# A Multiproduct Single-Period Inventory Management Problem under Variable Possibility Distributions

Authors: Zhaozhuang Guo; Shengnan Tian; Yankui Liu
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

In multiproduct single-period 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 interval-valued (PIV) fuzzy variables, and the analytical expressions about the mean values and second-order moments of selection variables are established. Taking second-order moment as a risk measure, a new credibilistic multiproduct single-period inventory management model is developed under mean-moment 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 mean-moment 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.


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## Body

## 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 stock-out or excess inventory will occur. The two possibilities should be considered during the decision-making 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 Lagrangian-based numerical solution procedure for the MSIMP. When the conditions of closed-form 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, Abdel-Malek 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 profit-maximization models. Abdel-Malek 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, Özler 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 mean-variance and utility function approaches, Van Mieghem [13] studied multiproduct single-period networks’ 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 closed-form formulas for the worst-case and best-case order quantities. Shu et al. [17] considered the distribution-free single-period inventory management problem by borrowing an economic theory from transportation disciplines. Moon et al. [18] found the differences between normal distribution approaches and distribution-free 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 [20–24].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 shelf-space 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 fuzzy-stochastic 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 single-period 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 single-product single-period 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 interval-valued 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 second-order 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 L-S multiple integral, two new optimization indexes, mean and second-order 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, Section2 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 [33–36].LetΓ be the universe of discourse, P(Γ) the power set of Γ, and P~os: P(Γ)↦R([0,1]) a fuzzy possibility measure. The triplet (Γ,P(Γ),P~os) is called a fuzzy possibility space.Letξ be a type 2 fuzzy variable defined on the space (Γ,P(Γ),P~os). If, for any r∈R, the secondary possibility distribution function μ~ξr=P~osξ=r is a subinterval [μξL(r;θl),μξU(r;θr)] of [0,1], then ξ is called a PIV fuzzy variable, where θl,θr∈[0,1] are two parameters characterizing the degree of uncertainty that ξ takes the value r.A type 2 fuzzy variableξ is called a generalized PIV normal fuzzy variable [36], if its secondary possibility distribution is the subinterval(1)1-θle-r-μ2/2σ2,e-r-μ2/2σ2+1-e-r-μ2/2σ2θrof [0,1] for r∈R, where μ∈R, σ>0 and θl,θr∈[0,1] are two parameters characterizing the degree of uncertainty that ξ takes on the value r. When θl=θr=0, the corresponding fuzzy variable is denoted by ξn, whose possibility distribution is called thenominal possibility distribution of ξ. In the following, ξ~n(μ,σ2;θl,θr) means that ξ is a generalized PIV normal fuzzy variable.A type 2 fuzzy variableη is called a generalized PIV triangular fuzzy variable [36], if its secondary possibility distribution is the subinterval [r-r1/r2-r1-θlr-r1/r2-r1,r-r1/r2-r1+θrr2-r/r2-r1] of [0,1], for r∈[r1,r2], and the subinterval [r3-r/r3-r2-θlr3-r/r3-r2,r3-r/r3-r2+θrr-r2/r3-r2] of [0,1] for r∈[r2,r3], where r1<r2<r3 are real numbers and θl,θr∈[0,1] are two parameters characterizing the degree of uncertainty that η takes on the value r. When θl=θr=0, the corresponding fuzzy variable is denoted by ηn, whose possibility distribution is called the nominal possibility distribution of η. In the following, η~Tri(r1,r2,r3;θl,θr) means that η is a generalized PIV triangular fuzzy variable.For a PIV fuzzy variable, its lambda selection is defined in [34]. Assume that ξ is a PIV fuzzy variable with the secondary possibility distribution μ~ξ(r)=[μξL(r;θl),μξU(r;θr)]. For any λ∈[0,1], a fuzzy variable ξλ is called a lambda selection of ξ if ξλ has the following generalized parametric possibility distribution:(2)μξλr;θl,θr=1-λμξLr;θl+λμξUr;θr.Obviously, the possibility distribution of lambda selection variable depends on the parameterλ. 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 [0,1].Based on L-S integral [37], the mean value of a fuzzy variable ξ is defined as(3)Eξ=∫-∞,+∞rdCrξ≤r,where the credibility Crξ≤r is computed by(4)Crξ≤r=12supx∈Rμξx;θl,θr+supx≤rμξx;θl,θr-supx>rμξx;θl,θr.In addition, the second-order moment of a fuzzy variableξ is defined as(5)Mξ=∫-∞,+∞r-Eξ2dCrξ≤r,where E(ξ) is the mean value of ξ defined by (3).For lambda selection variable, its mean value and second-order 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.
Letξλ be a lambda selection of the generalized PIV normal fuzzy variable n(μ,σ2;θl,θr). Then the mean value of the lambda selection ξλ is(6)Eξλ=ωμ,where ω=1-(1-λ)θl-λθr.Theorem 2.
Letηλ be a lambda selection of the generalized PIV triangular fuzzy variable (r1,r2,r3;θl,θr). Then the mean value of the lambda selection ηλ is(7)Eηλ=1-1-λθlr1+2r2+r34+14λθrr1-2r2+r3.Theorem 3.
Letξλ be a lambda selection of the generalized PIV normal fuzzy variable n(μ,σ2;θl,θr). Then the second-order moment of the lambda selection ξλ is(8)Mξλ=ω2σ2+ω-12μ2.Theorem 4.
Letηλ be a lambda selection of the generalized PIV triangular fuzzy variable (r1,r2,r3;θl,θr). Then the second-order moment of the lambda selection ηλ is(9)Mηλ=12λθrr1-m2+r3-m2+1-1-λθl-λθr6r2-r1r2-m3-r1-m3+1-1-λθl-λθr6r3-r2r3-m3-r2-m3,where m=E(ηλ).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 Parameters
n: number of products
i: product index, i=1,2,…,n
c
i: procurement cost for unit product i
g
i: goodwill cost for unit unmet demand of product i
p
i: retailer’s sales price for unit product i
s
i: salvage value for unit residual product i
θ
l
i: downward perturbation degree of nominal possibility distribution for product i
θ
r
i: upward perturbation degree of nominal possibility distribution for product i
λ
i: lambda selection parameter of demand distribution for product i
μ
λ
i: mean value of the lambda selection variable for product i
D
i: largest market demand for product i
B: total investment amount
K: total carbon emission allowance from government
β: predetermined confidence level
N
+: the set of nonnegative integersDecision Variables
Q
i: retailer’s order quantity for product iUncertain Parameters
ξ
i: uncertain market demand with variable possibility distribution
η
i: uncertain carbon emission due to logistic activities for product i
π: 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 quantityQi for product i to satisfy customer demand for each product. For product i, the distribution information of uncertain demand is only partially known based on the experts’ experiences or subjective judgments. Assume that the uncertain demand for product i(i=1,2,…,n) is characterized by generalized PIV normal fuzzy variable ξi=n(μi,σi2;θli,θri), i=1,2,…,n, and the largest market demand for product i is no more than Di. At the end of the period, if Qi≥ξi, then Qi-ξi units are salvaged for a per-unit revenue si, and if Qi<ξi, then ξi-Qi units represent lost sales cost for a per-unit cost gi.The profit for the retailer stemming from the sales of producti is represented as(10)πQi,ξi=pi-ciξi-ci-siQi-ξi+-pi-ci+giξi-Qi+for i=1,2,…,n, respectively.The profit function for producti 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 π(Qi,ξi) is computed by(11)EπQi,ξi=∫0,DiπQi,rdCrξi≤r.Sinceξi has an interval-valued possibility distribution, robust optimization method (see [38–41]) can be used to model the MSIMP.In this paper, the lambda selection variableξiλ is employed to represent the generalized PIV fuzzy variable ξi. In this case, the mean value of profit π(Qi,ξiλ) is computed by(12)EπQi,ξiλ=∫0,DiπQi,rdCrξiλ≤r=pi+gi-cihiQi-pi+gi-si∫0QiCrξiλ≤rdr-giμλi,where(13)hi=Crξiλ≤Di,μλi=∫0,DirdCrξλ≤r=DiCrξiλ≤Di-∫0DiCrξiλ≤rdr.Furthermore, the second-order moment of profitπ(Qi,ξiλ) is computed by(14)MπQi,ξiλ=∫0,DiπQi,r-EπQi,r2dCrξiλ≤r=hci-pi-gi2Qi2+2giDici-pi-giQi+gi2Di2+2cipi-si+si2Qi-pisi∫0QiCrξiλ≤rdr+2giQipi+gi-ci∫QiDiCrξiλ≤rdr-2pi-si2+gi2∫0QirCrξiλ≤rdr-2gi2∫QiDirCrξiλ≤rdr+mi2hi-2,where mi=E[π(Qi,ξiλ)].As a result, the total profit of the retailer in MSIMP is(15)ΠQ,ξλ=∑i=1nπQi,ξiλ.Based on L-S multiple integral, the mean total profit of the retailer is computed by(16)EΠQ,ξλ=∫⋯∫RnΠQ,ξλdCrξ1λ≤r×⋯×Crξnλ≤r,while the second-order moment of the total profit is computed by(17)MΠQ,ξλ=∫⋯∫RnΠQ,ξλ-EΠQ,ξλ2dCrξ1λ≤r×⋯×Crξnλ≤r.In order to find the optimal order quantityQi, the retailer should take into account the allocation of emission allowance K, which will be received before the selling season. It is well-known that transportation mode has a significant influence on carbon emission per ton-mile. For product i, it is usually difficult to determine the exact carbon emission during logistic activities. Based on the retailer’ experience, assume that the carbon emission for product i(i=1,2,…,n) is characterized by generalized PIV triangular fuzzy variable ηi~Tri(r1i,r2i,r3i;θli,θri).According to [36], ∑i=1nQiηi is also a generalized PIV triangular fuzzy variable; its lambda selection variable is denoted as (∑i=1nQiηi)λ.Under mean-moment optimization criterion, a new parametric credibilistic optimization model for the MSIMP is formally built as(18)maxQEΠQ,ξλ-γMΠQ,ξλE-M1(19)s.t.Cr∑i=1nQiηiλ≤K≥β(20)∑i=1nciQi≤B(21)0≤Qi≤Di,i=1,2,…,n(22)Qi∈N+,i=1,2,…,n.Objective function (18) in model (E-M 1) is to maximize the tradeoff between the mean total profit and the standard second-order moment of the total profit, where γ is some nonnegative constant that reflects the decision maker’s degree of risk aversion. Constraint (19) means that the carbon emission due to logistic activities is less than the total carbon emission K with a predetermined confidence level β. 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 Qi(i=1,2,…,n) are nonnegative integers in a reasonable range.In order to solve model (E-M 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.
Letηi be mutually independent fuzzy variables. Then model (E-M 1) is equivalent to the following deterministic programming model:(23)maxQΠeQ-γΠmQE-M  2(24)s.t.∑i=1nQiηiinfλβ≤K(25)∑i=1nciQi≤B(26)0≤Qi≤Di,i=1,2,…,n(27)Qi∈N+,i=1,2,…,n,where(28)ΠeQ=∑i=1nmi∏k≠ihk,hi=Crξiλ≤Di,mi=EπQi,ξiλ,ΠmQ=∑i=1n∏k≠ihk∫0,DiπQi,r2dCrξiλ≤r+∏k≠ihkmi2∏k=1nhk-2+∏k≠i≠jhk∑j≠imimj1-∏k=1nhk2.In Theorem5, model (E-M 2) is a parametric programming model with respect to parameter λ. 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 λ 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 [0,1]. For any given λ∈[0,1], the corresponding integer programming model (E-M 2) can be solved by conventional optimization software.


### 3.3. Domain Decomposition Method

Note that the analytical expressions ofΠe(Q) and Πm(Q) include the integral ∫0QiCr{ξiλ≤r}dr. According to the definition of Cr{ξ≤r}, the integral ∫0QiCr{ξiλ≤r}dr is a piecewise function with respect to Qi. Since decision makers do not know in advance which subregion the global optimal solution locates in, to solve 2n submodels by optimization software to obtain 2n local optimal solutions is required. By comparing the objective values of the obtained local optimal solutions, the global optimal solutions, Qi∗, i=1,2,…,n, can be found. Given the values of distribution parameters θli,  θri, and λi, the process of domain decomposition method is summarized as follows.Step 1.
Solve parametric programming submodels of model (E-M 2) by software Matlab 7.1. Let0≤Qi1≤μi,  μi≤Qi2≤Mi,  i=1,2,…,n. Given a set of values Qitk,  k=1 or 2,  i=1,2,…,n,  t=1,2,…,2n, denote the corresponding local optimal solutions as Qit∗,  i=1,2,…,n.Step 2.
Compare the local objective valuesvt=E[π(Q,ξλ)] at local optimal solution Qit∗ and find the global maximum profit by the following formula:(29)vl=max1≤t≤2nvt,where E[π(Q,ξλ)] is the mean profit of π(Q,ξλ).Step 3.
ReturnQil∗ as the global optimal solution to model (E-M 2) with the global optimal value E[π(Qil∗,ξλ)].In the next section, the effectiveness of the proposed domain decomposition method is demonstrated by a practical multiproduct single-period inventory management problem.


## 3.1. Notations

Fixed Parameters
n: number of products
i: product index, i=1,2,…,n
c
i: procurement cost for unit product i
g
i: goodwill cost for unit unmet demand of product i
p
i: retailer’s sales price for unit product i
s
i: salvage value for unit residual product i
θ
l
i: downward perturbation degree of nominal possibility distribution for product i
θ
r
i: upward perturbation degree of nominal possibility distribution for product i
λ
i: lambda selection parameter of demand distribution for product i
μ
λ
i: mean value of the lambda selection variable for product i
D
i: largest market demand for product i
B: total investment amount
K: total carbon emission allowance from government
β: predetermined confidence level
N
+: the set of nonnegative integersDecision Variables
Q
i: retailer’s order quantity for product iUncertain Parameters
ξ
i: uncertain market demand with variable possibility distribution
η
i: uncertain carbon emission due to logistic activities for product i
π: 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 quantityQi for product i to satisfy customer demand for each product. For product i, the distribution information of uncertain demand is only partially known based on the experts’ experiences or subjective judgments. Assume that the uncertain demand for product i(i=1,2,…,n) is characterized by generalized PIV normal fuzzy variable ξi=n(μi,σi2;θli,θri), i=1,2,…,n, and the largest market demand for product i is no more than Di. At the end of the period, if Qi≥ξi, then Qi-ξi units are salvaged for a per-unit revenue si, and if Qi<ξi, then ξi-Qi units represent lost sales cost for a per-unit cost gi.The profit for the retailer stemming from the sales of producti is represented as(10)πQi,ξi=pi-ciξi-ci-siQi-ξi+-pi-ci+giξi-Qi+for i=1,2,…,n, respectively.The profit function for producti 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 π(Qi,ξi) is computed by(11)EπQi,ξi=∫0,DiπQi,rdCrξi≤r.Sinceξi has an interval-valued possibility distribution, robust optimization method (see [38–41]) can be used to model the MSIMP.In this paper, the lambda selection variableξiλ is employed to represent the generalized PIV fuzzy variable ξi. In this case, the mean value of profit π(Qi,ξiλ) is computed by(12)EπQi,ξiλ=∫0,DiπQi,rdCrξiλ≤r=pi+gi-cihiQi-pi+gi-si∫0QiCrξiλ≤rdr-giμλi,where(13)hi=Crξiλ≤Di,μλi=∫0,DirdCrξλ≤r=DiCrξiλ≤Di-∫0DiCrξiλ≤rdr.Furthermore, the second-order moment of profitπ(Qi,ξiλ) is computed by(14)MπQi,ξiλ=∫0,DiπQi,r-EπQi,r2dCrξiλ≤r=hci-pi-gi2Qi2+2giDici-pi-giQi+gi2Di2+2cipi-si+si2Qi-pisi∫0QiCrξiλ≤rdr+2giQipi+gi-ci∫QiDiCrξiλ≤rdr-2pi-si2+gi2∫0QirCrξiλ≤rdr-2gi2∫QiDirCrξiλ≤rdr+mi2hi-2,where mi=E[π(Qi,ξiλ)].As a result, the total profit of the retailer in MSIMP is(15)ΠQ,ξλ=∑i=1nπQi,ξiλ.Based on L-S multiple integral, the mean total profit of the retailer is computed by(16)EΠQ,ξλ=∫⋯∫RnΠQ,ξλdCrξ1λ≤r×⋯×Crξnλ≤r,while the second-order moment of the total profit is computed by(17)MΠQ,ξλ=∫⋯∫RnΠQ,ξλ-EΠQ,ξλ2dCrξ1λ≤r×⋯×Crξnλ≤r.In order to find the optimal order quantityQi, the retailer should take into account the allocation of emission allowance K, which will be received before the selling season. It is well-known that transportation mode has a significant influence on carbon emission per ton-mile. For product i, it is usually difficult to determine the exact carbon emission during logistic activities. Based on the retailer’ experience, assume that the carbon emission for product i(i=1,2,…,n) is characterized by generalized PIV triangular fuzzy variable ηi~Tri(r1i,r2i,r3i;θli,θri).According to [36], ∑i=1nQiηi is also a generalized PIV triangular fuzzy variable; its lambda selection variable is denoted as (∑i=1nQiηi)λ.Under mean-moment optimization criterion, a new parametric credibilistic optimization model for the MSIMP is formally built as(18)maxQEΠQ,ξλ-γMΠQ,ξλE-M1(19)s.t.Cr∑i=1nQiηiλ≤K≥β(20)∑i=1nciQi≤B(21)0≤Qi≤Di,i=1,2,…,n(22)Qi∈N+,i=1,2,…,n.Objective function (18) in model (E-M 1) is to maximize the tradeoff between the mean total profit and the standard second-order moment of the total profit, where γ is some nonnegative constant that reflects the decision maker’s degree of risk aversion. Constraint (19) means that the carbon emission due to logistic activities is less than the total carbon emission K with a predetermined confidence level β. 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 Qi(i=1,2,…,n) are nonnegative integers in a reasonable range.In order to solve model (E-M 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.
Letηi be mutually independent fuzzy variables. Then model (E-M 1) is equivalent to the following deterministic programming model:(23)maxQΠeQ-γΠmQE-M  2(24)s.t.∑i=1nQiηiinfλβ≤K(25)∑i=1nciQi≤B(26)0≤Qi≤Di,i=1,2,…,n(27)Qi∈N+,i=1,2,…,n,where(28)ΠeQ=∑i=1nmi∏k≠ihk,hi=Crξiλ≤Di,mi=EπQi,ξiλ,ΠmQ=∑i=1n∏k≠ihk∫0,DiπQi,r2dCrξiλ≤r+∏k≠ihkmi2∏k=1nhk-2+∏k≠i≠jhk∑j≠imimj1-∏k=1nhk2.In Theorem5, model (E-M 2) is a parametric programming model with respect to parameter λ. 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 λ 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 [0,1]. For any given λ∈[0,1], the corresponding integer programming model (E-M 2) can be solved by conventional optimization software.


## 3.3. Domain Decomposition Method

Note that the analytical expressions ofΠe(Q) and Πm(Q) include the integral ∫0QiCr{ξiλ≤r}dr. According to the definition of Cr{ξ≤r}, the integral ∫0QiCr{ξiλ≤r}dr is a piecewise function with respect to Qi. Since decision makers do not know in advance which subregion the global optimal solution locates in, to solve 2n submodels by optimization software to obtain 2n local optimal solutions is required. By comparing the objective values of the obtained local optimal solutions, the global optimal solutions, Qi∗, i=1,2,…,n, can be found. Given the values of distribution parameters θli,  θri, and λi, the process of domain decomposition method is summarized as follows.Step 1.
Solve parametric programming submodels of model (E-M 2) by software Matlab 7.1. Let0≤Qi1≤μi,  μi≤Qi2≤Mi,  i=1,2,…,n. Given a set of values Qitk,  k=1 or 2,  i=1,2,…,n,  t=1,2,…,2n, denote the corresponding local optimal solutions as Qit∗,  i=1,2,…,n.Step 2.
Compare the local objective valuesvt=E[π(Q,ξλ)] at local optimal solution Qit∗ and find the global maximum profit by the following formula:(29)vl=max1≤t≤2nvt,where E[π(Q,ξλ)] is the mean profit of π(Q,ξλ).Step 3.
ReturnQil∗ as the global optimal solution to model (E-M 2) with the global optimal value E[π(Qil∗,ξλ)].In the next section, the effectiveness of the proposed domain decomposition method is demonstrated by a practical multiproduct single-period inventory management problem.


## 4. Numerical Experiments

### 4.1. Problem Statement

In order to illustrate the proposed credibilistic optimization model (E-M 2), a two-product single-period inventory problem is provided with generalized PIV normal demand variables. The retailer’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: air-conditioning (Product1) and evaporative air cooler (Product 2). The retailer is interested in determining the order quantity of air-conditioning Q1 and the order quantity of evaporative air cooler Q2 to satisfy customer demand. For product i(i=1,2), the distribution information of uncertain demand is partially available based on the experts’ experiences. Suppose that the uncertain demand ξi for product i follows generalized PIV normal possibility distribution n(μi,σi2;θli,θri), i=1,2. Based on the practical background of inventory problem, the largest market demand for product i is no more than Di. At the end of the period, if Qi≥ξi, then Qi-ξi units are salvaged for a per-unit revenue si, and if Qi<ξi, then ξi-Qi units represent lost sales cost for a per-unit cost gi. In view of the carbon emission constraint, the retailer receives the allocation of emission allowance K=251000 grams before the summer. For product i(i=1,2), the distribution information about the unit carbon emission during logistic activities is partially available based on the experts’ experiences. Assume that the unit carbon emissions for two products follow generalized PIV triangular possibility distributions Tri(85,100,110;0.25,0.15) and Tri(40,50,65;0.25,0.15), respectively. Due to logistic activities, the sum of emissions is less than the predetermined total emission K with confidence level β=0.9. Additionally, the available maximum investment for the retailer is B = $432000. The other pertinent data for the products are given in Table 1.Table 1
Parameters for a two-product single-period inventory problem.
Producti
c
i  ($)
p
i($)
s
i($)
g
i($)
ξ
i
D
i
Product 1
220
300
205
90
n
(
800,5
5
2
;
0.3,0.25
)
3000
Product 2
105
160
90
55
n
(
2400,7
5
2
;
0.15,0.2
)
6000


### 4.2. Computational Results

In numerical experiments, it is assumed thatλ1=0.6, λ2=0.8, and γ=0.3. 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 Qi, i=1,2. 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) Dual-Core E5700 3.00 GHz CPU and RAM 4.00 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 (Q1∗,Q2∗)=(813,2410) is found with the maximum mean total profit 117491.83.Table 2
The local optimal solutions with different domains ofQ1 and Q2.
The range ofQ1
The range ofQ2
Q
1
∗
Q
2
∗
E
∗
0
≤
Q
1
≤
800
0
≤
Q
2
≤
2400
800
2378
115885.79
0
≤
Q
1
≤
800
2400
≤
Q
2
≤
6000
800
2400
116558.62
800
≤
Q
1
≤
3000
0
≤
Q
2
≤
2400
814
2400
117240.53
800
≤
Q
1
≤
3000
2400
≤
Q
2
≤
6000
813
2410
117491.83


### 4.3. Sensitivity Analysis for Parameter Lambda

By the meanings of parametersλ1 and λ2, 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 λ1 and λ2 based on his experience or knowledge. If the decision maker cannot identify the values of parameters λ1 and λ2, he may generate randomly their values from some prescribed subintervals of [0,1]. In our experiments, to identify the influence of perturbation distribution on solution results, the optimal solutions are first computed by adjusting the selection parameter λ1 in the optimization problem with fixed λ2=0.6. When λ1 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 λ1 varies. Specifically, the optimal order quantity of product 1 is monotone decreasing with respect to parameter λ1, while the optimal order quantity of product 2 is monotone increasing with respect to parameter λ1. As a result, the mean total profit is monotone increasing with respect to parameter λ1.Figure 1
The optimal order quantities of product 1 and product 2 withλ2=0.6.Figure 2
The mean total profit.In the following, the optimal solutions are computed by adjusting the selection parameterλ2 in the optimization problem with fixed λ1=0.6. When λ2 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 λ2, while the optimal order quantity of product 2 is monotone decreasing with respect to parameter λ2. As a result, the mean total profit is monotone decreasing with respect to parameter λ2.Figure 3
The optimal order quantities of product 1 and product 2 withλ1=0.6.Figure 4
The mean total profit.The above computational results demonstrate that the optimal order quantities of product 1 and product 2 depend heavily on the location parametersλ1 and λ2. That is, the optimal order quantities of our multiproduct single-period 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 distributionsN(800,552) and N(2400,752), 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 189530. 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 toθli=θri=0, i=1,2. 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 parametersθli and θri. The computational results are reported in Tables 3–6, 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 θl or θr; 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 λ and perturbation parameter θ. For practical inventory management problems, if decision makers cannot identify the values of parameters λ and θ, they may generate randomly their values from some prescribed subintervals of [0,1]. The computational results demonstrate the advantages of variable possibility distributions over fixed possibility distributions.Table 3
The influence of perturbation parameterθl1 with θr1=0.25, θl2=0.15, and θr2=0.2.
θ
l
1
Q
1
∗
Q
2
∗
The mean total profit
The robust value
0.05
810
2415
134003.40
49745.45
0.10
809
2416
130918.55
52830.30
0.15
809
2417
127886.71
55862.14
0.20
808
2417
124771.49
58977.36
0.30
806
2417
118548.70
65200.15Table 4
The influence of perturbation parameterθr1 with θl1=0.3, θl2=0.15, and θr2=0.2.
θ
r
1
Q
1
∗
Q
2
∗
The mean total profit
The robust value
0.05
818
2400
131554.32
52194.53
0.15
816
2406
125167.66
58581.19
0.18
813
2409
123169.52
60579.33
0.25
806
2417
118548.70
65200.15
0.35
800
2421
111953.16
71795.69Table 5
The influence of perturbation parameterθl2 with θl1=0.3, θr1=0.25, and θr2=0.2.
θ
l
2
Q
1
∗
Q
2
∗
The mean total profit
The robust value
0.10
807
2419
121531.99
62216.86
0.15
806
2417
118548.70
65200.15
0.20
804
2414
115493.36
68255.49
0.25
800
2411
112352.60
71396.25
0.30
800
2406
109349.89
74398.96Table 6
The influence of perturbation parameterθr2 with θl1=0.3, θr1=0.25, and θl2=0.15.
θ
r
2
Q
1
∗
Q
2
∗
The mean total profit
The robust value
0.10
800
2422
126656.59
57092.26
0.15
800
2420
122469.12
61279.73
0.20
806
2417
118548.70
65200.15
0.25
812
2412
114563.42
69185.43
0.30
816
2400
110287.01
73461.84The comparison studies described in Sections4.4.1 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.


## 4.1. Problem Statement

In order to illustrate the proposed credibilistic optimization model (E-M 2), a two-product single-period inventory problem is provided with generalized PIV normal demand variables. The retailer’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: air-conditioning (Product1) and evaporative air cooler (Product 2). The retailer is interested in determining the order quantity of air-conditioning Q1 and the order quantity of evaporative air cooler Q2 to satisfy customer demand. For product i(i=1,2), the distribution information of uncertain demand is partially available based on the experts’ experiences. Suppose that the uncertain demand ξi for product i follows generalized PIV normal possibility distribution n(μi,σi2;θli,θri), i=1,2. Based on the practical background of inventory problem, the largest market demand for product i is no more than Di. At the end of the period, if Qi≥ξi, then Qi-ξi units are salvaged for a per-unit revenue si, and if Qi<ξi, then ξi-Qi units represent lost sales cost for a per-unit cost gi. In view of the carbon emission constraint, the retailer receives the allocation of emission allowance K=251000 grams before the summer. For product i(i=1,2), the distribution information about the unit carbon emission during logistic activities is partially available based on the experts’ experiences. Assume that the unit carbon emissions for two products follow generalized PIV triangular possibility distributions Tri(85,100,110;0.25,0.15) and Tri(40,50,65;0.25,0.15), respectively. Due to logistic activities, the sum of emissions is less than the predetermined total emission K with confidence level β=0.9. Additionally, the available maximum investment for the retailer is B = $432000. The other pertinent data for the products are given in Table 1.Table 1
Parameters for a two-product single-period inventory problem.
Producti
c
i  ($)
p
i($)
s
i($)
g
i($)
ξ
i
D
i
Product 1
220
300
205
90
n
(
800,5
5
2
;
0.3,0.25
)
3000
Product 2
105
160
90
55
n
(
2400,7
5
2
;
0.15,0.2
)
6000


## 4.2. Computational Results

In numerical experiments, it is assumed thatλ1=0.6, λ2=0.8, and γ=0.3. 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 Qi, i=1,2. 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) Dual-Core E5700 3.00 GHz CPU and RAM 4.00 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 (Q1∗,Q2∗)=(813,2410) is found with the maximum mean total profit 117491.83.Table 2
The local optimal solutions with different domains ofQ1 and Q2.
The range ofQ1
The range ofQ2
Q
1
∗
Q
2
∗
E
∗
0
≤
Q
1
≤
800
0
≤
Q
2
≤
2400
800
2378
115885.79
0
≤
Q
1
≤
800
2400
≤
Q
2
≤
6000
800
2400
116558.62
800
≤
Q
1
≤
3000
0
≤
Q
2
≤
2400
814
2400
117240.53
800
≤
Q
1
≤
3000
2400
≤
Q
2
≤
6000
813
2410
117491.83


## 4.3. Sensitivity Analysis for Parameter Lambda

By the meanings of parametersλ1 and λ2, 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 λ1 and λ2 based on his experience or knowledge. If the decision maker cannot identify the values of parameters λ1 and λ2, he may generate randomly their values from some prescribed subintervals of [0,1]. In our experiments, to identify the influence of perturbation distribution on solution results, the optimal solutions are first computed by adjusting the selection parameter λ1 in the optimization problem with fixed λ2=0.6. When λ1 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 λ1 varies. Specifically, the optimal order quantity of product 1 is monotone decreasing with respect to parameter λ1, while the optimal order quantity of product 2 is monotone increasing with respect to parameter λ1. As a result, the mean total profit is monotone increasing with respect to parameter λ1.Figure 1
The optimal order quantities of product 1 and product 2 withλ2=0.6.Figure 2
The mean total profit.In the following, the optimal solutions are computed by adjusting the selection parameterλ2 in the optimization problem with fixed λ1=0.6. When λ2 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 λ2, while the optimal order quantity of product 2 is monotone decreasing with respect to parameter λ2. As a result, the mean total profit is monotone decreasing with respect to parameter λ2.Figure 3
The optimal order quantities of product 1 and product 2 withλ1=0.6.Figure 4
The mean total profit.The above computational results demonstrate that the optimal order quantities of product 1 and product 2 depend heavily on the location parametersλ1 and λ2. That is, the optimal order quantities of our multiproduct single-period 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 distributionsN(800,552) and N(2400,752), 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 189530. 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 toθli=θri=0, i=1,2. 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 parametersθli and θri. The computational results are reported in Tables 3–6, 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 θl or θr; 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 λ and perturbation parameter θ. For practical inventory management problems, if decision makers cannot identify the values of parameters λ and θ, they may generate randomly their values from some prescribed subintervals of [0,1]. The computational results demonstrate the advantages of variable possibility distributions over fixed possibility distributions.Table 3
The influence of perturbation parameterθl1 with θr1=0.25, θl2=0.15, and θr2=0.2.
θ
l
1
Q
1
∗
Q
2
∗
The mean total profit
The robust value
0.05
810
2415
134003.40
49745.45
0.10
809
2416
130918.55
52830.30
0.15
809
2417
127886.71
55862.14
0.20
808
2417
124771.49
58977.36
0.30
806
2417
118548.70
65200.15Table 4
The influence of perturbation parameterθr1 with θl1=0.3, θl2=0.15, and θr2=0.2.
θ
r
1
Q
1
∗
Q
2
∗
The mean total profit
The robust value
0.05
818
2400
131554.32
52194.53
0.15
816
2406
125167.66
58581.19
0.18
813
2409
123169.52
60579.33
0.25
806
2417
118548.70
65200.15
0.35
800
2421
111953.16
71795.69Table 5
The influence of perturbation parameterθl2 with θl1=0.3, θr1=0.25, and θr2=0.2.
θ
l
2
Q
1
∗
Q
2
∗
The mean total profit
The robust value
0.10
807
2419
121531.99
62216.86
0.15
806
2417
118548.70
65200.15
0.20
804
2414
115493.36
68255.49
0.25
800
2411
112352.60
71396.25
0.30
800
2406
109349.89
74398.96Table 6
The influence of perturbation parameterθr2 with θl1=0.3, θr1=0.25, and θl2=0.15.
θ
r
2
Q
1
∗
Q
2
∗
The mean total profit
The robust value
0.10
800
2422
126656.59
57092.26
0.15
800
2420
122469.12
61279.73
0.20
806
2417
118548.70
65200.15
0.25
812
2412
114563.42
69185.43
0.30
816
2400
110287.01
73461.84The comparison studies described in Sections4.4.1 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.


## 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 distributionsN(800,552) and N(2400,752), 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 189530. 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 toθli=θri=0, i=1,2. 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 parametersθli and θri. The computational results are reported in Tables 3–6, 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 θl or θr; 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 λ and perturbation parameter θ. For practical inventory management problems, if decision makers cannot identify the values of parameters λ and θ, they may generate randomly their values from some prescribed subintervals of [0,1]. The computational results demonstrate the advantages of variable possibility distributions over fixed possibility distributions.Table 3
The influence of perturbation parameterθl1 with θr1=0.25, θl2=0.15, and θr2=0.2.
θ
l
1
Q
1
∗
Q
2
∗
The mean total profit
The robust value
0.05
810
2415
134003.40
49745.45
0.10
809
2416
130918.55
52830.30
0.15
809
2417
127886.71
55862.14
0.20
808
2417
124771.49
58977.36
0.30
806
2417
118548.70
65200.15Table 4
The influence of perturbation parameterθr1 with θl1=0.3, θl2=0.15, and θr2=0.2.
θ
r
1
Q
1
∗
Q
2
∗
The mean total profit
The robust value
0.05
818
2400
131554.32
52194.53
0.15
816
2406
125167.66
58581.19
0.18
813
2409
123169.52
60579.33
0.25
806
2417
118548.70
65200.15
0.35
800
2421
111953.16
71795.69Table 5
The influence of perturbation parameterθl2 with θl1=0.3, θr1=0.25, and θr2=0.2.
θ
l
2
Q
1
∗
Q
2
∗
The mean total profit
The robust value
0.10
807
2419
121531.99
62216.86
0.15
806
2417
118548.70
65200.15
0.20
804
2414
115493.36
68255.49
0.25
800
2411
112352.60
71396.25
0.30
800
2406
109349.89
74398.96Table 6
The influence of perturbation parameterθr2 with θl1=0.3, θr1=0.25, and θl2=0.15.
θ
r
2
Q
1
∗
Q
2
∗
The mean total profit
The robust value
0.10
800
2422
126656.59
57092.26
0.15
800
2420
122469.12
61279.73
0.20
806
2417
118548.70
65200.15
0.25
812
2412
114563.42
69185.43
0.30
816
2400
110287.01
73461.84The comparison studies described in Sections4.4.1 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 second-order moment have been established.(ii) Two new indexes, mean and second-order moment, about the total profit were defined based on L-S 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 makers’ 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.



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