Evolutionary Algorithms Applied to Agribusiness Scheduling Problem

Andre Noel, Jos

e Magon Jr. and Ademir Aparecido Constantino

Computer Science Department, Maringa State University, Av. Colombo, 5790, Maringa, PR, Brazil

Keywords:

Operational Research, Scheduling, Evolutionary Algorithms, Genetic Algorithms.

Abstract:

This paper addresses a scheduling problem in agribusiness context, especiﬁcally about chicken catching. To

solve that problem, memetic algorithms combined with local search in a two-phase algorithm were proposed

and investigated. Four versions of memetic algorithms were implemented and compared. Also, to apply

local search, k-swap and SRP is proposed and experimented. At last we analyze the results, comparing perfor-

mances. The obtained results show a good improvement in solutions, especially when compared to the manual

scheduling actually performed by the company that provides the data to this study.

1 INTRODUCTION

Scheduling Problem (SP) is a common name to a

computing problem set, often at NP-hard class, which

has the purpose of allocate event or resource sets

throughout a time period, satisfying a set of con-

straints, usually to optimize a ﬁtness function. That

resource might be persons, machines, vehicles, phys-

ical location, etc. The SP class arises from real prob-

lems on industry and organizations. These problems

are also observed on software engineering, according

(Xiao et al., 2013). The main challenge is usually

to ﬁnd a computational methodology to solve these

problems in a efﬁcient and effective way, making pos-

sible to generate computational systems to automate

real-problem resolutions.

In this paper, we examine a SP class variant, the

Agribusiness Scheduling Problem (ASP), at a Poultry

Industry context. It is a combinatorial optimization

problem with great computational complexity which

has been poorly addressed (Hart et al., 1999; Con-

stantino et al., 2011). The ASP is shown as a NP-

complete problem, as the mostly of scheduling prob-

lems (Bodin, 1983), without any polynomial-time

complexity algorithm that solves it with an optimal

solution.

The purpose of this work is to compare some evo-

lutionary algorithms applied to the ASP. So, in Sec-

tion 2 is explained what do we expect from a shift at

the chicken factory. In Section 3 the problem is de-

scribed in a way we can develop a solution, which is

proposed in Section 4. In Section 5 we discuss the ex-

periments and the obtained results. At the end, some

conclusions in Section 6.

2 CHICKEN CATCHING SQUAD

SCHEDULING

Our problem occurs on agribusiness context, dealing

with chicken transport to the factories. As chickens

are fragile animals, they can’t stay longer at the lor-

ries, exposed to high temperatures. So, the squad’s

shift for transport and discharging must be careful

planned to not create long queues at the factories.

There are different farms, which has different dis-

tances to the factories. When they get at the right age,

they will be catched and sent to slaughter.

Also, we have the catcher squads, who are respon-

sible to catch the chickens to transport them to the

factories. To generate the schedule, we need to ob-

serve labor law and constraints about work time and

rest time.

At last, the vehicles to transport the chickens must

be considered, calculating the capacity, velocity and

availability. All that factors may change the problem

modeling.

3 PROBLEM DESCRIPTION

The problem described here is a real-world schedul-

ing problem of a brazilian company, which daily has

to catch chicken at different farms and carry them to

three factories (slaughterhouses).

489

Noel A., Magon Jr. J. and Constantino A..

Evolutionary Algorithms Applied to Agribusiness Scheduling Problem.

DOI: 10.5220/0004895104890496

In Proceedings of the 16th International Conference on Enterprise Information Systems (ICEIS-2014), pages 489-496

ISBN: 978-989-758-027-7

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

The farms are located far from until 150 km of

each factory and different farms can be added to the

problem. So, the schedule has to be daily generated

according all different variables.

There are some squads to travel from their base to

the farms. A squad must travel to one farm, catch the

chickens, load the lorries and travel to the next farm

to do the same.

The lorries and drivers will be considered as an

unique entity, so we’ll just think about the lorry,

which may have known capacity and average velocity.

Also, the distance between each farm to the factories

is known.

Our work has to determine what load will be de-

signed to each factory and the schedule to minimize

the costs of idle factories and paid time. So, the

catches and the travels has to be calculated in a way

to avoid long queues at the factories and to catch the

chickens at the right moment.

The factories has a ventilated hangar where the

lorries can wait a determined period. Obviously it has

a limited size, so our goal is to not admit more lorries

at the queue than the hangar capacity. And even be-

ing ventilated there is a time limit that the the chickens

can wait.

Figure 1 illustrates how it all works. At this exam-

ple, we have two squads to catch the chickens to three

factories. The continuous lines represent the travel

sequence of each squad and the dashed lines repre-

sent the lorries’ travels carrying the chickens from the

farms to the factories. Also, the squads’ bases are in

different locations from the factories. Thus, we’re in-

terested on ﬁnd some informations as the sequence of

the farms to each squad, the squads’ initial and ﬁnal

work times, the time of lorries’ load and the destina-

tion of each load.

Figure 1: Daily the squads go to the farms, to catch the

chickens and load the lorries, which carry them to the facto-

ries. When they ﬁnish the load, the squad travel to the next

farm at the schedule.

To get clear, our purpose is to minimize the costs,

minimizing the work time and the idle time. To

achieve this, we have to observe some restrictions and

local laws, but we won’t take time explaining they at

this work.

4 PROPOSAL

In our approach the problem is divided into several

subproblems. At this section we present the ﬁrst

subproblem solved, that is the assignment problem,

which assign each farm load to one factory. After, we

present the subproblems of the initial solutions and

the local search. At last, the general algorithm to gen-

erate solutions to the general problem.

4.1 The Assignment Problem

To solve the load’s destination subproblem, we use

a mathematical model. This model is similar to the

Zero-One Knapsack Problem (Kellerer et al., 2004).

We get the loads by dividing the number of chick-

ens by the lorry capacity. Here we suppose an equal

capacity for the lorries. Each load is associated to:

• Origin (farm);

• Destination (factory);

• Load start time;

• Load duration;

• Responsible squad; and

• Number of chickens.

As we know the farm set and their informations, as

location and number of chickens, the number of loads

and load duration are automatically retrieved. So, we

still need to discover the destination, the squad and

the start time to catch the chickens.

Then, to minimize the costs, a fundamental phase

is to ﬁnd the assignment of the loads to the factories.

To do this, we use a binary integer linear program-

ming model which is solved using LP-Solve software.

But this model solves just part of the problem, mini-

mizing transport costs only. Still, the scheduling has

to be solved by using heuristic algorithms.

The evolutive algorithms in this paper use the lin-

ear assignment model on two phases: at the con-

struction phase and at the improvement phase (local

search). The assignment problem (AP), denoted by

PD([C

i j

]), is the following:

Min

∑

i=1

∑

j=1

i j

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Sub jectto :

∑

i=1

i j

= 1, ∀ j = 1, . . . , n

∑

j=1

i j

= 1, ∀i = 1, . . . , n

i j

∈ 0, 1, ∀i = 1, . . . , n

, ∀ j = 1, . . . , n

Let be x

i j

is a binary decision variable {0, 1},

where x

i j

= 1 if i is associated to j, or x

i j

= 0 oth-

erwise. And let be c

i j

the cost to associate i to j.

The n

value must be the same as n

value. At the

ﬁrst phase, n

represents the number of squads and

the number of loads. As the number of loads is

always greater than the number of squads, thus ﬁcti-

tious squads are created to get a squared matrix [c

i j

At improvement phase, n

and n

are always the max-

imum number of squads (ne

max

) allowed for schedul-

ing. Even though the algorithm takes the max number

of available squads, the goal is to minimize the global

cost of human resources. This model is used both on

construction and improvement phases.

In order to generate the scheduling, we can de-

ﬁne it as an activity sequence to be taken throughout

the day, with average lorry loading time of about one

hour. So, to simplify our problem, we can discretize

the day on one-hour time slot. Thus, we consider t as

the time index, with t = 1,...,24, where t = 1 represents

the ﬁrst time slot of the day, when some squad starts

its work.

4.2 Construction Phase

We can use the assignment model to create the initial

solution. Initially, all sequence of each squad (from

max

squads) are empty. So, the loads are succes-

sively assigned to the squads until assign to all squads,

using t = 1, starting the squad work at the ﬁrst time

slot. As we always have more loads than squads, to

the others time slots (t > 1) the assignment is solved

using the assignment problem.

After the initial process, we get a initial schedule,

as illustrated by Figure 2.

Figure 2: Example of an initial load distribution among the

squads.

4.3 Local Search

From initial solution, local search is a good method

to get improved solutions. In this work we use search

procedures based on VND (Variable Neighborhood

Descent), a particular case of Variable Neighborhood

Search (Hansen and Mladenovi

c, 2001).

4.3.1 k-swap

The ﬁrst local search method we used is the k-swap,

which consists on slice our schedule on k time slots,

starting at a t time, generating activity blocks. So, k-

swap investigates the swaps between the blocks to get

a improved solution.

The investigation uses successive assignment

problems, calculating the cost of assignment to dif-

ferent instances. Considering st as the starting time at

the squads schedules and et the end time, or the last

time slot of work, we can write the k-swap algorithm

as follows.

start;

z∗ ← ∞;

for t ← (st + 1) to (et - k - 1) do

Creates a cost matrix [c i j] to load ba(k, t);

Solves assignment problem AP([c i, j]);

if z < z∗ then

z∗ ← z; //saving the best z value till now

t∗ ← t; //saving the best t value till now

end if

end for

Associates loads to squads according AP([c i, j])

on t∗;

end

Figure 3 illustrates an example of 1-swap for t

= 4, using a graph to represent the activities that 4

squads would perform in a seven-hour period to load

18 loads. The labeled vertices represent the loads,

while blank vertices represent time slots without any

loading (rest or travel). The continuous edges rep-

resent the predeﬁned activity sequence and dashed

edges represents the possible swaps.

Figure 3: 1-swap considering only one interval at t = 4.

At Figure 4, we observe an example of 2-swap,

considering times t = 4 and t = 5, using consecutive

time slots.

EvolutionaryAlgorithmsAppliedtoAgribusinessSchedulingProblem

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Figure 4: 2-swap considering the consecutive intervals t = 4

and t = 5.

4.3.2 Split and Recombination Procedure

Another local search investigated is the Split and Re-

combination Procedure (SRP). This procedure works

slicing the schedule between two work times, dividing

the schedule on two parts, and recombines it using the

assignment problem. Figure 5 illustrates that process.

Figure 5: SRP applied at interval between t = 1 and t = 2.

4.3.3 bl-VND General Algorithm

The bl-VND algorithm is based on VND (Variable

Neighborhood Descent), which explores the solution

space with systematic neighborhood changes (Hansen

and Mladenovi

c, 2001), if current solution weren’t

improved on a particular neighborhood, a next neigh-

borhood is explored and so on. Let’s state R as the

neighborhood set, N

, N

, . . . , N

. On our algorithm,

we use R = 6, where N

= 1-swap, N

= 2-swap, . . . ,

= 5-swap and N

= SRP.

Each iteration of bl-VND explores all neighbor-

hoods and the algorithm stops when no improvement

occurs on any of iterations. The algorithm is pre-

sented below.

start

Initialization:

start

selects a neighborhood structure set N,

(r = 1, ..., 6);

selects a initial solution S;

end

repeat

r ← 1;

repeat

Local search: Find s’ solution from s

(s’ ∈ N r(s));

if f1(s’) < f1(s) then

s ← s

;

r ← 1;

end if

else

r ← r + 1;

end else

until r = R;

until satisfies break criteria;

end

4.4 Local Search at Queues

The construction phase and the previous local

searches aim to minimize the costs with paid time.

So, until now we didn’t consider the impact on the

factories’ work, as idle time or lorries’ wait time at

the queues.

To take care of that problem a penalty was intro-

duced to evaluate the idle time under our objective

function.

Then, to minimize the idle time and the wait time

at the hangars, a local search algorithm, named bl-

Queue, was designed. The main role of it is to change

the initial time of each squad’s schedule, in order to

don’t have many loads waiting at the same time on the

same factories.

bl-Queue algorithm then evaluates the impact of

each schedule starting at the times t = 1, . . . ,t

max

where t

max

is the max time allowed to unload the loads

on time to slaughter. This is done successively to each

of the squads. The below algorithm presents how bl-

Queue works.

start;

for s ← 1 to sn do

Finds a t time to starts the schedule

to s squad with minor cost;

Changes s initial time to t;

end for

end

To determine the time of the queues, a simulation

is performed, using the 3-phases method (Medina and

Chwif, 2007). In short, this method does:

• Checks the time of all events in schedule to select

the one who occurs ﬁrst. Update the simulation

clock to the selected time.

• Executes the selected event and the entities (loads)

are moved to the waiting queues to the next activ-

ity.

• Searchs the entities at the queue and starts the

events that satisﬁes current conditions. Moves the

entities from queues to activities and calculates

the time each entity stayed at the queues.

The loads analyzed before can be the entities

whose realizes the simulation events. Once we know

the squads initial work time, is possible to deter-

mine at what time the loads will arrive at the desti-

nation. So, we can calculate how many time each en-

tity stayed at queue. At the end of simulation we get

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statistics about the queues, as total time of wait and

average queue size.

4.5 Memetic Algorithms

Memetic Algorithms belongs to the populational al-

gorithms class. It uses several solutions searching

feasible solutions in a search space. The main fea-

ture of that approach is to uses hybridization of differ-

ent algorithmic techniques to a speciﬁc problem (Neri

et al., 2011).

That class of algorithms was introduced by

Moscato (Moscato, 1989) describing an evolutionary

process that uses local search as a decisive part of the

evolution. It may be seen as an local reﬁnement inside

a search space, in a way an agent can has his adaption

level increased after a reﬁnement stage.

The fundamentals of Memetic Algorithms con-

sists on combine different meta-heuristics concepts

and strategies, as the population-based search with lo-

cal search, intending to join the advantages of them

(Neri et al., 2011).

Four different versions of Memetic Algorithms

were developed in this work. They’re evolutive al-

gorithms with construction procedures, local search

and other techniques. The new algorithms are: Hi-

erarchical Memetic Algorithm (HMA), Hierarchical-

generational Memetic Algorithm (HGMA), Alternate

Memetic Algorithm (AMA) and Coevolutive Cooper-

ative Memetic Algorithm (CCMA).

5 EXPERIMENTS AND RESULTS

Some experiments were performed in order of an-

alyze and compare different implementations. The

four memetic algorithms of previous section were

used, as the existing version proposed by (Con-

stantino et al., 2011).

First of all, we deﬁne the parameters. Then we

analyze the impact of local search. At last, we show

the results using real data.

5.1 Deﬁning Parameters

To perform the experiments, we had to set up all the

parameters in a way to get a better performance using

genetic algorithms.

As observed, the crossing genetic operator gener-

ates many infeasible solutions. So, empirically, was

deﬁned a Crossover Rate of 100% to force the algo-

rithm to always use that operator.

After some experiments to calibrate our parame-

ters and some statistical analysis, we adjust the algo-

rithm parameters as showed on Table 1.

Table 1: Initial parameters for genetic algorithm.

Parameter Value

Population size 120

Mutation tax 3%

Number of iterations (MaxIt) 2000

Ω 50%

5.2 Local Search Impact

To identify the real contribution of local search to ob-

tain the solutions, a experiment was performed with

the aim to quantify the acting of the search at each

current best global solution, comparing to the other

solutions obtained by genetic algorithm.

The Figures 6, 7 and 8 show graphs generated

from tests using real data for test cases A, B and C.

Figure 6: Obtained results using GA and GA + Local

Search - Test case A.

Two scales were used for cost axis, because there

is a great distance between costs of using only GA or

GA combined with local search. After analyze these

graphs, we observed that the genetic algorithms have

the task to create potential good solutions, not nec-

essarily having good costs, whilst local search algo-

rithms are responsible to get solutions with acceptable

costs.

Another observation at these graphs is that in none

of the cases the solution was improved after 1000 it-

erations. Even though, there are some cases where

higher values of iterations has proved to be useful.

5.3 Computional Results

The experiments were performed using a CPU Intel

EvolutionaryAlgorithmsAppliedtoAgribusinessSchedulingProblem

493

Figure 7: Obtained results using GA and GA + Local

Search - Test case B.

Figure 8: Obtained results using GA and GA + Local

Search - Test case C.

Core I5, 2.3 GHz, under Ubuntu 11.04 64 bits, ker-

nel Linux 2.6.38, using C++ language, compiled with

GCC, from GNU. These experiments used real data,

collected from a real company over a three months

operation period (62 data instances). This period in-

volves 112 catchers, divided on squads of 14 persons

(including the driver).

To solve the mathematical model we used the LP-

Solve, a solver used to solve linear models, non-linear

models and integer mathematical programming, as

the proposed model.

A implementation of Kuhn-Munkres algorithm

(Kuhn, 1955), also known as Hungarian Algorithm,

was used to solve the Assignment Problem.

At Table 2 shows the instances used to the experi-

ments. Each instance represents a work day. So, at the

table is possible to observe the problem size, observ-

ing how many farms (fq) and loads (oq) are involved.

Table 2: Instances of the problem used on the experiments.

Instance fq oq Instance fq oq

1 10 48 32 7 52

2 9 46 33 6 52

3 9 47 34 5 44

4 8 44 35 5 38

5 9 50 36 7 54

6 9 48 37 7 26

7 10 46 38 5 42

8 6 37 39 9 48

9 7 52 40 8 55

10 6 43 41 9 53

11 7 47 42 8 45

12 5 37 43 10 57

13 6 46 44 12 55

14 5 37 45 9 43

15 5 49 46 8 35

16 5 38 47 9 42

17 4 49 48 12 41

18 7 49 49 8 45

19 7 43 50 10 45

20 10 44 51 7 52

21 5 44 52 9 47

22 2 45 53 5 49

23 4 46 54 7 46

24 4 39 55 6 50

25 7 41 56 7 54

26 7 53 57 6 49

27 5 52 58 7 44

28 4 52 59 7 46

29 3 37 60 3 26

30 4 46 61 2 16

31 7 54 62 8 54

Overall, the average of farms visited by day is 6.8,

with an average of 45.4 loads transported.

5.4 Explained Results

Table 3 shows the obtained results separated by

the different implementations of memetic algorithms.

Explaining, we have:

• The results about total distance traveled are the

same in all versions because it’s obtained from the

same mathematical model.

• Although the computational time has vary among

the versions, in no one the time has increased in a

way that prejudices any version. It all ﬁnished in

a feasible computational time.

• The Manual column has the cost of the original

scheduling, manually generated by the company.

• The HMA column shows the results using the

HMA algorithm with initial parameters conﬁgu-

ration.

• The AHMA column shows the results using the

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Table 3: Results summary from performed implementations.

Manual HMA AHMA HGMA AMA CCMA

Total paid

4060.25 3843.45 3766.54 3672.29 3837.28 3656.51

time (hours)

Total dist.

370436.8 347834.4 347834.4 347834.4 347834.4 347834.4

travelled (km)

Total idle

60.12 17.15 14.68 6.03 17.21 0.00

time (hours)

Avg lorries

4.01 2.35 2.14 2.48 3.74 2.24

at queue

Avg wait

119.04 4.33 4.33 26.82 26.24 4.93

time (minutes)

Lorries overﬂow

92 9 8 2 10 0

at queue

Infeasible schedule

51 7 7 4 13 0

quantity

Computational

- 115 130 160 120 180

time (minutes)

Table 4: Qualitative results for some instances of the problem.

Manual HMA AHMA HGMA AMA CCMA

Instance A

Factory got stopped? Y Y N N Y N

Got lorries overﬂow? Y Y Y N Y N

Was schedule infeasible? Y Y Y Y Y N

Instance B

Factory got stopped? Y Y Y Y Y N

Got lorries overﬂow? Y Y N N Y N

Was schedule infeasible? Y Y Y Y Y N

Instance C

Factory got stopped? Y Y Y N Y N

Got lorries overﬂow? Y Y Y Y Y N

Was schedule infeasible? Y Y Y N Y N

Adjusted-HMA algorithm, which has the parame-

ters adjusted, as shown at section 5.1 (Table 1)

• From HGMA column, we observe that the Gen-

erational approach provides better results, despite

of a increasing at the wait time at the factories.

• The AMA column shows the results using the

AMA algorithm, based on joining both phases of

HMA algorithm, but it didn’t improves the results,

getting worse at some aspects.

• The CCMA column shows the results using the

CCMA algorithm, based on coevolutive and pre-

vious versions to get a version that presented the

best results. It shows evidences that this approach

is more ﬂexible and makes a better exploration of

the search space.

5.5 About CCMA Version

The main contribution of CCMA version is the better

performance to generate scheduling to the test cases

previous versions was not able to ﬁnd good solutions.

Table 4 shows qualitative results about proposed

algorithms to some instances of the problem.

Table 5 compares CCMA results with the manual

scheduling generated by the company, evaluating the

difference between the solutions.

The results present a signiﬁcant reduction of oper-

ational costs. The paid time is related to the squads,

so the cost must be multiplied by 14, the number of

persons at each squad.

Also, it reduces infeasible duties, which are sched-

ules with erroneous calculation of travel time between

farms. It generates extra costs to the company. 100%

of these cases were reduced using CCMA. Still it re-

moves some undesirable schedules, with long work

time without rest.

The factories idle time was also reduced to zero.

The cost of an idle factory is usually very high, reach-

ing the value of US$6000.00/hour.

It still reduces the number of lorries parked in the

factories’ hangars in about 50% and the average wait

time was reduced in more than 95%. Also, a reduction

of 100% of the lorries overﬂow. When more lorries

EvolutionaryAlgorithmsAppliedtoAgribusinessSchedulingProblem

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Table 5: Comparison between manual and CCMA computational scheduling.

Manual CCMA Reduction % reduction

Total paid

4060.25 3656.51 403.74 9.94

time (hours)

Total dist.

370436.8 347834.4 22602.4 6.1

travelled (km)

Total idle

60.12 0.00 60.12 100

time (hours)

Avg lorries

4.01 2.24 1.77 44.1

at queue

Avg wait

119.04 4.93 114.11 95.86

time (minutes)

Lorries overﬂow

92 0 92 100

at queue

Infeasible schedule

51 0 51 100

quantity

are stopped than the factory capacity, the lorry has to

go to some unappropriated place, which may cause

chickens mortality and increase of operational costs.

6 CONCLUSIONS

This work presented a study on heuristic algorithms

to solve a scheduling problem in agribusiness context.

After all study and experiments, we can conclude:

• Memetic algorithms had an important role in this

work due their ﬂexibility and facility to incor-

porate new procedures. Also, they ease solving

problems with hard mathematical modeling.

• Performed experiments had demonstrated the fea-

sibility of using computational systems to auto-

mate the schedule generation to solve real prob-

lems.

• Local search had a great importance to reduce the

costs on obtained solutions.

• Comparing the obtained solutions with manual

solutions of the company, evolutive algorithms

had a very great reduction on operational costs,

generating schedules in feasible computational

time. The ﬁnal solutions presents improvements

both on quality and in reduction of the costs.

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