On the combined maintenance and routing optimization problem

https://doi.org/10.1016/j.ress.2015.09.016Get rights and content

Highlights

  • The paper focuses on planning and scheduling maintenance operations.

  • We present a joint maintenance and routing model for scattered machines.

  • We propose a two-step iterative approach to find a solution.

  • We illustrate the performance of this model on a case in oil and gas.

Abstract

This work focuses on the problem of planning and scheduling maintenance operations for a set of geographically distributed machines, subject to non-deterministic failures with a set of technicians that perform preventive maintenance and repair operations on the machines at the customer sites within a specific time window. This study presents a two-step iterative approach. In the first step, a maintenance model determines the optimal time until the next preventive maintenance operation, its frequency, and the time window for each customer, while minimizing the total expected maintenance costs. In the second step, a routing model assigns and schedules maintenance operations to each technician over the planning horizon within the workday. This two-step iterative process balances the maintenance cost, the failure probabilities, and waiting times at each customer. The novelty of this work lies in the integration of maintenance scheduling and a routing model that considers several machines.

Introduction

It is well known that thoughtful planning and scheduling of maintenance operations leads to significant improvements in the reliability of an industrial installation or a distribution network [16]. Maintenance planning determines the set of operations, time intervals, and resources (staff, supplies, and spare parts) necessary to conduct maintenance operations [6]. Companies often delegate maintenance planning to staff with experience, trusting in their intuition and knowledge. When they are faced with scheduling maintenance operations manually, even the most experienced planners can only consider a limited number of possibilities. Moreover, it is often the case that to manually generate a feasible schedule, they need to invest a significant amount of time.

When the machines are geographically distributed, the problem becomes even more complex because in addition to allocating maintenance operations to the workforce (e.g., crews) it is necessary to sequence their visits. The combinatorial optimization problem of finding the best set of routes (sequence of visits) for a workforce crew is known as the vehicle routing problem [21]. In a broad sense, this problem determines the best set of routes to be performed by a set of vehicles (crews) in order to serve a set of geographically-spread customers (machines) subject to some operational constraints. Among several variants of this problem, the vehicle routing problem with time windows (VRPTW) is closely related to planning and scheduling maintenance operations. In the VRPTW, the crews have a limited capacity (e.g., workday) to serve customers for whom maintenance operations must be started within given time windows.

Several applications for the combined maintenance and routing problem arise naturally in the oil and gas industry, telecoms, public utilities, health care, and the financial sector. For example, daily operations in an upstream oil and gas company involve managing a network of interconnected pumping stations. Because of the prohibitive costs of stopping the operation, the company incurs in an overly expensive maintenance policy that ensures the highest service levels at the intermediate stations. However, this policy translates into excessive technician visits to the stations, spread throughout a vast region. This type of company would be interested in other options that explore the tradeoff between service level and operational costs. Another application arises in maintenance operations of security hardware (e.g., video cameras) in a network of automatic teller machines (ATM). Most banks outsource these maintenance operations to a third party, who commits to perform periodical preventive maintenance within certain time windows. Upon occurrence of failures or accidents (e.g., vandalism), the technicians must go to the ATMs on a tighter time frame. The objective consists of reducing the operational costs, while maintaining a service level stipulated in the contract.

In this paper we consider a set of machines that are geographically distributed over a region. These machines are subject to unforeseen failures that result in downtime and hence loss of productivity. To reduce the occurrence of these unforeseen downtimes, a Preventive Maintenance (PM) operation is scheduled with a certain frequency. A set of technicians who are missioned to visit these machines to perform PM operations is also considered. When a machine fails suddenly, the maintenance crew has to perform a repair operation, called Corrective Maintenance (CM). Each machine starts in as good as new condition and after the repair or PM procedure the machine is assumed to be in as good as new condition again. Both operations, CM and PM, are assumed to take a certain but distinct amount of time to be performed, where the CM duration is higher than PM duration. Each PM operation is associated with a hard time-window dependent of the total expected cost, i.e., the time-window is calculated using a fixed amount of cost in which the optimal maintenance cost may increase to provide a flexibility for PM operation´s date. The purpose of the time-window is to generate a time interval within which a PM operation can occur, and in which the technician can start the PM operation. A technician must arrive before the upper limit of the time window. If the technician arrives before the lower limit of its time window, he or she must wait to start the PM operation at the beginning of the time window. Given a planning horizon, this problem consists of determining a joint best routing-maintenance policy for all technicians and machines. The problem aims to minimize the total expected maintenance cost (CM and PM costs as well as the cost of unavailability). The novelty of our approach lies in combining maintenance and routing models that determine when and how many PM operations are to be performed on the whole group of machines, and determine, for each technician the operations to perform and the sequence in which these PM operations must be carried out.

To solve this problem we propose a solution approach which integrates these two models iteratively, called the Combined Maintenance and Routing Model (CMR). Firstly, with the assumption of zero waiting times, the maintenance policy is optimized in order to determine the optimal time when each PM operation should be executed along with its corresponding time window, minimizing the total expected maintenance costs. Secondly, using the output from the maintenance model, the routing model is solved in order to determine the PM operations to be performed by each technician and the start time of service at each customer (or machine). After this, the expected waiting time is calculated for each customer and the maintenance model is solved again with new information as input in order to determine when each PM operation should be executed. The process continues until there is no improvement in the objective function for a fixed number of iterations or the iterative process reaches the maximum allowed run time.

The remainder of this paper is organized as follows. Section 2 reviews the literature related to maintenance and routing problems. Section 3 introduces the notation and the problem definition. Section 4 presents the maintenance model. Section 5 is dedicated to the mathematical formulation of the routing problem. Section 6 states the connection between the maintenance model and the routing model. Section 7 shows the benchmark procedure, a simple procedure to generate a feasible maintenance schedule in order to test the relevance of integrating the routing model within the maintenance policies. Section 8 provides some numerical experiments on random test instances where our CMR approach is compared to the benchmark procedure. Section 9 illustrates the performance of our proposed approach in an example inspired by a real-world case in the Oil and Gas industry. Finally, Section 10 concludes this work and provides possible research directions.

Section snippets

Literature review

A maintenance policy is the combination of inspections (for monitoring purposes), preventive and corrective operations intended to restore a machine to a state in which it can perform its required functions. Lugtigheid et al. [12], Crespo [4] and Wang and Pham [22] describe the inputs of a maintenance policy model as the failure model. Based on the available information, the failure model determines the probability that a machine fails during a certain period of time, the maintenance operation

Notation and problem definition

We consider a system with a set of customers VC={1,2,,N} geographically distributed over a region, where each customer has one machine subject to unforeseen failures. The machines are mutually independent regarding their failure behavior and hence it is possible to determine an optimal maintenance policy for each machine separately. To reduce the occurrence of these unforeseen failures, preventive maintenance (PM) operations are scheduled with a certain frequency for each machine. We consider

Maintenance Model (MM)

The notation of the MM for a single machine is summarized as follows:

  • T: planning horizon

  • T: random variable of the time to failure

  • f(T): probability density function (p.d.f.) of T

  • F(T): cumulative distribution function of T

  • δ: PM period at the beginning of the cycle

  • w: expected waiting time before the beginning of the CM operation

  • s: start time of a PM operation

  • M(δ): conditional expected time to failure when the PM period is δ knowing that the failure occurs before δ

  • TCM: mean time required to perform

Routing Model (RM) for PM operations scheduling

In this section the routing problem is formulated as a mixed-integer program to determine the set of routes to be performed by a set of technicians who execute all PM and CM operations over the planning horizon. First, we introduce the mathematical formulation of the routing problem as a distance-constrained vehicle routing problem with time windows with a nonlinear objective function and probabilistic constraints. Then, we present an approximation to a deterministic case and a piecewise linear

Connection between the MM and the RM

In Section 4 we present the maintenance model (MM) to determine the optimal time between PM operations and the number of PM operations to perform on each machine over the planning horizon. The routing model (RM) presented in Section 5 takes this information and modifies the original graph to determine the routes that each technician must follow to perform all the PM operations. In this section we present the proposed procedure that links the two models.

From the routing model, sjk is the start

Benchmark

In order to illustrate the relevance of integrating the routing model within the maintenance optimization problem, we developed another simple procedure referred as to the benchmark procedure, based on the intuition of maintenance planners. This procedure first computes the set of maintenance operations (VM) to be performed along the planning horizon by solving the MM model with zero waiting times. This first step provides the ideal date (ϕj) and cost per unit time (Cj(ϕj)) for each operation j=

Numerical results

In this section we report the computational results of the CMR model against those obtained with the benchmark procedure on randomly generated instances. All tests in this work were run using Java 7 and Xpress-MP 7.4 on a Windows 8 64-bit machine, with an Intel i5 3337 processor (2×1.8 GHz) and 6 GB of RAM. For the MM we use the Java Statistical Classes (JSC) library (available at http:// http://www.jsc.nildram.co.uk).

Illustrative case study in the oil and gas industry

We illustrate the performance of our proposed approach with an example based on a real-world scenario presented by Ortiz et al. [13], where daily operations in an upstream oil and gas company involve managing a network of interconnected pumping stations. These pumping stations are geographically distributed over a region and are subject to unforeseen failures that result in downtime. The failures can be of several causes, for example damage to electrical components, mechanics devices, or other

Conclusions and future work

This work presents a combined maintenance and routing problem for a set of geographically distributed machines subject to non-deterministic failures. The contributions of this paper are twofold: from the routing point of view, the problem exhibits some distinctive aspects like the presence of a nonlinear cost function, the need to synchronize resources (technicians), and the presence of probabilistic time-window constraints due to the random arrival times of the technicians who perform

Acknowledgments

We thank Fair Isaac Corporation (FICO) for providing us with Xpress-MP licenses under the Academic Partner Program subscribed with Universidad de los Andes and Universidad Distrital Francisco Jose de Caldas (Colombia). The first author would like to thank the Universidad Distrital Francisco Jose de Caldas for their assistance in providing a research scholarship for his M.Sc. thesis. Last, but not least, the authors would like to thank the comments of the anonymous referees that significantly

References (23)

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