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Author, year, reference | The current study | Arikan (2014) [2] | Nazari-Shirkouhi et al. (2013) [57] | Kang and Lee (2010) [58] | Razmi and Maghool (2010) [59] | Amid et al. (2009) [46] | Torabi and Hassini (2009) [16] | Torabi and Hassini (2008) [60] |
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Discount scheme(s) | All units (incremental, volume) | All units | All units | All units | All units, incremental, volume | All units | Lead time-dependent | Lead time-dependent |
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Period | Single | Single | Single | Multiple | Multiple | Single | Multiple | Multiple |
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Item | Multiple | Single | Multiple | Single | Multiple | Single | Multiple | Multiple |
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Sources of fuzziness | Aspiration levels of objectives and demand level | Aspiration levels of objectives and demand level | Aspiration levels of objectives | Aspiration levels of objectives and fuzzy triangular numbers in FAHP | Fuzzy capacity and demand levels | Aspiration levels of objectives and demand level | Aspiration levels of objectives and capacities and demand levels | Aspiration levels of objectives demand, capacity, quality, and service levels |
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Objectives | To minimize (1) the total production and ordering costs, (2) the total number of rejected units, (3) the total number of late delivered units | To minimize (1) the total monetary cost, (2) the total number of rejected units, (3) the total number of late deliveries | To minimize (1) the total purchasing and ordering costs, (2) the total number of defective units, (3) the total number of late delivered units | (1) To minimize the total cost (2) To maximize the yield rate (3) To fix the replenishment to a desired rate | (1) To minimize the total purchasing cost (2) To maximize the total value of purchasing | To minimize (1) the total purchasing cost, (2) the number of rejected items, (3) the number of late delivered units | To minimize (1) the total cost of logistics, (2) the total value of purchasing, (3) the number of defective items, (4) the late deliveries of purchased items | To minimize (1) the total cost of logistics, (2) the total value of purchasing |
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Solution approach | Interactive approach which utilizes Tiwari et al.’s [17] additive fuzzy model; Lai and Hwang’s [18, 19] augmented max-min model; Chen and Tsai’s [20] fuzzy model | Two phased additive approach [39] which utilizes Tiwari et al.’s [17] additive fuzzy model; Chen and Tsai’s [20] fuzzy model | Interactive approach including fuzzy goal programming representation with piecewise linear membership functions, max-min operator, and fuzzy add operator | Two models constructed: (1) Zimmermann max-min approach [41] (2) Tiwari et al.’s [17] weighted additive fuzzy model | Fuzzy constraints were converted to crisp constraints and the augmented ε-constraint method is performed in which objectives are considered lexicographically | Tiwari et al.’s [17] weighted additive fuzzy model | Interactive approach including fuzzy goal programming, the weighted average method for defuzzification, and fuzzy model defined by Werners’ [61] fuzzy or operator | Interactive approach including the weighted average method for defuzzification, Lai and Hwang’s augmented max-min model [18, 19], and fuzzy model defined by Werners (1988) fuzzy or operator |
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Software | GAMS | GAMS | GAMS | LINGO | GAMS | LINDO/LINGO | GAMS | GAMS |
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