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A Lagrangian Relaxation for a Fuzzy Random EPQ Problem with Shortages and Redundancy Allocation: Two Tuned Meta-heuristics

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Abstract

This paper develops an economic production quantity model for a multi-product multi-objective inventory control problem with fuzzy-stochastic demand and backorders. In this model, the annual demand is represented by trapezoidal fuzzy random numbers. The centroid defuzzification and the expected value methods are applied to defuzzify and make decisions in a random environment. In the case where the warehouse space is limited, the Lagrangian relaxation procedure is first employed to determine the optimal order and the maximum backorder quantities of the products such that the total inventory cost is minimized. The optimal solution obtained by the proposed approach is compared with that obtained by the traditional deterministic method. Moreover, a sensitivity analysis presents the rationality of the solution. Then, the model is extended to a multi-objective integer programming problem in which the optimal numbers of redundant production machines are determined to maximize the production system reliability. In the second proposed model, several constraints are considered to fit real-world situations. As the second model is developed for an NP-hard problem and hence cannot be solved using exact methods in a reasonable computational time, a multi-objective evolutionary algorithm called non-dominated sorting genetic algorithm-II (NSGA-II) is employed to provide Pareto front solutions. Due to non-availability of benchmark in the literature, another multi-objective evolutionary algorithm called non-dominated ranking genetic algorithm is implemented as well to validate the obtained results and evaluate the performance of NSGA-II. In addition, the Taguchi method is used to calibrate the parameters of both algorithms for better performance.

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References

  1. Simchi-Levi, D., Kaminsky, P., Simchi-Levi, E., Shankar, R.: Designing and Managing the Supply Chain: Concepts, Strategies and Case Studies, 3rd edn, p. 531. McGraw-Hill, New York (2007)

    Google Scholar 

  2. Sunil Chopra, P.M.: Supply Chain Management: Strategy, Planning, and Operation, 3rd edn, p. 536. Pearson Prentice Hall, Upper Saddle River (2007)

    Google Scholar 

  3. Axsäter, S.: Inventory Control, 2nd edn, p. 336. Springer, New York (2010)

    Google Scholar 

  4. Pentico, D.W., Drake, M.J.: A survey of deterministic models for the EOQ and EPQ with partial backordering. Eur. J. Oper. Res. 214(2), 179–198 (2011)

    Article  MathSciNet  Google Scholar 

  5. Montgomery, D.C., Bazaraa, M.S., Keswani, A.K.: Inventory models with a mixture of backorders and lost sales. Nav. Res. Logist. Q. 20(2), 255–263 (1973)

    Article  MATH  Google Scholar 

  6. Millar, H.H., Yang, M.: Lagrangian heuristics for the capacitated multi-item lot-sizing problem with backordering. Int. J. Prod. Econ. 34(1), 1–15 (1994)

    Article  Google Scholar 

  7. Darwish, M.A.: EPQ models with varying setup cost. Int. J. Prod. Econ. 113(1), 297–306 (2008)

    Article  Google Scholar 

  8. Farzadpour, F., Faraji, H.: A genetic algorithm-based computed torque control for slider–crank mechanism in the ship’s propeller. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 228(12), 2090–2099 (2014)

    Article  Google Scholar 

  9. Shahvari, O., Logendran, R.: Hybrid flow shop batching and scheduling with a bi-criteria objective. Int. J. Prod. Econ. 179, 239–258 (2016)

    Article  Google Scholar 

  10. Teymourian, E., Kayvanfar, V., Komaki, G.M., Zandieh, M.: Enhanced intelligent water drops and cuckoo search algorithms for solving the capacitated vehicle routing problem. Inf. Sci. 334–335, 354–378 (2016)

    Article  Google Scholar 

  11. Tavana, M., Li, Z., Mobin, M., Komaki, M., Teymourian, E.: Multi-objective control chart design optimization using NSGA-III and MOPSO enhanced with DEA and TOPSIS. Expert Syst. Appl. 50, 17–39 (2016)

    Article  Google Scholar 

  12. Teymourian, E., Kayvanfar, V., Komaki, G.M., Khodarahmi, M.: An enhanced intelligent water drops algorithm for scheduling of an agile manufacturing system. Int. J. Inf. Technol. Decis. Mak. 15(02), 239–266 (2016)

    Article  Google Scholar 

  13. Shahvari, O., Logendran, R.: An enhanced tabu search algorithm to minimize a bi-criteria objective in batching and scheduling problems on unrelated-parallel machines with desired lower bounds on batch sizes. Comput. Oper. Res. 77, 154–176 (2017)

    Article  MathSciNet  Google Scholar 

  14. Mondal, S., Maiti, M.: Multi-item fuzzy EOQ models using genetic algorithm. Comput. Ind. Eng. 44(1), 105–117 (2003)

    Article  Google Scholar 

  15. Pasandideh, S.H.R., Niaki, S.T.A.: A genetic algorithm approach to optimize a multi-products EPQ model with discrete delivery orders and constrained space. Appl. Math. Comput. 195(2), 506–514 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pasandideh, S.H.R., Niaki, S.T.A., Yeganeh, J.A.: A parameter-tuned genetic algorithm for multi-product economic production quantity model with space constraint, discrete delivery orders and shortages. Adv. Eng. Softw. 41(2), 306–314 (2010)

    Article  MATH  Google Scholar 

  17. Pirayesh, M., Poormoaied, S.: GPSO-LS algorithm for a multi-item EPQ model with production capacity restriction. Appl. Math. Model. 39(17), 5011–5032 (2015)

    Article  MathSciNet  Google Scholar 

  18. Ross, T.J.: Fuzzy Logic with Engineering Applications, 2nd edn, p. 628. Wiley, New Jersey (2004)

    MATH  Google Scholar 

  19. Shapiro, A.F.: Fuzzy random variables. Insur. Math. Econ. 44(2), 307–314 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kwakernaak, H.: Fuzzy random variables—I. Definitions and theorems. Inf. Sci. 15(1), 1–29 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  21. Puri, M.L., Ralescu, D.A.: Fuzzy random variables. J. Math. Anal. Appl. 114(2), 409–422 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lee, H.-M., Yao, J.-S.: Economic production quantity for fuzzy demand quantity, and fuzzy production quantity. Eur. J. Oper. Res. 109(1), 203–211 (1998)

    Article  MATH  Google Scholar 

  23. Chang, H.-C., Yao, J.-S., Ouyang, L.-Y.: Fuzzy mixture inventory model involving fuzzy random variable lead time demand and fuzzy total demand. Eur. J. Oper. Res. 169(1), 65–80 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Björk, K.-M.: A multi-item fuzzy economic production quantity problem with a finite production rate. Int. J. Prod. Econ. 135(2), 702–707 (2012)

    Google Scholar 

  25. Das, K., Roy, T.K., Maiti, M.: Multi-item stochastic and fuzzy-stochastic inventory models under two restrictions. Comput. Ind. Eng. 31(11), 1793–1806 (2004)

    MATH  Google Scholar 

  26. Dutta, P., Chakraborty, D., Roy, A.R.: Continuous review inventory model in mixed fuzzy and stochastic environment. Appl. Math. Comput. 188(1), 970–980 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Xu, Y., Hu, J.: Random fuzzy demand newsboy problem. Phys. Procedia 25, 924–931 (2012)

    Article  Google Scholar 

  28. Islam, S., Roy, T.K.: A fuzzy EPQ model with flexibility and reliability consideration and demand dependent unit production cost under a space constraint: a fuzzy geometric programming approach. Appl. Math. Comput. 176(2), 531–544 (2006)

    MathSciNet  MATH  Google Scholar 

  29. Islam, S., Roy, T.K.: Fuzzy multi-item economic production quantity model under space constraint: a geometric programming approach. Appl. Math. Comput. 184(2), 326–335 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Bag, S., Chakraborty, D., Roy, A.R.: A production inventory model with fuzzy random demand and with flexibility and reliability considerations. Comput. Ind. Eng. 56(1), 411–416 (2009)

    Article  Google Scholar 

  31. Chakrabortty, S., Pal, M., Nayak, P.K.: Intuitionistic fuzzy optimization technique for Pareto optimal solution of manufacturing inventory models with shortages. Eur. J. Oper. Res. 228(2), 381–387 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mezei, J., Björk, K.-M.: An economic production quantity problem with fuzzy backorder and fuzzy demand. Adv. Inf. Syst. Technol. 206(1), 557–566 (2013)

  33. Lee, S.-D., Yang, C.-M.: An economic production quantity model with a positive resetup point under random demand. Appl. Math. Model. 37(5), 3340–3354 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Çelebi, D.: Inventory control in a centralized distribution network using genetic algorithms: a case study. Comput. Ind. Eng. 87, 532–539 (2015)

    Article  Google Scholar 

  35. Chakraborty, D., Jana, D.K., Roy, T.K.: Multi-item integrated supply chain model for deteriorating items with stock dependent demand under fuzzy random and bifuzzy environments. Comput. Ind. Eng. 88, 166–180 (2015)

    Article  Google Scholar 

  36. Sadeghi, J., Niaki, S.T.A., Malekian, M., Sadeghi, S.: Optimising multi-item economic production quantity model with trapezoidal fuzzy demand and backordering: two tuned meta-heuristics. Eur. J. Ind. Eng. 10(2), 170–195 (2016)

    Article  Google Scholar 

  37. Shahvari, O., Salmasi, N., Logendran, R., Abbasi, B.: An efficient tabu search algorithm for flexible flow shop sequence-dependent group scheduling problems. Int. J. Prod. Res. 50(15), 4237–4254 (2012)

    Article  Google Scholar 

  38. Tersine, R.J.: Principles of Inventory and Materials Management, 4th edn, p. 608. Prentice Hall, New Jersey (1993)

    Google Scholar 

  39. Kiusalaas, J.: Numerical Methods in Engineering, 2nd edn, p. 431. Cambridge University Press, New York (2010)

    Book  MATH  Google Scholar 

  40. Fattahi, P., Hajipour, V., Nobari, A.: A bi-objective continuous review inventory control model: Pareto-based meta-heuristic algorithms. Appl. Soft Comput. 32, 211–223 (2015)

    Article  Google Scholar 

  41. Deb, K., Agrawal, S., Pratap, A., Meyarivan, T.: A Fast Elitist Non-dominated Sorting Genetic Algorithm for Multi-objective Optimization: NSGA-II. In: Schoenauer M. et al. (eds) Parallel Problem Solving from Nature PPSN VI. PPSN 2000. Lecture Notes in Computer Science, vol 1917, pp. 849–858. Springer, Berlin (2000)

  42. Deb, K.: Multi-objective Optimization Using Evolutionary Algorithms, 1st edn, p. 518. Wiley, New York (2001)

    MATH  Google Scholar 

  43. Jadaan, O.A., Rajamani, L., Rao, C.R.: Non-dominated ranked genetic algorithm for solving multi-objective optimization problems: NRGA. J. Theor. Appl. Inf. Technol. 4(1), 60–67 (2008)

    Google Scholar 

  44. Gen, M., Cheng, R.: Genetic Algorithms and Engineering Design, 1st edn, p. 432. Wiley, New York (2000)

    Google Scholar 

  45. Roy, R.: A Primer on the Taguchi Method, p. 247. Society of Manufacturing Engineers, New York (1990)

    MATH  Google Scholar 

  46. Taguchi, G., Chowdhury, S., Wu, Y.: Taguchi’s Quality Engineering Handbook, p. 1662. Wiley, New Jersey (2005)

    MATH  Google Scholar 

  47. Hwang, C.L., Yoon, K.: Multiple Attribute Decision Making: Methods and Applications: A State-of-the-Art Survey. Springer, Berlin (1981)

    Book  MATH  Google Scholar 

  48. Yoon, K.P., Hwang, C.L.: Multiple Attribute Decision Making: An Introduction. Sage, Thousand Oaks (1995)

    Book  Google Scholar 

  49. Naderi, B., Zandieh, M., Roshanaei, V.: Scheduling hybrid flowshops with sequence dependent setup times to minimize makespan and maximum tardiness. Int. J. Adv. Manuf. Technol. 41(11), 1186–1198 (2009)

    Article  Google Scholar 

  50. Hanss, M.: Applied Fuzzy Arithmetic: An Introduction with Engineering Applications, p. 256. Springer, New York (2005)

    MATH  Google Scholar 

  51. Dutta, P., Chakraborty, D., Roy, A.R.: A single-period inventory model with fuzzy random variable demand. Math. Comput. Model. 41(8–9), 915–922 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  52. Liu, B.: Uncertainty Theory: An Introduction to Its Axiomatic Foundations. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  53. Hatami-Marbini, A., M. Tavana, V. Hajipour, F. Kangi, and A. Kazemi.: An extended compromise ratio method for fuzzy group multi-attribute decision making with SWOT analysis. Appl. Soft Comput. 13, 3459–3472  (2013)

  54. Hatami-Marbini, A., Tavana, M., Hajipour, V., Kangi, F., Kazemi, A.: An extended compromise ratio method for fuzzy group multi-attribute decision making with SWOT analysis. Appl. Soft Comput. 13(8), 3459–3472 (2013)

    Article  Google Scholar 

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Acknowledgements

The authors are thankful for constructive comments of anonymous reviewers that significantly improved the presentation of the paper.

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Correspondence to Javad Sadeghi.

Appendix

Appendix

Definition 1

A number referring to several values with varying weights between 0 and 1 are the fuzzy number, for which the weight is called degrees of membership [50].

Thus, the membership function for a trapezoidal fuzzy number defined by four parameters {a, b, c, d}, is

$$\mu_{T} (x) = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {x < {\text{a}}} \hfill \\ {{\raise0.7ex\hbox{${(x - a)}$} \!\mathord{\left/ {\vphantom {{(x - a)} {(b - a)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${(b - a)}$}},} \hfill & {a \le x \le b} \hfill \\ {1,} \hfill & {b \le x < c} \hfill \\ {{\raise0.7ex\hbox{${(d - x)}$} \!\mathord{\left/ {\vphantom {{(d - x)} {(d - c)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${(d - c)}$}},} \hfill & {c \le x < d} \hfill \\ {0,} \hfill & {x > d} \hfill \\ \end{array} } \right.$$
(29)

where \(0 \le \mu_{\,T} (x) \le 1\) is the membership degree of x in the fuzzy set \(\tilde{T}\) and x is fuzzy member. Figure 8 illustrates a trapezoidal fuzzy number.

Fig. 8
figure 8

A trapezoidal fuzzy number

Definition 2

Transferring a fuzzy number to a number is called defuzzification, for which the centroid method (or center of gravity) is a common defuzzification technique [18]. The following algebraic expression illustrates how the centroid method defuzzifies a fuzzy number.

$$T^{*} = \frac{{\smallint \mu_{{\tilde{T}}} (x) \cdot xd_{x} }}{{\smallint \mu_{{\tilde{T}}} (x)d_{x} }}$$
(30)

where \(\smallint\) denotes an algebraic integration. Figure 9 shows a graphical representation.

Fig. 9
figure 9

The centroid defuzzification method

Definition 3

A FRV (\(\hat{T}\)) is a random variable that takes fuzzy values [19].

Note 1 To make a FRV non-random, the expected value method can be applied so that the expectation of \(\hat{T}\) is obtained by

$$E\hat{T} = \int {\hat{T}{\text{d}}P = \left\{ {{{\left( {\int {\tilde{T}_{\alpha }^{ - } {\text{d}}P,} \int {\tilde{T}_{\alpha }^{ + } {\text{d}}P} } \right)} \mathord{\left/ {\vphantom {{\left( {\int {\tilde{T}_{\alpha }^{ - } {\text{d}}P,} \int {\tilde{T}_{\alpha }^{ + } {\text{d}}P} } \right)} {0 \le \alpha \le 1}}} \right. \kern-0pt} {0 \le \alpha \le 1}}} \right\}}$$
(31)

If \(\hat{T}\) is a discrete FRV, assuming \(P(\hat{T} = \tilde{t}) = p_{i} ,i = 1,2,3, \ldots ,z\), its fuzzy expectation is written as [51]

$$E\hat{T} = \sum\nolimits_{i = 1}^{z} {t_{i} p_{i} }$$
(32)

Definition 4

\(\hat{D}_{i}\) is a trapezoidal FRV satisfying the following equation [19].

$$\hat{D}_{i} = \left\{ {\begin{array}{*{20}c} {(d_{i1}^{1} ,d_{i2}^{1} ,d_{i3}^{1} ,d_{i4}^{1} );{\text{with}}\;{\text{probability}}\;\Pr_{\text{i}}^{1} } \\ {(d_{i1}^{2} ,d_{i2}^{2} ,d_{i3}^{2} ,d_{i4}^{2} );{\text{with}}\;{\text{probability}}\;\Pr_{\text{i}}^{2} } \\ \ldots \\ {(d_{i1}^{z} ,d_{i2}^{z} ,d_{i3}^{z} ,d_{i4}^{z} );{\text{with}}\;{\text{probability}}\;\Pr_{\text{i}}^{z} } \\ \end{array} } \right. \Rightarrow \hat{D}_{i} = \left( {d_{i1}^{j} ,d_{i2}^{j} ,d_{i3}^{j} ,d_{i4}^{j} } \right);\Pr_{\text{i}}^{\text{j}}$$
(33)

where z is number of fuzzy sets for random variable with j = 1, 2, …, z and \(\sum\nolimits_{j = 1}^{z} {\Pr_{\text{i}}^{\text{j}} = 1}\).

Simple application Considering two TrFRVs \(\hat{D}_{1}\) and \(\hat{D}_{2}\) with parameters greater than zero [52]:

$$\hat{D}_{1} = \left\{ {\begin{array}{*{20}c} {\left( {d_{11}^{1} ,d_{12}^{1} ,d_{13}^{1} ,d_{14}^{1} } \right);\Pr_{1}^{1} } \\ {(d_{11}^{2} ,d_{12}^{2} ,d_{13}^{2} ,d_{14}^{2} );\Pr_{1}^{2} } \\ \end{array} } \right.$$
(34)
$$\hat{D}_{2} = \left\{ {\begin{array}{*{20}c} {\left( {d_{21}^{1} ,d_{22}^{1} ,d_{23}^{1} ,d_{24}^{1} } \right);\Pr_{2}^{1} } \\ {(d_{21}^{2} ,d_{22}^{2} ,d_{23}^{2} ,d_{24}^{2} );\Pr_{2}^{2} } \\ \end{array} } \right.$$
(35)

The basic arithmetic operations and their expected value are as follows.

$$\hat{D}_{1} + \hat{D}_{2} = \left\{ {\begin{array}{*{20}c} {\left( {d_{11}^{1} + d_{21}^{1} ,d_{12}^{1} + d_{22}^{1} ,d_{13}^{1} + d_{23}^{1} ,d_{14}^{1} + d_{24}^{1} } \right);\Pr_{1}^{1} \Pr_{2}^{1} } \\ {\left( {d_{11}^{1} + d_{21}^{2} ,d_{12}^{1} + d_{22}^{2} ,d_{13}^{1} + d_{23}^{2} ,d_{14}^{1} + d_{24}^{2} } \right);\Pr_{1}^{1} \Pr_{2}^{2} } \\ {\left( {d_{11}^{2} + d_{21}^{1} ,d_{12}^{2} + d_{22}^{1} ,d_{13}^{2} + d_{23}^{1} ,d_{14}^{2} + d_{24}^{1} } \right);\Pr_{1}^{2} \Pr_{2}^{1} } \\ {\left( {d_{11}^{2} + d_{21}^{2} ,d_{12}^{2} + d_{22}^{2} ,d_{13}^{2} + d_{23}^{2} ,d_{14}^{2} + d_{24}^{2} } \right);\Pr_{1}^{2} \Pr_{2}^{2} } \\ \end{array} } \right.$$
(36)
$$\begin{aligned} E(\hat{D}_{1} + \hat{D}_{2} ) & = \left( {d_{11}^{1} + d_{21}^{1} ,d_{12}^{1} + d_{22}^{1} ,d_{13}^{1} + d_{23}^{1} ,d_{14}^{1} + d_{24}^{1} } \right) *\Pr_{1}^{1} \Pr_{2}^{1} \\ & \quad + \left( {d_{11}^{1} + d_{21}^{2} ,d_{12}^{1} + d_{22}^{2} ,d_{13}^{1} + d_{23}^{2} ,d_{14}^{1} + d_{24}^{2} } \right) *\Pr_{1}^{1} \Pr_{2}^{2} \\ & \quad + \left( {d_{11}^{2} + d_{21}^{1} ,d_{12}^{2} + d_{22}^{1} ,d_{13}^{2} + d_{23}^{1} ,d_{14}^{2} + d_{24}^{1} } \right) *\Pr_{1}^{2} \Pr_{2}^{1} \\ & \quad + \left( {d_{11}^{2} + d_{21}^{2} ,d_{12}^{2} + d_{22}^{2} ,d_{13}^{2} + d_{23}^{2} ,d_{14}^{2} + d_{24}^{2} } \right) *\Pr_{1}^{2} \Pr_{2}^{2} . \\ \end{aligned}$$
(37)
$$\hat{D}_{1} - \hat{D}_{2} = \left\{ {\begin{array}{*{20}c} {\left( {d_{11}^{1} - d_{24}^{1} ,d_{12}^{1} - d_{23}^{1} ,d_{13}^{1} - d_{22}^{1} ,d_{14}^{1} - d_{21}^{1} } \right);\Pr_{1}^{1} \Pr_{2}^{1} } \\ {\left( {d_{11}^{1} - d_{24}^{2} ,d_{12}^{1} - d_{23}^{2} ,d_{13}^{1} - d_{22}^{2} ,d_{14}^{1} - d_{21}^{2} } \right);\Pr_{1}^{1} \Pr_{2}^{2} } \\ {\left( {d_{11}^{2} - d_{24}^{1} ,d_{12}^{2} - d_{23}^{1} ,d_{13}^{2} - d_{22}^{1} ,d_{14}^{2} - d_{21}^{1} } \right);\Pr_{1}^{2} \Pr_{2}^{1} } \\ {\left( {d_{11}^{2} - d_{24}^{2} ,d_{12}^{2} - d_{23}^{2} ,d_{13}^{2} - d_{22}^{2} ,d_{14}^{2} - d_{21}^{2} } \right);\Pr_{1}^{2} \Pr_{2}^{2} } \\ \end{array} } \right.$$
(38)
$$\begin{aligned} E(\hat{D}_{1} - \hat{D}_{2} ) & = \left( {d_{11}^{1} - d_{24}^{1} ,d_{12}^{1} - d_{23}^{1} ,d_{13}^{1} - d_{22}^{1} ,d_{14}^{1} - d_{21}^{1} } \right) \times \Pr_{1}^{1} \Pr_{2}^{1} \\ & \quad + \left( {d_{11}^{1} - d_{24}^{2} ,d_{12}^{1} - d_{23}^{2} ,d_{13}^{1} - d_{22}^{2} ,d_{14}^{1} - d_{21}^{2} } \right) \times \Pr_{1}^{1} \Pr_{2}^{2} \\ & \quad + \left( {d_{11}^{2} - d_{24}^{1} ,d_{12}^{2} - d_{23}^{1} ,d_{13}^{2} - d_{22}^{1} ,d_{14}^{2} - d_{21}^{1} } \right) \times \Pr_{1}^{2} \Pr_{2}^{1} \\ & \quad + \left( {d_{11}^{2} - d_{24}^{2} ,d_{12}^{2} - d_{23}^{2} ,d_{13}^{2} - d_{22}^{2} ,d_{14}^{2} - d_{21}^{2} } \right) \times \Pr_{1}^{2} \Pr_{2}^{2} . \\ \end{aligned}$$
(39)

The product and the division of two trapezoidal fuzzy numbers \(\hat{D}_{1}\) and \(\hat{D}_{2}\) are as follows.

$$\left( {\hat{D}_{1} \times \hat{D}_{2} } \right)(z) = \sup \left\{ {\hbox{min} \left\{ {\mu_{{\hat{D}_{1} }} (x),\mu_{{\hat{D}_{2} }} (y)} \right\}:x \times y = z} \right\}$$
(40)
$$\left( {{{\hat{D}_{1} } \mathord{\left/ {\vphantom {{\hat{D}_{1} } {\hat{D}_{2} }}} \right. \kern-0pt} {\hat{D}_{2} }}} \right)(z) = \sup \left\{ {\hbox{min} \left\{ {\mu_{{\hat{D}_{1} }} (x),\mu_{{\hat{D}_{2} }} (y)} \right\}:{x \mathord{\left/ {\vphantom {x y}} \right. \kern-0pt} y} = z} \right\}$$
(41)

As Eqs. (38, 39) do not exactly result in trapezoidal fuzzy numbers, some equivalence relations have been introduced in the literature to obtain trapezoidal fuzzy numbers [53]. This paper employs the equivalence relation in [54] for the division and the product of two trapezoidal fuzzy numbers as:

$$\hat{D}_{1} \times \hat{D}_{2} = \left\{ {\begin{array}{*{20}c} {\left( {d_{11}^{1} \times d_{21}^{1} ,d_{12}^{1} \times d_{22}^{1} ,d_{13}^{1} \times d_{23}^{1} ,d_{14}^{1} \times d_{24}^{1} } \right);\Pr_{1}^{1} \Pr_{2}^{1} } \\ {\left( {d_{11}^{1} \times d_{21}^{2} ,d_{12}^{1} \times d_{22}^{2} ,d_{13}^{1} \times d_{23}^{2} ,d_{14}^{1} \times d_{24}^{2} } \right);\Pr_{1}^{1} \Pr_{2}^{2} } \\ {\left( {d_{11}^{2} \times d_{21}^{1} ,d_{12}^{2} \times d_{22}^{1} ,d_{13}^{2} \times d_{23}^{1} ,d_{14}^{2} \times d_{24}^{1} } \right);\Pr_{1}^{2} \Pr_{2}^{1} } \\ {\left( {d_{11}^{2} \times d_{21}^{2} ,d_{12}^{2} \times d_{22}^{2} ,d_{13}^{2} \times d_{23}^{2} ,d_{14}^{2} \times d_{24}^{2} } \right);\Pr_{1}^{2} \Pr_{2}^{2} } \\ \end{array} } \right.$$
(42)
$$\begin{aligned} E(\hat{D}_{1} \times \hat{D}_{2} ) = & \left( {d_{11}^{1} \times d_{21}^{1} ,d_{12}^{1} \times d_{22}^{1} ,d_{13}^{1} \times d_{23}^{1} ,d_{14}^{1} \times d_{24}^{1} } \right) \times \Pr_{1}^{1} \Pr_{2}^{1} \\ & + \left( {d_{11}^{1} \times d_{21}^{2} ,d_{12}^{1} \times d_{22}^{2} ,d_{13}^{1} \times d_{23}^{2} ,d_{14}^{1} \times d_{24}^{2} } \right) \times \Pr_{1}^{1} \Pr_{2}^{2} \\ & + \left( {d_{11}^{2} \times d_{21}^{1} ,d_{12}^{2} \times d_{22}^{1} ,d_{13}^{2} \times d_{23}^{1} ,d_{14}^{2} \times d_{24}^{1} } \right) \times \Pr_{1}^{2} \Pr_{2}^{1} \\ & + \left( {d_{11}^{2} \times d_{21}^{2} ,d_{12}^{2} \times d_{22}^{2} ,d_{13}^{2} \times d_{23}^{2} ,d_{14}^{2} \times d_{24}^{2} } \right) \times \Pr_{1}^{2} \Pr_{2}^{2} . \\ \end{aligned}$$
(43)
$$\hat{D}_{1} /\hat{D}_{2} = \left\{ {\begin{array}{*{20}c} {\left( {d_{11}^{1} /d_{24}^{1} ,d_{12}^{1} /d_{23}^{1} ,d_{13}^{1} /d_{22}^{1} ,d_{14}^{1} /d_{21}^{1} } \right);\Pr_{1}^{1} \Pr_{2}^{1} } \\ {\left( {d_{11}^{1} /d_{24}^{2} ,d_{12}^{1} /d_{23}^{2} ,d_{13}^{1} /d_{22}^{2} ,d_{14}^{1} /d_{21}^{2} } \right);\Pr_{1}^{1} \Pr_{2}^{2} } \\ {\left( {d_{11}^{2} /d_{24}^{1} ,d_{12}^{2} /d_{23}^{1} ,d_{13}^{2} /d_{22}^{1} ,d_{14}^{2} /d_{21}^{1} } \right);\Pr_{1}^{2} \Pr_{2}^{1} } \\ {\left( {d_{11}^{2} /d_{24}^{2} ,d_{12}^{2} /d_{23}^{2} ,d_{13}^{2} /d_{22}^{2} ,d_{14}^{2} /d_{21}^{2} } \right);\Pr_{1}^{2} \Pr_{2}^{2} } \\ \end{array} } \right.$$
(44)
$$\begin{aligned} E(\hat{D}_{1} /\hat{D}_{2} ) & = \left( {d_{11}^{1} /d_{24}^{1} ,d_{12}^{1} /d_{23}^{1} ,d_{13}^{1} /d_{22}^{1} ,d_{14}^{1} /d_{21}^{1} } \right) \times \Pr_{1}^{1} \Pr_{2}^{1} \\ & \quad + \left( {d_{11}^{1} /d_{24}^{2} ,d_{12}^{1} /d_{23}^{2} ,d_{13}^{1} /d_{22}^{2} ,d_{14}^{1} /d_{21}^{2} } \right) \times \Pr_{1}^{1} \Pr_{2}^{2} \\ & \quad + \left( {d_{11}^{2} /d_{24}^{1} ,d_{12}^{2} /d_{23}^{1} ,d_{13}^{2} /d_{22}^{1} ,d_{14}^{2} /d_{21}^{1} } \right) \times \Pr_{1}^{2} \Pr_{2}^{1} \\ & \quad + \left( {d_{11}^{2} /d_{24}^{2} ,d_{12}^{2} /d_{23}^{2} ,d_{13}^{2} /d_{22}^{2} ,d_{14}^{2} /d_{21}^{2} } \right) \times \Pr_{1}^{2} \Pr_{2}^{2} . \\ \end{aligned}$$
(45)

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Sadeghi, J., Niaki, S.T.A., Malekian, M.R. et al. A Lagrangian Relaxation for a Fuzzy Random EPQ Problem with Shortages and Redundancy Allocation: Two Tuned Meta-heuristics. Int. J. Fuzzy Syst. 20, 515–533 (2018). https://doi.org/10.1007/s40815-017-0377-z

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