runs. Lower total ofﬂine error is

better because it shows the algorithm was closer to the P(x∗)for

each iteration. Note that when the problem becomes more uncertain,

the feasible region (without violating the probabilistic constraint) be-

comes more restrictive and the ofﬂine error will be increased.

Statistical comparisons are carried out by using the Kruskal-Wallis

test with 95% conﬁdence interval integrated with the posteriori Bon-

ferroni test to compare multiple solutions [5]. The stat column shows

the rank of each algorithm in the instances; If two algorithms can be

compared with each other signiﬁcantly, X(+) denotes that the cur-

rent algorithm is outperforming algorithm X. Likewise, X(−)signi-

ﬁes the current algorithm is worse than the algorithm Xsigniﬁcantly.

Otherwise, X(∗)shows that the current algorithm is not different sig-

niﬁcantly with algorithm X. For example, numbers 1(+),3(∗),4(−)

denote the pairwise performance of algorithm (2). The numbers show

that algorithm (2) is statistically better than algorithm (1); it is not

different from algorithm (3) and it is inferior to algorithm (4).

Table 2 lists the results when ris 500. We observe that when the

environment of the problem becomes more complex, ﬁnding a so-

lution which has a close distance to the optimal solution is harder.

As τdecreases, δincreases and αbecomes tighter, the ofﬂine error

for both (1+1)-EA and POSDC increases. However, as the problem

becomes more difﬁcult to solve, POSDC obtains solutions with a

lower total ofﬂine error and lower standard deviation in comparison

with (1+1)-EA. We also ﬁnd that the algorithms which use Chernoff

bound outperform other algorithms which use the Chebyshev’s in-

equality.

Table 3 lists our results when ris 2000. We observe that POSDC

can obtain better solutions in comparison with (1+1)-EA. When we

consider a bigger magnitude of changes in the constraint bound, the

population size of non-dominated solutions in POSDC is bigger than

when ris 500; because ηis equal to r, POSDC covers a bigger range

of solutions which leads to a bigger population.

Therefore, when the changes occur faster (smaller τ), POSDC has

less time to evolve its population. POSDC only mutates one indi-

vidual chosen randomly in its population, leading to a lower chance

of choosing the best individual for the mutation in its population.

In contrast, (1+1)-EA only handles one individual, mutates and im-

proves it on all iterations. Introducing our second objective function

for the bi-objective optimization approach helps POSDC to tackle

all these drawbacks and outperform its counterpart single-objective

approach; because trade-off solutions contain more information in

principle of ﬁnding better solutions.

For further investigation of our bi-objective optimization, we also

apply the Non-dominated Sorting Genetic Algorithm (NSGA-II) [8],

which is a state of the art multi-objective EA when dealing with

two objectives. We run NSGA-II with a population size of 20 using

Chebyshev and Chernoff inequality tails which are algorithms (5)

and (6), respectively in Table 4. Table 4 shows the results of NSGA-

II when ris 2000 for uncorrelated and bounded strongly correlated

instances and compares the performance of NSGA-II with POSDC.

For brevity, we only report stats of comparison between NSGA-II

and POSDC.

To have a fair comparison, we modify NSGA-II to keep the best-

obtained solution for the given knapsack bound Cin each iteration.

Table 4 shows that in most of the instances, NSGA-II performs as

good as POSDC when using the Chernoff bound. However, POSDC

can outperform NSGA-II in instances where δ= 25 and α= 0.01,

which is the most straightforward instance. The main difference be-

tween NSGA-II and POSDC is the selection mechanism. NSGA-II

uses the crowding distance sorting to maintain diversity through the

evolution of its population. This comparison can point out the possi-

ble research line to further investigate state-of-art non-baseline EAs

and multi-objective EAs solving DCCKPs.

5 CONCLUSIONS

In this paper, we dealt with the dynamic chance-constrained knap-

sack problem where the constraint bound changes dynamically over

time, and item weights are uncertain. The key part of our approach

is to tackle the dynamic and stochastic components of an optimiza-

tion problem in a holistic approach. For this purpose and to apply bi-

objective optimization to the problem, we developed an objective C∗

which calculates for a given solution xthe smallest possible bound

for which xwould meet the chance constraint. This objective func-

tion allows keeping a set of non-dominated solutions with different

C∗where an appropriate solution can be used to track the optimum

after the dynamic constraint bound has changed. As it is hard to cal-

culate the bound C∗(x)in the stochastic setting exactly, we have

shown how to calculate upper bounds for C∗(x)based on Chernoff

bound and Chebyshev’s inequality. We evaluated the bi-objective op-

timization for a wide range of chance-constrained knapsack prob-

lems with dynamically changing constraint bounds. The results show

that the bi-objective optimization with the introduced additional ob-

jective function can obtain better results than single-objective opti-

mization in most cases. Note that we also applied NSGA-II to the

problem to point out possible improvements by using state of the art

algorithms. It would be interesting for future work to extend these

investigations. In addition, our approach is not limited to dynamic

chance-constrained knapsack problems and the formulation can be

adapted to a wide range of other problems where we would formu-

late a similar second objective to deal with the chance constraint.

ACKNOWLEDGEMENTS

This work has been supported by the Australian Research Council

through grant DP160102401 and by the South Australian Govern-

ment through the Research Consortium ”Unlocking Complex Re-

sources through Lean Processing”.

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