Using Goal Programming on Estimated Pareto Fronts to SolveMultiobjective Problems

Rodrigo Lankaites Pinheiro1,2, Dario Landa-Silva2, Wasakorn Laesanklang3

and Ademir Aparecido Constantino4

1Webroster Ltd., PE1 5NB, Peterborough, U.K.2ASAP Research Group, School of Computer Science, The University of Nottingham, U.K.

3Department of Mathematics, Faculty of Science, Mahidol University, Thailand4Departamento de Informatica, Universidade Estadual de Maringa, Maringa, Brazil

Keywords: Multi-criteria Decision Making, Goal Programming, Pareto Optimisation.

Abstract: Modern multiobjective algorithms can be computationally inefficient in producing good approximation setsfor highly constrained many-objective problems. Such problems are common in real-world applications wheredecision-makers need to assess multiple conflicting objectives. Also, different instances of real-world pro-blems often share similar fitness landscapes because key parts of the data are the same across these instances.We we propose a novel methodology that consists of solving one instance of a given problem scenario usingcomputationally expensive multiobjective algorithms to obtain a good approximation set and then using GoalProgramming with efficient single-objective algorithms to solve other instances of the same problem scena-rio. We propose three goal-based objective functions and show that on a real-world home healthcare planningproblem the methodology can produce improved results in a shorter computation time.

1 INTRODUCTION

Modern multiobjective algorithms struggle to findgood approximation sets on highly constrained pro-blems or when the number of objectives is high Gi-agkiozis and Fleming (2012). Decision-makers of-ten require a single solution and the Pareto front isonly useful to enable them to choose the preferred so-lution(s). There are a few techniques that use dom-ain knowledge to estimate Pareto fronts. Giagkio-zis and Fleming (2014) propose a technique to es-timate the Pareto front of continuous multiobjectiveproblems and then use the estimated front to obtainvalues for the decision variables of interesting soluti-ons. Their technique is particularly useful because itcan be applied to subregions of the objective space.Another technique is Bayesian multiobjective optimi-sation, which consists of using a Bayesian model tolearn computationally expensive objective functionsand use the estimation model to quickly explore thesearch space Feliot et al. (2016).

In this research we tackle a workforce schedulingand routing problem (WSRP). A WSRP solution is aplan in which skilled mobile workers are scheduled to

visit locations that are geographically scattered. Real-world applications of WSRP usually involve multipleobjectives. However, most approaches in the litera-ture tackle this problem using a weighted sum to com-bine the multiple objectives into one. Even genera-ting single-objective solutions for the WSRP has beenshown to be difficult for problem instances of consi-derable size Castillo-Salazar et al. (2014, 2016); Lae-sanklang et al. (2015a). The WSRP arises in severaldomains, for example when scheduling home healt-hcare workers, technicians, and security personnelamong others. Here, we consider the problem of sche-duling nurses and care workers to visit and providecare services to patients in their homes. Works in theliterature tackling home healthcare planning problemsusually use data from real-world scenarios Fikar andHirsch (2017). This is also the case in the presentwork. We employ data from six distinct home healt-hcare scenarios provided by Webroster Ltd., who pro-vides enterprise resource planning software for homehealthcare companies.

The quality of solutions to the WSRP consideredhere can be measured using several criteria like: in-curred costs, travel distance, satisfaction of staff pre-

132Pinheiro, R., Landa-Silva, D., Laesanklang, W. and Constantino, A.Using Goal Programming on Estimated Pareto Fronts to Solve Multiobjective Problems.DOI: 10.5220/0006718901320143In Proceedings of the 7th International Conference on Operations Research and Enterprise Systems (ICORES 2018), pages 132-143ISBN: 978-989-758-285-1Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

ferences, satisfaction of patients preferences, etc. Dueto the high number of constraints and objectives inthis problem, even state-of-the-art multiobjective al-gorithms struggle to find good approximations to thePareto front within reasonable computation time. Inproblems like this where each instance is a repetitionof the same situation but in different days, parts ofthe data repeat across multiple instances of the sameproblem scenario. For example, the same set of staffis available for different days with only minor chan-ges regarding staff avilability. Also, there is a set ofrecurring visits that need to be scheduled in each in-stance of the problem scenario. This happens in otherproblems like vehicle routing and nurse schedulingwhere instances of the same problem scenario exhi-bit similar fitness landscapes (η-dimensional surfacerepresenting the Pareto front, where η is the numberof objectives). This issue was investigated in Pinheiroet al. (2015, 2017) which proposed a technique to ana-lyse and visualise complex objective relationships andfitness landscapes in multiobjective problems. Thus,this paper proposes a methodology that exploits thissimilarity between instances of a multiobjetive opti-misaton problem, in order to solve instances of theproblem more efficiently.

The proposed methodology works as follows. Fora given dataset consisting of multiple instances of theproblem scenario, a single pilot instance is solvedusing multiobjective algorithms or the best availablemethod in order to produce an approximation to thePareto optimal set. A decision-maker uses this ap-proximation set to choose target solutions for the re-maining instances of the dataset. The chosen targetsolutions represent the desired trade-off between themultiple objectives. Then, goal programming is app-lied with an efficient single-objective solving methodto obtain solutions that in quality are close enough to(or better than) the chosen target solutions. Hence,the main contributions of this work are:1. A novel methodology that employs an approxi-

mation to the Pareto optimal front of a pilot pro-blem instance, to solve other similar instances ofthe same problem with goal programming and ef-ficient single-objective algorithms.

2. The assessment of three approaches to define thegoal programming objective function, one usesa Chebyshev objective function, another uses aweighted function derived from the target solu-tion, and the third one uses an objective functionthat seeks to minimise the largest deviation to ob-jective values in the target solution.The remaining of this paper is structured as fol-

lows. Section 2 outlines the Workforce Schedu-ling and Routing Problems Project, which is used

to illustrate the application of the proposed metho-dology. Section 3 presents the proposed methodo-logy. Section 4 presents the experimental configu-ration while section 5 presents the results. Finally,section 7 concludes this work.

2 THE WORKFORCESCHEDULING AND ROUTINGPROBLEM

The WSRP is an optimisation problem that combinesa scheduling and a routing problem. A set of m wor-kers {w1,w2, . . . ,wm}, must perform tasks on a set ofn visits {v1,v2, . . . ,vn}, which are distributed acrossdistinct geographical locations. This work considersa specific case of this problem: the home healthcareplanning problem, where workers are nurses, doctorsand care workers and the purpose of the visits is toprovide care to patients in their homes. For more in-formation about this WSRP, please refer to Castillo-Salazar et al. (2012, 2016).

The WSRP studied in this work contemplates theplanning of a single day. Several hard constraints areconsidered in the problem: visits require a set of skillsand only workers possessing the required skills can beassigned; workers can only be assigned to visits if al-lowed by their respective contracts; every visit mustcommence at a specified time in which workers mustalready be at the visits’ premises; no time clashes areallowed, i.e. a worker cannot be assigned to simulta-neous visits and there must be enough time for a wor-ker to travel between visits. Also, two soft constraintsare considered: workers may have a list of time inter-vals in which they are available; locations are groupedby geographical areas and workers may have a list ofareas in which they are available.

The objectives considered in the problem are:

Z1 denotes the sum of the distances travelled by theworkers, including commuting from their hometo the first assignment, between each assignment,and from the final assignment back to their home;

Z2 denotes the sum of monetary assignment costs(workers wages for the assigned visits);

Z3 denotes the sum of skill preferences in respect ofassigning workers with skills that are desirable butnot necessary;

Z4 denotes the sum of patients preferences in respectof assigning preferred workers; and

Z5 denotes the sum of workers preferences in respectof assigning workers to visits in their preferredarea.

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(a) Instance B1 (b) Instance B3 (c) Instance B5

(d) Instance D2 (e) Instance D4 (f) Instance D6

Figure 1: Scatter plots of three instances of datasets B and D. In each case, the x axis shows the value of objective Z4 and they axis shows the values of each of the other objectives in different colours.

Data from four different home healthcare compa-nies of different sizes and characteristics was provi-ded and six problem scenarios or dataset were gene-rated. Each dataset (A, B, C, D, E and F) is composedof 7 instances for a total of 42 problem instances. SetA has small instances (number of visits and workers)while set F has the largest instances.

Other works in the literature have also tackled theWSRP datasets considered here:

• Laesanklang et al. (2015a) presented a computa-tional study on the suitability of using exact met-hods to solve the problem. It was concluded thatonly the smaller instances can be solved to opti-mality by a mathematical solver. Later, Laesan-klang et al. (2015b) proposed to decompose theproblem by geographical locations in order to em-ploy the mathematical solver on the sub-problemsand then use heuristic algorithms to integrate thepartial solutions.

• Algethami and Landa-Silva (2015) conducted astudy of various genetic operators applied to thisproblem. It was observed that the high number ofconstraints and large size of the instances presen-ted a considerable challenge to the standard ge-netic algorithm implemented. Later, Algethamiet al. (2016) proposed a knowledge-based indirectrepresentation and genetic operators to improvethe efficiency of the genetic algorithm for tacklingthe problem.

• More recently, Pinheiro et al. (2016) presented aVariable Neighbourhood Search (VNS) metaheu-ristic to tackle the WSRP, incorporating two no-

vel heuristics tailored to the problem-domain. Thefirst heuristic restricts the search space using a pri-ority list of candidate workers and the second heu-ristic seeks to reduce the violation of specific softconstraints. Also, two greedy constructive heuris-tics were used to give the VNS a good startingpoint. Since that VNS is the best-known algo-rithm to tackle these WSRP instances, that algo-rithm is employed here within the proposed goalprogramming framework.

As preliminary work, we applied the analysistechnique proposed by Pinheiro et al. (2015, 2017) onall instances considered here. The technique consistsof performing four steps: first the global pairwise re-lationships are analysed using the Kendall correlationmethod; then the ranges of the values found on the gi-ven approximation front are estimated and assessed;next these ranges are used to plot a map using Graycode, similar to Karnaugh maps, that has the abilityto highlight the trade-offs between multiple objecti-ves; and finally local relationships are identified usingscatter plots. The conclusion is that in the studied da-tasets, instances of the same datset indeed present si-milar fitness landscapes. Figure 1 provides evidenceof that and shows the scatter plots of instances B-01,B-03, B-05, D-02, D-04, and D-06 (arbitrarily cho-sen). Each datapoint corresponds to normalised ob-jective values. For reasons of space, we omit the otherdatasets and the scatter plots of all instances. The re-semblance between the fitness landscapes of the pre-sented instances of datasets B and D is evident. Also,it is clear that different datasets present distinct fitness

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landscapes, even though they correspond to the sameproblem formulation. Finally, it is important to emp-hasise that datasets A, B and C present similar fitnesslandscapes with solutions well spread throughout theobjective space while datasets D, E and F present dis-tinct fitness landscapes with complex objective relati-onships and gaps.

3 PROPOSED METHODOLOGY

Figure 2 presents the overall concept of the proposedmethodology. We use the aforementioned WSRP toillustrate the concept. Following, we explain each ofthe four numbered steps indicated.

Figure 2: Overview of the proposed methodology. Thenumbered steps are explained below.

1. A pilot instance within the dataset is selectedby the decision-maker and solved using multiob-jective algorithms to obtain the best possible non-dominated approximation set. The pilot instancemust be representative and present similar featu-res to other instances.

2. The decision-maker chooses an appropriate solu-tion t from the obtained non-dominated set. Thischosen solution is known as the target solutionand its objective-vector is denoted by

~Zt = (Zt1,Z

t2,Z

t3,Z

t4,Z

t5).

3. Each remaining instance in the dataset may nowbe solved by the VNS (Pinheiro et al., 2016) usinga modified objective function (goal programmingvariant) that attempts to reach the target objectivevector in the current dataset.

4. The final solution obtained by the modified VNSis presented.

The modified objective function of step 3 hasan important role in the methodology. Following,we present three approaches for determining the ob-jective function. The first is the well known Cheby-shev approach. The second aims to derive a weight-vector from the target solution and the approximationset of the pilot instance. The third approach mini-mises the Euclidean distances to the target objective-vector.

3.1 Chebyshev Goal Programming

Chebyshev goal programming (Flavell, 1976) is oneof the most widely employed goal programmingtechniques that does not necessarily rely on the deci-sion maker to set priorities between objectives (Jonesand Tamiz, 2016). The technique aims to obtain a ba-lanced solution by minimising the gap to the target ofthe objective that presents the highest gap. Hence, ifthe target goals for the objectives are similarly diffi-cult to be reached, the Chebychev GP technique canobtain a balanced solution. However, if at least onetarget objective value is more difficult to achieve (i.e.the target goal is too optimistic), the quality of thatobjective can be a bottleneck for the remaining ob-jectives because the search will solely focus on im-proving that objective. We define the Chebyshev ob-jective function for the WSRP as follows:

Minimise λ (1)Subject to

Z1

Zt1≤ λ (2)

Z2

Zt2≤ λ (3)

Z3

Zt3≤ λ (4)

Z4

Zt4≤ λ (5)

Z5

Zt5≤ λ (6)

The Chebyshev objective function given by Eq.(1) is used as the objective function for the WSRP.The main objective is now to minimise λ, thus fin-ding a well-balanced solution regarding reaching thetarget values. If all targets are reached, λ can assumefractional values and a solution that shows balancedimprovements on all objectives may be obtained.

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3.2 Derived Weight Vector

One problem with the Chebyshev approach is that itdoes not guarantee Pareto efficiency. However, theoptimal solution of a weighted sum (where weightsare not simultaneously null) objective function is al-ways Pareto efficient. To derive a weight vector fromthe target solution, we first convert the approxima-tion set of the pilot instance into a system of linearinequalities. Considering that the approximation setis composed of n objective-vectors (~Z1, ~Z2, . . . , ~Zn),the linear inequalities system can be defined as fol-lows where the aim is to determine the values of~w = (w1,w2,w3,w4,w5):

~w~Zt ≤ ~w~Z1

~w~Zt ≤ ~w~Z2

...~w~Zt ≤ ~w~Zn

(7)

There is no guarantee that the system of linear in-equalities has a solution because the fitness landscapeis non-convex, meaning that no set of weights can beset to achieve some points in the front. Therefore,instead of finding a solution for the system, we aim tofind a weight vector ~w that satisfies the largest num-ber of inequalities. Hence, we define the problem offinding the best weight vector as the following MIP(mixed-integer programming) minimisation problem.

Minimisen

∑j=1

x j (8)

subject to

~w~Zt −~w~Z j ≤Mx j , j = 1, . . . ,n (9)

wi ∈ (0,1],x j binary ,

{i = 1, . . . ,5j = 1, . . . ,n

(10)

Note that M is a very large constant.

The objective function given by Eq. (8) aims tofind a weight vector ~w that minimises the number oflinear inequalities in the system (7) which do not ful-fill the condition ~w~Zt ≤ ~w~Zi expressed by constraint(9). Note that constraint (10) guarantees that zero can-not be chosen as a weight-value as we do not wantcriteria to be removed.

Finally, the weight vector ~w obtained from theMIP model is used in the objective function for theWSRP as given by Eq. (11).

Minimise5

∑i=1

wiZi (11)

3.3 Euclidean Distances

We propose an alternative based on the Euclidean dis-tances to the target vector. In essence, this is a met-hod that considers all objectives as equally impor-tant. Hence, minimising the Euclidean distances al-one does not guarantee Pareto efficiency. In order toreduce this issue, the proposed method consists of mi-nimising the distances to the target vector for the ob-jectives that are worse than the target. If the currentdistance for the objectives that are worse than the tar-get vector is small (< ε), then the aim is to maximisethe distances of the objectives that are better than thetarget vector.

Henceforth, the objective function in Eq. (12) be-comes the objective function for the WSRP.

Minimize

{z , if z > ε−z′ ,otherwise

(12)

where

z =

√√√√ 5

∑i=1

zi (13)

z′ =

√√√√ 5

∑i=1

z′i (14)

zi =

{(Zi−Zt

i )2 , if Zi > Zt

i0 ,otherwise

(15)

z′i =

{(Zi−Zt

i )2 , if Zi ≤ Zt

i0 ,otherwise

(16)

In summary, when the Euclidean distances of theobjectives that are worse than the target vector are lar-ger than the given parameter ε, the objective functionconsists of minimising the Euclidean distances (z).Otherwise, when z≤ ε, the objective consists of max-imising the distances for the objectives that are betterthan the target solution (z′). Thus, if the solution hasnot reached the target, the objective function attemptsto close the gap to the target. If the solution is close orbetter than the target, the objective function attemptsto further improve it.

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4 EXPERIMENTALCONFIGURATION

We applied the proposed methodology to the WSRPdatasets mentioned previously. The first instance ofeach dataset (A-01, B-01, C-01, D-01, E-01 and F-01) was selected as the pilot instance of its respectivedataset. We used the approximation sets obtained asdescribed next. Fifteen target vectors were randomlyselected (uniformly distributed) from each approxi-mation set and the same target vectors were used forthe Derived Weight Vector (WV) objective function,the Euclidean Distances (ED) objective function, andthe Chebyshev (CV) objective function.

For each remaining instance in each dataset,the Full-VNS algorithm proposed by Pinheiro et al.(2016) was applied to the fifteen selected target vec-tors from the pilot instance of its respective datasetusing all three objective functions. For each targetvector of each instance, we ran the Full-VNS algo-rithm eight times for one minute each. The resultsshown comprise the aggregate data.

Multiobjective algorithms struggle to find goodapproximation sets in combinatorial problems withmany objectives (more than three) (Giagkiozis andFleming, 2012). Hence, we resort to a tailored pro-cedure to obtain an improved approximation set. Gi-agkiozis and Fleming (2014) state that most multiob-jective algorithms can be classified as either Pareto-based or decomposition-based. This study utilisesNSGA-II (Deb et al., 2002) as the Pareto-based al-gorithm and MOEA/D (Zhang and Li, 2007) as thedecomposition-based. Thus, for each problem in-stance the approximation set was obtained as follows:1. run both the NSGA-II and MOEA/D for one mil-

lion objective evaluations on each possible bi-objective vector (Z1, Z2), (Z1, Z3), . . . (Z4, Z5);

2. run both the NSGA-II and MOEA/D for one mil-lion objective evaluations on each possible three-objective vector (Z1, Z2, Z3), (Z1, Z2, Z4), . . . (Z3,Z4, Z5);

3. run both the NSGA-II and MOEA/D for one mil-lion objective evaluations on each possible four-objective vector (Z1, Z2, Z3, Z4), (Z1, Z2, Z3, Z5),. . . (Z2, Z3, Z4, Z5);

4. create an archive composed of the non-dominatedsolutions found in the previous three steps;

5. generate a population of individuals where half ofthe elements are randomly generated and the otherhalf are randomly drawn from the archive built inthe previous step;

6. run both the NSGA-II and MOEA/D four timeseach, for two million objective evaluations, using

the initial population generated in the previousstep and the five-objective vector; and

7. compile an approximation set with all non-dominated solutions found in all steps.The number of solution vectors obtained for each

pilot instance was as follows: 1162 for A-01, 1302for B-01, 1830 for C-01, 2470 for D-01, 3689 for E-01 and 1500 for F-01.

5 EXPERIMENTAL RESULTS

First, we show the effectiveness of the derived weightvector obtained from the MIP model presented in Eq.(8)–(10). The effectiveness of a weight vector ~w isgiven by the percentage of solutions (in the pilot in-stance of the approximation set) in which ~w~Zt ≤ ~w~Zi,i = 1, . . . ,5. Hence, if the effectiveness is 100%, itmeans that the MIP model found a solution for theinequalities system in Eq. (7).

A-01 B-01 C-01 D-01 E-01 F-0190%

92%

94%

96%

98%

100%

Dataset

Wei

gth

Vec

torE

ffec

tiven

ess

Figure 3: Average percentage of the solutions in the approx-imation set of each target instance such that ~w~Zt ≤ ~w~Zi.

Figure 3 presents the results of the effectivenessanalysis of the weight vectors obtained. In all pilot in-stances, the effectiveness surpassed 90% on average.Pilot instance D-01 presented the worst overall ef-fectiveness, with an average of 93.5%, while pilot in-stance A-01 presented the most effective weight vec-tors averaging 99.5%. In all cases, the MIP modelprovided good weight vectors and these were appliedto the WV objective function in Eq. (8).

Next, we show the results for each dataset in threecharts. The target achievement chart displays the per-centage of solutions, in the given dataset, that achie-ved the target value in each objective. The gap to tar-get chart contains the average gap to the target so-lutions for the solutions that did not reach the tar-get. Finally, the overall comparison chart displays theaverage quality of solutions where positive values in-dicate that, on average, the solutions found are betterthan the target solution and negative values indicatethat the solutions are worse than the target solution.

Using Goal Programming on Estimated Pareto Fronts to Solve Multiobjective Problems

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Z1 Z2 Z3 Z4 Z5

0%

20%

40%

60%

80%

100%

WV ED CV

Figure 4: Dataset A – target achievement.

Figures 4–6 present the results of applying theVNS algorithm with the WV, ED and CV objectivefunctions to the remaining instances of dataset A. Re-sults comprise the average values of eight runs foreach target vector of each problem instance for eachobjective function.

Z1 Z2 Z3 Z4 Z5

0%

10%

20%

30%

40%

WV ED CV

Figure 5: Dataset A – gap to the target.

In Figure 4, the percentage of achievement of in-dividual target objective is shown. Note that there areno values for Z3 because dataset A does not presentpreferred skills information. It is clear that the CV ob-jective function presented better overall target achie-vement on Z1 and Z2, followed by WV, while the EDobjective function is better on Z4 and Z5, whilst CVclearly underperforms.

Figure 5 shows that the average gap of the solu-tions that did not reach the target is always smallerthan 16% for WV and ED, but high for CV. Also, inconformity with the previous chart, the CV objectivefunction presented better values for the objectives Z1and Z2 while the ED objective function obtained bet-ter values for Z4 and Z5. The CV objective functionpresented large gaps on the two latter objectives. Fi-nally, in Figure 6, it is clear that on average solutionsare at most 10% off the target for WV and ED, butthe CV approach clearly presents a stronger bias to-wards Z1 and Z2 causing the quality of Z4 and Z5 todeteriorate.

Z1 Z2 Z3 Z4 Z5

−40%

−20%

0%

20%

WV ED CV

Figure 6: Dataset A – overall comparison.

Z1 Z2 Z3 Z4 Z5

0%

20%

40%

60%

80%

100%WV ED CV

Figure 7: Dataset B – target achievement.

The results for dataset B are presented in Figures7–9. In Figure 7, it is clear that the WV objectivefunction obtains better target achievement than theED on Z1, Z2 and Z3. On Z4 the ED has a small advan-tage. Also, when compared to dataset A, the averageoverall achievement of the targets is higher for WVand ED, but not for CV. The CV objective functionshows competitive results on Z1 and Z2, but presentsextremely low achievement rates on Z4 and Z5.

Z1 Z2 Z3 Z4 Z5

0%

10%

20%

30%

WV ED CV

Figure 8: Dataset B – gap to the target.

Figure 8 shows that on all objectives, the WV ob-jective function obtained smaller average gaps for thesolutions that have not met the target. Conversely, the

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Z1 Z2 Z3 Z4 Z5

−20%

0%

20%

WV ED CV

Figure 9: Dataset B – overall comparison.

CV objective function presents large gaps on Z4 andZ5. Figure 9 displays that on average, the objectivefunction WV can provide improved objective-valueson all objectives. However, the objective function EDpresents improved results on Z1, Z2 and Z5 only, fai-ling to deliver good results on Z4, while CV presents asimilar behaviour as on dataset A: good performanceon Z1 and Z2 but bad performance on Z4 and Z5. Thus,in this dataset, the WV objective function presents abetter alternative to solve the problem.

Z1 Z2 Z3 Z4 Z5

0%

20%

40%

60%

80%

100%WV ED CV

Figure 10: Dataset C – target achievement.

Figures 10–12 present the results for dataset C.Note that since that dataset does not have distances in-formation, Z1 is always zero. In Figure 10, it is clearboth the objective functions WV and ED are balan-ced, with WV presenting better results for Z2 and Z3while the ED has the edge on Z4 and Z5. However, theED objective function presents deteriorated results onZ2 and exceptional results on Z5. The CV objectivefunction underperforms compared to the other met-hods.

On Figure 11, it is clear that the ED objectivefunction not only struggles to reach the target for Z2,but it also misses it by 40% on average. This infor-mation is also reflected in Figure 12. Overall, the WVobjective function presents a more consistent choicefor this dataset as only on Z4 the overall results arenegative and the average overall performance is su-

Z1 Z2 Z3 Z4 Z5

0%

20%

40%

WV ED CV

Figure 11: Dataset C – gap to the target.

Z1 Z2 Z3 Z4 Z5

−40%

−20%

0%

20%

40%WV ED CV

Figure 12: Dataset C – overall comparison.

perior. Again, the CV objective function presents theworst results as it can be seen that for every objectivethere is a deficit in the overall quality obtained.

Also, it is important to notice that this dataset pre-sents a high variance in the sizes of the instances.Therefore, because C-01 is a large instance, the tar-get solution simply cannot be achieved on the smallerinstances of that dataset (C-02, C-04, C-05 and C-07).

Z1 Z2 Z3 Z4 Z5

0%

20%

40%

60%

80%

100%WV ED CV

Figure 13: Dataset D – target achievement.

The results for dataset D are presented by Figu-res 13–15. In Figure 13 we see an overall increase onthe target achievement when compared to the previ-ous datasets (A, B and C). As with dataset C, the EDobjective function displayed poor performance for Z2

Using Goal Programming on Estimated Pareto Fronts to Solve Multiobjective Problems

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but had better performance on three other objectives(Z1, Z4 and Z5). The CV objective function presentedcompetitive results on this dataset.

Z1 Z2 Z3 Z4 Z5

0%

20%

40%

WV ED CV

Figure 14: Dataset D – gap to the target.

Z1 Z2 Z3 Z4 Z5

−40%

−20%

0%

20%

40%

WV ED CV

Figure 15: Dataset D – overall comparison.

Figure 14 shows that, except for the ED and CVobjective functions on Z2, the gap for the solutionsthat have not met the target is small. Also, except forthat objective, the overall average results (Figure 15)show that all three objective functions present goodresults, with the ED surpassing the WV on all ob-jectives except on Z2 and the CV presenting resultsslightly worse than the ED. Still, the WV objectivefunction manages to present a higher consistency thanthe other objective functions, and it also manages topresent good results for all objectives.

Z1 Z2 Z3 Z4 Z5

0%

20%

40%

60%

80%

100%WV ED CV

Figure 16: Dataset E – target achievement.

Dataset E results are presented in Figures 16–18.Like with dataset D, a high overall target achievementrate is shown (Figure 16). The WV objective functionpresent exceptional results on Z1 and Z2, but fail tobeat the other objective functions on the remainingobjectives.

Z1 Z2 Z3 Z4 Z5

0%

5%

10%

15%WV ED CV

Figure 17: Dataset E – gap to the target.

Z1 Z2 Z3 Z4 Z5

0%

20%

WV ED CV

Figure 18: Dataset E – overall comparison.

It is clear from Figure 17 that the ED and CVobjective functions present higher gaps to the targetsolutions. Figure 18 shows that all objective functi-ons perform well on this dataset. The WV objectivefunction presents better performance on Z1 and Z2,while the ED objective function presents better resultson Z4. The CV objective function presents the best re-sults on Z3.

Lastly, the results for dataset F are presented in Fi-gures 19–21. The target achievement rates (Figure 19)show that all three objective functions perform wellon this dataset. However, on Z2 the WV objectivefunction is noticeably better.

Figure 20 reinforces that claim as the gap for ob-jective Z2 is much larger for the ED and CV objectivefunctions. Finally, Figure 21 shows that on average,the solutions provided by the WV objective functionare better than the target solutions for all objectives,and, while the ED and CV objective functions presentbetter results than the WV on Z1, Z3, Z4 and Z5, on Z2the average results are worse than the target values.

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Z1 Z2 Z3 Z4 Z5

0%

20%

40%

60%

80%

100%WV ED CV

Figure 19: Dataset F – target achievement.

Z1 Z2 Z3 Z4 Z5

0%

10%

20%

30%

WV ED CV

Figure 20: Dataset F – gap to the target.

Z1 Z2 Z3 Z4 Z5

0%

50%

WV ED CV

Figure 21: Dataset F – overall comparison.

6 DISCUSSION

It was shown that both the WV (derived weighted vec-tor) and ED (Euclidean distances) objective functionsprovide good results on the majority of the scenarios,with the WV objective function being slightly bet-ter than the ED objective function on average. Also,for the first three scenarios, the CV objective functionclearly presented inferior results. It is noticeable thatthe results were better on the largest datasets D, E andF. In Section 2 we explained that the smallest datasets

are also the ones with solutions well spread throug-hout the objective space. Also, it was mentioned thatdatasets D, E and F present unique fitness landsca-pes with several regions containing no solutions andseveral local relationships. Therefore, in the smal-lest datasets, if the objective function is not accurateenough (regarding reaching the target solution), thesearch can deviate from the region of interest moreeasily because the objective space is smoother. On thelargest datasets, because the objective space containsgaps and local relationships, it may be enough for thesearch (driven by the objective function) to reach thesame region as the target solution.

Take for instance the example in Figure 22. Theblue dots represent the Pareto front, the green dot isthe target solution, the red dot is the solution found bythe search and the yellow arrow is the search directiongiven by the objective function. Note that the searchdirection is not accurate, hence it does not point to-wards the target solution (this is a valid assumptiongiven that we are either estimating the weight vectoror approximating the Euclidean distances). In Figure22a, we have a well spread Pareto front and becausethere exists a solution aligned with the search vector(red dot), that solution is taken as the best solution. Inthe given example, the chosen solution is better thanthe target regarding the objective presented in the xaxis, but it is worse than the target regarding the ob-jective represented by the y axis.

In Figure 22b, we have a Pareto front with gapsand local relationships. The search direction is thesame as before but in that direction, there are no so-lutions, hence the search seeks a closer solution that,in this case, is one that dominates the target solution(surpasses the target in all objectives).

A reason for the ED objective function to under-perform when compared to the WV is the differentranges of objective-values. Distances and costs canrange from about a few thousand units while the pre-ferences are measured in dozens of units. This wouldlead us to believe that the ED objective functionwould provide better results on objectives Z1 and Z2because those are the objectives with high ranges, butthat does not happen, and the algorithm frequentlyprovides better results on Z3, Z4 and Z5. An explana-tion for this phenomenon lies in Eq. (15). When thesearch reaches local optima, it becomes very difficultfor the algorithm to escape it because of the conditionin that equation. If the search reaches local optimaand a few (but not all) objectives fulfil the conditionZi >Zt

i , then, slightly disturbing the solution to escapelocal optima might result in other objectives fulfil-ling the condition and making the new solution muchworse. In summary, the VNS used was tailored for the

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(a) Well-spread Pareto front. (b) Pareto front with gaps.Figure 22: Example of an inaccurate objective-function on a Pareto front that is well-spread (a) and on a Pareto front withgaps (b).

WSRP using a weighted function. The ED objectivefunction drastically changes the objective space andthe algorithm loses performance. The phenomenon iseven worse for the CV objective function.

Additionally, the overall target achievement ratefor all objective functions was higher on the largerdatasets. This happened because the size of those da-tasets was probably too large for the multiobjectivealgorithms to find near-optimal solutions while theVNS was able to provide improved solutions.

Nonetheless, it is clear that estimating the Paretofront for problem instances that have similar fitnesslandscape to the pilot instance in the same dataset,is an effective way to tackle the problem. While themultiobjective algorithms required up to four hours toobtain the approximation set for hepilot instance of adataset, the VNS managed to find competitive solu-tions in five minutes. For the majority of the experi-ments, targets were achieved and the overall qualityof results was high.

7 CONCLUSION

In this work we proposed a solving methodology touse efficient single-objective algorithms to solve amultiobjective WSRP problem. The methodologyconsists of solving an instance of the WSRP usingmultiobjective techniques (which are typically com-putationally expensive) to obtain an approximationset and then having the decision-maker to chooseadequate target compromise solutions. Then, em-ploy goal programming to solve other instances of thesame dataset using the selected solution as target. Ad-ditionally, we used three different objective functionsto guide the algorithms to reach the target. The firstone is the well-known Chebychev approach. The se-cond one attempts to obtain a weight vector from thetarget solution. The third on uses Euclidean distancesto the target solution.

The proposed methodology can potentially be ap-plied to other multiobjective problems where instan-ces present similar fitness landscapes. In other real-world problems, instances of the same scenario havethe same partial data as each instance is a repeti-tion of the same situation for a different time frame.For example, instances of a give vehicle routing pro-blem may have the same fleet every day and recur-ring deliveries. Similarly, instances of a given edu-cational timetabling scenario may have the same fa-cilities and teachers for different terms. Hence, thetechnique proposed i this paper can be of practicaluse in that type of problems. In addition, the mul-tiobjective analysis technique proposed by (Pinheiroet al., 2015, 2017) offers an effective tool to evaluatewhether using goal programming to solve the problemis applicable or not.

We showed that the methodology proposed hereis capable of finding good trade-off solutions in afraction of the time required by multiobjective algo-rithms to process the approximation set. Also, be-cause of the strong performance by the chosen single-objective algorithm in this case, it was common forsolutions to present better objective values than thetarget solution.

The fact that instances of the same dataset presentsimilar fitness landscapes in the WSRP tackled here isnot surprising because many features are recurring inthese home healthcare planning scenarios. The sameor very similar pool of workers is used for differentdays and many visits are recurring each day (patientsneed daily help to get out of bed, take baths, take me-dications, etc).

Proposed future work could investigate adaptiveobjective functions that can change the search di-rection according to the current state of the search andpossibly reach improved results.

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