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OPTIMIZATION AND BALANCE OF MALL STORES' WINDOWSCLEANING ROUTES USING SIMULATED ANNEALING

Thassyo G. Pereiraa and Yvelyne I. Santosb

aLaboratory of Logistics, Centre of Thechnology and Natural Sciences, Par State University, Tv. Dr.

Enas Pinheiro, 2626, Marco, 66095-015, Belm, PA, Brazil, http://paginas.uepa.br/engprod

bDepartment of Industrial Engineering, Centre of Thechnology and Natural Sciences, Par State

University, Tv. Dr. Enas Pinheiro, 2626, Marco, 66095-015, Belm, PA, Brazil,

http://paginas.uepa.br/engprod

Keywords: Vehicle Routing Problem, Simulated Annealing, Stores' Windows Cleaning.

Abstract. The cleaning operations of stores' windows is an important support activity to thecommercial sector. In the city of Belm of Par, many stores chose to outsource this activity,mainly to the company studied in this paper. In the malls, the scenario in which the windowcleaning operations are performed can be seen as the Vehicle Routing Problem: the cleaning

staff are the vehicles, the time window given by the mall administration to the operations is thecapacity, and the window's perimeter are the demands to be covered. In this paper, we collectthe actual routes of the cleaning staff in one of the three malls in which the company is present,and apply the Simulated Annealing metaheuristic to reduce the distance traveled by thecleaning staff and improve the workload balance among them. The results of the algorithmreduced in 35% the average traveled distance and improved in 93,84% the workload balance.


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1 INTRODUCTION

Service business is a central economic activity for any post-industrial society, the

development of this sector is a natural process to reach such a phase (Fitzsimmons andFitzsimmons, 2004). In the last few decades, service-related activities suffered an remarkableincrease, if compared to other economic sectors (Normann, 2002), gaining the attention ofoperations management researchers since the 1970's (Corra, 2003).

In Brazil, 60% of all income generation is related to the service and commerce sector(Corra, 2003). Specifically in the State of Par, where our study takes place, an increaseoccurred in the number of cities with service-based economics, summing 125 in 2007, against9 cities agribusiness-based and 9 with industrial economic base (Par, 2007). As the capital,Belm holds most of the service activities of the State (Par, 2007), with approximately 71%of its 2007's Gross Domestic Product (GNP) derived from the service and commerce sector(Brazil, 2007). The strong presence of the commercial activity is shown by the great amount of

clothing and accessories stores, many of which are concentrated in the three Malls of the cityand in streets devoted almost exclusively to commercial activities.

An important support activity to the city's commercial sector is the cleaning of the stores'windows, because the windows are an important mean of marketing, publicity (propagandaand discount adhesives are placed on they), display (they allow the costumers to quickly

browse the products that are inside the store), and as a measurement of the general level of theapparency and cleanness of the store, causing impact on the service level expected by thecostumer, giving that they are responsible, in most cases, for the first impression that thecostumer gets of the store.

Thanks to the dynamic routine of the stores, the window cleaning operations have to meet agiven time window, what makes it similar to the Vehicle Routing Problem (VRP), in which the

cleaning workers staff are the vehicles, the time window is their capacity, the stores are thecostumers, and the windows' perimeters are the demands to be covered. It must be pointed,though, one distinguish characteristic of the window cleaning routing: the human factor, in away that the simple minimization of the total traveled distance is not enough, being alsonecessary to level the workload distribution among the cleaning workers, since they hold thesame job level and are paid equally. This vehicle routing problem variation has not receivedmuch attention in the literature, being studied separately by few authors. Our study applies theSimulated Annealing metaheuristic (SA) with problem-oriented modifications to perform therouting of the window cleaning operations of one company present in Belm city. By doingthat, we aim at not only to improve the company competitiveness, but also contribute to the

problem's literature with a novel solution technique.

The paper remainder of the paper organized as follows. The VRP formulation is describedin Section 2, where we also make an literature review about the specific VRP variation understudy (Section 2.1). The mechanism of SA is described in Section 3. Section 4 presents theactual state of the cleaning operations along with the input data used in Section 5, which

presents the conditions and computational results of the developed algorithm. Finally, Section6 concludes the paper with reflections on the obtained results and suggestions for futureworks.

2 THE VEHICLE ROUTING PROBLEM

The VRP was first introduced by Dantzig and Ramster (1959) and consists in determiningthe optimal set of routes for one fleet of vehicles to serve a given number of costumers

with known locations and demands. Thanks to the large number of possible applications, anequally large number of objectives and constraints may arise (Christofides, Mingozzi, and


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Toth, 1979). Common constraints are: the vehicle's capacity, time windows, limits on thedriver's working time, limits on the distance traveled by the vehicles, multiple depots,heterogeneous fleet, possibility of both pick-up and delivery in the same route, etc.

According to Garey and Johnson (1979), the VRP (a generalization of the TravelingSalesman Problem, TSP) is NP-Complete, being hardly solved to optimality, causing manyresearchers to develop heuristics approaches to the problem, creating an extensive literature onthe topic. A broad review on these approaches can be found at Toth and Vigo (2002).

Mathematically, the classical VRP is defined as a complete weighted graph G = (V, A, d),where the set of nodes is represented by V = {v0, v1, v2, v3, , vn} andA = {(vi, vj) : i j} isthe set of arcs. The central depot, where all the vehicles start their routes, is represented as v0and the other nodes in Vare the costumers. The distance between the costumers vi and vj is thenon-negative weight dij, associated with the arc (vi, vj). The costumervi is associated with thenon-negative values qie i that represent, respectively, the demand and the service time (q0 = 0,0 = 0) that must be covered by a vehicle with capacity C, driving a distance not superior to L

(including service times).

2.1 The vehicle routing and balancing problem

As we mentioned, the vehicle routing problem in which the balance is also necessary has notreceived much attention in the literature. Thanks to that, a handful of authors have separatelystudied it under different names, such as balanced cargo vehicle routing problem (Kritikosand Ioannou, 2010), vehicle routing problem with load-balancing (Lee and Ueng, 1999), andvehicle routing problem with route balancing (Jozefowiez, Semet, and Talbi, 2009).

Despite the lack of attention, the variety of names is also a result of two different targets forthe balancing: (i) the route lengths, and (ii) the cargo load of the vehicles. In this sense, we willrefer to the problem as the vehicle routing and balancing problem (VRBP) as a broad category

that branches into:1) the vehicle routing and length balancing problem (VRLBP)2) the vehicle routing and cargo balancing problem (VRCBP)

As we will model the problem in Section 5, it will fit in both categories, being reasonable toclassify it as the broad VRBP type.

3 SIMULATED ANNEALING METAHEURISTIC

The SA metaheuristic was originally proposed by Kirkpatrick, Gelatt, and Vecchi (1983)based on techniques from the Statistical Mechanics. SA uses the procedure of Metropolis et al.(1953), which simulates a set of atoms in equilibrium in a certain temperature, using the MonteCarlo method for multidimensional numerical integration.

Using the objective function to represent the system's energy and a set of parameters torepresent the configurations, Kirkpatrick, Gelatt, and Vecchi (1983) demonstrated that it is

possible to apply the procedure of Metropolis et al. (1953) to generate a population ofdifferent configurations to a given optimization problem in a effective temperature, which is asimple a control parameter with the same measurement unit as the objective function.

The SA proposed by Kirkpatrick, Gelatt, and Vecchi (1983) consists in, first, melt thesystem to be optimized in a high enough temperature, and then slowly low the temperature,

phase by phase, until the moment at which the system freezes (when no more changes occur).In every temperature, the simulation must happen until the system reaches a stable state. Thesequence of temperatures and the number of tested rearrangements to reach the equilibriumcan be considered as a process of annealing. They recognized the need of three ingredients

for the algorithm: a good problem description; a generator or rearrangements in the system; aquantitative objective function, representing the compensations that must be made; a cooling


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schedule for the temperature and a time span in which the system must evolve.

3.1 Detailing and pseudocode of the algorithm

The algorithm reads the control parameters related to initial temperature (T0), cooling rate() and number of iterations (n), and generates some initial solution s0 (process described inSection 3.2). Then, starts its loops: (1) External while loop, controlled by the temperature,evolves until the temperature reaches zero; (2) Internal while loop, controlled by the number ofiterations. One internal loop is performed at each iteration of the external loop.

At each iteration of the internal loop, the generator or rearrangements defines one neighbor(sv) (procedure demonstrated in Section 3.3) of the current solution (s). If the evaluationfunction f(sv) of the neighbor is better than f(s), sv is accepted as the current solution. If it isalso better than the best solution (s*),sv is accepted as the system's solution. If it is not betterthans, it might be accepted as the current solutions with e-E/T probability (for minimization of

fand eE/Tfor maximization off), where e is the Euler's number,E = f(sv) f(s) represents the

variation in the evaluation function's value given by the neighbor, and Tis a control parameterin the same unit asf, called temperature and that decays at each iteration of the external loop ata rate of , non-dimensional parameter called cooling rate. The external loop runs until thefreezing, phase in which Treachs zero (a very small user defined number). The best result ofthe evaluation function f found during the algorithm's progress (s*) is taken as the system'sheuristic solution. The pseudocode for the algorithm is shown at Fig. 1.

Figure 1: SA's pseudocode


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3.2 Initial solution generation

For the generation of the initial solution (generate_greedy_solution()), the nearestneighbor based heuristic (NNBH) proposed by Tavakkoli-Moghaddam, Safaei, and Gholipour

(2006) will be used. It is a constructive process that initiates the route rof one vehicle at arandom point, moving to the nearest neighbors until that there are no more points left in the listof remaining points (Vremaining) or its capacity Cused is reached, when the route of one newvehicle is started in a new random point, repeating the process. The algorithm terminatesreturning the set of constructed routes (s0), its pseudocode is shown at Fig. 2.

Figure 2: NNBH's pseudocode

3.3 Mechanism for neighbor generation

Osman (1993), pioneer in the application of SA to solve VRP, created two generatormechanisms based on the-optof Lin (1965), to which he called-interchange. Taking = 1,the two possible moves to be performed in two different routes are:

1) A shift process: one point is shifted from one route to another.2) An interchange process: one point from one route is exchanged with one point of

another route.The procedure generate_neighbor(s) chooses randomly between 1) and 2), performs the

operation in s and returnssv.

4 CASE STUDY

The studied company has been on the market for more than 10 years, performing cleaningservices of commercial windows, floors and fronts, as well as sales representation of cleaning


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supplies in the city of Belm, Par State. The window cleaning staff is composed of 6employees, divided among the three malls of the city, commercial streets, and companies withindependent establishments. There are about 260 window cleaning contracts, being this activity

the company's main source of revenue.The cleaning staff is divided as 3, 2, and 1 workers in the so called Mall A, Mall B, and MallC, respectively. After the cleaning of the Malls A and B, the workers are drove to the stores ofthe near commercial streets.

4.1 Clients

The Malls can be seen as condos of stores, to obtain space in one of them it is necessaryto buy a property, to pay a fee similar to a condo fee, and to obey the rules of the MallManagement. Among those rules, there are the establishment of common promotion periods,common opening period, and the mandatory maintenance of a good general appearance,mainly a clean window, what makes many store owners outsource this service. To fail to obey

any rules means the payment of fines.The stores of commercial streets have a similar institutional environment, existing stores

owners associations acting in a similar way as the malls administrations.

4.2 Window cleaning environment

The malls open from 10:00 to 22:00 hours, the mall management establishes the timebetween 6:00 and 9:45 hours for activities like stock replenishment, organization and cleaning.This time is also used by the management for the maintenance of the mall's common spaces.

The commercial streets are open at the same period, but the stores owners associations doesnot establishes specific periods for the stores' support activities, which usually occur during theopened period.

It can be seen that, in the cleaning operations of the malls, the studied company has a timewindow of 3:45 hours, and that the violation is subjected to fines, what does not occur withthe stores outside the malls. The company establishes that the working journey of theemployees must start at 06:30, with a 15-minute tolerance. It is important to point that thewindow cleaning contracts stipulates not only a daily external cleaning, but also a monthlyinternal cleaning. The latter takes longer, given the need to move the dummies and other itemsleaned in the window, what makes the cleaning staff finish all the external cleanings first, andthen move to the internal cleanings. It will be considered that the external cleanings must finishat 08:45. The time window between 06:45 and 08:45 is 2 hours, or 7,200 seconds.

4.3 Current state of the cleaning operations and input data

There is no prior defined route for the external cleaning, causing the workers to waste a lotof time moving from one store to another instead of actually cleaning. To measure the

potential productivity of the cleaners, a time and motion study was performed with them,obtaining a standard time of 39.28 seconds per meter of window perimeter and a 1 m/swalking speed was considered. These numbers allow us to express the worker's capacity inmeans of time, which, for malls, will be considered as 7,200 seconds, as discussed in Section4.2.

Therefore, this problem will fit in both the categories we defined in Section 2.1, given thatwe transformed cargo and route length into the same measurement unit. So, it can be classifiedas a broad VRBP type.

Table 1 presents the localization of the Mall A's stores ( Fl means floor, Ug representsthe underground floor, and X and Y are the Cartesian coordinates, in meters) and its

perimeters (P, in meters), 0 means the entrance/exit, considered as the depot. The distance


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between two stores will be considered as the euclidean distance multiplied by a circuit factor ofk= 1.2, common in the literature. In the case of change of floors, it will be used the stairs thatoffer the shortest distance (Table 2), considering 6 meters per floor as the traveled distance in

the stair. It must be pointed the special case of the stores 8 to 10, that represent only one storethat has been divided into 3 because the workers currently do so and are not willing to changethat, given the size of the store.

Table 1: Data of the stores of the Mall A

Fl Store X Y P Fl Store X Y P Fl Store X Y P

Ug 0 52.62 97.87 0 1st 30 115.86 47.90 2.89 3rd 60 87.65 49.55 9.13

Ug 1 44.78 109.03 7.00 1st 31 111.86 48.77 6.80 3rd 61 80.95 51.95 5.97

Ug 2 46.94 101.46 8.23 1st 32 106.77 49.86 3.01 3rd 62 69.85 52.55 4.15

Ug 3 49.62 96.26 3.23 1st 33 95.60 37.28 2.60 3rd 63 63.79 59.71 8.13

Ug 4 49.42 93.32 2.91 1st 34 94.23 29.76 5.89 3rd 64 57.64 66.08 8.31

Ug 5 60.76 84.05 5.81 2nd 35 90.01 39.10 7.16 3rd 65 68.16 63.35 5.99

Ug 6 75.01 88.03 5.10 2nd 36 81.93 51.72 3.29 3rd 66 74.54 63.18 5.93

1st 7 85.68 11.44 2.76 2nd 37 79.11 51.69 2.68 3rd 67 94.05 63.32 2.96

1st 8 86.75 17.37 5.00 2nd 38 76.23 51.71 2.89 3rd 68 105.01 58.41 5.08

1st 9 86.75 17.37 5.00 2nd 39 64.74 59.99 9.89 3rd 69 109.73 57.52 4.24

1st 10 86.75 17.37 5.00 2nd 40 57.44 74.26 3.50 3rd 70 113.73 57.52 3.38

1st 11 90.81 45.34 2.74 2nd 41 65.27 67.87 4.19 3rd 71 117.43 57.52 4.03

1st 12 88.62 50.95 9.84 2nd 42 68.12 63.82 6.41 3rd 72 121.94 57.52 4.32

1st 13 83.02 55.17 4.17 2nd 43 88.79 42.65 4.39 3rd 73 128.65 57.52 5.89

1st

14 75.12 53.19 5.89 2nd

44 96.85 62.93 2.88 3rd

74 137.78 57.52 4.101st 15 64.56 60.74 9.11 2nd 45 99.97 62.72 3.53 3rd 75 141.23 57.52 2.63

1st 16 57.65 77.45 8.41 2nd 46 108.89 56.96 2.99 3rd 76 140.80 46.24 6.13

1st 17 60.91 74.36 4.13 2nd 47 127.43 57.05 4.40 3rd 77 134.83 46.24 5.89

1st 18 64.10 71.38 4.36 2nd 48 125.71 46.28 5.89 3rd 78 127.37 46.24 2.89

1st 19 66.80 66.00 10.51 2nd 49 121.27 46.28 2.90 3rd 79 124.41 46.24 2.89

1st 20 75.00 63.56 6.00 2nd 50 118.26 46.28 2.87 3rd 80 121.35 46.24 2.89

1st 21 103.06 58.13 3.41 2nd 51 115.28 46.28 2.93 3rd 81 118.36 46.24 2.89

1st 22 106.20 55.86 3.83 2nd 52 105.98 46.29 3.13 3rd 82 109.34 46.24 2.83

1st 23 117.31 58.42 5.89 2nd 53 92.75 27.11 2.88 3rd 83 103.17 46.24 2.70

1st 24 122.54 58.61 4.28 2nd 54 91.93 23.31 6.08 3rd 84 97.04 44.57 10.6

1st 25 128.27 58.81 4.50 2nd 55 108.24 51.67 7.00 3rd 85 94.06 33.27 2.89

1st 26 127.89 48.26 2.89 3rd 56 87.82 28.94 7.11 3rd 86 92.76 27.33 2.89

1st 27 124.91 48.17 2.88 3rd 57 88.94 34.29 2.92 3rd 87 111.59 51.68 1.99

1st 28 121.89 48.08 2.89 3rd 58 89.72 37.79 4.41 3rd

1st 29 119.07 48.00 2.85 3rd 59 90.23 41.76 2.82 3rd

Table 2: Data of the stairs of the Mall A

Axis Stair 1 Stair 2 Stair 3 Stair 4 Stair 5

X 58.71 72.13 98.42 114.97 125.76

Y 85.91 57.84 18.04 53.12 66.99

The previous methodology was used to obtain the current state of the operations by


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collecting 3 routes of the workers during 3 randomly chosen days. Table 3 and the Figs 3 and 4present that data. It must be pointed that the route is not prior defined, so the worker mighttravel different routes daily, although it is believe that the operation's general state is well

represented by this sample.Table 3: Current routes

Worker Route Ocupation (s) Traveling Cleaning

1 0, 64, 65, 66, 67, 69, 70, 71, 72, 73, 76, 77,78, 79, 80, 81, 82, 83, 84, 85, 86, 8, 7, 34, 33,

11, 32, 14, 0

5,062 8.11% 91.89%

2 0, 62, 68, 74, 75, 47, 46, 55, 45, 44, 43, 53,54, 42, 41, 6, 5, 4, 3, 2, 1, 9, 31, 30, 28, 27,

26, 25, 24, 0

5,640 12.86% 87.14%

3 0, 16, 17, 18, 19, 15, 20, 13, 12, 10, 21, 22,29, 23, 48, 49, 50, 51, 52, 35, 36, 37, 38, 39,

40, 63, 61, 60, 59, 58, 57, 56, 87, 0

7,166 8.40% 91.60%

Mean - 5,956 9.73% 90.27%

Figure 3: Cleaning staff's current occupation

Figure 4: Cleaning staff's current routes

Taking a closer look to the data, we can see that an average of 9.73% of the staff's time is


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spent in the walking activity, and that there is an unbalance in the workload, with an 29.36%difference between the most loaded worker and the least loaded.

It can be seen, then, the need to improve the workload balance among the workers, making

it more even, and reduce the time wasted with walking, given that it does not aggregates valueand generates wastes for the company. Being so, the developed routing algorithm must becapable of solve these two problems.

5 OPTIMIZATION

First of all, it is necessary to state the following:s = {r1, r2, r3, r4, , rn}: solution for the problem, given by the nri routesTs: standard time for the complete cleaning of 1 meter of window perimeter, given in

seconds per meter.Vel: worker's walking velocity, in m/s.k: circuit factor.

space(ri): total euclidean distance walked in a given route ri, in meters.perimeter(ri): sum of the perimeters of all stores in ri, in meters.ocup(s): total time spent ins for the cleaning of all stores, given by Eq. 1.

ocup (s )=Tsi

perimeter(ri)+k

Veli

space (ri ) (1)

ocup(s): mean time spent in all routes ofs.MD(s): mean deviation of the spent time in all routes ofs, given by Eq. 2.

MD (s )=1

n

i

ocup (s )TPperimeter( ri )+k

Velspace ( ri ) (2)

The fact that the number of stores and its perimeters do not change during the algorithm'srun, allow us to see state that, given two solutionss ands', we have the relation of Eq. 3.

i

perimeter(ri)=i

perimeter(r'i ) (3)

In such a way that the difference amongs ands'is given only by the total distance walked,so, in the moment to choose one VRP evaluation function to implement in the SA, we can pickEq. 1 in a full or its second part for the minimization of the total amount of work performed.But it is better to pick the whole equation, because, for each worker, the capacity is expressedin means of total time, and is subjected to the constraint of the Eq. 4.

Tsperimeter(ri )+k

Velspace ( ri )C i (4)

Using only Eq. 1, though, does not consider the workload balance among the workers. Tosolve this problem, we added to the objective function the measure of dispersion MD(s), whichwill be multiplied by an parameter b < 1, aiming to control its impact in the quality of thesolutions. The evaluation function used in our work, then, is shown in Eq. 5.

f (s ) =ocup (s ) +bMD (s ) (5)

During the SA's generation of neighbors, nothing is constraining the algorithm to reachunfeasible regions of the solution space, in which the constraint at Eq. 4 is violated. A verycommon solution for this problem is the penalization of these violations, making the unfeasible

solutions not likely to be chosen over the iterations. However, the evaluation function in Eq. 5makes the penalization unnecessary because, naturally, capacity violations will generate anincrease in the mean deviation, followed by a reduction of the solution's quality, acting


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similarly to the penalization.

5.1 Computer results

Table 4 presents the computer results of the algorithm. We used T= 0.1, = 0.95, zero =10-30 e b = 0.2, and varied the iterations number as shown, performing 10 runs at each level.

The algorithm was codified in Python language and executed in the Ipython IDE. TheOperational System was the 9.10 version of the Linux distribution Ubuntu. The hardware wasan Intel CoreTM2 Duo CPU E7300 processor with a 2.66 GHz clock, and 2 GB RAM.

Table 4: Overview of the computer results

Iterations fmean fbest fworseAverage run time

(seconds)

1 17,622.28 17,485.42 17,745.90 0.57

10 17,421.77 17,303.48 17,552.18 5.62

100 17,373.66 17,286.79 17,501.63 56.36

1,000 17,365.45 17,227.14 17,542.86 553.79

As can be seen, the computer time increases linearly with the number of iterations, withoutan equivalent decrease in the evaluation function value. What makes a large number ofiterations unwise.

The new routes to be implemented by the company are the better solution found, with a17,227.14 seconds evaluation function, detailed in Table 5 and Figures 5, and 6.

Table 5: Optimized routes

Worker Route Ocupation (s) Traveling Cleaning

1 0, 64, 63, 65, 61, 60, 83, 84, 59, 58, 85, 86,

56, 57, 62, 38, 37, 36, 43, 35, 53, 54, 7, 8, 9,10, 34, 33, 11, 0

5,724 6.42% 93.58%

2 0, 1, 2, 3, 4, 5, 16, 17, 18, 20, 13, 12, 32, 31,30, 29, 28, 27, 26, 25, 24, 23, 47, 48, 49, 50,

51, 52, 55, 22, 21, 0

5,800 5.64% 94.36%

3 0, 19, 15, 14, 66, 67, 68, 69, 70, 71, 72, 73,74, 75, 76, 77, 78, 79, 80, 81, 82, 87, 46, 45,

44, 42, 39, 41, 40, 6, 0

5,695 6.91% 93.09%

Mean - 5,740 6.32% 93.68%

Figure 5: Cleaning staff's optimized occupation


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Figure 6: Cleaning staff's optimized routes

Table 5 shows that the algorithm managed to reduce the mean time wasted in walk from9.73% to 6.32%, a 35% improvement, approximately, and increased the time devoted tocleaning from 90.27% to 93.68%. It reduced the mean workers' occupation in almost 4%,

from 5,956 seconds to 5,740. The workload balanced improved notably, as can be seen in Fig.5, with a 93.84% improvement in the difference between the most loaded worker and the leastloaded, going from 29.36% to 1.81%.

6 CONCLUSIONS

In our paper, we studied a window cleaning company and its employees' routes in a mall.The problem was modeled as a vehicle routing problem where, given the human factorinvolved, the workload balance also had to be optimized. Those characteristics lead us toclassify the problem as a vehicle routing and balancing problem (term that we coined), whichhas been scarcely studied in the literature, mostly using multi-objective approaches.

The current state of the operations shows that, in average, 9.73% of the cleaning staff's timeis wasted in walking activities, with a big unbalance (29.36%) among the greater and the lesserloaded workers. We developed a Simulated Annealing approach using the Nearest NeighborBased Heuristic to generate the initial solution, the lambda interchange as the operator, and anobjective function that considers the balance of the routes. It managed to reduce the walkeddistance in 35%, the overall worked time in 4%, and the greater/lesser workload unbalance in93.84%, which reached a 1.81% level.

As we managed to improve the company's operations and also contributed to the literatureof the VRBP with a novel approach, we must make some considerations to guide future worksand studies on the topic: the utilization of other heuristics methods to solve this VRP variation(in which may not be cleaver to use constructive heuristics, given that they have anunbalancing method), and to improve the evaluation function we developed, using a techniqueto control the unbalancement other than the parameter b, because it becomes another user-defined parameter and increases the complexity of the design of the computer tests and its


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tuning for applications.

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