Microsoft Word - 4-565-ed.doc Engineering, Technology & Applied Science Research Vol. 5, No. 3, 2015, 811-817 811 www.etasr.com Adebimpe et al.: Preventive Maintenance Interval Prediction: a Spare Parts Inventory Cost… Preventive Maintenance Interval Prediction: a Spare Parts Inventory Cost and Lost Earning Based Model O. A. Adebimpe Dept of Industrial and Production Engineering, University of Ibadan, Ibadan, Nigeria sebolic@yahoo.com V. O. Oladokun Dept of Industrial and Production Engineering, University of Ibadan, Ibadan, Nigeria victordokun@yahoo.com O. E. Charles-Owaba Dept of Industrial and Production Engineering, University of Ibadan, Ibadan, Nigeria oecharlesowaba@yahoo.com Abstract— In this paper, some preventive maintenance parameters in manufacturing firms were identified and used to develop cost based functions in terms of machine preventive maintenance. The proposed cost based model considers system’s reliability, cost of keeping spare parts inventory and lost earnings in deriving optimal maintenance interval. A case of a manufacturing firm in Nigeria was observed and the data was used to evaluate the model. Keywords-Preventive Maintenance; Maintenance Cost; Spare Parts Inventory. I. INTRODUCTION The term maintenance can be defined as all actions appropriate for retaining an item/part/equipment in, or restoring it to a satisfactory condition [1]. More specifically, maintenance includes the repair of broken equipment, the preservation of equipment conditions and the prevention of their failure, which ultimately reduces production losses and downtime and also reduces environmental and associated safety hazards. With the increasing pressure of high competition and stringent environmental and safety regulations, maintenance has shifted from being perceived as a “necessary evil” to being recognized as an effective tool for increased profitability. Maintenance has become an integrated part of the production process rather than a supporting or peripheral activity. Developing effective and optimum maintenance strategies and models has thus become a subject of research both in academic and in industry. Preventive maintenance tends to inspect, adjust, replace, lubricates etc the machine components to avoid any likely failure. This involves the usage of consumables and spare parts. Some of the components that need replacement have to be replaced, the ones that need adjustment have to be adjusted properly etc in accordance with manufacturers’ specification. In [2], a preventive maintenance system is indicated to consist of routine actions taken in a planned manner to prevent breakdowns and to ensure smooth operational accuracy. Industries carry out preventive maintenance (PM) on machinery and equipment to prevent or slow down deterioration. It is important to note that PM is justified only when it is cost effective, reduces the occurrences of failure and extends the useful life of the equipment. The spare parts involved in this activity need to be planned for and well managed to avoid shortages and also unnecessary tying down of capital. The question of how, when and what spare parts to be ordered becomes a challenge to be dealt with by the maintenance team. Failure to make proper planning on the spare parts inventory has led to disrupted or poor maintenance operations resulting in economic loss for many organizations. Maintenance activities are increasingly becoming complex in view of the rapidly expanding product diversification and the challenge for improved efficiency of manufacturing systems. Hence, existing models have considered a lot of ways to predict the optimal preventive maintenance interval, but many have excluded some of the pertinent costs associated with maintenance activities. In order to reflect reality there is the need to capture cost components associated with the spare parts inventory and lost earnings during the preventive maintenance intervals. This study is aimed at determining the optimal preventive maintenance interval that minimizes the cost of spare parts inventory and sum of lost earnings during preventive maintenance activities as a result of loss of production. Numerous researchers have demonstrated great efforts towards the study of the preventive maintenance models. An (s, S) PM policy in which the PM work-orders are performed as soon as inventory reaches a certain threshold value, S was described in [3]. PM and safety stock strategies were formulated assuming that the PM tasks are scheduled every M units of time and derived M by minimizing expected cost per unit time. A control policy where the system undergoes repair once it breaks down was described in [4]. The production resumes immediately after repair, continuing until the inventory level reaches a threshold value. A multi-criteria preventive maintenance optimization model to find the optimal preventive maintenance intervals of components in a production system was developed in [5]. An economic manufacturing quantity (EMQ) model was proposed in [6] and an improvement to a stochastic PM model to find an optimal Engineering, Technology & Applied Science Research Vol. 5, No. 3, 2015, 811-817 812 www.etasr.com Adebimpe et al.: Preventive Maintenance Interval Prediction: a Spare Parts Inventory Cost… policy, where manufacturing quantity and safety stock are derived minimizing the cost per unit time was described in [7]. The basic cost-based approach to maintenance planning that was developed by Jardine in 1973 was studied in [8] and was later extended in [9]. Machine reliability and preventive maintenance planning were considered for a cellular manufacturing system for an improved performance. A simulation model to find the best preventive maintenance strategy in semiconductor manufacturing plants was described in [10]. An analytical model for the joint determination of an optimal age-dependent buffer inventory and PM policy in a production environment that is subject to random machine breakdowns was formulated in [11]. The optimal preventive maintenance schedules by considering two modes of failure (maintainable and non-maintainable) and the number of PM tasks dependent on different failure rates of the system was investigated in [12]. A development of a combined opportunity cost and reliability model, taking into consideration a factor that was not considered in [8], was described in [13]. A computer-aided preventive maintenance program was developed in [14] in order to improve system availability and maintenance resources. The PM labor force requirement was, the availability of the equipment, reliability and total maintenance cost were modeled. Genetic algorithms were employed to find optimum solution for different maintenance planning and scheduling with multi-objective optimization research on multi parallel machine in [15] made. The reliability and time were the two main constraints and due to that, a lower bound for reliability was introduced. An optimal preventive maintenance model for steam turbines where Weibull distribution was applied to study the problems of a thermal power station was presented in [2] and it was shown that the total cost of replacement was decreasing as the cycle length was increasing. The model didn’t consider how the components to be replaced are being ordered or the cost which is attached to it. The preventive maintenance strategy and minimal repair were incorporated into the production process in the traditional integrated inventory model in [16] but didn’t take into consideration the inventory of the spare parts needed for the preventive maintenance activities itself rather, they tried to address the problem involved with the supply chain and production. It should be noted that past works on models of preventive maintenance are yet to incorporate the cost of the spare parts inventory cost and the lost earnings during preventive maintenance. Consequently, this work develops a model to incorporate these using Weibull distribution. II. MODEL DEVELOPMENT In this section, the procedure for identifying preventive maintenance cost related parameters for a manufacturing firm is presented. Also, outlined are the preventive maintenance cost function development process and the definition and solution procedure to the preventive maintenance optimization problem. To identify preventive maintenance cost parameters, relevant literature was reviewed, some manufacturing firms in Nigeria were visited, relevant personnel interviewed, maintenance process observed and relevant records examined. A. Model Notations This model uses the cost based approach to minimize total maintenance cost that assures the desired level of machine reliability. The following notations of the model are defined: 1. tpm is the preventive maintenance interval. 2. TC is the total maintenance cost for a planning horizon... 3. oC is the fixed cost of carrying out a planned preventive maintenance (preparation cost). 4. pmC is the estimated average preventive maintenance cost per maintenance to return machine to the “as-good- as-new” condition. 5. fC is the breakdown maintenance cost during the interval tpm 6. )(tpmH is the average number of machine failures during the interval tpm 7. mD is the estimated average duration for a breakdown maintenance. 8. tD is the estimated duration of time used for a preventive maintenance activities. 9. lP is the estimated profit loss per hour by the company due to downtime. 10. ocC is the cost of placing an order for the materials that are used in carrying out preventive maintenance activities. 11. hcC is the holding cost of the materials used for preventive maintenance activity in a year. 12. r is the demand of the preventive maintenance material per year. 13. α and β are respectively the scale parameter and the shape parameter of the Weibull distribution for the machine in question. 14. T is the total operating time of the machine in a planning horizon. B. Model Assumptions a. The failure of the machine is characterized by Weibull distribution (increasing rate of failure with age). b. There is a fixed cost for carrying out preventive maintenance and it is deterministic c. The cost of executing one breakdown repair is constant and deterministic. d. The preventive maintenance spare parts are ordered in bulk and received in bulk e. There is minimum acceptable reliability of the machine. Engineering, Technology & Applied Science Research Vol. 5, No. 3, 2015, 811-817 813 www.etasr.com Adebimpe et al.: Preventive Maintenance Interval Prediction: a Spare Parts Inventory Cost… f. The ordering cost and holding cost for the bulk preventive maintenance materials are deterministic. g. The replenishment size of the preventive maintenance material is constant and replenishments are made whenever the inventory reaches the prescribed zero level. h. Demand is deterministic at a constant rate of r . i. The duration for preventive maintenance is deterministic. j. The depreciation of the equipment is negligible since we assumed that the machine is as good as new each time they are maintained. k. The labors used for the maintenance are the employees of the company so they are always available. C. The Cost Function Development Expanding on the cost function of [8], the cost component of this model can be summarized as follows: developing the total maintenance cost function for a planning horizon, where total maintenance cost is defined as the summation of all the costs involved in the maintenance made up of the following components. Total maintenance cost={Preparation cost of preventive maintenance+Cost of executing preventive maintenance+Cost of executing breakdown maintenance+Lost earnings during breakdown maintenance+Lost earnings during preventive maintenance}+Cost of Preventive Maintenance Materials Inventory Management We now describe and develop mathematical expression for each cost component. 1. Preparation cost for the preventive maintenance activities oC : this is a fixed cost per preventive maintenance encompassing all cost for mobilizing resources, equipment into site, and cost of all preparatory paper works. 2. Cost of executing preventive maintenance pmC : This includes costs of tools materials and labour (include portion of employees’ salary for the involvement in preventive maintenance) 3. Cost of executing breakdown maintenance bmC : This includes cost of materials, labour, tools used, for breakdown maintenance within an interval tpm expressed as  )( tpmCC fbm  where, fC is the cost of breakdown maintenance per machine failure and )(tpmH the average number of machine failures during a preventive maintenance interval. Note:  )()( tpmtpmH   )(tpmCC fbm  4. Lost Earnings during Breakdown Maintenance bmLE : This accounts for earning opportunities forfeited during downtime due to sudden failure:  )( tpmPDLE lmbm    5. Lost Earnings during Preventive Maintenance pmLE : This accounts for earning opportunities forfeited during downtime associated with preventive maintenance activities: ltpm PDLE  6. Cost of spare parts Inventory to support Preventive Maintenance tc : The inventory cost has two components, the holding cost and the ordering cost based on the economic order quantity inventory model. That is, holding cost 2qChc and ordering costs qrCoc . If )(Itc is the total cost of inventory per unit time then. qrCqCItc ochc  2)( (1) Since rqt  which is the schedule period of ordering then, trq  (2) From (1) and (2) tCrtCItc ochc  2)( (3) Since (3) is the unit cost per unit time, therefore the total unit cost over the planning horizon T. Then: tCTrtTCtcTITC ochc  2.)( Note: t is actually equivalent to tDtpm  for a preventive maintenance cycle Hence: )(2)(.)( tocthc DtpmCTrDtpmTCtcTITC  (4) Therefore: )()( ITCLELECCCTTC bmpmpmbmo  i.e tocthclt lmfpmo DtpmCTrDtpmTCPD tpmPDtpmCCCTTC   2)(} )()({)(   (5) Therefore, a cost minimization problem can be defined based on the above cost function to determine the optimal values of the preventive maintenance decision variables as follows. Engineering, Technology & Applied Science Research Vol. 5, No. 3, 2015, 811-817 814 www.etasr.com Adebimpe et al.: Preventive Maintenance Interval Prediction: a Spare Parts Inventory Cost… Min ( ) { ( ) ( ) ( ) 2 } (6) o pm f t t t m l t l hc t t t oc t T T T TC T C C C tpm D tpm D tpm D tpm T T D P tpm D P TC tpm D r D tpm D tpm T C D tpm b b a a = + + + + + + + + + + + + Subject to    1)(1{ tpmRIntpm  (7) UpperBoundtpmR )( (8) Where tpmD T t  (9) The model when solved computes the optimum cost and PM interval combination using the objective function (i.e., the total cost TC (T)), the reliability of the machine and the determined values of α and β. The procedure can be repeated for various values of reliability. A series of preventive maintenance intervals and total cost combinations corresponding to different reliability probabilities will be generated to provide a basis for choosing the appropriate maintenance interval that will best suit the needs of the organization depending on the reliability level desired for the concerned machine. D. Solution Procedure Step 1. Specify the values of the parameters oC , pmC , fC , tD , lP , hcC , r , Coc , UpperBound ,  and  Step 2: Compute the optimum tpm using the reliability function of the model developed in (7) Step 3: Substitute the tpm in the cost objective equation to compute the total cost TC in the model at a minimum reliability and the preventive interval tpm computed. Step 4: Repeat this for a range of reliability and check for the minimum cost. III. APPLICATION, RESULTS AND DISCUSSION The manufacturing firm under study is a cash crop processing company in South-West of Nigeria. This company is one of the famous beverages company in Nigeria whose products are widely consumed on daily basis across the country and in the neighbouring countries as well. Because of the high demand and competitive market in which the firm is involved, there is need for a fact based determination of a realistic planned preventive maintenance interval to avoid downtime experienced as a result of frequent failures, particularly of the critical machine under consideration. The reliability of this machine is of importance as its failure could cause a serious production problem. We shall now apply the firm’s internal data to the developed model to test its validity and realism, and in the process establish an optimal preventive maintenance interval for the critical machine under study. We begin by establishing values of the Weibull distribution parameters α and β and other parameters as specified in the solution procedure. The data used are shown in Table I. TABLE I. MANUFACTURING FIRM DATA USED Item No. Parameters Values 1 MTBF (hours) 187 2 β 1.21 3 α 199.61 4 Co (Naira) 1400 5 Cpm (Naira) 2560 6 Cf (Naira) 15190 7 (Naira) 4500 8 Dm (hours) 2.5 9 T (hours) 8760 (1year) 10 tD (Hours) 8 (9.13x10-4year) 11 ocC (Naira) per order 1650 12 hcC (Naira)per quantity 250 13 r 700 IV. DISCUSSION OF RESULTS The results of the model are as shown in Table II and Figures 1 and 2. This work first considered the inclusion of the spare parts inventory cost which is shown in Table II. The results in Table II show that there is a minimum point at the Total Maintenance Cost (TC) at which the preventive maintenance could take place. This minimum point for the total maintenance cost at 100 percent inventory cost consideration (i.e I=1) is 1,349,105 Naira at tpm 173.58 for a planning horizon shown in Figure 1 for different percentage of inventory cost. Table II shows that at the 40 percent and 50 percent consideration of inventory cost the tpm are the same though with different Total Maintenance Costs. Also, for 70 percent and 80 percent inventory. This implies that there could be a uniform preventive maintenance interval tpm for the machines in a production line even when they have different cost implication, so far the minimum total maintenance cost is considered. Considering the lost earning due to preventive maintenance, the results are shown in Table III and described in Figure 2. As shown, there is an inverse relationship between the preventive maintenance interval tpm and the Total Maintenance Cost (TC). Considering the lost earnings due to preventive maintenance, the Total Maintenance Cost (TC) decreases continually with an increase in the preventive maintenance interval tpm and consequently reducing the reliability of the machine without giving a concise answer of when the maintenance could be done should the cost be given a priority. This suggests that the preventive maintenance interval should not be attached to the money that could be probably lost during the period of maintenance as this could lead to more catastrophic failures if the lost is prioritized. Engineering, Technology & Applied Science Research Vol. 5, No. 3, 2015, 811-817 815 www.etasr.com Adebimpe et al.: Preventive Maintenance Interval Prediction: a Spare Parts Inventory Cost… Fig. 1. Graph showing the relationship between the Total Maintenance Cost and Preventive Maintenance Interval of a Manufacturing Firm with Spare Parts Inventory Cost Consideration at different Percentage (i.e. I = 0.1 to 1) Fig. 2. Graph showing the relationship between the Total Maintenance Cost and Preventive Maintenance Interval of a Manufacturing Firm with Lost Earning at 100 percent Spare Parts Inventory Cost Consideration (i.e. I= 1) Engineering, Technology & Applied Science Research Vol. 5, No. 3, 2015, 811-817 816 www.etasr.com Adebimpe et al.: Preventive Maintenance Interval Prediction: a Spare Parts Inventory Cost… TABLE II. RESULTS OF TPM AND TOTAL COST OF MAINTENANCE OF A MANUFACTURING FIRM. Min R PM Interval tpm Total Cost of Maintenance (Naira) at I=0.1 I=0.2 I=0.3 I=0.4 I=0.5 I=0.6 I=0.7 I=0.8 I=0.9 0.75 78.66 1297552  1316005  1334459  1352913  1371367  1389821  1408275  1426729  1445182  0.73 84.84 1286017  1303139  1320260  1337382  1354503  1371625  1388746  1405868  1422989  0.71 90.6 1277878  1293923  1309967  1326011  1342055  1358099  1374143  1390187  1406232  0.69 96.11 1271966  1287101  1302236 1317371 1332506 1347641 1362776  1377911 1393046 0.67 101.5 1267620  1281962  1296304 1310646 1324987 1339329 1353671  1368013 1382355 0.65 107.75 1264032  1277554  1291076 1304598 1318120 1331642 1345164  1358686 1372208 0.63 113.81 1261768  1274582  1287396 1300210 1313024 1325837 1338651  1351465 1364279 0.61 119.61 1260519  1272723  1284927 1297130 1309334 1321538 1333742  1345945 1358149 0.59 *125.18 1260021  1271692  1283364 1295036 1306707 1318379 1330051  1341722 1353394 0.57 *131.58 1260155  1271271  1282388  1293504  1304620  1315737  1326853  1337970  1349086  0.55 *138.69 1261036  1271597  1282157  1292717  1303278  1313838  1324398  1334959  1345519  0.53 *145.46 1262463  1272545  1282627  1292709  1302791  1312874  1322956  1333038  1343120  0.51 *151.91 1264262  1273929  1283595  1293262  1302929  1312595  1322262  1331928  1341595  0.49 *158.94 1266627  1275880  1285133  1294385  1303638  1312891  1322144  1331396  1340649  0.47 *166.45 1269537  1278387  1287237  1296087  1304937  1313787  1322637  1331487  1340337  0.45 *173.58 1272601  1281102  1289602  1298102  1306603  1315103  1323603  1332104  1340604  0.43 181.83 1276444  1284574  1292705  1300836  1308967  1317098  1325229  1333359  1341490  0.41 189.63 1280314  1288126  1295937 1303749 1311561 1319372 1327184  1334996 1342807 0.39 193.7 1282410  1290065  1297721 1305376 1313032 1320687 1328343  1335999 1343654 0.37 206.66 1289359  1296559  1303760 1310961 1318161 1325362 1332562  1339763 1346963 0.35 215.78 1294445  1301359  1308273 1315187 1322101 1329015 1335929  1342843 1349757 TABLE III. RESULTS TPM AND TOTAL COST OF MAINTENANCE OF A MANUFACTURING FIRM WITH INVENTORY CONSIDERATION AND LOST EARNING. Min R PM Interval tpm Total Cost (Naira) Min R PM Interval tpm Total Cost (Naira) Min R PM Interval tpm Total Cost (Naira) I=1.0 0.75 78.66 5220784 0.61 119.61 4006922  0.47 *166.45 3243810  0.73 84.84 5157225  0.59 *125.18 3884318 0.45 *173.58 3165904 0.71 90.6 4903070  0.57 *131.58 3756919 0.43 181.83 3083988 0.69 96.11 4689421  0.55 *138.69 3629927 0.41 189.63 3013647 0.67 101.5 4503691  0.53 *145.46 3521220 0.39 193.7 2979394 0.65 107.75 4312505  0.51 *151.91 3427228 0.37 206.66 2880148 0.63 113.81 4148027  0.49 *158.94 3334047 0.35 215.78 2818160 V. SUMMARY AND CONCLUSION The maintenance records of an existing manufacturing firm reveals that a structured way of determining the preventive maintenance interval will help manufacturing companies to appropriate maximum benefits from application of preventive maintenance policy. The identification of parameters helped improve the existing model and will enhance the practitioners’ idea in capturing the comprehensive costs that are involved in the preventive maintenance activities of the firm. The developed model is suitable for predicting an optimal preventive maintenance interval in a profit oriented manufacturing or service organization, where emphasis is laid on carrying out the preventive maintenance activities at a minimum cost. The model is able to predict an optimum interval for preventive maintenance at a minimum cost for the company and can be used as a decision support system. Also it can help determining the actual maintenance cost for each machine and thus evaluate the excesses of the company’s maintenance team. REFERENCES [1] B. S. Dhillon, Engineering Maintenance: A Modern Approach , CRC Press. Boca Raton FL, 2002 [2] M. G. Sobamowo, S. O. Ismail, S. J. Ojolo, A. A. 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