Maritime inventory routing: recent trends and future directions

A single‐product maritime inventory routing problem (MIRP) is studied in which the production and consumption rates vary over the planning horizon. The problem involves a heterogeneous fleet and multiple production and consumption ports with limited storage capacity. Two discrete time formulations are developed: an original model and a reformulated model that is a pure fixed charge network flow (FCNF) model with side constraints. Mixed integer sets arising from the decomposition of the formulations are identified. In particular, several lot‐sizing relaxations are derived for the formulations and used to establish valid inequalities to strengthen the proposed formulations. Until now, the derivation of models and valid inequalities for MIRPs has mainly been inspired by the developments in the routing community. Here, we have developed a new model leading to new valid inequalities for MIRPs obtained by generalizing valid inequalities from the recent lot‐sizing literature. Considering a set of instances based on real data, a computational study is conducted to test the formulations and the effectiveness of the valid inequalities. The FCNF formulation is generally much stronger than the original formulation. The developed valid inequalities reduce the integrality gap significantly for both formulations. By using a branch‐and‐bound scheme based on the strengthened FCNF formulation, most of our test instances are solved to optimality. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 62(4), 297–314 2013

We present two decomposition algorithms for single product deep-sea maritime inventory routing problems (MIRPs) that possess a core substructure common in many real-world applications. The problem involves routing vessels, each belonging to a particular vessel class, between loading and discharging ports, each belonging to a particular region. Our algorithms iteratively solve a MIRP by zooming out and then zooming in on the problem. Specifically, in the “zoomed out” phase, we solve a first-stage master problem in which aggregate information about regions and vessel classes is used to route vessels between regions, while only implicitly considering inventory and capacity requirements, berth limits, and other side constraints. In the “zoomed in” phase, we solve a series of second-stage subproblems, one for each region, in which individual vessels are routed through each region and load and discharge quantities are determined. Computational experience shows that an integrated approach that combines these two algorithms is vastly superior to solving the problem directly with a commercial mixed-integer programming solver.

We consider an inventory routing problem (IRP) in the liquefied natural gas (LNG) supply chain, called the LNG-IRP. Here, an actor is responsible for the LNG production and inventory management at the liquefaction plants, the routing and scheduling of a heterogeneous fleet of LNG ships, as well as the inventories and sales at the regasification terminals. Furthermore, all ports have a limited number of berths available for loading and unloading. The LNG-IRP is more complicated than many other maritime inventory routing problems because a constant rate of the cargo evaporates in the tanks each day and is used as fuel during transportation. In addition, a variable number of tanks are unloaded at the regasification terminals. We introduce a new path flow formulation for this problem arising from a novel decomposition scheme based on parts of a ship schedule, called duties. A ship schedule for the entire planning horizon can be divided into duties consisting of a visit to a liquefaction plant, then one or two visits to a regasification terminal before ending in a liquefaction plant. The solution method suggested is based on a priori generation of duties, and the formulation is strengthened by valid inequalities. The same problem was previously solved by a branch-price-and-cut algorithm for a schedule-based formulation. Computational results show that the new formulation provides tighter bounds than the previous schedule-based formulation. Furthermore, on a set of 27 benchmark instances, the proposed algorithm clearly outperforms the previous branch-price-and-cut algorithm both with regard to computational time and the number of problems solved within a 10-h time limit.

The choice of a suitable time representation plays a major role in designing a computationally efficient planning/scheduling framework. For the time representation to be effective, it needs to capture the nuances of the actual process, while ensuring that the resultant model is tractable and computationally efficient. In this paper, a novel time representation scheme, titled hybrid time discretization, which combines both continuous- and discrete-time representations in a single framework, is proposed and used to solve a complex multiproduct maritime inventory routing problem (MIRP) with undedicated compartments. Several continuous- and discrete-time-based frameworks have been proposed to solve the MIRP over the years. While single-/multigrid discretized frameworks do efficiently capture the time-sensitive nuances of the problem (such as active/inactive jetty hours and hourly/daily variations in the demands), they struggle to solve larger problem instances to the required optimality gap in a reasonable time. This is primarily due to the large disparities among the timings of different task components in the model. On the other hand, continuous-time representation can significantly reduce the problem size and computational effort; however, modeling the entire problem using the continuous-time-based framework could have problems in representing some time-sensitive intricacies in the demands and jetty/port activities. To allay the above-mentioned drawbacks, in this paper, a novel hybrid time representation is proposed, which combines a continuous-time representation with a multigrid discrete-time framework. The proposed novel framework is cast in a rolling horizon strategy and tested on various scenarios generated for the above-mentioned maritime transportation problem, which are shown to be significantly more complex than those presented in the literature. The proposed MILP framework is able to solve most of the test cases to the required optimality in a reasonable time without the need for heuristic or meta-heuristic assists. The proposed hybrid time representation could find significant utility in other lateral applications such as batch scheduling and inventory routing problems.

We consider a single product maritime inventory routing problem in which the production and consumption rates are constant over the planning horizon. The problem involves a heterogeneous fleet and multiple production and consumption ports with limited storage capacity. Maritime transportation is characterized by high levels of uncertainty, and sailing times can be severely influenced by varying and unpredictable weather conditions. To deal with the uncertainty, this paper investigates the use of adaptable robust optimization where the sailing times are assumed to belong to the well-known budget polytope uncertainty set. In the recourse model, the routing, the order of port visits, and the quantities to load and unload are fixed before the uncertainty is revealed, while the visit time to ports and the stock levels can be adjusted to the scenario. We propose a decomposition algorithm that iterates between a master problem that considers a subset of scenarios and an adversarial separation problem that searches for scenarios that make the solution from the master problem infeasible. Several improvement strategies are proposed aiming at reducing the running time of the master problem and reducing the number of iterations of the decomposition algorithm. An iterated local search heuristic is also introduced to improve the decomposition algorithm. A computational study is reported based on a set of real instances.

In This Issue

We consider a maritime inventory routing problem for Norway’s largest salmon farmer both producing the feed at a production factory and being responsible for fish farms located along the Norwegian coast. The company has bought two new ships to transport the feed from the factory to the fish farms and is responsible for the routing and scheduling of the ships. In addition, the company has to ensure that the feed at the production factory as well as at the fish farms is within the inventory limits. A mathematical model of the problem is presented, and this model is reformulated to improve the efficiency of the branch-and-bound algorithm and tightened with valid inequalities. To derive good solutions quickly, several practical aspects of the problem are utilized and two matheuristics developed. Computational results are reported for instances based on the real problem of the salmon farmer.

SUMMARY In this paper we set forth an approach to queueing and inventory problems which is based on the theory of semi-Markov processes. The method is applicable to a wide variety of models in continuous or discrete time. After developing two basic theorems, we illustrate the method by determining the limiting distribution of a “batch service” queue, in which K customers are always served at a time. The batch service queue is also shown to be equivalent to a classical single-server queue with gamma interarrival distribution and general service distribution, and to an inventory model of the popular “s, S” type. Waiting-time distributions and expected lengths of busy periods are shown for the queues, and the application of the method to finite queues is indicated.

Supply chain management (SCM) is defined as a set of approaches utilized to efficiently integrate suppliers, manufacturers, distributors and retailers to make sure that merchandise is produced and distributed at the right quantities, to the right locations, and at the right time, in order to minimize system-wide costs while satisfying customer requirements. As a long-term cooperation strategy among enterprises, supply chain requires stable operations as necessary conditions for optimization and long-term competitive advantage of the whole supply chain. Stability, therefore, becomes an important indicator in supply chain management. Research on supply chain stability began in early 1960s. Based on system dynamics method, Forresterう1961え proves the presence and the cause of the bullwhip effect. He indicates that bullwhip effect could be weakened by improving decision-making behavior. From then on, the bullwhip effect and supply chain stability have been studied by many researchers. Most of these studies employed an ordering policy known as the APIOBPCS model (John et al., 1994; Disney and Towill, 2003). APIOBPCS policy is optimal in that, the policy minimizes the variance of the inventory levels with a sequence of forecast errors of demand over the lead time given (Vassian, 1954). In discrete time domain, pure time delays are readily handled by the z-transform and many results are known about its stability (Disney and Towill, 2002), variance amplification properties (Disney and Towill, 2003; Disney et al., 2004), and dynamic performance (Dejonckheere et al., 2003). Disney et al. (2006) investigate the stability of a generalized OUT (Order Up To) policy for the step response in both discrete and continuous time. Disney et al. (2007) prove that discrete and continuous Bullwhip Effect expressions have similar structures and show that the two domains are managerially equivalent and each domain can be used to study a supply chain in practice. The pure time delay causes complications in the time domain differential/difference equations. In the frequency domain, such equations generate an infinite number of zeros in the transfer function. Certain progress has been made on the stability of such systems by recasting the policy as a Smith Predictor (Warburton et al., 2004; Riddalls and Bennett, 2002). However, little is known about variance amplification issues or other aspects of the model’s dynamic performance. The use of Lambert ω function to solve such problems has recently gained some popularity. Several authors have studied the production and inventory control problem using continuous mathematics; and in the solution process the Lambert ω function is used (Warburton, 2004a, b; Warburton et al., 2004).

American Option Pricing with Discrete and Continuous Time Models: An Empirical Comparison

This work is motivated by the need to solve the inventory routing problem when implementing a business practice called vendor managed inventory replenishment (VMI). With VMI, vendors monitor their customers′ inventories and decide when and how much inventory should be replenished at each customer. The inventory routing problem attempts to coordinate inventory replenishment and transportation in such a way that the cost is minimized over the long run. We formulate a Markov decision process model of the stochastic inventory routing problem and propose approximation methods to find good solutions with reasonable computational effort. We indicate how the proposed approach can be used for other Markov decision processes involving the control of multiple resources.

We consider an inventory routing problem with stochastic demand where the demand is expressed in a probabilistic sense and it is dynamic in which the demand is gradually revealed at the end of each period. The problem is known as dynamic and stochastic inventory routing problem (DSIRP). DSIRP distribution network consists of a supplier and a set of retailers is considered. A limited number of single product is assumed to be produced by the supplier and a holding cost per unit product is incurred at both the supplier. The demand is assumed to follow the binomial probability distribution. We adopt an order-up-to level inventory policy and each unit of positive inventory at the retailer is incurred a holding cost, while the unmet demand (stock-out) is penalized. We assumed that the transportation of the product from supplier to the retailers is handled by a third party. The aim of DSIRP is to minimize the total expected cost which includes the inventory cost at supplier, inventory cost at retailers, stockout cost at retailers and transportation cost. The DSIRP is modeled as a Markov decision process (MDP) where the state in period t is represented as the inventory level for all retailers. A hybrid rollout algorithm where a Mixed Integer Linear Programming of the problem is used to obtain the control in each state. We also proposed an Artificial Bee Colony (ABC) algorithm to obtain a feasible and better controls. Computational experiments is conducted using an existing benchmark problem.

A genetic algorithm-Taguchi based approach to inventory routing problem of a single perishable product with transshipment

On the Consistency of the Lucas Pricing Formula

Modeling a green inventory routing problem with a heterogeneous fleet