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Background
Background
The delivery apps started dealing with high demand due to COVID-19 within groceries, medicine and food ordering.
The delivery apps started dealing with high demand due to COVID-19 within groceries, medicine and food ordering.
The demand increased since March 2020.
The demand increased since March 2020.
Example: the meal delivery orders (delivered by Uber Eats) of a restaurant in Mexico
Example: the meal delivery orders (delivered by Uber Eats) of a restaurant in Mexico
SIR Model
SIR Model
We build a traditional SIR model [2] to predict the virus spread.
We build a traditional SIR model [2] to predict the virus spread.
Prediction Result:
Prediction Result:
Real-world COVID-19 data in Nuevo Leon including cumulative infected cases, death tolls, and recovery cases, from April 2 to October 19, 2020.
Real-world COVID-19 data in Nuevo Leon including cumulative infected cases, death tolls, and recovery cases, from April 2 to October 19, 2020.
The log-plot of It versus t, using infection data from September 1 to October 19, 2020 (we believe the infection and recovery rates keep unchanged during these periods). We can estimate that β - γ = 0.0073.
The log-plot of It versus t, using infection data from September 1 to October 19, 2020 (we believe the infection and recovery rates keep unchanged during these periods). We can estimate that β - γ = 0.0073.
The prediction for the three subgroups by the SIR model for future 350 days starting from September 1.
The prediction for the three subgroups by the SIR model for future 350 days starting from September 1.
Companion between the actual versus predicted infected and recovered subgroups from September 1 to Oct 19.
Companion between the actual versus predicted infected and recovered subgroups from September 1 to Oct 19.
ARMA Model
ARMA Model
We build an ARMA model to predict the daily order number:
We build an ARMA model to predict the daily order number:
where the independent variables are weekdays (presented as dummy variables), the log value of the infections during previous days, the historical number of orders (referred to as AR variables), and the historical noise terms (referred to as MA variables). By estimating the coefficients of the ARMA model from the historical order data and infection data, we can predict the future number of orders with the infections, which are forecasted by SIR model, through the equation:
where the independent variables are weekdays (presented as dummy variables), the log value of the infections during previous days, the historical number of orders (referred to as AR variables), and the historical noise terms (referred to as MA variables). By estimating the coefficients of the ARMA model from the historical order data and infection data, we can predict the future number of orders with the infections, which are forecasted by SIR model, through the equation:
where part of the independent variables are also forecasted values. We estimate the coefficients using the order data from April 2 to October 19, 2020 collected from a restaurant located in Nuevo Leon, Mexico. The results show:
where part of the independent variables are also forecasted values. We estimate the coefficients using the order data from April 2 to October 19, 2020 collected from a restaurant located in Nuevo Leon, Mexico. The results show:
- There is no significant difference between the number of orders received in Saturday and Sunday.
- The model performs best when only including the log value of the daily infection in the day before yesterday, due to the latency between the time of notifying the infected cases and the time of making online takeout food orders.
- The model has the best performance when only including the number of takeout orders of yesterday and the log value of the number of daily infected cases of the day before yesterday.
Therefore, the model we built for this specific restaurant is:
Therefore, the model we built for this specific restaurant is:
Tactical Decisions
Tactical Decisions
We formulate a two-stage stochastic integer programming model. The first-stage decisions include a binary variable x, where x = 1 denotes partnering with third-party platforms and x = 0 otherwise, and a non-negative integer variable y denoting the number of drivers that will be hired during the targeted periods. For every hired driver, the restaurant needs to pay a fixed base salary cv per day. The driver will also receive an additional stipend, proportional to the travel time and distance of each delivery, at unit cost rates cτ and cd, respectively. Each driver can deliver at most Δ number of orders per day. We consider a finite number of scenarios, generated by Monte Carlo sampling approach. For each order, the correspondent decision variable u=1 means the order is delivered by drivers hired by the restaurant and v=1 means the order is delivered by the platform.
We formulate a two-stage stochastic integer programming model. The first-stage decisions include a binary variable x, where x = 1 denotes partnering with third-party platforms and x = 0 otherwise, and a non-negative integer variable y denoting the number of drivers that will be hired during the targeted periods. For every hired driver, the restaurant needs to pay a fixed base salary cv per day. The driver will also receive an additional stipend, proportional to the travel time and distance of each delivery, at unit cost rates cτ and cd, respectively. Each driver can deliver at most Δ number of orders per day. We consider a finite number of scenarios, generated by Monte Carlo sampling approach. For each order, the correspondent decision variable u=1 means the order is delivered by drivers hired by the restaurant and v=1 means the order is delivered by the platform.
Random Scenario Generation: We randomly generate locations for the predicted customers within a radius of 10 km from the location of the restaurant. Afterwards, we used Google Maps API to estimate the travel time and the distance between the generated points and the restaurant. We generate the sale of orders through a truncated normal distribution with mean and standard deviation estimated from historical order data.
Random Scenario Generation: We randomly generate locations for the predicted customers within a radius of 10 km from the location of the restaurant. Afterwards, we used Google Maps API to estimate the travel time and the distance between the generated points and the restaurant. We generate the sale of orders through a truncated normal distribution with mean and standard deviation estimated from historical order data.
Partner with Third-party Platform: For each order received from the platform, the restaurant pays δ portion of the sale to the platform. We consider a threshold Y for the number of drivers hired by the restaurant. If the restaurant hires more than Y number of drivers, then we believe that at least portion of the customers will order food through the platform rather than directly contacting the restaurant. If the number of hired drivers is smaller than Y , we assume that more customers will order food through the service platform because of the lack of service capacity of the own delivery team for the restaurant. In particular, we assume that another Γ portion of customers will use the service platform once there is 1 reduction of hired drivers from Y .
Partner with Third-party Platform: For each order received from the platform, the restaurant pays δ portion of the sale to the platform. We consider a threshold Y for the number of drivers hired by the restaurant. If the restaurant hires more than Y number of drivers, then we believe that at least portion of the customers will order food through the platform rather than directly contacting the restaurant. If the number of hired drivers is smaller than Y , we assume that more customers will order food through the service platform because of the lack of service capacity of the own delivery team for the restaurant. In particular, we assume that another Γ portion of customers will use the service platform once there is 1 reduction of hired drivers from Y .
Therefore, we formulate this decision making problem as:
Therefore, we formulate this decision making problem as:
We present part of our results under different parameter settings here:
We present part of our results under different parameter settings here:
References:
References:
[1] Banskota, S., Healy, M., and Goldberg, E. M. (2020). 15 smartphone apps for older adults to use while in isolation during the covid-19 pandemic. The western journal of emergency medicine , 21(32302279):514–525.
[1] Banskota, S., Healy, M., and Goldberg, E. M. (2020). 15 smartphone apps for older adults to use while in isolation during the covid-19 pandemic. The western journal of emergency medicine , 21(32302279):514–525.
[2] Biswas, K., Khaleque, A., and Sen, P. (2020). Covid-19 spread: Reproduction of data and prediction using a sir model on Euclidean network.
[2] Biswas, K., Khaleque, A., and Sen, P. (2020). Covid-19 spread: Reproduction of data and prediction using a sir model on Euclidean network.
[3] Fuller WA (2009) Introduction to Statistical Time Series, volume 428 (John Wiley & Sons).
[3] Fuller WA (2009) Introduction to Statistical Time Series, volume 428 (John Wiley & Sons).
[4] Jia, H., Garcıa, J., and Shen, S., (2020) Partner with a Third-Party Delivery Service or Not? – a Prediction-and-Decision Tool for Restaurants Facing Takeout Demand Surges During a Pandemic, Available at SSRN.
[4] Jia, H., Garcıa, J., and Shen, S., (2020) Partner with a Third-Party Delivery Service or Not? – a Prediction-and-Decision Tool for Restaurants Facing Takeout Demand Surges During a Pandemic, Available at SSRN.
Student Contributors:
Student Contributors:
Huiwen Jia
Huiwen Jia
PhD Student
PhD Student
Department of Industrial and Operations Engineering
Department of Industrial and Operations Engineering
University of Michigan
University of Michigan
Jorge Alberto Ramírez García
Jorge Alberto Ramírez García
Undergraduate major in Industrial and Operations Engineering, minor in Data Analytics
Undergraduate major in Industrial and Operations Engineering, minor in Data Analytics
University of Monterrey
University of Monterrey
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