Mixed integer programming based maintenance scheduling for the Hunter Valley coal chain

Abstract

We consider the scheduling of the annual maintenance for the Hunter Valley Coal Chain. The coal chain is a system comprising load points, railway track and different types of terminal equipment, interacting in a complex way. A variety of maintenance tasks have to be performed on all parts of the infrastructure on a regular basis in order to assure the operation of the system as a whole. The main objective in the planning of these maintenance jobs is to maximize the total annual throughput. Based on a network flow model of the system, we propose a mixed integer programming formulation for this planning task. In order to deal with the resulting large scale model which cannot be solved directly by a general purpose solver, we propose two steps. The number of binary variables is reduced by choosing a representative subset of the variables of the original problem, and a rolling horizon approach enables the approximation of the long term (i.e. annual) problem by a sequence of shorter problems (for instance, monthly).

1 Introduction

The Hunter Valley Coal Chain (HVCC) consists of mining companies, rail operators, rail track owners and terminal operators, together forming the world’s largest coal export facility. In 2008, the throughput of the HVCC was about 92 million tonnes, or more than 10 % of the world’s total trade in coal for that year. The coal export operation generates around $15 billion in annual export income for Australia. As demand has increased significantly in recent years and is expected to increase further in the future, efficient supply chain management is crucial. Our industry partner, the Hunter Valley Coal Chain Coordinator Limited (HVCCC), was founded to enable integrated planning and coordination of the interests of all involved parties, so as to improve the efficiency of the system as a whole. More details on the HVCC can be found in Boland and Savelsbergh (2012).

In this paper, we are concerned with the infrastructure that is necessary to bring the coal all the way from the mining areas in the Hunter Valley onto vessels transporting it to the final destination. The coal has to go by rail to one of the terminals in the port of Newcastle, where it is assembled and finally loaded onto a vessel. There is a natural subdivision of the chain into three parts.

• 1. The rail network between the load points at the mines and the terminals.

• 2. The inbound part of the terminals. The coal is unloaded from the trains at dump stations, transported to the stockyard via conveyor belts and stacked onto pads in the stockyard.

• 3. The outbound part of the terminals. The coal is reclaimed from the stockyard, and loaded onto the vessel at the berth.

We discuss the annual maintenance planning process carried out by the HVCCC. Supply chain components such as railway track sections and terminal equipment have to undergo regular preventive and corrective maintenance, causing a significant loss in system capacity (up to 15 %). The HVCCC had observed that careful scheduling of the maintenance jobs—good alignment of them—could reduce the impact of maintenance on the network capacity, and established a regular planning activity to carry it out, called “capacity alignment”. Currently capacity alignment for the approximately 1,500 maintenance jobs planned each year is a labour-intensive, largely manual process, achieved by iterative negotiation between the HVCCC and the individual operators. The items whose maintenance is considered in the process come in three groups corresponding to the above partition of the coal chain:

• 1. Railway track sections,

• 2. Terminal inbound, in particular, dump stations and stackers, and

• 3. Terminal outbound, in particular, reclaimers, ship loaders and berths.

The HVCCC currently uses an impact calculator written in a business rules management system to evaluate the quality of proposed maintenance schedules. This calculator is integral to the HVCCC Capacity Model, a software developed by the HVCCC. For a given set of maintenance activities, it determines three numbers: a rail track impact, a terminal inbound impact and a terminal outbound impact. The total system impact is taken to be the maximum of these three numbers. Summing over the time intervals of constant maintenance yields a single total impact for the full time horizon. In-depth analysis of rules for the terminal impacts and the HVCC coal handling system revealed that the rules can be well captured by solving maximum flow problems in certain networks. The arcs in these networks represent the different terminal machines, and a maintenance job simply means that the corresponding arc cannot carry any flow for the duration of the job. The railway track network is represented very coarsely in the HVCCC impact calculator. Basically, the impact of a track section outage is taken to be the sum of the expected demands (scaled to the duration of the outage) of the load points L with the property that the outage prevents trains from going between L and the terminals. This is motivated by the fact that, due to the tree structure of the rail network, there is a unique route from each load point to each terminal, and the terminals are very close to each other. This means that as long as the railway is not a bottleneck the sum of the affected load point demands corresponds indeed to the reduction of the system capacity. In the future, with increasing demands the rail network will become a bottleneck, so it is more accurate to model it also as a network, the nodes representing junctions and the arcs representing track sections. The additional conceptual advantage of having network models for all parts of the system is that they can easily be connected to capture the interaction between different parts. In particular, the link between the inbound and the outbound part of the coal chain is an important feature of our proposed model that is missing in the current impact calculator. So the buffering function of the stockyard, where the coal typically stays for between three and ten days, can be taken into account. This has the potential to enable more efficient coordination of inbound and outbound outages.

The maintenance jobs are scheduled initially according to standard equipment requirements, which typically dictate particular types of maintenance jobs to be performed at particular time points. Initial schedules are produced by the providers (track owners and terminal operators) more or less independently of each other. Based on this an iterative process begins, in which the coordinator evaluates the schedule, works out options for modifications that can release capacity, and negotiates these proposed changes with the providers. The purpose of our model is to support this process by providing an efficient way to explore and evaluate a variety of different rescheduling options.

Our paper is organized as follows. After a brief review of the relevant literature in Sect. 2, Sect. 3 contains a precise definition of the considered maintenance scheduling problem. This includes the underlying network model, the form of the given initial maintenance schedule, and the rescheduling rules that have to be followed by the optimizer. In Sect. 4, a mixed integer programming (MIP) formulation is given, and Sect. 5 contains two heuristic reduction steps necessary in order to make the problem computationally tractable. Computational results for two different real world data sets are presented in Sect. 6, and the final Sect. 7 contains concluding remarks and some directions for further investigation.

2 Literature review

Maintenance in general is an essential activity of many production and transportation systems, and in many cases accounts for a significant part of the cost of the operation. Maintenance is usually categorized as either preventive or corrective. The former is a regular, planned activity, whereas the latter is carried out in response to a failure or breakdown. A third category, known as predictive maintenance, concerns the use of measurement to predict and diagnose equipment condition. We refer the reader to Sharma et al. (2011) for an overview of maintenance activities and their optimization. In this work, we are concerned with planning preventive maintenance activities.

Much of the literature on preventive maintenance concerns identification of maintenance policies: how often to perform each type of maintenance, or under what conditions. In addition to the review of Sharma et al. (2011), we refer the reader to the review of Budai et al. (2008), which recognizes the significant advantages that can be realized by taking into account the impact on production when planning maintenance. This paper categorizes approaches as either taking into account the production impact of maintenance in maintenance planning, taking into account resource implications (e.g. manpower) in maintenance scheduling, and production planning subject to maintenance requirements. Three streams of research are identified: those focused on costing maintenance activity, those investigating the impact of carrying out maintenance at opportune moments (when a breakdown or other interruption has occurred), and those which schedule maintenance in line with production. The HVCCC process has elements of the first and third: (i) the “cost” of a maintenance plan is assessed in terms of its impact on throughput, and is used to compare alternative plans; and (ii) by seeking to re-time maintenance (align it, discussed further in the next paragraph), the HVCCC is seeking to schedule maintenance in line with production.

In our work, the preventive maintenance policy is already set, which yields initial schedules for all components of the system. Rail and terminal equipment have initial maintenance schedules largely directed by their corresponding maintenance policy. However, the maintenance policies do not determine the times for individual maintenance events exactly. There is some flexibility which can be exploited to re-time certain jobs, or align them, so as to reduce the impact of the maintenance on the system capacity. Most of the literature that considers scheduling maintenance in line with production addresses maintenance policy setting. Exceptions occur in power generation, when maintenance outages must be scheduled, for example, as in Frost and Dechter (1998). These are primarily concerned with minimizing the overall cost of maintenance and generation, while ensuring generation is sufficient to meet demand; in our context, by contrast, the cost of maintenance activities is fixed, and we want to maximize system capacity. Other exceptions can be found in the transport sector, for example, buses or airlines. These usually focus on the resource implications of maintenance, as in Haghani and Shafahi (2002) and Keysan et al. (2010), or on how to cover the scheduled transport operations while still meeting maintenance requirements, as in Barnhart et al. (1998). The focus is on scheduling maintenance for the transport equipment, as opposed to the infrastructure over which it moves; the latter is closer to our case of interest.

Rail track and road maintenance does address transport infrastructure, and a number of interesting studies are available, particularly for rail maintenance, see, for example, Budai and Dekker (2002). In these cases, the focus is on minimizing the disruption to scheduled activities, not on maximizing the number of trains that could be pushed through the system; the latter would be closer to the situation of interest to us. However, there are still some relevant papers: we discuss the most closely relevant and recent one, that of Budai et al. (2006), in some detail further below.

In production systems, which perhaps more closely resemble the HVCCC situation, multi-component system maintenance would appear to be relevant, since it is the interaction of subsystems and their maintenance schedules that produces the benefit of alignment. Multi-component maintenance models are defined to be those in which a system consists of multiple units (equipment or machines) that may depend on each other economically, stochastically or structurally (see Nicolai and Dekker 2008 and references therein). Economic dependence is focussed on the direct cost of the maintenance activity, and whether or not carry out maintenance on multiple units simultaneously can decrease costs, e.g. via economies of scale, or increase them, e.g. via the need to employ extra resources. Stochastic dependence concerns the failure probabilities and whether or not these are related across units. Structural dependence applies, for example, when the part needing maintenance is inside or connected to other components, so in order to have maintenance on one component, others may also need maintenance, or at the least dismantling. Clearly, these latter two types are not relevant to the HVCCC situation: here the failure probabilities are already accounted for in the preventive maintenance policies, which are set, and any structural connection between maintenance tasks simply implies a constraint, indicating that some maintenance tasks must be scheduled together. Neither of these addresses the primary mechanism of interest: that the alignment of maintenance tasks (by re-timing) can release capacity in the system. Clearly, economic benefit is realized by releasing capacity, thus we would hope that work categorized under economic dependence would address the HVCCC context. However, that does not seem to be the case, and all work highlighting the economic benefits of maintenance scheduling in multi-component systems focusses on the direct costs of the maintenance activities themselves, not on the consequent production benefits. We thus believe that a third type of economic dependence warrants attention, and that is when scheduling maintenance tasks either together or apart affects the production capacity, and hence realizable revenue, of the system.

The only work we are aware of which articulates the benefits of alignment as identified in our context is that of Budai et al. (2006) on preventive scheduling of railway track maintenance. As in our context, maintenance is planned over a finite time horizon. They consider maintenance of two types: routine work and projects. The former are cyclic, with given period, and so are not similar to the situation we consider. The projects are similar: input data is a list of maintenance projects, and for each, its duration, together with its earliest and latest start time. However, these are relatively infrequent (for each type, performed once every six months to a year), and a major part of the contribution of their paper is to explore the interaction between projects and routine work, which is an interesting challenge. Their objective is to minimize the combination of track possession costs, which reflect the train schedule disruption, and the direct costs of the maintenance activities. In our setting, the latter are irrelevant, but the former could be viewed as a proxy for track capacity. They recognize that by clustering jobs at the same time, the track possession cost may be minimized: this corresponds to one type of alignment in our context. Since their model focusses on a single link in the network, the other type is not noticed. They develop an integer programming model, and for the scale of problem they consider (15 types of maintenance over a 3–4 year horizon), the solution times are too long, indeed for their randomly generated problems over 2-year horizons, less than 30 % of instances are solved within three hours. Thus they consider a restricted integer program, and four heuristics, to get good quality solutions in reasonable time.

In this paper, we do not have cyclic maintenance tasks, but we do consider simultaneous scheduling over the whole system, not a single link. Most significantly, the cost of the schedule is not a simple linear function such as the track possession cost. Instead, because of the system-wide effects, and the interdependence of subsystems in achieving system capacity, we require solution of an optimization model to determine the impact of a combination of maintenance tasks at scheduled times. Also important is the treatment of time. Budai et al. (2006) are able to use a fairly coarse time discretization (weeks), needing only the order of 100 periods in their model. In our context, maintenance tasks have start time specified to the nearest 15 minutes in some cases, (to the nearest hour in others), and are of the order of hours to days in duration. Thus we need to address the challenge of how to handle time without leading to an extremely large number of variables.

A simplified version of the problem considered in the present work has been introduced in a more abstract setting in Boland et al. (2012b). That paper omits some of the complicating constraints specific to the coal chain maintenance scheduling to arrive at a problem that might be applied in different network related contexts. Even more simplified special cases (unit processing time jobs with arbitrary start times) are studied from a computational complexity viewpoint in Boland et al. (2012a). In contrast, the present work focuses on modelling the actual problem of the planners at HVCCC as closely as possible, and proposes solutions of immediate practical relevance.

To conclude, we believe our paper makes a quite unique contribution to the maintenance optimization literature. It considers a multi-component system, where maintenance activities on the components can be scheduled so as to produce economic benefit of a new type, not previously considered. It exploits a relationship between production and maintenance in what appears to be a new way. There does not seem to be any prior work that considers scheduling maintenance activities so as to maximize the production capacity of the system, unless one interprets minimizing track possession costs in rail track maintenance in that light, and in that case we see our paper makes further significant new contributions. In particular, maintenance is scheduled system-wide and the objective is a complex function of the interaction of the system as a whole, not a simple linear function. Furthermore, we provide insights and ideas for handling scheduling problems of this type when maintenance tasks are of widely varying durations, and their start times are fine-grained relative to the planning time horizon.

We expect the models and methods presented here to have broader applicability beyond the HVCC: they could be applied in any setting in which production is reasonably modelled as a flow in a network, throughput (total production rate) is the key objective to be maximized, and regular maintenance needs to be scheduled on network components. Most mining supply chains, whether for coal, iron ore, or other minerals, well fit this description. Applications to other bulk goods supply chains, such as for fertilizer or wheat, are also likely to be possible.

3 Problem description

In this section, we set the scene. The first subsection contains details on the underlying network while the second subsection introduces the actual scheduling problem.

3.1 The network model

The network representing the HVCC consists of subnetworks for railway track and for terminals:

• 1. The railway track network has nodes for load points and for junctions, and the arcs represent rail track sections.

• 2. There are two terminal networks whose arcs correspond to terminal equipment.

The full network is shown in Fig. 1. The ellipses indicate the terminals and the part outside of them is the rail network, where square nodes represent junctions while circle nodes are load points. The load point nodes are included in Fig. 1 just for illustration. In the actual model, they are identified to form a single source node, and for every load point there is a corresponding arc linking the source to the respective junction. The capacities of these load point arcs are the demand forecasts, which may vary over the time horizon. Arcs between junctions have capacities determined by the number of trains that can pass the corresponding piece of track per day.

Fig. 1

The HVCC network. The terminal subnetworks are indicated by ellipses. For the rail network, rectangular nodes represent junctions while load points correspond to circle nodes

For the terminal modelling, we focus on Terminals 1 and 2, as Terminal 3 was commissioned only recently, and we did not implement a more detailed model, yet. Both of the first two terminals have a column of four larger nodes in the middle, representing the pads on the stockyard where coal can be stored. In fact, real world operation can be reflected quite accurately by requiring that the coal has to stay on the pad for a certain time. This dwelling time is restricted to be between two parameters D ₁ and D ₂ which might be taken to be three and ten days to capture what typically happens in practice. The inbound and the outbound part of the terminals are to the left and to the right of the pad nodes, respectively. Labelled arcs represent terminal machines where the labels “D”, “S”, “R”, “SL” and “B” stand for dump station, stacker, reclaimer, ship loader and berth, respectively. Note that in the Terminal 1 network there are arcs labelled D2 and D2′, both corresponding to the same dump station, and similarly for dump station 3 and stackers 3 to 6. This is necessary to capture the following aspect of the operational practice at the terminal. The six stackers are grouped into four stacker streams: {1,2}, {3}, {4} and {5,6}. A stream is available if at least one of its stackers is available. The inbound capacity is determined by the following rules, where c ₁=85, c ₂=17.5 and c ₃=15 are capacity parameters.

• 1. Every dump station can be combined with every stacker stream.

• 2. The basic capacity of a (dump station, stacker stream)-pair is c ₁.

• 3. Every combination of dump station 2 or 3 with a stacker stream different from {1,2} (high-throughput pair) releases an additional capacity of c ₂.

• 4. If the number of available dump stations is greater than or equal to the number of available stacker streams, and exactly one of the stackers 1 and 2 is available, there is a reduction of c ₃ due to inefficiency.

For the given values of the capacity parameters, the inbound capacity of Terminal 1 can be characterized more formally as follows. For i∈{1,2,3} and j∈{1,2,…,6}, let a _i and b _j indicate the availability of dump station i and stacker j, respectively, i.e.

Then the inbound capacity equals the optimal value of the integer program (1)–(10):

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

By (2) and (3), x ₁₂ and x ₅₆ are the indicator variables for the availability of the streams {1,2} and {5,6}, respectively. Constraints (4) and (5) make y ₁ a bound for the number of available (dump station, stacker)-pairs, and similarly, y ₂ bounds the number of available high-throughput pairs by (6) and (7). Finally, (8) ensures that y ₃=1 if and only if the inefficiency condition holds, because in this case

$$2\geqslant(a_1+a_2+a_3)-y_1\geqslant0\quad\text{and}\quad2x_{12}-(b_1+b_2)=1. $$

In order for this characterization to be valid (i.e. the equivalence between the four inbound capacity rules and the integer program (1)–(10)), certain assumptions on the parameters c ₁, c ₂ and c ₃ are necessary. For instance, if c ₃ were larger than c ₁, the availability of stacker 1 does not enforce x ₁₂=1: In the situation

$$a_1=a_2=a_3=b_1=b_3=b_4=1,\quad\quad b_2=b_5=b_6=0, $$

the optimal solution of (1)–(10) is

$$x_{12}=x_{56}=y_3=0,\qquad y_1=y_2=2 $$

with objective value 2(c ₁+c ₂), while the capacity determined according to the rules is 3c ₁+2c ₂−c ₃: There are three (dump station, stacker stream)-pairs available, two of them high-throughput, but one of them inefficient.

We conclude that the operational terminal logic is captured by the following capacities on the inbound arcs in the network for Terminal 1:

• Capacity 85 for arcs D1, D2, D3, S3, S4, S5, S6,

• Capacity 70 for arcs S1 and S2, and

• Capacity 17.5 for arcs D2′, D3′, S3′, S4′, S5′, S6′.

Using these types of considerations, where the one explained in detail is by far the most involved, the full terminal network structure including capacities can be derived.

3.2 Maintenance scheduling

The initial schedules from the track owners and the terminal operators are given as a list of maintenance jobs. Every entry of this list consists of the name of the asset to be maintained, the start and the end time of the maintenance activity, and possibly an additional entry indicating the type of work to be done. At the terminals an outage simply makes the corresponding arcs unavailable for the job’s duration. In most cases that means the deletion of a single arc, but in some cases for Terminal 1 two arcs may be affected as described in the previous subsection. For the rail network a single asset can be associated with a sequence of arcs in the network, and the effect of the outage on the capacity can be specified for each of the affected arcs separately. For instance, if there is a double track available, an outage might reduce the capacity to 50 %, while on a single track it is reduced to zero. There are also certain track inspection jobs that do not block the track completely for their whole duration, but still cause delays. The effect of these jobs is taken into account by small capacity reductions: the exact value of this reduction is given as input data based on the practical experience of maintenance planners at HVCCC and lies typically between 10 and 20 %.

For a fixed schedule, we can collect the start and end times of the jobs and order the list of all these times. This defines a partition of the time horizon into intervals of constant maintenance activity. We call this partition the time slicing associated with the schedule, and the intervals time slices. In order to measure the quality of the schedule, we construct a time expanded network containing one copy of the basic network per time slice. Flows in this network represent total tonnes of coal transported during the time slice. The network copies for consecutive time slices are connected via arcs linking the corresponding copies of pad nodes. Flows on these linking arcs represent coal present on the pads at the transition time between the time slices. The arc capacities in a time slice are taken to be the capacities of the basic network—which express upper bounds on the rate of flow in each arc—discounted by the capacity reduction factor for any maintenance job occurring on the arc during the time slice, and multiplied by the length of the time slice. The results are upper bounds on the amount of coal that can move along each arc during the time slice. We solve a max flow problem in this expanded network, the value of which is interpreted as a measure for the total system capacity. We take this as our primary optimization objective. In fact, it is more complicated than a pure max flow problem, as there are side constraints from the requirement that the coal stays on the ground for some time.

In discussions with the maintenance planners, it emerged that they would be prepared to move the jobs, usually for intervals of plus or minus 7 days. We initially expected there would be some inter-maintenance constraints, for example, that a type of job carried out at 4-weekly intervals could not be carried out more than 5 weeks apart. But the maintenance planners were not concerned about this issue, and preferred the simple assumption that jobs could not deviate more than some fixed number of days around their initial scheduled time. The arising optimization problem is to take an input schedule and modify it according to certain rules such that the total system capacity is maximized. The scheduling rules can be summarized as follows:

• 1. No job can be moved by more than 7 days.

• 2. Major track outages (rail outages with a duration of more than 24 hours) must not be moved.

• 3. The track inspection jobs and jobs with certain specific work type tags must not be moved.

• 4. Rail jobs initially scheduled on a weekday (Monday to Friday) have to stay on a weekday.

• 5. Rail jobs initially scheduled between 7:00 am and 4:30 pm have to stay in this time window.

• 6. Jobs on the same item that do not overlap in the initial schedule are not allowed to overlap in the optimized schedule.

• 7. Some jobs on stackers and reclaimers have an associated so-called washdown job which immediately precedes them. These job pairs can only be moved together.

Our primary objective is to maximize the throughput, but from a practical point of view it is also desirable not to deviate too much from the initial input schedule, as this was the result of independent decision processes of the providers. So the final goal should be to treat the maintenance scheduling in a bi-objective framework with the objectives total throughput and (weighted) number of job movements. As a first step in that direction, we propose a lexicographic optimization: in a two-phase approach, we first maximize the total throughput and then minimize the number of moved jobs subject to a lower bound on the throughput.

4 A mixed integer programming formulation

In this section, we present an MIP formulation of the maintenance scheduling problem described in Sect. 3. Let (N,A,s,s′,u) denote the network with node set N, arc set A, source s and sink s′, and capacities u _a ∈ℝ₊ for a∈A. We denote the set of incoming and outgoing arcs of a node v∈V by δ ⁻(v) and δ ⁺(v), respectively. Recall that the source s replaces the load point nodes in Fig. 1. In addition, let N _P ⊆N denote the set of nodes corresponding to the pads on the terminal stockyards. They are special in that they allow the storage of flow, and each of them has associated upper and lower capacities $u^{\mathrm{lower}}_{v}$ and $u^{\mathrm{upper}}_{v}$, representing the acceptable variation of the amount of coal on the pad. We represent the considered time horizon by a real interval [0,T] where time is measured in days, so for the complete annual problem, T=365. A maintenance job j is specified by

• Its arc set A _j ⊆A with associated capacity reduction factors ρ _ja ∈[0,1] for a∈A _j , meaning that during the processing of job j the capacity of arc a is reduced to (1−ρ _ja )u _a ,

• A processing time p _j ∈ℝ₊,

• A finite set of possible start times $\mathcal{S}_{j}\subseteq [0,T]$, and

• An initial start time $S_{j}^{0}\in\mathcal{S}_{j}$.

Note that the first five scheduling rules listed in Sect. 3.2 will not appear as constraints in the MIP as they can be enforced by simply restricting the sets $\mathcal{S}_{j}$ appropriately. Scheduling a job to start at time $S_{j}\in\mathcal{S}_{j}$ reduces the capacity of arcs a∈A _j in the time interval [S _j ,S _j +p _j ]. We have to schedule a set J of maintenance jobs in such a way that the total throughput over the interval [0,T] is maximized. We denote the set of job pairs that are not allowed to overlap due to rule 5 by $R\subseteq\tbinom{J}{2}$, and we call a maximal clique $\mathcal{C}$ in the graph with vertex set J and edge set R a conflict clique. Then the scheduling rule just says that for any conflict clique $\mathcal{C}$, at any point of time at most one of the jobs in $\mathcal{C}$ can be processed.

In order to formulate the MIP, we need some more notation. Let $\mathcal{T}=\{0=t_{0}<t_{1}<\cdots<t_{M}=T\}$ be the set of all times relevant for the problem, i.e.

$$\mathcal{T}=\big([0,T]\cap \mathbb {Z}\big)\cup\bigcup_{j\in J}\bigcup_{a\in A_j}\mathcal{S}_j\cup\left(\mathcal{S}_j+p_j\right). $$

Note that this implicitly defines M, the number of possible time slices that could occur in the resulting maintenance schedule. We require $\mathcal{T}$ to contain all integers in the time horizon in order to control the daily balances of in- and outflow at the stockyard. As for a fixed schedule, this is a time slicing. In fact, it is a refinement of the time slicing associated with the initial schedule (assuming that the initial start time of job j is contained in $\mathcal{S}_{j}$ for every job j). Now we can define the variables of the model.

• For a∈A and i∈[M], x _ai ∈ℝ₊ is the flow on arc a in the ith time slice [t _i−1,t _i ].

• For v∈N _P and i∈[0,M], x _vi ∈ℝ₊ is the pad level at time t _i .

• For v∈N _P and d,d′∈[T] with d′−d∈{D ₁,D ₁+1,…,D ₂}(modT), $X_{v}^{dd'}\in \mathbb {R}_{+}$ is the amount of flow entering node v on day d and leaving on day d′.

• For j∈J and $t\in\mathcal{S}_{j}$, y _jt is the indicator variable for job j starting at time t, i.e.

  $$y_{jt}= \begin{cases}1 & \text{job $j$ starts at time }t,\\ 0 & \text{otherwise.} \end{cases} $$

• For a∈A, i∈[M] and impact factor γ, the variable $w_{a\gamma}^{i}\in\{0,1\}$ indicates if in time slice i arc a is affected by a job with impact γ, i.e.

  $$w_{a\gamma}^i\in\{0,1\}= \begin{cases} 0 & \text{if arc $a$ between $t_{i-1}$ and $t_{i}$ is affected} \\ & \text{by a job with reduction factor $\gamma$},\\ 1 & \text{otherwise}. \end{cases} $$

In order to formulate the constraints it is convenient to denote the set of relevant jobs for an arc a∈A by J _a , i.e. J _a ={j∈J:a∈A _j }, and the set of possible capacity discount (impact) factors by Π _a ={ρ _ja :j∈J _a }. We can now write down an MIP for the maintenance scheduling problem. At first, our objective is to maximize the total flow

(11)

subject to the following constraints.

Flow conservation constraints

Except at source and sink and at the pad nodes, where flow between time slices is possible, we have flow conservation per time slice. For every node $v\in N\setminus\left (N_{P}\cup\{s,s'\}\right)$ and every i∈[M], we have

(12)

and for v∈N _P , i∈[M]

(13)

We also include periodic boundary conditions

(14)

in this group of constraints.

Capacity constraints

There are arc capacities

(15)

for all a∈A, i∈[M] and impact factors γ∈Π _a . Note that this constraint and the use of the w variables exposes the fixed charge network flow structure in the problem. The job start indicator variables are linked to the impact indicators via constraints

(16)

for all arcs a∈A, all impact factors γ∈Π _a , all time slice indices i∈[M] and all jobs j∈J with ρ _ja =γ. In addition, we have node capacities

(17)

for all v∈N _P and i∈[M]. Note that these are the only lower bounds on flow in the model, so a flow of zero on all arcs, and a flow which is between these bounds and identical for all variables linking a storage node across time slices, provides a feasible solution.

Scheduling constraints

Every job has to be scheduled exactly once, and the processing periods of incompatible jobs must not overlap.

(18)

(19)

Dwell time constraints

The values of the flow variables x _ai determine the total daily in—and outflows at the pad nodes. Now the inflow of day d has to leave between day d+D ₁ and day d+D ₂. This is enforced by the nonnegativity of the variables $X_{v}^{dd'}$ and the constraints

(20)

(21)

for v∈N _P and d∈{1,2,…,T}.

Variable domains

The flow variables are nonnegative reals and the job start indicators are binary:

(22)

(23)

(24)

In a second phase, we change the objective function to maximize the number of jobs starting at their initially scheduled start time $S_{j}^{0}$:

(25)

and we add a lower bound for the total throughput, i.e. a constraint of the form

(26)

where the bound B is a function of the best objective value obtained in the first phase. For our computational experiments, we just multiplied the maximal throughput by 0.999.

5 Solution strategies

In Sect. 4, we formulated a large scale MIP for the maintenance scheduling of the HVCC. In this section, we present some strategies for coping with this large problem. We focus on Phase 1 of the lexicographic optimization, i.e. on the maximization of the total throughput, as this is the primary objective in practice. For the annual planning more than 2,000 jobs have to be scheduled. After some preprocessing taking into account the rules described in Sect. 3.2 (fixed jobs, washdowns, etc.), there are still about 1,000 jobs contributing binary variables y _jt . In practice, jobs are scheduled by the half-hour or on even finer time scales. If this is accurately modelled, allowing every half-hour in the time window as a potential start time, a job without additional daylight or weekday constraints has about 14×48=672 potential start times, corresponding to binary variables. In Sect. 5.1, we describe a heuristic method to choose a representative subset of the start times. Even with these reduced candidate start time sets, initial computational test reveal that the problem for the complete annual time horizon cannot be solved by simple application of a commercial MIP solver (in our case CPLEX 12.3). On the other hand, we obtain promising results if the problem is restricted to a shorter time horizon. This motivates a rolling horizon approach to the full problem as is described in Sect. 5.2.

5.1 Reducing the number of potential start times

Intuitively, focussing on two jobs j and j′, it seems one would always try to minimize their overlap or to maximize their overlap. Consider for example the three arcs in Fig. 2. If j and j′ operate on arcs (1,3) and (3,4), respectively, they should be scheduled to overlap as much as possible, while jobs on arcs (1,3) and (2,3) should be separated. To give a more specific example, suppose the processing times are p _j =2 and p _j′=3 and the candidate start time sets are $\mathcal{S}_{j}=\{1,2\}$ and $\mathcal{S}_{j'}=\{2,3\}$.

• 1. If the arcs for j and j′ are (1,3) and (3,4), it is a good idea (at least locally) to schedule both jobs to start at time S _j =S _j′=2, giving a total capacity of 2×12+3×0+12=36 over the time interval [0,6].

• 2. If the arcs for j and j′ are (1,3) and (2,3), the local analysis suggests to schedule the jobs to start at times S _j =1 and S _j′=3 with a total capacity of 1×12+2×9+3×7=51 over the time interval [0,6].

Of course, the situation becomes more complicated with more jobs involved. But still, the intuition is that there should be an optimal schedule with the property that every job j starts at its earliest or at its latest possible start time, or its start or completion time coincides with the start or completion time of some other job j′. For networks without storage at nodes, this intuition can be converted into a rigorous argument, which is the subject of ongoing research. For the present work, we adopt a more pragmatic viewpoint and generate candidate start time sets $\mathcal{S}_{j}$ by a heuristic based on the described intuition. Let J ₀⊆J be the set of jobs that are not fixed by a scheduling rule. For j∈J ₀, we initialize $\mathcal{S}_{j}$ with the initial start time $S_{j}^{0}$ and the earliest and latest possible start times, i.e. $S_{j}^{0}\pm7$. For j∈J∖J ₀, clearly $\mathcal{S}_{j}=\{S_{j}^{0}\}$. Then we iteratively add candidate start times to the sets $\mathcal{S}_{j}$ that could potentially align job j with the start or end of some job j′. In order to keep the sets $\mathcal{S}_{j}$ reasonably small, we ensure that every job gets at most 2 candidate start times per day. The details of this procedure are given in Algorithm 1.

Fig. 2

Three arcs from a network. Capacities are indicated by arc labels

Algorithm 1

Generating start time sets

5.2 A rolling horizon approach

Computational tests revealed that even after the reduction of the number of binary variables the complete annual problem is very hard. As the performance on restrictions of the problem to shorter time horizons is better, the iterative solution of smaller subproblems is a natural approach to solving the problem, which is also supported by the intuition that rescheduling of a job should have mainly local effects: the system’s performance in September should be largely independent of rescheduling of jobs in March. This suggests the following approach. Fix the binary start indicator variables for all jobs outside a time window [t,t′] and fix all flow variables outside a slightly larger time window [t−δ,t+δ], solve the resulting MIP, shift the time windows by some value σ, and iterate. The whole procedure is described more precisely in Algorithm 2.

Algorithm 2

The rolling horizon algorithm

6 Computational results

In this section, we present computational results for two real world data sets. We use the raw schedules for the years 2010 and 2011 as inputs. The capacities for the load point arcs are determined from the annual capacity forecast numbers for these years. They specify for every load point on a quarterly level the expected daily demands. For 2010 this amounts to a total daily demand of 375 kt and for 2011 it is 475 kt in total. After preprocessing, the 2010 schedule contains 1,277 jobs (197 of them fixed), and for 2011 there are 986 jobs (159 of them fixed). Figure 3 shows the distribution of the processing times.

Fig. 3

Distributions of the job processing times

All our computations are done on a Dell PowerEdge R710 with dual hex core 3.06 GHz Intel Xeon X5675 Processors and 96 GB RAM running Red Hat Enterprise Linux 6. CPLEX v12.3 is used in deterministic mode with a single thread.

From the 2010 and 2011 data we obtain eight quarterly and two annual instances. For each of these instances, we compare the performance of CPLEX with default settings on the complete problem to the rolling horizon heuristic in Algorithm 2 with parameters w=15, δ=10 and σ=10 (all times in days). For the quarterly problems, we impose a time limit of six hours for each phase. In the rolling horizon algorithm, the time limit of six hours is the stopping criterion for the while loop in Algorithm 2, and for each CPLEX call we use a time limit of 25 minutes. For the annual problems, we increase the time limit per phase to 24 hours, and the time limit per CPLEX call to 40 minutes. The results are presented in Tables 1 and 2. Table 1 shows the relative improvements in throughput that are obtained by the two methods as well as an upper bound for the throughput gain. More precisely, if the total throughput for the initial schedule is z ^init, the best schedules for the two methods give throughput values z ^full and z ^rolling, respectively, and the best upper bound from CPLEX on the whole quarterly problem is z ^bound, then the reported numbers are

Note that the total throughput is about 120 megatonnes for 2010 and 140 megatonnes for 2011, so an improvement of 1 % corresponds to an additional throughput of 1.2 million tonnes and 1.4 million tonnes, respectively. So the improvement obtained with our model makes a significant difference in practice. Table 2 contains results on the numbers of moved jobs. We make the following observations. For all ten instances the rolling horizon approach yields a significantly larger total throughput than CPLEX on the complete problem. For the full problem, the number of moved jobs can be reduced significantly in Phase 2, while for the rolling horizon method in nine of the ten instances, we do not find any alternative schedule with a smaller number of moved jobs that yields at least 99.9 % of the throughput achieved in Phase 1. In Phase 2, the difference between “full problem” and “rolling horizon” is that the “full problem” has a weaker lower bound on the throughput from the result of Phase 1. A comparison of the lower bound columns “LB” in Table 2 indicates that there might be a tradeoff between throughput and the number of moved jobs which should be studied in more detail in future work.

Table 1 Throughput improvements in Phase 1. All numbers are relative improvements compared to the initial schedule that is used as a start solution. The last column contains upper bounds for the throughput gains

Table 2 Numbers of moved jobs. We report the total number of jobs (column “Total”), the numbers of moved jobs for the final schedule after Phase 1 and Phase 2 (columns “P1” and “P2”) and the lower bound for the number of moved jobs to achieve the throughput that is enforced in Phase 2 (column “LB”)

7 Concluding remarks

In this paper, we present an MIP model for the maintenance scheduling at the HVCC, where the primary objective is to maximize the throughput, and the secondary objective is to minimize deviations from a given initial schedule. The resulting large scale problems cannot be solved directly, so efficient solution strategies are needed. We propose an iterative approach based on solving subproblems obtained by fixing most of the binary variables. Our computational tests show that this rolling horizon approach outperforms plain CPLEX in terms of the obtained total throughput. Also the experimental results indicate a tradeoff between throughput and the number of moved jobs.

Clearly, more work is necessary in order to improve the performance of the model. Our experiments indicate that the initial LP bounds are rather weak, so adding appropriate cutting planes is a promising approach. Another option for reducing the complexity of the problem is to put an explicit bound on the number of moved jobs. This seems to be a reasonable direction, especially as one outcome of discussions with the maintenance planners was that the option to add more specific constraints on the set of allowed job movements is a desirable feature. Another step towards a tool that is applicable in practice is the development of a true bi-objective framework to find (or at least approximate) the set of efficient solutions for the objectives “total throughput” and “number of schedule modifications”.