Fuzzy multi-criteria decision model for evaluating reconfigurable machines

ARTICLE IN PRESS Int. J. Production Economics 117 (2009) 1–15

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Fuzzy multi-criteria decision model for evaluating reconfigurable machines M.R. Abdi  Bradford University School of Management, Emm Lane, Bradford, West Yorkshire BD9 4JL, UK

a r t i c l e in fo

abstract

Article history: Received 5 April 2006 Accepted 3 June 2008 Available online 11 October 2008

The design and configuration of manufacturing equipment require crucial decision considering optimum capacity and functionality. The equipment selection problem might be involved with choosing between large-capacity machines versus a greater number of machines with smaller capacities, and/or dedicated facilities versus multiproduct facilities. This paper investigates reconfigurable machining system characteristics in order to identify the crucial factors influencing the machine selection and the machine (re)configuration. Furthermore, changeover cost and changeover time while switching from one product to the other are taken into account. In particular, a fuzzy analytical hierarchical process (FAHP) model is proposed to integrate the decisive factors for the equipment selection process under uncertainty. The expected values of the normalised fuzzy sets are determined to identify the preference values of the alternative machines. The fuzzy multi-criteria model is analysed within the fuzzy domains of the operational characteristics along with economic, quality and performance criteria. The proposed model is examined using monitoring sensitivity analysis through a case study. As a result, the alternative machines are prioritised with consideration of the inconsistency ratios. The relative performances of the alternative equipment in view of interactions of process reconfigurability and cost, and capacity and functionality are graphically illustrated. & 2008 Elsevier B.V. All rights reserved.

Keywords: Manufacturing Reconfigurable machine Decision making Fuzzy analytical hierarchical process (FAHP)

1. Introduction Advanced manufacturing systems need the capability to meet the fast-changing market demands quickly and cost effectively. The efficiency of the former paradigm of mass production was based on stability and control over the product types. In turn, new paradigm of mass customisation emphasises on creating variety and customisation (Chick et al., 2000). Consequently, the efficiency of such manufacturing systems is based on flexibility and quick responsiveness. The emphasis on high product variety forces manufacturers to customise the products, be flexible enough to produce a variety of products on the

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same system, be responsive to switch from one product to another. For the contemporary and future production facilities, the ability to launch new product types must be incorporated with new process technologies within existing systems. Fixed automation needs a high investment on special purpose equipment i.e. the dedicated transfer lines. On the other hand, flexible equipment is designed and configured to produce a range of product types with known manufacturing operations. In contrast, reconfigurable manufacturing system (RMS) is designed at the outset for rapid changes in hardware and software components in order to quickly adjust to production capacity and functionality within a part family in response to sudden changes in market or in regulatory requirements (Koren et al., 1999). In this manner, RMSs are open-ended systems and described by five key

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characteristics: modularity, integer-ability, convertibility, diagnosability and customisation (Mehrabi et al., 2000). Classical objectives for evaluating manufacturing systems have been known as low cost and high quality. However, cost, quality, responsiveness and performance are progressively becoming the four tangible cornerstones on which every manufacturing company can survive in the current/future dynamic market. Accordingly, the decision-making model must consider the benefits, costs, opportunities and risks for each alternative machine. The element must be clustered and evaluated along with their interactions in favour of alternative equipment. The benefits/opportunities cluster yields the alternative equipment with the most benefits/opportunities such as the highest reconfigurability/responsiveness. The cost/risk cluster yields the alternative equipment with the least costly/risky equipment. Reconfigurable equipment can provide sufficient flexibility to produce a number of product types grouped into families on the same system by means of different configurations. In addition, the equipment must be designed with certain qualitative and quantitative chara cteristics to achieve exact flexibility (no more no less) in response to fluctuations in demands. Therefore, a modularity-based structure in the equipment design stage as well as the process design stage will allow manufacturing systems to produce high product variety (Huang and Kusiak, 1997). The design and configuration of a manufacturing facility requires crucial decisions concerning product mix and capacity (Karmarkar and Kekre, 1989) that could deal with choosing between dedicated and multi-product facilities and/or large-capacity machines versus a group of smaller capacity. In the most conventional studies concerned with equipment selection, the consideration of process reconfigurability, product variety and new product introduction have been ignored. Therefore, such approaches must be restructured to obtain a reconfigurable decision support system with consideration of products variety, changeover time and changeover cost, and uncertain conditions. Due to the complexity and uncertainty of the decision process interacting elements, a fuzzy multi-criteria decisionmaking approach, which has been developed by a number of researchers such as Yager (2002) and Fenton and Wang (2008), is required to support managers in selecting the appropriate equipment choice. The objective of this paper is to develop a decision-making process for equipment selection whilst considering quantitative/qualitative objectives and measuring manufacturing process requirements and financial criteria using the fuzzy sets.

2. Fuzzy set: theory and application A fuzzy set, introduced by Zadeh (1965), is described by a membership function, which reflects the membership degree, generally between 0 and 1, for all the possible values. Therefore, a fuzzy set consists of values with different degrees of membership showing how much each value belongs to that interval. A typical triangular fuzzy number (TFN) can be defined as x _ ¼ ðl; m; uÞ with in

which l is the lower value, m is the mid-value and u is the upper value. The membership function mðx _ Þ can then be defined as 8 loxpm > < x=ðm  lÞ  l=ðm  lÞ

mðx_ Þ ¼

> :

x=ðm  uÞ  u=ðm  uÞ

moxpu

0

otherwise

(1)

Fuzzy set theory is a powerful tool for dealing with uncertainty existing in reconfigurable manufacturing due to the continuous changes inside the system caused by unpredictable market demands. The equipmrent configuration is changed everytime a new (diffeent) product family entered the system. In most resaerches dealing with evaluating alternative equipment configurations, reconfiguration time and reconfiguration cost have been ignored. However, these parameters associated with each equipment configuration will emerge a different system performance in terms of quality, cost, time and efficiency. The performance measures influencing the equipment selection problem are concerned with uncertain events, and are vague. In particular, capacity and reconfiguration time vary frequently and can be expressed by fuzzy sets. Furthermore, all the other quantitative and qualitative criteria with certain or fuzzy values must be taken into account. The appropriate membership functions for the most affected parameters i.e. capacity and reconfiguration time are investigated in the paper. Consequently, an integrated structure of the fuzzy membership functions expressed with the linguistic fuzzy terms and the AHP is developed while considering both quantitative and qualitative criteria. 3. Reconfigurable process Manufacturing processes can make a major impact on equipment configurations. Therefore, the process must be fully identified and managed prior to equipment selection in terms of monitoring new and/or unexpected conditions occurred during the production. In order to operate effectively, manufacturer must be able to capture, represent and reconfigure the processes in the environment, where the constraints and requirements are constantly evolving. As a result, alternative processes must be considered and the most suitable process must be selected. The alternative manufacturing processes must be found according to the four main requirements as follows: 1. Operation details: Producing a part requires many tasks. Some tasks can be accomplished by a variety of processes. A number of requirements must be evaluated to determine the appropriate process. 2. Product design: The product design characteristics such as tolerance, geometry and material will affect operations required for the product. 3. Machine specifications: Alternative processes can be generated based on the machine specifications such as max horsepower and the operator expertise and experience.

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4. Departments: The selection of different manufacturing departments within the existing system to operate on different products with different operational requirements will affect manufacturing processes. The selected department with a high capability and a greater performance will improve existing processes. The limitation and possible changes of each process must be taken into account in order to reconfigure the processes.

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with cutting tools. REs stand for a set of available machine tools in a manufacturing system in which relative motions exist between parts and cutting tools. Each REkj represents a collection of cutting tools and specific geometry k including the exclusive and the shared capability between all the available machines j, operated in a manufacturing facility. Machf ¼

n m X 1X 1 EREkj Pm2REk n k¼1 REkj j¼1

(2)

j¼1

The requirements consideration helps to reconfigure the process due to the change of manufacturing systems such as buying a new machine or developing a new process and/or installing a new manufacturing system. The changes need to be re-evaluated at each configuration stage in order to efficiently reconfigure the processes. Furthermore, the specifications of machines types are different and will influence the processes reconfigurations. Alternative processes must be generated and selected according to the type of the assigned machines. Having found the capacity requiresd, the number of machines needed would simply be equal to the demand for the machine divided by the machine capacity. Even for the same type of machines, the process capabilities in terms of capacity and functionality depend on the status of the machine such as the typical specifications in which each machine can tolerate. 4. Reconfigurable machine Having identified the manufacturing processes, the machines must be designed, assigned and configured for customised flexibility. This customisation may result in an optimal hybrid machine of a dedicated machine type and a multi-tool computerised numerical control (CNC) machine. The reconfigurable machines could be the CNC machines with reconfigurable tools. The machines should have a modular and changeable structure enabling adjustment to the resources such as different spindle units. Process management could vary from a basic level for dedicated facility, an advanced level for multi-purpose facility and a complex level for reconfigurable facility due to dynamic changes of requirements. In order to find out the type and number of machines, the required manufacturing operations must be determined. Accordingly, products must be grouped into families before manufacturing based on their operational similarities through a reconfiguration link between market and manufacturing (Abdi and Labib, 2004a). Machine reconfigurabilty can be comprehended by flexibility and/or convertibility, which are concerned with capacity and functionality to enhance different operational requirements. In contrast, convertibility is related to the responsiveness level to changes in the production. A mathematical expression to calculate machine flexibility (Machf) based on resource elements (REs), is presented by Gindy and Saad (1998) by the use of Eqs. (2) and (3). REs are defined as ‘facility-specific capability units, which capture information relating to the distribution (commonly and uniqueness) among available machines

where REkj is the resource element k on machine j, k ¼ 1,2, y, n and j ¼ 1,2, y, m; m the number of machines; n the number of different resource elements; EREkj the efficiency of resource element k on machine j; and also EREkj ¼

ðSREkj Þmin ðPREkj Þmin  SREkj PREkj

(3)

where P REkj is the processing time required by resource element k on machine j; SREkj the set-up time required by resource element k on machine j. Each SRE represents the set-up time for the corresponding RE (machine tool) on a specific machine, whereas each PRE represents the operation time for the identical machine tool. The efficiency of each RE measures the degree to which reconfiguration and operation time might affect the maximum throughput, which is concerned with the minimum set-up time and the minimum operation time Therefore, the minimum SREkj and PREkj are divided by the actual SREkj and P REkj in order to determine the set-up time efficiency and the operation time efficiency, respectively. As a result, the overall ERE is determined by multiplication of the two efficiency ratios. Simulation can be applied to measure the operational system performance and test the model’s ability when internal and external disturbances such as machine breakdown are allowed. It can be seen that machine flexibility has a direct relationship with the efficiency, which reflects the appropriate level of capacity and functionality. In turn, Machf has an inverse relationship with the number of resource elements and the cumulative resource elements over machines. This can be expressed as convertibility in terms of product variety and changeover time/cost. 4.1. Machine reconfiguration Due to the complexity of manufacturing operations, machines with different configurations cause different impacts on the manufacturing system in term of capacity, functionality and the changeover time/cost. For example, a machining center that performs both drilling and milling processes could perform the operations along with the product changes at a shorter time than when the two different machining centers separately operate. The value of using existing design configuration for a new product type is an economic factor called ‘reusability’, which can reduce extra investment for system reconfiguration (Abdi and Labib, 2003). For each product family a specific configuration of equipment may apply. The products required by

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customers are grouped into families indexed as 1,2, y, m. Production planners will then involve with X ¼ {x1,x2, y, xm}, where xi is the number of production rate or demand for product family i. Therfore, family i (iom) corresponds to one of the configuration stages stated as as si. The products belonging to family i are produced by the corresponding configuration. The set of configurations will be S ¼ {s1,s2, y, sm}. The equipmrent configuration is changed everytime a diffeent family is selected. A reconfiguration time and a reconfiguration cost are incurred when changes from si to sj (iaj). The alternative processes for the alternative equipment configurations must be evaluated. Since reconfigurable equment must adapt to demand variations, its design may vary over time in order to be able to change with new requirements of market and manufacturing in terms of changes in product mix and volume and fast introduction of new products. 4.2. Equipment selection approaches Manufacturing machine selection for a given planning horizon has mostly been investigated using two main approaches: (1) analytical approaches and (2) simulation. Most analytical approaches are based on mathematical programming considering production requirements and capacity volumes such as De Matta et al. (1999), and an integer-programming model with a heuristic algorithm such as Chen (1999). Multi-attribute utility theory (MAUT) as a mathematical approach can be applied for the equipment selection problem considering strategic infrastructure factors as proposed by Stading et al. (2001) based on the linear weighting rule model of Keeney and Raiffa (1976). On the other hand, a Monte Carlo simulation model could be applied for designing and selecting inspection systems and equipment choices (Delurgio et al., 1997). The AHP developed by Saaty (1980) is one of the multicriteria decision-making approaches that decomposes a complex problem to a hierarchical order and has been used by various researchers for decision-making in production systems such as (Kengpol and O’Brien, 2001), (Yusuff et al., 2001), Abdul-Hamid et al. (1999), and Hafeez et al. (2002), Simulation could be linked to the AHP in order to deal with uncertainty for evaluation of machines configurations (Chan and Abhary, 1996). The configuration selection of RMSs can be obtained by AHP with consideration of the multiple performance criteria of productivity, quality, convertibility and scalability (Maier-Speredelozzi and Hu, 2002). Due to deal with vague data a fuzzy set can also be connected to the conventional approaches. Kulak et al. (2006) developed the axiomatic design for the equipment selection problem using fuzzy sets while considering both crisp and fuzzy criteria. A fuzzy benefit/cost approach is used for uncertain data such as interest rate and determination of present value of cash flow (Kahraman et al., 2000). However, such classical approaches could only evaluate the cost effects on a manufacturing system such as capital, running and overhead costs without providing the details of cost information of products and manufacturing facilities. The fuzzy logic could be added to

the classical AHP for capturing optimum degree of capacity utilisation while considering demand uncertainty (Monitto et al., 2002; Weck et al., 1997). The fuzzy number must be normalised for the criteria of the judgment matrix to evaluate the alternatives (Chang, 1996; Zhu and Jing, 1999). The literature review unveils that there is still a lack of analytical approaches capable of evaluating the vague data of qualitative and quantitative performance criteria of a manufacturing system with different equipment configurations. 4.3. Machines selection criteria Manufacturing processes and resources are determined mostly at the early stage of the component design Kusiak and Lee (1997). The level of reusability and unused capacity to handle the uncertainties should also be considered. In addition to capital cost, operational criteria related to reconfigurability such as capacity and functionality, and particularly changeover cost and time imposed on the system while switching from one product to the other should be taken into account (Abdi, 2005). There are two main aspects of the equipment selection criteria as follows: (a) manufacturing requirements such as process/product requirements and (b) market requirements such as product type and volume, cost and customer satisfaction. In order to evaluate different alternative equipment the required specifications must be identified according to the manufacturing operational requirements. Machine specifications are determined based on the part features and the process requirements. These specifications are classified and expressed with respect to operation (capacity and functionality) and time (reconfiguration). The operation-related specifications are focused on operative requirements dealing with products volumes and types, whereas the time related ones are associated with machine response while switching from a product to the other product. Having considered the machine specifications in connection with operation and time and the criteria of cost, quality and performance, the machine selection procedure can be structured. As illustrated in Fig. 1, firstly, the selected products are designed according to the market and manufacturing requirements. Secondly, the process requirements are determined according to the part features of the product(s) design. The processes are considered from two aspects of time and operational requirements in order to determine the reconfigurable machine characteristics. Thirdly, the criteria for evaluating reconfigurable machines are set up according to the operational and economic attributes. Finally, the appropriate machine is selected from a set of feasible available machines. 5. Fuzzy multi-criteria decision approach An integrated configuration of the equipment selection criteria can determine the overall judgment of the machining capability with respect to reconfigurability, cost, quality, performance and risk. Evaluation of the

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Product(s)-machine specifications Market requirements

Product(s) design

Manufacturing requirements

Product(s) standardisation Part features

Process requirements

Operationoriented specifications

Time-oriented specifications

Reconfigurable machine characteristics

Economic attributes

Criteria for reconfigurable machine selection

Operational attributes

Economic feasibility

Alternative feasible machines

Operational feasibility

Machine selection Fig. 1. Reconfigurable machine-selection procedure.

Identification of factors

Structuring relationship

Quantification/ fuzzification of crisp/fuzzy factors

Defuzzification of fuzzy factors

Overall performance

Sensitivity analysis

Fig. 2. The fuzzy multi-criteria decision approach.

alternative manufacturing equipment requires quantification and aggregation of the performance criteria with different measure units. Pound can measure cost, quality may be measured by percentage of defects, reconfiguration time may be measured by minutes, and the number of product types and range of production volume can measure variety and capacity, respectively. In addition to uncertainty of the performance values different decision makers may have different understandings of the performance criteria, and therefore they may prioritise them differently. As a result, the equipment evaluation necessitates tolerating vague data. As shown in Fig. 2, the fuzzy multi-criteria decision approach consists of the following six steps: 1. Identification of the fuzzy/crisp factors affecting performance of reconfigurable machines; 2. Structuring the relationship between the factors; 3. Quantification of the crisp factors and fuzzification of the fuzzy factors influencing the machining system performance in comparison with the other factors; 4. Defuzzification of the fuzzy attributes; 5. Overall performance of each alternative equipment; and 6. Sensitivity analysis of the equipment selection policy.

5.1. The FAHP model The proposed model is intended to provide an integrated hierarchical structure of quantitative and qualitative criteria along with the current/future requirements of the internal/external factors influencing reconfigurable machine selection. Accordingly, the citeria are transferred into a common linguistic score for their evaluation. Quantitative fuzzy elements are defined using triangular/trapezoid fuzzy numbers. The fuzzy numbers are then transferred to the fuzzy preference values as the input to the AHP model. In addition, sensitivity analysis for the crucial attributes in the case study is undertaken within their fuzzy domains. The evaluation process is proposed as follows: 1. 2. 3. 4.

Set strategic goal(s); Set the objectives and criteria in a hierarchical order; Identify available machine/configuration options; Trade-off the objectives and criteria with fuzzy/crisp values to evaluate the alternative machines within the domains of the fuzzy attributes; and 5. Select the most preferred machine (configuration). The strategic goal is to find the most appropriate equipment or configuration that can be linked to the other

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objectives such as customer satisfaction, introducing new products, increasing market share, and the need to get the products to market more rapidly with greater quality and reduced costs. The main objectives of the proposed fuzzy AHP model are defined as manufacturing reconfigurability, cost, quality and performance. The objectives have been broken into sub-objectives as criteria leading to the feasible equipment alternatives. Manufacturing capacity and manufacturing functionality are then considered as the distinctive criteria. The model is hierarchically structured as depicted in Table 1. The main goal (level 0) is to select the most preferred equipment configuration based on four main objectives at level 1: manufacturing reconfigurability, cost, quality and performance. Manufacturing reconfigurability and cost are decomposed into criteria positioned at level 2, which may themselves consist of sub-criteria (level 3). The hierarchy will lead to the alternative equipment configurations EC1, EC2 and EC3 at level 4. Accordingly, the AHP model consists of the five hierarchical levels including quantitative and qualitative criteria as follows: Level 0: The main goal, which is the equipment selection for the manufacturing system. Level 1: The main objectives, which are manufacturing reconfigurability (PROCESS), cost (COST), quality (QUALITY) and performance (PERFORM). Levels 2 and 3: The criteria with their corresponding sub-criteria if there is any.

Level 4: The alternative equipment configurations: EC1, EC2 and EC3 for the case study.

Once the hierarchy is structured, the quantitative evaluation through pairwise comparisons is performed for all elements at each hierarchy level with respect to the next higher-level elements. All the criteria or alternatives are evaluated in support of the favourable contribution (positive effect value) with respect to the higher-level criterion or objective. The comparisons are undertaken in favour of higher reconfigurabilty, higher quality and higher performance and lower cost. For example, a machine with a higher capacity and a higher price has more contribution to manufacturing reconfigurability, but an inferior contribution to capital cost. It is important to note that if the production system needs a new machine, the criteria can be applied for the evaluation of feasible and available alternative machines to be purchased. The installation of new machine(s) may create the need for a new machining system design. In contrast, if the existing marching system can cope with new requirements in terms of capacity and functionality, the existing machines must be reconfigured to reflect new requirements. The system reconfiguration can appear by means of (inter)changing process routes, relocation of machines, sharing machines, retooling machines and/or using multi-directional material handling systems.

Table 1 Hierarchy levels of the decision elements Level 1: Objectives

Level 2: Criteria

Level 3: Sub- criteria

Level 4: Alternatives

Manufacturing process Reconfigurability (PROCESS)

Capacity (CAPACITY)

Set up time (SETUP) Changeover time (CHANGET) Varity (VARIETY) New product introduction (NEW PROD) Mobility (MOBILITY) Volume (VOLUME)

EC1

Functionality (FUNCTION)

EC2 Cost (COST)

Operating cost (OPERATE)

Labour (LABOUR) Maintenance (MAINTAIN) Work in process (WIP) Changeover cost (CHANGEC)

Capital cost (CAPITAL)

Price (PRICE) Install (INSTAL) Tools and fixtures (TOOLS)

Overhead (OVERHEAD) Quality (QUALITY)

Convenience of use (CONV) Reliability (RELIAB) Accuracy (ACCUR) Compatibility (COMPAT)

Performance (PERFORM)

Efficiency (EFFIC) Risk (RISK) Safety (SAFE)

EC3

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5.2. Quantification of reconfigurable machine capacity using fuzzy sets Since the operational requirements for each product have different levels of machine utilisation, the machine capacity/production rate varies over time. As shown in Fig. 3, machine capacities for the three products A, B and C are different as indicated as follows: CA ¼ machine capacity for product A; CB ¼ machine capacity for product B; CC ¼ machine capacity for product C. It can be seen that each unit of product A requires more time than product B to be operated by the individual allocated machine. In contrast, each unit of product B needs less operational time than product C being processed on the same machine. As a result, the machine capacity for product B is greater than the machine capacity for product A, and the machine capacity for product A is greater than the machine capacity for product C (CB4CA4CC). The capacity changes for the different product types may take place in three sub-periods as follows (Abdi and Labib, 2004b): ti1 (i ¼ A,B,C): Set-up time i.e. the time required to introduce a new product type and/or change products within the product family. It includes machines set-up time including retooling and operators reassignments. ti2 (i ¼ A,B,C): Steady-state time i.e. the time that machine is processing without any changes in the current product type. Therefore, no manufacturing reconfiguration occurs and production rate can rise up to the maximum level (machine capacity). ti3 (i ¼ A,B,C): Switch-off time i.e. the time required to switch a machine from an existing product to the next one within the product family determined in the production range. During set-up time and switch-off time, reconfigurable machines may still be operating before completely finish processing the existing product type and/or starting processing the next product type. As a result, these two sub-periods can be recognised as machine reconfiguration times, which may vary over configuration stages.

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The reconfiguration/production process for each product over the sub-periods can be heuristically expressed as a trapezoid fuzzy number as illustrated in Fig. 4. Each edge of the trapezium presents a start (end) of a sub-period of the reconfiguration/production process for the corresponding product. The vertical axis is the ‘membership function’ value of the fuzzy reconfiguration/production process, where the horizontal axis is the ‘time’ value. The duration of set-up time and switch-off time will be a fuzzy variable indicated as ti1 and ti3 (i ¼ A,B,C) and can be defied as a trapezoid fuzzy defined in Section 5.1. Fuzzy variables quantifying the sub-period times and their membership functions are applied to put emphasis on their effects on manufacturing capacity whilst settingup and/or switching-off a product. The set-up and switchoff times for the products within a family will have a tighter mean with a lower tolerance compared with those in different families. The capacity during off-reconfiguration sub-period posses the maximum capacity over the reconfiguration/production process with more stability than the other two sub-periods. The triangular membership functions m(C(t)) for the fuzzy machine capacity C(t) can be expressed by Eq. (4). The triangular membership functions m(C(t)) is a function of capacity C(ti), which varies between 0 and 1 for the configuration period t including the sub-periods mentioned before. C(t) can be denoted as (Cl(t), Cm(t), Cu(t)), where Cl(t)pCm(t)pCu(t) and are the lower, modal and upper values of C(t), respectively. If Cl(t) ¼ Cm(t) ¼ Cu(t), then C(t) is a non-fuzzy (crisp) number by principle. As shown in Fig. 5, during the production/reconfiguration periods, the possibility of Cl(t) and Cu(t) are zero and possibility of Cm(t) equals one. The slope between Cl(t) and Cm(t) is increasing whereas the slope between Cm(t) and Cu(t) is decreasing. 8 0 > > > > CðtÞ C l ðtÞ > > > < C m ðtÞ  C ðtÞ  C m ðtÞ  C ðtÞ l l mðCðti ÞÞ CðtÞ C u ðtÞ > > >  > > C m ðtÞ  C u ðtÞ C m ðtÞ  C u ðtÞ > > : 0

CB CA Product A

Set-up time

Product B

C l ðtÞoCðtÞpC m ðtÞ C m ðtÞoCðtÞpC u ðtÞ CðtÞ4C u ðtÞ (4)

Machine production rate

CC

CðtÞpC l ðtÞ

Product C

Switch-off time

Steady-state time Fig. 3. Machine production over production/reconfiguration periods.

Time

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Membership function (µ)

1

Product A

Set-up time

Product B

Product C

Time (t)

Switch-off time Steady-state time Fig. 4. Fuzzification of production/reconfiguration sub-periods.

Membership function (µ)

µ(C(t)

1 increasing slope

decreasing slope

1 Cl(t)

Cm(t)

Cu(t)

C(t)

Fig. 5. Quantification of machine-capacity utilisation using TFN.

Assuming the machine production rate for product A during a configuration stage can be in a range of 250–400 units a day with a mode of 300. Therefore, Cl(t), Cm(t), Cu(t) are 250, 300 and 400 units, respectively. It then appears that the fuzzy capacity variable Cl(t) can be fuzzified by (250, 300, 400). Therefore, the fuzzy membership function m(C) will be obtained as below 8 0 Cp250 > > > < ðC  250Þ=ð300  250Þ 250oCp300 (5) mðCÞ ¼ > ð400  CÞ=ð400  300Þ 300oCp400 > > : 0 C4400

5.3. Quantification of machine production/reconfiguration time using trapezoidal fuzzy sets To determine manufacturing facilities required for the system design, a time study over manufacturing operations and reconfigurations is required to determine reconfiguration times between product changes. In most studies related to reconfigurable systems, the reconfiguration time has been ignored such as Yigit et al. (2002), Zhao et al. (2001) and Koren et al. (1999), as it is assumed to be very short. Considering reconfigurable machine specifications described in Section 4, time-oriented specifications associated with convertibility must be taken into account.

tl

tm

tn

tu

Time (t)

Fig. 6. Quantification of production/reconfiguration time using trapezoidal fuzzy number.

Therefore, set-up time, production time (steady state) and switching time between two sequential products are crucial factors for achieving an accurate evaluation of reconfigurable machines. The reconfiguration time is trapezoidal (Fig. 6); hence, trapezoidal fuzzy variables could be employed for quantifying the setting-up and/or switching-off of a product with their membership functions to put emphasis on their impacts on the manufacturing capacity in the reconfiguration stages. Accordingly, the two subperiods of set-up time (t1) and switch-off time (t3) can be defined individually by fuzzy variables as formulated in Eq. (6), where m(ti) is a trapezoidal membership function for sub-period i; (i ¼ 1,3) within the three intervals [til, tim, tiu]. 8 0 > > > > i > > ðt  t il Þ=ðt im  t il Þ > < i mðt Þ ¼ 1 > > > > ðt iu  t i Þ=ðt iu  t in Þ > > > :0

t i pt il t il ot i pt im t im ot i pt in t in ot i pt iu t i 4t iu

(6)

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Products must be grouped into families only if there are reasonable operational dissimilarities among families. Therefore, switching to a product belonged to a different family causes an increase in set-up time. In contrast, setup time might be approximately equivalent to a switch-off time whilst changing products among similar products within a family. Practically, similar products are regularly altered for processing, and subsequently a number of frequent similar reconfiguration processes with a shorter reconfiguration time take place. As a result, the set-up and switch-off times for the products within a family will have a tighter expected value with a lower tolerance compared with those in different families.

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µ (x) Membership functions defined by linguistic priorities l m n u L H VH EQ M

1

0

1

5

3

9

7

x

Fig. 7. Fuzzy linguistic priorities.

5.4. The fuzzy preference scale Having structured the classical AHP hierarchy, due to uncertainty of the importance weights of pairwise comparisons, the elements of matrix A (see Appendix A) are characterised by the fuzzy membership functions for both fuzzy quantitative and qualitative criteria. The first step of the proposed FAHP model is to identify the relative importance of each pair factors in the same hierarchy level. By using triangular fuzzy numbers via pairwise ˆij elements comparisons, the fuzzy evaluation matrix A of a ˆij ¼ (l, m, u) is the importance of is constructed, in which a element i over element j under a certain criterion with lower (l), mean (m), and higher (h) values, respectively. The value (ul) represents a fuzzy degree of judgment. The greater (ul), the fuzzier the degree; when ul ¼ 0, the judgment is a non-fuzzy number with m importance value. Fuzzy consistency can be described as the existence of relative weights within the region (Leung and Cao, ˆij1 ¼ (1/u, 1/m, 2000). Subject to the fuzzy consistency, a 1/l) represents the fuzzy importance of element j over element i with lower value (1/h), mean (1/m) and higher value (1/l). As a result, the fuzzification increases the complexity of computational operations for synthesis judgments, which are basically performed on the fuzzy ˆij s). Similarly, the linguistic trapezoidal elements (a fuzzy number for reconfiguration time is represented by ˆij ¼ (l, m, n, u), which denotes the importance of element a i over element j under a certain criterion with low (l), medium (m), notable (n) and ultra (u) values, respectively. To simply evaluate the FAHP model through the typical pairwise comparisons, all the fuzzy values are standardised into a single-pattern fuzzy set dealing with both linguistic and quantifiable criteria. Accordingly, the importance weights are defined with five fuzzy sets ^ 3; ^ 5; ^ with their corresponding lower, mean, and ^ 7; ^ 9, 1; upper values defined as 8 ^ > > < 1; 2 ð1; 1; 3Þ (7) a^ ij ¼ x^ ; 2 ðx  2; x; x þ 2Þ; 1oxo9 > > : 9; ^ 2 ð7; 9; 9Þ ^ 3; ^ 5; ^ are ^ 7; ^ 9) As shown in Fig. 7, the fuzzy range of (1; used to express linguistic priorities for both quantitative and qualitative criteria. These model criteria can be compared and measured by using the fuzzy linguistic priorities in terms of equal (EQ), low (L), medium (M),

Table 2 Linguistic priorities and the quantified values between two criteria i and j Criterion i

VH

H

M

L

EQ

L

M

H

VH

9

7

5

3

1

3

5

7

9

Criterion j

high (H) and very high (VH). For instance, the linguistic trapezoidal value for l, m, n and u can be represented by L, M, H and VH, respectively. The pairwise comparison can be then undertaken by using the preference bar as outlined in Table 2. For example, the quantified preference values of EQ and VH for an element to another element are distinguished by 1 and 9, respectively. For example, if value 5 is assigned to criterion (cj) at the right side of the bar, the criterion cj will be more important than ci with a moderate degree. Similarly, if value 5 is assigned to criterion (ci) at the left side of the bar, the criterion ci will be more important than cj with a moderate degree. The corresponding values of the preferences can also appear between any two fuzzy linguistic preferences. In addition, if one of the non-zero numbers is assigned to criterion (ci) when compared with criterion (cj), then cj has the reverse preference value when compared with ci. For example, if functionality retains moderate importance value 3 comparing with capacity with respect to manufacturing process, then capacity possess the relative importance value 1/3 to functionality. Similarly, the reverse fuzzy value for 1 the linguistic possess score 3^ can be represented by 3^ ^ or (1=3). The synthesis judgement degree of the FAHP model can be derived from the expected values of the normalised fuzzy numbers as illustrated in the following example. Assuming that there are three equipment choices: EC1, EC2 and EC3 for a reconfigurable manufacturing system, and the alternatives are pairwise compeered with respect to manufacturing capacity as shown in Table 3. Values 1 on the pairwise matrix diameter are not fuzzy. However, to simplify the calculation for the synthesis judgement degrees, those can be represented by a fuzzy set (1,1,1), as shown in Table 4. The three-value judgments in the decision matrix could be normalised in the range of [0, 1], and then

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Table 3 The fuzzy sets of pairwise comparison matrix of the alternative machines with respect to machine capacity

Table 5 The normalised fuzzy numbers for the alternative machines with respect to machine capacity

Alternative

Alternative

Alternative EC1

EC2

EC3

EC1

1

EC2

1=3^ 1=1^

3^ 1

1^ 5^ 1

EC3

1=5^

Table 4 The fuzzy numbers for the alternative machine with respect to machine capacity Alternative

EC1 EC2 EC3

EC1 EC2 EC3

EC2

EC3

(1/1, 1/1, 1/1) (1/3, 1, 1) (1/3, 1, 1)

(1, 3, 5) (1, 1, 1) (1/7, 1/5, 1/3)

(1, 1, 3) (3, 5, 7) (1, 1, 1)

EC1

EC2

EC3

(1, 1, 1) (0.11, 0.33, 0.33) (0.11, 0.33, 0.33)

(0.2, 0.6, 1) (0.2, 0.2, 0.2) (0.03, 0.04, 0.07)

(0.14, 0.14, 0.43) (0.43, 0.71, 1) (0.14, 0.14, 0.14)

Table 6 The fuzzy performance judgement of the example Alternative

Alternative EC1

Alternative

EC1 EC2 EC3

Alternative EC1

EC2

EC3

EV

1 0.28 0.28

0.6 0.2 0.05

0.21 0.71 0.14

0.52 0.36 0.12

configurations, which can meet the operational requirements with the total production rate 4P units/h as follows: transformed to the performance matrix. The normalising transformation for the performance of benefit criteria would be different from that for the cost criteria. The normalised triangular fuzzy numbers, proposed by Fenton and Wang (2006), are used in the research as defined as n^ ij

8 > < a^ ij =M;

M ¼ a^ ij

max

with respect to a benefit criterion

> : N  a^ ij =N;

N ¼ a^ ij

max

with respect to a cost criterion

i i

(8) Since the machine capacity is a benefit criterion, the normalised fuzzy values are obtained as shown in Table 5. ˆij ¼ (al, am, au) is a According to Liu and Liu (2002), if a triangular fuzzy variable, then the expected value will be ˆij ¼ (al, am, an au) is a equal to (al+2am+au)/4. Similarly, If a trapezoidal fuzzy variable, then the expected value will be equal to (al+am+an+au)/4. Accordingly, the expected value of the normalised fuzzy values from Table 5 can be calculated along with the corresponding engine value (EV) for each alterative as illustrated in Table 6. As a result, EC1 with EV ¼ 0.52, is preferred to EC2 with EV ¼ 0.36, and EC2 is preferred to EC3 with EV ¼ 0.12. 6. A case study A case study is carried out for company A in order to demonstrate the application of the proposed FAHP model. The decisive factors of the model are considered as to be directly or indirectly linked to reconfigurability of alternative machines. The balance weights of the preference values are the model input by using the Expert Choice software (Expert Choice, 1999). The monitoring sensitivity analysis will then be performed within the fuzzy domain of the fuzzy criteria. Four highly demanded product families A, B, C and D are selected in the production range. The production rate for each product family is simplified as to be P units/h. There are three feasible different alternatives of machine

1. EC1: four machines each of capacity P units/h. Each machine can produces all product types within the four-product families A, B, C and D. A minor set up is needed if a product type is changed. 2. EC2: two machines each of capacity 2P units/h. One machine can produces product types within the product families A and B, and the other one can produce models within product families C and D. 3. EC3: Only one large machine of capacity 4P units/h. The machine can produces all product types within the four-product families A, B, C and D. A major set up is needed while switching from a product type to the other type. 6.1. The analysis of solutions An important benefit gained from the FAHP model is that the interaction of the factors can be clearly identified and expressed in quantitative terms. This identification will bring us one step forward in understanding the dynamic behaviour of market/manufacturing factors affecting the performance criteria. The priorities and criticality of the criteria and sub-criteria will change as the manufacturing and market environments change. Therefore, it is important to rapidly recognise these changes in order to reconfigure the model through restructuring and re-evaluating the criteria. In contrast, priority changes of some elements up to a certain level in the fuzzy range would have no significant effects on the selection policy. In this respect, the current selected equipment and configuration can be still remained unchanged. As shown in Fig. 8, the model is hierarchically built in the software with the balance defuzzified priorities of objectives, criteria and alternatives. The objectives and criteria are evaluated and prioritised by manufacturing system designers of the company. Some of the preferences

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11

Fig. 8. The hierarchy of the elements weighed for the case study.

OVERALL INCONSISTENCY INDEX = 0.05 EC3 EC1 EC2

0.354 0.336 0.310 Fig. 9. Overall solution (preferred alternative equipment).

values are estimated based on data, while the others are based on verbal judgments according to the current/future strategy of the company. All the preference values of qualitative and quantitative are expressed by the fuzzy linguistics values, and then ranked between 1 and 9. The alternative machines priorities are derived by synthetic judgment of the whole elements of the hierarchy and stated in terms of the percentage scale (%). The initial synthesis of the hierarchy produces the alternative machines priorities. As shown in Fig. 9, the alternative equipment configurations are prioritised with respect to the goal (the best equipment choice). Accordingly, EC1 is preferred to EC3, and EC3 is preferred to EC2. In other words, EC14EC34EC2 with overall weights 0.381, 0.325 and 0.294, respectively. The inconsistency

ratio of the initial judgment is 0.05 and acceptable since it is less than the standard level 0.10 according to Saaty (1994). A linear presentation of the alternative machines priorities against a single-criterion facilitates the analysis of solutions within the fuzzy range of the criterion. Fig. 10 presents a single-criterion graph which highlights the alternative equipment priorities with respect to process reconfigurability (PROCESS). The alternative priorities are shown on Y-axis, while the PROCESS priority is shown on X-axis. At the current priority level of PROCESS (42%) shown by a solid vertical line, the equipment alternatives are preferably ordered as EC34EC14EC2. This means that machine 3 is preferred to machine 1, and machine 1 is preferred to machine 2. The two vertical dash lines located

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on the left/right side of the solid line present the critical lower/higher limit of the PROCESS weights. These dash lines reflect the possible changes in the equipment selection while changing the priority of PROCESS in comparison with the other objectives at the first level of the hierarchy i.e. cost, quality and performance. The left vertical dash shows that a decrease in the current PROCESS priority with regard to the goal will result in changing the solution order to EC14EC34EC2. It can be seen that if the PROCESS priority drops under 32% (tradeoff point with the dash), the most preferred alternative will be changed to EC1. On the other hand, the right vertical dash shows that an increase in the PROCESS priority being over 58% (trade-off point with the dash) will result in the solution order indicated as EC34 EC24EC1. As shown in Fig. 11, the current solution is concerned with the current preference of the reconfiguration time of the alternative machines (solid vertical line nearby priority 18%. An increase in the relative preference of the reconfiguration time over 46% in comparison with the

other criteria at the same level will changes the solution to EC14EC34EC2. This solution change highlights the fact that EC1 is less concerned with negative impacts caused by configuration time while switching from one product to the other. The analyses of the two graphs above were based on changing one criterion while all the others are remained unchanged. In reality, the relative preferences of two or more criteria at the same level may change at the same time. A two-criterion graph coordinating the fuzzy preference values of two selected criteria can facilitate the assessment with simultaneous changes. In this section, the effects of two specific criteria on the alternative machines are evaluated at the same time by using a projection graph. Fig. 12 shows the interactions of the PROCESS and COST priorities affecting the alternative machines. The current priorities result in the solution order being EC34EC14EC2 (shown by small circles). In the case of high priority for either COST or PROCESS, the solution order will be changed to EC34EC24EC1 (shown by big circles).

0.40

Alternative (%)

0.30

0.20

0.10

0.00 0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 0.8 Priority of process reconfigurability (PROCESS)

0.9

1

Fig. 10. Alternative solutions with respect to manufacturing reconfigurability.

0.60

Alternative (%)

0.50 0.40 0.30 0.20 0.10 0.00 0

0.1

0.2

0.3

0.4

0.5 0.6 0.7 Priority of CON TIME

0.8

0.9

Fig. 11. Analysis of the alternative machines with respect to the reconfiguration time.

1

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13

0.40

COST

0.30

0.20

0.10

0.00 0.00

0.10

0.20 PROCESS

0.30

0.40

Fig. 12. Simultaneous evaluations of alternative equipment based on COST and PROCESS.

0.50

CAPACITY

0.40

0.30

0.20

0.10

0.00 0.00

0.10

0.20

0.30 FUNCTION

0.40

0.50

Fig. 13. Simultaneous evaluations of alternative equipment considering capacity and functionality.

Similarly, capacity and functionality have interactive preferences with respect to process reconfigurability (see Fig. 13). At the current preference levels of capacity and functionality shown by the upward line, the small circles on the line present the defuzzified alternative priorities. The projected value of each alternative on the horizontal axis (FUNCTIONALITY) determines the corresponding priority of the alternative with respect to FUNCTIONALITY. In contrast, the projected value of each alternative on the vertical axis (CAPACITY) determines the corresponding priority of the alternative with respect to the CAPACITY. Any changes of the preferences result in changing the solution. For example, EC1 seems more attractive while increasing capacity rather than functionality. In turn, EC3 appears to be more beneficial to the production system while increasing functionality (rather than capacity) to respond product variety.

7. Conclusion The FAHP model is developed to take both quantitative and qualitative criteria with fuzzy/crisp values into

account. Reconfigurable-machine characteristics are classified and expressed with respect to operations (capacity and functionality) and reconfiguration time as the distinctive factors. Machine capacity and production/ reconfiguration time are quantified by using triangular or trapezoid fuzzy numbers. The fuzzy linguistic scale is then developed to establish a common preference values for the pairwise comparisons. The expected values are utilised to defuzzifiy the preference values and reach the synthesis judgments. The feasible machine types are considered as the alternative solutions of the model. The alternatives are compared and analysed via monitoring sensitivity analysis within their fuzzy domains of the defined criteria through a case study. Single-criterion graphs are presented to facilitate the analysis of solutions within the fuzzy range of a single criterion. In comparison, two-criterion graphs are illustrated to asses the alternative machines with simultaneous changes of the criteria within their fuzzy ranges. Using the expected values has simplified the evaluation process and offered meaningful results. This has significantly reduced the computational time and effort required

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to perform the sensitivity analysis. However, the assumption might limit the validation of the proposed model particularly when all criteria are allowed to change within their fuzzy ranges simultaneously. Although the proposed model is generic and could be applicable in many firms, the hierarchy structure can differ from one company to another because of differences in: the product families in the production range, operational requirements, feasible technology and available budget for investment on equipment. In addition, the cost of material handling system and the available floor space are the other vital factors, which could be added to the criteria. As a further research, the cost of handling material and floor space must be addressed while considering a new facility or the redesign of the existing facilities layout. The model can be developed using a group decisionsmaking approach with conducting manufacturing engineers, plant managers, operators and suppliers. Analytical Network Process (ANP) can be developed while accepting interactions among the high level elements and the lower level ones via a super matrix. The author intends to proceed this research in order to explore reconfigurable layout characteristics and the key elements of workstation formation i.e. machines, tools, material handling systems, floor space and operators.

Acknowledgement The author would thank an anonymous referee for the constructive comments, which helped to improve the paper. Appendix A. The AHP and the FAHP theories The AHP decomposes a complex problem to a hierarchical structure. Pairwise weighing among n elements in each level leads to an approximation to the ratio of aij ¼ wi/wj, which is the weight of element i divided by the weight of element j. The estimated weight vector w is found by solving the following eigenvector problem: Aw ¼ lmax w

(A.1)

where matrix A consists of aijs, lmax is the principal eigenvalue of A. If there is no inconsistency between any pairs of elements then aij is equal to 1/aji for any i and j, and we have Aw¼nw

(A.2)

In reality, consistency does not usually take place and the formulation (A.2) can be expressed as Aw ¼ lmaxw ¼ E, where E is the principal eigenvalue, a value around n (the total number of elements in the same level). To estimate eigenvector (E), each column of A is first normalised and then averaged over its rows. The vector is used to find the relative importance of each element with respect to the higher level of hierarchy. The inconsistency ratio (IR) is given as by (lmaxn)/(n1), which is the variance of the error incurred in estimating matrix A. As if an inconsistency becomes more than 10%

the problem and judgements must be investigated and revised. To construct a fuzzy judgment matrix A we denote comparison quantitatively with the triangular fuzzy numbers as aij ¼ (lij, mij, uij) in which mij is mid-value and can be valued from one to nine (1, 2, y, 9) or in reverse (1/9,1/8, y,1) as usually used in the AHP method. The synthesis judgement degree of a triangular fuzzy number located on the kth layer can be derived from the formula (A.3) proposed by Chang (1996). Ski

¼

n X

0 akij

@

j¼1

n X n X

11 akij A

;

i ¼ 1; 2; . . . ; n

(A.3)

i¼1 j¼1

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