Spatial Resource Allocation: Distance vs. Delay

Abstract

Both travel distance to a service location and service delay (wait time) after arriving at the location jointly influence the total turnaround time of getting a service and are widely considered as two main factors affecting demand in the service industry. We consider a service provider with a limited number of servers (also referred to as service capacity) that serves a pool of customers who live uniformly across a region and travel to the closest station to receive the service. We aim to determine the optimal number of service stations, each of which may consist of multiple servers,  and the service fee to maximize revenue (or social welfare) by balancing the trade-off between distance and delay. With an exogenous service fee, we demonstrate that a full-capacity pooling strategy is optimal if the service fee is low enough; however, the service provider should adopt a partial capacity pooling strategy and move towards a fully decentralized (one-station-one-server) policy as the service fee increases. With an endogenous service fee, we show that given a high service reward and high capacity, partial capacity pooling with moderate pricing is optimal. With low capacity and a high service reward, a full-capacity pooling strategy is optimal, while low capacity and a low service reward would lead to the optimality of the fully decentralized policy. Moreover, we compare the service provider's optimal strategy for delay-and-distance-sensitive customers with that for distance-sensitive-only or delay-sensitive-only customers. Compared to that for delay-sensitive-only customers, we show that the provider should increase the number of stations and reduce the service fee only if customers are highly sensitive to distance and the service capacity is low. Compared to that for distance-sensitive-only customers, the service provider should reduce the number of service stations and lower the service fee. Finally, we conduct a case study using data from MRI services performed in a region in Ontario, Canada, and investigate the impact of different parameters on the optimal number of service stations. For example, we observe that as the variability in the service duration increases, the optimal number of stations decreases to have more machines at each station and reduce the negative impact of variability on the service delay.