Selecting Genetic Operators to Maximise Preference Satisfaction in a

Workforce Scheduling and Routing Problem

Haneen Algethami

, Dario Landa-Silva

and Anna Mart

ınez-Gavara

School of Computer Science, ASAP Research Group, The University of Nottingham, Nottingham, U.K.

Estad

ıstica y Investigaci

on Operativa, Universidad de Valencia, Valencia, Spain

Keywords:

Genetic Operators, Constraints Satisfaction, Scheduling and Routing Problem, Home Health Care.

Abstract:

The Workforce Scheduling and Routing Problem (WSRP) is a combinatorial optimisation problem that in-

volves scheduling and routing of workforce. Tackling this type of problem often requires handling a consider-

able number of requirements, including customers and workers preferences while minimising both operational

costs and travelling distance. This study seeks to determine effective combinations of genetic operators com-

bined with heuristics that help to ﬁnd good solutions for this constrained combinatorial optimisation problem.

In particular, it aims to identify the best set of operators that help to maximise customers and workers pref-

erences satisfaction. This paper advances the understanding of how to effectively employ different operators

within two variants of genetic algorithms to tackle WSRPs. To tackle infeasibility, an initialisation heuristic

is used to generate a conﬂict-free initial plan and a repair heuristic is used to ensure the satisfaction of con-

straints. Experiments are conducted using three sets of real-world Home Health Care (HHC) planning problem

instances.

1 INTRODUCTION

The workforce scheduling and routing problem

(WSRP) involves scheduling and routing of work-

force to visit customers at different locations in order

to complete a set of tasks or activities. The problem

arises in real-world scenarios, such as home health

care, security guard routing and rostering, mainte-

nance personnel scheduling among other worker al-

location problems (Castillo-Salazar et al., 2016).

The WRSP is a combination of two combinato-

rial optimisation problems, personnel scheduling and

routing, which are known to be NP-hard problems

(Lenstra and Kan, 1981). The scheduling aspect al-

locates workforce to customers in order to fulﬁl work

demands as well as satisfying their preferences. The

routing aspect requires generating routes for workers

to visit customers across various locations and within

given time windows. Researchers have reported that

real-world instances of the WSRP are large and difﬁ-

cult to solve (Mısır et al., 2015; Castillo-Salazar et al.,

2016). Hence, there is a need to develop efﬁcient al-

gorithms to solve this type of problem.

Preliminary work evaluated a set of genetic oper-

ators within a steady-state genetic algorithm applied

to a few instances of a real-world home health care

(HHC) problem (Algethami and Landa-Silva, 2015).

That work produced evidence that some operators ob-

tain better results than others when used within the

steady-state genetic algorithm for WSRP scenarios.

The present paper conducts a more comprehensive

study in order to achieve a deeper understanding of

the behaviour and performance of the various genetic

operators when applied to the WSRP.

The aim in this paper is to identify the best combi-

nation of genetic operators for each WSRP instance,

in order to maximise the satisfaction of customers and

workers preference constraints. Twelve genetic oper-

ators are considered in different combinations. The

two genetic algorithms (GA) applied in this study are

a steady state GA and a generational GA.

2 RELATED WORK

The routing component of the WSRP is related to

variants of the classical vehicle routing problem

(VRP) and in particular to the vehicle routing problem

with time windows (VRPTW) (Toth and Vigo, 2014).

Many GA applications have been used to tackle VRP

including hybrid approaches incorporating heuristic

methods and problem-speciﬁc operators to avoid pre-

416

Algethami H., Landa-Silva D. and MartÃ nez-Gavara A.

Selecting Genetic Operators to Maximise Preference Satisfaction in a Workforce Scheduling and Routing Problem.

DOI: 10.5220/0006203304160423

In Proceedings of the 6th International Conference on Operations Research and Enterprise Systems (ICORES 2017), pages 416-423

ISBN: 978-989-758-218-9

mature convergence of the GA (Prins, 2004; Chang

and Chen, 2007). In addition, the study by (Prins,

2004) suggested the best genetic components for an

efﬁcient GA to tackle VRP problems. According to

that study, order crossover (OX) is the most suitable

operator for VRP-like problems.

Genetic algorithms (GAs) have been effective in

providing good solutions relatively quickly, partic-

ularly when addressing real-world scheduling prob-

lems (Kotecha et al., 2004; Aickelin and Dowsland,

2004). It has been argued that this success is a result

of the GA’s capability to solve different segments of a

problem simultaneously (Rothlauf, 2003).

A number of studies have applied GAs to real-

world problems where scheduling and routing are

combined. Examples include (Cowling et al., 2006;

Mutingi and Mbohwa, 2014). In those works, the fo-

cus has been on algorithm design in order to obtain

good solutions. However, well-known operators and

repair heuristics were used to deal with infeasibility

issues. Far too little attention has been given to in-

troducing new genetic operators to reduce the overall

cost. To date, the impact of selecting compatible op-

erators for tackling WSRP instances has not yet been

investigated.

The focus of this paper is not to produce the

most competitive genetic algorithm, but to advance

the understanding of how different combinations of

genetic operators perform when tackling preference

constraints in instances of the WSRP. The problem

instances used in this study were also tackled in

(Laesanklang and Landa-Silva, 2016; Pinheiro et al.,

2016). This work seeks to identify effective combina-

tions of genetic operators for tackling preference con-

straints in WSRP instances to then inform the design

of competitive GAs to tackle this difﬁcult problem.

3 PROBLEM DESCRIPTION

A WSRP solution is a daily plan of visits, i.e. a set

of workers W = {w

,... ,w

|W |

} assigned to per-

form a set of tasks T = {t

,. .. ,t

|T |

} for customers

at different locations. The assignment of a worker to

travel to a customer location in order to perform a

task is called a visit. Thus, x

i, j

is a binary decision

variable that indicates if a path connects two nodes

(visit i and visit j) or not. The assignment x

i, j

= 1

means that worker w travels from visit i to visit j,

thus w makes both visits. For visit j, if x

i, j

= 1 then

≤ r

−1 where r

is the number of workers required

for visit j and y

is an integer decision variable indi-

cating the number of unsatisﬁed assignments, hence

∑

w∈W

∑

i∈T

∑

j∈T

i, j

+ y

= r

This paper tackles a home health care (HHC) plan-

ning problem, in which workers are nurses, doctors,

health carers, etc., and customers are patients receiv-

ing health care at their home. Several features have

been identiﬁed as important in solutions to HHC sce-

narios, such as distance travelled and customers’ and

workers’ requirements and preferences (Mankowska

et al., 2014). A good quality plan for an HHC plan-

ning problem should have a low operational cost as

well as assigning workforce. Thus, a solution requires

all tasks to be assigned while satisfying some require-

ments. That is, assigning tasks according to workers’

skills and avoiding time conﬂicts in respect to work-

ers’ time and area availability. A time conﬂict oc-

curs when a worker is assigned to visits overlapping in

time. Additional preferences include workers prefer-

ring to work in certain geographical areas, customers

requiring workers with special skills or preferring cer-

tain workers to perform a task.

Table 1 lists WSRP objectives and constraints

considered here. See (Laesanklang and Landa-Silva,

2016) for details of the MIP model of this WSRP.

Note that in (Laesanklang and Landa-Silva, 2016),

unassigned visits constraint is considered as a soft

constraint. However, here this is a hard constraint,

hence all visits must be assigned. Additionally, in this

paper, time-conﬂict constraint is introduced, while

the study by (Laesanklang and Landa-Silva, 2016)

avoided conﬂicts. The decomposition method divided

a problem into sub-problems, then available workers

were updated so that no conﬂicting assignments ex-

ists.

A solution S is a set of assignments to workers in

order to make visits. The objective function includes

the operational cost and the penalty cost. The oper-

ational cost is the accumulated cost d

i, j

+ p

, where

i, j

is the distance travelled between visit i to visit j

and p

is the cost of assigning worker w to visit j, i.e.

wages plus journey costs for all workers, as calculated

by the service provider in our HHC scenarios.

The penalty cost is the accumulated penalty for

the violations on constraints. An assignment can be

written as a tuple x

i, j

,ψ

,θ

,τ

. Where, a

the arrival time of a worker w to the location of a visit

j. The assignment is also composed of binary deci-

sion variables indicating an assignment of worker w to

visit j with violations on area availability (ψ

), time

availability (θ

) and conﬂicting assignments (τ

The non-satisfaction of preferences is also in-

cluded in the penalty cost. There are three types

of preferences including preferred worker-customer

pairing, worker’s preferred region and customer’s pre-

ferred skills. There is a degree of satisfaction for these

preferences when assigning a worker w to a task j and

Selecting Genetic Operators to Maximise Preference Satisfaction in a Workforce Scheduling and Routing Problem

417

Table 1: Objectives and constraints in the WSRP.

Objectives Hard constraints Soft constraints

Minimise the operational cost Assign all visits Respect workers area availability

Minimise the penalty cost Respect visit time (No time-conﬂicts) Respect workers time availability

Respect max working time per week Assign preferred workers to visits

Respect min working time per week Assign preferred workers with a speciﬁc skill

Assign qualiﬁed workforce Assign workers to preferred areas

is given by ρ

which has a value that ranges between



0,3



. For each assignment, the satisfaction value for

each preference ranges between



0,1



, from not sat-

isﬁed to satisﬁed. The satisfaction level is reverted to

a penalty by subtracting it from the full satisfaction

score, which is 3r

for a visit j.

f (S) = λ

∑

w∈W

∑

i∈T

∑

j∈T

i, j

+ p

i, j

+ λ

∑

w∈W

∑

i∈T

∑

j∈T

(3r

− ρ

i, j

+ λ

∑

w∈W

∑

j∈T

(ψ

+ θ

)

+ λ

∑

j∈T

+ λ

∑

w∈W

∑

j∈T

(1)

The best solution should have: the least operational

cost and the least penalty cost. A weighted sum

is proposed to combine the objectives into a single

scalar value (Pinheiro et al., 2016; Algethami et al.,

2016). The objective function is written as in equation

(1), where weights λ

,. .. ,λ

are deﬁned to establish

priority between objectives (more about the weights

used here later in the paper).

4 ALGORITHMS AND

OPERATORS

This paper investigates the behaviour of various

genetic operators when tackling instances of the

WSRP by analysing their performance within rela-

tively straightforward implementations of two varia-

tions of GAs. A simple solution representation al-

lows direct implementation of genetic operators on

the genotype (Rothlauf, 2003). Thus, a direct repre-

sentation scheme is used for chromosome encoding.

A vector of integers of length equal to the number of

visits, |T|, represents a one-day plan. Hence, all visits

are assigned. Indexes of the chromosome correspond

to the set of tasks T , for example the i

gene in the

chromosome means the corresponding visit with in-

dex i ∈ T . In order to increase the possibility of ob-

taining a feasible initial plan, indexes in the chromo-

some are associated to visits in non-decreasing order

of visit start time. In this way, index 1 is for the visit

with the earliest start time and index |T| is for the visit

with the latest start time. For each visit in the vector,

a worker w is selected at random from W . A worker w

may undertake more than one visit, and some workers

may not be utilised as a part of a particular one-day

plan.

4.1 Genetic Algorithms (GA)

Two GAs are implemented in this study: a steady-

state genetic algorithm (SSGA) and a generational ge-

netic algorithm (GenGA) (Vavak and Fogarty, 1996).

Such relatively simple algorithms were selected in

order to analyse the emergent behaviour and perfor-

mance of the operators on a straightforward GA im-

plementation.

Initially, a time conﬂict reduction (TCR) opera-

tor is applied to each individual in the initial popu-

lation in order to reassign visits and reduce the num-

ber of time conﬂicts. After that, the evolutionary pro-

cess for each GA is executed as follows. For the

SSGA, there is only one population P of size M dur-

ing the whole evolutionary process, where parents are

selected and the offspring is inserted; thus, no gen-

erations required. Two parents i, j are selected by

tournament selection from the parent lists L

and L

each of size M/2. To create a parent list, six different

individuals are selected at random from P and split

into two groups of three; the best individual of one

group is added to L

and the best individual of the

other group is added to L

. This process is repeated

until the two lists of parents are complete. Then, for

i, j from 1, .. ., M/2, parent i in L

and parent j in L

are combined through crossover, producing two off-

spring. The next step is to apply mutation operator.

A mutation operator is applied with some probability

to the generated offspring. That is, if the mutation is

applied to an individual k, the mutated individual k

replaces k, regardless of the objective function value.

The recombination plus mutation process is imple-

mented on the two parents; the best two individuals

out of the two parents and the two children are added

into P so that the population size remains constant.

For the GenGA a new population is created at the

ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems

418

start of each generation. The recombination plus mu-

tation process is repeated M/2 times until the new

population P

is complete. At this point, individuals in

are sorted in non-decreasing order of their ﬁtness.

The best 10% of solutions found are never removed

from the population. However, the worst 10% of in-

dividuals in P

are replaced by randomly generated

individuals to introduce diversity onto the population.

After this, the new population replaces the old one,

i.e. P = P

. Then, the WSR operator (described be-

low) is applied onto infeasible individuals within the

population based on hard constraints violations shown

in Table 1. Finally, population P

is passed to the next

generation.

4.2 Repair Operators

A time conﬂict reduction (TCR) operator works as

follows. Each pair of visits, i and j, are compared

to identify any time conﬂicts, i.e. the same worker w

being assigned to the two visits at the same time. If

there is a time conﬂict, worker w is replaced in visit

j by another worker w

, selected from the list of pre-

ferred workers for visit j, if that list exists, or selected

at random otherwise. Because TCR is applied only

once on an individual, by removing one pair of visits

at a time, it cannot ensure that all time conﬂicts are

removed, but it does reduce their number.

The worker suitability repair (WSR) operator

seeks to improve the suitability of workers for each

visit, and works as follows. For each visit in an in-

dividual, the assigned worker is checked against the

skills requirements, maximum hours constraints and

time conﬂicts (i.e. the hard constraints listed in Ta-

ble 1). If the worker does not satisfy these require-

ments, the operator aims to ﬁnd another worker who is

feasible for that visit. If no such worker can be found,

the operator leaves the original worker in place. Thus,

the WSR operator cannot ensure that all visits have a

suitable worker, but it does improve the overall as-

signment with respect to the constraints.

4.3 Genetic Operators

The aim of this study is to select the best conﬁguration

of crossover and mutation operators that can tackle

the WSRP within the SSGA and GenGA. Twelve op-

erators are implemented, ten well-known operators

plus two cost-based operators tailored for the problem

tackled here.

Ten well-known operators were chosen after a lit-

erature survey of operators applied in WSRP-related

problems (Algethami and Landa-Silva, 2015). The

operators selected are divided into two groups. One

Algorithm 1: Cost-based uniform crossover (CBUX).

Require: parent individuals p

and p

Ensure: offspring individuals o

and o

1: o

← new empty individual

2: for i ← 1 to |T | do

3: Let w

be worker assigned to visit i ∈ p

4: Let w

be worker assigned to visit i ∈ p

5: if w

and w

are both available for visit i then

6: o

← o

∪ w

7: o

← o

∪ w

8: else

9: if one of w

or w

, called w

, is available for

visit i then

10: o

← o

∪ w

11: o

← o

∪ w

12: else

13: o

← o

∪ w

14: o

← o

∪ w

15: end if

16: end if

17: end for

group has ﬁve scheduling operators: single-point

crossover (1PX), uniform crossover (Mitchell, 1998),

two-point crossover (2PX) (Hartmann, 1998), half-

uniform crossover (HX) (Eshelman, 1991) and ran-

dom swap mutation (RSM) (Cicirello and Cernera,

2013). The other group has ﬁve routing operators:

order crossover (OX) (Zheng and Wang, 2003), cy-

cle crossover (CX) (Oliver et al., 1987), partially

matched crossover (PMX) (Zhu, 2000), inversion mu-

tation (IM) (Eshelman, 1991) and scramble mutation

(SM) (Cicirello and Cernera, 2013).

Two cost-based operators operators have been

purposely designed to improve the satisfaction of soft

constraints in the WSRP, even at the expense of hav-

ing a larger total cost in the solution.

One of these operators is a cost-based uniform

crossover (CBUX) shown in Algorithm 1. Cost-based

crossovers have been applied in the literature to pro-

duce improved results by restricting mating to the fea-

sible region only (Kotecha et al., 2004). This CBUX

operator works as follows. Each position i for the two

parents, corresponding to the worker assigned to visit

i, is processed one at a time (line 2). The availabil-

ity, in terms of time and area, of the worker assigned

to visit i is examined for each parent. If both parents

have an available worker in that position, their gene is

copied to the corresponding offspring (lines 5–7). If

only one of the parents has an available worker in that

position, that gene is copied to both offspring (lines

9–11). If no parent has an available worker in that

position, offspring 1 gets the gene from parent 2 and

offspring 2 gets the gene from parent 1 (lines 13–14).

Selecting Genetic Operators to Maximise Preference Satisfaction in a Workforce Scheduling and Routing Problem

419

Algorithm 2: Cost-based mutation (CBM).

Require: individual k

1: Choose a random mutation point i ∈ k

2: Let w

be the worker assigned to visit i

3: Let Ω

be the list of preferred workers for i

4: if w /∈ Ω

then

5: Choose a random worker w

∈ Ω

6: Replace w with w

in k for visit i

7: end if

A cost-based mutation (CBM) is shown in Algo-

rithm 2. This operator seeks to ensure that workers

assigned to visits are among those considered as pre-

ferred workers for that visit. As part of the input data

in the problem instances considered here, a list of pre-

ferred workers is given for each visit as deﬁned by the

patients. Then, for position i in the individual, CBM

tries to assign one of the preferred workers for that

visit i, only if the one already assigned is not a pre-

ferred worker (lines 4–6).

5 EXPERIMENTS AND RESULTS

Table 2: Parameter settings used in the experiments.

Parameter Settings

Population size M 100

Crossover operators 1PX, 2PX, UX, HX, PMX, OX, CX, CBUX

Crossover rate P

10%, 50%, 100%

Mutation operators RSM, IM, SM, CBM

Mutation Rate P

1%, 10%, 30%

Running time 5 minutes

Experiments were conducted to evaluate the perfor-

mance of different operators on real-world WSRP

scenarios. The GAs described in Section 4 were im-

plemented with different algorithm conﬁgurations as

stated in Table 2. Values for mutation and crossover

rates are taken from previous parameter tuning exper-

iments.

The best suitable combinations of operators, each

combination is one crossover operator with one muta-

tion operator, might be later embedded in a more ef-

ﬁcient approach. To this end, the experimental study

focused on comparing the performance of the vari-

ous genetic operator combinations aimed at satisfying

customers’ and workers’ preference constraints. Nev-

ertheless, one of the major issues in the random di-

rect representation is allowing infeasible individuals

throughout the search process, so that the end result

can have individuals with high number of constraint

violations. Thus, reducing hard constraint violations,

such as skills required, maximum hours requirements

and time conﬂicts, is required to maintain feasibility

in WSRP solutions. As explained in Section 4, the

WSR mechanism is applied to individuals that present

hard constraint violations. WSR implementation oc-

curs in two stages of the GA: after the TCR, and then

again after the mutation operator in the optimisation

process.

For each GA, 32 mutation-crossover combina-

tions with 9 different rates were applied. Thus 288×2

algorithm conﬁgurations and each one was executed 8

times, all seeded with the same initial population. The

best cost solution was obtained from each set with the

same amount of computation time. The implementa-

tion was in Java running on a PC with I7 four-core

processor with hyper-threading enabled and 16GB of

RAM.

5.1 Problem Instances

Problem instances from three UK real-world HHC

scenarios are used as instances of WSRP. The

instances data and weights used here (blue set-

ting) are available at https://drive.google.com/open?

id=0B2OtHr1VocuSNGVOT2VSYmp6a2M. In this

study, three scenarios were used with 7 problem in-

stances each, for a total of 21 instances. Table 3 shows

the main features of each problem instance.

Scenario A instances are considered the smallest,

while instances in scenario B are larger. Problem

instances in scenario C are very different to the in-

stances in the other 6 scenarios in that the number of

workers is much larger than the number of visits.

Table 3: Features of the WSRP instances.

Instance A1 A2 A3 A4 A5 A6 A7 Mean

Number Visits 31 31 38 28 13 28 13 26

Number Workers 23 22 22 19 19 21 21 21

Number Areas 6 4 5 4 4 8 4 5

Instance B1 B2 B3 B4 B5 B6 B7 Mean

Number Visits 36 12 69 30 61 57 61 47

Number Workers 25 25 34 34 32 32 32 31

Number Areas 6 5 7 5 8 8 7 7

Instance C1 C2 C3 C4 C5 C6 C7 Mean

Number Visits 177 7 150 32 29 158 6 80

Number Workers 1037 618 1077 979 821 816 349 814

Number Areas 8 4 7 8 6 11 6 7

5.2 Performance of Operators

The ﬁrst set of experiments was designed to select

the best combination of the operators that maximise

customer/worker requirements and preferences sat-

isfaction. To do so, all operators listed in Table 2

were examined by statistical analysis to determine

their performance. There are four mutation operators

(RSM, IM, SM, CBM) and eight crossover operators

divided into three different groups: routing crossovers

ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems

420

Table 4: Performance Comparison of Crossover Operators Grouped by Category (Routing, Scheduling, Cost-Based) Under a

Mutation Operator.

Mutation RSM SM IM CBM

SSGA

Crossover OX UX CBUX OX UX CBUX OX UX CBUX OX 1PX CBUX

Dev. 0.09% 1.66% 4.99% 0.07% 1.23% 4.90% 0.04% 1.47% 4.24% 0.66% 1.37% 3.23%

#Best 0.71 0.19 0.00 0.76 0.14 0.00 0.86 0.05 0.00 0.48 0.38 0.05

Score 0.90 0.57 0.57 0.93 0.60 0.12 0.98 0.50 0.17 0.76 0.60 0.29

GenGA

Crossover PMX UX CBUX PMX 1PX CBUX PMX 1PX CBUX PMX UX CBUX

Dev. 0.87% 0.70% 8.98% 2.28% 1.17% 9.82% 2.90% 1.92% 9.65% 0.65% 3.12% 5.19%

#Best 0.38 0.38 0.14 0.43 0.38 0.14 0.29 0.52 0.19 0.57 0.19 0.14

Score 0.67 0.74 0.24 0.74 0.69 0.24 0.67 0.74 0.31 0.81 0.50 0.33

(OX, CX, PMX), scheduling crossovers (1PX, 2PX,

UX, HX) and cost-based crossovers (CBUX). Each

crossover was combined with one mutation at a time

for a total of 32 combinations, however only the best

performing crossover operators from each group is

presented in Table 4. The GA was executed for 5 min-

utes using the highest rate values, i.e. P

= 100% and

= 30% to ensure that the operators were utilised.

The following three metrics were used to measure

the performance of the combinations of operators:

Dev. average percentage deviation from the best pref-

erence value (the three preferences satisfaction value

of all the conﬁgurations applied). Best fraction of in-

stances in a set for which a conﬁguration matches the

best preference value. This performance metric is ab-

solute and can be compared across existing results in

different tables. Score fraction of the instances for

which the current method produces better solutions

than the other conﬁgurations, i.e. ‘win’. This score is

calculated as ((q × (p− 1)) − r)/(q × (p − 1)), where

p is the number of conﬁgurations compared, q is the

number of problem instances, and r is the number of

instances in which the p −1 competing conﬁgurations

ﬁnd a better result. Hence, the best score value is

1, when r = 0, and the worst score value is 0, when

r = q × (p − 1). This is a relative measure of per-

formance. Hence, these values are only meaningful

within one table and not across different tables.

The results presented in Table 4 are the perfor-

mance metrics values that correspond to each of the

four mutation operators, including the CBM operator

proposed in this study. These values are calculated

based on the average preferences satisfaction values

for each run. The crossover categories are: the best

routing operator, the best scheduling operator and the

CBUX operator proposed in this study. Thus, each

mutation operator has three comparable crossover op-

erators values, and the best crossover out of the three

is highlighted in bold. The aim is to identify the best

crossovers for each mutation with respect to the pref-

erences value by grouping crossovers based on their

category, thus mutation operators are not comparable

in this table.

Table 5: Performance Comparison Between the Best Com-

binations of Operators ( Mutation - Crossover ).

Procedur Dev. #Best Score

SSGA

RSM-OX 0.01% 0.52 0.87

SM-OX 0.03% 0.29 0.82

IM-OX 0.15% 0.00 0.50

CBM-OX 0.23% 0.10 0.45

GenGA

RSM-UX 0.92% 0.57 0.84

SM-PMX 7.57% 0.05 0.43

IM-1PX 2.73% 0.24 0.68

CBM-PMX 18.36% 0.05 0.25

For SSGA, OX provides the best scores, with the

highest number of best values and the lowest devia-

tion when combined with all mutation operators. For

GenGA, UX provides the best scores for RSM with

the highest number of the best values and the low-

est deviation. Even though UX is selected as the ﬁrst

competing crossover, PMX obtained the same number

of the best values for RSM; the winning crossovers

are considered in the next overall comparison. Addi-

tionally, 1PX provides the best score value with the

highest number of best solutions and lowest on devi-

ations among the compared crossovers for IM, while

PMX provides the best score value for both SM and

CBM, with the lowest deviation obtained for CBM

only.

Table 5 shows a comparison between the cho-

sen combinations of operators (mutation - crossover)

from Table 4. The aim is to identify the best combina-

tion for each GA by using the same performance mea-

surement matrices explained above. The best combi-

nation is highlighted in bold.

The results indicate that RSM–UX and RSM–UX

obtain the highest score and the smallest deviation

value among all methods, with the maximum frac-

tions of the best solutions of 0.87 and 0.84 for SSGA

and GenGA respectively. Interestingly, cost-based

methods failed to achieve good results in compari-

son to the generic operators. This result might be due

to the search space restrictions that led to infeasible

areas. However, when combined with more generic

Selecting Genetic Operators to Maximise Preference Satisfaction in a Workforce Scheduling and Routing Problem

421

Table 6: Results of the best f (S) produced by different combinations of operators, crossover probabilities P

and mutation

probabilities P

SSGA GenGA

Instance Procedure P

f (S) C pt(s) Procedure P

f (S) Cpt(s)

1 SM-PMX 0.5 0.3 5.1 176.6 RSM-PMX 1 0.3 3.5 180.8

2 SM-OX 1 0.3 4.4 184.4 SM-UX 1 0.3 2.8 190.7

3 SM-UX 1 0.3 6.1 240.8 SM-1PX 1 0.3 3.3 212.3

4 SM-UX 0.5 0.3 2.3 159.9 SM-UX 1 0.3 1.4 114.5

5 SM-1PX 1 0.1 3.2 50.7 SM-HX 1 0.3 2.4 52.4

6 SM-2PX 1 0.1 4.4 198.6 RSM-HX 1 0.3 3.6 109.0

7 SM-2PX 1 0.1 4.2 32.7 RSM-HX 1 0.3 3.7 83.2

1 SM-UX 1 0.3 2.1 239.7 RSM-PMX 1 0.3 1.7 255.0

2 SM-OX 1 0.1 2.4 50.7 RSM-HX 1 0.3 1.8 14.7

3 RSM-PMX 0.5 0.3 2.8 292.9 RSM-PMX 0.5 0.3 1.9 274.1

4 SM-2PX 1 0.3 2.9 75.3 SM-2PX 1 0.3 2.1 138.0

5 RSM-UX 0.5 0.3 3.8 243.5 RSM-2PX 1 0.3 2 243.2

6 RSM-UX 1 0.3 2.5 226.1 RSM-2PX 0.5 0.3 1.7 229.2

7 RSM-UX 0.5 0.3 3.2 231.8 RSM-1PX 0.5 0.3 1.9 288.2

1 CBM-OX 1 0.1 5454.5 285.0 CBM-OX 1 0.3 159418.6 304.8

2 SM-HX 1 0.01 4.8 15.0 SM-HX 1 0.3 3.2 152.7

3 RSM-2PX 1 0.3 3270.7 299.4 SM-OX 1 0.3 82582 293.9

4 RSM-OX 1 0.01 22.2 210.0 IM-HX 1 0.3 17.6 269.8

5 SM-PMX 1 0.1 20.1 172.6 IM-HX 1 0.3 16 267.3

6 CBM-PMX 1 0.01 20776.5 266.6 CBM-OX 1 0.01 94335.9 297.8

7 SM-UX 1 0.3 4.9 0.7 SM-UX 1 0.3 4.3 15.9

operators, in the case of CBM, they generate more di-

verse individuals that led up to high deviation among

all mutations.

5.3 Computational Results for Different

Instance Sizes

Table 6 presents the best objective values obtained for

all instances. The columns under SSGA and GenGA

provide the best values obtained for each GA under

the stopping criterion for each combination. All val-

ues are averaged and only the best values are pre-

sented. The remaining column C pt shows the compu-

tation time where the best value is found in seconds.

Two issues were considered to compile this table: the

GAs performance on each instance and the best per-

forming combinations/settings under each GA with

the minimum computation time.

It appears that GenGA provides better results

than SSGA on 85.71% of all instances. The best-

performing operators under the methods applied are

PMX, UX and HX across all instances when com-

bined with RSM and SM. However, CBM managed

to obtain some of the best results, especially for sce-

nario C instances. This indicates that problem domain

knowledge needs to be incorporated in operators for

more complicated instances. The average computa-

tion times for the best solution found for all instances

are as follows: SSGA, and GenGA are 173.954 s and

189.88 s respectively. In terms of convergence speed,

both SSGAs used here converged earlier to a local

minima, with poor results in problem sets A and B.

For problem set C, more computation-time provided

better results when using GenGA.

Despite the fact that the cost values still need to

be improved, this study has helped to understand the

performance of various combinations of genetic op-

erators executed with different probability rates and

implemented on simple steady-state and generational

GAs.

6 CONCLUSION

This paper has investigated the suitability of a set of

genetic operators when applied within a steady-state

genetic algorithm (SSGA) and a generational genetic

algorithm (GenGA) to tackle the workforce schedul-

ing and routing problem (WSRP). Twelve operators

were considered in this study including two operators

incorporating problem domain knowledge, and ten

well-known operators (three mutation operators and

ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems

422

seven crossover operators) from the literature. From

the experimental results, existing operators such as

RSM and UX perform the best. Future research will

look at investigating the performance of the repair op-

erators, parameter setting of the operators and the de-

sign of an improved evolutionary approach informed

by the better understanding achieved in this paper.

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