Innovative Applications of O.R.A risk-constrained time-dependent cash-in-transit routing problem in multigraph under uncertainty
Graphical abstract
Introduction
Globally, cash continues to be a widely used payment instrument for transactions. This instrument is the ultimate resource for financial transactions, especially for small-value purchases. The recent report of the Federal Reserve on payment habits and transactions of the United States population shows that 26% of all transactions, and 40% of payments from $10 to $25 in 2019 were in cash (Federal Reserve, 2020). In spite of lawmakers’ support in utilizing non-cash transactions, cash in circulation (CIC) has a consistent growth (from 4% to 7% per year) in the recent five years. Statistics accentuate that CIC remains high in developed countries. For example, in Singapore which stands as a mature and developed market in electronic payment mechanisms, CIC is still high and six of ten transactions are performed by cash (Capgemini, 2019). Referring to Xu, Li, Szeto and Li (2019), the rate of the CIC in the United States has increased by 82.1% between 2007 and 2016. In other countries including China, the amount of cash swirling is still high. In the past decade, these countries are faced by CIC growth rate of nearly 124% (Xu et al., 2019). Meanwhile, CIC grows drastically in countries undergoing severe distress such as Ukraine, Myanmar, and Mozambique (Capgemini 2019). The consistent growth of CIC across countries led the related organizations to put their efforts on managing the banknote supply chains in an effective way. CIT routing problem is generally studied as one of the application areas in the VRPs for transportation of valuable goods including cash and jewelry in populated cities and metropolitan areas.
In spite of employing better-armored vehicles and technological development (e.g., weapons on board, vehicle tracking), providing safe and protected transportation is still a challenging operation for CIT companies. Security considerations in route planning of CIT are pivotal in preserving the operation from heists and robberies (Smith and Louis, 2010; Yan, Wang & Wu, 2012). However, security analysis for minimizing the risks and threats may require a high-level planning and generally affect the operating costs considerably. Thereby, the safe and efficient routing scheme should be executed for the success of CIT operations. To this end, minimizing travel costs along with minimizing risks and threats should be considered simultaneously (Talarico, Sörensen & Springael, 2017a).
Traffic congestion is a major issue for both commuters and logistics companies in urban environments. This growing phenomenon considerably affects travel speeds during the rush hours. To capture the impact of traffic congestion, the time-dependent VRP (TDVRP) is introduced in the literature. This problem considers the presence of traffic patterns during the day. In this approach, time-varying speed pattern is predicted to determine an average traveling speed from one period to another. To the best of our knowledge, none of the previous works on CIT problems studied time-dependency of travel times. However, due to the importance of time factor in CIT, ignoring the traffic congestion may bring imprecise planning decisions.
Another deficiency of traditional VRPs is that many researchers consider only one edge between each pair of nodes by a simple graph network representation. By this, many feasible solutions may be discarded from the solution space. Due to the complexity of urban network, it is more plausible to reach one node from another one by multiple parallel links. These links are usually differentiated by multiple inherent attributes (e.g., travel time, distance, and risk). According to Garaix, Artigues, Feillet and Josselin (2010) and Ticha, Absi, Feillet and Quilliot (2017), simple graph network is not capable of handling routing problems when several attributes are considered on the links. To handle this, multigraph network is suggested to maintain all non-dominated available links among each pair of nodes in the network (Garaix et al., 2010). In addition, in a time-dependent environment, choosing a suited arc among the nodes of a multigraph, not only depends on traveling distance but also on the actual departure time at the origin (Setak, Habibi, Karimi & Abedzadeh, 2015). In the CIT problem, the presence of alternative links among the key-locations is extremely important and helps accelerating the operations and accessing better solutions.
Considering the multigraph network, this research aims to investigate a time-dependent CIT routing problem, which is abbreviated as a TD-CITRM. This is the first study that incorporates the effect of congestion for the CIT problems. Ignoring the speed variations not only affects logistics costs but also increases the exposure in transportation of valuable goods to a robbery in congested arteries and streets. The proposed problem is studied under both deterministic and stochastic travel speeds. In what follows, the deterministic version of the problem is called TD-CITRM-DT and the stochastic one is called TD-CITRM-ST. Due to the time-dependency, stochasticity, nonlinearity, and multigraph structure, it is computationally challenging to solve the TD-CITRM-DT and TD-CITRM-ST. To do this, we focus on proposing effective strategies to cope with the uncertainty of traffic congestion for a non-linear version of TDVRP. Finally, a real case study on cash transportation of an organization in Isfahan metropolis in Iran is provided to assess the applicability of the presented models and associated solution algorithms.
The content of this paper is structured as follows. Section 2 is dedicated to the relevant literature. Section 3 provides the problem description and preliminaries on the TD-CITRM. We formulate both deterministic and stochastic versions of the problem in Section 4. In Section 5, the procedures of the proposed algorithms are described in detail. Section 6 includes the results of various computational experiments for the models on a transportation case in Isfahan and generated instances. This section provides a sensitivity analysis on some parameters of the model and presents managerial insights and effective guidance for decision makers. Finally, in Section 7, conclusions and future research perspectives are listed.
Section snippets
Literature review
We now present the literature review in three subsections. Section 2.1 provides a brief review of time-dependent VRP studies. In Section 2.2, we overview the works that evoke multigraph representation in VRPs. A subsequent Section 2.3 reviewed the studies that focused on the route planning for CIT operations. Finally, we elaborate that how the current study contributes to the literature.
Preliminaries and problem description
The problem described in this paper deals with the transportation of valuable goods (e.g., money) in urban environments. The proposed VRP models determine the best sequence of nodes and the best link between each pair of nodes (according to the time of a day). An example for showing the necessity of considering alternative links in a CIT routing problem is provided in Fig. 1. In this example, there are three options to traverse the distance between two specific nodes. The travel time and
Mathematical formulations
In this section, we introduce the TD-CITRM-DT as a mixed-integer non-linear programming and the related stochastic version TD-CITRM-ST using two-stage stochastic programming method. The following assumptions have been made to formulate the problem: (i) Vehicles are arranged to begin their carrier operations at the starting points of time intervals; (ii) if the expected operation time or cumulative risk on each route exceeds the given threshold values, the solution becomes infeasible; (iii)
Solution approaches
Due to the complexity and non-linearity features of the investigated problem, exact commercial solvers cannot even handle small-sized instances in reasonable computation times. Previous related papers confirmed the high computational challenges of solving the TDVRP in a multigraph. For example, Setak et al. (2015) managed to solve a simple and linear TDVRP model in a multigraph with only six nodes in more than 7.5 hours using the CPLEX solver. While, the solution methods in the current study
Numerical experiments
We first provide the applicability of TD-CITRM-ST by studying a real-life transportation case. Then, the performance of the proposed models and solution algorithms are evaluated using generated instances.
Conclusions
In this paper, we have formulated a risk-constrained time-dependent VRP for the transportation of physical currency. This study explicitly strives to fill the gap in the relevant literature by:
- (i)
employing multigraph representation in routing plans of CIT sector,
- (ii)
investigating the problem under both deterministic and stochastic time-varying traffic congestion. Ignoring the concept of traffic congestions in CIT transportation may bring imprecise decisions in such problems.
- (iii)
introducing a new risk
Acknowledgements
We thank the Editor and three reviewers for their valuable suggestions and helpful comments which helped us to significantly improve the manuscript.
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