Research article Special Issues

Scheduling deferrable electric appliances in smart homes: a bi-objective stochastic optimization approach


  • In the last decades, cities have increased the number of activities and services that depends on an efficient and reliable electricity service. In particular, households have had a sustained increase of electricity consumption to perform many residential activities. Thus, providing efficient methods to enhance the decision making processes in demand-side management is crucial for achieving a more sustainable usage of the available resources. In this line of work, this article presents an optimization model to schedule deferrable appliances in households, which simultaneously optimize two conflicting objectives: the minimization of the cost of electricity bill and the maximization of users satisfaction with the consumed energy. Since users satisfaction is based on human preferences, it is subjected to a great variability and, thus, stochastic resolution methods have to be applied to solve the proposed model. In turn, a maximum allowable power consumption value is included as constraint, to account for the maximum power contracted for each household or building. Two different algorithms are proposed: a simulation-optimization approach and a greedy heuristic. Both methods are evaluated over problem instances based on real-world data, accounting for different household types. The obtained results show the competitiveness of the proposed approach, which are able to compute different compromising solutions accounting for the trade-off between these two conflicting optimization criteria in reasonable computing times. The simulation-optimization obtains better solutions, outperforming and dominating the greedy heuristic in all considered scenarios.

    Citation: Diego G. Rossit, Segio Nesmachnow, Jamal Toutouh, Francisco Luna. Scheduling deferrable electric appliances in smart homes: a bi-objective stochastic optimization approach[J]. Mathematical Biosciences and Engineering, 2022, 19(1): 34-65. doi: 10.3934/mbe.2022002

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  • In the last decades, cities have increased the number of activities and services that depends on an efficient and reliable electricity service. In particular, households have had a sustained increase of electricity consumption to perform many residential activities. Thus, providing efficient methods to enhance the decision making processes in demand-side management is crucial for achieving a more sustainable usage of the available resources. In this line of work, this article presents an optimization model to schedule deferrable appliances in households, which simultaneously optimize two conflicting objectives: the minimization of the cost of electricity bill and the maximization of users satisfaction with the consumed energy. Since users satisfaction is based on human preferences, it is subjected to a great variability and, thus, stochastic resolution methods have to be applied to solve the proposed model. In turn, a maximum allowable power consumption value is included as constraint, to account for the maximum power contracted for each household or building. Two different algorithms are proposed: a simulation-optimization approach and a greedy heuristic. Both methods are evaluated over problem instances based on real-world data, accounting for different household types. The obtained results show the competitiveness of the proposed approach, which are able to compute different compromising solutions accounting for the trade-off between these two conflicting optimization criteria in reasonable computing times. The simulation-optimization obtains better solutions, outperforming and dominating the greedy heuristic in all considered scenarios.



    The paradigm of smart cities aims at increasing resource efficiency in several daily activities that citizens perform in urban environments. In the case of energy management, this aim is not only related to the amount of energy consumed, but also to the infrastructure required to distribute the energy [1]. The capacity of this infrastructure is often conditioned by peak consumption, as it should be able to distribute the energy during the periods of high demand without producing power outages. However, if consumption of a certain area is remarkably unbalanced (having important variations along the day), this would required a large investment in infrastructure that will be idle the most of the time [2].

    Time-of-Use (ToU) pricing for households contributes to the overall efficiency of the electrical system. ToU incentives citizens to have a smoother consumption patron, shifting the usage of electric appliances from expensive peak hours to relatively cheaper off-peak hours. This behavior reduces the maximal instant power consumption of an urban area and, therefore, cuts back the required infrastructure investment to handle the peak and the risk of power outages [2]. However, usually off-peak hours, in which electricity is cheaper, are not preferred by users for using their appliances. This effect, which is known as inconvenience due to timing [3], can affect the well-being of the users. Therefore, there is a trade-off between both criteria, i.e., electricity cost and users satisfaction. Intelligent computer-aid tools may help users in the decision-making process of scheduling their deferrable appliances [4,5].

    This article proposes a novel mixed integer programming model for scheduling deferrable electric appliances in households, which simultaneously considers minimizing the electricity cost and maximizing the users satisfaction. Users satisfaction measures to what extend the starting time and duration for appliances usage scheduled by the model match the users preferences, which is estimated through the analysis of historical data [6,7,8]. However, since this parameter can show certain variability between different days, stochastic resolution approaches that consider this uncertain behaviour are devised. Therefore, the main contributions of the research reported in this article include: i) a novel mathematical formulation for the household energy planning problem based on integer programming that improves upon previous work by reducing the number of variables and constraints, ii) two resolution approaches for handling uncertain users preferences and the conflicting goals of minimizing the electricity cost and maximizing the users satisfaction, which have not been used before in the context of this problem, and iii) experimental evaluation over instances based on real-world data and a thorough analysis of the results. This article extends our previous conference article "A simulation-optimization approach for the household energy planning problem considering uncertainty in users preferences", presented at 10th International Conference of Production Research-Americas. New content and contributions in this extension include a novel greedy heuristic for addressing the bi-objective household energy planning problem and extended experiments, including building-like instances. These instances consist of a unique building that has inside several housing units or users and, thus, besides of respecting the maximum power contracted per individual household, the energy planning of all the households of the building has to respect the overall power consumption contracted by the building.

    The article is structured as follows. Section 2 presents the mathematical formulation of the problem, the resolution approaches and the related works. Section 3 presents the computational experimentation, including the description of the used instances, the implementation details and the obtained results. Section 4 discusses the main results obtained. Finally, Section 5 formulates the conclusions and describes the main lines of future research.

    The household energy planning problem addressed in this article aims at reducing expenses of electricity in households while enhancing users satisfaction. This last objective was estimated by considering in which part of the day users prefer to use the appliances (inferred from historical data).

    The household energy planning problem addressed in this article is modelled as a mixed-integer programming (MIP) model considering the following elements:

    Sets:

    ● a set of users U=(u1u|U|), each user represents a household;

    ● a set of time slots T=(t1t|T|) in the planning period;

    ● sets of domestic appliances Lu=(lu1lu|L|) for each user u;

    Parameters:

    ● a penalty term ρu applied to those users that surpass the maximum (electric) power contracted;

    ● a parameter Dul that indicates the average time of utilization for user u of appliance lLu;

    ● a parameter Ct that indicates the cost of the power in time slot t in the ToU pricing system;

    ● a parameter Pul that indicates the power consumed by appliance l;

    ● a binary parameter UPult that is 1 if user u prefers to use the appliance lLu at time slot t, 0 in other case;

    ● a parameter Eu that indicates the maximum power contracted by user u;

    ● a parameter Ejoint that indicates the maximum power that the (whole) set of users U are allowed to consume, which is used in building-like instances;

    Variables:

    ● a binary variable xult that indicates if user u has appliance lLu turn on at time slot t;

    ● a binary variable δult that indicates if the appliance lLu of user u is turn on from time slot t up to a period of time that its at least equal to Dul;

    ● a binary variable ψut that indicates if user u is using more power than the maximum power contracted Eu.

    ● a binary variable Ψut that indicates if user u is using more power than 130% of the maximum power contracted Eu.

    The problem aims at finding a planning function X={xult} for the use of each household appliance that simultaneously maximizes the users satisfaction (given the users preference functions) and minimizes the total cost of the power consumed. The mathematical formulation is outlined in Eqs (1)–(10).

    maxF=uUlLut1Tt|T|Dul(δult1(t2Tt1t2<t1+DulUPult2)) (1)
    minG=tTuU(lLuxultPulCt+ρu(0.3ψut+0.7Ψut)) (2)

    Subject to

    δult1Dul(t2Ttt1<t+Dulxult1)Dul,  uU,lLu,tT (3)
    ψutlLuPulxultEulLuPul,  uU,tT (4)
    ΨutlLuPulxult1.3EulLuPul,  uU,tT (5)
    uUlLuPulxultEjoint,  tT (6)
    ψut{0,1}, uU tT (7)
    Ψut{0,1}, uU tT (8)
    δult{0,1},  uU,lLu,tT (9)
    xult{0,1},  uU,lLu,tT (10)

    Equation (1) aims at maximizing the users satisfaction according to their preferences. Equation (2) aims at minimizing the energy expense budget, which include the charge for power consumption and the penalization for exceeding the maximum power contracted. Equation (3) enforces δult to be one when the length of time an appliance will be on is equal or larger than the required by the user. Equation (4) enforces ψut to be one if the user exceeds the maximum power contracted. Equation (5) enforces Ψut to be one if the user exceeds the maximum power contracted for more than 30%. For building-like instances, Eq (6) enforces that the joint electric consumption by the set of users do not surpass a the maximum power allowed to the building. Equations (7)–(10) establishes the binary nature of the variables.

    Real-world data shows that considering users preferences (UP) as a deterministic parameter does not adjust to reality [9]. Users satisfaction is modelled more accurately if uncertainty is taken into account for preferences in the model. Therefore, this article develops a resolution approach that considers this stochastic behaviour.

    In order to handle the bi-objective nature of the optimization problem presented in Section 2.1, a weighted sum optimization approach is applied. The weighted sum is a traditional method in the multiobjective optimization literature which has extensively been used in many applications, including other household energy planning related problems [3]. Applying this approach, Eqs (1) and (2) are jointly optimized with Eq (11), where α and wβ are the relative weights given to users satisfaction and cost criteria by the decision-maker.

    maxH=αFFbestFbestFworstβGGbestGworstGbest (11)

    One of the main drawbacks of this method is to know the actual best and worst values of each objective within the set of non-dominated solutions which are used for normalization (i.e., Fbest and Gbest, Fworst and Gworst in Eq (11), respectively). In this article, for addressing this issue, the procedure proposed in Rossit [10] and applied in Rossit et al. [11] is used. This is a two step procedure. In the first step, the best and worst values of each objective are approximated by solving the single objective problem of each of the criteria involved. These values, which are likely to be dominated, are improved in the second step of the procedure. In this second phase, these best and worst values are used in the weighted sum formula (Eq (11)) along with a biased combination of weights. This is, two different problems are solved, one problem using α>>β>0 and the other problem using β>>α>0. Finally, from the solutions of these last two multiobjective problems, the new best and worst values are obtained.

    Formally, in a stochastic optimization problem with a probabilistic objective function, the expected value of this function should be optimized. In the case of the formulation described in Section 2.1, if parameters UP are considered stochastic, Eq (1) should be replaced by Eq (12).

    e=EP[F(Δ,UP)]. (12)

    In Eq (12), UP is the random vector of the stochastic users preferences and Δ is the vector of decision variables δ described in Section 2.1. In order to optimize Eq (12), all the possible realizations of vector UP with its corresponding probability should be considered. Taking into account that the model of Section 2.1 uses a finite set of time slots, the set of possible realizations of UP is also finite. Particularly, there are |T|uU|Lu| realizations of this vector, each one constituting a possible scenario for the stochastic problem. For example, consider an instance in which the day is split in intervals of 30 minutes, i.e., |T|=48, there are two users (households) and each user has only two appliances (|Lu1|=|Lu2|=2). Then, the number of possible scenarios would be 484=5,308,416.

    For the cases in which the large number of scenarios of real-world instances makes impractical to compute the exact expected value of Eq (12), the expected value is approximated with an independently and identically distributed (i.i.d.) random sample. This technique is called the "sample-path optimizatio [12]" or "sample average approximation [13]". Thus, Eq (13) is an estimator of the expected value of Eq (12).

    ˆe=1NNj=1F(Δ,UPj) (13)

    As aforementioned, the set of values UP1,...,UPN, is an i.i.d. random sample of N realizations of the stochastic vector parameter UP. The optimization problem obtained when Eq (13) is used instead of Eq (12), is the sample average approximation optimization problem (hereafter SAA) and can be solved deterministically with commercial solvers. Clearly, the solution of the SAA problem depends on the realizations UP that are included in the random sample. Moreover, the larger the size of the sample (N), the smaller is the difference between Eq (12) and its estimator Eq (13). Particularly, when N, ˆee [14].

    Different samples of size N (i.e., different set of realizations of the stochastic vector parameter UP) allow shaping different forms of Eq (13). Therefore, all algorithms based on sample average usually solve the SAA problem several times with different samples and after that the most promising solution is selected according to a given (predefined) criteria as the final solution.

    Let ˆe1N,ˆe2N,...,ˆeMN be the values of Eq (13) when solving M SAA problems, each one with a different sample of size N. Moreover, considered that ˆs1N,ˆs2N,...,ˆsMN are the solution (values of decision values) obtained for each of the aforementioned M SAA problems. An intuitive criteria for selecting the final solution among the M possibilities, would be to pick the solution with the best ˆeN value. In this article, a more sophisticated procedure to select the final criteria, which was proposed in Norkin et al. [15] and implemented in Verweij et al. [16], is used. This procedure is described as follows. First, an independent sample of size N with N>>N is built to evaluate the M solutions using this sample. Then, the solution with the best value as it is expressed in Eq (14) for a maximization problem is selected.

    ˆsN=argmax{ˆeN(ˆsN):ˆsNˆs1N,ˆs2N,...,ˆsMN} (14)

    The previously described idea takes advantage from the fact that even though using the large sample size N for the optimization phase is very time consuming (specially in NP-hard problems as the one addressed in this paper), using it for just for evaluation of the objective function Eq (13) is achievable in reasonable computing time [14]. The pseudocode of the proposed SAA approach is outlined in Algorithm 1.

    Algorithm 1 Schema of a the Sample Average Approximation approach.
    1: procedure SO (pult,N,M,α,β)
    2:  initialize list S of size M
    3:  for m0,m++,mM do
    4:    for n0,n++,nN do
    5:      for all uU do
    6:        for all lLu do
    7:           for all tT do
    8:            initialize trandom(0,1)
    9:            if tpult then UPult=1
    10:            else  UPult=0
    11:    S[m]Solve MDR(α,β,UP))
    12:    return S

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    A greedy heuristic is proposed as reference baseline for results comparison. Greedy algorithms are conceived to heuristically obtain a global good solution to a problem by making locally optimal decisions by a repetitive procedure [17]. These heuristics have been efficiently applied in other energy planning problems by our research group [18,19,20]. The pseudocode of the greedy heuristic is outlined in Algorithm 2.

    Algorithm 2 Greedy algorithm for household appliances planning
    procedure BestPrefInterval(tm, ui, lk, X)
      pref 0; duration 0
      for (tn=tm; tn<t|T|; tn++) do
        if duration <D(lk,ui) then
          if lLuixult×Pl+Plk<Eui then
            pref += UP(ui,tn,lk)
            duration += tntn1
          else
            pref 0
            duration 0
        else
          return [tn, pref -1]                    ▷ interval found
        return [t|T|, pref]                     ▷ no interval was found
    procedure BestCostInterval(tm, ui, lk, X, UPN)
      cost 0; duration 0; pref 0
      for (tn=tm; tn<t|T|; tn++) do
        if duration <D(lk,ui) then
          if lLuixultm×Pl+Plk<Eui then
            cost += Plk×C(tn)
            duration += tntn1
            pref += UP(ui,tn,lk)
          else
            cost 0
            duration 0
            pref 0
        else
          return [tn, cost, pref]                     ▷ interval found
      return [t|T|, cost BigM, pref 0]              ▷ no interval was found
    procedure Greedy (UPN)
      X 0; minPref π
      for (ui=u0; ui<u|U|; ui++) do                ▷ for each user
        for (lk=l0; lk<Lu|K|; lk++) do                ▷ for each appliance
          pref 0; bestPref -1 ▷ esearch best interval for pref
          for (tm=t0; tm<t|T|; tm++) do
            [tm, pref] = BestPrefInterval(tm, ui, lk, X, UPN)
            if pref > bestPref then
              bestPref pref
          if bestPref < 0 then
            break                                ▷ no feasible solution found by the greedy
          pref 0; cost 0; bestCost -1                ▷ search best interval for cost and min pref
          for (tm=t0; tm<t|T|; tm++) do
            [tm, cost, pref] = IntervalMaxPrefCost(m, dk, ui, X)
            if cost < bestCost & pref > bestPref * minPref then
              bestCost cost
              tbestmintm
          for (tm=tbestminD(lk,ui); tmtbestmin; tm++) do
            xuilktm 1                          ▷ set appliance ON
    return X

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    The main goal of the proposed greedy algorithm is to build low-cost solutions (according to Eq (2)). However, it also considers a threshold level of user satisfaction that must be fulfilled. For this reason, function BestPrefInterval() seeks the interval with the maximum user satisfaction for each appliance. Then, function BestCostInterval() seeks the interval that minimizes the cost given that the user satisfaction is not smaller than a percentage (0<π<1) of the maximum user satisfaction for the same user and appliance and that the maximum power contracted by the user is not exceeded. Thus, sets the appliance as switched ON starting from that time slot (up to the time slot in which expected duration is fulfilled). Within each user, appliances are processed in descending order of power consumption. Similarly to the SAA approach, BestPrefInterval() uses average user preferences (UP) given a certain number of realizations of this stochastic parameter. The greedy heuristic is also applied to M different samples of size N of the preferences vector and the final solution is selected using the same procedure as in the SAA.

    Household energy planning has been considered as a complex problem in the related literature. This article focuses on the stochastic version of the problem. A more general review of the topic was presented by Lu et al. [21].

    The deterministic version of the household energy planning problem is associated with bin packing [22], a well-known NP-hard problem. The inclusion of uncertainty increases the complexity of the problem [23]. Several articles have addressed stochastic versions of this problem, by considering uncertainty in different parameters. Chen et al. [24] considered uncertainties in the power consumed by the appliances and the renewable solar energy gathered by a photovoltaic array. A three-stages resolution process was proposed. First, Chen et al. solve a deterministic linear programming optimization model considering mean values for the appliances consumption and maximum solar power generation. Then, they apply a stochastic procedure based on Monte Carlo simulation was applied to the resulting solution. The simulation considers different energy consumption rates of appliances and selects the consumption rate that minimizes the probability of shortcuts, which occurs when the overall consumption of electricity surpass a certain threshold value. Finally, an online adjustment of the previous (offline) solution was applied, which monitors the instant solar power generation and the consumption of appliances in real-time, compensating the household electric balance of the offline solution with a larger power storage in the battery or purchase from the grid. Hemmati and Saboori [25] proposed a particle swarm optimization algorithm to deal with uncertainty of photovoltaic panels in a similar problem. Assuming that the energy generated in the panels has a Gaussian probabilistic distribution, a Monte Carlo simulation was used each time the stochastic function has to be evaluated to obtain a sample of the generation values.

    Other researchers have used robust optimization, which aims at minimizing the impact of the worst-case scenario, considering that random parameters have a bounded probabilistic distribution [3]. Jacomino and Le [26] presented a robust optimization approach to simultaneously minimize energy cost and maximize the comfort of users. They considered uncertainty in two aspects: the outdoor temperature and the solar radiation related to weather forecast -that affect the energy to be consumed to satisfy the required indoor temperature-, and users decisions related to not programmable services, i.e., despite the scheduled starting time and duration of the appliances the user can modified these conditions when actually using them. For handling uncertainty on users behaviour, a decomposition approach based on estimating the probability of occurrence of each scenario was used. Wang et al. [27] proposed a robust optimization approach for dealing with photovoltaic energy generation in household planning by using a mixed integer quadratic programming model, and Wang et al. [28] for dealing with uncertainty in hot water utilization and outdoor temperature that influences the usage of heating and air conditioning systems. Judge et al. [29] proposed a robust optimization model to manage uncertainties associated with thermal loads such as heating and air conditioning and solved combining Harris Hawks' optimization [30] and linear programming. {Hosseini et al. [31] presented a robust optimization approach to minimize the energy cost while satisfying certain comfortability restrictions considering uncertainty from two different sources: the decisions of user of when using each appliance and the intermittency of renewable energy sources. Another work that uses robust optimization for handling uncertainty of renewable sources of energy was performed by Shi et al. [32]. Other published material deals with this problem as a control problem by using a closed-loop approach such as Scarabaggio et al. [33], who used a sample average approximation based on a probability density function to cope with uncertainty in wind power availability, or Nassourou et al. [34], in which a control strategy that is divided into an open-loop system that manage the dependent control inputs and a closed-loop system that uses local feedback control for the independent inputs.

    From the analyzed works, it can be concluded that fine grained energy consumption data collection from smart homes considering uncertainty has shown to be a powerful tool to define more efficient and reliable electricity services. However, the collection and exchange of information raise concerns about consumer privacy. The collected data could be used to infer activities and behavior patterns of consumers or an attacker could create fake power information to jeopardize the power system [35]. In order to deal with these privacy issues, Tonyali et al. developed a meter data obfuscation scheme to protect consumer privacy from eavesdroppers and the utility companies while preserving the utility companies' ability to use the data for state estimation [36]. Mohammed et al. proposed an approach based on adding noise to the reading data so no one can obtain the meters' individual data, however, the total readings of the meters can be known by the utility [37]. In line with the work presented in this article, the problem of enhancing the decision making processes in demand-side management has been addressed by adding a specific optimization objective related to preserving users' privacy. Thus, there have been proposed multiobjective optimization approaches that have proposed the minimization of the energy consumption cost while maximizing users' privacy by masking the energy consumption profile of the user [38,39]. Chang et al. defined load variation as the privacy metric and scheduled inflexible and unshiftable appliances, flexible appliances, and shiftable appliances [38].

    Other authors, although without considering uncertainty in their models, have explored the trade-off that usually exists between electricity cost and users satisfaction through linear mathematical programming approaches, as it is performed in this article. Among them, Yahia et al. [40] modeled a bi-objective problem considering these two objectives, which were combined by means of a linear weighted sum to form a unique objective function. Authors solved two single-household instances, i.e., a real South African case study and an artificial large instance, using LINGO. Additionally, they performed an extensive analysis of the sensitivity of the results to the modifications of certain parameters. Authors extended the approach by considering the reduction of the peak load as a third objective [41]. Moreover, an instance considering several households simultaneously was solved. Three different multiobjective approaches were compared: lexicographic optimization, normalized weighted sum and compromise programming. Our previous articles explored the trade-off between the users satisfaction and energy cost in a deterministic version of the problem using evolutionary algorithms [19,42].

    This article contributes to the literature in several aspects. Firstly, a novel linear mathematical formulation of the household planning energy optimization problem that explicitly considers users satisfaction as an objective function is presented. Approaches like that are not common in the related literature [40]. Moreover, this is an novel mathematical formulation compared to the one presented in our previous article [19] for a similar conceptual model, but improving upon it by having a smaller number of variables and constraints that eases its solvability. Secondly, this article considers stochastic users preferences, which differentiates it from other linear programming applications in the related work [40,41]. This leads to a novel scientific contribution of the work, which is the application of the simulation-optimization Sample Average Approximation method to handle the uncertainty which has not been applied to this specific problem before.

    This section presents the computation experimentation, including a description of the instances that were used, the experiment design and the main results of the experimentation.

    The instances addressed were generated using realistic information and expanding the REDD dataset [9] via a urban data analysis approach [43]. One of the key parameters to estimate in the household energy planning model presented in this work are the users preferences. For estimating this, historical information retrieved from the REDD dataset about the power consumption of the selected appliances on each household was analyzed. This task involved cleaning the data from comparatively very small power consumption that are related to stand-by operation mode of each appliance, for example, small screen leds. After this, for each combination of user and appliance, a probability of usage for each time slot was estimated (pult). With this probability, M instances were constructed for each sample size N as is described in Section 3.2. Additionally, from the REDD dataset, the mean power consumption of each appliance in KW (Pul) and the duration of the average time of utilization of each appliance (Dul) were estimated. The weekend period was considered to introduce noticeable differences in the instances, a behaviour that is usual for household users [44]. Thus, instances were grouped into two categories: weekdays and weekends. Parameters Eu (maximum electric power contracted for each household) and Ct were obtained from the National Electricity Company, Uruguay, as reported in the ECD-UY dataset [45].

    Besides the weekly separation (noted as wd and we for weekday and weekend, respectively), instances with increasing sizes were also defined, as already described in the methodology of the experimental evaluation of previous works [18]:

    ● small (s.wd and s.we), modeling scenarios with one household with seven deferrable appliances.

    ● large (l.wd and l.we), modeling scenarios having two households with six and seven deferrable appliances, respectively.

    ● building (b.wd and b.we), modeling scenarios with four households with six and seven deferrable appliances, respectively.

    Electric appliances are classified in deferrable and non-deferrable appliances [46]. Deferrable appliances are those devices that can be controlled by the user and deferred to be switched on in different time-slots on the scheduling horizon, without a critical result in the comfort of users [47]. Conversely, non-deferrable appliances are those which its standard operation time cannot be shifted without a significant impact on the comfort of users, since they are critical for users to accomplish basic everyday activities, such as lighting. The scheduling approach proposed in this article considers deferrable appliances. Few works in the related literature have included non-deferrable appliances in smart home planning systems, mainly because they do not provide flexibility to compute accurate schedules, and even slight shifts of their operation times cause severe penalizations on user-comfort related objectives. This article considers both non-interruptable deferrable appliances, i.e., microwave, washer dryer, dishwasher and refrigerator, and interruptable deferrable appliances, i.e., electric stove and air conditioning.

    In both small and large size instances, the constraint defined by Eq (6) was not applied, since the considered households are independent and, thus, the constraints in Eqs (4) and (5) already allow limiting the maximum consumed power. The instances b.wd and b.we have to meet not only the maximum power contracted per individual household, but also the overall power consumption contracted by the building.

    After preliminary calibration experiments, the following sample sizes were chosen N = 1000, 2000, 3000, 5000, and 10000. Within each sample size, the number of independent samples (M) was set to 100. The evaluation sample size (N) was set to 100, 000.

    In order to apply the SAA approach, the bi-objective optimization procedure introduced in Section 2.2 was used. This optimization procedure requires estimating both the ideal and nadir values for the weighted sum function defined in Eq (11). The estimation of the ideal and nadir value was performed for each sample size N applying the two step procedure presented in Section 2.2: initially they are estimated through single-objective optimization and, later, they are improved applying the weighting sum method with a biased combination of weights. Then, five weight vectors (α,β) were used for exploring different trade-off combinations between the objectives of energy cost and users satisfaction: (0.99, 0.01), (0.25, 0.75), (0.5, 0.5), (0.75, 0.25), and (0.01, 0.99). In the SAA method, for each weight vector a MIP problem is solved using Gurobi [48] through Pyomo as modelling language [49]. In the case of the greedy heuristic three aspiration levels were considered (π): 0.60, 0.75, and 0.90.

    The experiment was divided in two parts. Firstly, the random realizations or samples of vector UP were generated and secondly the optimization algorithms were applied to these random samples. This separation was performed because of two reasons: i) to study the impact of the generation of random samples of vector UP in the overall efficiency of the algorithm and ii) to apply both algorithms over the same set of random samples to provide a more fair comparison avoiding differences in the results because of this random procedure. Then, for each instance and size N, a set of 100 (M) independent realizations of vector UP were generated. Table 1 reports the computational times demanded for generating the realizations of vector UP. The execution times indicate that the average time increases linearly with the sample size N. This is connected to the trade-off between having a large sample size N which is computationally expensive but provides a better estimation of the real expected value (Eq (12)) by Eq (13) or a smaller sample size N which is lees time-consuming but provides a worse approximation of the real expected value.

    Table 1.  Computing times of the realizations of vector UP.
    Instance N Time (s) Instance N Time (s) Instance N Time (s)
    Avg Std Avg Std Avg Std
    s.wd 1000 0.2098 0.0011 l.wd 1000 0.3911 0.0011 b.wd 1000 0.7325 0.0020
    2000 0.4197 0.0012 2000 0.7887 0.0021 2000 1.4828 0.0129
    3000 0.6033 0.0017 3000 1.1764 0.0031 3000 2.2112 0.0047
    5000 1.0601 0.0050 5000 1.9600 0.0080 5000 3.6860 0.0081
    10000 2.1305 0.0092 10000 3.9047 0.0056 10000 7.3938 0.0150
    s.we 1000 0.2074 0.0005 l.we 1000 0.3882 0.0014 b.we 1000 0.7621 0.0018
    2000 0.4174 0.0003 2000 0.7860 0.0025 2000 1.5276 0.0037
    3000 0.60241 0.0006 3000 1.1806 0.0038 3000 2.3114 0.0056
    5000 1.0444 0.0010 5000 1.9669 0.0079 5000 3.8473 0.0067
    10000 2.0955 0.0016 10000 3.9177 0.0160 10000 7.6511 0.0134

     | Show Table
    DownLoad: CSV

    This section presents the main results of the computational experimentation with SAA and the greedy heuristic. Detailed results about all the runs performed can be depicted in the Appendix B. To condense the outcome of the proposed approach into a suitable indicator that measures the quality of the results, the deviation to the ideal vector is used. This is computed using the L2 distance norm according to Eq (15).

    Σ=oO(valuebestobesto100%)2 (15)

    In the definition of the Σ metric in Eq (15), O is the set of objectives, (for the considered problem, O={F,G}), and besto is the best value achieved for each objective evaluated over N in all the experiments performed for that instance. Thus, from all the solutions, the solution with the smallest distance is the best comprising solution, as graphically represented in Figure 1.

    Figure 1.  Best compromising solution.

    Another relevant aspect that should be analyzed when controllable deferrable loads are shifted collectively, is the peak rebound effect that can be associated to a drastic increment of the consumption during low priced hours. The metric of the load factor is usually used in the related works to measure this aspect [50,51]. The load factor is defined as the ratio of the average energy consumption to the maximum energy consumption in the planning horizon. A higher load factor implies a more stable consumption which can help to avoid problems in the electric grid [50]. Thus, the load factor for all the users (Lf) is reported for the presented solutions, calculated according to Eq (16).

    Lf=(uUlLutTPulxult)/|T|maxtT{uUlLuPulxult} (16)

    The results of the SAA are presented in Table 2. This table reports for each instance, the sample size N, the combination of weights (α,β), the average execution time in seconds, the values of F and G of the best solution, i.e., the solution that has the minimal value of function H (Eq (11)), and the deviation of the solution to the ideal vector Σ (Eq (15)). In turn, the experimental results of the the greedy heuristic are reported in Table 3. The table presents for each instance, the sample size N and the aspiration preference level π, the same results as for the SAA. As aforementioned, the computing times in Tables 2 and 3 do not include the time to generate the N random realizations of the user preferences vector UP.

    Table 2.  Results of the SAA.
    N (α,β) Avg. Time (s) F(HNbest) G(HNbest) Σ Lf Avg. Time (s) F(HNbest) G(HNbest) Σ Lf
    s.wd s.we
    1000 (0.99, 0.01) 0.0138 3.2925 113.3289 26.78% 0.1854 0.0081 1.4763 36.7695 58.62% 0.2363
    (0.01, 0.99) 0.1625 2.1814 89.3876 33.75% 0.2068 0.0452 1.0148 23.1810 31.26% 0.1685
    (0.50, 0.50) 0.0911 3.1368 98.2011 10.94% 0.1650 0.0122 1.4363 28.8151 24.46% 0.2363
    (0.75, 0.25) 0.0453 3.1368 98.2011 10.94% 0.1650 0.0099 1.4363 28.8151 24.46% 0.2363
    (0.25, 0.75) 0.1436 2.5555 90.8443 22.44% 0.1650 0.0189 1.2623 24.7508 16.00% 0.1873
    2000 (0.99, 0.01) 0.0137 3.2925 113.3289 26.78% 0.1854 0.0085 1.4763 36.7695 58.62% 0.2363
    (0.01, 0.99) 0.1619 2.1813 89.3876 33.75% 0.1663 0.0475 1.0149 23.1810 31.25% 0.1685
    (0.50, 0.50) 0.1051 3.1368 98.2011 10.94% 0.1650 0.0149 1.4363 28.8151 24.46% 0.2363
    (0.75, 0.25) 0.0486 3.1368 98.2011 10.94% 0.1650 0.0118 1.4363 28.8151 24.46% 0.2363
    (0.25, 0.75) 0.1421 2.5558 90.8443 22.43% 0.1650 0.0290 1.2627 24.7508 15.97% 0.1873
    3000 (0.99, 0.01) 0.0136 3.2925 113.3289 26.78% 0.1854 0.0084 1.4763 36.7695 58.62% 0.2363
    (0.01, 0.99) 0.1624 2.1819 89.3876 33.73% 0.1781 0.0476 1.0148 23.1810 31.26% 0.1873
    (0.50, 0.50) 0.0963 3.1368 98.2011 10.94% 0.1650 0.0148 1.4363 28.8151 24.46% 0.2363
    (0.75, 0.25) 0.0497 3.1368 98.2011 10.94% 0.1650 0.0117 1.4363 28.8151 24.46% 0.2363
    (0.25, 0.75) 0.1351 2.5558 90.8443 22.43% 0.1650 0.0289 1.2627 24.7508 15.97% 0.1873
    5000 (0.99, 0.01) 0.0133 3.2925 113.3289 26.78% 0.1854 0.0088 1.4763 36.7695 58.62% 0.2363
    (0.01, 0.99) 0.1607 2.1815 89.3876 33.74% 0.1758 0.0475 1.0151 23.1810 31.24% 0.1699
    (0.50, 0.50) 0.1038 3.1368 98.2011 10.94% 0.1650 0.0146 1.4363 28.8151 24.46% 0.2363
    (0.75, 0.25) 0.0499 3.1368 98.2011 10.94% 0.1650 0.0115 1.4363 28.8151 24.46% 0.2363
    (0.25, 0.75) 0.1359 2.5558 90.8443 22.43% 0.1650 0.0288 1.2622 24.7508 16.01% 0.1994
    10000 (0.99, 0.01) 0.0131 3.2925 113.3289 26.78% 0.1854 0.0088 1.4763 36.7695 58.62% 0.2363
    (0.01, 0.99) 0.1616 2.1814 89.3876 33.75% 0.1663 0.0477 1.0150 23.1810 31.25% 0.1782
    (0.50, 0.50) 0.1030 3.1368 98.2011 10.94% 0.1650 0.0146 1.4363 28.8151 24.46% 0.2363
    (0.75, 0.25) 0.0497 3.1368 98.2011 10.94% 0.1650 0.0116 1.4363 28.8151 24.46% 0.2363
    (0.25, 0.75) 0.1340 2.5558 90.8443 22.43% 0.1650 0.0282 1.2622 24.7508 16.01% 0.1873
    l.wd l.we
    1000 (0.99, 0.01) 0.0226 6.4528 194.4567 47.98% 0.2158 0.0216 8.1457 277.5776 40.50% 0.3929
    (0.01, 0.99) 0.2624 4.4226 131.4108 31.46% 0.2059 0.2226 5.9167 197.5689 27.37% 0.3178
    (0.50, 0.50) 0.0769 6.0373 145.3292 12.40% 0.1471 0.1225 7.4650 211.7288 11.01% 0.2744
    (0.75, 0.25) 0.0485 6.4120 185.3487 41.05% 0.2158 0.0746 7.9526 251.6011 27.45% 0.4529
    (0.25, 0.75) 0.1768 5.7316 140.2288 13.04% 0.1471 0.4123 7.0347 205.0788 14.16% 0.2448
    2000 (0.99, 0.01) 0.0232 6.4528 194.4567 47.98% 0.2158 0.0211 8.1457 277.5776 40.50% 0.3929
    (0.01, 0.99) 0.2596 4.4229 131.4108 31.46% 0.1962 0.2229 5.9174 197.5689 27.36% 0.3178
    (0.50, 0.50) 0.0920 6.0373 145.3292 12.40% 0.1471 0.1166 7.4650 211.7288 11.01% 0.2744
    (0.75, 0.25) 0.0631 6.1053 150.3087 15.36% 0.2223 0.0718 7.9525 251.6011 27.45% 0.4529
    (0.25, 0.75) 0.2310 5.3944 136.7499 16.90% 0.1405 0.4118 7.0815 205.6305 13.69% 0.2744
    3000 (0.99, 0.01) 0.0232 6.4528 194.4567 47.98% 0.2158 0.0206 8.1457 277.5776 40.50% 0.3929
    (0.01, 0.99) 0.2582 4.4230 131.4108 31.46% 0.2059 0.2226 5.9168 197.5689 27.36% 0.2871
    (0.50, 0.50) 0.0876 6.0373 145.3292 12.40% 0.1471 0.1181 7.4650 211.7288 11.01% 0.2744
    (0.75, 0.25) 0.0632 6.1049 150.3087 15.36% 0.2223 0.0769 7.9423 250.4978 26.91% 0.4529
    (0.25, 0.75) 0.2334 5.3944 136.7499 16.90% 0.1405 0.4140 7.0349 205.0788 14.16% 0.2448
    5000 (0.99, 0.01) 0.0230 6.4528 194.4567 47.98% 0.2158 0.0207 8.1458 277.5776 40.50% 0.3929
    (0.01, 0.99) 0.2591 4.4230 131.4108 31.46% 0.1962 0.2265 5.9175 197.5689 27.36% 0.2871
    (0.50, 0.50) 0.0781 6.0373 145.3292 12.40% 0.1471 0.1180 7.4650 211.7288 11.01% 0.2744
    (0.75, 0.25) 0.0578 6.3393 176.3038 34.21% 0.2403 0.0779 7.9526 251.6011 27.45% 0.4529
    (0.25, 0.75) 0.1886 5.7325 140.2288 13.02% 0.1471 0.4271 7.0815 205.6305 13.69% 0.2744
    10000 (0.99, 0.01) 0.0224 6.4528 194.4567 47.98% 0.2158 0.0203 8.1458 277.5776 40.50% 0.3929
    (0.01, 0.99) 0.2592 4.4224 131.4108 31.47% 0.1962 0.2236 5.9171 197.5689 27.36% 0.2788
    (0.50, 0.50) 0.0846 6.0373 145.3292 12.40% 0.1471 0.1155 7.4650 211.7288 11.01% 0.2744
    (0.75, 0.25) 0.0630 6.1053 150.3087 15.36% 0.2223 0.0783 7.9526 251.6011 27.45% 0.4529
    (0.25, 0.75) 0.2232 5.6829 139.6771 13.49% 0.1471 0.4185 7.0817 205.6305 13.69% 0.2744
    b.wd b.we
    1000 (0.99, 0.01) 0.0661 13.8077 606.4009 139.60% 0.2112 0.0588 15.1327 673.2478 109.23% 0.2774
    (0.01, 0.99) 0.4757 8.5678 253.0874 37.98% 0.1999 0.4897 9.7527 321.7733 35.56% 0.2827
    (0.50, 0.50) 0.2532 11.7540 278.1533 17.91% 0.2012 0.3244 12.9297 350.6130 17.10% 0.3877
    (0.75, 0.25) 0.1234 13.4735 483.9198 91.24% 0.2157 0.1389 15.0719 617.7601 91.99% 0.2996
    (0.25, 0.75) 0.3049 11.6152 273.2761 17.81% 0.2012 0.3772 12.6639 342.2254 17.51% 0.3016
    2000 (0.99, 0.01) 0.0682 13.8106 597.8174 136.21% 0.2112 0.0585 15.1319 673.3213 109.25% 0.2774
    (0.01, 0.99) 0.4740 8.5677 253.0874 37.98% 0.1957 0.4966 9.7521 321.7733 35.56% 0.3041
    (0.50, 0.50) 0.2471 11.7545 278.1533 17.90% 0.2012 0.3175 12.9298 350.3693 17.06% 0.3877
    (0.75, 0.25) 0.1327 13.4746 483.5656 91.10% 0.2157 0.1269 15.0740 617.9178 92.04% 0.2996
    (0.25, 0.75) 0.3104 11.5933 272.8259 17.87% 0.2012 0.3765 12.6648 342.2354 17.51% 0.3341
    3000 (0.99, 0.01) 0.0682 13.8148 623.0884 146.19% 0.2112 0.0591 15.1316 672.6650 109.05% 0.2774
    (0.01, 0.99) 0.4765 8.5678 253.0874 37.98% 0.1957 0.5026 9.7524 321.7733 35.56% 0.3041
    (0.50, 0.50) 0.2564 11.7642 278.7050 17.97% 0.2037 0.3136 12.9297 350.3693 17.06% 0.3877
    (0.75, 0.25) 0.1280 13.7706 541.3700 113.91% 0.1884 0.1285 15.0718 617.6007 91.94% 0.2996
    (0.25, 0.75) 0.3070 11.6260 273.5103 17.78% 0.2012 0.3773 12.6652 342.2354 17.51% 0.3341
    5000 (0.99, 0.01) 0.0686 13.8122 597.9184 136.25% 0.2112 0.0599 15.1321 672.5696 109.02% 0.2774
    (0.01, 0.99) 0.4751 8.5677 253.0874 37.98% 0.1992 0.5123 9.7533 321.7733 35.55% 0.2994
    (0.50, 0.50) 0.2510 11.7639 278.7050 17.97% 0.2037 0.3166 12.9315 350.3693 17.05% 0.3877
    (0.75, 0.25) 0.1248 13.7818 543.5414 114.76% 0.1882 0.1314 15.0722 617.6647 91.96% 0.2996
    (0.25, 0.75) 0.3250 11.6264 273.5103 17.78% 0.2012 0.3760 12.6666 342.2354 17.50% 0.3341
    10000 (0.99, 0.01) 0.0692 13.8113 597.9185 136.25% 0.2112 0.0597 15.1336 673.4111 109.28% 0.2774
    (0.01, 0.99) 0.4754 8.5684 253.0874 37.98% 19.57% 0.5109 9.7526 321.7733 35.56% 0.3041
    (0.50, 0.50) 0.2528 11.7652 278.7050 17.96% 0.2037 0.3121 12.9307 350.3693 17.05% 0.3877
    (0.75, 0.25) 0.1278 13.7709 541.2802 113.87% 0.1884 0.1272 15.0720 617.5108 91.91% 0.2996
    (0.25, 0.75) 0.3183 11.6256 273.5103 17.78% 0.2012 0.3636 12.6665 342.2354 17.50% 0.3341

     | Show Table
    DownLoad: CSV
    Table 3.  Results of the greedy heuristic.
    N π Avg. Time (s) F(HNbest) G(HNbest) Σ Lf Avg. Time (s) F(HNbest) G(HNbest) Σ Lf
    s.wd s.we
    1000 0.6 0.0107 3.0527 99.0290 13.01% 0.1592 0.0051 1.3479 29.1914 27.35% 0.2067
    0.75 0.0107 3.2114 99.4425 11.52% 0.1592 0.0051 1.4285 30.1426 30.21% 0.2363
    0.9 0.0106 3.2052 115.4524 29.28% 0.2168 0.0051 1.4364 30.1426 30.15% 0.2363
    2000 0.6 0.0106 3.0242 99.0290 13.52% 0.1592 0.0052 1.3290 29.1914 27.78% 0.2067
    0.75 0.0105 3.2032 115.7901 29.66% 0.2135 0.0051 1.4254 30.1548 30.28% 0.2363
    0.9 0.0104 3.1869 115.7901 29.71% 0.2168 0.0051 1.4452 30.8174 33.01% 0.2363
    3000 0.6 0.0107 3.0248 99.2357 13.69% 0.1592 0.0049 1.3206 29.1914 27.99% 0.2067
    0.75 0.0108 3.1768 115.7901 29.75% 0.2135 0.0050 1.3887 30.1548 30.66% 0.2363
    0.9 0.0107 3.1863 127.0964 42.31% 0.1854 0.0050 1.4004 30.8174 33.34% 0.2363
    5000 0.6 0.0106 3.0242 99.2357 13.70% 0.1592 0.0051 1.3110 29.1914 28.24% 0.2067
    0.75 0.0107 3.1757 115.7901 29.75% 0.2135 0.0052 1.3976 30.8174 33.37% 0.2363
    0.9 0.0105 3.1863 127.0964 42.31% 0.1854 0.0050 1.4004 38.7719 67.45% 0.2363
    10000 0.6 0.0107 3.0248 99.4425 13.88% 0.1592 0.0052 1.3198 30.1548 31.90% 0.2165
    0.75 0.0106 3.1046 115.7901 30.08% 0.2135 0.0051 1.3812 30.8174 33.57% 0.2363
    0.9 0.0106 3.1774 127.0964 42.33% 0.1854 0.0052 1.4004 38.7719 67.45% 0.2363
    l.wd l.we
    1000 0.6 0.0182 5.9512 161.8788 24.45% 0.1919 0.0226 7.1588 224.5650 18.26% 0.3413
    0.75 0.0182 6.0822 181.0020 38.17% 0.2403 0.0227 7.4886 243.7498 24.73% 0.4593
    0.9 0.0183 6.2173 192.9641 46.98% 0.2403 0.0227 7.7724 261.7975 32.83% 0.4784
    2000 0.6 0.0182 5.9512 162.0856 24.60% 0.1919 0.0226 7.1588 226.5650 19.03% 0.3413
    0.75 0.0185 6.0525 192.6460 47.01% 0.2403 0.0224 7.4688 241.7776 23.87% 0.4593
    0.9 0.0183 6.2166 192.9641 46.98% 0.2403 0.0227 7.7741 261.7975 32.83% 0.4593
    3000 0.6 0.0185 5.9101 162.2924 24.96% 0.1919 0.0225 7.1386 226.7717 19.27% 0.3413
    0.75 0.0184 6.0494 192.6460 47.02% 0.2403 0.0225 7.4384 253.1568 29.45% 0.4593
    0.9 0.0183 6.2044 199.0992 51.65% 0.2403 0.0228 7.7340 273.1469 38.59% 0.3977
    5000 0.6 0.0183 5.9512 176.0916 34.88% 0.2545 0.0227 7.1588 226.7717 19.11% 0.3413
    0.75 0.0183 6.0484 192.6460 47.02% 0.2403 0.0227 7.3845 253.1568 29.65% 0.4593
    0.9 0.0183 6.2228 204.2703 55.56% 0.2158 0.0225 7.7340 273.4415 38.73% 0.3977
    10000 0.6 0.0186 5.9101 178.4332 36.76% 0.3422 0.0227 7.1298 243.1194 26.21% 0.4529
    0.75 0.0184 6.0392 192.6460 47.04% 0.2403 0.0225 7.3742 253.1568 29.69% 0.4593
    0.9 0.0184 6.1931 204.2703 55.59% 0.2158 0.0226 7.7340 273.4415 38.73% 0.3977
    b.wd b.we
    1000 0.6 0.0308 11.3678 417.6331 67.39% 0.2377 0.0345 11.8833 481.3362 111.62% 0.3905
    0.75 0.0311 11.8052 450.8393 79.48% 0.2832 0.0345 12.3141 497.0862 113.88% 0.4711
    0.9 0.0309 12.1265 502.4882 99.30% 0.2832 0.0346 13.0252 561.2765 124.66% 0.3823
    2000 0.6 0.0310 11.4068 417.4538 67.24% 0.2377 0.0345 11.8965 481.6732 111.67% 0.3905
    0.75 0.0309 11.6766 450.8393 79.65% 0.2817 0.0345 12.2864 498.2471 114.05% 0.4648
    0.9 0.0310 12.0744 508.8815 101.85% 0.2832 0.0343 12.9924 561.9514 124.79% 0.4208
    3000 0.6 0.0309 11.4068 417.6606 67.32% 0.2377 0.0345 11.8256 482.9548 111.84% 0.3861
    0.75 0.0309 11.6062 450.8393 79.75% 0.2817 0.0345 12.2977 503.0244 114.77% 0.4711
    0.9 0.0309 12.0749 509.1201 101.94% 0.2832 0.0346 12.9632 561.9898 124.79% 0.3823
    5000 0.6 0.0312 11.4258 419.8494 68.12% 0.2377 0.0344 11.8763 483.9182 111.98% 0.3905
    0.75 0.0311 11.6031 450.8393 79.76% 0.2832 0.0345 12.2460 523.8958 118.09% 0.5698
    0.9 0.0311 12.0838 514.0527 103.87% 0.2672 0.0346 12.9826 569.6496 126.23% 0.3823
    10000 0.6 0.0310 11.4643 434.2840 73.59% 0.2781 0.0343 11.8366 483.9182 111.98% 0.3905
    0.75 0.0308 11.6123 476.5378 89.72% 0.2817 0.0344 12.2854 526.3093 118.49% 0.5698
    0.9 0.0310 12.0540 514.1322 103.93% 0.2672 0.0343 12.9655 569.9058 126.28% 0.3823

     | Show Table
    DownLoad: CSV

    This section discusses the results obtained in the computational experimentation, considering different aspects, including the impact of sample size and objective biased in algorithms efficiency and the quality and distribution of solutions in the Pareto front. Finally, the analysis of an illustrative case study is presented, to proper evaluate the quality of service provided to citizens.

    The obtained experimental results allow concluding that the methods are robust with respect to the size of the sample, since the increment of N has a limited effect on the performance. In both objectives, the increment in N generally reduces the standard deviation of the computed values. However, the average and best value only varies slightly (Tables A1 and A2). Moreover, results of the greedy heuristic using larger sizes of N are systematically worse in terms of distance to the ideal vector than than those computed using smaller sample sizes.

    The SAA problems were solved to optimality by Gurobi, being able to find solutions with 0% MIPGap for the compact mathematical formulation presented in Section 2.1 in relatively short computing times (less than 1 s for all instances). The analysis of execution time shows that schedules that are biased towards minimizing the cost objective (with higher values of β) are more difficult to solve for Gurobi, which requires a much larger computing time to solve the instances. In regard to the greedy heuristic, the algorithm is very fast to solve the instances, as all average computing times are less than 0.1 s. Moreover, unlike SAA, which is sensitive to the bias among objectives, the computing times of the greedy heuristic are independent of the aspiration level used since computing times do not vary.

    Another relevant aspect is that when considering the computing time of the whole resolution process, i.e., the generation of the random samples of the user preferences (reported in Table 1) and solving the optimization problem (either by the greedy heuristic or the SAA), the most time consuming stage is the generation of the random samples. Additionally, the time of generating the random sample increases approximately proportional to the size of N, whereas the average time of solving the optimization problem is almost constant for any size of N.

    Regarding solution quality, Table 4 reports the minimum, average and maximum value of the distance to the ideal vector for each instance and each algorithm.

    Table 4.  Minimum, average and maximum distance to the ideal vector.
    Instance SAA Greedy heuristic
    min avg max min avg max
    s.wd 10.94% 20.72% 33.75% 11.52% 25.63% 42.33%
    s.we 15.97% 29.80% 58.62% 27.35% 35.52% 67.45%
    l.wd 12.40% 25.24% 47.98% 24.45% 41.91% 55.59%
    l.we 11.01% 24.45% 40.50% 18.26% 28.07% 38.73%
    b.wd 17.78% 65.43% 146.19% 67.24% 84.19% 103.93%
    b.we 17.05% 51.96% 109.28% 111.62% 117.68% 126.28%

     | Show Table
    DownLoad: CSV

    Results in Table 4 indicate that SAA is able to obtain, on average, better solutions than the greedy heuristic. SA computed the smallest average distance on instance s.wd (20.72%). The best average distance for the greedy heuristic was obtained on the same instance s.wd (25.63%). Both algorithms obtained the worst results in terms of distance to the ideal vector for the building like instances. Results show that instances in which different users have to coordinate the use of appliances to not surpass the overall power consumption contracted by the building are more difficult to solve than those instances where a single user is considered. Regarding the best compromise solution, i.e., the solution that has the smallest distance to the ideal vector, it was obtained in instance s.wd in both algorithms. The smallest distance to the ideal vector computed by SAA was 10.94%, achieved using two different weights vectors, (0.5, 0.5) and (0.75, 0.25) for all the sample sizes N. Finally, regarding the greedy heuristic, the smallest computed distance is 11.52%, computed using π=0.75 and sample size N=1000.

    Regarding the load factor, Table 5 indicates that the greedy heuristic is able to obtain better results in all the instances either considering the average or the maximum load factor. Although maximizing the load factor was not part of the optimization problem, it is a relevant characteristic of the greedy heuristic since, as aforementioned, higher load factors are associated with a more stable functioning of the electric grid.

    Table 5.  Minimum, average and maximum load factor.
    Instance SAA Greedy heuristic
    min avg max min avg max
    s.wd 0.16500 0.17181 0.20680 0.15920 0.18660 0.21680
    s.we 0.16850 0.21462 0.23630 0.20670 0.22709 0.23630
    l.wd 0.14050 0.18641 0.24030 0.19190 0.23509 0.34220
    l.we 0.24480 0.33610 0.45290 0.34130 0.41636 0.47840
    b.wd 0.18820 0.20232 0.21570 0.23770 0.26829 0.28320
    b.we 0.27740 0.31824 0.38770 0.38230 0.42965 0.56980

     | Show Table
    DownLoad: CSV

    For better depicting the distribution and trade-off between the objective function values of the computed solutions, the Pareto fronts of the experiments with the larger sample size are presented in Figure 2. For the SAA, a total of 100 solutions are computed. These solutions were calculated using evenly separated weight vectors (α, β) with α+β=1. Regarding the greedy heuristic, 35 solutions with evenly separated aspiration level π with π[0.6,0.95] were computed. The SAA is able to better explore the search space, whereas the greedy heuristic finds, in general, solutions that have relatively large costs. From the analysis of the figures, it can be inferred that several runs of the SAA obtain similar solution (or even the same solution). The possibility of obtaining repeated solutions, i.e., obtaining the same solution for two different weight vectors, is a known disadvantage of using the weighting sum method for handling the multiobjective nature of an optimization problem [11]. To overcome this problem, more sophisticated multiobjective approaches, such as the augmented ε-constraint method, can be used. In regard to Pareto dominance, the solutions of the SAA usually dominates the solutions of the greedy heuristic.

    Figure 2.  Solutions for instances with sample size N=10000.

    This subsection presents an illustrative case study for one of the solved problem instances, to provide an insight on the scheduled power consumption in each time slot of the planning period, computed by the two studied methods. Figures 3 and 4 report the power consumption (in KW) in each time slot of representative solutions computed by the proposed approaches of the building-like scenario discriminated per user on weekday (b.wd) and weekend (b.we), respectively. Each time slot represents an interval of thirty minutes and they are numbered subsequently (e.g., time slot 0 represents the first thirty minutes of the day, and so on). Additionally, the cost of electricity foe each time slot is plotted as a line in the Figures (expressed in Uruguayan pesos per KW). Three solutions of the SAA and one solution of the greedy heuristic are presented. Selected solutions for the SAA correspond to the two extreme solutions and a balanced solution: one solution biased towards users satisfaction using vector (0.99, 0.01), a second solution biased towards cost reduction using vector (0.01, 0.99), and the third solution equally weighting the problem objectives, using vector (0.5, 0.5). The selected solution of the greedy heuristic is the one with an aspiration level of 0.75. These solutions are representative of different optimization results for both studied methods and provides diverse trade-offs between the problem objectives.

    Figure 3.  Power consumption per time slot for representative solutions of the b.wd instance and sample size N=10000.
    Figure 4.  Power consumption per time slot for representative solutions of the b.we instance and sample size N=10000.

    The analysis of Figure 3 allows concluding that users have a preference for using electric appliances at the end of the day. Thus, the solution that prioritizes user satisfaction has a large power consumption during the evening and night (Figure 3(a)). This is a common habit when users return to their homes after work at the end of the day, and they perform the majority of the activities in these hours. However, this part of the day corresponds to the peak hours, in which electricity price is more expensive and, thus, the solution presented in Figure 3(a) is rather expensive. Conversely, solutions that have a smaller total cost are biased towards using the appliances at the beginning of the day (as presented in Figure 3(c)). As expected, the solution presented in Figure 3(c), which was computed using a more balanced weight vector, defers the use of some appliances to the middle hours of the day. However, the cost objective has a greater influence than the user preferences, since a large part of the consumption is still allocated at early hours, where the electricity price is lower. Regarding the solution computed by the greedy heuristic (presented in Figure 3(d)), the energy consumption patter is rather similar to the one proposed by the solution of the SAA using a large weight for the user satisfaction objective (Figure 3(a)). However, the utilization of appliances is more distributed throughout the day. As a consequence, the peak consumption, i.e., the time slot with the highest consumption, is smaller for the greedy solution (8 KW) than for the SAA solution (10 KW).

    In any problem considering the scheduling or planning of human-related activities, the normal lifestyle and the timeline of daily actions limit the possibility of displacing the considered activities to some convenient, but dead periods. This is also the case for the studied problem, since deferring the use of electric appliances from peak hours to off-peak hours with lower electricity prices is not always possible, since several off-peak hours usually coincide with the time the users are resting at night. Another element that prevents users from taking advantage of lower electricity prices of off-peak hours are the normal working timetables, since usually during the morning and the noon the user is out of home, at work. However, a different scenario happens during weekends, when users remain more time at home and, therefore, they can perform some household tasks during off-peak hours.

    Regarding the computed solutions, the different situation that happens during weekends is depicted when performing pairwise comparisons between the four representative solutions of the weekend scenario (Figure 4) and the corresponding solutions of the weekday scenario (Figure 3). In the four cases, the solutions of the weekend scenario have a more distributed power consumption throughout the day, having a larger consumption in the middle hours of the day and a smaller peak consumption. The reduction of the peak consumption is particularly important for the greedy solution, i.e. from 8KW in the weekday solution (Figure 3(d)) to 4KW in the weekend solution (Figure 4(d)). Since the ToU pricing bill applied by the electricity company is the same for weekdays and weekends, the reason of the differences among weekdays and weekends solutions relies on the differences in users preferences. Although users still prefer to use appliances at the evening (as is evidenced in the solution that prioritizes user satisfaction of Figure 4(a)), users are also more willing to use the appliances in the middle of the day, allowing the resolution algorithms to better distribute the power consumption.

    As aforementioned, a more distributed power consumption throughout the day as occurs on the weekends results, benefits both users and electric companies. On the one hand, users are able to take advantage of the relatively cheaper off-peak hours. On the other hand, the reduction in the peak consumption, when considered in the city aggregated level, allows reducing the required infrastructure investment that electric companies have to perform to handle peak consumption and also allows significantly reducing the risk of power outages. In line with these benefits for the system, the recent rise of home office that has occurred due to the COVID-19 pandemic is as a great opportunity to balance the energy utilization by households, since users remain more time at home. However, to better analyze this possibility, new datasets should be gathered to incorporate the changes on the lifestyle of users of the pandemic. In line with this goal, the project 'Computational intelligence for the analysis of residential electricity consumption' is carried out in Uruguay, to gather relevant data from residential customers. The most relevant result of this project has been the generated ECD-UY dataset [45].

    Energy management is a crucial issue in modern societies, since an increasingly number of urban activities rely on an efficient electricity service. In order to improve energy management, it is not only required to improve the offer of electricity supply by companies, but also to enhance the demand-side of the system.

    This article addressed the household energy planning problem, aiming at improving the efficiency of the consumed energy. For achieving this goal, an optimization model was proposed for scheduling deferrable appliances considering two conflicting objectives: reducing the total cost of electricity paid by households (in a context of ToU pricing in electricity bills) and enhancing the users satisfaction with the energy consumed. To account for a realistic model, able to be implemented in practice, the restriction of the maximum allowable power consumption contracted by the user (to the electric company) was incorporated.

    The users satisfaction was estimated through a data-analysis model, studying historical data of households in order to determine the preferred time slots for using each appliance. Since considerable variations of these preferences were identified for different users, a stochastic resolution approach was applied to consider the uncertainty of this parameter.

    For solving the problem, two different algorithms were devised: a Sample Average Approximation method, which is a simulation-optimization approach that combines Monte Carlo simulation and deterministic mixed integer programming, and a greedy heuristic, which attempts at obtaining good global solutions by making locally optimal decisions repeatedly. The algorithms were tested on realistic instances. The instances comprehend scenarios with a single household, several households and building-like scenarios (in which diverse households or users has to coordinate the usage of appliances so the overall power consumption of the building does not surpass a certain joint threshold value). The results of the computational experimentation show the competitiveness of the proposed approach which are able to compute different compromising solutions accounting for the trade-off between these two conflicting optimization criteria in reasonable computing times The Sample Average Approximation method systematically outperformed the solutions obtained by the the greedy heuristic. However, the heuristic is much faster. The building-like instances were the more challenging for both algorithms requiring larger computing times. At least, for the analyzed cases, the size of the sample of the user preferences seems to not affect largely the performance of the algorithm. The results also allowed analyzing the different users behaviour between the weekdays and the weekend, finding that during weekends the appliance usage is more distributed throughout the day.

    The main lines for future work are related to expand the computational experimentation of the proposed model and algorithms, by including more households, e.g., instances that represent an apartment building or a gated community. In turn, the proposed model can be expanded by considering non-controllable appliances and renewable power generators within the household, e.g., solar or wind power generators. In relation to the input data, it would be useful to gather updated information in order to analyze if the variations in the lifestyle of users due to the pandemic and home office have substantially alter the user preferences, and compute accurate planning for this new situation too. Regarding the resolution algorithms, two future lines of work are will be consider. On the one hand, SAA can be improved by replacing the bi-objective approach based on weighting sum with a more advanced exact multiobjective method (e.g., augmented ε-constraint method) to avoid obtaining repeated solutions. On the other hand, population-based explicit multiobjective optimization methods, such as multiobjective evolutionary algorithms, can be implemented to better explore the trade-off among objectives. {Regarding preserving users' privacy, the proposed model can be extended by including appliance shifting and scheduling to control battery charging and discharging. Finally, an interesting research line to explore in the future is the comparison with other stochastic and/or robust resolution approaches.

    D. Rossit was supported by the program "Estancias de investigadores de reconocido prestigio en la UMA'' (ayuda D.3) of the Vicerrectorado de Investigación y Transferencia of the Universidad de Málaga and the research projects 24/J084 and 24/J086 of the Universidad Nacional del Sur. The work of S. Nesmachnow is partly funded by ANII and PEDECIBA, Uruguay. J. Toutouh was funded by European Union's Horizon 2020 research under the Marie Skłodowska-Curie grant agreement No. 799078. We would like to thank the anonymous reviewers for their insightful comments that led us to improve the article.

    All authors declare no conflicts of interest in this paper.

    This Section presents the detailed experimental results of the SAA and the Greedy heuristic. The details of the SAA are presented in Table A1. This table reports for each instance, the sample size N, the combination of weights (α,β), and the average and standard deviation of five relevant metrics:

    ● the execution time;

    ● the users satisfaction function F evaluated over N;

    ● the cost function G evaluated over N;

    ● the values of F and G of the best solution, i.e., the solution that has the minimal value of function H, as defined in Eq (11);

    ● the deviation of the solution to the ideal vector Σ, computed using the L2 distance norm, according to Eq (15).

    In turn, the detailed experimental results of the the greedy heuristic are reported in Table A2. This table presents for each instance, the sample size N, the aspiration preference level π, and the average and standard deviation of the five metrics also reported for the SAA. The computing times in Tables A1 and A2 do not include the time to generate the N random realizations of vector UP.

    Table A1.  Detailed results of the SAA.
    N (α,β) Time (s) FN GN F(HNbest) G(HNbest) Σ
    Avg Std Avg Std Avg Std
    Small instance weekday (s.wd)
    1000 (0.99, 0.01) 0.0138 0.0016 3.2803 0.0139 113.3180 1.8047 3.2925 113.3289 26.78%
    (0.01, 0.99) 0.1625 0.0108 2.1790 0.0012 89.3876 0.0000 2.1814 89.3876 33.75%
    (0.50, 0.50) 0.0911 0.0167 3.1311 0.0108 98.2122 0.1354 3.1368 98.2011 10.94%
    (0.75, 0.25) 0.0453 0.0077 3.1455 0.1290 100.7468 2.6877 3.1368 98.2011 10.94%
    (0.25, 0.75) 0.1436 0.0184 2.5557 0.0404 90.8810 0.3302 2.5555 90.8443 22.44%
    2000 (0.99, 0.01) 0.0137 0.0017 3.2870 0.0068 113.0862 0.5205 3.2925 113.3289 26.78%
    (0.01, 0.99) 0.1619 0.0134 2.1789 0.0012 89.3876 0.0000 2.1813 89.3876 33.75%
    (0.50, 0.50) 0.1051 0.0168 3.1336 0.0079 98.1928 0.0828 3.1368 98.2011 10.94%
    (0.75, 0.25) 0.0486 0.0069 3.1482 0.0247 99.3211 1.8497 3.1368 98.2011 10.94%
    (0.25, 0.75) 0.1421 0.0323 2.5525 0.0087 90.8388 0.0388 2.5558 90.8443 22.43%
    3000 (0.99, 0.01) 0.0136 0.0015 3.2892 0.0051 113.1910 0.4158 3.2925 113.3289 26.78%
    (0.01, 0.99) 0.1624 0.0124 2.1789 0.0013 89.3876 0.0000 2.1819 89.3876 33.73%
    (0.50, 0.50) 0.0963 0.0131 3.1347 0.0058 98.1956 0.0552 3.1368 98.2011 10.94%
    (0.75, 0.25) 0.0497 0.0055 3.1371 0.1149 99.5480 1.9951 3.1368 98.2011 10.94%
    (0.25, 0.75) 0.1351 0.0119 2.5538 0.0021 90.8443 0.0000 2.5558 90.8443 22.43%
    5000 (0.99, 0.01) 0.0133 0.0012 3.2916 0.0027 113.2958 0.2010 3.2925 113.3289 26.78%
    (0.01, 0.99) 0.1607 0.0090 2.1791 0.0012 89.3876 0.0000 2.1815 89.3876 33.74%
    (0.50, 0.50) 0.1038 0.0117 3.1301 0.0626 98.2011 0.0000 3.1368 98.2011 10.94%
    (0.75, 0.25) 0.0499 0.0051 3.1378 0.0737 98.9091 1.5854 3.1368 98.2011 10.94%
    (0.25, 0.75) 0.1359 0.0113 2.5544 0.0010 90.8443 0.0000 2.5558 90.8443 22.43%
    10000 (0.99, 0.01) 0.0131 0.0012 3.2920 0.0014 113.3179 0.1103 3.2925 113.3289 26.78%
    (0.01, 0.99) 0.1616 0.0117 2.1790 0.0012 89.3876 0.0000 2.1814 89.3876 33.75%
    (0.50, 0.50) 0.1030 0.0095 3.1293 0.0706 98.2011 0.0000 3.1368 98.2011 10.94%
    (0.75, 0.25) 0.0497 0.0043 3.1421 0.0201 98.7403 1.3909 3.1368 98.2011 10.94%
    (0.25, 0.75) 0.1340 0.0122 2.5544 0.0006 90.8443 0.0000 2.5558 90.8443 22.43%
    Small instance weekend (s.we)
    1000 (0.99, 0.01) 0.0081 0.0003 1.4628 0.0154 37.1724 2.1527 1.4763 36.7695 58.62%
    (0.01, 0.99) 0.0452 0.0005 1.0122 0.0024 23.1810 0.0000 1.0148 23.1810 31.26%
    (0.50, 0.50) 0.0122 0.0005 1.4190 0.0621 28.8515 0.0989 1.4363 28.8151 24.46%
    (0.75, 0.25) 0.0099 0.0005 1.4341 0.0208 30.4370 3.1578 1.4363 28.8151 24.46%
    (0.25, 0.75) 0.0189 0.0018 1.2663 0.0196 24.9394 0.3013 1.2623 24.7508 16.00%
    2000 (0.99, 0.01) 0.0085 0.0005 1.4715 0.0068 36.9087 0.1834 1.4763 36.7695 58.62%
    (0.01, 0.99) 0.0475 0.0018 1.0126 0.0014 23.1810 0.0000 1.0149 23.1810 31.25%
    (0.50, 0.50) 0.0149 0.0008 1.4238 0.0256 28.6683 0.60120 1.4363 28.8151 24.46%
    (0.75, 0.25) 0.0118 0.0006 1.4309 0.0071 28.8746 0.0970 1.4363 28.8151 24.46%
    (0.25, 0.75) 0.0290 0.0026 1.2576 0.0120 24.7408 0.0998 1.2627 24.7508 15.97%
    3000 (0.99, 0.01) 0.0084 0.0003 1.4740 0.0029 36.8783 0.1410 1.4763 36.7695 58.62%
    (0.01, 0.99) 0.0476 0.0017 1.0125 0.0010 23.1810 0.0000 1.0148 23.1810 31.26%
    (0.50, 0.50) 0.0148 0.0010 1.4333 0.0041 28.8258 0.0395 1.4363 28.8151 24.46%
    (0.75, 0.25) 0.0117 0.0005 1.4335 0.0040 28.8658 0.0727 1.4363 28.8151 24.46%
    (0.25, 0.75) 0.0289 0.0027 1.2590 0.0023 24.7508 0.0000 1.2627 24.7508 15.97%
    5000 (0.99, 0.01) 0.0088 0.0006 1.4743 0.0011 36.8685 0.1362 1.4763 36.7695 58.62%
    (0.01, 0.99) 0.0475 0.0018 1.0126 0.0010 23.1810 0.0000 1.0151 23.1810 31.24%
    (0.50, 0.50) 0.0146 0.0006 1.4342 0.0012 28.8243 0.0367 1.4363 28.8151 24.46%
    (0.75, 0.25) 0.0115 0.0004 1.4342 0.0012 28.8612 0.0709 1.4363 28.8151 24.46%
    (0.25, 0.75) 0.0288 0.0027 1.2596 0.0012 24.7508 0.0000 1.2622 24.7508 16.01%
    10000 (0.99, 0.01) 0.0088 0.0005 1.4740 0.0014 36.8696 0.1290 1.4763 36.7695 58.62%
    (0.01, 0.99) 0.0477 0.0022 1.0124 0.0010 23.1810 0.0000 1.0150 23.1810 31.25%
    (0.50, 0.50) 0.0146 0.0007 1.4340 0.0014 28.8151 0.0000 1.4363 28.8151 24.46%
    (0.75, 0.25) 0.0116 0.0004 1.4339 0.0014 28.8335 0.0503 1.4363 28.8151 24.46%
    (0.25, 0.75) 0.0282 0.0031 1.2594 0.0014 24.7508 0.0000 1.2622 24.7508 16.01%
    Large instance weekday (l.wd)
    1000 (0.99, 0.01) 0.0226 0.0014 6.4349 0.0196 202.8882 12.5886 6.4528 194.4567 47.98%
    (0.01, 0.99) 0.2624 0.0178 4.4163 0.0047 131.4108 0.0000 4.4226 131.4108 31.46%
    (0.50, 0.50) 0.0769 0.0067 6.0279 0.0123 145.5251 0.4130 6.0373 145.3292 12.40%
    (0.75, 0.25) 0.0485 0.0054 6.3453 0.0706 179.6205 8.1416 6.4120 185.3487 41.05%
    (0.25, 0.75) 0.1768 0.0169 5.7133 0.0674 140.2097 1.0259 5.7316 140.2288 13.04%
    2000 (0.99, 0.01) 0.0232 0.0014 6.4460 0.0071 198.2309 10.5549 6.4528 194.4567 47.98%
    (0.01, 0.99) 0.2596 0.0184 4.4182 0.0040 131.4108 0.0000 4.4229 131.4108 31.46%
    (0.50, 0.50) 0.0920 0.0093 6.0262 0.0351 145.2319 0.1326 6.0373 145.3292 12.40%
    (0.75, 0.25) 0.0631 0.0059 6.1017 0.0300 150.7660 2.8692 6.1053 150.3087 15.36%
    (0.25, 0.75) 0.2310 0.0666 5.3985 0.1048 136.9402 0.7238 5.3944 136.7499 16.90%
    3000 (0.99, 0.01) 0.0232 0.0016 6.4477 0.0054 198.3285 10.7203 6.4528 194.4567 47.98%
    (0.01, 0.99) 0.2582 0.0164 4.4193 0.0030 131.4108 0.0000 4.4230 131.4108 31.46%
    (0.50, 0.50) 0.0876 0.0083 6.0310 0.0064 145.2613 0.1481 6.0373 145.3292 12.40%
    (0.75, 0.25) 0.0632 0.0057 6.1123 0.0357 151.8834 3.8517 6.1049 150.3087 15.36%
    (0.25, 0.75) 0.2334 0.0648 5.4604 0.1090 137.4749 1.1065 5.3944 136.7499 16.90%
    5000 (0.99, 0.01) 0.0230 0.0013 6.4498 0.0037 198.1247 10.1350 6.4528 194.4567 47.98%
    (0.01, 0.99) 0.2591 0.0181 4.4200 0.0021 131.4108 0.0000 4.4230 131.4108 31.46%
    (0.50, 0.50) 0.0781 0.0052 6.0342 0.0037 145.3265 0.0936 6.0373 145.3292 12.40%
    (0.75, 0.25) 0.0578 0.0053 6.3499 0.0715 178.4627 8.2341 6.3393 176.3038 34.21%
    (0.25, 0.75) 0.1886 0.0178 5.7109 0.0757 140.1015 0.2313 5.7325 140.2288 13.02%
    10000 (0.99, 0.01) 0.0224 0.0010 6.4504 0.0019 196.0937 8.7299 6.4528 194.4567 47.98%
    (0.01, 0.99) 0.2592 0.0135 4.4204 0.0011 131.4108 0.0000 4.4224 131.4108 31.47%
    (0.50, 0.50) 0.0846 0.0067 6.0333 0.0043 145.2682 0.1123 6.0373 145.3292 12.40%
    (0.75, 0.25) 0.0630 0.0038 6.1074 0.0125 150.8103 1.2606 6.1053 150.3087 15.36%
    (0.25, 0.75) 0.2232 0.0136 5.6141 0.1047 139.0050 1.0691 5.6829 139.6771 13.49%
    Large instance weekend (l.we)
    1000 (0.99, 0.01) 0.0216 0.0015 8.1262 0.0179 279.8359 4.5634 8.1457 277.5776 40.50%
    (0.01, 0.99) 0.2226 0.0117 5.9123 0.0060 197.5689 0.0000 5.9167 197.5689 27.37%
    (0.50, 0.50) 0.1225 0.0258 7.4217 0.1720 211.8408 0.2304 7.4650 211.7288 11.01%
    (0.75, 0.25) 0.0746 0.0069 7.9338 0.1105 252.6845 7.3916 7.9526 251.6011 27.45%
    (0.25, 0.75) 0.4123 0.0161 7.0213 0.0979 205.1840 0.60202 7.0347 205.0788 14.16%
    2000 (0.99, 0.01) 0.0211 0.0010 8.1375 0.0075 278.7941 1.5364 8.1457 277.5776 40.50%
    (0.01, 0.99) 0.2229 0.0098 5.9143 0.0025 197.5689 0.0000 5.9174 197.5689 27.36%
    (0.50, 0.50) 0.1166 0.0072 7.4556 0.0706 211.8562 0.3389 7.4650 211.7288 11.01%
    (0.75, 0.25) 0.0718 0.0068 7.9747 0.0401 255.2160 4.7386 7.9525 251.6011 27.45%
    (0.25, 0.75) 0.4118 0.0219 7.0625 0.0377 205.4796 0.4686 7.0815 205.6305 13.69%
    3000 (0.99, 0.01) 0.0206 0.0011 8.1408 0.0051 279.2213 1.4356 8.1457 277.5776 40.50%
    (0.01, 0.99) 0.2226 0.0151 5.9147 0.0010 197.5689 0.0000 5.9168 197.5689 27.36%
    3000 (0.50, 0.50) 0.1181 0.0074 7.4424 0.1168 211.7536 0.0945 7.4650 211.7288 11.01%
    (0.75, 0.25) 0.0769 0.0066 7.9613 0.0346 253.3056 4.1294 7.9423 250.4978 26.91%
    (0.25, 0.75) 0.4140 0.0173 7.0567 0.0242 205.3864 0.2758 7.0349 205.0788 14.16%
    5000 (0.99, 0.01) 0.0207 0.0010 8.1421 0.0036 279.0386 1.4598 8.1458 277.5776 40.50%
    (0.01, 0.99) 0.2265 0.0097 5.9146 0.0010 197.5689 0.0000 5.9175 197.5689 27.36%
    (0.50, 0.50) 0.1180 0.0055 7.4597 0.0335 211.7412 0.0546 7.4650 211.7288 11.01%
    (0.75, 0.25) 0.0779 0.0064 7.9604 0.0312 253.0517 3.7191 7.9526 251.6011 27.45%
    (0.25, 0.75) 0.4271 0.0202 7.0579 0.0721 205.4672 0.2488 7.0815 205.6305 13.69%
    10000 (0.99, 0.01) 0.0203 0.0009 8.1433 0.0018 278.9856 1.3821 8.1458 277.5776 40.50%
    (0.01, 0.99) 0.2236 0.0095 5.9147 0.0011 197.5689 0.0000 5.9171 197.5689 27.36%
    (0.50, 0.50) 0.1155 0.0058 7.4602 0.0333 211.7288 0.0000 7.4650 211.7288 11.01%
    (0.75, 0.25) 0.0783 0.0051 7.9709 0.0357 254.1777 4.3063 7.9526 251.6011 27.45%
    (0.25, 0.75) 0.4185 0.0211 7.0515 0.0992 205.4698 0.2461 7.0817 205.6305 13.69%
    Building instance weekday (b.we)
    1000 (0.99, 0.01) 0.0661 0.0040 13.7769 0.0218 599.3116 21.1016 13.8077 606.4009 139.60%
    (0.01, 0.99) 0.4757 0.0277 8.5596 0.0065 253.0874 0.0000 8.5678 253.0874 37.98%
    (0.50, 0.50) 0.2532 0.0481 11.7499 0.0317 279.7284 2.0997 11.7540 278.1533 17.91%
    (0.75, 0.25) 0.1234 0.0134 13.5128 0.1569 500.0129 26.3286 13.4735 483.9198 91.24%
    (0.25, 0.75) 0.3049 0.0429 11.5921 0.0194 273.2586 0.2954 11.6152 273.2761 17.81%
    2000 (0.99, 0.01) 0.0682 0.0030 13.7903 0.0135 603.8349 20.2691 13.8106 597.8174 136.21%
    (0.01, 0.99) 0.4740 0.0220 8.5617 0.0055 253.0874 0.0000 8.5677 253.0874 37.98%
    (0.50, 0.50) 0.2471 0.0280 11.7560 0.0282 279.5247 2.0091 11.7545 278.1533 17.90%
    (0.75, 0.25) 0.1327 0.0147 13.4178 0.2435 484.8300 23.8387 13.4746 483.5656 91.10%
    (0.25, 0.75) 0.3104 0.0328 11.5977 0.0160 273.1843 0.3000 11.5933 272.8259 17.87%
    3000 (0.99, 0.01) 0.0682 0.0024 13.7980 0.0116 606.0593 19.6059 13.8148 623.0884 146.19%
    (0.01, 0.99) 0.4765 0.0183 8.5616 0.0051 253.0874 0.0000 8.5678 253.0874 37.98%
    (0.50, 0.50) 0.2564 0.0314 11.7610 0.0281 279.5719 1.9144 11.7642 278.7050 17.97%
    (0.75, 0.25) 0.1280 0.0108 13.6629 0.1275 522.8374 24.9542 13.7706 541.3700 113.91%
    (0.25, 0.75) 0.3070 0.0412 11.6026 0.0145 273.2208 0.2718 11.6260 273.5103 17.78%
    5000 (0.99, 0.01) 0.0686 0.0028 13.8027 0.0095 608.3604 17.3050 13.8122 597.9184 136.25%
    (0.01, 0.99) 0.4751 0.0223 8.5625 0.0046 253.0874 0.0000 8.5677 253.0874 37.98%
    (0.50, 0.50) 0.2510 0.0281 11.7857 0.0352 281.0278 2.5287 11.7639 278.7050 17.97%
    (0.75, 0.25) 0.1248 0.0093 13.7560 0.0360 540.0094 6.6175 13.7818 543.5414 114.76%
    (0.25, 0.75) 0.3250 0.0286 11.6083 0.0148 273.2866 0.2900 11.6264 273.5103 17.78%
    10000 (0.99, 0.01) 0.0692 0.0021 13.8061 0.0047 605.8480 14.9975 13.8113 597.9185 136.25%
    (0.01, 0.99) 0.4754 0.0222 8.5646 0.0033 253.0874 0.0000 8.5684 253.0874 37.98%
    (0.50, 0.50) 0.2528 0.0229 11.7888 0.0351 280.8773 2.5132 11.7652 278.7050 17.96%
    (0.75, 0.25) 0.1278 0.0089 13.7618 0.0253 540.4495 3.6847 13.7709 541.2802 113.87%
    (0.25, 0.75) 0.3183 0.0326 11.6065 0.0140 273.1783 0.2917 11.6256 273.5103 17.78%
    Building instance weekend (b.wd)
    1000 (0.99, 0.01) 0.0588 0.0044 15.1073 0.0162 668.9267 20.4477 15.1327 673.2478 109.23%
    (0.01, 0.99) 0.4897 0.0424 9.7430 0.0068 321.7733 0.0000 9.7527 321.7733 35.56%
    (0.50, 0.50) 0.3244 0.0300 12.9151 0.0170 351.0687 0.90856 12.9297 350.60130 17.10%
    (0.75, 0.25) 0.1389 0.0159 14.9392 0.1861 598.2614 32.5772 15.0719 617.7601 91.99%
    (0.25, 0.75) 0.3772 0.0285 12.6216 0.1420 342.1330 0.5688 12.6639 342.2254 17.51%
    2000 (0.99, 0.01) 0.0585 0.0032 15.1182 0.0092 669.5280 16.9211 15.1319 673.3213 109.25%
    (0.01, 0.99) 0.4966 0.0523 9.7451 0.0062 321.7733 0.0000 9.7521 321.7733 35.56%
    (0.50, 0.50) 0.3175 0.0259 12.9235 0.0150 351.0239 0.90798 12.9298 350.3693 17.06%
    (0.75, 0.25) 0.1269 0.0135 15.0654 0.0712 618.6035 12.5092 15.0740 617.9178 92.04%
    (0.25, 0.75) 0.3765 0.0419 12.6483 0.0177 342.1749 0.2827 12.6648 342.2354 17.51%
    3000 (0.99, 0.01) 0.0591 0.0024 15.1207 0.0086 670.8186 17.6587 15.1316 672.6650 109.05%
    (0.01, 0.99) 0.5026 0.0533 9.7466 0.0051 321.7733 0.0000 9.7524 321.7733 35.56%
    (0.50, 0.50) 0.3136 0.0227 12.9253 0.0148 350.90454 0.90633 12.9297 350.3693 17.06%
    (0.75, 0.25) 0.1285 0.0130 15.0703 0.0676 618.8199 11.9090 15.0718 617.6007 91.94%
    (0.25, 0.75) 0.3773 0.0313 12.6503 0.0229 342.1598 0.2545 12.6652 342.2354 17.51%
    5000 (0.99, 0.01) 0.0599 0.0027 15.1249 0.0061 673.6397 15.1150 15.1321 672.5696 109.02%
    (0.01, 0.99) 0.5123 0.0468 9.7481 0.0045 321.7733 0.0000 9.7533 321.7733 35.55%
    (0.50, 0.50) 0.3166 0.0224 12.9256 0.0088 350.60527 0.60831 12.9315 350.3693 17.05%
    (0.75, 0.25) 0.1314 0.0113 15.0658 0.1927 621.1851 3.9240 15.0722 617.6647 91.96%
    (0.25, 0.75) 0.3760 0.0230 12.6550 0.0200 342.2071 0.2067 12.6666 342.2354 17.50%
    10000 (0.99, 0.01) 0.0597 0.0024 15.1262 0.0047 675.4433 11.6835 15.1336 673.4111 109.28%
    (0.01, 0.99) 0.5109 0.0424 9.7488 0.0039 321.7733 0.0000 9.7526 321.7733 35.56%
    (0.50, 0.50) 0.3121 0.0251 12.9263 0.0069 350.5282 0.4890 12.9307 350.3693 17.05%
    (0.75, 0.25) 0.1272 0.0126 15.0861 0.0197 621.3305 3.9752 15.0720 617.5108 91.91%
    (0.25, 0.75) 0.3636 0.0299 12.6598 0.0065 342.2288 0.0662 12.6665 342.2354 17.50%

     | Show Table
    DownLoad: CSV
    Table A2.  Detailed results of the greedy heuristic.
    N π Time (s) FN GN F(HNbest) G(HNbest) Σ
    Avg Std Avg Std Avg Std
    Small instance weekday (s.wd)
    1000 0.60 0.0107 0.0011 2.9225 0.0677 112.1103 6.5536 3.0527 99.0290 13.01%
    0.75 0.0107 0.0012 3.0518 0.0669 116.3108 4.0727 3.2114 99.4425 11.52%
    0.90 0.0106 0.0011 3.1621 0.0295 125.7762 3.5871 3.2052 115.4524 29.28%
    2000 0.60 0.0106 0.0011 2.9230 0.0527 114.3086 4.4380 3.0242 99.0290 13.52%
    0.75 0.0105 0.0010 3.0444 0.0552 116.4233 2.3194 3.2032 115.7901 29.66%
    0.90 0.0104 0.0011 3.1637 0.0182 126.8703 1.5908 3.1869 115.7901 29.71%
    3000 0.60 0.0107 0.0013 2.9387 0.0514 114.5011 4.2024 3.0248 99.2357 13.69%
    0.75 0.0108 0.0012 3.0365 0.0420 115.8806 0.90045 3.1768 115.7901 29.75%
    0.90 0.0107 0.0011 3.1663 0.0112 127.0964 0.0000 3.1863 127.0964 42.31%
    5000 0.60 0.0106 0.0010 2.9308 0.0392 115.2916 2.3066 3.0242 99.2357 13.70%
    0.75 0.0107 0.0010 3.0293 0.0395 115.9710 1.2727 3.1757 115.7901 29.75%
    0.90 0.0105 0.0011 3.1676 0.0105 127.0964 0.0000 3.1863 127.0964 42.31%
    10000 0.60 0.0107 0.0011 2.9291 0.0401 115.4509 1.6186 3.0248 99.4425 13.88%
    0.75 0.0106 0.0012 3.0216 0.0302 115.7901 0.0000 3.1046 115.7901 30.08%
    0.90 0.0106 0.0011 3.1661 0.0101 127.0964 0.0000 3.1774 127.0964 42.33%
    Small instance weekend (s.we)
    1000 0.60 0.0051 0.0009 1.2498 0.0476 30.0738 0.5013 1.3479 29.1914 27.35%
    0.75 0.0051 0.0010 1.3340 0.0364 32.4094 3.3157 1.4285 30.1426 30.21%
    0.90 0.0051 0.0009 1.3813 0.0267 37.9895 2.4309 1.4364 30.1426 30.15%
    2000 0.60 0.0052 0.0008 1.2567 0.0349 30.0826 0.3731 1.3290 29.1914 27.78%
    0.75 0.0051 0.0009 1.3467 0.0324 32.7214 3.5060 1.4254 30.1548 30.28%
    0.90 0.0051 0.0010 1.3904 0.0158 38.5697 1.3731 1.4452 30.8174 33.01%
    3000 0.60 0.0049 0.0009 1.2591 0.0254 30.2018 0.2190 1.3206 29.1914 27.99%
    0.75 0.0050 0.0008 1.3386 0.0241 31.7521 2.6078 1.3887 30.1548 30.606%
    0.90 0.0050 0.0010 1.3843 0.0145 38.6282 1.1224 1.4004 30.8174 33.34%
    5000 0.60 0.0051 0.0009 1.2617 0.0201 30.1916 0.1976 1.3110 29.1914 28.24%
    0.75 0.0052 0.0010 1.3406 0.0183 31.2152 1.7424 1.3976 30.8174 33.37%
    0.90 0.0050 0.0009 1.3864 0.0117 38.7811 0.0367 1.4004 38.7719 67.45%
    10000 0.60 0.0052 0.0009 1.2569 0.0173 30.1747 0.1136 1.3198 30.1548 31.90%
    0.75 0.0051 0.0009 1.3395 0.0137 30.8970 0.7954 1.3812 30.8174 33.57%
    0.90 0.0052 0.0010 1.3853 0.0109 38.7734 0.0154 1.4004 38.7719 67.45%
    Large instance weekday (l.wd)
    1000 0.60 0.0182 0.0011 5.6026 0.1344 184.4582 10.1460 5.9512 161.8788 24.45%
    0.75 0.0182 0.0011 5.7627 0.1197 193.1504 4.0661 6.0822 181.0020 38.17%
    0.90 0.0183 0.0012 6.0024 0.1231 209.5946 6.7438 6.2173 192.9641 46.98%
    2000 0.60 0.0182 0.0011 5.6043 0.1253 186.9391 7.6256 5.9512 162.0856 24.60%
    0.75 0.0185 0.0010 5.7550 0.1187 192.9317 1.5485 6.0525 192.6460 47.01%
    0.90 0.0183 0.0011 5.9891 0.1029 209.3575 4.8414 6.2166 192.9641 46.98%
    3000 0.60 0.0185 0.0011 5.6218 0.1203 187.1703 7.3182 5.9101 162.2924 24.96%
    0.75 0.0184 0.0010 5.7628 0.1152 193.1102 1.9787 6.0494 192.6460 47.02%
    0.90 0.0183 0.0011 5.9927 0.1063 210.3703 4.1216 6.2044 199.0992 51.65%
    5000 0.60 0.0183 0.0012 5.6197 0.1198 188.2678 6.4818 5.9512 176.0916 34.88%
    0.75 0.0183 0.0010 5.7412 0.1094 192.6548 0.0250 6.0484 192.6460 47.02%
    0.90 0.0183 0.0011 5.9957 0.1040 209.6794 2.0785 6.2228 204.2703 55.56%
    10000 0.60 0.0186 0.0011 5.6375 0.1188 189.7967 5.5210 5.9101 178.4332 36.76%
    0.75 0.0184 0.0010 5.7535 0.1043 192.6468 0.0080 6.0392 192.6460 47.04%
    0.90 0.0184 0.0012 6.0035 0.1041 210.1639 1.3449 6.1931 204.2703 55.59%
    Large instance weekend (l.we)
    1000 0.60 0.0226 0.0011 6.8773 0.1228 240.7261 6.6997 7.1588 224.5650 18.26%
    0.75 0.0227 0.0012 7.1466 0.1317 256.4022 4.7405 7.4886 243.7498 24.73%
    0.90 0.0227 0.0012 7.5502 0.1127 275.5612 3.8706 7.7724 261.7975 32.83%
    2000 0.60 0.0226 0.0011 6.8760 0.1061 241.9364 5.5255 7.1588 226.5650 19.03%
    0.75 0.0224 0.0011 7.1351 0.1091 255.5291 3.1638 7.4688 241.7776 23.87%
    0.90 0.0227 0.0011 7.5550 0.1038 276.3039 3.4833 7.7741 261.7975 32.83%
    3000 0.60 0.0225 0.0011 6.8761 0.1075 243.5329 3.5692 7.1386 226.7717 19.27%
    0.75 0.0225 0.0011 7.1217 0.1051 256.0430 2.2317 7.4384 253.1568 29.45%
    0.90 0.0228 0.0011 7.5352 0.0886 276.7002 2.8177 7.7340 273.1469 38.59%
    5000 0.60 0.0227 0.0011 6.9074 0.1146 244.0033 2.1013 7.1588 226.7717 19.11%
    0.75 0.0227 0.0011 7.1456 0.1071 255.6718 1.0131 7.3845 253.1568 29.65%
    0.90 0.0225 0.0014 7.5650 0.1014 277.2199 2.6652 7.7340 273.4415 38.73%
    10000 0.60 0.0227 0.0012 6.8757 0.0973 244.0110 1.1620 7.1298 243.1194 26.21%
    0.75 0.0225 0.0011 7.1262 0.0972 255.7413 0.7002 7.3742 253.1568 29.69%
    0.90 0.0226 0.0012 7.5460 0.0911 277.5050 2.5468 7.7340 273.4415 38.73%
    Building weekday (b.wd)
    1000 0.60 0.0308 0.0012 11.0079 0.1488 441.6881 10.60020 11.3678 417.6331 67.39%
    0.75 0.0311 0.0013 11.3737 0.1292 485.5665 16.4973 11.8052 450.8393 79.48%
    0.90 0.0309 0.0012 11.8555 0.1035 520.1808 7.5063 12.1265 502.4882 99.30%
    2000 0.60 0.0310 0.0011 11.0445 0.1526 443.0685 8.6618 11.4068 417.4538 67.24%
    0.75 0.0309 0.0010 11.3566 0.1163 492.4860 15.3699 11.6766 450.8393 79.65%
    0.90 0.0310 0.0011 11.8440 0.1085 520.0559 3.8396 12.0744 508.8815 101.85%
    3000 0.60 0.0309 0.0012 11.0273 0.1593 442.9396 8.0281 11.4068 417.6606 67.32%
    0.75 0.0309 0.0010 11.3437 0.1138 491.7616 14.6810 11.6062 450.8393 79.75%
    0.90 0.0309 0.0013 11.8308 0.1045 519.7176 4.1177 12.0749 509.1201 101.94%
    5000 0.60 0.0312 0.0012 11.0649 0.1446 444.8930 7.4917 11.4258 419.8494 68.12%
    0.75 0.0311 0.0011 11.3348 0.1147 496.9334 11.7561 11.6031 450.8393 79.76%
    0.90 0.0311 0.0010 11.8272 0.0990 519.9015 1.4629 12.0838 514.0527 103.87%
    10000 0.60 0.0310 0.0011 11.0605 0.1447 445.3209 6.5916 11.4643 434.2840 73.59%
    0.75 0.0308 0.0011 11.3351 0.1219 500.7149 6.1324 11.6123 476.5378 89.72%
    0.90 0.0310 0.0012 11.8402 0.1021 520.0757 1.0513 12.0540 514.1322 103.93%
    Building weekend (b.we)
    1000 0.60 0.0345 0.0011 11.5544 0.1442 487.6129 5.0430 11.8833 481.3362 111.62%
    0.75 0.0345 0.0011 12.0346 0.1199 538.6840 16.4980 12.3141 497.0862 113.88%
    0.90 0.0346 0.0012 12.7948 0.1120 573.4305 5.4551 13.0252 561.2765 124.66%
    2000 0.60 0.0345 0.0011 11.5617 0.1353 486.5699 2.8961 11.8965 481.6732 111.67%
    0.75 0.0345 0.0013 12.0380 0.1076 544.3418 14.9656 12.2864 498.2471 114.05%
    0.90 0.0343 0.0013 12.7841 0.1015 573.4661 3.1760 12.9924 561.9514 124.79%
    3000 0.60 0.0345 0.0013 11.5590 0.1208 486.1981 2.8327 11.8256 482.9548 111.84%
    0.75 0.0345 0.0012 12.0298 0.1109 545.2023 12.5249 12.2977 503.0244 114.77%
    0.90 0.0346 0.0012 12.7696 0.1007 572.7407 3.1134 12.9632 561.9898 124.79%
    5000 0.60 0.0344 0.0012 11.5714 0.1331 485.9443 1.5795 11.8763 483.9182 111.98%
    0.75 0.0345 0.0011 12.0225 0.1021 548.0941 9.8610 12.2460 523.8958 118.09%
    0.90 0.0346 0.0016 12.7725 0.0988 573.2314 2.8112 12.9826 569.6496 126.23%
    10000 0.60 0.0343 0.0013 11.5650 0.1236 485.9842 1.7995 11.8366 483.9182 111.98%
    0.75 0.0344 0.0012 12.0144 0.1069 551.8443 3.8878 12.2854 526.3093 118.49%
    0.90 0.0343 0.0014 12.7717 0.0953 573.6158 2.7063 12.9655 569.9058 126.28%

     | Show Table
    DownLoad: CSV


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