Discrete Optimization
A heuristic for emergency operations scheduling with lead times and tardiness penalties

https://doi.org/10.1016/j.ejor.2015.10.005Get rights and content

Highlights

  • We model an emergency operations scheduling problem with tardiness penalties.

  • The lead times (waiting, production and transportation times) are considered.

  • Through a structure analysis of the problem, solvable cases are identified.

  • A heuristic algorithm based on a mixed integer programming is developed.

  • The algorithm runs very fast and has small errors in numerical experiment.

Abstract

We study the operations scheduling problem encountered in the process of making and distributing emergency supplies. The lead times of a multi-echelon process, including shipping time, assembly time, and waiting time for raw materials must be explicitly modeled. The optimization problem is to find an inventory allocation and a production/assembly plan together with a shipping schedule for inbound supplies and outbound deliveries so that the total tardiness in customer order fulfillment is minimized. We define the problem as a mixed integer programming model, perform a structure analysis of the problem, and then propose a new search heuristic for the problem. This proposed heuristic finds a feasible solution to the problem by solving a series of linear programming relaxation problems, and is able to terminate quickly. Observations from an extensive empirical study are reported.

Introduction

Natural disasters such as hurricanes, earthquakes, wildfires and tornados have occurred frequently in recent years. According to the National Earthquake Information Centre of the US Geological Survey (http://www.usgs.gov/), the number of earthquakes above the magnitude 4.0 exceeded 158,000 times worldwide between 2000 and 2012. Emergency supply chains, formed upon the needs for disaster relief, collect and distribute rescue supplies to affected areas, and operate differently from commercial supply chains. Holguín-Veras, Jaller, Van Wassenhove, Pérez, and Wachtendorf (2012) compared the commercial logistics and humanitarian logistics in emergencies from seven aspects. In addition to the high expectation on responsiveness, emergency supply chains also face challenges such as poor information/ communication, uncertainties in network capacity, limited resource availability, lack of coordination, and frequent last-minute priority change in the content, quantity, and destinations of shipments. According to Sheu (2007a), the goal of emergency logistics is to meet the urgent needs of the affected people under emergency conditions, which is reflected in our study by using delivery tardiness minimization as the objective function.

Rescue kits have been commonly used in various real life disaster-relief operations. A rescue kit typically consists of multiple components (e.g., emergency trauma dressing, latex gloves, blood-stoppers, bandages, alcohol wipes, etc.) from various suppliers. Since different areas hit by a natural disaster may experience different levels/types of damages, both common-purpose (i.e., standard) rescue kits and area-dependent (i.e., customized) rescue kits are usually needed. In general, only standard kits are inventoried in advance in the network, while various customized kits are provisioned during and after a disaster since the contents of customized kits are highly dependent on the types of disaster, damages, season, and areas, and are therefore not built to inventory.

In this study, we consider and analyze an integrated replenishment, production and distribution problem defined upon an emergency supply chain for both standard and customized rescue kits. The hypothetical supply chain network consists of component suppliers, manufacturers, regional distribution centers, and customer demand points. The bill of materials for assembling standard kits is identical regardless of customers, while the bill of materials for assembling customized kits is customer/area-dependent. Each customer may order a standard kit, or a customized kit, or both, and specifies the preferred time and quantity for each product to be delivered. The order for a standard kit may be fulfilled from either existing inventory in the network or a newly produced batch by a manufacturer. There is no inventory for customized kits, which are usually ordered/specified by the customers according to their local needs for the disaster relief. The order lead-time, including shipping time, assembly time and waiting time (for the component supplies) must be explicitly considered and modeled. The optimization problem is to find an integrated inventory allocation and a production/assembly plan together with a shipping schedule for inbound component supplies and outbound product deliveries so that the total tardiness in customer order fulfillment is minimized.

Emergency logistics has been studied extensively over the past two decades. Altay and Green (2006) discussed the potential applications of operations research to disaster operations management. Caunhye, Nie, and Pokharel (2012) reviewed the applications of optimization applied to emergency logistics, and pointed out the lack of results for tardiness minimization because of the potential complexity for tracking the response times. The authors also mentioned that the resulting computational inefficiency is a main reason for the absence of comprehensive optimization models for emergency logistics. The review by Galindo and Batta (2013) is a continuation of Altay and Green (2006), and evaluated how research in disaster operations management has evolved since then. There have been quite a few studies on specific disaster relief operations in practice. To list a few: Haghani and Oh (1996) reported a mixed integer programming model for a logistical problem in disaster relief management; Barbarosoglu, Özdamar, and Cevik (2002) developed a mathematical model for helicopter mission planning during a disaster relief process; Özdamar, Ekinci, and Küçükyazici (2004) studied a dynamic time-dependent transportation problem and proposed an approach to solve the problem repetitively during an emergency logistic planning, which was further extended by Yi and Özdamar (2007); and Sheu (2007b) reported a study on relief distribution during the crucial rescue period after a disaster.

There also exist many studies for integrated operations planning and scheduling. The available literature results can be classified into the following five categories: single commodity models (Ahuja et al., 2007, Karakitsiou and Migdalas, 2008, Lei et al., 2006, Lei et al., 2009, Wang et al., 2010, Yilmaz and Catay, 2006); multiple commodity models Dogan and Goetschalckx, 1999, Eksioglu et al., 2007, Gen and Syarif, 2005, Jang et al., 2002, Park, 2005, Yung et al., 2006; models with bidirectional flows Chen and Hall, 2007, Kannan et al., 2010, Lei et al., 2009; stochastic models Aliev et al., 2007, Lejeune and Ruszczynski, 2007, Liang, 2008, Liang and Cheng, 2009, Peidro et al., 2009; and models involving due date related constraints Geismar et al., 2008, Mohan and Ritzman, 1998, Zhong et al., 2010. Most models in the aforementioned papers have the objective of minimizing the total cost. Two comprehensive reviews in this area can be found in the work by Thomas and Griffin (1996) and Mula, Peidro, Diaz-Madronero, and Vicens (2010).

The main difference between the focus of our study and those considered in the literature for integrated operations planning is that our problem involves multi-stage lead time of a supply chain network, and that our objective is to minimize the delivery tardiness instead of cost minimization, which together introduce new challenges in modeling and algorithm designs. The remaining parts of this paper are organized as follows. A formal mathematical model for our problem, together with the assumptions and notations, are given in Section 2. In Section 3, a partial linear programming (LP) relaxation-based heuristic is proposed to solve the general version of the problem. An extensive empirical study for the proposed heuristic is reported in Section 4, where part of the geographical data (i.e., traveling distances among hospitals) from affected areas during 2012 Hurricane Sandy was used. Finally, in Section 5, we conclude the study and discuss its future extensions.

Section snippets

A formal problem definition

Our problem is defined upon a three-stage supply chain network (see Fig. 1) encountered in a real life project performed by Rutgers Center for Supply Chain Management (Lei, Handley, Melamed, Qi, & Stamatellos, 2012). The network consists of customer demand points, regional distribution centers (DC), manufacturers, and suppliers of components used in the rescue kit assemblies. The network produces and delivers both single type standard kits and customer-dependent customized kits subject to

A LP-relaxation based heuristic for P

While the general version of P is difficult to solve optimally, there exist special cases that can be solved efficiently. The optimal solutions to such special cases can be used to construct heuristics for P in future studies. We examine two such solvable cases in the Appendix for readers with interest.

We introduce in this section an iterative algorithm for solving P, by which the search process is guided by a sequence of linear programming (LP) relaxation solutions. Each iteration starts with

The empirical study

In this section, we discuss the implementation of LPR, and compare its computational performance with the best solutions obtained by the commercial optimizer Gurobi (on Intel Core Duo CPU, 2.10 gigahertz). Table 2 below summarizes the values of parameters used in our experiments.

In these experiments, the shipping times (see Table 2) were randomly generated proportional to the scale of the affected area of Hurricane Sandy (see Fig. 3), and the order delivery due dates were generated based on the

Conclusion and extensions

We study the operations scheduling problem defined upon a multi-stage network with non-negligible lead times between the stages, with an objective to minimize the total tardiness in customer order fulfillment. Two solvable cases of this problem are analyzed, and a heuristic search algorithm is proposed. The proposed heuristic solves a series of partial linear programming relaxation problems, and is able to terminate quickly with a near-optimal solution. Observations from an extensive empirical

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