An extensive operational law for monotone functions of LR fuzzy intervals with applications to fuzzy optimization

The operational law proposed by Zhou et al. (J Intell Fuzzy Syst 30(1): 71–87, 2016) contributes to developing fuzzy arithmetic, while its applicable conditions are confined to strictly monotone functions and regular LR fuzzy numbers, which are hindering their operational law from dealing with more general cases, such as problems formulated as monotone functions and problems with fuzzy variables represented as fuzzy intervals (e.g., trapezoidal fuzzy numbers). In order to handle such cases we generalize the operational law of Zhou et al. in both the monotonicity of function and fuzzy variables in this paper and then apply the extensive operational law to the cases with monotone (but not necessarily strictly monotone) functions with regard to regular LR fuzzy intervals (LR-FIs) (of which regular LR fuzzy numbers are special cases). Specifically, we derive the computational formulae for expected values (EVs) of LR-FIs and monotone functions with regard to regular LR-FIs, respectively. On the other hand, we develop a solution scheme to dispose of fuzzy optimization problems with regular LR-FIs, in which a fuzzy programming is converted to a deterministic equivalent one and a newly devised solution algorithm is utilized to get the deterministic programming solved. The numerical experiments are conducted using our proposed solution scheme and the traditional fuzzy simulation-based genetic algorithm in the context of a purchasing planning problem. Computational results show that our method is much more efficient, yielding high-quality solutions.


Introduction
In real-life cases, uncertainty on input parameters involved in some optimization problems is inevitable due to the unpredictable natural factors. In this regard, fuzzy set theory initiated by Zadeh (1965) as one of the popular ways coping with uncertainties is applied to practical optimization process of various research fields, such as supply chain management (Ke et al. 2018), transportation (Büyüközkan et al. 2018), and finance investment (Stoklasa et al. 2021). Under the fuzzy circumstance, uncertain parameters are commonly assigned to be fuzzy variables, in which fuzzy numbers and  (Dubois and Prade 1979)) are two frequently used types. The main difference between them is that the modal value of fuzzy number is a point value, while the set of modal values of fuzzy interval is an interval. From the mathematical viewpoint, we can consider fuzzy numbers as a particular situation of fuzzy intervals. Therefore, the emphasis of research in this paper is placed on fuzzy intervals.
With regard to the parametric representation of fuzzy interval, Prade (1979, 1988) defined a well-known L-R representation, in which L and R are shape functions representing the left and right sides of membership function, respectively. From another point of view, Goetschel and Voxman (1986) proposed an equivalent representation named as L-U representation in accordance with the lower and upper branches, which define the two endpoints of an α-cut. Stefanini and Guerra (2017) suggested ACF-representation to describe a fuzzy interval by using a new defined average cumulative function on the basis of the possibility theory. It should be mentioned that the typical L-R representation in Dubois and Prade (1988) is utilized to delineate fuzzy interval in this paper, and the corresponding fuzzy interval is termed as LR-FI accordingly.
Fuzzy arithmetic plays an essential role in processing fuzzy variables, which has attracted interests of many researchers. Zadeh (1975) initially extended the common arithmetic operations for real numbers to fuzzy intervals via the proposed extension principle on the basis of a triangular norm (t-norm). As for the t-norm-based arithmetic operations, an important feature is to offer a way for controlling the rise on uncertainty in the process of computations and avoid variables shifting off their most vital values simultaneously. However, the practical use of Zadeh's extension principle is a little complicated owing to the involved nonlinear operators. Subsequently, Dubois and Prade (1979) proposed some analytical calculations including the basic arithmetic addition, subtraction, multiplication, and division among LR-FIs together with some properties. Meanwhile, there is some other literature focusing on algebraic operations of LR-FIs. Hwang and Lee (2001) studied the sum for LR-FIs in accordance with a given nilpotent t-norm. For the sake of preserving the shapes of fuzzy intervals during the practical computation, some shape-preserving operations on fuzzy intervals with sigmoid and bell-shaped membership functions (Dombi and Gyorbiro 2006;Hong 2007) were investigated. With the same aim, Mako (2012) constructed the real vector space of LR-FIs, and then presented the algebraic forms and the associated application. Based upon the definition of unrestricted LR-FI (Kaur and Kumar 2012), Kaur and Kumar (2013) presented the product of unconstrained LR-FIs, thereby formulating a Mehar's method to deal with the linear programming problems. Recently, Abbasi and Allahviranloo (2021) proposed new fuzzy arithmetic operations on LR type flat fuzzy numbers based on the transmission average. Additionally, some arithmetic operations on a specific type of fuzzy intervals such as trapezoidal fuzzy numbers (Shakeel et al. 2019a, b), and pseudo-octagonal fuzzy numbers (Abbasi and Allahviranloo 2019) were also discussed.
As a particular kind of LR-FIs, LR fuzzy numbers have got quite a few attentions because of the good interpretability and easy performing for usual operations since they were introduced by Dubois and Prade (1978). So far there have been many relevant studies on arithmetic operations of LR fuzzy numbers (see, e.g., Ban et al. 2016;Garg 2018;Garg and Ansha 2018;Ghanbari et al. 2022). In particular, on account of the credibility measure pioneered by Liu (2002), Zhou et al. (2016) proposed an operational law targeting at strictly monotone functions with regard to LR fuzzy numbers. Based on this, a crispy solution framework for the fuzzy programming was formulated, which reduces the computation complexity a lot. Given the effectiveness of the operational law in Zhou et al. (2016), it has been gradually employed to handle different optimization problems.
For example, Wang et al. (2018) developed a revised hybrid intelligent algorithm to solve a green-fuzzy vehicle routing problem. Fang et al. (2020) devised an analytical method to tackle a newly established fuzzy quality function deployment model for product design in multi-segment markets. Besides, the research findings in Zhou et al. (2016) were also used for diagnosis of prostate cancer (Kar and Majumder 2017), reliability analysis (Dutta 2019), preventive maintenance scheduling problem (Zhong et al. 2019;Wang et al. 2020), location problem (Soltanpour et al. 2019;Yang et al. 2019), and so on.
From the review of existing research on the fuzzy arithmetic, the majority of research studied arithmetic calculations on fuzzy variables and presented useful fuzzy arithmetic operations, while Zhou et al. (2016) focused on handling functions with fuzzy variables and proposed the operational law for computing the inverse credibility distributions (ICDs). Nonetheless, their operational law aims at strictly monotone functions with regard to regular LR fuzzy numbers. As we know, many optimization problems (e.g., classical newsvendor problem), in practice, cannot be modeled using strictly monotone functions, and there are some problems where LR-FIs (e.g., trapezoidal fuzzy numbers) defined by Liu et al. (2020) are more appropriate to represent fuzzy variables. In such cases, the operational law in Zhou et al. (2016) is unable to come into use. Therefore, it is necessary and valuable to make an extensive study for Zhou et al. (2016). The purpose of this paper is to propose an extensive operational law based on the one proposed by Zhou et al. (2016) so that more fuzzy optimization problems modeled using monotone (but not necessarily strictly monotone) functions and regular LR fuzzy intervals (LR-FIs) can be handled.
The main contributions of this paper to the field of fuzzy arithmetics and fuzzy optimization are fourfold.
1. We propose the ICD of an LR-FI based on credibility measure, which is a generalization for the ICD of regular LR fuzzy number defined in Zhou et al. (2016), and verify two equivalent conditions of regular LR-FIs. 2. We present an extensive operational law on monotone functions with regard to regular LR-FIs, which generalizes the operational law in Zhou et al. (2016) from both the function monotonicity and the type of fuzzy variables. Concretely, the strictly monotone functions are extended to be monotone (but not necessarily strictly monotone) functions, and the regular LR fuzzy numbers are generalized to regular LR-FIs such as trapezoidal fuzzy numbers. 3. We develop calculation formulas for EVs of LR-FIs and monotone functions with regard to regular LR-FIs based on the extensive operational law. In accordance with the calculation formulas, the EVs of monotone functions of regular LR-FIs can be derived directly by means of the corresponding ICDs. 4. We construct a solution strategy including a newly devised heuristic algorithm with a new effective simulation for the fuzzy chance-constrained programming (CCP) with monotone objective and constraint functions regarding regular LR-FIs. Then we illustrate the better performances of our method on both solution accuracy and efficiency in comparison with another traditional heuristic algorithm through a purchasing planning problem.
The remaining of this paper is organized as follows. Section 2 recalls some fundamental notions regarding the LR-FI, defines its ICD in the light of the credibility distribution, and then derives the equivalent conditions of regular LR-FIs. In Sect. 3, we explore the property of monotone functions with regard to regular LR-FIs, propose a new operational law and then discuss the EVs of LR-FIs and monotone functions in regard to regular LR-FIs. In Sect. 4, a solution strategy for the fuzzy CCP is formulated based on the new operational law. To exhibit the effectiveness of our strategy, some numerical experiments are implemented by using our method and a traditional heuristic method, respectively, in the context of a purchasing planning problem. Finally, the main findings are concluded in Sect. 5. The conceptual framework of our study is demonstrated in Fig. 1.

LR fuzzy interval and its inverse credibility distribution
In this section, some elementary conceptions in relation to LR-FIs and the credibility distribution of fuzzy variable are reviewed first. We subsequently define the ICD of an LR-FI, and derive its mathematical expression. After that, we introduce the definition of the regular LR-FI and prove its two necessary and sufficient conditions.

LR fuzzy interval and its credibility distribution
The well-known LR-FI was initially proposed by Dubois and Prade (1988), in which L and R are the decreasing left and right shape functions from [0, ∞) → [0, 1] with L(0) = 1 and R(0) = 1, respectively. The LR-FI is the most general class of fuzzy intervals, and the LR fuzzy number can be seen as a special case of LR-RI with unique modal value. The LR-FI is also a kind of fuzzy parameters commonly and frequently used and covers most of fuzzy parameters used in the fuzzy optimization. Different from other fuzzy intervals, the LR-FI is one type of unimodal fuzzy intervals and its corresponding membership function can be expressed by two decreasing shape functions L and R with four parameters, in which the left side of membership function is monotonically increasing and right side of membership function is monotonically decreasing.
Definition 1 (Dubois and Prade 1988) A fuzzy interval M defined on universal set of real numbers R is said to be an LR-FI if it has the membership function with shape functions L, R and four parameters c, c, ρ > 0, σ > 0 as To measure fuzzy events in the fuzzy world, Zadeh (1978) suggested the possibility measure. However, it lacks selfduality. To overcome this deficiency, Liu and Liu (2002) defined the credibility measure based on the possibility measure and proved its self-duality. A self-dual measure is needed in this study as only when the measure satisfies self-duality can we use inverse distribution to draw some important inferences about fuzzy arithmetic. Hence the credibility measure is adopted in this paper rather than the possibility measure.
Suppose that ζ is a fuzzy variable with membership function ν and t is a real number. The credibility of fuzzy event {ζ ≤ t} is defined by Liu and Liu (2002) as To describe a fuzzy variable, credibility distribution as a carrier of incomplete information of this variable is defined by Liu (2004) as follows.
Definition 2 (Liu 2004) If ζ is a fuzzy variable, then its credibility distribution ψ : R → [0, 1] is defined by In accordance with the mathematical properties of credibility measure, it is known that the credibility distribution ψ is non-decreasing on R, in which ψ(−∞) = 0 and ψ(+∞) = 1.
Example 1 If an LR-FI (c, c, ρ, σ ) L R has the shape functions L(t) = R(t) = max{0, 1 − t}, then it is called trapezoidal fuzzy number denoted by T (c, c, ρ, σ ) L R with the membership function as depicted in Fig. 2a, b, respectively.

Example 3
If an LR-FI (c, c, ρ, σ ) L R has the shape functions L(t) = max{1 − t, 0} and R(t) = max{1 − t 2 , 0}, denoted by B(c, c, ρ, σ ) L R , then it has the membership function otherwise and the credibility distribution as depicted in Fig. 4a, b, respectively.

Example 4
If an LR-FI (2, 4, 2, 4) L R has the following shape functions then it can be deduced that it has the membership function as depicted in Fig. 5a, b, respectively.

Inverse credibility distribution of LR fuzzy interval
For our purpose, the ICD of an LR-FI is defined as below, which will play an important role then.

Definition 3 Let ζ be an LR-FI. A multi-valued function
where Remark 2 For simplicity, F(δ) is denoted by ψ −1 (δ), which differs from the inverse function of ψ(t).

Theorem 1 Let ψ(t) be the credibility distribution of an LR-FI ζ , and D ψ be the domain of values of ψ(t). Then the ICD of ζ is deduced as
Proof The proof is provided in Appendix B.

Regular LR fuzzy interval
It is worth noting that the credibility distributions ψ(t) (or the ICDs ψ −1 (δ)) of LR-FIs in Examples 1-3 (or in Examples 5-7) are continuous and strictly increasing on the domain {t| 0 < ψ(t) < 0.5 or 0.5 < ψ(t) < 1}. For the sake of describing such kind of LR-FIs, we first introduce the definition of regular LR-FI proposed in Liu et al. (2020) and then verify two equivalent conditions. Definition 4 (Liu et al. 2020) If the shape functions L and R of an LR-FI ζ are continuous and strictly decreasing on the domains {t| 0 < L(t) < 1} and {t| 0 < R(t) < 1}, respectively, then the LR-FI is regular.
As regards regular LR-FIs, it seems clear that its L and R shape functions are both continuous and strictly decreasing on respective domains, which also means that there exist the inverse functions of shape functions, i.e., L −1 and R −1 . Thus it follows from the above analysis and the definition of regular LR-FI that the ICD of a regular LR-FI can be deduced directly.
Proof This theorem follows from Eq. (4) and Theorem 1.
Definition 5 (Liu et al. 2016;Liu 2010) A real-valued function f (t 1 , t 2 , . . . , t n ) is called monotone function if it is increasing regarding t 1 , t 2 , . . . , t k and decreasing regarding t k+1 , t k+2 , . . . , t n , that is, for any t i > s i for i = 1, 2, . . . , k and t i < s i for i = k + 1, . . . , n, then it is said to be strictly monotone.
then it is called strictly increasing (decreasing) function.

Example 9
The following functions are strictly monotone, Example 10 The following functions are monotone but not strictly monotone, In some practical optimization problems, the objective functions of the formulated optimization models are usually monotone but not strictly monotone, such as the well-known newsvendor problem, inventory problem, project scheduling problem, etc. Considering the generality of monotone functions in practical applications, we will proceed to analyze the property of continuous and monotone (but not necessarily strictly monotone) functions of regular LR-FIs.
Theorem 4 Let ζ 1 , ζ 2 , . . ., ζ n be independent regular LR-FIs and f : R n → R a continuous and monotone function. Then Proof The proof is provided in Appendix C.
is increasing but not strictly increasing. Then the credibility distribution of f 1 (ζ 1 ) is obtained as follows as depicted in Fig. 10. It can be concluded from Fig. 10 that is not a regular LR-FI.

Operational law
Based on the extensive applications of monotone functions and regular LR-FIs in optimization problems, a new operational law is proposed in this subsection, which can be considered as an extension to the one developed in Zhou et al. (2016).

Remark 4
It should be noted that the operational law proposed by Zhou et al. (2016) is used to compute the ICDs of strictly monotone functions regarding regular LR fuzzy numbers. However, the new operational law provides a convenient and powerful approach to computing the ICDs of monotone (but not strictly monotone) functions regarding regular LR-FIs, which are unable to be obtained by the operational law in Zhou et al. (2016).

Example 12
Let ζ 1 be a trapezoidal fuzzy number denoted by T (2, 4, 2, 3) L R with the credibility distribution ψ 1 . Then the ICD of f 1 (ζ 1 ) with f 1 defined in Eq. (8) is deduced from Theorem 5 as Fig. 10 The credibility distribution of f 1 (ζ 1 ) in Example 11

Example 13
Let ζ 2 be a trapezoidal fuzzy number denoted by T (1, 2, 2, 1) L R with the credibility distribution ψ 2 . Then by using the operational law in Theorem 5, it is easy to deduce the ICD of f 2 (ζ 2 ) where the function f 2 is defined as Since f 2 is decreasing, in the light of Theorem 5, the ICD of f 2 (ζ 2 ) is derived as (see Fig. 12) Fig. 12 The ICD of f 2 (ζ 2 ) in Example 13 Example 14 Let ζ 1 and ζ 2 be two independent trapezoidal fuzzy numbers denoted by T (2, 4, 2, 3) L R and T (1, 2, 2, 1) L R with the ICDs ψ −1 1 and ψ −1 2 , respectively. As the function with f 1 and f 2 defined in Eqs. (8) and (11), respectively, is increasing regarding t 1 and decreasing regarding t 2 , in accordance with Theorem 5, the ICD of ζ = f (ζ 1 , ζ 2 ) = f 1 (ζ 1 ) + f 2 (ζ 2 ) is obtained as Then, on account of Eqs. (10) and (12), we can obtain that as depicted in Fig. 13.

Expected value
Expected value (EV) is the mean value of all possible values of a fuzzy variable in the sense of fuzzy measure. Based on the EV of a fuzzy variable defined by Liu and Liu (2002), a calculation formula of the EV of an LR-FI is presented. Fig. 13 The ICD of f (ζ 1 , ζ 2 ) in Example 14 Definition 6  If ζ is a fuzzy variable, then its EV is defined as suppose that at least one of the two integrals is finite.

Theorem 6 If the EV of an LR-FI ζ exists, then
Proof The proof is provided in Appendix E.
Following from Theorems 5 and 6, a theorem that can be able to calculate EVs of monotone functions with regard to regular LR-FIs is proposed.

Fuzzy programming
Fuzzy programming is a type of mathematical models to address optimization problems involving fuzzy parameters, which has been studied by many researchers from different points of view (see Liu 1998;Liu and Iwamura 1998a, b;Liu and Liu 2002;Zhou et al. 2016). In this section, we discuss the fuzzy CCP model in Zhou et al. (2016) containing monotone but not necessarily strictly monotone objective and constraint functions with regular LR-FIs and then develop a solution framework.
Owing to the fuzziness of the objective function f (t, ζ ), it is hard to be minimized directly. As an alternative way, it is quite natural to minimize its EV, i.e., E[ f (t, ζ )]. In addition, as to the fuzzy constraints h v (t, ζ ) ≤ 0, v = 1, 2, . . . , w, since there is no deterministic feasible set defined by them, Liu and Iwamura (1998a) suggested that it should be desirable that the solutions satisfy the fuzzy constraints at predetermined confidence levels δ 1 , δ 2 , . . . , δ w , that is, In this way, a fuzzy CCP model to minimize the EV of objective function under a series of chance constraints was constructed by Zhou et al. (2016) subject to: When fuzzy parameters in the fuzzy CCP model (15) are regular LR-FIs, and the objective and constraint functions are both continuous and monotone with regard to these fuzzy parameters, model (15) can be converted to a deterministic counterpart, which is verified in the following theorem.
Proof The proof is provided in Appendix F.

Solution methods
It is worth noting that there exists an integral in the objective function of model (16), which means that the fuzzy model (15) cannot be solved directly by well-developed software packages after translation. In order to solve model (15), Liu (2002) designed a fuzzy simulation-based genetic algorithm, called hybrid intelligent algorithm (HIA), by integrating a stochastic discretization algorithm (SDA) into a classical genetic algorithm. However, Li (2015) and Liu et al. (2020) pointed out that SDA has poor performance both on accuracy and computational time over simulating the EV. Liu et al. (2020) subsequently proposed a numerical-integral based algorithm, but it is not applicable to monotone but not strictly monotone functions with regard to regular LR-FIs. Thus this paper proposes a new numerical integration algorithm (NIA) to fill the gap.
With regard to the basic principle of NIA for simulating E[ f (t, ζ )], on account of Theorem 8, we know that (1 − δ)). Based on the definition of definite integral in mathematics, we partition the closed interval [0, 1] into S equal parts and take the value of the right of each equal part as the integration variable, that is, δ = s/S for s = 1, 2, . . . , S. When the number of integration points, S, is set to be sufficiently large, we can obtain that The NIA is given as Algorithm 1.
Step 4. If s ≤ S, go to Step 2. Otherwise, return the E.
To illustrate the performance of NIA on accuracy and efficiency, comparisons between NIA and SDA for simulating the EV of monotone functions over some numerical experiments of two examples are conducted.
For each case, after using SDA and NIA (5000 points for simulation) to calculate EVs of the functions, respectively, the experimental results of two examples covering exact value, simulation value and running time are all listed    in Tables 2 and 3, respectively. To facilitate comparing the differences between the simulation values obtained by two algorithms and the exact value obtained based on the extensive operation law, a parameter named Error is introduced, which is derived from the formula | simulation value -exact value | / exact value×100%. From Tables 2 and 3, it can be seen that there are slight differences over errors (i.e., ≤ 0.05%) between the EVs obtained by NIA and the exact values, and the errors are still small as the number of fuzzy intervals and the complexity of monotone functions increase. When it comes to the function f 4 , however, the largest error between the EVs derived by SDA and the exact values is up to 12.82%. In addition, as the number of fuzzy intervals and the complexity of monotone functions increase, the errors may further increase. Therefore, NIA is more reliable and stable in terms of the accuracy of solutions compared with SDA. On the other hand, the running time of NIA can be negligible, while the running time of SDA is more than 120 times slower than NIA's. Overall, NIA outperforms SDA and is probably able to get an accurate value in a relatively short time. Based on the above analyses, we embed NIA used for simulating E[ f (t, ζ )] into a classical genetic algorithm, thereby formulating a new algorithm (NIA-GA) to dispose of model (16), whose performance will be compared with HIA algorithm and evaluated on a set of numerical experiments from a purchasing planning problem in the following section.

Numerical example
Provided that there is a dealer selling n types of products, he would like to determine the optimal order quantity to satisfy customer demands for products with the aim of maximizing the total profit. In order to have a better understanding for this problem, some assumptions are given as follows and some relevant notations are shown in Table 4 where the parameter values are summarized in Table 5.

Assumption
1. The customer demands are uncertain and characterized by regular LR-FIs. 2. Any leftover inventory can be salvaged at a unit value, which is lower than the selling price. 3. The total cost of purchasing products from supplier is not more than budget.
According to the assumptions and notations, the total procurement cost, total opportunity loss and the total profit are

respectively.
Assuming that the total budget on the procurement is C 0 and the biggest opportunity loss the dealer can undertake is S 0 . C 0 is mainly determined based on the dealer's financial performance, and S 0 is subjectively determined by the decision maker according to his risk-averse attitude and the available budget C 0 . Then it follows from the idea of fuzzy CCP model (15) that a fuzzy CCP model for this problem is constructed as Apparently, both the total profit Π(t, ζ ) and the total opportunity loss S(t, ζ ) are increasing but not strictly increasing with respect to ζ i . Here we assume that the retailer has three types of the products, and other parameter values are all summarized in Table 5. Then based on Theorem 8, the fuzzy CCP model (17) can be translated into a deterministic programming model where ψ −1 i is the ICD of ζ i , which can be derived from Theorem 1.
Afterward, 40 test problems are generated by increasing C 0 from 61000 to 70000 with an increase of 1000 and decreasing S 0 from 6500 to 2000 with a decrease of 500 simultaneously under the confidence level fixed at 0.6, 0.7 0.8 and 0.9, respectively. For each problem, HIA and NIA-GA (5000 points for simulation and 300 generations in genetic algorithm) are run accordingly to solve the models (17) and (18), respectively. Considering the randomness of results obtained by metaheuristic algorithms, we implement each test problem for 10 times and then select the optimal solution with the best target value as the final solution. Then the optimal solutions, the corresponding target values, E[Π(t * , ζ )], and the average time for running 10 times are shown in Table  6. Moreover, in accordance with the poor performance of SDA in HIA as illustrated in Sect. 4.2, a new EV of profit, E[Π(t * , ζ )] * , is computed by substituting the optimal solution acquired by HIA into NIA, which is listed in the last column of Table 6. So, it makes sense to judge the quality of the optimal solutions obtained by HIA and NIA-GA by com- Table 6. Furthermore, for the sake of visualizing the differences better, the target values, E[Π(t * , ζ )] obtained by two solution methods and E[Π(t * , ζ )] * in Table 6 are plotted in Fig. 14a-d.
From Table 6, it can be seen that NIA-GA has an outstanding advantage over HIA in terms of running time. Concretely, the running time of NIA-GA is almost one hundred times faster than that of HIA. The reason is that the fuzzy simulation (SDA) in HIA for the EV of profit and chance constraint of opportunity loss is time-consuming. In the meantime, as for the quality of solutions found by two methods, we can conclude that the solutions derived  , ζ )] is more precise than SDA, which has been proved in Examples 17 and 18. All in all, compared with HIA, NIA-GA not only has excellent performance on running time but also can obtain a better target value. In order to see the impact of parameter δ 0 on the expected profit, the sensitivity analyses of parameter δ 0 with the total budget on the procurement C 0 under a fixed S 0 at 3000 and with the biggest opportunity loss the dealer can undertake S 0 under a fixed C 0 at 60000 are conducted separately, which are visualized in Fig. 15a, b, respectively. From Fig. 15a, b, we can see that the expected profit decreases with δ 0 but the pace of declines becomes slower as C 0 and S 0 increase. For the fixed C 0 and S 0 , the increase of δ 0 means that more products with less procurement cost but high opportunity loss need to be purchased to meet opportunity loss constraint, but those products have lower profit rate, which leads to the reduction of order quantity for other products with higher profit rate due to the limited budget. Thus the expected profit decreases with δ 0 . If C 0 or S 0 increases, more products with high profit rate can be purchased, and thus the expected profit decreases slowly with δ 0 .
In addition, it is clear that the expected profit increases with C 0 since more products can be purchased to meet customer demands as the C 0 increases. However, as the C 0 further increases, the dealer cannot increase the order quantity exceeding customer demands. In such a situation, the expected profit has no change. Similarly, the increase of S 0 can also improve the expected profit since more products with high profit rate can be purchased. For the higher S 0 , the opportunity loss constraint is not bound and has no impact on the expected profit.
In summary, this section discusses fuzzy CCP involving monotone objective and constraint functions of regular LR-RIs and the performances of two solution methods. One is the HIA for solving the fuzzy CCP model. As to the second one, it is proved that the fuzzy CCP can be translated into crisp one based on the operational law and then NIA-GA is designed to handle the crisp model. The main difference between two methods is that HIA employs fuzzy simulation to handle the objective and constraint functions while our method utilizes NIA to deal with the objective function and reduces the fuzzy simulation on constraint function. Subsequently, two methods are used to deal with a fuzzy purchasing planning problem and their performances are evaluated by a set of numerical experiments. The experimental results demonstrate that NIA-GA not only has excellent performance on running time but also can obtain a better target value compared with HIA. Finally, sensitivity analysis for some parameters are carried out. In particular, the parameter δ 0 has negative impact on the expected profit and its influence decreases with the increase of C 0 and S 0 , while C 0 and S 0 have positive impact on the expected profit and their influences fade if C 0 or S 0 is large enough.

Conclusion
Fuzzy arithmetic is of great importance as an advanced tool to be used in fuzzy optimization and control theory. In this research field, Zhou et al. (2016) proposed an operational law to exactly calculate the ICDs of strictly monotone functions regarding regular LR fuzzy numbers, which facilitates the development of fuzzy arithmetic both in theory and application. Although the operational law is rather useful to handle many fuzzy optimization problems, restrictions on the strictly monotone functions and regular LR fuzzy numbers block its applications to some problems modeled by monotone functions with LR fuzzy intervals, such as the classical newsvendor problem with fuzzy demands represented by trapezoidal fuzzy numbers. Thus, this paper aims at generalizing the operational law in Zhou et al. (2016) and exploring the generalized operational law's applications to fuzzy arithmetic and fuzzy optimization problems.
The main findings of this study are summarized as follows. First, the ICD of an LR-FI in view of the credibility measure was defined and accordingly its calculation formula was suggested. Following that, some equivalent conditions of the regular LR-FI were proved. Next, an extensive operational law on exactly calculating ICDs of monotone functions with regard to regular LR-FIs was proposed. Then an equivalent formula for calculating the EV of an LR-FI and a theorem for calculating the EVs of monotone functions were proposed. Subsequently, a solution strategy for the fuzzy CCP with monotone functions of regular LR-FIs was formulated, where the fuzzy model was translated into a crisp equivalent one first and then a new heuristic algorithm called NIA-GA which integrates NIA with a standard genetic algorithm was devised. Finally, we used a purchasing planning problem to illustrate the performance of proposed solution method by comparing with HIA over a set of numerical experiments. The computational results revealed that our method outperforms HIA in both solution accuracy and efficiency. In summary, this paper made a contribution to fuzzy arithmetics on regular LR-FIs and provided a general approach to handling fuzzy optimization involving monotone functions regarding regular LR-FIs, which can be utilized to deal with any fuzzy optimization problem satisfying the corresponding conditions.
Even so this paper still has some limitations may open opportunities for future research. First, we just applied the theoretical findings to solve a fuzzy purchasing planning problem. As a matter of fact, the proposed extensive operational law is a general approach able to handle different fuzzy optimization problems involving monotone objective and constraint functions concerning regular LR-FIs. In practice, there are many such type of optimization problems, e.g., simultaneous delivery and pickup problem, project scheduling problem and reliability optimization problem under fuzzy environment, so the theoretical findings can be further applied to deal with those fuzzy optimization problems in future. Second, we suggested a new solution method to address the fuzzy optimization involving monotone function with respect to a special kind of fuzzy variables called regular LR-FIs. Apart from LR-FIs, there also exist other types of fuzzy variables, such as type-2 fuzzy numbers, intuitionistic fuzzy numbers and hesitant fuzzy numbers, which are also used by many scholars to represent the uncertainty in some situations. Future research can further extend the study to those kinds of fuzzy variables in theory so that more optimization problems can be worked out easily. Third, we devised a new heuristic algorithm called NIA-GA which integrates NIA with a standard genetic algorithms. Nevertheless, the genetic algorithm is not the only choice and many classical algorithms, such as simulated annealing algorithm, evolutionary algorithm, particle swarm optimization, etc, may be considered as an alternative. Future research can integrate NIA with other heuristic algorithms so as to handle fuzzy optimization problems effectively.
Author Contributions M. Zhao contributed to conceptualization, formal analysis, and methodology; M. Zhao and Y. Han contributed to writing-original draft; J. Zhou contributed to writing-review and editing, funding acquisition, project administration, supervision, and resources; and Y.Han contributed to software, validation, and visualization.
Funding This work was supported in part by grants from the National Natural Science Foundation of China (Grant No. 71872110).

(22)
In accordance with Eqs. (21) and (22), we get For another thing, since f is increasing for ζ 1 and decreasing for ζ 2 , it can be deduced that Following from Eq. (24), we can get In terms of the increase of the credibility measure Cr, it can be attained that Then it can be derived that In view of Eqs. (25) and (26), we have Cr {ζ ≤ f (t 1 , t 2 )} ≤ δ, ∀ f (t 1 , t 2 ) ∈ ψ −1 (δ).