Solving the ship inventory routing and scheduling problem with undedicated compartments

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Abstract

This paper presents a ship inventory routing and scheduling problem with undedicated compartments (sIRPSP-UC). The objective of the problem is to find a minimum cost solution while satisfying a number of technical and physical constraints within a given planning horizon. In this problem, we identify four sub-problems that need to be decided simultaneously: route selections, ship selection, loading, and unloading activity procedures. To solve this problem, first, we developed a mixed integer linear programming model. We then developed a one-step greedy heuristic, and then based on this heuristic, we propose a set of heuristics. Each heuristic has a combination of rules for each sub-problem. A number of problem instances are used to compare the solutions of the two approaches. We applied selected good combinations of rules to solve each problem using the heuristic approach. The results show that 8 out 12 of the considered problem instances have no gap with MILP solution solved using LINGO. We also find that the average gap is 1.96%. In contrast when we consistently use the same combination for all iterations, there are no dominant combinations of heuristics that can find good solutions for all the problem instances.

Introduction

In this paper, we present a ship inventory routing and scheduling problem with undedicated compartments (sIRPSP-UC). This problem is a class of inventory routing problem (IRP) which considers an integration of an inventory management problem and a routing problem. In maritime literature, this problem can be categorized as industrial shipping as the same company controls both the inventory levels and the transportation of products. A recent survey on ship scheduling research is given in Christiansen et al. (2007).

Likes other IRP, sIRPSP-UC is concerned with the repeated distribution of a set of liquid bulk products from a number of production depots to a number of demand nodes over a given planning horizon. Unlike the vehicle routing problem (VRP) where a company assigns vehicles to meet customer orders, there are no direct orders from its customers. The company assigns a fleet of ships to maintain the stock level of the commodities within their limits at minimal cost during a given planning horizon. In this problem, we need to determine the type and the quantity of products to be loaded, the ship routing and delivery schedules, and the type and the quantity of products to be unloaded at the destinations ports simultaneously.

This paper is a variation to the model considered by Christiansen, 1998, Christiansen et al., 2007, Christiansen and Fagerholt, 2009, Al-Khayyal and Hwang, 2007. Christiansen, 1998, Christiansen et al., 2007, Christiansen and Fagerholt, 2009 considers a single product model. Al-Khayyal and Hwang (2007) extends the problem to have multiple non-intermixable products. The heterogeneous ships distributing these products have several compartments to keep products separately. However, they assume that compartments are dedicated for certain products. This means that it is not permissible to assign a product to a compartment that has been used previously by other products.

As a variation to the model considered by Al-Khayyal and Hwang (2007), the sIRPSP-UC relaxes the problem to consider an assignment of multi-undedicated compartments to products. We call this assignment as multi-product loading assignment (MPLA). MPLA exists in practice in delivering of oil products as discussed, for example by Bruggen et al., 1995, Cornillier et al., 2008a, Cornillier et al., 2008b, Cornillier et al., 2009. Furthermore, we also found a real world problem in transporting oil products in the South East Asia region. The problems discussed in Bruggen et al., 1995, Cornillier et al., 2008a, Cornillier et al., 2009 basically are VRP because they concern single period problems. Cornillier et al. (2008b) extends their previous research to deal with a multi period problem. All of these papers assume that partially unloading is not permitted. This assumption is common when we discuss oil product delivery using land-based transportation, such as a truck. In our problem, maritime transportation, the capacity of compartment of a ship is mostly larger compared to the quantity to be unloaded. Because of that, we assume that partial unloading is permitted. The detailed comparisons between land-based and maritime transportation are described in Ronen, 2002, Christiansen et al., 2007. In addition to the above papers, MPLA has also been discussed in many papers, such as in Yuceer, 1997, Bukchin et al., 2004, Bukchin et al., 2006, Smith, 2004.

In this paper, sIRPSP-UC problem is formulated as a mixed integer linear programming (MILP) model. Considering the difficulty of solving large problems, we have developed a multi-heuristics based approach. The methodology is validated against MILP solution using LINGO for a number of problem instances.

The remainder of this paper is organized as follow. A mathematical model of sIRPSP-UC is presented in Section 2. A one-step greedy heuristic method is described in Section 3. In Section 4, heuristics for solving the problem is described. In Section 5, an illustrated example is discussed. The result of both the mathematical model and heuristic methods are presented and compared in Section 6, and this is followed by conclusions in the last section.

Section snippets

Problem description

We present the sIRPSP-UC that involves the delivery of multiple bulk liquid products which cannot be mixed. This delivery uses heterogeneous types of ships in term of capacity, traveling cost and time. The ships also differ in number of compartments. These compartments are not dedicated to specific products. This means that each compartment can load any type of product but only one at a time. In each delivery trip, a ship can visit more than one port. However, it is assumed that all products

Mathematical model

The development of this model is similar to Christiansen, 1998, Christiansen et al., 2007, Christiansen and Fagerholt, 2009, Al-Khayyal and Hwang, 2007, but with significant modifications to account for loading (or unloading) activities in undedicated compartments of ships. The sIRPSP-UC model can be formulated into five-parts: routing, loading and unloading, scheduling, inventory and objective function, as shown below:

Multiple-heuristics for the problem

We have solved the mathematical model derived in the last section using LINGO. Considering the excessive computational time in solving this large scale problem, we proposed a multiple-heuristic based approach for solving the sIRPSP-UC. As stated by Cowling et al., 2002, Chakhlevitch and Cowling, 2008, individual heuristic may perform well for certain problem instance, but it is unlikely that the heuristic may be applied successfully to a different problem.

In this approach, we consider a set of

An illustrative example

An illustrative example is presented in this section. The model consists of one production port (H1) and two consumer ports (H2 and H3). Each port has two products (P1 and P2) and two storages (S*1 and S*2) where (*) denotes the port’s number. S*1 stores product P1, while S*2 stores P2. H1 has 55 and 50 unit daily production rates for storages S11 and S12, respectively. The maximum capacity of both its storages is 1000 units. At the beginning of the period, the initial level of the storages of

Computational result and discussions

We have conducted experiments to test the applicability and performance of the proposed method. We use six generated datasets which represent two groups of problems of different dimensions. The first five datasets belong to the first group that have (2, 3, 2, 2) configurations of the number of ships, ports, products and compartments, while the second dataset has (3, 4, 2, 2) configuration. Each dataset is run for 10 and 15 days planning horizon. The problem instances can be found from 9000/~z3231348/SIRP-data-instances

Conclusion

In this paper, we developed a mathematical model for solving a ship inventory routing and scheduling problem with undedicated compartments (sIRPSP-UC). The objective of the problem is to find a minimum cost solution while satisfying a number of technical and physical constraints within a given planning horizon. The problem is an extension of the work in the area in that the problem considers an assignment of multi-undedicated compartments to products. Considering the excessive computational

Acknowledgement

The authors thank the anonymous reviewers for their comments and suggestions that improved the presentation of the paper.

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