Addressing Uncertainty in Meter Reading for Utility Companies using RFID Technology: Simulation Experiments Christof Defryn Maastricht University, School of Business and Economics e-mail:
[email protected]
Debdatta Sinha Roy, Bruce Golden University of Maryland, Robert H. Smith School of Business email:
[email protected],
[email protected]
Introduction Utility companies read the electric, gas, and water meters of their residential and commercial customers on a regular basis. A short distance wireless technology called the radio-frequency identi cation (RFID) technology is used for automatic meter reading (AMR). An AMR system has two parts: an RFID tag and a vehicle-mounted reading device. The RFID tag is connected to a physical meter and encodes the identi cation number of the meter and its current reading into a digital signal. The vehicle-mounted reading device collects the data automatically when it approaches an RFID tag within a certain distance. Utility companies would like to design routes for the reading vehicles such that all customers (meters) in the service area are covered and the total length (cost) of the routes is minimized. The problem to be solved is a close-enough vehicle routing problem (CEVRP) over a street network. There are issues with RFID technology that gives the meter reading problem a great amount of inherent uncertainty. The signal transmitted by an RFID tag occurs at regular time intervals and not continuously, for extending the battery life of the RFID transmitters. This leads to the possibility of a missed capture of a signal if the truck with the receiver is within the range of the meter only for a short time. Also, the signal range of a meter can vary from the distance speci ed by the manufacturer of the radio frequency transmitters and receivers due to weather conditions, surrounding obstacles, and decreasing battery life of the transmitters. These unplanned missed reads can lead to increased costs for a utility company, because another vehicle has to be sent at a later time to read the missed meters.
Research Goal Published papers have not considered the issues with RFID technology and thereby have not taken into account the inherent uncertainty in the meter reading problem. The main contribution of our research is to address the issues of RFID technology by generating robust routes for the CEVRP that minimize the number of missed reads and to demonstrate these ideas by simulation experiments using real-world data from utility companies. We want to design routes that are shorter in length and are better at capturing the uncertain signals from meters.
Integer Programming Formulation We formulate the meter reading problem with RFID technology as a two-stage integer program (IP). The Stage 1 IP nds the street segments that are to be traversed for reading each meter with a pre-speci ed chance of being read from the full route. The Stage 2 IP solves a mixed rural postman problem that adds deadhead segments to the solution of the Stage 1 IP to obtain the full route. The deadhead segments added in the Stage 2 IP increase the likelihood of reading the meters. 1
Let E be the set of the edges and A be the set of the arcs in a street network. We denote the mixed graph by G = (V, E ∪ A), where V is the set of nodes. Let cj ≥ 0 be the cost (length) of street segment j . Let I be the set of the meters. Let pij be the probability that meter i is read at least once from street segment j . Let Li ∈ [0, 1] be the speci ed likelihood of reading meter i from the full route. We de ne xj to be the decision variable denoting whether or not street segment j should be traversed in the full route. We consider a single meter reading vehicle. The Stage 1 IP formulation is given by the following. min
X
xj
(1)
cj xj
(2)
j∈E∪A
min
X j∈E∪A
s.t.
Y
(1 − pij )xj ≤ (1 − Li ) ∀i ∈ I
(3)
j∈E∪A
xj ∈ {0, 1}
∀j ∈ E ∪ A
(4)
This is a bi-objective formulation. Objective functions (1) and (2) minimizes the total number of street segments and the total cost (length) respectively. Constraint (3) selects the values of the decision variables (xj ) accordingly so that the probability of reading meter i from the full route is at least Li . Constraint (4) restricts the decision variables to 0 and 1. Note that constraint P (3) can be linearized in the decision variables j∈E∪A xj × log(1 − pij ) ≤ log(1 − Li ) for every meter i ∈ I yielding a linear Stage 1 IP.
Regression and Bayesian Updating In order to solve the Stage 1 IP, we need to estimate the values of the pij 's. We use a regression model. The data are in the form of 1 and 0, where 1 indicates that meter i is read from street segment j and 0 indicates that meter i is not read from street segment j . The predicted values of the dependent variable in the regression model have to be between 0 and 1, which will denote the probability pij . Based on the type of the data we have and our requirements on the dependent variable, logit and probit models are considered. Every time the meter reading vehicle collects readings, it adds more data to the previous readings. The more data we have, the better will be the estimates of the pij 's. Therefore, the routes generated by the two-stage IP will be of a higher quality. They will be better at capturing the uncertain signals thereby reducing the number of missed reads. There are some serious issues if we use regression to update the estimates of the pij 's at every stage with the new data. Suppose in time period 1 we observe the rst set of i.i.d. data (denoted by y1 ). We run the regression on y1 . In time period 2, we observe a second set of i.i.d. data (denoted by y2 ), independent of the rst set. We run the regression on y1 and y2 together as a single data set, and so on. We are regressing on the older data sets repeatedly which makes this process of updating ine cient. Data sets from dierent time periods are given equal weights in the regression which should not be the case in practice. If we update the estimates of the pij 's at every stage when new data comes in using concepts from Bayesian statistics, then we can avoid the two drawbacks faced while updating using regression. Bayesian updating to estimate the pij 's can be done for both logit and probit models. A more complex method for estimating the pij 's is to perform Bayesian updating for hierarchical probit models. The estimates of the pij 's from hierarchical probit models are more accurate but di cult to calculate as compared to the logit and probit models. Hierarchical probit models account for the uncertain behavior of each meter separately while also accounting for the similarity between meters.
Simulation Experiments and Conclusions Bayesian updating helps us to solve the two-stage IP at every time period when new data comes in, thereby potentially helping to produce more robust routes by updating the estimates of the pij 's. The main contribution of our research is combining vehicle routing with Bayesian statistics and data analytics to address the uncertainty in the meter reading problem. As mentioned, we will demonstrate these ideas and compare the performance of the dierent Bayesian updating models using real-world data and simulation experiments.
