Vol. 3, No. 2 | Jul – Dec 2019 SJCMS | P-ISSN: 2520-0755| E-ISSN: 2522-3003 © 2019 Sukkur IBA University – All Rights Reserved 46 Solution of Maintenance Strategy Selection Problem by using modified Fuzzy TOPSIS for of Material Handling Equipment Alia Kausar1, Sana Akram1, Muhammad Farhan Tabassum2, Adeel Ahmad3, Shahzaman Khan4 Abstract: In today's competitive production environment, maintenance is one of the most important issues. Today in manufacturing plants productive methods are used to improve operating capacities that change environmental factors that lead to a competitive market. There is thus an important role to play in achieving those goals in selecting and explaining an optimal maintenance plan. Furthermore, there is a lack of an integrated model consisting of available requirements and choices, a systematic approach to maintenance instructions and strong maintenance decisions. In this paper modified Fuzzy TOPSIS method has been used for the solution of maintenance strategy selection problem. Linguistic variable and triangular fuzzy number have been used for modification in multi-criteria decision-making to solve maintenance strategy selection problem. Five experts have been considered for six types of maintenance strategy and ten decision criteria have been used in this problem. In this paper breakdown maintenance strategy best one out of all maintenance strategy for material handling equipment. Keywords: Maintenance strategy selection; Multi-criteria decision-making; Fuzzy TOPSIS method; Triangular fuzzy number; Linguistic variables. 1. Introduction Proper plant maintenance will significantly reduce total operating costs while increasing plant profitability. In order to provide maintenance personnel with improved technological and management skills [1], new technology and management experience must be emerging. In many companies, there is a strong motivation to optimize their equipment and plants ' life. This means that plants and 1 Department of Mathematics, Lahore Garrison University, Lahore, Pakistan. 2 Department of Sports Sciences & Physical Education, Faculty of Allied Health, University of Lahore, Lahore, Pakistan. 3 Department of Mathematics, Lahore Garrison University, Lahore, Pakistan. 4 Department of Physical Education and Sports Sciences, Sukkur IBA University, Sukkur, Pakistan. machinery can go further than their original design life. Damage and efficiency analysis has therefore recently become a critical tool to improve maintenance strategy, ensure safety and reduce costs [2]. Maintenance is an unavoidable cause of costs for many businesses. The maintenance function of these companies is corrective and is carried out only under emergency conditions. In view of certain key aspects such as product quality, protection of installations Alia Kausar (et al.), Solution of Maintenance Strategy Selection Problem by using modified Fuzzy TOPSIS for of Material Handling Equipment (pp. 46 - 54) Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 47 and rises in maintenance costs between 15% and 70% of total cost of production, this type of interference can now be no longer accepted. [3]. Many plants have different reliability, risk, and failure criteria for different machines. It is therefore evident that different maintenance strategies for different machines must be defined in a proper maintenance program. This makes it possible to preserve production plants ' reliability and availability at an acceptable level and to prevent unnecessary investments to implement an improper maintenance strategy [4]. In the evaluation of related factors the problem of choosing maintenance approaches is a multi-criteria decision-making (MCDM). A number of solutions were proposed using fuzzy principles to solve this problem. The paper suggests a new maintenance strategy that can be the most effective maintenance strategies, depending on the level of uncertainty and also the nature and importance of the maintenance requirements [5]. Of fact, making the best possible alternative is the way to make choices. In some cases, decision-making problems are the main problem given other parameters. TOPSIS describes ideal solutions as positive and ideal solutions as negative. For the cost criteria and for the benefit criteria the positive solution is minimum and the negative ideal solution is minimum for benefit criteria and maximum for cost criteria [4]. In other types, the best possible values for the criteria consist of the optimal solution and all the worst values for the acquisition of the criteria are negative ideal solutions. As TOPSIS is an effective MCDM tool, a number of researchers have already used TOPSIS to solve many decision-making issues. Some of them de-fuzzy scores and weight in crisp values, other information has been lost during de-fuzzification. [6-8]. For the positive, ideal solution and the ideal negative solution based on criteria, Chen developed normalized values [9]. Standardized values are always (0, 0, 0) and (1, 1, 1) respectively for the ideal negative solution and ideal positive solution on parameters. (0, 0, 0) and (1, 1, 1) are extraordinary values that could be a long way from true min and high values, so that TOPSIS ' maximum and minimum estimates were not capable of extraordinary values. In Chen's work weight criteria are classified as triangular fuzzy numbers and triangular weight fuzzy numbers depending on loss of unusual values [5]. Chan's estimation is therefore very simple but incorrect triangle fuzzy numbers cannot be expressed. We have appointed fuzzy TOPSIS for the decision- making process of several criteria for a fluffy environment in order to avoid these problems [10]. The Fuzzy Technique for Order Preference by Similarity to Ideal Solution method (FTOPSIS) for the evaluation of maintenance strategies is used by some modification, In FTOPSIS Triangular Fuzzy Number (TFN) is used to model the uncertainty in the selection and the maintenance strategy selection problem is based on a fuzzy linguistic approach [11]. The remaining of this paper is organized as follows: in Section 2, the comprehensive detail of linguistic variable, fuzzy sets, and fuzzy numbers. In Section 3 the proposed fuzzy TOPSIS and the concepts behind it are introduced in details. In Section 4, the modified fuzzy TOPSIS ethod has been implemented for the solution of maintenance strategy selection problem. Finally, conclusions are given in Section 5. 2. Some Basic Concepts Zadeh [12] first proposed the fuzzy set theory in order to deal with the vagueness of human thinking. A fuzzy collection is a class of things in which membership rates are constant. Such items are marked with a membership function that assigns a range of zero membership to one object. [12]. A linguistic variable is a variable with linguistic values. Intuitively easy-to-use linguistic terms were found to convey a Alia Kausar (et al.), Solution of Maintenance Strategy Selection Problem by using modified Fuzzy TOPSIS for of Material Handling Equipment (pp. 46 - 54) Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 48 decision-maker's subjectivity and/or conceptual inaccuracy [13]. Depending on the situation, different fuzzy numbers can be used. It is often convenient for applications to work with triangle fuzzy number (TFN) and useful for promoting the representation and processing of information in a sophisticated environment because of its computational simplicity [14]. Definition 1. If all the decision makers are the highest in terms of performance of an alternative with respect to certain 𝑐𝑛 parameters, then the alternative is the GPIS criteria 𝑐𝑛 , named 𝐺 +. Definition 2. If all the decision makers are the lowest in terms of performance of an alternative with respect to certain 𝑐𝑛 parameters, then the alternative is the GPIS criteria 𝑐𝑛 , named 𝐺 − . In many MCDM implementations, such as supplier evaluation and selection, group decision-making and performance analysis, Fuzzy TOPSIS has been described as a dominant method [15]. Junior et al. (2014) conducted an analysis of the merits of the two techniques for MCDM problems between the fuzzy TOPSIS and the fuzzy analytical hierarchy (AHP) and found Fuzzy TOPSIS is the best way to find an ideal solution [16]. It can also be used to simplify the selection process and resolve ambiguities and uncertainty. Therefore, for a multi-criteria group decision-making scenario, we described Fuzzy TOPSIS process. In this study TFNs are adopted in the fuzzy TOPSIS methods. 3. Proposed Fuzzy TOPSIS Method Through 𝑀𝐴𝑋 and 𝑀𝐼𝑁 operations, we can find the positive ideal and negative ideal solutions, though against the criteria of positive ideal and negative ideal solution, this fuzzy number can be picked up by 𝑀𝐴𝑋 and 𝑀𝐼𝑁 operations which have been conceivable alternative cannot be found on rating. In this paper, another generally selected 𝑇𝑂𝑃𝑆𝐼𝑆 that replaces 𝑈𝑝 and 𝐿𝑜 operations for 𝑀𝐴𝑋 and 𝑀𝐼𝑁 operations. By 𝑈𝑝 and 𝐿𝑜 operations, a set of fuzzy numbers is ranked quickly. On this occasion, we effectively find the ideal solution and negative ideal solution, and the fuzzy number can be found extra on these conceivable alternatives against the criteria of the positive ideal and negative ideal solution. The various steps of Fuzzy TOPSIS method are presented as follows: STEP 1: Firstly, performance rating and weight are estimated with verbal terms. It represents the performances under linguistics classification, standardized by specialists, are: 𝑣𝑒𝑟𝑦 𝑙𝑜𝑤 (𝑉𝐿), 𝑙𝑜𝑤 (𝐿), 𝑚𝑒𝑑𝑖𝑢𝑚 𝑙𝑜𝑤 (𝑀𝐿), 𝑚𝑒𝑑𝑖𝑢𝑚 (𝑀), 𝑚𝑒𝑑𝑖𝑢𝑚 ℎ𝑖𝑔ℎ (𝑀𝐻), ℎ𝑖𝑔ℎ (𝐻) 𝑎𝑛𝑑 𝑣𝑒𝑟𝑦 ℎ𝑖𝑔ℎ (𝑉𝐻). Choosing committee of experts for decision- making. (𝐸𝑘 ; 𝑘 = 1, 2 … 𝑛) and then to alternative 𝑀𝑖 against choosing the criteria (𝐶𝑗 ; 𝑗 = 1, 2 … 𝑚) where 𝐺𝑖𝑗𝑘 = (𝑔1𝑖𝑗𝑘 , 𝑔2𝑖𝑗𝑘 , 𝑔3𝑖𝑗𝑘 ) is a triangular fuzzy number. STEP 2: By using extension principle find the average performance rating of alternative 𝑀𝑖 against criterion 𝑐𝑗 , 𝑖𝑠 𝐺𝑖𝑗 as 𝐺𝑖𝑗𝑘 = (𝑔1𝑖𝑗𝑘 , 𝑔2𝑖𝑗𝑘 , 𝑔3𝑖𝑗𝑘 ) = 1 𝑝⁄ × (𝐺𝑖𝑗1 + 𝐺𝑖𝑗2 + ⋯ + 𝐺𝑖𝑗𝑝 ) By the extension principle, we have 𝑔1𝑖𝑗 = ∑ 𝑔1𝑖𝑗𝑘 𝑝 𝑝 𝑘=1 𝑔2𝑖𝑗 = ∑ 𝑔2𝑖𝑗𝑘 𝑝 𝑝 𝑘=1 𝑔3𝑖𝑗 = ∑ 𝑔3𝑖𝑗𝑘 𝑝 𝑝 𝑘=1 STEP 3: The performance ratings of alternative 𝑀1, 𝑀2 , 𝑀3 … composed a decision matrix that is, 𝐺 = [𝐺𝑖𝑗 ]𝑚𝑋𝑛, [𝐺𝑖1, 𝐺𝑖2, . . 𝐺𝑖𝑛 ] are the performance ratings of alternative 𝑀𝑖 . Alia Kausar (et al.), Solution of Maintenance Strategy Selection Problem by using modified Fuzzy TOPSIS for of Material Handling Equipment (pp. 46 - 54) Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 49 Let 𝑀− and 𝑀+ are the negative ideal solution and positive ideal solution respectively; Thus 𝑀− = [𝐺1 −, 𝐺2 −, … 𝐺𝑛 −] and 𝑀+ = [𝐺1 +, 𝐺2 +, … 𝐺𝑛 +] where 𝐺𝑗 − = 𝐿𝑜 [𝐺𝑖1, 𝐺𝑖2, . . 𝐺𝑖𝑚 ] and 𝐺𝑗 + = 𝑈𝑝[𝐺𝑖1, 𝐺𝑖2, . . 𝐺𝑖𝑚 ] for 𝑗 = 1, 2, . . . , 𝑛. STEP 4: Next find the distance from alternatives to the negative ideal solution (or positive ideal solution). Let 𝑑𝑖𝑗 − and 𝑑𝑖𝑗 + be the distance from 𝐺𝑖𝑗 to 𝐺𝑗 − and 𝐺𝑗 +respectively; where 𝑖 = 1, 2, . . . , 𝑚; 𝑗 = 1, 2, . . . , 𝑛. Let 𝐴 = (𝑎1, 𝑎2, 𝑎3) and 𝐵 = (𝑏1, 𝑏2, 𝑏3) be two triangular fuzzy numbers. So by the definition of distance, 𝑑(𝐴, 𝐵) = √ 1 3 [(𝑎1 − 𝑏1) 2 + (𝑎2 − 𝑏2) 2 + (𝑎3 − 𝑏3) 2 STEP 5: Let 𝑊𝑗𝑘 = (𝑤1𝑗𝑘 , 𝑤2𝑗𝑘 , 𝑤3𝑗𝑘 ) are weight evaluated by expert and 𝐸𝑘 under criterion 𝐶𝑗, where 𝑗 = 1, 2, . . . , 𝑛; 𝑘 = 1, 2, . . . , 𝑝 Suppose 𝑊𝑗 be the average weight on criterion 𝐶𝑗 is 𝑊𝑗 = (𝑤1𝑗 , 𝑤2𝑗 , 𝑤3𝑗 ) = 1 𝑝⁄ × (𝑊𝑗1 + 𝑊𝑗2 + ⋯ + 𝑊𝑗𝑝) where 𝑗 = 1, 2, . . . , 𝑛. From the extension principle, 𝑤1𝑗 = ∑ 𝑤1𝑗𝑘 𝑝 𝑝 𝑘=1 𝑤2𝑗 = ∑ 𝑤2𝑗𝑘 𝑝 𝑝 𝑘=1 𝑤3𝑗 = ∑ 𝑤3𝑗𝑘 𝑝 𝑝 𝑘=1 STEP 6: The weight distance of alternative 𝑀𝑖 to negative ideal solution 𝑀 −and ideal solution 𝑀+ respectively are find 𝐷𝑖 − and 𝐷𝑖 +. 𝐷𝑖 − = ∑ 𝑊𝑗 × 𝑑𝑖𝑗 −𝑛 𝑗=1 and 𝐷𝑖 + = ∑ 𝑊𝑗 × 𝑑𝑖𝑗 +𝑛 𝑗=1 , where 𝑖 = 1, 2, . . . , 𝑚. STEP 7: Weighted distance of 𝑀𝑖 can be find by [𝐷𝑖 −, 𝐷𝑖 +] ; 𝐿𝐷− = 𝐿𝑜({𝐷1 −, 𝐷2 −, … 𝐷𝑚 − }) 𝑈𝐷− = 𝑈𝑝({𝐷1 −, 𝐷2 −, … 𝐷𝑚 − }) 𝐿𝐷+ = 𝐿𝑜({𝐷1 +, 𝐷2 +, … 𝐷𝑚 + }) 𝑈𝐷+ = 𝑈𝑝({𝐷1 +, 𝐷2 +, … 𝐷𝑚 + }) STEP 8: From these two operations Lo and Up, the negative ideal solution is [ 𝐿𝐷−, 𝑈𝐷+] and the ideal solution is [𝑈𝐷−, 𝐿𝐷+] find for weighted distance of all alternatives. STEP 9: Let 𝑀𝑖 −are the distance from [ 𝐷𝑖 −, 𝐷𝑖 +] to [ 𝐿𝐷−, 𝑈𝐷+], and 𝑀𝑖 +denote the distance from [ 𝐷𝑖 −, 𝐷𝑖 +] to [𝑈𝐷−, 𝐿𝐷+]. Define 𝑀𝑖 − = 𝑑(𝐷𝑖 −, 𝐿𝐷−) + 𝑑(𝐷𝑖 +, 𝑈𝐷+) And 𝑀𝑖 + = 𝑑(𝐷𝑖 −, 𝑈𝐷−) + 𝑑(𝐷𝑖 +, 𝐿𝐷+) where 𝑖 = 1, 2, . . . , 𝑚. STEP 10: Closeness coefficient 𝑀𝑖 ∗ of alternative 𝐴𝑖 is defined as: 𝑀𝑖 ∗ = 𝑀𝑖 − 𝑀𝑖 −+𝑀𝑖 + where 𝑖 = 1, 2, . . . , 𝑚. If 𝑀𝑖 ∗ = 0, alternative 𝑀𝑖 will be the poorest. And 𝑀𝑖 ∗ = 1 𝑀𝑖 is the best alternative. 4. Maintenance Strategy Selection Problem for Material Handling Equipment In this problem their five experts, such as: 𝐸1 = 𝐸𝑥𝑝𝑒𝑟𝑡 1 𝐸2 = 𝐸𝑥𝑝𝑒𝑟𝑡 2 𝐸3 = 𝐸𝑥𝑝𝑒𝑟𝑡 3 𝐸4 = 𝐸𝑥𝑝𝑒𝑟𝑡 4 𝐸5 = 𝐸𝑥𝑝𝑒𝑟𝑡 5 And six maintenance strategies: 𝑀1 = corrective maintenance, 𝑀2 = 𝑝𝑟𝑒𝑣𝑒𝑛𝑡𝑖𝑣𝑒 𝑚𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒, 𝑀3 = condition based maintenance 𝑀4 = opportunistic maintenance 𝑀5 = predictive maintenance 𝑀6 = breakdown Maintenance Alia Kausar (et al.), Solution of Maintenance Strategy Selection Problem by using modified Fuzzy TOPSIS for of Material Handling Equipment (pp. 46 - 54) Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 50 Also this problem includes 10 evaluation criteria, such as: 𝐶1 = quality 𝐶2 = spare parts inventories 𝐶3 = purchasing cost of spare parts 𝐶4 = 𝑚𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒 𝑙𝑎𝑏𝑜𝑢𝑟 𝑐𝑜𝑠𝑡 𝐶5 = reliability 𝐶6 = safety 𝐶7 = maintenance time 𝐶8 = facilities 𝐶9 = cost of supporting equipment 𝐶10 = environment The different weights of priority of each criterion and strategy are calculated using the fuzzy TOPSIS method following the construction of the hierarchy. Linguistic variables are represented in Table 1. Table I. Fuzzy numbers and corresponding linguistic variables Linguistic Variables Fuzzy Number 𝑉𝑒𝑟𝑦 𝑙𝑜𝑤 (𝑉𝐿) (0.0, 0.0, 0.1) 𝐿𝑜𝑤 (𝐿) (0.0, 0.1, 0.3) 𝑀𝑒𝑑𝑖𝑢𝑚 𝑙𝑜𝑤 (𝑀𝐿) (0.1, 0.3, 0.5) 𝑀𝑒𝑑𝑖𝑢𝑚 (𝑀) (0.3, 0.5, 0.7) 𝑀𝑒𝑑𝑖𝑢𝑚 ℎ𝑖𝑔ℎ (𝑀𝐻) (0.5, 0.7, 0.9) 𝐻𝑖𝑔ℎ (𝐻) (0.7, 0.9, 1.0) 𝑉𝑒𝑟𝑦 𝐻𝑖𝑔ℎ (𝑉𝐻) (0.9, 1.0, 1.0) Step 1: The performance ratings of the six maintenance strategies in linguistic term is presented in Table 2. Table II. Experts have assigned the correct rating in terms of linguistic variables for each criterion 𝐶𝑗 𝑀1 𝑀2 𝑀3 𝑀4 𝑀5 𝑀6 𝐶1 𝐻, 𝑉𝐻, 𝐻, 𝐻, 𝑉𝐻 𝑀, 𝑀𝐻, 𝑀, 𝑀, 𝑀𝐻 𝐻, 𝐻, 𝑉𝐻, 𝑉𝐻, 𝑀𝐻 𝑀, 𝑀𝐻, 𝐻, 𝐻, 𝑀𝐻 𝑀𝐿, 𝑀, 𝑀𝐻, 𝑀𝐻, 𝑀 𝑉𝐻, 𝐻, 𝑉𝐻, 𝐻, 𝑉𝐻 𝐶2 𝐻, 𝑀𝐻, 𝑀, 𝑀𝐻, 𝐻 𝑀, 𝑀𝐿, 𝑀, 𝑀𝐿, 𝑀 𝐿, 𝑉𝐿, 𝐿, 𝑀𝐿, 𝐿 𝑀𝐻, 𝐻, 𝐻, 𝑀𝐻, 𝑀 𝐿, 𝑉𝐿, 𝑀𝐿, 𝑀𝐿, 𝐿 𝑉𝐻, 𝐻, 𝐻, 𝐻, 𝑀𝐻 𝐶3 𝐻, 𝑀𝐻, 𝑀, 𝑀𝐻, 𝐻 𝑀𝐿, 𝑀, 𝑀, 𝑀𝐿, 𝐻 𝑀𝐿, 𝑀, 𝑀𝐻, 𝑀, 𝑀𝐿 𝑀𝐻, 𝐻, 𝑀, 𝑀𝐻, 𝑀𝐻 𝐿, 𝑀𝐿, 𝑀, 𝑀𝐿, 𝑀 𝐻, 𝐻, 𝑉𝐻, 𝑉𝐻, 𝐻 𝐶4 𝑀𝐿, 𝑀, 𝐻, 𝐻, 𝑀 𝐻, 𝑀, 𝐿, 𝑀𝐿, 𝑀𝐿 𝑀, 𝑀𝐻, 𝑀, 𝑀𝐿, 𝑀𝐿 𝑀𝐿, 𝑀𝐿, 𝑀, 𝐿, 𝑀 𝐿, 𝑀𝐿, 𝑀𝐿, 𝑀𝐿, 𝑀 𝐻, 𝑉𝐻, 𝑉𝐻, 𝐻, 𝑉𝐻 𝐶5 𝐻, 𝑉𝐻, 𝑉𝐻, 𝐻, 𝑀𝐻 𝑀𝐿, 𝑀, 𝑀, 𝑀𝐻, 𝑀𝐻 𝐻, 𝐻, 𝑉𝐻, 𝐻, 𝑀𝐻 𝑀𝐻, 𝐻, 𝑀, 𝑀, 𝐻 𝑀𝐿, 𝑀, 𝑀𝐻, 𝑀, 𝑀𝐿 𝐻, 𝐻, 𝑉𝐻, 𝑉𝐻, 𝐻 𝐶6 𝐻, 𝑀𝐻, 𝑀𝐻, 𝐻, 𝑀𝐻 𝑀𝐻, 𝑀, 𝑀, 𝑀𝐻, 𝐻 𝐻, 𝐻, 𝑉𝐻, 𝐻, 𝐻 𝐻, 𝑀𝐻, 𝑀, 𝑀, 𝑀𝐻 𝑀, 𝑀𝐻, 𝐻, 𝑀𝐻, 𝑀 𝐻, 𝐻, 𝑉𝐻, 𝑀𝐻, 𝑀𝐻 𝐶7 𝐻, 𝑀𝐻, 𝐻, 𝐻, 𝑀𝐻 𝑀, 𝑀, 𝑀𝐿, 𝑀𝐿, 𝐻 𝑀, 𝑀𝐿, 𝑀𝐿, 𝐻, 𝑀𝐿 𝑀, 𝑀𝐿, 𝑀𝐻, 𝐻, 𝑀𝐻 𝑀𝐻, 𝑀, 𝑀𝐿, 𝑀𝐿, 𝐿 𝑉𝐻, 𝐻, 𝑉𝐻, 𝑉𝐻, 𝑉𝐻 𝐶8 𝐻, 𝑉𝐻, 𝐻, 𝑀𝐻, 𝑀𝐻 𝑀𝐻, 𝑀, 𝑀, 𝑀𝐻, 𝑀 𝑉𝐻, 𝐻, 𝑉𝐻, 𝐻, 𝑉𝐻 𝑀, 𝑀𝐿, 𝐿, 𝑀𝐻, 𝑀𝐿 𝐻, 𝑀𝐻, 𝐻, 𝑀𝐻, 𝑀𝐻 𝑀𝐻, 𝐻, 𝑉𝐻, 𝑉𝐻, 𝑉𝐻 𝐶9 𝐻, 𝑀, 𝑀𝐻, 𝑀, 𝑀𝐻 𝑀, 𝑀𝐿, 𝑀𝐿, 𝐻, 𝑀𝐻 𝑀, 𝑀𝐻, 𝑀𝐻, 𝑀𝐿, 𝑀𝐻 𝑀𝐿, 𝑀𝐿, 𝑀, 𝑀, 𝑀𝐿 𝑀𝐿, 𝐿, 𝑀𝐿, 𝑀𝐿, 𝑀 𝑉𝐻, 𝑉𝐻, 𝐻, 𝑉𝐻, 𝑉𝐻 𝐶10 𝑀, 𝐻, 𝑀𝐻, 𝐻, 𝑉𝐻 𝐻, 𝑀𝐻, 𝑀𝐻, 𝐻, 𝐻 𝑉𝐻, 𝐻, 𝑉𝐻, 𝑉𝐻, 𝐻 𝑀, 𝐻, 𝑀𝐻, 𝑀, 𝑀𝐻 𝐻, 𝐻, 𝐻, 𝑀𝐻, 𝑀𝐻 𝑉𝐻, 𝐻, 𝐻, 𝑉𝐻, 𝐻 Step 2: The average ratings of the six maintenance strategies is in Table 3. Table III. Average ratings of maintenance strategies 𝐶𝑗. 𝑀1. 𝑀2. 𝑀3. 𝑀4. 𝑀5. 𝑀6. 𝐶1 0.78,0.94,1.00 0.38,0.58,0.78 0.74,0.90,0.98 0.54,0.74,0.90 0.34,0.54,0.74 0.82,0.96,1.00 𝐶2 0.54,0.74,0.40 0.16,0.34,0.54 0.02,0.12,0.30 0.54,0.74,0.90 0.02,0.16,0.34 0.70,0.88,0.98 𝐶3 0.54,0.74,0.90 0.30,0.50,0.68 0.26,0.46,0.66 0.50,0.70,0.88 0.16,0.34,0.54 0.78,0.94,1.00 𝐶4 0.42,0.62,0.78 0.24,0.42,0.60 0.26,0.46,0.66 0.16,0.34,0.54 0.12,0.30,0.50 0.82,0.96,1.00 𝐶5 0.74,0.90,0.98 0.34,0.54,0.74 0.70,0.88,0.98 0.50,0.70,0.86 0.26,0.46,0.66 0.78,0.94,1.00 𝐶6 0.58,0.78,0.94 0.46,0.66,0.84 0.74,0.92,1.00 0.46,0.66,0.84 0.46,0.66,0.84 0.66,0.84,0.96 𝐶7 0.62,0.82,0.96 0.30,0.50,0.68 0.26,0.46,0.64 0.42,0.62,0.80 0.20,0.38,0.58 0.86,0.98,1.00 𝐶8 0.66,0.84,0.96 0.38,0.58,0.82 0.82,0.96,1.00 0.20,0.38,0.58 0.58,0.78,0.98 0.78,0.92,0.98 𝐶9 0.46,0.66,0.84 0.34,0.54,0.72 0.38,0.58,0.78 0.18,0.38,0.58 0.12,0.30,0.50 0.86,0.98,1.00 Alia Kausar (et al.), Solution of Maintenance Strategy Selection Problem by using modified Fuzzy TOPSIS for of Material Handling Equipment (pp. 46 - 54) Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 51 𝐶10 0.62,0.80,0.92 0.62,0.82,0.96 0.82,0.96,1.00 0.46,0.66,0.84 0.62,0.82,0.96 0.78,0.91,1.00 Step 3: By the performance ratings of alternative composed a decision-making matrix 𝐺 is in Table 4. Table IV. Decision-making matrix 𝐺1 + =0.82,0.96,1.00 𝐺6 + =0.74,0.92,1.00 𝐺1 − =0.34,0.54,0.74 𝐺6 − =0.46,0.66,0.84 𝐺2 + = 0.70,0.88,0.98 𝐺7 + =0.86,0.98,1.00 𝐺2 − =0.02,0.12,0.30 𝐺7 − =0.20,0.38,0.58 𝐺3 + =0.78,0.94,1.00 𝐺8 + =0.82,0.96,1.00 𝐺3 − =0.16,0.34,0.54 𝐺8 − =0.20,0.38,0.58 𝐺4 + =0.82,0.96,1.00 𝐺9 + =0.86,0.98,1.00 𝐺4 − =0.12,0.30,0.50 𝐺9 − =0.12,0.30,0.50 𝐺5 + =0.78,0.94,1.00 𝐺10 + = 0.82,0.96,1.00 𝐺5 − =0.26,0.46,0.66 𝐺10 − =0.46,0.66,0.84 Step 4: The distance values for the six maintenance strategies on 10 criteria is in Table 5. Table V. Distance values for maintenance strategies 𝐶𝑗 𝑀1 𝑀2 𝑀3 d (𝐺1𝑗 , 𝐺𝑗+ ) d (𝐺1𝑗 , 𝐺𝑗− ) d(𝐺2𝑗 , 𝐺𝑗 + ) d(𝐺2𝑗 , 𝐺𝑗− ) d(𝐺3𝑗 , 𝐺𝑗 + ) d(𝐺3𝑗 , 𝐺𝑗− ) 𝐶1 0.08246 0.3747 0.3589 0.0400 0.0589 0.3402 𝐶2 0.1311 0.5816 0.5089 0.2046 0.7077 0.0000 𝐶3. 0.1894 0.3804 0.4189 0.1469 0.4533 0.1137 𝐶4 0.3286 0.3004 0.5125 0.1137 0.4758 0.1536 𝐶5 0.0346 0.4189 0.3747 0.0800 0.0589 0.3968 𝐶6 0.1275 0.1137 0.2392 0.0000 0.0000 0.2392 𝐶7 0.1681 0.4141 0.4642 0.1071 0.5033 0.0673 𝐶8 0.1178 0.4350 0.3514 0.2082 0.0000 0.5469 𝐶9 0.3098 0.3468 0.4252 0.2269 0.03824 0.2735 𝐶10 0.1549 0.1311 0.1428 0.1479 0.0000 0.2859 𝐶𝑗 𝑀6 𝑀6 𝑀6 d(𝐺4𝑗 , 𝐺𝑗+ ) d(𝐺4𝑗 , 𝐺𝑗− ) d(𝐺5𝑗 , 𝐺𝑗 + ) d(𝐺5𝑗 , 𝐺𝑗− ) d(𝐺6𝑗 , 𝐺𝑗 + ) d(𝐺6𝑗 , 𝐺𝑗− ) 𝐶1 0.2135 0.1876 0.3977 0.0000 0.0000 0.3977 𝐶2 0.1311 0.5816 0.6808 0.0327 0.0000 0.7077 𝐶3 0.2239 0.3468 0.5645 0.0000 0.0000 0.5540 𝐶4 0.5864 0.0400 0.6260 0.0000 0.0000 0.6260 𝐶5 0.2277 0.2274 0.4533 0.0000 0.0000 0.4532 𝐶6 0.2392 0.0000 0.2392 0.0000 0.0693 0.1701 𝐶7 0.3479 0.2269 0.5692 0.0000 0.0000 0.5692 𝐶8 0.5469 0.0000 0.1736 0.3934 0.0346 0.5125 𝐶9 0.5770 0.7394 0.6481 0.0000 0.0000 0.6481 𝐶10 0.2859 0.0000 0.1428 0.1479 0.0258 0.2622 Alia Kausar (et al.), Solution of Maintenance Strategy Selection Problem by using modified Fuzzy TOPSIS for of Material Handling Equipment (pp. 46 - 54) Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 52 Step 5: The linguistic weights and average ratings for 10 criteria are in Table 6-7. Table VI. Linguistic weights for criteria 𝐶𝑗 𝐸1 𝐸2 𝐸3 𝐸4 𝐸5 𝐶1 𝑉𝐻 𝐻 𝑉𝐻 𝑉𝐻 𝐻 𝐶2 𝑀𝐻 𝑀 𝑀𝐿 𝐿 𝑉𝐿 𝐶3 𝑀 𝑀𝐿 𝑀 𝑀𝐻 𝑀𝐿 𝐶4 𝑀𝐻 𝐻 𝑀𝐿 𝑀 𝑀 𝐶5 𝑉𝐻 𝐻 𝐻 𝑉𝐻 𝑉𝐻 𝐶6 𝐻 𝑉𝐻 𝐻 𝑉𝐻 𝑉𝐻 𝐶7 𝑀𝐻 𝑀 𝐻 𝑀𝐻 𝐻 𝐶8 𝐻 𝑀𝐻 𝑉𝐻 𝐻 𝑀𝐻 𝐶9 𝐿 𝑀𝐿 𝑀 𝑀 𝑉𝐿 𝐶10 𝑀𝐻 𝐻 𝑉𝐻 𝐻 𝑀𝐻 Table VII. Average ratings 𝐶𝑗 𝐸1 𝐸2 𝐸3 𝐸4 𝐸5 𝐶1 0.9,1.0,1.0 0.7,0.9,1.0 0.9,1.0,1.0 0.9,1.0,1.0 0.7,0.9,1.0 𝐶2 0.5,0.7,0.9 0.3,0.5,0.7 0.1,0.3,0.5 0.0,0.1,0.3 0.0,0.0,0.1 𝐶3 0.3,0.5,0.7 0.1,0.3,0.5 0.3,0.5,0.7 0.5,0.7,0.9 0.1,0.3,0.5 𝐶4 0.5,0.7,0.9 0.7,0.9,1.0 0.1,0.3,0.5 0.3,0.5,0.7 0.3,0.5,0.7 𝐶5 0.9,1.0,1.0 0.7,0.9,1.0 0.7,0.9,1.0 0.9,1.0,1.0 0.9,1.0,1.0 𝐶6 0.7,0.9,1.0 0.9,1.0,1.0 0.7,0.9,1.0 0.9,1.0,1.0 0.9,1.0,1.0 𝐶7 0.5,0.7,0.9 0.3,0.5,0.7 0.7,0.9,1.0 0.5,0.7,0.9 0.7,0.9,1.0 𝐶8 0.7,0.9,1.0 0.5,0.7,0.9 0.9,1.0,1.0 0.7,0.9,1.0 0.5,0.7,0.9 𝐶9 0.0,0.1,0.3 0.1,0.3,0.5 0.3,0.5,0.7 0.3,0.5,0.7 0.0,0.0,0.1 𝐶10 0.5,0.7,0.9 0.7,0.9,1.0 0.9,1.0,1.0 0.7,0.9,1.0 0.5,0.7,0.9 Calculated average weights against the 10 criteria from Table 7 are: 𝑊1 = (0.82,0.96,1.00) 𝑊2 = (0.18,0.32,0.50) 𝑊3 = (0.26,0.46,0.66) 𝑊4 = (0.38,0.58,0.76) 𝑊5 = (0.82,0.96,1.00) 𝑊6 = (0.82,0.96,1.00) 𝑊7 = (0.54,0.74,0.90) 𝑊8 = (0.66,0.84,0.96) 𝑊9 = (0.14,0.28,0.46) 𝑊10 = (0.66,0.84,0.96) Step 6: The weighted distance values of six maintenance strategies on 10 criteria are presented as follows. 𝐷1 + = (1.3209,1.7071,1.9826) 𝐷2 + = (1.8293,2.4644,2.9810) 𝐷3 + = (0.8680,1.3035,1.7613) 𝐷4 + = (1.6803,2.2561,2.7167) 𝐷5 + = (2.0081,2.7557,3.3931) 𝐷6 + = (0.0969,0.1175,0.1276) 𝐷1 − = (1.7075,2.2854,2.7531) 𝐷2 − = (0.5413,0.7561,0.9483) 𝐷3 − = (1.5127,1.9045,2.1538) 𝐷4 − = (0.7764,1.1422,1.5094) 𝐷5 − = (0.4161,0.5593,0.6831) Alia Kausar (et al.), Solution of Maintenance Strategy Selection Problem by using modified Fuzzy TOPSIS for of Material Handling Equipment (pp. 46 - 54) Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 53 𝐷6 − = (2.2559,3.0780,3.7704) Step 7: The values of weighted distance are given below 𝐿𝐷 + = (0.0969,0.1175,0.1276) 𝑈𝐷 + = (2.0081,2.7557,3.3931) 𝐿𝐷 − = (0.4161,0.5593,0.6831) 𝑈𝐷 − = (2.2559,3.0780,3.7704) Step 8: Positive and negative ideal solutions from weighted distance are given below 𝑑(𝐷1 +, 𝑈𝐷+) = 1.0893 𝑑(𝐷1 +, 𝐿𝐷+) = 1.5774 𝑑(𝐷2 +, 𝑈𝐷+) = 0.8378 𝑑(𝐷2 +, 𝐿𝐷+) = 2.3559 𝑑(𝐷3 +, 𝑈𝐷+) = 1.4224 𝑑(𝐷3 +, 𝐿𝐷+) = 1.2477 𝑑(𝐷4 +, 𝑈𝐷+) = 0.5209 𝑑(𝐷4 +, 𝐿𝐷+) = 2.1435 𝑑(𝐷5 +, 𝑈𝐷+) = 0.0000 𝑑(𝐷5 +, 𝐿𝐷+) = 2.6629 𝑑(𝐷6 +, 𝑈𝐷+) = 2.6629 𝑑(𝐷6 +, 𝐿𝐷+) = 0.0000 and 𝑑(𝐷1 −, 𝑈𝐷−) = 0.8091 𝑑(𝐷1 −, 𝐿𝐷−) = 1.7255 𝑑(𝐷2 −, 𝑈𝐷−) = 2.3532 𝑑(𝐷2 −, 𝐿𝐷−) = 0.2039 𝑑(𝐷3 −, 𝑈𝐷−) = 1.2306 𝑑(𝐷3 −, 𝐿𝐷−) = 1.3134 𝑑(𝐷4 −, 𝑈𝐷−) = 1.9191 𝑑(𝐷4 −, 𝐿𝐷−) = 0.6198 𝑑(𝐷5 −, 𝑈𝐷−) = 2.5338 𝑑(𝐷5 −, 𝐿𝐷−) = 0.0000 𝑑(𝐷6 −, 𝑈𝐷−) = 0.0000 𝑑(𝐷6 −, 𝐿𝐷−) = 2.5338 Step 9: From these above distance values, 𝑀𝑖 + and 𝑀𝑖 − can be calculated: 𝑀1 + = 2.3865 𝑀1 − =2.8148 𝑀2 + = 4.7091 𝑀2 − =1.0417 𝑀3 + =2.4783 𝑀3 − =2.7358 𝑀4 + =4.0626 𝑀4 − =1.1407 𝑀5 + = 5.1967 𝑀5 − =0.0000 𝑀6 + =0.0000 𝑀6 − =5.1967 Step 10: Finally, all the results are evaluated, final Scores and ranks are given in Table 8. 𝑀1 ∗ =0.5412 𝑀2 ∗ =0.1811 𝑀3 ∗ =0.5247 𝑀4 ∗ =0.2192 𝑀5 ∗ =0.0000 𝑀6 ∗ =1.0000 Table VIII. Final Scores and ranks of strategies Strategy Final Scores Ranks 𝑀1 0.5412 2 𝑀2 0.1811 5 𝑀3 0.5247 3 𝑀4 0.2192 4 𝑀5 0.0000 6 𝑀6 1.0000 1 Clearly, from Table 8 the ranking order is 𝑀6 > 𝑀1 > 𝑀3 > 𝑀4 > 𝑀2 > 𝑀5 Therefore, performances of 𝑀6 is the best. 5. Result and Discussion The result reported in [17] the same problem has been solved with the help of fuzzy SAW method the order ranking of maintenance strategy for material handling equipment. The results are as follows 𝑀6 > 𝑀1 > 𝑀3 > 𝑀4 > 𝑀2 > 𝑀5. And from the above experimentation we have the same results which showed that breakdown maintenance 𝑀6 is the best maintenance strategy for material handling equipment and predictive maintenance 𝑀5 is the poor maintenance strategy for material handling equipment. 6. Conclusion In this paper modified Fuzzy TOPSIS method has been used for the solution of maintenance strategy selection problem. Linguistic variable and triangular fuzzy number have been used for modification in multi-criteria decision- making to solve maintenance strategy selection Alia Kausar (et al.), Solution of Maintenance Strategy Selection Problem by using modified Fuzzy TOPSIS for of Material Handling Equipment (pp. 46 - 54) Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 54 problem. Five experts have been considered for six types of maintenance strategy and ten decision criteria have been used in this problem. Two operators 𝑈𝑝 and 𝐿𝑜, which satisfied fuzzy numbers of partial ordering relations for generalized of TOPSIS, these two operations are used to determine the negative ideal and positive ideal solutions in a fuzzy environment. Ultimately, it is concluded that the malfunction management approach for equipment for material handling is best of all maintenance strategies. REFERENCES [1] Bertolini, M. and M. Bevilacqua, A combined goal programming—AHP approach to maintenance selection problem. Reliability Engineering & System Safety, 2006. 91(7): p. 839-848. [2] Yatomi, M., et al., Application of risk-based maintenance on materials handling systems. IHI engineering review, 2004. 37(2): p. 52-58. [3] Sarkar, A., D.K. Behera, and B. Sarkar, The maintenance strategy selection of a gas turbine power plant system. Journal of Information and Operations Management, 2011. 2(1): p. 9. [4] Akhshabi, M., A new fuzzy multi criteria model for maintenance policy. Middle-East Journal of Scientific Research, 2011. 10(1): p. 33-38. [5] Saeed, M. and M. Tabassum, COMPARISON BETWEEN FUZZY SOFT MATRIX (FSM) AND INTERVAL VALUED FUZZY SOFT MATRIX (IVFSM) IN DECISION MAKING. Science International, 2016. 28(5). [6] Saeed, M., et al., Generalization of TOPSIS from Soft Set to Fuzzy Soft Sets in Decision Making Problem. Scientific Inquiry Review, 2017. 1(1): p. 11-18. [7] Tsaur, S.-H., T.-Y. Chang, and C.-H. Yen, The evaluation of airline service quality by fuzzy MCDM. Tourism management, 2002. 23(2): p. 107-115. [8] Ahmad, A., et al., Multi-Criteria Decision- Making for Airport Operation Performance Using Triangular Fuzzy Numbers. Scientific Inquiry Review, 2019. 3(3): p. 01-15. [9] Chen, C.-T., Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy sets and systems, 2000. 114(1): p. 1-9. [10] Gani, A.N. and S.M. Assarudeen, A new operation on triangular fuzzy number for solving fuzzy linear programming problem. Applied Mathematical Sciences, 2012. 6(11): p. 525-532. [11] Omar, M.N. and A.R. Fayek, A TOPSIS‐based approach for prioritized aggregation in multi‐ criteria decision‐making problems. Journal of Multi‐Criteria Decision Analysis, 2016. 23(5- 6): p. 197-209. [12] Zadeh, L.A., Fuzzy sets. Information and control, 1965. 8(3): p. 338-353. [13] Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning-III. Information sciences, 1975. 9(1): p. 43-80. [14] Ertuğrul, İ. and N. Karakaşoğlu, Comparison of fuzzy AHP and fuzzy TOPSIS methods for facility location selection. The International Journal of Advanced Manufacturing Technology, 2008. 39(7-8): p. 783-795. [15] Bottani, E. and A. Rizzi, A fuzzy TOPSIS methodology to support outsourcing of logistics services. Supply Chain Management: An International Journal, 2006. 11(4): p. 294- 308. [16] Junior, F.R.L., L. Osiro, and L.C.R. Carpinetti, A comparison between Fuzzy AHP and Fuzzy TOPSIS methods to supplier selection. Applied Soft Computing, 2014. 21: p. 194-209. [17] Sagara, M.K., P. Jayaswala, and K. Kushwahc, Exploring Fuzzy SAW Method for Maintenance Strategy Selection Problem of Material Handling Equipment. 2013.