Mitigating Hard Capacity Constraints in Facility Location Modeling

Abstract

In many real-world settings, the capacity of processing centers is flexible due to a variety of operational tools (such as overtime, outsourcing, and backlogging demand) available to managers that allow the facility to accept demands in excess of the capacity constraint for short periods of time. However, most capacitated facility location models in the literature today impose hard capacity constraints that don’t capture this short term flexibility. Thus, current capacitated facility location models do not account for the operational costs associated with accepting excess daily demand, which can lead to suboptimal facility location and demand allocation decisions.

To address this discrepancy, we consider a processing distribution system in which demand generated on a daily basis by a set of demand sites is satisfied by a set of capacitated processing facilities. At each demand site, daily demands for the entirety of the planning horizon are sampled from a known demand distribution. Thus, the day to day demand fluctuations may result in some days for which the total demand arriving at a processing facility exceeds the processing capacity, even if the average daily demand arriving at the processing facility is less than the daily processing capacity. We allow each processing facility the ability to hold excess demand in backlog to be processed at a later date and assess a corresponding backlog penalty in the objective function for each day a unit of demand is backlogged.

This dissertation primarily focuses on three methods of modelling the aforementioned processing distribution system. The first model is the Inventory Modulated Capacitated Location Problem (IMCLP), which utilizes disaggregated daily demand parameters to determine the subset of processing facilities to establish, the allocation of demand sites to processing facilities, and the magnitude of backlog at each facility on each day that minimizes location, travel, and backlogging costs. Whereas the IMCLP assumes each demand site must be allocated to exactly one processing facility, the second model relaxes this assumption and allows demand sites to be allocated to different processing facilities on various days of the week. We show that such a cyclic allocation scheme can further reduce the system costs and improve service metrics as compared to the IMCLP.

Finally, while the first two models incorporate daily fluctuations in demand over an extended time horizon, the problems remain deterministic in the sense that only one realization of demand is considered for each day of the planning horizon. As such, our final model presents a stochastic version of the IMCLP in which we assume a known demand distribution but assume the realization of daily demand is uncertain. In addition to assessing a penalty cost, we consider three types of chance constraints to restrict the amount of backlogged demand to a predetermined threshold. Using finite samples of random demand, we propose two multi-stage decomposition schemes and solve the mixed-integer programming reformulations with cutting-plane algorithms.

In summary, this dissertation mitigates hard capacity constraints commonly found in facility location models by allowing incoming demand to exceed the processing capacity for short periods of time. In each of the modelling contexts presented, we show that the location and allocation decisions obtained from our models can result in significantly reduced costs and improved service metrics when compared to models that do not account for the likelihood that demands may exceed capacity on some days.