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Showing papers in "Journal of the Operational Research Society in 1999"


Journal ArticleDOI
TL;DR: The application of discrete-event simulation modeling to health care clinics and systems of clinics (for example, hospitals, outpatient clinics, emergency departments, and pharmacies) and future directions of research and applications are discussed.
Abstract: In recent decades, health care costs have dramatically increased, while health care organisations have been under severe pressure to provide improved quality health care for their patients. Several health care administrators have used discrete-event simulation as an effective tool for allocating scarce resources to improve patient flow, while minimising health care delivery costs and increasing patient satisfaction. The rapid growth in simulation software technology has created numerous new application opportunities, including more sophisticated implementations, as well as combining optimisation and simulation for complex integrated facilities. This paper surveys the application of discrete-event simulation modeling to health care clinics and systems of clinics (for example, hospitals, outpatient clinics, emergency departments, and pharmacies). Future directions of research and applications are also discussed.

891 citations


Journal ArticleDOI
TL;DR: In this article, a hybrid ant colony system coupled with a local search is applied to the quadratic assignment problem, which uses pheromone trail information to perform modifications on QAP solutions.
Abstract: This paper presents HAS–QAP, a hybrid ant colony system coupled with a local search, applied to the quadratic assignment problem. HAS–QAP uses pheromone trail information to perform modifications on QAP solutions, unlike more traditional ant systems that use pheromone trail information to construct complete solutions. HAS–QAP is analysed and compared with some of the best heuristics available for the QAP: two versions of tabu search, namely, robust and reactive tabu search, hybrid genetic algorithm, and a simulated annealing method. Experimental results show that HAS–QAP and the hybrid genetic algorithm perform best on real world, irregular and structured problems due to their ability to find the structure of good solutions, while HAS–QAP performance is less competitive on random, regular and unstructured problems.

710 citations


Journal ArticleDOI
TL;DR: A 0–1 mixed integer programming model is presented that simultaneously solves for the location of remanufacturing/distribution facilities, the transshipment, production, and stocking of the optimal quantities of re manufactured products and cores.
Abstract: Recoverable product environments are becoming an increasingly important segment of the overall push in industry towards environmentally conscious manufacturing. Integral to the recoverable product environment is the recoverable manufacturing system that focuses on recovering the product and extending its life through remanufacture or repair. Remanufacturing provides the customer with an opportunity to acquire a product that meets the original product standards at a lower price than a new product. The flow of materials and products in this environment occurs both from the customer to the remanufacturer (reverse flow), and from the remanufacturer to the customer (forward flow). Since most of the products and materials may be conserved, essentially this forms a closed-loop logistics system. We present a 0–1 mixed integer programming model that simultaneously solves for the location of remanufacturing/distribution facilities, the transshipment, production, and stocking of the optimal quantities of remanufactured products and cores. We also discuss the managerial uses of the model for logistics decision-making.

544 citations


Journal ArticleDOI
TL;DR: This paper reviews the rapidly growing literature on single machine scheduling models with time dependent processing times and attention is focused on linear, piecewise linear and non-linear processing time functions for jobs.
Abstract: In classical scheduling theory job processing times are constant However, there are many situations where processing time of a job depends on the starting time of the job in the queue This paper reviews the rapidly growing literature on single machine scheduling models with time dependent processing times Attention is focused on linear, piecewise linear and non-linear processing time functions for jobs We survey known results and introduce new solvable cases Finally, we identify the areas and give directions where further research is needed

471 citations


Journal ArticleDOI
TL;DR: This work investigates an extension to the classical insertion-based heuristic for the vehicle routing problem with backhauling (VRPB) based on the idea of inserting more than one backhaul at a time, with encouraging results.
Abstract: We investigate an extension to the classical insertion-based heuristic for the vehicle routing problem with backhauling (VRPB). It is based on the idea of inserting more than one backhaul at a time. This method is tested on data sets with single and multiple depots with encouraging results at no additional computational burden. This approach can also be useful in generating good starting solutions for the more computer-intensive meta-heuristics.

304 citations


Journal ArticleDOI
TL;DR: This paper focuses on the effect of the backlogging rate on the economic order quantity decision and Numerical examples are presented to illustrate the model.
Abstract: In the classical economic order quantity model, it is often assumed that the shortages are either completely backlogged or completely lost. However, in some inventory systems, it is more reasonable to assume that the backlogging rate is dependent on the length of the waiting time for the next replenishment. The longer the waiting time is, the smaller the backlogging rate would be. In this paper, we focus on the effect of the backlogging rate on the economic order quantity decision. Numerical examples are presented to illustrate the model.

299 citations


Journal ArticleDOI
TL;DR: This book is an important reference volume and an invaluable source of inspiration for advanced students and researchers in discrete mathematics, computer science, operations research, industrial engineering and management science.

266 citations


Journal ArticleDOI
TL;DR: It is suggested that within studies the true power of system dynamics to address problem solving lies in a judicious blend and intertwining of both qualitative and quantitative ideas, aimed at addressing as broad an audience as possible whilst remaining sufficiently rigorous to be useful.
Abstract: This paper addresses the issue of what are the wise uses of qualitative mapping and what are the conditions that require formal quantitative modelling within System Dynamics. The background to the evolution of qualitative and quantitative system dynamics is explored. This analysis recognises that, although the history of feedback thought repeatedly contains the assertion that formal, quantitative models are essential for understanding the dynamics of complex systems, the need for quantification is relative and depends on the purpose of analysis, which, in turn, is related to the methods used and the audience addressed. The central theme of the paper is to examine the strengths and weaknesses of qualitative and quantitative system dynamics and to relate these to their respective tool sets. The paper also focuses on evidence from the author’s extensive recent use of qualitative and quantitative system dynamics in education, training, research and consultancy studies of the way in which qualitative and quantitative system dynamics can be linked together to consolidate management learning, both in projects and in organisations. The paper concludes that both qualitative and quantitative system dynamics are important to management problem solving and related to the purpose of analysis. It is suggested that within studies the true power of system dynamics to address problem solving lies in a judicious blend and intertwining of both qualitative and quantitative ideas, aimed at addressing as broad an audience as possible whilst remaining sufficiently rigorous to be useful. A process referred to as intertwined project learning (IPL) is outlined for this purpose. Within organisations it is suggested that there is a need to link the use of qualitative system dynamics in management development programmes and quantitative system dynamics modelling projects into an integrated organisational learning activity. A process referred to as accelerated business learning (ABL) is outlined for this purpose.

265 citations


Journal ArticleDOI
TL;DR: A new approach based on the use of Saaty's Analytic Hierarchy process method is illustrated, which provides a more systematic way of deriving the weights to be used and for scoring the performance of vendors.
Abstract: With collaborative purchasing programmes where one of the aims is to develop suppliers, vendor rating is important not only in supplier selection and in deciding how to allocate business but also to determine where scarce development effort is best applied. This paper describes a case study into vendor rating for a government sponsored Entrepreneur Development programme in Malaysia. The paper reviews current methods for vendor rating and finds them wanting. It illustrates a new approach based on the use of Saaty's Analytic Hierarchy process method, which was developed to assist in multi-criteria decision problems. The new method overcomes the difficulties associated with the categorical and simple linear weighted average criteria ranking methods. It provides a more systematic way of deriving the weights to be used and for scoring the performance of vendors.

248 citations


Journal ArticleDOI
TL;DR: Three heuristic algorithms and a branch and bound algorithm are proposed to minimise total completion time of jobs and the problem is proved to be NP-hard in the strong sense.
Abstract: This paper considers a single machine scheduling problem with preventive maintenance. In many cases, a machine must be maintained after it continuously works for a period of time. But most papers in the literature ignore non-availability of the machine. For this reason, this paper studies the problem of scheduling processing of jobs and maintenance of machine simultaneously. The objective is to minimise total completion time of jobs. The problem is proved to be NP-hard in the strong sense. Three heuristic algorithms and a branch and bound algorithm are proposed. Computational experiments are done to evaluate the effectiveness of the algorithms.

216 citations


Journal ArticleDOI
TL;DR: This paper seeks to contribute to the orchestration of coherent pluralism in management science by outlining the form that pluralism needs to take if it is to be both theoretically defensible and provide the greatest benefit to practitioners.
Abstract: This paper seeks to contribute to the orchestration of coherent pluralism (i.e. the use of different methodologies in combination) in management science. Its starting point is the eagerness of the applied disciplines to embrace pluralism and some thoughts on why this should be the case. This is followed by a summary of two contributions which have sought to define the nature of pluralism in management science, as well as to nurture its development. A brief history of the establishment of some important landmarks in pluralist thinking in management science is then provided. The paper goes on to try to build on these sections by outlining the form that pluralism needs to take if it is to be both theoretically defensible and provide the greatest benefit to practitioners. Finally, a hopefully coherent version of pluralism, which seems to meet these criteria, is offered. A conclusion suggests an agenda for future research in the area.

Journal ArticleDOI
TL;DR: This study looks at the question when will borrowers default not if they will default and suggests that some of the reliability modelling approaches may be useful in this context and may help identify who will default as well as when they may default.
Abstract: Credit scoring systems are based on Operational Research and statistical models which seek to identify who of previous borrowers did or did not default on loans. This study looks at the question when will borrowers default not if they will default. It suggests that some of the reliability modelling approaches may be useful in this context and may help identify who will default as well as when they may default.

Journal ArticleDOI
TL;DR: This paper describes several insertion-based savings heuristics for the fleet size and mix vehicle routing problem with time window constraints and found that heuristic with the consideration of a sequential route construction parameter yielded very good results.
Abstract: This paper describes several insertion-based savings heuristics for the fleet size and mix vehicle routing problem with time window constraints. A certain number of candidate fleet compositions are recorded in the construction phase, followed by applying a composite improvement scheme on them to enhance the solution quality. Computational results on 168 sample problems are reported. We found that heuristics with the consideration of a sequential route construction parameter yielded very good results. In addition, results on the 20 benchmarking problems for the fleet and mix vehicle routing problem with no time window constraints also demonstrate the effectiveness of our heuristics.

Journal ArticleDOI
TL;DR: The proposed zero-one integer linear programming model handles some of the issues that frequently arise in real world applications but are not addressed by previously suggested models, such as situations in which the amount of available and consumed resources varies in different periods.
Abstract: A zero-one integer linear programming model is proposed for selecting and scheduling an optimal project portfolio, based on the organisation's objectives and constraints such as resource limitations and interdependence among projects. The model handles some of the issues that frequently arise in real world applications but are not addressed by previously suggested models, such as situations in which the amount of available and consumed resources varies in different periods. It also allows for interactive adjustment following the optimisation process, to provide decision makers a method for controlling portfolio selection, based on criteria that may be difficult to elicit directly. It is critical for such a system to provide fast evaluation of alternatives the decision makers may want to examine, and this requirement is addressed. The proposed model not only suggests projects that should be incorporated in the optimal portfolio, but it also determines the starting period for each project. Scheduling considerations can have a major impact on the combination of projects that can be incorporated in the portfolio, and may allow the addition of certain projects to the portfolio that could not have been selected otherwise. An example problem is described and solved with the proposed model, and some areas for future research are discussed.

Journal ArticleDOI
TL;DR: The cumulative knowledge and experience of delay time modelling is reviewed, and case sketches present practical experience in using the DT concept to model actual plant, to assess the benefits obtained, and to validate modelling and parameter assessment.
Abstract: Delay time (DT) analysis is a pragmatic mathematical concept readily embraced by engineers which has been developed as a means to model maintenance decision problems. Attention is focused upon the maintenance engineering decisions of what to do, as opposed to the logistical decisions of how to do it. This paper reviews the cumulative knowledge and experience of delay time modelling. The decision environment within which delay time (DT) models are intended as decision aids is briefly reviewed, and the initial development of simple DT models for a repairable component and a complex plant presented. Variations on the basic model are outlined and discussed including perfect and non-perfect inspection, steady state and non-steady state conditions, and homogeneous and non-homogeneous Poisson arrival rate of defects. Attention is given to the parameter estimation process, and both subjective and objective estimation techniques are outlined. Case sketches present practical experience in using the DT concept to model actual plant, to assess the benefits obtained, and to validate modelling and parameter assessment.

Journal ArticleDOI
TL;DR: This paper suggests four alternative purposes of DEA modelling, and offers four measures of the quality of a DEA model which reflect those purposes, and explores the performance of DEA under a wide variety of assumptions.
Abstract: The user of data envelopment analysis (DEA) has little available guidance on model quality. The technique offers none of the misspecification tests or goodness of fit statistics developed for parametric statistical methods. Yet, if a DEA model is to guide managerial policy, the quality of the model is of crucial importance. This paper suggests four alternative purposes of DEA modelling, and offers four measures of the quality of a DEA model which reflect those purposes. Using Monte Carlo simulation methods, it explores the performance of DEA under a wide variety of assumptions. It notes that four issues will have an important influence on model results: the distribution of true efficiencies in the study sample; the size of the sample; the number of inputs and outputs included in the analysis; and the degree of correlation between inputs and outputs. The paper concludes that any judgement about the reliability of model results must be dependent on the objective of the analysis.

Journal ArticleDOI
TL;DR: The analysis shows how DEA can complement the profitability measure currently used at the bank and achieve the branches' most productive scale size through the elimination of scale inefficiencies, with minimal changes to branches' scale size.
Abstract: This paper describes an application of data envelopment analysis (DEA) to the performance assessment of Portuguese bank branches. The analysis shows how DEA can complement the profitability measure currently used at the bank. The use of an efficiency-profitability matrix enabled the characterisation of the branches' performance profile. Consistent with the bank's development objectives, the analysis focused on the relation between branch size and performance. Two alternative target setting strategies were explored. One eliminates pure technical inefficiencies by focusing on the selection of appropriate benchmarks. The other attains the branches' most productive scale size through the elimination of scale inefficiencies, with minimal changes to branches' scale size.

Journal ArticleDOI
TL;DR: Proposals to use Data Envelopment Analysis as a tool for Multiple Criteria Decision Making violate simple normative properties that are commonly accepted, according to a simple model.
Abstract: The purpose of this paper is to study proposals to use Data Envelopment Analysis (DEA) as a tool for Multiple Criteria Decision Making (MCDM). We first recall, using a simple model, the equivalence between the concept of ‘efficiency’ in DEA and that of ‘convex efficiency’ in MCDM. Examples are then used to show that various techniques that have been proposed in the DEA literature to deal with MCDM problems violate simple normative properties that are commonly accepted. We conclude with some remarks on the possible areas of interaction between DEA and MCDM.

Journal ArticleDOI
TL;DR: A review is made of some system dynamics models which address health care issues and suggestions are made for the types of role which these models should take, bearing in mind the strategic orientation of system dynamics modelling.
Abstract: Taking a European perspective, a review is made of some system dynamics models which address health care issues. Suggestions are made for the types of role which these models should take, bearing in mind the strategic orientation of system dynamics modelling. Examples are described of qualitative models where influence diagrams are the main analytical tool. Quantitative system dynamics models have a contribution to make in epidemiological studies and have been used to analyse the AIDS epidemic. A detailed example of one aspect of model formulation is given. This concerns the AIDS incubation time distribution and shows how real-world complications arising from virological staging and treatment effects are handled in a model of AIDS spread.

Journal ArticleDOI
TL;DR: In this paper, a case intervention, undertaken at a National Health Service hospital outpatients department, utilised a soft systems approach, to address gaps between the customers' and providers' expectations, and simulation was used within this framework.
Abstract: This paper is an account of a case intervention, undertaken at a National Health Service hospital outpatients department. The intervention utilised a soft systems approach, to address gaps between the customers' and providers' expectations, and simulation was used within this framework. A procedure to reduce unexpected non–attendance of patients has been implemented. Patient bookings are now scheduled according to simple rules, with the result that in–clinic waiting times have been reduced. A model of the intervention process has been accepted by the key stakeholders as a sensible framework for continuing investigations. More widely, the participants increased their knowledge of their own systems. The results from the case suggest that the time spent following this approach may be worthwhile in some interventions, but there are concerns relating to the time needed, and the cost of such an approach. The paper provides a critique of the framework and the intervention, and concludes that this approach is probably viable in unpaid, collaborative, investigations where there is little or no cost. However, time and cost may prove prohibitive in the case of paid consultancy.

Journal ArticleDOI
TL;DR: This paper is motivated by the experiences since 1990 with developing system simulation models to help UK companies in the restructured electricity industry understand the radically different market within which they must become competitive.
Abstract: This paper is motivated by our experiences since 1990 with developing system simulation models to help UK companies in the restructured electricity industry understand the radically different market within which they must become competitive. When public utilities such as electricity have been restructured, deregulated and/or privatised, the process has often been associated with a major change in the competitive environment. As a consequence, the strategic and regulatory uncertainties ahead for these companies are unprecedented. In such a market there has been no historical evolution and all the participants including the regulatory institutions have very little understanding of how it will operate in the short term and evolve in the future. In this situation, the use of system dynamic models appears to offer an attractive way of gaining insights into how aspects of the competitive market might evolve. In the absence of real experience and relevant analogies, learning from models assumes a key role. Such models cannot be validated empirically, but can be developed to represent how the system is designed to operate. From such a prototypical basis, sensitivity analysis can generate insights on the strategic opportunities created by failings in the market design, or its potential instability to shocks and market imperfections.

Journal ArticleDOI
TL;DR: Four preference disaggregation methods, namely the UTADIS method and three of its variants, are compared to three well-known multivariate statistical and econometric techniques, namely discriminant analysis, logit and probit analyses to explore the applicability of multicriteria analysis to predict business failure.
Abstract: Business failure prediction is one of the most essential problems in the field of financial management. The research on developing quantitative business failure prediction models has been focused on building discriminant models to distinguish among failed and non-failed firms. Several researchers in this field have proposed multivariate statistical discrimination techniques. This paper explores the applicability of multicriteria analysis to predict business failure. Four preference disaggregation methods, namely the UTADIS method and three of its variants, are compared to three well-known multivariate statistical and econometric techniques, namely discriminant analysis, logit and probit analyses. A basic (learning) sample and a holdout (testing) sample are used to perform the comparison. Through this comparison, the relative performance of all the aforementioned methods is investigated regarding their discriminating and predicting ability.

Journal ArticleDOI
TL;DR: A computerised system based on heuristic algorithms was developed for vehicle routing with random demand which represent breakdowns that require a repair crew and an exponential smoothing method was implemented for the prognosis of breakdowns.
Abstract: Chilean companies have only recently begun developing and implementing computerised systems to aid in decision making, related to the assignment and routing of vehicles. This paper presents the problem of assigning and routing repair vehicles for the Emergency Services Division of Chilectra S.A., the electricity utility for the city of Santiago, the capital of Chile. A computerised system based on heuristic algorithms was developed for vehicle routing with random demand which represent breakdowns that require a repair crew. An exponential smoothing method was implemented for the prognosis of breakdowns. Evaluation of the system's performance showed a 16% improvement in service quality as measured by the time required for servicing breakdowns under regular conditions and a 53% improvement under adverse climate conditions.

Journal ArticleDOI
TL;DR: A goal programming (GP) model which aids in allocating a health-care system's information resources pertinent to strategic planning and facilitates decision-making planning process and managerial policy in health- care information resources planning and similar planning settings is presented.
Abstract: This paper presents a goal programming (GP) model which aids in allocating a health-care system's information resources pertinent to strategic planning. The model is developed based on the data obtained from a major health-care system in the United States. The overall objective is to design and evaluate a model for effective information resource planning in a health-care system. The proposed model: (1) utilizes a GP approach to reflect the multiple, conflicting goals of the health-care system; (2) employs a GP solution process to reflect multi-dimensional aspects of the resource allocation planning; and (3) allows for some degree of flexibility of decision-making with respect to resource allocation. The goals are decomposed and prioritized with respect to the corresponding criteria using the analytic hierarchy process (AHP). The model result is derived and discussed. This GP model facilitates decision-making planning process and managerial policy in health-care information resources planning and similar planning settings.

Journal ArticleDOI
TL;DR: A tabu search algorithm is presented that systematically explores feasible machine cells configurations determining the corresponding part families using a linear network flow model.
Abstract: As part of the cellular manufacturing design process, machines must be grouped in cells and the corresponding part families must be assigned. Limits on both the number of machines per cell and the number of parts per family can be considered. A weighted sum of intracell voids and intercellular moves is used to evaluate the quality of the solutions. We present a tabu search algorithm that systematically explores feasible machine cells configurations determining the corresponding part families using a linear network flow model. The performance of this tabu search is benchmarked against two simulated annealing approaches, another tabu search approach and three heuristics: (ZODIAC, GRAFICS and MST).

Journal ArticleDOI
TL;DR: The objective is to simultaneously optimise the order quantity, ordering cost, reorder point and lead time in modified continuous review inventory systems involving variable lead time with a mixture of backorders and lost sales.
Abstract: This paper investigates the impact of ordering cost reduction on the modified continuous review inventory systems involving variable lead time with a mixture of backorders and lost sales. The objective is to simultaneously optimise the order quantity, ordering cost, reorder point and lead time. We first assume the lead time demand follows a normal distribution, then relax this assumption to consider the distribution free case where only the mean and variance of lead time demands are known. An algorithm procedure of finding the optimal solution is developed, and two numerical examples are given to illustrate the results.

Journal ArticleDOI
TL;DR: A comparison of the performance of a simple moving average with an exponentially weighted moving average (EWMA) is made and it is shown that, for a steady state model, the variance of the forecast error is typically less than 3% higher than the appropriate EWMA.
Abstract: Simple (equally weighted) moving averages are frequently used to estimate the current level of a time series, with this value being projected as a forecast for future observations. A key measure of the effectiveness of the method is the sampling error of the estimator, which this paper defines in terms of characteristics of the data. This enables the optimal length of the average for any steady state model to be established and the lead time forecast error derived. A comparison of the performance of a simple moving average (SMA) with an exponentially weighted moving average (EWMA) is made. It is shown that, for a steady state model, the variance of the forecast error is typically less than 3% higher than the appropriate EWMA. This relatively small difference may explain the inconclusive results from the empirical studies about the relative predictive performance of the two methods.

Journal ArticleDOI
TL;DR: A linear programming based heuristic is considered for a two-stage capacitated facility location problem with single source constraints to find the optimal locations of depots from a set of possible depot sites.
Abstract: In this paper, a linear programming based heuristic is considered for a two-stage capacitated facility location problem with single source constraints. The problem is to find the optimal locations of depots from a set of possible depot sites in order to serve customers with a given demand, the optimal assignments of customers to depots and the optimal product flow from plants to depots. Good lower and upper bounds can be obtained for this problem in short computation times by adopting a linear programming approach. To this end, the LP formulation is iteratively refined using valid inequalities and facets which have been described in the literature for various relaxations of the problem. After each reoptimisation step, that is the recalculation of the LP solution after the addition of valid inequalities, feasible solutions are obtained from the current LP solution by applying simple heuristics. The results of extensive computational experiments are given.

Journal ArticleDOI
TL;DR: This paper presents a matrix geometric method for deriving the steady-state probabilities, using which various system performance measures that can be obtained are obtained based on assumed numerical values given to the system parameters.
Abstract: We consider the machine repair problem in which failed machines balk (do not enter) with a constant probability (1 – b) and renege (leave the queue after entering) according to a negative exponential distribution. A group of identical automatic machines are maintained by R servers which themselves are subject to breakdowns. Failure and service times of the machines, and breakdown and repair times of the servers, are assumed to follow a negative exponential distribution. Each server is subject to breakdown even if no failed machines are in the system. This paper presents a matrix geometric method for deriving the steady-state probabilities, using which various system performance measures that can be obtained. A cost model is developed to determine the optimum number of servers. The minimum expected cost, the optimal number of servers, and various system performance measures are provided based on assumed numerical values given to the system parameters. Also the sensitivity analysis is investigated.

Journal ArticleDOI
A Bolat1
TL;DR: An interactive optimum and heuristic procedures are proposed to cope with the major changes in disrupting the initial gate-assignments, and outperforms the existing practice on the number of remote served aircraft and towed aircraft.
Abstract: Unexpected changes in the flight schedules may disrupt the initial aircraft-gate assignments, and result in congestions and delays in getting aircraft onto gates. A mathematical model is developed to assign the flights with the minimum range of unutilised time periods of gates, subject to the level of service offered to passengers and other physical and managerial considerations. (The assignments are expected to be flexible enough to absorb the minor modifications in the flight schedules.) Interactive optimum and heuristic procedures, both utilising lower bounds on the ranges of future solutions, are proposed to cope with the major changes in disrupting the initial gate-assignments. Over randomly generated schedules, 74 flights can be optimally assigned to seven gates within 17 seconds when the gates are re-utilised within 30 minutes after each departure. The heuristic reaches the optimal solution after evaluating at most 20 partial solutions at one level. Over data obtained from Riyadh’s International Airport, the heuristic outperforms the existing practice: On average, 72.03% and 54.28% improvements are obtained on the number of remote served aircraft and towed aircraft, respectively.