Research Article  Open Access
Zhaozhuang Guo, Shengnan Tian, Yankui Liu, "A Multiproduct SinglePeriod Inventory Management Problem under Variable Possibility Distributions", Mathematical Problems in Engineering, vol. 2017, Article ID 2159281, 14 pages, 2017. https://doi.org/10.1155/2017/2159281
A Multiproduct SinglePeriod Inventory Management Problem under Variable Possibility Distributions
Abstract
In multiproduct singleperiod inventory management problem (MSIMP), the optimal order quantity often depends on the distributions of uncertain parameters. However, the distribution information about uncertain parameters is usually partially available. To model this situation, a MSIMP is studied by credibilistic optimization method, where the uncertain demand and carbon emission are characterized by variable possibility distributions. First, the uncertain demand and carbon emission are characterized by generalized parametric intervalvalued (PIV) fuzzy variables, and the analytical expressions about the mean values and secondorder moments of selection variables are established. Taking secondorder moment as a risk measure, a new credibilistic multiproduct singleperiod inventory management model is developed under meanmoment optimization criterion. Furthermore, the proposed model is converted to its equivalent deterministic model. Taking advantage of the structural characteristics of the deterministic model, a domain decomposition method is designed to find the optimal order quantities. Finally, a numerical example is provided to illustrate the efficiency of the proposed meanmoment credibilistic optimization method. The computational results demonstrate that a small perturbation of the possibility distribution can make the nominal optimal solution infeasible. In this case, the decision makers should employ the proposed credibilistic optimization method to find the optimal order quantities.
1. Introduction
The MSIMP is a classical inventory management problem. In order to maximize (minimize) the total expected profit (cost), the decision makers have to make the optimal order quantities at the beginning of the period. At the end of the selling period, either stockout or excess inventory will occur. The two possibilities should be considered during the decisionmaking process. The popularity of the MSIMP is due to its applicability in retailing and manufacturing industries. Hadley and Whitin [1] first considered a MSIMP with storage capacity or budget constraints and proposed a dynamic programming solution procedure to find the optimal order quantities. Since then, many researchers have developed stochastic MSIMP. For instance, Nahmias and Schmidt [2] discussed the MSIMP under the linear and deterministic constraints on budget or space. H.S. Lau and A. H. L. Lau [3] extended the MSIMP to handle multiconstraint and presented a Lagrangianbased numerical solution procedure for the MSIMP. When the conditions of closedform expressions did not hold, Erlebacher [4] proposed an effective heuristic solution. Moon and Silver [5] dealt with the MSIMP subject to not only a budget constraint on the total value of the replenishment quantities but also fixed costs for nonzero replenishment. Furthermore, AbdelMalek et al. [6] considered a MSIMP under a budget constraint with probabilistic demand and random yield. Zhang [7] considered the MSIMP with both supplier quantity discounts and a budget constraint and formulated it as a mixed integer nonlinear programming model. In order to deal with the possible shortage of limited capacity, Zhang and Du [8] discussed zero lead time outsourcing strategy and nonzero lead time outsourcing strategy. They also developed the structural properties and solution procedures for their profitmaximization models. AbdelMalek and Montanari [9] proposed a methodology for studying the dual of the solution space of the MSIMP with two constraints and introduced an approach to obtain the optimal order quantities of each product. In addition, Huang et al. [10] studied a competitive MSIMP with shortage penalty cost and partial product substitution. In view of risk preference, Ă–zler et al. [11] proposed the MSIMP under a Value at Risk constraint. Van Ryzin and Mahajan [12] reviewed the contributions to multiproduct inventory problem with demand substitution. Under meanvariance and utility function approaches, Van Mieghem [13] studied multiproduct singleperiod networksâ€™ problems in probabilistic framework.
When the exact probability distribution of demand is unavailable, probabilistic robust optimization method [14] is a tool to deal with the corresponding uncertainty in inventory management problem. Based on the assumption that demand was described by discrete or interval scenarios, Vairaktarakis [15] discussed several minimax regret formulations for the MSIMP with a budget constraint. When the distribution of demand had known support, mean, and variance, Kamburowski [16] presented the theoretical foundations for analyzing the inventory management problem. They derived the closedform formulas for the worstcase and bestcase order quantities. Shu et al. [17] considered the distributionfree singleperiod inventory management problem by borrowing an economic theory from transportation disciplines. Moon et al. [18] found the differences between normal distribution approaches and distributionfree approaches in four scenarios with mean and variance. Under interval demand uncertainty, Solyali et al. [19] proposed a new robust formulation which could solve the intractability issue for large problem instances. As for recent development in stochastic inventory management problems, the interested reader may further refer to [20â€“24].
Most of the extensions of inventory management problem have been made in the probabilistic framework, where uncertain parameters are characterized by random variables. However, in some cases, there are not enough data to determine the exact probability distribution of random variable because of economic reason or technical difficulty. In such a case, the variable is approximately specified based on the experiences and subjective judgments of the experts in related fields, so fuzzy inventory management problem is also an active research area. Fuzzy set theory was applied in the early inventory management literature [25, 26]. In the area of fuzzy MSIMP, Mandal and Roy [27] considered a multiproduct displayed inventory model under shelfspace constraint in fuzzy environment, where the demand rate of a product was considered as a function of the displayed inventory level. Under fuzzy demand environment, Ji and Shao [28] studied the MSIMP and formulated three kinds of models. Dutta [29] formulated a fuzzy MSIMP model whose objective was to maximize the total profit by considering fuzzy demands. In fuzzystochastic environment, Saha et al. [30] developed multiproduct multiobjective supply chain models with budget and risk constraints, where the manufacturing costs of the items were fuzzy variables and the demands for the products were random variables. Based on credibility measure, Guo [31] proposed two singleperiod inventory models, where the uncertain demands were characterized by discrete and continuous possibility distributions, respectively. Tian and Guo [32] formulated a credibilistic optimization model for a singleproduct singleperiod inventory problem with two suppliers.
The work mentioned above studied inventory management problem under the assumption that the exact possibility distribution of fuzzy variable was available, which motivates us to study the MSIMP from a new perspective. The motivation of this paper is based on the following considerations. First, shorter product life cycles and growing innovation rates make the market demand extremely variable. In this case, the distribution information about market demand is only partially available. It is reasonable to assume that the exact possibility distribution is embodied in a zonal area for a practical MSIMP, so the intervalvalued fuzzy variable is introduced to characterize uncertain market demand. Second, the optimal order quantities for different products are heavily influenced by the carbon emission constraint. In some practical inventory management problems, it is difficult to determine the exact carbon emission during logistic activities. Under credibilistic carbon emission constraint, a parametric credibilistic optimization model is developed for MSIMP. To the best of our knowledge, this issue has not been addressed in the literature.
This paper studies MSIMP by parametric credibilistic optimization method, where uncertain market demand and uncertain carbon emission are characterized by generalized PIV possibility distributions. Decision makers can make informed decisions based on a tradeoff model between the mean total profit and the secondorder moment of total profit under budget constraint and uncertain carbon emission constraint. The strength of the proposed method is that the distributions of market demand and carbon emission can be tailored to the partial information at hand. That is, when the distribution information about uncertain parameters is partially available, the proposed method is more convenient for modeling uncertain demand and carbon emission in a practical MSIMP. The proposed credibilistic optimization method differs from the existing MSIMP literature in the following several aspects. (i) A novel method is introduced to model the perturbation distributions of uncertain demand and carbon emission, which is different from the existing literature. (ii) For PIV fuzzy variable, its lambda selection variable is introduced as its representative; the possibility distribution of lambda selection can traverse the entire support of the PIV fuzzy variable as the lambda parameter varies its values. (iii) On the basis of LS multiple integral, two new optimization indexes, mean and secondorder moment, about the total profit are defined to build a parametric credibilistic optimization model under credibilistic constraint of carbon emission. (iv) A domain decomposition method is designed to divide the original credibilistic optimization model into several equivalent parametric programming submodels, which can be solved by conventional optimization software.
The remainder of this paper is organized as follows. After introducing some basic concepts in fuzzy possibility theory, Section 2 discusses the properties about generalized PIV fuzzy variable and its selection variable. In Section 3, a new parametric credibilistic optimization model is first developed for MSIMP, where uncertain demand and uncertain carbon emission are characterized by variable possibility distributions. Then the equivalent deterministic model of the proposed parametric credibilistic optimization model is discussed in this section. A new domain decomposition method is also designed in this section to find the optimal order quantities. In Section 4, some numerical experiments are conducted to demonstrate the validity of the proposed credibilistic optimization method. Section 5 gives the conclusion of the paper.
2. Generalized PIV Fuzzy Variables
First, in this section, some basic concepts in fuzzy possibility theory are recalled [33â€“36].
Let be the universe of discourse, the power set of , and : a fuzzy possibility measure. The triplet is called a fuzzy possibility space.
Let be a type 2 fuzzy variable defined on the space . If, for any , the secondary possibility distribution function is a subinterval of , then is called a PIV fuzzy variable, where are two parameters characterizing the degree of uncertainty that takes the value .
A type 2 fuzzy variable is called a generalized PIV normal fuzzy variable [36], if its secondary possibility distribution is the subintervalof for , where , and are two parameters characterizing the degree of uncertainty that takes on the value . When , the corresponding fuzzy variable is denoted by , whose possibility distribution is called the nominal possibility distribution of . In the following, means that is a generalized PIV normal fuzzy variable.
A type 2 fuzzy variable is called a generalized PIV triangular fuzzy variable [36], if its secondary possibility distribution is the subinterval of , for , and the subinterval of for , where are real numbers and are two parameters characterizing the degree of uncertainty that takes on the value . When , the corresponding fuzzy variable is denoted by , whose possibility distribution is called the nominal possibility distribution of . In the following, means that is a generalized PIV triangular fuzzy variable.
For a PIV fuzzy variable, its lambda selection is defined in [34]. Assume that is a PIV fuzzy variable with the secondary possibility distribution . For any , a fuzzy variable is called a lambda selection of if has the following generalized parametric possibility distribution:
Obviously, the possibility distribution of lambda selection variable depends on the parameter . That is, the possibility distribution of lambda selection variable can traverse the entire support of PIV fuzzy variable as the lambda parameter varies its value in the interval .
Based on LS integral [37], the mean value of a fuzzy variable is defined aswhere the credibility is computed by
In addition, the secondorder moment of a fuzzy variable is defined aswhere is the mean value of defined by (3).
For lambda selection variable, its mean value and secondorder moment are important optimization indices in the MSIMP. The following theorems establish their analytical expressions, which will be used in the rest of the paper. For the sake of presentation, the proofs of the following theorems are provided in the appendix.
Theorem 1. Let be a lambda selection of the generalized PIV normal fuzzy variable . Then the mean value of the lambda selection iswhere
Theorem 2. Let be a lambda selection of the generalized PIV triangular fuzzy variable . Then the mean value of the lambda selection is
Theorem 3. Let be a lambda selection of the generalized PIV normal fuzzy variable . Then the secondorder moment of the lambda selection is
Theorem 4. Let be a lambda selection of the generalized PIV triangular fuzzy variable . Then the secondorder moment of the lambda selection iswhere .
In the next section, the distribution information about uncertain demand and uncertain carbon emission is partially available and characterized by generalized PIV normal fuzzy variable and triangular fuzzy variable, respectively.
3. Credibilistic Optimization Model for MSIMP
In order to model MSIMP, some necessary notations are provided in the following subsection.
3.1. Notations
Fixed Parametersâ€‰: number of productsâ€‰: product index, â€‰: procurement cost for unit product â€‰: goodwill cost for unit unmet demand of product â€‰: retailerâ€™s sales price for unit product â€‰: salvage value for unit residual product â€‰: downward perturbation degree of nominal possibility distribution for product â€‰: upward perturbation degree of nominal possibility distribution for product â€‰: lambda selection parameter of demand distribution for product â€‰: mean value of the lambda selection variable for product â€‰: largest market demand for product â€‰: total investment amountâ€‰: total carbon emission allowance from governmentâ€‰: predetermined confidence levelâ€‰: the set of nonnegative integers
Decision Variablesâ€‰: retailerâ€™s order quantity for product
Uncertain Parametersâ€‰: uncertain market demand with variable possibility distributionâ€‰: uncertain carbon emission due to logistic activities for product â€‰: uncertain profit for retailer
3.2. Credibilistic Optimization Model and Its Equivalent Deterministic Form
In this subsection, a MSIMP is studied, where the uncertain demand and uncertain carbon emission are characterized by generalized PIV fuzzy variables. At the beginning of selling season, the retailer is interested in determining the order quantity for product to satisfy customer demand for each product. For product , the distribution information of uncertain demand is only partially known based on the expertsâ€™ experiences or subjective judgments. Assume that the uncertain demand for product is characterized by generalized PIV normal fuzzy variable , , and the largest market demand for product is no more than . At the end of the period, if , then units are salvaged for a perunit revenue , and if , then units represent lost sales cost for a perunit cost .
The profit for the retailer stemming from the sales of product is represented asfor , respectively.
The profit function for product cannot be directly maximized because it is a fuzzy variable. In order to transform the fuzzy objective into a crisp one, the mean profit of is computed by
Since has an intervalvalued possibility distribution, robust optimization method (see [38â€“41]) can be used to model the MSIMP.
In this paper, the lambda selection variable is employed to represent the generalized PIV fuzzy variable . In this case, the mean value of profit is computed bywhere
Furthermore, the secondorder moment of profit is computed bywhere .
As a result, the total profit of the retailer in MSIMP is
Based on LS multiple integral, the mean total profit of the retailer is computed bywhile the secondorder moment of the total profit is computed by
In order to find the optimal order quantity , the retailer should take into account the allocation of emission allowance , which will be received before the selling season. It is wellknown that transportation mode has a significant influence on carbon emission per tonmile. For product , it is usually difficult to determine the exact carbon emission during logistic activities. Based on the retailerâ€™ experience, assume that the carbon emission for product is characterized by generalized PIV triangular fuzzy variable .
According to [36], is also a generalized PIV triangular fuzzy variable; its lambda selection variable is denoted as .
Under meanmoment optimization criterion, a new parametric credibilistic optimization model for the MSIMP is formally built as
Objective function (18) in model ( 1) is to maximize the tradeoff between the mean total profit and the standard secondorder moment of the total profit, where is some nonnegative constant that reflects the decision makerâ€™s degree of risk aversion. Constraint (19) means that the carbon emission due to logistic activities is less than the total carbon emission with a predetermined confidence level . Constraint (20) represents the fact that the investment amount on total production cost has an upper limit on the maximum investment. Constraints (21) and (22) ensure that decision variables are nonnegative integers in a reasonable range.
In order to solve model ( 1), its equivalent deterministic model is discussed in the following theorem. For the sake of presentation, the proof of the following theorem is also provided in the appendix.
Theorem 5. Let be mutually independent fuzzy variables. Then model (EM 1) is equivalent to the following deterministic programming model:where
In Theorem 5, model ( 2) is a parametric programming model with respect to parameter . The value of parameter lambda determines the location and shape of the possibility distribution of selection variable. According to the definition of lambda selection variable, parameter may change its value from 0 to 1. It is highlighted that the possibility distribution of lambda selection variable can traverse the entire support of PIV fuzzy variables as the lambda parameter changes its value in the interval . For any given , the corresponding integer programming model ( 2) can be solved by conventional optimization software.
3.3. Domain Decomposition Method
Note that the analytical expressions of and include the integral . According to the definition of , the integral is a piecewise function with respect to . Since decision makers do not know in advance which subregion the global optimal solution locates in, to solve submodels by optimization software to obtain local optimal solutions is required. By comparing the objective values of the obtained local optimal solutions, the global optimal solutions, , , can be found. Given the values of distribution parameters ,â€‰â€‰, and , the process of domain decomposition method is summarized as follows.
Step 1. Solve parametric programming submodels of model (EM 2) by software Matlab 7.1. Let ,â€‰â€‰,â€‰â€‰. Given a set of values ,â€‰â€‰ or ,â€‰â€‰,â€‰â€‰, denote the corresponding local optimal solutions as ,â€‰â€‰.
Step 2. Compare the local objective values at local optimal solution and find the global maximum profit by the following formula:where is the mean profit of .
Step 3. Return as the global optimal solution to model ( 2) with the global optimal value .
In the next section, the effectiveness of the proposed domain decomposition method is demonstrated by a practical multiproduct singleperiod inventory management problem.
4. Numerical Experiments
4.1. Problem Statement
In order to illustrate the proposed credibilistic optimization model (EM 2), a twoproduct singleperiod inventory problem is provided with generalized PIV normal demand variables. The retailerâ€™s optimal strategy will be obtained by the proposed credibilistic optimization method. Before a hot summer, the retailer needs to order two kinds of products: airconditioning (Product ) and evaporative air cooler (Product ). The retailer is interested in determining the order quantity of airconditioning and the order quantity of evaporative air cooler to satisfy customer demand. For product , the distribution information of uncertain demand is partially available based on the expertsâ€™ experiences. Suppose that the uncertain demand for product follows generalized PIV normal possibility distribution , . Based on the practical background of inventory problem, the largest market demand for product is no more than . At the end of the period, if , then units are salvaged for a perunit revenue , and if , then units represent lost sales cost for a perunit cost . In view of the carbon emission constraint, the retailer receives the allocation of emission allowance grams before the summer. For product , the distribution information about the unit carbon emission during logistic activities is partially available based on the expertsâ€™ experiences. Assume that the unit carbon emissions for two products follow generalized PIV triangular possibility distributions and , respectively. Due to logistic activities, the sum of emissions is less than the predetermined total emission with confidence level . Additionally, the available maximum investment for the retailer is = $432000. The other pertinent data for the products are given in Table 1.

4.2. Computational Results
In numerical experiments, it is assumed that , , and . According to the proposed domain decomposition method, the feasible region of the above inventory management problem can be decomposed into four disjoint subregions of decision variables , . Matlab 7.1 optimization software is employed to solve the corresponding parametric programming submodels. The numerical experiments are conducted on a personal computer (Lenovo with Intel Pentium(R) DualCore E5700 3.00â€‰GHz CPU and RAM 4.00â€‰GB) by using the Microsoft Windows 10 operating system. The computational results are reported in Table 2. By comparing the obtained local optimal solutions, the global optimal solution is found with the maximum mean total profit .

4.3. Sensitivity Analysis for Parameter Lambda
By the meanings of parameters and , the two parameters determine the location and shape about the possibility distribution of selection variable in the support of uncertain demand and uncertain carbon emission. A decision maker may prescribe the values of parameters and based on his experience or knowledge. If the decision maker cannot identify the values of parameters and , he may generate randomly their values from some prescribed subintervals of . In our experiments, to identify the influence of perturbation distribution on solution results, the optimal solutions are first computed by adjusting the selection parameter in the optimization problem with fixed . When increases its value from 0.1 to 1 with step 0.1, the computational results about the optimal order quantities of product 1 and product 2 are plotted in Figure 1, and the corresponding mean total profits are plotted in Figure 2. From Figures 1 and 2, it is found that the optimal order quantities and mean total profit vary while the selection parameter varies. Specifically, the optimal order quantity of product 1 is monotone decreasing with respect to parameter , while the optimal order quantity of product 2 is monotone increasing with respect to parameter . As a result, the mean total profit is monotone increasing with respect to parameter .
In the following, the optimal solutions are computed by adjusting the selection parameter in the optimization problem with fixed . When increases its value from 0.1 to 1 with step 0.1, the computational results of the optimal order quantities of product 1 and product 2 are plotted in Figure 3, and the corresponding mean total profits are plotted in Figure 4. From Figures 3 and 4, it is concluded that the optimal order quantity of product 1 is monotone increasing with respect to parameter , while the optimal order quantity of product 2 is monotone decreasing with respect to parameter . As a result, the mean total profit is monotone decreasing with respect to parameter .
The above computational results demonstrate that the optimal order quantities of product 1 and product 2 depend heavily on the location parameters and . That is, the optimal order quantities of our multiproduct singleperiod inventory problem depend heavily on the possibility distribution of uncertain demand.
4.4. Comparison Study
4.4.1. Comparing with Stochastic Optimization Method
In this subsection, the credibilistic optimization method is compared with stochastic optimization method, where the stochastic demands of product 1 and product 2 follow normal probability distributions and , respectively. According to the stochastic optimization method for MSIMP, the optimal order quantities for product 1 and product 2 are 815 and 2407 with the maximum mean profit . The solution result is totally different from our credibilistic optimal solutions reported in Figures 1 and 3. Compared with our credibilistic optimal solutions, the optimal solutions 815 and 2407 to stochastic model are not feasible solutions to the deterministic programming model in Theorem 5. That is, the stochastic optimal solution does not satisfy carbon emission constraint (24) and the investment amount constraint (25).
4.4.2. Comparing with Fuzzy Optimization Method under Fixed Possibility Distribution
In this subsection, the credibilistic optimization method is compared with fuzzy optimization method, where the uncertain demands of product 1 and product 2 follow fixed possibility distributions. For the sake of comparison, the fixed possibility distributions are taken as the nominal possibility distributions of uncertain demands corresponding to , . By solving the fuzzy optimization model, the obtained nominal optimal order quantities are 800 and 2400 with the nominal maximum mean total profit 183748.85. Obviously, the nominal maximum mean total profit is larger than the optimal mean total profits obtained in Figures 2 and 4. The computational results imply that a small perturbation of the nominal possibility distribution may heavily affect the quality of optimal solution.
To further analyze the influence of the perturbation parameters, some additional experiments are conducted with different values of perturbation parameters and . The computational results are reported in Tables 3â€“6, in which the robust value is defined as the reduction from the nominal optimal profit to the optimal profits with different values of perturbation parameters. The computational results imply that the robust value is increasing with respect to perturbation parameters or ; that is, the larger the perturbation parameter, the larger the uncertainty degree embedded in the generalized PIV possibility distribution of uncertain demand. The decision makers can adjust the values of perturbation parameters according to their obtained distribution information. As a consequence, the considered MSIMP depends heavily on the location parameter and perturbation parameter . For practical inventory management problems, if decision makers cannot identify the values of parameters and , they may generate randomly their values from some prescribed subintervals of . The computational results demonstrate the advantages of variable possibility distributions over fixed possibility distributions.




The comparison studies described in Sections 4.4.1 and 4.4.2 lead to the following observations.
Firstly, stochastic optimization method for MSIMP is based on the assumption that the market demands are of stochastic nature, and the probability distributions of uncertain parameters are available. When the probability distributions of uncertain market demands cannot be determined, the stochastic optimization method cannot be used to determine the optimal order quantities.
Secondly, in fuzzy MSIMP, it is usually assumed that the nominal possibility distributions of uncertain parameters can be determined exactly and a small perturbation of nominal possibility distribution will not affect significantly the solution quality. The comparison study shows that the robust value is increasing with respect to perturbation parameters. The decision makers should adjust the values of perturbation parameters according to their obtained distribution information.
Finally, it should be highlighted that under given perturbation parameters the optimal order quantities depend heavily on the values of location parameter lambda. The proposed parametric credibilistic optimization method is capable of detecting cases when perturbation distributions can heavily affect the quality of the nominal solution. In these cases, the decision makers should employ the proposed credibilistic optimization method to find the optimal order quantities, which may immunize against the effect of perturbation distribution.
5. Conclusions
In this paper, the MSIMP has been studied from a new perspective. The major new results include the following several aspects.
(i) When the distribution information about uncertain demand and uncertain carbon emission was partially available, these uncertain parameters were characterized by generalized PIV fuzzy variables. For their selection variables, the analytical expressions of the mean and secondorder moment have been established.
(ii) Two new indexes, mean and secondorder moment, about the total profit were defined based on LS multiple integral, and their analytical expressions have been established. Furthermore, a new parametric credibilistic optimization model was developed for MSIMP.
(iii) The equivalent deterministic model of the proposed credibilistic MSIMP has been established. According to the structural characteristics of the equivalent deterministic model, a domain decomposition method was designed to find the optimal order quantities.
(iv) In numerical experiments, the proposed optimization method was compared with stochastic optimization method and fuzzy optimization method under fixed possibility distribution. The computational results demonstrated that a small perturbation of the demand distribution could make the nominal optimal solution infeasible and thus practically meaningless. In this case, the decision makers should employ the proposed credibilistic optimization method to find the optimal order quantities, which may immunize against the effect of perturbation distribution.
The developed parametric credibilistic optimization model for MSIMP addressed the effect of perturbation possibility distributions. In the model process, the generalized PIV fuzzy variables were represented by their lambda selections. For a practical MSIMP, based on the uncertain distribution sets of generalized PIV fuzzy variables, distributionally robust optimization method will be studied in our future research. Extension to considering decision makersâ€™ risk tolerance fuzziness [42] for the MSIMP is another interesting research direction. In addition, hybrid uncertainty and their solution method [43] can be introduced to tackle the MSIMP.
Appendix
Proofs of Main Theorems
Proof of Theorem 1. Since , the generalized possibility distribution of isAccording to the definition of credibility measure [44], the credibility is computed bywhich can generate a measure using the method discussed in [45]. By calculation, one hasAccording to (3) and (A.3), one hasThe proof of theorem is complete.
Proof of Theorem 2. Since , the generalized possibility distribution of the lambda selection variable isAccording to the definition of credibility measure, one hasBy calculation, one hasThe proof of theorem is complete.
Proof of Theorem 3. According to (5) and (6), the secondorder moment of is computed as follows:The proof of theorem is complete.
Proof of Theorem 4. In the following, the mean value of is denoted as ; that is, . According to (5) and (7), the secondorder moment of the lambda selection variable is computed byThe proof of theorem is complete.
Proof of Theorem 5. First, objective function (18) is equivalent to maximizing . By calculation the multiple LS integrals, one hasSimilarly, the secondorder moment of the total profit is computed byNext, the equivalence between constraint (19) and constraint (24) is discussed. Since are mutually independent generalized PIV fuzzy variables, according to [36], one haswhere and .
Since the credibility is a monotone increasing function, is equivalent tothat is, . The proof of theorem is complete.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (no. 61773150).
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Copyright © 2017 Zhaozhuang Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.