A matheuristic for the prepositioning of emergency supplies

In this paper, we describe the matheuristic we developed for the problem of pre-positioning emergency supplies that aims to increase disaster preparedness by making the relief items readily available to the people in need. To solve the pre-positioning problem is to develop a strategy that determines the location and size of storage facilities, the quantities of various types of emergency supplies stocked in each facility, and the distribution of the supplies to demand locations after an event, under uncertainty about demands, survival of pre-positioned supplies, and transportation network availability. The matheuristic employs iterated local search techniques to look for good location and inventory configurations, and uses CPLEX to optimize the aid distribution. Numerical experiments on a number of case studies and random instances demonstrate the effectiveness and efficiency of the matheuristic, which is shown to be particularly useful for tackling larger instances that are intractable for exact solvers. The matheuristic can therefore be used by both academics and practitioners to further study the pre-positioning problem and to support the planning of pre-positioning strategies. 1 Motivation and literature review Between 2008 and 2017, the world has witnessed 5 782 disasters that resulted in 787 096 deaths and affected almost 2 billion people [1]. The agility and readiness in the distribution of critical relief commodities (such as water, food, or medicine) are crucial, especially in the first 72 hours after the event, so that rescue teams can begin their activities and victims can thus stabilize their lives. The importance of pre-positioning relief supplies was demonstrated when Hurricane Katrina devastated New Orleans in 2005. The lack of pre-positioned materials and the delay in arrival of these supplies hampered further relief to the victims [10,22]. The policies and models developed for commercial supply chains most often cannot be directly applied to manage humanitarian inventories, due to the unique characteristics of the humanitarian setting. For example, even though both customer service and low logistics costs are important for humanitarian organizations and business enterprises, efficiency is crucial for commercial supply chains, whereas satisfying beneficiary needs is always the utmost priority for humanitarian organizations. In a commercial setting, the demands are known or can be easily predicted, whereas in most humanitarian settings, there is a high level of uncertainty associated with location, type and amount of demand. The network infrastructure is generally stable and reliable in a commercial supply chain, whereas post-disaster network may be damaged and involve uncertainties [6]. Attempts to adapt analytical formulations originally developed for the commercial sector to the humanitarian context therefore have major limitations; specialized models and solution procedures need to be developed to capture the full complexity of these problems [20]. 1 Article Instance Solution Mathematical model Heuristic Various instances M u lt ip le fa ci li ty ca te g o ri es M u lt ip le co m m o d it y ty p es U n ce rt a in ty a b o u t d em a n d U n ce rt a in ty a b o u t a id su rv iv a b il it y U n ce rt a in ty a b o u t n et w o rk d a m a g e F a ci li ty d ec is io n s In v en to ry d ec is io n s D is tr ib u ti o n d ec is io n s Distribution formulation Objective function [3] 3 3 3 3 7 3 3 Network flow problem Sum of logistics and penalty costs 3 3 [4] 7 3 3 7 3 3 3 3 Maximum covering problem Met demand 7 7 [7] 7 3 7 7 7 3 3 3 Maximum covering problem Coverage; Sum of logistics and penalty costs 7 7 [8] 7 3 3 7 7 3 3 Network flow problem Sum of logistics and penalty costs 7 7 [9] 7 3 3 3 7 3 3 Network flow problem Sum of logistics and penalty costs 7 7 [10] 7 3 3 7 3 3 3 3 Transportation problem Sum of logistics and penalty costs 7 7 [12] 7 3 3 7 3 3 3 3 Network flow problem Sum of logistics and penalty costs 3 3 [13] 7 3 3 7 7 3 3 3 Transportation problem Demand-weighted time 7 7 [16] 3 3 3 3 3 3 3 3 Network flow problem Sum of logistics and penalty costs 7 7 [18] 7 7 7 7 7 3 7 3 Network flow problem Logistics costs; Met demand 7 3 [23] 3 7 3 7 3 3 3 3 Network flow problem Logistics costs 7 7 [24] 3 3 7 3 7 3 3 Network flow problem Sum of logistics and penalty costs 7 7 [25] 7 3 7 7 7 3 3 3 Network flow problem Sum of logistics and penalty costs 7 7 [26] 7 7 3 7 7 3 7 3 Assignment problem Minimum and average weight of open facilities; Distance 7 3 [27] 7 3 3 7 3 3 3 3 Network flow problem Sum of logistics and penalty costs 7 7 [31] 3 3 3 3 3 3 3 3 Network flow problem Sum of logistics and penalty costs 3 7 [32] 3 3 3 3 3 3 3 3 Network flow problem Sum of logistics and penalty costs 7 7 [33] 7 3 3 7 3 3 7 3 Routing problem Sum of met demand utility and residual budgets 7 3 [34] 7 3 3 3 3 3 3 3 Transportation problem Demand-weighted time; Sum of logistics and penalty costs 7 7 [35] 7 3 7 7 7 3 3 Network flow problem Sum of logistics and penalty costs 7 7 [36] 3 3 3 3 3 3 3 3 Network flow problem Logistics costs; Time; Sum of penalty costs 3 7 [37] 7 7 3 7 7 3 3 3 Routing problem Logistics costs; Met demand 7 7 [40] 7 3 3 7 7 3 3 Network flow Demand-weighted distance 3 7 [41] 7 7 7 7 7 3 7 3 Routing problem Sum of logistics costs and coverage reliability 7 7 This article 3 3 3 3 3 3 3 3 Assignment problem Met demand; Time 3 3 Table 1: Literature review on the problem of pre-positioning emergency supplies shows that most of the articles do not consider all the problem aspects, often minimize costs and only employ a commercial solver to solve a single case study. 2 In the growing body of literature on the topic of pre-positioning emergency supplies (for a recent literature survey, see [6, 17]), we have identified a number of areas of improvement. Firstly, there are many formulations of the pre-positioning problem that fail to consider the crucial complexities of the problem. In this paper, we adopt the problem definition introduced in [31], and consider the problem of pre-positioning emergency supplies that decides on location and size of storage facilities, the quantities of various types of emergency supplies stocked in each facility, and the distribution of the supplies to demand locations after the disaster, under uncertainties about the particularities of the disaster. We assume uncertainties about demands, survival of pre-positioned supplies, and transportation network availability, and describe them through a finite set of possible disaster scenarios. Most of the papers do not consider all the afore-mentioned aspects of the problem: some do not consider multiple facility categories (that can be opened at any potential facility location) or commodity types, some do not consider uncertainties about demand, survivability of pre-positioned aid or network damage, some do not incorporate facility or inventory decisions (Table 1). Furthermore, in some articles the authors assume uncapacitated facilities or, e.g., assume a single open facility, or a single demand location in each disaster scenario. These assumptions significantly simplify the mathematical formulation of the problem, and in particular the algorithms that solve the problem. Next to the underlying problem assumptions mentioned, the mathematical models that are used to describe the pre-positioning problem vary greatly with respect to the formulation of the aid distribution sub-problem (Table 1). Most of the authors model the aid distribution as a network flow problem per commodity, most probably due to existence of efficient algorithms that solve it, e.g., [14, 15, 29]. Such formulation over-simplifies the distribution problem as it does not allow to easily take into account the capacity nor the number of vehicles needed to transport different commodities. In most cases, a model is used that only provides the flow amounts between vertices without specifying the destined path of the flow, making the solution difficult to implement in a real-world system. Furthermore, serving a demand location from multiple facilities is operationally overly complex (and might also pose significant risks) for a chaotic setting after a disaster, e.g., carrying out a plan where 17% of demand of one commodity and 54% of demand of another commodity of a vertex are served by one facility, and the remainder by another (or more) facilities (Figure 1). The latter is the reason why we also did not choose the formulate the aid distribution as a transportation problem. On the other hand, the formulation of the aid distribution sub-problem as a routing problem is a waste of computational effort. Indeed, in the preparedness phase before a disaster, one is only interested in deciding where to open the facilities and what to store there; the aid distribution sub-problem is only solved to provide an evaluation of the quality of the pre-positioning facility and inventory decisions. Once a disaster happens, it is highly unlikely that it will completely match one of the considered disaster scenarios, implying that the optimized routing schemes would be of no use. In our formulation, we consider the aid distribution as an assignment problem, deciding which demand locations are served by which open facilities. Making the decisions binary (a vertex is assigned to a facility or not) is also more suitable for heuristic procedures that are necessary for complex real-world problems like the problem of pre-positioning emergency supplies. Furthermore, although the objective of humanitarian relief is to minimize human suffering [21] (what is an important distinction from commercial supply chains, as mentioned above), cost minimization is the common objective in the pre-positioning problem formulations. Since meeting all demand after an emergency is rarely possible, the objective function is usually defined to be the sum of logistics costs and different type of penalty costs, e.