a sensitivity

International Journal of Quality & Reliability ManagementEmerald Article: A sensitivity analysis of an optimal Gantt charting maintenance scheduling modelS.A. Oke, O.E. Charles-Owaba

Article information:

To cite this document: S.A. Oke, O.E. Charles-Owaba, (2006),"A sensitivity analysis of an optimal Gantt charting maintenance scheduling model", International Journal of Quality & Reliability Management, Vol. 23 Iss: 2 pp. 197 - 229

Permanent link to this document: http://dx.doi.org/10.1108/02656710610640952

Downloaded on: 19-07-2012

References: This document contains references to 63 other documents

Citations: This document has been cited by 1 other documents

To copy this document: [email protected]

This document has been downloaded 2481 times since 2006. *

Users who downloaded this Article also downloaded: *

James DeLisle, Terry Grissom, (2011),"Valuation procedure and cycles: an emphasis on down markets", Journal of Property Investment & Finance, Vol. 29 Iss: 4 pp. 384 - 427http://dx.doi.org/10.1108/14635781111150312

Shana Wagger, Randi Park, Denise Ann Dowding Bedford, (2010),"Lessons learned in content architecture harmonization and metadata models", Aslib Proceedings, Vol. 62 Iss: 4 pp. 387 - 405http://dx.doi.org/10.1108/00012531011074645

Sandrine Roginsky, Sally Shortall, (2009),"Civil society as a contested field of meanings", International Journal of Sociology and Social Policy, Vol. 29 Iss: 9 pp. 473 - 487http://dx.doi.org/10.1108/01443330910986261

Access to this document was granted through an Emerald subscription provided by OXFORD BROOKES UNIVERSITY For Authors: If you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors service. Information about how to choose which publication to write for and submission guidelines are available for all. Please visit www.emeraldinsight.com/authors for more information.

About Emerald www.emeraldinsight.comWith over forty years' experience, Emerald Group Publishing is a leading independent publisher of global research with impact in business, society, public policy and education. In total, Emerald publishes over 275 journals and more than 130 book series, as well as an extensive range of online products and services. Emerald is both COUNTER 3 and TRANSFER compliant. The organization is a partner of the Committee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for digital archive preservation.

*Related content and download information correct at time of download.

A sensitivity analysis of anoptimal Gantt charting

maintenance scheduling modelS.A. Oke

Department of Mechanical Engineering, University of Lagos,Lagos, Nigeria, and

O.E. Charles-OwabaDepartment of Industrial and Production Engineering, University of Ibadan,

Ibadan, Nigeria

Abstract

Purpose – The purpose of this paper is to work on an analytical approach to test sensitivity of amaintenance-scheduling model. Any model without sensitivity analysis is a “paper work” withoutadvancing for wider applications. Thus, the simulation of simultaneous scheduling of maintenanceand operation in a resource-constrained environment is very important in quality problem andespecially in maintenance.

Design/methodology/approach – The paper uses an existing model and presents a sensitivityanalysis by utilising an optimal initial starting transportation tableau. This is used as input into theGantt charting model employed in the traditional production scheduling system. The degree ofresponsiveness of the model parameters is tested.

Findings – The paper concludes that some of these parameters and variables are sensitive tochanges in values while others are not.

Research limitations/implications – The maintenance engineering community is exposed tovarious optimal models in the resource-constraint-based operational and maintenance arena. However,the models do lack the sensitivity analysis where the present authors have worked. The work seemssignificant since the parameters have the boundary values so the user knows where he can apply themodel after considering the constraints therein.

Originality/value – The underlying quest for testing the sensitivity of the model parameters of amaintenance scheduling model in a multi-variable operation and maintenance environment withresource constraints is a novel approach. An optimal solution has to be tested for robustness,considering the complexity of the variables and criteria. The objective to test the model parameters is arather new approach in maintenance engineering discipline. The work hopefully opens a wide gate ofresearch opportunity for members of the maintenance scheduling community.

Keywords Maintenance, Production scheduling, Sensitivity analysis, Cost analysis

Paper type Research paper

1. IntroductionFor decades, the intellectual activities in the maintenance scheduling community havebeen successful (Enscore and Burns, 1983; Fwa et al., 1999, Garver, 1972, Haghani andShafahi, 2002; Hall, 2000; Higgins et al., 1999; Marwali and Shahidehpour, 1998a, b;

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/0265-671X.htm

The authors would like to express their gratitude to both referees for their valuable commentsand constructive criticism.

Analysis ofoptimal Gantt

charting

197

Received October 2005Revised April 2005

Accepted April 2005

International Journal of Quality &Reliability Management

Vol. 23 No. 2, 2006pp. 197-229

q Emerald Group Publishing Limited0265-671X

DOI 10.1108/02656710610640952

Stremel and Jenkins, 1981; Walker et al., 2001). The central focus has been on thedevelopment of schedules with heuristic approaches since optimal schedules proveddifficult due the non-polynomial (NP) hard inherent problems associated with it(Atkinson et al., 2003; Billinton and Pan, 1998; Charest and Ferland, 1993;Chattopadhyay et al., 1995; El-Sheikhi and Billinton, 1984). A number ofpre-determined criteria are used to develop frameworks within which the practisingmaintenance engineering or research scholar could work (Burke et al., 1998; Burke andSmith, 1999a, b). A breakthrough in research resulted in the development of optimalsolutions for single machine systems (El-Sheikhi and Billinton, 1984; Lund, 1990).Variants of this optimal solutions have however, been introduced (Christiaanse, 1973;Dapazo and Merrill, 1975; Yamayee et al., 1998; Zurn and Quintana, 1977). Some ofthese are single machine job sequencing with precedence constraint, single machinescheduling to minimise weighted sum of completion time with secondary criterion,single machine sequencing with ordered constraint, etc. (Chen and Toyoda, 1990;Yellen et al., 1992). With increased technological sophistication and internationalcompetitiveness maintenance professionals are compelled to revisit the state-of-the-artof maintenance scheduling at that time (Oke, 2004a; Yamayee, 1982).

Thus, researchers were challenged to expand the theoretical frameworks for singlemachine systems to two and three machines system due to increased complexity ofsystems. As development in technology increased, researchers and practitioners soonrealised that the scheduling problem is more complex than that of scheduling a fewnumber of machines. In modern industrial and service systems, the scheduling ofseveral tenths and hundredths of machines or facilities may be the challenge (Contaxiset al., 1989; Coudert et al., 2000; Dahal and McDonald, 1997; Duffuaa and Al-sultan,1997; Egan et al., 1976; Joshi and Gupta, 1996; Lake and Ferreira, 2001; Marwali andShahidehpour, 1998a, b). Examples of such cases exist in fleet of aircraft, fleet ofvehicles, vessels of ships, and a number of machines. Complication is introduced incases where non-identical facilities are to be scheduled. Research documentationreveals success in the maintenance scheduling of such multiple machines or facilitiesunder various conditions. Successful efforts have also been made in the scheduling ofsuch multiple facilities for operations (Christiannse and Palmer, 1991; Hariga, 1994;Khatib, 1979; Kurban, 1999; Stremel, 1981).

Unfortunately, only of late is there a successful effort that considers thesimultaneous scheduling of resource-constrained operations and maintenance. Thecredit for the development of such a novel work is due to Charles-Owaba (2002). Theintegration of operations scheduling models with maintenance scheduling models isimportant in view of the fact that the two functions – operations and maintenance, areviewed as important to the profit generation of the organisation. Traditionally, theoperations department is viewed as the profit yielding function for the organisationwhile the maintenance department is currently viewed as the value adding functioninstead of the former viewpoint of being a “bottomless pit of expenses”.

While it has been established that optimal solutions could be formulated for thesimultaneous scheduling of resource-constrained maintenance and operations, it seemsthat no documentation exist on how to test the robustness of the model or its sensitivityanalysis. The issue of sensitivity analysis is of prime importance to us in this work,and is thereby addressed. The specific problem whose formulated solution is tested forsensitivity is defined as follows:

IJQRM23,2

198

Given a set of M machines for preventive maintenance and operations in T contiguousperiods, limited periodic maintenance capacity (Aj), duration (Bj) per maintenance visit,arrival periods (ki) and number of visits per machine (Ni) select the periods for alternatepreventive maintenance and operations such that the total preventive maintenance cost isminimum.

A theoretical framework for the model is built and described with the assumptions thata cycle of activities occur in maintenance – operation order, the total number ofmaintenance-operation period (T) are fixed and contiguous, and an arriving machinehas its maintenance activities commenced only when resources are available. Thecriterion is the minimisation of preventive maintenance cost subject to the constraintthat maintenance capacity is limited (see Figure 1) (Bar-Noy et al., 1998; Chattopadhyayet al., 1995; Lake et al., 2000; Silver and Murphy, 1994). Figure 1 shows the conceptualmodel of the problem whose model is tested for sensitivity.

The fundamental question of the model sensitivity to small variations in parametersand variable is addressed. The sensitivity analysis identifies the model aspects mostsusceptible to uncertainties in the model development by varying each parameter bysome accounts and run the model in each case to see which parameters have thegreatest impact on model performance.

The introductory part of the paper is concluded by noting that in recent years,there has been a gradual change of attitude of company executives and otherpractitioners on the role of maintenance in adding value to the organisation’sproducts and services. The viewpoint towards the maintenance function haschanged from “a bottomless pit of expenses” to the life wire of the organisation. Asa result of this, conscious efforts are made by maintenance practitioners towardsimproving their efficiency and hence optimising the organisational performance. Inan effort to achieve these objectives, several techniques have been implemented inpractice. One of these widely accepted techniques is the use of maintenancescheduling models and tools (Khan and Hadara, 2003; Kin et al., 1997; Kobbacy et al.,1997; Satoh and Nara, 1991; Sriskandarajah et al., 1998; Wang et al., 2002; Zurn andQuintana, 1975). At the early stage of research, a number of scholars were perhapsfrustrated with the non-polynomial (NP) hard problems that are seemingly difficultto solve. Fortunately there seems to be a recent breakthrough in the formulation andsolution of maintenance scheduling models. One of the models proposed, which istreated in this work is christened Charles-Owaba’s Gantt charting (OGC) model. Asa further step toward improvement of the model there is currently a great need foran article that investigates into the sensitivity of some of the proposed models in

Figure 1.Structural model of

Charles-Owaba’s OGCmodel


charting

199

this domain. This aim is pursued in the current paper. In particular, the degree ofresponsiveness of the model’s variables and parameters to changes in values areinvestigated.

2. Related literatureThe literature on sensitivity analysis is wide and all encompassing, covering severalscientific fields. Even in the engineering fields, extensive work has been done on thesubject. Unfortunately, there seems no documentation on cross-studies betweensensitivity analysis and maintenance scheduling, which is the main focus of thepresent work. As such, the present paper appears to be a front-line study in this area.The review of the literature is approached by carrying out an extensive survey on thework done so far on maintenance scheduling (Mosley et al., 1998; Nguyen and Murthy,1981; Percy et al., 1997, 1998; Ram and Olumolade, 1987; Sarker and Yu, 1995). The aimis to identify important studies, relate them to the current, and conclude on the need forthe present work. Thus, this work is for the benefit of maintenance professionals andscholars in the area of maintenance scheduling.

For this purpose, the work carried out by Oke (2004a) may be a helpful reference toauthenticate our claim that there is a great need for an article presenting a scientificviewpoint of sensitivity analysis as it relates to maintenance scheduling. Oke (2004a)reviewed a variety of applications in maintenance scheduling, notably, aircraftmaintenance, process industry, pavement maintenance, highway maintenance,refinery, and production facilities.

On aircraft, Alfares (1999) investigated the scheduling of an aircraft maintenanceworkforce. The paper describes an aircraft maintenance labour scheduling study. Thestudy’s objectives is to determine the optimum maintenance workforce schedule tosatisfy growing labour requirements with minimum cost. The main recommendationof the study is to switch from a five- to a seven-day workweek for aircraft maintenanceworkers. A new integer programming formulation, used to obtain an optimumseven-day work schedule with no increase in workforce size, is presented. Incomparison to the existing five-day schedule, switching to a seven-day workweek isexpected to produce savings of about 13 per cent, or $100,000 annually. Unfortunately,the integer programming formulation proposed on the work was not tested forsensitivity. This is an important deficiency of the work.

In a related work Anily et al. (1998) study a discrete problem of scheduling inmachines, M, . . . , Mm, in an infinite time horizon discrete time. The cost of operating amachine at any given period, C(t, p), is a liner cost structure where each machine i isassociated with a constraint ai and the cost of operating the machine in the jth periodafter the last maintenance of that machine is jai for j $ 0. The problem is to find anoptimal policy such that the average cost over period 1, t is minimum. Although themodel proposed appears appealing and beneficial to the maintenance schedulingcommunity, no record is shown on the test of sensitivity of the model. Without this, itbecomes difficult to conclude on a wide applicability of the model.

Yet in another work, Anily et al. (1999) consider a situation in which theperformance of a machine deteriorates over time until it receives a maintenance serviceand there are three machines, M1, M2 and M3 to be served and each with the sameservice time of one time unit. Associated with each machine, mi, is a cost constant, ai..The cost of operating Mi during a period in which it is serviced is 0. The cost of

IJQRM23,2

200

operating it in the jth period after its last service is jai. The cost of operating Mi in thejth period after the last maintenance of that machine is jai, for i ¼ 1, 2, 3 and j $ 0. Apolicy p to the m-machine problem is a sequence, ii, i2 . . . , where ii 1 {1, . . . , m} forK ¼ 1, 2 . . . donates the machine scheduled for service during the kth period. Forexample a cyclic service sequence with a basic cycled 1123, the average cost of thepolicy is 3a1 þ 6a2 þ 6a3ð Þ=4 for a policy P, let C(t, p) denote the average cost overperiods 1, . . . , t. The average cost Cð pÞ ¼ Tim Cðt; pÞ.

Anily et al. (1999) observed that for the two-machine problem, with a1 $ a2, there isan optimal cyclic solution in which M2 is serviced exactly in a basic cycle and thereexists an optimal basic cycle of length t2, which is the unique integer satisfyingðtðt2 2 1Þt2 # 2a1=a2 , t2ðt2 þ 1ÞÞ. Here, the minimum average cost is given byC12 ¼ a2 t2 2 1ð Þ=2 þ a1=t2. They also compared the average cost of S0 to that of S toobtain:

C sð Þ2 C s0� �

¼a2t2 t 2 t2ð Þ2 a3t2 t3 2 t2ð Þ

T sð Þ$

a2t2 t 2 t2ð Þ2 a3t3 t 2 1 2 t2ð Þ

T sð Þ

¼a2t2 þ a2 2 a3ð Þt2 t 2 1 2 t2ð Þ

T sð Þ$ a2t2 . 0:

Unfortunately, the paper by Anily et al. (1999) did not address the issue of sensitivityanalysis of the model parameters. This important gap is addressed in the current work.

The same anomaly was observed in the work by Bar-Noy et al. (1998). The authorsolved a general problem of scheduling machines {1 . . . m} for maintenance over aninfinite discrete time horizon, in which at most M machines can be scheduled in eachtime slot. The schedule with m machines is S ¼ S1, S2 . . . where St # {1, . . . m} and=St= # i for all t $ 1. Here, i 1 St means that machine I is scheduled for maintenance attime-slot t. The maintenance cost at time-slot t is

Pi1StCi. The operating cost of ( j) of

machine j at time-slot t is ajðt 2 t1 þ bÞ for integer b $ 0, where t1 # t is the largesttime-slot at which j 1 St1. (We assume that all machines are maintained at time-slot 0 inS). Not that the operating cost is incurred in all time-slots but could be zero at time slot tif i 1Est and b ¼ 0. We want to find a schedule S minimising:

1

h

lim

n!a

Xn

t¼1 i[ST

XCi þ

XM

J¼I

Oi jð Þ

0@

1A:

Also, no indication about the treatment of sensitivity model was discussed. Anotherarticle of prominent importance to maintenance scheduling is based on heuristicoptimisation algorithm (Adzakpa et al., 2004). The work centres on an heuristic-basedoptimisation algorithm for online scheduling and assignment of preventivemaintenance jobs to processors, to minimise under availability constraints, on agiven time-window, and the total cost of the maintenance operatives of a distributedsystem. This algorithm minimises the cost of carrying out preventive maintenancetasks or jobs, while assigning the tasks along with balancing the processor load. Thiswork also has not considered the important concept of sensitivity in the test of themodel.


charting

201

In Samanayake and Yu (1995), an investigation into the application of unitarysoftware was made. The computer programme is composed of the critical path method(CPM), materials requirements planning (MRP) and production activity control (PAC)techniques, to the management of large-scale maintenance activities (specially aircraftmaintenance). The structure is reported to have been previously applied to themanufacturing (i.e. assembly) process, but lately applied to the maintenance problem.Although the authors claim a wide application of the model, no documentation existson the sensitivity analysis of the model.

In concluding the literature review, the current authors re-visit the specific studieson optimal Gantt charting. Since the development of the optimal Gantt charting someyears ago, a small number of studies have extended the knowledge in this domain. Theprincipal studies are due to Oke (2004a, b). In one of the studies, a literature review ismade (Oke, 2004a). In the other a reformulation of the model to incorporate an inflationfactor was conducted. Thus, we complement the efforts of the original proponent of themodel (i.e. Charles-Owaba, 2002) and the other investigator (Oke, 2004a, b) to analyzescientifically the sensitivity test of this increasingly acceptable model. This isobviously a research gap in the literature that is closed in the current work.

3. Model frameworkThe original OGC model proposed by Charles-Owaba (2002) is based on an integratedframework of the traditional transportation tableau and the Gantt chart. It wasdeveloped on the premise that an initial feasible solution could be obtained from thecalculations of cost in the transportation model. This optimal cost, which woulddisplay the assignments of machines at various time intervals would then be blotted ona Gantt chart which would show the schedule of activities, the possible optimalpreventive maintenance cost, the operations period, the maintenance period and theidle time. The transportation tableau (see Appendix, Figure A1) is a structuredmethodology used for distribution problems, but has its importance and application inmaintenance scheduling. The original model by Charles Owaba is a linearised form ofthe transportation model. The demand and supply points of the traditionaltransportation model are converted into maintenance capacity and durationconstraints. Using an optimal solution approach (Vogel’s Approximation Method),the pursuit in the calculation is to find the minimum transportation cost from sourcesto destinations. The same idea was adopted in Charles Owaba’s model with sourcesand destinations substituted for number of vehicles and the period to be spent by eachvehicle for preventive maintenance. For the OGC model, the cost to be minimised iscalled the total preventive maintenance cost.

This is the sum of the cost that the maintenance of vehicles would be at the assignedperiods. The individual cost of preventive maintenance for each vehicle is referred to asthe period-dependent cost function. Since the traditional transportation model could betransformed to a linear function, Charles-Owaba found it convenient to express themodel in a linearised form. The model has the objective function of minimising the totalpreventive maintenance cost (see equation (1) below) subject to two constraints ofmaintenance capacity and duration (see equations (2) and (3) below respectively).Mathematically, we could express the OGC model as follows.

IJQRM23,2

202

Minimise

C T;M ;Yij

� �¼

XM

j¼1

XT

j¼1

CijY ij ð1Þ

subject to:

XM

j¼1

Yij # Aj ðmaintenance capacity constraintÞ ð2Þ

and

XT

i¼1

Yij ¼XNi

r¼1

Bri : ð3Þ

In order to lay a good foundation for explaining the structure of the OGC model, thefollowing notations used in the OGC model and the sensitivity analysis that follows aredefined as:

T total number of periods in planning horizons.

M total number of machines in maintenance system.

Yij the binary Gantt charting variable.

i index indicating machine identify.

j index indicating period.

Cij the unit cost of maintaining machine i at period j.

Bri the number of periods needed to maintain vehicle i at the rth visit.

Ni the total number of visits vehicle I can make for maintenance within the timehorizon T.

Aj the maintenance capacity at period j (number of vehicles that can bemaintained in period j.

To a curious reader, the question may be: “How do I relate equations (1), (2) and (3) withthe transportation tableau for the Vogel’s Approximation Method (VAM) shown in theAppendix, Figure A1?”. The C(T, M, Yij) function on the right-hand side of equation (1)is the sum of the individual cells where allocations for maintenance servicing for eachvehicle is made. Mathematically, it is:

15:23 £ 1ð Þ þ 35:63 £ 1ð Þ þ 39:03 £ 1ð Þ þ 42:43 £ 1ð Þ þ 45:33 £ 1ð Þ½

þ 22 £ 1ð Þ þ 29:8 £ 1ð Þ þ 37:6 £ 1ð Þ þ :::þ 44:4 £ 1ð Þ �:

This gives a value of N311.45 (where N is the Nigerian currency). The data in thetransportation tableau are the same has those presented in the original work of CharlesOwaba but with the application of VAM. From the right hand side of equation (1), the


charting

203

expressionPM

j¼1

PTj¼1CijY ij is all the elements that we have added together to obtain

N311.45. The Cij, which is the period-dependent cost function are shown in the boxes as15.23, 18.63, 22.03 . . . respectively in cells (1, 6), (1, 7) and (1, 8). The Yij which has abinary values of 0 and 1 shows at what period the vehicle is to be maintained. For thefirst vehicle, the solution obtained (see the Appendix, Figure A1) suggests that thepreventive maintenance to be carried out on vehicle (1) should be done in periods 6, 12,13, 14 and 15 while the vehicle should not be scheduled for maintenance at periods 7 to11. It should be clear that in boxes where Cij are not assigned, the cost is not feasible atthat period. This is represented with cells not having allocations but with themathematical symbol / (infinity or non-feasible).

Equations (2) and (3) are constraints for the problem at hand while equation (2)represents the maintenance capacity constraints, equation (3) reflects the durationconstraints. Maintenance capacity in the context of this paper refers to the maximumnumber of vehicles the system could service in view of the limitations of manpowerand space resources. Therefore, any extra vehicle beyond this capacity may have towait until the number of vehicles on queue is reduced to a level that waiting vehiclescould be accommodated. For a numerical interpretation of expression (2), consider arow. For row (1),

PMi¼1Yij is equal to 2. This should be less or equal to Aj. For

i ¼ dai/dt ¼ ddi/dt ¼

1 ð5 2 3Þ=ð10 2 12Þ ¼ 21 ð0 2 0Þ=ð10 2 12Þ ¼ 02 ð3 2 3Þ=ð12 2 5Þ ¼ 0 ð0 2 7Þ=ð12 2 5Þ ¼ 213 ð3 2 4Þ=ð5 2 11Þ ¼ 21=6 ð7 2 0Þ=ð5 2 11Þ ¼ 27=44 ð4 2 2Þ=ð11 2 10Þ ¼ 2 ð0 2 3Þ=ð11 2 10Þ ¼ 235 ð2 2 5Þ=ð10 2 9Þ ¼ 23 ð3 2 1Þ=ð10 2 9Þ ¼ 26 ð5 2 2Þ=ð9 2 13Þ ¼ 23=4 ð1 2 0Þ=ð9 2 13Þ ¼ 1=47 ð2 2 3Þ=ð13 2 12Þ ¼ 21 ð0 2 0Þ=ð13 2 2Þ ¼ 08 ð3 2 3Þ=ð12 2 12Þ ¼ 1 ð0 2 0Þ=ð12 2 12Þ ¼ 19 ð3 2 3Þ=ð12 2 11Þ ¼ 0 ð0 2 1Þ=ð12 2 11Þ ¼ 21

10 ð3 2 3Þ=ð11 2 11Þ ¼ 0 ð1 2 0Þ=ð1 2 11Þ ¼ 111 ð3 2 4Þ=ð11 2 4Þ ¼ 21=7 ð0 2 7Þ=ð11 2 4Þ ¼ 21

Table II.Analysis of sensitivity

Vehicle Unit cost (Cij) ai di t

1 305.3 5 0 102 778.3 3 0 123 672 3 7 54 1,068.75 4 0 115 160.2 2 3 106 783 5 1 97 869 2 0 138 643.5 3 0 129 452.4 3 0 12

10 726.6 3 1 1111 1,042.58 3 0 1112 666 4 7 4

Table I.The values of Cij, ai, di

and t for the vehicles

IJQRM23,2

204

% change in cost, Cij % change in cost parameter, a Cost, Cij Cost parameter, a

262.5 2100 12 0259.375 295 13 1256.25 290 14 2253.125 285 15 3250 280 16 4246.875 275 17 5243.75 270 18 6240.625 265 19 7237.5 260 20 8234.375 255 21 9231.25 250 22 10228.125 245 23 11225 240 24 12221.875 235 25 13218.75 230 26 14215.625 225 27 15212.5 220 28 1629.375 215 29 1726.25 210 30 1823.125 25 31 190 0 32 203.125 5 33 216.25 10 34 229.375 15 35 2312.5 20 36 2415.625 25 37 2518.75 30 38 2621.875 35 39 2725 40 40 2828.125 45 41 2931.25 50 42 3034.375 55 43 3137.5 60 44 3240.625 65 45 3343.75 70 46 3446.875 75 47 3550 80 48 3653.125 85 49 3756.25 90 50 3859.375 95 51 3962.5 100 52 4065.625 105 53 4168.75 110 54 4271.875 115 55 4375 120 56 4478.125 125 57 4581.25 130 58 4684.375 135 59 4787.5 140 60 4890.625 145 61 49

(continued )

Table III.Cost, Cij and cost

parameter, a


charting

205

equation (3), Bi is a reflection of the total duration available for servicing vehicles thatcome into the workshop or machines in production.

After the development of the transportation tableau (the Appendix, Figure A1) thenext stage is to construct the Gantt chart as shown in the Appendix, Figure A2. Thelocations where “1” are filled in the transportation tableau would have to be shaded“black”, while vehicle operations periods are also shaded but on a lighter scale.However, the space for the idle time in the Gantt chart is left blank. For a goodinterpretation of the transformation from the transportation tableau to the Gantt chart,consider the activities of vehicle 1. Since periods 6, 12, 13, 14 and 15 have theallocations of “1”, we have shaded these in Appendix, Figure A2 with a “black” shade.Vehicle 1 is expected to be in operation between periods 7 and 11. On the other hand,the vehicle would be idle between periods 1 and 5.

By revisiting the transportation tableau, we could easily identify the cost functionthat is at the northeast corner of every cell referred to as period dependent cost function(Cij). In the model suggested by Charles-Owaba, this cost function is regarded as alinear expression. This is the main variable of interest to us in the current work. Thus,the sensitivity of the components of these cost functions are examined. Mathematically,the following expression for the period dependent cost function is true:

Cij ¼ ai þ di j 2 kð Þ: ð4Þ

Since the concept of sensitivity analysis is based on the gradient theory, it may beinteresting to find out the application of this theory in equation (4) above. Therefore, inthe calculations that follow, we shall consider finding the slope of expression (4) bydifferentiating Cij with respect to its component parameters. The first attempt would beby differentiating Cij with respect to “a”. Here, we apply a partial derivative to obtain›Cij=›a ¼ 1 þ 0 2 0 þ 1. Therefore, Changes in Cij=Changes in a ¼ 1. By consideringthe next parameter, di, the partial differential of Cij with respect to di gives amathematical expression stated as ›Cij=›di ¼ 0 þ j 2 k. Furthermore, ondifferentiating Cij with respect to “k” we have ›Cij=›k ¼ 0 ¼ þ0 2 di ¼ di also,›Cij=›j ¼ 0 þ 0 ¼ di . Since the situation that we are considering is a plannedmaintenance system when preventive maintenance activities are scheduled andimplemented for specific maintenance periods, then the values of “j” and “k” are

% change in cost, Cij % change in cost parameter, a Cost, Cij Cost parameter, a

93.75 150 62 5096.875 155 63 51100 160 64 52103.125 165 65 53106.25 170 66 54109.375 175 67 55112.5 180 68 56115.625 185 69 57118.75 190 70 58121.875 195 71 59125 200 72 60Table III.

IJQRM23,2

206

% change in cost, Cij % change in cost parameter, d Cost, Cij Cost parameter, d

237.5 2100 20 0235.625 295 20.6 0.3233.75 290 21.2 0.6231.875 285 21.8 0.9230 280 22.4 1.2228.125 275 23 1.5226.25 270 23.6 1.8224.375 265 24.2 2.1222.5 260 24.8 2.4220.625 255 25.4 2.7218.75 250 26 3216.875 245 26.6 3.3215 240 27.2 3.6213.125 235 27.8 3.9211.25 230 28.4 4.229.375 225 29 4.527.5 220 29.6 4.825.625 215 30.2 5.123.75 210 30.8 5.421.875 25 31.4 5.70 0 32 61.875 5 32.6 6.33.75 10 33.2 6.65.625 15 33.8 6.97.5 20 34.4 7.29.375 25 35 7.511.25 30 35.6 7.813.125 35 36.2 8.115 40 36.8 8.416.875 45 37.4 8.718.75 50 38 920.625 55 38.6 9.322.5 60 39.2 9.624.375 65 39.8 9.926.25 70 40.4 10.228.125 75 41 10.530 80 41.6 10.831.875 85 42.2 11.133.75 90 42.8 11.435.625 95 43.4 11.737.5 100 44 1239.375 105 44.6 12.341.25 110 45.2 12.643.125 115 45.8 12.945 120 46.4 13.246.875 125 47 13.548.75 130 47.6 13.850.625 135 48.2 14.152.5 140 48.8 14.454.375 145 49.4 14.7

(continued )

Table IV.Cost, Cij and cost

parameter, d


charting

207

constants. Now, we would consider the total derivative since it would take care of bothindependent variables “a” and “d”. Then, we have a new equation:

dCij ¼›Cij

›ai

:dai þ›Cij

›di

:ddi: ð5Þ

We can either substitute the values of the partial derivatives ›Cij=›ai and ›Cij=›di intoequation (5) or further observe equation (5) would vary with time. By considering thelatter option, we would have a new expression as follows:

dCij

dt¼

›Cij

›ai

:dai

dtþ

›Cij

›di

:ddi

dt: ð6Þ

By substituting the values of the partial derivatives obtained earlier, then expression(6) changes its form to (7) below:

dCij

dt¼ ði Þ

dai

dtþ j 2 kð Þ

ddi

dt: ð7Þ

In order to apply equation (7) to the data tested in this work, we utilise Table I. Table Iconsists of the unit cost incurred for each of the vehicles (Cij), ai, di and t.

Using the data in Table I, the slopes for both cost parameters i.e. dai=dt; ddi=dt wasevaluated as shown below. This is based on the fact thatdai=dt ¼ Da=Dt ¼ change in a=change in t. These steps apply to ddi=dt. Thefollowing results are obtained in the analysis shown in Table II.

However, on getting the value of the slopes, it was discovered that some values wereinfinity. Statistics will therefore be applied. The mean of the defined values will betaken to get a representative value for each of dai=dt and ddi=dt. Therefore, using theformula for arithmetic mean (ignoring the infinite slopes).

% change in cost, Cij % change in cost parameter, d Cost, Cij Cost parameter, d

56.25 150 50 1558.125 155 50.6 15.360 160 51.2 15.661.875 165 51.8 15.963.75 170 52.4 16.265.625 175 53 16.567.5 180 53.6 16.869.375 185 54.2 17.171.25 190 54.8 17.473.125 195 55.4 17.775 200 56 18Table IV.

IJQRM23,2

208

% change in cost, Cij % change in maintenance period, j Cost, Cij Maintenance period, j

293.75 2100 2 0289.0625 295 3.5 0.25284.375 290 5 0.5279.6875 285 6.5 0.75275 280 8 1270.3125 275 9.5 1.25265.625 270 11 1.5260.9375 265 12.5 1.75256.25 260 14 2251.5625 255 15.5 2.25246.875 250 17 2.5242.1875 245 18.5 2.75237.5 240 20 3232.8125 235 21.5 3.25228.125 230 23 3.5223.4375 225 24.5 3.75218.75 220 26 4214.0625 215 27.5 4.2529.375 210 29 4.524.6875 25 30.5 4.750 0 32 54.6875 5 33.5 5.259.375 10 35 5.514.0625 15 36.5 5.7518.75 20 38 623.4375 25 39.5 6.2528.125 30 41 6.532.8125 35 42.5 6.7537.5 40 44 742.1875 45 45.5 7.2546.875 50 47 7.551.5625 55 48.5 7.7556.25 60 50 860.9375 65 51.5 8.2565.625 70 53 8.570.3125 75 54.5 8.7575 80 56 979.6875 85 57.5 9.2584.375 90 59 9.589.0625 95 60.5 9.7593.75 100 62 1098.4375 105 63.5 10.25103.125 110 65 10.5107.8125 115 66.5 10.75112.5 120 68 11117.1875 125 69.5 11.25121.875 130 71 11.5126.5625 135 72.5 11.75131.25 140 74 12135.9375 145 75.5 12.25

(continued )

Table V.Cost, Cij and maintenance

period, j


charting

209

Mean slope ofdai

dt¼

1

n

X9

i¼1

dai

dt

� �

¼1

92 21ð Þ þ 0 þ 20:167ð Þ þ 2 þ 23ð Þ þ 20:75ð Þ þ 21ð Þ þ 20:14ð Þð Þ

¼20:4057

9¼ 20:0451:

Similarly:

Mean slope ofddi

dt¼

1

n

X9

i¼1

ddi

dt

� �

¼1

90 21ð Þ þ 21:75ð Þ þ 23ð Þ þ 2 þ 0:25 þ 0 þ 21ð Þ þ 21ð Þð Þ

¼25:5

9¼ 20:61:

From the values obtained, both slopes are negative but the slope for the cost parameterdi i.e. ddi=dt

� �was observed to be steep (since ddi=dt

�� . 0:5).In a further analysis, the percentage changes in cost (Cij), cost parameters a and d,

maintenance period, j, and arrival period k (see Tables III-VI) are calculated byvarying the value of interest by 5 per cent over a range of 2100 per cent to 200 percent. The summary of further analysis of the work is presented in Table VII. Itshould be noted that the initial starting solution used is based on the VAM that isdemonstrated in the example in the table obtained from Charles-Owaba (2002). Thedata of Charles-Owaba was used to observe the differences in results when a newmethod different from what he proposed is used. The final values obtained are asshown in the Tables III-VII.

From our analysis two parts of the discussions are noted. The first part consists ofthe results presented in Tables III-VI. The second part consists of the analytical results

% change in cost, Cij % change in maintenance period, j Cost, Cij Maintenance period, j

140.625 150 77 12.5145.3125 155 78.5 12.75150 160 80 13154.6875 165 81.5 13.25159.375 170 83 13.5164.0625 175 84.5 13.75168.75 180 86 14173.4375 185 87.5 14.25178.125 190 89 14.5182.8125 195 90.5 14.75187.5 200 92 15Table V.

IJQRM23,2

210

% change in cost, Cij % change in arrival period, k Cost, Cij Arrival period, k

56.25 2100 50 053.4375 295 49.1 0.1550.625 290 48.2 0.347.8125 285 47.3 0.4545 280 46.4 0.642.1875 275 45.5 0.7539.375 270 44.6 0.936.5625 265 43.7 1.0533.75 260 42.8 1.230.9375 255 41.9 1.3528.125 250 41 1.525.3125 245 40.1 1.6522.5 240 39.2 1.819.6875 235 38.3 1.9516.875 230 37.4 2.114.0625 225 36.5 2.2511.25 220 35.6 2.48.4375 215 34.7 2.555.625 210 33.8 2.72.8125 25 32.9 2.850 0 32 322.8125 5 31.1 3.1525.625 10 30.2 3.328.4375 15 29.3 3.45211.25 20 28.4 3.6214.0625 25 27.5 3.75216.875 30 26.6 3.9219.6875 35 25.7 4.05222.5 40 24.8 4.2225.3125 45 23.9 4.35228.125 50 23 4.5230.9375 55 22.1 4.65233.75 60 21.2 4.8236.5625 65 20.3 4.95239.375 70 19.4 5.1242.1875 75 18.5 5.25245 80 17.6 5.4247.8125 85 16.7 5.55250.625 90 15.8 5.7253.4375 95 14.9 5.85256.25 100 14 6259.0625 105 13.1 6.15261.875 110 12.2 6.3264.6875 115 11.3 6.45267.5 120 10.4 6.6270.3125 125 9.5 6.75273.125 130 8.6 6.9275.9375 135 7.7 7.05278.75 140 6.8 7.2281.5625 145 5.9 7.35

(continued )

Table VI.Cost, Cij and arrival

period, k


charting

211

obtained for slope calculation using the gradient formula. While the first part of theanalysis is for the equation:

CðT;M ;Y Þ ¼ mz T a þ dT

22 ki

� �� 2 ki a 2

d

2ki

� �� :

The second part is for the simplified equation: Cij ¼ ai þ dið j 2 kÞ.

% change in cost, Cij % change in arrival period, k Cost, Cij Arrival period, k

284.375 150 5 7.5287.1875 155 4.1 7.65290 160 3.2 7.8292.8125 165 2.3 7.95295.625 170 1.4 8.1298.4375 175 0.5 8.252101.25 180 20.4 8.42104.0625 185 21.3 8.552106.875 190 22.2 8.72109.6875 195 23.1 8.852112.5 200 24 9Table VI.

Slope Gradient % change in gradient Modulus of gradient Modulus of % change in gradient

Cij=a 1 0.625 1 0.625Cij=d 2 0.375 2 0.375Cij=j 6 0.9375 6 0.9375Cij=k 26 20.5625 6 20.5625

Table VII.Summary of results

Figure 2.Variation of Cij whenj ¼ 5:5, k ¼ 5:5

IJQRM23,2

212

From Table IIIThe lowest cost of Cij tends to be 12 while its lowest percentage change in cost of Cij

appears to be 2625 going down the table at where the cost of Cij appears to be 32. Itspercentage change in cost becomes 0. Considering the cost of parameter “a” from thesame table. Its value appears to be 20 when its percentage change in cost of parameter“a” is 0. The lowest value for cost of parameter “a” is 0 while its percentage change incost at this point appears to be 2100. Now going down the table to the highest cost ofCij, it appears to be 72 while its percentage change in cost of Cij for this highest value is

Figure 3.Variation of Cij when

j ¼ 5:0, k ¼ 10:0


j ¼ 10:0, k ¼ 5:0


charting

213

125. Looking at parameter “a” at its highest value, it tends to be (cost) 60, while itspercentage change in cost for this value is 200. So it can be seen that the percentagechange in cost for both Cij and parameters “a” tends to go from negative to positive aswe go down the table.

From Table IVThe lowest cost of Cij can be seen to be 20 while its percentage change in cost for thisvalue is 2375 as we go down the table the percentage change in cost Cij moves to 0

Figure 5.Variation of Cij whenj ¼ 0:5, k ¼ 8:5

Figure 6.Variation of Cij whenai ¼ 2:5; k ¼ 3:0

IJQRM23,2

214

when the cost is 32. Going further down the table the cost of Cij rises to 56 whichappears to be its highest value, here its percentage change in cost is now seen to be 75.Considering parameter “d” from this same table its percentage change in cost forparameter “d” is 2100 for its lowest cost value, which is, 0. Moving down the table,where the percentage change in cost moves to 0, its cost now rises to 6. As we gofurther down the table to see the highest value the percentage change in cost ofparameter “d” is now 200, when its cost appears to be 18 for its highest value.


ai ¼ 3:5; k ¼ 3:0


ai ¼ 4:5, k ¼ 3:0


charting

215

From Table VComparing the cost Cij to the maintenance period “j”. Looking at Cij from this table, itcan be seen that its cost is 2 for the lowest value, here its percentage change in cost is293.75, going down the table where the percentage change in cost of Cij appears to be0, the cost now rises to the value of 32. As we go further down the table the highest costfor Cij is 92 when its percentage change in cost is 187.5.



IJQRM23,2

216

Considering the maintenance period “j”, it is 0 while its percentage change inmaintenance period for this value is 100. As we look down the table its percentagechange in maintenance period “j” is now 0 where its maintenance period for this valueis seen to be 5 which shows an increment as we go down the table, looking at thehighest value of the maintenance period “j” it is seen to be 15 which means its increasedby a value of 10 for the percentage change in maintenance period to now be 200 for itshighest value.


ai ¼ 7:5; k ¼ 3:0


ai ¼ 8:5; k ¼ 3:0


charting

217

From Table VIComparing the cost Cij to the arrival period “k”. Looking at the cost Cij its value here isseen to be 50 while its percentage change in cost Cij for this value is 56.25 which is seento be a positive value. Going down the table where the percentage change in cost in Cij

is 0 its cost appears to be 32 going down the table the cost Cij is now 24 for the finalvalue while its percentage change in cost is 2112.5 which means that the values aredecreasing we go down the table. Looking at the arrival period “k” its lowest value iszero up the table at which its percentage change in arrival period for this value is 2100going down the table the percentage change in arrival period “k” is zero, where itsarrival period “k” for this value is 3. Going down the table the highest value of arrivalperiod “k”, is seen to be 9 where its value is 200 which shows that the arrival period “k”increases down the table.


Figure 14.Variation of Cij whenai ¼ 5:5, k ¼ 1:0

IJQRM23,2

218

In order to extend the degree of testing the model, we simulate data with the aid ofMatlabq software and plot the results as graphs as shown in Figures 2-24. However,the expression used is of the form Cij ¼ ai þ di j 2 kð Þ

2. This done so in order to testthe model in the widest ranges possible.

4. ConclusionsWider applications of mathematical models are usually made if the sensitivity of themodel parameters is known, this would help in advancing the frontier of knowledge inthe scientific discipline concerned. It would also bring much practicality into the modelapplication instead of “paper work” with little significance to practice. In this paper, ananalysis of the sensitivity of model parameters of a maintenance scheduling model is


ai ¼ 5:5, k ¼ 2:0


ai ¼ 5:5, k ¼ 3:0


charting

219

presented. The work is motivated by the importance of simulating the simultaneousscheduling of maintenance and operation in a resource constrained environment. Thishas linkages with quality problems and maintenance performance as a whole. Second,although the maintenance engineering community is exposed to various optimalmodels in the resource constrained-based operational and maintenance arena, themodels do lack sensitivity analysis that the current work has investigated. We haveaddressed the issue of sensitivity analysis by explaining some vital points, whichinclude the establishment of the relationship among the parameters of the model. Fromdata, we employed graphical tools to demonstrate the sensitivity. Finally, we discusson the basis of the scale of maintenance cost and size in order to demonstrate thepractical application of the approach stated in this work.



IJQRM23,2

220

The initial basic solution approach applied in this work is the Vogel’s approximationmethod in transportation modelling. Based on this, the principle of Gantt charting wasused to define the problem. The conclusions drawn from the extensive testing of themodel are as follows:

. parameter “a” is more sensitive than parameter “d”;

. parameter “j” is more sensitive than parameters “a” and “d”, parameter “k” is lesssensitive to parameter “j” and “a” but more sensitive than parameter “d”.


ai ¼ 5:5, k ¼ 6:0


ai ¼ 8:5, j ¼ 0:5


charting

221

The results obtained here are for the model of the form:

CðT;M ;Y Þ ¼ mz T a þ dT

22 ki

� �� 2 ki a 2

d

2ki

� �� :

However, for the simplified equation of the form Cij ¼ ai þ diðj 2 kÞ.

Figure 21.Variation of Cij whenai ¼ 8:5, j ¼ 3:5

Figure 22.Variation of Cij whenai ¼ 8:5, j ¼ 8:5

IJQRM23,2

222

The following conclusions are valued “j” is the most sensitive out of “a”, “d”, “j” and “k”.This is followed by “a”. The next sensitive item is “d” while the least sensitive item is “k”.

There are a number of fertile research areas on the sensitivity analysis of the modelpresented. If the original model could be fuzzified, comparative analysis could be madebetween the result obtained in this work and the expected results from the fuzzifiedexercise. The same line of experimentation could be conducted on sensitivity analysiswhen other soft computing tools are applied to the original model. Some interestingresults may be observed when soft computing tools such as genetic algorithm, artificialneural-network and neuro-fuzzy are applied to the original model. Further extension ofthe frontier of knowledge on sensitivity analysis with respect to the maintenance


ai ¼ 1:0, j ¼ 8:5


ai ¼ 5:5, j ¼ 8:5


charting

223

scheduling modelling could be made if researchers engaged the collaborative efforts ofour statistics trained experts. Apart from the “goodness of fit” which is a widely usedconcept in sensitivity test analysis, some more advanced tools could be adopted to thebenefit of maintenance engineering community. It is possible to focus some attention onthe important area of model calibration. This, the present model could be calibratedunder a wide range of circumstances and assumptions. This would add great value to thepool of knowledge in the sensitivity analysis area of maintenance scheduling model.

References

Adzakpa, K.P., Adjallah, K.H. and Yalaoui, F. (2004), “On-line maintenance job scheduling andassignment to resources in distributed systems by heuristic-based optimization”, Journalof Intelligent Manufacturing, Vol. 15 No. 2, pp. 131-40.

Alfares, H.K. (1999), “Aircraft maintenance workforce scheduling: a case study”, Journal ofQuality in Maintenance Engineering, Vol. 5 No. 2, pp. 78-88.

Anily, S., Glass, C.A. and Hassin, R. (1998), “The scheduling of maintenance service”, DiscreteApplied Mathematics, Vol. 82, pp. 27-42.

Anily, S., Glass, C.A. and Hassin, R. (1999), “Scheduling of maintenance services to threemachines”, Annals of Operations Research, Vol. 86, pp. 375-91.

Atkinson, E., Elango, K., Mohan, S., Fadda, G. and Cinus, S. (2003), “A rational approach toscheduling main-system maintenance”, Irrigation and Drainage Systems, Vol. 17 No. 4,pp. 239-61.

Bar-Noy, A., Bhatia, R., Naor, J. and Schieber, B. (1998), “Minimizing service and operation costsof periodic scheduling”, paper presented at SODA’98, Holiday Inn Golden Gateway Hotel,San Francisco, CA, 25-27 January.

Billinton, R. and Pan, J. (1998), “Optimal maintenance scheduling in a two identical componentparallel redundant systems”, Journal of Reliability Engineering and System Safety, Vol. 59No. 3, pp. 309-16.

Burke, E.K. and Smith, A.J. (1999a), “A multi-stage approach for the thermal generatormaintenance scheduling problem”, Proceedings of the Congress on EvolutionaryComputation (CE’99), Washington DC, USA, Vol. 2, IEEE Press, Chicago, IL, pp. 1085-92.

Burke, E.K. and Smith, A.J. (1999b), “A memetic algorithm to schedule grid maintenance”,Proceedings of the International Conference on Computational Intelligence for ModellingControl and Automation: Evolutionary Computation and Fuzzy Logic for IntelligentControl, Knowledge Acquisition and Information Retrieval, Vienna, IOS Press,Amsterdam, pp. 122-7.

Burke, E.K., Clark, J.A. and Smith, A.J. (1998), “Four methods for maintenance scheduling”,Proceedings of the International Conference on Artificial Neural Networks and GeneticAlgorithms, Springer, New York, NY, pp. 264-9.

Charest, M. and Ferland, J.A. (1993), “Preventative maintenance scheduling of power generationunits”, Annals of Operational Research, Vol. 41, pp. 185-206.

Charles-Owaba, O.E. (2002), “Gantt charting multiple machines’ preventive maintenanceactivities”, Nigerian Journal of Educational Research and Development, Vol. 1 No. 1,pp. 60-7.

Chattopadhyay, D., Bhattacharya, K. and Pariksh, J. (1995), “A system approach to least-costmaintenance scheduling for an interconnected power system”, IEEE Transactions onPower Systems, Vol. 10 No. 4, pp. 2000-7.

IJQRM23,2

224

Chen, L.N. and Toyoda, J. (1990), “Maintenance scheduling based on two level hierarchicalstructure to equalize incremental risk”, IEEE Transactions on Power Systems, Vol. 5 No. 4,pp. 1510-6.

Christiaanse, W.R. (1973), “A program for calculating optimal maintenance schedulesrecognising constraints”, 1973 PICA Conference Proceedings, 1973, pp. 230-9.

Christiannse, W.R. and Palmer, A.H. (1991), “A technique for automatic scheduling of themaintenance of generating facilities”, IEEE Transactions on Power Apparatus andSystems, PAS-91, No. 1, pp. 319-27.

Contaxis, G.C., Kavatza, S.D. and Vournas, C.D. (1989), “An interactive package for riskevaluation and maintenance scheduling”, IEEE Transactions on Power Systems, Vol. 4No. 2, pp. 389-95.

Coudert, T., Grabot, B. and Archimede, B. (2000), “Integration of maintenance constraints inscheduling: fuzzy modelling and multi-agent approach, 4th IEEE/IFIP InternationalConference on Information Technology for Balanced Automation Systems in Productionand Transportation (BASYS 2000), Berlin, Allemagne”, in Camarinha-Matos, L.M.,Afsarmanesh, H. and Erbe, H.H. (Eds), Advances in Networked Entreprises, Kluwer,New York, NY, pp. 297-304.

Dahal, K.P. and McDonald, J.R. (1997), “Generational and steady state genetic algorithms forgenerator maintenance scheduling problems”, Proceedings of the International Conferenceon Artificial Neural Networks and Genetic Algorithms, pp. 260-4.

Dapazo, J.F. and Merrill, H.M. (1975), “Optimal generator maintenance scheduling using integerprogramming”, IEEE Transactions on Power Apparatus and Systems, PAS-94, No. 5,pp. 1537-45.

Duffuaa, S.O. and Al-sultan, K.S. (1997), “Mathematical programming approaches for themanagement of maintenance, planning and scheduling”, Journal of Quality in MaintenanceEngineering, Vol. 3 No. 3, pp. 163-76.

Egan, G.T., Dillon, T.S. and Morsztyn, K. (1976), “An experimental method of determination ofoptimal maintenance schedules in power system using branch and bound techniques”,IEEE Transactions on Man. and Cybernetics. SMC-6, No. 8, pp. 538-47.

El-Sheikhi, F.A. and Billinton, R. (1984), “Generating unit maintenance scheduling for single andtwo interconnected systems”, IEEE Transactions on Power Apparatus and Systems,PAS-103, No. 5, pp. 1038-44.

Enscore, E.E. and Burns, D.L. (1983), “Dynamic scheduling of a preventive maintenanceprogramme”, International Journal of Production Research, Vol. 21 No. 3, pp. 357-68.

Fwa, T.F., Cheu, R.L. and Muntasir, A. (1999), “Scheduling of pavement maintenance to minimisetraffic delays”, Transportation Research Record 1650, Transportation Research Board,Washington, DC, pp. 28-35.

Garver, L.L. (1972), “Adjusting maintenance schedule to levelize risk”, IEEE Transactions onPower Apparatus and Systems, PAS-91, No. 5, pp. 2057-63.

Haghani, A. and Shafahi, Y. (2002), “Bus maintenance systems and maintenance scheduling:Model formulation and solutions”, Transportation Research Part A: Policy and Practice,Vol. 36 No. 5, pp. 453-8.

Hall, R.W. (2000), “Scheduling transit railcar maintenance and facility design”, TransportationResearch, No. 34A, pp. 67-84.

Hariga, H. (1994), “A deterministic maintenance-scheduling problem for a group of non-identicalmachines”, International Journal of Operations & Production Management, Vol. 14 No. 7,pp. 27-36.


charting

225

Higgins, A., Ferreira, L. and Lake, M. (1999), “Scheduling rail track maintenance to minimiseoverall delays”, in Ceder, A. (Ed.), Transportation and Traffic Theory, Pergamon,Kidington and Oxford, Ch. 10, pp. 779-96.

Joshi, S. and Gupta, R. (1996), “Scheduling of routine maintenance using production schedulesand equipment failure history”, Computer and Industrial Engineering, Vol. 10 No. 1,pp. 11-20.

Khan, F.I. and Hadara, M.M. (2003), “Risk-based maintenance (RBM): a quantitative approach formaintenance/inspection scheduling and planning”, Journal of Loss Prevention in ProcessIndustries, Vol. 16, pp. 561-73.

Khatib, H. (1979), “Maintenance scheduling of generating facilities”, IEEE Transactions onPower Apparatus and Systems, PAS-98, No. 5, pp. 1604-8.

Kin, H., Hayashi, Y. and Nara, K. (1997), “An algorithm for thermal unit maintenance schedulingthrough combined use of GA, SA and TS”, IEEE Transactions on Power Systems, Vol. 12,pp. 329-35.

Kobbacy, K.A.H., Fawzi, B.B., Percy, D.F. and Ascher, H.E. (1997), “A full history proportionalhazards model for preventive maintenance scheduling”, Quality and ReliabilityEngineering, Vol. 13, pp. 187-98.

Kurban, M. (1999), “The maintenance schedule optimisation in an interconnected power systemusing the levelised risk method”, IEEE Budapest Power Tech ’99 Conference, Budapest,Hungary, No. 386, pp. 26-40.

Lake, M. and Ferreira, L. (2001), “Considering the risk of delays in scheduling railway trackmaintenance”, Proceedings of the 7th International Heavy Haul Conference, InternationalHeavy Haul Association Inc., Virginia Beach, VA, USA, pp. 367-72.

Lake, M., Ferreira, L. and Murray, M. (2000), “Minimising costs in scheduling railway trackmaintenance”, in Allan, J., Hill, R.J., Brebbia, C.A., Scuitto, G. and Sone, S. (Eds), Computerin Railways VII, WIT Press, Southampton, Ch. 13, pp. 895-902.

Lund, J.R. (1990), “Scheduling maintenance dredging on a single reach with uncertainty”, Journalof Waterway, Port, Coastal, and Ocean Engineering, ASCE, Vol. 116 No. 2, pp. 211-321.

Marwali, M. and Shahidehpour, M. (1998a), “Coordination of short-term and long-termtransmission maintenance scheduling in a deregulated system”, IEEE Power EngineeringLetters, Vol. 1 No. 1, pp. 46-8.

Marwali, M. and Shahidehpour, M. (1998b), “Integrated generation and transmissionmaintenance scheduling with network constraints”, IEEE Transactions on PowerSystems, Vol. 13 No. 3, pp. 1063-8.

Mosley, S.A., Teyner, T. and Uzsoy, R.M. (1998), “Maintenance scheduling and staffing policiesin a wafer fabrication facility”, IEEE Transactions on Semiconductor Manufacturing,Vol. 11 No. 2, pp. 316-23.

Nguyen, D.G. and Murthy, D.N.P. (1981), “Optimal repair limit replacement policies withimperfect repair”, J. Opl. Res. Soc., Vol. 32, pp. 409-16.

Oke, S.A. (2004a), “Maintenance scheduling: description, status, and future directions”, SouthAfrican Journal of Industrial Engineering, Vol. 15 No. 1, pp. 101-17.

Oke, S.A. (2004b), “A re-examination of the optimal Gantt charting model for maintenancescheduling”, Proceedings of MIMAR 2004, the 5th IMA International Conference, 5-7 April2004, Salford, United Kingdom, pp. 219-24.

Percy, D., Kobbacy, K. and Ascher, H.T. (1998), “Using proportional-intensities models toschedule preventive maintenance intervals”, IMA Journal of Mathematical Applied inBusiness and Industry, Vol. 9, pp. 289-302.

IJQRM23,2

226

Percy, D.F., Kobbacy, K.A.H. and Fawzi, B.B. (1997), “Setting preventive maintenance scheduleswhen data are sparse”, International Journal of Production Economics, Vol. 51, pp. 223-34.

Ram, B. and Olumolade, M. (1987), “Preventive maintenance scheduling in the presence of aproduction plan”, Production and Inventory Management, Vol. 8, pp. 81-9.

Samanayake, B.R. and Yu, J. (1995), “A balanced maintenance schedule for a failure-pronesystem”, International Journal of Quality & Reliability Management, Vol. 12 No. 9,pp. 183-91.

Sarker, B.R. and Yu, J. (1995), “A balanced maintenance schedule for a failure-prone system”,International Journal of Quality & Reliability Management, Vol. 12 No. 9, pp. 183-91.

Satoh, T. and Nara, K. (1991), “Maintenance scheduling by using the simulated annealingmethod”, IEEE Transactions on Power Systems, Vol. 6, pp. 850-7.

Silver, E.A. and Murphy, G.F. (1994), “Cost analysis and extension of a simple maintenancescheduling heuristic”, NRL (US), Vol. 41 No. 7, pp. 945-58.

Sriskandarajah, C., Jardine, A.K.S. and Chan, C.K. (1998), “Maintenance scheduling of rollingstocks using a genetic algorithm”, Journal of Operational Research Society (UK), Vol. 49,pp. 1130-45.

Stremel, J.P. (1981), “Maintenance scheduling for generation system planning”, IEEETransactions on Power Apparatus and Systems, PAS-100, No. 3, pp. 4010-9.

Stremel, J.P. and Jenkins, R.T. (1981), “Maintenance scheduling under uncertainty”, IEEETransactions on Power Apparatus and Systems, PAS-100, No. 2, pp. 460-5.

Walker, L., Bryan, U. and Turner, K. (2001), “Scheduling preventive maintenance of transmissioncircuits”, paper presented at the 4th IMA Conference on Modelling in IndustrialMaintenance and Reliability Decision Support in the New Millennium, April.

Wang, Y., Cheu, R.L. and Fwa, T.F. (2002), “Highway maintenance scheduling using geneticalgorithm with microscopic traffic simulation”, Proceedings of the 81st Annual Meeting ofthe Transportation Research Board (on CD-ROM).

Yamayee, Z.A. (1982), “Maintenance scheduling: description, literature survey, and interfacewith overall operation scheduling”, IEEE Transactions on Power Systems, PAS-101, No. 8,pp. 2770-9.

Yamayee, Z.A., Sidenblad, K. and Yoshimura, M. (1998), “A computationally efficient optimalmaintenance scheduling method”, IEEE Transactions on Power Apparatus and Systems,Vol. 102 No. 2, pp. 330-8.

Yellen, J., Al-Khamis, J., Vemuri, S. and Lemonidis, L. (1992), “Unit maintenance scheduling withfuel constraints”, IEEE Transactions on Power Systems, Vol. 7, pp. 726-33.

Zurn, H.H. and Quintana, V.H. (1975), “Generator maintenance scheduling via successiveapproximations dynamic programming”, IEEE Transactions on Power Apparatus andSystems, Vol. 94 No. 2, pp. 665-71.

Zurn, H.H. and Quintana, V.H. (1977), “Several objective criteria for optimal generator preventivemaintenance scheduling”, IEEE Transactions on Power Apparatus and Systems, PAS-96,No. 3, pp. 884-922.

Further reading

Marwali, M.K.C. and Shahidehpour, S.M. (1999), “A probabilistic approach to generationmaintenance scheduler with network constraints”, International Journal of ElectricalPower and Energy Systems, Vol. 21 No. 8, pp. 533-45.


charting

227

Appendix

Figure A1.Transportation tableau forthe Vogel’sApproximation Method(VAM)

IJQRM23,2

228

Corresponding authorS.A. Oke can be contacted at: [email protected]

Figure A2.The preventive

maintenance Gantt chartfor VAM


charting

229

To purchase reprints of this article please e-mail: [email protected] visit our web site for further details: www.emeraldinsight.com/reprints

a sensitivity

Documents

emerald publication

emerald subscription

emerald group publishing

university of ibadan

university of lagos

metadata models

quality problem

oxford brookes university