Building a maintenance policy through a multi-criterion decision-making model

Faghihinia, Elahe; Mollaverdi, Naser Article Building a maintenance policy through a multi-criterion decision-making model Journal of Industrial Engineering International Provided in Cooperation with: Islamic Azad University (IAU), Tehran Suggested Citation: Faghihinia, Elahe; Mollaverdi, Naser (2012) : Building a maintenance policy through a multi-criterion decision-making model, Journal of Industrial Engineering International, ISSN 2251-712X, Springer, Heidelberg, Vol. 8, pp. 1-7, https://doi.org/10.1186/2251-712X-8-14 This Version is available at: https://hdl.handle.net/10419/78583 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/2.0/ ORIGINAL RESEARCH Open Access Building a maintenance policy through a multi-criterion decision-making model Elahe Faghihinia 1* and Naser Mollaverdi 2 Abstract A major competitive advantage of production and service systems is establishing a proper maintenance policy. Therefore, maintenance managers should make maintenance decisions that best fit their systems. Multi-criterion decision-making methods can take into account a number of aspects associated with the competitiveness factors of a system. This paper presents a multi-criterion decision-aided maintenance model with three criteria that have more influence on decision making: reliability, maintenance cost, and maintenance downtime. The Bayesian approach has been applied to confront maintenance failure data shortage. Therefore, the model seeks to make the best compromise between these three criteria and establish replacement intervals using Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE II), integrating the Bayesian approach with regard to the preference of the decision maker to the problem. Finally, using a numerical application, the model has been illustrated, and for a visual realization and an illustrative sensitivity analysis, PROMETHEE GAIA (the visual interactive module) has been used. Use of PROMETHEE II and PROMETHEE GAIA has been made with Decision Lab software. A sensitivity analysis has been made to verify the robustness of certain parameters of the model. Keywords: Preventive maintenance, Age-dependent PM policy, PROMETHEE II, Bayesian approach, PROMETHEE GAIA Background Global trade, higher levels of automation, and the desire to apply lean production are some factors that increase the demand for effective maintenance (Salonen and Deleryd 2011). In recent decades, industrial and service systems have realized that establishing a proper maintenance policy plays an essential role in achieving their objectives (Cholasuke et al. 2004; van der Meulen et al. 2008). It can also lead to maximizing their profits (Alsyouf 2009). One of the most important reasons of considering maintenance as a crucial concept can be its large contribution of operating budget in organizations with heavy investments in machinery and equipment (Tsang et al. 1999). Moreover, because of the development of technology, competitive industrial and service systems should make use of more advanced machinery which need higher levels of maintenance because they are more complex and more difficult to control (Alsyouf 2009). The role of maintenance in modern manufacturing systems is becoming even more important with companies adopting maintenance as a profitgenerating business element (Sharma and Yadava 2011). In order to avoid failures at random times and the effect of such failures on the performance of systems that appear as a reducing production rate and loss of quality of the products, maintenance management is required to reduce the loss of system operating time and the number of defective parts produced (Tsarouhas 2011). Maintenance is becoming a critical functional area in most types of organizations and systems including construction, manufacturing, transportation, etc. This increasing role of maintenance is reflected in its high cost, which is estimated to be around 30% of the total running cost of modern manufacturing and construction businesses. As such, planning for maintenance is becoming an essential part of planning for the whole organization (Al- Turky 2011). Therefore, common practices like repairing a system when there is a problem have to be substituted by monitoring the system condition and planning the maintenance intervals (Cavalcante and De Almedia 2007). Also, effective maintenance can extend the equipment life, * Correspondence: [email protected] 1 Department of Industrial Engineering, Islamic Azad University of Najafabad, Isfahan 8514143131, Iran Full list of author information is available at the end of the article © 2012 Faghihinia and Mollaverdi; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Faghihinia and Mollaverdi Journal of Industrial Engineering International 2012, 8:14 http://www.jiei-tsb.com/content/8/1/14 improve equipment availability, and restore the equipment to a good condition (Swanson 2001). Well-defined maintenance system will ensure optimal performance of the machineries (Oberschmidt et al. 2010). Therefore, it can not only improve the quality of goods and services but also satisfy and rather exceed customers’demands especially in service sectors (Oke and Charles-Owaba 2006). The importance of running proper maintenance policies in organizations has led researchers to define maintenance in several ways. For example, Tsarouhas (2011) defines maintenance as a tool whose objectives are to increase the time to failure and reduce the repair time of equipment. Al-Turky (2011) defines it as the activities related to maintaining a certain level of availability and reliability of the system and its components and the system ability to perform with a standard level of quality. Still, maintenance can be defined as the combination of all technical efforts which can retain an item or equipment, or restore it to an acceptable operating condition (Dhillon 2002; British Standards Institute Staff 1993). With regard to the critical role of maintenance in improving reliability, preventing unexpected system failures and reducing maintenance costs, maintenance and replacement problems have been widely studied from different perspectives in the literature, and several models have been proposed (Wang 2002). All of them seek to elaborate on different maintenance problems and propose more rational solutions. This paper proposes a multi-criterion decision-aided maintenance model with regard to three criteria important to selecting the best maintenance policy. They are maintenance costs, reliability, and maintenance downtime criteria. This model not only considers the various aspects of a maintenance problem but also attends to the preference of a decision maker. Furthermore, Bayesian approach has been applied to overcome failure data shortage. Finally, a sensitivity analysis has been made to verify the robustness of certain parameters of the model. Preventive maintenance Complex equipment and machinery systems used in the production of goods and delivery of services constitute the vast majority of capital invested in industry (Savsar 2011). As time passes, the machines age and unplanned failures occur, causing the system performance to drift away from its initial state. In fact, no piece of equipment or system can continue to function without failure forever; however, carefully it might have been designed and manufactured (Samar Ali and Kannan 2011). System deteriorationisoftenreflectedinhigherproductioncostsand lower product quality. Therefore, the function of the system must be periodically restored to the desired level; this is practically achieved by maintenance operations. Proper maintenance can increase the reliability of a piece of equipment or a system at regular intervals (Samar Ali and Kannan 2011). Such maintenance is known as preventive maintenance (PM); it is done periodically before the failure of the system; hence, it is different from corrective or repair maintenance, which is carried out only after the failure of the item or the system (Savsar 2011). To keep production costs down while maintaining good product quality, PM is often performed on systems subject to deterioration (Savsar 2011). The probability of failure would increase as a machine is aged, and it would sharply decrease after a planned PM is implemented (Savsar 2011). It should be pointed out quickly that the maintenance actions which are normally classified as corrective maintenance (CM) include all actions performed as a result of a failure to restore an item to a specified working condition, while PM includes all actions performed on an operating equipment to restore it to a better condition (Oberschmidt et al. 2010). Moreover, making use of CM could be costly for organizations because most of the time, CM takes a long time to have an acceptable effect on a failed system or component (Nakagawa 2005). Thus, it can be disastrous for some systems where failures and interruptions could be dangerous. For example, we can consider military systems, aircraft, and health systems where a small mistake can lead to a horrible disaster (Cavalcante and De Almedia 2008). Also, the costs of applying CM in organizations are usually three or four times bigger than applying PM (Chitra 2003). So, it would be more rational to study PM models as a basic concept for the purpose of proposing an optimum maintenance model. In addition, PM policies are used for contexts where the component failure rate increases by age and usage (Cavalcante and De Almedia 2008). PM models Although a lot of maintenance models have been created during the past decades, there are few maintenance policies on which all the other maintenance models can be based (Wang 2002). There is a categorization proposed by Wang (2002). According to him, there are seven categories of maintenance policies, of which five are preventive. They are age-dependent PM, periodic PM, failure limit, sequential PM, and repair limit. According to age-dependent PM policy, a unit is replaced at the predetermined time Tor in the case of failure, whichever occurs first, where Tis a constant (Barlow and Hunter 1960). The given time Tis measured from the time of the last replacement (Wang 2002). According to periodic PM policy, a unit is preventively maintained at fixed time intervals independent of the failure history of the unit and repaired at intervening failures where Tis a constant. Faghihinia and Mollaverdi Journal of Industrial Engineering International 2012, 8:14 Page 2 of 15 http://www.jiei-tsb.com/content/8/1/14 According to failure limit policy, PM is performed only when the failure rate or other reliability indices of a unit reach a predetermined level, and intervening failures are corrected by repairs. According to sequential PM policy, a unit is preventively maintained at unequal time intervals under the sequential PM policy. Usually, the time intervals become shorter and shorter as time passes, considering that most units need more frequent maintenance with increased ages. According to repair limit PM policy, when a unit fails, the repair cost is estimated and repair is undertaken if the estimated cost is less than a predetermined limit; otherwise, the unit is replaced. He also indicates that the age-dependent policy can be the most common and popular PM. In several recent works, age-replacement policy was extensively studied. The age-replacement policy and its extensions belong to the age-dependant policy (Wang 2002). Therefore, by taking a look at PM models, we can realize that there are a large variety of preventive maintenance models and their extensions, so it would be necessary to specify a given problem to resolve in this context. Therefore, the age-replacement policy has been chosen as the basis for this research. Also in this paper, it is assumed that the replacement of a piece of equipment or part gives the system a goodas-new performance. In addition, there are two requisites for PM implementation in each system where (Cavalcante and De Almedia 2008): 1. The replacement cost of a component (c p ) before failures should be less than the cost of replacement due to failures (c f ). 2. The component failure rate should increase by age and usage. This paper proposes a multi-criterion decision-aided model with three criteria which deals with the problem of the replacement times. It determines the best timing and frequency for replacing components by taking into account three criteria, which are the total cost of maintenance per unit of time, the reliability, and the total maintenance downtime per unit of time. Thus, after choosing the policy followed by this research, it is important to describe the importance of these three criteria in maintenance decision making. Maintenance has become one of the most important issues in the manufacturing industry due to high costs involved (Savsar 2011). In production systems, maintenance managers concentrate on reducing maintenance costs (Cavalcante and De Almedia 2008). In manufacturing organizations, maintenance-related costs are estimated to be 25% of the overall operating cost (Cross 1988). According to Maggard and Rhyne (1992), the maintenance can represent between 10% and 40% of the production cost in a company. Coetzee (2004) means that the numbers should be 15% to 50%. Bevilacqua and Braglia (2000) state that maintenance costs can represent as much as 15% to 70% of the total production cost. So, it seems plausible that the maintenance costs may very well represent over 15% of the total production cost in industry (Salonen and Deleryd 2011). These findings show that maintenance cost cannot be ignored by maintenance managers. But there are several situations in some organizations where other criteria like reliability, availability, downtime, etc., play critical roles in systems. Earlier, researchers were using the optimization criteria as minimizing system maintenance cost rate, ignoring the reliability performance. In fact, minimizing system maintenance cost rate may not imply maximizing the system reliability measures. Sometimes, when the maintenance cost rate is minimized, the system reliability measures are also so low that they are not acceptable in practice (Sharma and Yadava 2011). According to Cavalcante and De Almedia (2008), in the services sector, the decision maker can show a preference for minimizing undesirable consequences which are difficult to measure in financial units. Because in this context, the customer is in direct contact with the production, and frequent interruption in the service can negatively affect the desire of the customer to enter into a new contract with that supplier or can lead the customer to cancel the current contract, which is unacceptable in competitive markets today. Generally, managers would like to see their system run as planned, and an unscheduled event such as a machine failure will disrupt the smooth running of the plant. Sometimes, the marketing department brings emergency product orders for important customers, and a system failure may result in severe losses (Chareonsuk et al. 1997). Therefore, looking at the cost criterion as the most important factor to establish an optimum maintenance modelisaverydangerousperspectiveforindustrial and service systems, especially for systems where failures and interruptions could be disastrous. It is moreover impossible to capture all of a system’seffectsina cost function. In some systems, the reliability criterion plays an essential role and which must be taken into account when an optimum maintenance model is to be established. Therefore, in a number of situations, maintenance managers mean to consider reliability as a separate criterion (Chareonsuk et al. 1997). Reliability, R(t), is the probability that a component or system will perform its design function for a specified mission time, given the operating conditions. Faghihinia and Mollaverdi Journal of Industrial Engineering International 2012, 8:14 Page 3 of 15 http://www.jiei-tsb.com/content/8/1/14 Table 1 Preference functions (adapted from Brans and Mareschal 2005) Generalized criterion Definition Parameters to fix PdðÞ¼ 0d≤0 1d>0 - PdðÞ¼ 0d≤q 1d>q Q PdðÞ¼ 0d≤0 d p0≤d≤p 1d>p 8 > < > : P PdðÞ¼ 1d≤q 1 2q<d≤p 1d>p 8 > < > : p,q PdðÞ¼ 0d≤q dq pqq<d≤p 1d>p 8 > < > : p,q PdðÞ¼ 0d≤0 1ed2 2e2d>0 8 < : S s=(p+q)/2. Faghihinia and Mollaverdi Journal of Industrial Engineering International 2012, 8:14 Page 4 of 15 http://www.jiei-tsb.com/content/8/1/14 There is another factor added to the model maintenance downtime. A question might be in order here. How important is this factor in maintenance decision making? Chareonsuk et. al (1997) considered a situation where preventive maintenance cost is not high, so in order to keep the system at a high level of reliability, the maintenance department decides to run maintenance programs very frequently. This can lead to very frequent shutdowns where the production department will be reluctant to attend to the preventive maintenance programs. Therefore, they will either cause forced postponed preventive maintenance or schedule when there is no production (for example, at night when it would be inconvenient for maintenance people). This problem can postpone maintenance programs. Therefore, in practice, maintenance programs cannot be fully maintained (Chareonsuk et al. 1997). Besides, downtime is very important and must not be neglected because the minimum downtime for a piece of equipment could result in undesirable consequences (Cavalcante and De Almedia 2008). With respect to the importance of taking into account the criterion of cost per unit of time, the reliability criterion and the maintenance downtime in making a proper maintenance decision need to be integrated in considering maintenance scheduling in a multi-criterion environment. This paper seeks to determine PM intervals during which the three criteria are in their best compromise with each other. Bayesian approach Mathematics has had an important role to extend maintenance models. Stochastic mathematical models have been developed to improve system reliability, prevent unexpected failures, and reduce maintenance costs (Zhang 2005). The use of mathematical modeling for this purpose is well established in the literature (Sortrakul and Cassady 2007). A number of surveys have been published by some authors in this area. McCall (1965) proposes a survey of researches on maintenance policies subject to stochastic failure. Pierskalla and Voelker (1976) also present a survey on maintenance models for deteriorating systems. Sherif and Smith (1981) review various maintenance models subject to failure and propose a classification. Sharma and Yavada (2011) present a survey on maintenance optimization models. More surveys can be found in the studies of Valdez-Flores and Feldman (1989), Cho and Parlar (1991), Dekker (1996), and Wang (2002). This part explains the mathematical requirements. In order to plan a maintenance program in this research, a failure distribution is needed which has a wear-out characteristic, namely the failure rate should increase with age. The Weibull model is a most prevalent distribution that satisfies this perquisite. It can be shown to be of the form: fðtÞ¼β η ⋅ t η  β1 β;η>0;t>0ð1Þ Table 2 Model’s parameters c a ($) c b ($) T f (days) T p (days) β 1 η 1 β 2 η 2 1,000 250 3 0.5 3.40 4.15 2.80 2200 Figure 1 GAIA plane. Figure 2 Decision model flowchart. Faghihinia and Mollaverdi Journal of Industrial Engineering International 2012, 8:14 Page 5 of 15 http://www.jiei-tsb.com/content/8/1/14 where ηis called the scale parameter; β, the shape parameter. In order to establish optimum maintenance intervals, we need to recognize the failure behavior of system or component. Thus, the parameters of the failure distribution of system or component should be estimated. In order to estimate the distribution function parameters, historical dataareoftenused;therefore,alargequantityofdatais needed to obtain reliable estimates. But because of the rapid growth of industry, often sufficient historical information about the components or system failures is not available (Chen and Popova 2002). Often, only a few failure data are available, and in some cases where there are enough data, they are not reliable (Scarf 1997). Therefore, estimation parameters from failure data is another difficulty in maintenance program (Cavalcante and De Almedia 2008). However, during the process of the system production and its operating time, reliability engineers and specialists find out by intuition about its failure behavior (Chen and Popova 2002). Combined with actual observations, this information can provide better assessment of the failure rate parameters. Bayesian analysis is one way to enter this information into the decision-making process in order to make a more objective decision. Therefore, a major advantage of Bayesian analysis is when only a few data are available. Bayesian statistics provides a way to incorporate specialist advice about a system into the maintenance model. The Bayesian maintenance models have been used frequently to establish maintenance policies in recent decades (Cavalcante and De Almedia 2008). Some authors such as Jorgenson et al. (1967), McCall (1965), Dayanlk and Gurler (2002), Wilson and Benmerzouga (1995), Sheu et al. (2001), Juang and Anderson (2004), Kallen and Van Noortwijk (2005), Makis and Jardine (1992), McNaught and Chan (2011), and many others have used this approach in different maintenance models (Oberschmidt et al. 2010). Finally, using a Weibull distribution to model failure in cases of incomplete data, specialist knowledge can be used. Therefore, the Weibull distribution parameters are considered random variables with a priori distributions representing specialist knowledge: μ(η) and μ(β). Figure 3 Criteria relationships. Table 3 Performances of alternatives (g i (t)) T(days) R(t)C(t)D(t) 200 0.9904 1.2874 0.0030 300 0.9797 0.8890 0.0018 400 0.9643 0.6996 0.0014 500 0.9441 0.5944 0.0013 600 0.9192 0.5315 0.0012 700 0.8899 0.4928 0.0011 800 0.8566 0.4690 0.0011 900 0.8198 0.4551 0.0011 1,000 0.7802 0.4478 0.0011 1,100 0.7383 0.4452 0.0012 1,200 0.6947 0.4458 0.0012 1,300 0.6502 0.4489 0.0012 1,400 0.6051 0.4534 0.0012 1,500 0.5604 0.4591 0.0013 1,600 0.5162 0.4654 0.0013 1,700 0.4730 0.4722 0.0013 1,800 0.4314 0.4792 0.0014 1,900 0.3916 0.4861 0.0014 2,000 0.3538 0.4930 0.0014 2,100 0.3182 0.4996 0.0014 2,200 0.2849 0.5060 0.0015 2,300 0.2541 0.5120 0.0015 2,400 0.2258 0.5179 0.0015 2,500 0.1998 0.5230 0.0015 2,600 0.1762 0.5274 0.0016 2,700 0.1549 0.5318 0.0016 2,800 0.1358 0.5357 0.0016 2,900 0.1187 0.5393 0.0016 3,000 0.1034 0.5425 0.0016 Table 4 Criteria thresholds Criteria Thresholds pQ Reliability 0.0300 0.1000 Cost 0.0150 0.2000 Downtime 0.0011 0.0001 Faghihinia and Mollaverdi Journal of Industrial Engineering International 2012, 8:14 Page 6 of 15 http://www.jiei-tsb.com/content/8/1/14 After estimating the Weibull parameters, evaluation of the cost, reliability, and maintenance downtime criteria can be obtained using the model, and then a multi-criterion decision with PROMETHEE methods can be made. PROMETHEE: one of the multi-criterion decision-making methods By taking a look at decision-making problems in the real world, it can be seen that most of them are multicriterion. Decision making in many contexts depends on several criteria not just on one criterion. This can be seen in many fields such as industries, economics, finance, or politics. Making decisions in maintenance programs can be a multi-criterion decision problem. According to Shyjith et al. (2008), selecting a maintenance policy based on a few factors makes it unrealistic. There is a need to consider maintenance problems as multi-criterion. This outlook can give a comprehensive view to maintenance management. So, it can be critical to consider maintenance problems as multi-criterion especially in systems that take into account only the cost criterion for making a maintenance decision because in some systems with special conditions, it could result in disasters. If the maintenance department only wants to look at the cost criterion, it could lead it to ignoring other criteria like reliability or maintenance downtime. A multi-criterion problem is mathematically defined as (Brans and Mareschal 1994 a,b; Brans et al. 1984): Max g1aðÞ;g1aðÞ;...;giaðÞ;...;gkaðÞ⋅⋅⋅aєA jg ; ð8Þ where Ais a finite set of npossible alternatives {a 1 , a 2 ,... ,a n }, and {g 1 (.), g 2 (.), ... ,g k (.)} is a set of evaluation -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 200 300 400 500 600 700 800 900 1000 1100 Scenario1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Scenario 6 Scenario 7 Scenario 8 Scenario 9 Scenario 10 Figure 5 Net flow values. Table 5 Criteria weights Criteria Weights (%) Reliability 35 Cost 40 Downtime 25 Figure 4 PROMETHEE II ranking 1. Faghihinia and Mollaverdi Journal of Industrial Engineering International 2012, 8:14 Page 7 of 15 http://www.jiei-tsb.com/content/8/1/14 criteria. They are called the basic data of a multi-criterion problem (Brans and Mareschal 1994a,b; Brans et al. 1984). In recent years, many decision-aid methods have been proposed. The PROMETHEE methods are one group of these methods consisting of seven. PROMETHEE methods developed by Brans are one of the best known and most widely used outranking approaches in many applications (Makis and Jardine 1992). A comprehensive overview of applications can be found in Behzadian et al. (2010). In general, outranking approaches are based on comparisons of pairs of alternatives (Oberschmidt et al. 2010). The PROMETHEE methods have been frequently used in many fields, and their success is due to their mathematical processes and the fact that they are easy to use by decision makers (Brans and Mareschal 1994a,b; Brans et al. 1984). The input required concerns the evaluation of the criteria for all of the alternatives considered as well as the weightings needed to reflect their relative importance. In order to apply PROMETHEE, first, the performance of the alternatives regarding all criteria needs to be determined. Then, alternatives are compared in pairs for each criterion based on generalized preference functions. Based on the weighted sum of single criterion preferences, positive and negative outranking flows are calculated as a measure of dominance of alternatives. Criteria weights reflect the subjective relative importance of the criteria. Based on positive and negative outranking flows, a partial preorder of alternatives can be defined according to PROMETHEE I. The net outranking flow can also be calculated to avoid incomparabilities and define a complete preorder on the set of alternatives according to PROMETHEE II (Oberschmidt et al. 2010). After that, PROMETHEE III that ranks alternatives based on intervals and PROMETHEE IV, the continuous case, were developed by Brans and Mareschal. They also proposed the visual interactive module GAIA in 1988, which provides an interesting graphical view to support the PROMETHEE methodology. In 1992 and 1994, Brans and Mareschal extended these two types: PROMETHEE V, an extension of PROMETHEE I and II where a subset of alternatives has to be selected by considering a set of constraints, and PROMETHEE VI, an extension of the results from PROMETHEE I and II that provides the decision maker with the freedom to think of the weight in terms of intervals, rather than of exact values (Brans and Mareschal 1994a,b; Brans et al. 1984; Cavalcante and De Almedia 2008). The PROMETHEE II method has been chosen for outranking results in this research, and the PROMETHEE GAIA has been chosen for a visual realization and sensitivity analysis of the results in this research. The reasons for selecting these methods are fast use, easy-to-analyze results, and a flexible comparison process (Cavalcante and De Almedia 2008). Moreover, the information which needed to use PROMETHEE and GAIA is easy and clear to define for decision makers (Brans et al. 1984; Brans and Mareschal 1994b). To make use of PROMETHEE methods, first, the two following phases should be passed (Brans and Mareschal 1994a,b; Brans et al. 1984; Cavalcante and De Almedia 2008): 1. Calculating the evaluation of each alternative for each criterion : g i (a); and 2. Calculating the differences between the evaluations of the alternatives within each criterion: dia;bðÞ¼giaðÞ–gibðÞ ð9Þ We also need two types of additional information to run PROMETHEE (Brans and Mareschal 1994a): 1. The information between the criteria that consists of the relative importance of the different criteria and which depends on the preferences of a decision maker. They are shown by w j ,j=1,2,... ,k.They are considered as norm weights. 2. The information within the criteria is referred to assign a preference function to each criterion. 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Submit your manuscript to a journal and benefi t from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the fi eld 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com Faghihinia and Mollaverdi Journal of Industrial Engineering International 2012, 8:14 Page 15 of 15 http://www.jiei-tsb.com/content/8/1/14