CP methods for scheduling and routing with time-dependent task costs

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Abstract

A particularly difficult class of scheduling and routing problems involves an objective that is a sum of time-varying action costs, which increases the size and complexity of such problems. Solve-and-improve approaches, which find an initial solution for a simplified model and improve it using a cost function, and mixed integer programming (MIP) are often used for solving such problems. However, constraint programming (CP), particularly with lazy clause generation (LCG), has been found to be faster than MIP for some scheduling problems with time-varying action costs. In this paper, we compare CP and LCG against a solve-and-improve approach for two recently introduced problems in the area of maritime logistics with time-varying action costs: the liner shipping fleet repositioning problem (LSFRP) and the bulk port cargo throughput optimisation problem (BPCTOP). We present a novel CP model for the LSFRP, which is faster than all previous methods and outperforms a simplified automated planning model without time-varying costs. We show that a LCG solver is faster for solving the BPCTOP than a standard finite domain CP solver with a simplified model. We find that CP and LCG are effective methods for solving problems with time-dependent task costs and are worth investigating for other scheduling and routing problems that are currently being solved using MIP or solve-and-improve  approaches, even when customized global constraints are not available. We also investigate a novel approach to solving the BPCTOP—converting the problem into a vehicle routing problem (VRP) and solving using an existing VRP solver.

Introduction

Scheduling problems typically aim to select times for a set of tasks so as to optimise some cost or value function, subject to problem-specific constraints. Traditional scheduling problems usually aim to minimise the makespan, or total time, of the resulting schedule. More complex objective functions, such as minimising the total weighted tardiness, may vary with time (Smith 2005). Routing problems have many similarities with scheduling—both may have resource constraints and setup time constraints, both have actions that need to be scheduled in time, and both may have complex time-dependent cost functions for actions.

In a number of important, real-world scheduling problems, such as the liner shipping fleet repositioning problem (LSFRP) (Tierney and Jensen 2012; Tierney et al. 2012b) and the bulk port cargo throughput optimisation problem (BPCTOP) (Kelareva et al. 2012a, c), the objective is a sum of time-varying costs or values for each task. Additional problems include net present value maximization in project scheduling (Russell 1970); satellite imaging scheduling (Lin et al. 2005; Wolfe and Sorensen 2000); vehicle routing with soft time windows (Sexton and Choi 1985; Figliozzi 2012); ship routing and scheduling with soft time windows (Fagerholt 2001; Christiansen and Fagerholt 2002); and ship speed optimisation (Fagerholt et al. 2010; Norstad et al. 2011).

Mixed integer programming (MIP) is a standard approach used to solve many scheduling and routing problems. Solve-and-improve approaches are also commonly used to solve scheduling and routing problems with complex constraints or complex objective functions, such as objective functions that are the sum of time-dependent task costs. Solve-and-improve approaches initially solve a simplified problem, then improve the solution using the objective function and constraints of the full problem.

However, for some problems, such as the BPCTOP (Kelareva et al. 2012a), Constraint programming (CP) (see, e.g., Rossi et al. 2006) has been shown to be more effective than using a MIP. CP is also a very flexible method that can be used to model a wider variety of constraints than MIP, which is limited to linear constraints. A number of recent approaches have combined CP with other techniques such as large neighbourhood search for vehicle routing (Kilby and Verden 2011), SAT (Ohrimenko et al. 2009) and MIP (Achterberg 2009) to combine the flexibility of CP with fast algorithms for specific problems. CP approaches may be worth investigating for other problems that have traditionally been modelled with MIP. One CP technique in particular which has been found to be effective on a number of scheduling problems is lazy clause generation (LCG) (Ohrimenko et al. 2009)—a method for solving CP problems which allows the solver to learn where the previous search failed. A CP solver that uses LCG was found by Schutt et al. (2012) to be more efficient than traditional finite domain CP solvers for a number of scheduling problems (Schutt et al. 2012; Feydy and Stuckey 2009).

When faced with a new problem with time-dependent task costs, or when trying to add such costs to an existing problem, it is not clear which approaches will yield good results. In this paper, we try to provide guidance for researchers and practitioners dealing with problems with such costs. We do this through an investigation of several key problems in the literature and the introduction of models for two maritime transportation problems. To this end, this paper contains the following novel components: (1) a review of methods for solving problems with time-dependent task costs, (2) a CP model of the LSFRP that outperforms all previous approaches, (3) a comparison of LCG and solve-and-improve approaches on the LSFRP, (4) a solve-and-improve approach for the BPCTOP, (5) and a vehicle routing based modelling of the BPCTOP that finds high-quality solutions even on large BPCTOP instances. We note that this article is an expanded version of the conference paper by Kelareva et al. (2013).

In “Background”, we review a number of scheduling and routing problems with time-varying action costs, present a summary of problem characteristics and solution approaches, and discuss techniques that may be generalisable between applications, and problems that may benefit from being extended with time-varying action costs.

In the rest of this paper, we adapt several techniques from other applications to two recent scheduling and routing problems with time-varying action costs in the field of maritime transportation. The main contribution of this paper is to compare the effectiveness of CP and LCG against traditional MIP and solve-and-improve techniques for the BPCTOP and LSFRP. “BPCTOP” presents a summary of earlier work on the BPCTOP, including CP and MIP models, as well as a VRP formulation. We compare these models against the new techniques presented in this paper. “LSFRP” summarises earlier work on the LSFRP, including a MIP model and an automated planning approach. This section also presents a novel CP model for the LSFRP and shows that this model is faster than existing approaches. In “Lazy clause generation”, we show that a CP solver with LCG is more effective at solving both the BPCTOP and LSFRP than a traditional finite domain CP solver. In “Solve-and-improve”, we describe solve-and-improve approaches for the LSFRP and BPCTOP that simplify the time-varying cost function to find an initial solution. Finally, in “Summary”, we solve our BPCTOP VRP model using an existing VRP solver, obtaining solutions much more quickly than any previous method, but at the expense of solution optimality.

Section snippets

Background

This section presents a review of a number of scheduling and routing problems with time-dependent action costs, as well as approaches that have been used to solve these problems. A variety of optimal techniques exist for solving routing and scheduling problems, but it is often unclear which technique will offer the best results given a problem with time-dependent task costs. Our goal is to give a detailed overview of the approaches that are available in the literature before we investigate

BPCTOP

Kelareva et al. (2012a) presented CP and MIP models for the bulk port cargo throughput optimisation problem (BPCTOP) and found that CP with a good choice of search strategy was much faster than MIP. However, the CP model solution time was highly dependent on the choice of modelling approach and search strategy used—a number of different modelling approaches and search strategies were investigated in (Kelareva et al. 2012a, c). This section briefly summarises the results presented in (Kelareva

LSFRP

In this section, we describe the LSFRP and present a novel CP model. We compare the CP model against the MIP and automated planning models introduced in Tierney et al. (2012b) on a dataset of 39 instances based on data from our industrial collaborator. We show that CP greatly outperforms previous approaches. We begin the section with an overview of the previous approaches, followed by our CP model and experimental results.

MIP model

Despite the success of LTOP in solving LSFRP instances, the question remains as to how it performs versus more traditional combinatorial optimization techniques. To find out, we create a MIP model of the LSFRP that considers the activities that a vessel may undertake and connects activities based on which ones can feasibly follow one another temporally. The structure of the LSFRP is embedded directly into the graph of the MIP, meaning that it is unable to model general automated planning

A novel CP model

In contrast to our MIP model for the LSFRP, which is based on an activity graph, our CP model exploits the fact that SoS and SE activities cannot be chained together. The CP model uses a number of variables to store the state of each vessel in each stage of its repositioning and connects these stages through logical constraints.

Let Ov be the set of possible phase-out actions for the vessel v and let P be the set of possible phase-in ports for the new service. The decision variable ρP is the

LSFRP CP results

To compare our CP model for the LSFRP against an earlier MIP model and against the LTOP planner (Tierney et al. 2012b), we use the 11 AC3 problem instances from Tierney et al. (2012b) and augment them with 27 new instances based on a real-world scenario provided by an industrial collaborator. The problem instances contain up to 9 vessels, with varying SoS and sail-with-equipment opportunities that may be used to reduce repositioning costs.

The LSFRP CP model was formulated in the MiniZinc 1.6

Lazy clause generation

Lazy clause generation (LCG) (Ohrimenko et al. 2009) or CP with learning has been found to be effective on a number of scheduling problems (Schutt et al. 2012; Feydy and Stuckey 2009; Chu et al. 2010). LCG combines a finite domain CP solver with a SAT solver by mapping finite domain propagators to clauses in a SAT solver.

Mapping a complete CP problem to a SAT problem often results in a very large SAT problem which is intractable for even the best modern SAT solvers. LCG gets around this

Solve-and-improve

Many scheduling and routing approaches initially solve a simplified model and then use the constraints and objective function of the full problem to improve it. For routing and scheduling problems with time-dependent action costs, removing the time dependence of the objective function is one way to simplify the problem for the first step of a solve-and-improve approach, as used by Lin et al. (2005).

In this section, we solve simplified models for both problems which ignore time-varying action

Conversion to vehicle routing

We solve the BPCTOP without tug constraints from “Conversion to vehicle routing” using the Indigo VRP solver, which is able to consider a wide variety of side constraints (Kilby and Verden 2011). One limitation of the solver is that it is only able to handle soft time windows with linear penalties, whereas the draft function can vary non-linearly outside the peak draft windows. Using linear penalty functions to approximate our draft functions may lead to schedules that are not entirely optimal

Conclusions and future work

While scheduling and routing problems have usually been solved using mixed integer programming (MIP) and solve-and-improve approaches, constraint programming (CP) with a good choice of model and search strategy, as well as recent techniques such as lazy clause generation (LCG), have been found to be faster for some problem types. In this paper, we reviewed scheduling and routing problems with time-varying action costs, which increase the complexity of a problem, across a number of applications,

Acknowledgments

The authors would like to acknowledge the support of ANU and NICTA at which Elena Kelareva is a PhD student. NICTA is funded by the australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the australian Research Council through the ICT Centre of Excellence program. They would also like to thank OMC International for their support for the research into the port optimisation problem. The research into the fleet repositioning problem is

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