Material Handling Equipment Selection using Fuzzy Axiomatic Design Principles

Transcription

1 Material Handling Equipment Selection using Fuzzy Axiomatic Design Principles Anant V. Khandekar 1*, Shankar Chakraborty 2 1* Department of Mechanical Engineering, Government Polytechnic, Bandra (East) Mumbai , Maharashtra, India, anantkhandekar2709@gmail.com 2 Department of Production Engineering, Jadavpur University Kolkata , West Bengal, India, s_chakraborty00@yahoo.co.in Abstract Effective movement of raw material, semi-finished and finished products plays an important role in successful operation of any manufacturing organization. Material movement from one workstation to another accounts for about 30 to 40% of the cost of final product. Proper methods adopted for material movement are also important for the overall safety of the personnel involved in the manufacturing processes. So, selection of the appropriate material handling equipment (MHE) is a vital task for improving the productivity of an organization. In today s technological era, numerous varieties of MHEs are readily available to carry out a desired task. At the same time, depending on the type of material to be moved, there are many conflicting factors influencing the MHE selection decision. For MHE selection, these factors are generally stated in both quantitative and qualitative terms. Hence, the problem of selecting the right type of MHE for a given task can be solved using a multi-criteria decision-making (MCDM) approach capable of dealing with a combination of crisp and fuzzy data. In this paper, an MCDM method based on fuzzy axiomatic design principles is applied for selecting the most appropriate MHE. Trapezoidal fuzzy numbers are employed for representing the qualitative attributes and subsequently converting them into crisp values. As a measure of suitability, total information content is calculated for each MHE alternative. The MHE alternative with the least total information content is regarded as the best choice. Two real time MHE selection problems from the literature are solved to establish the applicability and potentiality of the adopted approach. Keywords: Axiomatic design principles, Material handling equipment, Fuzzy set theory 1 Introduction Efficient and economical movement of all kinds of materials within a manufacturing unit results in the production of quality goods. Hence, the MHEs used for this critical activity assume a great deal of importance. Consequently, the proper type of MHE is to be chosen taking into consideration all the relevant parameters and functions. In addition to the main function of transportation, MHEs also carry out other auxiliary but equally important functions, like positioning, unit formation and storage. While performing all these functions, there should not be any obstruction to the other processes and safety of the concerned personnel should also be maintained. Selection of the MHE should be optimal as it affects each and every parameter of the manufacturing process, like lead time, efficiency of labor force, facility utilization and hence, the productivity of the organization (Onut et al., 2009). It is observed that there are about 50 different types of MHEs and they are characterized by about 30 different attributes. For MHE selection, most of the influencing attributes are expressed linguistically and use of fuzzy set theory comes as an aid for obtaining the final solution. Besides this, some selection criteria are of beneficial type, i.e. higher values are desired and others are non-beneficial where lower values are preferred. In addition, factors contributing to the complexity of MHE selection process are the constraints imposed by the layout of the existing facility, type of the material to be handled, variety of process requirements, wide range of MHEs available for a specific purpose and uncertainty in the operational environment. In the present paper, the application of fuzzy axiomatic design (FAD) principles is demonstrated while solving two MHE selection problems having vague and imprecise information. The derived results are largely in agreement with those obtained by the earlier researchers employing other MCDM methodologies, which vehemently establishes the potentiality of FAD principles as an efficient MCDM tool. As it involves less mathematical steps, its low computation time favors for its application. It has also a systematic and scientific base which helps to obtain accurate ranking results. 2 Literature Review During the period of last one and half decade, many researchers have dealt with the selection of MHEs while applying different mathematical approaches. About 25% of those research works are based on the successful deployment of various MCDM techniques integrated with fuzzy set theory. Deb et al. (2002) adopted a fuzzy 21-1

2 Material Handling Equipment Selection using Fuzzy Axiomatic Design Principles MCDM method to aggregate rating attitudes of the decision makers (DMs) and trade-off various selection criteria to find out values of fuzzy suitability indices for final ranking of MHE alternatives. A decision support system was developed by Kulak (2005) to identify the most appropriate MHE among the alternatives of the same type using fuzzy information axiom of axiomatic design (AD) principles. Mirhosseyni and Webb (2009) developed a hybrid method for selection and assignment of the most appropriate MHE for a given operation. At first, the developed system would select the most appropriate MHE for every material handling operation using an expert system consisting of crisp and fuzzy rules, and in the second phase, a genetic algorithm would search throughout the feasible solution space, constituting of all possible combinations of feasible equipments, in order to determine the optimal choice. Onut et al. (2009) proposed a combined MCDM methodology for evaluation and selection of MHEs for a steel construction company in Turkey. Fuzzy analytic network process (ANP) was utilized for assigning criteria weights and fuzzy technique for order preference by similarity to ideal solution (TOPSIS) was applied for deciding the best MHE type. Sawant and Mohite (2009) applied fuzzy TOPSIS method for assessing and ranking of automated guided vehicles, and studied the effect of varying the impreciseness of the criteria values on the suitability ranking of the alternatives. Ulubeyli and Kazaz (2009) applied ELECTRE III (ELimination and Et Choice Translating REality) method as an MCDM tool for selection of concrete pumps while collecting and analyzing the real time data from about 70 civil engineering firms having various construction equipments. Bazzazi et al. (2011) proposed a new fuzzy MCDM model for selecting the optimal type of open pit mining equipment while considering the objective, critical and subjective factors as encountered in real time situations. Lashgari et al. (2012) selected the optimal fleet of loading and hauling equipments to be used in an open pit iron mine at Gole Gohar, Iran while employing a hybrid approach of fuzzy analytic hierarchy process and fuzzy TOPSIS method. From the literature review, it becomes apparently clear that the applicability of FAD principles is still not sufficiently explored to analyze the variety of existing MHE selection problems. 3 Methodology The adopted methodology is an integrated approach comprising of the principles of AD theory and fuzzy set theory, as detailed out below. 3.1 Principles of AD theory This theory was originally proposed by Suh (1990) as a systematic and scientific approach for the design of products. It is based on logical thinking and takes into consideration the requirements of the end users of the products. Till that time, the process of designing of products was carried out simply by trial and error method. There used to be lots of iterations before getting the final design. While designing a product, the customer needs or attributes related to that product are determined first. The functional requirements (FRs) represent the features/characteristics of the product which are able to satisfy the customer needs. These FRs are then mapped into the final design parameters (DPs) of the product. In the present context of decisionmaking, FRs represent different criteria with respect to which the feasibility of an alternative is judged for its intended function. In AD theory, the DPs are expressed in terms of range of values. Usually, this range is set by the designer or DM for a certain DP and is known as design range (DR). The range of values of DPs for different available alternatives is known as system range (SR). Naturally, in order to satisfy a certain set of FRs, there can be different combinations of DPs. In other words, the best combination, i.e. the design solution from various alternatives available needs to be chosen. This is the logic behind using AD theory as an MCDM tool. It is based on two axioms, stated as below Independence axiom It stresses on maintaining the independence of FRs. The meaning of independence axiom is that a particular FR should be satisfied independently by a certain DP without affecting the other FRs (Kulak, 2005). In real time problems too, a given complex design or a decision task is decomposed into smaller components and the independent solution for each of them is sought. So, the independence axiom supports this analogy Information axiom The design solutions satisfying the independence axiom are further analyzed by the information axiom. According to this axiom, the information content (IC) of each alternative design solution is determined and the alternative with the minimum IC value is treated as the optimal choice (Suh, 2001). The IC is related in its simplest form to the probability of satisfying a given FR. It determines that the design with the highest probability of success is the best choice. The IC i value for a given FR i is defined using the following equation: 1 IC i = log 2 (1) p i where p i is the probability of satisfying the functional requirement FR i. The information is expressed in units of bits. The logarithmic function is chosen so that the IC values will be additive when there are many FRs that must be satisfied simultaneously (Suh, 2001), and the 21-2

3 logarithm is based on 2 which is the unit of bits of information. The DR is decided by the designer or DM and it is the ideal range of values to be tried to achieve in the design process. The SR denotes the capability of the available manufacturing process. As shown in Figure 1, the overlap between the designer-specified DR and the SR is known as common range (CR), where the acceptable solutions exist. or imprecisely, it is also expressed by TFN or TrFN according to fuzzy set theory. For example, if it is stated that the horizontal distance traveled by an MHE is approximately equal to 300 units, it becomes difficult for the DM to ascertain the exactness of the data for further processing. Therefore, this imprecise value is represented by a TrFN as (270,300,300,330), taking into consideration 10% fuzziness. On the other hand, another MHE property with a value approximately between 360 and 400 can be denoted by a TrFN as (324,360,400,440) with 10% fuzzification. A desired MHE cost smaller or equal to about USD 20K can be represented as (0,0,20,22) because the minimum cost is zero. Figure 2 exhibits a trapezoidal fuzzy membership function. Some basic concepts about TrFNs are briefly stated here-in-under. Figure 1 Design range, system range and common range of FR Therefore, in the case of uniform probability distribution function, the value of p i is given as follows: CR p i = (2) SR So, the value of IC can now be expressed as below: SR ICi = log2 (3) CR 3.2 Fuzzy set theory Fuzzy set theory states that, in a universe of discourse X, a fuzzy subset A ~ of X is defined by a membership function ~ ( x), which maps each element f A x in X to a real number R in unit interval of [0,1]. The function value f ~ A ( x) represents the grade of membership of x in A ~. The larger the value of f ~ A ( x), the stronger is the grade of membership for x in A ~. A certain MHE may be specified to have higher positional accuracy and moderate lifting speed as compared to the others. Many a times, the DMs come across such type of imprecise and vague information as opposed to definitive type of information, while taking real time decisions. Fuzzy set theory (Zadeh, 1965) was developed to handle such type of imprecise information in an efficient manner to arrive at the logical conclusion in a more scientific manner. It is used to convert imprecise linguistic terms, such as high and moderate into numerical values using triangular fuzzy numbers (TFNs) or trapezoidal fuzzy numbers (TrFNs). Similarly, when a quantitative property is stated vaguely Figure 2 Membership function of trapezoidal fuzzy number A fuzzy number A ~ is a TrFN if its membership function is expressed as follows: f ~ A ( x) = (x α)/(β α), α x β = 1, β x τ (4) = (x δ)/(τ δ), τ x δ = 0 otherwise with α β τ δ. The TrFN, as given above, can be denoted as (α,β,τ,δ). Let us assume a homogeneous group of k decision makers (D 1,D 2,,D k ), evaluating the suitability of m number of MHEs (A 1,A 2,,A m ) with respect to n number of criteria (C 1,C 2,,C n ). Let ~ (i =1,2,,m; j = 1,2,,n; t = 1,2,...,k) be the fuzzy rating assigned to MHE A i by the DM D t for criterion C j. Therefore, the rating of an MHE A i with respect to criterion C j can be aggregated as follows: ~ 1 t k x [ ~ 1 x ~ 2 = + x ~ x ~ x ] (5) k In order to ensure compatibility between evaluation of objective criteria and subjective criteria stated in linguistic ratings, a linear scale transformation is used to convert various criteria values into a comparable unit. t x 21-3

4 Material Handling Equipment Selection using Fuzzy Axiomatic Design Principles Let a j = min (α, β, τ, δ ) for all the MHE alternatives and cost criteria, then the aggregated rating of an MHE A i with respect to cost criterion C j can be formulated as below: = a j a j a j a j r ~,,, δ τ β α (6) 3.3 Fuzzy axiomatic design principles In most of the MHE selection problems, the alternative and criteria ratings are expressed in semantic scales, like low, medium, high or disagree, little disagree etc. Such type of unquantifiable data can be well dealt using fuzzy set theory. An integrated approach of FAD principles is now effectively used to solve the MCDM problems involving both imprecise and crisp data (Kulak and Kahraman, 2005). In FAD approach, the SR and DR are usually denoted using fuzzy numbers, i.e. TFNs or TrFNs due to the presence of imprecise information. In Figure 3, the TrFNs of SR and DR are respectively denoted by (α 1,β 1,τ 1,δ 1 ) and (α 2,β 2,τ 2,δ 2 ). Mathematically, the area of common range (CR) between SR and DR can be evaluated as follows: CR = 2 ( δ1 α2 ) ( δ τ + β ) α2 (7) The IC value as estimated applying FAD principles is then expressed as below: IC = log 2 (System area/area of common range between SR and DR) (8) Figure 3 Common range between system and design ranges 4 Illustrative examples Two examples from the real time MHE selection environment are solved here employing FAD principles to prove its universal applicability as an effective MCDM tool. 4.1 Example 1 An industrial organization from Turkey, manufacturing agricultural equipments, especially tractors, was in need of purchasing industrial trucks (IT) for its warehouse facility (Tuzkaya et al., 2010). For this, a decision-making team (DMT) was formed, consisting of warehouse manager, night and day shift warehouse superiors, five academicians and five expert labours responsible for the related warehouse activities. At first, the DMT shortlisted a feasible set of six IT alternatives, and four evaluation criteria as operational considerations (C 1 ), economical considerations (C 2 ), environmental considerations (C 3 ) and strategic considerations (C 4 ). The linguistic scale used by the DMT for evaluating the alternatives is presented in Table 1, showing the linguistic values and their corresponding TrFNs. Table 1 Linguistic scale for evaluation of alternatives Linguistic value TrFN Strongly disagree (SDA) (0,0.07,0.07,0.15) Disagree (DA) (0,0.15,0.15,0.30) Little disagree (LDA) (0.15,0.30,0.30,0.50) No comment (NC) (0.30,0.50,0.50,0.65) Little agree (LA) (0.50,0.65,0.65,0.80) Agree (A) (0.65,0.80,0.80,1) Strongly agree (SA) (0.80,0.9,0.9,1) The DMT then evaluated all the available alternatives with respect to the shortlisted criteria in terms of the linguistic values, as shown in Table 2. Table 2 Linguistic evaluations of alternatives Alternative C 1 Criteria C 2 C 3 C 4 IT 1 SA NC LA LA IT 2 SA LA A A IT 3 SDA DA DA DA IT 4 SA LDA A SA IT 5 A A SA A IT 6 DA DA A A Based on the experience and knowledge of the members of the DMT, the DR of each of the four criteria is decided on a scale of 0 to 1 and is shown in Table 3. These values are also expressed in terms of TrFNs and the performance of each alternative is evaluated against these values for computing its information content. Table 3 Design range of criteria Criteria Design range Operational considerations (C 1 ) (0.4,0.6,0.8,1) Economical considerations (C 2 ) (0.4,0.6,0.8,1) Environmental considerations (C 3 ) (0.4,0.6,0.85,1) Strategic considerations (C 4 ) (0.4,0.6,0.8,1) According to FAD methodology, the IC value is now calculated for each alternative-criterion pair. For example, the IC value for pair IT 1 -C 1 is computed with the help of Figure

5 Figure 4 System range, design range and common range for IT 1 -C 1 pair In this figure, triangle PQR represents the area of CR between SR and DR. Taking (α 1,β 1,τ 1,δ 1 ) and (α 2,β 2,τ 2,δ 2 ) as (0.4,0.6,0.8,1.0) and (0.8,0.9,0.9,1.0) respectively, and using Eq. (7), the area of CR is calculated as The area corresponding to TFN of SR is 0.1. Therefore, the value of IC is calculated using Eq. (8) as IC = log 2 (System area/area of CR between SR and DR) = log 2 (0.1/0.0667) = In this way, after determining the IC value for each alternative-criterion pair, the total IC for each alternative is derived by adding up their individual values, as shown in Table 4. It is observed that alternative IT 3 is not at all suitable for the desired function as it does not satisfy any of the four DRs of the considered criteria. Similarly, alternative IT 6 does not satisfy the first two criteria (C 1 and C 2 ) and hence, is ultimately rejected. The remaining four industrial trucks are now ranked according to their increasing values of total IC. It is observed that the ranking order of the four selected alternative trucks is exactly in agreement with that obtained by the past researchers. The two alternatives (IT 3 and IT 6 ) which are identified as unsuitable using FAD principles are also adjudged as the last choices by the past researchers while applying a combined ANP and PROMETHEE (Preference Ranking Organization METHod for Enrichment Evaluations) method. 4.2 Example 2 This MHE selection problem (Deb et al., 2002) consists of four alternatives. A team of two DMs (DM 1 and DM 2 ) evaluated each alternative against four dependent criteria. Among these, three criteria (supervision, safety and environment) are subjective in nature, whereas, the operating cost criterion is objective in nature. The DMs adopted a linguistic rating scale, as shown in Table 5, to evaluate the suitability of four MHE alternatives with respect to each subjective criterion. The membership functions in terms of TrFNs for these linguistic ratings are also subsequently developed. Table 5 Linguistic ratings of alternatives Rating TrFN Very poor (VP) (0,0,0,0.2) Poor (P) (0,0,0.2,0.4) Fair (F) (0,0.2,0.4,0.6) Good (G) (0.2,0.4,0.6,0.8) Very good (VG) (0.4,0.6,0.8,1) Similarly the set of linguistic values used for defining the DR of criteria as (Low, Medium, High) are expressed using TrFNs as (0,0.2,0.4,0.6), (0.2,0.4,0.6,0.8) and (0.4,0.6,0.8,1.0) respectively. The linguistic ratings of MHE alternatives for all the criteria are shown in Table 6. As decided by the DMs, the linguistic DR values and their corresponding TrFNs are shown in Table 7. Table 7 DR of criteria Criteria Linguistic rating TrFN C 1 High (0.4,0.6,0.8,1.0) C 2 High (0.4,0.6,0.8,1.0) C 3 High (0.4,0.6,0.8,1.0) C 4 Medium (0,0.2,0.4,0.6) In Table 6, it is observed that each alternative is evaluated against each subjective criterion by the two DMs separately. Therefore, these two ratings are aggregated using Eq. (5), as shown in Table 8. The Table 4 Total IC and ranking of alternatives Alternative IC C1 IC C2 IC IC C3 IC C4 IC Total Rank IT IT IT 3 Infinite Infinite Infinite Infinite Infinite Rejected IT IT IT 6 Infinite Infinite 0 0 Infinite Rejected Table 6 Ratings of MHE alternatives MHE C 1 C 2 C 3 C 4 DM 1 DM 2 DM 1 DM 2 DM 1 DM 2 DM 1 DM 2 MHE 1 F G P F G VG (18,22,28,32) 21-5

6 Material Handling Equipment Selection using Fuzzy Axiomatic Design Principles MHE 2 G F F G F P (22,26,32,38) MHE 3 P G G VG P F (22,25,28,33) MHE 4 VG G F F P G (16,20,28,33) Table 8 SR data of alternatives Alternative Supervision (C 1 ) Safety (C 2 ) Environment (C 3 ) Operating cost (C 4 ) MHE 1 (0.1,0.3,0.5,0.7) (0,0.1,0.3,0.5) (0.3,0.5,0.7,0.9) (0.5,0.57,0.73,0.89) MHE 2 (0.1,0.3,0.5,0.7) (0.1,0.3,0.5,0.7) (0,0.1,0.3,0.5) (0.42,0.5,0.62,0.73) MHE 3 (0.1,0.2,0.4,0.6) (0.3,0.5,0.7,0.9) (0,0.1,0.3,0.5) (0.48,0.57,0.64,0.73) MHE 4 (0.3,0.5,0.7,0.9) (0,0.2,0.4,0.6) (0.1,0.2,0.4,0.6) (0.48,0.57,0.8,1.0) Table 9 Total information content of alternatives Alternative IC C1 Information content IC C2 IC C3 IC C4 IC Total Rank MHE MHE MHE MHE operating cost criterion values are also normalized applying Eq. (6) and are given in Table 8. Thus, all the alternative ratings expressed in terms of TrFNs in the range of 0 to 1 are exhibited in Table 8 as the corresponding SR values. Now, considering the DR and SR data from Table 7 and Table 8 respectively, the IC value for each MHE alternative is computed in Table 9. Now, the MHEs are ranked based on their total IC values, indicating the superiority of MHE 4 over the other alternatives. 5 Conclusions The FAD principles as an MCDM tool are applied here for solving two MHE selection problems. In the first problem of industrial truck selection, two of the six alternatives not satisfying one or more design criteria, are adjudged to be unsuitable for the intended function. The suitability ranking order of the remaining MHE alternatives is exactly in conformance with that of the past researchers. The second problem of MHE selection consists of four alternatives and their ratings are expressed in qualitative as well as quantitative terms. The derived solutions from the two considered problems clearly confirm the applicability and flexibility of the adopted methodology to provide accurate MHE selection decision. Application of FAD principles enhances the creativity of DMs, eliminates the random search process involving trial and error techniques. Also, the less number of computational steps involved in this approach is an added advantage over the other MCDM methods. It can be equally applicable to other domains of manufacturing technology, like project selection, supplier selection and so forth. References Bazzazi, A.A., Osanloo, M. and Karimi, B. (2011), A new fuzzy multi-criteria decision making model for open pit mines equipment selection, Asia-Pacific Journal of Operational Research, Vol. 28, pp Deb, S.K., Bhattacharyya, B. and Sorhkel, S.K. (2002), Material handling equipment selection by fuzzy multicriteria decision-making methods, Proc. of Int. Con. on Fuzzy Systems (AFSS 2002), India, pp Kulak, O. (2005), A decision support system for fuzzy multi-attribute selection of material handling equipments, Expert Systems with Applications, Vol. 29, pp Kulak, O. and Kahraman, C. (2005), Multi-attribute comparison of advanced manufacturing systems using fuzzy vs. crisp axiomatic design approach, International Journal of Production Economics, Vol. 95, pp Lashgari, A., Abdolreza, Y.C., Fouladgar, M.M., Zavadskas, E.K., Shafiee, S. and Abbate, N. (2012), Equipment selection using fuzzy multi-criteria decision making model: Key study of Gole Gohar iron mine, Engineering Economics, Vol. 23, pp Mirhosseyni, S.H.L. and Webb, P. (2009), A hybrid fuzzy knowledge-based expert system and genetic algorithm for efficient selection and assignment of material handling equipment, Expert Systems with Applications, Vol. 36, pp Onut, S., Kara, S.S. and Mert, S. (2009), Selecting the suitable material handling equipment in the presence of vagueness, International Journal of Advanced Manufacturing Technology, Vol. 44, pp Sawant, V.B. and Mohite, S.S. (2009), Investigations on benefits generated by using fuzzy numbers in a TOPSIS model developed for automated guided vehicle selection problem, Proc. of 12 th Int. Con. on Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, 2009, India, pp Suh, N.P. (1990), The Principles of Design, Oxford University Press, New York. 21-6

7 Suh, N.P. (2001), Axiomatic Design: Advances and Applications, Oxford University Press, New York. Tuzkaya, G., Gulsun, B., Kahraman, C. and Ozgen, D. (2010), An integrated fuzzy multi-criteria decision making methodology for material handling equipment selection problem and an application, Expert Systems with Applications, Vol. 37, pp Ulubeyli, S. and Kazaz, A. (2009), A multiple criteria decision making approach to the selection of concrete pumps, Journal of Civil Engineering and Management, Vol. 15, pp Zadeh, L. A. (1965), Fuzzy sets, Information and Control, Vol. 8, pp